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Or
OF MIC
THE UNIVERS,
MICHIGAN
THE
1811
LIBRARIES
FLIGHT
TEST
MANUAL
VOLUME 1
Performance
VOLUME II Stability and Control
VOLUME III
Instrumentation Catalog
VOLUME IV
Instrumentation Systems
GENERAL EDITOR
COURTLAND D. PERKINS
Professor and Chairman
Aeronautical Engineering Department, Princeton University
ASSOCIATE EDITOR
DANIEL O. DOMMASCH, 1953-56
ENOCH J. DURBIN, 1956-
Aeronautical Engineering Department, Princeton University
Mr
P
Hej for and on behalf of AGARD
Advisory Group for Aeronautical Research and Development,
North Atlantic Treaty Organization.
1959
PERGAMON PRESS
NEW YORK LONDON
PARIS
LOS ANGELES
LIMITEDSITV OC MICUICAN LIDDADIES
19
PERGAMON PRESS, INC.
122 East 55th Street, New York 22, N. Y.
P.O. Box 47715, Los Angeles, California
PERGAMON PRESS LTD.
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Engin. Long
TL
671.7
.N57
1959
V.
PERGAMON PRESS, S.A.R.L
24 Rue des Ecoles, Paris Ve
Second revised edition 1959
Copyright
1959
ADVISORY GROUP FOR
AERONAUTICAL RESEARCH AND DEVELOPMENT
NORTH ATLANTIC TREATY ORGANIZATION
This volume was prepared under U.S.
Air Force Contract 18(600)-1323,
administered by the Air Force Office
of Scientific Research of the Air
Research and Development Command.
1
Library of Congress Card No. 59-13097
Lithographed in the United States by Edwards Bros., Ann Arbor, Mich.
87-189163
THE AGARD FLIGHT TEST PANEL
(May 1959)
Present Members
Mr. Alec F. ATKIN (United Kingdom)
Mr. R. P. DICKINSON (United Kingdom)
Mr. M. N. GOUGH (United States)
Lt. Col. J. J. BERKOW, USAF (United States)
Ten. Col. Dott. Ing. Enzo BIANCHI (Italy)
Prof. F. HAUS (Belgium)
F/Lt. H. E. BJORNESTAD (Canada)
Mr. Jean IDRAC (France)
Mr. Yavuz KANSU (Turkey)
Prof. Dr. phil. H. BLENK (Germany)
Prof. Dr. G. BOCK (Germany)
Lt. Col. P. N. BRANDT-MOELLER (Denmark)
Mr. P. LECOMTE (France)
Cdr. W. H. LIVINGSTON, USN (United States)
Col. de BUEGER (Belgium)
Dr. Anton J. MARX (Holland)
Ing. G. CIAMPOLINI (Italy)
Mr. Tor MIDTBO (Norway)
Col. D. CHRISSAITIS (Greece)
Mr. A. D. WOOD (Canada)
Panel Executive: Lt. Col. J. A. WOIDA, USAF (June 1958 to present)
Past Members
Major H. AASS (Norway)
Major H. NESSET (Norway)
Major W. RICKERT, USAF (United States)
Ing. G. CERZA (Italy)
Col. Dr. Ing. F. COLUMBA (Italy)
Lt. Col. J. L. RIDLEY, USAF (United States)
Mr. Handel DAVIES (United Kingdom)
Cdr. L. M. SATTERFIELD, USN (United States)
Ing. en Chef B. DAVY (France)
Cdr. R. J. SELMER, USN (United States)
Lt. Col. G. B. DOYLE, USMC (United States)
Mr. T. E. STEPHENSON (Canada)
Mr. R. R. DUDDY (United Kingdom)
Brig. Gen. M. STRATIGAKIS (Greece)
Major E. TUSTER (Norway)
Ing. en Chef J. FOCH (France)
Mr. N. E. G. HILL (United Kingdom)
Brig. Gen. Fuat ULUG (Turkey)
Mr. P. A. HUFTON (United Kingdom)
Major H. UNSAL (Turkey)
Panel Executives: Col. J. J. DRISCOLL, USAF (May 1952 March 1954)
Cdr. Emil P. SCHULD, USN (September 1954 - 1958)
Acting Panel Executive: Mr. R. A. WILLAUME (France) (April 1954 - August 1954)
RECORD OF REVISION
(This sheet is prepared for your convenience to keep
a record of number and date of revisions.)
REVISION NUMBER:
DATE:
ENTERED BY:
Science and technology have a big part to play in our efforts to improve
the economic and military strength of NATO. The work of the Advisory
Group for Aeronautical Research and Development is therefore of great
importance.
It is always difficult to achieve a harmony of view between technical
experts of many nations. The Flight Test Manual is a sigaal example of
how these difficulties can be overcome, and a happy augury for the future.
Isman
THE RIGHT HONORABLE LORD ISMAY
Vice-Chairman, North Atlantic Council
Secretary General, North Atlantic Treaty
Organization
Air Force-USAPE, Voba, Ger-15-7
FOREWORD
In this first volume of the NATO Flight Test Manual, prepared under the auspices of
the Advisory Group for Aeronautical Research and Development, I am pleased to note
the cooperative etfort of the best scientific talent of NATO nations coacemed with the
problems of flight testing. I am strongly convinced that a cooperative effort of this na-
ture will accomplish two general purposes: it will give opportunity to individuals, in a
country where scientific and technical accomplishments have been attained, for mutual
exchange of ideas with their counterparts in other countries; and it will fumish inval.
uable guidance to flying personnel and engineers of NATO countries as yet without real
opportunity to develop their own theories and practices.
a
I sincerely trust that this venture in NATO technical cooperation will set an encour
aging example for similar efforts in other domains of aeronautical activity.
Famii hortet
taco
LAURIS NORSTAD, General, United States Air Force
Air Deputy, Supreme Headquarters, Allied Powers, Europe
NATO
NORTH ATLANTIC TREATY ORGANIZATION
(Organisation du Traite de l'Atlantique Nord)
FLIGHT TESTING
VOLUME I
PERFORMANCE
Edited by
Daniel O. Dommasch
Princeton University
PREFACE TO THE SECOND EDITION
The second edition of the AGARD Flight Test Manual is being brought out under
new "ground rules" which will undoubtedly go a long way towards improving the develop-
ment and distribution of new material in the years to come. This new edition published
by Pergamon Press will be available for public sale and will have the great advantage of
a well-organized procedure for getting out new material at regularly scheduled intervals
and an efficient system for ensuring that the users of the manual can receive this new
material when it is available. This new edition contains all of the additions and correc-
tions introduced into the first edition from time to time, as well as nearly seven hundred
pages of new material, bringing it up to date as of the fall of 1959.
COURTLAND D. PERKINS
Princeton University
28 September 1959
PREFACE
The Advisory Group for Aeronautical Research and Development of NATO found there
was a great need for a flight test manual covering performance, stability and control, and
instrumentation of aircraft chat could be used by design, development or research engineers,
test pilots, and instrumentation personnel of the participating nations in order to expand
their knowledge, improve their methods, and standardize their techniques. Although various
member nations of NATO have their own separate publications covering the subjects con-
tained herein, AGARD recognized the need for the compilation, revision, and enlargement
of this material for the benefit of all NATO nations.
The authors generously contributed their time and knowledge in the writing of the various
chapters. The high quality of their contributions to this manual will insure its success and
will further AGARD's mission in the field of flight testing and instrumentation. AGARD was
fortunate to find in the person of Professor Courtland D. Perkios ao editor of high competency
in the field, who was willing to devote time and effort to consolidate the individual contri.
butions of the authors into an integrated techoical publication. To authors and editors I
wish to extend my gratitude and appreciation on behalf of AGARD.
The members of the Flight Test Panel, together with AGARD Executives Colonel Jobo
J. Driscoll and Mr. Rolland Willaume, and the AGARD Clerical Staff are to be congratulated
for their cireless efforts, andl especially wish to express my thanks to Messrs. Jacques Foch
Bernard Davy, and Jean Idrac, of France; P.A. Hufton, and N.E.G. Hill, of the United
Kingdom; Le. Col. G.B. Doyle, United States Marine Corps; Ms. A.J. Marx of the Netherlands;
Li. Col. J.L. Ridley and Major Walter Rickert, of the United States Air Force, all of whom
participated in the final review of the text of this manual and supplied valuable suggestions
as to the suitability of the material.
THEODORE VON KARMAN
Chairman
Advisory Group for Aeronautical
Research and Development
Acknowledgements
In the preparation of a book of this type, a great deal of work must be carried out by
people whose names cannot be prominently displayed in the final published product, but
without whom the book could never have been assembled. I should like therefore to express
my appreciation to the members of the small editorial group at Princeton who worked very
hard to prepare the manuscript under the pressure of a very right time schedule. In partic-
ular I would like to express my gratitude to Mrs. Dorothy S. Webster, who typed most of the
rough and final copy and acted as editorial assistant; Mr. Robert Westover, who prepared a
very excellent set of illustrations and to Mr. Sylvester Hight for his help in translating sev-
eral of the French papers,
A great deal of credit must go to Professor D.O. Dommasch who acted as volume editor
for this project. It was only through a tremendous amount of effort on his part that this first
volume of the Flight Test Manual was brought into final form within the scheduled period of
time. Finally, I would like to express my appreciation to Dr. Carl L. Fredericks and his
staff who prepared the final manuscript copy and cooperated with our editorial group at all
times.
Courtland D. Perkins
VOLUME I, PERFORMANCE
TABLE OF CONTENTS
Page
PREFACE
ACKNOWLEDGEMENTS
INTRODUCTION TO VOLUME I
Chapter 1
AIRSPEED, ALTITUDE AND TEMPERATURE MEASUREMENTS
TERMINOLOGY
CHAPTER FOREWORD
1:1
1:2
1:3
1:4
1:5
1:6
1:7
1:8
1:9
1:10
1:11
1:12
Introductory Comments
The NACA and ICAN Standard Atmospheres
Altitude Definitions
Speed Definitions
Machmeters
Speed Measurement
Pressure and Temperature Pickups
Development of Airspeed Equations
Methods of Measurement and Errors
Methods of Calibrating the Airspeed System
Pressure Measurement
The Determination of Speed and Pressure Altitude From The
Instrument Readings
The Establishment of The Pressure Error Correction From
Flight Test Results
The Estimation of The Pressure Error Under Other Than Test
Conditions
Examples of Measured Pressure Errors
Temperature Measurement
Theoretical Considerations
Flight Calibration of Temperature Measuring Systems
Concluding Remarks on Temperature Measurement
Introduction to Lag Measurements in Pressure Systems
Development of Equations
Determination of The Lag Constant
Design of Systems
Concluding Remarks
Sample Determinations of AV cal and hp
1:1
1:1
1:3
1:4
1:9
1:10
1:11
1:12
1:16
1:17
1:18
1:19
1:13
1:21
1:14
1:15
1:16
1:17
1:18
1:19
1:20
1:21
1:22
1:23
1:24
1:25
1:24
1:28
1:35
1:35
1:37
1:38
1:39
1:40
1:50
1:54
1:54
1:55
REFERENCES
1:57
TABLE OF CONTENTS
(Continued)
Page
Chapter 2
THRUST AND POWER DETERMINATION
TERMINOLOGY
2:1
2:2
2:3
2:4
2:5
2:6
2:7
2:8
2:1
2:2
2:6
2:7
2:8
2:9
2:9
2:9
2:10
2:11
2:12
2:13
2:14
2:15
2:16
2:17
2:18
2:19
2:20
Introductory Comments
Power Measurement; Internal Combustion Engines
Theory of Reciprocating Engine Power Corrections
Corrections Applicable to Power Charts
Torquemeter Power Corrections
Power Measurement - Compound Engines
Jet Thrust Measurement - Introductory Comments
Discussion and Region of Application of the Methods of Jet
Thrust Measurement
The Jet Flow Measurement Method
The Climb Performance Method
The Measurement of the Useful Thrust of Turbo-Propeller Engines
Measurement of Shaft Horsepower
The Estimation of Jet Thrust
Concluding Remarks on Turbo-Propeller Thrust Determination
Ramjet Thrust Measurement
Concluding Remarks on Ramjet Thrust Measurement
Measurement of Rocket Thrust in Flight
General Analysis of Jet Thrust Measurement
The Measurement of the Thrust of Jet Engines
Thrust Measurements with Afterburning
2:10
2:10
2:13
2:15
2:17
2:17
2:18
2:18
2:21
2:22
2:23
2:28
2:32
REFERENCES
2:35
Chapter 3
A SURVEY OF PERFORMANCE REDUCTION METHODS
SUMMARY
TERMINOLOGY
3:1
3:2
3:3
3:4
3:5
3:6
Introduction
The Problems
Methods of Performance Reduction
Experimental Methods
Analytical Methods
Summary of Conclusions
REFERENCES
3:1
3:1
3:2
3:2
3:5
3:7
3:8
Chapter 4
PERFORMANCE OF TURBOJET AIRPLANES
TERMINOLOGY
4:1
4:2
Introductory Comments
Introduction to Level Flight Performance Testing of Turbojet
Airplanes
4:1
4:1
TABLE OF CONTENTS
(Continued)
Page
4:3
4:4
4:5
4:6
4:7
4:8
4:9
4:10
4:11
4:1
4:6
4:13
4:16
4:19
4:23
4:26
4:32
Turbojet Engine Parameters
Airplane-Engine Combination Characteristics
Reduction of Thrust Required Data to Basic Form
Thrust Available
Range and Endurance in Level Flight
The Cruise Climb
The Nominal Best Rate of Climb Airspeed
Time to Climb for Jet Aircraft
Determination of Maximum Energy Storage Schedule and Time to
Climb to a Given Energy Height
Proof of Construction Used to Determine Optimum Energy Climb
Speed from Acceleration Run Data
Descent Performance of Turbojet Aircraft
Measurement and Correction of Turbojet Take-off and Landing
Characteristics
Turning Performance of Turbojet Aircraft
Concluding Remarks
4:37
4:12
4:40
4:42
4:13
4:14
4:15
4:16
4:42
4:43
4:46
REFERENCES
4:47
Chapter 5
PERFORMANCE REDUCTION METHODS FOR TURBO-PROPELLER
AIRCRAFT
SUMMARY
FOREWORD
5:1
5:2
5:3
5:4
5:5
5:6
5:7
5:8
5:9
Introduction
Differential Reduction Methods
The Basic Performance Equations
Fuel Flow in Level Flight
Equations for Climb Reduction
Data Required in Addition to the Basic Performance Quantities
Required Accuracy of the Coefficients
Possible Simplifications of the Method
Reduction of Level Speeds and Fuel Consumptions by Performance
Analysis
Concluding Remarks
5:1
5:1
5:2
5:7
5:9
5:11
5:13
5:14
5:14
5:16
5:10
REFERENCES
5:16
TABLE I
5:17
TABLE II
5:18
APPENDIX, USE OF LOGARITHMIC DIFFERENTIALS IN
PERFORMANCE REDUCTION
5:19
TABLE OF CONTENTS
(Continued)
Page
Chapter 6
DATA REDUCTION AND PERFORMANCE TEST METHODS FOR
RECIPROCATING ENGINE AIRCRAFT
TERMINOLOGY
-
6:1
6:2
6:3
6:4
6:5
6:6
6:7
6:8
6:9
6:10
6:11
6:12
6:13
Introductory Comments
Level Flight - Power Required
Power Available in Level Flight
Determination of V max
Range and Endurance Testing (Cruising Flight)
Pilot Technique in Level Flight Performance Testing
The Instrumentation Required for Level Flight Testing
Climb and Descent Testing - Sawtooth Climbs
Climbs to Ceiling
Pilot Technique for Climb Tests
Measurement and Correction of Take-off Characteristics
Measurement and Correction of Landing Characteristics
Concluding Remarks
6:1
6:1
6:5
6:16
6:17
6:25
6:26
6:26
6:29
6:35
6:37
6:45
6:46
REFERENCES
6:47
Chapter 7
TOTAL ENERGY METHODS
FOREWORD
Part I
OPTIMUM CLIMB THEORY AND TECHNIQUES OF DETERMINING
CLIMB SCHEDULES FROM FLIGHT TEST
7:1
7:2
7:2
7:1
7:2
7:3
7:4
7:5
7:6
7:7
7:8
7:9
7:10
Introduction
A Simplified Theory
End Conditions
An Exact Approach
Miscellaneous Cases
Effect of Operating Conditions
Derivation of the Optimum Climb Schedule for the Particular Case
Instrumentation and Analysis
Standardization of Rate-of-Climb Curves
References
7:5
7:5
7:8
7:9
7:9
7:11
7:15
7:15
-
APPENDICES
1. Flight Test Technique for Accelerated Levels
II. Analysis of Airspeed Indicator Time History from Level
7:16
Acceleration
7:17
TABLE OF CONTENTS
(Continued)
Page
Part II
ANALYSIS OF THE OPTIMUM ENERGY CLIMB SCHEDULE
7:19
7:11
7:12
7:13
7:14
7:15
7:16
7:17
7:18
7:19
Introduction
Definitions
Theoretical Analysis of the Optimum Climb Schedule
Geometric Representation
Properties of the Graph
Discussion of Assumptions
Corrections
Practical Determination of the Optimum Energy Climb Schedule
Conclusions
7:19
7:19
7:20
7:21
7:23
7:24
7:26
7:27
7:31
Part III
CORRELATION OF THE ENERGY CLIMB ANALYSES
7:32
7:32
7:20
7:21
Review of Preceding Analyses
The Third Method of Expressing the Conditions for Optimum
Energy Storage
Presentation of the Third Set of Conditions in Terms of V and h
Summary
7:22
7:23
7:33
7:35
7:36
APPENDIX A
Procedures for Determining the Energy Climb Performance of
Turbojet Aircraft
7:38
Chapter 8
TAKE-OFF AND LANDING PERFORMANCE
TERMINOLOGY
Part I
FLIGHT TEST ANALYSIS
Summary
Take-off Definitions
Take-off Measurements
Basic Take-off Equations under No Wind Conditions
Parameter Theory
Take-off Performance Reduction
Summary of Equations
Landing Definitions
Landing Measurements
Basic Landing Equations
Landing Performance Reduction
Summary of Equations
Concluding Remarks
8:1
8:2
8:3
8:4
8:5
8:6
8:7
8:8
8:9
8:10
8:11
8:12
8:13
8:1
8:1
,8:2
8:4
8:7
8:9
8:12
8:15
8:15
8:16
8:17
8:18
8:19
TABLE OF CONTENTS
(Continued)
3
Page
Part II
THEORY
8:14
8:15
8:16
8:17
8:18
8:19
8:20
8:21
8:22
8:23
General Principles
The Ground Run
The Transition and Climb
Piloting Technique in the Airborne Phase
Application to Flight Test Procedure
Critical Speed During Take-off
The Landing Maneuver
Estimation of Landing Distance
The Choice of the Landing Approach Airspeed
Sources of Variation in Total Landing Distance
8:20
8:20
8:22
8:28
8:30
8:30
8:32
8:32
8:36
8:38
REFERENCES
8:39
Chapter 9
SPECIAL TESTS
TERMINOLOGY
9:1
9:2
9:3
9:4
9:5
9:6
9:7
9:8
Introductory Comments
Introductory Comments on Air Flow Visualization
Methods of Flow Investigation
Flight Testing
Other Techniques
Calibration of an Angle of Attack Measuring System
Measurement of Sideslip Angle
Airbrake Evaluation
9:1
9:1
9:2
9:8
9:10
9:11
9:12
9:14
REFERENCES
9:21
Chapter 10
PERFORMANCE TESTING OF HELICOPTERS
SUMMARY
TERMINOLOGY
Introductory Comments
Flight Test Program
Ground Resonance
Control and Center of Gravity Limits
Airspeed Calibration
Level Flight Performance Measurement
Climb Performance Determination
Descent Performance
Take-Off and Landing Performance
Miscellaneous Tests
10:1
10:2
10:3
10:4
10:5
10:6
10:7
10:8
10:9
10:10
10:1
10:2
10:4
10:4
10:6
10:7
10:12
10:17
10:18
10:23
TABLE OF CONTENTS
(Continued)
Page
10:11
10:12
10:13
Calculation Methods
Blade Stall Limitations
Concluding Remarks
10:26
10:30
10:31
REFERENCES
10:32
Chapter 11
THE EFFECT OF THE GROUND ON THE PERFORMANCE
OF A HELICOPTER
TERMINOLOGY
11:1
11:2
11:3
11:4
11:5
Introductory Comments
Theoretical Analysis for a Single Rotor lelicopter
Theoretical Analysis for Tandem Rotor Helicopters
Flight Tests
Comparison of Theory and Experiment
11:1
11:1
11:5
11:6
11:7
REFERENCES
11:9
Chapter 12
THE TRANSITION PERFORMANCE OF A HELICOPTER
FOLLOWING A SUDDEN LOSS OF POWER
TERMINOLOGY
12:1
12:2
12:3
12:4
Introductory Comments
Theoretical Analysis for Single-Engine Helicopters
Multi-Engine Helicopters
Test Techniques
12:1
12:1
12:6
12:7
REFERENCES
12:10
APPENDICES TO VOLUME I
A-1
A-3
A-5
Appendix 1 Part I, ICAN Standard Atmosphere
Part II, Properties of the NACA Standard Atmosphere
Appendix II Graphical Relations Between Calibrated Airspeed, Mach Number
and Pressure Altitude for Subsonic Speeds (NACA Standard Atmosphere
for Altimeter and Airspeed Indicator Calibration)
Appendix III Graphical Relation Between True Airspeed, Altitude and Calibrated
Airspeed Under Standard Conditions (NACA Standard Atmosphere)
Appendix IV Mach Number Temperature Relations for Various Recovery Factors
Appendix V Relations Between True Speed V, Calibrated Airspeed V Mach
Number M, and Pressure Altitude h
р
A-8
A-9
cal
A-10
VOLUME I, PERFORMANCE
3
Contributing Authors
Idrac, J.
Centre d'Essais en Vol
France
Bottle, David W.
Aeroplane & Armament Experimental
Establishment, United Kingdom
Cheeseman, I. C.
Aeroplane & Armament Experimental
Establishment, United Kingdom
Davy, Bernard
France
Langdon, G. F.
Aeroplane & Armament Experimental
Establishment, United Kingdom
Lean, D.
Royal Aircraft Establishment
United Kingdom
Davidson, T. W.
Naval Air Test Center
United States Navy
Douwes Dekker, F. E.
National Aeronautical Research Institute
The Netherlands
LeDuc, Rene
France
Lightfoot, Ralph B.
Sikorsky Aircraft Division
United Aircraft Corporation
Dommasch, Daniel O.
Princeton University
Down, H, W.
Naval Air Test Center
United States Navy
Lush, Kenneth J.
Air Force Flight Test Center
United States Air Force
Foch, Jacques
Centre d'Essais en Vol
France
Moakes, John K.
Aeroplane & Armament Experimental
Establishment, United Kingdom
Renaudie, J. F.
Centre d'Essais en Vol
France
Gray, W.E.
Royal Aircraft Establishment
United Kingdom
Gregory, J. D, L.
Aeroplane & Armament Experimental
Establishment, United Kingdom
Schwarzbach, Jerome M.
Naval Air Test Center
United States Navy
Guenod, M.
Centre d'Essais en Vol
France
Shields, Robert T.
Aeroplane & Armament Experimental
Establishment, United Kingdom
Soisson, Jean
Centre d'Essais en Vol
France
Hesse, Walter J.
Naval Air Test Center
United States Navy
Hufton, Phillip A.
Royal Aircraft Establishment
United Kingdom
Utting, Ivan E.
Aeroplane & Armament Experimental
Establishment, United Kingdom
i
AGARD FLIGHT TEST MANUAL
VOLUME I
INTRODUCTION
Flight testing of piloted aircraft is conduc-
ted with several fundamental purposes in
mind:
(a) To determine the actual characteris-
tics of the machine (as contrasted to the
computed or pedicted characteristics);
field has lain buried in various manuals,
reports and the like, to be dug up only as a
last resort. This situation has made it diffi-
cult for the engineer just entering the field
to find his way and has, in certain instances,
produced an air of mystery where none should
have existed. Certainly the lack of a compre-
hensive work on the subject has frequently led
to unnecessary and wasteful duplication of
effort.
(b) To provide developmental informa-
tion;
(c) To obtain research information.
Thus, in assembling this volume, it is felt
that we not only serve the purpose of collec-
ting and assimilating the offerings of the
several NATO nations, but also serve the
purpose of providing as unified as possible
a basic reference work in the flight testing
field.
These three purposes have remained the
reasons for flight testing work since the day
the Wright brothers made their first flight
about a half century ago, and there seems to
be little reason that they will change in the
future. Indeed, as “the proof of the pud-
ding lies in the eating”, so also must the
proof of an airplane lie in the flying, for in
spite of what developments are yet to come,
engineering will always remain in part an
empirical science, subject to verification by
actual test. Thus, although the aeronautical
sciences have undoubtedly enjoyed vast devel-
opment during the past fifty years, this
development has not eased, but rather intensi-
fied the need for flight resting work.
In the original plan for preparation of
these NATO flight test manuals, the material,
which now appears in this first volume and
the stability and control work of the second
volume, were to have been spread over three
volumes; the first dealing with fundamental
analyses; the second, with engineering appli-
cation; and the third, with the routine conduct
of tests. It was later decided that the lack of
a unified approach resulting from such a
spread was not desirable, and accordingly,
all work on performance testing has been
combined to form a single volume.
Since flight testing is a necessary and an
integral part of aeronautical development, the
science of flight testing has not stood still
and, along with other things aeronautical, has
made great strides forward during the past
five decades. Unlike other, and perhaps better
defined branches of aeronautics, the field of
flight testing has not been the subject of
numerous anthological texts or handbooks,
and the great wealth of information in this
Moreover, since the manner in which the
routine test procedures are organized in
detail (if any flight test program can really be
considered a routine matter) depends to a
great extent on the type of engineering per-
sonnel available at a given test center and,
of course, on the equipment available, it was
decided to minimize emphasis on such things
ii
source by the airplane's or pickup's pressure
field).
as data analysis sheets and detailed plans of
approach to any given problem. Care has
been taken, however, to point out certain
pitfalls into which inexperienced personnel
may be lured if they are not careful.
The second problem, that of reducing test
data to standard conditions, is closely related
to the third problem involving the design of
the test program since the conditions under
which the data are obtained and the selection
of these conditions determines the manner of
reducing the data. Thus, the second and third
problems are related and are generally con-
sidered simultaneously.
Most of the material, contributed by the
various authors, pertained to what might be
called acceptance flight testing of aircraft;
that is to say, it was more concerned with
determining what the flight characteristics of
a given airplane actually were, rather than
with the breakdown of these characteristics
into data useful directly in research and
developmental work. It should be apparent,
however, that the same procedures useful
for acceptance tests also are applicable to
research and developmental testing, the
major difference being the manner in which
the final corrected data are presented. Thus,
it has been unnecessary to differentiate here
between the ultimate purposes for conducting
flight tests.
These latter two problems are subject to
theoretical analysis using several different
techniques. When the number of variables
which cannot be controlled during the test
is small, test methods based on dimensional
analysis prove very useful provided the analy-
sis considers all the variables which are
significant under all circumstances.
1
Finally, we should like to comment on the
major problems encountered in flight testing
work, broadly these are:
(a) Measurement of the actual values
obtained during test;
In certain instances when a well-defined
relationship exists among two or more of the
test variables, the so-called analytic methods
of anaysis may be used as a basis for a
test procedure, as is the case with recipro-
cating engine airplanes. Where other methods
are not usable, finite difference techniques
are available; however, when these are
employed, one must generally schedule tests
to obtain data initially under conditions as
close as possible to standard to reduce the
size of the corrections to a minimum.
(b) Determination of what the measured
values would have been under some arbitrary
set of standard conditions;
(c) Design of the test program to provide
the desired results for the least cost in time
and money within the limitations of avail-
able manpower and equipment.
When reducing data, it is generally pre-
sumed that one or more of the test variables
such as pressure altitude, density altitude,
temperature altitude, velocity (true, calibra-
ted or equivalent) or Mach number are
"standard", and the data reduction process
then revolves about these standards.
has given rise to such terms as density
altitude methods, etc., which usage, in some
cases, is not too descriptive of the actual
reduction process involved.
In solving problem (a), we must consider
the nature of the instrumentation (its accura-
cy, reliability, size, sensitivity, etc.), the
nature of the quantity to be measured (whether
it exhibits local variations, whether auxiliary
conditions affect the instrumentation, and so
forth) and the effects of disturbances pro-
duced by the airplane or by the measuring
equipment on the measured quantity (for
example, the errors produced at a static
In planning any test program, it is neces-
sary to consider the variables of pilot tech-
nique and of weather. Obviously, one cannot
control atmospheric conditions; however, it is
iii
this procedure is the lack of knowledge con-
cerning the functioning of the power plants
during the tests, since no data are available
determine if the engines are actually
performing according to specification.
necessary to consider how such disturbances
as air turbulence, temperature inversions and
so forth affect the test data and plan accor-
dingly. With regard to pilot technique, this
variable produces the most pronounced scat-
ter of data when the pilot is asked to control
a number of quantities simultaneously, such
as during landing and take-off maneuvers.
Thus, if the test program is planned so that
the pilot has to maintain rigorously only one
item such as altitude, less error due to
pilot technique may be expected than if he
must control altitude, airspeed, flight path
angle, and so forth, all simultaneously.
Consequently, no special attention is given
to the topic of side by side testing, but
rather, in the first eight chapters of this
volume, the three basic problems of flight
testing are considered and, in the last
chapter (nine), certain special topics are
investigated.
It has frequently been proposed that one
way to eliminate certain variables and to
simplify the task of data reduction, when com-
parison only of airplane characteristics is
concerned, is to fly two airplanes simultane-
ously through a given set of maneuvers, and
then perform the same set of maneuvers
interchanging pilots. The major drawback to
Because the flight testing field is not a
static one, it is anticipated that new and
revised ideas will present themselves in the
future and for this reason, a loose leaf
binding has been adopted for this book to
permit changes to be made readily as neces-
sary. Comments and submission of new
material to the editor by all readers of this
text will be welcomed.
iv
)
(
VOLUME I, CHAPTER 1
CHAPTER CONTENTS
Page
TERMINOLOGY
CHAPTER FOREWORD
1:1
INTRODUCTORY COMMENTS
1:1
1:2
THE NACA AND ICAN STANDARD ATMOSPHERES
1:1
1:3
ALTITUDE DEFINITIONS
1:3
1:4
SPEED DEFINITIONS
1:4
1:5
MACHMETERS
1:9
1:6
SPEED MEASUREMENT
1:10
1:7
PRESSURE AND TEMPERATURE PICKUPS
1:11
1:8
1:12
DEVELOPMENT OF AIRSPEED EQUATIONS
METHODS OF MEASUREMENT AND ERRORS
1:9
1:16
1:10
METHODS OF CALIBRATING THE AIRSPEED SYSTEM
1:17
1:11
PRESSURE MEASUREMENT
1:18
1:12
THE DETERMINATION OF SPEED AND PRESSURE ALTITUDE
FROM THE INSTRUMENT READINGS
1:19
1:13
THE ESTABLISHMENT OF THE PRESSURE ERROR CORRECTION
FROM FLIGHT TEST RESULTS
1:21
1:14
THE ESTIMATION OF THE PRESSURE ERROR UNDER OTHER THAN
TEST CONDITIONS
1:24
1:15
EXAMPLES OF MEASURED PRESSURE ERRORS
1:28
1:16
TEMPERATURE MEASUREMENT
1:35
1:17
THEORETICAL CONSIDERATIONS
1:35
1:18
FLIGHT CALIBRATION OF TEMPERATURE MEASURING SYSTEMS
1:37
1:19
CONCLUDING REMARKS ON TEMPERATURE MEASUREMENT
1:38
1:20
INTRODUCTION TO LAG MEASUREMENTS IN PRESSURE SYSTEMS
1:39
1:21
DEVELOPMENT OF EQUATIONS
1:40
1:22
DETERMINATION OF THE LAG CONSTANT
1:50
1:23
DESIGN OF SYSTEMS
1:54
1:24
CONCLUDING REMARKS
1:54
SAMPLE DETERMINATIONS OF AVcal and Ahp
1:25
1:55
REFERENCES
1:57
TERMINOLOGY
с
Speed of Sound in Free Stream
co
Speed of Sound at Sea Level
Ср
Specific Heat at Constant Pressure (perfect gas)
CP
Pressure Error Coefficient (= - AP/¿PV?)
-
Су
Specific Heat at Constant Volume (perfect gas)
f( )
Function of
h
Tapeline Altitude
ho
Density Altitude
hp
Pressure Altitude
ht
Temperature Altitude
Ahp
Altimeter Pressure Error Correction (= hp-hip)
J
Mechanical Equivalent Heat
L
Tube Length, Ft.
M
Mach Number ( = V/c)
n
Normal Load Factor or Exponent for Polytropic Process
P
Pressure
P;
Impact Pressure (equal to total pressure for M< 1)
Ps
Static Pressure
PI
Total Pressure, Lbs./Sq.Ft.
APS
Static Pressure Error Correction (Ps -p's)
AP
Pitot Pressure Error Correction (Pt-P't)
Q
Instrument Volume, Cu, Ft.
a
Incompressible Dynamic Pressure (= {P V2)
9c
P1-Ps = "Compressible" Dynamic Pressure
"'
TERMINOLOGY
(Continued)
ДЧc
Dynamic Pressure Error Correction (9c-9'c)
R
Perfect Gas Constant
r
Tube Inside Radius, Ft.
s
Entropy
S
Wing Area
T
Absolute Temperature
t
Time, Sec.
Tt
Total Temperature
V
True Airspeed
Vcol
Calibrated Airspeed
Ve
Equivalent Airspeed (= Vo)
Vi
Indicated Airspeed
AVcal
A.S.I. Pressure Error Correction ( = Vcal-V'cal)
W
Aircraft Weight
A
Increment
8
Relative Pressure (= Ps/Pso)
Y
Ratio of Specific Heats (Cp/cv = 1.4 for air) )
X
Lag Constant, (sometimes called time constant or characteristic time, corre-
sponds to RC in a simple RC electrical circuit), Sec.
Viscosity of Air, Lb./Sec. per Sq. Ft.
T
W
Pi
Relative Density ( = P/ Po)
Acoustic Lag, Sec.
$
Temperature Ratio ( = T/To)
P
Air Density, Slugs/Cu, Ft.
TERMINOLOGY
(Continued)
Superscripts and Subscripts
Observed or indicated conditions are signified by the use of a prime (1) superscript
whereas the true or free stream conditions carry no superscripts. The subscript (o) rep-
resents standard sea level conditions and the subscript (1) represents conditions aft of a
shock wave.
(t)
Denotes at a given time t
(t + o)
Denotes at a given time t to
(9)
First derivative of a quantity with respect to time.
FOREWORD
This first chapter of Volume 1 of the AGARD Flight Test Manual is devoted to the im-
portant topic of the speed, pressure and temperature measurements required in flight testing
work.
The chapter is divided into five basic parts:
(1) Introduction, Standard Atmosphere, Altitude and Airspeed Definitions
(2) Speed Measurement
(3) Pressure Measurement
(4) Temperature Measurement
(5) The Effects of Lag Error
The various portions of the chapter have been contributed by several nations, and slight
differences in the manner of approach to the problems by each group are illustrated in dif-
ferent sections. No attempt has been made to reduce the presentations to a single approach;
however, to avoid confusion, the terminology has been standardized as far as possible.
A certain amount of overlapping of the work of the various contributors exists; however,
the retention of this overlap is felt to be a desirable feature of the chapter since it adds to the
clarity of the individual sections.
Detailed instructions on the mechanics of conducting calibration checks are not included
in this chapter. However, it should be noted that before checking any system in the air, a
careful prior ground check should be conducted to determine if system leakage exists, whether
the instruments are functioning properly, and so on.
Moreover, as is the case with any type of flight test work, careful planning and proper
test scheduling can save much time, money and effort during the actual conducting of the
tests. Items such as these are the responsibility of the test engineering staff, and the decision
as to what comprises the proper method of approach in a given case depends primarily on the
equipment available and the accuracy required.
Finally, turbulent air conditions should be avoided during calibration checks of pressure
sensing systems, since there is no way to correct for the effect of turbulence.
1:1
INTRODUCTORY COMMENTS
In this first chapter we shall consider
the theory and the procedures required for
determination of speeds, pressures and tem-
peratures as well as the definition of the two
fundamental standard atmospheres.
1:2 THE NACA AND ICAN STANDARD
ATMOSPHERES
The performance characteristics (and in
certain cases, the handling qualities) of an
airplane depend on the nature of the atmos-
phere through which it flies. Because the
nature of the actual atmosphere changes
from day to day, it is not possible to define
airplane performance precisely without first
exactly defining the state of the atmosphere
at the time we expect to realize the specified
performance. This thought leads quite nat-
urally to the idea that to standardize per-
formance information we must work with
some type of standardized atmosphere where-
in there exist given values of temperature,
density and pressure at each altitude.
Both the NACA and ICAN standard atmos-
pheres approximate the standard conditions
existing at latitude 40° N and are identical
up to a standard altitude of 35,332 feet.
Above this altitude, the temperature is as-
sumed to be a constant -67°F (-55° C) in the
NACA atmosphere. On the other hand, in
the ICAN atmosphere it is assumed that the
temperature does not stabilize until an alti-
tude of 36,088 feet is reached, at which point
the temperature is taken to be -69.7°F
(-56.5° C).
Several standard atmospheres have been
defined, and two are in use by the NATO
nations. These are the NACA (National Ad-
visory Committee for Aeronautics) standard
atmosphere (used in the United States) and
the ICAN (International Commission of Air
Navigation) standard atmosphere used by
most of the European nations.
In both standard atmospheres the sea
level conditions are taken to be
Po
0.002378 slugs/cu.ft.
Po
2116 lbs. per sq. ft. = 760 mm.
Hg = 29.921 in. Hg.
T.
=
59°F : 518.4°R = 288° K = 15°C
Because standard atmospheres are, of
necessity, somewhat arbitrary statistical
averages, the conditions in the actual atmos-
phere seldom, if ever, agree with those of
a “standard" atmosphere. This means that
measurement of such items as ambient tem-
perature or pressure at a point in the actual
atmosphere cannot by themselves define (for
instance) the altitude of the airplane. Indeed,
as we shall see, only in the standard atmos-
phere are we able to define a “standard
altitude."
80
32.17 ft./sec2
In both, it is assumed that the equation of
state or perfect gas law holds
р
р
RT •
1:1
The major part of this first volume of
the NATO Flight Test Manual is devoted to
methods of determining from flight test in
the actual atmosphere, what the performance
would be at standard altitudes in the stand-
ard atmosphere. For this purpose, it is
first necessary to determine the ambient
conditions in the actual atmosphere as well
as such items as true airspeed and Mach
number of the airplane relative to the air.
where T is in absolute temperature units.
With p in lbs. per sq. ft., p in slugs per cu.
ft. and T in Rankine units, we have R 1718
ft.? /sec20R.
-
Again, in both atmospheres it is assumed
that the temperature lapse rate is 3.57°F
per 1000 feet of standard altitude. This
1:1
provides the equation
In the ICAN atmosphere we have
T(°F) = 59° -(3.57°h/1000)
1:2
р
8 :
Po
: e-d,
1:11
Both standard atmospheres are assumed
to be dry (relative humidity zero) and non-
turbulent. Therefore, the equation of balance
is
Р
: 1.33e-
Po
1:12
o
To
: 0.7519
1:13
dp = -pg dh
:
1:3
where
From Eqs. 1:1, 1:2, and 1:3 we find that
up to the isothermal region the following
equations hold for either standard atmos-
phere
-
di : 1.498 +
h-36088
20790
1:14
6.256
sen - (17)***
р
Po
T
To
(0) 5.256
(1-0.00000689h)5.256
1:4
Eqs. 1:4 through 1:14 completely define the
ICAN and NACA standard atmospheres. Tab-
ulated data for these atmospheres are given
in the appendices to this volume.
4.256
- (5.) .
(0)*.286
=(1-0.00000689h) *.266
1:5
2
T
= 1-0.00000689h
To
1:6
In addition to the two basic standard at-
mospheres considered above, certain varia-
tions of these are also used on occasion.
The variations simulate either hot or cold
days and in general presume the same pres-
sure variation with altitude as exists in the
basic standard atmosphere, but with different
temperature and density variations. The
location of the tropopause (or start of the
isothermal region) is, of course, different
from the standard locations when the varia-
tions are considered.
In the NACA atmosphere at altitudes in
excess of 35,332 feet, we have
р
-0-1
1:7
1.32e-1
1:8
Although it is well known that the iso-
thermal or stratospheric region of the actual
atmosphere is of limited extent, no account
of this is taken in the definitions of the ICAN
or NACA atmospheres, and it is assumed
that isothermal conditions extend indefinitely.
A
3
Foro
: 0.7508
Actually the assumption of constant tem-
perature is satisfactory up to altitudes some-
what greater than 20 miles. Since present
day piloted aircraft do not attain altitudes as
great as these, the existing standard atmos-
phere definitions are adequate as of this
time and for the purposes of this manual.
1:9
where
d=1.452 +
n-35332
20950
1:10
1:2
1:3
ALTITUDE DEFINITIONS
At any given altitude in the standard at-
mosphere there exist fixed values of tem-
perature, pressure, and density, which are
the standard values for that altitude. In the
actual atmosphere, however, we cannot ex-
pect to find (except by coincidence) that any
of the quantities (temperature, pressure, or
density) at a given distance above sea level
will have values equal to those of the stand-
ard atmosphere at the same height.
1:12 for h, the resulting relations would de-
fine an altitude which is actually a measure
of ambient density. This altitude is called
density altitude, symbolized by hd or Hd.
Because the power required characteristics
of a given airframe at speeds lower than
drag divergence are a function only of density
(insofar as the atmospheric conditions are
concerned) density altitude is frequently
used as a base reference when the perform-
ance of the propeller-driven airplanes is
being considered.
(c) Temperature Altitude, he
Therefore, since many performance char-
acteristics depend on ambient conditions
rather than on actual or tape line altitude,
we frequently find that we are not concerned
with true height, but rather with such things
as pressure altitude, density altitude and
temperature altitude. Before defining these
quantities, we note that tape line altitude is
significant in evaluating climb performance
and obstacle take-off and landing perform-
ance.
The altitude in the standard atmosphere
corresponding to a given ambient tempera-
ture is called temperature altitude, sym-
bolized by hy or Ht. In common with pres-
sure and density altitude, temperature alti-
tude is not a measure of actual altitude,
se, but rather is a measure merely of am-
bient temperature,
(a) Pressure Altitude
(d) Standard Altitude
The altitude read on the dial of an alti-
meter set to 29.92 inches of mercury is
called pressure altitude. In the United States,
altimeters are calibrated according to Eqs.
1:4 and 1:7. Therefore, if we solve these
equations for h, the resulting relations de-
fine the pressure altitude in terms of ambient
pressure. Pressure altitude, symbolized by
hp or Hp, is actually a measure only of am-
bient pressure and clearly is not a precise
measure of the actual distance above sea
level.
Any altitude in the standard atmosphere
is, by definition, a standard altitude; i.e.,
the pressure, density and temperature all
have their proper values simultaneously. In
the actual atmosphere we would be at a stand-
ard altitude, if for a given pressure altitude,
we had the standard atmosphere value of
temperature altitude (and of course, by virtue
of the equation of state under these circum-
stances, we would also have a density alti-
tude equal to the pressure altitude).
In the NATO European countries, alti-
meters are calibrated according to Eqs. 1:4
and 1:11; however, the significance of pres-
sure altitude is precisely the same as in the
United States.
This does not mean that we would know
how far we actually were above sea level.
Rather it simply means that we could find
some altitude in the standard atmosphere
where the conditions were the same as those
existing about us in the actual atmosphere.
As we have noted previously, it would be the
result of pure coincidence if we were actually
to locate a standard altitude point in the
actual atmosphere.
(b) Density Altitude, hd
If we were to solve Eqs. 1:5 and 1:8 or
1:3
difficult to measure air density while aboard
an airplane, and accordingly, it is general
practice to measure only temperature and
pressure. The density, if required, is then
computed using the equation of state or suit-
able charts prepared for this purpose,
Because we cannot generally hope to con-
duct tests at "standard altitudes" even
though we must finally present our test data
in terms of "standard altitude", we must
devise ways and means to correct data ob-
tained in the actual atmosphere to standard
altitude conditions. For instance, if we con-
ducted our tests at a given pressure altitude,
in general, both temperature and density
would be non-standard, and if we wished to
consider that the pressure altitude was the
standard pressure altitude, we would have
to make data corrections for the non-standard
density and temperature.
1:4 SPEED DEFINITIONS
Similarly, if we presumed that we could
consider the density altitude as standard,
corrections would have to be imposed for
non-standard temperature and pressure.
Just as the definitions of altitude given
above are associated with standard atmos-
pheric conditions, so also are the definitions
of measured airspeed.
The conventional airspeed indicators in
use by all the NATO nations are actually
differential pressure gages calibrated ac-
cording to the law of frictionless adiabatic
flow, presuming that the ambient conditions
are those of the standard atmosphere at sea
level. For subsonic flow the calibration
equation is derived from the isentropic flow
relations and for supersonic flow the calibra-
tion equation used assumes the existence of
adiabatic flow with a normal shock located
just forward of the total pressure pickup.
For subsonic speeds, therefore,
>
A data reduction procedure is frequently
identified by the atmospheric quantity held
constant during correction. Thus, the first
case of the preceding paragraph represents
a “pressure altitude" method of data re-
duction, whereas the second represents a
“density altitude" procedure. Obviously,
we also have a third basic procedure and this
is the "temperature altitude" method.
v2
/
20
veoma o 1) 516-1)
[ +-
Dp-Ds
Ds
(e) True, Tape Line or Geometric Alti-
tude
1:15
and for supersonic speeds
-
Dri-Ds
Ps
(y+l) v27 4114-1)
YIY
2c2
The actual distance of the airplane above
sea level is called either true, tape line or
geometric altitude. Normally, true altitude
can only be determined by radar equipment.
It cannot be established by instantaneous
reading of the pilot's flight instruments,
although it can be computed with some pre-
cision on the basis of data recorded at
various altitudes by a climbing airplane.
This latter topic is discussed in Chapters 4
and 7 of this volume in connection with en-
ergy climb computations.
[**]
[v-
)
Y+1
1-y+2
1
]
*zrya)"17-1
1:16
.
Pol
Because the equation of state relates
pressure, density and temperature, the meas-
urement of any two of these quantities auto-
matically determines the third. It is rather
when Pr : free stream total pressure
: total pressure aft of normal shock
Ps
: free stream static pressure
Y : specific heat ratio
free stream sonic speed
8
1:4
or
YPSO /Vcal
Because the airspeed indicator only senses
the quantity Pt-Ps or Pt, -Ps, the indicator
dial is calibrated according to the assump-
tion that c has its sea level value of co and
that Ps, when it appears alone, has the value
Pso Therefore, taking Psoas 2116 psf,
,
y = 1.4 and co : 1117 fps. The calibration
equations are
.
oli
)
1/Vcal
+
4
Со
ac
2
Со
.
But
.2855
Pt-ps
Vcal
: 2495
+1
-1
YPSO
2116
ce
Po
1:17
2
so that
for subsonic flow where Vcal is in feet per
second, and
Povcal
Vcal
.2835 Véal
4c =
-Cina
2
4
Pt, Ps
2116
1272.5
1:19
cal
1:18
for supersonic flow.
Eq. 1:19 is an excellent approximation to
1:17 throughout the Mach number range to
which it applies. Therefore, for practical
purposes the British and American defini-
tions of calibrated airspeed may be con-
sidered the same. Inasmuch as the other
members of NATO use the same basic cali -
bration equations presented herein, we con-
clude that except for the units used to mark
off the dial, the term calibrated airspeed
has the same meaning in all the NATO
countries.
In Great Britain a simplified version of
Eq. 1:17 is employed. The equation is ob-
tained from 1:15 as follows:
1
Let
ac :
Dr-Ds.
Then from Eq.1:15, letting c =
= co and p = PSO
.
-
ac
pt -ps
Y-1
Vcal
27 Y/17-1)
DSO
o {[+ they }
(
+
2
Со
1:18A
Several other airspeeds are frequently
referred to in flight test literature. These
are: (1) equivalent airspeed, (2) true air-
speed and (3) indicated airspeed. Before
Mach numbers in excess of about 0.4 were
attained, equivalent airspeed was generally
referred to as indicated airspeed. However,
since the indicator reading at higher Mach
number differs considerably from the equiv-
alent airspeed, the two definitions (indicated
airspeed and equivalent airspeed) are no
longer interchangeable. The relationship be-
tween the various airspeeds is given below,
V; • Indicated airspeed as shown on
the instrument dial, when cor-
rected for lag errors, instru-
ment error and position error,
becomes calibrated airspeed.
We next expand the bracketed quantity
using the binomial expansion, whence, re-
taining three terms only,
2
y
Vcal
1 +
()
2
Со
ac :
pso
@ly
+
y (Vcal
8 CO
Wca * -
1:5
Vcal - Calibrated airspeed, when cor-
rected for the difference be-
tween sea level pressure and
actual ambient pressure,
be.
comes equivalent airspeed.
Normally it is quite difficult to measure
lag errors while in flight, particularly for
the case of acceleration runs, Ground checks
are generally relied upon to establish sep-
arately the lag constants for the Pitot and
static systems.
Ve : Equivalent airspeed, when cor-
rected for the difference be-
tween ambient and sea level
density, becomes true airspeed.
VT
: True airspeed
At this point we shall consider briefly
the relation between the speeds listed above.
A detailed treatment is given later in the
chapter.
Since the lag in a given system depends
on the rate-of-change of pressure being
applied to the system, and since in flight
this rate of change must be established on
the basis of indicator readings, it is gen-
erally not possible to assume that the over-
all lag error correction can be made with
a precision of more than 80 to 90%. Good
design practice, however, can reduce the
system lag errors under any set of con-
ditions to a small percentage of the quantity
being measured, in which case precise cor-
rections are not required for practical work.
The topic of lag errors and their determina-
tion is considered in detail in sections 1:20
to 1:25 of this chapter.
9
Relation Between V; and Vcal
On the basis of a calibration conducted
on the ground, the inherent error in a given
airspeed indicator or altimeter may be de-
termined as a function of the indicator read-
ing. This error is called instrument error,
and is subject to change due to wear of the
instrument and other similar factors. Since
instrument error changes with time, frequent
calibration is required if precise results are
desired.
Once corrections for instrument and lag
errors have been imposed, position error
may be accounted for and suitable corrections
made. In this connection we note that under
steady level flight conditions, there are no
lag errors and when correcting data for
cases such as these, position error correc-
tions are made immediately following the
instrument error corrections.
Corrections for instrument error should
be imposed prior to any other correction.
Following these corrections, the effects of
lag should be considered and suitable cor-
rections made. In this connection we note
that lag errors are present only when the
airplane is changing altitude or speed so
that the flow about the airplane is of an
unsteady nature.
For most test work at subsonic speeds it
is presumed that all of the position error
originates at the static source (static pres-
sure pickup). This is a reasonable assump-
tion inasmuch as flow perturbations due to
the airplane which can affect a properly
located total head probe are very nearly
isentropic in nature, and accordingly, do not
effect the total pressure.
Under these circumstances, there is some
question as to the exact value of the position
error since it is known that the value of the
pressure coefficient is affected by unsteady
flow conditions. Provided the position error
has a reasonably small value, any change
due to unsteady flow will be of second order
and may be ignored.
Of course, if the total head tube is located
behind a propeller, in the wing wake, in the
boundary layer or in a region where the flow
has been influenced by transonic shock for-
mation or in a region of localized supersonic
flow (such as in the case where the total
head tube is located on a wing boom of a
1:6
)
swept-wing airplane where localized super-
sonic flow regions may extend past the nose
of the boom), there will exist total head
position error which must be accounted for.
Now calibrated airspeed (Eds. 1:17 or
1:19) is a function only of ac; consequently,
at a given pressure altitude, there is a unique
relation between calibrated airspeed and
Mach number so that at a given pressure
altitude
Normally, however, it is possible to lo-
cate the total head pickup properly and thus
avoid these difficulties. It is most desirable
to do this inasmuch as such things as lo-
calized supersonic flow regions produce
rather erratic readings.
Ap = f(nw, Vcal).
:
This is also true under supersonic con-
ditions since for Y : 1.4, Eq. 1:16 becomes
y =
Pt, Ps
166.9 M2
1.
2.5
Ps
Position error is determined principally
by the value of the pressure coefficient at
the location of the static source. As is well
known, the pressure coefficient in turn is
determined by the value of the airplane lift
coefficient and the flight Mach number. For
calibration purposes, however, it is neces-
sary to determine the relation between posi-
tion error and calibrated airspeed. This is
accomplished as follows:
M2
1:22
We have
Comparing this with Eq. 1:18, we see that
under supersonic as well as subsonic con-
ditions, there exists a unique relation be-
tween Vcal and Mat a given pressure altitude.
Frequently, weight and load factor effects
are neglected when presenting position error
data; however, for airplanes carrying large
fuel loads and whose weight accordingly may
change markedly during the course of a
flight, the "W" effects should be taken into
account.
Ср
q
4 = m)
뿡 ​f(CL,M)
f
M]ofrow,p, m)
nW
M
Үрм?
S
2
1:20
where Ap is the position error in lbs. per
sq. ft. and n is load factor acting on the air-
plane. Since q = y pM²/2, we see that Ap is
itself a function only of nW, p and M.
Under the assumption that the position
error is produced only by pressure coeffi -
cient variation at the static source, it is
possible to relate altimeter position error
directly to airspeed indicator position error
(since in most installations both utilize the
same static source). This thought is the
basis of the altimeter depression method
used to calibrate airspeed systems by meas-
uring the altimeter error as a function of
calibrated airspeed at given pressure alti-
tudes.
The equation for Mach number under sub-
sonic flow conditions is obtained by dividing
both sides of Eq. 1:15 by c?, whence
2
P-Ps
/y
M?
1 0 +,7-10,-
Y-1
Ps
1:21
Provided we are dealing with position
errors of small magnitude (which is nor-
mally the case when the magnitude of these
errors is limited by design specifications),
it is possible to develop rather simple equa-
tions relating altitude, calibrated airspeed
and Mach number errors to one another at a
and we have that the Mach number is deter-
mined by only two quantities, ac and ps.
1:7
Using Eq. 1:23 for Ap, we obtain a direct
relation between AVcal and shp,
given pressure altitude. The relations be-
tween pressure altitude and calibrated air-
speed position errors are developed below.
Similar relations for Machmeter error are
given in section 1:5.
2
2.5
Vcal
Anp
AVcal
:
[1+0.2 (copy)"
go
Со
For small errors, we may write the at-
mospheric balance equation in the form
1:25
Δρ - - P9 Δhp
pg
- -
1:23
where the density ratio o has the standard
atmosphere value corresponding to the test
pressure altitude.
where p has its standard atmosphere value
for the indicated test pressure altitude. This
is the differential form of altimeter calibra-
on equations written as a finite difference
equation.
We may next obtain the differential form
of the airspeed calibration equation from
Eq. 1:18A. If we presume that the position
error arises from static source error only,
then the total pressure in 1:18A is a constant
and taking the differential of both sides of
Eq. 1:18A leads to
From Eqs. 1:24 and 1:25 it is seen that
the relation between Ap and AVcal is a func-
tion only of Vcal whereas the relation be-
tween A hp and Vcal depends also on the
actual test pressure altitude.
It is of course possible to write “exact"
equations relating the various position er-
rors, and chart solutions of these "exact"
equations are available (see Ref. 26, for in-
stance). However, greater precision than
provided by Eqs. 1:23, 1:24 and 1:25 is not
required except when dealing with exces-
sively large position errors. The topic of
pressure errors is covered in detail by the
section of this chapter prepared by Messrs.
Shields and Utting and the discussion here
serves merely to introduce the topic.
Y-
1
dps :
PSO
fiziki
Y-1 (Vcal
2 Со
Y-1
Vcal dVcal.
co
For
y
: 1.4 this gives
Once the increment AVcal has been ob-
tained, the indicated speed is corrected ac-
cordingly and Vcal is obtained. In practice,
position error is determined using the alti-
meter depression method, a trailing static
bomb, a swiveling static head, the speed
course procedure, radar tracking, pacing
airplanes and the like. By far the most pop-
ular procedure is the altimeter depression
method. At the present time, it is generally
assumed that no position error exists at su-
personic speeds for a properly located static
source.
2.5
dps
dVcal
povcal 1 + 0.2
Ро
[1+(
or for small finite errors
2.5
Δρ
Vcal
voodit
ve (
[
Po Vcal 1 +0.2
Relation Between Vcal, Ve and V
(ve
Alcal
The equivalent airspeed is of importance
for it is a direct measure of the free stream
со
1:24
1:8
1:5 MACHMETERS
8
dynamic pressure (q=1/2pV': 1/2 po Ve )
and is frequently used as a basis for reducing
flight test data for piston-engined airplanes
(see Chapter 6). Equivalent airspeed is de-
fined by the relation
Ve : Vo
In the United States direct reading Mach-
meters are available for use as standard
panel instruments. These units are similar
to existing airspeed indicators with the ex-
ception that an additional element is added
which senses ambient pressure. This aneroid
diaphragm is connected to the instrument
linkage in such a fashion that the instrument
is capable of solving Eqs. 1:21 and 1:22
directly.
1:26
where Ve
s
equivalent airspeed
V =
true airspeed
o: P/Po where p is the actual am-
bient density.
Consequently from Eq. 1:15
Because the Machmeter is a somewhat
more complex instrument than the airspeed
indicator, less reliance is placed on it for
test work than on the airspeed indicator. How-
ever, the Machmeter is nonetheless a valuable
instrument.
22: (* )3=117-
-
-1)/
+
Y -
1:27
Just as the airspeed indicator and alti -
meter are subject to position error, so also
is the Machmeter. To determine the rela-
tion between the static source pressure
error and Mach number error, we proceed
as follows:
and from Eq. 1:17 we have
Dr-ps
1)/y
Solving Eq. 1:21 for Pr/Ps, we have for
ve
ps
Y: 1.4
+.)'=17-1
x
8
2
Vool
Pt
(09 -Ps +.)""
-1)/y
3.6
PSO
PI
. (1+0.2002"
[
1:28
where 8 = Ps/Pso.
ps
and taking differentials of both sides hold-
ing Pt constant, we have
)
dps
1.4 MPS
.
OM
1 +0.2M2
From Eq. 1:28 we see that to determine
equivalent airspeed all we need to know are
pressure altitude and calibrated airspeed.
To determine true speed, Eq. 1:26 is ap-
plied to the value of Ve obtained from Eq.
1:28 (provided the ambient pressure and
temperature are known). Charts relating
Ve to Vcal are given in an appendix to this
volume. Other approaches to the problem
of determining true airspeed are given in
the portion of this chapter prepared by Mr.
Davy.
or as a finite difference equation
Ap
1.4 MP s
il
AM
1 + 0.2 M2
1:29
1:9
and
Using Eq. 1:23 in Eq. 1:29, we obtain the
relation between Ahp and AM,
voor
ca.
Ahp
YPs
P
1.4 MPs
M2
AM
(1 + 0.2MP)pg
1:35
1:30
where
e has the standard atmosphere value
for the test pressure altitude.
From Eqs.1:29 and 1:24 we have
In the following sections of this chapter,
calibration of the airplane's speed, pressure
and temperature measuring systems is con-
sidered from several different viewpoints.
It should be noted that as far as airspeed
and altimeter systems are concerned, the
prime object of calibration tests is to deter-
mine position or pressure error.
AM
ΔVcal
AM. Ap
.
Ap Alcal
Polcal
.(1 + 0.2M2
Ncal 12/2.3
we [+02
1:6 SPEED MEASUREMENT
1.4 MPS
Со
1:31
The purpose of this portion of Chapter 1
is to explore the currently used methods of
determining airplane speed. Although other
procedures will eventually be developed, we
shall here consider only those which depend
on pressure measurements on board the
plane and, in some cases, the measurements
of ambient air temperature.
Relations between Mach number, Vcal, V
and Ve are also readily obtainable. To re-
late Vcal to M, we take the ratio of Eqs.1:17
and 1:21, whence
Pt-Ps
-1)/y
In this presentation the following defini-
tions apply:
+1
.)%/y
cal
pso
- có
(a) True Airspeed, V
M2
(Po-Ps Jy-1)/y
)
Ps
+1
1.)="12-
1:32
The speed of the airplane's center of
gravity with respect to still air is called
true airspeed or simply true speed. The
true speed is also known as the relative
velocity* or aerodynamic speed. We note
that airplane ground speed is the vector sum
of the wind velocity and the true airspeed.
Similarly for Ve and V we have
9 = { pove? - ¿pv? : { xpsMe
:
1:33
(b) Calibrated Airspeed, Vcal
from which
The true speed which an airplane would
ve?
ур
-
CO
M2
Ро
* Editor's note: The reverse of this velocity
is referred to as the relative wind,
1:34
1:10
1:7
PRESSURE AND TEMPERATURE
PICKUPS
have at sea level under standard conditions
(hp : 29.92 in.Hg, T = 59°F) for the same
difference between total and static pressures
(P7-Ps) as exists for a given flight is called
calibrated airspeed.
In section 1:8 we shall see that to deter-
mine speed it is necessary to measure the
total pressure Pt, the ambient static pres-
sure Ps, and also an air temperature indica-
tive of ambient conditions.
Calibrated airspeed is of interest for the
following reasons:
(a) Static Pressure
(1) It is measured aboard the air-
plane by a simple,precise instrument amount-
ing to a differential pressure gage (assum-
ing that the correct total and static pres-
sures exist at the pickups). Measurement
of any other quantity, such as V, Ve (equiv-
alent airspeed) or M requires a more com-
plex system.
Static or ambient pressure (as would be
measured by a balloon, for example) is dif-
ficult to determine with precision aboard
an airplane, and the pickup or static heads
which measure it must be calibrated in
flight as discussed in section 1:9.
(b) Total Pressure, Pt
(2) Calibrated airspeed is equal to
true speed at sea level under standard con-
ditions.
(3) It provides a positive measure of
take-off and landing speeds regardless of the
altitude of the ground or the local barometric
pressure (provided the weight is known).
Total pressure exists at the end of the
total head or Pitot tube, i.e., where the rel-
ative velocity has been reduced to zero and
stagnation conditions prevail. It is supposed
that frictionless flow of the air exists up-
stream of the stagnation point, and therefore
the stagnation or total pressure represents
the maximum pressure encountered at any
point on the airplane. This supposition may
be used to verify the accuracy of the total
head probe since, if two different Pitot tubes
give the same pressure for all cases of
stabilized flight (in which case, there is no
lag), we may be quite sure that both are
correct.
(4) It provides the pilot with a meas-
4
ure of the dynamic pressure, qc, on which,
above all, depend the high speed structural
problems.
(c) Mach Number
The ratio of true speed to the speed of
sound in the ambient air (V/c) is called the
Mach Number. In meters per second,
c: 20.05VT where T is the absolute (Kelvin)
с :
ambient temperature measured in degrees
Centigrade. In feet per second with T in
degrees Rankine, c : 49.1VT.
If possible, the total head tube should be
directed into the airstream; however, it is
known from experience that even consider-
able angles of yaw do not appreciably affect
the measurement of Pitot pressure. On the
other hand, care must be taken to insure
that the upstream flow is undisturbed and
that the total head pickup is not located aft
of a shock wave*, in the slipstream or in any
At this point we merely mention the equiv-
alent airspeed Ve, which is a measure of the
incompressible dynamic pressure (9 :
1/2 Pole?). Ve is related to true speed by
the relationship Ve : vo. It is not pos-
sible to measure Ve directly.
*At supersonic speeds a shock always forms
ahead of the total head tube; we shall see in
section 1:8 how this fact is accounted for.
1:11
region where the total pressure is modified
by boundary layer effects, etc.
Writing this equation between the free
stream and a stagnation point
(c) Total Temperature, To
v2/2 + JCpT
Јерте
or
va
2Jcp (Ty-T).
.
1:37
Total temperature is related to static
temperature, T, and results from adiabatic
compression of the air to stagnation or total
conditions. We shall see later how total and
static temperature are related to each other.
Static or ambient temperature can be ob-
tained from meteorological surveys or meas-
ured by the airplane.
This formula relates the temperature
rise* (T-T) to speed and theoretically allows
us to determine speed by measurement of
two temperatures; however, this cannot easily
be done aboard the plane.
1:8 DEVELOPMENT OF ALRSPEED
EQUATIONS
(a) Equations for V, M and Vcal at Sub-
sonic Speeds
In this section the classical thermody-
namic notation given below is employed.
If no normal shock exists between the
free stream and the stagnation point, the
compression is isentropic and therefore
obeys the law
p : Gas pressure
P = Density
2
constant.
T: Absolute temperature
1:38
J: Mechanical equivalent heat
Since air may be considered a perfect
gas, the perfect gas law
Ср
Specific heat at constant pres-
sure (perfect gas)
р
RT
1:39
CV : Specific heat at constant vol-
ume (perfect gas)
applies. On eliminating p between Eq8. 1:38
and 1:39, we have (between two points 0 and
1)
Y
: Cp/cv : 1.4 for air
To
(),
Mo
Y
Y-1
Y
8
R = Perfect gas constant = J(cp-cv)
:
: 1718 f?/sec2 °R
It is assumed that the energy equation
(which holds for an adiabatic process (re-
versible or not) applies; i.e.,
1:40
(1) Equations for true airspeed
vº/2 + JCPT : constant.
We may also write Eq. 1:37 as
1:36
ve 2 Jept (+-
:
)
* Editor's note: The original paper used the
word isentropic, but since total temperature
is the same as long as a process is adiabatic,
this more general term was substituted.
*Since this applies to any adiabatic process,
Eq.1:37 holds subsonically as well as su-
personically.
1:12
1
and substituting Eq. 1:40
With y : 1.4, we find that
DA
V*:23 697 [C:)* -]
= (1+0.2 M2)3.5
Ps
v
: Cp
т
.
1:46
1:41
Similarly from Eq.1:5
This equation gives the true speed in terms
of
Ps, PP
and T. If we know the total tem-
perature, we may also write
T+
= 1+0.2M2.
T
1:47
1
-
8
?: 2 J Cp To
Ti
pt: [:-(43)*""]
Ps
(/
P+
Eq8. 1:46 and 1:47 show that the Mach
number can be obtained in terms of a single
parameter, which is
1:42
From the foregoing we see that V is a
function of the two parameters, T (or T.)
and (Ps/pp)
either the ratio of total to static
pressure
(2) Calculation of Mach number
or
the ratio of total to static
temperature.
By definition, M : V/c where c is the
speed of sound in the free stream (pressure
Ps, temperature T, density p), and
(3) Equations for calibrated airspeed
ca
=
you
(Y: 1.4)
By definition, calibrated airspeed is a
function of the total and static pressures,
P, and Ps, and corresponds to the true speed
under sea level standard conditions.
since
Eq. 1:44 may be written
ale
2
RT
Dp-ps
v2 =
- 2.2.1 ( A * +, 97-117-11
]
2 y
RT
Y-1
Ps
c2 : YRT.
1:43
Ср
Letting 9c * PA-Ps and replacing T by To and
p by Po, we have the definition of calibrated
airspeed
In Eq. 1:41 we shall replace J by
(Y/(y-1) R, which is its equivalent. Thus,
Р
Poly-1)/y
v2 :
27
RT
ac
2yRTO
2.4
57-1
vCal
8
C +.) 3061-10
Y
Y-1
Ро
1:44
1:48
or, using Eq. 1:43
It is easy to show that
2
WAY
M2
Ps
2y
vě = (
•v%). * : 016-1
(s [
8
yol
Po
1:45
1:13
V2
(b)
M2
and it may be seen that Ve cannot be deter-
mined in terms of a single parameter as
=
YRT
can Vcal
(c)
Yol
jylly-1)
De +7
Po
ps
M2
(b) Practical Considerations in the Sub-
sonic Range
2
(d)
Y-1
T.I
Assuming that the pressures Pr and Ps are
known and have been corrected for position
and lag errors, we may compute ac : Pt-Ps
which gives us Vcal directly (using a table
or graph representing Eq.1:48). Using the
ratio Pr/Ps, we may similarly obtain M (from
a table or graph representing Eq. 1:46).
T+
T
1+
블
​M2
2
Therefore, we have three independent
variables.
The entropy is defined as
It may be noted that the pressure altitude
hp, calibrated airspeed Vcal and M are all
determined by the two quantities P, and Ps;
consequently, a relationship exists between
hp, Vcal and M. A chart graphically present-
ing this relationship is given in the appendix
to this volume.
(e) s = Cy log(Ps/py) + constant.
The properties of the fluid across the
shock are related by
(1) PV : PV, (continuity)
(c) Equations for M, V and Vcal at Su-
personic speeds (Normal shock in an adia -
batic flow of a perfect fluid)
(g)
Y-1
2y
Ps
my verloren y vp + PSI
Ya lo psi
2y
Considering the uniform one-dimensional
flow ahead of a normal shock (no subscript)
and behind the shock (subscript 1), the in-
variant characteristics of the fluid in these
two flows are the two known specific heats,
Cp and cy, and as usual we assume
(conservation of energy)
(h)
(PV) V + ps = 1, VSV, Psi
(م)
R: Jlcp - cv) = constant
R
y : Cp - CV
(conservation of momentum).
In each of the two flows, the seven quan-
tities (speed V, density p, static pressure P,
total pressure Pt, static temperature (abs) T,
total temperature Ty and Mach number M)
are separately related by the four relations
Considering (d), relation (g) shows that
Ty : Tt, ; consequently, if we know at least
three of the upstream characteristics, re-
lations (f), (g) and (h) permit determination
of all of the characteristics of the down-
stream flow, as functions of given properties
of the upstream flow. (This is merely a
matter of algebra.)
(a)
Ps
2
وا
RT
(for a perfect gas)
Using the down stream relations, it is
1:14
demonstrated in many text books that the
following relations hold across a normal
shock.
pickup measures "P,.”, and the static source
“Ps”. Thus, using the Rayleigh Pitot For-
mula, the flight Mach number M may at once
be determined.
V =
) °
i = v'y-1)M? +2
(-) M?
If the ambient air temperature is known,
the true airspeed is obtained from
V: MY RT
PI
Ро
(y + 1)M?
(y -1) M° +2
T may be determined by atmospheric
soundings or by using a properly calibrated
low lag total temperature pickup and the
equation
2 ym2-ly-1)
Psi Ps
Y+1
T :
Ti
M2
(with y : 1.4).
I +
ylly-1)
y+l
5
M2
2
Pt, Ps
-
2)
[ + "]"!
Consequently, the true airspeed is
2y
.(M2-1)
M
(which is known as the Rayleigh Pitot
Formula)
SYRTI
M2
5
2ly-1)
(M2-1)(1+y)
T
- r[+
+
(x + 1)^
M?
Th
Tt
As far as the airspeed indicator is con-
cerned, it should be noted that at supersonic
speeds it measures (Pt, -Pr) where Pr differs
from the true static pressure by the position
error. The indicator does not measure the
difference between free stream total pres-
sure Pr or Pi
and P, as in the subsonic case
where the impact pressure and total pressure
are identical (since there is no shock sub-
sonically).
(x-1) M? + 2
MP, -
2 Mº-tr-)
Calibrated airspeed is always referred
to as an indicator of the difference between
Pt and Ps: This correspondence between
speed and pressure difference is based on
the fact that calibrated airspeed is the same
As is well know, the last equation of the
above group demonstrates that if a normal
shock exists, then the flow upstream must
be supersonic and the flow downstream sub-
sonic. In practice, if a standard airspeed
system is used with a total head pickup lo-
cated well ahead of the airplane and approxi-
mately along its centerline, and if a static
pressure probe, corrected for lag and posi-
tion error, provides the correct ambient
static pressure, we shall have that the total
* Editor's note: Supersonic Machmeters us-
ing the Rayleigh formula for calibration
: 1.4 are available for direct Mach
number indication.
with y
1:15
parameters, angle of attack and the Mach
number M. We shall disregard the effect
of Reynolds number.
as true speed under standard sea level con-
ditions, assuming a perfect instrument, no
position error and no lag. Thus, up to Mach
(1), under standard sea level conditions
(P1-Ps = 905 mb - 1117 psf), the corre-
spondence must be based on the isentropic
or subsonic relation between pressure and
speed, whereas above P.-Ps : 1117 psf, we
.
must use the Rayleigh Pitot Formula.
Presuming that a suitable location of the
static orifice has been determined which
minimizes errors as much as possible, we
may determine the amount of residual posi-
tion error in terms of Q and M.
1:9 METHODS OF MEASUREMENT AND
ERRORS
Pr
Let Ps be the true static pressure and
be the pressure sensed by the pickup.
We then define the static position error as
dps =
3
Pr - Ps.
To determine the pressures P, and Ps,
pressure pickups or probes are used. We
shall now discuss the characteristics of
these pickups.*
We further define
(a) Total Pressure
Ap
.
Pr - Pr
and then form the ratio
dos
Measurement of total pressure presents
no particular problems provided the total
pressure probe is placed ahead of any shock
waves formed by the airplane, itself. This
condition is essential, for it is difficult to
imagine any way of correcting for the errors
which would result were it not fulfilled. The
shock wave due to the pickup itself is, of
course, considered in the calibration equa-
tions discussed in the preceding section.
Δρ
which we designate as “relative static er-
ror." We recognize that
(b) Static Pressure
dps
Δρ
: fla,M)
and a dimensional analysis establishes that
The static pressure pickup may be located
on the side of the fuselage or on a suitable
probe. Unfortunately, it is seldom possible
to find a location where true ambient pres-
sure is sensed because the pressure field
at all points in the vicinity of the airplane
(at subsonic speeds)** is generally a func-
tion of speed and altitude, and the secondary
dps
mon,
)
Ap
Pr
Δρ
SPT
1:49
where m =
airplane mass
g :
gravitational acceleration
* The topic of lag errors due to changing
pressures is discussed in sections 1:20
through 1:25 of this chapter. Here we con-
sider only stabilized conditions.
** Editor's insert
S
-
wing area
n
: load factor.
1
1:16
The foregoing chart may then be used as
the basis of correcting for variations in
weight and load factor.
This relation may be deduced from dps/Ap
: f(a,m), since P/P, is directly related to
the uncorrected Mach number and for each
value of Ap/Pr, mgn/Spr is directly related
to CL and hence too.
Eq. 1:14 may also be written
(c) Total Temperature
dps
mgn
S
Δρ Pr
mgn'mon
S S
10.
The total temperature pickup may be
placed at almost any location on the airplane
but not in the boundary layer. Since total
temperature is invariant through a shock, it
is not necessary to avoid the formation of
airplane shock in the air ahead of this pickup.
1:50
For a given airplane, S is a constant, and
a standard mass, mo, may be established
for the given set of tests. Variation of weight
will then be accounted for in the load factor,
n, which under these circumstances repre-
sents the angle of attack, Pr, and Ap which
defines altitude and speed. Thus we may
write
A sufficient flow of air must be maintained
through the pickup to avoid thermal lag. How-
ever, the flow must not be large enough to
prevent the existence of essentially stagnant
conditions at the pickup.
dps
F
ܝ 42 ܝ
은
​용​)
O.
1:10 METHODS OF CALIBRATING
THE AIRSPEED SYSTEM
Consequently, when the relationship be-
tween P, P, Ps, and n has been determined
by means of the calibration procedure of
section 1:10, the following chart may be
prepared,
When speed is determined by measure-
ment of the quantities, Pp, Ps and Ty, we
must, in theory, provide a calibration for
each of these parameters. At this point we
shall consider the quantities Pt
and
Ps.
Two different procedures are used to
provide pressure calibration. These are:
(1) The speed course method
(2) The altimeter depression method
wherein we determine the static pressure
error dps by determining the flight altitude
of the airplane and then measure the true
static pressure for compariso: with the in-
dicated static pressure.
The combination of these two methods may
be used to determine total pressure error;
however, total head errors (other than lag
error) are normally insignificant and gen-
erally it is assumed that all position error
originates on the static side.
-16
Fig. 1:1
1:17
(a) The Speed Course Method
This method may be applied at low or high
altitudes using some means of precisely de-
termining the time of flight between two fixed
points. The method is subject to numerous
.
errors, the worst being due to wind effect
which alters the ground-airspeed relation-
ship. Runs in opposite directions tend to
minimize the wind errors.
run through the test air mass is first con-
ducted to establish the relation between tape
line and pressure altitude at the time of the
tests, Obviously, the pacing procedure is
also useful, particularly at low speeds with
the calibrated pacing airplane having a lower
stalling speed than the test vehicle.
Thus we may determine positive error
under various conditions of speed and alti -
tude at load factors near one. Test calibra-
tion for load factors greater than one is pos-
sible by using radar altitude determination
during turning flight.
We must also have knowledge of the true
temperature of the air through which the
plane flies (see Eqs, 1:41 and 1:42), since
temperature has an important effect. For
these reasons it is often simpler to obtain
a precise calibration using the altimeter
depression procedure.
1:11
PRESSURE MEASUREMENT
(b) Calibration of Static System for Po-
sition Errors
To determine the speed and pressure al-
titude at which an aircraft flies, the "free
stream" values of dynamic and ambient
'
static pressure are required. The pressures
registered by the aircraft pitot-static sys-
tem will, in general, differ from these be-
cause:
To determine the true static pressure at
the test altitude, different procedures are
employed which depend on the test altitude
and speed.
(a) the pressure field of the aircraft
causes the local pressures at the measuring
sources to differ from the “free stream"
values,
there are errors in registration of
the local pressure by the source used (nor-
mally a pitot-static head).
At low altitudes, we employ tower runs
wherein the airplane flies past a tower of
known height, and its height with respect to
the tower is established by photographic or
other means. Comparison of the “actual”
pressure altitude at which the airplane is
flying, determined by adding the pressure
altitude at the top of the tower to the height
above the observer, with the airplane indi-
cated pressure altitude provides the static
error. (This procedure is satisfactory as
long as the pilot flies by at approximately
the level of the observer. Otherwise, serious
errors may result.) The runs are made at
various speeds to obtain a plot of position
error as a function of speed,
The resulting errors are called pressure
or position errors and the associated cor-
rections to be added to the recorded values
to obtain "free stream" values are pres-
sure error corrections. Thus we have
tion :
(1) Static pressure error correc-
Ps-P's : APS
At altitude the same principle is utilized
with the test airplane flying past a calibrated
"base" airplane. Radar methods may also
be employed to determine high speed errors
at altitudes, provided a low speed calibration
of the airplane is first made and a low speed
(2) Pitot pressure error correc-
tion - Pt-P'; : AP
(3) Dynamic pressure error correc-
tion : 9c-9c = APA-APS
-
49c:
1:18
(b) The Determination of Equivalent Air
speed from the A.S.1. Reading
Generally, static pressure error may be
due to both (a) and (b) above, but pitot pres-
sure error, when it exists, will be due only
to (b) except at supersonic speeds when shock
losses occur in front of the head, for which
theoretical allowances can be made.
The calibration equation for instruments
in current use in the United Kingdom is
V's.
a's + +po. Vacol (i + +
2
:
+
1:51
The corresponding errors produced in
the readings of the Air Speed Indicator (A.S.I.)
and altimeter are termed the A.S.I. pressure
error and altimeter pressure error, the
latter of course arising only from the static
pressure error and the former from dynamic
pressure error and hence from both static
and pitot pressure errors.
where
Po
and
Co are the density and speed
of sound under standard sea level conditions,
Vcol and co being here in ft. per sec.
The reading of the instrument in the ab-
sence of pressure error is thus given by
Thus we have
(1) A.S.l. pressure error correc-
Vcal - V'cal : AVcal
tion :
96 - tro Vol('++ vente
9c =
+
co
1:52
(2) Altimeter pressure error cor-
rection : hp-h'oAnp
In general these errors must be estab-
lished by a flight calibration made under the
appropriate conditions. In some cases, how-
ever, it is possible to extrapolate over a wide
range of conditions from a calibration at one
height (conveniently ground level) and meth-
ods of doing this are considered.
From these equations, therefore, the A.S.I.
pressure error correction, AVcal can be
obtained in terms of the dynamic pressure
error, A9c and Vcal or Vicol. This relation
is plotted in Fig. 1:2 as AV'cal V8.A 9c for
various values of Vical.
To obtain the equivalent airspeed Ve *
from the "free-stream" instrument reading
Vcal, the exact relation between dynamic
pressure and speed (of which the calibration
equation is an approximation) must be con-
sidered.
1:12 THE DETERMINATION OF SPEED
AND PRESSURE ALTITUDE FROM
THE INSTRUMENT READINGS
(a) General
The relation between equivalent airspeed
and pressure follows from Bernoulli's equa-
tion for adiabatic compressible flow
The instruments used in the aircraft to
record speed and height are of course pres-
sure gages, though scaled in knots and feet
respectively. A unique relation, therefore,
exists according to the scale or calibration
equation between the applied pressure and
the instrument reading.
{ve
+
Ps
P
constant
뿅
​* These comments are based mainly on Ref.
1 and use the approach of that report.
* The equivalent airspeed Ve is defined by
Ve - Vo where V is the true airspeed and
o the relative density.
1:19

soots
A.SI
PRESSURE
ERROR
CRARECTION
20
Alcal.
KNOTS
SOOTS
o
u
-70
-30
-10
OI
OS
70
20
40
SO
GO
DYNAMIC PRESSURE ERROR CORRECTION APLB./SQ FT
Fig. 1:2 A.S.1, Pressure Error Correction
-10
bos
soo
BASCO ON
(+ Alcal
c?
12
Apa por con1
spot mai ves ta veri3 [ + 4 (Won t execu] - Firavaillon te
-20
cal
{
1 (+
Vcal
c2
)
400
25
Vcal = 200KTS.
300
OOI
-30
1:20
which gives (under isentropic conditions)
where Po is the pressure under standard sea
level conditions and the air density is related
to the pressure altitude hp according to the
scale on which the altimeter is calibrated,
For U.K. instruments the ICAN standard
atmosphere* values of Po and p are used.
YI(Y-1)
Pp B (var de har
.
:
+
Y-1
2
va
8 COP
."
1:53
From Eq. 1:56 it follows that
where 8 is the ratio of the ambient static
pressure Ps to the sea level standard pres-
sure. This equation relates 9c to Ve and S,
and is often expanded as a power series in
Ve/8 co? to give
hp
Aps - Ps - D's
8
s
gp dh
hp
1:57
2
4c = { Po ve? (i++
Ve
Ve?
1. Ve
Sco? '40 8?com
+
:)
1:54
From this equation the altimeter pressure
error correction Ahp can be obtained in terms
of static pressure error correction Ap's and
h'p or hp. This relation is plotted in Fig.
1:4 as A hp vs Ap's for various values of hp.
Eqs. 1:52 and 1:53 or 1:54 give the rela-
tion between Ve and Vcal , viz.,
Vcal?
Vcol²
Vcal?(1 +
(1+ t. vcov
:)
1:13 THE ESTABLISHMENT OF THE
PRESSURE ERROR CORRECTION
FROM FLIGHT TEST RESULTS
Experimental Methods
vo(+ tv
I
Vez
+
Ve
40 8200
8 CO
1:55
Various experimental methods are avail-
able for the determination of the pressure
error correction. They may be divided into
two main classes:
This relation, which includes the calibra-
tion relation, Eq. 1:52, and which through is
a function of the altitude, may conveniently
be called the calibration-altitude relation.
It is given in Fig. 1:3 as a chart of Ve - Vcal
against Vcal at a series of pressure altitudes
hp.
(a) A comparison of applied pitot and
static pressures, P't and p's with an experi-
mental determination of the “free stream"
pressures,
(b) A comparison of the indicated cali-
brated airspeed Vical with an experimental
determination of the equivalent airspeed Ve.
(c) Determination of Pressure Altitude
from the Altimeter Reading
The calibration equation for the altimeter
is of the form
* Editor's note: The NACA standard atmos-
phere is exactly the same as the ICAN at-
mosphere up to 35,332 ft. altitude at which
point the stratosphere is assumed to start
and the temperature is taken as -67°F. In
the ICAN atmosphere the temperature of the
stratosphere is taken as -56.5°C (-69.7°F).
h
Ps - Do
--S"
9p dh
1:56
1:21

INDICATED CALIBRATED AIRSPEED, VCal
KNOTS
6
100
200
300
400
500
600
SEA LEVEL
-5
01-
Sooo
-15
Fig. 1:3 Scale - Altitude Relation , Ve-Vcal ,KNOTS
M=1.0
10000
PRESSURE ALTITUDE hp FEET
20
60000
15000
55000
1500
-25
750000
20000
40009
25000
30000
-30
Se 1 + 1) en (over 1 + 1). No casve
+ ਕਰ
ve?
8c2
+ ) *(1 + i en helt
(----+
1:22

PRESSURE ALTITUDE INDICATEO
SY ALTIMETER hP FEET
20.000
40,000
50,000
60,000
2500
20,000
2000
ALTIMETER
PRESSURE
ERROR
CORRECTION
Ahp
FEET.
0.000
1500
SCALEVE
H000
Fig. 1:4 Altimeter Pressure Error Correction (ICAN Scale)
500
sos
0000t
0.000
210.000
SEA LEVES
ooo2
F30 000
-90
-70
-60
-50
-40
-30
-20
-10
-80
STATIC
20
30
40
SO
PRESSURE CORRECTION Aps:LB/SQ.FT.
SEA LEVEL
TORI-1
het hp
-1000
BASED ON APS
se.dh
20.000
hp
-1500
6000,
30000
- 200g
-0.000
1:23
Comparison of Speeds
In the first method, the pressures may be
measured either directly on a pressure scale,
or in terms of pressure altitude on the alti-
meter. These two methods will now be con-
sidered and it will be shown how the pressure
error corrections Vcal and A hp may be
deduced from the flight test results.
In this method the airspeed may be deter-
mined by timing the aircraft over a measured
speed course, by radar tracking (for both of
which methods an allowance for windspeed
is required) or by flying in formation with
a calibrated aircraft,
Comparison of Pressures
This is the basis of the standard method
at present used in the U.K. for ground level
calibrations. The static pressure error is
determined by the aneroid method which con-
sists of comparing the reading of a sensitive
pressure gage connected to the aircraft static
source with another gage registering the air
pressure at a suitable reference point past
which the aircraft flies,
The equivalent airspeed Ve deduced from
these tests enables Vcal to be found from
the scale altitude relation of Eq. 1:55 plotted
in Fig. 1:3. The A.S.I. pressure error is
then given directly by
AV'cal
Vcal
· V'col
To derive the altimeter pressure error
correction, the reverse process of section
1:14 is then applied.
The pitot pressure error is usually meas-
ured by a differential pressure gage connec-
ted between the aircraft pitot pressure source
and a venturi pitot head mounted at some suit-
able position on the aircraft clear of slip-
stream and other disturbances.
1:14 THE ESTIMATION OF THE PRESSURE
ERROR UNDER OTHER THAN TEST
CONDITIONS
General
Other experimental methods of deter-
mining the static pressure error correction
by pressure comparison include the compari-
son of hip with the pressure altitude meas-
ured by flying the aircraft in formation with
a calibrated aircraft, by using a trailing
static head (Ref.2), or by flying in a part of
the atmosphere recently calibrated (Ref. 3).
While it is desirable to measure the pres-
sure errors under the conditions to be oper-
ationally obtained, this will not always be
convenient in practice. By making appropri-
ate assumptions, estimates of the error at
other conditions of height, weight or normal
acceleration may sometimes be made from
a calibration over the speed range at one
height.
According to the scale on which the pres-
sure errors are measured, Eqs. 1:51 and
1:57 or Figs. 1:2 and 1:4 may then be used
to obtain the instrument pressure error
corrections. Thus, if the static pressures
are compared on the altimeter scale, Aho is
given directly from the experimental results.
In general, from conditions of flow sim-
ilarity, we should expect
AP
Cpo teve
Ave-flCL, M) or a leaf (CL, m)
of ,M.
The appropriate A.S.1, correction may then
be found by using Fig. 1:4 to deduce Ap's from
Oh's and hence obtain the dynamic pressure
р
error Aq'c-Ap', -Ap's (the pitot pressure error
being determined independently, e.g., by
venturi-pitot). The A.S.I, pressure error
correction is then obtained from Fig. 1:2.
1:58
To derive this relation experimentally
for direct application to any flight condition
1:24
angle of attack
would thus require level flight calibrations
at several heights over the full height range.
The appropriate assumptions on which pre-
dictions to other conditions can be made from
tests at one height depend on the Mach num-
ber and are considered below for several
ranges of this parameter.
CLT-M2
= constant
and
CP/I-M2 = constant.
.
Low Mach Number
Therefore
For M < 0.4, the effects of compressi-
bility on pressure error are usually neg-
ligible, and without introducing serious er-
rors, Eq. 1:58 may be simplified to
Vel/nW
1 - M2
• constant
en geef ICU) or favor
Δη
Vcal
VnW
nW
and
AP
: constant.
nW
From this relation established over the
CL range at one height, the instrument pres-
sure error corrections at any relevant height,
weight and normal acceleration can be ob-
tained.
These relations can be used to obtain the
pressure error at any altitude, weight and
normal acceleration in the region where the
Prandtl-Glauert rule holds from a calibra-
tion over the full speed range at one height.
(b) Change of Altitude
There is a unique relation between AVcal
and Vcal at all heights for a given weight
and normal acceleration. However, Ahp vs.
Vcal will depend on height and can be ob-
tained by use of Fig. 1:4.
Medium Subsonic Mach Numbers
(a) General
For higher speeds, account must be taken
of compressibility effects. Allowance may
be made for these up to a moderate Mach
number (say, M = 0.75) by use of the Prandtl-
Glauert relation derived for thin aerofoils
at small angles of attack provided no shock
waves have appeared.
If the relationship between AVcal and Vcal
had been established by test at one height
h, it is possible to calculate this relation-
ship appropriate to another height h2. Thus
Ap = constant and ve? W1-M2 = constant at
constant angle of attack, weight and normal
acceleration,
The process is as follows:
(1) Select a value of v'cal, and its
1
appropriate AV'cal, giving
Vcali Včali + AVcali
and obtain Ap from Eq. 1:52.*
From this relation we have at constant
* Editor's note: As used here, the quantity Ap represents the overall position error of the
airspeed system and therefore is taken equal to 9c-a'c. Consequently from Eqs.1:51 and 1:52
Ap=9c-4c = {povéal [i++
#mo{Vēl [i++. Po
prvi [*
{ 1
VI?
112
cal
cal
+
-1}
1:25
(2) Obtain Ve, from Vcal, and Eq.
1:55.
(3) Find the value of Ve ? W1-M2 and
from this determine the value of Vez at the
new altitude.
.
constant M and Cl, the quantity nW/p is
constant and there is a change of height.
This may be read from Fig. 1:5 using the
constant Mach number lines (dotted) and the
ordinate scale in Vcol. Correction back to
the original height, if the Prandtl-Glauert
relation is assumed, can be made from Fig.
1:5 by following a constant Mach number line
to the corrected Vcol and returning along
a constant Vcalo line to the original altitude.
(4) From Vez and Ap obtain Včala
and AVCal2 by reversing the process of (1)
and (2).
Repetition of this process over the range
of Vcal gives the new relationship between
Vcal and AVcal at any required height.
(d) Applicability of Method
Stage 3 involves a successive approxima-
tion and to simplify the calculation, the re-
lationship between Vcali and Vcale with
altitude at constant angles of attack has been
plotted in Fig. 1:5 (full lines which give
the change in Vcol with hp at a constant
pressure error correction Ap).
This application of the Prandtl-Glauert
relation has been found to give fairly good
agreement with experiment for several types
of wing tip and nose-boom pitot-static in-
stallations up to about M = 0.75 at low CL
values (< 0.2), though discrepancies have
occurred from about M = 0.5 at high CL's? .
.
It is thus desirable to check experimen-
tally the applicability of the Prandtl-Glauert
relation to a particular type of installation
by calibrations at high and low altitudes be-
fore using it above about M = 0.6.
Unless the pressure error is large, this
will give sufficient accuracy when used with
applied pressures, i.e., to give the variation
of Veal with n'p. Note that although Ap is
unchanged when converting a Ap - Vcal chart
to another altitude, AVcal at a given Vcal will
change as it depends on Vcal as well as on
Ap.
.
Transonic Speeds
(c) Change of Weight and Normal Ac-
celeration
Generally in the transonic range the full
generalized expression must be considered,
i.e.,
At constant angle of attack and Mach num-
ber we have as for the incompressible case
Cp = f(CLM).
.
Ve?
nW
: constant
and
Δρ
nW
= constant.
The effects of Mach number in this range
are not predictable by any simple theory and
it is unlikely that any reliable method can
be devised permitting extrapolation to be
made from measurements at lower speeds
into the mixed flow region just below M : 1.
Pressure error calibrations must therefore
be made at high altitude up to the maximum
attainable Mach number and in general at
several heights to permit separations of the
effects of CL and M.
These relations follow without any as-
sumptions about the effects of compres-
sibility. Since this correction is made at
1:26

BASED ON
v2
= CONSTANT
600
oog
1-M2
THE
AND THE SCALE-ALTITUDE
RELATION OF FIG. 3
BETWEEN Ve & Vcal
500
Imzodo
M=0.85
M0-89
SEA LEVEL. CALIBRATED AIRSPEED, Vcal., KNOTS
500
400
=o75
LLLLY || || IL
CALIBRATED AIRSPEED, Vcal , KNOTS
ME 0-65
400
M:0.66
300
FM-o 50
M=0.40
300
-2004
M0301
200
v
M20.20
-10,000
o
+30.000
+50,000
+60poo
+19,000 +20000
+40,000
PRESSURE ALTITUDE, hp, FEET
Fig. 1:5 Variation of Calibrated Airspeed with Altitude at Constant
Attitudes Assuming the Glauert Law
1:27
by the Rayleigh formula.
Y/ (Y-1)
Mach number effects become increasingly
predominant as height is increased and CL
effects less important so that a plot of Cp vs.
M obtained at one height might be generally
applicable (above, say, M = 0.7) over a mod-
erate CL range. It is desirable, however,
to check the validity of this assumption for
each installation by tests at widely different
heights. Since Cp : Ap/1/2yPsM? these
calibrations can be presented for use with
Figs. 1:2 and 1:4 in terms of A p/Ps vs. Mcal.
( )
M2
اه
Y+1
2
2Y
M2
Y+1
D's
Y-I\1/17-1)
Y+1
7-
or for air with Y: 1.4
-
Supersonic Speeds
PH
pia
- 5/2
166.9 M' ( 7M2-1)
At supersonic speeds, pressure error
problems may be expected to be of a less
serious nature than in the transonic range.
This equation may thus be used to extend
the scale-altitude relation of Fig. 1:3 to su-
personic speeds.
(a) Static Pressure
1:15 EXAMPLES OF MEASURED
PRESSURE ERRORS
Provided the static source is ahead of the
fuselage bow wave (as for a pitot-static head
on a nose boom) the pressure at the source
will be unaffected by the aircraft pressure
field and any pressure error will be due to
failure of the head itself to register local
"free stream" pressure.
General
The calibration can thus be derived from
wind tunnel tests on the head alone or from
flight measurements with the same head on
another aircraft. Available evidence sug-
gests that for a static tube with the pressure
holes more than 8 10 diameters behind the
nose the effects of the nose shock wave have
mainly died out so that the pressure error
may be very small at supersonic speeds.
Some experimental data extracted from
various sources are presented in Figs. 1:6,
1:7, 1:8, 1:9 and 1:10 to indicate the effects
of several parameters and illustrate the
magnitude and variation of pressure error
which may be expected. To facilitate com-
parisons, all the errors are presented in the
form of a pressure coefficient, Cp :
Ap/1/2pV?.
Additional data on these and other aspects
of pressure error problems together with a
comprehensive list of earlier references
are given in Ref. 4.
(b) Pitot Pressure
Errors Arising From the Head Itself
(a) Pitot Pressure Error
Because of the local shock formed ahead
of the pitot entry, the pressure registered by
the pitot will be less than that given by Eq.
1:53. Assuming a normal shock at the entry
the pitot pressure is given in terms of "free
stream" Mach number and static pressure
Subsonically, when the airılow up to the
head is isentropic, there will be no effect of
the aircraft pressure field on the local total
1:28
ANGLE OF YAW OR INCIDENCE -DEGREES
o
5
10
15
PITOT
-0.02
PRESSURE
ERROR
• 12 pv?
M=0.4
=
се
-0.04
M:07
M
1:06
0.06
-0.08
PITOT PRESSURE ERROR. FIG. 5A.

ANGLE OF YAW OR INCIDENCE - DEGREES
5
15
M: 0.6
-0.02
STATIC
PRESSURE
ERROR
12 pv2
M: 0.7
\ M:04
= cp
-0.04
-0.06
-0.08
STATIC PRESSURE ERROR. FIG. 5B.
Fig. 1:6 Effects of Inclination to Flow and Mach Number on Pressure
Registered by British Mark VIII A. Pitot Static Head
1:29
DISTANCE OF STATIC HOLES FROM NOSE.
METHOD OF TEST
NOSE DIAMETER.
GROUND LAUNCH POCKET.
8-7
8.3
TRANSONIC TUNNEL
15.0
9-7
}
}
+ 0.03
20-0+
from Various Heads at Zero Yaw
Fig. 1:7 Comparison of Static Pressure Errors at High Mach Numbers
Cp
+ 0.01
10.0.
0.8
0.9
0-1
1-1
1-2
1-3
1.4
MACH NO
1:30
.12
•08
YA CHORD
04
YZ CHORD
Cp
34 CHORD
I CORD
Distances Ahead of the Aircraft Wing
Fig. 1:8 Calibrations of a Static Tube Located at Various
THAT
2 CHORD
112 CHORD.
--04
-2
192
2x
80.-
9
OD
1-0
o
1-2
À
tul
9-1
8-1
w
1:31
8.1
9.1
1.4
12 DIA.
1/2 DIA
2.1
I DIA
୯
0.1
CL
8.
9.
2
.12
80
VO
-04
80.-
dy
Fig. 1:9 Calibrations of a Static Tube Located 0.5, 1.0 and 1.5 Body
Diameters Ahead of the Fuselage Nose
1:32
HEAD POSITION
TYPE OF ARCRAFT
POSITION OF STATIC SOURCE
ALONG SPAN LENGTH OF BOOM
A WING LEADING EDGE SWEPT WING
.91 s
•63c
8
W
.715
660
N
1:05
1.470
0
11
DELTA WING
1.05
.
· 155
.
NOSE BOOM
N
.950
SWEPT WING
1.810
WHERE
S: SEMI-SPAN.
CE LOCAL CHORO. O: FUSELAGE DIAMETER
+0.15
+0.10
-
-
-
-
А
+0.05
STATIC
PRESSURE
ERROR
/Pv²
B
= Cp
-0.05
-0-10
-0-15
0.5
0.9
1.0
1.1
0.7
0.8
TRUE MACH NUMBER
Fig. 1:10 Some Examples of Static Pressure Errors
Measured in Flight at High Mach Numbers
1:33
pressure, and any pitot pressure error will
be due to the failure of the head to register
the local total pressure, usually because of
inclination to the local flow.
is normally zero except at high angles of
attack. Static pressure error, however,
varies considerably with head location and
speed, and it is not possible to make gen-
eralizations applicable to all installations
and flight conditions. Individual calibrations
are the only reliable method of obtaining
static pressure errors. However, some ex-
amples are given to illustrate the effects of
some of the parameters.
The effect of incidence or yaw and Mach
number on pitot reading is illustrated in Fig.
1:6A by wind tunnel data obtained on the RAF
Mark VIII pitot-static head (data from Ref.
5). The error is negligible up to about 5° and
in practice it is generally found that aircraft
pitot pressure errors are zero above about
200 knots IAS. The effects of yaw may be
delayed by modifying the nose geometry to
give a sharper-edged entry but for this par-
ticular head the aim has been to minimize
the dynamic pressure variation with yaw, and
the pitot pressure error variation with yaw
is similar to that for the static pressure
error.
(b) Effects of Head Location
These effects are illustrated in Figs. 1:8
and 1:9, which represent flight data (Ref.6)
obtained up to moderate Mach numbers, on
straight-wing aircraft, the results being
plotted in terms of CP vs. CL.
(b) Static Pressure Error
(c) Wing-tip Leading Edge Position
The effects of inclination to the flow and
of Mach number on the pressure registered
by the static holes are illustrated by wind
tunnel data (from Ref. 5) for the RAF Mark
VIII head in Fig. 1:6B. The effects of Mach
number shown up to M = 0.7 are not large
and other similar evidence indicates a sim-
ilar trend for this type of head up to M = 0.9.
Fig. 1:8 shows Cp vs. CL for several
lengths of boom at the wing-tip (Ref. 6). At
low CL's the static pressure error is fairly
small for a boom length of 1/2 chord or more
and negligibly small at 2 chords. At high
CL's the error becomes negative in all cases
and tends to a fairly high value. Part of this
trend is evidently the angle of attack effect
on the head itself, illustrated in Fig. 1:6B.
(d) Nose Boom Position
The effects of Mach number in the tran-
sonic and supersonic ranges are illustrated
in Fig. 1:7 for several heads with zero in-
clination to the local flow. These indicate
that with the static holes located well back
down the tube (more than 8 diameters) the
error is small at supersonic speeds but that
there is a region around M : 1 where local
shock waves affect the registered pressure
and the error may become moderately large.
There are no data at present available in
U, K, on the effects of yaw at supersonic
speeds.
Fig. 1:9 shows similar data for nose
booms. The error tends to be large for a
boom less than about 1-1/2 fuselage dia-
meters in length, the variation with CL show-
ing a trend similar to the wing position data.
(e) Effects of High Mach Number
Pressure Error for Aircraft Installation
Examples of measured static pressure
error on several installations over the high
Mach number range are plotted in Fig. 1:10
as Cp vs. M, all the data being from meas-
urements in level flight between 30,000 and
35,000 feet.
(a) General
As has been stated, pitot pressure error
1:34
1:17 THEORETICAL CONSIDERATIONS
Assuming that the flow is adiabatic, the
energy relation
Few systematic trends can be discerned
but the static error tends to a high value near
M : l in all these cases, though it appears
possible to delay the rise by lengthening the
supporting boom. The two nose-boom in-
stallations shown have a constant pressure
error coefficient over a large Mach number
range (one up to M : 0.96) while the wing
installations exhibit various trends. It is
possible that wing fences are in some cases
affecting these. *
D V2
y
Y-Ip 2
y
.
= 9cp To
: constant
applies to the flow past the temperature pick-
up provided it is located at a point on the
airplane where the energy content of the air
is the same as in the free stream. In the
above equation
The limited evidence here above M : 1
confirm that the error reduces again to a
small value as indicated for heads alone in
Fig. 1:7.
Y
р
• Cp/cy : ratio of specific heats
= ambient pressure
- ambient density
1:16 TEMPERATURE MEASUREMENT
P
11
Ср
specific heat at constant pres-
sure
T,
At the speeds and altitudes at which air-
planes are capable of flying it is possible to
assume (with insignificant resulting error)
that the flow ahead of any temperature pick-
up is adiabatic in nature and that the air
obeys the perfect gas law. Thus in theory,
at least, the measurement of air temperature
poses no particular problems. However,
problems do arise in the practical measure-
ment of temperature, principally because of
equipment limitations.
g
: total temperature (absolute
scale)
= gravitational acceleration con-
stant
We shall here consider the problem of
measuring temperature, using equipment
carried aboard the airplane, and merely
mention that radiosonde equipment or similar
means may be used also in certain instances,
provided the time lag between the measure-
ment and the conducting of the actual tests
is small.
If the speed is low (of the order of 200 feet
per second or less) the term [y/67-1)). (p/p)
in Eq. 1:59 is predominant, and little error
is introduced by ignoring the V2/2 term.
Under these circumstances the total and am-
bient temperatures may be assumed the same
for all practical purposes. At higher speeds
the velocity must be considered.
A convenient relation for evaluating ve-
locity effects is developed below. The per-
fect gas law may be written
р
:
SRT
P
1:60
* Editor's note: The eccentricities in wing
boom data shown may possibly be due to the
formation of shocks ahead of these booms
even at subsonic free stream Mach numbers.
This phenomenon results from the outward
extension of localized supersonic flow regions
which build up along the fuselage sides under
transonic conditions,
where g = gravitational constant
I
.
R = Gas constant (53.3 ft/ºr using
English engineering units).
1:35
1
Combining Eqs. 1:59 and 1:60, we have
the flow past a pickup element which projects
into the air stream,
ܙܘo(-,)
Y
y?
(RT) +
2
: gop te
There is at least one stagnation point on
the temperature head where total conditions
exist. Moreover, a boundary layer is built
up around the sensing element, and as a first
approximation, total temperature exists in
the air particles comprising the base of the
layer.
where T : ambient temperature,
But by definition
:
Ср
=
R.
Thus, at first glance one would imagine
that the pickup would sense total temperature.
Actually, the sensing element as a whole
never does achieve stagnation temperature
because of heat flow from the thin boundary
layar to the passing air and because of heat
flow from the pickup itself, to surrounding
structure. Thus, most pickups measure a
temperature somewhat less than the total,
but greater than ambient.
Thus, on dividing by gcp and rearranging,
-1
T:
-
To
-6.
v2.
R
1:61
Thus the total temperature differs from
the ambient by a constant multiplied by the
true airspeed squared.
The losses from the pickup are usually
accounted for by inserting a correction (or
recovery factor) "e" in Eq. 1:62, which then
assumes the form
If we note that YgRT c? (the ambient
speed of sound squared), Eq. 1:61 may be
written
To - f[it (39)w:]
: +
EM
1:63.
To
=
- T[i+(") ne]
1:62
where M = Mach number,
For most test work it is assumed that e is
a constant, independent of temperature, and
generally this is a satisfactory assumption.
However, it seems apparent that at super-
sonic Mach numbers will become quite
temperature dependent,
Since Mach number can readily be deter-
mined when total and ambient pressures are
known (see section 1:4 of this chapter), Eq.
1:62 is a most convenient relation between
total and ambient temperatures for use in
flight test work.
Excepting for the recently developed axial
flow vortex thermometer (Ref. 7), existing
flight test temperature pickups unfortunately
measure neither the total nor ambient tem-
perature, but rather something in between. To
understand the factors which cause this situa-
tion, we shall consider briefly the nature of
Although the value of € (which varies
from about 0.60 to 0.95 for typical installa-
tions) may be determined from wind tunnel
tests of a given type installation, it is desir-
able to determine e from actual flight runs.
The procedure used to accomplish this is
discussed in section 1:18.
Although vortex thermometers are not at
present used at major flight test activities,
1:36
1:18 FLIGHT CALIBRATION OF TEM-
PERATURE MEASURING SYSTEMS
they have certain features worth discussing
here. The vortex thermometer is based on
the idea that the pressure and density, and
consequently, the temperature, decrease to-
ward the center of a free vortex (provided,
of course, we do not consider the vortex core
itself). This phenomenon is called the Ranque
effect and is the basis of operation of the
Hilsch tube which may be used to provide
small amounts of refrigeratej air.
The major factor to determine by flight
calibration of temperature measuring sys-
tems is the element recovery factor. Nor-
mally there are no position errors to worry
about, although lag errors do exist and may
be appreciable during dives or acceleration
runs at supersonic Mach numbers. Under
subsonic conditions any lag errors which
may exist are customarily ignored. Here
we shall concern ourselves only with the
determination of recovery factors.
Referring now to Eq. 1:63, this may be
written
Toi
۲۰(7) 1
Te M 2
In a vortex thermometer, spin is intro-
duced into the air passing the temperature
sensing element and an approximation to free
vortex motion is obtained. The spin or vor-
ticity is introduced by a rotating vane (gen-
erally rotated by the passage of air past the
vane) designed to produce sufficient circula-
tion at various forward speeds to produce a
temperature drop equal to the difference
between the total and ambient temperatures
of the air stream. Accordingly, a properly
designed vortex thermometer reads ambient
air temperature regardless of forward speed
(at least in the subsonic speed region).
1:64
where Tt;
: indicated total temperature.
At constant ambient air temperature T,
Eq. 1:64 describes a linear relationship be-
tween Tt; and M’, provided e is constant.
Therefore, if a series of flight runs is made
at various flight speeds (covering the speed
range of the airplane at a given altitude), a
plot of Tti vs. M? has the general nature
shown in Fig. 1:11.
Two types of vortex thermometers have
been investigated, the tangential type and the
axial flow type.
According to Ref. 7, the
axial flow type may be designed to give sat-
isfactory results up to 500 mph true air-
speed; however, little information is avail-
able describing functioning at higher air-
speeds.
Siode
2
The sensing elements in present use for
test work are of two types:
(
O
(a) the
shielded resistance-type con-
nected to a voltage compensated Wheatstone
bridge;
(b) the shielded thermocouple-type con-
nected to a galvanometer circuit (with am-
plification as required).
Minimum Test Speed
M2
These units are theoretically satisfactory
at all speeds provided the recovery factor is
known.
Fig. 1:11
1:37
As may be seen from the figure, extra-
polation of the test data curve to M? : 0
determines the ambient temperature of the
air mass in which tests were run. Knowing
the ambient temperature, the slope of the
curve of Tr; vs. M2 may be used to deter-
mine €. If m = slope, and y is taken as 1.4,
then
.
5m
T
1:65
If the test data exhibit curvature, extra-
polation to M2 = 0 will still give the ambient
temperature; however, the curvature indi-
cates that e = f(M), and, therefore e is not
a constant.
In conducting temperature system cali -
bration tests, it is important that the pilot
make his runs at a given altitude and remain
in the same air mass during all runs. Tests
should be conducted as rapidly as possible
to avoid the possibility of the ambient tem-
perature changing during the test runs. Nor-
mally no more than four points will be re-
quired at any one altitude, and it is not
important that the airspeed be absolutely
stabilized when data are taken, inasmuch as
relatively slow rates of change of speed
produce negligible lag errors either in the
temperature pickup or the airspeed system,
Tests should be conducted at several alti-
tudes to determine the effect of height on the
heat loss characteristics of the pickup.
Data to be obtained consist of indicated
total temperature, total pressure and static
pressure. Data may be recorded by the
pilot or if a photopanel or transmitting
equipment are installed, these may be uti-
lized. No special pilot technique is required,
and, if convenient, the calibration may be
carried out in conjunction with other tests
such as level flight runs.*
.
Provided the curvature is not excessive,
a straight line approximation to the actual
data may be used to determine an average
value of e, which may then be used for cor-
recting data to standard. The amount of
curvature which may be ignored in a given
instance is a factor which the test engineer
must decide upon for himself.
1:19 CONCLUDING REMARKS ON
TEMPERATURE MEASUREMENT
If the test data exhibit excessive curva-
ture, a convenient calibration curve may
still be prepared on the basis of the equa-
tion obtained by dividing both sides of Eq.
1:64 by T, thus
Measurement of air temperature using
equipment installed within the airplane is a
relatively simple process, and the major
problem is the determination of the system
recovery factor. If the recovery factor is
constant or nearly so, it is readily found
from a plot of indicated total temperature
against M?.
Ti.1+() cm
(
+
€M2
1:66
where the ambient temperature T is obtained
by first plotting the test data as in Fig.1:11
and extrapolating to Me : 0.
For a variable recovery factor a tempera-
ture calibration plot should be prepared in the
form of a graph of Tti/T vs. M?.
Knowing T, the ratio Tt;/T may then be
plotted against M or M?, and the result is
the calibration curve. We note that if e is
insensitive to altitude variation, data from
all altitudes should fall along the same line
when plotted according to this latter scheme.
* Note: If temperature system calibrations
are conducted simultaneously with accelera-
tion runs, proper account should be taken of
airspeed system lags which normally are
considerably greater than any lags exhibited
by the temperature system.
1:38
1:20
INTRODUCTION TO LAG MEASURE-
MENTS IN PRESSURE SYSTEMS
application might be the measurement of
surface pressures of an oscillating airfoil.
Equations describing the system for this case
are quite complex. Refs. 8 and 9 analyze
this case.
Both theory and experience show that
when a pressure gage is connected by means
of tubing to a source of pressure, it does not
respond instantly to a change in pressure at
the source but lags behind the pressure at
the source by an amount called the lag error.
(b) Measurement of constantly increasing
or decreasing pressures or of oscillating
pressures of extremely low frequency. The
second major division is further divided into
two subdivisions:
The pressure at the source is transmitted
through the tubing with lag due to: (1) pres-
sure drops in the tubing resulting from the
viscous friction between the flowing air and
the tubing wall; (2) Pressure drops across
orifices and restrictions; (3) inertia of the
air; (4) the finite time required for a pres-
sure disturbance to travel the length of the
tube.
(1) Measurements where the pres-
sure lag is large compared to the pressure
in the tube. This would be the case in an
intermittent wind tunnel where there is a
change of several atmospheres from one
pressure to another in essentially step func-
tion fashion, Refs. 10 and 11 analyze this
case,
1
.
After the pressure disturbance reaches
the gage, the instrument diaphragm must
deflect and move the indicating needle or
actuate some other output generator. This
results in additional lag due to the instrument
damping, inertia and viscous and coulomb
friction.
(2) Measurements where the pres-
sure lag is small compared to the pressure
in the tube.
Airspeed, altitude and Mach number meas-
uring systems currently used in airplanes
and considered herein consist of a gage
(which is usually of the diaphragm type) con-
nected to sources of total and/or static
pressure by means of tubing. Sometimes
other instruments, orifices and/or restric-
tions are in the system.
From the above considerations and the
fact that in actual installations there is heat
transfer between the system and the sur-
rounding medium, it is readily apparent that
a general mathematical treatment of the
response of such a complicated system would
involve at least simultaneous second order
partial differential equations. Even if solu-
tions could be found for these equations, the
results would not be usable because of the
lack of knowledge of the factors (heat trans-
fer characteristics, temperatures along the
tubing, etc.) entering into the equations.
Analysis of flight test data from these
systems comes under the category (b (2)
above) of constantly increasing or decreas-
ing pressure changes where the lag is small
compared with the pressure in the tube, Since
lag exists in airspeed, altitude and Mach
number measuring systems, either correc-
tions must be made for the lag errors, or it
must be shown that for a particular case the
error is small and can be neglected.
In view of the complicated nature of the
problem, analyses of pressure measuring
systems are usually divided into several
specialized categories which allow simplifi-
cations to be made in each particular case.
The two major divisions are:
It is indeed fortunate that drastically
simplified approaches have been found which
will supply adequate lag corrections over a
large range of flight conditions encompassing
those presently encountered in performance
testing. These simplified approaches will
(a) Measurement of comparatively high
frequency oscillating pressures.
Such an
1:39
be considered in connection with the follow-
ing:
(a) The influence of system design and
operation on lag.
(b) Methods of correcting flight data for
lag errors.
(c) Present examples of lag corrections
to flight data.
As noted earlier in this chapter, lag is
only one of the several errors for which
corrections are required to observed values
of airspeed, pressure altitude and Mach
number. These various corrections are
applied in the following order:
>
and of ambient conditions only, and by ground
tests determine the lag as a function of these
variables. A lag correction can then be ap-
plied to the flight data for each flight value of
ambient condition and rate of change of pres-
sure, This process takes into account pos-
sible changes in the character of flow in the
tubing from laminar at low rates of change of
pressure to turbulent at high rates. However,
the flow in the tubing is laminar over the
usual flight test regions for almost all air-
speed, altitude and Mach number measuring
systems.
The assumption that the flow is laminar
together with the neglect of air inertia and
the assumption that the pressure lag is small
compared with the system pressure, permits
the use of a simple expression to describe
the system. The simplified equation describ-
ing the response of a pressure measuring
system which is used for airspeed, altitude
and Mach number flight data corrections,
has been derived from two somewhat different
physical pictures.
It is observed that the performance of a
typical system is similar to a one-degree-of-
freedom damped oscillator in that there are
conditions under which the system can be
underdamped, critically damped or over-
damped. Therefore, one approach, described
in Refs. 12 and 13, is to neglect the instru-
ment lag entirely and to assume that the equa-
tion for the oscillator applies to the pressure
system. Solutions of this equation for vari-
ous forcing functions are discussed in Ref.
14.
(a) Instrument scale error.
(b) Lag error.
(c) Position error.
In this section of Chapter 1, V'cal is used
to designate the indicated airspeed corrected
for instrument error.
1:21 DEVELOPMENT OF EQUATIONS
The most accurate method of correction
of flight measurements of airspeed, altitude
or Mach number for lag error would be to
set up the actual airplane system in the lab-
oratory and by a trial-and-error process
determine the true airspeed, altitude or Mach
number history corresponding to the indicated
history and ambient conditions of the flight.
While this procedure is possible, it is not
feasible since it would require excessive time
and effort, and, to the authors' knowledge,
has not been attempted except for the simple
case of a steady dive.
Systems are underdamped when the tubing
is short and the altitude is low. If buffeting
or oscillating pressures are being measured,
the underdamped system will exaggerate the
amplitude of the oscillation although mean
values will usually be given correctly. Also,
an underdamped system will be excessively
sensitive to step input type disturbances
such as would result from encountering a
gust. Therefore, underdamped systems are
not usually employed for flight testing work.
A somewhat less accurate but much sim-
pler process is to assume that the lag is a
function of the rate of change of pressure
For both underdamped and overdamped
1:40
systems which are subjected to constant
rate of change forcing functions, it is found
that, after the transient has died out, the
system behaves as with zero mass with the
indicated pressure lagging behind the source
pressure by a time a. In this case the sys-
tem is described by the equation:
to
+ p': p(t)
1:67
Another factor to be considered is the
acoustic lag T (the time required for a pres-
sure disturbance at one end of the tube to
reach the other). For example, if a step
function type pressure change is made to the
source end of the tubing, time elapses be-
fore the effect of this pressure change is
first felt at the other end of the tubing and
the gage pressure responds according to
Eq. 1:67.
.
where p' is the indicated pressure
p is the applied pressure
P(t) = (constant) (t) for a steady rate of
(
pressure change.
For p(t) = (constant) (t) = kt, the partic-
ular solution (steady state solution) to Eq.
1:67 is
i in seconds is the tube length divided by
the speed of pressure propagation which is
given in Ref. 15 as 1000 ft. per sec, for small
diameter tubing.
The acoustic lag produces a phase shift
and is included in Eq. 1:67 to yield:
p': klt - 1) = p-ö'd when Ø = k
do'(t+ 7) + p'lt + r) - plt)
1:69
so that
P-pupa.
with the solution for p(t) =kt; p'= k[(1+r)-
k -1].
1:67 A
On the basis of an analogy between elec-
trical and pneumatic systems, the lag con-
stant is theoretically derived for laminar
flow in the tubing as:
Ref. 15 presents a comparison between the
measured response of a system (consisting
of a section of tubing and an instrument) to
a very rapid change in pressure and the re-
sponses computed with and without consider-
ing the acoustic lag. When the acoustic lag
was included, the calculated response more
nearly agreed with the actual response.
0
8μ.
d:
nor?
[
L +
#p2
1:68*
* Editor's note: The important thing to note
here is that i depends on and pas atmos-
pheric variables. This can be directly es-
tablished by dimensional analysis since we
know that i = f (Pw, r, L, Q) for laminar
1
flows with small pressure gradient. Since
has the dimension of time, our dimensional
equation is
In most flight test applications, however,
the acoustic lag contribution is small and
can be ignored; it is considered here only
for the sake of completeness. It is a simple
matter to estimate the acoustic lag contri-
bution for a particular system and compare
this with the pressure lag to determine
whether or not the acoustic lag can be ig-
nored. Sample calculations ignore the acous-
tic lag.
M
T:
M\b
(L) (L
from which
()° (Lvºrnier de
of [66%)
. 60W
Another approach, given in Refs. 16, 17
and 18, is to compute the lag on the basis
that conditions are steady; i.e., P = constant.
Instrument effects are neglected. The de-
rived equation is exactly the same as the
equation derived through the first approach
except that the factor (L + Q/#r?) becomes
:
1:41
(L/2+ Q/tr?) and again t can be ignored
for steady flow.
The approximate relation between compres-
sible dynamic pressure, total and static
pressure, and calibrated airspeed for sub-
sonic flow is:
ac - -ps
2
= { POV calli+
Pov?
allit (...
:
1 ( cal
4 Со
+
1:70
The further assumption is made that Eqs.
1:67 and 1:69 can be applied to forcing func-
tions other than constant rate of change. A
very simple method of correcting pressure
data for lag results from this assumption.
The correction, in terms of pressure, de-
pends only on the indicated pressure, the
time rate of change of indicated pressure
and the lag constants, i and . The time
rate of change of indicated pressure and the
indicated pressure are measured in flight
while and are determined from ground
tests.
Applying Eq. 1:69 to the static and total
pressures of Eq. 1:70 and defining the lag
correction as Aq
9c-4c gives:
Aqc aclt+r)-akt)
+A4 t + )
+(4, -asip's It + )
The lag of the instruments alone has been
examined by various investigators. For ex-
ample, Ref. 19 has an analytic evaluation of
altimeter lag and Ref. 17 has an analytic
evaluation of airspeed indicator lag. Ref.
16 deduces the airspeed indicator and alti-
meter lag from test measurements while
Ref. 20 gives lags of altimeters actually
subjected to pressure conditions simulating
dives and climbs.
1:71
where it
.
lag constant of total pressure
system
and
λς
lag constant of static pressure
system.
The sum and substance of all these anal-
yses and tests are that by proper design and
selection of the instruments, the instrument
lag can be made small and may therefore be
neglected. However, care must be taken to
insure that the instruments actually used
have low lag. Instrument lag is partially
taken into account by making the necessary
ground calibrations on the complete systems
including the instruments actually used for
flight measurements.
This equation shows that the lag correc-
tion in terms of Aqc at a given time (t) can
be obtained from measurements of a'c at
that time and of ac, P's and ac at time
(t + ) plus the values of the total and static
pressure lag constants.
>
It then remains to transform Eq. 1:69,
which is expressed in terms of pressures,
into terms of altitude, airspeed or Mach
number the desired quantities.
The values a'c and q'c are computed from
airspeed measurements and Eq. 1:70. The
ambient pressure and time rate of pressure
change are obtained from altitude time his-
tories, knowing the altimeter calibration
equation and the scale error correction. The
lag constants are obtained with the relation:
(a) Airspeed
X
1=106)
(。
:
λο
The airspeed correction could be ex-
pressed in terms of either compressible dy-
namic pressure, qc, or indicated airspeed.
which for the total and static lines become
1:42
(see Eq. 1:68)
1:70, we have
DSO
V'2
19 = d0 666)
1
cal
.
da'c = { po 2v Cal dVcal
Ps + ac
va •
2V caldv.cal
+ Ź pov?com
is
..(post
)
Aso leo
Po
col 4
Ро
Ps
Со
1:72
1:74
whence
Fig. 1:12 presents the variation of the
ratio is so and 14/1tol with standard al-
titude and speed. iso and 11. are obtained
from ground tests as described later in the
chapter.
da's
i'ca
sevceloveolit dhe
p
dv
1+
2
Со
and
Usually the acoustic lag is small compared
to the total lag and may be ignored under
these circumstances. The terms of Eq.1:71
in the rectangle drop out and the lag correc-
tion at time t is equal to the sum of the re-
maining terms on the right-hand side of the
equation taken at time (t) instead of at time
(t+), i.e.,
cal
ac
da's
dt
on
Povcalcal
+
2
Со
1:75
Also, from the atmosphere equation of
balance
Aqc= d70'c + (14 - Asos'
=doen- isüs'.
do's : - godno
gpaho
1:76
so that
1:73
p's -- gprie
1:77
Letting dac and dVcal become finite dif-
ferences in Eq. 1:75, and substituting with
Eqs. 1:75 and 1:77 in Eq. 1:73, we have
Avcol : dqV cal
Eq. 1:71 should be used for airspeed lag
corrections when the acoustic lag is to be
taken into account. For the more usual case
where the acoustic lag is small enough to be
ignored, it is usually convenient to use an
equation expressed directly in terms of air-
speed and pressure altitude and their time
derivatives (as well as the lag constants).
The equation is based on the assumption of
a small error where differentials are equal
to differences, and while sufficiently accurate
for small corrections has an increasingly
large error as the corrections become large.
The equation is derived as follows:
(Ar-dsigphp
Ncoll?
Percoilith
,
Со
1
11
.
gorp (Ag-As)
- dovical
/Vcal
cal
2
1:78
where o has its standard atmosphere value at
the test pressure altitude.
Voorlitz
.
1
Taking differentials of both sides of Eq.
1:43
6
5
ASL
4
λ./λεsL
.L.
a /
S.L.
STATIC λ./λς.
TOTAL
3
001
175
200
250
2
300
350
400
450
500
550
600
Z
650
C
Vcal. - kts.
SEA LEVEL
10,000 20,000 30,000 40,000 50,000
mp, PRESSURE ALTITUDE - FT.
Fig. 1:12
1:44
Fig. 1:13 presents a plot of
and for y: 1.4
272.5
go
cal
dac
[tomonga
(
+0.2
= f(hp, Vald
P.Vcal d Vcal
Vcor (
i + Vesty
2
со
1:80
so that
272.5
as a function of calibrated airspeed and
pressure altitude.
Vcal
ac po Vcal
-
boxeal [0.2 (en in
Vcal
со
1:81
Still more accurate but somewhat more
complex equations are obtained by differ-
entiating the actual calibration equation
and Eq. 1:73 gives us
goholupds!
.
:
z face + 93-067 -
-1)/y
AVcal = lyVcal
vCal
.
-1
[)
Ро
Vcoito ay may
1:82
whence
We note that the quantity
-
dac
2 Vcal d Vcal
2 Y PO / 9c
Yol Po
+1
Vy
Y-
Y
go
Ро
Ро
1272.5
Ncall
Vcal I+0.2
1
ol[
Со
or
ac
has very nearly the same value as the ap-
proximation to it f(hp, Vcal) used in Eq.
1:78.
:
+)"
dac - povcal d Vcal
ilo
Ро
1:79
Now, from the calibration equation
y-1)
[(y-1) Vºcal
ac
Po
(b) Altitude
The pressure altitude may be obtained in
terms of either ambient pressure or feet of
altitude, If an exact difference method is
desired, the altimeter calibration (Ref. 21)
is operated upon in the same manner as the
dynamic pressure calibration in the previous
section, Included herein is a differential
correction method which provides a correc-
tion in terms of altitude. The equation is
+1:
+
ya
28
so that Eq. 1:79 becomes
1/(Y-1)
dac - In the Moon )+]"
[
Anplt) = holt + o) - hp (t)
+ishpltto).
Со
· Po Vcal d Vcal
1:83
1:15
650
Vcal [1+į i vode]
go
VARIATION OF
i hp, Vcol)
Vcal
co
WITH PRESSURE ALTITUDE AND CALIBRATED AIRSPEED
.20
.040
Fig. 1:13 Variation of
.18
.035
.16
.030
.14
.025
Vcalli+ė (col
:.12
Altitude and Calibrated Airspeed
Со
go
f( hp, Vcol.)
..10
fl hp, Vcal.
으
​.020
.08
SEA LEVEL (Pressure Altitude)
.015
10,000
000's
+--/--/-
.06
20,000
.010
10,006
15,000
20,000
SEA LEVEL (Pressure Altitude)
= f(hp, Vcall with Pressure
2017
.04
25,000
30,000
30,000
35,000
.005
.02
35,000
40,000
45,000
50,000
40,000
45,000
50,000
55,000
55,000
60,000
60,000
100
150
200
250
300
300
350
400
450
500
550
600
CALIBRATED AIRSPEED, Vcal, KNOTS
1:46
This equation is differentiated and solved
for dM,
The correction given by Eq. 1:83 is added
to the indicated value of pressure altitude
to obtain the corrected value.
Again, the
first two terms of the right side of Eq. 1:83
represent the acoustic lag and may be ignored
in most cases; in those cases,
Dolly-1)/y
-1
i
dpt
dps
dm :
(
IPs
YM
Do
Ps
1:85
Ane : isho.
Дhp
1:84
Thus the lag correction can be obtained
from a history of indicated altitude and the
ground measured lag constant.
Since the assumption is made that the
lag is small compared to the pressures, dM
is small compared to M, dps is sn:all com-
pared to Ps, and dp, is small compared to Pt.
Eq. 1:67 gives the lag of the total and static
pressure sides of the system as:
(c) Mach Number
Ds CD's - Aps - dps
.
-
λsβς
Dr-p'- Apr : dpt = lopte
1:86
Since the Mach number is a function of
the two pressures P, and Ps, it is evident
that a lag correction to Mach number must
include functions of p, and Ps and their time
derivatives. Calibrated airspeed and pres-
sure altitude, or Mach number and cali-
brated airspeed, or Mach number and pres-
sure altitude might be used. If the Mach
number is computed from measurements of
calibrated airspeed and pressure altitude,
as is usual in flight testing, the lag correc-
tions should be applied to these quantities
individually prior to the Mach number cal-
culation.
Eqs. 1:86 are substituted into Eq. 1:85 to
give the Mach number lag in terms of total
and static pressures and their time deriva-
tives, namely,
Ps
AM:
("
)
Pty-1/7
Ps
。
is DS
YM
1:87
To determine py/Pa and Ps/Ps we solve
Eq. 1:84 for pp/Ps to obtain
If, however, the Mach number is obtained
from a direct reading Machmeter, the most
convenient lag correction would include the
indicated Mach number and indicated alti-
tude terms. If the Mach number is less than
one, the equation, assuming small lag and
neglecting the acoustic lag, is obtained as
follows:
PT
Y-1
ylly-1)
.
MS
블 ​]
+1
Ps
2
1:88
and on differentiating,
At subsonic speeds the Mach number is
related to the static and stagnation pressure
(page 36 of Ref. 21) by the equation
dpt
dps
PA
Ps
2
Ps?
-1)
Y
11
M2
y
M2 =
2
Y-1
Pilly-1)/y
IPs
- "
( 23 ) +1]***
(I)
(2 MdM)
1:84
2
1:47
:
or
also, from the equation of state p: P/RT so
that
dpt
dps
៩
Ps
ghip
hip
-1
Ds
Ps
RT
53.3T
= y MOM
".(و ،
(2) lu«(). )"+".
/(
+1
1:91
where T is in degrees Rankine.
Using Eq. 1:88 for Ps/P7
In the standard atmosphere
dpt
T: To-.00357h = 518.4-.00357h
y MdM
dps
+
Ps
Po
Y1
[
m2 (4:1)
up to the stratosphere,
whence
T: 392.4° R at 35,332 feet and above
Dr. PS
умм
in the NACA atmosphere and
PT
+
Ps
*[w(37)+]
M2
2
T : 389.7° R at 36,088 feet and above
1:89
in the ICAN atmosphere.
Substituting Eq8, 1:88 and 1:89 in Eq.1:87,
we have
AM:
M2 +1
CH
X
2
YM
M
Inasmuch as the altimeter calibration
equation assumes standard density and tem-
perature at a given pressure altitude, the
temperature in Eq. 1:91 is the standard at-
mosphere value at the test pressure altitude.
Substituting Eq. 1:91 in Eq. 1:90 and setting
y = 1.4, we find that (letting M = M' and
hphp)
Ps
AtY Mм
(dy-1s) +
Ps
1039
Y-1
2
M2+
1:90*
Now we have the atmospheric balance
equation as
AM = {1+.2M") (:2719
[
х
1.4MMA
dp = - pgdhp
hip
(λι - λς)
53.3 T
بود
2
1 +.2M2
1:92
and
Ø : - pghp ;
where, as we have already noted, T has the
standard atmosphere value corresponding to
hip, and 14 and is are obtained by means of
ground calibrations and Eq. 1:72.
* Editor's modification from here to Eq. 1:101.
For the supersonic condition, the Mach-
meter is calibrated according to Eq. 1:22,
1:48
which may be written
Now
OPII
dporu
Pri
166.9 M2
Ps
2.5
dps
Asps
(78)
so that 1:95 becomes
1:93
Taking differentials of both sides, we have
>
ܐ
is
dm =
M[7M2- 1] [Arena
7(2 M2-1)
Pol
ps
dpti
Pr, dps
1:96
Ps
PS
2 MdM(166.9)
Moreover, from Eq. 1:94
2.5(166.9)M2
Х
2.5
3.5
1-
7
-
(-)
7
M2
M2
PS
Elő
-
7M ( 2M2 - 1)
(7M2 1)
M
Ps
2 DM
M3
)
or
or
eti
Pς
7 M ( 2M2
|dpt
dos
-1)
1)
Pti
Ps
M
( 7M2
1:97
Pri
Ps
Ps
2 d M(166.9)
2.5
M-
M
and on substituting Eq. 1:97 in 1:96 and sim-
plifying
2.5
PRI
7
M
M2
OM :
D
Mar-as) ( 7M? - 1)
Ps 712M2-1)
+
үм
1:98
Using Eq. 1:93, this simplifies to
Using Eq. 1:91, which holds at any speed
dpt
dps
1)
DM ( 2M2
7
M ( 7M2
dm = may
hip M14-238( 7M2 - 1 )
(-as) ( 1
53.37 17/ 12M2-0)
PHI
Ds
1)
1:94
1:99
or
Letting dM go to AM and writing this in
terms of indicated quantities
dpt
dps]
dM
M (7M? - 1]
712M2-1)
PH
Ps
HOM Atels
AM= 'p-
(
373.11 2M2-1
( 7M2 - 1)
1:95
1:100
1:49
where T has the standard atmosphere value
corresponding to hp .
Eq. 1:68 can be used to compute the lag
constants of simple systems. Usually a
value of n = 1.0 is used for the computation.
Ref. 16 presents equations for computing the
lag constants of more complicated systems.
Whereas at subsonic Mach numbers, it is
permissible to use ground calibration values
of Ito and iso together with Eqs. 1:62 to
obtain it and is , considerable error may
result if this procedure is employed super-
sonically for the reason that the stagnation
temperatures are considerably greater than
ambient in the latter case.
Experience has shown that the actual lag
constants are related to the calculated lag
constants by the relation (actual) = KA
1
=
(calculated) where K is a correction factor
which accounts for the unknown effects of
orifices and roughness. This factor varies
from 1 for a system consisting of a single
instrument and a straight tube to as high as
5 for complicated systems with rough tubing.
A possibly satisfactory procedure under
supersonic conditions would be to use the
indicated total temperature for computation
of the absolute viscosity coefficient of the
airspeed system. If
Hti
is this coefficient,
then
(a) Step Function Input Test Method
λς : λςο
:
iso
Centro
vtipso
PSO
p's
)
and
РО
The step function input test method con-
sists (as may be surmised from the name)
of applying a step function input to the source
end of the system and measuring the response.
Eq. 1:67 is solved for a step function input
and the solution rearranged to obtain;
PSO
dy a dro
Pti
t
1:101
d:
loge
where pt, is the total pressure existing aft
of a normal shock computed from the indi-
cated Mach number reading.
(p - p') at t=0
(p-platt=1
1:102
The corrections given in Eqs, 1:92 and
1:100 are added to the indicated values to
obtain the corrected Mach numbers.
where p is the pressure p' approaches.
1:22
.
DETERMINATION OF THE LAG
CONSTANT
Eq. 1:102 shows that the lag constant is
the slope of the response curve, P-P' vs. t,
plotted on semi-log paper and is the time for
the pressure difference to drop to 1/e (.368)
of its initial value.
First, let it be clearly understood that
although there are methods of computing the
lag constant, flight corrections should never
be based on computations but should always
be based on experimentally determined val-
ues. The computed lag constant is useful
only as a means of roughly estimating be-
forehand the approximate magnitude of lag
error for a given system and flight condition.
If the system is underdamped and thus not
accurately defined by Eq. 1:102 or because of
the instrument contribution, actual time-
histories can have an oscillation superim-
posed on the logarithmic decay somewhat as
shown in Fig. 1:14.
1:50
A sample lag constant determination is
shown in section 1:24.
(b) Steady v Test Method
For these cases the lag constant is obtained
from a mean line drawn through the oscil-
lation. If the oscillation is large in com-
parison to the time constant, large errors
will creep in and the lag constant should be
determined by the steady time rate of change
of pressure method described below.
Tests may be made with step functions of
different magnitudes and the resulting lag
constant plotted against step input magnitude
and extrapolated to zero magnitude. Other-
wise, the step should be made small enough
to avoid violating the assumptions that the
flow is laminar and that the pressure change
is small compared to the pressure in the
tube. Ref. 17 gives the maximum pressure
drop per foot of line for which laminar flow
can exist as
Although this method has not been used
to date by the authors, apparatus for these
tests is now being built. The method has
been used by others, (Refs. 22 and 23) and
is useful for underdamped systems; there-
fore, it will be briefly described.
A constant rate of change of pressure
= k) is applied to the source end of the
system. Under these conditions, Eq. 1:67A
describes the steady state variation of p',
and from this equation we have
Δρ
3.745.10-6
8p3
Ib. per sq. ft.
P-p': pa.
L
1:103
Also, Ap/p (ambient) should be less than
.03.
Thus a plot of p and p' or t will provide
two parallel lines as shown in Fig. 1:15, once
the transients have been damped out.
Pop'
D
AP
p-p'
P
اد
(D-D')
TIME, 1
TIME, 1
Fig. 1:14
Fig. 1:15
1:51
From Fig. 1:15 we see that from similar
triangles Ap = p-p' for steady state motion
so that Eq. 1:67A becomes
replaces the instrument by an equivalent
volume (Ref. 22) is shown in Fig. 1:16.
Another apparatus which includes the test
instruments in the airplane (Ref. 23) is
shown in Fig. 1:17.
Др л
Δρ :
At
(c) Sample Corrections to Flight Data
or
At:1.
Hence a measurement of the time separa-
tion of the curve of p and p' at any given
pressure value determines the lag constant.
Application of the constant p test technique
has been limited in the past because of the
necessity of removing the airspeed (or alti-
meter or Machmeter) system from the air-
plane for the test. One such apparatus which
Sample lag corrections to airspeed and
altitude test data are given in section 1:25.
The calculations do not include the acoustic
lag correction which is small for the ex-
amples presented.
It is emphasized that, to correct for lag,
time histories of the quantities are needed
which in turn requires a photopanel or os-
cillograph installation in the airplane. No
sample Machmeter Mach number lag cor-
rection is included since no actual examples
are available at present.
System Source
Pressure Gage For Measuring
Source Pressure
System Tubing
Tubing 7 7
SOTO Atmosphere
To Vacuum Pump
I Used To Evacuote System)
Added Volume
VIIMT
수
​AD
U Tube Water Manometer for Measuring
Pressure Log AD
Fig. 1:16
1:52
Enclosure
Static Source
रु.
Total
Source
Altimeter
Counter
Pitot Head
Photopanel
Storage
Tank
Needle
Valve
Camera
b
To Vacuum
Pump
Counter
Intervelometer Controlling
Counters
Altimeter
Airspeed
Indicator
Photopanel
Switch Controlling
Intervelometers And
Cameras
Camera
Fig. 1:17
1:53
(d) Accuracy of the Correction
Although an accurate error analysis has
not been made, it is felt that the estimate in
Ref. 22 of £20% accuracy is probably quite
close to the truth. This accuracy means that
the system should be proportioned to make
a 20% error in correction lie within the
required accuracy of airspeed, altitude or
Mach number. For example, if the allowable
error is airspeed due to lag is 1 knot, then
the lag correction must be held to 5 knots
or less.
relative to the instrument. In some cases
it is possible to locate the instruments (or
associated recording equipment) close to the
sources of pressure as, for example, lo-
cating a photopanel with airspeed meter in
a fuselage nose close to a nose boon pitot-
static pickup. In other cases it is not pos-
sible to locate the instruments close to the
pressure sources as, for example, when a
trailing static head is required to hang a
wingspan or more below and behind the fuse-
lage.
1:23 DESIGN OF SYSTEMS
In all cases the tubing should follow as
direct a path as possible and should be free
of sharp bends. The tubing should have a
smooth interior finish, the diameter should
be as large as practical, and the pressure
orifices should have areas equal to the tubing
diameters,
The general arrangement and location of
the static and total pressure sources of air-
speed, altitude or Mach number measuring
systems are usually dictated by require-
ments other than low lag. However, within
the specified general arrangement, the de-
signer should arrange the variables under
his control to either reduce the lag to obtain
the required accuracy of the correction, or,
more preferably, to eliminate the need for
a correction. The system designer usually
has some control over the following variables
that affect lag:
Controlling the relative lag of the total
and static pressure sides of the system can
be a powerful means of reducing the airspeed
and Mach number lag in flight during specified
maneuvers. For example, if the lags of the
total and static sides are equal, there would
be no airspeed lag during steady speed climbs
and descents. If the lag of the total side is
negligible, the airspeed lag would be negligi-
ble during constant altitude increasing speed
tests regardless of the static pressure lag
error. Ref. 15 considers the effect of relative
lag on the Mach number lag error. The alti-
meter lag depends only on the lag of the static
pressure system.
(a) Instrument characteristics.
(b) Number of instruments in the system.
(c) Tubing length, inside diameter and
surface finish.
.
(d) Relative lag of total and static pres-
sure sides of system (this influences the
airspeed and Mach number lag).
1:24 CONCLUDING REMARKS
Ref. 17 shows that instrument lag is mini-
mized by selecting instruments with a high
natural frequency, low damping and low fric-
tion. Ref. 20 discusses altimeter lag. In
general, the number of instruments in the
system should be kept to a minimum in order
to reduce the volume and complexity of the
system (and the lag).
The lag errors in airspeed, altitude and
Mach number measurements have been dis-
cussed and examples of corrections for these
errors presented. Maneuvers during which
significant lag errors have been encountered
have included stalls and high speed dives. Lag
errors must be taken into account either by
showing that the corrections are small and
can be discarded or by correcting the flight
data. The pitot-static system should be de-
signed with the view of minimizing lag errors
insofar as this is practical.
The minimum tubing lengths are usually
set by the locations of the pressure sources
1:54
1:25 SAMPLE DETERMINATIONS OF AVcal AND Ahp
Lag Correction to Altitude:
Lag Correction to Airspeed:
AT
AVcal =
it
Anp = + so
is hip
Aso
to
+
Vcal + (aso
,
hp [fhp. Vcal) ]
ito
As - xto xto
iso
PLAA MANEUVER
STALL
TIME
QUANTITY
SEC
UNITS
FLIGHT DATA
FROM
STEP
0
1
2
3
1
5
6
7
1
ho
Ft
Flight Data
5, 020 4,850 4,695 | 4, 580 | 1, 535, 4, 740 5,0155, 330 10,006
2
Ft
Instr. Corr.
5, 024 4,853 4,698 4, 582 4, 537| 1, 7131 5,019 5, 334 10,000
Sec
Lag Check #
.556
.556
. 556
. 556
. 556
. 556
. 556
.556
. 50
4
1.15
1. 15
1.17
1.14 1.13
1.14
1.15
1.16
1.36
1:
3 150
+ /
λς/λso
hip
Ahp
hp
5
Ft/sec. Flight Data
-190
-170
-130
-100
0
+300
+320
+320
-20*
6
Ft
3 x 4 x 5
-121
-109
-82
-63
0
+190
+ 205
+206
-14
7
Ft
2 + 6
4, 903 1, 744 4,616 4,519 4, 537 4,933! 5, 2245, 540
9, 986
8
Kts
V cal
Flight Data
558.0 558.0 557.0 553.5 542.0 531.5 523. 1 510.0
100.8
9 V'cal
Kts
Instr. Corr.
555.8 556.3 554.8 551.3 539.8 529.0 520.3 507.3 100.0
01-1
10
V'col
+3.0
0
-3.0
-9.0 -11.0 -10.8 -11.50 -13.0
-1.0*
Kts/sec | Flight Data
Sec Lag Check
11
. 086
.086
.086
.086
.086
.086
. 086
.086
.086
12
.69
.69
.68
.68
.71
.72
.73
.74
1. 34
Ato
λ/λfo
dVca! /dhp
dy
13
.0127.0128.0130 .0131 .0136 .0139.0145 .0145
.089
1/sec
Kts
14
V cal
10 x 11 x 12
+. 178
0 -. 175 -. 527 -.672 -. 669 -.722 -.829
-.115
xto
ܘܪܐ ܘܪܬ
15
λς.
so
Kts
3 x 4 x 5 x 13
-1.537 1.395 -1.066 -. 825
0 +2.641 +2. 973 +2.987 -1.246
iso
As hip
it no
d Vcal
dhp
dVcal
hp
dhp
16
-Xto
Kts
-5x11x 12 x 13 +. 143 +. 129 +.099 +.077
01 -. 258 -. 291 -. 295
+. 205
ito ito
17
AV.
cal
Kts
14+ 15 + 16
-1.2
-1.3
-1.1
-1.3
-.7
+1.7
+2.0
+1.9
-1.2
18
Kts
Vcal
9 - 17
554.6 555.0 553.7 550.0 539.1 530.7 522. 2 508. 2
98.8
* Obtained from plotted time history of stall
#Similar to Fig. 1:18
1:55
100
90
80
70
C
STATIC SYSTEM
60
TOTAL SYSTEM
50
)
(
AT .368
STEP FUNCTIONS
2
58.0 LBS. / FT.
= 21.3 LB/FT:2
λς
= 0.52 SEC.
AT .368 ( 64.5 LBS./F1.2,
:) = = 23.7 LB/FT.9
40
30
is=0.07 SEC.
.07 SECA
-.52 SEC
20
(9:-9c4 LBS./FT.2
10
9
8
7
6
NOTE :
5
RESPONSE MEASURED BY
AIR SPEED INDICATOR
4
3
O
0.25
0.50
0.75
1.00
1.25
1.50
TIME
- SEC.
Fig. 1:18
1:56
REFERENCES
1. Weaver, “The Calibration of Airspeed and Altimeter Systems,"U, K. Ministry of Supply
Report No. AAEE/Res/244, August, 1949.
2. Smith, “The Measurement of Position Error at High Speeds and Altitude by Means of a
Trailing Static Head," U. K. Ministry of Supply, RAE Technical Note No. Aero, 2163,
June, 1952.
3. Zalovcik, “A Radar Method of Calibrating Airspeed Installations on Airplanes in Maneu-
vers at High Altitudes and at Transonic and Supersonic Speeds," U.S.A. NACA Tech-
nical Note 1979.
4. Huston, "Accuracy of Airspeed Measurements and Flight Calibration Procedures,”
U.S.A, NACA Report No. 919, 1948.
5. Rogers and Berry, "Tests On The Effects Of Incidence On Some Pressure Heads At
High Subsonic Speeds,” U. K. ARC Report No. 13,263, July, 1950.
6. Gracey and Scheithauer, “Flight Investigation of the Variation of Static Pressure Error
of a Static Pressure Tube with Distance Ahead of a Wing and a Fuselage," U.S.A. NACA
Technical Note 2311, March, 1951.
.
7. Ruskin, R. E., Schecter, R. M., Dinger, J. E. and Merril, R, D., "Development of the
NRL Axial Flow Vortex Thermometer,” NRL Report No. 4008, September 4, 1952.
8. Tabach, Israel, “The Response of Pressure Measuring Systems to Oscillatory Pres-
sures," NACA Technical Note No. 1819, February 1949.
9. Iberall, Arthur S., "Attenuation of Oscillatory Pressures in Instrument Lines," U.S.
Department of Commerce, NBS Research Paper RP2115, Journal of Research of the
NBS, Vol. 45, July, 1950.
10. Kendall, J. M., “Time Lags Due to Compressible-Poiseuille Flow Resistance in Pres-
sure-Measuring Systems," NOL Memo No. 10677, May 4, 1950.
11. Sinclair, Archibald R. and Robins, Warner A.,“A Method for the Determination of the
Time Lag in Pressure Measuring Systems Incorporating Capillaries," NACA Technical
Note No. 2798, September, 1952.
12.
Weidemann, Hans, "Inertia of Dynamic Pressure Arrays,” NACA Technical Memo No.
998, December, 1941.
13. DeJuhasz, Kalman J., “Graphical Analysis of Delay of Response in Air-Speed Indicators,”
Journal of Aeronautical Sciences, Vol. 10, No. 3, March, 1943.
14. Draper, C. S. and McKay, Walter, "Instrument Analysis," MIT, 1943-44.
15. Huston, Wilbur B., “Accuracy of Airspeed Measurements and Flight Calibration Pro-
cedures,” NACA Technical Note No. 1605, June, 1948.
1:57
REFERENCES
16. Charnley, W. J., "A Note on a Method of Correcting for Lag in Airspeed Pilot-Static
Systems," RAE Report No. Aero 2156, September 1946.
17. Schwarzbach, J. M., “Lag Correction to Flight Measurement of Airplane Stall Speed,"
M. S. Thesis for the University of Maryland, 1953.
18. Wildhack, W. A., "Pressure Drop in Tubing in Aircraft Instrument Installations," NACA
Technical Note No. 593, February 1937.
19. Head, R. M., "Lag Determination of Altimeter Systems," Journal of Aeronautical Sci-
ences, Volume 12, No. 1, January 1945.
20. Johnson, Daniel P., "Calibration of Altimeters Under Pressure Conditions Simulating
Dives and Climbs,” NACA Technical Note No. 1562, March, 1948.
21. Dommasch, D. O., Sherby, S, S., and Connolly, T. P., "Airplane Aerodynamics," Pitman,
New York, 1951.
22.
Smith, K. W., "Pressure Lag in the Piping of the MK V Trailing Static Head," RAE
Technical Memo No. Aero 258, May, 1952.
23. Herrington, Russel M. and Schoemacher, Paul E., "Flight Test Engineering Manual,"
USAF Technical Report No. 6273, May, 1951.
24.
Schaefer, Herbert, "Machmeters for High-Speed Flight Research,” Journal of the Aero-
nautical Sciences, Vol. 15, No. 6, June, 1948.
>
25. Swanson, W. E, and Gray, A, K., "Methods of Flight Test Performance Data Reduction
for Turbojet Propelled Airplanes, "North American Aviation, Inc., Report No.NA-47-1033
of October 24, 1947.
26. Dommasch, D. O., et al, “Flight Test Manual," Part I, Revised Edition, Preliminary
Copy, NATC, Patuxent River, August, 1953.
1:58
AGARD FLIGHT TEST MANUAL
VOLUME I, CHAPTER 2
THRUST AND POWER DETERMINATION
Ву
Daniel O. Dommasch
Princeton University
Section 2:1
Walter J. Hesse
Naval Air Test Center
United States Navy
Sections 2:2, 2:3, 2:4, 2:5, and 2:6
Jean Soisson
Centre d'Essais en Vol, France
Sections 2:7, 2:8, 2:9, and 2:10
John K. Moakes
Aeroplane and Armament Experimental Establishment
United Kingdom
Sections 2:11, 2:12, 2:13, and 2:14
Rene Le Duc
France
Sections 2:15 and 2:16
Kenneth J. Lush
Air Force Flight Test Center
United States Air Force
Section 2:17
David W. Bottle
Aeroplane and Armament Experimental Establishment
United Kingdom
Sections 2:18, 2:19, and 2:20
VOLUME I, CHAPTER 2
CHAPTER CONTENTS
Page
TERMINOLOGY
2:1
INTRODUCTORY COMMENTS
2:1
2:2
POWER MEASUREMENT; INTERNAL COMBUSTION ENGINES
2:2
2:3
THEORY OF RECIPROCATING ENGINE POWER CORRECTIONS
2:6
2:4
CORRECTIONS APPLICABLE TO POWER CHARTS
2:7
2:5
TORQUEMETER POWER CORRECTIONS
2:8
2:6
POWER MEASUREMENT - COMPOUND ENGINES
2:9
2:7
JET THRUST MEASUREMENT - INTRODUCTORY COMMENTS
2:9
2:8
DISCUSSION AND REGION OF APPLICATION OF THE METHODS OF JET
THRUST MEASUREMENT
2:10
2:9
THE JET FLOW MEASUREMENT METHOD
2:10
2:10
THE CLIMB PERFORMANCE METHOD
2:13
2:11
THE MEASUREMENT OF THE USEFUL THRUST OF TURBO-PROPELLER
ENGINES
2:15
2:12
MEASUREMENT OF SHAFT HORSEPOWER
2:17
2:13
THE ESTIMATION OF JET THRUST
2:17
2:14
CONCLUDING REMARKS ON TURBO-PROPELLER THRUST
DETERMINATION
2:18
2:15
RAMJET THRUST MEASUREMENT
2:18
CONCLUDING REMARKS ON RAMJET THRUST MEASUREMENT
2:16
2:21
2:17
MEASUREMENT OF ROCKET THRUST IN FLIGHT
2:22
2:18
GENERAL ANALYSIS OF JET THRUST MEASUREMENT
2:23
2:19
THE MEASUREMENT OF THE THRUST OF JET ENGINES
2:28
2:20
THRUST MEASUREMENTS WITH AFTERBURNING
2:32
REFERENCES
2:35
TERMINOLOGY
N
Engine RPM
Q
Torque Pressure
K
Torque Constant
Pb
BMEP: Brake Mean Effective Pressure
L
Length of Stroke
A
Piston Area
Pm, MAP
Absolute Manifold Pressure
BHP
Brake Horsepower
P
Air Density
po
Sea Level Standard Density
Ga
Weight or Mass Air Flow Rate
а
(f/a)
Weight Fuel-Air Ratio
H.V.
Heating Value of Fuel
Пт
Thermal Efficiency
e
Volumetric Efficiency
Pm
Manifold Density
F
Thrust
Fo
Gross Thrust
F
Net Thrust
N
р
Static Pressure
А.
Nozzle Area
V
True Airspeed
Q
Mass Flow
y
Specific Heat Ratio C
с/сь
Gas Constant
R
TERMINOLOGY
(Continued)
T
Ambient Temperature
8
Pressure Ratio
A
Effective Nozzle Area
F
pre
Pre-Entry Thrust
F
int
Internal Thrust
F
post
Post-Exit Thrust
F
SN
Net Standard Thrust
FSG
Gross Standard Thrust
F JN
Net Jones Thrust
FJG
Gross Jones Thrust
Fв
Trunnion or Bearer Thrust
u
V cos • : Velocity Component in Free-Stream Direction
V
Resultant Velocity
6
Angle between Flow and Free-Stream Direction
Local Swirl Angle
Subscripts
m
Manifold Conditions
e,ex
Exit Conditions
t
Total Conditions, also Tailpipe
8
Free-Stream
i
Intake
e
Exit
W
Far Downstream
Superscripts
(') (prime) Test Conditions
6
在
​@
2:1
INTRODUCTORY COMMENTS
(3) Effects of engine cooling on climb
and range performance,
In conducting flight tests, the power plant
is used not only as a means of propelling
the airplane, but also as a means for measur-
ing the drag characteristics of the airframe.
Thus, when we measure thrust and/or power,
we generally have two separate purposes in
mind:
(4) Effects of piston engine exhaust
stack thrust on apparent power required
curve,
(5) Effects of induction system on
power available.
(1) The determination of the capabil-
ities of the propulsive system as installed in
the airplane;
(b) Jet Type Aircraft
(2) The measurement of the drag
characteristics of the airframe in the pres-
ence of the operating propulsive system.
(1) Effects of entrance air spillage
on drag characteristics of airframe (sub-
sonic).
(2) Effects of external induced flow
produced by internal engine flow.
Because the airplane (considered here as
composed of an airframe and a powerplant)
functions as a unit, we cannot, in a general
sense, completely divorce engine and air-
frame characteristics from one another. This
holds true for performance analyses and also
for stability and control investigations where
engine operation, particularly at low airspeed
at full throttle may profoundly affect the air-
plane's moment balance.
(3) Effects of inlet design on flow
characteristics in jet intake and compres-
sor.
(4) Effects of entrance shock waves
on airframe drag (supersonic).
(5) Effects of yawing and pitching
moments produced by the changing direction
of airflow passing through the engine(s).
Although interaction effects may be small
in many cases, they are a possible cause of
failure to obtain generalized data using reduc-
tion processes which ignore them. Thus, even
though possible interaction effects are not
considered in detail in all portions of this
manual, they should be kept in mind and made
the subject of investigation in cases where
one suspects that poor performance is attrib-
utable to such effects. A partial list of inter-
actions is as follows:
(a) Propellered Airplanes
(1) Effects of airplane configurations
on propeller efficiency (cowl flaps open or
closed, etc.).
(2) Effects of propeller on airplane
drag characteristics, stalling characteris-
tics, stability characteristics, airspeed sys-
tem calibration, etc.
In some cases, the method of conducting
tests and of reducing the data tacitly takes
into consideration interaction effects, at
least in part.
For instance, this is the
case for the reduction procedures described
in Chapter 6 for piston engine aircraft, where-
in the brake horsepower required rather than
the thrust horsepower required character-
istics are determined so that direct meas-
urement of airframe drag is avoided. Meth-
ods such as these are generally satisfactory,
provided attempts are not made to gener-
alize data over too great a spread of al-
titudes or other operating conditions.
2:1
power output should be for the engine as
installed in the airplane and such charts
should be used with care by the test en-
gineer.
The measurement of items such as shaft
brake power of a piston engine or the gross
thrust of a jet engine is a rather simple pro-
position and direct measurements are possi-
ble. On the other hand, measurement of
thrust power of the piston-engine-airplane
combination or of the net thrust of a jet en-
gine is not so simple a matter because inter-
action effects are likely to enter the picture.
Indeed, for thrust power developed by a piston
engine we must normally assume a propeller
efficiency since we cannot readily measure
the efficiency in flight.
In the following sections of this chapter,
the detailed analyses of thrust and power
measurements for various types of engines
are discussed. In editing this work, which
has been contributed by several of the NATO
nations, the principal modifications have been
to standardize terminology.
2:2 POWER MEASUREMENT; INTERNAL
COMBUSTION ENGINES
Furthermore, propeller charts prepared
either from computations or from wind tunnel
tests very seldom represent accurately the
propeller operation as it actually occurs in
the presence of the airplane. In the case
of a jet engine, the question arises as to
whether or not the possible drag changes
due to engine operation should be subtracted
from the engine thrust or added to the air-
frame drag. The question further arises as
to the proper value of ram drag which is
normally computed on the basis of all the
engine intake air obeying the laws of one-
dimensional flow and on the basis that there
is no spillage of the entering air.
The power of an aircraft internal combus-
tion engine is determined either by a torque-
meter or by the use of power curves (prepared
by the engine manufacturer, or by a suitable
engine testing laboratory). The preferred
method is actually to measure the power
with a torquemeter. For engines not equipped
with torquemeters, reasonably accurate
values of power can be obtained from the
power charts.
In a standard torquemeter installation, the
torque force is balanced by an oil pressure
which acts on small pistons located around
the periphery of the reduction gear housing.
The brake horsepower is then given by the
following equation:
BHP =
KNQ
2:1
For practical purposes, it is generally
sufficient to work on the basis that the ram
drag of the engine is equal to the rate of
change of momentum of the air entering the
engine, and that the net thrust of the engine
is equal to the difference of the gross thrust
produced aft of the turbine and the ram drag
computed as noted above. This may produce
difficulties in drag data generalization if air
spillage exists under some circumstances
and not under others. However, since air
spillage is an indicator of faulty design, the
fact that the tests reveal its existence should
not be construed as a deficiency in the test
method or of the definition of terms used.
Finally, a general word of caution: en-
gine charts prepared on the basis of ground
tests and/or computations are seldom a
sufficient basis for extrapolating test data
to determine what the standard thrust or
where N = the engine RPM
Q Q = torque pressure
-
K = torque constant.
From Eq. 2:1 it is evident that it is a
simple matter to obtain the brake horsepow -
er of an engine when the torquemeter oil
pressure and the engine RPM are known.
This method of measuring power has been
2:2
used very successfully, and when the torque-
meter equipment is properly installed and
maintained, precise power measurements
may be obtained.
horsepower by replacing the indicated mean
effective pressure with the brake mean effec-
tive pressure, Pb; i.e.,
POLAN
Brake horsepower can be expressed by an
equation similar to the equation for indicated
BHP :
.
РЫ
.NK.
33,000
2:2
BRAKE HORSEPOWER
FULL THROTTLE LINE
2,600 R.P.M.
2,400 R.P.M.
2,200 R.P.M.
2,000 R.P.M.
9
MAP- ABSOLUTE MANIFOLD PRESSURE
Fig. 2:1 Sea Level Calibration Curve
2:3
Thus the torquemeter oil pressure is a
direct function of the brake mean effective
pressure. Because of this fact, many torque-
meter installations are calibrated so that the
torquemeter oil pressure indicates values of
brake mean effective pressure instead of the
actual oil pressure. This alternate presenta-
tion is used to advise the pilot of the BMEP
that his engine is developing so that he may
know its relation to the operating limitations.
It should be noted that when the instrument
dial is calibrated in terms of BMEP, the
constant K will differ from that defined by
Eq. 2:1.

MAXIMUM POWER LIMIT
RESTRICTED REGION
FULL THROTTLE, CONSTANT R.P. M. LINES
BRAKE HORSEPOWER
2,600 R.P.M
2,400 R.P.M.
2,200 R.P.M.
42
i 2,000 R.P.M.
40
FULL THROTTLE,
CONSTANT MAP LINES
38
MANIFOLD PRESSURE, INCHES OF HG.
32
30
28
26
60
10
20
30 40 50
ALTITUDE. THOUSANDS OF FEET
Fig. 2:2 Altitude Calibration Curve
2:4
given engine can be determined from them
for given values of RPM and manifold pres-
sure. A full throttle line obtained by connect-
ing the end points of the RPM curves is also
shown on the chart.
As with other instruments, care must be
exercised to insure that the torquemeter func-
tions properly. In addition to the normal in-
strument problems of lag, hysteresis, needle
fluctuation, etc., the torquemeter readings
are subject to error at low ambient temper-
atures because of the congealing of oil in
the line from the engine to the instrument
panel. This problem is usually overcome by
incorporating an oil bypass line with a valve
in the system which is left open except when
readings are actually taken.
The altitude calibration chart shown in
Fig. 2:2 can be: (1) plotted from equations
or (2) obtained from actual data determined
by flight tests or simulated flight test con-
ditions. The latter scheme is, of course,
preferable; however, the equations which
are used are based on data which have been
collected over a period of years using many
reciprocating engines, and the results ob-
tained are reasonably accurate,
As stated previously, when an engine is
not equipped with a torquemeter, the power
developed by the engine must be obtained
from a power curve as prepared by either
the engine manufacturer or other engine test
laboratory. The use of power charts, although
not really an accurate procedure, does give
a fair indication of the power being delivered
(usually within 5%).
The equation which relates horsepower at
altitude to power at sea level is
P
BHP : ВнР.
sl
. [.3245 -0.1324]
Po
2:3
The reciprocating engine performance
charts consist of two plots, a sea level cal-
ibration chart (or some other fixed altitude
chart) and an altitude calibration chart. The
former is always obtained by actual test; the
latter may be calculated by certain equations
or may be determined from actual or simu-
lated flight test conditions.
The altitude scale of Fig. 2:2 is usually
based on a linear density ratio scale in order
that the power lines may be straight. The
solid lines of the figure represent full throttle
operation for a given supercharger condition.
The two charts when combined present
the whole performance story on a specific
engine for any type of operating conditions.
Therefore, in addition to being used for power
determination, these charts can be used to
find the potentialities and the limitations
of a given engine.
When the data of the sea level calibration
curves are combined with the data of the al-
titude calibration curves, the combined charts
are presented as in Fig.2:3. The reader is
referred to Ref. 1 for a detailed discussion
on the construction and use of such perform-
ance charts,
For the sea level calibration tests, the
engine is connected to a dynamometer or some
similar device whereby the power can be
measured. Runs are made at constant values
of RPM with varying manifold pressure to
provide charts such as illustrated in Fig.
2:1.
Although engine performance charts are
not now used to the extent they were, they
still serve a useful purpose in that they al-
ways provide a check on torquemeter read-
ings, and furthermore, they give the perform-
ance characteristics of a specific engine for
any combination of RPM, manifold pressure,
,
and altitude conditions. From these charts
a pilot can determine what performance he
may expect from his engine and therefore
achieve the best settings for a specific flight.
Such charts are applicable only for stand-
ard sea level conditions. The power of a
2:5
2:3 THEORY OF RECIPROCATING ENGINE
POWER CORRECTIONS
If we assume that all terms are essen-
tially constant for a given engine, with the
exception of the air density and volumetric
efficiency, we obtain the equation
BHP : constant (pm)(e).
The theory of corrections is based on
certain simplifying assumptions, which in
most cases are logical. Although some as-
sumptions are made that are not exactly
correct, the overall result is usually within
allowable flight test limits. The theory is
based on the fact that BHP is proportional
to the amount of fuel burned; i.e.,
Since we do not normally measure the man-
ifold density, we must express the BHP in
terms of quantities we do measure. Ref. 2
shows that the BHP is closely approximated
by the relation
BHP : Ga
GO
(6-) .
(H.V.)ny const.
2:4
BHP : const pm
Vio
Expressing the airflow through the engine
in terms of displacement, RPM, manifold
air density, and volumetric efficiency,
where Ta = ambient air temperature
=
-
Pm = manifold pressure.
Go - (Displacement)(N)(e)lem).
:
2:5
Therefore, the correction equation can be
.

46
44
+42
B140
FULL THROTTLE, CONSTANT MAP
LINES
FULL THROTTLE,
CONSTANT R.P.M. LINES
2,400 R.P.M.
BRAKE HORSEPOWER
BRAKE HORSE POWER
| 2,200 R.P.M.
2,400 R.P.M.
12,600 R.P.M.
2,200 R.P.M.
2,000 R.P.M.
38.
36
1,800, R.P.MA
1
34.
320-
1,800 R.P.M.
30
2850
261
24
22
10
20
30
40
50
24 28 32 36 40 44
MANIFOLD PRESSURE,
IN INCHES OF HG.
ALTITUDE, THOUSANDS OF FEET
Fig. 2:3 Combined Sea Level - Altitude Calibration Curves
2:6
written as
(b) Outside Air Temperature
CAT'
MAP
OAT'
BHP
BHP
-
MAP
MAP'
.
CAT
MAP
ОАТ
2:6
when
CAT - Carburetor Air Temperature
OAT : Outside Air Temperature
A correction must be applied if the outside
air temperature is not standard for the par-
ticular altitude. The brake horsepower var-
ies about 1% for a change of 6 degrees cent-
igrade (or 10 degrees Fahrenheit). A colder
temperature means that the air is more dense.
Consequently, more power is developed at
lower ambient temperatures. The formula
for correcting to standard conditions for a
deviation of outside air temperature for a
given manifold pressure is obtained indirectly
from Eq. 2:6 as
8
and the primes represent test conditions and
the terms without superscript represent
standard conditions. Eq. 2:6 is the relation-
ship normally used for power corrections.
HP
OAT'
2:4 CORRECTIONS APPLICABLE TO
POWER CHARTS
HP
ОАТ
2:7
(c) Manifold Air Pressure
Engine performance charts are always
drawn for standard altitude conditions without
ram. Therefore, if a given engine is opera-
ted under nonstandard conditions or operated
with the benefit of ram pressure, certain
corrections must be made to adjust the engine
performance to correspond to the perform-
ance charts applicable to the engine,
Sometimes a given problem requires the
computation of the manifold pressure neces-
sary to produce a given brake horsepower un-
der standard conditions. In such cases, the
observed brake horsepower reading is as-
sumed invariant and we correct the manifold
pressure by applying a correction on outside
air temperature. Since manifold pressure
is a direct function of horsepower, its cor-
rection equation appears as
(a) Outside Air Pressure or Altimeter
Reading
OAT
MAP
MAP
.
OAT'
No correction need be made to the altitude
reading (except for instrument and position
errors) provided that the altimeter has been
set to 29.92" Hg. and that pressure altitude
is to be used as the reference for all correc-
tions.
2:8
(d) Ram Pressure
There is available a choice in procedure.
For example, suppose we have data on alti-
tude, OAT, MAP, and BHP. The question
arises, "Should we correct altitude and MAP
to give us the same BHP that we would obtain
on a standard day or should we correct the
MAP and BHP to values which we would have
obtained at the observed altitude reading on a
standard day?" The usual practice is to use
the latter scheme; therefore, we start our
corrections on the basis of the altimeter read-
ing which is a measure of ambient pressure.
The basic performance charts are com-
puted for the condition of zero ram and it is,
therefore, necessary to make allowances for
these effects of ram pressure on engine oper -
ation, During part throttle operation, the
amount of ram pressure which may exist
cannot be obtained from the performance
chart. It must actually be measured. This
.
is true because for part throttle operation
the performance charts present data on RPM,
MAP, BHP, and altitude, but not throttle
position. When an engine installation provides
positive ram pressure, the same performance
2:7
condition may be obtained as for an engine
without ram, but at a lower throttle setting.
because all we do is specify some reference
state at which we make our corrections.
(b) Correction of BHP to the Observed
MAP
For full throttle operation, the story is
quite different because the throttle position is
fixed (full); therefore an engine which oper -
ates with ram will be able to operate at
altitudes and powers beyond those predicted
by the performance charts. These differences
can readily be evaluated from the flight data
and the perforinance chart data. It is there-
fore possible to obtain the amount of ram using
a performance chart for full throttle opera-
tion. Ram pressure is usually expressed in
feet of altitude,
This type of correction answers the fol-
lowing question: "What power will I obtain
on a standard day for the same MAP that
I observe today?" Since in this method we
are correcting to the observed manifold pres-
sure, we automatically state the following:
MAP: MAP'.
*
2.5. TORQUEMETER POWER
CORRECTIONS
•
Torquemeter power corrections are usu-
ally divided into two categories, namely, part
throttle and full throttle. Both methods are
based however, on Eq. 2:6, the difference
being that in part throttle corrections a
simplification is made by making the ratio
MAP/MAP or BHP/BHP' equal to unity.
This simplification is realistic in part throttle
corrections because the engine can be op-
erated at the same MAP or BUP on a test
day and a standard day.
When this relation is substituted into our
basic correction equation 2:6, we obtain the
following expression for the corrected BHP:
OAT'
CAT'
BHP = BHP'
BHP'
ОАТ
CAT
2:9
If it is desired to correct the carburetor
air temperature to the standard day condition,
this can be accomplished by the following
relationship:
CAT = CAT' + OAT -OAT '
2:10
Correction of MAP to the Observed
(c)
BHP
(a) Part Throttle Corrections
(Any Type Supercharger)
This type of correction answers the follow-
ing question: "What MAP would I need on a
standard day to obtain the observed BHP that
I read today?" For this particular type of
correction we state that the standard BHP is
equal to the observed BHP, and write
Whenever performance data are obtained
at part throttle operation, we do one of two
things, namely: (1) correct the BHP to stand-
ard conditions at the observed value of mani-
fold pressure; or (2) correct the manifold
pressure to standard conditions at the ob-
served value of BHP. Thus, we either spec -
ify the manifold pressure or the BHP, to be
the standard value.
BHP : BHP'.
-
We may then correct the manifold pressure
by inserting the above into the basic correc-
tion Eq. 2:6 to obtain
ОАТ
CAT
In addition to this, the normal practice
is to make all corrections at the observed
pressure altitude. Consequently, we also
specify the altitude to be a standard value.
No errors are introduced when doing this
MAP = MAP
✓
= MAP
OAT'
CAT'
2:11
2:8
2:6
POWER MEASUREMENT -
COMPOUND ENGINES
Again, the carburetor air temperature can
be corrected to standard condition as shown
before by Eq. 2:10. It should be noted that
using the above procedure, we always make
our correction at the observed value of the
pressure altitude. This is the simplest way
of performing power corrections.
(d) Full Throttle Corrections
The full throttle corrections are more
difficult to make because it is necessary to
correct both MAP and BHP for a given set of
flight conditions, We saw earlier that for
part throttle operation we needed to correct
only one variable, but for full throttle opera-
tion it is necessary to correct both MAP and
BHP. We must do this because, when opera-
ting at full throttle, the only thing that we
can duplicate from day to day in actual flight
testing is the throttle position, namely full.
For example, if one flies at full throttle and
desires to know what MAP and what BHP
would be obtained on a standard day, it would
be necessary to correct both quantities to
standard conditions. Then, theoretically, if
one flew these settings on a standard day, all
data would coincide exactly.
The compound engine is merely a modi-
fication of the basic reciprocating engine in
that it is equipped with exhaust gas turbines
designed to extract power from the exhaust
gases and feed this power back to the crank -
shaft. The compound engines utilized by the
U.S.Navy incorporate three exhaust blowdown
turbines which are connected to the engine
crankshaft by means of a fluid coupling.
Since there is no definite speed ratio be-
tween the turbines and engine crankshaft, the
use of engine power charts is even less re-
liable for the compound engine; therefore,
practice thus far has dictated the use of a
torquemeter to determine compound-engine
power. The application of the torquemeter
to power measurement of the compound en-
gine is the same as illustrated in previous
sections; thus, the procedures will not be
repeated here.
2:7
JET THRUST MEASUREMENT -
INTRODUCTORY COMMENTS
If one corrected only BHP, however, and
assumed the MAP to be the standard value,
conceivably part throttle conditions would be
represented for a standard day. Thus, it is
evident that since operationally we can only
duplicate the throttle position for full throttle
operation, we must make corrections to both
MAP and BHP.
The acceptance test of a jet engine should
include the measurement of thrust, fuel con-
sumption and tailpipe temperature which all
are functions of the airplane speed, the alti-
tude and the engine RPM.
Of this group
of characteristics, only the thrust cannot be
measured in flight using the same techniques
employed on the ground test stand,
It is possible to measure the inflight
thrust in several different ways. Two methods
presently used in France which give close
agreement between their results are de-
scribed here: the first is based on a study
of the flow in the engine aft of the turbine, and
the second on a study of the airplane perform-
ance in a climb.
We shall consider first the principles on
which the methods are based and the cases to
which they may be applied. Subsequently, the
calculations required to reduce the flight data
In addition to correcting these for full
throttle operation, we must also take into
account the ram pressure which is available
to the engine and the number of supercharger
stages. Thus, the full throttle power correc-
tions become much more complex than the
part throttle corrections. The procedure for
accomplishing full throttle corrections is
contained in Chapter 6.
2:9
to terms of thrust are considered.
2:9 THE JET FLOW MEASUREMENT
METHOD
2:8
If FG =
Gross thrust
DISCUSSION AND REGION OF
APPLICATION OF THE METHODS
OF JET THRUST MEASUREMENT
Qe
Il
Mass flow through tailpipe
(a) The Method of Jet Flow Measurement
Vex = Exit velocity
Ae
Exit area
This method is based on the fact that the
thrust delivered by a jet engine depends on the
nature of the entering and exhaust flow through
Pe =
Exit static pressure
the engine.
p -
-
Ambient static pressure
Q.
Entering mass flow
The net thrust FN of a jet engine is deter-
mined by the tailpipe nozzle pressure ratio,
the total tailpipe temperature Ttt , the amb-
ient static pressure p
the nozzle area Ae
and the true flight speed V.
V
True flight path speed and
FN =
=
Net thrust,
The method of measuring engine thrust in
terms of the aforementioned parameters is
rapid and simple to use; however, initial
ground thrust stand calibrations are required
to correlate the theoretical equations with the
actual operating conditions. It is desirable
to verify the calibration coefficients de-
termined on the ground by an alternate in
flight measuring technique,
we have the customary gross and net thrust
definitions given below:
FG = Qe Vex + Ae (pe -p)
(
2:12
FN = FG
- QV.
2:13
For subcritical or unchoked flow, Pe = p.
(b) The Climb Performance Method
For sonic (choked) conditions at the exit
of the tail cone, the exit pressure Pe is re-
lated to the total tailpipe pressure Ptt , by
the equation,
Pe
2 Y/17-1)
+62)
.
This method is particularly applicable to
interceptor-type aircraft having high rates of
climb at maximum or climbing RPM. It is
only necessary to measure the classic air-
plane parameters: time t, static pressure p,
ambient air temperature T, true airspeed V,
and the fuel consumption. From these we
may determine the true and energy heights,
the actual and energy climb rates, and the
airplane weight at any instant.
Ptt
+1
2:14
where Ptt, the total tailpipe pressure, may
be measured at any point in the tailpipe, if
we assume isentropic flow between the tur-
bine outlet and the exit section,
Similarly for choked flow, we have the
temperature relation
The climb is made under the conditions
specified in Chapter 7 for the maximum ener-
gy climb. As we shall see later, this proce-
dure almost entirely eliminates the influenc
of engine flow on drag, which later inter-
action factor may cause errors in perform-
ance determinations as noted in section 2:1.
2
Te
T +7
Y+1
2:15
2:10
where Te
= exit static temperature
Tu
= tailpipe total temperature.
Editor's insert: In the United States a
direct reading gross thrustmeter has been
developed based on Eq. 2:19, expressed in
a modified form. To obtain the gross thrust -
meter relation for unchoked flow, we note
that Eq. 2:19 may be written
FG 2y
Ptolly-lly
-1)/y
If the exit conditions are subsonic (un-
choked), the exit velocity is given theoret-
ically by the isentropic relation
(
-.
p
2 YR
(- 1)/Y
Ae
Yol
༢༣༨
.
2:20
Vex -
[
(697) vor]
Y-1
=
and introducing 8 = p/po; where po = stand-
ard sea level static pressure, we obtain
2:16
FG
2y
For sonic exit conditions
(
0-2)
-
8 AE
Y-1
2:21
Vex : ✓YRTE
.
2:17
from which we see that the quantity FG/Ae is
a function only of the pressure ratio Pu/p
and does not depend directly on either the
ambient pressure or the total tailpipe pres-
sure, but only on their ratio.
The exit mass flow rate differs from the
entering flow rate because of the addition of
fuel. Normally, the fuel weight is ignored,
since it is a small quantity in comparison
to the total mass of air passed through the
engine in a given time. Under these circum-
stances, Qe and Q may be assumed equal,
both being given by the relation
Pe
Q = pe Alvex - Ae Vex
2:18
Moreover, Eq. 2:21 is identical in form to
the Machmeter calibration formula except for
a radical. Therefore, a Machmeter connected
on one side to the total tailpipe pressure pick-
up and on the other to the static source may
be calibrated to read in terms of the para-
meter FG/Aed.
RTe
1
For the unchoked nozzle
1
Pro \y-1)/y
ly .)y
T: (+1
:)
Pe :p and
Te
Actually, it has been found that a simple
differential pressure gage measuring the dif-
ference between Plt and p may be directly
calibrated to read gross thrust in pounds
with the calibration being precise at sea level
and only slightly in error at altitude. For
precise work at altitude, simple corrections
may be applied.
and the gross thrust equation becomes
FG
QVE ZYR
.
-TT
х
x
Ае
Ае
Y-1
For sonic flow at the exit, we have
FG
Pe
(y-1)/y
: V
V?
P
+ (De-o)
lex RTE
11/1-G2-20%*
A)?
10K
(0)"
Ae
RT17
2:22
or
FG
2y
р
외
​Pit
and, using Eq. 2:17,
FG
+ De p = (y + 1)pe-p;
Ае
Y-1
۲۹۱)
= y Pe
-
2:23
2:19
Ae
2:11
then from Eq. 2:14
and
FG
DIE
2
FG
Y/17-1)
-D
= ly+1) P17 lyt
-
КРО
1.26
for choked flow
Ae
Y+1
Ae8
D
2:28
2
ylly-1)
р
PAT
( y +1)
where K = f(Pt/p) in both cases.
f
For gross thrustmeter purposes, Eq.2:24
may be written
ylly-1).
F G
Рft
of(
+1)
2
Y +1
-1
Ae
р
For choked flow, K is essentially constant
with a value of between 0.95 and 1.00. For
subcritical flow this coefficient decreases
gradually as Plt/p decreases. We note that
the pressure Pt, should be determined by some
sort of averaging total head tube which ad-
equately samples the exhaust gases. Tocal-
culate the net thrust it is necessary to deter-
mine the mass flow rate Q. The equation for
mass flow rate depends on the nature of the
flow (subcritical or choked).
2:25
Since the tailpipe flow is at an elevated
temperature, y~1.33 and
FG
L.
1.26
SA
For subcritical flow, Eq. 2:18 gives
2:26
pe
.
Ае
IRT e
2 YR
р
For normal nozzles, the flow becomes
choked at Plt/p~1.85; therefore, Eq. 2:21
applies to lower ratios and Eq. 2:26 to higher
ratios.
-1/
T
1,16) =117.
Y-!
Pit
and p = Pe,
27
Q
Since Te = Ttt (p/Pr)(y-1lly
PRE
/т,
=
Х
For practical purposes it must be realized
that isentropic, one-dimensional flow con-
ditions (as assumed in the derivation) do not
actually exist except on the average, and
therefore, an efficiency factor should be ap-
plied to all equations. This factor must be
determined by calibration procedures.
Ае
(Y-IR
р
217
(y+1)/y
Il
( 4 )
67.82***
۲۹۱
Dat
2:29
Because the real nature of the flow is a
function of the pressure ratio, the K factor is
also a function of the pressure ratio and
should be determined accordingly.
or, alternately,
Q
р
27
Х
AD
VTT
ly-IR
Thus, we may write
F G
2y
Pylly-illy
=
Ae8
* (34). [-
.)(x1
Pt 12ly-llly
Prilly
9-17
-1/y.
2:27
for subcritical flow
2:30
2:12
Eq. 2:30 may be written
and for choked flow,
Q TT
2Y
Х
outte
SA e
(-1)R
-
SA
PTH2ly-1)/y
Ptt
Pfly-1)/y
0:2243
6) -Y
КQPo
()
VGA
Y 2 \6y+1)/(y-1)
/1
R V+
-)(x+
p
2:31
2:35
For choked flow, Eq. 2:18 gives
Pe
y
ola
YRTE
-
Pe
Ае
RTe
RTe
and using Eqs. 2:14 and 2:15,
+1)/y-1)
ola
Sen +264)
where KQ = f(Pty/p).
We might compare this method of deter-
mining mass flow in terms of exit conditions
to the possibility of measuring conditions at
the intake duct where the temperature dis-
tribution is far more uniform than in the
tailpipe.* In this latter case, the nature of
the entrance flow and the pressure distribu-
tion therein can be markedly influenced by
changing flight conditions and possible flow
separation, particularly when split ducts are
employed. Moreover, the calibration of the
entrance cone is a laborious proposition,
whereas the calibration of the tail cone is
quite conveniently accomplished.
(67)671
R
yt
2:32
or, in another form,
($)/,(). ܙܙ*/ܘ
Q
art
PIA
2
2 \r+1)/(x-1)
R+
1(+1(.
SAe
2:33
We note finally that the term (V in level
flight (at subsonic speeds) has a value about
30% as great as the gross thrust, so that
greater errors are permissible in the deter-
mination of the ram drag than in the gross
thrust.
2:10 THE CLIMB PERFORMANCE METHOD
Since the mass flow rate depends on the
temperature as well as on the nozzle pres-
sure, an additional source of error is intro-
duced here since one must properly measure
the total tailpipe temperature. To account
for the fact that the flow is not actually of
the isentropic one-dimensional type and for
the difficulty of accurately sampling pres-
sures and temperatures, a nozzle discharge
factor should be added to Eqs. 2:29, 2:30,
2:32 and 2:33, whence for unchoked flow,
The most severe criticism which can be
made of the preceding method is the fact that
its accuracy depends on the proper deter-
mination of the coefficient K and KQ, which
must be established by ground calibrations.
Even using nozzles more convergent than the
flight tailpipes, it is not possible on the
ground to obtain the pressure ratios, Plt/p,
encountered at altitude. Also in the region
where the curves of K and KQ as functions
27
KOPO
х
Ae
ly-I)R
PT
*+y
2ly-1)/y
Pilly-1)/Y
- 0)
* See also Eq. 2:46 for determination of mass
flow rate from entrance conditions.
2:34
2:13
of the ratio Puc/p do not require extrapolation,
there is no guarantee that the ground caiibra-
tion will hold al all altitudes.
the same values at other altitudes, it is suf-
ficient to determine the relationship between
FN, and FN at some higher altitude, first
using the "jet method" and then using the
“performance method”, which directly gives
the ratio FN/FN.
Attempts have been made to correlate the
"jet method" of thrust determination with
data obtained from accelerated and decel-
erated level flights (with corrections imposed
for slight inadvertent climb or descent rates).
These tests were based on the fact that the
net thrust equals the drag plus the inertia
force.
Thus, this latter method compares the
thrust obtained at the same engine RPM un-
der the same conditions of climb, but at dif-
ferent altitudes. It essentially eliminates
the drag variations which might be produced
by changes of entrance duct flow. We shall
sce also that the method allows us to deter-
mine the increase in thrust due to after-
burning.
Two flight conditions were considered:
the first one at high RPM (for which we
wish to measure the thrust), and the second
at a reduced RPM (nearly closed throttle) and
at the same true speed as the first. The
difference between the two values of accel-
eration at the same airplane speed gives the
variation between the thrust at the reduced
RPM and at high RPM. The reduced RPM
thrust can be considered as a corrective term,
and may be determined, for example, from
the "jet method" because even a larger error
in this term introduces only a small error in
the thrust measured at high RPM.
In the following work, the subscript o rep-
resents low altitude and symbols without
subscripts refer to high altitudes. Drep-
resents the aerodynamic drag; w, the rate of
change of energy height; m, the airplane
mass, and g, the gravitational acceleration.
From the flight equations, we have
(FN-DV = mgw.
It follows that
FN-D
<
313
313
$TS
FNO-
vo
2:36
The procedure described above has not
provided satisfactory results and it is be-
lieved that the major source of error is the
“a priori' assumption that at a given alti-
tude and airplane speed, the airplane drag is
independent of engine RPM. Wind tunnel
model tests have shown that variations of
CD as great as 25% may be produced by
variations of flow rate through the intake,
and for this reason the outlined procedure
was abandoned.
from which
FN
W
FNO
mo Wo
la
D
W
Do
+
D。
mo wo vo
Wo
FNO
2:37
We now seek a simple method which per-
mits us to establish that the coefficients K
and KQ are functions only of pressure ratio
and not of altitude itself.
We may assume that the values of K and
KQ, established by ground tests, are correct
when the airplane is at its best climbing
speed at low altitude, and use the "jet method"
to compute the net thrust FN, under these
conditions. To establish that K and KQ have
During a climb, the first term of Eq. 2:37
decreases. As measured during recent tests
on an interceptor at the Flight Test Center
(France), the first term had an initial value
of 1 at sea level; 0.6 at near 20,000 ft.; 0.4 at
30,000 ft.; and 0.2 at 40,000 ft. The second
term, on the other hand, varied from 0 at
the ground to 0.04 at 20,000 ft.; 0.09 at
2:14
30,000 ft.; and 0.1 at 40,000 ft. Thus, the
second term may be considered as a correc-
tion term at altitudes less than 30,000 feet
for the case considered.
In practice, at altitudes where the ratio
w/wo exceeds 0.5, one may consider the
second term as of second order in comparison
to the first; therefore, this term can be com-
puted from wind tunnel data for the particular
airplane without introducing serious errors,
Knowing the airplane weight and the airspeed,
we thus compute D and Do.
and some prior knowledge of propeller ef-
ficiencies under the encountered range of
operating conditions. Methods of measuring
jet thrust, and useful propeller thrust are
discussed in other sections of this chapter.
In this section these two measurements are
considered in relation to the turbo-propeller
engined aircraft, by investigating the ac-
curacy to which net jet thrust and shaft power
should be measured, and thence discussing
current and possible methods available for
making these measurements.
Accuracy Requirements
As for the thrust FNo, this may be deter-
mined by the "jet method", since at low al-
titudes the sea level calibrations cannot be
far from correct.
Aircraft performance characteristics
must be defined according to some standard
of accuracy.
In the United Kingdom the
standard sought after is that the level speed
performance should be defined within 1%. *
In applying this method through altitude
intervals during which w/wo remains greater
than 0.5, the results have been found to be
in excellent agreement with the "jet method”,
less than 3% variation having been found
among all tests.
This procedure also allows us to deter-
mine the increase in thrust due to afterburner
operation. In this case, we may still apply
Eq. 2:37 but with subscript o referring to the
engine alone and the absence of a subscript
to conditions with the afterburner operating.
If similar climb schedules are chosen, the
first term of Eq. 2:37 remains the important
one over a large altitude range, so that the
use of wind tunnel data for determination of
the second term is permissible, notwithstand-
ing the possible variations in drag produced
by afterburner operation.
At first glance, it would appear possible
to achieve this without considering the ac-
curacy of determination of engine power and
thrust, merely relying upon the accuracy of
the airspeed instrumentation. It is, however,
usually quite impracticable to measure the
aircraft performance under standard condi-
tions, hence some form of performance re-
duction must be used (the exact form is im-
material) and for this, engine powers and
thrusts (or their non-dimensional equiva-
lents) must be measured. Also, if airframe
drag data are determined from power and
thrust measurements, these must be meas-
ured with an accuracy equivalent to that
specified for the level speeds.
2:11 THE MEASUREMENT OF THE USE-
FUL THRUST OF TURBOPROPELLER
ENGINES
Some distinction can be made here be-
tween random and systematic errors; for
instance, if the random errors are within
the limits to be prescribed, considerably
larger systematic errors in engine meas-
urements can be tolerated if airframe drag
is not required. However, inasmuch as sys-
tematic errors are usually easier to detect
and to make allowance for than random er-
rors, this distinction is at best academic,
As with other types of power plants, in-
formation on airframe drag and aircraft
performance under standard conditions is
likely to be required for turbo-propeller
engined aircraft. This necessitates meas-
urement of net jet thrust, engine shaft power,
* 95% probability of a single observation.
2:15
and systematic and random errors are, there-
fore, usually grouped together.
stallation to installation of a particular en-
gine type.
By differentiating the performance equa-
tion in an identical manner to that adopted
in the development of differential perform-
ance reduction methods, equations can be
developed similar to Eqs. 8 and 12 of Ref. 3,
from which the accuracy requirement for
engine measurements to meet the specified
accuracy in performance definition can be
determined. Using this method, and insert-
ing experimentally determined values of the
coefficients of Vi/Vi, AP/P, etc., in these
equations, the results shown in the table below
were obtained for a particular installation in
1953.
The climb data have been included to il-
lustrate their more stringent accuracy re-
quirements which tend to zero error as the
ceiling is approached. However, since climbs
are equally dependent for consistency on
pilot technique and atmospheric conditions,
it is not usual to base instrumentation ac-
curacy requirements on them.
Basing the requirements on level speed,
it would appear, therefore, that 1.5% for
power and 12.5% for thrust could be safely
adopted as individual contributions to errors
in airspeed. Grouped together with other
sources of error, such as air temperature
and pressure, it is necessary to reduce these
values still further.
This can be done if it is assumed that all
errors are random, and 0.5% error in speed
is allowed for each of the seven possible
variants, namely: air temperature, static
pressure, dynamic pressure, aircraft weight,
jet thrust, engine torque, and engine speed.
This results in an overall random error of
1.32% in speed on a 95% probability basis
(if this is the standard adopted for defining
the accuracies of measurement), and enables
final and fairly realistic determination of
rounded-off limits of accuracy for engine
speed, engine torque and jet thrust to be
quoted, respectively 0.75% 0.75% and 6%
The level speed values hold for an engine
where the jet thrust power is about 10% of
the total thrust power, which is fairly rep-
resentative of current engines. Reduction in
this ratio (which is the direction in which
engine development should proceed) will re-
sult in a slightly greater accuracy being re-
quired for the engine power measurement,
and a relatively greater reduction in the
accuracy required for the thrust measure-
ment. The climb values depend as much
upon the ratio of total thrust power to drag
power at the best climbing speed, as upon
the ratio of jet thrust power to total thrust
power. They may, therefore, vary from in-
PERFORMANCE
ITEM
X
ENGINE
PARAMETER
Y
Accuracy Required of Instrumentation
Percent Per 1% in X
X AY
Y AX
Near Sea Level
Near Useful Ceiling
(登​·錢​)
Level
Shaft power
2.5
1.4
Speed
Jet thrust
25.0
12.5
Climb
Shaft power
0.55
0.28
Climb
Jet thrust
5.0
2.6
2:16
2:12 MEASUREMENT OF SHAFT HORSE-
POWER
advisable to consider whether in fact the
pressure gage records this,
In practice,
both absolute and differential pressure gages
are used, and the latter may be vented to
aircraft static or cabin pressure.
This is achieved, as in the piston engine,
by separate measurements of torque and en-
gine speed. Engine speed indicators meet-
ing the accuracy requirement of 0.75% can
usually be selected from equipment which is
generally available.
Allowances may have to be made for the
differences in back pressure which occur
between ground calibration, and in flight, both
for the instrument and the torquemeter piston.
All ambiguity and unnecessary calculation
can be obviated if a differential pressure
gage is used to measure the direct differential
across the torqucmeter pistons, both during
the ground calibration and similarly in flight.
The usual torquemeter on turbo-propeller
engines is connected to the propeller epicyclic
reduction gear train, in which the movement
of an annular gear (which would be fixed in
the absence of a torquemeter) is opposed by
oil pressure acting on a piston and connecting
rod assembly. By a system of ports in the
cylinders, the annular gear is constrained to
remain, within limits, in a predetermined
position, so that the geometry of the linkage
does not change, and the oil pressure nec-
essary to maintain this position is taken as
a measure of the torque.
2:13 THE ESTIMATION OF JET THRUST
Because of the relatively low standard of
accuracy required in the estimation of net
jet thrust for a turbo-propeller engine, it
is not necessary to adopt such precise meth-
ods as when dealing with the turbo-jet engine,
The usual approximation is to assume the
effective area remains constant at the value
for the cold nozzle.
Shortcomings encountered in torque meas-
urement have been:
(1) Layshaft type. Change in cali -
bration with oil temperature.
(2) Epicyclic type. Excessive lag,
and change in torquemeter constant with
torque, in epicyclic versions where the con-
necting rods make an appreciable angle with
lines of action of the pistons.
There are little data available to suggest
what error is involved here, because to date
little interest has been shown in establishing
effective nozzle areas for these engines. The
error will depend upon the jet pipe configura-
tion and the location of the tailpipe pitot head,
and in general any carefully placed pitot
should sample the mean total head to within
10%. This has been shown to be true by rake
measurements on at least one engine, which
produced discharge coefficients ranging be-
tween 1.04 and 0.92 respectively at the low-
est and highest pressure ratios achieved in
a ground run.
(3) All oil pressure types.
It is
necessary to consider whether the indicator
is a differential or absolute pressure gage.
Items (1) and (2) can only be rectified in
the design stage. It is advisable, however,
to consider each installation on its own
merits, and to arrange the program of ground
calibration of the torquemeter so as to as-
certain whether these effects are significant.
The measurement of intake momentum
should present no additional difficulty, be-
cause similar assumptions with regard to the
effective area can also be made here, with
an even more generous latitude in general
in the measurement of jet pipe total tem-
perature.
Item (3) involves the consideration that the
torque is balanced by the differential pres-
sure across the torquemeter piston, and it is
2:17
It will be observed that the final nozzle
pressure ratios on turbo-propeller engines
are much lower than those for turbo-jet en-
gines, and differential pressure gages in-
tended for measuring the total head should
be of suitable range, a full scale value of
2 psi sufficing for all current engines,
coefficients which are used to describe ram-
jet performance. Later, we shall investigate
how these coefficients may be determined by
flight tests; however, we must postulate at
the beginning that certain of the engine char-
acteristics should first be determined by
wind tunnel tests wherein the effects of com-
bustion are artificially simulated. (Under
these conditions the wind tunnel model pro-
duces some thrust and the mass flow through
the jet simulates the mass flow encountered
in actual flight.)
As a final word of warning, turbo-propeller
engine jet pipes are frequently not parallel
to the longitudinal axis of the aircraft, and
allowances should be made for this where
necessary.
2:14 CONCLUDING REMARKS ON TURBO-
PROPELLER THRUST DETERMINA-
TION
The wind tunnel data may then be com-
pared and correlated with flight test data.
These data should establish whether or not
the assumptions of the analysis (one-dimen-
sional flow all passing through the engine
without spillage) are appropriate.
Consider any thrust-producing device as
illustrated in Fig. 2:4,
Assuming that Po and P4 are equal, we
have
EN : QIVA - Vol
FN
2:38*
where Q
= mass flow per unit time
Vo
= free stream velocity
In the measurement of useful propulsive
thrust of turbo-propeller engines, torque and
shaft speed should be estimated to 0.75% of
the total speed and net jet thrust should be
estimated to 6% of the total thrust, if it is
required to define aircraft forward speed to
within an accuracy of 1%.
Care is required in the design of torque-
meters and in the selection of engine speed
indicators, if they are to meet this require-
ment. The estimation of jet thrust within a
tolerance of 6% implies that in all normal
installations, a rake calibration of the single
pitot installation is unnecessary, and that the
effective area associated with the final nozzle
pressure ratios deduced from the single tail-
pipe pitot readings can be assumed to be the
same as the cold nozzle area.
It is advisable to examine the torquemeter
installation and its pressure gage, if it be
hydraulic, in order to ensure that due allow -
ance is made for possible effects of altitude
and cabin pressure on the torquemeter cali-
bration. The torquemeter calibration should
be arranged to investigate lag and tempera-
ture effects.
V4
= ultimate wake velocity
Eq. 2:38, of course, is the classic second
law of Newton that force is equal to rate of
change of momentum.
If p represents gas density and A rep-
resents area at a given station, then
Q = PAYPAYA
4 4 4
2:39
оо
so that from Eq. 2:38
FN
v2
Р A
444
-Po
A
2:40
2:15 RAMJET THRUST MEASUREMENT
We shall first examine the nature of the
* Neglecting the mass of the fuel.
2:18
NA
Fig. 2:4
A4
A3
Ао
AI
A2
( Section Aft Of
Combustion Area )
2:19
or
air passing through the engine has been sub-
jected (compression, combustion, expansion).
Povo
FN
2(KA4 – A
2
Measurement of CT in Flight
2:41
where
Since the areas A3 and A1 are known, it
is only necessary to measure the pressures
p3 and pi to determine the areas A4 and Ao.
2
PAVA
4
M2
4
K =
(since pa po
Pol
-D
2
P
va
M
°
2:42
It is easy to show that if we know the
pressures existing at two stations such as
A2 and A3, the ultimate wake Mach number,
M4, is determined.* If only compression and
expansion were involved, K could theoreti-
cally be one, but for any possible combustion
process, K is less than 1. Then, knowing Mo,
with M -
Mach number.
If S is the wing area of the airplane
powered by the ramjet, then we may write
ma
4
ov
K =
FN : CT
S
2
2:43
may be found.
where Cris the ramjet thrust coefficient.
Comparing Eqs. 2:41 and 2:43, we have
СТ
.
2
S
를
​(KA4 - Ao).
)
2:44
Eq. 2:44 is quite general, applying not only
to ramjets but to any propulsive system. For
a turbojet, K is greater than 1. Moreover,
the quantities Ao and A4 vary with speed so
that Eq. 2:44 is not too useful when applied
to this type of engine. On the other hand,
when applied to a ramjet with a given relative
heating parameter
T2-Ti
A =
TI
Editor's insert: It was not clear from
Mr. LeDuc's paper in just what fashion the
areas A4 and Ag and the ultimate wake Mach
number are to be determined. Indeed, there
seem to be several possible approaches to
this problem depending on whether the free
stream conditions are subsonic or super-
sonic and on what assumptions are made in
the analysis. Presumably the laws of con-
servation of momentum and mass are applied
to give:
+ pove) AO
= 10, + Piva; (momentum)
IPO
2
PAYO
A? V
ve
(mass)
2
2
PA
from which
where T = absolute total temperature, it is
found that the parameters in Eq. 2:44 are
essentially independent of speed. Thus, the
thrust coefficient CT is of prime importance
in the study of ramjets.
PA?v
pode A
(Po + povola.
(61
+
AL
PA
2:45
In an ideal ramjet without losses, Kis
equal to 1. The actual value of K, which is
of course less than 1, is a measure of the
overall efficiency of the process to which the
Assuming that combustion is completed
within the engine.
2:20
from conventional aerodynamics
For subsonic free stream conditions, we
may assume isentropic compression between
stations 0 and 1, in which case
2
M4
3)(y-1)/y
109) 03-17
Vy
2:47
roles
Pi
where Pt3
so that Eq. 2:+5 becomes
= total pressure at station 3,
and Y has a value somewhat less than 1.4
(approximately 1.33).
I + y MS
y
PY
Aě - 6 month aga, eta
Ро
P.1 (Y+1)/ Y
The static pressure at station 3 depends
on whether the flow is choked or unchoked.
For unchoked flow the exit pressure is equal
to atmospheric whereas for choked flow, the
pressure exceeds atmospheric by a given
amount.
PO
O
2
Y
у м.
2:16 CONCLUDING REMARKS ON RAMJET
THRUST MEASUREMENT
and consequently
ty
MO
Ao
гума
Pilly
-
A, C
(a) It is pointed out that the determination
of Ma is a relatively delicate process. This
is so because the equations presented so far
are written for one-dimensional or average
flow through the engine, and it is well known
that the actual flow may differ considerably
from the assumed one-dimensional type,
particularly for nozzles having a large value
of the ratio A3/A max
2
2
Vy
I +YMS
2YME
AL
A,
(
Pilly+1)/y
Po
emo
2:46
Flight tests conducted on two French ram-
jets indicate that for one of these, the equa-
tions seem to give the correct results where-
as for the other, discrepancies of the order
of 5% are found between flight and tunnel
tests based on a comparison of Ct and Cp in
steady level flight with CD obtained from tun-
nel data.
so that if we know Mo, Al, Pi and Po, then
A. may be determined.
For supersonic flow, the design of the
entrance section would have to be considered
and the nature of the entry shocks evaluated
for a precise analysis.
Eq. 2:46 may be written for the exit flow
by substituting sub 4 for subo and sub 3 for
sub 1. The value of the tailpipe total pres-
sure may be used to evaluate the theoretical
ultimate wake velocity. Assuming an isen-
tropic expansion from station 3 to 4, we have
(b) It is appropriate at this point to com-
ment on the validity of Eqs. 2:41 and 2:44.
It should be understood that these relations
give the overall net thrust which, as is well
known, may be produced by pressures acting
both on the internal and external surfaces of
the ramjet vehicle (particularly the external
surface in the vicinity of the entrance sec-
tion).
2:21
where Q
-
rate of gas mass flow
If we consider only the thrust on the in-
ternal walls, we have
Fi QI V3 - V.) +103-P) A3 -10,-pola,
Af
.
area at end of nozzle
2:48
P
.
ambient air pressure
and if we write
Pf
gas pressure at end of nozzle
AF :
FN -Fi
Vf
Ve = exhaust velocity (relative to mo-
tor)
2:49
The mass flow is given by the equation:
it should be recognized that in some cases
AF may comprise 15% of the total thrust,
FN.
Q : CoPc At
2:51
where At
= throat area of nozzle
Finally, we point out that the design of the
pressure pickups at station 3 is very im-
portant since they must provide a proper av-
erage value of the static pressure (and of the
total pressure if Eq. 2:47 is used) while at
the same time be resistant to high tempera-
tures. *
Cp : discharge coefficient
Pc : chamber pressure
Combining (1) and (2), we have
F: (Co.Ag.Vp) Pc + Pf. Ap-P.Ar
2:52
(Editor's note: Mr. LeDuc's original
paper also included a section on fuel con-
sumption characteristics of ramjet engines,
which has been deleted here.)
2:17 MEASUREMENT OF ROCKET THRUST
IN FLIGHT
With supersonic flow in the divergent part
of the nozzle, the internal flow will be in-
dependent of external pressure. Also, the
chamber temperature is independent of cham-
ber pressure being determined by the chem -
istry of the combustion process. It follows
that for a given rocket, the term Pf Af will
be proportional to the chamber pressure Pc
and hence that
€
We will consider only cases in which the
exhaust gases are under-expanded, the gas
pressure at the end of the nozzle being no
less than the pressure of the ambient air.
This will usually be so, as over-expansion
is uneconomical. The thrust of the rocket
motor is given by the equation
F
BPc - Pfaf
2:53
F : QVp + (Pp - Polat
where B is a constant of the particular de-
sign and is determined from test runs on a
ground rig.
.
2:50
* It is, of course, also important to obtain
a satisfactory average value of the intake
static pressure Pi.
It is customary to define the in-flight
thrust as that which the rocket would give if
the ambient air pressure at the end of the
nozzle were equal to that of the undisturbed
air. Any departure from this condition which
2:22
results from the presence of the airplane is
classified as an interference effect in drag.
This classification is arbitrary, but it has
the substantial advantage that airframe ef-
fects are attributed to the airframe and that
the thrusts of the same motor in different
installations are immediately comparative.
consider several possible methods of meas-
uring jet thrust and shall also consider
several definitions used to describe jet
thrust of an engine as installed in an air-
frame. In the following work we assume
that the engine axis is aligned with the air-
flow direction so that there are no effects
of pitch or yaw on the thrust developed.
It is convenient to define the thrust of a
jet engine in terms of a simple ducted body
whose axis is parallel to the direction of
the undisturbed motion, i.e., the flight path.
Fig. 2:5 shows the internal flow ahead of,
through, and downstream of such a ducted
body, though the flow downstream is idealized
in that no mixing of internal and external
flow is shown.
With assumptions of ideal gases and so
on, it is possible to analyze the factor B in
terms of nozzle geometry and relate it to
more fundamental quantities such as the
discharge coefficient. This, however, is
hardly worthwhile inasmuch as the empiri-
cal approach suggested above avoids unnec-
essary idealizing assumptions. However, a
theoretical treatment of nozzle flow for an
ideal gas may be found in Ref. 4.
>
From momentum considerations the thrust
between any two planes* perpendicular to the
2:18 GENERAL ANALYSIS OF JET
THRUST MEASUREMENT
*Such planes will be referred to as a "sta-
tion" with a reference to its position, e. g.,
station i is the plane at the engine inlet,
In this portion of the chapter we shall
Station At Infinity
Upstream
Entry
Station
i
Exit
Station
Position of
Effective Area
f
Station At Infinity
Downstream
W
Free-stream
Direction
Boundaries Of Pre-entry
Stream Tubo
Tinclination Of Stream-
line To Free-stream
Direction)
Boundaries Of Equivalent
Post-exit Stream Tube
FIG. 2:5 DIAGRAMMATIC REPRESENTATION OF THE FLOW THROUGH A
DUCTED BODY
Fig. 2:5 Diagrammatic Representation of the Flow through a Ducted Body
2:23
the internal thrust
direction of motion is given by (Refs. 5 and 6)
the change between the two planes in the
quantity
Fint -
'pe u + Pe -Poo I dag
)
Jude+fie-Polo
(p
) DA.
SMU? +P; - POI DA,
U²
i
2:54
2:58
and the post exit thrust
Fpost
تا ۔
w
W
Applying this to the thrust of an installed
jet engine, the integration is limited to the
internal flow, and the difference in the
quantity must be taken between stations and
w, where station is sufficiently far up-
stream for the pressure of the internal flow
to be that of the undisturbed stream, and
the internal stream tube is therefore parallel
to the direction of the undisturbed flow.
Station w similarly is far enough downstream
for the pressure to be uniformly that of the
undisturbed stream.
In en
Pw up y daw
-fil revé + Pet
+ Pe - Pool dhe
2:59
so that
FN = Fpre + Fint + Fpost ·
2:60
The thrust is then defined as
The pre-entry thrust represents the force
exerted in the upstream direction on the
external flow by the pre-entry internal stream
tube and this force acts on the engine instal-
lation in the form of an external pressure
mainly near the lips of the entry.
F
FN-Sen u dane
U
W
vô
2
U
А,
2:55
where FN is the net thrust and represents
the net force of the internal flow on the body
in the upstream direction.
Also the gross thrust FG is defined by
The internal thrust is the force exerted
in the upstream direction by the internal
flow on the internal surfaces of the air ducts
and engine.
The post-exit thrust is the
force exerted in the upstream direction by
the post-exit internal stream tube on the
external flow, and this force acts on the
engine installation in the form of an external
pressure mainly near the jet exit.
Fou Semua DA
few
dAw
2:56
(a) Practical Definitions of Thrust
from which it follows that as a special case
under static conditions, the gross thrust
represents the force of the internal flow on
the body in the upstream direction.
The net thrust can be considered as the
sum of three parts as follows:
The definitions of net and gross thrust
(Eqs. 2:55 and 2:56), while strictly correct,
depend on integration over the internal flow
at station w far downstream of the body. In
a practical case such an integration cannot
be made because the distance is inconven-
iently far from the exit, and more important,
due to mixing in the wakes it is not practical
to separate the engine thrust and external
drag at station w. We must therefore
The pre-entry thrust
Fpre = S, 12. ups + P PO IDA, - PO MO AO"
(P.
P.
u²
1
2:57
2:24
Similarly, for choked flow
attempt to relate the conditions at station w
to those at or near the jet exit where meas-
urements are possible. Practical definitions
of thrust depend on the accuracy of such
relationships in determination of the post-
exit thrust.
FSG
Ро
Pe
Sac[y
Lorena
She
hoe ]
-1) June
Y
+
cos
Pa
Ρω
2:63
The standard thrust is obtained by choos-
ing the exit station e or "effective area"
station f as the downstream reference sta-
tion and by assuming that there is no contri-
bution to the thrust from the flow between
that station and station w far downstream
(i.e., no post-exit thrust).
The standard thrust is thus directly ap-
plicable if the jet discharges into a region
where the pressure is equal to Po and the
post-exit thrust is therefore zero. It is the
thrust normally specified in engine manu-
facturers' brochures and may be quoted
either as a gross or net standard thrust.
If we make the customary assumption
that for unchoked flow, the exit pressure
is equal to the free-stream pressure, we
find that for $=0°.
Y-1
2Y
Y
dA
Y-1
2:64
(Pe
FSG
Poo
5 [ {)
$
-1} dae
Ae
which coincides with Eq. 2:20 except that
average conditions have not been assumed.
For $ = 0º, Eq. 2:63 for choked flow be-
comes
-
FSN
Sleeve +PC - Pe) dAQ - muco
Sipe
Fusce fe. [v+1 -1 Dome
Pe
FSG
Doo
So
(Y+)
Pe
POO
DA
2:61
and
Y
Pe
An equivalent expression for this net
thrust is obtained from thermodynamic con-
siderations, in the following fashion:
The differential standard gross thrust
Pet
ܪ
(
2
Y-
(from Eq. 2:14)
Fsg is
.
2ں
so that
IDAE
o FSG = l pe + Po-Polda
Pe
=lpe ve cos ide + Pe - Polda
2
Pet
FSG - SA
Sell
(
1.26
--]
I dae
POO
2:65
for
Following the derivation of Eq. 2:20, but
assuming that exit and free-stream condi-
tions are different, we have
2Y
2
Y
y
Pe ve ²
Y
y = 1.33.
Y-1
noveerimine :)
(
er
Pe
--}
Subtracting the entering rate of change of
momentum, QUO gives the standard net
thrust; thus from Eq. 2:62,
where
Y-1
Y
Y
Pet
27
2Y
Pe
Y- POO
Poo
Pe
FSG
1937,
tot )
dae
force
FsN" S[ 4 Pe{
[ (738-1 cos't
}
+ -)
POO
Pe
Pe
-1
COS
dhe-muco
०२
POO
2:62
2:66
2:25
(b) The Jones Thrust
constant y between w and c
2
De
Y
y Pe
Y1
The
Y Y Pw
Y
+
Y- Pe
Y-1 PW
ใช้
Y-1
hence
2
2
Vå avě + 2 los
27
+
Y-1
-
Powe
and
27
onlen
For convergent nozzles (subsonic or sonic
velocity at exit) which exhaust at a local
pressure which differs from Po the assump-
tion can be made that the jet expands or con-
tracts adiabatically and isentropically to
the undisturbed pressure Poat station w
without any transfer of energy or momentum
through mixing in the wake. This model of
the flow enables the conditions at station w
to be calculated from those measured at
station e, the jet exit. The thrust so ob-
tained is designated the Jones thrust since
it represents an extension of Sir Melvill
Jones' original method of obtaining the drag
of a body (Ref. 7).
For these assumptions the conditions at
station w can be calculated from those at
station e using the condition of continuity,
of mass flow between stations e and w, and
the energy equation assuming isentropic flow.
The resulting expression for the jet Jones
thrust in terms of the measured quantities
at station e is derived as follows:
By definition, the gross Jones thrust is
= 1 +
)
(7-1) ve
Pe
Pe
-
but
2Y
ve
Pe
Pe
-
pe
>
so that
Pe Pw
Pe Pw
Por
Pe
1 PW De
oulsen
W
= 1 +
: 1 +
Pet
Pe
Pet Pe
Pe
2
Per
Per
Pe
PwYw
Fuo San
daw.
JAW
Now, since we have assumed the existence
of isentropic flow at constant y throughout
If we assume that angle of swirl y is neg-
ligible and that isentropic flow without mixing
exists between stations w and e, then conti-
nuity provides*
Ý
(Pe
Y-
Pe
会​”:
Pe
D
i
فوای
.
De
w
let
Pele cos de dae
PW Yw daw
or
and
2
2
pv
dAw
Pe
ve
Vw
cos de dae
Ve
dA
.
(Pw H
Y1
7
Vw
22
Assuming isentropic flow conditions and
Ver
+
२९५)
Pe
-1
since
*If the swirl angle is not small and the jet
diameter changes appreciably between sta-
tions e and w, cos de must be replaced by
cos de cos yw, where yw is obtained from
We by the law of conservation of angular
momentum.
Y-
H
Pe ve = 21 Pe
24
2**
et
Pe
Y-1
2:26
and
FUG Sam Povo
66
If we presume a constant value of re-
covery factor for the thermocouple tempera-
ture pickup, then from Eq. 1:63,
Vw
3
cos de dae
Ten
• Te + al Team
we have
Y-
(Pet
Il
FUG'S. 44 po{ le :) -}
F18
2Y
Y-1
Pe
and from Eq. 1:62
Pe
1/2
(2
Te m² = Tex - Te
| -
Pe.
.
cos de: {! +
dae
so that
'Pet H
Pe
:
2:67
Ter
;
Te + el Ter - Te) = c Ter + (1-6) Te
and
and
FUN= (Eq. 2:67) - QUOD
2:68
Teti
her
Te
€ + (1-€)
ose
Tex
for unchoked flow.
but
Y-1
1 Pe
If we assume that for the choked flow,
we have isentropic conditions between e and
w (which assumes the absence of shock out-
side the engine) then,
Pet
Ž
2
17-18
(
Pe
for isentropic flow, hence
and Eq. 2:67 becomes
Y-1
Tex
Pe
ket
+ (1-6)
pet
Fuo = Son 7 Pe cos de [
[1+ ka {1-
:
* {1-63 By
Ae
2:71
Ae
2:69
From Eq. 2:30
Pe
on tform
фе
cos de VATRE
From Eq. 2:70 the actual tailpipe total
temperature may be obtained from a tailpipe
temperature indicator and the nozzle pres-
sure ratio, provided a properly calibrated
pickup is utilized. Using Eqs. 2:70 and 2:61,
we see that mass flow may be determined
from measurements of exit static pressure,
total exit pressure, and total exit tempera-
ture,
Tex
2Y
17-1R
P.
Y-1
fe]
dae
2:70
If the exit pressure equals ambient static
pressure, then the Jones thrust is identical
to the standard thrust. If Pet Poo the two
thrusts are essentially the same up to a
ratio of Per/Pe > 2, but at higher pressure
where Ter is the exit total temperature,
obtained from a calibrated thermocouple.
1
2:27
ratios, the difference increases, and is no
longer negligible particularly for net thrusts
at high flight Mach numbers.
will be obtained by measuring the momentum
change of the internal flow. However, there
are certain possibilities for direct measure-
ment which will be discussed first.
(a) Direct Force Measurements
Since the assumption of isentropic flow
between stations w and e is open to some
question, the equation for Jones thrust should
be modified if more information, either
theoretical or from measurements is avail-
able in particular cases.
(c) Application of Thrust Definitions to
Particular Engine Installations
The definitions of the preceding para-
graphs may not be immediately applicable
to particular installations. For instance,
if the jet axis is at an angle to the direction
of motion then this must be taken into ac-
count in the application of the momentum
equations. In most normal installations this
angle will be small and the corrections to
the definitions as quoted will be unimportant.
Again, the jet pipe exit may be cut off ob-
liquely. In this case the plane e will have to
be arbitrarily defined to suit the particular
case and any jet deflection will have to be
taken into account. Some engine installations
have cooling flow ducted round the engine and
exhausting round the final nozzle as indicated
in Fig. 2:6. In such cases, it is arbitrary
whether the cooling drag including ejector
losses be debited to engine thrust or added
to the airframe drag.
(1) Static Gross Thrust of Bare En-
gine on Test Bed. By definition the gross
thrust of a simple jet engine under static
conditions, FGTB , can be measured directly
on an uncowled engine by a force balance on
a ground level test bed (Fig. 2:6A). The
test bed must be arranged so that entry air
can be drawn from a source of uniform total
pressure; jf windage over the engine is not
negligible or if there are any obstructions
such as silencers in the path of the hot jet,
these effects must be allowed for by correc-
tions. The gross thrust obtained by direct
force measurements includes the post-exit
thrust appropriate to static conditions. Com-
parison of thrusts measured by balance, with
the Jones thrust obtained from pitot static
traverse at station e showed agreement in a
series of tests on a centrifugal engine to
within 2% which is of the order of accuracy
of the tests.
(2) Static Gross Thrust of Engine In-
stalled in an Airframe. By suitably suspend-
ing an aircraft on a thrust balance, the static
gross thrust of an installed jet engine can be
measured. Such direct force measurements
may disagree with static test bed results
due to:
2:19 THE MEASUREMENT OF THE
THRUST OF JET ENGINES
Drag of parts of aircraft in
flow induced by the jet at exit.
General
b. Thrust or drag of cooling air
(Fig. 2:6) and air bled from the compressor
which is sometimes ducted to exit round the
final nozzle, or to engine and aircraft equip-
ment.
Methods of measuring the in-flight thrust
of jet engines installed in aircraft will now
be considered. The thrust under static con-
ditions of the bare engine and of the engine
installed in the airframe will also be con-
sidered briefly insofar as such thrusts are
used as a basis to evaluate thrusts from
flight measurements. From the definitions
previously presented it follows that thrust
Changes in ram ratio and
entry total pressure distribution due to in-
efficiency of flight entry under static con-
ditions (Fig. 2:6B).
2:28
Slove Entry
I Flow Induced
! By Jet
Y 6 6
Jogo que
P
p.
8
A) BARE ENGINE ON TEST BED
Pressure Around
Engine, P
P
Entry Condition
Stotic
e
PLENUM
CHAMBER
CENTRIFUGAL
ENGINE
OD
444
Entry Condition
in Flight
B) SIMPLE INSTALLATION OF ENGINE IN NACELLE
Thrust
Measuring
Trunnions
Pressure Around
Engine, p
Flexible
Seal
DIFFUSER
BURNERS
EXHAUST
537ZZON
C) ENGINE IN NACELLE WITH COOLING FLOW
AND AFTERBURNING
Fig. 2:6
Diagrammatic Sketches of Jet Engine Conditions for Thrust Measurement
2:29
(e. g., force an entry air ducting) is given by
Fan-F85010,-U00) + A, 10, + Pool
Such force balance measurements made
on installed engines must, therefore, be used
with due precaution if absolute values of
thrust are required. They can be usefully
used to check the thrust of a particular
engine relative to the standard for a given
aircraft engine combination.
2:73
from which it can be seen that the net
standard thrust can be obtained from the
trunnion thrust if the conditions at station 1
(the entry to the compressor) are measured. *
(3) Thrust of a Jet Engine on its
Trunnions or Bearers. It might be conven-
ient in some cases, for example, for engines
using reheat, if the thrust in flight could be
obtained from measurements of the thrust
force of the engine on its trunnions or
bearers. For centrifugal engines with plenum
chambers the relation between net standard
thrust and trunnion thrust is given to a first
approximation by
FSN - Fg is normally less than 35% of
B
FSN so that the error in FSN due to errors
in the compressor entry conditions can be
kept small,
Trunnion thrust F : F
B
SG - Ae (p - Po)
A
е
In addition, Fe must be corrected for
engine incidence and aircraft acceleration.
Measurement of thrust of axial engines in
flight by the trunnion method is therefore
possible, though it should be considered as a
last resort in view of the difficulty of the
installation, i.e., the engine with no restraints
other than the trunnions, the force measuring
cells in the trunnions, and the pitot static
measuring gear at the compressor entry
with its danger to the engine if any parts
broke away.
2:72
(b) Momentum Measurements
from which it is seen that FB depends on
the force of the engine due to the local static
pressure p around it (see Fig. 2:6B).
This force would be very difficult to meas-
ure to the required degree of accuracy and
trunnion thrust measurements for the centri-
fugal type of engine with plenum chambers
therefore are of little value for aircraft
performance purposes. Tests by N.G.T.E.*
have shown that the force acting through the
engine trunnions can be measured in flight
though the installation is not easy, but as
shown above, this force is not simply re-
lated to the thrust defined for aircraft per-
formance purposes.
Thrust by Integration of Measurements
at the Final Nozzle. The standard (or Jones)
thrust which must be measured if thrust and
drag are to be obtained from flight tests,
can be determined from measurements at the
final nozzle exit. From the expressions for
thrust previously developed, it follows that
the standard or Jones gross thrust can be
obtained precisely from traverses over the
jet exit area measuring the following:
For axial type engines (Fig. 2:6) the force
on the engine due to the local static pressure
round it is likely to be smaller than in the
case of centrifugal engines and may be neg-
ligible. With this assumption the difference
between trunnion thrust and net standard
thrust, that is, the thrust taken on the air-
frame other than through the engine bearers
(1) The total pressure in the direc-
tion of the local flow, PT.
(2) The static pressure, p.
* An approximation from wind tunnel or en-
gine brochure data could not be relied on to
give FSN to within 5% in all cases.
*Results unpublished.
2:30
(3) The direction of the local flow
to the flight path $, where will be deter-
mined from the local angle of swirl and the
radial component,
Also for the net thrust
therefore necessary, particularly for un-
choked jets, to measure static pressure at
least at the center and walls.
Under static conditions on a test bed, the
Jones thrust measured by pitot static inte-
grations can be compared with direct force
measurements. In such tests (Ref. 8) the
thrust by integration was 2% higher than
balance thrust, and a similar result was
obtained on unpublished tests on an axial
engine. This difference is often of the order
of the experimental error and it has not been
possible to trace its source.
(4) The local total temperature.
Such transverses must take account of
the jet boundary layer and any alteration of
jet size with temperature.
From the results of Ref. 8, it may be
concluded that absolute measurements of
thrust can be made in flight using a pitot
static comb to an accuracy of 2%, though at
the cost of some complication of installation.
To make such complete measurements
would clearly be an arduous task in flight,
and acceptable simplification will be con-
sidered. The local angles of swirl and pitch
effect both the measurement of total head
and static pressure and also the component
of momentum in the thrust direction. If
Prandtl type pitots and statics are used set
parallel to the nozzle axis, then for conver-
gent nozzles an error of 0.5% in thrust
would be caused by a value of $ of approxi-
mately four degrees. Under normal circum-
stances in a well-designed jet engine o will
be less than four degrees so that it need not
be measured.
(c)
The Single Pitot Method*
A very much simpler application of the
momentum method of measuring thrust is
the well-known single pitot method (Refs. 8
and 9), which on account of its simplicity
has been used in Great Britain up to the
present time for almost all routine meas-
urements of thrust in flight in connection.
with performance measurements.
From Eq. 2:62 it can be seen that
FSG
8
sao bilo
()
dae
Ае
Poo Ao
A
The need to measure the static pressure
distribution over the jet exit has been dis-
cussed in Refs. 8 and 9. In Ref. 8 it is
shown that on an engine with a conventional
final nozzle, values of (Pe - Po)/(Pte - Po )
varying from 0 at the wall to 0.3 at the
center with elliptic distribution were meas-
ured on the ground and in flight, i.e., the
exhaust gases do not fully expand in the
final nozzle. If no measurements of Pe had
been made and the thrust estimated assum-
ing Pe - Poo below the choke and Pte/Pe - 1.85
above the choke, then an error in thrust of
2% would occur with the engine just choked.
This
error increased to some 5% at
Ptę/Pe = 1.2 and fell to 0.5% at Pte/Pe above
2.0.
2:74
In the single pitot method the relationship
between FSG and the single pitot pressure
Pér is taken to be of the form of Eqs. 2:64
and 2:65, i.e.,
FSG
: fe
PO Aė
Boo
2:75
Static tests made on an axial engine gave
similar results (unpublished). If accurate
absolute measurements of thrust by the
momentum method are required, it is
*Editor's note: An averaging total head tube
can be used to improve accuracy.
2:31
where he is an effective jet exit area which
is obtained as a function of pressure ratio
pe/Pc from measurements of FSG and
per/Po normally made on a test bed at
ground level conditions.
Comparison of Eqs. 2:74 and 2:75 shows
that the accuracy of the single pitot method
in measuring thrust in flight depends on the
inherent assumptions that the final nozzle
total pressure distribution, static pressure,
and static pressure distribution at a given
value of per / Po is the same on the test bed
and in flight.
methods, analagous to the single pitot method
in that they depend on the measurement of
jet effective area de but he is based on
other measurements. First, it is possible
to base thrust measurements on wall static
pressure measurements for example, in the
jet pipe, upstream of the final contraction,
or at the nozzle guide vanes, Such wall
static pressures are steadier than single
pitot pressures and are therefore easier to
measure accurately, but they are more liable
to consistent errors if there is any buckling
of the wall or local deviation from smooth-
ness near the static hole.
A more reliable result can be obtained
from the mean of several wall statics at a
given station.
Further, it is often necessary to extrapo-
late values of Ae obtained on the test bed to
higher values of pressure ratio per / Pc only
obtained in flight. This is normally done
assuming that above the choke per/P00 – 1.85,
Ae remains constant, although 'altitude test
bed measurements on a centrifugal jet engine
showed an increase of a'e with pe/poo above
1.85 (Fig. 2:7A).
2:20 THRUST MEASUREMENTS
WITH AFTERBURNING
(a) General
The definitions of thrust and principles
of measuring it previously outlined are
identical when afterburners are installed in
the jet pipe.
The magnitude of errors in thrust meas-
urement of the single pitot method due to the
above assumptions has been measured in
flight on a particular centrifugal engine by
comparing single pitot thrusts with thrust
from a pitot static rake which showed that
errors in single pitot thrust up to 6% were
encountered in some conditions in flight
(Fig. 2:7A).
The application of methods of measure-
ment may have to be modified due to (1) the
very high temperature at the final nozzle,
(2) the advantage of changing the final nozzle
area when the afterburner is switched on,
(3) the less uniform conditions in a final
nozzle with afterburning, and (4) the larger
volume of cooling air with large resulting
ejector losses at the final nozzle.
The single pitot method has the great
advantage of simplicity and ease of instal-
lation as the thrust is obtained from only
one additional pressure reading. Errors
due to the inherent assumptions should how-
ever be watched, especially where flight
conditions as regards air entry conditions,
Reynolds number, nozzle pressure ratio,
and exit static pressure (Fig. 2:7B) differ
appreciably from the conditions of the engine
calibration tests.
Insufficient experience has been obtained
in Great Britain to establish the best method
for measuring thrust with afterburning, so
the methods available can only be enumer-
ated.
(b) Thrust by Integration of Momentum
at the Final Nozzle
(d) Static Pressure Method
Brief mention may be made of thrust
This method is the ideal if a very full
2:32
1
.
+
1
1.0
1
의
​EFFECTIVE AREA, A.
GEOMETRICAL COLD AREA
Slope Checks With
Altitude Test Bed
0.5
1
1
!
.
1.2
1.5
2.0
2.5
pre
1
PRESSURE RATIO,
Pe
Ground Level Test Bed
Ground Level Test Bed Extrapolated
Calibrated Against Roke In Flight
Fig. 2:7A Variation of Single Pitot Effective Jet Area with Pressure Ratio.
Centrifugal Engine. (Ref. 8)
1
N
0.3
CENTER
C
0.2
Pe-Poo
- DOO
0.1
WALLY
O
1.2
*
2.0
1.5
2.5
Pre
Poo
Simple Theory Convergent Nozzle Choked
Centrifugal Engine Flight Tests
х Axial Engine Ground Tests
Fig. 2:7B Static Pressure in Jet Final Nozzle Measured in Flight. (Ref. 8)
2:33
investigation of the thrust is required, but
the difficulties of finding a material capable
of withstanding the temperatures alterna-
tively, or of cooling the rake, are formidable
and would probably only occasionally be
justified. A more hopeful alternative would
be a single pitot static traversed relatively
quickly across the jet. A temperature
transverse would again be necessary to
establish net thrust unless the engine mass
flow were measured at another station, or
was based on test bed measurements of
engine characteristics.
the turbine and the afterburner flameholders.
Because losses between this plane and the
jet final nozzle are a function of attitude and
pressure ratio, this method is subject to in-
accuracies due to this cause, unless the
losses can be measured. For net thrusts,
engine mass flows would have to be derived
from a temperature calibration or engine
characteristics.
(d) Thrust Measurement by Aircraft
Performance Measurements
(c) Methods Based on Engine Calibration
Methods such as the single pitot in the
final nozzle, or static head upstream of the
final nozzle, identical in principle to those
already discussed for the non-afterburning
engine dependent on static engine thrust
calibrations are attractively simple, but are
likely to be even less accurate than for
non-afterburning engines, due to the more
uneven distribution.
A possible method of estimating thrust
with afterburning would be to establish the
drag of the aircraft, without afterburning,
by measuring the thrust by one of the
methods given in section 2:19, measure the
increase in performance with afterburner
in operation and hence determine the thrust
with afterburning.
This method is not likely to be very ac-
curate, and a correction would have to be
applied for any change in drag due to change
of jet final nozzle area. The thrust could
be measured by this method up to the highest
speed with afterburning by measuring the
drag without reheat in decelerating level
flight (Ref. 11).
The problem of air cooling a single pitot
at the jet exit is thought to be soluble, but
wall static measurements upstream of the
final nozzle would be more attractive and
should be tried out for reliability.
(e) Trunnion Thrust
An alternative method (Ref. 10) is to
calibrate the thrust in terms of pitot and
static measurements in the diffuser between
As a last resort, the trunnion thrust of
axial flow engines might be measured.
1
2:34
REFERENCES
1.
PWA 01.60 by Pratt & Whitney Aircraft, “The Use of Operating Curves," December,
1946.
2.
U. S. Navy Dept., Bureau of Aeronautics Power Plant Memo #16, "Aircraft Engine
Horsepower Correction Methods," November, 1945.
3.
Lush, K. J., “Differential Performance Reduction Methods for Turbo-Propeller Air-
craft," U. K. Ministry of Supply Report No. AAEE/Res/275.
4.
Liepman, H. W., and Puckett, A, E., “Introduction to Aerodynamics of a Compressible
Fluid," Chapters 1, 2, and 3, John Wiley & Sons, New York; Chapman & Hill Ltd.,
London.
5.
Jakobsson, Bengt, "Definitions and Measurement of Jet Thrust,” British Journal of
Royal Aeronautical Soc., April, 1951.
6.
A report in the British A. R. C. Series on the definitions of the thrust of a jet engine
and of the internal drag of a ducted body will be issued shortly.
7.
The Cambridge Univ. Aeronautics Lab., "The Measurement of Profile Drag by the Pitot
Traverse Method,” Br. R. and M. 1688, January, 1936.
8.
)
Stephenson, J., Shields, R. T., and Bottle, D, W., "An Investigation into the Pitot Rake
Method of Measuring Turbo Jet Engine Thrust in Flight,” Br. Report AAEE/Res/265,
December, 1952.
9.
Wright Field Eng. D. W. Tech Note No. TN-T SEAL-2-6, “Determination of Turbo Jet
Engine Thrust in Flight."
10.
Flight Test Engineering Manual, USAF Tech. No. 6273, 1953.
11.
Lush, K. J., "Preliminary Performance Assessment from Brief Flight Tests, Consid-
erations of a New Technique,” Br. Report AAEE/Res/267, July, 1952.
2:35
AGARD FLIGHT TEST MANUAL
VOLUME I, CHAPTER 3
A SURVEY OF PERFORMANCE REDUCTION METHODS
Ву
Kenneth J. Lush, AFFTC
United States Air Force
and edited by
John K. Moakes
Aeroplane & Armament Experimental Establishment
United Kingdom
I
CHAPTER CONTENTS
Page
SUMMARY
TERMINOLOGY
3:1
INTRODUCTION
3:1
3:2
THE PROBLEMS
3:1
3:3
METHODS OF PERFORMANCE REDUCTION
3:2
3:4
EXPERIMENTAL METHODS
3:2
(a) General
3:2
(b) Application to Simple Jet Aircraft
3:3
(c) Application to More Complex Cases
3:4
(d) The Effect of Interconnection
3:4
(e) The Assumptions Required
3:5
3:5
ANALYTICAL METHODS
3:5
(a) General
3:5
(b) Application of the Methods
3:5
(c) Control Systems
3:6
3:6
SUMMARY OF CONCLUSIONS
3:7
REFERENCES
3:8
SUMMARY
The purposes of performance reduction are dis-
cussed and the available methods classified. Argu-
ments for and against each type method are made
with reference to their application to various types
of reduction problems. The classification is broadly
as follows:
(a) Experimental methods, which require no ad-
vance numerical data from other sources, but because
of the stringent requirements for their validity and
convenience in use may be applicable only in certain
cases.
(b) Analytical methods, which do require advance
numerical data from other sources. These may be
further subdivided into:
(1) Differential methods, based on linearized
relationships suitable for small corrections, which
depend for their existence on generalized information
on engine behavior and airframe drag. These can
only be applied when the generalized information is
both available and convenient to use; thus, their use
is virtually prohibited when compressibility effects
are present. These methods are convenient for the
reduction of isolated performance points, provided
the necessary requirements for their applicability
are met.
(2) Performance analyses, which only require
advance Information on engine behavior. These are
possibly the best substitute (at least for tests at high
altitude or Mach number) if experimental methods
are either impracticable or very inconvenient. They
may also be more convenient than differential methods,
when performance data are required over a wide range
of the variables, or when drag data are of primary
importance.
TERMINOLOGY
b
Aircraft Span
coe
Profile Drag Coefficient
с
Engine Parameter
L
Length Dimension in L.M.T. System
Engine Speed
z z
Ni
Speed of Compressor
N2
Speed of Power Turbine
р
Ambient Air Pressure
Q
Fuel Flow
ун
T
Time Dimension in L.M.T. System
V
True Air Speed
Ve
Equivalent Air Speed - V.To
8
W
Aircraft Weight
T
Ambient Air Temperature
) regarded
as
) enthalpy
Tj
Jet Pipe Total Temperature
Po
Air Density at Sea Level
P
Air Density at a Particular Pressure Height
o
Relative Density : P/PO
PIPO
8
D
Typical Length Representative of the Problem
3:1
INTRODUCTION
RPM, manifold pressure, air tenperature,
air pressure, and aircraft weight.
Performance reduction is the term given
to methods for deducing aircraft performance
under standard conditions from tests made
on the aircraft under non-standard conditions.
With regard to the necessity for these cor-
rections, it suffices to state that due to
uncontrollable variations in atmospheric con-
ditions, and aircraft and engine settings, it
is usually impossible to make aircraft per-
formance tests under precisely standard con-
ditions.
In operational use the air pressure, engine
RPM and manifold pressure are easily varied
over the working range. These quantities
are, therefore, retained as variables or pa-
rameters in the presentation of results; the
steady level speed being plotred against air
pressure (height) at each of several repre-
sentative combinations of engine speed and
manifold pressure (e.g., at the maximum for
continuous weak mixture running and for
combat).
It follows that reduction methods are neces-
sary to rationalize the test results so that
they may be used for valid prediction and
comparison.
Performance reduction methods take many
forms and the selection of a given form de-
pends upon the nature of the tests and the
amount and range of information which is
required, as much as upon the nature of the
power plant concerned.
As regards air temperature, it is not gen-
erally possible to obtain a selected value
on any given occasion because the air tem-
perature associated with a given air pressure
varies from day to day. The practice* is,
therefore, to plot the performance against
pressure (height) for a “standard atmos-
phere"; in this standard atmosphere" the
air temperature associated with any given air
pressure (or “pressure height'') is approx-
imately the mean value for the conditions in
which the aircraft is to operate.
Before choosing a method it is desirable,
therefore, to examine the purposes which
performance reduction methods serve and the
types of methods available for solution of
the various kinds of problems.
The aircraft weight as a variable is slight-
ly different again, in that it bears some re-
lation to the weight at take-off which is con-
trollable. Therefore, it is usual to quote
steady speeds at some standard weight which
is related to the take-off weight: One such
standard weight is 95% of the take-off weight,
which is roughly the weight after climbing
to operational altitude.
3:2 THE PROBLEMS
The performance of an aircraft is a func-
tion of a considerable number of independent
variables. The number of these variables
is sufficient to cause difficulty both in the
description of the capabilities of the aircraft
and the establishment of these capabilities
by flight tests. In the description of the
performance capabilities of the aircraft the
difficulties are reduced by taking average
values of those variables which are not con-
trollable in practice.
In general, the practice is to quoce perform-
ance over suitable ranges of those variables
which can be controlled, and for specified
"standard" values of those variables which
cannot be controlled.
If the test programn could be so arranged
that the values of the uncontrolled variables
Consider, for example, an aircraft with
piston engine and propeller, for which the
steady level speed will depend on the engine
* Editor's note: This represents British
practice and should not be construed as
representative of all cases.
3:1
3:4
EXPERIMENTAL METHODS
(a) General
were randomly distributed about the standard
values, their effect would appear as scatter
and the required data could be obtained
directly, given sufficient test results. Such
a technique would, however, be very expen-
sive in flying hours and would take many
months. For general purposes, it is quite
impracticable.
It is, therefore, necessary to have methods
of deducing the performance under standard
conditions from tests made under non-stand-
ard conditions. This process is known as
performance reduction.
We have observed (section 3:2) that the
independent variables on which the perform-
ance of an aircraft depends may be control -
lable or uncontrollable. It is convenient to
distinguish further between:
(1) variables which can be adjusted in
flight, but not exactly to any predetermined
value;
(2) variables which can be adjusted in
flight, with sufficient precision, to any pre-
determined values in their working ranges.
For brevity, type (1) will be referred to
as adjustable and type (2) as fully control-
lable.
The information required from a particu-
lar series of tests may be anything from the
steady level speed at one particular height,
weight and engine rating to air speed and
fuel flow data for the whole practical range
of working conditions. The choice of a
reduction method may depend on the type and
extent of the data required and the flight
test program may depend on both; therefore,
it may be that no one method is superior
in all circumstances.
This distinction is of importance when re-
sults are required only at particular values
of controllable variables. It is of prime
importance in an experimental method of
reduction that the required numerical data
be obtained directly from the performance
tests, so that if results are required at a
particular value of any variable, the tests
must either be made at that value (as can
be done with a fully controllable variable) or
must cover a range about that value (as must
be done with an adjustable variable).
3:3 METHODS OF PERFORMANCE
REDUCTION
Methods of performance reduction may
conveniently be divided into two classes
which we will refer to as the 'experimental"
and the "analytical" classes of method.
In the extreme case, for example, where
the performance is required at one partic-
ular value of each of n controllable varia-
bles, a single test would give an answer if
all the variables were fully controllable,
whereas 2n tests would be required if all
were merely adjustable.
Experimental methods will be those in
which the performance under standard con-
ditions is deduced directly from the flight
tests on the particular aircraft with no nu-
merical data being called up from other
sources.
If results are required over a range of
all variables, - the distinction between fully
controllable and adjustable variables is not,
of course, of much importance; it affects
only ease of analysis and interpretation of
results.
Analytical methods will be those in which
numerical
rical data from other sources are
called up in deducing the performance under
standard conditions from the tests under
non-standard conditions.
Thus, if a particular variable is adjust-
able but not fully controllable, it may cause
3:2
From this we may deduce by dimensional
analysis that
N
W
inconvenience but does not make experimen-
tal methods impossible. The difficulty intro-
duced by a completely uncontrollable variable
such as air temperature is more fundamental.
We have already (section 3:2) dismissed as
quite impracticable, arranging the tests so
that the values of all uncontrolled variables
were randomly distributed about their stand-
ard values.
talut
)
:O
*
р
3:2
In Eq. 3:1 we had four independent var-
iables, two of which (T and W) could not be
conveniently controlled. In Eq. 3:2, after
the application of dimensional analysis, we
have only two independent variables, both of
which can be controlled during any test where
N and p can be varied.
A conceivable alternative would be to
make enough tests to give results over a
range of each uncontrollable variable and
deduce the standard performance by inter-
polation. This would avoid the requirement
that the test values of the variables should
be randomly distributed about the standard
values, but a large test program would still
be required and waiting for the weather to
produce a suitable range of air tempera-
tures would be very objectionable except in
fortuitously favorable circumstances.
In this example, dimensional analysis has
reduced the number of independent variables
by two and so made control of all inde-
pendent variables possible. Dimensional a-
nalysis cannot reduce the number of inde-
pendent variables in this case by more than
two.
By recourse to dimensional analysis,
however, it is often possible to reduce the
first difficulty by decreasing the number of
variables involved, and to remove the second
difficulty by associating the uncontrolled var-
iables with variables which can be controlled.
If now these independent variables (N/T
and W/p) are fully controllable we can make
level speed runs at the N/T and W/p which
correspond to the values of N, T, W and p
at which the steady level speed is required
and deduce V/T (and thence V) with very
little trouble.
This is the approach used in current
experimental methods.
Let us consider first its application to
jet aircraft, which is familiar; subsequently,
we shall consider more complex cases.
In practice with jet aircraft it has been
found that W/p can be fully controlled by
adjustment of p, but full control of N/T
has not usually been attempted because it is
usually difficult to correct the thermometer
reading accurately for the effects of forward
speed while the tests are in progress.
(b) Application to Simple Jet Aircraft
It is usual to assume that the steady
level speed (V) is a unique function of the
engine speed (N), the aircraft weight (W),
the air temperature (T) and pressure (p).
The air temperature is treated as an en-
thalpy (dimensions L /T2 in the L.M.T.
system), other quantities associated with
temperature (such as viscosity) being neg-
lected. Then we may write
* Upon inspection, some of these variables
appear not to be nondimensional. Powers of
a typical length (D) of the problem have
been omitted, because they are constant to
the problem. In the general case the var-
iables would be
V
ND
W
TTi' Dap
filV, N, W, T, p) : 0
5
3:1
3:3
It is usual, both for level speeds and
climbs (for which a similar relation may be
deduced), to make tests over a range of N
sufficient to cover the required range of
N/VT and deduce the answer by interpola-
Independent control of N and p is,
of course, essential to this method.
As an illustration of this type case, let
us consider a turbo-propeller aircraft for
which the steady level speed V is a function
of N, W, T and p and the jet pipe temper -
ature Tj, so that we may write
tion.
f31V, N, W, T, P, Tj) = 0
.
3:3
(c) Application to More Complex Cases
In the above case, we have the following
features:
After application of dimensional analysis,
we may write
( TT
(1) Dimensional analysis has reduc-
ed the numbers of independent variables by
two, thus making the test technique prac-
ticable, as this was the number of the orig-
inal independent variables which could not
be controlled.
(2) Independent control of these two
remaining independent variables is essential.
With these in mind let us consider more
complex engines, i. e., engines with one or
more independent variables over and above
those which were present in the case of the
simple turbojet. It will generally be found
that dimensional analysis will still reduce the
number of independent variables only by
two, but this does not matter if the re-
maining independent variables can be fully
controlled.
If they are "adjustable" only, they may
be a source of embarrassment; if so, the
embarrassment increases with the number of
adjustable variables. To give a single level
speed under standard conditions, for example,
at least two, four or eight tests would be
required if there were one, two or three
adjustable variables respectively. If on the
other hand, full curves of performance against
all variables were required, no embarrass-
ment would result from adjustable variables.
V N W N
f4
= 0
PT;
3:4
If now, N/WT; (or any other convenient
non-dimensional group containing the addi -
tional variable T;) is fully controllable, inde-
pendently of N/VT etc., the advent of the
additional variables does not cause trouble.
However, tests must be made over a range
of this variable if it is adjustable only.
If it cannot be varied independently of
N/T and W/P, experimental methods cannot
be used unless its test value is fortuitously
exactly the required standard value or be-
cause the controls have been so intercon-
nected as to give this effect (see par. d
below). This is equivalent in our case to
saying that Tj must be varied independently
of N and p; the pilot must have two indepen-
dent engine controls.
(d) The Effect of Interconnection
Some engines may, however, have control
interconnection schemes such that the oper-
ator will not have independent control of
the interconnected variables, and will not,
for normal operational purposes, need it.
If these control interconnections impose
non-dimensional relationships*, the indepen-
d
* An additional difficulty is that one cannot
make a test on a hot day at the N/T
corresponding to combat engine speed on a
standard day without exceeding the engine
limitations. This has caused surprisingly
little trouble.
* The description "non-dimensional" is here
applied to terms whose dimensions can be
referred to standards from within the prob-
lem. For example, W/p can be compared
with some area, such as the wing area,
associated with the aircraft.
3:4
dent variables will be reduced by the number
of such relationships and the installation can
still be treated experimentally.
sometimes at a high weight (and low alti -
tude) and sometimes at a low weight (and
high altitude), any weaknesses in these as -
sumptions can be detected and precautions
taken. Large corrections should be avoided
if possible.
9
If, on the other hand, as seems to occur in
many cases, a non-dimensional relationship
is not introduced, the pilot has lost indepen-
dent control of the variables, and precise
values of these variables (corresponding to
standard conditions) cannot be applied. It
is then necessary to use analytical methods
of performance reduction.
3:5 ANALYTICAL METHODS
(a) General
(e) The Assumptions Required
Use of a purely experimental method of
performance reduction is only practicable if
the number of independent variables is small,
Inasmuch as there are usually many indepen-
dent variables which affect the performance
to some extent, it is necessary in using
such a method to be able to assume that
only a few of these are of importance.
We have classified as "analytical" all
methods of performance reduction which re-
quire numerical data from sources other than
the flight tests on the particular aircraft.
Within this class there is a wide range of
methods and the choice of the type of method
to be used will be influenced by the informa-
tion required from the tests and by the data
available about the characteristics of propul-
sive unit and airframe.
For example, it is usual to assume that
the level speed and the associated mass fuel
flow of a simple jet aircraft are a function
of engine speed, aircraft weight, air pressure
and air temperature (considered as an en-
thalpy) only. This implies that the following
quantities, among others, can be omitted from
consideration as independent variables:
It is conventient to divide the methods into
two sub-classes and to examine the merits
and demerits of each sub-class of method
with its associated flight test technique;
further advance into detail is inappropriate
to the present discussion.
(1) The viscosity of the air;
Methods of the first sub-class are essen-
tially methods involving fairly small correc-
tions and are commonly referred to as
differential methods. One makes perform-
ance tests under conditions approximating
standard as closely as practicable and uses
a reduction method to adjust individual results
for the relatively small discrepancy between
test and standard conditions.
(2) The temperature rise of working
fluid per unit mass of fuel injected.
This implication is probably correct when
the test and standard conditions do not differ
greatly, provided that the aircraft and its
engine are not near a “critical Reynolds
number" or in a region where combustion
efficiency is sensitive to working conditions,
and provided that the working fluid is not
changed (e.g., by dilution with water or
ammonia).
Methods of the second sub-class involve
a performance analysis, the test program
being designed to provide data suitable for
analysis rather than to give test results
under near-standard conditions.
(b) Application of the Methods
If the test program is so arranged that
tests at a given N/T and W/p are made
To estimate the corrections required for
3:5
the first type method, one usually linearizes
the relations between the variables and esti-
mates the slopes from an analysis of data
from other sources, preferably some fund of
generalized data. In other words, one must
have prior information about the perform-
ance response to changes in air temperature
and weight, but the information need not be
very precise as it is only used to estimate
relatively small corrections.
For methods of this type, the test program
should cover a fairly wide range of air speed
at one altitude at least. This assists with
the estimation of the effect on airspeed of
changes in thrust at constant altitude. The
test program should be designed primarily
to provide the required drag data.
Such methods may be more convenient than
the first type if the drag data required for
the differential method are not available,
or if the drag and thrust data cannot be put
into an easily used form. Also, if drag
data are required anyway, it may be more
economical to merge the performance reduc-
tion with this and to design the test program
accordingly. Again, if the first type method
is not much easier to use than the second,
it may be considered worth while to go to
a little more trouble to get drag data en
passant, even though these are not a primary
requirement.
If such information is available and easy
to use,* this
this type method can be very
convenient. It need impose no restriction
on the test program. It does not, however,
provide any internal check of the accuracy
of the assumed data, nor does it provide
drag data as part of the reduction process.
To use the second type method known as
“performance analysis”, one requires prior
information about the performance of the
propulsive unit and its response to changes
in air temperature but not about the drag
characteristics of the airframe, as drag
data are obtained during the analysis. Obser-
ved or estimated jet thrusts and/or shaft
powers combined with estimated propeller
efficiencies where appropriate give estimated
drag or power curves.
Estimates of the thrust or thrust power
which would be available under standard
conditions then lead to the standard perform-
ances. A comparison of the drag curves
obtained at various altitudes gives some
check of the assumed performance of the
propeller, etc.
(c) Control Systems
**
It is necessary with either type of analyt-
ical method to be able to estimate how the
thrust (which the power plant under test
gives) varies with air temperature*. This
requires a knowledge of the behavior not
only of the basic power plant but also of
its control system,
* In the method used in the United Kingdom
for piston-engine aircraft (Ref. 1), one engine
parameter () and two airframe parameters
(CDe and ebé) suffice to adapt generalized
formulae to the particular aircraft. The
method, however, is unsuitable in the pre-
sence of compressibility effects on airframe
drag and propeller efficiency.
For instance, curves of (thrust power)
I o/w..5) may be plotted against Ve VW
for various values of W/p.
* The aim is to show what the performance
of the aircraft under test would be under
standard conditions. It is not, for instance,
to show what the performance would be if
the engine did what its designer intended it
to do. One will therefore need either to de-
duce the engine performance under standard
conditions from its observed performance
under test conditions, or to estimate its per-
formance under test conditions from power
or thrust charts drawn up for standard condi-
tions. The actual approach chosen will de-
pend on the reliance which can be placed on
the power charts, and the availability of suit-
able instrumentation.
3:6
.
methods). For these methods to be prac-
ticable and economical, some fund of gener-
alized data on the response of performance
to changes in air temperature and aircraft
weight is required from other sources from
which linearized corrections are deduced.
If the control system is such that the
fuel flow at a given engine speed and intake
pressure is independent of air temperature,
the jet temperature would probably increase
with air temperature; the engine output might
then be limited by jet temperature at high
air temperatures and by engine speed at low
air temperatures. On the other hand, if the
control system kept the jet temperature
associated with a given speed constant, this
change of regime would not occur, as the
same limitation would always be the deter-
mining one. A similar change of regime can
occur near the full throttle height of a piston
engine. It need not be troublesome.
This information need not be very precise
as it is only used to estimate relatively
small corrections. If such information is
available and easy to use, as it is in the case
of the conventional piston-engined aircraft
(Ref. 1), the method can be very convenient,
particularly when reduction is required for
flight tests covering only a small part of
the full performance envelope.
3:6 SUMMARY OF CONCLUSIONS
3
We have seen that performance reduction
methods exist in two broad groups, exper-
imental and analytical.
Similar but more flexible methods have
been developed for turbo-propeller-engined
aircraft (Ref. 2), which at the moment are
somewhat inconvenient to use, due to the
impracticability of generalizing the perform-
ance characteristics of these engines at
this introductory stage.
Neither of these
small correction methods is applicable, how-
ever, in the presence of compressibility
effects on airframe drag, nor may either
be used with confidence in the region of
the speed for minimum drag or drag power.
(a) Experimental methods which require
no prior numerical data for engine or air-
frame are:
(1) Practicable only if any automatic
control or control linkage imposes only di-
mentionally correct relationships;
(2) Convenient if data are required
over a range of all variables or if enough
of the non-dimensional groups used in the
reduction process are fully controllable.
(2) Performance Analyses. Where
experimental methods are either impractic-
able or inconvenient, or where differential
methods cannot be applied because of com-
pressibility effects, the approach will neces-
sarily be from performance analyses. This
method also shows advantage over the differ-
ential method when data are required through-
out a wide range of all variables or when
drag data are required.
If these conditions are met, as they are
in the case of the simple turbojet-engined
aircraft and possibly some types of more
complex turbine engine, the potential advan-
tage of the tolerance shown by experimental
methods towards ignorance of the aircraft and
engine is decisive, especially at high alti-
tude or high Mach number.
The essence of the method is the estab-
lishment of drag or drag power curves
covering the speed range of the aircraft
from measurements of observed thrusts
and/or shaft powers combined with estimated
propeller efficiencies where appropriate.
From prior information about the perform-
ance of the propulsive unit and its responses
to changes in temperature and forward speed,
estimates of thrust or thrust power which
(b) Analytical methods which require ad-
vance numerical data from other sources:
(1) Small correction analytical
methods (also referred to as differential
3:7
would be available under standard conditions
then lead to standard performance.
which they are required, and the amount
of generalized data already available on the
type of aircraft and power plant, in addition
to the nature of the power plant and the
degree of complexity of its control system.
In certain problems a combination of experi-
mental and differential methods may possibly
be the best solution.
The selection of the most suitable method
for a particular application may therefore
depend upon the amount of data involved,
the region (height, Mach number, etc.) at
REFERENCES
1. Cameron, D., "British Performance Reduction Methods for Modern Aircraft," U. K.
Ministry of Supply Report No, AAEE/Res/170.
2. Lush, K, J., “Differential Performance Reduction Methods for Turbo-propeller Aircraft,"
U, K. Ministry of Supply Report No. AAEE/Res/275.
3:8
AGARD FLIGHT TEST MANUAL
VOLUME I, CHAPTER 4
PERFORMANCE OF TURBOJET AIRPLANES
By
Daniel O. Dommasch
Princeton University
Sections 4:1, 4:5 through 4:12,
4:14 and 4:16
W. J. Hesse
Naval Air Test Center
United States Navy
Sections 4:2, 4:3, and 4:4
T. W. Davidson
Naval Air Test Center
United States Navy
Section 4:13
Phillip Hufton
Aeroplane & Armament Experimental Establishment
United Kingdom
Section 4:15
TERMINOLOGY
VOLUME I, CHAPTER 4
CHAPTER CONTENTS
4:1
INTRODUCTORY COMMENTS
4:2
Page
4:1
INTRODUCTION TO LEVEL FLIGHT PERFORMANCE TESTING OF
TURBOJET AIRPLANES
4:1
4:3
TURBOJET ENGINE PARAMETERS
4:1
4:4
AIRPLANE-ENGINE COMBINATION CHARACTERISTICS
4:6
4:5
REDUCTION OF THRUST REQUIRED DATA TO BASIC FORM
4:13
4:6
THRUST AVAILABLE
4:16
4:7
RANGE AND ENDURANCE IN LEVEL FLIGHT
4:19
4:8
THE CRUISE CLIMB
4:23
4:9
THE NOMINAL BEST RATE OF CLIMB AIRSPEED
4:26
4:10
TIME TO CLIMB FOR JET AIRCRAFT
4:32
4:11
DETERMINATION OF MAXIMUM ENERGY STORAGE SCHEDULE
AND TIME TO CLIMB TO A GIVEN ENERGY HEIGHT
4:37
4:12
PROOF OF CONSTRUCTION USED TO DETERMINE OPTIMUM ENERGY
CLIMB SPEED FROM ACCELERATION RUN DATA
4:40
4:13
DESCENT PERFORMANCE OF TURBOJET AIRCRAFT
4:42
4:14
MEASUREMENT AND CORRECTION OF TURBOJET TAKE-OFF
AND LANDING CHARACTERISTICS
4:42
4:15
TURNING PERFORMANCE OF TURBOJET AIRCRAFT
4:43
4:16
CONCLUDING REMARKS
4:46
REFERENCES
4:47
TERMINOLOGY
CD
هی هم بن
CDG
D
E
e
Drag Coefficient
Lift Coefficient
Gross Drag Coefficient
Parasite Drag Coefficient
Thrust Specific Fuel Consumption lbs./lb. thrust-hour
Characteristic Engine Diameter
Endurance
Span Efficiency Factor
FG
Gross Thrust
FN
Net Thrust
FR
Ram Drag
G
Gf
hp
Weight Airflow Rate
Fuel Flow Rate as an Energy Rate
Pressure Altitude
Energy Height
he
L
Length
M
Mach Number
Z 3
p
Pn
Q
q
R
Mass
Engine RPM
Ambient Air Pressure
Pressure at Any Engine Station
Mass Airflow Rate
Dynamic Pressure (pv² =ypM²/2)
Range
1
TERMINOLOGY
(Continued)
RC
T
Tn
t
V
Vcal
Wf
W
μ
պ
♡
P
Y
σ
Δ
Subscripts
Rate of Climb
Ambient Air Temperature
Temperature at Any Engine Station (as enthalpy)
Time
True Airspeed
Calibrated Airspeed
Weight Fuel Flow Rate
Time Rate of Change of Energy Height
Absolute Coefficient of Viscosity
Efficiency
Pressure Ratio P/Po
Temperature Ratio T/To
Air Density slugs/ft.3
Cp/cv Specific Heat Ratio
P/PO Density Ratio
Increment of
Sea Level
t
Total Conditions
Superscripts
'(prime) Test Observed Value
4:1
INTRODUCTORY COMMENTS
Because it is not possible to presume that
a fixed (independent of Mach number) rela-
tion between C₁ and Cô exists throughout the
speed range covered by jet-propelled air-
craft, the reduction procedures convention-
ally employed with reciprocating engine air-
planes are not applied to jet aircraft. Rather,
the procedures used depend either on dimen-
sionless parametric relations or on analyses
of particular conditions of flight.
It is possible to present jet performance
in terms of several different sets of para-
meters, each set having advantages in a
given case depending on such things as equip-
ment available, time available, accuracy re-
quired, and so forth. The various procedures
are discussed in the body of the chapter and
it is up to the test engineer to determine
which procedure should be used in a given
case.
The material in this chapter is essentially
the contribution of people presently or re-
cently associated directly with naval aviation
and, for the most part, is similar to material
appearing in an existing preliminary edition
of the Naval Air Test Center Flight Test
Manual (Ref. 4). However, certain portions
of the material have been rewritten, taking
into account simplified techniques or im-
provements which have made their appear-
ance subsequent to the publication of Ref. 4.
4:2 INTRODUCTION TO LEVEL FLIGHT
PERFORMANCE TESTING OF TURBO-
JET AIRPLANES
The theory of reducing the level flight
performance data for any flight vehicle is
based on the nature of its power required
and available curves, where these curves are
defined in terms of thrust, power, or some
engine parameter such as RPM.
The parameters used for the reduction of
data from turbojet-engine-powered aircraft
stem partially from an analysis of the engine
characteristics and partially from an anal-
ysis of airframe characteristics. The en-
gine parameters will be considered first and
then combined with the airplane parameters
to produce the combined engine airplane
parameters which are actually used for level
flight performance data presentation.
4:3 TURBOJET ENGINE PARAMETERS
The performance characteristics of a tur-
bojet engine depend on many variables. These
variables include the engine size, engine
RPM, flight velocity or Mach number, am-
bient pressure, ambient temperature, and the
internal efficiencies of the component parts.
To present the performance characteris-
tics of a turbojet engine in a practical way,
it is necessary to collect the aforementioned
variables into a minimum number of signif-
icant groups so that the number of test flights
or wind tunnel tests and plots required to
present the complete performance picture
of the engine may be considerably reduced.
This grouping is accomplished by a di-
mensional analysis of the variables which
affect engine performance.
If there are n variables and r fundamental
units (such as length L, Mass m, and time t)
then, as is well known, the equation relating
the variables can be expressed in terms of
(n − r) dimensionless numbers. The working
procedure is as follows:
·
(a) Select r variables which include a
group of all the fundamental units from the
list of n variables, then
(b) Set up dimensionless equations com-
bining the variables in (a) with each of the
others in turn.
It should be emphasized that the equation
which results from this procedure is no
better than the original assumptions made;
i.e., if one variable is originally left out, it
will also be left out in the final answer. The
variables and performance quantities perti-
nent to the turbojet engine with symbols, units,
and dimensions are listed in Table 4:1.
1
4:1
TABLE 4:1 - VARIABLES AND PERFORMANCE QUANTITIES
OF THE TURBOJET ENGINE
CHARACTERISTICS
SYMBOLS
UNITS
DIMENSIONS
VARIABLE
Flight Speed
V
ft./sec.
Ambient Air Temperature
T
°R
Ambient Air Pressure
P
lb./in.2
Engine RPM
N
Engine Size or Diameter
D
ft.
Viscosity
μ
Component Efficiencies
η
η
none
rad./sec.
slug/ft.-sec.
L
m/Lt
none
L/t
L2/t2
m/Lt²
1/t
PERFORMANCE QUANTITY
Thrust (FG, FR, or FN)
F
lb.
mL/t2
Air Flow*
QG
lb.m/sec.
m/t
Fuel Flow (energy input)
Gf
ft.-lb./sec.
mL2/t3
Pressure at Any
Engine Section
Pn
lb./in.2
m/Lt2
Temperature at Any
Engine Section**
In
°R
L²/t²
2
*Note that this is a mass flow rate expressed in pounds mass per unit time rather than the
customary slugs mass per unit time.
** Temperature is here considered as a measure of enthalpy.
Our purpose is now to express F, Ga, Gf,
Pn, and Th as functions of the listed variables,
the functional relationship to be expressed
in an equation made up of a group of terms,
each of which is a dimensionless number.
We shall first consider the thrust.
From an analysis of the variables which
affect thrust, we can write
F = f ( V, T, P, N, D, H,
nr. 7c. 7b. nt. nn)
Eq. 4:2 consists of six variables. Since
we have three fundamental units, Bucking-
ham's Pi Theorem states that we can express
these variables in an equation made up of
three dimensionless numbers. Performing
the analysis, we have:
V
PD²= f( ND · √T).
ND
4:1
Since the component efficiencies are pri-
marily functions of V, T, p, N, and D, and μis
primarily a function of T and p, we can sim-
plify the above equation and write thrust as
a function of its prime variables as
F = f ( V, T, p, N, D )
4:2
*For simplicity, viscosity has been omitted
in this relation. However, it is pointed out
that at altitude, the Reynolds number term
which would result from a consideration of
viscosity may produce important effects.
4:2
For convenience we may recast the above
relation in a somewhat different form. Any
of the ratios can be turned upside down (the
ratio is still dimensionless) and ND in a
number can be replaced by V or T because
of similar dimensions. Changing the above
relation by these procedures gives
F
f
BD₂ = 1 (Y, ND.).
PD2
4:3
It is conventional to replace the ratio
V/✓T by Mach number*, to eliminate Dfrom
the equation for an engine of fixed size (this
elimination cannot be made on engines with
variable exit nozzles or variable inlet guide
vanes) and to refer the pressures and tem-
peratures to standard sea level conditions
by using the following technique:
F
PD2
=
F
14.750** 14.780** (const) 8
14.7pD2
The above relation represents an implicit
function of the variables which affect thrust.
The purpose of flight testing, wind tunnel
tests, or theoretical calculations is to solve
this equation explicitly. The solution is best
shown in graphical form as indicated in Fig.
4:1.
This one plot (Fig. 4:1) represents the
thrust performance of an engine at all alti-
tudes and all Mach numbers over the range
shown. If parameters were not used, it would
require many curves and curve sheets to
present this same information. Therefore,
it is evident that the use of suitable para-
meters saves much time and money when
the performance characteristics of a turbo-
jet engine are determined by flight tests,
wind tunnel tests, or calculations.
Table 4:2 on the following page presents
a summary of the functional relations among
the turbojet engine performance parameters,
obtained by dimensional analysis.
and
ND
√519 ND
√519√T
NO
√519 √8
where = P/P。 and 8 - T/T。.
Thus, Eq. 4:3 can be written
통 ​+ (M. 씀​).
N
(const)√8
* Illustrated in the following equation:
1
V
√gy R
V
4:4
√gy RT
= (const) M
4:3
F/8
M = 1,0
M = 0.5
M = 0.0
N/√6
Fig. 4:1 Thrust Parameter Variation
Thrust
TABLE 4:2 - FUNCTIONAL RELATIONSHIP OF THE TURBOJET
ENGINE PERFORMANCE PARAMETERS
Engine Parameter
General Form
Obtained From
Dimensional Analysis
Relation Referred
or Corrected to
Standard Sea Level
Conditions for Constant
Diameter Engine*
F
PD2
+ (ND, M)
F
f
8
(N,M
0
Air Flow
Fuel Flow
Ga
PD2
Specific Fuel
Consumption
Gf
FJT
f(ND, M)
T
Gov. 1(MD.M)
=
ND
DO² / T = f( ND, M)
Gf
PD2
T
=
N, M
G₂+(M)
8
816 • 1 (A.M)
Gf
8√8
0%· ·(N·M)
Gf
F√O
Fuel-Air Ratio
Gf
Go T
f
+ (ND, M)
Gf
Ga
Glo··(MM)
N
Pressure**
(at any section)
Pn
f
P
(ND, M)
Pn
8
(NO,M)
•
Temperature**
f
푸
​Tp = + (ND, M)
In
꼼 ​= + (M)
(at any section)
*When the engine diameter is not constant such as with an afterburner installation, it is
necessary to retain the engine size, D, within the functional relation.
Subscript n refers to any section.
The parameters developed by dimensional
analysis can also be used to correct or refer
observed data to standard sea level conditions
or to any specified altitude. This fact is, of
course, extremely useful in the test work of
engines and it has led to giving the names
"corrected RPM to N/√@" and "corrected
thrust to F/8", etc.
4:4
Table 4:3 presents a collection of the corrected or referred turbojet engine parameters.
The table with its footnotes is self-explanatory; however, it should be noted that when it is
desired to correct or refer data to some altitude other than sea level, the following changes
should be made:
Replace 8 with 8 where 8 = p observed/p at selected altitude and replace with a
where = T observed/T at selected altitude.
8
TABLE 4:3 - TURBOJET ENGINE CORRECTED OR REFERRED PARAMETERS
Corrected or Referred Quantities
Quantity
Observed
Value
General
Form
Corrected to Corrected to
SSL Variable SSL Constant
Diameter Engine** Diameter Engine
*
F
Thrust
F
PD2
F
842
ико
Air Flow
Qa
QO√T
Q0√o
Qo√8
PD2
842
8
Gf
Gf
Fuel Flow
Gf
PD2 √T
802√0
Gf
8√8
Specific Fuel
Gf
Gf
Gf
sfc
Consumption
F√T
F√8
Gf
Gf
Gf
Fuel-Air Ratio
f/a
Ga T
Gae
Gae
Pressure***
Pn
(at any section)
Pn
8
Temperature***
Tn
T
(at any section)
Engine RPM
ND
N
T
2/3 0/500/50
Pn
NA
---
N
7/6
V
V
V
Flight Velocity
V
Flight Mach Number
M
M
M
M
* SSL - Standard Sea Level Conditions.
**▲ in this column is defined as the ratio of the nozzle exit diameter at any operating condi-
tion to nozzle exit diameter at standard sea level condition. We note that these parameters
will not produce generalized data when changes in flow regime are produced by diameter
changes.
*** Subscript n refers to any section.
4:5
an aircraft from these plots, but the stipula-
tion is for a given aircraft weight and stand-
ard altitude conditions; To obtain the char-
acteristics at different airplane weights, one
would require another set of these four plots,
or to determine the characteristics at some
pressure altitude where the temperature is
not standard, one would require still another
set of plots.*
One set of plots is required for every
change in weight and/or altitude deviation
from standard values.
The plots are valuable within their limita-
tions, namely, the performance story of an
aircraft at one weight and under standard
altitude conditions. The method of obtaining
these plots depends, of course, upon the tech-
nique of applying fundamental dimensionless
parameters to the airplane-engine combina-
tion as was done for the engine in the previous
section.
(a) Airplane Parameters
The thrust required by an aircraft in steady
level flight is given by
F = CD ½ PV²S = (CD₂ + C²²) ½ PV's.
TEAR
Since the lift coefficient can be expressed
in terms of aircraft weight and speed, the
above expression may be written in para-
metric form as
2
F
POYS
2
4:4 AIRPLANE-ENGINE COMBINATION
CHARACTERISTICS
Our discussion thus far has been devoted
to engine characteristics to show where cer-
tain of the parameters used in the reduction
of turbojet aircraft data originate. Our pur-
pose now is, to examine the performance
characteristics of the overall vehicle, the
airplane-engine combination. When we con-
sider such a combination, the airplane vari-
ables such as aircraft weight, angle of attack,
etc., enter into the overall functional relation-
ships.
Before developing the equations necessary
to handle the airplane-engine combination,
it is desirable to examine the real purpose
and goal of the level flight performance eval-
uation of a turbojet-powered airplane. The
aircraft must first be flight tested over the
complete range of speed and altitude, suffi -
cient and accurate data must be obtained
during these tests, and then the data must be
reduced and assembled for logical presenta-
tion,
Fig. 4:2 presents, four plots which rep-
resent a typical method of presenting the
most important performance characteristics
of an airplane equipped with a fixed geometry
jet engine.
The data for Fig. 4:2 are based upon a
given airplane weight and for standard con-
ditions at all altitudes. These plots show:
(1) variation of standard altitude with
flight Mach number for various engine RPM's;
(2) variation of standard altitude with
fuel flow for various engine RPM's;
(3) specific range (miles/lb. fuel)
variation with flight Mach number; and
(4) endurance (hours) variation with
flight Mach number.
It is evident that one can determine the
complete level flight performance picture of
4/00
CDe M² +
14.77 S(EAR) (~)* (M²)
4:5
* As discussed later, the use of dimensional
parameters sharply reduces the number of
required plots.
4:6
MILES PER POUND OF FUEL
STANDARD ALTITUDE,FT.
80% R.P.M.
90% R.P.M.
95% R.P.M.
100% R.P.M.
STANDARD ALTITUDE, FT.
MACH NUMBER, M
FUEL FLOW RATE
40,000 FT.
30,000 FT.
20,000 FT.
10,000 FT.
ENDURANCE, HOURS
MACH NUMBER, M
MACH NUMBER, M
Fig. 4:2 Four Plots Showing Important Level Flight Performance
Characteristics of Turbojet
4:7
40,000 FT.
30.000
20.000
10.000
80% R.P.M.
90% R.P.M.
95% R.P.M.
100% R.P.M.
1
The above equation is in general explicit
for flight below the drag divergence Mach
number; however, for flight above this Mach
number, CDe becomes a variable and is a
function of M and W/8. Thus in general, Eq.
4:5 must be written as (ignoring as before
the effects of viscosity)
f
t.
등​• + (M. 뿡​)
W).
(b) Combined Parameter Relations
4:6
From an analysis of the engine alone we
have (from the last section for a fixed geo-
metry engine)
4/00
·
Gf
}
(M)
4:7
in the airplane dimensional analysis, Rey-
nolds number will appear in the equations,
just as it would appear in Eqs. 4:7 and 4:8
if viscosity were included in the engine anal-
ysis.
When viscosity is ignored, the airplane
parameters, like the engine parameters,
will not generalize completely. Thus, low
altitude data cannot be used to predict ac-
curately the high altitude characteristics,
and it is necessary to obtain performance
data over the complete range of altitudes.
Inasmuch as in steady level flight, net
thrust is equal to drag, Eq. 4:7 can be equated
to Eq. 4:8 and, since the fuel flows in Eqs.
4:10 and 4:11 are equivalent, these equations
can also be equated. Equating either or both
sets of equations shows that
M =
·(*. *)
4:11
4:8
A similar dimensional analysis of the
variables which affect the drag of an airplane
or the analysis shown above yields a drag
parameter for the airplane which is
이쁨
​M
m).
This equation merely states that the flight
Mach number of a turbojet-powered aircraft
with fixed geometry engine is a function only
(for the assumptions made) of the parameters
N/√ and W/8. From this equation we can
also write
= f
(~, M) or I = f (W,M) *
4:12
and
Gf
N
+ (~~/ M) O
Gf
or
(WM)*
4:9
8√Ā
4:13
A further analysis of the airplane, that
of equating the heat energy input as a func-
tion of the airplane drag energy, yields the
following fuel flow parameter
G₁
4:10
We note that in the above equations, air-
craft weight appears in one of the parameters.
It is pointed out that if viscosity is included
*It should be noted that the latter forms of
Eqs. 4:12 and 4:13 apply to any type of engine
geometry, fixed or variable, because they
come directly from Eqs. 4:9 and 4:10, which
were derived from airframe considerations
only.
4:8
}
W
8
Level flight performance testing of air-
planes with fixed geometry engines has been
accomplished most frequently in the past on
the basis of solving experimentally the two
following functional relationships
and
M = f
(学​)
Gf
8/10
(NM).
A typical graphical solution of these
equations is sketched in Fig. 4:3.
It is readily apparent that the level flight
performance characteristics curves (the 4
plots of Fig. 4:2) can be obtained from the
two plots of Fig. 4:3. In fact, any given num-
ber of these sets of four plots corresponding
to different airplane weights or non-standard
altitude conditions can be obtained from Fig.
4:3.
Actually, all the level flight performance
data are generalized and available on the two
engineering plots of Fig. 4:3. The four plots
of Fig. 4:2 are merely convenient plots and
are special cases because they are applicable
only to a certain specified airplane weight.
We can thus state that the purpose of level
flight performance testing is to obtain suf-
ficient data so that engineering plots (such
as Fig. 4:2) can be constructed.
M = 0.5
M = 0.6
M = 0.7
M = 0.8
Gf
8/M
Lines of Constant W/S
2/10
N
e
N
Ꮎ
Fig. 4:3 Graphical Solution of the Performance Relations
of the Turbojet Powered Airplane with Fixed Geometry Engine
4:9
7
It should be noted from the above plots
that thrust, which is a very basic parameter,
does not appear. For the fixed geometry
engine, the performance characteristics have
not been measured in terms of thrust until
recently for several reasons. First, thrust
has been more difficult to measure than the
parameters N/√, M, and W/8 and second,
service aircraft are not equipped with thrust
meters. Therefore, it is desirable to present
performance characteristics in terms of
variables which can be evaluated by the pilot
in the cockpit.
It is pointed out, however, that the per-
formance characteristics of the aircraft can
be presented in terms of the thrust para-
meter. Employing this system, plots of the
following parameters are made.*
= f
풍​(뿔​M)
W
Gf
= f
3/8·1 (2·M).
•
4:14
FG/8
FG values at the
minimum points are
about the same.
W/8 - 80,000
W/8=50,000
W/8=20,000
Mmox at 100% RP.M.
MACH NUMBER, M
4:15
Plots of these functions will also yield the
performance story of an aircraft. The above
thrust parameter may be expressed either
in terms of net thrust or gross thrust. At
the Naval Air Test Center (NATC), gross
thrust is preferred because it is easier to
measure than net thrust and accurate results
are obtained with a relatively simple gross
thrust meter installation (see Ref. 1).
*Editor's note: The use of RPM as a re-
placement for airplane drag or thrust re-
quired has not been entirely satisfactory for
the reason that the thrust RPM relationship
varies somewhat from engine to engine and
with time due to engine wear. Similarly, fuel
flow data are easier to generalize when pre-
sented in terms of Eq. 4:15 rather than Eq.
4:8.
Fig. 4:4 Typical Thrust Required Curves
for Turbojet Airplane Engine Combination
Fig. 4:4 presents the typical variation of
the gross thrust parameter with flight Mach
number for several values of W/8.
These curves (Fig. 4:4) represent the
gross thrust required curves of the basic
airplane-engine combination. (See Ref. 2 or
Chapter 2 for satisfactory methods of deter-
mining the gross thrust available curves for
an airplane-engine combination.) Thus, when
the thrust available and thrust required
curves are combined, the level flight per-
formance characteristics are known.
It should be recognized that when variable
geometry engines are performance tested in
conjunction with an aircraft, it is advanta-
geous to use the thrust parameter function
because the parameter Eq. 4:11 then contains
an additional term.*
* See Chapter 3 for a more detailed discus-
sion of this type of problem.
4:10
:
Eq. 4:11 for a variable geometry engine
then becomes
f
W
M-1 (M··A)
= (NO
8
*
and, for a dual rotor engine, becomes
(***):
米
​N1 N2
M = f
4:16
4:17
The thrust parameter relationship even
for the variable geometry engine or the dual
rotor engine is still given by Eq. 4:14, which
contains only three terms.
Thus far, two basic procedures have been
used at the NATC for presenting level flight
performance data of jet-powered aircraft,
namely, the required curves in terms of the
RPM parameter (Fig. 4:3) and the required
curves in terms of the thrust parameter
(Fig. 4:4).
Emphasis has recently been placed on the
thrust method of presenting the data because
of its more direct approach, elimination of
inherent engine tachometer errors in the
data reduction procedures, its simplicity for
variable geometry engines, and, in general
its application to the solution of more varied
types of level flight performance problems.
A very simple gross thrust meter is de-
scribed in detail in Ref. 1.
Consider, for example, the gross thrust
available and required plots of Fig. 4:5.
The thrust required curves of Fig. 4:5
are obtained from readings of the thrust
meter mentioned above (to eliminate the nec-
essity of cross plotting, flights may be at-
tempted at constant values of W/8), and the
* In these equations, ▲ is the ratio of vari-
able area to some reference area; N₁ and
N2 are the two rotor speeds in a dual rotor
engine.
thrust available curves from the methods
outlined in section 4:6.
Once these curves are found, the level
flight performance story is told, namely:
(1) maximum Mach number;
(2) minimum thrust;
(3) stall speed;
(4) minimum level
flight speed
(which is important for catapult launching of
heavy jet aircraft because it has been found
that in some aircraft the minimum catapult
end speed is dictated by this point instead of
the following point);
(5) the aerodynamic stall.
(c) External Store Drag
As mentioned before, the (gross) thrust
method of performance presentation is more
versatile than the corrected RPM method.
One of the newer uses of the gross thrust
method is its application to the evaluation
of drag of external stores such as mines,
bombs, rockets, etc.
FG/8
Gross Thrust Available
Gross Thrust Required-
Gross Thrust Required, Low W/8
High W/8
MACH NUMBER, M
Fig. 4:5 Thrust Available and
Required Curves
4:11
1
i
The reduction theory for the evaluation of
external store drag can be developed from
an analysis of the two gross thrust required
curves shown in Fig. 4:6.
The curves shown in Fig. 4:6 correspond
to the airplane clean configuration and the
configuration with some external store, both
at the same value of W/8. Consider now
points (1) and (2) at the same Mach number
and assume operation at sea level (to sim-
plify the equations).
For the two configurations, the gross
thrust is given by
FG₁ = D+FRI
FG₂ = D+FR₂+DS
The change in ram drag, FR, can be ex-
pressed in terms of gross thrust and air
flow by the following analysis. The rate of
change of ram drag with respect to gross
thrust is given by
ΔΕR
AFG
=
which for V
or
=
(G2V2 - G|V₁)
9 AFG
V2 (see Fig.4:6) becomes
AFR=Y AFG
AFG
g
•
4:18
where FR is the engine ram drag, D the
basic airframe drag, and DS the external
store drag including interference effects.
The change in gross thrust is given by
FG = FR+DS
4:19
AFR
(AFG)
(AFG)
4:20
The term AFG/AG can readily be obtained
from engine specifications curves or approxi-
mated by 2Vef/g where Vef is the effective
exhaust velocity.
Inserting the results of the above equation
in Eq. 4:19 gives
FG/8
With External Store
Clean
Some W/8
V
1
g
AFGI
AFG
=DS
AG
Θ
10
4:21
MACH NUMBER, M
Fig. 4:6 Gross Thrust Required Curves for
the Clean and External Store Configuration
(AFG and V from flight data and ▲ FG/AG
from specification curves.)
The above equation thus provides a method
of determining the actual drag of the external
store.
Eq. 4:21 can also be extended in certain
cases to calculate the performance of an air-
plane with some store that has been previously
4:12
tested on another airplane. Suppose, for ex-
ample, airplane x has been tested in the clean
configuration and with stores, and the flight
data are presented as shown in Fig. 4:6, and
suppose further that airplane y has been
tested only in the clean configuration. Now
if the external store drag Ds is the same on
both airplanes (Dsx = Dsy), we can write
वै
AFGy
=
(AFG)
FGX
| +
AFG)
AG
X
airframe drag and the ram drag in steady
level flight divided by the dynamic pressure
and wing area. This coefficient is useful if
combined airplane-engine characteristics
only are being considered and, as we shall
show, the gross drag coefficient is in special
cases a function of lift coefficient and Mach
number as is the airplane drag coefficient.
To prove this latter statement, we con-
sider the relation already established that
FG = f(WM)
(for steady
level flight)
and consider the nature of the parameter
W/8 at constant Mach number during one g
flight.
In steady level flight, we know that
W =
YPM2
DM² CLS = Po. YSM². CLS.
2
Therefore,
W = (PO M²) · (CLS)
2
4:22
Since all terms in the above equation are
known except ▲ FGy, we can calculate this
quantity and apply it to the clean configura-
tion curve to obtain the gross thrust required
curve for the external store configuration.
It is pointed out that careful judgment must
be used before one can assume that the in-
terference drag is the same for airplanes
x and y. When such an assumption can be
made, considerable testing time and money
can be saved.
4:5 REDUCTION OF THRUST REQUIRED
DATA TO BASIC FORM
For the purpose of simply determining if
an airplane meets its design specifications,
it is not often necessary to reduce thrust
required data to its most fundamental form;
i.e., plots of drag coefficient as a function of
lift coefficient and Mach number. However,
when precise corrections of climb data and
the like are required, a knowledge of drag
variation characteristics becomes important
and, of course, such information is also of
great interest to design engineers. We shall,
therefore, briefly consider here methods of
reducing flight test data to terms of aerody-
namic coefficients.
First, we note that it is possible to define
a new drag coefficient, the gross drag coef-
ficient, which is simply the sum of the
and since po, Y, and S are constants
Therefore,
W = (M²CL) times a constant
FG = f (M² CL, M) = f(CL, M).
=
If we measure FG in steady level flight,
then FG equals the sum of the airplane drag
4:13
and the ram drag, so that
FG
qS
8f(M, CLL.
= CDG
урма
2
S
where = free stream density
ρ
V = true flight speed
A = stream tube area far ahead of
the engine,
the value of C
CDR
becomes
Thus,
CDG
f(M, CL)
(M²) (constant)
= f (M, CL)
4:23
CDR
00
DR
PAV²
q S
PV2
S
2
الله
where CDG
=
gross drag coefficient.
4:25
As discussed in Chapter 2, we may readily
measure gross thrust and we may also de-
termine net thrust from flight tests, although
the necessity of measuring total tailpipe tem-
perature somewhat complicates the net thrust
determination.*
In this connection, Eq. 4:23 allows us to
establish that the area of the stream tube of
entering air far ahead of the jet intake has
a cross sectional area which is also a func-
tion only of CL and M for the steady level
flight case. For this case we have
CDR
where CD airplane drag coefficient and,
since
= ram drag coefficient = CDG - CD
CDG = f(CL, M) and CD = f(CL, M)
CDR = √2(CL, M).
Now, inasmuch as
4:24
DR = ram drag = pAV² = FR in steady
level flight
* For many practical purposes, the standard
tailpipe temperature gage (properly cali-
brated) will provide a sufficiently accurate
temperature reading.
From Eqs. 4:25 and 4:24,
A = f(CL, M).
4:26
Eq. 4:26 shows that the engine mass flow
during steady level flight depends on C₁ and
M and thus, in theory, we may relate required
engine RPM to fundamental aerodynamic
parameters.
The foregoing work serves to illustrate
the interdependence of airplane-engine char-
acteristics under steady flight conditions and
shows that steady level flight data may be
presented in terms of various equivalent sets
of parameters.
Once we deviate from the special case of
steady level flight, however, it is apparent
that the equivalence of the parameters ceases
to exist. Consequently, curves of airplane
performance presented in terms of engine
RPM or gross thrust or gross drag coeffi-
cient cannot be considered sufficiently gen-
eral for a thorough-going performance eval-
uation because results expressed in terms of
parameters such as these have meaning only
under certain restricted circumstances.
If we wish to evaluate airplane perform-
ance during climbs, descents and under con-
ditions of normal and longitudinal accelera-
tion, and moreover, if we wish to correct
such flight data to standard, it is essential
that we be able to separate specifically
4:14
provide the following information for each
test point
(a) Drag, D
(b)
Gross weight, W
(c) Ambient pressure, p
(d) Mach number, M
In turn, we may then compute for each
point
D
со
=
YPM2
S
2
CL
W
урма
S
2
where y = 1.4
S = wing area.
This provides us with the plots of Fig.4:7
with each curve representing a series of
runs at one nominal altitude.
LIFT COEFFICIENT, CL
MACH NUMBER, M
airplane and engine characteristics. For-
tunately, we have the means to do this, and
level flight tests serve to provide part of
this means for they allow us to determine
the nature of the airplane lift-drag polar by
actual measurements.
As explained in Chapter 2, the net thrust
may be measured in flight, and if it is meas-
ured in steady level flight, it equals the air-
plane drag. Accordingly, we may make a
series of level flight runs at various altitudes
and obtain data in the form of drag and weight
versus Mach number for each set of runs.
For each test point it is important that
the aircraft be stabilized; however, succeed-
ing points need not be at exactly the same
pressure altitude. Tests should be conducted
starting at Vmax and each succeeding run
made at a lower speed until the minimum
speed for stabilized level flight is reached.
Data to be recorded consist of pressure
altitude, ambient temperature, total pres-
sure, total tailpipe pressure, and tempera-
ture and fuel flow. These data are used to
DRAG COEFFICIENT, CD
MACH NUMBER, M
Increasing
Altitude
Fig. 4:7
4:15
Increasing
Altitude
→
DRAG COEFFICIENT, Co
Increasing CL
MACH NUMBER, M
Fig. 4:8
These data may now be cross-plotted to
give curves of Cp at constant C vs. M as
shown in Fig. 4:8 (or cross-plot may be
prepared of Co vs. CL at constant M).
Since the data of Fig. 4:8 have been re-
duced to a basic form, they may be used to
compute the airplane drag under varying
situations and are not restricted to the spe-
cial case of level flight.
We note that in special instances, the in-
terference flow produced by engine operation
may alter the relations of Fig. 4:8 depend-
ing on the airplane configuration; however,
such cases should make themselves apparent
in the scatter of data in the figure, in which
case further detailed tests will be required
to determine the cause of this interference.
We now turn to the topic of determining
standard engine thrust available at a given
altitude.
4:6 THRUST AVAILABLE
Because the engine is used merely as a
measuring instrument when thrust required
data are obtained, engine corrections to
standard conditions need not be made for
thrust required runs. However, to deter-
mine Vmax, we must know: (1) what thrust
the engine should have given under test con-
ditions; or (2) what thrust it would give
under standard conditions.
In Chapter 2 and in the appendix to Chapter
7, the problem of standard thrust available
is briefly considered and it is pointed out
that engine charts are not too reliable a
source of standard information, inasmuch
as they do not necessarily represent properly
the operation of the engine as installed in
the airplane, nor, in some instances, are the
specified operational limits actually obtain-
able in the air. Moreover, it is well recog-
nized that not all engines of a given model
provide the same value of F/8 at a fixed set
of values of M and N/√8.
Accordingly, for really precise work, the
engine thrust characteristic should be tied
down by a comprehensive set of flight runs,
which procedure, if attempted, would be
rather costly as regards both time and money.
If one is satisfied with something "less than
the best," the maximum speed at any altitude
may be determined using several approxi-
mate procedures, two of which are considered
here.
In the absence of thrust measuring equip-
ment, thrust required data are presented
using the parametric relation
N/√ē = f(M, w/8)
discussed earlier in this chapter.
Curves of N/√ē vs. W/8 for various Mach
number values are readily obtainable from
tests similar to those described in section
4:5 for the determination of drag coefficient
variations. To obtain the desired curves, we
first prepare curves of N/√ vs. M and W/8
vs. M for each set of test runs and then
cross-plot to obtain curves of N/√ vs.
W/8 for constant Mach number, as shown in
Fig. 4:9
4:16
N/√6
Now suppose we wish to know Vmax at
some weight W at an altitude where the pres-
sure ratio is 8, and the temperature is 8. We
first compute W/S locating point (1) on Fig.
4:9. Then for 100% RPM, we compute N/ē,
thus locating point (2). Erecting perpendic-
ulars to points (1) and (2) locates point (3),
and by interpolation on the figure, the Mach
number corresponding to Vmax is at once
established.
As we have already pointed out, presenta-
tion of test data in the form of Fig. 4:9, al-
though frequently a satisfactory procedure,
is far from the optimum. A somewhat better
method is to utilize the thrust required re-
lation
Now we also know that the thrust avail-
able equation may be written
FN
f(M, NO).
Thus, for small corrections
A(FN/8) =
where
FNN AIN√O)
+ F
FNMAM
4:27
FN
8
f(M, W/8).
FNN
=
J(FN/S)
a(N/√ē
and
a (FN/S)
FNM
am
On the basis of this relation, test data
may be plotted in the form of Fig. 4:10 (using
a cross-plotting technique to obtain the form
shown).
HIGH M
LOW M
0
w/8
Since Fig. 4:10 must be obtained by cross
Fig. 4:9 Determination of Vmax
4:17
03/27
Increasing W/8
Lines of Constant W/8]
(W/S)
(W/8)2
MACH NUMBER, M
Fig. 4:10
· ΔΜ
plotting test data*, a test data point at the
full throttle conditions obtainable during test
might be at position (a) when plotted on Fig.
4:10.
Since in steady level flight, thrust
available and thrust required are the same
thing at full throttle, point (a) represents a
test thrust available point which may be cor-
rected to standard conditions using 4:27. To
make the correction, we proceed as follows:
For the test conditions (correcting at con-
stant pressure altitude), we know the value
of N actually obtained. Call this value
(N/√). If we presume that 100% of rated
RPM is obtainable under standard conditions
(which we must normally assume unless test
results indicate the contrary), then we may
compute the standard N/√ value using the
standard atmosphere value for at test pres-
sure altitude. Consequently,
^(~~) - (~) - (~~)'
이승
​We now refer to the engine charts for the
value of FNN at the test value of FN/8 and
determine the increment
A(FN), FNNA (ME)
=
Consequently, we may determine the Mach
number increment as illustrated in Fig.4:10.
Using this increment and the value of FNM
from the engine charts, we now compute
(FN)₂ = (FNM)AM.
A 2
ΔΜ.
This moves points (c) to (d), and we know
that points (b) and (d) are points on the thrust
available curve.
Thus, since we have already linearized
the problem, we connect points (b) and (d) by
a straight line which is extended until it in-
tersects the (W/8) line through point (a).
This intersection, point (e), gives us the
standard thrust available and also the maxi-
mum Mach number for the (W/8) value con-
sidered.
If we correct all full throttle test points by
the procedure outlined above and connect
them by a single curve, this curve deter-
mines the maximum Mach number for any
value of (W/8).
This latter correction procedure is con-
sidered more accurate than the first and can,
of course, be employed just as well for
gross
thrust data as for net thrust data. The only
drawback to the procedure is its dependence
on engine charts. However, since these
charts are more accurate as regards slopes
of curves than they are as regards actual
values of the engine parameters, the proce-
dure gives satisfactory results, provided the
corrections are kept small.
We close this first part of Chapter 4 by
noting that for level flight tests of jet air-
craft (including fuel consumption data), we
need to obtain test information defining the
following physical quantities:
(1) P
Ambient pressure
(2) V
-
True airspeed
This moves points (a) to (b), for the case
where test conditions are warmer than stand-
ard. Now at the test value of W/ §, the incre-
mental increase in thrust produced by the
change in N/√ would cause the airplane to
speed up to point (c) which is on the constant
(W/8) line through point (a) drawn parallel
to the (W/8), and (W/8)2 lines as shown in
the figure.
*By scheduling or programming the runs
according to prior computations relating
pressure altitude to fuel used, it is usually
possible to conduct runs at constant W/8,
the success of this technique depending on
the skill and cooperation of the pilot.
4:18
.
(3) M
- Mach number
-
(4) OAT - Outside ambient air tempera-
ture
(5) N - Engine RPM
(6) W - Weight fuel flow rate
Since W/S and M determine a given value
of C₁, the following expressions may also
be written for the case of steady level flight:
Wf
8√8
=
f(CL, M)
4:30
(7) Ptt
- Total tailpipe pressure for net
thrust determination
Wf
f(CL, M)
8√ M
4:31
(8) Ttt - Total tailpipe temperature for
net thrust determination
Standard instrumentation may be used for
measurement.
W{
8√8 M
W
(*,M).
4:32
As regards pilot technique, the most im-
portant thing in level flight testing of jets is
the achievement of stabilized flight conditions
for each test point. This is a more difficult
problem than when piston-engine airplanes
are being tested because of the more rapid
change in weight (due to burning of fuel) to
which the jet is subject in comparison to the
piston airplane. Other than this, pilot tech-
nique is essentially that described in Chap-
ter 6.
4:7 RANGE AND ENDURANCE IN
LEVEL FLIGHT
Using the techniques of dimensional anal-
ysis, it is easy to demonstrate that if Wf is
the weight fuel flow rate (weight per unit
time) for a fuel of constant heating value,
then we have the parametric relations
N
Although any of the foregoing parametric
representations may be used to present fuel
flow data, the disadvantages of using RPM
where it is unnecessary have resulted in
standardization of terms of Eqs. 4:29 and
4:32. The quantities W/8√ and W/8√M
in these equations are called the specific
endurance and specific range parameters,
respectively.
Ref. 3 presents explicit equations for jet
range and endurance from which we may
derive definite equations for specific range
and endurance. These equations are given
below.
specific range
dR
d W
C1/2
= 1.414
CD CI√PS
W1/2
4:33
Wf
・
+(1/ M)
4:28
dE
= specific endurance
Wf
dw
이쁨
​3/1/0 (MM).
<!-
1. CL
W ct CD
4:29
4:34
(We note that these equations are equivalent
in steady, level flight.)
where R is in feet, E is in seconds, and c is
4:19
the specific fuel consumption in lbs./(lb.
thrust-second).
We note further that
dR dE
·(V)
dw dw
so that (dR/dW) max
to (dE/dw)
max
4:35
does not correspond
From Eqs. 4:33 and 4:34 we see that for
any given weight and altitude the criterion
for maximum range is that the product
(C2 / Cp) times (1/c) be a maximum,
whereas, for maximum endurance, (CL/CD)
times (1/c) must be a maximum.
1/2
For flight testing work the parametric
equations are far more convenient than the
explicit relations shown directly above. These
latter equations were introduced merely to
show what factors are involved in range and
endurance work, and the manner of their
involvement.
Fuel consumption test data should be
obtained concurrently with other level flight
SPECIFIC ENDURANCE PARAMETER, W¢/S√@,
Increasing Altitude
Note: Each Curve Represents A
Series Of Runs At A Nominal
Test Pressure Altitude
High Altitude
performance data, and no discussion of the
manner of taking data is required here. The
test data are converted to the desired para-
metric form by using appropriate values of
8 and 8.
The converted test data are first plotted
in the form shown by Figs. 4:11 and 4:12.
Using Figs. 4:11 and 4:12, a cross-plot
is prepared as shown in Fig. 4:13.
Fig. 4:13 is the generalized plot of the
fuel flow or specific endurance parameter.
To obtain generalized plots of the specific
range parameter, Fig. 4:13 may be used di-
rectly or one may proceed to prepare a test
data plot of Wf/MƐƑ similar to Fig. 4:11
which, together with Fig. 4:12, provides a
cross-plot of Wf/M8√ as a function of M
and W/S. Normally, a plot of the quantity
1/(Wf/M&√ē) = Md√/Wf is prepared rather
than the direct plot; however, the choice of
whether to plot the reciprocal or the direct
function is rather immaterial, since either
procedure yields the same answer. The
reciprocal plot gives a presentation directly
related to specific range, and this is one of
the reasons for its use.
Low Altitude
MACH NUMBER, M
Fig. 4:11
4:20
RATIO OF WEIGHT TO PRESSURE, W/8
Increasing Altitude
Note: Each Test Curve Corresponds To
A Curve In Figure 4:11
High Altitude
Low Altitude
MACH NUMBER, M
Fig. 4:12
T
Regardless of the procedure followed, the
resulting specific range parameter plot ap-
pears as in Fig. 4:14.
From Fig. 4:14 the maximum range Mach
number for any set of operating conditions
is readily determined by drawing a line
through the peaks of the individual W/8
curves.
Using Fig. 4:14, the specific range under
any given set of operating conditions may be
readily computed. For example, suppose
that an operation requires flight at a given
Mach number, weight, and altitude, and we
wish to know what the specific range is. To
find our answer, we first compute W/8.
Then, knowing M, we read the corresponding
values of Md√/Wf from Fig. 4:14.
If We is expressed in lbs, per sec., multi-
plication of MS/Wf by √ToygR/8 = Vc。 ồ
gives the specific range in feet per pound of
fuel (note that √gy RT
Vco). When Wf is
in lbs. per hour and we wish to determine
the specific range in nautical miles per
pound we have
=
dR
d W
= S. R. in nautical miles per lb. of fuel
Md√ Vco
3,600
Wf
8
6,080
and since Vco
9/23/3
a constant, we have
= S. R. in nautical miles per lb. of fuel
661 Md√ē
S Wf
4:36
Using Eq. 4:36, the specific range is at
once obtained for the given values of 8 and
MS √б/Wf.
Next consider the problem of computing
maximum range at any given fixed altitude
More
Wp
SPECIFIC RANGE PARAMETER =
Increasing (W/8)
Lines
of Constant
(W/8)
MACH NUMBER,
Fig. 4:14
SPECIFIC ENDURANCE PARAMETER, W‡/ S√☎
Lines of Constant Mach Number
W/S
Fig. 4:13
4:21
MAXIMUM RANGE
MACH NUMBER
Σ
M
MAXIMUM RANGE MACH NUMBER
W/8
SPECIFIC RANGE PARAMETER
Μόνθ'
Wp
'max.
MAXIMUM VALUES OF SPECIFIC RANGE PARAMETER
W/8
Fig. 4:15
from the data of Fig.4:14. By cross-plotting
the curve of maximum range Mach number
we obtain an optimum Mach number schedule
as a function of W/S. This is as shown in
Fig. 4:15, and a corresponding curve of
maximum MS✓√ē/Wf vs. W/d is shown in
Fig. 4:16.
For any given constant 8, Fig. 4:15 can
Fig. 4:16
be used to give a schedule of Mach number
vs. fuel remaining in the airplane. Assum-
ing now that we follow this schedule, and
know how much fuel we may use, we next
prepare a table listing the maximum values
of the specific range parameter as a func-
tion of airplane gross weight, using Fig.
4:16 as the source of our information. The
table is shown below.
W W/8
Wf
(MSVE)
661
S.R.=
M
8
/max
Wf max
Maximum specific
range parameter
for (W/8)
from Fig. 4:16
4:22
Mach number for
maximum range
for W/8
from Fig. 4:15
The data from the foregoing table are
plotted as shown in Fig. 4:17.
NAUTICAL MILES
POUNDS OF FUEL
dR
MP
=
SPECIFIC RANGE
Since
Wfinish
start
W = AIRPLANE WEIGHT IN POUNDS
Fig. 4:17
R =
dw,
Wstart
(d) aw
Wfinish
4:37
the shaded area under the curve of Fig. 4:17
gives the total range for the altitude con-
sidered.
The search for ways to operate an air-
plane most effectively leads quite naturally
to the concept of the so-called cruise climb
which is discussed in the following section.
4:8 THE CRUISE CLIMB
It is well known that the specific range of
a jet airplane increases with increasing al-
titude. The reason for this may be seen from
an examination of Eq. 4:33 which shows that
dR/dW varies inversely as the square root
of the density, so that as long as ct, the
}
thrust specific fuel consumption, does not
increase markedly, a continuous gain of dR/dW
is experienced as altitude is gained.
Actually, up to the stratosphere, c† tends
to decrease for most engines, so that greater
gains in range are obtained than would be
found if & were assumed constant. Moreover,
at low altitudes, inefficient part throttle op-
eration further increases the obtainable ct,
producing an additional decrease in dR/dw.
At high altitudes (above 35,000 to 40,000 ft.),
ct starts to increase so that present test
data reveal a "leveling off" in dR/dW data
for stratospheric conditions.* As airplanes
fly higher, this leveling off should result in
an optimum best range altitude for any given
gross weight, above which altitude, decreases
in dR/dW will be encountered.
Because increases in altitude result in
increases in specific range, it may be rea-
soned that a gradual climb should increase
overall range, provided that the climb were
made at close to the optimum aerodynamic
speed for best range and close to the engine
throttle setting for best thrust specific fuel
consumption. This thinking leads at once to
the concept of the cruise climb.
The cruise climb amounts first to setting
the airplane to fly at the optimum range
Mach at a given value of W/8. Then, as fuel
is used up and W decreases, allow a gradual
climb so that the ratio of W/8 is kept con-
stant as the Mach number is also held con-
stant. This amounts to flight at the optimum
Mach number for the initially selected W/8
value.
Actual tests have established that the
cruise climb procedure results in improved
range performance over and above that ob-
tained by flight at constant altitude (varying
W/S and M).
* Also, of course, the effects of encountering
drag divergence at a speed lower than the
speed for best range (indicated) obtained from
low altitude tests necessarily limit the high
altitude best range speed.
4:23
We note here that the cruise climb does
not necessarily result in the best possible
range characteristics for any given mission;
rather it gives an improvement over con-
stant altitude flight.
To provide a visual means of examining
the foregoing statements, we may prepare
a plot of maximum specific range in nautical
miles per lb. of fuel as a function of gross
weight and W/8. To obtain this plot, we
refer to Fig. 4:16 and tabulate the data of
this figure as shown below.
For various values of weight W, W₂,
W3..., we then compute the specific range
corresponding to the W/8 value in column
(1) as shown in the table. The results plot
as in Fig. 4:18.
In practice, the cruise climb is accom-
plished by starting out at some value of W/8,
say (W/8), and establishing the optimum
Mach number for that W/8 from data such
as presented in Fig. 4:15. Given a schedule
of weight vs. altitude presented in some con-
venient form, the pilot then climbs the air-
plane so as to maintain a constant W/8 and
M as the gross weight decreases. This means
that he flies from point (1) to (2) of Fig. 4:18,
with the net range being given by the shaded
area of the figure.
In this same figure, flight at constant
altitude is illustrated by the dashed line be-
tween (1) and (3), with the increase in range
of the cruise climb shown as the area
bounded by the lines 1-2, 2-3, 3-1.
In the foregoing we have made the assump-
tion that the fuel consumption data obtained
in level flight will be adequate to describe
conditions in the cruise climb even though
the cruise climb does not represent level
flight conditions. Experiments have demon-
strated that this assumption is satisfactory,
at least for present day airplanes, for the
simple reason that the climb rates are quite
small, and in themselves produce negligible
errors. Accordingly, cruise climb fuel con-
sumption characteristics may legitimately
be computed on the basis of data obtained
during level flight runs.
Once Fig. 4:18 has been obtained, the
cruise climb range between points (1) and
(2) may easily be calculated without resort
to graphical integration. The formula for
doing this is derived below.
Eq. 4:36 may be rewritten
199 =MP
(뽕​)(
(W). (MONO). W.
WF
4:38
In the cruise climb all quantities on the
right side of 4:38 are constants (by definition
of the cruise climb) except W; therefore,
or
R = 661
W
8
(MSVE).
Rcruise climb = 661
loge
Wstart
dw
W
Wfinish
) (MSN)>
Wstart
Wf
W finish
4:39
2
3
4
5
3'
4'
5'
W/8
MS√A
(3)
661
(3')
W₁ 8 =
S.R.=
(2)
Wf
(1)
(4)
W2 8:
661
=
S.R.=
(1)
(4)
4:24
(2)
ہیں
W3
SPECIFIC RANGE, NAUTICAL MILES PER POUND
Lines of Constant (W/8)
2
(W/S)
3
Increasing (W / 8 )
CRUISE CLIMB
CONSTANT ALTITUDE CRUISE
Wfinish
GROSS WEIGHT, W
Fig. 4:18 The Cruise Climb
4:25
Wstart
(a) Sawtooth Climbs as a Means of De-
termining the Speed for Best Climb
The oldest flight test procedure used to
determine the speed for best rate of climb
at any given altitude is called the sawtooth
method. This procedure is discussed thor-
oughly in Chapter 6 in connection with the
performance of piston-engine aircraft and the
reader is referred to that chapter for details
of the method.
(b) Acceleration Runs as a Means of De-
termining the Speed for Best Climb
Another means of determining the speed
for the best rate of climb is to employ level
flight acceleration run data. If we assume
the angle of climb at the speed for best rate
of climb to be under 15°, then the power
required to overcome drag in level constant
altitude flight is essentially the same as the
power required to overcome drag in a climb
at the same airspeed.*
Accordingly, the speed for maximum sep-
aration of the power available and power re-
quired curves in level flight will be about
the same as the speed for this maximum
separation in a climb, and it follows that we
can obtain this speed from level flight runs.
It is emphasized that all this holds true
only for climb angles of less than 15°. For
greater angles the assumption of lift equal
to weight ceases to be realistic.
Now, on the basis of the above assumption
(that the flight path angle is less than 15°) we
may write that the excess horsepower avail-
able over and above that required to main-
tain level constant altitude flight may be used
either to produce a rate of change of potential
or kinetic energy, i.e.,
Pav - Preq Pxs d(KE)
=
=
dt
+
d(PE)
dt
Before completing our discussion of level
flight range and endurance testing, it is worth-
while mentioning the use of Eq. 4:28 for the
presentation of range and endurance infor-
mation,
Within the framework of the assumptions
used to develop. Eqs. 4:28 and 4:29, both
should in theory provide equivalent results.
However, this has not proved to be the case
in practice. Indeed if one prepares curves
of test data in the form of W/dē vs. N/√ē
for constant Mach number values, it is found
that the high and low altitude data tend to
form individual curves for any given Mach
number. Accordingly, the airframe para-
meter relations 4:29 and 4:32 should be
used in preference to the engine parameter
equation (Eq. 4:28).
4:9 THE NOMINAL BEST RATE
OF CLIMB AIRSPEED
Specification-wise, the climb performance
of an airplane is generally described in
terms of its maximum sea level rate of
climb, its service or operating ceiling, and
the time to climb to a given altitude. Recent
investigations have shown the most realistic
approach to climb performance testing is
obtained when one considers the time to climb
to a given energy level rather than merely
to a given height, since it is apparent that
if an airplane arrives at an altitude at a very
low airspeed an additional amount of time is
required to accelerate it to fighting speed
for that altitude before the airplane's mis-
sion can be performed. For this reason we
shall consider the topic of maximum energy
climbs as well as the simpler topic of nom-
inal maximum rate climbs.
In the following sections we will first
review the aerodynamic theory of climbs
and develop the data reduction procedures,
after which the special pilot techniques re-
quired in climb testing will be discussed.
4:40
* Because lift and weight may be assumed
equal in both cases.
4:26
where Pav
Preq
KE
PE
Thus
= power available
=
power required
kinetic energy (W/g) times
(V²/2)
potential energy = Wh.
Pxs =
W
d(V)2
2g
dt
+ W
dt
or, noting that
Pxs
19
d(V)2
v dv
dt
W
2g
v dvo
+ W
dt
dt
4:41
4:42
4:43
Now, as shown by Eqs. 4:40, 4:41, and
4:42, excess power may be used to produce
a change of either kinetic energy, potential
energy, or both. In a climb at constant true
airspeed the only change is in potential en-
ergy, whereas in an acceleration run the
only change is in kinetic energy. Since either
rate of change will be a maximum at the
speed for maximum excess power, this speed
may be determined either by running constant
speed climbs over short altitude bands (as
in the case of sawtooths) or by making ac-
celeration runs from which the speed for
maximum rate of change of KE may be de-
termined.
?
In making acceleration runs, for purposes
of obtaining climb data, we are not interested
in learning about the acceleration character-
istics in the immediate vicinity of the stall
or Vmax but rather in the region of the
best climb airspeed. The approximate value
of this speed may be estimated by trial ac-
celeration runs during preliminary evalua-
tion, from contractor's test data, or by using
the estimation procedure discussed below.
(c) Estimation of Speed for Best Rate of
Climb
In the absence of other data, estimates
of the speed for best rate of climb may be
obtained from the relations developed below.
First consider the case of altitudes close
to the ceiling of the aircraft. In this case
the power available and power required
curves will appear as in Fig. 4:19.
From Fig. 4:19 we see that the aerody-
namic speed for best range (given by the
point of tangency between a line through the
origin and the power required curve) is only
slightly less than the speed for best rate of
climb.
Thus, as a first estimate to the speeds
for best rate of climb at high altitudes, we
may use the aerodynamic best range speeds
at these altitudes (aerodynamic best range
speed is the speed at which a jet airplane
must fly to attain the maximum value of
(c/CD), and is the speed determined by
the tangent to the power required curve
which passes through the origin.)
The conditions determining the aerody-
namic best range speed are simply
T = D (Thrust
H
Drag)
d (TV)
dv
d(dv)
dv
4:44
(slopes of power available and power re-
quired curves are the same).
Assuming that dT/dV = 0, and simplify-
ing, we have
dD
v
but D = ½/pV²CDS,
CL = 2W/pV²s,
and CD = CDe +CL/TEAR,
4:27
POWER
YBR
so that
v.dD = pv²C Des
-
POWER
MAXIMUM
EXCESS
4W2
pV2STEAR
Solving for V4,
= 0.
V₁ =
4W2
CDe TEAR
POWER REQUIRE
POWER AVAILABLE
TANGENT THROUGH ORIGIN
SPEED FOR BEST RATE OF CLIMB
VELOCITY, V
Fig. 4:19 Power Curves at High Altitude
4:28
or
VBR:
derived equation for V times dD/dV, we ob-
tain
༢
2W
PS.
свет
CDTE AR
v² Vimax
V4-
4W2
=0.
3
3p² V² S² πe AR CDE
4:45
4:47
(approximation to best rate of climb speed
at high altitudes).
At low altitudes near sea level, Eq. 4:45
predicts a best climb airspeed much lower
than the actual, and therefore, at lower alti-
tudes a different approximation is required.
This is obtained in the following manner.
The speed for maximum rate of change
of energy is given by the condition that the
slopes of the power required and full throttle
power available curves are the same; i.e.,
using the second of Eqs. 4:44, we have
V
dT
dv
+ T = V
dD
dv
Using Eq. 4:45, this becomes
V4_
V² Vmax VÅR
-
3
and on solving for V,
= 0
3
V =
Vmax
6
+
Vmax
36
+
VBR
R
3
4:48
(approximation to best rate of climb airspeed
at low altitudes)
For typical jet fighter or attack aircraft
with aspect ratios between 3 and 5,
VBR=1
3W
PS
and using this in Eq. 4:48 we have
2
Vfps
max
6
+
Vmax!
36
+
ㅎ
​(3W)
2
(for low altitudes).
4:49
4:50
Since the term 1/3(3W/pS)² in Eq. 4:50
is quite small at low altitudes compared to
Vimax /36, Eq. 4:50 has the simplified form
at low altitudes
V =
Vmax.
1.732
+ D.
Again presuming dT/dV = 0, we obtain
dD
V
dv
+ D — T = 0·
4:46
At low altitudes, the drag at Vmax of a
jet airplane is due essentially to the parasite
drag coefficient CD and the induced drag is
entirely of second order. (We note this is
not true at high altitudes.)
Thus, in Eq. 4:46
T = ½ pV max CDe S.
Substituting this in Eq. 4:46, together
with the relation for drag and the previously
4:51
4:29
This equation, in common with Eqs. 4:45
and 4:49, predicts a speed for best rate of
climb which is lower than the actual.
Summarizing, we have the following:
(1) At high altitudes (near the air-
craft ceiling) the speed for best rate of climb
will be somewhat greater than
Vfps
3W
PS
During the run, continuous recordings of
the following items should be maintained at
intervals of not more than ten seconds.
h'p
V'
·
-
t
-
Observed pressure altitude (with
altimeter set to 29.92" Hg.)
Observed airspeed
Time
OAT' Outside air temperature
-
RPM'- Engine RPM
4:52
M'
-
(for AR between 3 and 5 for fighter or attack
types)
(2) At low altitudes the speed for best
rate of climb will be somewhat greater than
V =
Vmax
Fuel
-
Mach number (if Machmeter is
available)
Fuel remaining
A photopanel or other automatic record-
ing system is most desirable for recording
the data; however, it is not absolutely essen-
tial, and data may be transmitted verbally
via radio by the pilot to a ground recording
station.
At this point a few words concerning the
accuracy of the data seem appropriate. If
the altitude is held constant during the ac-
celeration run, a smooth variation of V' vs.
time should be obtained; however, the smooth-.
ness of the data by itself does not indicate
the absence of error.
At low altitudes, the maximum accelera-
tions obtained are quite high, producing a
rapid rate of change of dynamic pressure
which results in flow in the pitot system,
and therefore, a lag in the airspeed indicator
reading. Except for a minor flow in the static
system due to change of position error with
changing speed, all the flow is in the pitot
system, and therefore, balancing the pitot
and static systems against one another can-
not alleviate this lag, and may actually in-
crease it.
Normally, the rate of change of dynamic
pressure obtained during low altitude accel-
eration runs is considerably greater than
1.732
4:53
The foregoing formulae determine the
approximate speeds for best rate of climb
at various altitudes. To apply these equations
for preliminary estimation of the climb
schedule, we determine the approximate
speeds for best rate of climb at sea level
and at the operational ceiling, and assume
a linear variation of speed with altitude be-
tween these speeds.
(d) Range of Speeds Covered in Accel-
eration Runs
The starting velocity for the acceleration
runs should be selected (at any altitude) as
somewhat less than (at least 50 knots) the
speeds given by the preliminary estimate or
contractors' data, as the case may be. Dur-
ing the runs it is essential that altitude be
maintained as constant as possible from a
speed of about 50 knots less than the start-
ing speed to a speed at least 50 knots, and
preferably more than this, above the esti-
mated speed for best rate of climb.
4:30
that obtained in stall tests where lag is con-
sidered an important factor, and it follows
that lag errors in acceleration runs are a
large possible source of error. These lag
errors may be minimized by using a low
volume pitot system with large diameter
tubing; however, they cannot be eliminated
altogether.
Corrections for these errors may, of
course, be imposed provided the lag con-
stant for the pitot system is first deter-
mined by ground checks.
If corrections are not imposed, errors of
the order of magnitude of 10 knots are pos-
sible with high performance aircraft unless
special low lag design systems are used.
Once the data have been obtained, the first
step is to convert observed pressure alti-
tude to true pressure altitude, observed air-
speed to true airspeed, and observed OAT to
true OAT.
Thus far in tests at the Naval Air Test
Center, Patuxent River, no corrections have
been made to account for non-standard en-
gine thrust, nor for the time rate of change
in engine airflow rate resulting from air-
plane acceleration. The magnitude of the
errors produced by omitting these correc-
tions is unknown; however, the overall climb
results produced by the acceleration methods
have been satisfactory, so present con-
clusions are that these factors produce er-
rors of small magnitude.
Once we have the corrected true airspeed
variation with time, the points of maximum
energy storage rate at constant altitude may
be obtained in two fashions.
2
First, we may plot a curve of VAS vs.
t, and graphically differentiate this curve to
give a plot of d(VTAS )/dt. The peak of this
curve represents the conditions for maximum
energy storage rate. Second, since d(Vias >/
dt =
2VTAS times d(VTAS)/dt we may ob-
tain this same result by plotting a curve of
V TAS vs. t, and by graphical differentiation
2
TAS
obtain a curve of dVTAS /dt. Using these
two curves, a third plot of VTAS times
dVTAS/dt is obtained whose peak point rep-
resents the condition for maximum energy
storage rate.
In both of these procedures we are pre-
suming that the weight change experienced
during a particular run is negligible, and
that the test weight is close enough to the
standard weight so that the effects of non-
standard weights are negligible.
Weight changes during the run actually
may be accounted for rather easily and may
be important when afterburner operation is
being evaluated. To account for such weight
changes, recall that we are solving for the
speed at which we have d(KE)/dt a maximum
so that d(W/g times V2/2)dt is a maximum,
With g and 2 being constants, this means
that
d(Wv2)
dt
maximum .
4:54
VBEST CLIMB
Fig. 4:20 Typical Speed-Altitude Schedule
as Determined by Acceleration Runs. Case
of TAS Variation Illustrated.
4:31
(1) Pressure altitude, h'p
(2) Time, t
(3) Observed airspeed, V'
(4) OAT
(5) Gross weight, W
(6) Tailpipe temperature
(7) Thrust meter reading (if in-
stalled)
During the climb it may be discovered
that the tailpipe temperature limits RPM,
If so, the fact should carefully be noted, and
an investigation initiated to determine if this
limitation can be removed.
Theory of Time-to-Climb Corrections
When a climb to a given altitude or energy
level is considered, the major factor which
we want to determine is the time required
to achieve the height or energy height.
It is well known that should we compute
time to climb on the basis of rate data ob-
tained from sawtooth or acceleration runs,
the time obtained would be optimistic; i.e.,
too small. This is true because in general
the true speed for best climb rate increases
with altitude, producing an increase in air-
craft kinetic energy not accounted for in in-
stantaneous rate data obtained, for instance,
by sawtooth runs. For this reason alone,
regardless of the climb schedule followed,
the time to climb should be determined by
a continuous climb to ceiling.
Now, the climb schedule which we present
to the pilot and which he will follow as closely
as possible during tests is normally given
in terms of calibrated airspeed versus pres-
sure altitude, since these are quantities which
he can observe from his cockpit instruments.
However, although we present the schedule
Thus, if we prepare a plot of WV² vs. t
and graphically differentiate this plot, the
peak on the resulting curve of d(WV²)/dt
represents the conditions for maximum en-
ergy storage with variable weight considered.
Acceleration run data obtained at various
altitudes are reduced by the methods just
described. Using the reduced data, the speed
for best climb at each test pressure altitude
is determined and a plot similar to Fig.4:20
prepared.
By flying according to the schedule of
Fig. 4:20, the actual time to climb and ac-
tual rates of climb may be determined.
Reduction of these data to standard is
discussed below.
4:10 TIME TO CLIMB FOR JET AIRCRAFT
Climbs to operational ceiling (that alti-
tude where the maximum rate of climb is
500 fpm) or to other ceilings in jet aircraft
may be made in accordance with the schedule
obtained from the sawtooth procedure, the
acceleration run procedure, or according to
the maximum energy schedule discussed in
section 4:11 of this chapter.
Both the sawtooth method and the accel-
eration method should produce the same
curve of nominal best climb speed* vs. alti-
tude; i.e., the curve shown in Fig. 4:20.
During the climb, regardless of the sched-
ule followed, the following data are recorded
at short intervals with a photopanel or other
means.
* In considering the energy climb schedule
in section 4:11, we shall see that this nom-
inal best climb speed actually does not cor-
respond to maximum rate of climb conditions
because it does not account for flight path
accelerations.
4:32
in these terms, what we really seek is a
schedule of true airspeed versus tapeline
altitude under standard atmospheric con-
ditions.
If the pilot were flying in the standard
atmosphere, our Vcal vs. hp schedule would
provide the proper "true speed-tapeline
altitude" schedule, but, since the standard
atmosphere is only a statistical mean, the
actual flight must deviate from that which
we desire. Accordingly, we must make cor-
rections both on climb rate, flight path speed,
and necessarily for non-standard engine
thrust.
First, let us consider the nature of the
speed correction:
Provided the pilot follows the prescribed
schedule, he will be at the correct calibrated
airspeed at a given pressure altitude. There-
fore, we have these two items as constants
during correction (i.e., Vcal, and hp). It is
shown in text books (Ref. 3, for instance)
that there exists only one value of equivalent
airspeed (Ve) for a given set of values of
Vcal and hp and it follows at once that the
airplane's equivalent airspeed is also a con-
stant in the correction process, since it is
the standard value. However, the true speed
is not standard and accordingly, the kinetic
energy content of the airplane is also not
standard. In Chapter 7 it is pointed out that
time to climb to a given energy height is in-
sensitive to small deviations from the pre-
scribed schedule, and therefore, it is pos-
sible to presume that the true airspeed of
the climb may be held invariant during the
correction process for energy climbs.
On the other hand, if we are interested
in time to climb to a given tapeline altitude
only with a jet-propelled airplane, changes in
true speed are important, for we are in-
terested only in obtainable climb rates rather
than the sum of the kinetic and potential en-
ergies of the aircraft. Knowing the obtain-
able climb rate for standard conditions, it
is a simple matter to compute standard time
to climb.
Thus for time to climb determinations,
we have that the following quantities should
be held constant during correction:
hp, pressure altitude
Vcal, calibrated airspeed
Ve, equivalent airspeed
Also, since Mach number depends only on
Vcal and pressure altitude, we have constant
Mach number.
Now, if we hold as constants the Mach
number and the equivalent airspeed, then we
also have that the airplane drag is constant
during correction and the increment rate of
climb correction* depends only on the change
in power available between test and standard
conditions.
This change in power available is due to
two factors:
(a) The increment in true airspeed
(b) The increment in thrust.
Thus, if RC is the rate of climb correc-
tion,
ARC = f(AV, AF).
4:55
Normally no correction for weight changes
is imposed for correction of climb to ceiling,
because if we start at the proper weight at
sea level, the change in weight during the
climb should, for practical purposes, simu-
late the standard variation.
To evaluate the quantity ARC, we first
determine AV and then AT. Consider the
quantity AV:
If V' is the test true airspeed and V is
* Dimensionless methods of analysis are also
available for climb data reduction, but gen-
erally require obtaining considerably more
flight data than the differential method pre-
sented here.
4:33
the standard or desired true airspeed, then
Now,
v' + AV = V.
during correction, we have
where
AF
OF
(1/8)
N
Δ
√o
so that
V
V =
داد
Ve
To test
Ve
√ostd
Ρ
std
To std
Otest
P test
+△(7)
()
Δ
N
The quantity aF/¿(N/√ğ) is readily ob-
tainable from engine charts for the test values
of (N/√@)', p and M. (Note if the charts are
in terms (F/S) = ƒ(N/√☎ and M), we may
write
AF =
a(F/8)
Ə(N√√ē)
(승​).8
or since pressure is held constant, the equa-
tion of state (perfect gas law)gives
where & has its test value.)
Knowing AV and ▲F, we may write
Test Excess Power Available
V' (F' -D)
=
Pxs
V
Trest
Tstd
'ז
T
Standard Excess Power Available
Pxs = V(F-D) = (V'+ AV) ((F'+ AF)-D
where D = airplane drag.
Hence, neglecting the term AV AF, we have
APxs - P'xs = V'AF + F'AV - DAV
= v'AF + Av [F'-D].
Now, it is well known that in a steady
climb
V(F-D)
RC
W
4:56
If F' is the test value of thrust and F the
standard value, then
F' + AF = F.
If thrust measuring equipment is avail-
able, F' can be accurately determined, and
if standard conditions of nozzle pressure
ratio and total temperature are specified,
standard thrust can be determined as dis-
cussed in the appendix to Chapter 7.
If thrust measuring equipment is not
available, we may compute the changeAN/
between test and standard conditions. Then,
since the Mach number is held constant
4:34
Therefore,
V'(F'-D)
(RC)' =
W
but
T
1+
יז
and
so that
F'D
(RC)'W P'xs
v'AF
V'
RC = (RC)'
+
W
4:57
4:59
Thus, if we assume that the acceleration
change along the flight path is negligible
between standard and test conditions (which
it normally will be), we have
and
AV(RC)' W
APxs
V'AF+
From Eq. 4:57
(RC)'
P'
XS
W
V'
4:58
=
We next must determine the relation be-
tween (RC)', the test tapeline rate of climb,
and the pressure rate of climb, (RC)p. After
correcting the raw altimeter data for posi-
tion and lag error, we may determine the
pressure rate of climb, (RC)p dhp/dt from
a plot of hp vs. time. During the climb a
continuous record of temperature vs. height
should be taken and after reducing the ob-
served temperature to ambient, we have a
record of T' vs. pressure altitude.
From our (RC)p and T' vs. hp data, we
may compute (RC)', the test value of tape-
line rate of climb, using the following con-
siderations:
P XS
P'xs + APxs
W
is
RC =
W
so that
RC
= 1 +
(RC)'
APxs
P'xs
From Eqs. 4:58 and 4:57
APxs
VAF
AV
+
P'xs
W(RC)'
V'
Hence,
RC
= 1+
AV
V'AF
+
The equation of balance in the atmosphere
dp = -pg dh
and the altimeter is calibrated using this
equation, presuming that p has its standard
atmosphere value. Therefore, for calibra-
tion
-
dp = pg dhp
(RC)'
W(RC)'
whereas actually,
dp = -p'g dh.
On equating the two expressions for dp,
we have
dh =
ρ'
// dhp
4:35
and at constant pressure altitude
dh =
'ז
T
dhp.
On dividing by dt, we have
(RC)' = (RC)p()
Using this in Eq. 4:59, we find that
4:60
(6) Tailpipe total pressure for net
thrust determination
(7) Tailpipe total temperature for
net thrust determination
(8) Engine RPM
(9) Test configuration
RC =
(RC)p
√E+
V'AF
W
4:61
Normally tests are not conducted under
gradient wind conditions because of the dif-
ficulty of actually determining the wind gra-
dient. However, wind gradient corrections
may be made using energy principles as
discussed in Chapter 7, or preferably runs
made on reciprocal headings to cancel or
average wind effects. Under no circum-
stances may the effects of wind gradient be
neglected.
The pilot technique required in conducting
timed climbs in jet aircraft is essentially
the same as required with piston-engine
aircraft and this topic is considered in Chap-
ter 6. The minimum test data required for
time to climb determination are:
time
(1) Time
(2) Pressure altitude variation wit
(3) Ambient temperature variation
with time
(4) Calibrated airspeed variation
with time
In addition to recording the foregoing
data, the pilot should record the existence
of operating limitations such as limited RPM
due to excessive tailpipe temperature and
the like.
Corrections for instrument error, posi-
tion error, etc., are made according to the
standard procedures considered elsewhere
in the manual and corrections to standard
are made using Eq. 4:61. The actual time
to climb is determined by plotting 1/RC vs.
STANDARD ALTITUDE,
h or H
ABSOLUTE CEILING (BY EXTRAPOLATION)
OPERATIONAL CEILING
TIME TO CLIMB
RATE OF CLIMB
(5) Fuel flow data (to determine the
variation of weight with time or pressure
altitude during the climb)
STANDARD TIME TO CLIMB, t
STANDARD RATE OF CLIMB, (RC)
Fig. 4:21
4:36
altitude and then graphically evaluating the
integral.
h
dh
t = time to climb to altitude h =
(RC)
4:62
climbs is considered in detail in Chapter 7
and here we shall only briefly discuss the
theory of the energy climb. As shown in
Chapter 7, the problem to solve is the maxi-
mization of the quantity w as a function of
time where
Final data are presented in the form of
Fig. 4:21.
dhe
V²
W =
1
and he (energy height) = h +
2g
dt
?
4:11 DETERMINATION OF MAXIMUM
ENERGY STORAGE SCHEDULE
AND TIME TO CLIMB TO A GIVEN
ENERGY HEIGHT
As we have mentioned, the nominal best
rate of climb schedule does not generally
produce the minimum time to climb to a
given geometric altitude, and certainly does
not produce the minimum time to climb to
a given energy height. This is so because
two factors are not considered in the con-
ventional climb analysis:
which is the mechanical energy content of
the airplane per unit weight.
The analysis in Chapter 7 reveals that
one way to express the condition for maxi-
mum energy storage is the equation
dw
aw
ah
a
(2)
2g
(1) The effects of changing flight
path speed and, hence, kinetic energy with
height.
(2) The possibility of storing more
energy at lower altitudes where the separa-
tion of the power required and power avail-
able curves is a maximum.
When conducting a timed climb to ceiling,
the effects of changing kinetic energy are
automatically accounted for by the tests
and accordingly, the rate of climb actually
achieved is less than predicted by either
sawtooth climbs or acceleration runs. By a
more precise analysis of either sawtooth or
acceleration run data, it is possible to pre-
dict actual rates of climb with fair accuracy
by accounting for kinetic energy changes with
altitude. However, the most satisfactory
procedure is always to determine time to
climb by conducting a climb to ceiling ac-
cording to the desired schedule.
The general topic of maximum energy
4:37
4:63
Accordingly,* if we prepare a plot of h
vs. V²/2g for lines of constant w, we shall
have aw/ǝh = dw/d (V²/2g) at points on the
w = constant curves where the slopes of the
tangents to the curves are 135°. This is the
method of determining the maximum energy
climb schedule on the basis of acceleration
runs.
The optimum schedule may also be de-
termined from continued climb to ceiling
made according to several different sched-
ules. In this case, we plot he vs. time and
from this set of curves prepare cross-plots
of w vs. V for constant values of he. From
this second set of plots, we may readily de-
termine the value of V for maximum w, and
knowing he, we may determine the schedule
V = (h). The procedure is fully described in
Chapter 7.
At the NATC, the maximum energy sched-
ule has, in the past, been obtained by con-
ducting acceleration runs and then analyzing
* See section 4:12.
the data to determine the optimum energy
storage speeds. The analysis procedure ac-
tually used has been to solve for the climb
schedule by an iterative method; however,
since a direct solution is available, this will
be considered here.
To obtain the raw data, acceleration runs
are made at a number of altitudes (at least
four) and data are recorded as discussed in
section 4:9. Although weight corrections are
not normally imposed on acceleration run
data, as noted in section 4:9, it is possible
to account for weight variations during the
runs. Where an excessive time rate of
weight change is encountered (as is possible
with afterburner-equipped aircraft), accel-
eration runs should not be used since it is
not possible to predict a priori what weight
will exist at a given altitude. In cases such
as this, the energy schedule should be ob-
tained using the method of continued climbs
discussed in Chapter 7.
Acceleration runs should be conducted at
constant altitude; however, the effects of
minor changes in height during the run may
readily be accounted for by correcting the
observed kinetic energy variation with speed
for any small potential energy changes en-
countered.
After the observed data have been reduced
to the form of true airspeed vs. time, the
derivative w = dhe/dt is readily obtained by
graphical differentiation of the velocity-time
curve (note that at constant altitude dhe/dt
1/2g times d(V²)/dt = V/g times dV/dt).
=
data to obtain the plot (Fig. 7:8) of h vs.
V²/2g, and drawing the 135° tangents, deter-
mine the variation of V²/2g vs. h. These
data are reduced to the form of true speed
variation with altitude using obvious pro-
cedures.
Our next problem is the determination of
time to climb according to the schedule ob-
tained above. The timed energy climb is
conducted in the same fashion as any timed
climb and the same data are recorded. This
topic is discussed in section 4:10. However,
the corrections to impose on the energy climb
are not the same as those required for the
simple climb to altitude discussed in section
4:10.
Because of necessity, the pilot follows a
schedule of Vcal vs. altitude and the only
quantities which he can correctly maintain
during the climb are:
W
-
(weight) (we assume this to be
standard)
Vcal (calibrated airspeed)
hp
-
Ve
-
M
D
(pressure altitude, and hence, am-
bient pressure)
(equivalent airspeed)
-
(Mach number)
·
(airplane drag)
The non-standard quantities are:
T
V
-
F
as discussed
in
section 4:10
-
(ambient air temperature)
-
(true airspeed)
(engine thrust)
Atmospheric turbulence
When determining time to climb to a given
geometric altitude (see section 4:10), cor-
rections were imposed to account for the
Because tests are run at constant pres-
sure altitude and because no corrections are
normally made for non-standard temperature
or non-standard thrust, the pressure alti-
tude may be considered the standard altitude
of the test. We then obtain a plot of w vs.
V (or V²/2g) for various constant standard
altitudes by plotting the w vs. V curves for
the test pressure altitudes as shown in Fig.
7:6 of Chapter 7.
Using Fig. 7:6, we next cross-plot the
4:38
variation between the true airspeed of the
test and the standard atmosphere true air-
speed called for by the climb schedule;
i.e., corrections were made only to the climb
rate. Here we shall impose our correc-
tions on the rate of energy climb which
will simultaneously account for changes in
flight path speed and in rate of climb. The
correction procedure will be developed pre-
suming that the calibrated airspeed-pres-
sure altitude schedule is followed by the
pilot as accurately as is operationally pos-
sible according to the stipulations of sec-
tion 6:10 of Chapter 6.
The corrections for the case where it is
assumed that the pilot maintains the proper
true airspeed-pressure altitude relation are
considered in the appendix to Chapter 7. At
this point, we shall proceed on the assumption
that the proper calibrated airspeed is main-
tained. Under these circumstances we have
WW = V(F-D)
w' W = V'(F'-D)
where the notation is as in section 4:10. Be-
cause the equivalent airspeed and Mach num-
ber are standard, D is unaltered during
correction. Hence,
V = V' + Av
F = F' + AF
and neglecting the product AF AV, we have
on dividing wW by w'W
AV
AF
1+
+
F'-D
also that
so that
w'W
F'-D =
V'
W = 'w'
T V'AF
+
W
יז
4:64
As mentioned previously, it is undesirable
to conduct tests under gradient wind con-
ditions although corrections can be imposed
if the gradients are known. As shown in the
appendix to Chapter 7, the wind correction
for a horizontal wind with vertical gradient
(dV/dh)wind is
dw =
dh
anwind dt
To determine the quantity w', we must
determine h', the true altitude obtained up
to the time t. For this purpose we utilize
the relation developed in section 4:10,
dh =
Idhp
and prepare a plot of T'/T vs. hp as obtained
from the test data.
This plot is as shown in Fig. 4:22.
Then
hpi
푸
​ni₁ = s = dhp.
!
4:65
We next reduce the test calibrated air-
speed to true airspeed by conventional means
(which is easy to do since V= f(Vcal, P, T)
and all quantities on the right side are known).
Then
Now we know that
AV
1+
4:39
ne = n'+
2g
T
dhp
we have
Via
he = he + (hp-h') +
2g
(-1).
4:66
Knowing w and he, from the relation
we have
dhe
W
dt
dt =
dhe
W
про
or
hp PRESSURE ALTITUDE
Fig. 4:22
We next prepare a plot of h'e vs. time
and by graphical differentiation or other
means obtain
w'
dh'e.
dt
W and AF are obtained from fuel flow and
engine data, and substituting in Eq. 4:64, we
obtain w, the corrected rate of change of
energy. The corrected energy height is ob-
tained from the following considerations:
The test energy height is he = h'+ (V² /2g),
and since pressure altitude is used as a con-
stant during correction
he
=
and on subtracting
V²
hp +
2g
· + 29 (V² - Vi².
hehe hp-h' +
Now, since
√
t =
ne
dhe.
W
V = V'
Hence, if we plot the quantity 1/w as a
function of he, the standard time to climb to
any given energy height is readily obtainable.
4:12 PROOF OF CONSTRUCTION USED TO
DETERMINE OPTIMUM ENERGY
CLIMB SPEED FROM ACCELERA-
TION RUN DATA
The condition for maximum "nominal"
rate of climb at a given constant altitude is
that the climb should occur at the speed for
maximum rate of change of kinetic energy
at that altitude. Consequently, if we plot a
curve of d(V²/2g)/dt as obtained from accel-
eration run data vs. V or V²/2g, the speed
for "nominal" best rate of climb is deter-
mined by drawing a horizontal tangent to our
curve.
As is quite apparent, this construction
locates the speed for maximum separation
of the power required and available curves
and would actually give the speed for best
rate of climb at the given altitude were it
4:40
not for the fact that in the actual climb the
true flight path speed is changing along with
the altitude.
Since the time to climb to an energy height
is given by
t =
s
dhe
W
in comparison to the conventional time to
climb formula
s
dh
RC
it is apparent that what we want is the maxi-
mum value of w at any given he.
Therefore, if we could run tests at con-
stant energy height, a plot of w vs. V or
v²/2g obtained at constant he could be used
to obtain the speed for best energy climb.
In this case, we would merely have to draw
a horizontal tangent again to the new curve
of w vs. V²/2g to determine the speed. Ac-
tually a curve of w vs. V²/2g at constant
he can be constructed from acceleration run
data; it is simpler, however, to make use of
the procedures of Chapter 7.
Since w = f(h, v²/2g), we could find the
maximum of w at any given fixed value of
energy height by plotting w vs. h for fixed
he and by then locating the height for the
horizontal tangent.
The conditions for maximizing w at any
given value of he are that
aw = O at constant he
ah
v²
2g
= O at constant he⚫
Either operation yields the same result,
namely
35 / 53
11
a
aw
2
V²
2g
The foregoing equation defines any line in
a plane perpendicular to the h, V²/2g plane
and including the line h = V²/2g (presuming
an orthogonal set of coordinates consisting of
w, h and V²/2g).
Since we are seeking to maximize w at
constant he, we also know that the other con-
ditions required are that aw/a (V/2g) = 0 at
constant he and aw/ah = 0 at constant he.
These latter two conditions amount to the
same thing, since in the first case we are
considering the projection of a curve of w at
constant he in the w, V²/2g plane and, in the
second, the projection of this same curve on
the w,
h plane.
Thus, regardless of how we project the
curve of w at constant he, at the maximum
point, we have a tangent parallel to planes
of constant w. If we consider the surface
defined by w = f(V²/2g, h), the planes of
he = constant have the equations h+ (V²/2g)
= constant and, therefore, the tangents to the
curves of w at constant he lie in planes of
constant w along lines of h + (V²/2g) con-
=
stant.
Now, therefore, if we plot our acceleration
run data as indicated in section 4:11, we shall
come up with curves of constant w projected
in the h, v²/2g plane and the tangents, which
maximize w at constant he, will appear as
lines having slopes of 135° as shown in Fig.
7:8 of Chapter 7.
This completes our discussion of energy
climb theory for this chapter. For a more
complete analysis, the reader is referred to
Chapter 7.
4:41
4:13 DESCENT PERFORMANCE OF
TURBOJET AIRCRAFT
The descent data reduction process theory
for turbojets is exactly the same as that for
the climb data; however, at high rates of
descent, consideration can be given to omit-
ting small corrections such as the thrust
correction made for temperature variation
from standard and fuel consumption weight
corrections.
Tactical employment of a particular air-
craft usually defines the type of descent nec-
essary. Such controlling items as engine
RPM, Mach number, airspeed, fuel pressure,
and configuration have to be scheduled with
altitude before proper evaluation can take
place. Thus, determination of a descent
schedule is much more difficult (and im-
portant) than the reduction of descent data
once they are taken.
In general, the maximum range flight for
jet aircraft includes cruise or climb-cruise
at high altitude and a high speed descent at
minimum power and/or minimum fuel con-
sumption.
4:14 MEASUREMENT AND CORRECTION
OF TURBOJET TAKE-OFF AND
LANDING CHARACTERISTICS
Take-off and landing performance of tur-
bojet aircraft is governed by the same fac-
tors which govern this performance of pro-
peller-driven aircraft.
The equations used by the NATC to re-
duce take-off data to standard are developed
in Chapter 6 (section 6:11) and need not be
given here. Rather, at this point we shall
briefly review a few considerations peculiar
to jet-propelled aircraft.
When dealing with reciprocating-engine
aircraft, accurate estimation of thrust is
quite difficult; however, when thrust meas-
uring equipment is available for jets, it is
a simple matter to determine the test values
of thrust developed. Thus, thrust corrections
•
can generally be made quite accurately when
we deal with jet aircraft. Moreover, if we
do not have thrust measuring apparatus
aboard the airplane, the increment between
measured static thrust and standard static
thrust may be used for correction purposes
without generally introducing serious errors.
Since the power developed by a jet air-
plane depends on its speed, it is sometimes
inadvisable to take off at speeds close to the
stall for the reason that insufficient power
may be available for climb out immediately
after the aircraft has moved out of ground
effect. Several serious accidents have oc-
curred due to take-offs at low airspeeds with
marginally powered aircraft. The optimum
take-off speed at a given gross weight should
be determined by a series of trial runs start-
ing at a speed considerably greater than the
stall speed.
In correcting data for take-offs over ob-
stacles, it is of help to have a knowledge of
the airplane's lift-drag relation. As ex-
plained earlier in this chapter, provided
thrust measuring equipment is available,
this relation may be obtained from level
flight tests of jet aircraft.
The landing run of turbojet aircraft is
generally lengthened by the effects of residual
thrust produced by the engines at idle RPM.
Since this residual thrust is frequently ap-
preciable, the use of such devices as jet de-
flectors, drogue chutes and the like must be
resorted to when minimum landing distances
are sought. Generally, the braking systems
of modern aircraft are not designed to dis-
sipate the heat produced by continued full
use of the brakes during the landing run and
this factor must be considered when tests
for minimum landing run distance are con-
ducted.
For further information on the take-off
and landing tests, the reader is referred to
sections 6:11 and 6:12 of Chapter 6 and to
the detailed analysis of Chapter 8.
4:42 -
4:15 TURNING PERFORMANCE OF
TURBOJET AIRCRAFT
The turning performance of any airplane
is limited aerodynamically by several fac-
tors, namely, the thrust or power available,
the drag characteristics, the maximum lift
coefficient available and the controllability.
In this discussion we shall limit our thinking
principally to the effects of these factors on
level (constant altitude) turns. We shall
consider first the combination of thrust and
drag characteristics which collectively pro-
vide what is called the thrust limitation.
and for constant wing area we have
D = 8M² f(M,a, Ni√ē)
Sm² f¡(M, CL, FN/8)
4:69*
Similarly, the quantity nW, being the sum
of the thrust component normal to the flight
path and the lift, may be written
nW = 8M² f₂ (a,M) +8f3 (M, N/√) sin a
(a) Thrust Limitation
Consider an airplane in a level turn pro-
ducing a normal load factor "n" and accel-
erating along the flight path with an accel-
eration "a". If we sum the flight path forces,
we have
ola
and from Eq. 4:70
f4(nW/8,M,N/√@).
4:70**
4:71
Substituting Eq. 4:71 in Eq. 4:69, we have
FN cos a
W
D
D = pM² f5 (M, nW/d, N/√ē)
W
4:67
4:72
where FN
=
net thrust
also
FN cos a = 8f6(M,nW/8,N/√).
Using Eqs. 4:72 and 4:73 in Eq. 4:67
8
== f (M, nw/8, N/√8)
g
W
4:73
a = angle of attack of the thrust line
D = airplane drag
W gross weight
If we wish to consider the effects of en-
gine spillage flow on airplane drag, then in
general,
f(M, a N/√8)
CD=
f(M, CL, NAB)
f(M, CL, FN/8)
4:68
4:43
& f5 (M, nw/8, N/√ē),
4:74
* Where the terms N/√ or FN/8 account for
the effects of entrance airflow on drag.
**The last term being the component of
thrust normal to the flight path.
hence on dividing by n and placing 8/nW in-
side the function
℗(nW/8, M, N/√ē).
ng
4:75
This equation, of course, applies when-
ever there is an acceleration along the flight
path (in a turn or otherwise). Thus, if we
consider a steady climb (n = 1), we have
(for small climb angles)
y = = = (W/8, M, N/√Ƒ).
g
4:76
In developing the foregoing relations, the
following assumptions, which are generally
accepted in estimating jet turning perform-
ance, have been made, namely:
(a) Reynolds number effects are negli-
gible both for the engine and airplane.
(b) Effects of absolute pressure on the
engine are negligible (i.e., the effects of ab-
solute pressure on combustion are disre-
garded).
(c) The drag coefficient in turning flight
is identical with that obtained in level unac-
celerated flight at the same angle of attack
and Mach number (i.e., there are no effects
due to varying load distribution, local ve-
locity variations and the like).
In cases where these assumptions hold,
we have developed a simple relation for
turning performance. In particular, for the
case of the steady turn a = 0, we have
0 = $(nW/8, M, N/√ē )
or
8
f7 (M, N/√8).
W
From Eq. 4:77 we see that at constant
W, M and N/√ (or FÑ/8 as a replacement
for N/√) the normal load factor n is a di-
rect function of the pressure ratio 8; thus,
steady level turning performance at one al-
titude may be simply related to the perform-
ance at any other altitude. For instance, if
we consider that steady level flight repre-
sents a one g turn (infinite radius), then a
plot of maximum level flight Mach number
vs. altitude or pressure ratio may be used
to determine the maximum turning accel-
eration in level flight at various altitudes
as shown in Fig. 4:23.
Tests at the A. and A.E.E. have shown
that the generalization of data predicted by
the equations is not always obtained, presum-
ably due either to the effects of Reynolds
number or of absolute pressure on the air-
plane and engine performance. Nonetheless,
because of the great effort involved in mak-
ing complete turning performance tests di-
rectly at all altitudes, it is considered well
worthwhile to use the process described in
an attempt to generalize turning data.
As previously demonstrated in this chap-
ter, a function f(M, N/√) is equivalent to
f(M, FN/8); consequently Eq. 4:75 may be
written
a
nW
8
M,
FN
8
ng
4:78
The use of thrust rather than N, may well
avoid the difficulties mentioned above, since
it is hard to imagine large and sudden varia-
tions of aircraft drag being produced by Rey-
nolds number effects and Eq. 4:78 is correct
excepting for the omission of these effects.
(b) Lift Limitation
If we analyze the CLmax or lift boundary
of an airplane, arguments similar to those
presented above lead to the relations
nLW
♡
= f(M, N/√8)
4:77
4:79
4:44
or
ոլ W
8
f(M, FN/8)
4:79
according to whether we use N/√ or FN/8
to define the engine operating conditions.
These equations allow for the upward com-
ponent of the jet thrust and again permit a
generalization of data obtained under a vari-
ety of altitude conditions. Clearly, these
equations should be equally applicable to
buffet boundaries as well as the true CL, or
lift boundary.
max
However, generalization cannot be ex-
pected in cases where the lift limitation is
1
Maximum Level Flight
Mach Number (n = 1)
8,
(Test Curve)
+th
+18
Mmax for n=2
Obtained from
n=1 Data
28, = 82
M
Mmax at 8, for n=l
Mmax at 8₂ for n=2
ni
8
n2
82
Fig. 4:23 For Contant M and N/√ ;
4:45
set, say by instability or lack of control. In
such cases, additional factors including nat-
urally the c.g. position would need to be in-
troduced.
(c) General Comments
It may be noticed that we have presented
the foregoing work in terms of the load fac-
tor “n” rather than in terms of actual turn-
ing radius. This was done because the
simplest relations are obtained in this way.
It can be shown, however, that the steady
level flight radius of turn is given by
are of course interchangeable and it is not
necessary to go into further detail here.
Correction methods for turning perform-
ance can readily be developed on the basis
of the foregoing relationships. One simple
procedure is to conduct level acceleration
tests at measured thrust values and then
correct the measured acceleration for the
increment of thrust between the measured
and standard thrust using the simplest of
expressions
Aa = g
•
ΔΕ
W
8
f(M, W/d, N/√ē) or f(M,W/8,FN/8)
4:80
which follows at once from the fact that in a
steady level turn
(constant) TM2
R =
9 n²
-
√n² -1.
4:16 CONCLUDING REMARKS
In this chapter the general base of the
analysis of jet aircraft performance has been
considered and the undamentals of the data
reduction procedures required for these air-
craft have been presented.
The methods described are essentially
those in use at the NATC at the time of this
writing; however, the procedures for the re-
duction of energy climb data have been pre-
sented in a modified form to take into account
a simplified data analysis procedure de-
scribed by Mr. Davy in Chapter 7.
where T = absolute temperature.
Thus, using Eq. 4:77 for n, Eq. 4:80 is
readily obtained.
It is similarly possible to obtain expres-
sions for the climbing or descending turns
which involve the climb or descent angle y.
Flight path acceleration and flight path angle
Special topics, such as carrier suitability
testing and the like, have not been considered;
however, it is hoped to cover these items in
future volumes.
REFERENCES
1. Hesse, W. J., "A Simple Gross Thrust Meter Installation Suitable for Indicating Turbo-
jet Engine Gross Thrust in Flight," TPT, TR 2-52, NATC, Patuxent River, Maryland.
2. Hesse, W. J., "Determination of Level Flight Thrust Available Curves of Turbojet
Powered Aircraft," TPT, TR 1-53, NATC, Patuxent River, Maryland.
3. Dommasch, D. O., Sherby, S. S.,and Connolly, T. F., "Airplane Aerodynamics," Pitman,
1951.
4. Dommasch, D. O., (Editor), "Flight Test Manual, Part I," Revised Edition, Preliminary
Copy, Patuxent NATC, August, 1953.
4:46
AGARD FLIGHT TEST MANUAL
VOLUME I, CHAPTER 5
PERFORMANCE REDUCTION METHODS
FOR TURBO-PROPELLER AIRCRAFT
By
Kenneth J. Lush
AFFTC, United States Air Force
and
John K. Moakes
Aeroplane & Armament Experimental Establishment
United Kingdom
VOLUME I, CHAPTER 5
CHAPTER CONTENTS
Page
SUMMARY
FOREWORD
5:1
INTRODUCTION
5:1
5:2
DIFFERENTIAL REDUCTION METHODS
5:1
5:3
THE BASIC PERFORMANCE EQUATIONS
5:2
Level Flight
5:5
5:4
FUEL FLOW IN LEVEL FLIGHT
5:7
5:5
EQUATIONS FOR CLIMB REDUCTION
5:9
(a) Reduction Process
5:10
(b) Pressure Height and Pressure Rate of Climb
5:10
(c) Longitudinal Acceleration
5:11
5:6
DATA REQUIRED IN ADDITION TO THE BASIC PERFORMANCE
QUANTITIES
5:11
5:7
REQUIRED ACCURACY OF THE COEFFICIENTS
5:13
5:8
POSSIBLE SIMPLIFICATIONS OF THE METHOD
5:14
5:9
REDUCTION OF LEVEL SPEEDS AND FUEL CONSUMPTIONS
BY PERFORMANCE ANALYSIS
5:14
(a) Introduction
5:14
(b) The Approach Considered
5:14
(c) Standard Power
5:15
5:10
CONCLUDING REMARKS
5:16
REFERENCES
5:16
TABLE I
5:17
TABLE II
5:18
APPENDIX - USE OF LOGARITHMIC DIFFERENTIALS IN
PERFORMANCE REDUCTION
$
TERMINOLOGY
b
Wing Span
Cp
Propeller Power Coefficient = P/pN3d5
CDe
Effective Parasite Drag Coefficient
D
Drag
d
Propeller Diameter
e
NACA Efficiency Factor
Vemp
AB3
+
4
(the
f(Ky) =
Ky4 + 3
Ky
271
(Kv3 + 3/KV) 2.23./W(LD)max
Dmin
2./3 Kv2
D
KyA + 3
f1(Ky)
6
f2) =
Ky
12(ky) = 1/2 (3 - par ) - Kvents
-
(3
FKV JKY
Ky+ 3
h
Rate of Increase of Height
hd
DV/W = Rate of Climb Equivalent of Drag Power
hp
Pressure Height
We
'F
Fuel Flow
J
Propeller Advance Ratio V/Nd
Ky
Veremp
C
L
Lift
N
Propeller Rate of Rotation (RPM)
P
Shaft Power
XVe
r
=
Pvto+
XVe
S
Gross Wing Area
T
Ambient Air Temperature OK
TA
Jet Pipe Temperature OK
V
True Airspeed
TERMINOLOGY
(Continued)
Ve
Equivalent Airspeed = V po
Vemp
Equivalent Airspeed for Minimum Drag Power
W
Aircraft Weight
X
Net Exhaust Thrust
Cp
n
δη
аСр
Jan
B
JIT
7 OJ
Propeller Efficiency (Propulsive)
s 6
Density Ratio (Relative Density)
pipo
P
Air Density
Air Density at Sea Level Standard Temperature and Pressure
Po
Air Pressure
р
Suffix S
Standard Value
T
Test Value
SUMMARY
behavior. Thus, the methods are flexible at the
expense of simplicity in application.
Reduction methods for level speed, climb
and fuel consumption tests are presented,
which are, in the absence of compressibility
effects, suitable for use with turbo-propeller
aircraft at the present stage. The methods re-
quire individual treatment of the particular
engine installation, inasmuch as it is not prac-
ticable at present to generalize about engine
It is anticipated that it will eventually
prove possible to simplify the methods when
experience has been gained on these engines,
and the design of their control systems has
been stabilized.
EDITOR'S FOREWORD
accomplish fairly complex interacting con-
trol functions.
As far as airframe characteristics are
concerned, turboprop airplanes are similar
to other propeller-driven airplanes. More-
over, at the present time, turboprops are
generally designed to fly at speeds lower
than drag divergence so that the assumption
of a lift-drag relationship independent of
Mach number may be made in analyzing their
performance.
As noted in the body of this chapter, un-
til engine design has been more or less
stabilized, it is not safe to assume that any
of the existing variables may be neglected
and, therefore, the analysis presented here
is rather complex since all variables of pos-
sible importance are considered.
On the other hand, the testing of turbo-
prop aircraft is complicated by the fact that
only very limited experience has been gained
in testing these machines and, moreover, by
the fact that power is produced both by a
propeller and by direct jet thrust. Thus, the
determination of both power required and
power available is not as simple a matter
as with piston engine aircraft. Similarly,
it is not easily possible to determine stand-
ard fuel consumption characteristics because
of the large number of variables which must
theoretically be accounted for in data anal-
ysis.
Unlike other chapters of this volume,
which deal with fairly well developed pro-
cedures, this chapter describes what must
be considered (as of this time) to be experi-
mental techniques. This fact should be taken
into account in test planning and it should
not be anticipated that the turboprop test
program can be conducted in a routine man-
ner with much hope of collecting standardized
data,
The possibility of obtaining really repeat-
able data from flight tests of turboprop air-
planes depends in part on the design of the
engine and propeller controls and governing
systems. Simple mechani
Simple mechanical linkages with
easily predictable characteristics tend to
provide better data than servo systems which
The testing of airplanes with mixed-type
power plant installations (combination of
reciprocating engine with turbojets) presents
problems quite similar to those encountered
in the testing of turboprop airplanes. Thus,
although the testing of mixed power plant
aircraft is not considered in detail in this
volume, it is noted that the differential an-
alysis techniques presented in this chapter
are adaptable to this work.
5:1 INTRODUCTION
are convenient where level speed perform-
ance data are required over a large range
of the variables.
It has been shown that for many turbo-
propeller installations, the application of
experimental methods of performance reduc-
tion, such as those used for turbojet aircraft
(which require no advance numerical data)
will be either impracticable or impossible.
Alternative analytical methods which are
available are:
(1) "Differential methods”, based on
linearized relationships between the varia-
bles, suitable for small corrections to ob-
served data. Here the performance items
per se, viz., level speed or climb points,
are corrected for deviations from standard
of such parameters as aircraft weight or
air temperature.
The common difficulty in the application
of either of these methods to turbo-propeller
aircraft is the lack of knowledge about the
response of the engines to changes in air
temperature and forward speed and the
associated difficulty in the prediction of the
power which would be available under stan-
dard conditions. This will only be solved
when considerable experience is gained with
these engines and their performance charac-
teristics can be generalized. Until then each
installation must be considered on the basis
of its own characteristics, which in most
cases must be established prior to or during
the measurement of the primary aircraft
performance quantities.
The method requires advance numerical
data regarding the behavior of the airframe
as well as the propulsive unit, perferably
some fund of generalized data in a conven-
ient form. The information need not be
very precise inasmuch as it is only used
to estimate relatively small corrections,
In the paragraphs which follow, a differ-
ential reduction process is developed which
is sufficiently flexible to be adaptable to
any turbo-propeller installation, provided
that compressibility effects on airframe drag
and propeller efficiency are not present.
The methods are convenient for the cor-
rection of isolated performance points where
compressibility effects are not present, and
at the present level of performance are
thus quite appropriate to turbo-propeller
aircraft.
The developments are presented in detail
to illustrate the principles involved (which
have been current practice in the U.K, for
the past ten years for piston-engined per-
formance reduction) and to indicate the
most likely direction in which simplification,
based upon generalization of engine charac-
teristics, will take place.
(2) “Performance analyses'', where
observed or estimated jet thrusts and shaft
powers combined with estimated propeller
efficiencies give estimated power curves.
From prior information about the perform-
ance of the propulsive unit, and its response
to change in air temperature, estimates of
thrust power which would be available under
standard conditions then lead to standard
performance.
Also given is a brief outline of an approach
based on the performance analysis of method
(2), from which could be developed a routine
similar to that proposed in this volume for
piston engine level speed reduction.
5:2 DIFFERENTIAL PERFORMANCE
REDUCTION METHODS
Prior information about airframe drag
is not required inasmuch as drag data are
obtained during the analysis. Such methods
In developing the present
differential
methods, it was decided that because devel-
opment of turbo-propeller engines and their
5:1
control systems is in its early stages, it
was not possible to provide the degree of
generalization concerning engine behavior
which is at present practiced in similar
reduction methods applied to reciprocating
engines (Refs. 1 and 2) and it would be
preferable to provide for the insertion of
more engine parameters, and if necessary
propeller parameters particular to the type
under test.
on the power and fuel flow of turbo-propeller
engines. Inclusion of this effect in the
reduction process presents difficulty; it is,
therefore, only included in the correction of
level speeds at particular ratings, e.g., at
maximum continuous power. The correction
process for fuel consumption tests at lower
powers avoids changes in ram by making
corrections to fuel flow at constant equiva-
lent airspeed.
It is assumed throughout that compressi-
bility effects on airframe drag and propeller
efficiency may be ignored.
To a certain extent the test program can
be fitted to the reduction process.
If the
tests are carried out at the required air
pressure, corrections need only be made for
the effects of the differences in air temper-
ature, engine speed and aircraft weight from
their standard values.
5:3 THE BASIC PERFORMANCE
EQUATIONS
If we use consistent units and neglect
longitudinal acceleration, we have from the
law of energy conservation
In determining the performance with the
engines operating at some limiting output,
e.g. “maximum level speed” or “climb
power", some difficulty may arise due to a
change of these operative limitations with
a change in air temperature. For example,
at low air temperatures the engine may
operate at maximum possible engine speed
but with the jet temperature below the maxi-
mum and vice versa at high air temperatures.
OPVO + XVe - Dve + Whio
:
5:1
Editor's insert: If we suppose that the
lift drag polar can be approximated by the
relation
CL
CO :
.
Сое
+
TEAR
To handle this difficulty, only the “external
features of the engine performance (i.e.,
shaft power, thrust and engine (or propeller)
speed) are inserted in the performance equa-
tions, their variations with changes in air
temperature being considered separately
from their effect on performance. In the
case of fuel consumption tests, the engine
will be operating frequently within the limi-
tations and this difficulty will not arise.
where Coe
Effective parasite drag coef-
ficient
e : Span efficiency factor
AR : Aspect ratio,
then we have
The effects of weight changes are consid-
ered in deriving the equations for level
flight and fuel flow, but in the climb the
rate of loss of weight with increase of
height is insensitive to air temperature.
Therefore, no correction for weight has
been applied. If these are required, however,
the methods of Ref. 1 or 2 may be adopted.
DECOS:9CDES
cias
+
TAR
Changes in ram may have a marked effect
5:2
-
and, since q= Ź pove (see Chapter 1) and
ci = W/?S? , we have
q
2
Po
In terms of Vemp the equation for Dve
may be written
?
BW
D - Ave +
Ve
BVemp
:
emp
DVE - AV .
+
W?.
vemo
Ve
Vemp
5:2
where A : CDEP.5/2
or
B :
2/P. STEAR
BW?
1
Dve
.
AKÝ.vemp +
+
so that
KV Vemp
BW
Dve : AVO+
Ve
+ BV
Kuvamo [axi. vemp + Bw')
]
5:3
We next introduce a dimensionless veloc-
ity, Ky, defined as
but
BW?
ку
2
Ve
Vemp
Vemp
:
ЗА
5:4
so that, after simplification
where Vemp is the equivalent airspeed cor-
responding to minimum airplane power re-
quired. Vemp is readily found in terms of
the parameters A and B from the follow-
ing considerations:
BW?
Dve
Kỳ +3
ку
3 Vemp
The power required is
DV :
(avo + ) v. avě to +
BW2
+
Ve
or
8
BW? To
Ve
BW'
"2
Kỷ +3
For minimum power
Dve
W
1372
3 Vemp
ку
5:6
(DV)
a Ve
1:05
= 0 = 3AV2 vo
BW? Jo
ve
Now substituting for Vemp from Eq. 5:5,
we obtain
whence
BW
Dve
AB'
Ky 3
(209)* ( ** + 3).
vieme
WVz
27
3A
Ку
5:5
5:7
5:3
and Eq. 5:1 may be written
Another useful alternate form of Eq. 5:6
or 5:7 is obtained by writing
Pro + x Ve = W3/2 f(KV) + who,
BW1/2
/
BWV2 Vemp
3 vemp
5:10
3 Vemp
2
8
BW2
where D = drag - Ave +
va
i
so that from Eq. 5:5
2
BWV2
CDEPOS
A :
2
; B
A12B1/2 Veme
3 W
:
Po STAR
3 Vemp
P: shaft power
5:8
X : exhaust net thrust
Now for maximum C/Co we have
ń
: rate of climb
Kv: Ve /emp
d
0
O CL
.
CL
CL
CDe+
Te AR
(
Vemp
equivalent air speed for minimum
power required
(22"). (+3)
:
whence at maximum C/Co
AB
f(KV) = 4
27
KV
(Kỳ+3
23 W (L/D)max ку
Vemp
Ľ
CDe
.
TTAR
and
If the air pressure is maintained constant
during the reduction process, we also have
CDeTTAR
(
CL
co Imax
:
2 CDe
do
dT
.
T
글
​e
1 NEAR
2
JCDe
2 AV281/2
Consequently, Eq. 5:6 becomes
5:11
where T is the ambient air temperature in
°K or °R.
Kº +39
Dve
-
Vemp
CL
23 W
W 3/2
Menomax
KV
If there are no compressibility effects on
propeller efficiency, this efficiency is a
function of CP and J only; hence, using
5:9
5:4
logarithmic differentials*
where N is the propeller rotational speed.
du
don
η
.
CP on dCp
ПдСР СР
.
+
J.on.
η δυ
zlo
Also, as Vemp is, in the absence of
compressibility effects on drag, proportional
to VW (see Eq. 5:5), from the definition of
Ky, we have
dP
30N
Cp on
тәСР
{
dT
+
T
}
P
N
dky
| dw
-
dve
Ve
+g
2 W
j.on
nou
{
dve I dTdN)
+
Ve
2 T N
ку
5:12
5:13
Level Flight
An expression for the change in equivalent airspeed which will result from changes in
weight, air temperature, shaft power and thrust is obtained from Eq. 5:10 with h = 0; thus,
dn
dP
| DT
ηΡσ
novo +X Ve
+
액
​XVe
+
nPro +xve
{
dx
dve
+
х
Ve
}
3 0W Ky Of(ky) dky
+
f(ку дку KV
η
p 2 T
2 W
1
1
1
Using Eq. 5:1, this may be written more neatly
1
1
3 dW
(-1){432 * am
)
}
ку
dP
+
P
I OT
2 T
-
x sax
X
d Ve
+
+
DUX
of(kv) dk v
ку
Ve
2
W
f(KV)
дку
From Eq. 5:13 the right hand side becomes
|
W {
Ку
f(KV)
afiky)
ку
(} +
+
aky
f(кудку
)
!
3
of(KV) dve
2
Ve
5:14
Substituting for the propeller efficiency from Eq. 5:12 we have (denoting Cp/n times
ondo Cp and J/n times on/DJ byaand Brespectively)
X SOP
:ll
X 10X
+
+
D х
6Ne)
| -
Ve
Ve
{la
ce + a)+ 4(a+B/2-)
°F (2+8/2- ) - (32+B+B over
뿌
​IN (3a+81+8 om du er en av
!
OW
d
1
2
Kv. Of(KV)
f(ky) aky
3
.
over ove story of (y)
Ку
fKV)
W
Ve
дку
5:15
* The use of logarithmic differentials is discussed in the appendix to this chapter.
5:5
In the above equation, dP and dx are
the total changes in power and thrust, inclu-
ding ram effects. Application of the equation
is simplified if the ram effects are separated
from the effects of changes in air temperature
or engine speed.
Assuming as before that
ci
CD : CDe +
TEAR
8
we have that
:
BW2
D = Ave +
ve
and for minimum drag
Let us rewrite Eq. 5:15 in the approximate
form for finite changes and write AP & AX
for the changes in P and X which would
result at constant equivalent airspeed Ve
from the excesses AT and AN of the test
air temperature and engine speed over stand-
ard. Following Ref. 2, we take AN, AT, and
AW as "errors" in the test conditions, that
is, the amounts by which the test values of
N, T and W exceed the standard or desired
values Ns, Ts and Ws and Ave as the
corresponding correction to the observed
equivalent airspeed Ve. We replace dP/P
and dX/X by
이
​aD
ду
: 0:2 Ave - 2
BW?
vel
hence vemd
Equivalent speed for mini-
mum drag = Bw / A and Dmin : 242 B" W.
Also from Eq. 5:6
dP
AP
-
Ve
OP Ave
pove Ve
P
Kit
+ 3
D:
BW?
3 Vemp Ve
BW KV+3 ]
KO
5:16
ку
3vémp
dx
AX
-
Ve ox
ax Ave
Δve
X dve ve
Х
and using Eq. 5:5
5:17
BV2AVZW
Kỳ + 3
D
We will also write
3
K?
V
fi (ky) :
Dmin
D
2./3K
(Kỳ +3)
Consequently,
Dmin
2/3KÝ
f2(ky) :
ਨੂੰ (3 -
ку
f(KV)
of(KV)
OKV
2
D
Kỳ + 3
6
(Kỳ 13)
)
To obtain fz(Kv), we have from the defini-
tion of f(Ky) that
where Dmin is the minimum value of D
(at standard W).
afiky
return
ABVa
3k -3
ску
27
Kv
Editor's insert: These relations are ob-
tained as follows:
5:6
and
ку
f(KV)
large
-)
27
AB?
**
+ 3
Note that X/D can also be written as
XV/(XV+ 7 P) and as fi(Kv) (X/Dmin). The
latter forms may sometimes be the more
convenient. Also, in the expression AP/P,
P may be the observed or standard power
whichever is the more convenient.
hence
Kv. Of(kw)
f(ку оку
3 Kº-3
Kỳ + 3
5:4 FUEL FLOW IN LEVEL FLIGHT
and
6
falkvi
2
3K-3
3-
KV + 3
kỳ +3
It is convenient here to distinguish between
"maximum level speed" fuel flows, such as
those at the maximum steady level speed
attainable without exceeding the engine limi-
tations for continuous operation, and fuel
flows under conditions not requiring operation
at maximum continuous power. Fuel flows
under the latter set of conditions must be
precisely determined in order that the opti-
mum conditions for cruising flight may be
defined at any altitude other than the cruising
ceiling, and a suitable reduction method is
developed under (a) below.
Then we have, after substituting Ave for
dve
*
AT ΔΝ
+ AT
Ts
Ave
: AP
AV
용
​- AN
-
AND
Ve
+
AX
AW
fi(ky) - -f2(kv)
Dmin
WS
5:18
where
Op
Pave
Ve ox
+
X ove
+2f2(KV) - 3
Generally, it is not necessary to determine
"maximum level speed” fuel flows with the
same degree of precision as those at part
power; moreover, the reduction process at
full throttle is complicated by ram effects.
The problem is considered in general terms
in (b) below but no correction is derived
because the main part of the correction is
likely to be purely an engine matter. The
remaining part of the correction involves
the airframe, but as the correction is likely
to be small even in extreme cases, it is
concluded that in most applications it can
be legitimately neglected. A method of
correction is, however, indicated based on
certain assumptions about
the engine
performance.
Av = (-1){B+(1+0) We coup
a } + $ *
x
:)
(-7)+1+a)
-5
AT = (- )la +B12 – 12)
AN(-)13a+B)
Ap :
(a)
:
+B)
:
5:19
(a) Fuel Flows at Less Than Maximum
Power
* By definition
AP
TOP
POT
N OP AN
+7 P'ON' N
P- 4T+
ox. At ox. AN
It is proposed to correct the fuel flows
for the direct effect of the departure of the
test temperature and weight from standard
at constant equivalent air speed and pressure
ax
AX :
.
ΔΤ
ON
X
ΔΝ
.
5:7
altitude.
becomes
From Eq. 5:18, at constant Ve and p
we obtain
AP
Ap
P
AT
-AT
TS
AN
NS
312
ㄹ
​AN
AX
+
Dmin
fiſkv)
AN
Дх
AP AT
Ар + AT
р
TS
쁨
​씀
​AN
filky)
NS
Dmin
Δw
+
wsz/KV) = 0.
-
AW
.f2(ky)
WS
= 0
5:21
Consider now how changes in P, X and N
are associated with changes in fuel flow.
We may write:
5:20
&
P: (ve, p, WF, T)
x = 2(Verp, WF, T)
where the terms may be regarded as either
all “corrections'' or all "errors''.
If we
now regard AP and AX as the
"corrections to the engine shaft power and
thrust required to give the observed equiva-
lent air speed at standard temperature, and
AN the associated correction (if any) to the
engine speed, and also regard AT and AW as
"errors" in the test conditions, Eq. 5:20
N: 03 (Ve, P, WF, T)
where $3 is determined by the engine control
system and will in certain cases be a con-
stant.
Then at constant Ve and p
AP
WE OP
AWE
TOP AT
P T TS
.
POWF
WE
ax AWF
AX : WF
OWF WE
WF
ax AT
T:
OT TS
Any change of N with temperature at constant
Ve, P and WF is small and has been neglec-
>
ted.
AWF
AN
WE ON
WF
δN
N N
OWF
.
WF
5:22
9
where AP, AX and AN are "corrections" and AT is an "error" as in Eq. 5:21; AWF is the
correction to be applied to the observed fuel flows.
Substituting in Eq. 5:21 we obtain
AWE
WE OP
WE
ON
AN:
NOWF
WF. ox
WF
p OWF
+ fi (ky):
Dmin OWE
{ap.
{
AT
}
AW
fs (KV)
WS
To op
Ta ox
AT +AP
+ fi(KV).
TS
Рота
Dmin ata
OTO
AWE
AW
i.e., RF
f2 (ky)
WE
TS WS
AT
RT
씀
​5:23
5:8
where
WE
WE OP
(1 + a).
ax
(-)" +
RE:
WE ON
-(3a+B).
NOWF
-
POWF
+ fi (ky):
Dmin OWE
TOP
T
O x
RT:
(-[(a+p12-3)(1+21
0)
1 + a)
+ f(ky):
Р
OT
Dmin OT
5:24
The associated correction to engine speed
(if any) follows from Eq. 5:22,
It will often be possible to determine
WF/P times do/a WF, WF/N times oN/OWF,
WF times ox/ oWF experimentally.
(b) Fuel Flow at Maximum Speed
The reader will have noted that the method
of reduction of fuel flows at powers below
maximum continuous avoids the influence of
ram on power, thrust and fuel flow by making
corrections at constant equivalent airspeed,
the fuel flow correction being deduced from
the change in engine shaft power and thrust
necessary to maintain the test equivalent
airspeed under conditions of standard weight
and temperature,
(2) A correction for the indirect
effect on fuel flow of the change in equivalent
airspeed associated mainly with the power
and thrust alterations caused by (1). This
of course involves the airframe but inves-
tigation suggests that the correction is un-
likely (even in extreme cases) to exceed
2%. This latter correction will be needed
for under optimum range conditions; i.e.,
flying at the cruising ceiling appropriate to
maximum continuous power. However, for
"maximum level speed" fuel flows at other
heights it would seem legitimate to ignore
this part of the correction for the errors
introduced by its omission will usually be
within the limits of engine to engine charac-
teristic variations.
It may even be possible, in view of the
reduced accuracy requirements for rated
fuel consumptions, to concede that the effects
of realistic changes in ram and air tempera-
ture on the power-fuel flow relationship are
negligible. If so, the specific range figures
can be obtained by correcting the level
speeds to standard conditions by means of
Eq. 5:18 and adjusting the fuel flows in
proportion to the AP derived for the reduc-
rion process.
When dealing with “maximum level speed"
fuel flows, where the engine is operating
under conditions subject to an arbitrary
limitation (which may itself change with air
temperature), it is no longer possible to
correct at constant equivalent airspeed be-
cause the required engine shaft power and
jet thrust are now fixed. The correction
to the fuel flow, therefore, consists of two
parts:
(1) A correction at constant equiva-
lent airspeed for the direct effect on fuel
flow of the departure of the test temperature
from standard with the engines operating at
a constant limitation. This is purely an
engine matter.
5:5 EQUATIONS FOR CLIMB REDUCTION
It is proposed to correct the climb per-
formance for changes in air temperature,
shaft power and thrust at constant air pres-
Bure and equivalent airspeed, Weight correc-
tions to rate of climb are not normally
required; if they are, the methods of Ref. 1
or 2 may be used.
5:9
(a) Reduction Process
From Eq. 5:1, we may deduce that at constant Ve
.
ηΡ
n Pro + xve
dP
+
P
| dTV Xve dx
+
2 T no vo +XVe X
w dlho)
DVE + Who
din o)
D
Ve thivo
W
.
5:25
If we now write
XVe : rmpro +xve)
and hd
• DV/W=
rate of climb equivalent to the drag power, we have
1
dT
(1-7).
in
dP
+
η
Р
T)
dx
tr
х
dh
hath
1 | DT h
2 T hd the
2
T
5:26
-
where we note that 1-r = TP Tomp Vo + X Ve and that at constant pressure altitude Eq. 5:11
gives dolo : -dT/T.
Substituting for dn/n from Eq. 5:12 we have
rdx
dI
h
«-1{ "1+a) + $? (a+B12 - ) - ( 3a+8} +
) +
dh
hid thi
-
х
2 т.
nath
5:27
and the approximate equation for small
finite difference is
where Ah is the correction to be applied
to the observed rate of climb and AN, AP,
AT and AX are "errors" in the test condi-
tions.
Δx
Δή
rigth
AN
BN
NS
ΔΡ
Bp
Р
-
AT
Вт
Ts
х
(b) Pressure Height and Pressure Rate
of Climb
5:28
where BN :
(1-1) (3a+B)
Вр (1-r) (1+0)
BT = (1-1) (a+B/2 - + )
h
1
2
A complication results from the fact that
heights and rates of climbs are commonly
deduced from pressure measurements, the
th
5:29
5:10
following:
2w\V2
pressure gage being calibrated in height
above sea level
sea level in the ICAN or NACA
standard atmospheres. Thus, when the local
temperature at a particular pressure height
differs from standard, the pressure rate of
climb will differ from the true rate of
climb, their ratio being equal to the ratio
of the corresponding air densities.
Vemp - Company "13C0q7d'esi-vo
Po
for the evaluation of Ky and
12
SCDe
Dmin : 2W
The most straight forward way of dealing
with this difficulty is to correct the observed
pressure rate of climb if desired.
(
пь°e
(c) Longitudinal Acceleration
for Eqs. 5:18, 5:19 and 5:24.
In the foregoing, longitudinal accelera-
tions during the climb have been ignored.
This is legitimate for performance reduction
work on turbo-propeller aircraft. If desired,
however, a more rigorous treatment may
be deduced by substituting for h and Ah in
Eqs. 5:1, 5:10 and 5:28, etc. he and Ane
These can usually be estimated with
sufficient accuracy from estimates of CDe
and the assumption of an average value
for e (0.77 to 0.80), but if suitable flight data
are available, these data should be used.
Engine
he = ŕ it
h +
19 dh
.
+ Gento
+ 5
Until further experience is gained, the
engine derivatives listed below should be
measured at each pressure altitude for which
standard performance is required.
ht V
9
5:30
(a) With respect to forward speed,
In this case it is most desirable to
transform the observed pressure rate of
climb into a true rate of climb as a prelim-
inary to the reduction process. Inclusion
of the acceleration term is considered un-
necessary for turbo-propeller aircraft, but
it is routine for turbojet aircraft which
climb at high true airspeeds with the result
that the term V/8 times dv/dh is appreciable.
Ve /p times oP / O Ve , Ve /X times oX/a Ve
are required at constant throttle setting hp
and T for use in Eq. 5:19. These quantities
are most conveniently measured on single-
engined aircraft by conducting accelerated
and decelerated level flight runs. On multi-
engined aircraft they can also be determined
more reliably (if less conveniently) by holding
a constant throttle setting on one engine while
varying the settings on the remaining engine.
5:6 DATA REQUIRED IN ADDITION TO THE
BASIC PERFORMANCE QUANTITIES
(b) With respect to air temperature,
Airframe
The only airframe prerequisites are the
T/P times OP/OT, T/X times ox/aT are
required
5:11
(d) Fuel flow RPM linkage
(1) at constant hp, Ve and We for
evaluating A WF/WF in Éq. 5:23 (powers
below rated')
(2) at constant hp, Ve and throttle
setting for evaluating AP/P, X/X in Eq8.
5:18 and 5:28 for "rated" powers
WE/N times on/ oWe represents the way
in which engine speed and fuel flow are
linked; it is thus peculiar to the type of
engine and will be established during other
measurements without particular investiga-
tion. It is required in the fuel consumption
reduction Eq. 5:24 and in conjunction with
WF/P times oP / oWF, WF/X times oX/OWE
in the establishment of N/P times DP/ON
and N/X times OXON for AP/P and AX/X
N
//
of Eqs. 5:18 and 5:28.
(3) at constant hp, Ve and tailpipe
temperature for powers below rated, for
evaluating AP/P, AX/X in Eq. 5:18 if it
is required to correct the level speeds of
the fuel consumption runs to standard condi-
tions 80 that their consistency with the
"rated" level speeds may be checked (see
(e) below).
(e) General
C
The most critical measurement is (2),
the determination of the rated powers over
a representative range of air temperatures,
During these measurements the effects of
operative limitations should be investigated.
This will reveal whether any change in oper-
ative limitation such as is described in
section 5:2 is present and the air temperature
at which such changes in regime occur.
Any such changes are likely to have a
marked effect on the temperature derivatives
(see section 5:8).
It may often be either impracticable or
impossible to obtain precise control over
those parameters which are intended to be
held constant during the determination of a
particular derivative. Should this occur,
corrections can usually be made quite safely
for small errors in Ve , from foreknowledge
of Ve /P times oP / Ove , particularly if the
latter is small. Similar adjustments may
be made for small errors in setting up other
parameters.
(1) and (3) can be obtained by covering
a range of jet tailpipe temperatures and fuel
flows at constant Ve and hp at each air
temperature and cross plotting as required.
Limited experience has shown that the
functioning of the automatic engine control
system may not be entirely consistent. Fur-
ther, the measurement of jet tailpipe temper-
ature may not be very accurate. These,
and the normal causes of scatter in perform-
ance tests, make it desirable to compare
all the level speed data on a common basis,
to check the consistency of performance at
rated and below rated conditions,
(c) With respect to fuel flow,
WF/P times DP / oWF, WF/X times o X/OWE
are required at constant hp, Ve and T. These
will generally be a by-product from the runs
required at each temperature level to estab-
lish the temperature derivatives of (b), and
thus no special provision is required. They
are needed at powers below "rated" for
application in the fuel consumption reduction
Eq. 5:24,
In a constant speed engine, for example,
where the operative limitation is jet tailpipe
temperature, a convenient basis of compari-
son is to plot aircraft speed against tailpipe
temperature at standard air temperature and
aircraft weight. To achieve this, the differ-
ential method of section 5:3 may be used
for the correction of the level speeds of
the fuel consumption runs of section 5:4 as
well as for the rated conditions for which
it is specifically intended.
5:12
In correcting "non-rated" level speeds
it is convenient to assume that the jet pipe
temperature (T4) remains constant and an
appropriate power derivative at constant T4
is used. This can be deduced from plots of
observed power against T4 at two or more
fixed air temperatures (T) if such data be
available.
sary until sufficient experience is gained to
enable generalization to be made with relia-
bility. It is desirable therefore that experi-
mental results on derivative measurements
on turbo-propeller engines of all types should
be forwarded with details of the engine control
system for digestion by a central agency, so
that simplification may be expedited.
5:7 REQUIRED ACCURACY OF THE
COEFFICIENTS
If level speed data are available at air
temperatures more or less randomly dis-
posed about two or more distinct tempera-
tures, it is advisable to correct these to the
nearest standard air temperatures, and to
present the results in a carpet plot of Ve
against T and 14. Such a carpet is valuable
for showing whether the limited information
at the rated conditions fits into the general
picture.
This problem was considered in detail in
Ref. 3. The broad conclusions that were
reached therein were that the accuracies
of estimation of the various parameters of
the equations should be as shown in the
accompanying table in order to meet the
accuracies of correction shown in parenthe-
sis.
Derivative measurements will be neces-
Level Speeds
Rate of Climb
Performance
Item
(Ave
Fuel Flow
AWF
WF
to 0.01
(
to 0.01
)
Ve
(An to 0.01 hor
1/2 ft. per sec.)
Reduction
Parameter
Ay
+0.01
RT/RF
+0.1
Вр
+0.05
a
Cp on
ПОСР
+0.1
+0.1
+0.05
B
j.on
.
+0.1
+0.1
+0.1
η δυ
ӘР
ove
+0.2
محمد داد
cel
OX
+0.2
X ove
5:13
These limits are fairly wide and should
enable single values or a small number of
values to be used for the coefficients Ap,
AN, Bp, BN, etc.
if the present evidence that power is propor-
tional to fuel flow, and independent of air
temperature at a given fuel flow and pressure
height, is substantiated by tests on other
engines. This asumption is probably already
valid in the case of “maximum level speed”
fuel flows.
5:8 POSSIBLE SIMPLIFICATIONS OF THE
METHOD
If these features are repeated on other
engines, there would appear to be a reason-
able possibility of generalizing on engine
performance and simplifying the method
considerably.
Since its development, the method has been
used intensively on one aircraft only. In
case the engine control system was
such that the engine speed was held constant,
and it was found that fixed values of the
propeller terms could be taken on the basis
of the accuracies quoted in section 5:7 and,
of course, terms in ON/aWF in Eq. 5:24
and in OP/ON in establishing AP could
be ignored.
5:9
REDUCTION OF LEVEL SPEEDS AND
FUEL CONSUMPTIONS BY PERFORM-
ANCE ANALYSIS
(a) Introduction
Also, although not actually used, single
values of the power and thrust derivatives
with respect to aircraft speed could have
been adopted without objectionable errors,
and perhaps a single value of X/D. Thus
Av (Eqs. 5:18 and 5:19) was dominated by
Ky, and could be satisfactorily defined by
Ky independently of altitude and engine set-
ting.
It is possible that the need may arise
for adopting the alternative method for level
speed and range reduction. An outline of
a possible approach is presented here, the
approach being similar to that of methods
developed elsewhere for the reduction of
level speeds for reciprocating engine air-
planes. It is emphasized that the method
has not yet been applied to a turbo-propeller
aircraft and thus unforeseen problems may
be encountered.
The derivatives of rated engine power with
respect to air temperature fell into two
classes:
(1) where the jet tailpipe tempera-
ture varied with air temperature (T/P times
OP/aT effectively zero )
(2) where the jet tailpipe tempera-
ture was held constant by adjusting fuel
flow. In this case, T/P times DP/T had
an appreciable value, but could be assumed
constant (2-3) independent of engine setting
and altitude without leading to objectionable
errors in the particular case examined,
provided the temperature range through which
reduction was made was not excessive
( 5 10°C).
(b) The Approach Considered
It will be assumed that Reynolds number
and Mach number effects on airframe drag
and propeller efficiency are negligible and
that, at a given pressure altitude, ambient
temperature effects on specific fuel consump-
tion can likewise be ignored.
The first requirement is a universal power
required curve independent of weight and al-
titude. With the usual notation, in stabilized
level flight
W2
Dož pové scde+
PV-be
2
1
In fuel flow reduction the range reduction
equations may later be materially simplified
5:31
5:14
Then
drag power
DVelo
and
flow. Thus, if the shaft power-fuel flow
relationship is established for each pressure
altitude under test conditions, such a relation-
ship can be applied to standard conditions
and range performance deduced.* Unfortu-
nately, the jet thrust fuel flow relationship
is more sensitive, and allowances may have
to be made for its variation with air tempera-
ture, if its effect on performance is not
sufficiently small.
W2
1
drag power to = 'po SCD Vo+
Ź
-
+
2
Ve
& Boote v
3
drog power:vo
Ve
Roscoe
SCDe
W 3/2
(c) Standard Power
1
w
+
121b²e
Ve
or
To implement the equations previously
developed, it is necessary to know the power
available under standard conditions, at the
test equivalent air speed and pressure alti-
tude. Thus
3
mp Jo + xve.gov tuz ve
XV
ki
+k2
w
Ve
W 3/2
w
novo + x ve
W
3/2
S
5:32
needs to be evaluated for the required engine
settings.
Whence all flight results can be represen-
ted as a single functional plot against
Ve W. This plot should provide a check
a
on the assumptions made, including the values
Now if AP, AX are the required correc-
tions to PT, XT
of n.
OP ON
ОР
Ps: PT + AP=PT+ AT +
ОТ
ON
XS = XT + Ax = XT+ OX AT +
=
ax
OT
ox
AN
ON
5:33
During the development of this power
required curve, measurements of installed
powers, thrusts and fuel consumptions, will
be obtained for a range of forward speeds
at selected pressure altitude, at tempera-
tures not very different from standard, and
speeds differing from those obtainable under
standard conditions by virtue of the differ-
ences in test air temperature and aircraft
weight from standard. From these test data,
standard installed power available curves
will be developed, in conjunction with test
measurements or estimates of OP/T,
DP / Ove, etc., and with due allowance for
propeller efficiency, this leads to standard
performance.
at the test Ve. If the magnitudes of oP/a Ve ,
ax/a Ve be known, then
(mp Vo + XVels
w
1.5
S
the power available parameter, may be
Evidence to date suggests that for these
engines, at a given pressure altitude shaft
power is proportional to fuel flow, and in-
dependent of air temperature at a given fuel
* It has already been remarked that such a
state of affairs will also bring about a simpli-
fication in the reduction of fuel consumption
data by the differential method.
5:15
where T
-
test air temperature
plotted over the necessary small range of
Ve Ws, for each rating and the rated
performance arrived at by comparing these
power available curves at each setting with
the power required.
TR = air temperature at which change
of regime occurs
Ts
.
standard air temperature
T4 jet pipe temperature. .
In applying these temperature corrections
to power, it must be appreciated that oP/OT,
etc., may change significantly with air tem-
perature, as a result of an operative limita-
tion being encountered. Thus part of the
correction may be, for instance, at constant
jet tailpipe temperature where oP / OT will
be appreciable, and part where the tailpipe
temperature is dependent upon T and OP/OT
is small. It is necessary, therefore, to
know the air temperature at which such
changes of regime occur.
Just as much prior information about
engine behavior is therefore needed to utilize
this method as is necessary for the differ-
ential method, and thus the auxiliary program
on derivative measurements outlined in
section 5:6 still has to be undertaken.
5:10
CONCLUDING REMARKS
Thus AP may be equal to
(Tp-to)(87) Tg = 117
+
f[T]
TRT
(TS-TR) (OM)
The methods given should provide a foun-
dation on which reduction routines can be
built as and when the need arises. The
flexibility necessary at the present stage
makes them rather more tedious to use than
the methods used for piston engined aircraft,
but there are grounds for hoping that experi-
ence will permit simplifications.
OP
+ AN
ON
(T4=const.)
REFERENCES
1. Herrington, Russell M., Shoemacher, Paul E., “Flight Test Engineering Manual,”
Edwards, California, USAF Technical Report No. 6273, 1953.
2. Cameron, D., "British Performance Reduction Methods for Modern Aircraft," U, K.,
Ministry of Supply Report No. AaEE/Res/170, 1942.
3. Lush, Kenneth J., “Differential Performance Reduction Methods for Turbopropeller Air-
craft," U, K., Ministry of Supply Report No. AAEE/Res/275, 1952.
€
5:16
TABLE I
fi(Kv) = = 2 V3 Kv2/(Kv 4 + 3) for values of Ky
.02
.07.08.09
Ky
.00
.01
.03.04
.05
.06
1.3
1.00
8
1.00
1.00
1.00
1.00
1.00
1.00
.99
.99
.99
1.4
.99
.99
.99
.99
.98
.98
.98
.98
.97
.97
1.5
.97
.96
.96
.96
.95
95
.94
.94
.94
.93
1.6
.93
.93
.92
.92
.91
.91
.90
.90
.89
.89
1.7
.88
.88
.87
.87
.86
.86
.85
.85
.84
.84
1.8
.83
.83
.82
.82
.81
.81
.80
.80
.79
.79
1.9
.78
.78
.77
.77
.76
.76
.75
.74
.74
.73
2.0
.73
.72
.72
.71
.71
.70
.70
.69
.69
.68
2.1
.68
.68
.67
.67
.66
.66
.65
.65
.64
.64
2.2
.63
.63
.62
.62
.62
.61
.61
.61
.60
.60
2.3
.59
.59
.58
.58
.58
.57
.57
.56
.56
.56
2.4
.55
.55
.55
.54
.54
.53
.53
.53
.52
.52
2.5
.52
.51
.51
.50
.50
.50
.49
.49
.49
,48
2.6
.48
.48
.47
.47
.47
.46
.46
.46
.46
.45
2.7
.45
.45
.44
.44
.44
.44
.43
.43
43
.42
2.8
.42
.42
.42
.41
.41
.41
.41
.40
.40
.40
2.9
.40
.39
.39
.39
.39
.38
.38
.38
.38
.37
Ку
.0
.1
.2
.3
.4
.5
.6
.7
.8
.9
3
.37
.35
.33
.31
.29
.28
.26
,25
.24
.22
5:17
TABLE II
f2(Ky) = 6/(Ky4 + 3) for values of Kv
Ky
.00
0 1
.01
.02
.03
.04
.05
.06
.07
.08
.09
1.3
1.02 | 1.01
.99
.98
96
.95
.93
.92
.91
.89
1.4
.88
.86
.85
.84
.82
.81
.80
.78
.77
.76
1.5
.74
.73
.72
.71
.70
.68
.67
.66
.65
.64
1.6
.63
.62
.61
.60
.59
.58
.57
.56
.55
.54
1.7
.53
.52
.51
.50
.49
.49
48
.47
.46
.45
1.8
.44
.44
.43
.42
.42
.41
.40
.39
.39
.38
1.9
.37
.37
.36
.36
.35
.34
.34
.33
.33
.32
2.0
.32
.31
.31
.30
.30
29
29
28
.28
.27
2.1
.27
.26
26
.26
25
.25
.24
.24
.24
.23
2.2
.23
.22
.22
22
.21
21
.21
20
.20
.20
Ky
,0
.1
2
.3
.4
.5
.6
.7
.8
.9
2
.32
.27
.23
.19
.17
.14
.12
.11
.09
.08
3
.07
.06
.06
.05
.04
.04
.04
.03
.03
.03
4
.02
5:18
APPENDIX
USE OF LOGARITHMIC DIFFERENTIALS IN PERFORMANCE REDUCTION
δY
a2
X1
- AQ,X, X2
Αα
дXI
al
= Q, Y
3
In performance reduction equations it is
often simpler and neater to use what are
sometimes called "logarithmic differ-
entials."
(a) General Form:
A:6
If y: f (X1, X2, ... )
Y
A:1
XI OY
YOX1
:
al
logarithmic differentials take the general
form
A:7
dY
XI DY XI
X2 OY axz
Y OXI XI YOX2 X2
+
+...
and similarly, for X2, X3, and so on. Hence
A:2
DY
dXI
dX2
t..
ta2
al
X1
This relation is fairly easily derived,
since by definition
X2
A:8
OY
dY :
OY
ax
dX +
dX2 ...
In particular, since
axedxe+
A:3
1/2
-1/2
n Pro:nP
2
())
TSL
and if we divide both sides by Y and re-
arrange the terms a little, the logarithmic
form is obtained.
A:9
(b) Products of Powers:
where PSL and TSL are constants, we may
write
The logarithmic form is neat when
the function is a product of powers of the
independent variables. For example, if
dimpto)
d,
din
dP
+
7 Р
+
| OP.
2 P.
| dTo
2 Ta
ηΡσ
@
Y : AXI
a 2
X2
A:10
A:4
where A is a constant, we have
This form is not only neat, but it enables
one to work with the proportional changes of
n and P without needing to know their
absolute values, which is often very conven-
ient. Proportional changes in power and also
drag, can often be generalized, enabling the
same relations to be used for all engines or
airframes of a class irrespective of size
or weight.
a 2
δY
ox
2
Aa, x,ºr-1
AQ
X2
A:5
5:19
(c) Sums of Functions:
In particular, if
이
​Y = x;"' +
02
+X2
+::
A:14
The logarithmic form is less conven-
ient for sums, but may sometimes be desir-
able, when dealing with a mixture of sums
and products or powers.
Suppose that
then
dy
Y
t,
Y= f,(X1) + f2(x2)+ ..
22+
Xl.of. dx/
X1
+.
xiol + x
fi oxi
A:11
A:15
then
x, a,
ofl.dXI
Qi'
dX
+
X1
62
x, al + x2
+
dY :
axl
A:16
A:12
For example, the drag equation can often
be written
and we can write
W2
dY
Y
X1 of dxi
XI
DE AVē+B
Y OXI
V2
A:13
A:17
60
W2
B
dD
dve
2
ve
dve
+
dW
2
W
2
AV?
W2
AVS-B
ve
D
Ve
W2
AVS+B
Ve
-
В
A:18
With the aid of Eqs. A:8 and A:16 as "standard forms”, many performance equations can
be differentiated “logarithmically” by inspection.
5:20
AGARD FLIGHT TEST MANUAL
VOLUME I, CHAPTER
DATA REDUCTION AND PERFORMANCE TEST METHODS
FOR RECIPROCATING ENGINE AIRCRAFT
By
Daniel O. Dommasch
Associate Professor of Aeronautical Engineering
Princeton University
3
VOLUME I, CHAPTER 6
CHAPTER CONTENTS
Page
TERMINOLOGY
6:1
INTRODUCTORY COMMENTS
6:1
6:2
LEVEL FLIGHT - POWER REQUIRED
6:1
6:3
POWER AVAILABLE IN LEVEL FLIGHT
6:5
6:4
DETERMINATION OF Vmax
6:16
6:5
RANGE AND ENDURANCE TESTING (CRUISING FLIGHT)
6:17
6:6
PILOT TECHNIQUE IN LEVEL FLIGHT PERFORMANCE TESTING
6:25
6:7
THE INSTRUMENTATION REQUIRED FOR LEVEL FLIGHT TESTING
6:26
6:8
CLIMB AND DESCENT TESTING - SAWTOOTH CLIMBS
6:26
6:9
CLIMBS TO CEILING
6:29
6:10
PILOT TECHNIQUE FOR CLIMB TESTS
6:35
6:11
MEASUREMENT AND CORRECTION OF TAKE-OFF CHARACTERISTICS
6:37
6:12
MEASUREMENT AND CORRECTION OF LANDING CHARACTERISTICS
6:45
6:13
CONCLUDING REMARKS
6:46
REFERENCES
6:47
TERMINOLOGY
Units
CD
Drag Coefficient
Numeric
Сре
Effective Parasite Drag Coefficient
Numeric
CL
Lift Coefficient
Numeric
e
Span Efficiency Factor
Numeric
AR
Aspect Ratio
Numeric
S
Wing Area
Square Feet
P
Air Density
Slugs/Cu.Ft.
Density Ratio
P/ Po
Numeric
H, h
Altitude
Feet
Hp
Pressure Altitude
Hpc
Pressure Altitude Corrected for Position Error
Feet
P, Pa
Ambient Pressure
In. Hg. abs.
PCO
Carburetor Deck Static Pressure
In. Hg. abs.
Pmo, MAP
Absolute Intake Manifold Pressure
1
In, Hg, abs,
9
Dynamic Pressure
tev?
Lbs. per Sq. Ft.
W
Weight
Pounds
WF
Weight Fuel Flow Rate
Lbs./Hr.
t
Time
Seconds
V
Velocity
Feet/Second
Vo
Observed Indicated Airspeed
Knots or mph
Vc , CAS
Calibrated Airspeed (Vo corrected for
instrument, position and lag errors)
Knots or mph
Ve
Equivalent Airspeed
Knots or mph
1.A.S.
Indicated Airspeed
Knots, mph
TERMINOLOGY
(Continued)
Units
Voc
Best Climb Airspeed
Knots, mph
Vmax
Maximum Level Flight Airspeed
Knots, mph
Vstall
Minimum Level Flight Airspeed
Knots, mph
Vg
Take-off Speed (true)
Knots, mph
VW
Wind Velocity (true)
Knots, mph
Vtas
True Airspeed
Knots
Vo
Touch Down Speed (true)
Knots
Q
Torque
Foot-pounds
n, RPM
Revolutions Per Minute
RPMT
Turbine RPM
7p
Propeller Efficiency
Numeric
Tco, CAT
Carburetor Air Temperature
OR
Tas
Temperature at Standard Altitude
OR
Tao, OAT
Observed Outside Ambient Air Temperature
R
THP
Thrust Horsepower
Horsepower
BHP
Brake Horsepower
Horsepower
BHPao
Observed Brake Horsepower Available
Horsepower
BHPro
Observed Brake Horsepower Required
Horsepower
BMEP
Brake Mean Effective Pressure
Pressure units
EBP
Exhaust Back Pressure
In. Hg.
bsfc
Brake Specific Fuel Consumption
Lbs./(BHP) (Hr.)
MSD
Metering Suction Differential
In. Hg.
R
Range
Miles (Feet)
TERMINOLOGY
(Continued)
Units
E
Endurance
Hours (Seconds)
dR/W
Specific Range
Miles/Pound
dE/W
Specific Endurance
Hours/Pound
RC
Rate of Climb
(usually fpm)
RCT
True Rate of Climb
(usually fpm)
RCE
Equivalent Rate of Climb
(usually fpm)
8
Take-off Distance
Feet
SW
Take-off Distance Through Air
Feet
sg
Ground Distance
Feet
so
Distance Travelled by Air
Feet
Fm
Mean Accelerating Force
Lbs.
Y
Angle of Flight Path with Horizontal
Degrees
K1, K2, kl, k2
Constants
fpm
Feet Per Minute
Subscripts
e
Equivalent
S
Standard
O
Sea Level (also used for observed quantity)
req
Required
av
Available
ew
"ew" System Value
T
Test Value (also turbine)
obs
Observed
6:1 INTRODUCTORY COMMENTS
Our basic assumption is that
CL
co
Сре
+
Te AR
Consequently, the thrust horsepower re-
quired to maintain level flight at a given
airspeed may be written
Because of the nature of the reciprocat-
ing engine-propeller combination, airplanes
equipped with this means of propulsion are,
for the most part, designed to fly at speeds
less than drag divergence. Although Mach
number effects are present and do modify
the lift slope at velocities less than that for
drag divergence, there are no major modi-
fications in the lift-drag relationship until
drag divergence is encountered. Thus, for
all the important phases of level flight per-
formance testing of reciprocating engine
aircraft, we have the following fundamental
relation valid
9 Co. Sv
W?V.as
THP rea
+
550
q?S?FEAR(550)
6:2
:
where a 1/2p Vrepresents dynamic pres-
sure and CL has been replaced by its equal
W/qs.
CL
co
Со: Сре
Coe +
PCDE S.V3
THPrea
:
TeAR
W?
+
PV STEAR(275)
6:1
1100
where
Ср
8
Airplane Drag Coefficient
Сре
Effective Parasite Drag Co-
'efficient
Since we cannot conveniently measure thrust
horsepower, it is desirable to express power
required in terms of brake horsepower, and
for this purpose we know that
CL
8
:
Lift Coefficient
THPrea
BH Prea
тр
e
8
Airplane Efficiency Factor
where
тр
= propeller efficiency.
AR :
Aspect Ratio
Thus
pcdes.vo
BHPreg
1100p
In our further discussions, we shall assume
that Eq. 6:1 is valid in all cases for air-
planes with reciprocating engines, and that
normally coe and e remain constant at
all times during the data correction process.
Where it is important, the effect of variations
of CDe and e will be noted.
W?
+
PVS TAR( 275)7p
6:3
6:2 LEVEL FLIGHT - POWER REQUIRED
Now letting
Coes
2
KI
1100 mp
6:4
The most convenient method of presentation
of power required data for piston engine
airplanes known to the author is called the
"ew" system which presents data from all
altitudes on a common sea level-standard
weight plot. The basis for this method and
a procedure for fairing the data and at the
same time finding the generalized plot are
given below.
and
TEARS(275)p
K2
6:5
6:1
and multiplying through by V, we obtain
Taking ratios of horsepowers and velocities,
THPreglew
3
K2W
(BH Preq)(V) = p KiVa +
= THP rea
VIN
Ws
W
Р
6:6
Ws
THPreq vo
)。
6:7
and
.
Examining Eq. 6:6, we see that if we
plotour power required data at a given
weight and density altitude in the form
(BHPreq) (V) vs. V“, we should obtain a
straight line with slope pK and intercept
K2W/p: However, we shall not always
obtain this straight line because of minor
variation in e and CDe and somewhat more
important variations in np during tests.
Vew : Vo
vo
WS
W
6:8
Now let us return to Eq. 6:3. This may
be written
Ka W}
:
BHProq = 8,0 KAVO+ Cm
()
pov
Ws
Eqs. 6:3 and 6:6 can be generalized so
that data taken at various altitudes fall on
a single curve.
To show how this comes
about, consider flight at the same value of
CL at sea level under standard conditions,
at standard weight, and at some altitude
where the density is p and the weight differs
from standard. At sea level, the density is
Po
and the weight is Ws , so that letting
V
: sea level standard velocity,
ew
or, on multiplying by
Molo
po
W
Nola
BHP reqvo
( m ) .
( ) .K, 9.(oyť vo (
) Cn
WS
W
Ws
W
2ws
Vew
S
K2 W}
W
+
Povov
Ws
Copvos
1
2w3co
THPrenew
1100
550
Posc
However, from Eqs. 6:7 and 6:8 this is (con-
verting from THP to BHP)
BHPregew - Kip. Vow +
K₂ W}
Povew
6:9
At altitude (flying at the same CL) we must
have the same value of co as at sea level
according to the basic assumption of this
analysis; therefore, at altitude
and on multiplying by Vew
.
Kzw.
New = Kip Vew +
Ро
6:10
2W
(BHP regew
V:
✓
PCLS
Since
Po
and We are constante,
Copy's
THPreg
1
550
2wc
(BHP reqew
Vew = kivew + kz
8
1100
PSCE
6:11
6:2
7
where
k, poki
P. Coes
11001p
where stabilized flight had not been achieved
during tests or where instrument malfunction
produced erroneous readings. Test data
plotted in accord with Eq. 6:10 will appear
as in Fig. 6:1. Points such as "a" in Fig.
6:1 are immediately seen to represent un-
satisfactory data.
6:12
K2W
k2
WS
TEARS(275)7 P.
PO
6:13
Since altitude is not a factor in Eq. 6:10,
test data of power required versus velocity
from all altitudes converted to terms of
BHPregew and Vew, should plot into a
single curve which is approximately a straight
line. This curve may be used to conveniently
fair the test data and to eliminate points
As we have previously noted, CDe and e
may not actually be constant throughout the
range of lift coefficients and Mach numbers
encountered in the tests, and, therefore, the
"straight line” of Fig. 6:1 may actually
exhibit curvature, particularly at its ends.
At the high speed end (low CL), curvature
may be produced by Mach number effects,
whereas at the low speed end (high CL),
Сое
(BH Preq.! Vew
slope
Pocoes
-
Central Portion
:K,
T100 10
k
Extrapolated From Central Portion
Minimum Flight Speed
vaw
2
Intercept
.
WS
TEARS-2757pp.
=k2
Fig. 6:1 General Plotting Form of Equation 6:10
6:3
curvature is the result of tire relation
CZ
Co-CDe+
:
TAR
MO
BHP
not fitting the actual CD - CL curve at the
high lift coefficients. This low speed end
curvature is quite common, and is duc es-
sentially to the fact that CDe itself becomes
a function of CL (or a) at lift coefficient
values of about one. Curvature is also in-
troduced by variations in propeller efficiency
which were disregarded in the analysis. Not-
withstanding the fact that our "straight line"
may be a curve, the use of the procedure
described is highly recommended because
it provides such an excellent means of fair-
ing data, and moreover provides a means of
estimating CDe and e, provided the central
portion of the curve is used as a basis for
such computation. Eqs. 6:12 and 6:13 pro-
vide the required relations between
e, and the slope and intercept (obtained by
extrapolation) as illustrated in Fig. 6:1.
Vow
,
Fig. 6:2 Gencralized Power Required Curve
If Ve is expressed in units of feet per
second, Eqs. 6:12 and 6:13 provide the fol-
lowing
1.53 WS
From Fig. 6:2, the actual power required
curve at any altitude may readily be ob-
tained, for from Eqs. 6:7 and 6:8 we have
Vew
W
пkzь* пр
T
V =
6:14
Ws
and
and
6:16
BHP
ew
:
coe
Pos
1100np
ki
BHP
✓o
W.
6:15
where
b
wing span in feet
пр
propeller efficiency (85% on the
average).
From examination of Fig. 6:2 the reader
will note that no data are shown below the
speed for minimum power. This does not
represent an omission, but rather reflects
the fact that it is almost impossible to ob-
tain stabilized level flight test data at speeds
below the minimum power speed in what is
known as the reverse command region. Ac-
tually it is desirable to obtain at least one
data point in this region to provide a clear
definition of the speed for minimum power;
however, due to the difficulty of achieving
stabilized level flight, data are generally
Once the test data have been reduced and
faired in accord with Fig. 6:1,
Fig. 6:1, a generalized
horsepower required plot may be prepared
using the faired curve of Fig. 6:1. This
curve appears as in Fig. 6:2.
6:4
taken while the airplane is either slightly
climbing or descending, and a correction is
then imposed to account for rate of change of
altitude (during the test run it is necessary,
of course, that the speed, at least, be stabi-
lized).
testing and reduction of data have many ad-
vantages for they bring about savings in both
time and money.
Our next topic is the measurement of power
output and power available.
6:3 POWER AVAILABLE IN LEVEL FLIGHT
Because power is rate of change of energy,
we know that the descending airplane requires
less power from the engine than when in level
flight, and the converse holds true for the
climbing airplane. The correction to be
applied is from fundamental considerations
On military aircraft, brake power devel-
oped is generally measured using a torque-
meter, which measures the torque developed
at the propeller shaft, and a tachometer which
permits determination of shaft RPM. The
brake power is given by the relation
W
dH
A BHPrea
( )
33,000
dt
np
6:17
2nQ
BHP :
33,000
where dh/dt is positive for a climb and is
assumed (in Eq. 6:17) to be expressed in
feet per minute.
6:19
where Q : torque in foot pounds
: RPM.
n
The increment of Eq. 6:17 is subtractive
from the observed power in a climb and is
additive in a descent. Thus
(BHP reqlevel - (BHPreglobserved
W
Although it would be desirable to measure
thrust and then determine thrust horsepower
from the relation THP : TV/550, (where T
is in pounds and V in feet per second), there
is unfortunately no simple way to accomplish
this for the well-known eason that the
thrust in the propeller shaft is not the actual
overall thrust produced by rotation of the
propeller in the presence of the airframe.
+
()
dH
dt
33,000
6:18
In obtaining data in the reverse command
region, rate of climb or sink should be re-
stricted to about 50 fpm, in which case the
rate of change of pressure altitude may be
assumed equal to the actual rate of change
of true altitude without introducing serious
error.
On smaller engines no provision may have
been made to permit the installation of a
torquemeter. In these cases it is necessary
to refer to engine characteristic charts for
the estimation of power output delivered by
the engine. Engine charts are usually pre-
pared in two parts, one being a sea level
test calibration in terms of BHP under stand-
ard sea level conditions as a function of MAP
for various values of RPM, and the other, a
computed set of curves for altitude conditions
(although these altitude curves may also be
obtained from tests).
The instrumentation required, flight tech-
niques, and tabular data reduction forms for
reducing level flight data will be discussed
later in the chapter. However, before we
.
become involved with these we should first
consider the topics of power available deter-
mination, and range and endurance testing.
As we shall show, the actual flight testing
for all of these may be done simultaneously,
as may the data reduction. Such simultaneous
The use of power charts is not entirely
satisfactory inasmuch as in their prepara-
tion it is generally presumed that such factors
6:5
as oil pressure, oil temperature, cylinder
head temperature, etc., either remain con-
stant or vary in some predetermined manner.
Similarly it is normally assumed that the
fuel-air ratio is properly maintained by the
carburetor and that the ignition is perfect.
Since the assumed conditions are seldom
satisfied in actual test work, the power charts
obviously only provide an approximation to
the truth and should be used to determine
the power delivered only when torquemeters
are not available,
it is a good idea to define precisely what we
mean by the term. By definition, critical
altitude is the altitude at which the throttle
must be fully opened to develop rated power
at rated RPM. An unsupercharged engine
(or a ground boosted engine) has a critical
altitude of zero feet (i.e., at sea level),
whereas a supercharged engine, which is
capable of exceeding engine limits at sea
level, will have a critical altitude consider-
ably above sea level. Aircraft with multi-
speed, multi-stage superchargers will have
several different critical altitudes depending
on the possible blower combinations.
The use of power charts to determine ac-
tual power output is discussed in several
text books and need not be considered in
further detail here. However, these charts
have another use, namely, the determination
of standard power available and of critical
altitudes and blower shift points for aircraft
equipped with gear-driven superchargers.
These topics we must consider in some detail.
Because we do not directly measure power
output of an engine, critical altitude is fre-
quently defined either in terms of limiting
BMEP or MAP at rated RPM, and tests are
conducted using either BMEP or MAP as
the determining factor.
If a BMEP gage
(torquemeter calibrated in terms of BMEP)
is available, the procedure for determining
critical altitude with a single-speed, single-
stage supercharger is as follows:
Constant BMEP Tests for Critical Altitude
No corrections to standard are required
for points determining the power required
curves, since here we are merely using an
engine as a means of measuring the drag
requirements of the airframe. Thus, assum-
ing a torquemeter is available, the meter
reading together with the tachometer reading
directly gives us the required information
in determining power required data. Above
the highest critical altitude, full throttle will
be used to determine the test Vmax points
at the various test altitudes; these points,
therefore, may also be considered test points
for the power available vs. altitude curves
for the airplane in level flight. We note that
due to differences in ram at the air intake,
and possibly due to differences in engine
cooling characteristics, a different power
available curve will probably exist for the
climbing airplane than for the craft in level
flight. Consequently separate tests are nor-
mally required for level flight and climb
power available characteristics. The deter-
mination of critical altitudes, shift points
and standard power available as functions
of altitude is discussed below.
Depending on whether military or normal
rated power data are desired, the propeller
controls are set to maintain the proper
limiting RPM. Starting out at low altitude,
level flight runs are made holding BMEP at
its limiting value by throttle control. Runs
are made at part throttle until an altitude
is reached where full throttle is required to
give rated BMEP at rated RPM. Above this
altitude a series of full throttle runs is made,
and developed BMEP and MAP are recorded
along with carburetor deck static pressure,
carburetor air temperature, pressure alti-
tude, and outside (ambient) air temperature.
If plotted, the test data will appear as in
Fig. 6:3.
To determine the standard critical alti-
tude, no corrections need be made to the
part throttle curve; however, the full throttle
curve must be corrected by means of the
procedure outlined below or some simi-
lar method. Assuming that the tests are
Since we must test for critical altitude(s),
6:6
Full Throttle Curve
( Constant RPM)
dH
Observed Critical Altitude
lo
PRESSURE ALTITUDE ,

Part Throttle Curve
( Constant RPM )
BMEP (Torquemeter Reading)
Fig. 6:3 General Plotting of Pressure Altitude vs. BMEP
6:7
To: OAT, °R
conducted on a day warmer than the standard
day, the observed and corrected data will
appear as in Fig. 6:4.
In. H20
RAM: (Pco-Pal, In. Hg. or
13.6
BHPMeasured
torquemeter horse-
power
Computations
Corrected Full Throttle Curve
(1) Determine observed supercharger
pressure ratio
Test Dota For Worm Day
Pmo
H.
( )。
Pm
PC
Standord Critical
Altitude
(pen + RAM)
.
(2) Determine carburetor air tempera-
ture corrected to standard conditions
LTITUDE
RE ALT
PRESSUR
.
Tes = Too
Tao + Tas
(3) Determine the corrected pressure
ratio for standard conditions, (Pm/Pc's, from
Fig. 6:5 by entering the curve at observed
pressure ratio and carburetor air tempera-
ture, Tco, and following the slope of the
nearest slanting line to the carburetor air
temperature for a standard day, Tcs.
BRAKE MEAN EFFECTIVE PRESSURE
Fig. 6:4
(4) Determine corrected manifold pres-
sure by the relation Pms
+ RAM)
(PM/Pc)s : Pco (Pm/Pc's
(POO
(5) Determine corrected BHP
A most convenient procedure to follow to
correct full throttle power available data
to standard is the semi-empirical Wright
Case II Method. The procedure for the case
of a single-stage supercharger is as follows:
BHP
BHPO (Pms/Pmo) (Tco /Tcs JE
Data Required
When making these single-stage correc-
tions it is usually more convenient and less
time-consuming to use the special charts
as given in Figs. 6:6 and 6:7.
PCO
• Carburetor deck static pressure,
In.Hg.abs.
-
Pmo
• Manifold pressure, In. Hg. abs.
Тco
• Carburetor air temperature, °R
If an engine is equipped with a single-
stage two-speed supercharger (gear shift
type), there will be two critical altitudes.
The procedure for determining the lower
altitude in this case is the same as described
above with the engine operating in low blower.
To determine the second (higher) critical
altitude, the procedure is as follows:
Ро
• Ambient pressure, In. Hg.
Tas
: Standard altitude temperature
corresponding to Po (standard
atmosphere tables), OR
Starting at an altitude a little higher than
6:8
MAIN AND AUXILIARY STAGE
PRESSURE RATIO CORRECTION CHART
EXAMPLE : a.) PM/Pc = 1.840 at 90° F. CAT (c)
= 1.865 of 60°F. CAT
°
b.) PM/Pc
m/Pc or Pcr po
3.2
3.0
2.8
2.6
2.4
2.2
SUPERCHARGER COMPRESSION RATIO,
2.0
D
C!
B
1.8
1.6
1.4
1.2
-60 -40 -20 O 20 40 60 80 A 100 120
DEGREES F. AIR INLET TEMPERATURE, To or To
Fig. 6:5 Main and Auxiliary Stage Pressure Ratio Correction Chart
6:9
BRAKE HORSEPOWER CORRECTION TO STANDARD CONDITIONS
SHEET 1 OF 2
MANIFOLD ABSOLUTE PRESSURE INS. HG.
20 24 28 32 36 40
44
48
WRIGHT CASE
DEVIATION OF OAT FROM STD-'c
25 20_15_10_5 5 10 15 20 25
COLDER THAN WARMER THAN
STD.
STD
SUPERCHARGER EFFICIENCY CORRECTION
1.03
1.01 1.00 0.99 0.98 0.97 0.96
32
1.04
102
3.1
30
-3.0
8
01.
12
-2.9
2.9
14
91
18
2.8
2.8
20
2.7
2.7
2.6
2.6
-2.5
2.5
OBSERVED SUPERCHARGER
2,4
14 CARBURETOR DECK PRESSURE – INS. HG.
COMPRESSION RATIO
C
Fig. 6:6 Brake Horsepower Correction to Standard Conditions
OBSERVED SUPERCHARGER COMPRESSION RATIO
2
2.3.
2.2
2.2-
_01
12
2.1
97
2.0
2.0
20
22
24
26
28
30
30.000
10.000
poo
10,000
Log
52
56
1.08
1.071
1.04
-
1.03
DUE TO DENSITY
* 1.06
21.03
1.04
1.03
1.02
21.01
1.06 FINAL BHP 'CORRECTION
10.99
-0.98
70.97
0.96
/ 0.95
0.94

20
24 28 32 36 40 44 48
MANIFOLD ABSOLUTE PRESSURE INS. HG.
EXAMPLE
OBSERVED DATA
ALTITUDE 15,000 FT.
OAT +1
MAP 40.5 IN. HG.
CARD. DECK PRESS. 17.0 IN. HG.
CALCULATED DATA
OAT - STD. TEMP. 14--148). 16.3°C (AT)
DERIVED DATA FROM CHART
CORRECTED MAP 41.4 IN. MG. POINT (F)
BHP CORRECTION FACTOR (MULTIPLYING)
POINT (J) 1.088
bec
20 15 10 5
COL DER THAN
STD
10 15 20
1.00
WARMER THAN
0.99 STD.
SL-8,000
0.98
BHP CORRECTA
0.93/
.
000 or
0.97
0.94
10,000
30,000
18,000
6:10
0.91
SHEET 2 OF 2
BRAKE HORSEPOWER CORRECTION TO STANDARD CONDITIONS
WRIGHT CASE I
MANIFOLD ABSOLUTE PRESSURE – INS. HG.
20 24 28 32 36 40
DEVIATION OF OAT FROM STD.- 'c
COLDER THAN WARMER THAN
STD.
STD.
25 20 15 10 5 0 5 10 15 20 25
48
44
SUPERCHARGER EFFICIENCY CORRECTION
1.04 103 102 101 100 0.99 0.98 0.97 0.96
52
1.9.
9
1.8
97
1.7
18
20
24
SUPERCHARGER COMPRESSION RATIO
26
OBSERVED SUPERCHARGER
28
,
CARBURETOR DECK PRESSURE - INS. HG.
1.6
COMPRESSION RATIO
X30
1.5
32.
Fig. 6:7 Brake Horsepower Correction to Standard Conditions
1.3
1.2
OBSERVED
1.2
30,000
20,000
18,000
10,000
36
52
20 24 28
32
40 44 48
MANIFOLD ABSOLUTE PRESSURE INS. HG.
1.04
1.09
1.08
1.07!
- 1.06
/ 1.05
1.04
/ 103
1.02
/LOI
1.00 FINAL BHP CORRECTION
10.91
0.98
0.97
-996
/ 0.95
-0.94

JL - 6,000
DUE TO DENSITY
11.02
koo
WILO
29 18 10 3 10 15 20
COLDER THAN WARMER THAN
STD.
0.99 STD.
1
0.98
1
0.97
1
0.96
20,000
1,000
BHP CORRECTION
000'or
| 0.93
0.92
10.000 - 10.000
6:11
Wright Case II Method outlined previously
may be extended to provide the following cor-
rection. The procedure is as follows:
Obtain
the low blower critical, the shift to high
blower is made and the throttle retarded to
hold the engine to its limiting BMEP at full
RPM. We note here that the limiting BMEP
in high blower will usually be less than the
low blower value, and that the engine speci-
fications should be consulted to determine
the proper value. Tests are now conducted
in the same manner as for determination
of the low blower critical altitude, the final
corrected plot of test data appearing as in
Fig. 6:8.
Ро
Тоо
Tas
:
Ambient pressure, In. Hg.
Ambient air temperature, 'R
Standard air temperature, ºr
Measured auxiliary supercharger
inlet static pressure, In. Hg.
Measured auxiliary supercharger
inlet temperature, 'R
po
Too
:
PCO
Measured carburetor deck static
pressure, In. Hg.
TCO
Measured carburetor air tempera-
High Blower
Critical Altitude
ture, 'R
pmo
Snitt Point
Measured manifold pressure, In.
Hg.
(P. - Po), In. Hg.
Measured torquemeter power
RAM
*
PRESSURURE ALTITUDE,
Lou Blover
Critical Altitude
BHPO
Procedure
(1) Determine actual auxiliary super-
charger pressure ratio
BRAKE MEAN EFFECTIVE PRESSURE
Pc
DCO
Рco
)
Fig. 6:8
po
(pao +
+ RAM)
poo
From these data the best blower shift
point may also be determined as shown in
the figure.
(2) Determine auxiliary supercharger
inlet temperature under standard conditions
as follows:
T'ago:tc+ Tos - Too
The case of the two-stage two-speed blower
is handled in the same manner as described
above, and these data appear as shown in
Fig. 6:8 excepting that there are now three
critical altitudes and two shift points.
(3) Determine corrected auxiliary su-
percharger blower ratio for standard con-
ditions (PC/Pa)s, from curve Fig. 6:5 by
entering the curve at observed pressure ratio
and auxiliary supercharger entrance temper-
ature To., and following nearest slant line
to the corrected auxiliary supercharger en-
trance temperature Tos.
Two Stage Systems
To correct to standard power available
when using a two-stage supercharger, the
6:12
(4) Determine corrected carburetor deck
pressure from the relation
is similar to the turbo-driven unit which has
no definite critical altitude, and has its per-
formance limited by structural consider-
ations.
PCS
= (po + RAM)
Menos
3
-
par le monde
(5)
Determine observed main super-
charger pressure ratio (Pm/Pc).
(6) Determine carburetor air temper-
ature corrected to standard conditions,
Tos
: Tco
Tao + Tas
(7) Determine corrected main super-
charger pressure ratio by entering Fig. 6:5
with observed main stage pressure ratio
(Pm/Pc), and observed carburetor air tem-
perature Tco. Follow slope of nearest slant
line to corrected carburetor air temperature
Tcs, and read corrected main supercharger
compression ratio (Pm/Pc's.
More recently the compound engine has
made its appearance. As presently used,
this engine is equipped with a single-stage
two-speed supercharger together with blow-
down exhaust turbines whose output is fed
back to the crankshaft through fluid cou-
plings. Because the blow-down turbines are
not directly used for supercharging, but have
the purpose of converting exhaust gas energy
to shaft horsepower, the operation of the
compound engine is quite similar to that of
a conventional reciprocating engine. Indeed,
it has been found that the same correction
procedure used to determine power available
characteristics of the standard engine (as
outlined above) also applies to the compound
units.
(8) Determine final corrected manifold
pressure by multiplying corrected main su-
percharger pressure ratio by carburetor
deck pressure as follows:
Pm
Pms
Dc
(9) Determine corrected BHP from re-
lation
Pms
BHP= BHP.
:
els
DCS
In the United States, military engines of
2000 cu. in. displacement and above are
required to have provisions for the instal-
lation of torquemeters (AN 9500a, para.D-8)
which also function as BMEP gages. For
test work, such gages are always installed.
However, this is not true operationally in
all cases, and frequently the operational
pilot is provided only with a tachometer and
a manifold pressure gage for use in deter-
mining engine power output. For this rea-
son, data on power available are frequently
presented in terms of MAP rather than
BMEP. To convert the BMEP data to MAP
for the part throttle region, we have the
constant RPM relation
8
(一​) (==)
(
The special charts contained in Figs. 6:6
and 6:7 can be used with modification for
the two-stage system. However, this is not
usually done because the method becomes
rather complex,
BMEP ~ MAPCOAT)
6:20*
In addition to the types of superchargers
described above, we also have the two-stage
automatic-shift variable-speed type, which
has no definite critical altitudes, and which
automatically regulates the engine to prevent
development of excessive BMEP values.
With this latter type of installation, it is
extremely difficult to obtain repeatable data
because of the unknown variation of super-
charger RPM. This type of supercharger
* This relation immediately follows from
the well-known equation BHPov = (constant)
(
(MAP). (TjV2 since at constant RPM,
BMEP is directly proportional to BHPov.
.
.
6:13
Constant MAP Tests For Critical Altitude
For the further condition of constant
BMEP, Eq. 6:20 immediately provides the
relation
OAT
test
MAP
std
- MAP test
OAT std
6:21
Because of decreasing exhaust back pres-
sure at altitude, less MAP is required to
maintain a given BMEP at altitude than at
sea level. Therefore, a condition of constant
BMEP with varying altitude corresponds to
a decreasing MAP with altitude. This is il-
lustrated in Fig. 6:9. As noted, the observed
test values of MAP for part throttle operation
are corrected according to Eq. 6:21; full
throttle MAP test values are corrected ac-
cording to the Wright Case II Method de-
scribed in this chapter with the corrected
MAP data appearing as in Fig. 6:9.
Constant MAP tests are conducted in ex-
actly the same way as constant BMEP tests
except that the MAP is kept constant rather
than BMEP. Runs are continued until full
throttle operation will not give rated MAP, and
the altitude(s) where rated MAP corresponds
to full throttle is (are) recorded. If plotted,
the raw test data appear as shown in Fig.6:10.
To reduce the test data to standard, no
corrections are imposed on the part throttle
region. However, the full throttle curves are
corrected by the Wright Case II Procedure.
The corrections will shift the location of the
critical altitude(s), but the general nature of
the curves will still be as shown in Fig. 6:10.
The corrected plot of MAP vs. Ho deter-
mines the critical altitudes, but does not show
the location of the shift points, nor does it
directly yield information on the variation
of power available with altitude. To obtain
these data, in the absence of a torquemeter,
it is necessary to refer to engine charts.
It should be recognized that the use of
these charts is not a very satisfactory
a
procedure, but rather is a necessary al-
ternative in the event of unavailability of a
When BMEP gages are not available for
test work (as is the case when dealing with
the smaller engines), critical altitude deter-
mination is based entirely on MAP and RPM
readings. The procedure is described below.
Full Throttle
Lines of Constant
BMEP
High Blower
Critical Altitude
TH
Full Throttle
Part Throttle
PRESSURE ALTITUDE, Mp
PRESSURE ALTITUDE
----
Low Blower
Critical Altitudo
ABSOLUTE MANIFOLD PRESSURE
ABSOLUTE MANIFOLD PRESSURE
Fig. 6:9 Variation of MAP with Altitude for
Constant BMEP Part Throttle Operation
Fig. 6:10
6:14
torque nose installation. Using the charts,
the MAP, altitude and RPM data are con-
verted to a plot of BHP av vs. altitude, ap-
pearing as in Fig. 6:11. The increase in
power available with the altitude shown in
the figure is due to the fact that constant
MAP operation causes an increase in BHP
available with altitude.
To obtain power available as a function of
altitude, runs are made (which may be made
as part of the power required test) at various
altitudes at rated RPM and either at limiting
MAP or turbine RPM. At low altitudes MAP
limits are maintained by retarding throttle
and partially bypassing the turbine. At the
higher altitudes the throttle will be fully open,
and allowable turbine RPM restrictions will
prevent obtaining allowable MAP limits.
Full Throttle
Part Throtile
For wide open throttle, we may suppose
that the MAP obtained at a given test pres-
sure altitude and fixed value of engine RPM
is the same as under standard conditions.
Then as long as we have not exceeded turbine
RPM limits (even though the observed tur-
bine RPM is not the same as the standard
for the test altitude) we shall have the re-
quirement that corrections must be imposed
for:
Critical
Altitudes
Shift
Altitudo
PRESSURE ALTITUDE , He
Full Throttle
(1) Non standard outside air tempera-
ture
Port Throttle
(2) Non standard carburetor air tem-
perature
BRAKE HORSEPOWER
(3) Non standard turbine RPM
Fig. 6:11
(4) Non standard exhaust back pressure
(5) Non standard BHP (power obtained
from torquemeter)
Engines With Turbosuperchargers
To correct for the above deviations, we
proceed as follows:
As previously mentioned, turbosuper-
charged engines display no well-defined criti-
cal altitude; however, the determination of
their power available as a function of altitude
is still an important matter.
From the engine characteristic charts (ob-
tained by test or computation) we may obtain
a value of the partial derivative
When flying with a turbosupercharged in-
stallation the pilot must observe three basic
limits on engine operation. These are limits
on:
a(CAT)
O (RPM)T
(1) Engine RPM
(2) MAP or BMEP
at the test value of turbine RPM where
CAT = carburetor air temperature and
(RPM)T • turbine RPM. Knowing A(RPM)T,
we may compute A(CAT)t, effective change
in CAT due to non standard turbo RPM, from
(3) Turbine RPM
6:15
2 in. Hg. decrease in EBP. Further the
relation
AICATII
(CAT)
a(RPM)
.
A(RPM)T.
Trest
There also exists another increment in CAT
due to non-standard OAT. This is
BHPS - BHP
at constant MAP
Tstd
A(CAT)2 = (OAT)std -(0AT)test.
.
holds for turbosupercharged engines so that
finally
The total change in CAT is
САТ,
BHP std - BHPtest
LE
test )
CAT std
A(CAT) = A(CAT)2 + A(CAT)T
-
-
(OAT)std
+.005 [EBPtest - EBP
Pstal
(OAT)test
+
(CAT)
A(RPM)T.
O (RPM)T
Now
CAT Std
.
The effects of changing exhaust back
pressure are approximately given by the re-
lation that the power increases 1% for each
CATtest + (OAT)std
O(CAT)
- (OAT)test +
O(RPM),
A(RPM)T.
Therefore
CAT test
BHP std - BHP test
(CAT)
CATtest + (OAT)std - (OAT)test + J(RPM)
)
.(RPM)T
OT
+.005 (E8Ptest – EBP sta]}
where n ~.5
6:22
6:4 DETERMINATION OF Vmax
Eq. 6:22 may be used as a basis for cor-
recting full throttle turbosupercharged en-
gine data to standard. At less than full
throttle at constant MAP the power may be
presumed to vary in accord with Eq. 6:21.
At a selected number of altitudes which
should include sea level and the critical and
shift altitudes the generalized power re-
quired curve BHPew vs. Vew is used to ob-
tain actual curves of standard BHP required.
Knowing the variation of standard power
available with altitude as determined in sec-
tion 6:3, the intersections of the power avail-
able and power required curves may be at
Having determined the standard BHP avail-
able, our next step is to obtain the variation
of standard Vmax with altitude. This is dis-
cussed in section 6:4 below.
6:16
for specific range and endurance given below:
once determined. These intersections are
the standard Vmax points for the altitudes
considered. A typical plot of Vmax Vs.
altitude for an airplane equipped with a two-
speed single-stage supercharger is shown
in Fig. 6:12.
specific range
dR
η
C
1
dw
Со
CO W
6:23
specific endurance
3/2 1/2
dE
CL S
P
dw с
CD
2
np.
.
1
3/2
W
6:24
where R
• Range in feet
Critical Altitude
W
• Weight in lbs.
с
: bsfc
Shift Point
р
• Air density
PRESSURE ALTITUDE, Hp
E
: Endurance in seconds
Critical Altitude
ap
: Prop efficiency
CL
• Lift coefficient
CD
: Drag coefficient
Vmax
S: Wing area
Fig. 6:12
Recognizing that
np
and c are variables,
the quantities to maximize are, respectively,
3/2,
6:5 RANGE AND ENDURANCE TESTING
(CRUISING FLIGHT)
03 ) and ( 7 et ).
CO
Range and endurance test data are fre-
quently obtainable along with power required
and available data. Therefore, the theory of
range and endurance testing will be covered
at this point followed by a brief discussion
of general data reduction form and required
instrumentation.
Because the optimum values of "p, c and
CL/CD or C32/ co may not occur simul-
taneously, it is necessary during tests to
consider variations of each of these quanti-
ties.
Ideally, if engine bsfc and prop efficiency
were constants, power required data could
be used to obtain directly specific range and
endurance information. In any actual case,
however, variations of these quantities pro-
hibit this simple an approach. To see what
is required, consider the Breguet equations
At any given weight and altitude, the values
of the aerodynamic parameters CL/CD and
CL"/2/ Co are dictated by the flight speed.
The variations in speed alone suffice to tie
these variables down. Engine bsfc is a func-
tion of BMEP and RPM which together de-
fine power output. Normally at any given
values of RPM, bsfc will tend to be least at
maximum allowable BMEP, although this is
6:17
not an inflexible rule. Propeller efficiency
is a function of the quantity V/nD (advance
ratio) and therefore, changes with both for -
ward speed and RPM. This is true even
though the propeller is of the constant-speed
type because such a propeller adjusts its
blade angle to produce enough torque to
govern RPM rather than to achieve maximum
prop efficiency. Therefore, maximum prop
efficiency is not necessarily attained at the
same RPM required for limiting engine
BMEP. Similarly, optimum engine and pro-
peller operation may be unobtainable at the
aircraft aerodynamic speeds for best range
and endurance. Changes in weight and alti -
tude change the aerodynamic speeds for best
range and endurance, which in turn alter
prop efficiency and power required, and
therefore, the effects of weight and altitude
must be considered along with the effects of
speed changes, RPM, and BMEP setting.
To attempt to account fully for all the
effects of all variables in testing for range
and endurance characteristics is generally
not practical from the standpoint of time and
effort involved. Moreover, the actual varia-
tions encountered in carburetor functioning
are great enough to make any attempts to
attain absolute precision ridiculous. Thus,
the procedures prescribed herein are di-
rected toward achieving the best results con-
sistent with reasonable expenditure of time
and effort, rather than toward the achieve-
ment of meticulous accuracy.
Normally, we are required to obtain fuel
consumption data for several different alti-
tudes and weights rather than for all possible
combinations of these. Because power re-
quired and available data must usually be
obtained at the same altitudes and weights
as those chosen for fuel consumption testing,
data for these latter items are most easily
obtained simultaneously with fuel consump-
tion data.
the given RPM (as specified by the engine
manufacturer). Speed is reduced by reduc-
ing MAP (and the BMEP) until minimum
speed is obtained. Normally, 6 to 10 points
will suffice to define the curve between the
minimum and maximum speed points. The
procedure is repeated for different RPM
values. Usually, from the point of view of
convenience in making the test, it is desir-
able to start first with maximum rated RPM
and BMEP and then work down with subse-
quent runs at lower RPM's. The lowest RPM
used is determined by generator cutout speed
or similar considerations. Data taken during
tests are readings of:
MSD Carburetor metering suction
differential (a measure of fuel
flow)
RPM Engine rotational speed
MAP Absolute intake manifold
pressure
BMEP Or torquemeter reading
ОАТ Outside air temperature
Observed airspeed
Fuel flow
VO
Fuel quantity
Pressure altitude
Carburetor deck pressure
Carburetor air temperature, cylinder
head temperature, mixture setting,
blower setting, throttle position (part
or full), external configuration,
When two-stage superchargers with inter-
coolers are used, additional data as listed
below should be obtained to permit proper
reduction of power available data simultan-
eously with the range and endurance data:
Pressure at inlet of auxiliary super-
charger stage
Temperature at inlet of auxiliary su-
percharger stage
The procedure is as follows:
At the specified altitude(s), pressure, and
weight(s), a series of level flight runs is
made at a constant RPM setting, starting
out with the maximum allowable BMEP for
Pressure at outlet of auxiliary super-
charger stage
6:18
Temperature at outlet of auxiliary su-
percharger stage
Reduction of Range and Endurance and Power
Required Data
Lines of Constant RPM
Because existing fuel flowmeters and car-
buretors are rather erratic in their behavior,
the first step in reducing the test data is
correlation of MSD with observed fuel flow.
For this purpose, data on these two quan-
tities obtained at all weights and altitudes
are plotted as in Fig. 6:13 and an average
curve relating MSD to fuel flow is deter-
mined.
METERING SUCTION DIFFERENTIAL
Separato Plot Required for
Each Pressure Altitude
BRAKE HORSEPOWER
Fig. 6:14
METERING SUCTION DIFFERENTIAL
o
Doc
Loon Limit
Ayorage
Rich Limit
WF WEIGHT FUEL FLOW RATE
WF"
• WEIGHT FUEL FLOW RATE

Fig. 6:13
Lings of Constont RPM
Separato Plot Required for
Eoch Pressuro Altitudo
Following the preparation of the above
plot, all the data (at varying weights and
RPM's) at one pressure altitude are plotted
as shown in Fig. 6:14.
BRAKE HORSEPOWER
Fig. 6:15
Using the data of Fig. 6:14 and the faired
curve of Fig. 6:12, prepare the cross plot
of Fig. 6:15 which represents the faired fuel
flow vs. BHP data for various values of
RPM at the selected altitude.
The foregoing procedure is repeated for
all test altitudes.
6:19
TABLE 6:1 - FLIGHT DATA ANALYSIS SHEET
Std. Gross Wt.
Model
Configuration
* Observed Data Corrected for Instrument Error
車
​.
(1) Flight No.
(2) Vo
(3) CAS
(4) Ve
(5) * Press. Alt.
(6) Pos. Error
(7) lipc
(8) TOAT
(9) Po P
( (10) *MAP Pmo
|(11)
*RPM
(12) *Torque Q
(13) BHIPO
|(14) *Fuel Flow 157hr
(15) GPH
|(16) bsfc
(17) * MSD
(18) Gross Wt. -
WO
(19) WoW
|(20) (W6Wsia
W.7W
'
(21) (W/W) 3/2
(22) Vew for wt.
|(23) BHP s1
(24) (BHP)w for wt.
|(25) Vew
|(26) (BHP)si Vew
(27) BMEP
(28) *Pco In. Hg.
(29) *Polo In.Hg.
(30) (Pc/Polo
(31) Too
(32) Too
(33) Tag
(34) To's
(35) (Pc/Pa')s
(36) Pcs
(37) (Pm/Pcle
(38) * TCO
(39) TCS
(40) (Pm/Pc)s
(41) Pms
(42) BHPs
(43) *Head Temp. °C
6:20
>
Following the fairing of fuel flow data, the power required data are reduced to standard using
the procedures developed in section 6:2. A typical data analysis sheet for this purpose, which
also includes the steps required for reduction of power available data, is given as Table 6:1.
Use of Flight Data Analysis Sheet
1. Number of flight
2. Observed data
3. (2) and airspeed calibration
4. (3) corrected for pressure altitude
5. Observed data
6. (2), (2) - (3), (5), (8) and correction chart for position error
7. (5) 1 (6)
8. Observed data, corrected for airspeed
9. (7) and (8) and atmosphere charts
10. Observed data
11. Observed data
12. Observed data
13. (11) and (12) and torque constant
14. Observed data
15. (14) + 6(lb/gal.)
16. (14) + (13)
17. Observed data
18. Computed from T. O, weight and fuel burned
19. (18), - standard weight
20. (19)
21. (20)
22. (4) + (20)
23. (13) + (9)
24. (23) : (21)
25. (22)*
26. (22) times (24)
27. (12) times conversion constant
28. Observed data, carburetor deck pressure
29. Observed data, auxiliary inlet pressure
30. (28) · (29) observed auxiliary compression ratio
31. Observed data, auxiliary inlet temperature
32. (8), outside air temperature
33. Standard OAT for (7)
34. [(31) - (32)] + (33) standard auxiliary inlet temperature
+
35. (30), (31), and (34) and compression ratio curve
36. (29) times (35) standard carburetor deck pressure
37. (10) + (28) observed main stage compression ratio
38. Observed data, CAT
39. (38) - (31) + (34), standard CAT
40. (37), (38), and (39) and compression ratio curve of engine standard main stage com-
pression ratio
41. (36) times (40) standard MAP
(38)460
៤
42. (13) times (41) = (10) times
standard BHP
(39) 460]
43. Observed data, cylinder head temperature
6:21
Using the data from Figs. 6:15 and 6:16,
plots of fuel flow vs. Ve or CAS are now
prepared for each standard weight and al-
titude as shown in Fig. 6:17.
If the standard test weights specified differ
from one another appreciably, as they may
in the case of patrol or bomber type aircraft,
it is good practice to account for unknown
variations of propeller efficiency by prepar-
ing more than one "standard” weight plot
so that two or more generalized "ew"power
required curves are obtained as shown in
Fig. 6:16.
From Fig. 6:17 the best endurance speed
for any given set of circumstances is selected
as the speed for minimum fuel flow. The
engine RPM setting for this speed is also
given.
If we now take each curve of Fig. 6:17 and
compute the ratio of true flight speed to fuel
flow, we obtain data relating specific range
to Ve or CAS (the specific range will have
units of miles per pound, with V in MPH and
fuel flow in lbs./hr.). These data may be
plotted up in the form of Fig. 6:18 to deter-
mine optimum range information.
High Gross Weight Tests
BHP
Low Gross Weight Tests
From Fig. 6:18 the speed and RPM setting
for max range may be determined, as well
as the engine setting for optimum range, at
any selected airspeed. Provided small var-
iations from standard weight (of the order
of 10% of G.W.) are involved, it is possible
to use the data of Figs. 6:17 and 6:18 to
determine range and endurance character-
istics at other weights,
vow
Fig. 6:16
The assumption we must make here is that
for small weight changes, the quantity 7/c
in Eqs. 6:23 and 6:24 does not vary appreci-
ably, in which case we have
(Specific rangel, Wz
6:25
Specific rangela
wi
3/2
Web
(Specific endurance),
(Specific endurancela
)
6:26
WI
Having obtained the curves of Fig. 6:16
(which we note could just as well have been
obtained entirely separately from the fuel
consumption data), we are in a position to
determine corrected curves of fuel flow as
functions either of true or equivalent air-
speed. We note here that no attempt is made
to correct engine characteristics themselves
to standard, and that the corrections merely
involve correction of the power required
data. The procedure is justified on the basis
that from a practical standpoint the engine
corrections are small in the first place, and
in the second, normal variations in carburetor
performance produce larger unknown errors
than could be corrected for.
with the equivalent airspeed remaining con-
stant during correction,
Thus, to correct the data of Figs. 6:17 and
6:18 for small changes in standard weight
each fuel flow or specific range point has
its ordinate changed according to Eqs. 6:25
and 6:26 while the abscissa values remain
unchanged.
6:22
RATE
( Note: Curves Required For Each Standard Weight
And Altitude )
WEIGHT FUEL FLOW
ines Of Constant RP
Envelope Curve
Horizontal Tangent
WF
Best Endurance Speed
EQUIVALENT AIRSPEED, V
Fig. 6:17
6:23

Horizontal Tangent
Envelope
Curve
SPECIFIC RANGE IN MILES/LB. FUEL
Max. BMEP Points
Note: Separate Plot Required
For Each Weight And
Altitude.
Speed For Best Range
EQUIVALENT AIRSPEED, Ve
Fig. 6:18
6:24
6:6 PILOT TECHNIQUE IN LEVEL FLIGHT
PERFORMANCE TESTING
(c) Achieving stabilized conditions
Now that we are aware of what is required
in the way of test data and have followed
through the procedure for reducing these
data, we are in a position to discuss the tech-
niques which should be used to obtain good
raw test data. These techniques are as fol-
lows:
.
Since it is easier to slow an airplane at a
given altitude value than it is to accelerate
it, all level flight performance tests should
be started at the Vmax speed. To achieve
this speed most conveniently, the airplane is
first climbed to an altitude several hundred
feet higher than the test altitude. At this
altitude the external configuration is set, the
position of each external item is recorded,
and the altimeter set to 29.92 in. Hg. to make
sure it is reading pressure altitude. The
engine controls are set to give maximum
power at the test altitude and a slow descent
is made to the test altitude during which
Vmax is attained for the level flight run.
(a) Planning the flight
Before the flight, make sure the pilot knows
exactly what data are going to be obtained
and under what conditions he is to obtain it.
This will normally entail his knowing:
(1) The test pressure altitude(s)
(2) The test gross weight range(s)
(3) The engine power setting(s) and
mixture control setting(s)
The airplane should then be flown at the
test altitude a sufficient period of time to
insure that stabilized flight has been achieved.
This will take from three to ten minutes.
During stabilization the altitude should be
held to within plus or minus 20 feet and the
airspeed to plus or minus one mile per hour.
After at least three minutes have passed and
the airplane is stabilized, data may be taken.
(4) The external configuration(s) to be
checked.
The data items to be recorded should be
noted, and a convenient flight data card pre-
pared to allow rapid data recording. If a
photopanel is being used along with the data
card, its operation should be checked prior
to flight.
We note here that efficient planning will
cut down the flight time required to obtain
data, and in this connection it is frequently
possible to plan flights so that at least two
extremes of altitude and gross weight may
be tested for in a single flight.
Following the Vmax run the engine controls
are adjusted for the next lower power setting.
The airplane is again stabilized and data are
taken. The procedure is continued until the
required range of speeds has been inves-
tigated. As data for each point are obtained,
note should be made of instrument fluctua-
tions, and the amplitude of the fluctuation
and average reading recorded. The apparent
cause of the fluctuation should be determined
and recorded. Change in configuration due to
creeping of the cowl flaps, bomb bay doors,
etc., should also be recorded as well as the
time allowed to obtain stabilized flight,
(b) Climb to test altitude
(d) Specialized technique for obtaining
low speed data
After take-off and during the climb to the
altitude for the first run, engine instrumenta-
tion and airplane operation should be checked
for satisfactory behavior. This avoids a
wasteful climb to altitude should the equip-
ment show signs of malfunctioning.
It is quite difficult to maintain stabilized
level flight over the range of airspeeds ex-
tending from the stall to slightly above the
minimum power required speed (known as
6:25

Horizontal Tangent
Envelope
Curve
SPECIFIC RANGE IN MILES/LB. FUEL
Max. BMEP Points
Note: Separate Plot Required
For Each Weight And
Altitude.
Speed For Best Range
EQUIVALENT AIRSPEED, Ve
Fig. 6:18
6:24
6:6
PILOT TECHNIQUE IN LEVEL FLIGHT
PERFORMANCE TESTING
(c) Achieving stabilized conditions
Now that we are aware of what is required
in the way of test data and have followed
through the procedure for reducing these
data, we are in a position to discuss the tech-
niques which should be used to obtain good
raw test data. These techniques are as fol-
lows:
Since it is easier to slow an airplane at a
given altitude value than it is to accelerate
it, all level flight performance tests should
be started at the Vmax speed. To achieve
this speed most conveniently, the airplane is
first climbed to an altitude several hundred
feet higher than the test altitude. At this
altitude the external configuration is set,'the
position of each external item is recorded,
and the altimeter set to 29.92 in. Hg. to make
sure it is reading pressure altitude. The
engine controls are set to give maximum
power at the test altitude and a slow descent
is made to the test altitude during which
Vmax is attained for the level flight run.
(a) Planning the flight
Before the flight, make sure the pilot knows
exactly what data are going to be obtained
and under what conditions he is to obtain it.
This will normally entail his knowing:
(1) The test pressure altitude(s)
(2) The test gross weight range(s)
(3) The engine power setting(s) and
mixture control setting(s)
The airplane should then be flown at the
test altitude a sufficient period of time to
insure that stabilized flight has been achieved.
This will take from three to ten minutes.
During stabilization the altitude should be
held to within plus or minus 20 feet and the
airspeed to plus or minus one mile per hour.
After at least three minutes have passed and
the airplane is stabilized, data may be taken.
(4) The external configuration(s) to be
checked.
The data items to be recorded should be
noted, and a convenient flight data card pre-
pared to allow rapid data recording.
If a
photopanel is being used along with the data
card, its operation should be checked prior
to flight.
We note here that efficient planning will
cut down the flight time required to obtain
data, and in this connection it is frequently
possible to plan flights so that at least two
extremes of altitude and gross weight may
be tested for in a single flight.
Following the Vmax run the engine controls
are adjusted for the next lower power setting.
The airplane is again stabilized and data are
taken. The procedure is continued until the
required range of speeds has been inves-
tigated. As data for each point are obtained,
note should be made of instrument fluctua-
tions, and the amplitude of the fluctuation
and average reading recorded. The apparent
cause of the fluctuation should be determined
and recorded. Change in configuration due to
creeping of the cowl flaps, bomb bay doors,
etc., should also be recorded as well as the
time allowed to obtain stabilized flight.
(b) Climb to test altitude
(d) Specialized technique for obtaining
low speed data
After take-off and during the climb to the
altitude for the first run, engine instrumenta-
tion and airplane operation should be checked
for satisfactory behavior. This avoids a
wasteful climb to altitude should the equip-
ment show signs of malfunctioning.
It is quite difficult to maintain stabilized
level flight over the range of airspeeds ex-
tending from the stall to slightly above the
minimum power required speed (known as
6:25
the region of reverse command); however,
data are still obtainable over at least part
of this region, the procedure being as follows:
excess speed, flying very smoothly at the
selected altitude, and stabilizing all the
controllable variables as rapidly as pos-
sible. Any deviation from standard tech-
niques should be noted in the final report of
the particular set of tests.
(1) Assume and hold a constant air-
speed at the test altitude.
(2) Adjust power for as near level flight
as possible (holding the rate of climb or
descent to less than 50 fpm).
6:7 THE INSTRUMENTATION REQUIRED
7
FOR LEVEL FLIGHT TESTING
(3) Make a five-minute run recording
altitude at the beginning and end of the run.
The table on the opposite page lists the
special test instrumentation required for
the level flight performance tests. A photo
recorder is desirable but it is not necessary
if the scope of the project does not warrant
such an installation,
(4) During the data reduction proce-
dures the observed brake horsepower re-
quired should be corrected for the time rate
of change of potential energy by the relation
(Rate of climb or descent) W
BHPrea
BHP I
6:8 CLIMB AND DESCENT TESTING
SAWTOOTH CLIMBS
33,000 np
with minus holding for climbing flight and
the plus for a descent.
Here
nap
is the prop efficiency which may
be assumed to be between 80% and 90%. We
note that excessive precision is not required
in making the above correction inasmuch as
the magnitude of the correction will normally
be small as long as the 50 fpm restriction
on climb or descent rates is observed.
Normally the speeds for best rate of climb
at various altitudes for reciprocating engine
aircraft are obtained by using the sawtooth
method of testing. Acceleration run tech-
niques may also be used, but are not as
satisfactory for piston engine aircraft as
for jets. Since these acceleration techniques
are discussed elsewhere in this volume, they
will not be considered further here.
Because this method is not as accurate as
the stabilized altitude and airspeed proce-
dure, it should not be used except when data
are not obtainable in any other manner.
Checking of descent characteristics of
piston engine aircraft is generally not ac-
complished in detail except for the landing
conditions where the rate of sink becomes
an important factor insofar as the structural
integrity of the airplane is concerned.
(e) General comments
The outlined procedures for making level
flight speed runs should be followed exactly,
unless special considerations of a particular
project require a different approach. It is
anticipated that a shortening of the duration
of the stabilization runs will be required
for airplanes with high fuel consumption and
relatively low fuel supply. This in turn will
require extreme care in letting down without
In this section we shall consider the nature
of sawtooth climbs, and in sections 6:9 and
6:10 the continued climb to ceiling and the
pilot techniques involved. It is noted that
the maximum energy climb concept is ap-
plicable to reciprocating engine aircraft as
well as jets. However, since this topic is
discussed elsewhere in the manual, no rep-
etition will be made at this point,
The sawtooth procedure is the oldest flight
test method used to determine the speed for
6:26
SPECIAL TEST INSTRUMENTATION REQUIRED
FOR LEVEL FLIGHT PERFORMANCE TESTS
NECESSARY FOR
INSTRUMENT
POWER
REQUIRED
POWER
AVAILABLE
FUEL
FLOW
Sensitive Tachometer
1
1
1
Torquemeter
(if installation permits)
Sensitive Manifold
1
1 1
1
1
1
1
Pressure
1
1
1
Sensitive Altimeter
1
1
1
1
1
1
2
2.
3
1
3
1
1
1
Sensitive Airspeed Indicator
Sensitive Outside Air Temperature
(OAT) Gage
Sensitive Carburetor Deck Pressure
(CDP) Gage
Carburetor Air Temperature
Gage (CAT)
Totalizer Type Fuel Quantity Gage
(desirable to measure gross weight
variation accurately)
Fuel Flowmeter
Metering Suction Differential Gage
Auxiliary Supercharger Pressure
Gage (inlet)
Auxiliary Supercharger Temperature
Gage (inlet)
Auxiliary Supercharger Pressure
Gage (outlet)
Auxiliary Supercharger Temperature
Gage (outlet)
3*
3
1
3*
3
1
2**
3**
3***
3***
1. Cockpit and Photo Recorder
2. Cockpit Only
3. Photo Recorder Only
For Fuel Consumption Only
** For Two-Stage Supercharger Only
*** For Intercooler Efficiency Survey Only, of Two-Stage Installation
6:27
best climb rate at a given altitude. In this
procedure, one first selects the altitude(s) at
which data are desired and then an altitude
band of 1000 to 4000 feet (surrounding the test
altitude) through which a series of climbs at
different stabilized airspeeds are made. For
each of these runs the time to climb through
the altitude band is measured and the aver-
age rate of climb through the band computed.
As an example, consider the test pressure
altitude of 20,000 feet. For this altitude the
test band could be chosen as 19,000 to 21,000
feet or possible 19,500 to 20,500 feet if the
rate of climb were low. Further assume
that we want to check the airspeed range from
150 knots IAS to 300 knots IAS.
.
Horizontal Tangent
RATE OF CLIMB
Speed For Maximum
Rate Of Climb
V
Fig. 6:19 Raw Sawtooth Climb Data at One Altitude
6:28
Our procedure would be as follows:
Starting at some altitude less than the
lower limit of the test band, full throttle is
applied and a climb initiated which is sta-
bilized at some airspeed close to 150 knots
by the time the lower band limit altitude is
reached. This stabilized airspeed is main-
tained through the test band and the time
required to climb through the band measured.
After this first climb, a descent is initiated
and the procedure repeated, say at 175 knots,
200 knots, etc., until the full range of air-
speeds has been investigated. The fact that
succeeding ascents and descents produce a
flight path resembling a sawtooth pattern,
leads quite naturally to the name “sawtooth
climbs.”
Now, as we have noted, the sawtooth pro-
cedure is used only to obtain the speeds for
best rate of climb at various altitudes, not
the climb rates themselves. Therefore, no
corrections are imposed for non-standard
power available or for non-standard gross
weight. This procedure is quite satisfactory
for piston engine equipped aircraft whose
fuel load is a relatively small percentage
of the gross weight. However, determina-
.
tion of the effects of weight change are
easily accomplished when necessary. The
procedure is to run the sawtooths at various
weights and from the data thus obtained to
plot a curve of speed for best rate of climb
at any altitude vs, weight. From this curve
the speed for maximum climb rate at any
standard weight may be directly obtained.
A rough and ready rule for estimating the
speed for maximum rate of climb for a pro-
pellered airplane is:
6:9 CLIMBS TO CEILING
Vbc - Vstall + ķxVmax - Vstall )
:
6:27
The sawtooth data obtained at various
pressure altitudes define a curve of CAS for
best rate of climb vs. altitude similar to
that shown in Fig. 6:20.
PRESSURE ALTITUDE, Hp
In conducting the tests, although the nom-
inal values of the airspeed may be 150, 175
knots, etc., stabilized airspeed values of 154
knots, 180 knots, etc., are just as satis-
factory as the nominal values as long as air-
speed is held constant during the climb, and
satisfactory data points are obtained. A typ-
ical plot of raw sawtooth data at one altitude
is shown in Fig. 6:19.
In obtaining sawtooth data, care must be
taken to insure that the airplane is not climb-
ing with or against a wind having a vertical
gradient; otherwise, the kinetic-potential en-
ergy balance will be modified sufficiently
to give false data. Although correction tech-
niques for gradient winds are available, in
general, insufficient information is available
to allow their proper application. We note
in passing that the sawtooth method is dif-
ficult to apply to aircraft with very high
rates of climb because of the small incre-
ments of time during which the airplane re-
mains in the test band and the difficulties
associated with achieving a stabilized climb
airspeed. Also, altimeter (static system)
lag at high ascent rates introduces errors
of frequently undetermined magnitude.
RATE OF CLIMB
CALIBRATED AIRSPEED FOR MAXIMUM
Fig. 6:20 CAS for Maximum RC
6:29
On the basis of the schedule shown in Fig.
6:20 continued climbs to service ceiling (al-
titude at which RC: 100 fpm) are made,
during which the following data are contin-
uously recorded either by photopanel or by
using a kneepad data card:
aircraft, the actual climb to service ceiling
is made in stages, and the data are presented
in such a manner as to simulate instantane-
ous blower shift. Although such a presenta-
tion is not entirely realistic, it is considered
to provide the most convenient representation
of climb performance for the type of air-
plane involved.
(1) Pressure altitude
(2) Time
(3) IAS
(4) CAT
(5) Torquemeter reading
(6) Engine RPM
In making a measured climb to ceiling,
the aircraft is first trimmed at the indicated
speed for best climb for the starting alti-
tude of the climb. With the propeller con-
trols set at the proper RPM for the climb,
sufficient MAP is used to maintain level
flight at the climbing speed. The throttle
is then opened to provide maximum permis-
sible power while the observed airspeed is
maintained constant until stabilized climbing
conditions are reached. The procurement of
useful data commences at the altitude and
time at which V, BHP, trim, and configura-
tion are stabilized. The altitude(s) at which
full throttle is (are) reached should be noted
carefully.
(7) Gross weight
(8) Carburetor deck pressure
(9) Manifold pressure
(10) Configuration including blower po-
sition
From the observed data are calculated
the following:
(1) Pressure rate of climb at various
altitudes
It is normal practice to continue a climb
in any one blower setting to an altitude as
much as 2000 feet above the blower shift
point. The climb is then discontinued and
the airplane altitude reduced to as much as
2000 feet below the shift point during the
time the shift is being accomplished. The
climb in the next higher blower is begun
below the shift point and continued up to the
shift point for the next higher blower. In
other words, the climb is made in as many
sections as there are blower ratios on the
airplane, and the various sections are inte-
grated to give the final climb curve.
(2) Brake HP developed by the engine
(3) Density ratio
(4) True rate of climb
(5) Equivalent BHP
(6) Equivalent rate of climb
(7) Standard BHP available vs, altitude
(8) Standard rate of climb vs. altitude
Once the basic test data have been obtained
as outlined above, the power available data
for the climb configuration are reduced to
standard by the procedures previously de-
scribed in this chapter to give a curve as
shown in Fig. 6:21 which defines the various
shift points and critical altitudes for the
climbing airplane,
For gear-driven supercharger-equipped
6:30
available and BHPro is the observed brake
power required to maintain flight speed only,
then from basic considerations the true rate
of climb is given by
Critical Altitudo
BHP. - BHP
RCT: np
Le)33,000
12)
ALTITUDE
Wo
6:29
Climb Shift Point
Critical Altitudo
STANDARD
where
пр
is the propeller efficiency. We
may define an equivalent rate of climb in
the same fashion that we defined equivalent
airspeed, i.e.
RCE : : RCT vo
6:30
and from Eqs. 6:29 and 6:30
BRAKE HORSEPOWER AVAILABLE
BHP. - BHPT
33,000.
RCE = np Vo
Fig. 6:21 Standard Power Available in Climb
.
Wo
6:31
Further, define RCew as
After the power available data have been
corrected, the observed rates of climb (ob-
tained from a plot of pressure altitude vs.
time, and which are actually the rates of
change of pressure altitude) are corrected
to tape line or true rates of climb by using
Eq. 6:28, which is obtained from the relation
Δp : -Pg Δh
RCT:vo.
Wsz
Ws
: RCe
Wo
Wo
6:32
then
33,000 mp
(
Mole
R Cew :
Ws
BHP
Wo
RC true
- RCobs
Rcobs (1905
)
6)
2) Ouroove
(i)
'obs
Ws
- BHProvo
B
6:28
Wo
6:33
where obs = observed and s: standard.
We may further define a term, equivalent
brake horsepower available as
From this point on data reduction con-
veniently follows what is called the “'Equiv-
alent Rate of Climb Analysis Procedure"
the basis for which is developed below.
Ws
BHPaoew
2
ВнРаос
Wo
and recall that
Development of the Equivalent Rate of Climb
Concept
W
BHP
Proew
BHProvo (en
If BHP,. is the observed brake power
6:31
Then, on multiplying both sides of Eq. 6:33
by Ws / Wo,
and, therefore, Eq. 6:35 does serve as a
basis for fairing test data from all altitudes.
Moreover, the “ew" presentation allows
correction to standard altitude conditions by
the following procedure.
RCew Ws = 33,000 mp BHP a Rew - BHProew)
6:34
Solving Eq. 6:34 and BHPoo ew
From the test data corrected in accord
with Eq. 6:35, prepare a plot of observed
equivalent power available vs. equivalent
rate of climb
+ BHP roew
BHPoCew 33.0001p
(RCew) Ws
(Note:
6:35
BHP 00 EW
,
BHP.
Ws
BHP ovo
Wo
(
;
- -).
Govo (w
RCTVO ( )
Now, if under any set of circumstances it
happened that the climb schedule provided for
a climb at constant propeller efficiency and at
constant CL then the quantities Ws /33000 mp
and BHProew would be constants leaving only
two variables in Eq. 6:35, namely, BHPCO ew
and RCew. In other words, we would have a
relation of the type y = mx + b where
Wsla
RCew :
Wo
similar to that shown in Fig. 6:22.)
EQUIVALENT POWER AVAILABLE, BHP.
HP ooow
y :
BHPaoev
RCew
Ws
m =
EQUIVALENT RATE OF CLIMB, RCOW
33,000 mp
b =
-
BHProew
Fig. 6:22 Equivalent BHP., vs. Equivalent RC
In such a case as this, a plot of BHPao ew
V8. RCew would be a straight line with data
taken from all altitudes falling along this
same line. The slope of the line would de-
termine the propeller efficiency and the in-
tercept the brake power required at sea level
for non-climbing flight at standard weight.
Actually, one must expect that neither np nor
CL will remain constant and, therefore, a
plot of Eq. 6:35 will exhibit curvature,
The curve of Fig. 6:22 is somewhat anal-
ogous to the brake horsepower required
curve, in that it represents the basic climb
performance of the airplane without regard
to the actual power available for climbing.
To use this curve, we proceed as follows.
Experimental results, however, reveal that
if the test data are plotted in the form pre-
scribed by Eq. 6:35 only one curve results,
At any selected standard altitude we refer
6:32
to our previously determined curve of stand-
ard power available in the climb and obtain
the standard power available at that altitude.
This value of BHPas is then converted to
equivalent power available by means of the
equation
Eqs. 6:36 and 6:37 assume that the equiv-
alent rate of climb vs. equivalent power
available curve has already been corrected
to standard weight, and, therefore, addi-
tional weight corrections are not required.
BHPasew - BHPasvo
6:36
Knowing the value of BHPasew , we may
enter our experimental curve of reduced
climb data (Fig. 6:22) to obtain the value of
RCew. Then, by virtue of Eq. 6:32, the true
standard rate of climb is given by
In some cases where the weight correc-
tions are of small magnitude, they may be
neglected throughout the entire data reduc-
tion procedure; however, if the observed
weight differs from standard by 5% or more,
corrections should be imposed.
Data reduction is most conveniently car-
ried out in tabular forms of the type illus-
trated below:
RCT :
RCew
To
2
6:37
TABLE 6:2 - DETERMINATION OF BHPaew vs. RCew
1.
Pressure Altitude
2.
OAT °C
3.
Tobs : OAT 273°
8
4.
Ts, °K (Standard temperature at pressure altitude)
5.
Tobs/Ts
6
. 7.
P
from (1) and (2)
vo: (plodle
BHP 40 (from torquemeter and tachometer)
8.
9
.
RCobs
dhp/dt
10.
RCT: (9) x (5)
11.
Wo from test data
12.
Ws/Wo; (Ws = standard weight)
13.
(Ws/W.)"
(Ws /W.) 8/2
312
14.
15.
RCew • (10) x (13)
(
16.
BHP.
'aoew
(8) X (14)
6:33
TABLE 6:3 - DETERMINATION OF STANDARD RC
1.
Standard Altitude
12.
BHPas (standard power available)
3.
To (standard density ratio)
14.
BHPasew = (2) x (3)
15.
RCews from plot of Table 8:1
16.
RC std = (5)/(3)
.
17.
1/RCstd
:
1/(6)
Following the determination of the standard
rate of climb, the standard time to climb
may be determined using row (7) of Table
6:3 To do this, first plot 1/RC std vs.
standard altitude. This gives a curve similar
to that of Fig. 6:23, which represents the
case of a single stage two-speed super-
charger installation.
Critical Altitude
.
(R.Custo)
Critical Altitude
Shift Point
I.
-ΔΗ
STANDARD ALTITUDE
Fig. 6:23
6:34
The specialized pilot techniques required
to obtain raw sawtooth and time-to-climb
data are discussed in section 6:10.
The area under the curve of Fig. 6:23
taken to some altitude, say H, is the time to
climb to that altitude, This area can be
found by any one of the graphical integration
methods, all of which will serve to evaluate
the integral
HI
dh
RC
.
6:38
6:10 PILOT TECHNIQUE FOR
CLIMB TESTS
(a) General
In considering climb tests of any type it
is essential that the test day provide smooth
air without high altitude temperature inver-
sions and without gradient wind. Attempts
to obtain data under unfavorable weather
conditions are almost certainly doomed to
failure and represent wasted effort and im-
proper planning.
The reduced climb data obtained as pre-
viously explained is generally presented in
a final composite plot similar to that shown
in Fig. 6:24.
STANDARD ALTITUDE - FEET
Time To Climb
Rate Of Climb
STANDARD BHP
AVAILABLE
RATE OF CLIMB CAS FOR BEST
& TIME TO CLIMB CLIMB RATE
Fig. 6:24 Composite Presentation of Climb Data
6:35
(b)
Sawtooth Climbs
least three different airspeeds in the vicinity
of the estimated speed for best climb, and
preferably for four or six speeds.
(c) Timed Climbs
When the sawtooth climbs are to be made,
the altitude increment or sawtooth depth to
be checked is determined by the rate of climb
characteristics of the airplane. For instance,
if at a given altitude the rate of climb of
one airplane were 700 FPM, a sawtooth depth
of 1000 feet would be the maximum to use;
however, if the rate of climb were 5000 FPM,
an increment of at least 2000 feet would be
required. The selection of the altitude band
is a matter of initial planning.
As noted previously, either sawtooths or
acceleration runs are used only to deter-
mine the schedule of speed vs. altitude for
conducting timed climbs to ceiling. The
procedure for conducting these climbs is the
same regardless of what schedule is used
or how it was determined.
In performing an individual sawtooth climb,
the actual climb should be started several
hundred feet below the lower limit of the
test altitude band. Prior to commencing the
climb, the configuration should be completely
checked and care should be taken to insure
that instruments and equipment are function-
ing properly.
The technique involved in performing
measured climbs requires more care and
attention to detail than any other part of
flight testing. However, this care is useless
if the proper choice of weather is not made
since climbing performance is very seriously
affected by turbulent air thermals, vertical
gradient in horizontal wind, and temperature
inversions. Climbs should be undertaken
only in the most ideal weather because data
taken during unsatisfactory weather condi-
tions tend to distort the remaining climb
data.
To avoid the possibility of actually zoom-
ing into the test band and to insure stabilized
flight through the band, entry into the saw-
tooth should be accomplished in the following
manner:
During the climb the pilot must:
(1) At an altitude several hundred feet
below the lower limit of the test band, set
the configuration and trim the airplane to
hold, in level flight, the airspeed desired in
the climb. t to wird
(1) Maintain the indicated airspeed
within plus or minus one knot of the specified
value and hold ball-in-center wings level
trim.
(2) Keeping airspeed constant, smooth-
ly advance the throttle until rated power for
the test is attained. The airplane will then
be climbing.
(2) Maintain the proper power setting
on the engine in the part throttle region.
This will generally mean the maintenance of
a prescribed RPM and torque.
(3) Observe and note the altitude at
which full throttle is reached.
(3) Make sure that the airplane is
stabilized in the climb as it enters the test
band and start recording data (through the
test band the speed variation should not ex-
ceed 1 knot). Data is best taken with a
photopanel; however, stop watch data plus
pilot instrument panel readings may be re-
corded on a kneepad to give adequate data
provided the climb rates are not excessive.
(4) Record the data specified in the
section on data reduction at specified inter-
vals except in the case where an engineer
observer is on board or a photopanel is
available for recording the data.
(4) The procedure is repeated for at
6:36
The technique of entering the climb is as
follows:
6:11 MEASUREMENT AND CORRECTION
OF TAKE-OFF CHARACTERISTICS
Provided the test conditions do not deviate
excessively (20% or more) from standard,
it is most convenient to correct take-off data
by semi-empirical methods.
The airplane is trimmed at the indicated
climbing speed for the starting altitude of
the climb with the propeller governor set at
the proper RPM for the climb and sufficient
manifold pressure to maintain level flight at
the climbing speed. The throttle is then
opened smoothly and the airplane speed is
maintained constant until stabilized climbing
conditions are reached.
Correction of take-off data is required to
account for
(1) Non-standard wind conditions
(2) Non-standard weight
(3) Non-standard air density
(4) Non-standard thrust
The procurement of useful data commences
at the altitude and time at which V., BHP,
trim and configuration are stabilized. If the
airplane is properly trimmed with the con-
trol tabs the problem of holding observed
airspeed within the allowable limits is made
much easier. In some of the larger airplanes
it has been found practical to utilize the auto-
matic pilot in climb, with the necessary ad-
justments to the longitudinal trim being
accomplished as speed changes are required.
The approximate numerical procedure
commonly used to correct for the above
listed deviations from standard is a ratio-
exponential procedure using empirically de-
termined exponents. An increment correction
procedure is also available. The procedures
are theoretically justified as follows:
From Ref. 3, Eq. 10:22, the approximate
take-off distance of an airplane traveling
against a wind of velocity Vw is given by
It is good practice to continue a climb in
any one blower setting to an altitude as much
as 2000 feet above the blower shift point.
The climb is then discontinued and the air-
plane altitude reduced to as much as 2000
feet below the shift point during the time
that the shift is being accomplished. The
climb in the next higher blower is begun be-
low the shift point and continued up to the
shift point for the next higher blower. In
other words, the climb is inade in as many
sections as there are blower ratios in the
airplane, and the various sections are inte-
grated to give the final climb curve.
s : Sw + sa
w (va - Video
.
Vw W
(Vg - Vw)
-
sg
Fm
gFm
W
[Vg - VW]'
29Fm
6:39
where
It is true that such a procedure supposes
an infinitely fast blower shift in the final
climb curve, but this procedure is considered
to be the best representation of the airplane
performance. It should be pointed out in
conjunction with this procedure that the time
lost in accomplishing the blower shift and
the amount of climb which is repeated in the
blower shift region should be kept to a mini-
mum to avoid the necessity for weight cor-
rection of the final observed climb data.
8 = take-off distance
Sw
- distance traveled through moving
air mass
• distance traveled by moving air
mass
Sa
6:37
W = gross weight
V,
VW
Fm
= take-off airspeed (getaway speed)
: wind velocity
: mean accelerating force during
take-off run
Next, consider the case of non-standard
weight, with
P and Fmw standard. In this
case V, in Eq. 6:33 is a dependent variable
and must be expressed in terms of W. Assume
that the data have already been corrected to
zero wind conditions, then Eq. 6:39 becomes
g : acceleration of gravity
WV
S
29Fm
or, since
2 w
Vå
Ideally, the parameter which is held con-
stant during all take-offs regardless of ex-
ternal conditions is the lift coefficient. There.
fore, the correction procedure is based on
the concept that the take-off lift coefficient
is a fixed quantity. To obtain the form of
the wind correction formula, consider for
the moment that the weight, density and mean
accelerating force are all fixed. Then, since
PCLS
6:42
w 2
S
gp CLSFM
6:43
woževi CLS
W
Using “sub o" for observed and “sub s"
for standard, Eq. 6:43 provides the relation
Ws
Ss
• So
where CL is the take-off lift coefficient, we
may express our constant W in terms of Vg
(also a constant for the conditions of this
analysis since Vg is fixed in value by Wip,
and CL).
)。
Wo
6:44
Thus, Eq.6:39 becomes
į PVO CLS
Eq. 6:43 also provides the relation for
correcting for non-standard air density since
Vg (which is a function of p as well as W)
has been expressed in terms of p and W and
does not appear in Eq. 6:43. Thus, assuming
the weight correction has already been ac-
complished and that Fm is not altered by
density
-
S:
[Vo - vu]'.
29Fm
6:40
If we let
Ss
. So
()
so : observed ground run distance and
Ss - standard ground run distance for
standard wind Vws
6:45
then, from Eq. 6:40, the ratio of ss to so is
2
Ss
or
so
Vg - Vws
Vg - VWO
Vg – Vws
Vg - vwo'
In the preceding development, Fm has been
held constant to permit isolation of the di-
rect effects of wind, weight, and air density.
Actually, changes in weight and wind produce
second order changes in Fm, while changes
in density produce a first order change in
Fm. The second order effects are custom-
arily accounted for by modifying the exponent
2 in Eqs. 6:41 and 6:43 to fit experimental
Ss
so
6:41
6:38
.
data. The proper values of these exponents
are discussed later in the chapter.
employed as a means of obtaining standard
power during test. The quantity Fmo itself
is determined directly from Eq. 6:39 by re-
writing this equation in the form
Wo
2
As noted before, changes in density pro-
duce a direct aerodynamic effect on take-
off distance, which is essentially independent
of the change in Fm due to density. Conse-
quently, it is possible to handle changes in
Fm separately from changes in the other
variables.
Fmo
:
[Vg. - Vw.)?
2950
where Wo, so, Vgo and Vwo are the observed
test values,
The important factor altering Fm is vari-
ation of engine thrust due to non-standard
atmospheric conditions, improper engine set-
tings, or any of the other causes which in-
fluence power plant operation. Proceeding
as before, the effects of variation of Fm are
best isolated by holding the quantities W,
Vw, and
and p constant and permitting Fm alone
to vary.
In this case, Eq. 6:34 provides
As noted, the procedures developed here
presume that the corrections made are all
relatively small, and within the framework
of this assumption, the foregoing work may
be employed for "JATO" take-offs as well
as normal take-offs even though the 'JATO"
unit firing time does not equal the take-off
time.
sofmo.
Fms
Ss
To conclude this brief analysis of take-off
characteristics, the preceding work is sum-
marized below in a collected form, and the
data reduction procedure outlined.
6:46
But Fms differs from Fmo by an amount
AFm, the deviation from standard, so that
AFm : Fms – Fmo or
Because the corrections for weight and
density were obtained on the basis that the
observed data had previously been corrected
to zero wind conditions, the wind correction
is made first and is followed by the remain-
ing corrections. From an examination of
E28, 6:41, 6:44, 6:45 and 6:47 it follows that
the complete correction formula to zero wind
standard is
Fms
8
Fmo + AFM.
Substituting in Eq. 6:46, we obtain
Vgo
Ws
1
Ss
so
-) ( )
Ss
: So
Voo
AFM
W
- vwo
AFm
1+
1+
Fmo
Fmo
6:48
6:47
where "sub o" indicated observed (as tested
quantities) and “sub s" the standard quan-
tities.
For practical purposes, the quantity AFm
may be taken as the difference between the
thrust available on a standard day and the
actual thrust available during the test. This
increment may be computed using engine
charts and standard engine correction pro-
cedures or torquemeter equipment may be
If it is desired to correct to a standard
wind velocity at a standard weight, an addi-
tional wind correction term is required
based on standard getaway speed Vgs -
6:39
Since
2 w
va
PCLS
The data on which the exponents of the
wind and weight correction terms are based
has been drawn from many sources and has
been accumulated over a long period of time.
Early test data (which are presented in a dif-
ferent form than used here) are given in Fig.
216 of Ref. 4 and computed data in Fig. 217
of the same reference.
By plotting the data of Fig. 216 in the form
log (so/ss) vs. log (1 - VW/Vs ) we obtain
the relation
we have that
Ws
Ves
Vg
.
Wo
Ps
)
i
1.85
Vgo
Ws2
:
Vgs
Vgo
ܘܘ
ss
so
wo
s
- Vwo
V90
6:49
6:51
(Note:
V, is not a function of Vw or Fm.)
Using Vgs defined by Eq. 6:49 and multi-
plying Eq. 6:48 by the additional correction
term
Vgs - VWS
Vgs
It is believed that these data were obtained
for tail skid type alighting gear having higher
ground friction coefficients than found in
present day airplanes. More recent data
indicate exponents of 1.8 to 1.9 with some
evidence favoring the value of 1.9. On the
basis that the corrections are of small mag-
nitude, using a value of 1.85 produces neg-
ligible errors.
Considering the weight correction term,
various references give values of from 2.1
to 2.6 for its exponent. For modern aircraft
a value of 2.2 or 2.3 seems most realistic.
Thus the equation
the following is obtained:
M90
Vgs - Vws
ss
so
Х
Vgs
Vgo
- two
Ws
1
)
6:50
Wo
ss
W.
2.2
AFM
- (
So
Fimo
Wo
6:52
where
Ws
Vgs
:
(
Wo
vo Cewek
(
en [V8o – vwol".
.
's
Wo
Fmo
2950
(6:50
cont'd)
In the preceding analysis it was mentioned
that second order variations in Fm due to
wind and weight effects had been ignored and
would be accounted for by modification of the
exponents of wind and weight correction
terms. These modified exponents are dis-
cussed below.
is recommended.
Ref. 5 indicates that the exponent of the
weight term of Eq. 6:49 should be .476 rather
than the theoretical 1/2. This indicates a
modification of pilot technique due to change
in gross weight rather than a change in any
other factor. The data determining the value
of .476 were, it is believed, obtained for
conventional gear reciprocating engine air-
craft, and as such are probably not directly
applicable to tricycle alighting gears. Ac-
cordingly, the factor of 1/2 is recommended
pending further investigations.
6:40
Summarizing, Eq. 6:50 with empirical exponents becomes
1.06
1.85
2.2
Vgo
Ws
Pobs
1
ss
so
Vgs - Vws
Vgo - Vwo
)
Vgs
Wo
DFm
Ps
1+
Fmo
6:53
where
Ws
Vesivo choses
) i
and
Fmo
8
Wo
2 g so
[vco -vo)
wo
Vgo
Wo
Ps
(6:53
cont'd)
Test Procedures
There are two basic approaches used to
obtain take-off data; these are:
(1) Determination only of total distance
and end speed
before take-off by causing the aircraft to
travel over tar strips whose separation dis-
tance is known. Knowing the separation
distance and the time taken to cover the
distance between adjacent strips, the speed
just prior to take-off can be computed; more-
over, the point of take-off is fixed between
two strips by the disappearance on the ac-
celerometer trace of blips and roughness
caused by passage over the tar strips and
runway surface.
(2) Determination of a space time rec-
ord of the entire run.
Various other schemes have been proposed
using limit switches on the gear to indicate
full extension and so forth. Most of these
can be made to work, if sufficient attention
is paid to detail.
For routine testing, the first procedure is
all that is required, and the data are obtain-
able in several fashions. The most obvious
procedure is to estimate first the take-off
distance from some fixed reference point,
and then to place a motion picture camera at
the approximate take-off point so that pictures
of the actual take-off are obtained. Knowing
the film speed (using a timing pendulum or
stop watch in the field of view) and the dis-
tance of the airplane from the camera, the
take-off ground speed is obtained. Problems
arise here in the determination of the exact
point at which the airplane leaves the ground,
and generally several observers must be
employed to obtain a reasonable estimate of
the take-off point.
When more information is desired, space-
time records can be obtained using a the-
odolite tracking camera with or without a
grid screen or some device similar to the
new Fairchild Associates distortion cor-
rected tracking camera.
In the event that such data as take-off
distance over a fifty-foot obstacle are re-
quired, the use of a tracking camera is al-
most mandatory.
Analysis of Obstacle Take-offs
Another procedure utilizes accelerometers
mounted on the shock struts as a basis for
determining the take-off point, and revolution
counters attached to the wheels. The method
is based on determining ground speed shortly
-
The problem of determining the ground
distance required to clear a fifty-foot ob-
stacle (or an obstacle of other height) under
6:41
The ground run portion of the data may
be corrected by the procedures previously
developed; however, the air distance re-
quires further consideration. For this anal-
ysis, the actual curved flight path shown in
Fig. 6:25 as a dashed line is represented
by a solid straight line also shown in the
figure.
The required wind correction on sa is
apparent from the figure. Suppose that the
take-off were conducted into a wind of ve-
locity Vw, and a time "q" was required to
cover the observed distance sao. The wind
has no effect on the time, as will be seen on
a moment's reflection. However, it shortens
so by an amount Vwt.
.
Thus, the observed air distance is cor-
rected to standard no wind conditions by the
relation
SOS
Sa. + Vwt.
500
6:55
standard conditions is more complex than
determination of standard ground run itself.
There are a number of reasons for this with
the principal one being the large possible
variation in pilot technique. However, pro-
vided one is satisfied with reasonable ap-
proximations it is possible to correct to
standard distance required to clear an ob-
stacle of a given height. The problem is
analyzed below.
First, it should be realized that the prob-
lem of achieving minimum distance from the
start of the run to over the obstacle is dif-
ferent from the previously analyzed problem
of minimum ground run. In achieving mini-
mum distance on the ground, the pilot is not
nearly as concerned with the initial climb
characteristics as he must be when he desires
to clear an obstacle at the end of the run-
way.
For good climb-out characteristics, it may
pay to remain on the deck longer when at-
tempting to clear an obstacle than when
seeking minimum ground run. The reasons
for this are simply that the speed for best
climb angle is generally somewhat greater
than the minimum getaway speed, and the
possibilities of performing a zoom maneuver
are better with more speed available.
This means that the ground run data ob-
tained for minimum take-off distance on the
deck is not usable for obstacle take-offs
even for the ground run portion of the ob-
stacle take-off runs.
The procedures for obtaining data may
vary; however, the use of some type of photo-
tracking device which provides a space time
record is mandatory. In addition, records
of wind velocity, air density, weight, etc.
should be kept,
The actual distance component parallel to
the ground from the start of take-off to over
the obstacle is best broken into two parts,
the ground run distance Sg and the projec-
tion of the flight path from getaway to over
the obstacle so. Thus, the total distance is
Similarly, the corrected value of the aver-
age flight path angle is given by
tan ys
.
h
Sao + Vwt
and since tan yo
Yo
:
h/800;
ton Yo
(1+
Vw
I
)
tony's
Sao
6:56
Assuming that the observed data have been
corrected for wind effects, the remaining
corrections can be made on the basis of
Eq. 9:25 of Ref. 3, which is rewritten below
in modified form.
T CO
T
CL
w
sin y :
Со
nCL
s.sg
+ sa
6:54
6:57
6:42
h
VW
so
Flight Path
Sg
a
h : Obstacle
Height
S
td
-Take-off Point
Fig. 6:25
where
9:26 of Ref. 3 for CL, whence
n* : 4 for values of y less than 30°
2 WCOS Y
CL
:
.
n
: 2 for values of y between 30º and
45°
pVPS
(Note that y is not a function of air density
except as this factor alters the thrust.)
The value of Co corresponding to CL may
be obtained from wind tunnel test data or
performance flight test data. For propeller
driven aircraft, the power required curve
serves the basis for determining the
equation for the lift drag polar,
as
)
In Eq. 6:57 the observed value of y is the
angle between the straight line approxima-
tion to the flight path and the horizontal and
is the approximate average value for the
actual flight path.
cz
CO :
CDE
+
TAR
The average value of CL, at which the
aircraft flies during the time the distance
So is traversed, is obtained by solving Eq.
from which CD is readily calculated for any
test value of CL.
* Note: This improves the approximation
given in Ref. 3.
The variation in observed thrust from
standard may be obtained from the engine
6:43
Procedure for Correcting Obstacle
Take-off Data
charts using a reasonable assumed value of
propeller efficiency. Letting AT be the thrust
increment, defined by the relation
Ts : To + AT
+
6:58
8
(a) First reduce the ground run data to
standard by means of the corrections dis-
cussed earlier in this section.
(b) Reduce the observed air distance and
flight path angle data to no wind conditions
by Eqs. 6:55 and 6:56
where
Ts :
standard thrust
To : observed thrust
SOS
Sao
+ Vwot
To + AT
Ws
tanys
• ton Yo
1
sinyo
Vwt
CO
CL
Со
n CL
1 +
soo
(c) Compute the average value of CL dur-
ing the climb-out using the relation
6:59
Then
2 Wocos Yo
CL
Pobs tas
sin Ys
To + AT
CD
Ws CL
то Ср
Wo CL
sinyo
-
6:60
where the value of Yo is the observed value
corrected to zero wind conditions.
(d) Knowing the average test value of
CL, obtain the value of the average drag
coefficient from level flight performance
data.
where Ws: standard gross weight,
W. = observed gross weight.
From Fig. 6:25
(e) Using Eq. 6:57 solve for the average
T/W ratio, and knowing the gross weight,
obtain the average thrust during the climb-
out.
h
Soo
tanyo
CO
().." sin yo ()
CO
+
Yo
1 -
пCL
CL
h
SOS
tonys
so that
(f) Determine the deviations from stand-
are thrust from engine charts and compute
the standard no wind value of y from Eq.6:60
TO + AT
Со
Ws
SOS
ton Yo
CL
3
Sinys
sin yo
500
To CD
Wo CL
tanys
6:61
6:44
(g) Compute the standard no wind air
distance from Eq. 6:61
for weight are attempted inasmuch as the-
oretical considerations alone do not seem to
give proper answers. Accordingly, landing
tests should be made at weights as close as
possible to standard.
tonyo
SOS
Soo tonys
(h) If data on obstacle take-offs into
standard wind are required, compute the
average true air speed under standard con-
ditions from
Since the landing run analysis is precisely
the same as the take-off run analysis, ex-
cepting that there is no effect due to chang-
ing engine operating conditions (except for
airplanes with propeller pitch reversing
mechanisms), Eq. 6:53 may be rewritten
for the ground run during landing in the
following form:
2 Wscos Ys
Voss
PS CLS
INDO
1.85
1,65
Sgs
sgo (VDs
VDS - Vws
VDO - VW
x
-
(i)
from
Compute the new air distance time
2.2
Ws
Pobs
Ps
Wo
6:62
sos
ts
:
Viass
when Sgs
8
: standard ground distance;
(j) Compute the wind correction sws
ts Vws and subtract from the distance in
step g to give air distance to clear an ob-
stacle in the standard wind
sgo
: observed ground distance,
For aircraft equipped for propeller pitch
reversal an additional thrust correction may
be added.
sas
SO 59
SWS
(k) The total corrected distance from the
start of motion to over the obstacle is given
by either the sum of a + g or a + j.
Eq. 6:62 provides for the ground run cor-
rection. If landings over obstacles are to
be analyzed, we must also consider the air
distance corrections. According to Ref. 2,
the only correction on air distance which can
properly be imposed is that accounting for
the effect of wind. Thus, if
6:12 MEASUREMENT AND CORRECTION
OF LANDING CHARACTERISTICS
sao
= observed air distance
-
Sas
: standard air distance
Actual test measurements of landing run
characteristics are accomplished using the
same equipment and similar procedure as
those used for take-off measurements.
LOO
: observed time for sa
VWO
= observed wind velocity
Because of the wide variation in pilot
technique encountered during landings, re-
peatable data for landing runs are more
difficult to obtain than for take-off runs.
This shows up particularly when corrections
Vws
: standard wind velocity, then
-
sog
soo
+ (Vwo
- Vws) too
6:63
6:45
Combining Eqs. 6:62 and 6:63
voo
1.85
Vws
1.85
Ws
2.2
Pobs
:
+ sos
Sgs
Ss
sgo \ VOS
vo var en meg) ** ()
VDO
Vwo
Wo
Ps
+ SOO
+ ( Vwo - VWs) too
6:64
The data reduction process required with
Eq. 6:64 is similar in all respects to that
required for landing analysis, and, therefore,
need not be discussed in further detail here.
utilized. In this case, the test data obtained
over a wide altitude band may be expected
to display scatter over and above that attrib-
utable to propeller efficiency variations and
similar effects considered earlier in the
chapter.
6:13 CONCLUDING REMARKS
In this chapter we have summarized in-
formation on performance testing of recip-
rocating engine aircraft, with all important
phases being analyzed. Special topics, such
as engine cooling investigation, were not in-
cluded since these are matters which depend
more on specifications than on fundamentals.
Finally, we note that although the effects
of thrust produced by the engine exhaust
stacks were not considered here, significant
amounts of power may be generated by this
thrust even though an ejector system is not
If desired, exhaust thrust may be con-
sidered in the data analysis using methods
similar to those described in Chapter 5.
However, it should seldom be necessary to
go to such extremes; rather, an attempt
should be made to separate high and low
altitude data by reducing all data taken above
20,000 feet (for instance) to a 20,000-foot
level and data obtained at altitudes lower
than 20,000 feet to sea level. This procedure
provides two sets of “generalized" data
curves and generally may be expected to
reduce the scatter of data.
(
6:46
REFERENCES
1. Dommasch, D. O., “Flight Test Manual, Part I," Revised Edition, Preliminary Copy,
NATC, Patuxent River, Maryland, August, 1953.
2.
Herrington, R. M. and Shoemacher, P.E., AFTR No. 6273, Revised January, 1953, "Flight
Test Engineering Manual.”
3. Dommasch, D. O., Sherby, S. S., and Connolly, T. F., “Airplane Aerodynamics," Pitman,
1951.
4. Dichl, W. S., “Engineering Aerodynamics,” Ronald, 1936.
5. Forry, J. E. and Horton, C. F., “Manual of Flight Test Methods and Procedures, Part I,"
Revised 1 June 1948, NATC, Patuxent River, Maryland.
6:47
1
AGARD FLIGHT TEST MANUAL
VOLUME I, CHAPTER 7
TOTAL ENERGY METHODS
Ву
Kenneth J, Lush
AFFTC, United States Air Force
PARTI
Bernard Davy
Centre d'Essais en Vol, France
PART II
D, O. Dommasch
Princeton University
FOREWORD and PART III
J. F. Renaudie
Centre d'Essais en Vol, France
APPENDIX A
3
VOLUME I, CHAPTER 7
CHAPTER CONTENTS
Page
FOREWORD
TERMINOLOGY
PART I, OPTIMUM CLIMB THEORY AND TECHNIQUES OF
DETERMINING CLIMB SCHEDULES FROM FLIGHT TEST
7:1
7:1
INTRODUCTION
7:2
7:2
A SIMPLIFIED THEORY
7:2
7:3
END CONDITIONS
7:5
7:4
AN EXACT APPROACH
7:5
7:5
MISCELLANEOUS CASES
7:8
7:6
EFFECT OF OPERATING CONDITIONS
7:9
7:7
DERIVATION OF THE OPTIMUM CLIMB SCHEDULE FOR THE
PARTICULAR CASE
7:9
7:8
INSTRUMENTATION AND ANALYSIS
7:11
(a) Climbs
7:11
(b) Level Accelerations
7:11
7:9
STANDARDIZATION OF RATE-OF-CLIMB CURVES
7:15
7:10
REFERENCES
7:15
APPENDICES
1
FLIGHT TEST TECHNIQUE FOR ACCELERATED LEVELS
7:16
II
ANALYSIS OF AIRSPEED INDICATOR TIME HISTORY FROM
LEVEL ACCELERATION
7:17
PART II, ANALYSIS OF THE OPTIMUM ENERGY CLIMB SCHEDULE
7:19
7:11
INTRODUCTION
7:19
7:12
DEFINITIONS
7:19
CHAPTER CONTENTS
(Continued)
Page
7:13
THEORETICAL ANALYSIS OF THE OPTIMUM CLIMB
SCHEDULE
7:20
7:14
GEOMETRIC REPRESENTATION
7:21
7:15
PROPERTIES OF THE GRAPH
7:23
7:16
DISCUSSION OF ASSUMPTIONS
7:24
7:17
CORRECTIONS
7:26
7:18
PRACTICAL DETERMINATION OF THE OPTIMUM ENERGY
CLIMB SCHEDULE
7:27
7:19
CONCLUSIONS
7:31
PART III, CORRELATION OF THE ENERGY CLIMB ANALYSES
7:32
7:20
REVIEW OF PRECEDING ANALYSES
7:32
7:21
THE THIRD METHOD OF EXPRESSING THE CONDITIONS FOR
OPTIMUM ENERGY STORAGE
7:33
7:22
PRESENTATION OF THE THIRD SET OF CONDITIONS IN
TERMS OF V AND h.
7:35
7:23
SUMMARY
7:36
APPENDIX A, PROCEDURES FOR DETERMINING THE ENERGY
CLIMB PERFORMANCE OF TURBOJET AIRCRAFT
7:38
7A:1
INTRODUCTION
7:38
7A:2
COMPUTATION OF TAPELINE OR GEOMETRIC ALTITUDE AND
ENERGY HEIGHT
7:38
7A:3
DETERMINATION OF W: dhe/dt
7:40
7A:4
CORRECTION PROCEDURES
7:41
7A:5
DETERMINATION OF CORRECTED w= dhe/dt AND CORRECTED
TIME-TO-CLIMB
7:45
7A:6
SUMMARY OF PROCEDURES
7:46
7A:7
CONCLUDING REMARKS
7:48
VOLUME I, CHAPTER 7
TOTAL ENERGY METHODS
FOREWORD
Starting with the original German concept of energy
height, separate analyses of the optimum energy climb
schedule have been carried out independently by the several
NATO nations. Because of different approaches to the
problem and the use of different notations, at first glance
there appear to be some basic variations in the suggested
procedures. However, the several analyses do arrive at
the same fundamental conclusions, and it is worthwhile to
point this fact out at the beginning of the chapter.
The actual correlation of the methods is presented in
Part III of this chapter.
In an effort to preserve originality, the papers, sub-
mitted by Mr. Lush of Wright Field (which also reflects
the British approach to the problem) and Mr. Davy of the
Centre d'Essais en Vol of France, have been left in essen-
tially the form submitted (although certain idiomatic changes
were of course required in the translation of the latter
paper). No attempt was therefore made to correlate the
two papers by modifying the papers themselves. Rather, it
was felt that a more satisfactory correlation could be
achieved by adding a third section to the chapter, which
would serve to summarize the joint thinking expressed by
both authors and to introduce a third approach to the prob-
lem.
)
The order of appearance of the first two papers in this
chapter was determined by the date of their arrival, rather
than by any other considerations and therefore no signifi-
cance should be attached to the sequence.
TERMINOLOGY
h
True Altitude
he, H
Energy Height
V
True Airspeed along Flight Path
E
Total Energy
W
Weight
m
Mass
W
dhe/dt
t
Time
F, T
Thrust, also Temperature
Ft
Excess Thrust
o
Flight Path Angle
N
RPM
T
Temperature
n
Load Factor
RO
Rate of Climb with Zero Flight Path Acceleration
PART I
OPTIMUM CLIMB THEORY AND TECHNIQUES OF
DETERMINING CLIMB SCHEDULES FROM FLIGHT TEST
ABSTRACT
Consideration is given to the problem of
determining the optimum technique by which
an airplane will pass from given initial con-
ditions to assigned final conditions in the
minimum time. It is shown that introduction
of the concept of energy height enables a
simple approach which takes proper account
of the kinetic energy of the airplane and
appeals to physical intuition.
It is concluded that on the quasi-steady
climb the speed height schedule should be
such that at each level of total energy level
attained, the speed is that which gives the
most rapid increase in total energy. This
is equivalent to the condition
ORO
av
V ORO
g on
.
Other climb cases are discussed briefly.
It is concluded that the climb technique for
minimum time will give a range performance
very close to the best possible. The problem
of achieving the greatest speed in a given
time is discussed briefly.
Determination of the optimum schedule
from flight test is considered. It is concluded
that this is most conveniently done from level
accelerations or from full climbs to a range
of speed height schedules. The latter have
advantages if it is reasonably practicable to
make a performance climb on most flights,
but it is essential to have reliable wind grad-
ient corrections, either from weather data or
from radar tracking on a plotting table,
Otherwise level accelerations, in which the
airplane accelerates horizontally in substan-
tially level flight with the engine at the re-
quired rating, are better.
Instrumentation and analysis for level ac-
celerations is discussed at length, methods
considered and the following conclusions
reached:
(a) Accelerometer. Convenient if satis-
factory instrumentation is achieved, but very
high sensitivity is required. As yet unproved.
(b) Airspeed Indicator and Clock. Incon-
venient, but available and in successful use.
Analysis routine given in appendix.
(c) Time Trace of Dynamic Pressure,
Available but unproved. Should work and is
much more convenient than (b) and probably
as good as (a).
(d) Rate-of-Climb Indicator Connected to
Total Pressure. Unproved. May be very
convenient for a quick check, but unlikely to
be suitable to give absolute values.
(e) Air Temperature Thermometers with
Different Recovery Factors. Unlikely to be
sensitive enough. Probably not sufficiently
superior to (c) to be worth developing.
where V is the true airspeed, h the tape-
line height, and Ro the rate-of-climb which
would be attained at the revelant speed and
height with zero longitudinal acceleration.
The transition between the end conditions
and the quasi-steady climb is discussed, and
it is concluded that it is sufficient to ensure
that these phases are as brief as possible.
This requires horizontal or near horizontal
acceleration at the end of a climb.
The above approximate approach suffices
until, with very fast climb, limited maneu-
verability, and low aspect ratio the technique
in the initial and final phases of the climb
becomes so important that it must be deter-
mined quantitatively. A more complex ap-
proach is then required. A possible method
is indicated, but it appears unlikely that any
general solution is possible, and numerical
solution is not attempted here.
7:1
7:1
INTRODUCTION
In the days of the biplane fighter the climb
consisted essentially of increasing the poten-
tial energy of the airplane, any incidental
changes in kinetic energy being small. It was
then customary, and entirely legitimate, to
ignore the kinetic energy of the airplane both
in selecting climb schedules and in predict-
ing climb performance.
these reduce to differential equations of the
second order and second degree in the load
factor (lift/weight) which have varying coef-
ficients. Routine solution of these equations
is not practicable, and does not give much
physical insight into the problem. For the
moment, therefore, we will consider a simple
theory.
The climb equation for an airplane in near
straight flight in still air may be written in
the following forms, assuming a consistent
system of units:
The advent of the monoplane of relatively
high wing loading increased climb speeds and
also increased the speed range between climb
speeds and maximum level speeds. This in-
crease was sharply accelerated by the adop-
tion of jet propulsion.
W
dV
V(F-D) : W.
dh
dt
S
+
·V.
g
dt
7:1
V
dV
dh
: W:
dt
(
(+)
9
dh
7:2
Consequently, the kinetic energy changes
associated with the acceleration from take-
off to climb speeds, with the quasi-steady
climb and with the passage from climb to, for
example, maximum level speed, became im-
portant. Attention was first directed to the
second of these, and it became normal prac-
tice to make allowance for the change in kin-
etic energy with altitude when predicting per-
formance in the quasi-steady climb. It was
not until German engineers introduced the
concept of "energy height", fairly late in
World War II, that a simple but satisfactory
presentation of climb performance was ob-
tained
d
: W
wit (nt
h +
o
dt
7:3
where V :
true airspeed
F = total net thrust
The aspect of climb performance of most
general interest is the achievement of a pre-
scribed change of height and speed in the
least practicable time. This will be consid-
ered at length below, and brief reference will
be made to other problems. Given a climb
technique, the establishment of climb per-
formance is simple enough. We will, there-
fore, study the theory of selecting climb tech-
niques.
D = drag
h
altitude above some arbitrary
datum
10
W = weight
7:2 A SIMPLIFIED THEORY
In a given atmosphere and at a given en-
gine rating, F and D are functions of W, V and
h. If initial gross weight is now fixed we
may omit W, as effect of climb technique on
weight variation with height would not be ex-
pected to have any appreciable effect on the
optimum technique.
It is possible to write down the exact
equations for the technique for minimum
time between specified end conditions, but
7:2
With the above in mind, we now write
and hence
dh
dV
ffv, moet on
n,
dhe
,
di
dh
11,2
hezo V2
x (V, he)
ne, ovi
7:6
and proceed to find the climb technique which
minimizes the time required to pass from a
speed V and a height h, to a speed V2 and
a height h2 by minimizing the integral
h2o V2
-
Minimization of this integral is in principle
very simple, and is achieved by making X as
large as possible. At those points on the
climb away from the influence of end condi-
tions, the necessary condition for maximiza-
tion of X is
ах
dh
1,2
hvi
f(
V, h,
)
dV
dh
o v
7:7
using the Euler-Lagrange conditions. How-
ever, a clearer physical picture is obtained
if we substitute for h in terms of the quantity
"energy height", he, defined by the equation
v
h +
where the partial differentiation is, of course,
made at constant he. That is, the schedule
of the quasi-steady climb is to be such that
at each level of total energy achieved, the
rate of increase of total energy is as large
as possible. (Such a schedule will not neces-
sarily give the most rapid increase of height,
or potential energy.)
:
he
siä
7:4
Energy height is clearly the height equiv-
alent of the total energy of the airplane. Its
use amounts to a recognition of the fact that
the kinetic and potential energies of an air-
plane are fairly readily interchangeable and
that it is the total energy, rather than either
the kinetic or the potential energy, which must
be increased as rapidly as possible to mini-
mize climb time. Substituting from Eq. 7:4
into Eq. 7:3 we have, of course,
This conclusion may be drawn semi-intu-
itively if it is first agreed, as indicated above,
that the division of the airplane's energy
between kinetic energy and potential energy
can be adjusted fairly rapidly by climbing or
diving, and is therefore of no consequence
except at the beginning and end of the climb.
dhe
Eq. 7:7 is simple in appearance, but is not
readily applied in practice as both calculated
and experimental values of X are most read-
ily obtained as curves of X against V at con-
stant height. It is therefore desirable to cast
the equation in another form,
VIF-D)
W
()
dt
With our assumption of constant weight,
F and D are functions of V and h only, and
hence, from Eq. 7:4, of V and he only. We
may therefore write
As X is a function of V and he, and he is
itself a function of h and V, we may write the
identity
dhe
dhe
= X(V, hel
x(he, V):
$(h, v).
dt
dt
7:5
7:3
But
We may now rewrite Eq. 7:7 in the more
convenient form
1
2
ola
he Sht
дф
у дф
• O
ду
9 on
7:14
By definition we therefore have
ох
ax
dV+ dhe = dx
.
av
ane
Given computed or experimental curves of
$ against true airspeed V for various heights
we may estimate 0/0 h, which is fairly
small, and thus deduce the slope of the rele-
vant curve at the optimum speed.
дф
дф
= d$ = dV+
av
dh
dh
7:8
and
It is of interest at this stage to compare
the schedules deduced from Eq. 7:14 with
those deduced by earlier methods. Two meth-
ods have been used. These are, in historical
order: (a) to take as the schedule speed, the
speed which gave maximum rate of climb in
a “sawtooth climb,” at constant indicated
airspeed, through the relevant height; and (b)
to take the speed at which o/a V is zero.
dhe = dh +
tř. av
dV
7:9
Substituting from Eq. 7:9 into Eq. 7:8 to
eliminate dhe , we have
ox
ах
Ох
V
+
9
.)
DV +
dh
ane
av
ahe
дф
дф
av
dV+
ola
dh.
Consider (b) first. With air breathing en-
gines 04/0h is negative, so the correct op-
timum speed as given by Eq. 7:14 will be
higher than that given by (b). In the absence
of compressibility effects, this difference
usually amounts to between 5% and 10% in
speed, which is insufficient to cause any no-
ticeable loss in climb performance. If the
optimum speed is dictated by compressibility
effects, the two conditions will give closely
similar speeds and the loss incurred by using
condition (b) is again very small.
7:10
Equating coefficients of dh and dV we have
ax
.
ola
ane
7:11
and
Method (b) is thus adequate and may be
used as a convenient approximation instead
of using Eq. 7:14. This, of course, does not
in any way detract from the energy height
approach to the climb problem, as it is only
by reference to the exact condition that we
can evaluate the approximate condition.
дф
ol
Vax
+
ov
9 dhe
av
7:12
Eliminating a X/he we now have
ах at v дф
However, condition (a), which was used for
many years with slower airplanes, is more
seriously in error. When climbing at constant
indicated airspeed, the proportion of the
available thrust power which goes into in-
creasing the kinetic energy of the airplane
ov
ду
9
on
7:13
7:4
increases with increase in forward speed.
Consequently, the sawtooth climb curve peaks
at a lower speed than the curve of $ against
V at constant height and condition (a) gives
an even lower speed than condition (b).
airplane approximates the required final
value. The airplane should then be pushed
over into a moderate dive so as to level out
at the required altitude and speed.
The increased negative error (for air-
planes with air breathing engines) leads to
an appreciable loss in climb performance of
the order of 5% to 10%. This is not tolerable.
Thus, condition (a) must be rejected. This
does not, of course, mean that sawtooth climbs
may not be used to determine optimum climb
schedules, only that correct interpretation is
required.
This is a vague statement, but fortunately
quite a moderate dive angle will result in a
short acceleration period and should, there-
fore, give results approximating the best
possible. If the required final speed is low,
or if the aim is to achieve altitude quickly
regardless of speed, the climb will, of course,
be ended by zooming and leveling off at the
required altitude.
7:3
END CONDITIONS
It may be noted that with some airplanes
which operate up to, but not past the transonic
drag rise, climb and maximum speeds may
be very close together and acceleration in
level flight rapid. It would then be sufficient
in many cases to level off at the required
altitude.
We have derived above a condition which
accurately determines the optimum schedule
for the quasi-steady part of the climb. This
simple theory cannot determine quantitatively
what the techniques of entering and leaving the
climb should be. Useful qualitative conclu-
sions may be drawn, however.
It is worth remarking that with present
military airplanes, no precise meaning can be
attached to the time to reach operational
altitude unless initial and final speeds are
quoted. The initial condition is probably best
fixed by quoting the time taken to reach
climb speed from the start of the ground roll.
The final condition is arbitrary. One may
quote time to reach altitude at climb sched-
ule speed provided that it is made clear what
the end condition is.
During these parts of the climb, the rate
of increase of energy is necessarily less than
the maximum possible. The time spent in
these unfavorable conditions should, there-
fore, be made as short as is practicable. The
extent to which this aim is realized is limited
by the maneuverability of the airplane. Its
acceleration cannot exceed that in a vertical
dive and is further limited immediately after
take-off by the presence of the ground, while
its deceleration is limited to that in a verti-
cal climb. Further, it can only change the
angle of its flight path at a limited rate, par-
ticularly at high altitude.
The above qualitative approach is likely to
suffice for airplanes of the class presently
operational,
For supersonic airplanes of very low
aspect ratio, however, it may be necessary
to adopt a more quantitative approach. This
will be discussed briefly in a later section.
7:4
AN EXACT APPROACH
It is clear that after take-off, the airplane
should be kept at as low an altitude as is
practicable until it approaches the climb
speed, then pulled up fairly quickly into the
climb. If the final speed is to be much great-
er than that given by the climb schedule at the
final altitude, the climb should be continued
past that altitude until the total energy of the
We will now examine briefly a more exact
approach to the problem, which will take into
account the limited longitudinal acceleration
7:5
and
and deceleration available and also the varia-
tion of induced drag with normal acceleration.
V. dy
dy
dt
8
용
​la
.
비
​9
dhe
9
dt
dhe
We will retain he as the variable instead
of h, inasmuch as he increases monotonical-
ly with time over the whole climb in almost
all cases of interest, and we thereby avoid
difficulties with double values.
$ (n - cos y,
7:18
We may write
We now denote total differentiation with
respect to he by a dash (for example,
dy/dhe = Y'). Also we write
dhe
V (F-D)
dt
W
5 + $ sin y
: f(V, he,n)
8
= - \ IV, V, he,n,y)
7:15
in a given atmosphere and at a given engine
rating, where n is the load factor, defined
as the ratio of the lift to the gross weight,
řx'- $ln - cos y)
= XIV, he, n, y, y').
>
Integrating between initial and final con-
ditions we may write
Then our problem is to minimize the
integral
dt
11,2
dhe
dhe
sød
$ dhe
3
subject to the additional conditions $=0=X.
s'o V, ne,n) dhe.
7:16
We write
F: + 4,W + 12x
We have auxiliary conditions concerning
the longitudinal acceleration and the rate of
pitch, which we
we may write in the form
7:19
dt
where , and 2 are functions of he which
are to be determined.
1
O
ov
dhe
DV
dt
g
dhe
D
$
{
Standard text books on variational calculus
show that a necessary (but not sufficient) set
of conditions is
sin y
{;
}
W
where y is the angle of climb
ola
: 0
av
dhe lov
olo to 15
• {+ -
()
()
d
$ sin y
= 0
an
dhe
on
7:17
7:20
7:6
d
af
d
dhe
ar
dhe
d
OF
ela
(
105) (125
. +42 (+ - + siny).
: 0
dhe
ду
x2
$
7:26
g
(7:20
cont'd)
Substituting from Eq. 7:21 through Eq. 7:26
into Eqs. 7:20 we have, after rearrangement,
where V, V', n, n', Y and y' are to be trea-
ted as independent during differentiation.
.
PLY
+ 1, sin y - 121n-cosy)
, ys}
av
Considering each function in turn, we have
+
t
OF
av
1.0.
+ 12 j In-cosy) -
дф
OV
+
+
sin Y
9
히 ​긍
​ov
7:27
+
0 %
-12
-
do
(n-cosy)
av
FE+ ( )
na iyo
o $ {1+11 sin 7-12 (n- cos x)}
, Y y
+d2. in-cos y),
1 + 1, sin y
-d21n-cos
coom) - 2 - 0
: 0
on
7:28
3
In-,
X2
1, $ cos y
7:21
- ه -
.
0.
V
7:29
d OF
dhe love
(0)
:
dhe
:
“Ai
7:22
Eq. 7:29 enables substitution for , in
terms of dę and 'g. After differentiation it
also enables substitution for ti. We then
have one equation in ' , l'2 and 12 and one in
X'q and 12. Both equations will have varying
coefficients. Solution of these equations in
practice would probably be best made by
some step-by-step process.
af
:
дф
on
Y
비​"
on
{{1+, sin )
- 121n- cos 1)} -$12,
7:23
It is of interest to apply to the above equa-
tions the simplifying assumptions used in the
preceding section. We assume that is in-
dependent of n, and deduce from Eq. 7:28
that
(5) 。
8
(0) : 0,
dhe
dhe
7:24
$d2 = 0,
OF
that is,
1, $cos y - 12$ sin y
ay
da :0.
7:25
7:30
7:7
As this is to be true throughout the climb,
we also deduce that
examination will be made which will sug-
gest that a precise treatment is not worth-
while,
dà
= 0.
7:31
Substituting in Eq. 7:27 and Eq. 7:29 we
then have
Consider a modification of the climb tech-
nique for minimum time in which the airspeed
at each point on the climb is increased in the
constant ratio (1 + i). The fuel flow at each
height will be sensibly unchanged, so the dis-
tance covered for each pound of fuel consumed
will be increased by approximately 1 times
average miles/lb. on climb.
дф
{1 + 1, sin Y}
1+ } -
o
g
ду
7:32
1,0 cos y = 0.
7:33
On the other hand, the time required to
reach cruising height and speed, and hence
the fuel used on the climb, will be increased
by an amount of the second order in . The
cruising range will be reduced by the product
of this increase in fuel used on climb with
the average miles per pound cruising.
From Eq. 7:33 either
COS Y
O,
y
: $ 90°
7:34
Or
, = 0
Thus it is evident that at least a small in-
crease in the climb speed schedule will be
profitable. Examination of particular cases
tends to show, however, that the most profit-
able increase in speed is only around 5%
giving an increase in range of rather less
than 5% of the distance covered on the climb,
which is itself only a fraction of the total
range.
and hence from Eq. 7:32
дф
: 0.
ду
7:35
We thus have the two solutions discussed
in the preceding section, namely
Thus it would appear that only small im-
provements in range can be achieved by de-
parting from the climb schedule for minimum
time and that a precise examination is hardly
worthwhile.
a. (oflav) = 0 during quasi-steady climb
b. vertical ascent or descent to pass be-
tween end conditions and quasi-steady climb.
7:5
MISCELLANEOUS CASES
A third problem which may arise is the
achievement of the maximum possible speed,
or Mach number, in a given time. This prob-
lem will be most likely to arise with air-
planes having high induced drags at high lift
for which a more exact approach, similar to
that outlined in Section 7:4 would probably be
required. It could be stated in the form of
maximizing the integral
Determination of the climb technique for
minimum time, considered above, is the prob-
lem of most general interest. We will, how-
ever, consider some other cases briefly.
11
First, we will consider the climb tech-
nique for best range. An exact treatment
will not be attempted, but an approximate
dt
(dt/ dv)
7:8
subject to the same sort of continuity condi-
tions as those of section 7:4. Solution of this
problem will not be attempted here.
Ro against speed. When this is done, the
optimum technique for standard conditions is
readily deduced. Even when this is not done,
it is evident from the above discussion that
standardization of the climb schedule is rare-
ly necessary. If it is required, it may be
done by methods given in R & M 2576.
7:6 EFFECT OF OPERATING CONDITIONS
It is of interest to know over what range
of operating conditions a fixed schedule of
indicated Mach number or indicated airspeed
against indicated height may be used without
a serious loss in value of performance.
7:7 DERIVATION OF THE OPTIMUM CLIMB
SCHEDULE FOR THE PARTICULAR
CASE
We are presently concerned with a quant-
itative determination of the optimum climb
schedule on the quasi-steady climb only.
This schedule is therefore defined by the
equation
This matter is considered in British Aero-
nautical Research Council Report R & M 2576
(Ref. 2). It is shown that the optimum climb
technique of a jet airplane is very insensitive
to gross weight, but that in the absence of
compressibility effects on drag, it does vary
considerably with thrust, and hence, with air
temperature. However, the curves of rate of
climb against airspeed are very flat-topped,
and considerable departure from the optimum
is possible before a serious loss of perform-
ance is incurred.
ORO
V ORO
av
g
ah
7:36
This is the exact condition. It is for con-
sideration by the test engineer whether he
will use this or the approximate condition
It is shown in R & M 2576 that within
approximately the ranges of gross weight,
engine speed and air temperature tabulated
below, the loss in performance which will
result from keeping to a fixed climb tech-
nique will not exceed the greater of 1% and
1/2 ft. per sec.
ORO
: 0.
av
7:37
Limits of Variation
Parameter
Low
Altitude
Near
Ceiling
Gross weight
Engine speed
Large
3%
15°C
12%
3%
10°C
Given a family of curves of Ro (or of ex-
cess thrust power) against airspeed or Mach
number for various heights, derived from per-
formance estimates or from flight tests, the
schedule defined by Eq. 7:37 is very readily
deduced. However, if the curves are against
true airspeed, the exact solution is itself
readily deduced by a rapidly convergent ap-
proximation process as follows:
Air temperature
The loss in performance varies as the
square of the departure of the operating condi-
tions from those appropriate to the schedule
used. When the optimum schedule is deter-
mined by the onset of compressibility effects,
It will be insensitive to changes of thrust.
(a) At each height, take the speed
for maximum Ro as a first approximation
to the optimum speed. Estimate Vg times
Rolah at this speed, by cross plotting if
desired. (This curve is often nearly linear.
Inasmuch as a rough value of V/g times Oro/dh
suffices, it would usually be satisfactory to
assume a Rolah = ARO/Ah, where ARO is
Some test units standardize the curves of
7:9
should, if possible, be flown perpendicular to
the wind gradient,
the change in height between adjacent curves
for heights separated by Ah.)
(b) Find the speed at each height at which
the slope of the Ro - V curve is equal to the
corresponding value of V/8 times o Rolah.
(c) Plot this speed against height. It will
be found that a second approximation to the
optimum speed is not necessary. For pre-
sentation to the pilot, the curve must be con-
verted to one of either indicated Mach number
or indicated speed against height.
An attractive technique of recording and
analysis, if suitable equipment were readily
available, would be to track the airplane by
radar on a plotting table and derive the rate
of change of true speed with height from the
radar record. This technique is being con-
sidered for check climbs.
In view of the above, the problem of the
flight test engineer is to deduce curves of Ro
against V, at constant values of height, from
flight tests. Such curves may be deduced
from sawtooth climbs, from level accelera-
tions (as is routine British practice for jet
airplanes), or from full climbs covering a
range of schedules which bracket the optimum
speed at each altitude.
Level accelerations avoid or reduce most
of the difficulties mentioned above in connec-
tion with sawtooth climbs. They need less
flying for equivalent accuracy, are negligibly
affected by vertical gradients in headwind, and
can be made satisfactorily when the available
rate-of-climb is very high. Pilots usually
require some time for familiarization, but
not a great deal. Bad runs are sometimes
obtained, often for no obvious reason, but once
the pilot has practiced the technique, the pro-
portion of bad runs is not too high,
Sawtooth climbs have been in regular use
for this purpose for many years, and can
still yield satisfactory data. However, they
have several disadvantages. For example:
For equivalent precision, the flying time
required was not more than one-half of that
required for sawtooth climbs. One level ac-
celeration run takes about as long as one saw-
tooth climb run, but four runs at most should
be sufficient at each altitude and configura-
tion. The main difficulties arise frm rather
unsatisfactory instrumentation, a problem
discussed in the next section.
(a) They require a long test program,
much of it at high engine output.
(b) On jet airplanes, which climb best at
high forward speeds, they are susceptible to
large errors from wind gradients or from
small irregularities in flying.
(c) They are not too practical at rates of
climb exceeding, say, 6000 ft. per min., be-
cause a large height range must be covered
on each climb to get good data.
The essential points of the test technique
are to fly as smoothly as possible through the
important part of the speed range, maintain-
ing the engine speed (or whatever determines
the engine output) as constant as is practic-
able. Tests are flown at constant indicated
pressure height to simplify flying and elimin-
ate static pressure lags, and correction made
for any inclination of the flight path resulting
from changes of position error with speed.
Objection (b) can be largely overcome by
suitable flight planning and analysis and by
careful flying. The airplane heading should be
reversed between runs, as this will show up
any wind gradient effects present as a nearly
symmetrical scatter about the mean. Care
must be taken to enter the sawtooth climb
with the speed thoroughly stabilized.
At low altitudes it may be convenient while
setting up engine speed either to extend dive
brakes or to climb from below the intended
test altitude. At high altitudes acceleration is
slow and the pilot will have ample time to
set up his engine speed.
If speed variations are present, the anal-
ysis should take account of them. The tests
7:10
Weight corrections for induced drag are
usually small except near the ceiling, but if
their value is in doubt, they can be eliminated
by flying the tests at a constant ratio of lift
to air pressure. With current operational
airplanes, the effect of flight path angle on
induced drag is small because high angles of
climb are available only at low lift coeffic-
ients where the induced drag is low.
acceptable. This is presently under consider-
ation by the U.S. Air Force Flight Test Center.
The analysis should preferably be based on
the rate of change of total energy of the air-
plane, allowing for test wind gradients and
test air temperature gradients. The latter
may vary considerably from standard near
the ground in very hot or very cold areas.
(b) Level Accelerations
For airplanes with which this is no longer
true (for example, rocket-powered airplanes
of low aspect ratio), it may be necessary for
this and other reasons to re-examine the
climb problem using the more exact approach
discussed in section 7:4. The flight test tech-
nique is discussed in slightly more detail in
Appendix I.
Instrumentation for measurement of level
accelerations presents some difficulties, in-
asmuch as there is not yet a proven method
which is entirely satisfactory. The choice of
method may depend on whether the tests are
solely to give optimum climb technique or
whether they are also to give standardized
curves of Ro against speed for inclusion in a
report.
Instrumentation which has been suggested
includes:
a.
Accelerometer
A further method which has been used a
little is to make full climbs to a range of
schedules sufficient to cover the interesting
range of airspeed at each height (or each ratio
of weight to air pressure). To fly such tests
specially would probably take about as long
as a program of sawtooth climbs. However,
when it is practicable to measure climb per-
formance on every test flight, they might be
economical. It would no longer be possible to
eliminate wind gradients by reversing the air-
planes heading between runs, but the method
would largely avoid the difficulty of stabiliz-
ing sawtooth climbs.
.
b. Airspeed indicator and clock
c.
Time trace of dynamic pressure
d. Rate-of-climb indicator connec-
ted to total pressure
e.
Air temperature thermometers
It is essential to have reliable wind gradi-
ent corrections, either from good weather
data or from radar measurements of ground
speed.
Accelerometer:
7:8 INSTRUMENTATION AND ANALYSIS
3
An accelerometer mounted with its axis
along the flight path has the considerable
merit of giving a direct measure of the excess
Te of thrust over drag, since it measures
(a) Climbs
dv
Te
+ sin y
dt
w
7:38
Instrumentation and analysis for sawtooth
climbs or full climbs require little comment.
Good height or pressure measurement with
small or predictable lag and hysteresis is
required. Radar tracking on a plotting table
would be very useful if enough tracking range
is available and the method is operationally
where y : angle of flight path to horizontal
and W =
gross weight.
7:11
have been recorded using an airspeed indica-
tor and a clock, usually on a photo-panel. This
method has been in routine use in England
for about five years.
It requires relatively little analytical
effort and does not require conversion of a
rate of change of IAS reading to a rate of
change of true speed, or a correction for any
inclination of the flight path resulting from
changes in altimeter position error with
speed. The main difficulty is that it is nec-
essary to measure very low accelerations and
provide high sensitivity. At a true forward
a
speed of 400 knots, an acceleration of 0.01 g
is equivalent to an unaccelerated rate-of-
climb of 400 ft. per min. Measurement of the
acceleration to within around 0.002 g, there-
fore, seems desirable.
Its main merits are availability and a
reasonable degree of reliability. The instru-
ments commonly form part of the standard
equipment of the test airplane. The airspeed
indicator is a robust and reliable instrument,
usually very consistent in behavior. As test
runs are made with the pressure at the static
source constant, lags can only arise from
pressure lag in the pitot lines (which will
be negligible inasmuch as the volumes of the
instruments are small) and instrument lag,
which should not be serious in the present-
case, with so simple an instrument.
It may be noted that it is not essential to
measure the inclination of the accelerometer
axis to the flight path, as the effect of this
may be determined from the accelerometer
readings in steady level flight at the relevant
Mach numbers and weight/pressure ratio.
.
The practicability of this method depends
on the availability of a suitable accelerometer
and the practicability of determining, with
sufficient precision, the inclination of the
accelerometer axis to the flight path, Pre-
sently the method is unproved, but there are
grounds for hope of its success.
Differentiation of the record of indicated
speed against time, to give the excess thrust
power, stretches the instrument to the limit
of its sensitivity and readability, but with
suitable techniques of analysis, satisfactory
results are obtained. Any errors give visible
scatter and are not hidden as accelerometer
errors might be.
The procedure for analyzing accelero-
meter records would be simple. It would be:
(1) Correct for instrument error.
The disadvantages arise from the fact
mentioned above, that the airspeed indicator
is barely sensitive enough and that much
film reading and computation are required. In
general, altimeter position error varies with
airspeed, so that an acceleration run made
at constant indicated height will be made along
an inclined path. If the technique of analysis
does not take account of this, errors will be
introduced in the absolute values and some-
times in the slopes of the curves.
(2) Correct for inclination B of ac-
celerometer axis to flight path, by subtract-
ing g sin B from the reading (B positive
if the accelerometer axis is pitched up rela-
tive to the direction of flight).
(3) Multiply the reading so corrected
by 101.3 VT (Vt being the true airspeed in
knots, accelerometer reading being in ft. per
sec.? ) to give the corresponding unaccelera-
ted rate-of-climb Ro in ft. per min.
The effect should be allowed for, if the
position error change is appreciable and if
accurate absolute values of the rate-climb
are required. If the tests are only to give
the optimum technique, the effect is less im-
portant,
Airspeed Indicator and Clock:
Two kinds of procedure are possible:
The great majority of level acceleration
runs which have been made up to the present
(a) Compute and plot total energy against
time, then take the slope as the rate of change
7:12
of energy
Tos
ICAO or NACA ambient air tem-
perature K
2.5
(b) Take increments of indicated airspeed
over suitable internals of time At and com-
pute the corresponding changes AE in total
energy, taking AE/At as the rate of change
of energy.
1 +0.2
Bulan
285.2
AI
-
gos
1 + 0.2M?
7:40
Experience suggests that procedure (b)
is preferable. In particular, it ensures that
any scatter in measurement and analysis is
present in the final plot and so enables an
intelligent appreciation of the agreement be-
tween repeat tests.
Az 1 +
{
(1 + 0.2M2)3.5 -1
(1 +0.2 M2)2.5
7:41
It has been found desirable to have a speed
change of at least 10 knots in calibrated
airspeed, and a time change of at least 0.2
minutes in each interval. It is, of course,
possible to overlap the intervals, if the total
speed range is small. About twenty such
intervals should be taken.
where M = Mach Number
a = speed of sound at 15° C, knots
os
• standard relative density at test
pressure altitude.
It is desirable to use automatic computa-
tion if it is available; however, if computing
is by hand, the unaccelerated rate-of-climb
Ro may be deduced from the change in cali-
brated speed Vc and pressure altitude Hp by
the relation (assuming an instrument cali-
brated to read correctly at sea level).
An analysis routine is outlined in Appendix
II.
A, V AVC
Tat
A time trace of airspeed and height would
materially reduce the labor of reading the
record, in that it would be possible to deter-
mine the rates of change of Vc and Hp much
more easily.
Ro
100
Δ:
Tos
+ A2A HP
7:39
where Ro - unaccelerated rate-of-climb, feet
It may be noted that the two-turn type of
airspeed indicator is likely to be better than
the multi-turn type of linearized scale, in
that it should have less friction and inertia.
8
per minute
At : time interval, minutes
Vc : calibrated airspeed, knots
Time Trace of Dynamic Pressure:
AVC = increase in Vc in interval At
:
AHp: increase in pressure height in
interval At
A time trace of dynamic pressure (given,
for example, by a transducer recording on an
oscillograph) is potentially a better and more
convenient form of record than photographs
of an airspeed indicator in that the slope of
the trace is roughly proportional to the de-
sired quantity.
Tay - test ambient air temperature
:
°K
7:13
dh
Ro : -Ri +12 +0.2MP)
:
We have, assuming a consistent system of
units,
1
d
(Pt-Pol
g
-
)
RTA
RO
8
(A2-1)
Py-Podt
R; + 2.1
approximately
RTO
1
dia
7:43
-A2
O
9
Po
dt
7:42
where A2 is defined by Eq.7:41 and
as dh/dt is small, If, on the other hand, the
air entering the instrument were at ambient
temperature, we would have
Po = ambient air pressure
Ro
: -(1 + 0.2 M?) Ri +(2+0.2M?)
2+0.2m2 (car)
dt
Pr: total air pressure
:
7:44
R : gas constant (= 3090 ft /sec? /ºK
in ft-sec units)
O
Neither of the above assumptions will hold
exactly. Only tests can show which is more
nearly true.
To: ambient absolute air tempera-
ture
t : time
It will be convenient when applying Eq.7:42
to have position error corrections to the in-
dicated pressures plotted against indicated
dynamic pressure.
It is worth noting that when the optimum
Mach number for climb is high, it will often
be sharply defined by a rapid drag rise. Use
of Eq.7:43 would then give a good indication
of the optimum speed. We may also note
that with many installations the pressure
error at the static pressure source will vary
closely as the square of the airspeed, making
dh/dt proportional to the rate of change of
the square of the speed. The term in dh/dt in
the above equation could then be omitted with-
out altering the position of the peak of the
curve.
Given suitable instrumentation, this would
seem to be a promising method of recording
level accelerations, particularly if the trace
displacement is linear with regard to the
dynamic pressure. Determination of d(Pt-Pa)
/dt is then fairly simple,
Rate-of-Climb Indicator Connected to Total
Pressure:
We may summarize by saying that a rate-
of-climb indicator connected to total pressure
is promising as a means of making a very
quick but rather rough assessment of the
optimum speed, but that it is as yet unproved.
A rate-of-climb indicator connected to
total pressure will give a reading which will
approximate, after a change in sign, the de-
sired quantity Ro. This makes it very attrac-
tive as a method of making a quick but fairly
rough check of the optimum speed,
Air Temperature Thermometer:
Let us assume that the instrument reads
correctly when used in the normal way. Then
if the temperature of the air entering the in-
strument were equal to the stagnation temper-
ature of the ambient air, we would have the
following relation for an instrument reading Ri:
An idea which is attractive at first sight
is to record temperatures from two thermo -
meters with widely different recovery fac-
tors, because the temperature difference is
proportional to the square of the true air-
speed. This is a time differential that would
lead fairly directly to the required data.
Further consideration suggests, however,
7:14
that such an approach may be impractical.
We would have
334
dh
RO-
OST, -Tzlt
K, -k2 dt
к
dt
7:45
where K1, K2 are the recovery factors
rate-of-climb R, is straight forward, using
established methods. If the aim is to pre-
sent curves of Ro as part of the published
test results, this must be done. If, on the
other hand, the tests are only to produce
data from which to determine a climb tech-
nique, it may not be considered worthwhile
to go through the whole standardization pro-
cedure when the test air temperatures do
not differ too much from standard (section
7:6). However, even then, it is probably
worthwhile to plot test rate-of-climb mul-
tiplied by the ratio of standard to test weight;
this makes an almost complete weight cor-
rection and so will bring repeat runs made
on the same day (and hence at the same air
temperature) together.
T1, T2 are the indicated tempera-
tures and
Ro
is unaccelerated rate-of-
climb, ft. per min.
.
The factor 334/(K,-K2) is unlikely to be
much less than 500. Therefore, to deter-
mine Roto within, say, 100 fr. per min, one
must determine d(T-T2)/dt to within 0.2°
per minute. This would be difficult. It would
seem that relative to the dynamic pres-
sure trace technique the temperature method
seems too unpromising to be worth develop-
ing, at least for subsonic and transonic air-
planes. It would, of course, be necessary to
determine (K1-K2) from flight test.
For airplanes with simple jet engines, an
alternative approach is to make the tests at
the desired value of NWT, and, possibly, of
the ratio of weight to air pressure. Correc-
tion for temperature would then consist only
of multiplying the test unaccelerated rate-of-
climb by Tor/Tas, where Toy and Tas are
respectively 'the test and standard ambient
air temperatures.
7:9 STANDARDIZATION OF
RATE-OF-CLIMB CURVES
If tests were at the required ratio of weight
to air pressure, no weight correction would
be required; otherwise, the usual weight cor-
rection procedure would be used.
Standardization of the test unaccelerated
7:10 REFERENCES
1. Lush, K, J., "A Review of the Problem of Choosing a Climb Technique, with Proposals
for a New Climb Technique for High Performance Aircraft," British Aeronautical Re-
search Council Report R & M 2557.
2. Lush, K, J., "The Loss in Performance, Relative to the Optimum, Arising from the Use
of a Practical Climb Technique,” To be published as British Aeronautical Research
Council Report R & M 2756.
7:15
APPENDIX I
FLIGHT TEST TECHNIQUE FOR ACCELERATED LEVELS
FLIGHT SCHEDULE
to the intended values. If photopanel record-
ing is used, records should be taken about
every five (5) seconds.
1. At least two and preferably four runs
should be made at each of three or four
heights, the heights being:
a.
Fairly near ground level
b.
At a medium altitude
2. At low or medium altitudes, start each
run at about the minimum drag speed (say
180 knots) at or below the desired altitude,
trim suitably, bring the engine to the required
conditions, level off at desired altitude
and let the airplane accelerate well past the
expected optimum speed. Start recording at
any convenient time before settling down to
the acceleration.
C. At a height at which the maximum
rate-of-climb is expected to be about 1000
ft. per min.
d. At the tropopause, if this is much
below (c). Tests at the two extreme heights
at least should include repeats made on dif-
ferent flights.
2. If a number of configurations are to
be checked, it may be sufficient to run tests
at two heights only for some of them. Also,
if the climb schedule is expected to become
one of constant Mach number at high alti-
tude, fewer tests may suffice.
3. It is important to have the airplane
under smooth control before the important
range of speed is reached. To give time to
set up the test conditions, it is often helpful
to extend the dive brakes during this pre-
paratory phase or alternatively, as indicated
above, to start below the intended test alti-
tude and set up the conditions while climbing.
The trim setting should be so chosen that
retrimming will not be necessary until late
in the run.
TECHNIQUE
1. Each run will consist of an accelera-
tion through a range of airspeed covering the
expected best speed at constant altimeter
reading, and engine speed (or other critical
engine parameter). Small changes in alti-
meter reading will not matter, but it is de-
sirable to hold the engine conditions closely
4. Runs made near the ceiling will differ
from the above in two ways. First, if the
initial speed is too low, the airplane may
decelerate instead of accelerating. If this
occurs, try again using a higher initial speed.
Second, acceleration is usually slow in terms
of indicated speed and the pilot has ample
time to make adjustments.
7:16
APPENDIX II
ANALYSIS OF AIRSPEED INDICATOR TIME HISTORY
FROM LEVEL ACCELERATION
altitude computed for the run as a whole.
This should approximate the mean altitude,
*
1. A procedure is outlined below by which
to analyze an airspeed indicator time history.
This type of recording is presently the only
one in established routine use; other methods
of recording promise to be more convenient,
but it would be somewhat premature to set
up procedures at this stage inasmuch as too
many details will vary between particular
cases, For similar reasons a procedure is
given for hand computing only, since machine
techniques vary too much with the type of
computer available.
Choose a speed interval of at least 10
knots, big enough to cover a time interval of
at least 15 seconds (very large time inter-
vals may be needed near the ceiling). Pick
out from the record some twenty intervals
of about the above length. These may be
consecutive or overlapping if necessary.
2. The basic equation (Eq. 7:39) is
At the beginning and end of each interval
read airspeed, altitude (if required), engine
speed or equivalent. Read air temperatures
with one set of readings around the middle of
the run.
A VCAVC
Ro
#azono
+
100
PREPARATION
where Ai is a function of calibrated speed
and Mach number and A2 a function of Mach
number only and
Correct the above readings for instrument
error and, where relevant, position error,
and so deduce, among other things, calibrated
airspeed Vc , calibrated altitude (that is, true
pressure altitude) Hp.
Vc : calibrated airspeed
AVC
: increment in Vc in time At
AHP : increment in pressure height in
time At.
At the speed at which the temperature
was read, evaluate:
1. Mach number, from Vc and Hp,
PRELIMINARY DATA
2. Ambient temperature, from in-
dicated temperature and Mach number, and
Preliminary data required are position
error corrections to indicated airspeed and
altitude, the air thermometer recovery fac-
tor and, of course, instrument calibration.
3. Test weight.
ANALYSIS
TEST DATA
If the height range covered during the run
is less than about 200 feet, the steps involv-
ing AHp may be omitted and one calibrated
Denote values of parameters at the begin-
ning and end of any interval by subscripts
1 and 2 respectively, the mean values by
subscripts m, and differences by A.
7:17
For each interval evaluate
(10) AI Vcm AVC
100
(1) AVc =
AVc = VC2
Vc2 - VCI
(11) A20Hp
(2) ΔΗρ: Ηρ2
(2)
Hp2- HAI
(12) AL VCM AVC
()
+ A2 AHp
100
(3) At
8
12-1
(4) Vom - ¿ (Vc, + Vcz)
(13) Al Vem AVC
100
{ ལ་
+ A2 Ho}()
6+
-
) (5) Hpm (Hp, + Hp2)
{
(6) Os
(14) Ro
(13)
Standard relative density at
height Hpm
()
To
Tos
(7) Mach numbers and Vts, the true speed
which would give the test Mach number on
a standard day, from Vcm and H Pm
Check that the engine speed (or other en-
gine operating parameter) was held substan-
tially constant in the region of peak rate of
climb (say, within 14% in the case of engine
speed). If not, evaluate the mean engine
speed į (N, + N2) or equivalent for each in-
terval and plot it against Vts. This assists
subsequent evaluation.
(8)
Aſ from Mach number and Vom and os
(9) A2 from Mach number
€
7:18
PART II
ANALYSIS OF THE OPTIMUM ENERGY CLIMB SCHEDULE
7:11
INTRODUCTION
7:12 DEFINITIONS
Total energy is defined as the sum of the
kinetic and potential energies and is given
by
For a given airplane with a given power-
plant, the climb schedule may be expressed
in the form V = f(h) which relates airspeed
:
to altitude. It is apparent that similar equa-
tions can be written in terms of the cali-
brated airspeed, Vcal, or the Mach number
M.
WV 2
E = Total energy - Wh +
29
7:46
where h is the geometric or “tapeline” alti-
tude and V the true airspeed of the airplane.
The origins for h and V are arbitrarily taken
as h = 0 and V :
0.
It is well known that for any type of air-
plane an increase in flight path speed during
a climb decreases the instantaneous rate
of climb and also that the converse is true,
In most cases the flight path speeds for
piston-engined airplanes are small compared
to those achieved by jet airplanes, and there-
fore, the effects of flight path speed are
more important in the case of jet aircraft
than for piston-engine planes.
For propeller-driven aircraft, the deter-
mination of the best climb schedule is simply
accomplished by using the "sawtooth” climb
technique to determine the speed for maxi-
mum climb rate.
Energy height (total height) is designated
by he and is defined as the total energy per
unit weight. Thus
v2
he-h +
slo
7:47
which is seen to be independent of the air-
craft mass.
Por jet-propelled airplanes, it is known
from analysis and experiment that the opti-
mum climb angles are less than those cor-
responding to the maximum rate of climb
at a given altitude, and thus represent greater
flight path speeds than the speeds for so-
called maximum rates of climb.
We next define w as the rate of change of
energy height, thus
dhe
W
dt
7:48
)
Under these conditions, the time required
to go from one altitude to another depends
on the variation of flight path speed between
the two altitudes, a factor which has gen-
erally been neglected in the climb analysis
of propeller-driven airplanes.
(Note that w is the instantaneous rate of
climb when the flight path speed is main-
tained constant.)
Explaining this thinking in terms of energy,
we see that when the flight velocity is large,
variations of kinetic energy (mV2/2) can no
longer be ignored in comparison to the po-
tential energy due to height.
At speeds in the neighborhood of the op-
timum climb speed, potential energy may be
transformed to kinetic energy, and vice-
versa, in a reversible fashion, i.e., without
appreciable loss in the conversion.
7:19
7:13 THEORETICAL ANALYSIS OF THE
OPTIMUM CLIMB SCHEDULE
This is not a definite integral, since we
have not as yet provided a specific relation
between w and he. We shall assume that
W = f(h,V). (This assumption is discussed
in section 7:16.)
We shall begin by making certain sim-
plifying assumptions with the understanding
that these assumptions must be re-examined
later to correlate the analysis with actual
operating conditions. (See sections 7:16 and
7:17.)
Thus, we have
w :flh,V)
7:50
and
y
he : h +
29
Our assumptions are as follows:
(1) The control of the engine is not at the
disposal of the pilot because the throttle is
either fixed or must be operated by the pilot
in some scheduled manner depending on speed
and altitude. For example, a jet engine must
be regulated to maintain a given constant
RPM or to hold a give tailpipe temperature.
7:51
or, on eliminating h
1
v (he,v).
W
7:52
(2) The climbs occur in a standard at-
mosphere.
Then
nez
'1,2
s
elhe, vidhe
.
(3) Atmospheric turbulence is negligible
in the case of measurement with respect to
the ground as reference, or any atmospheric
motions are in the form of a constant hori-
zontal wind, which is a function only of alti-
tude, in the case of measurements with re-
spect to the air mass.
nei
7:53
This integral will be minimized if V is a
function of he such that
(4) The climb is made in a single vertical
plane.
Oy
: 0
av
7:54
According to these assumptions, the sin-
gle parameter determined by the pilot during
the climb is the extent to which he can pre-
cisely control his plane to follow the pre-
scribed climb schedule, V = f(h) while av- .
eraging the load factor, a topic to which we
will return later. (See section 7:16.)
provided the function satisfies certain
conditions of regularity apparent in the geo-
metric representation of section 7:14.
Eliminating he from Eqs. 7:51 and 7:52,
we obtain an equation of the form
V = $(h)
Consider the two energy heights, he, and
hez with hez > hej i since w: dhe /dt,
dt = dhe/w, the time required to go from
he, to he becomes
7:55
ne2 dhe
*1,2 -
s
W
which is the condition under which the in-
tegral (Eq. 7:53) will be minimized. Accord-
ingly, the values of V between the limits
her and hez are fixed for all intermediate
values of he.
he
7:49
7:20
(2) By means of accelerated level flight
runs during which the height h is held con-
stant, in which case
d
v2
V dV
W =
The climb schedule of Eq. 7:55 is the op-
timum schedule, i.e., it gives the minimum
time required to go from one combination of
speed and altitude of the schedule to another
combination also satisfying Eq. 7:55. How-
ever, the problem of passage from one com-
bination of speed and altitude not satisfying
Eq. 7:55 to any other combination in minimum
time is not resolved. We shall see in sec-
tion 7:15 how this latter problem can be
solved practically.
dt \2 g
gdt
These two methods serve to provide a
good definition of the surface (E), which
for turbojet aircraft has the general form
illustrated in Fig. 7:1.
7:14 GEOMETRIC REPRESENTATION
Eq. 7:52 can be written, in the light of
Eq. 7:51,
From the foregoing, the optimum energy
climb schedule has its projection in the (h,
V2/2g) plane defined by the Eq. 7:55. The
optimum climb schedule can also be repre-
sented on the surface (£) by Eq. 7:54 or the
equivalent equation obtained by rewriting
Eq. 7:56 as
w : $ in, he-h).
W
on)
V 2
,
29
-
7:56
The condition for minimization of the in-
tegral in Eq. 7:49 becomes
This functional relationship can be de-
termined practically from flight tests during
which the assumptions made in the preceding
section are valid. Actually, it is sufficient
to measure V and h at each instant to obtain
don{tin, ne – n}= 0
ch
2
or
h
he
:ht
sla
29
od
dlhe-h)
dø dh
+
дь
dh
: 0.
and then
dlhe-h)
dh
dhe
W
dt
But at V =
constant
ane
:
dh
on the curve representing he as a function
of time.
or
0$
дф
dlhe-n)
O,
ah
Thus we obtain points on the surface (3)
represented by Eq. 7:56. In fact, we see
now that there are two methods which can
be used to obtain the surface (2):
and finally
дw
318
on
2
(1) By means of continued climbs follow-
ing schedules in the neighborhood of the op-
timum schedule.
olemas
29
7:57
7:21
4 W
Po
6
M
P.
slo
hezo
H=0
h
Z
2
1
: cte
29
21+
H, = 2,7 zig
V. constant
29
he, hi +
Hq = cte
: constant
nez
Fig. 7:1
This Eq. 7:57*, which defines the optimum
climb schedule (C) on the surface (E) of
Eq. 7:56, indicates that at each point of C,
the plane tangent to (E) is parallel to the
direction established by the intersection of
the planes w : 0 and
V2
*
constant.
h +
2g
* By interchanging variables, it can easily be
verified that Eq. 7:57 is identical to Eq.
7:54.
7:22
T
Geometrically this result is simply in-
terpreted and it follows that the integral
by the points of maximum w along intersec-
tions between the surface () and planes per-
pendicular to w : 0 erected through lines of
constant he
ne2
dhe
11,2
8
W
hey
7:15
PROPERTIES OF THE GRAPH
will be a minimum if for each value of he,
1/w is a minimum, i.e., w is a maximum.
(1) The projection (c) of the curve (C) on
the horizontal plane (w : 0) gives the opti-
mum climb schedule relation
V = f(h)
7:55
On projecting C onto the bisecting plane
h : V2/2g (perpendicular to the direction of
the line defined by w: 0, he = constant),
)
we obtain the following graph.
which is the practical definition as far as the
pilot is concerned.
.
From Fig. 7:2 it may be seen that the area
A between he, and he will be a maximum
when the curve w : f(he) is the projection of
the outline of (2) viewed parallel to the di-
rection of he constant lines in the plane
w = 0.
(2) Examination of Fig. 7:1 reveals that
if P is the maximum of the curve w f(V)
obtained with he = constant, and if M is the
maximum of the curve w = f(V) obtained for
h : .constant, then the point P corresponds
to a greater speed than that of M.
:
The curve (C) representing the optimum
energy climb schedule is, therefore, given
(3) In the neighborhood of the optimum
climb schedule w does not vary appreciably
from its maximum value along the he • con-
stant intersection. From this it follows that
small variations in speed from the optimum
schedule V = f(h), do not materially change
:
the value of the integral representing the
time for ascent from he, to hez (i.e., the
area A of Fig. 7:2 is not appreciably altered
by small changes in V from the optimum
schedule V = f(h)).
h
Therefore, at a given height, provided that
the speed does not vary too much from the
speed for best energy climb, a variation of
V is equivalent to a variation of h, each pro-
ducing the same variation in energy height.
That is to say, the kinetic and potential en-
ergies may be interchanged in a reversible
fashion provided the climb schedule does not
deviate too much from the optimum.* On
any other portion of the surface (E), it is
apparent that this reversibility does not exist.
H
H
H₂
por
* This fact is also expressed analytically by
Eq. 7:57.
Fig. 7:2
7:23
(1) Atmospheric Conditions
This condition of equivalence of energy
transformation permits us to deduce the
nature of the surface (E).
(4) Figs. 7:1 and 7:2 allow us to study
schedules other than those corresponding to
the optimum. If we depart from a point on
(2) not situated along C, and which repre-
sents an energy height he, it is evidently
impossible at first to follow a path along the
surface with he, a constant, since in this
a
case
We have assumed that the only atmospheric
disturbance present has been a horizontal
wind of constant velocity at any one altitude.
In the event that measurable vertical move.
ments exist, it is evident that the measure-
ment of w will be in error. It should be
possible to correct our test data for vertical
motions. However, the complications in-
volved cause rejection of this idea.
In the case where the horizontal wind grad-
ient with altitude is not zero, the airplane
will gain or lose energy in the climb. If the
horizontal wind gradient is known at all al-
titudes, we can compute the acceleration due
to the wind
dhe
W :
: 0.
dt
Rather, one should attempt to maximize
the area of Fig. 7:2 by approaching as closely
as possible to the intersection of the he. con-
stant plane and the curve (C) without increas-
ing the load factor sufficiently to risk a de-
crease in w.
()
dt wind
at any point of the climb and correct w ac-
cording to the relation
V dV
Aw:
g dt wind.
)
Therefore, it will generally be advan-
'tageous:
(a) If one is at a speed less than that of
constant he on (C), to speed up to the point
(C).
(b) If the airplane takes off from the
ground to gain speed close to the ground.
(c) If one wishes to obtain an altitude h2
and a speed V2 greater than the optimum,
to first climb to an altitude greater than h2
without exceeding hey and then to descend
slightly allowing the speed to increase to V2.
To insure that all data points form a sin-
gle surface (2), it is necessary to correct
data obtained under various observed con-
ditions to standard atmospheric conditions.
The correction is considered in section 7:17.
(2) Operation of the Airplane
In the maneuvers described above, it
should be realized that because of load factor
variation, there exists the risk of departing
from the region wherein our assumptions are
valid.
We have already noted that the engine
must be operated in accord with certain re-
strictions, which must be well defined for
each set of flight conditions. It is evident
necessary and sufficient condition
(from the point of view of airplane operation)
for the existence of a well-defined surface
(E) is that the engine limits be specified for
all values of V and h.* In section 7:17 we
7:16 DISCUSSION OF ASSUMPTIONS
To simplify the discussion, we have made
certain assumptions which we must inves-
tigate further to assure the validity of our
results.
* Since the atmospheric conditions are de-
fined in Part I of this section, the engine op-
erating restrictions in effect call for a spec-
ified variation of thrust with speed and alti-
tude,
7:24
shall examine the practicability of achieving
this in the case of a turbojet airplane,
The most important operating restriction
on the airplane involves the value of load
factor during the climb.* If we call F, the
resultant of the forces along the flight path
(thrust-drag), we have
In the event that drag varies little with n
(generally the case at low altitudes at the
optimum climbing speed for jet airplanes),
Fy is insensitive to n and no corrections
need be made for variable load factor. In
cases where large variations of drag are
encountered, F, is a function of n and cor-
rections may be required if level flight runs
are used to determine the climb schedule.
However, we shall not concern ourselves with
this topic here.
dv
m
dt
FT - (mg sin y)
where mg : airplane weight
y :
flight path inclination to the horizontal.
With Vsin y = dh/dt, we have, on multi-
:
plying the preceding equation by V,
To determine the influence of non w (which
is required for a rigorous definition of the
surface (£)) n must be defined for each value
of h and V. In practice, slight variations of
energy level are acceptable without producing
noticeable thickening of the surface (E) es-
pecially at low altitudes. This permits ac-
ceptance as points on the surface, data ob-
tained in different fashions with different
flight path slopes or curvatures, provided
the variations in n are small.
.
dh d
FTV
1
dt dt
la
mg
or
FTV.
W
mg
In general, the method of determining the
optimum energy climb schedule by continued
climbs (see section 7:18) will be more pre-
cise than the method of using accelerated
level flight runs, since the load factors at-
tained in the former case are quite close to
those of the actual climb; for instance, the
normal load factor in a steep climb differs
appreciably from the n : 1 of level flight.
Fy is the instantaneous excess thrust over
that required for level flight at the same
load factor, and w is, therefore, the instan-
taneous excess power per pound weight,
If we now consider a variation of load fac-
tor between two flights carried out under
otherwise identical conditions (same weight,
same h and same V), w will not vary from
one case to the other if it remains constant.
The difference in angle of attack in the two
cases generally produces negligible differ-
ences in thrust, but possibly noticeable dif-
ference in drag especially at high altitudes.
Finally, the weight variations due to burn-
ing of fuel can generally be accounted for as
a given function of altitude in a continued
climb, which provides another advantage of
the continued climb method.
)
(3) Analytic Conditions
* We shall not consider the variation of air-
plane mass during the climb due to burning
of fuel, this variable being readily handled
by determining the relation m = f(h).
)
When the preceding conditions are ful-
filled, the surface (E) is well defined, but
certain theoretical paths on the surface are
not possible without changing the load factor
in which case the actual path is no longer on
the surface.
7:25
Similarly, certain paths are analytically
impossible, since w - dh /dt, and, there-
fore, if w is positive, one cannot move ex-
cept to increase he (i.e., decreases in he are
not permitted).
Here we shall consider only the case of a
turbojet with a single variable parameter
(assumption 1, section 3).
.
(1) Correction for Operating Conditions
If we write
If during an actual climb the thrust of a
turbojet (determined as a function of RPM,
N) deviates from standard, a correction may
be applied. In effect, we have shown that
dhe
dhe dh
dhe
24)
dt
an
dt
V
dt
д
20
W
FTV
mg
.
If we now increase the thrust T by an
amount AT, the excess thrust Ft and, there-
fore, w will be increased proportionally.
Thus
we see that at a point on (2) where the par-
tial derivatives dhe/ah and one/a(V2/2g) are
defined at the same time as w, there remains
only one variable parameter (for example,
ascent speed or flight path slope). Further,
this parameter defines the tangents to the
curve representing the displacement on the
surface, or, since the maximum slope of the
flight path is vertical, one has the theoretical
extreme case corresponding to the values of
dh/dt and d(V2/2g)/dt in vertical flight, the
load factor corresponding to the point under
consideration.
Δw
V
AT .
mg
AT
-
It is sufficient then to know the variation of
thrust AT due to a correction AN on N to
obtain Aw.*
It follows that the analytic conditions re-
quire not only a positive variation of he for
w > 0, but also an angle of less than 180°
which corresponds to the slopes of the de-
fined surface. Without going into further
detail, we note that the possible paths on the
surface (E) are generally still further lim-
ited by load factor limitations.
To accomplish the correction, one gen-
erally refers to the engine manufacturers'
charts presenting thrust as a function of
RPM, speed and altitude. The accuracy ob-
tained (using such charts) is often sufficient
for correction procedures. However, it is
always possible to make precise corrections
provided the necessary engine parameters
are measured. This method is used when
certain anomalies are noted in the engine
operation or when complex engine designs
are employed.
7:17 CORRECTIONS
(2) Temperature Corrections
It is not possible to consider here all the
practical cases which require corrections to
make them conform to the theoretical cases
considered; i.e., where our assumptions are
satisfied in a convenient manner. In particu-
lar we shall not consider the important case
of correcting for load factor. This correc-
tion would require a special chapter by it-
self treating "reduced coordinates."
A variation of ambient temperature T,
.
* Editor's note: The use of RPM as a thrust
correction parameter is not an entirely sat-
isfactory procedure. However, it is con-
sidered acceptable for small variations from
standard.
Each type of power plant requires a spe-
cial study of the corrections applicable to it,
7:26
between two climbs through a given point
(h, V) effects w in two ways. It modifies
the engine thrust and also the airplane drag.
7:18 PRACTICAL DETERMINATION OF
THE OPTIMUM ENERGY CLIMB
SCHEDULE
We recognize that the variations of air-
plane drag with temperature are small. How-
ever, since the drag decreases with the tem-
perature, neglecting drag variations provides
optimistic data from trials in warm air, in
which cases it may be necessary to consider
drag effects in the analysis.
Let us now assume that we are able either
to conduct our flights according to the theo-
retical assumptions or are able to make the
necessary corrections to standard conditions.
Under these circumstances, the experimental
determination of the surface (E) is then pos-
sible from which the curve C (determined by
the intersections of the planes perpendicular
to w = 0 and including the lines he = constant)
is obtained. The projection (c) of (C) on the
(h, V) plane gives the desired optimum energy
climb schedule.
As far as the thrust is concerned, the
effects of temperature can be accounted for
by noting that thrust can be expressed as a
function of NWT where N = RPM and T - tem-
N
:
perature. Other factors being constant, the
same results are obtained if NVT = N'NT',
where N may differ from N' and T from T'.
Since the engine charts are presented in terms
of NWT, the RPM and temperature correc-
tions may be made simultaneously.
We shall now consider the two methods of
determining this climb schedule, which we
mentioned earlier in section 7:14.
(3) Weight Corrections
(1) The Method of Continued Climbs
Several (3 or 4) continued climbs to ceil-
ing are made during which the true airspeed
is linearly increased with altitude. Provided
the optimum climb schedule lies within the
range of speeds and altitudes tested for, the
surface (2) can be deduced from an analysis
of the continued climb data, the test data ap-
pearing as in Fig. 7:3.
If the mass m of the airplane differs from
the standard mo, the relation w = FTV/mg
shows that the simplest way of determining
Wo corresponding to mo will be to conduct
tests at mass m under conditions such that
Ft remains unchanged; i.e., by making the
test at the same h and V while maintaining
the same angle of attack by variation of the
load factor. The relation between the meas-
ured w and wo is then given by wo - w(m/m).
.
Finally let us recall that the wind can be
other than horizontal and constant. In sec-
tion 7:16 we have developed the corrections
required for the case of a horizontal wind of
known vertical gradient (naturally, with the
wind oriented along the flight path).
2
)
3
Corrections for vertical components of
known winds would be simple to make, but,
since vertical air currents are practically
always unknown, and, since their influence is
large, the only rule is not to conduct tests
under conditions when strong vertical air
movements exist.
Fig. 7:3
7:27
The test data are corrected to standard
conditions and we then obtain (for each con-
tinued climb) curves of he as function of time
(Fig. 7:4). Values of h and V as functions of
time can similarly be presented for each
climb.
H
M3
Hi
MI
If we now consider constant values of he,
we shall have, for example, at hej, inter-
sections M1, M2, and M3 (see Fig. 7:4) and
the slope of the curves he or t at each inter-
section may be at once determined at four
different values of V for the case shown.
Since w : dh /dt, we may cross plot the
slopes as a function of V as illustrated in
Fig. 7:5.
TEMPS
TIME
From Fig. 7:5 we may at once obtain the
optimum speed at the selected value of he
(he), and, therefore, at the corresponding
h, which gives us the optimum climb sched-
ule V : f(h). The value of w is also known,
80 that considering different values of che,
we may determine Wmax = f(he) and from
:
this the time to climb may be obtained by
direct integration. In practice the time to
climb should be verified by a climb made
according to the optimum schedule. This
time will be obtained very accurately, even
if the pilot is not very precise, provided the
speed remains close to that called for by the
optimum schedule.
Fig. 7:4
HOH
We note in passing that Fig. 7:4 is gen-
erally sufficient for the determination of the
optimum schedule, for the magnitude of the
slopes may be directly observed from this
figure without recourse to Fig. 7:5.
Max.
(2) Acceleration Run Method
We shall present here the principle of
this method*, which consists of making ac-
celerated level flight runs from a speed con-
siderably less than the optimum to a speed
We shall not consider the difficulties which
this type of testing often entails.
Fig. 7:5
7:28
approaching Vmax at various different alti-
tudes. From the data (since dh - 0), we may
calculate
d
W :
love you
V dV
gdt
di 12
.
At each altitude, h, at which tests were
conducted, we shall have cross sections of
the surface (E) averaging certain correc-
tions, in particular on the load factor as
shown in Fig. 7:6.
At each point on each curve, h and V are
known and, consequently, he is defined as a
function of h and V. Therefore, the data can
be cross plotted in the form w = f(he) as
shown in Fig. 7:7.
In this system (Fig. 7:7) the curves rep-
resent the projections of section of (a) on
the planes h = constant, and, therefore, the
optimum climb schedule is defined by the
envelope of the curve (see Fig. 7:7).
Another more precise method of determin-
ing the optimum schedule (C), utilizing the
information of Fig. 7:6, consists of con-
structing the cross sections w = constant by
cross plotting the data of Fig. 7:6 as shown
in Fig. 7:8.
2, 22 23
Ni na ng
Fig. 7:6
nz
(C)
NE
Hecto
ho constant
22
ܕܐܢ
ha
v2
29
Fig. 7:8
Thus we obtain the intersections of planes
of constant w with the surface (E) projected
on the (h, V2/2g) plane of Fig. 7:1. The
Fig. 7:7
7:29
slope of the line he : constant in this plane
is given by the normal to the bisector of the
axis system (i.e., by a line with a negative
45° slope) and the tangent parallel to this
direction determines points on the optimum
curve C.
nil. Moreover, the proper load factor and
weight conditions are very well simulated
during the test ascents. Also the required
data measurement is reduced to a minimum,
records of speed altitude and time generally
being sufficient (if one used meteorological
data describing the actual variation of tem-
perature with altitude).
If we plot the data in the (h, V) plane
rather than the (h, V2/2g) plane , the curves
of constant he will be parabolas displaced
from one another by translation along the h
axis, and in this case the points of tangency
will also be well defined thus establishing
the curve C.
The method of accelerated level flight runs
is desirable in spite of piloting difficulties
frequently encountered, if one wishes to de-
termine the optimum energy climb schedule
very quickly without resorting to the climbs
required by the other method. Temperature
corrections may then be ignored, for a first
approximation to the optimum speed at the
altitudes of the level flight runs (but obviously
not for the ascent time).
For these latter cases, we note that the
curves of Fig. 7:7 need not be obtained from
level flight runs. Any other family of curves
obtained holding some quantity constant, for
example constant load factor, could be pre-
sented in terms of w and h with the envelope
curve giving the optimum climb schedule. *
This thought is very useful for resolving cer-
tain difficulties, for example, when it is dif-
ficult for the pilot to maintain constant load
factor in level flight.
Level flight runs serve to provide other
data as well; such as, acceleration time
(obviously), maneuver boundaries* and max-
imum level flight speeds. Finally, certain
difficulties encountered in the operation or
control of complex engines or those having
poor regulation are avoided in level flight
runs so that useful data may be obtained
from these tests in cases where the con-
tinued climbs may produce inaccurate data,
(3) Choice Between the two Methods
The two methods of testing described
above are sufficiently different to justify
some indecision on the part of the test en-
gineer who must choose the one most adapt-
able to the given test program. Ideally both
methods should be used, but rarely are suf-
ficient time or justification available to per-
mit this procedure to be used. We shall,
therefore, present some of the advantages
and disadvantages of the two methods to pro-
vide some basis for a choice.
Summarizing, the corrections to be made
and the supplementary measurements re-
quired are determined by the particular
problem to be solved and these factors should
be considered when choosing one method in
preference to the other. The complications
of a given program and the total number of
tests required to supply the data (both for
the climb schedule and for other purposes)
should be carefully considered in selecting
which method is best in any given instance,
The method of continued climbs is de-
sirable when other tests require climbs under
similar conditions, since under these con-
ditions the loss of flight time is practically
* Editor's note: It would be necessary to
have h and V defined precisely in terms of
the given constant quantity to determine
V = f(h) from this scheme.
* Editor's note: It can be easily shown that
the ability to attain a given load factor with-
out loss of altitude is directly related to the
acceleration characteristics provided suf-
ficient maximum lift coefficient is available
to permit attainment of the load factor.
7:30
7:19 CONCLUSIONS
assumptions which establish the validity of
the theory. This should serve to provide a
basis for the further study required in each
particular case.
We have presented the essential prin-
ciples and also the methods of determining
the optimum energy climb schedule using the
energy height parameter. It is important to
recognize on one hand, the theoretical com-
plexity of the problems encountered in the
utilization of these methods, and on the other,
the large number of possible results obtain-
able, which give more than just a knowledge
of the optimum climb schedule.
Each type of power plant introduces new
problems because of the way it operates (the
rocket motor, for example) and also because
of the changes of order of magnitude of per-
formance which it is likely to produce. We
have considered here only the fundamental
When this study has been completed in any
particular case, the performance of the plane
may then be well defined and the influence
of the various parameters which affect its
performance clearly established. The man-
ner of most effective operation of the air-
plane (from the performance viewpoint) can
be determined from the tests, as well as the
effect of practical departures from the theo-
retical case. Finally, using energy methods,
we may solve problems hitherto not con-
sidered amenable to solution by flight testing
methods.
7:31
PART III
CORRELATION OF THE ENERGY CLIMB ANALYSES
7:20 REVIEW OF PRECEDING ANALYSES
maximize dhe/dt we required that
ow
ow
.
On
V 2
29
7:59
In both Parts I and II of this chapter it
has been demonstrated that the optimum en-
ergy climb schedule is determined by the
condition that dhe/dt shall be a maximum
at all times during the climb so that the air-
plane will have stored the maximum pos-
sible amount of mechanical energy (kinetic
plus potential) at the end of any given time,
where
dhe
키
​W
: fln,
(
dt
29
It is obvious that $ and w are the same
quantity so that Eqs. 7:58 and 7:59 are the
same relation expressed in different ways.
It is easy to verify the truth of this statement
for we know that
d
It has been pointed out that the optimum
schedule can be determined by two different
flight testing procedures; one involving con-
tinued climbs to ceiling and the other, level
flight acceleration runs conducted at con-
stant altitude. We note that at the onset of
the climb, it is not possible to follow the op-
timum schedule immediately because of the
necessity of accelerating the airplane from
rest to optimum speed. For climbs to low
maximum altitudes therefore, energy climb
techniques are of little value. On the other
hand, the use of these techniques applied to
high altitude interceptors provides signifi-
cant gains in operational performance. It
is this fact that explains the great emphasis
recently given to the study of the energy
height concept.
)
dV
29
9
or
VOV
d
:
)
29
g
7:60
so that Eq. 7:59 may be written
In Part 1 of this chapter, it was estab-
lished that since he is a function of both h
and V, we shall maximize dhe/dt when
V
ow
дw
.
gon
av
дф
V
дф
7:61
av 9 on
7:58
which is the same as Eq. 7:58 (since $ :
w).
where
dhe
V 2
olo
(
Thus we see that identically the same
conditions are specified in both Part 1 and
Part II for the determination of the optimum
energy climb schedule.
h +
dt
2g/.
Similarly in Part II, it was shown that to
In Part II an important geometrical in-
terpretation is made of Eq. 7:59; namely,
7:32
1
T
d²ne
d²ne
that this equation defines the maximum of
w for a given constant value of energy
height. Although the analytic development
of this fact was not detailed in Part II,
it is easy enough to establish the truth of
this interpretation. This is done as follows:
()
a )
level
climb
d12
di?
7:63
7:21 THE THIRD METHOD OF EXPRES-
SING THE CONDITIONS FOR OPTI-
MUM ENERGY STORAGE
Writing Eq. 7:58 in terms of w, we have
Eq. 7:59 defines any line which lies in
a plane parallel to a plane perpendicular
to w : 0 and
and oriented in the direction
h : V2/2g, w: 0. However, we also know
that this plane is tangent to the surface
(E) of Part II. The plane can only be tangent
to (E) and still include the line defined by
Eq. 7:59 when it passes through the maximum
of the curve obtained by cutting the surface
(E) along an he = constant line (which is
perpendicular to the direction of the line,
h : V2/2g).
31
V dw
gan
av
o
with
dhe
dt
Therefore, since the maximum of w along
a constant he line is given by a horizontal
tangent parallel to w : 0 and parallel to the
he h+V2/2g = constant lines, it follows
that the tangent plane, defined in part by
Eq. 7:59, includes the line parallel to w : 0
and parallel to lines of constant he. This
establishes the interpretation of Equation 7:59
given in section II.
now since dw/OV is obtained at constant
altitude, we have
д
dh
eleon
de
av
-181
av
d12 / level
(2)
hieveloped level
"
dt
l level
/
We shall show in the following paragraphs
that there is yet a third way of specifying
the optimum energy climb schedule which
involves time derivatives rather than deriva-
tives of h and V. Although this procedure
may be derived directly from physical consid-
erations, we shall obtain it using Eq. 7:58
as a starting point since, by so doing, we
also demonstrate that this third way of
expressing ourselves is merely a restatement
of what has already been developed.
where a : acceleration.
Similarly,
ow
.
Com
dhe
I
climb dh
on
dt
dt
This third specification is that the follow-
ing equations must be satisfied if the airplane
is to fly at the optimum energy storage
speed at a given altitude during a continuous
climb,
Therefore, our original Eq. 7:61 becomes
dhe
dhe
dh
dt
dhe
(2) climb
9 dhe
v dt? / level
)
dt
level
dt
climb
level
7:62
7:64
7:33
which is satisfied when
Now, since the excess power can be used
just as well, to produce a rate of change of
potential energy as to produce a rate of
change of kinetic energy, we have that at
a given speed and altitude
d²hel
d²ne
dt? I climb
dt2/level
7:67
d (Wh) :d
dt
dt
WV2
29
)
(he) level
dhe
-
dt I climb
7:68
where W : weight. Since the weight is the
same in both cases and since dw/dt will
also be about the same regardless of the
method of energy storage, we find that
which are the conditions we sought to esta-
blish in Eqs. 7:62 and 7:63.
dh
V dv
ole
.
.
10
dt
9
dt
Eq. 7:68 also expresses the reversible
interchangeability of kinetic and potential
energy at points in the neighborhood of
the optimum climb speed. (This was tacitly
implied when we set dh/dt = (dhe/dt)climb
in the development of Eq. 7:66).
7:65
but in the neighborhood of the optimum climb
schedule
To amplify this latter statement, we note
that since both true airspeed and altitude will
be simultaneously changing when the airplane
dh
dhe
.
dt
dt
I climb
and
dhe
dt
v
위은
​dhe
II
dt
g
(%。
dt level
di 'climb
so that Eq. 7:64 becomes
RATE OF CHANGE OF ENERGY HEIGHT
d²he
اههه
()
dhe
dt
dt? I climb
level
a²ne
dhe
1. time
()
d12
level
dt
climb
7:66
Fig. 7:9
7:34
is following the optimum energy climb sched-
ule, the quantity (dhe/dt)climb is a function
of both hand V, although in the region of
the optimum energy climb speed, (dhe/dt)
will be the same regardless of what method
is used to change he.
( dhe/dt)climb curve with this latter curve
being determined for the optimum energy
climb condition. Therefore, if the airplane
starts to climb at point (E), we shall have
at all times the maximum possible value of
( dhe/dt: w), which maximizes the value of
he as a function of time.
A physical interpretation of the conditions
7:62 and 7:63 may be obtained by referring
to Figure 7:9.
First, we note that relations 7:62 and 7:63
locate point (E) as the optimum energy climb
point, since only at point (E) are the speci-
fied conditions satisfied. The physical rea-
sons why this must be so are as follows:
We note that this analysis has ignored
the topic of load factor variation, which would
practically be involved if we were first to
accelerate to either point (A) or (E) of
Fig.7A:1 and then attempt to follow the given
climb schedule. It is apparent, however, that
the analysis is nonetheless valid for the
determination of point (E) when the airplane
is in a continued climb. In this case,
Fig. 7A:1 represents a true comparison of
the climbing airplane and the equivalent
airplane in level flight at some given altitude.
Suppose that an airplane accelerates in
level flight to point (A) (which determines
the speed for so-called maximum rate of
climb, conventionally determined in the past
by sawtooth climbs) and then starts to climb.
Due to the decrease of excess power with
altitude and the necessity of accelerating
along the flight path (dhe/dt)climb de-
creases initially at a greater rate than
(dhe/dt level
so that at some time ti,
the climbing airplane, will be at point (B).
Had it remained in level flight, it would have
been at point (C).
7:22 PRESENTATION OF THE THIRD SET
OF CONDITIONS IN TERMS OF
V AND A
To complete the energy climb picture,
it is desirable to write equations which
illustrate the relationship between w as ob-
tained in a climb and w as obtained in level
flight tests.
The energy height being the integral
By definition we know that
s
S'Care
) at
hen
dt,
v2
di
he
: h +
to
29
so that in level flight with h =
constant
we see that climb out at point A cannot
store the maximum amount of energy in
a given time, since in this case, he, is
the area under the curve DAB which is less
than the area DAC representing the air-
plane in a level acceleration run.
dhe
d
V dV
to
W :
)
-
en
dt
level
dt
29
gdt
7:69
In the general climb, both V and h are
variable, so that in the climb
It is easy to demonstrate that the
(dhe/dt climb curve is concave upward at
speeds exceeding those for maximum climb
rate, and moreover, that at speeds exceeding
the optimum energy
climb speed the
(dhe / dt )level curve falls below the
dhe
dh
$18
V
+
g
dV
dt
W
dt / level
7:35
Moreover, if we fly in accord with a
given schedule, V: f(h), then the preceding
=
equation becomes
from which it follows that Eqs. 7:67 and 7:68
may be written
V dV
dh
dt
Go
dhe
dt. I climb
(1 + ý
g dt level
W
:
dh
V dV dh
dh
V
+
7:72
do
1+
g dh
g
dh dt
dt
wdw
7:70
(i+)
dw
dt level
hanno
9 dh
dh
climb •
from which it follows that the actual rate of
climb is given by
7:73
dh
W
dt
1 +
V dv
g
dh
These relations may be used to determine
the optimum climb schedule using flight data
from acceleration runs conducted at various
altitudes. However, normally a trial and
error procedure similar to that suggested
in Part I is required.
7:71
Note that if we make dv/dh = 0, we will
then have that the rate of change of potential
energy equals the total rate of change of
energy. For in this case, there is no accel-
eration along the flight path.
7:23 SUMMARY
Moreover, if we make dv/dh negative,
then we can achieve a rate of climb greater
than the rate of change of energy height,
since we are converting kinetic to potential
energy and under these conditions the air-
plane is a "zoom". As noted previously,
provided we do not deviate excessively from
the optimum schedule, the energy interchange
may be made reversibly.
In this chapter we have shown that the
conditions for maximum energy storage may
be presented in several different fashions,
each having some advantages from the point
of view of visualizing the physical nature of
the problem.
Using the foregoing, we have
In Part II it was shown that if the quantity
dhe/dt was considered a function of h and
V42g, then a surface representing the rate
of change of energy height could be construc-
ted which allowed geometric interpretation
of the optimum energy climb requirements.
Using this surface as a basis of discussion,
direct procedures (not involving trial and
error) were developed for determining the
optimum energy climb schedule using several
different flight techniques.
(5)
dane
dw
v2
29
at²
-
d12 / level
dt /level
d
and
dw
d
dhe
***
le to climb
d+2 I climb
dt
In both Parts I and II, limiting cases
were considered which revealed that although
accelerated level flight runs were a satis-
factory means of determining the optimum
energy climb schedule in many cases, a
dt / climb
dhe
dh dt
dh
ols
()
climb
dt
7:36
more accurate technique was that of making
continued climbs. It is pointed out here
that this is particularly true when the climb
angles become large (as is the case with
certain modern high performance jets, par-
ticularly at low altitude). In such case,
the equality of lift and weight existing in
level flight is a rather poor approximation
to the actual conditions existing in a climb.
It is apparent that much more work is
yet to be done on investigating applications
of the energy height concept, both in tactical
evaluation of aircraft and in analytic treat-
ment of performance. However, it is felt
that here we have set down the fundamental
thoughts, and further amplification of the
topic of energy height does not come within
the scope of this manual.
.
7:37
APPENDIX A
PROCEDURES FOR DETERMINING THE
ENERGY CLIMB PERFORMANCE OF TURBOJET AIRCRAFT
Editor's Note
Because of a revision in the original concept of the AGARD
Flight Test Manual, it has been necessary to reduce the size of
Mr. J. F. Renaudie's excellent paper, “Procedures for Determin-
ing the Energy Climb Performance of Turbojet Aircraft." In
reducing the size, every effort has been made to maintain the es.
sential features of the original work.
D. O. DOMMASCH
7A:1 INTRODUCTION
to conduct flight tests at constant he to
provide curves directly such as shown in
Fig. 7A:2, we find that alternate schemes
for obtaining the optimum energy climb
schedule are employed. There are two basic
procedures, which are:
In Chapter 7 it was established that
a surface (E) may be constructed represen-
ting the overall climb performance of a given
airplane. In that chapter, the surface (E)
was presented in terms of a coordinate
axis system consisting of the orthogonal
rectangular coordinates, w: dhe/dt, V2/2g
and h.
(1) Level flight acceleration runs
(2) Continued climbs
It is apparent that this surface can also
be presented in terms of the set of coordi-
nates, w, he and V2/2g as shown in Figure
7A:1.
As discussed in Chapter 7, there are
advantages and disadvantages to both methods
and the selection of one depends on the test
circumstances. For any case, it is necessary
to compute the value of w: dhe/dt and, in
the case of continued climbs, the true tape-
line altitude. These items are considered
below.
The cross section of (E) at a given
value of energy height appears as in Figure
7A:2.
7A: COMPUTATION OF TAPELINE OR
GEOMETRIC ALTITUDE AND EN-
ERGY HEIGHT
From this figure we see that the speed
corresponding to maximum rate of change
of energy height is at once determined by
constructing a horizontal tangent to the cross
section curve. Since it is not convenient
To determine the true altitude we start
7:38
W
w
=h-h
29
(8)
constant
Section along
he=constant plane
h: constant
Villa
ILA
V for optimum
energy climb
Fig. 7A:1
Fig. 7A:2
with the well-known equations
dh
T
dhp
Tst
dp :-podh
7A:5
7A:1
P: PRT
7A:2
Either of Eqs. 7A:3 or 7A:5 may be in-
tegrated between the limits (1) and (2) to
give (for average values of temperature and
pressure)
where in the English engineering system
R : 1718 ft.? /sec? °R.
RT
Δη :
Ap
Combining these, we find that
9 P
7A:6
dp =
g p
RT
dh.
or
7A:3
т T
An :
Anp.
T st
In the standard atmosphere
9 р
RTst
dp
dhp
7A:4
The average temperature is (T, + T2)/2
and the average pressure (Pi + P2)/2 and
Ap : P2 - P. Thus Eq. 7A:6 becomes
when Tst • standard temperature at hp
R
an :
hp
filho Hom) -
pressure altitude.
Tat
2
+ T
+
PI
(P2 - DD
9
P2
Dividing Eq. 7A:3 by 7A:4
7A:7
7:39
and
R
ή : ΣΔ )
T2+ TI
R
Σ
g
El
Under subsonic conditions the Mach num-
ber may be determined from readings of im-
pact pressure and static pressure (suitably
corrected for instrument, position and lag
error) using the isentropic relation
P2 + P
(P2-PI).
Y/Y-1)
Y-1
Di
Ds
= (1+1=1 me) *Ar=1
부
​M2
7A:11
The pressure and temperature may be
measured at intervals of several thousand
feet of pressure altitude during the climb (or
these data may be determined from radio-
sonde transmissions) and Ah computed for
each interval. The sum of the interval heights
gives the true altitude corresponding to each
pressure altitude.
where y = 1.4.
7A:3 DETERMINATION OF w: dhe/dt
Graphical integration may of course also
be used to determine the altitude, since
2
T
ከ :
f
dhp
Tst
The value of w for any value of he is ob-
tained as the slope of the curve of he vs. t.
Although the slope of this curve is often
obtained graphically, greater precision may
sometimes be obtained using analytic pro-
cedures. The finite difference interpolation
formulae of Stirling or Bessel may be dif-
ferentiated to provide equations for obtaining
derivatives of experimental data. Consider
Fig. 7A:3, which is a plot of energy height vs.
time for a given test run.
7A:8
and the area under the curve of T/Tgqvs. hp
gives the true tapeline altitude.
The energy height being defined as
V2
ne : energy height = h +
29
We first break the time interval into con-
venient equal increments At. At some time
In the energy height, has a value hen. Simi-
larly at tn+i, we have hent, etc. The dif-
ferences hen - henti-hen are now computed
for each in and from these the coefficients
7A:9
-
is simply obtained by adding the tapeline alti-
tude to the value of V?/2g existing at a given
test point.
An
.
Ahe, Ahen-
Bn
+ Ahen-2
Ahen tl
Since we must know the true speed, we
make use of the equation given in Chapter 1,
V = Mc where V : true speed, M : Mach
number, C: sonic velocity. The sonic
velocity is given by
are determined.
The value of wn :
by the relation
dhen/dt is then given
c: KVT
т
7A:10
where K is a constant depending on the units
used for c and T.
7An -Bn
:
wo
12 AT
7A:12
7:40
During test, dE/dt is the excess power
given by
he
Ww' : (F'-D')
whereas, under standard conditions
henta
menti
WW
: (F-D) V
nen
subtracting
neni
hen-al
Wow' -w) = [F'-F-D'-D] v
or
Wdw : VIDF-dD)
7A:13
where dF is the variation of thrust from
standard and DD is the variation of drag
from standard.
+1 +2
Fig. 7A:3
We now consider the problem of reducing
the data to standard atmospheric conditions
using a constant weight, constant speed, con-
stant pressure altitude procedure. In cor-
rection the following are held constant
7A:4 CORRECTION PROCEDURES
pressure altitude
h
In general, flight data must be corrected
for the following variations from assumed
standard conditions
static pressure
P
true speed
V
(a) Thrust variations
(b) Non-standard atmospheric conditions
weight
W
The table on the following page gives a
listing of the relation between test and cor-
rected conditions.
(c) Thermal air current effects
(d) Gradient wind effects
Under standard conditions the energy
height is
he : hp +
ala
29
7A:14
In the following work a prime superscript
designates uncorrected data and the symbol
by itself represents corrected data, We
shall consider first the effects of variation
of thrust and drag brought about by a "non-
standard engine' and by atmospheric varia-
tions.
)
and during test
V2
he : h' +
šla
By definition
29
7A:15
de
wdhe
so that
: W w
di
do
where the notation is as in Chapter 7.
ne che : n'-ho.
7A:16
7:41
The corrected value of w may be com-
puted using relation 7A:13
The test Mach number may be obtained by
standard procedures; moreover, since we
are holding true speed constant during cor-
rection, the corrected Mach number is given
by
Wdw : VdF -dD)
7A:13
M : M
Normally, the value of dF greatly exceeds
that of DD 80 that the quantity dD may be ig-
nored. However, in certain instances it may
be necessary to compute dd to determine its
order of magnitude. This may be accom-
plished as follows:
ww
where Tg is the standard atmosphere tem-
perature corresponding to the test pressure
altitude.
Provided the lift-drag polars at various
Mach number are known (we note that it is
possible to obtain these from flight tests),
the quantity dD may be computed as follows:
The test lift coefficient is given by
dD :D-D
'
W
CL:
and
Yog MB
S
2
Pom'' sco
o
8
2
and the corrected lift coefficient by
Ps
M² SCO
D:
W
2
CL
YD, M
м?
where Ps : test static pressure : standard
static pressure.
S
2
Test Conditions
Corrected Conditions
Energy height time derivatives
w': W + dw
W
Energy height he
he
True altitude h'
h :
• hp
Ambient temperature
T': Tg+ dTg
Ts • standard temperature at hp
Mach number, M': M + dM
8
M
Thrust, F' : F + df, under the test
conditions P's, T's and N'
F : standard thrust at Ps, Ts and standard
RPM
7:42
Referring to the airplane polars, using
the known values of CL and M, the drag cor-
rection dD is readily obtained.
To obtain the thrust which the engine
would produce in a standard atmosphere,
standard engine correction procedures are
used. When thrust measuring equipment is
not available, engine manufacturers' data
may be used to compute standard thrust,
with the procedure being as follows:
operation within the airplane, although the
use of the derivatives of/a (N/00) and of/oM
does provide improved accuracy, since gen-
erally the slopes of the curves do not change
as much as the values of the functions them-
selves.
In the event that thrust measuring equip-
ment is available, the actual thrust developed
may be determined. This is accomplished
using the following considerations:
For choked or unchoked flow in the tail
pipe we find (from the De Laval equations)
that the gross thrust FG can be expressed
by the relation
The relation between thrust and the vari-
ables on which it depends for a standard jet
engine (without variable exit area or twin
spool arrangement) is the well known equa-
tion
N, M,
F : f
VO
7A:17
FG
8
18)
8 AB
Ps
7A:19
where FG
2
gross thrust
where 8 = Ts/Tso
0 To/Tso: ambient temperature/
sea level standard temperature
:
00
Ag : tail pipe exit area
8 P/P.
Pry
: total pressure in tail pipe aft of
turbine
M : Mach number ~ VIT
8 = Ps/P80
/
: pressure ratio
Ps
ambient static pressure
3
and
For this analysis, V : constant and Ps •
constant (therefore, 8: constant). However,
the temperature varies so that N/V and M
are variable. Thus,
《)
C**-(32)(7=-
: KD
for
unchoked
flow
af
N
so
dF : F'-F
4 )
(
AM
ам
DIN./0)
7A:18
7A:20*
and
: KD SO
so (1. 26 )ror
-1
Pt
1.26
Ds
for choked flow.
where AM is computed using the test speed
and the test temperature in comparison to
the test speed and standard temperature at
the test pressure altitude,
In Eq. 7A:18 we may correct at constant
N if this corresponds to the standard value
of N or A(NWO) may be computed for both
variations in N and Jo. Generally the de-
rivative df/oM is small and the first term
in Eq. 7A:18 is predominant.
Frequently engine charts do not present
too accurate a picture of the actual engine
7A:21*
.
* These equations have been rearranged
somewhat from their original form to permit
the utilization of the gross thrust meter de-
veloped at the USNATC.
7:43
()
PS (+1)/Y
:
27
* W = KQGVy-D)R
open your
ly1
Pr
The transition from choked to unchoked
flow occurs at about P1/Ps : 1/85 for con-
ventional nozzles. The factor Kappearing in
Eqs. 7A:20 and 74:21 is an efficiency fac-
tor accounting for non-isentropic conditions,
whose value(s) should be determined by static
thrust stand calibration as a function of
(Pty/Ps).
for unchoked flow
7A:27
Using the test values of (P/Ps) the test
value of FG may be computed from Eqs.
7A:19, 7A:20, and 74:21.
The constant Kog is the discharge effi-
ciency factor comparing the assumed isen-
tropic flow to the actual. For many nozzles
KOG * 0.9; however, its value should be de-
termined from ground tests as a function of
pressure ratio.
The gross thrust is defined as
FG
QVex
7A:22
where Q
8
mass flow per unit time
From the above considerations, we may
determine the mass flow through the engine
when values of Tto, Ps and Pr, are measured.
Therefore, using'Eq. 7A:24,' the value of F'
may be determined.
.
Vex : effective exit velocity
:
Now we are interested in the net thrust
which is
F: FG-FR
7A:23
where FR
8
: Ram drag : QV, and V is the
V
flight velocity. Thus,
The question now arises as to what are
the desired standard values of Pta Tt, and
Ae (for a variable tail pipe area installation).
The engine manufacturers' data are fre-
quently the only source of this information,
however, the specified values of tail pipe
pressure and temperature at a given RPM may
not actually be attainable in flight, so that
these data may be of limited use. Nonethe-
less, in the absence of other data, such as is
attainable by flight under nearly standard at-
mospheric conditions, the manufacturers'
data must be used.
.
F: FG - QV.
7A:24
To determine Q, we again turn to the
De Laval nozzle equations, from which
The
QTTE
AePt,
o
Regardless of the method followed to ob-
tain the standard values of pit, Tt, and Ae,
we have the following relations (assuming
Ps and V are constants):
7A:25
where Tit
• total tail pipe temperature
(a) From Eq. 7A:19
PT.
Pr : total tail pipe pressure
FG : Ap 84
Ap80 (:
FG = A' 84
sos
y
:
KOG
금 ​(유 ​$)**"
2 lly+1)/(Y-1)
y+1
R
:. dFG : FG-FG
constant for choked flow
Аеф
Ps
7A:26
7:44
Similarly,
7A:5 DETERMINATION OF CORRECTED
W = dhe/dt AND CORRECTED TIME
TO CLIMB
dFR = FR-F' : VIQ'-Q) - VdQ
) =
where, from Eq. 7A:25
The test or observed value of w must be
corrected for thrust, drag and horizontal
wind variations. From Eq. 7A:13 we have
that the change in w due to thrust and drag
variations is
AePtt
Q = y
Т.
dw =
(DF-dD)
W
and
Frequently dD is small and may be ig-
nored so that
V
dw : DF
W
Aept,
dQ
Aept
/те
W
(5)
TH
where dF is given either by Eqs. 7A:18 or
7A:26.
7A:28
Due to gradient wind effects, we have an
additional increment, (Eq. 7A:29)
so that dF :
dFG - VdQ.
V
dw =
Obviously, the correction may also be
accomplished using equations obtained by
differentiation of Eqs. 7A:19 and 74:25.
co
dV
dh
dh/wind dt
g
Thus,
dh
W
w
>13
dF
ch
in /wind
wind dt
7A:30
It remains here to consider the effects of
atmospheric disturbances. As previously
noted in Chapter 7, it is difficult to account
for vertical air motions since their measure-
ment is not easily accomplished. Therefore
tests should not be conducted under known
conditions of significant vertical atmospheric
motion. To account for a known vertical
gradient of horizontal wind during continued
climbs, we have the relation (from Eq. 7:16)
that
)
We have demonstrated that the standard
atmosphere energy height corresponding to
the test pressure altitude and speed is given
by Eq. 7A:16, which we rewrite as
ne-he-n' the
.
7A:31
v
dw :
co).
g
dt/ wind.
Using these two equations, 7A:30 and 7A:31,
each raw data point may be corrected to give
the standard variation of wvs. he. Then
since
hea
If the wind gradient is dv/dh and the ac-
tual climb rate is dh/dt then
dhe
I
hel
7A:32
dw :
o )
V dV
dh
dh wind dt
)
7A:29
we may prepare a plot of 1/w vs. he as shown
7:45
in Fig. 7A:4 and the shaded area represents
the time to climb from he, to
hez:
Data may either be recorded using a
photopanel or may be transmitted to a ground
station. Appropriate measures should be
taken to identify runs and to insure proper
functioning and continued calibration of the
instrumentation. Note should be made by
the pilot of operating limitations other than
those specified by the manufacturer (for ex-
ample, excessive build up of tail pipe temper-
ature preventing attainment of rated RPM).
After it has been ascertained that the
flights were properly made and that the in-
strumentation was functioning properly dur-
ing the tests, the data may be reduced to
standard, First, consider the case of the
continuous climb to ceiling. The raw data at
each point are first corrected for (a) instru-
ment errors, (b) position errors and (c) lag
errors using the procedures described in
Chapter 1 of this manual. Following these
corrections, the test data (again using the
procedures explained in Chapter 1) are con-
verted to:
nez
Fig. 7A:4
7A:6
SUMMARY OF PROCEDURES
(1) P's test ambient pressure
We shall here indicate briefly the proce-
dures to be followed in obtaining and reduc-
ing energy climb test data.
(2) hp test pressure altitude
First, we must provide means of meas-
uring the following basic quantities:
(3) Vcal test calibrated airspeed
(4) M' test Mach number
(a) Time
(5) T's test ambient temperature
(6) V' test true airspeed
(7) c' test speed of sound
(b) Static pressure (system calibrated at
all altitudes)
(c) Dynamic pressure
(d) Total temperature
(e) Engine RPM
(1) Fuel flow
(g) Normal acceleration
(8) W' test gross weight (from fuel flow
data).
We now compute the tapeline altitude for
the points in question using either Eq. 7A:7
For engine thrust determination the fol-
lowing should also be measured:
R
Alt.
h : Ean :
S.L.
(h) Tail pipe area
(i) Tail pipe total pressure
(j) Tail pipe total temperature
T2 + TI
P2 + PI
# =( Tato)
(P2-DID
7:46
2
or Eq. 7A:8
of (12) and (14)
h :
T
dhp •
To
st
(15) df
If it is considered necessary to correct
for gradient wind and drag effect, we com-
pute (dw)wind from Eq. 7A:29
For continued climbs, a systematic pro-
cedure may be worked up for calculating Ah
between succeeding points with h being given
by the sum of the Ah's to the given point.
Thus,
(dw)wind
V dV
dh
g dh /wind dt
(9) h, tapeline or geometric altitude.
Where dh/dt is obtained from (a) and (9);
dD is computed according to the method de-
scribed in section 7A:4. This gives
Knowing h and V, we compute he:
(10) he
h + V2/2g :
(16) dD
From a plot of he vs. time, or using Eq.
7A:12
(17) (dw)wind.
Tan
Bn
From (15) and (16) and Eq.79:13, we com-
pute dw = V/W (dF-dD)
wn
12 DI
(18) dw :
8
V/W (DF-dD) then
we compute w'
(19) w: w'
(18) - (17)
.
(11) w' . dhe /dt.
:
From Eq. 7A:31
From the engine data we next determine
the test values of net thrust using either Eq.
7A:18 or 7A:28
he
ne : the ch' the, thus
né -n't ho
(20) he = (10) - (9) + (2).
:
O
(12) F': engine net thrust.
We now correct to standard atmosphere
conditions holding the following quantities
constant:
If we are correcting data for continued
climbs to ceiling, the standard time to climb
must be determined. This is accomplished
using items (19) and (20) for all data points
up to the given point and Eq. 7A:32
2
dhe
1
hp : pressure altitude
3
V = true speed
:
W
W: weight
(21) t : corrected time to climb to he.
At hp and V we determine
After the data from all runs have been
corrected we proceed as follows:
() Τ
(13) Ts : standard temperature at he
(14) Pst: standard thrust at hp and V.
(a) For continued climbs to ceiling the
data from all runs are plotted in the form of
Fig. 7:4; i.e., in the form of he vs, time.
Cross plots are then prepared as in Fig. 7:5
We next determine dF from the difference
.
7:47
to determine the velocity V for wmax at
any value of he. Knowing V and he for wmax
we may compute the standard altitude for
Wmax from
7:8 then prepared. This immediately deter-
mines the climb schedule V = f(h). When
this method of testing is used, the time to
climb should be determined by a continued
climb to ceiling according to the optimum
schedule V = f(h) with the time to climb and
other data corrections being made by the
methods outlined in this section.
v2
h =
he
-
2
29
and continuing the process we thus deter-
mine the maximum energy climb schedule
V = f(h).
7A:7
CONCLUDING REMARKS
For pilot information this schedule should
be finally presented in terms of Vcal vs. f(h).
(b) If tests were conducted utilizing ac-
celeration runs, the corrected data are plotted
as in Fig. 7:6 and the cross plot of Fig.
Mr. Renaudie's paper contained consider-
ably more detail than this abbreviated ver-
sion; however, it is felt that the missing
details can readily be filled in by the flight
test engineer working with a particular
problem.
7:48
AGARD FLIGHT TEST MANUAL
VOLUME I, CHAPTER 8
TAKE-OFF AND LANDING PERFORMANCE
PART I
FLIGHT TEST ANALYSIS
By
F. E. Douwes Dekker
National Aeronautical Research Institute
Amsterdam, The Netherlands
PART II
THEORY
By
D. Lean
Royal Aeronautical Establishment
United Kingdom
ش ش کن
TERMINOLOGY
(Continued)
Lift Coefficient
Drag Coefficient
Power Coefficient
Ground Run Wind Correction Factor
Unit
Fw
FB
Ground Run Slope Correction Factor
rad.
rad.
Y
Flight Path Angle
rad.
rad.
B
Slope (uphill positive)
P
Air Density
kgs2/m+ (lb) (sec2)
ft+
σ
Density Ratio
relative to sea
8
Absolute Temperature Ratio
level standard
8
Pressure Ratio
μ
HB
Subscripts
0
1
2. S
. t
A
Propeller Efficiency
Coefficient of Rolling Resistance
Coefficient of Effective Braking Resistance
for All Wheels
Sea Level Standard (also standstill)
Take-off or Touch Down Point
At Obstacle Height
Standard
Test
Increment of Test Quantity to Reach Corrected or
Standard Quantity
A1, A2 etc. are Constants
8:12
8:13
(c) Lift Coefficient Corrections
(d) Landing Assistance
SUMMARY OF EQUATIONS
CONCLUDING REMARKS
CONTENTS
(Continued)
Page
8:18
8:18
8:18
8:19
PART II
THEORY
8:14
GENERAL PRINCIPLES
8:20
8:15
THE GROUND RUN
8:20
8:16
THE TRANSITION AND CLIMB
8:23
8:17
PILOTING TECHNIQUE IN THE AIRBORNE PHASE
8:29
8:18
APPLICATION TO FLIGHT TEST PROCEDURE
8:31
8:19
CRITICAL SPEED DURING TAKE-OFF
8:31
8:20
THE LANDING MANEUVER
8:33
8:21
ESTIMATION OF LANDING DISTANCE
8:33
8:22
THE CHOICE OF THE LANDING APPROACH AIRSPEED
8:37
8:23
SOURCES OF VARIATION IN TOTAL LANDING DISTANCE
8:39
REFERENCES
8:40
TERMINOLOGY
Unit
m/s
ft/s
RC
Rate-of-Climb
RS
Rate of Descent
D
Total Aerodynamic Drag = 0.5 v2 CDS
kg
lbs
E
Total Energy W/2g (V²+2gh)
kgm
lb ft
L
Total Aerodynamic Lift = 0.5 v2CLS
kg
lbs
N
Engine Speed
RPM
RPM
S
Wing Area
ft2
S
Horizontal Distance in Take-off or Landing
S
$2
Ground Distance
Air Distance
EE E
ft
ft
ft
V
True Airspeed
m/s
ft/s
<
True Airspeed at Take-off Point (getaway) or Touch Down
m/s
ft/s
V₂
True Airspeed at Obstacle Height
m/s
ft/s
W
All-up Weight or Gross Weight
kg
lbs
P
Power
BHP
Engine Brake Horsepower
F
Net Engine Thrust
kgm/s ft lb/s
kg
lbs
J
Advance Ratio
a
Acceleration
m/s2 ft/s2
d
Deceleration
m/s2
ft/s2
60 C
g
Acceleration Due to Gravity
m/s2
ft/s2
h
Obstacle Height
m
ft
t
Time
S
's
Vw
Surface Headwind
m/s
ft/s
TERMINOLOGY
VOLUME I, CHAPTER 8
CHAPTER CONTENTS
Page
PART I
FLIGHT TEST ANALYSIS
8:1
SUMMARY
8:1
8:2
TAKE-OFF DEFINITIONS
8:1
8:3
TAKE-OFF MEASUREMENTS
8:2
8:4
BASIC TAKE-OFF EQUATIONS UNDER NO WIND CONDITIONS
8:4
(a) Ground
8:4
(b) Air Phase
8:5
8:5
PARAMETER THEORY
8:7
8:6
TAKE-OFF PERFORMANCE REDUCTION
8:9
(a) Wind Corrections
8:9
(b) Slope Corrections
8:10
(c) Lift Coefficient Corrections
8:11
(d) Take-off Assistance
8:12
8:7
SUMMARY OF EQUATIONS
8:12
8:8
LANDING DEFINITIONS
8:15
8:9
LANDING MEASUREMENTS
8:15
8:10
BASIC LANDING EQUATIONS
8:16
(a) Ground Run
8:16
(b) Air Phase
8:17
8:11
LANDING PERFORMANCE REDUCTION
8:17
(a) Wind Corrections
8:17
(b) Slope Corrections
8:18
Although the choice of h is fixed by appli-
cable specifications, it is supposed for this
discussion that the air phase is a transition
maneuver, which begins when the aircraft
becomes airborne and continues until the
altitude h and speed V2 are attained simul-
taneously. It is assumed that take-off power
is maintained until the air phase is completed,
that the flap setting remains constant during
this phase and that the landing gear is
retracted as soon as safety permits.
To keep the scope of this chapter within
practical limits, the discussion will pertain
exclusively to land planes with take-off
weights of more than 2000 kg (4400 lbs)
operating from hard surface runways
(μ≈0.02). Light aircraft utilizing hard or
soft turf require special analysis for each
given case as do multi-engined airplanes
under simulated engine failure conditions.
The determination of critical engine failure
speed and accelerate-stop distances will not
be considered here.
The accuracy of a given measuring device
or instrument or method is assumed to be
such that there is a probability of 95%
that the difference between the actual and
measured values will be less than the stated
error (95% confidence limit).
Throughout the following work, suffixes
s and t are used for standard and test
conditions respectively; however, subscripts
are omitted in evident cases.
The increment A, added to the test
quantity of a variable, will give the standard
or corrected quantity, i.e., sc = S+As.
The definitions of the take-off mentioned
above are generally used for take-off per-
formance reduction. The formulae appearing
later in the chapter are based on these
definitions. However, a different approach
PART I
FLIGHT TEST ANALYSIS
8:1 SUMMARY
In this chapter the basic principles of
take-off and landing performance testing are
presented. The object, means and accuracy
of the measurements are discussed.
The basic equations underlying the
performance reduction methods are derived
and explained.
8:2 TAKE-OFF DEFINITIONS
A take-off is defined as the process by
which an airplane is safely brought from
standstill to a safe flight condition.
The safe flight condition is defined as
that point in a climb at which the airplane
has first reached a height of h meters or
feet (obstacle height) above the point of
departure from the ground and an instan-
taneous true airspeed of V2 meters per
sec. or ft. per sec.
The take-off is divided into two parts:
the ground run and the air phase. The air
phase begins when the airplane becomes
airborne.
In the past, the choice of the obstacle
height has been made somewhat arbitrarily.
Usually 15.2 m (50 ft) has been taken as
the required height above the runway, but
there is now a tendency to choose 10.6 m
(35 ft). The choice of h affects the reduction
method because it determines whether or not
a steady climb will be achieved before the
obstacle height is reached. Generally, high
average acceleration during take-off and/or
low obstacle height h eliminate the require-
ment for subdivision of the air phase into
transition and steady climb phases.
8:1
to the problem is presented in section 8:5.
8:3
TAKE-OFF MEASUREMENTS
The principal aim of flight testing is to
collect reliable data which may be used to
predict airplane performance under standard
conditions as well as under certain non-
standard conditions. Uncontrollable factors
such as pilot skill must be eliminated from
the tests as far as this is possible so that
the uncontrolled variation in the quantities
to be measured may be reduced to a minimum.
Take-off tests should be delayed until
a late stage of the prototype test program
in order that some agreement between pilots
regarding a standard take-off technique may
be arrived at. Moreover, for a take-off
test the airplane should be operated in such
a way that repeatability of data is assured,
rather than in a manner leading to the best
take-off performance.*
This means that during a take-off sudden
changes in altitude and configuration should
be avoided unless repeatability of the action
is assured. The whole maneuver should be
carried out gently and safely. The basic
features of take-off performance of the air-
plane are the topic to be investigated and not
the abilities of the pilot.
The accelerations obtained during the
ground run and air phase may be regarded
as fundamental characteristics of the take-
off. However, it is usual to measure the
take-off performance in terms of ground and
air phase distances under standard and other
conditions. These distances détermine the
mean accelerations defined later in the text.
Generally, the standard take-off distances
must be determined within about 5%. The
accuracy of the final reduced data is deter-
* Editor's note: This statement, it is believed,
applies generally to commercial aircraft
but not necessarily to all military machines.
mined mainly by the accuracy of the reduc-
tion process, the test accuracy and the number
of tests.
The test accuracy depends on the accuracy
of airspeed measurement at the point where
the airplane leaves the ground and at the point
where the airplane has reached the height of
h, and thus on the test equipment available.
Supposing a 4% natural scatter in the true
distance (due to uncontrollable factors), it
can be determined from a probability analysis
that with high accuracy test equipment a
minimum of five tests for one configuration
are required. Generally, a test accuracy of
2.5%, obtainable with six runs, should be
aimed for. One or two additional runs allow
a decrease in the required test accuracy to
3.6% and 5% respectively while maintaining
the overall 5% accuracy.
The actual ground speed at the point where
the airplane leaves the ground, the ground
speed at obstacle height, ground distance,
air distance, air phase time, surface head-
wind, air temperature and air pressure are
the principal quantities to be measured.
A full time history of the take-off is usually
accepted as a must, but accurate take-off
analysis shows that the measurement of the
speed and height at the two stations defined
above is sufficient for the purposes of this
analysis. In the latter case the test equip-
ment may be simplified considerably without
reducing the accuracy of the results.
Records of only the two portions of the
airplane's flight path are necessary: (1)
where the airplane becomes airborne; (2)
where it attains the obstacle height. For
detailed investigations, a complete record of
time and distance or acceleration is required.
All test methods require knowledge of
• actual net thrust at about 0.7 V and at
0.5 (V₁+ V₂).* This involves measurement
*V is the speed at which the airplane
leaves the ground and V2 is the speed at
the altitude h.
8:2
of at least the following: engine speed,
manifold pressure (jet tail pipe temperature),
air temperature, air pressure and humidity.
The power setting during take-off must be
recorded either by the pilot, observer or
automatic recording equipment.
For jet-
powered airplanes the use of the latter is
preferable to provide an accurate record of
engine speed. The actual net thrust as well
as the standard net thrust may be estimated
from engine manufacturer's data or may be
measured by means discussed elsewhere in
this manual.
All controllable variables should be held
as nearly standard as possible with maximum
operational take-off weight usually being
chosen as the standard weight. It will be
worth the effort required to refuel frequently
in order to hold each test take-off weight
within 5% of the standard value.
Tests on very hot or wet or iced runways
should be avoided: moreover, tests should
not be attempted under crosswind or gusty
conditions or when the windspeed exceeds
10% of V2.
The initial acceleration does not affect
the ground distance appreciably so that the
use of maximum power with the airplane
restrained by the use of brakes prior to
the start of the take-off is not required.
However, the throttle should be fully opened
as soon as possible after commencement of
the ground run. It is noted that initial
acceleration does affect ground run time;
however, the use of measured ground run
time is avoided in the following analysis.
Determination of the exact point at which
the airplane leaves the ground has always
been a difficult and rather confusing problem.
Direct observation, camera, or accelerome-
ter recordings do not provide the means
for accurately determining this point. Only
eventmarkers on the main undercarriage legs
might solve the problem. However, for this
analysis the exact location of the point at
which the airplane first becomes airborne is
considered unimportant, compared with the
location of the point where the airplane has
achieved the obstacle height and safe flight
conditions.
For the purpose of this anaysis, therefore,
it is emphasized that accurate measurement
of the ground speed near the point where
the airplane leaves the ground is more
valuable than determination of the exact
location of the point. The ground distance
required to achieve a certain kinetic energy
level, based on a safe true getaway speed
appropriate to actual wind, weight and density
conditions is considered a reliable perform-
ance parameter, as is the total horizontal
distance required to reach a certain total
energy based on obstacle height and minimum
safe equivalent climbing speed. Some pro-
posals based on this concept are referred
to in section 8:5.
Present day equipment used for actual
airport-level air pressure, air temperature,
humidity, wind speed and wind direction
measurement is adequate for purposes of ob-
taining take-off test data. Self-counting ane-
mometers giving the average wind speed
during the take-off are preferred.
The track or path of the airplane is usually
recorded by motion picture cameras placed
either at the side of the runway or at some
distance up or down the runway. Plate
cameras with a moving diaphragm and a very
accurate timing device are also in use. All
photographic systems, however, lose accu-
racy for very long take-off distances which
require more cameras and complicated syn-
chronization.
When ground and air distances under the
test conditions can be estimated well enough
beforehand, it may be sufficient to measure
the ground speed by photographic or photo-
electric means at only the two predetermined
stations, (1) and (2), in which case the
problem of synchronization can be simplified.
8:3
8:4
is:
BASIC TAKE-OFF EQUATIONS UNDER
NO WIND CONDITIONS
(a) Ground Run
The ground distance s, (no wind, no slope)
V₁
Sds =
s₁ = √
VI
s
ds ᏧᏙ
1
dt dt
dv
2
s
dv²
2a
=
vi
2a
8:1
The actual acceleration a during the
ground run is a function of V and decreases
from the initial value ao to a final value of
a₁. A mean acceleration ā, giving the same
overall speed-distance change as the actual
variable acceleration, can be defined by the
equation
For the average jet-powered airplane,
V/V₁ = 0.70±0 02 for a₁/= 0.8.
For special power units as well as mixed
power systems(turbo-prop) the value of V/V₁
can be obtained from Fig. 8:1 for an estimated
value of a₁/a。.
Using the above approach, actual solution
of the basic integral (Eq. 8:1) is avoided,
and the performance reduction can be based
on the use of the simplified concept of an
average acceleration a
When the coefficients of drag, lift and
rolling resistance are kept constant during the
reduction process, it can be stated that
V² = A₁ W/o
and
µ(W− ¯ ) +Ō = A₂W
10
= // [F - μ (W - [) - ō]
W
where A, and A2 are constants.
Then the expression for the
ground
distance becomes
11
A₁ W/O
2g
(틉​-A2
8:3
The difference between the ground run
distance obtained during test and the distance
existing under standard conditions of weight,
density and thrust may be approximated by
logarithmic differentiation of Eq. 8:3, whence
F
ASI
A(W/o)
W
SI
W/σ
8:2
where F, L and D are mean thrust, mean
lift and mean drag, respectively, at a mean
speed V.
The ratio V/V depends mainly on the
acceleration ratio a/a。 and the functional
relationship between a and V.
However, including all possible functions
a = f(V), i.e., linear, square, etc., it can be
stated that:
For the average reciprocating-engined
airplane,
V/V₁ = 0.74 ± 0.02 for ª¡/。 = 0.4.
a/ao
8:4
where
and
A(W/G)= W (AW - AG)
A (F/W)
-
F/AF AW
W F
W
s₁ = ground distance corrected for wind,
runway slope and C₁
V₁ = true ground speed at the point where
the airplane becomes airborne (cor-
rected for C₁)
W =
F
11
။
so that
ā =
Asi
Δω Δσ
Fg
+
SI
W
Wa
(AW - 4F)
ΔΕ
م
test weight
test density ratio
calculated mean thrust at V(for test
power setting and atmospheric con-
ditions)
v² /2s,
(b) Air Phase
The air phase distance s₂ (no wind, no
8:4
slope) is:
$2
S
s2 =
SI
The true speed V₁ will change by an amount
AV₁, consequently
۵۷۱
AVI
= 0.5
AW
W
Aσ
0.5
م
dE
dE
dt
1745 - 7 (1/4) E
ds =
as
013
E2
s
WEI
dE
v2 – v³ + 2gh
2ā
8:6
w/2g
where the total energy at point lis E₁ =
(Vi) and at the height h (point 2)
is E2 = W/2g (V² +2gh).
The energy increase dE/dt is equal to the
useful work accomplished per second (T-D)V,
where the excess thrust (T-D) is equal to
the product of mass and flight path accelera-
tion a, relative to the gravitational field.
The mean acceleration ā at the mean true
speed V can be expressed as
이
​g
W
[(F-D)].
8:7
As the speed increase during the air phase
is usually small, the mean speed V can be
8:5
It should be emphasized that Eqs. 8:4
and 8:5 are only valid for small increments
▲, and for cases where s and V have
been previously corrected for wind, runway
slope and C₁ as discussed in section 8:6.
Similar relations for larger corrections
and an anlysis of the relation between net
thrust and effective accelerating force are
presented in Ref. 1.
Generally, the increments AW, Ao and
AF should be determined with an accuracy
of about 0.5% of the test values of weight,
density ratio and mean thrust, respectively.
Summing up, the following test data are
required when the reduction formulae 8:4
and 8:5 are used:
8:5
When
taken equal to 0.5(V₁+ V2). The terms
F and D are mean net thrust and mean
aerodynamic drag, respectively, at V.
CL, CD and h are assumed constant during
the reduction process, the true speeds
squared V and V2 are proportional to
W/o, and the drag is proportional to W so that
v2 - v² + 2gh
+ 2gh = Az
W
+ 2gh
so that
As2
$2
v2 - v?
Vẻ - Vĩ + 2gh
and
Δσ
(AW - D=)
gF /Aw
AF
-(-)
W
8:9
Because the observed true speeds V₁ and
V2 also change by amounts AV, and ▲V2, we
have
D = A4W
where A3 and A4 are constants.
Accordingly, the expression for the air
distance becomes
52
11
A3+2gh
2g
W
29-A4
8:8
An expression approximating the change
in air distance for small variations in weight,
density and thrust is obtained by logarithmic
differentiation of Eq. 8:8, whence
A(W/o)
A(F/W)
As2
A3 2052
- g
52
where
and
A(W/σ) =
१|ह
Δω
Δσ
W
م
ā
Δω
・E (AF - AW)
W
A(F/W) =
AVI AV2
V2
= 0.5
AW
W
- 0.5
ala
8:10
Similar relations for large corrections and
details about the kinetic/total energy increase
ratio and the net thrust/accelerating force
ratio are given in Ref. 1.
The required accuracy for the increments
AW, Ao and AF can be taken to be 0.5% of
the test values of weight, density ratio and
mean thrust respectively.
Application of Eqs. 8:9 and 8:10 requires
the following test data:
s2 = air distance, corrected for wind,
runway slope, and CL
V₁
true getaway speed, corrected for CL
V₂ = true speed at obstacle height, cor-
rected for CL
W = test weight
b 14
test density ratio
F = calculated mean thrust at V, cor-
responding to test conditions
=
(V² − v² + 2gh)/282
8:6
8:5 PARAMETER THEORY
Dimensional analysis has often been used
to determine the controllable parameters af-
fecting engine performance, in particular
when dealing with jet engines. When the
results of such an analysis are combined
with the functional relations for the airplane
in a specific flight condition, valuable per-
formance reduction equations are obtained.
The basic ground run Eq. 8:3
SI
**
2g
A, W/O
F
W
1
A2
8:3
becomes, on substituting σ = 8/8
S1
AWS
8
F/8
29
W/8
- A2)
8:11
Thus, ignoring the influence of airspeed*
in this case, the general relationship will be
N
8
W
8
where Ñ = mean engine speed at V.
8:14
This latter Eq. 8:14 is for a specific
value of C at V and a fixed value of
runway friction coefficient μ. Also the dis-
tance s is the test value, corrected for
headwind, runway slope, and preferably for
C₁ at V₁.
A similar relationship applies to the air
distance (see Ref. 17)
$2² = 1. ( W )
f
8
8:15
or more generally
SI
= f
8
8
W
S
8:12
or for a jet airplane, ignoring speed (Mach
number) variation
52
= f
8
N
(뽕​)
which is a "non-dimensional" relationship,
with the constants S, CL, CD and g omitted.
For a jet-propelled airplane, Eq. 8:12 may
be combined with the well-known relationship
(see Ref. 6)
F
등 ​(.岩​)
N
8:16
8:13
It should be noted that poor intake effi-
ciencies at low airspeeds may affect the
validity of the basic Eq. 8:13. Some experi-
ence concerning this is reported in Ref. 17.
* Editor's note: The quantity V/ is actual-
ly proportional to Mach number, hence the
statement that we ignore the influence of
airspeed is equivalent, in this case, to
stating that the Mach number effects are
unimportant. This is, of course, a proper
assumption for take-off analysis.
8:7
For reciprocating-engine airplanes
and
F = "
P
V
n = f(Cp, J) = f
P
so that F/σ = f(P/o, N, V) and the mean
take-off thrust F/o= f(P/o, N,V).
For constant C, the mean speed V is
proportional to W/o and the take-off mean
thrust for the ground run or air phase is
N
W
f
σ
흥아 ​(흥​,지​, 뽕​)
8:17
Combining Eqs. 8:17 and 8:3 gives for the
ground run
(E. W. N) (P and N at V)
Ñ
S₁ = f
8:18
which equation also applies to the air phase
and pertains to specific C values at V₁ and
V2.
Thus, it is seen that the take-off distances
can be related to controllable parameters,
being combinations of controllable and free
variables. Theoretically it should be possible
to carry out every test at standard values of
these parameters, thus avoiding any correc-
tion. In practice, however, accurate control
of the parameters is sometimes not feasible,
and cross plotting is required.
Possibly the functions derived here might
be calculated during the design stages and
checked, as well as corrected, by take-off
tests. This, however, would require a great
number of tests, at least three for every
combination of two parameter values. The
advantage is that a three-dimensional "car-
pet" plot obtainable from the test data would
enable us to predict the take-off performance
for any non-standard condition in a relatively
simple way.
In the performance reduction method pre-
sented in the preceding sections, ground and
air distances must first have been corrected
to specific C₁ values at V and V2. This
means that exact determination of the test
distance at which the airplane becomes air-
borne is not really required. Rather, atten-
tion should be directed to the accuracy of
speed measurement at the point on the runway
where the kinetic energy is expected to reach
a level sufficient to lift the airplane from
the ground under the given conditions.
Following this same line of thinking,
accurate measurement of speed and height in
the air are required only at the point where
the total energy of the airplane is expected
to reach the standard desired value (at this
point speed and height separately should be
near their respective standard values). Cor-
rection for wind effects requires measure-
ment also of the time required to travel
between points 1 and 2.
If the philosophy of the analysis presented
here is applicable to the type of operation
for which the airplane is intended, the given
test reduction procedure leads to high accu-
racy at low cost. Fully normal and repeata-
ble test take-offs can be performed without
requiring attainment of the shortest possible
take-off distance. The forced pull-off climb
as well as the held-down zoom are not needed
as part of the take-off test techniques.
The test data are corrected to optimum CL
values at points 1 and 2. This requires
prior agreement about the standard optimum
take-off technique, and an investigation into
the efficiency of transformation of kinetic
energy into potential energy, and conversely.
8:8
8:6 TAKE-OFF PERFORMANCE
REDUCTION
(a) Wind
W
When the ground run is made into a
constant surface headwind of velocity Vw,
the initial condition of the airplane is such
that the ground speed is zero and the air-
speed is equal to the windspeed V. For
a still-air ground run, a ground distance Sw
is required to accelerate the airplane from
standstill to the airspeed Vw to reach the
initial condition mentioned above, assuming
that the coefficient of rolling friction μ is
independent of ground speed.
From this point on, the distance traveled
relative to the air to reach getaway air-
speed V₁ will be the same in both cases
with or without wind; the airplane accel-
erates from an airspeed Vw to an airspeed V₁.
In the first case, however, (ie., with wind)
the ground speed will always be Vw less than
under equivalent conditions during a still-air
ground run. Therefore, the ground distance
will also be Vwtw less, where tw is the time
required for the acceleration from Vw to V
true airspeed.
The influence of a headwind on the total
ground distance can be expressed by the
relation
SIW
11
=
|
Sw - Vw tw
8:19
or
or
W
V
VdV
a
VW
VdV
Vw
Vw
dv
(V-Vw) dV
a
11
Vw
VI-VW
(V_Vw) d (V-Vw)
(VI-Vw)²
2ā
where the mean acceleration ā occurs at
the mean ground speed (V/V₁ )(V¡ -Vw) or the
mean airspeed (V/V₁)(V₁ -Vw)+Vw for the
still air ground run (see section 8:4 for values
of V/V₁).
W
The basic equation for the still-air ground
run was
S
11
v²
2a
where a = acceleration at V.
Therefore, the ratio Fw of the ground
distances without and with headwind is
where
W
= ground distance with headwind V
2
V
W
の
​S|
ground distance in still air =s₁+Ası
Sw
tw
= ground distance from standstill to
speed Vw, still air
= time elapsed from speed Vw to VI
Fw =
5 | W
or in exponential form =
still air
8:9
8:20
where V₁ -V
=
C =
ground speed at getaway
with wind
Vw = surface speed at getaway
with wind
acceleration at (V/V₁){V₁−Vw)+Vw
acceleration at V
(for still air)
An analysis of the acceleration ratio C
for different acceleration characteristics re-
veals that the exponent p is practically in-
dependent of Vw and can be regarded as a
function of the final initial acceleration
ratio a/a。. In Fig. 8:2, Fw is given as a
function of Vw/(VI -Vw) and aι/ª。. If no
other data are available, the acceleration
ratio a/a。 can be assumed 0.8 (p = 1.92)
for jet airplanes and 0.4 (p = 1.68) for
propeller airplanes.
W
Thus, for purposes of this analysis, the
air distance correction for headwind is
simply
As₂ = Vw 12
8:21
For rather strong surface winds, the wind
gradient might be accounted for by taking
the magnitude of the effective surface wind
Vw to be 10% higher than the actual.
(b) Slope Corrections
The influence of a runway slope B on
the ground distance is given by
=
s₁ + As₁ = FB • S1
8:22
where Fg
ground run slope reduction factor
and s₁ = test ground run with slope ß.
An up-slope ẞ reduces the effective mean
excess thrust by the amount BW. This is
equivalent to an acceleration increment of
=
Aã = gß
where B mean runway slope at the test,
uphill positive.
So
From Eq. 8:1 it follows that
Asi
Δα
SI
ASI
2gs, B
v}
The influence of a surface headwind Vw
on the air distance can be divided into two
parts:
(1) The distance travelled by the
surface wind during the air phase time t2.
(2) The increase of effective accel-
eration, due to wind gradient.
With a headwind of speed Vw at the ground
and speed Vw+dVw at obstacle height h, the
acceleration due to wind gradient is roughly
equal to dVw/t2
According to the Prandtl wind gradient
equation, the wind at 15 m (50 ft) is equal
to 1.25 times the surface wind at 3 m (9 ft).
In which case, the decrease in air distance
due to the extra acceleration should be
roughly (Vw/8)†2, which is less than 1% of
the air distance when the surface wind is
less than 0.1 V2. Therefore, the influence
of the wind gradient is usually ignored;
moreover, actual conditions can differ appre-
ciably from the Prandtl law.
8:10
and
Thus
s₁ + As₁ = s
AS
(1-20618)
FB
= |
2gs, B
v²
2
8:23
correction to specified getaway and climb
speeds, respectively.
The differential equation for ground dis-
tance can be written
ds
의스
​dv₁
which when combined with the basic equation
v²/2ā gives
S
dsl ā dvi
2
S
8:25
The influence of B on the air distance
depends on the definition of h. There will of
course be no influence in case his defined as
the tape line altitude above the point where
the airplane becomes airborne.
Sometimes, however, h is measured per-
pendicular to an average runway plane with
slope ẞ. In that case, the excess thrust
reduction BW will in effect continue during
the airphase and can be included in AF
of the general reduction Eq. 8:9 or treated
separately
A52
52
Δα
gB
이
​For constant weight and density it may be
deduced from
W = 0.5p。 S CL, Vic
that
*CLI
CLI
11
- 2
consequently, for small increments
Δ
Asi
SI
a
ō ACLI
CL
8:26
Due to the change in C, the getaway
speed is changed by
ACLI
۵۷۱
- 0.5
CLI
8:24
(c) Lift Coefficient Corrections
Up to this point the performance reduction
equations have been developed keeping Cand
CD constant. In most cases, the pilot some-
what arbitrarily chooses the lift coefficients
at getaway and during the air phase, and it is
probable that the C values at getaway and
at the obstacle height will be different for
all test take-offs introducing some scatter in
the reduced performance data. Therefore,
ground and air distances should be corrected
to specific values of CL, which amounts to
8:27
Later on in this analysis when corrections
for weight and density are considered, the
standard value of CL will be held constant
and V₁ be further modified.
8:11
When no other data are available, the
factor a/a, in Eq. 8:26 can be taken as 1.55
for reciprocating-engine airplanes and as 1.1
for jet-propelled airplanes.
The influence of V2 on the air phase may
be expressed by the equation
V2
dV2 ·
ds2
02
8:28
air distance and so the total air phase correc-
tion is
As2
11
I
vi
v3 − v² + 2gh
ACL2
CLZ
S2 - As ·
Assuming the acceleration to be constant,
Eq. 8:28 can be combined with the basic
air phase equation
$2
11
v2 – v² + 2gh
20
giving
dV2
ds2
52
2v2
2
v3 − v² + 2gh
V2
and at constant weight and density
dCL2
CL2
Thus approximately
- 2
dV2
V2
As2
$2
v32
v2 − vi + 2gh
ACL2
CL2
8:29
8:31
Due to the CL2 correction, the airspeed at
obstacle height will be altered by
AV2
V2
-0.5
ACL2
CL2
8:32
(d) Take-off Assistance
Full duration take-off assistance and the
use of mixed power systems do not alter the
fundamental reduction formulae. However,
any change in the acceleration ratio a/a。
affects some constants in the equations and
this must be accounted for.
When part time assistance is used, it may
be necessary to subdivide the ground run and/
or the air phase for purposes of analysis.
In Ref. 1, it is proposed to use an effective
mean assisting thrust in the reduction equa-
tions, equal to the actual thrust multiplied
by the (actual operating/total phase time)
ratio.
Accurate space time recording at the time
of the start and finish of the use of assisting
thrust is required. Therefore, in some cases
a complete space time record may be neces-
sary.
8:30
To account for the kinetic energy change,
i.e., change in V₁ due to a change in CLI
the ground distance increment As, due to the
CL correction must be subtracted from the
8:7 SUMMARY OF EQUATIONS
(a) Ground Run
Measured values s₁, V₁, Vw, W, o, ß.
All corrections are to be applied in the
8:12
and $2
order given below; all values SI
appearing in the formulae are related to the
respective distances corrected for the pre-
vious effects.
(1) Wind correction
As
-1 = Fw - 1
(See
|| |
Fig. 8:1)
(2) Slope correction
Asi
2gs, B
SI
v}
FB-1
s, corrected for Vw
WIND CORRECTION FACTOR
Fw
1.4
1.3
1.2
1.1
5
1.0
0.9
0.8
10
-5
-5
-10
15
(3) CL, correction
ASI
SI
ā ACLI
CLI
AV
ACLI
– 0.5
V₁
CLI
reciprocating a = 1.55 a
jet
a = 1.10 a
s₁ corrected for V and B
GROUND RUN ACCELERATION RATIO
a
1.0
0.8
0.6
do
0.4
5
10
15
SURFACE HEADWIND
GETAWAY GROUND SPEED
VW
V₁ = Vw)%
EXPONENT
P
2.0
JET
1.8
MIXED
PROP
0.2
0.4 0.6 0.8
ACCELERATION RATIO
1.6
1.0
P
Fig. 8:1 The Wind Correction Factor Fw=
V1
Vw
and Exponent p as Functions
of Acceleration Ratio for the Ground Run
8:13
= 0.78
MEAN SPEED ▾
GETAWAY SPEED VI
0.74
PROP
MIXED
0.70
JET
ACCELERATION
LINEAR v2
ACCELERATION
LINEAR V
(ACCELERATION)
LINEAR v2
0.66
1.0
0.8 0.6 0.4
ACCELERATION RATIO
བད
"
Fig. 8:2 The Mean Speed as a Function of Acceleration Ratio
for Different Acceleration Characteristics During Ground Run
Combination of (1), (2) and (3) gives:
s, (corrected for wind, slope and CL)=
Fw.
- FB (1 - -1 - ACLI)
(b) Air Phase
Measured values s₂, V1, V2, Vw, σ, [2, W
(1) Wind correction
As₂ = Vw 12
S¡ test
where SI
test =
Slw
(2) CL2 correction
and
thrust
V를
​ACL2
52 - As
As2
v2 −vî + 2gh
CL2
(4) Weight, density
correction
ASI
AW Ao Fg [Aw
AF
+
م
Wa W F
SI
where a =
W
vi
251
where As is the correction on ground run
for ACLI
V₁ corrected for CL
s, corrected for Vw, B and CL,
F = mean thrust at V = 0.74 V₁ for
reciprocating engine and V= 0.70
V₁ for jet engine (see Fig. 8:2)
AV2
V2
= − 0.5
ACL2
CL2
82 corrected for Vw
8:14
(3) Weight, density and thrust
correction
As2
$2
where
AW
Δε
gF /Aw
+
wã \ w
VZ-V₁
v2-v²+2gh \ w
It
م
v2 − vi + 2gh
252
V₁ corrected for CL
V2 corrected for CL2
S2 corrected for Vw, C
CL
and
CL2
F mean thrust at 0.5 (V + V2)
AF
In Ref. 1, empirical constants are given
for a proposed simplification of the reduction
equations. These constants are based on
extensive tests of several groups of airplanes
having about the same characteristics.
somewhat different approach to take-off data
analysis is given in Chapter 6 of this manual.
A
8:8 LANDING DEFINITIONS
A landing is defined as the process in which
an airplane is safely brought from a safe
flight condition to a standstill.
The safe flight condition is defined as a
condition in which the airplane has a height
of h meters (feet) above runway level and
descends steadily with a true rate-of-descent
of RS meters per sec. (ft. per sec.) or main-
tains an angle of approach of y degrees with
a true airspeed of V2 meters per sec. (ft. per
sec.) in both cases.
The landing is divided into two parts: the
ground run and the air phase. The ground run
begins as soon as the airplane touches the
runway for the first time.
To simplify the problem, the following
assumptions are applied in the landing per-
formance reduction theory outlined here. The
air phase is considered as a transition
maneuver from the steady safe flight condi-
tion to touch down. It is assumed that the
engine setting is constant and that the lift
coefficient and deceleration increase until
touch down, at which point the engines are
throttled back. Moreover, it is supposed that
the flap setting remains unchanged throughout
the landing.
During the ground run the coefficient of
rolling friction is taken to be constant. The
possible reversal of propeller or jet thrust is
regarded as a safety feature and is not
considered in the standard landing perform-
ance analysis. Investigation of reverse thrust
requires separate tests, preferably with ac-
celerometers. The use of a brake or drogue
parachute might be taken as a standard landing
aid, provided that the moment of application
is controllable.
The theory presented here is limited to
land planes having more
more than 2000 kg
(4400 lbs.) landing weight with nose wheel or
tandem undercarriage and to landings on a
dry hard-surface runway. It is assumed that
only the main wheel brakes are applied.
Maximum allowable landing weight is re-
garded as standard, although it may be useful
to conduct tests at other weights as well. The
other variables, approach speed in particular,
should be held as close to their standard
values as possible.
Only power-on approaches, characterized
by a certain rate-of-descent or flight path
angle, will be considered. Glide approaches
for light aircraft are dealt with in Ref. 5.
8:9 LANDING MEASUREMENTS
Prior to the time when formal landing
8:15
obstacle height, groundspeed at touch down,
air phase time, air distance, ground distance,
wind speed, air temperature and air pressure.
Records must be kept of power settings, run-
way state, brake disc temperature, and tire
wear after landing.
As in the case of take-off tests, a full time
history of distance is not required; it is
only necessary to measure the speed at two
stations accurately, near the obstacle height
and at touch down. Since the point at which
the airplane is at obstacle height and the
point of touch down may differ greatly for
each test, the two recording cameras should
cover a sufficiently large area to insure that
records are obtained for the desired flight
path regions.
Landing tests should be divided into air-
phase and ground run tests. Only a few
ground runs (approximately three), utilizing
maximum brake capacity at standard weight,
should be performed to avoid excessive wear
and consequent replacement of the tires. At
least five air phase tests are necessary for
the chosen approach technique which provides
a given rate-of-descent or flight path angle.
8:10 BASIC LANDING EQUATIONS
(a) Ground Run
The equation for ground distance s (no
wind, no slope) is developed in the same
manner as the take-off ground run equation.
The average deceleration d, giving the
same overall speed-distance change from
touch down to standstill as the actual variable
deceleration, is given by
g
J = 1 [μg (W-C) + D-F]
W B
performance tests are conducted, the stan-
dard approach technique should be esta-
blished. The rates-of-descent of present-day
airplanes in power-off glides are so high
that the application of a considerable amount
of power during final approach has become
standard practice.
This means that the type of approach is
limited by predetermined values of rate-of-
descent or flight path angle, and not by specific
performance features of the airplane. Inas-
much as instrument approaches require the
longest total landing distances and determine
the required landing runway length, it is
obvious that this type of approach is of
primary importance. The airplane should
arrive at the obstacle height in a steady
descent with a predetermined true airspeed,
rate-of-descent or flight path angle.
On the other hand, to obtain the shortest
possible air distance, a different technique is
required, the success of which depends mainly
on the pilot's skill and his acceptance of
risks. These factors tend to introduce scatter
in the results.
To insure that consistent approach data are
obtained, the instrument type of approach is
considered here. Again it should be empha-
sized that for purposes of this analysis, there
is no need to obtain exceptional performance
data. The determination of the best standard
approach within the operational limits of the
airplane is the real aim of the tests.
During the ground run, the use of excessive
brake power should be avoided. In the past, it
has been difficult to achieve optimum brake
operation, and as of this time, automatic
wheel brakes or any equipment providing
direct control on brake power and tire condi-
tions have not been made standard installa-
tions. In this connection, it might prove
useful to provide the pilot with an indication
of horizontal deceleration as a guide to proper
use of the brakes.
During landing tests the following vari-
ables must be measured: groundspeed at
8:33
where L, D and F occur at a mean speed V.
8:16
Keeping the lift and drag coefficients con-
stant during the reduction process, an equa-
tion similar to Eq. 8:4 can be derived
Ast Δω Δσ
gFAF AW
+
SI
W
Wd
W
م
8:34
The last term of Eq. 8:34 is usually ignored
when the idling thrust is low, in which case
the deceleration ratio d/do is roughly 1.
Generally, the deceleration is a linear func-
tion of V². For the case of constant decelera-
tion the mean speed V has a limit value of
0.7 V₁ ·
(b) Air Phase
The air phase s2 (no wind, no slope) is
given by
dE V2 - Vi+2gh
$2
E2
g
ds =
52
W
d
E₁
d
8:35
tion principally of the relatively low lift-drag
ratio, whose value depends on ground effect
and the normal acceleration required to curve
the flight path. Weight and density corrections
at constant lift coefficient involve only
changes in the true approach and touch down
speeds, thus changing only the total energy
to be dissipated. Consequently, when weight
and density corrections are the only ones to
be made, the last term of Eg. 8:37 is zero.
In practice, the influence of weight and density
increments on flight path angle or rate-of-
descent is frequently abolished by changing
the power setting so that usually AW/W=AF/F.
Therefore, weight and density corrections
may often be omitted entirely.
8:11 LANDING PERFORMANCE
REDUCTION
(a) Wind Corrections
For the ground run the same correction
factor as given in section 8:6 may be applied.
However, because the deceleration is
generally constant, it may be assumed that
the exponent p equals 2 so that
The mean deceleration d at the mean speed
V of 0.5 (V₁+V½) is
=
g
W
2
Slw + Asi
8:36
Again keeping lift and drag coefficients
constant, an equation similar to Eq. 8:9 can
be derived, whence
As2 =
$2
v-vi
Δό
V²) (AW DO
\V−vi + 2gh,
:)
σ
14-15
wd
X
V₁
Fw=
SI.
8:38
where s₁₁+As
gound distance in still air
S| W
ground distance with head-
wind Vw
ΔΕ
AF AW
W
(+-)
8:37
V₁ -Vw
ground speed at touch down.
A headwind of magnitude Vw will shorten
the air distance by an amount equal to the
product of headwind velocity and the air phase
time; i.e.,
When the net thrust acting during the air
phase is low, the mean deceleration is a func-
8:17
As₂ = Vw 12
8:39
Again, the possible wind gradient might
be accounted for by using an effective wind
velocity in Eq. 8:39 equal to 1.1 times the
actual surface headwind velocity (see section
8:6).
(b) Slope Correction
A positive runway slope B increases the
effective deceleration during the ground run
so that
s₁ + As¡ = FB si
where RS is rate-of-descent at obstacle
height h above runway plane.
(c) Lift Coefficient Corrections
Usually it is not advisable to correct
for lift coefficient at obstacle height and
touch down. Rather, an attempt should be
made to keep airspeed close to a predeter-
mined value and to avoid excessive float
after transition. This is feasible with
airplanes having nose wheel or tandem landing
gears.
and
Ad =
- gß
where ẞ = mean slope of that portion of the
runway used for the test (uphill positive).
As in section 8:6, we may write
Asi
SI
or
Asi
$1
5
11
Ad gß 2gs,B
v²
2gs, B
FB-1 =
vi
8:40
No slope correction to the air distance is
applied when the obstacle height is related
to touch down level. However, when the ob-
stacle height is measured perpendicular to the
runway plane with slope B, the correction
As is
(d) Landing Assistance
The use of reverse thrust or a brake or
drogue parachute requires separate analyses
of those portions of the ground run during
which the assistance is operative and a
separate investigation of the effects of atmos-
pheric conditions. Accurate measurement of
effective change in deceleration and accurate
establishment of the time of application are
necessary.
8:12 SUMMARY OF LANDING EQUATIONS
(a) Ground Run
Measured values:
The distances s
s₁, V₁, Vw,o, B, W.
S¡, V¡,
and s2 appearing in the
formulae are corrected for the previous
effects in the given order.
ASI-
(1) Wind correction
As2
$2
11
V2
RS
B
2
-I = Fw - I
W
W
8:18
(See
Fig. 8:1)
(2) Slope correction
ASI
SI
2gs, B
29518 · FB-1
s₁ corrected for Vw. No CL, correction is
required.
Combination of 1 and 2 gives
A52
v-vi
/AW
AW Ao
$2
V2-V₁+2gh W
σ
SI corrected for wind and slope
=
Fw・ FB • SI test
(when s
Si test
S₁)
rections
where
(3) Weight, density and thrust cor-
Asi
SI
AW Ao Fg [AW AF
W
סו
| 6
+
251
wd w
F
S2 corrected for Vw
Reduction equations for different landing
techniques are given in Ref. 4 and in Chapter
6 of this manual.
SI corrected for Vw and F = mean thrust
(if any) at ▼ = 0.7 V
(b) Air Phase
Measured values: S2, VI, V2, Vw, W,σ, t2
rection
(1) Wind correction
As₂ = Vw t2
(2) Weight, density and thrust cor-
None, except for glide approach
8:13 CONCLUDING REMARKS TO PART I
In this part of the chapter, we have con-
sidered the general problem of take-off
and landing performance from the point of
view of what might be considered "conven-
tional operation" of airplanes. The cases of
minimum length take-offs and landings either
with or without obstacles in the flight path
have not been considered in this chapter.
However, the problem of minimum length
take-offs is considered in Chapter 6 as well
as in Part II of this chapter.
Important specialized analyses such as for
carrier-type take-offs and landings or the
performance of vertical take-off aircraft
wherein the stability and control, as well as
the performance characteristics of the air-
plane, enter the picture, are beyond the scope
of this present work. However, it is hoped
that such topics will be considered in future
volumes.
Finally, we note that, although the analysis
given here is applicable to turboprop air-
planes, as of this time little experience has
been obtained with these machines, so that
the most satisfactory method for correcting
for power variation is not yet known. This
is another topic warranting further con-
sideration.
8:19
PART II
THEORY
8:14 GENERAL PRINCIPLES
The take-off maneuver is usually divided
into three phases, which are illustrated in
Fig. 8:3:
(1) The ground run, from the start
of roll to the point at which the chosen take-
off speed is reached.
(2) The transition to climbing flight.
(3) The climb to some obstacle.
Throughout the ground run the aircraft is
assumed to remain close to, though not
necessarily in contact with, the runway. The
second and third phases often comprise a
single maneuver.
8:15 THE GROUND RUN
During this phase, the speed increases
progressively through a number of important
points which require definition. These speeds
are defined in the order in which they usually
occur on a typical multi-engined aircraft.
Control Speed on
Speed on the
(a) Minimum Control
Ground, VMCG
This is the speed above which, if sudden
failure of any engine occurs, the aircraft
being on the ground, the pilot is able to regain
control and thereafter maintain a straight
path parallel to the one originally intended,
without reducing thrust or retrimming. The
lateral divergence of the path, and the control
forces must be within defined limits.
(b) Critical Speed, Vcrit
If sudden failure of one engine occurs
during the take-off run at this speed, the
pilot is able either to stop, or to continue
the take-off to the 50-foot point, in the same
total distance. If power failure occurs
earlier, it is more economical of distance to
stop the aircraft, while if it occurs later
it is better to continue the take-off. Vcrit
must, therefore, be not less than VMCG.
(c) The Stalling Speed, Vs
This definition is largely self-explanatory.
In its estimation no account is taken of the
beneficial effects of the proximity of the
ground but it is open to question whether or
not the effect of engine thrust should be in-
cluded. In all other respects the configura-
tion is that appropriate to take-off.
(d) Minimum Control Speed in the Air,
VMCA
This is the speed above which, if sudden
failure of any engine occurs while the aircraft
is away from the influence of the ground,
the pilot is able, without reducing thrust or
retrimming, to regain control and thereafter
maintain straight, steady flight at the same
airspeed. The control forces and angular
divergence of the aircraft must lie within
certain limits. This speed is, of necessity,
greater than the stalling speed, though this
is not implied by the definition.
(e) Take-off Safety Speed, VTOS
This is the speed below which the airspeed
must not be allowed to fall after leaving the
proximity of the ground. It must, therefore,
exceed the stalling speed by an appropriate
margin, and must, in addition, exceed VMCA
by an agreed amount.
The relation of these speeds to the various
stages of the whole maneuver is shown (purely
diagrammatically) in Fig. 8:3.
8:20
TRANSITION
DISTANCE
SAFETY MARGIN OVER STALL
CONTROL
SAFETY
MARGIN
4
4
YCRIT. VS
GETAWAY
8:21
Fig. 8:3 Take-off Maneuver Illustrating Significance of Various Phases and Speeds
START
GROUND RUN
CONTROL
SAFETY
MARGIN
VNCO
—ACCELERATION AT FULL POWER
STOP, IF ENGINE FAILS
BEFORE THIS POINT
VMCA
VT.O.S
CLIMB DISTANCE
TRANSITION
ENDS
XGROUND
LEVEL
AIRCRAFT CAN TAKE-OFF TO 50 FEET WITH ONE ENGINE
"DEAD" OR STOP, IN THIS DISTANCE
CONTINUE TAKE-OFF IF ENGINE FAILS
AFTER THIS POINT
50 FT
The getaway point, i.e., the point at which
the aircraft becomes airborne, can occur at
any point after the speed Vs has been
exceeded, provided that sufficient angle of
attack can be achieved with the main wheels
on the ground. No appreciable height may
be gained, however, until the take-off safety
speed, VTOS, has been exceeded.
(f) Estimation of Ground-run Distance
Consider the aircraft to have reached a
speed V ft. per sec, in zero wind at a distance
s feet from the start. The aircraft, of weight
W lbs., is being accelerated by a forward
thrust F lbs. The retarding forces are
partly aerodynamic, arising from a total drag
coefficient CD and partly frictional, due to the
coefficient of rolling friction μ. The coeffi-
cients CD and CL are assumed to be held
constant throughout the ground run and it is
assumed that ground effects have been con-
sidered in evaluating these coefficients.
3/0
The equation of motion is therefore:
3 = F - S (W-CL
CD · — pV²s - µ (W - CL · — pv²s)
where s = forward acceleration,
ft. per sec.² = d²s/dt²
2
g = acceleration due to gravity
P
= air density, slugs per cu. ft.
8:41
off speed, where Fo is the static thrust and
k is a constant.
W
310
Eq. 8:12 may then be rewritten:
ï = (F。 − µW) - — pv²s (CD¯µCL+kFo/{{pS).
With the following substitutions,
Ŝ = V
ૐ :
ᏧᏙ
I d(v²)
ds
2
ds
A = 2g (Fo/W-μ)
PS
9 15 (CD - μCL + KFO / /
/ PS)
B = 9
W
8:43
we can put Eq. 8:43 into the integrable form:
d(v²)
ds
= A - BV².
8:44
Since we may be dealing with a take-off
during which the thrust undergoes discontin-
uous changes due to engine failure or the use
of rockets, it will be as well to integrate
Eq. 8:44 in the general form to give the
distance travelled in accelerating from a
speed V₁ to a higher speed V2. This distance
is then given by
I/B
S=
109,
A – BVi
A - BV2/
A - BV²
loge
A-BV2
8:45
It may be noted that when B tends to zero,
the expression for the distance becomes
simply
(vž – vi)
S =
A
S = wing area, square feet
t = time from start, seconds.
The thrust T is assumed to decrease with
speed according to the relation
F = F。 (l-kV²)
8:42
over the speed range from rest to the take-
8:22
8:46
and in this form an approximation may be
obtained for the effect of wind. The speeds
V₁ and V₂ become ground speeds, and the con-
stant A should be the mean acceleration,
in ft. per sec. throughout the run.
If we split the total drag coefficient Cp
into profile and induced drag coefficients so
that
CD = CDe
+ KC²,
8:47
then by differentiation of Eq. 8:45 with respect
to C, we find that the lift coefficient for a
minimum value of acceleration distance is
given by
CL = μ/2K ·
8:48*
Typical values of μ and K are 0.05 and
0.03 respectively, for a concrete runway and
for an aircraft of aspect ratio of about 6, in
the presence of the ground. The optimum
value of C₁ during the run is, therefore, as
high as 0.8, and would be even higher for a
take-off from, say, a grass surface or other
uneven surface. Shortest take-off runs will,
therefore, be achieved where the lift coeffi-
cient is brought as quickly as possible to a
value approaching the optimum.
Although this phase has been referred to as
* Differentiation of Eq. 8:45 leads to the re-
quirement that
dB
= 0
dCL
together with another extraneous condition
involving velocity. Discarding the extraneous
condition and evaluating
dB
dC
in terms of the definition of B leads to Eq.8:48.
8:23
the "ground" run, the last part, up to the
take-off safety speed, can in fact be made with
the wheels clear of the ground. This change
may produce a very little difference in the
total distance, for throughout this part of the
run there may be very little load on the
wheels, while the induced drag will be reduced
by the favorable effect of the ground. To lift
the aircraft clear of the ground at an early
stage may therefore have little or no effect
on the total ground run, and might even
increase it.
8:16. THE TRANSITION AND CLIMB
The airborne phase of the take-off maneu-
ver has to be considered in two separate parts,
or in one single stage, depending on the class
of aircraft and the type of take-off required.
As the take-off safety speed is reached, it
is assumed that the pilot increases angle of
attack and thereby applies an increment of
normal acceleration, causing the aircraft to
start to climb. When the angle of climb
reaches the value at which the aircraft is
to climb steadily, the lift coefficient is re-
duced to the value appropriate to the climbing
speed, and steady flight ensues. If the
standard 50-foot height has not been attained
at this stage, then the airborne phase falls
naturally into two parts. If, however, the
50-foot point is passed before steady condi-
tions can be achieved, the airborne phase is
obviously one continuous maneuver.
Our attention is therefore first directed to
that part of the maneuver in which the angle
of climb is being increased.
*If we consider an airplane at any point
in the transition flight path between take-off
and steady climb, we have the equilibrium of
forces shown in Fig. 8:4.
* Editor's note: This section to Eq. 8:55 has
been rewritten to provide the basis for the
transition equations given in the original
paper.
Summing forces perpendicular and paral-
lel to the flight path we have
D + ma
+ W siny - T = 0 ·
h
L = W cos y
+ CF
8:49
L
Flight Path S
(CG)-
T
ma
D
CF
W
Fig. 8:4 Forces During Transition
8:24
ハ
​Tangent_To_S_
X
8:50
To solve these equations we presume that
initially the airplane is flying at a speed Vg
and a lift coefficient CL. The pilot then
instantaneously applies a lift coefficient in-
crement ACL to start the curved flight path.
The transition lift coefficient CL。+ACL is
assumed constant during the maneuver as is
the difference between the thrust and drag.
Under these circumstances
Since Va is not too different from Vg,
this equation may be linearized by setting
Va= Vg in the right-hand member of the
equation, whence
where
dy
vå A - B = |
ds
L =
= 1/2 pv² (CL。 + ACL) S
where Va is the equivalent speed at the point
considered and
W =
Po
POV CLOS
where Vg is the equivalent getaway speed.
Thus,
CLO + ACL
A =
V& CLO
v
B
go
Similarly the second balance equation
becomes
I d (v²)
2 ds
T-D
+ gy
= 00 = 970
E
2
W
~ - (VO) (1 + ACL)
The centrifugal force is given by
WV2
wv2
CF =
wv2 dy
gR
g
dx
d2h
dx²
(assuming y and dy/dx to be small).
For small angles s~x and the first balance
equation becomes
W
11
CF
W
4/3
where a。 is initial flight path acceleration
ao
and
Yo is climb angle equivalent of ao or
d(va)
ds
+ 2 goy = 290 Y
giving us the pair of simultaneous linear
differential equations in Va and Y
v² - B. dy = 1
A ds
d(va)
+29σy = 29σYO
29070
ds
8:51
or
va
v
+1
els
ACL
v² dy
dy.
CLO
ds
об
8:25
These equations have solutions of the form
and provide the characteristic equation
2go
BA
YN
whence
λ = ±¡
2922
va
for ACL << CLO
2
1+
ACL
CLO
2
об
Since has no real part and since the
right-hand terms of the differential equation
set are constants, the general solution to
our equations may be written
va =
C₁ sin as + C₂ cos as + K₁
y = C3 sin as+ C4 cos as + K2
Υ
where K, and K2 are the particular solutions
and a =
√2go/V
Direct substitution gives
1
K₁
DI-
V& CLO
CLOTACL
This gives
K₂ = Yo
The initial conditions are:
Thus
(Valo = Vg
lylo = 0
d(va)
ds
dy
ds
C
لى
11
2go Yo
=
√2
JI
v
v²
Yo g
ACL
v
CLOTACL
ACL
C3 = √2CLO
CA
Yo'
v² va
V₂ = V₁ {√zy sin (12000) + C + AC
gos
ACL
CLO+ACL
COS
√2 gos
+
CLO
(12))
CLOT ACL
8:52
gos
Yo {1 - cos (√2 90s)} + ACL sin (VZ98)
of
Y = Yo
For small angles we have dh = yds
X
· ƒ yds = }
and
n =
X
{Y。 [ ! -
v²
- Yo [3- Vo
n = 0
S
1%
√2 CLO
v
√2 gos
ACL
COS
+
gos
va
√2CLO
ACL. V
CLo 290
sin (2003)] + ACL.
[
sin
(√2000))
COS
√2905
8:53
√290
8:54
8:26
By setting Y=Yo in Eq. 8:53, we find
that the limiting value of s for transition is
given by
S = V3
√290
arctan (yo√2 CLOACL).
8:55
If the height gained, given by Eq. 8:54, in
the distance s given by Eq. 8:55 exceeds the
obstacle height, then Eq. 8:54 would be used
to determine the airborne distance with the
substitution h equal to obstacle height.
If, on the other hand, steady conditions
are reached at a height less than that of the
obstacle, then we define the transition dis-
tance as the difference between the actual
distance to some point on the steady climb
path and the distance in which this point
would have been reached had the aircraft been
able to climb straight off the ground at the
steady climb angle Yo. Fig. 8:3 illustrates
this definition.
In this latter case the transition distance,
ST, which is not the same as that given by
Eq. 8:55 may then be written
ST=f. V² /√2.go
where the factor f is given by
f = sin 8
ACL
CLO
(1 - cos 8) /√20
8
= arctan (√2% CLO/ACL).
8:56
Fig. 8:5 shows the variation of the factor
f with yo for a range of values of ACL/CLO
from 0.05 up to 0.8. It is clear that the value
of ACL/CLO chosen has a marked effect on
the transition distance, and the effect is
equally important in Eq. 8:54.
We can combine Eqs. 8:52 and 8:54 to
give the total airborne distance to obstacle
height in the form
SA
Yo
va - va
2go
- + h)
8:59
which could, of course, have been obtained
quite simply by consideration of the changes
in total energy occurring during the maneu-
ver, with the assumption that variations in
drag are small.
Eq. 8:59 does not involve ACL directly
but it does require accurate information on
the change in speed, which appears as the
difference of the squares of two large quanti-
ties. This change in speed, given by Eq. 8:52,
in turn depends on ACL.
Where steady climb conditions are
achieved after the obstacle has been passed,
a simpler, though less accurate method may
be used in estimating the distance to the height
h. It is assumed that a mean equivalent lift
increment ACL is applied, and that the flight
path is an arc of a circle. If the root mean
square airspeed over this path is Vm, corre-
sponding to steady flight at a lift coefficient
CLm, then we have
Vi /Rgσ = ACL/CLm
8:60
where R is the radius of curvature of the
path. The distance SA to an h of 50 feet is
then given by
SA
200 W/S
PGACLI
8:57
The steady climb distance to h feet is then
h/Yo, assuming Yo is small, and the total
airborne distance to h feet becomes
SA = Sth/Yo
2500
8:58
8:61
8:27
FACTOR "f"
0.9
0.8
0.7
0.6
0.5
0.4
0.3
1.0
من الله
0.05
0.10
0.20
0.30
0.40
0.60
0.80
0.2
0.1
0.1
0.2
X
TRANSITION DISTANCE
A = +. V 2 2 1√2.80
T
V
f
•
= TAKE-OFF SPEED,
FT/SEC.
Y = LONGITUDINAL ACCN
AT TAKE-OFF 'g' UNITS
ACE LIFT COEFFICIENT
INCREMENT.
CLE
0.3
= LIFT COEFFICIENT
AT AIRSPEED Vg
0.4
0.5
Fig. 8:5 Effect of Normal Acceleration on Transition Distance
8:28
where W/S is the wing loading. C in Eqs.
8:60 and 8:61 includes the increase in lift
arising from any increase in speed above the
initial value Vg and is normally higher than
ACL used earlier.
Flight tests have shown that where the
airborne phase is, in fact, a continuous
maneuver, irrespective of whether the climb
angle at the obstacle height is greater or less
than the steady value, Eq. 8:61 is sufficiently
accurate for the purpose of calculating the
lift increment ACL.
8:17 PILOTING TECHNIQUE IN THE
AIRBORNE PHASE
From the foregoing it will be clear that
the pilot has a far greater influence on this
part of the take-off than he has on the ground
run, and it is here that we must attempt to
be precise in our requirements for pilot
technique if repeatable results are to be ob-
tained.
The important parameter is the lift coef-
ficient increment ACL or ACL and for predic-
tion purposes we need to know how the maxi-
mum increment used by the pilot varies with
such factors as speed margin over the stall,
acceleration or climb angle, etc. It has been
found that a skilled test pilot, asked to achieve
the shortest possible distance, can produce
consistent results for AC, derived by the
method of Eq. 8:61.
The total lift coefficient, (CLm+AC), was
found to be a linear function of (Vm/Vs)²,
reaching the expected value of CLmax at
(Vm/Vs)² = 1, and falling with increasing
speed. The reduction with speed is to be
expected since at the higher speeds the time
taken to apply the increment is proportionate-
ly a larger fraction of the total time to reach
50 feet, thus reducing the mean effective in-
crement.
There is therefore an optimum speed at
which the maximum value of ACL will be
applied. Below this speed, proximity to the
stall will limit the increment that can be used,
while at higher speeds there will not be time
to apply and use the available increment,
This optimum applies only to the airborne
phase, and the total distance from the start
to the 50-foot point will be a minimum for
take-off speeds much nearer the stall,
Some typical flight results are shown in
Fig. 8:6. The small scatter of the points,
one for each take-off, is indicative of the
accuracy with which this particular type of
airborne path can be repeated.
During the experiments from which these
results were obtained, no particular safety
limitations were imposed on the pilot apart
from his own knowledge of what was a safe
maneuver. In normal practice, and particu-
larly for civil aircraft, safety considerations
will generally limit the lift increment that
may be used. In particular, the airspeed
must never be allowed to fall below the take-
off safety speed and from this it follows that
the instantaneous climb angle ought not to
exceed the steady climb value.
Additional data from take-off measure-
ments on civil and military transport air-
craft have shown that the lift coefficient in-
crements used are about half the values that
would give the absolute minimum airborne
distance, and hence the airborne distances
involved will be roughly 50% greater than the
minima.
To determine analytically whether a par-
ticular lift coefficient increment can safely be
held until obstacle height is reached, the
distance defined in Eq. 8:55 is substituted in
Eq. 8:54 with the value of the height h. The
solution of the resultant equation for Yo gives
the longitudinal acceleration which will en-
sure that the climb angle at the obstacle
height is equal to the steady value.
If the thrust available gives an accelera-
tion less than this limiting value, the steady
climb should be started before the obstacle
height is reached. For example, at a take-
off safety speed of 160 knots, a value of
8:29
8:30
LIFT COEFFICIENT
TWIN-JET FIGHTER-MINIMUM TAKE-OFFS AT FULL THROTTLE
1.5
1.0
0.5
201
MAXIMUM LIFT COEFFICIENT
TAKE-OFF CONFIGURATION
TOTAL LIFT COEFFICIENT
USED DURING TRANSITION
LIFT COEFFICIENT
FOR LEVEL FLIGHT
LIFT COEFFICIENT
INCREMENT
1.0
1.5
2.0
2.5
3.0
3.5
4.0
(V.
Fig. 8:6 Lift Coefficients Used in Airborne Phase
NOTE - LIFT COEFFICIENT
INCREMENTS WERE CALCULATED
ON ASSUMPTION THAT FLIGHT
PATH WAS A CIRCULAR ARC
VM
ROOT MEAN SQUARE
AIRSPEED DURING TRANSITION
= STALLING SPEED, TAKE-OFF
CONFIGURATION
off at reduced power, or to close all throttles
and come to rest, in the same overall distance.
One
Analytically, the critical speed can be
estimated by the following process.
engine is assumed to fail at a range of speeds.
Up to this speed the ground run is made at
full power; thereafter it is continued at
reduced thrust to the obstacle height using the
procedure already described. The total dis-
tance thus obtained is plotted against the speed
at which the engine is assumed to fail, as in
Fig. 8:7.
The distance required to stop the aircraft
following engine failure is also estimated,
using the method of Eq. 8:45 with the modifi-
cations to the following sections on the esti-
mation of ground run during landing.
The total accelerate-stop distance is then
plotted on the same graph as shown in Fig.
8:7. Two such (dotted) curves are drawn,
indicating the effect of different amounts of
braking---either wheel brakes or air drag.
The points of intersection of these curves
with that for the take-off distance give the
corresponding critical speeds, and the dis-
tances involved.
In the
the example illustrated, improved
braking appears, at first sight, to have little
effect on the distance involved, but the im-
portant effect is the increase in critical speed.
The take-off can be continued following engine
failure only if the speed is above the minimum
control speed VMCG.
If, therefore, the critical speed is below
VMCG, the take-off must be abandoned, even
though, as shown in Fig. 8:7, a considerably
greater distance is required. With improved
braking, however, the critical speed may be
raised above VMCG giving a useful reduction
in the length of runway required.
(b) Flight Test Measurements
Measurements of the critical speed can
usefully be combined with those for the deter-
ACL/CL。, held at 0.2 until a 50-foot height
is reached would require a longitudinal ac-
celeration of about 0.1g to prevent the climb
angle from being too steep. At 80 knots, the
same increment would require an accelera-
tion of 0.26g, so that light aircraft will nor-
mally perform the airborne maneuver in two
parts, for lack of sufficient acceleration.
Similarly, using the distance given by Eq.
8:59 for no change in airspeed in Eq. 8:54
with h = 50 feet, we may determine the value
of Yo for no drop in speed at a 50-foot point.
Again, at a value of ACL/CL。 of 0.2, at 160
knots take-off speed an acceleration of 0.05g
is required while at 80 knots, 0.1g is neces-
sary. The requirement for no loss of air-
speed is therefore easier to meet than that
for no excessive climb angle.
8:18 APPLICATION TO FLIGHT TEST
PROCEDURE
The total distance to obstacle height may
be measured under what are considered to be
normal operating conditions, but it is often
difficult to produce consistent results, par-
ticularly for this airborne phase. It may be
more reliable, therefore, to request the pilot
to produce the shortest possible airborne dis-
tance consistent with safety and hence to
calculate the lift coefficient increment ACL
by the method of Eq. 8:61 for high perform-
ance aircraft, or Eqs. 8:56, 8:57, and 8:58
when the available thrust is relatively low.
Then, using, say, half this value of ACL,
the length of the airborne path can be cal-
culated either as a single maneuver, if the
acceleration is adequate, or in two separate
parts if it is not.
At least 3, preferably 4, take-offs should
be made in each case, and the average of
the best two used for reduction purposes.
8:19 CRITICAL SPEED DURING TAKE-OFF
(a) Estimation
If, on a multi-engined aircraft, one engine
fails during take-off, at the critical speed,
the pilot is able either to continue the take-
8:31
3,000
2,500
2,000
TOTAL TAKE-OFF
DISTANCE TO 50FT.
|
TOTAL DISTANCE, YARDS
1,500
NORMAL DISTANCE
TO 50 FT. (ALL
ENGINES OPERATING)
1,000
ACCELERATE - STOP
DISTANCE NORMAL BRAKES
500
25
50
ACCELERATE-STOP
DISTANCE WITH
IMPROVED BRAKING
75
CRITICAL SPEED
NORMAL BRAKES
YMCG
TAKE-OFF SAFETY SPEED
IMPROVED BRAKING |
CRITICAL SPEED
100
125
150
SPEED, KNOTS, AT INSTANT OF ENGINE FAILURE
Fig. 8:7 Critical Speed During Take-Off
mination of the minimum control speed on the
ground, VMCG. The additional complication
is mainly in the ground equipment, rather than
in the pilot's task.
Sudden engine failure should be simulated
as closely as possible, starting at a fairly
high speed, the take-off being continued in
each case, until a failure is simulated at a
8:32
speed at which the pilot is not able to keep
the angular and lateral divergences of the
aircraft within prescribed limits. The total
distance obstacle height is obtained in each.
case, and plotted against the speed at failure.
Similarly, a series of abandoned take-offs
should be made at gradually increasing
failure speeds, the accelerate-stop distances
being measured and plotted against the speed
at failure. Then, even though the minimum
control speed VMCG may be higher than the
critical speed, extrapolation of the two curves
towards each other will enable an intersection
to be obtained, from which the critical speed
may be derived. The greater the length of
runway available for the simulation of aban-
doned take-offs, the less will be the amount of
extrapolation required.
8:20 THE LANDING MANEUVER
The landing maneuver is, in many ways,
the exact opposite of the take-off, not the
least difference being the difficulty of achiev-
ing precise, consistent results when a quan-
titative study is undertaken. It may therefore
be profitable to discuss in general terms,
the reasons for this lack of precision, although
it is obvious that in this case the pilot's
skill and judgment plays a much larger part
than it does during a take-off.
The landing path can be divided into several
phases as in the case of the take-off. Fig.
8:8 illustrates these phases which are defined
as follows:
(a) The approach, at a steady rate of
descent which must, however, be reduced
before touch down. The approach distance is
that length of this steady path from obstacle
altitude to the point where that path would
intersect the ground.
(b) The flare, started at some height
above the gound, by an increase in angle of
attack, with the intention of reducing the
vertical velocity to as near zero as possible
by the time ground level is reached.
(c) The float, which follows the flare if
zero vertical velocity is achieved before the
wheels touch the gound. During this phase,
speed decreases until the aircraft attains an
angle of attack at which a touch down can be
made. With a nose-wheel type undercarriage,
the aircraft can usually touch down at the end
of the flare, without floating.
(d) The ground run, from the point of
touch down until the aircraft comes to rest.
8:21 ESTIMATION OF LANDING DISTANCE
It is usual to calculate (and to measure)
the landing distance from the point where the
aircraft is at some altitude h above the runway
to the point where the aircraft comes to rest.
While this procedure has the advantage of
precision, it does not indicate the length of
runway needed for a safe landing.
It may be argued that, for practical
purposes, the important distance is the length
of runway used in reducing the aircraft speed
to a value at which it may be said to be
taxying. The starting point is therefore at
the down-wind end of the runway, which may
be at a height which can vary widely from
one landing to the next, depending on the
pilot's skill and judgment.
The calculation will, in any case, follow
closely the procedure used for the take-off
distance, and the normal practice is to per-
form the calculation for a starting height of
50 feet, and to multiply the result by a factor
to allow for the inevitable inaccuracies in
height and glide path angle in crossing the
end of the runway.
(a) The Approach Distance
If his the height at the start of the approach
path (normally 50 ft.) and y is the steady
glide angle, then the approach distance is
simply h/y when Y is a small angle,
in radians.
(b) The Flare Distance
The comprehensive analysis of this portion
of the maneuver has recently been published
by Doenhoff and Jones (Ref. 18).
In this paper, a study is made of the ideal
landing flare. The aircraft is assumed to
8:33
START OF
HOLD-OFF
8:34
h = 50 FT.
APPROACH
FLARE
FLOAT
GROUND RUN
STOP
START OF
FLOAT
TOUCH-DOWN
AIM POINT OF
INITIAL GLIDE
Fig. 8:8 The Landing Maneuver
start the flare at an optimum speed such that
at the end of the flare the minimum possible
distance has been covered and the speed has
fallen to the stalling speed, with the flight
path tangential to the runway surface.
Because of the reduction in speed during the
flare, there is therefore a minimum speed
at the start if the flare is to end with the
speed equal to the stalling speed. This effect
has been discussed by Meredith (Ref. 19).
Doenhoff and Jones consider the effects of
the inevitable errors in the pilot's judgment
of his height, position and speed on the
approach, with reference to the length of the
flare, and conclude that the most effective
way of reducing the length of runway required
would be to provide some means of fixing the
point of the start of the approach glide. This
point is commented upon later.
An elementary analysis of the flare maneu-
ver will serve to show the source of the
variations in this distance in practice.
We assume the flare to start from a steady
glide at a speed Vo ft. per sec. and a glide
angle Y radians. Assuming Y to be small,
as usual, the rate of descent is therefore
VY ft. per sec. The flare is accomplished
by the use of a lift coefficient increment
▲С above the value CL required for steady
flight at the speed Vo. A normal accelera-
tion of g (ACL/CLO) ft. per sec? is therefore
produced, and, to a first approximation, a
circular flight path results.
The vertical velocity VY is thus reduced
to zero in a time tf seconds, where
tf = VYCLo /gACL (approx.).
The height lost in this time, which is
therefore the height at which the hold-off
should be started, is hf, where
hf = VZt+ f = V²y²CLo/2gACL;
f
i.e.,
hf
11
(x² W) 19
/gp ACL feet
where W/S is the wing loading, in lbs. per
sq. ft.
If the flare path is assumed to be a circular
arc, then the flare distance, as defined in
Fig. 8:8, is half the total distance occupied by
the flare maneuver. Thus the flare distance,
Sf, is given by
Vit = =
Sf=
2
v²yCLo/2gACL
YW
(TW)/9PACL feet.
S
Throughout this simple analysis we have
ignored the loss in speed during the flare,
and assumed that the initial speed is high
enough for the lift coefficient increment ACL
to be applied and maintained in safety. Actual-
ly the speed loss is of the order of 5%.
At a given approach speed, therefore,
a strictly limited value of ACL is available.
The flare distance is then proportional to the
glide angle at the start of the flare, while
the height at which the hold-off must be
started varies as the square of the glide
angle.
Since the pilot is unlikely to make use of
the full lift coefficient increment, for fear
of stalling while still well above the runway,
there inevitably will be variations in ACL
which will appear as further variations in the
flare distance.
Although the airspeed does not appear
directly in the above expressions for flare
distance and hold-off height, it does, of
course, affect directly the value of ACL to be
used. The normal approach airspeed will be
dictated by engine-cut safety requirements as
well as by the requirement for an adequate
margin over the stall for maneuvering, gusts,
8:35
etc.
The approach airspeed will generally be
near, if not actually less than, the minimum
drag speed. An inadvertent loss in airspeed
might therefore result in an excessive glide
angle close to the ground, and at the same
time reduce the lift increment that can be
used in safety. In these circumstances, if the
flare is to be completed in order to avoid
damage to the undercarriage, the hold-off
must be started at a much greater height
than usual, and will take a correspondingly
greater distance to complete.
Piloting skill, as well as the aerodynamic
characteristics of the aircraft, both play an
important part in fixing the distance occupied
by the flare and the type of touch down which
ends it.
(c) The Float and Ground Run
An aircraft with a nose-wheel type under-
carriage is able, by virtue of its relatively
low ground angle of attack, to touch down at
speeds well above the stall, and the float is
eliminated. However, whether the actual
touch down is made immediately at the end
of the flare, or later, the distance involved
in bringing the aircraft to rest, or reducing
its speed to the taxying speed can be estimated
by a method identical with that used for the
ground during take-off.
Eq. 8:45 gives the distance x required to
change the speed from V₁ to V2 ft. per sec.
in the form
X =
loge
B
A - BV12
A - BV₂²
2
B =
aps (CD - HCL + KTO / / Ps).
W
To
For landing run analysis we have
A = 2g
(To -H)
8:36
The thrust To normally becomes the idling
thrust of the engines with throttles closed;
if reversed thrust is produced, To will become
negative. The coefficient of friction μ is
now due to the braking action and must, of
necessity, be arithmetically greater than
To/W if To is positive. The constant A is
therefore negative during the landing ground
run.
For the constant B, the lift and drag coef-
ficients C and Cp are, of course, the appro-
priate values considering ground effect and
are again assumed constant throughout each
stage of the run. The constant k accounts
for the variation of the idling or reversed
thrust with airspeed and will differ from that
used for the take-off estimate. The drag
coefficient CD will include the effect of such
drag producing devices as parachutes or air
brakes, if used.
Since the initial speed V₁ will exceed the
final speed V2, while B is normally positive
(though not necessarily so, if the drag is low
while the wheel brakes are particularly
effective), the distance x will always be posi-
tive and the expression can be used without
modification.
Obviously, if the run from the end of the
flare to rest is made in stages over which
the means of
of deceleration are changed
discontinuously (for example, by release of a
parachute, or application of wheel brakes, or
a change from a true float to a run along
the ground) then any estimation of distance
using the above expression would have to be
made in similar stages.
A major difficulty in making reliable esti-
mates of this part of the landing run is in
the choice of an appropriate value for the
friction coefficient μ. Reliable information is
very meager, but such as it is, it confirms
that μ varies with speed, and that at high
ground speeds very much reduced values of μ
are appropriate.
Where pilot-operated brakes are used, the
pilot is unlikely to use much brake at the
start of the run, for fear of producing a
dangerous skid, and he may do little more
than give an occasional "jab" at the brakes
until the speed has fallen, and the wheel load
increased to a point where the wheels will
not skid.
Even if the wheels have automatic brakes,
the ground run must, therefore, be divided
into two parts. Over the first part, a coeffi-
cient of friction should be used which is
between the unbraked value (usually 0.05) and
the maximum attainable value. For the second
part, full braking action is assumed and the
braking force becomes constant, with slip
occurring in the brakes themselves.
The speed at which full brakes can be used
may be estimated graphically on a diagram
on which the friction coefficient μ is plotted
against ground speed V, as in Fig. 8:9. The
available value of μ is shown as a full line,
while the value of μ required to prevent skid-
ding is derived from the known maximum re-
tarding force which the brakes can apply,
divided by the instantaneous load on the
braked wheels.
In the example shown in Fig. 8:9, from the
touch down speed down to about 80 knots,
danger of skidding exists unless automatic
brakes are used. Without such brakes,
therefore, the usual technique is for the
pilot to hold the nose wheel up and to rely on
air drag until the brakes can be put full on,
when the angle of attack is reduced to a min-
imum to increase the wheel load.
By differentiation of the expression for
the deceleration distance it can be shown that
this distance is a maximum when the lift
coefficient used is equal to μ/2K, the value K
being the induced drag factor (CD CD+K C²).
=
No compromise between the use of wheel
brakes and air drag is therefore practicable,
and the pilot should concentrate on making
the maximum use of one or the other, but
not both.
8:22 THE CHOICE OF THE LANDING
APPROACH AIRSPEED
An essential part of any landing tests or
measurements is the choice of the minimum
approach airspeed. Since this depends crit-
ically on the stalling speed, it is assumed
that this speed, or the minimum practicable
flight speed if the aircraft has an unconven-
tional stall, will have been established before-
hand. The maximum lift coefficient increment
to be used for maneuvering must still leave
a margin over the maximum determined by
the preliminary tests.
The maneuvering requirements refer to
corrections to the flight path in both the
horizontal and vertical planes. The magnitude
of these corrections depends mainly on the
pilot's ability to detect errors in position and
rate of change of position.
We should expect that if the visual or other
form of guidance is improved, not only will
the accuracy of the final arrival be increased,
but also that the magnitude of the corrections
which the pilot applies will be reduced,
although he may make such corrections more
frequently.
Maneuverability is, of course, not the only
criterion on which the pilot will assess the
minimum approach airspeed, and he will be
influenced by the controllability of the aircraft
as regards attitude, forward speed and rate
of sink. The speed must also be above the
minimum at which full throttle can be applied,
with one engine inoperative, for an emergency
overshoot.
Control of the attitude of the aircraft,
e.g., correction for wing dropping, will de-
teriorate as the speed is reduced due to the
more sluggish response to control move-
ments. Further, on swept-wing aircraft the
lateral stability derivatives change markedly
with lift coefficient, and dutch-rolling may
occur. If the aspect ratio is low, the attitude
8:37
8:38
FRICTION COEFFICIENT M
0.8
0.6
0.4
0.2
ނ
AVAILABLE (VERY APPROXIMATE, ESPECIALLY ABOVE 80 KNOTS }
1
SPEED AT WHICH FULL
BRAKES CAN BE APPLIED
FB
REQUIRED =
WHERE:
W-L
FB
W
L
= MAXIMUM BRAKING
FORCE, LB.
= AIRCRAFT WEIGHT, LB.
= WING LIFT AT
APPROPRIATE SPEED, LB.
ROLLING FRICTION - NO_BRAKES
0.0
50
60
70
80
GROUND
90
RUNNING
100
110
SPEED, KNOTS
Fig. 8:9 Use of Wheel Brakes During Ground Run
120
130
140
of the aircraft may become excessively nose-
up making the view inadequate.
Control of airspeed and rate of sink depend
on the relation of the airspeed to the minimum
drag speed, VMD. If the airspeed is above
the minimum drag speed, the pilot is able to
make a correction to his height by elevator
movement only, and a glide path parallel to
the original can be flown, at the original
airspeed, without change in thrust.
If the airspeed is below VMD, however, then
a form of instability exists, in that if the
airspeed changes in the course of making
some correction, then that change will in-
crease with time unless a change in thrust
is made. Flight at speeds below VMD is pos-
sible---deck landing approaches are fre-
quently made at speeds in this region---but
the maintenance of a steady approach requires
constant adjustment of the power setting as
well as the elevator.
It is clear, therefore, that the minimum
approach airspeed is not amenable to calcula-
tion, and one can only indicate the general
principles which govern the pilot in his choice.
8:23 SOURCES OF VARIATION IN TOTAL
LANDING DISTANCE
Of all flight test experiments, the
measurement of total landing distance is
probably the least precise. Even when all
the test conditions are accounted for in the
reduction of the results, large residual varia-
tions must be expected, and a generous safety
factor has to be applied in determining the
minimum safe runway length. The fundamen-
tal inability of the pilot to follow the ideal path
in space and time is the main reason for this
inconsistency.
While part of this residual variation arises
from the variation in the use of the brakes
during the ground run, automatic brakes now
allow the maximum use to be made of the
brakes irrespective of piloting skill. Addi-
tional decelerating devices, such as brake
parachutes, further reduce the length of the
ground run. Nevertheless, the point on the
runway at which the aircraft is brought to
rest will continue to be ill-defined until the
pilot is given additional assistance in starting
his approach glide from the optimum point in
space.
This assistance can, in fact, be given in
a simple form, and need involve no additional
airborne equipment. The starting point for the
ideal glide will, presumably, be on the ex-
tended center line of the runway, and this is
fairly easy to indicate to the pilot except in
bad visibility, when radio aids may be re-
quired. In VFR conditions, however, markers
indicating the position of the runway center
line would suffice.
In the vertical plane the pilot needs an
indication of his position relative to the ideal
path, and this can be provided, in VFR con-
ditions, by two markers alongside the runway
at the aim point of the ideal glide. These
markers, separated by 150-200 feet and at
different levels, are arranged to lie on, and
thus to define, the path which the pilot's
eye should follow. He then has a direct
indication of his height error above or below
the ideal path.
This is essentially a short-range device,
covering the final mile or less of the approach.
Tests have shown that this form of indication
is readily appreciated by the pilot, and con-
sistent flight paths can be flown. The hold-off
and flare will still be required unless the
chosen glide angle is so shallow that the
undercarriage can safely absorb the vertical
component of the velocity.
With simple equipment of this sort, the
variations in height and glide angle at the
edge of the airfield are appreciably reduced,
and more consistent total landing distances
will result.
8:39
REFERENCES
1. Lush, K. J., "Standardization of Take-off Performance Measurements for Aeroplanes,"
USAF Technical Note R-12, 1952.
2.
"Review of Performance Techniques," A. & A.E.E. Discussion Memo 2 5821/PAH, 1952.
3. Herrington, R. M. and Schoemacher, P. E., "Flight Test Engineering Manual,” USAF
Technical Report 6273, 1951.
4. Royal Aeronautical Society Data Sheets, RC 2/1 & 2, 1950.
5. Perkins, C. D. and Hage, R. E., “Airplane Performance, Stability and Control," John
Wiley and Son, 1949.
6. Hamlin, Benson, "Flight Testing," MacMillan, 1946.
7. Hartman, E. P., "Considerations of the Take-off Problem," NACA TN 557, 1936.
8. Diehl, W.S., "The Calculation of the Take-off Run,” NACA TR 450, 1932.
9. John, G., “A Further Development in Calculating the Take-off to 50 Feet Distance of
an Aeroplane," Aircraft Engineering, April, 1948.
10. Gates, S. B., "Notes on a Method of Analysis of Ground Run During Take-off," R. & M.
1820, 1937.
11. Wetmore, J. W., "The Transition Phase in the Take-off of an Airplane," NACA TR 626,
1938.
12. Ewans, J. R., Hufton, P. A., "Note on a Method of Calculating Take-off Distance," A. &
A.E.E. B.A. Dep. Note Performance 20.
13. Garbell, M. A. and Young, W. M., "The Ground Run of Aircraft in Landing and Take-off,"
Garbell Aeron. Series 3, 1951.
14. Lovell, J. C., Lipson, S., "An Analysis of the Effect of Lift-drag Ratio and Stalling Speed
on Landing Flare Characteristics," NACA TN 1930, 1949.
15. Ribnitz, W., "Umrechnungsverfahren fur Start und Landewege," GDC 10/12022, D. V. L.
TB 3, 1939.
8:40
16. Multhopp, "The Problem of the Shortest Take-off Run," Translation GDC 16/57 T.
17. Quinn, J. J. & Lush, K. J., "Take-off Turbine-Jet-Propelled Aircraft, an Investigation
into the Technique of Measurement and of Reduction to Standard Conditions," A. & A.E.E.
Res/232, 1946.
18. Doenhoff and Jones, "An Analysis of the Power-off Landing Maneuver in Terms of the
Capabilities of the Pilot and the Aerodynamic Characteristics of the Airplane," NACA
Tech. Note 2967, August, 1953.
19. Meredith, "Note on the Minimum Speed from which the Direction of a Gliding Airplane
can be Changed to a Horizontal Path for Landing," R. & M., No. 993, British A.R.C., 1925.
8:41
AGARD FLIGHT TEST MANUAL
VOLUME I, CHAPTER 9
SPECIAL TESTS
By
D. 0. Dommasch
Princeton University
Section 9:1
W. E. Gray
Royal Aeronautical Establishment
United Kingdom
Sections 9:2, 9:3, 9:4, and 9:5
J. Idrac
Centre d'Essais en Vol, France
Section 9:6
M. Guenod
Centre d'Essais en Vol, France
Section 9:7
J. Foch
Centre d'Essais en Vol, France
Section 9:8
(
VOLUME I, CHAPTER 9
CHAPTER CONTENTS
Page
TERMINOLOGY
9:1
INTRODUCTORY COMMENTS
9:1
9:2
INTRODUCTORY COMMENTS ON AIR FLOW VISUALIZATION
9:1
9:3
METHODS OF FLOW INVESTIGATION
9:2
9:4
FLIGHT TESTING
9:8
9:5
OTHER TECHNIQUES
9:10
9:6
CALIBRATION OF AN ANGLE OF ATTACK MEASURING SYSTEM
9:11
9:7
MEASUREMENT OF SIDESLIP ANGLE
9:12
9:8
AIRBRAKE EVALUATION
9:14
REFERENCES
9:21
)
TERMINOLOGY
R.
Angle of Attack
CL
Lift Coefficient
F
Engine Thrust
m
Airplane Mass
8
Gravitational Acceleration
V
True Flight Speed
Vcal
Calibrated Airspeed
P
Air Density
9
Dynamic Pressure
M
Mach Number
Со
Drag Coefficient
ACO
Drag Coefficient Increment
Y
Flight Path Angle with Horizontal
9c
Compressible Dynamic Pressure
Pi
Indicated Pressure Difference
B
Sideslip Angle
9:1 INTRODUCTORY COMMENTS
problems. Measurements of pressures and
velocities have always been among the chief
methods of study, but as provisions are
seldom made in a given airplane design for
such measurements, the long established
method of studying the airflow visually with
light streamers has been much used in
flight test work.
It is useful when conducting stability and
control tests to know the values of the side-
slip angle and the angle of attack with some
degree of precision. Knowledge of these
parameters is also of importance when
evaluating the performance of the airplane
armament as well as take-off and landing
characteristics under critical operating con-
ditions (such as carrier take-offs). For
this reason several types of angle of attack
and sideslip angle measuring devices are in
present use, and in general, flight calibra-
tion of these is required before dependence
may be placed on their scale readings.
The flight calibration of equipment such as
this is classified here under the broad
heading of "special tests.
The use of streamers, or wool tufts, has
been invaluable in studying the stalling of
wings, and in showing where the flow at high
speed breaks down or separates at junctions
or behind shock waves and for these purposes
it is still widely used.
In addition to calibrations such as these,
we also must conduct special tests to deter-
mine the suitability and effectiveness of
such items as dive brakes or dive flaps
particularly in regard to the let-down char-
acteristics of very clean jet aircraft.
The steady increase in flight speeds and
improved aerodynamic design cleanness has
increased emphasis on skin friction drag, and
has led to a closer study of boundary layer
flow. While this flow can be studied by
measuring pressures and velocities, these
methods are clumsy when applied to a large
area and their use at only one location leaves
room for errors due to local surface imper-
fections. These latter may change the bound-
ary flow from the laminar to the turbulent
type, with a six-fold to ten-fold increase in
drag.
For research work and for developmental
flight testing, it is sometimes mandatory to
investigate the actual local airflow conditions
existing about a given airplane to establish
the causes of buffeting, control malfunction-
ing, flow breakaway and the like.
The topics listed above, although they bear
little direct relation to one another, have all
been grouped together in this chapter under
the heading of special tests. There are, of
course, a number of other special tests
which are conducted, and it is hoped in the
future that this chapter will be expanded
to include additional topics.
It is for these reasons that the visual
methods of studying the boundary layer over
whole areas, devised in Britain and used in
flight testing there for the past ten years,
have largely superseded the older methods.
The visual methods all rely on the basic
difference between laminar and turbulent
flow, viz., that the turbulence is accompanied
inevitably by a rapid vertical interchange of
air particles throughout the boundary layer
and so to the surface.
9:2
INTRODUCTORY COMMENTS ON
AIRFLOW VISUALIZATION
Whether the action used is a chemical one
or a physical evaporative one is a choice
dictated by the test conditions that have to
be met.
In either case only a very modest
amount of preparation is needed on the air-
craft that is to be tested to change the sur-
face finish on the test areas. This prepara-
tion usually takes from one or two hours to
one or wo days depending on the areas
A clear understanding of what the air is
doing as it passes over the surfaces of an
aircraft is always helpful and sometimes
vital in the investigation of aerodynamic
9:1
involved. These methods will greatly speed
the work of the scientist. The present chap-
ter will be directed mainly to a description
of them.
of stalling and of local breakaway of flow; it
cannot be used to show turbulence in the
boundary layer flow where the size of the
disturbances is of a totally different order.
(b) Chemical Methods
A study of shock waves in high speed flight
as distinct from the breakaway of flow due to
their action has recently been carried out.
The methods employed for this have made use
of the change in the refractive index of the
air at the shock wave as in the older wind-
tunnel techniques. These methods have al-
ready been widely described and will be
dealt with briefly here.
These methods (Ref. 1) of studying the
boundary layer flow make use of the turbu-
lence itself to indicate the turbulent areas.
To do this, the wing surface is sensitized
with a material that changes color on expo-
sure to an active gas or to impurities in the
atmosphere. The best results have been ob-
tained on a white surface, which is sprayed
before flight with a colorless sensitizing
solution, the composition of which is chcsen
to suit the concentration of gas to be used
or likely to be encountered.
9:3
METHODS OF FLOW INVESTIGATION
(a) Tufting
This well-known method indicates the
local direction of, and the directional steadi-
ness of the flow and to a slight extent its
velocity. It can be used to show the flow
both on and at a distance from the surface
and it relies on the relative lightness and
flexibility of the material used.
The passage of the aircraft through the
air brings much more gas or impurity into
contact with the surface where the flow is
turbulent and so causes a darkening in these
regions, the areas of laminar flow remaining
white. The only active gas that has been used
is chlorine in a concentration of about five
parts per million. This is safe for short
exposures and a flight of a few seconds
through it produces a picture.
If the material is too heavy, the tufts or
streamers can give a false indication of un-
steady flow by the oscillations they can ex-
hibit in a steady stream, a feature that must
always be guarded against. When the tufts
are mounted on masts, precautions must be
taken to avoid their getting entangled with the
supports during violently unsteady flow. The
result of taking such precautions may well
be excessive tuft weight and stiffness and
this also must be guarded against.
An example of such a test result is given
in Fig. 9:1. The sensitizing solution is com-
posed of starch, potassium iodide, sodium
thiosulphate and water, and an atmospheric
relative humidity of over 50% is needed to
give a reaction. The proportions used are
as follows:
Starch
1.0 gm.
Potassium iodide
3.0 gms.
Sodium thiosulphate
0.5 gm.
Water
100 cc
Properly arranged, however, tufts have
the great merit of showing the flow behavior
under changing flight conditions, during which
time they can be photographed continuously.
The chief use of this method is in the study
The solution is sprayed on the surface as
a fine "mist," the tiny droplets never being
allowed to merge into a wet film. To avoid
9:2

E
Fig. 9:1
Transition at 60% Chord on “King Cobra” Wing.
Chemical Method at Low Altitude.
this happening, several sprayings at short
intervals are made until a total of three or
four cubic centimeters have been sprayed on
each square foot of surface.
Two methods of producing a chlorinated
atmosphere have been used:
(1) by discharging gas into a high
factory chimney and flying through the smoke
from it, and
The action of the chlorine or the atmo-
spheric impurities is to oxidize the KI,
which discolors the starch. The function of
the sodium is to absorb the oxidizing agent to
the limit of its capacity and so give a chemical
"threshold," thus producing a better contrast
between turbulent and laminar areas.
(2) by releasing liquid chlorine from
an aircraft simultaneously with smoke to
mark its location; the liquid immediately
vaporizes into a gas.
The second method permits tests to be
done at altitude although it has not been used
at more than 7000 feet.
For oxidation by atmospheric impurities,
the sodium content is reduced by one-half
to keep the flying time within reasonable
bounds. Even so, the use of a town polluted
atmosphere is almost essential.
(c) Evaporative Methods (Dry and Wet)
These methods were developed from the
9:3
chemical method, and rely on the differing
rates of evaporation (instead of chemical
action) in the laminar and turbulent areas of
boundary layer flow.
black. It becomes white again as it dries and
this happens first where the boundary layer is
turbulent.
The dry or sublimation method (Ref. 2)
uses the evaporative properties of slightly
volatile solids to give a picture of the flow.
The solid is chosen to suit the test conditions
and is dissolved in a solvent that does not
affect the wing surface. The surface should
be black and it is sprayed with a solution in
what is best described as a just-wet state,
the solvent evaporating within a second or
two of reaching the surface.
When the method (Ref. 3) was first devel-
oped it was thought that the wetting liquid had
to have approximately the same refractive
index as the white coating to get good results.
Published descriptions of the method have
named several liquids as suitable, most of
which, if not all, must be sprayed on as they
soften the surface. This is a serious dis-
advantage in flight testing owing to the ad-
herence of grit and dust.
The coating thus formed has a "frosted"
appearance and can be seen and photographed
fairly easily. It is, however, "aerodynamic-
ally rough" owing to its crystalline nature
and it must be lightly rubbed down before
flight. This is best done with the palm of
the hand. An example of a test result using
this method is given in Fig. 9:2.
It has since been found that the refractive
index can be very appreciably different and
still give black and white pictures. This
permits kerosene to be used in conjunction
with a coating of china clay in diluted aero-
plane dope* and allows it to be wiped on just
before flight and any surplus removed. Al-
though it is more likely to collect grit than
the dry solid coatings, the “wer" surface
can be dusted down just before take-off.
Owing to the dry nature of the coating this
method has the advantage for flight work of
being less likely to collect dust and grit be-
fore take-off and that a final removal of any
grit is possible just before flight. It has
the disadvantage of being very sensitive to
changes of temperature, a factor which
limits its use to low level testing.
The variation of evaporative rate of kero-
sene with temperature is somewhat less
sensitive than with the volatile solids so
that when this method is used and the surface
wetted on the ground, tests can be made at a
somewhat greater altitude.
The solids that have proved most suitable
have been napthalene in cold weather, ace-
napthene, and azo-benzene under very hot or
high-speed conditions. Petroleum ethers
with boiling points between 80°C and 120°C
C
have proved to be suitable solvents, and a
5% solution (W/V) is convenient.
A recent development of this method for
use at very high altitudes employs what has
been called a "wetting at altitude" technique.
This overcomes the otherwise inherent diffi-
culty of applying a coating to a wing in warm
conditions at low altitudes and expecting it
to evaporate in the intense cold of the upper
air. The wetting liquid is therefore emitted
as a spray from a second aircraft and the
test wing is wetted by this spray at the
operating height by flying close behind the
tanker aircraft.
(d) The Wet or "China Clay'' Method
This method employs the evaporative
properties of liquids and uses a porous white
coating on the wing over a black undercoat.
This has the advantage of giving pictures in
black and white, for the white surface becomes
transparent when wet and therefore appears
*200 gm. fine kaolin, 250 cc clear dope and
750 cc thinners.
9:4

Fig. 9:2 Transition on a "Vampire" Wing. Sublimation
(Dry Evaporation) Method, at Low Altitude.
9:5
This development gives complete freedom
of choice of liquid, and as the picture dries
out on the test wing it is photographed by the
tanker aircraft. Several tests at various
speeds can be done on one flight even with
changes of altitude. An example of such a
test result is given in Fig. 9:3.
where flow was laminar during the first test
are still available to record a new "front,"
dried out during the second test. The first
result, of course, would be photographed in
flight. This procedure has not yet been
demonstrated as no case has arisen that
offered a possible saving of effort.
(e) Deposition Method
This method is the only visual one for
the study of the boundary layer that enables
a series of tests to be carried out during one
flight, although in the cases of the dry evapo-
ration and the foregoing "wetted before
flight” method, it may be possible to get two
results per flight.
light. This should be pos
if the transition “front' is known to move
forward on both the top and bottom surfaces
with change of test speed. The wet areas
The deposit of smoke particles in regions
of turbulent boundary layer flow has been
used occasionally in ground tests on rotating
airscrews. The deposition is due to the
velocity normal to the surface given to the
particles by the turbulence enabling them to
adhere by impact. The effect was first

Fig. 9:3 Transition at 50% Chord on''Vampire" Wing. M = 0.7 at 35,000 feet.
Wetting at Altitude (Wet Evaporation) Method.
9:6
noticed in high-speed tunnel tests where
city air was being drawn over models and
later on airscrews used in polluted atmo-
spheres and even on a glider after prolonged
towing.
Some tests have been made to assess the
practicability of the method as a flight
technique, for the action is mechanical in its
nature and therefore unaffected by tempera-
ture or humidity to any extent. The conclu-
sion reached was that the amount of smoke
required would be excessive and piloting
difficulties of seeing well enough to keep in
dense smoke might arise. Moreover, only
one test per flight could be made, probably
involving two aircraft. Therefore, the method
has not been pursued further.
density at a shock wave. Both employ light
passing in a spanwise or more or less span-
wise direction near the wing surface.
One of the methods is virtually the same
as has been used for some years in wind
tunnels with a refinement for indicating the
spanwise position of the shock waves re-
corded. It gives a photograph of the chord-
wise position and inclination to the surface
of a shock wave and the distance it extends
from the wing surface. The field covered
in one test, however, is rather limited both
in depth and in chordwise extent.
The other method records on a moving
film the chordwise and spanwise positions
of shock waves at a chosen distance from the
wing surface. It is much less limited in a
chordwise direction, but does not indicate
depth or slope of the waves. A full descrip-
tion of these techniques has already been
made public (Ref. 4).
(f) Optical Techniques
These have been used successfully in
flight to get photographic records of shock
waves and their positions. Two techniques
have been used, both making use of the re-
fractive effect of the sudden change in air
An example of the first method is shown
in Fig. 9:4 with shock waves extending
almost to the wing surface.

AleFLOW
Hlasa
TunZIY S2
WING SURFACE
1
Fig. 9:4 Shock Waves at M = 0.77, Extending Almost to Wing Surface (the Several Waves
are at Different Spanwise Position). Actual Depth of Photograph is 2.5 Inches.
9:7
9:4 FLIGHT TESTING
The application of the foregoing methods
of air flow visualization to flight test work
is covered in this section with special
reference to the limitations, difficulties, and
choice of techniques that have been en-
countered or that might arise in pushing the
methods to their limits.
(a) Tufts to Show Flow Separation
cumstances may often dictate the choice, as
may also the extent of the areas to be tested,
If the location is windy and sandy, then the
dry evaporation method is best because there
is less contamination by grit. If tests have
to be done in a fly-laden atmosphere, then
again the dry evaporation is best because the
surface can most easily be protected before
take-off (see par. 9:5(c)).
The operating factor of elapsed time from
preparation of the test surface till take-off
can be important in hot sunny conditions as
the black surfaces needed for dry evapora-
tion absorb heat rapidly and the coating evap-
orates from the upper surface.
Hangar protection is required for all
spraying if there is more than a light wind,
otherwise the uniformity of coating suffers
too much. A local wind-screen may be used
for dry evaporative coatings but hangar
coverage is nearly always required for
chemical method spraying so as to avoid dust
while the spray is drying.
(1) Stalling
Experience in the use of this method is
widespread. It is perhaps only necessary to
emphasize here the need to avoid the use of
material that is too heavy for the job in hand.
The risk is that the tufts may appear to show
unsteady flow which is in fact not there. Some
compromise between suitability and dura-
bility is usually required but the choice should
lean towards a fleeting truth rather than an
enduring untruth and the bother of renewing
tufts periodically or even frequently should
be faced.
Tuft behavior is usually recorded satis-
factorily by one or more fairly high-speed
motion picture cameras mounted on the test
aircraft. This may be supplemented by the
pilot's impressions, when obtainable.
The wet evaporation method has the ad-
vantage of easy application under windy con-
ditions immediately before take-off (if nec-
essary). This lessens the risk of collecting
dust on the surface which would otherwise
require a final wiping of the test surface. The
chemical method coating cannot be wiped over
after application but it can usually be omitted
from the forward part of a wing.
a
(2) High Speed Separation
The observations on stalling again apply,
but the durability problem is more acute.
Nylon tufts usually stand up well to the
tremendous battering imposed on them in a
breakaway region at high speed.
An important factor in the use of the evap-
oration methods for low altitude tests is that
the duration of the flying at the test speed or
condition must be quite considerably longer
than the duration of the unsteady conditions
of take-off, climb, acceleration, etc. If the
test speed is very high this problem is eased
slightly since the stagnation temperature
(which is a major factor) rises rapidly with
speed.
(b) Boundary Layer Observations
(Chemical and Evaporative Methods)
(1) Low Altitude Steady Conditions
(2) High Altitude, Steady Conditions
Here the choice is wide, as all the methods
will work under these conditions. Local cir-
Here there is no choice; the only possible
alternative to wet evaporation is a modified
9:8
1
chemical method which permits it to be
used in almost dry air; this has only been
tested in a laboratory (see par. 9:5(b)).
The wet evaporation must be of the
"wetted at altitude" kind, since no liquid
capable of evaporating at 30,000 or 40,000
feet in any acceptable time would remain
unevaporated during the climb. Evaporated
solids at 30,000 feet disperse at about 1/200
of the ground-level rate and, although liquids
are appreciably better, the falling off in rate
with height is still prohibitive.
If both aircraft can fly at the test speed
required and even wet at that speed, then a
fairly quick drying liquid is used and a
series of pictures of the drying is recorded.
Several tests can thus be done in rapid
succession. While the liquid is drying and
the surface is changing from black to white
in the turbulent regions, the film of liquid
not embedded in the porous coating in the
laminar areas is continually moving rear-
ward into the turbulent areas. This results
in a dribbly picture in the intial stages but
this "overflow” finally dries away to give a
clean transition line or "front."
The most satisfactory wetting arrange-
ment so far tried is to emit liquid at a rate
of about 10 gallons per minute from the wing
tip of the cooperating aircraft. This keeps
it as far away from the jet engine efflux as
possible and so avoids drying the test wing
accidentally during the wetting process. The
test aircraft is flown some 100 feet behind
with the test wing in the spray of liquid.
Owing to the angle of attack, the lower surface
gets wet first. Tests have been confined to
wings outboard of the engines, although
kerosene spray has accidentally gone into
jet engine intakes on some occasions without
noticeable effects.
If the tanker aircraft cannot attain the
speed required of the test aircraft, the
routine is more complicated. A slower
drying liquid must be used and the test run
lengthened so as to give time to accelerate
after wetting and to decelerate for photo-
graphy while permitting the full-speed run to
dominate the record. The two aircraft keep
within immediate reach of each other by
orbiting with the tanker on the inside.
The liquids that have been used for wetting
are aircraft kerosene at heights of about
20,000 feet and stagnation temperatures of
about -20°C and "white spirit" (kerosene,
boiling point about 180°C) at 35,000 feet and
stagnation temperatures of about -40°C.
In this more difficult adaptation the drying
time must be nicely judged so that at the end
of the test run at full speed there is no free
film of liquid on the wing. If there is, it mi.
blot out the real answer during the decelera.
tion. There is, fortunately, a good margin
of time between the stages when the picture
has become a good “clean” one and when it
becomes too faint to be photographed.
(3) Transient Conditions
3
If the pilot or crew of the test aircraft
cannot see the top wing surface (if the top
is wet, the bottom must be wet) the ''tanker"
pilot will have to verify whether the test sur-
face has been wetted,
It has been found best to equip the "tanker"
with two cameras and sights, one looking
obliquely upward and forward and the other
down and sideways. The pilot can thus "fly
formation" on the test aircraft and photograph
both surfaces.
When it is necessary for research pur-
poses to know the state of the boundary layer
flow under flight conditions which can only
be held for a very short duration, then the
original chemical method is the only work-
able visual technique. It has given satisfac-
tory results when the duration of the test
condition was only two or three seconds and
visual records have been obtained with ex-
posures of much less than one second.
9:9
conditions to ease the lighting problems.
These results are possible because there is
practically no chemical action except while
in the active gas and because the concentra-
tion of dilution of the gas can be controlled
to suit the experiment.
These methods require a moderate amount
of installation, and parts of the equipment
must of necessity be housed in fairings
external to a wing. Although one of the tech-
niques involves a traveling light source, it is
not unduly complicated and has been used
very successfully.
9:5 OTHER TECHNIQUES
The type of test with which this method
can cope is where the aircraft has to be
dived to maintain the test conditions (in
which case pressure readings become un-
reliable) or where, for instance, a wing's
range of angle of attack for laminar flow has
to be established in free air for comparison
with its corresponding range when tested in a
low-turbulence wind tunnel. A test of this
kind can involve flying at negative incidence
down a curved path. Such tests have been
done successfully. The marking, aligning,
and timing of such experiments requires
precision flying.
(a) For Separation
Two methods that might be useful in flight
to determine an area of breakaway flow under
special conditions such as where camera
positions for tuft photography are difficult,
are worth keeping in mind.
Since with this method the gas is released
into the laying aircraft's wake, it is essential
to break up the wing-tip vortex system by
flying with the wing flaps lowered. If this is
not done, the trail of gas and smoke must be
left for at least five minutes before the test
aircraft flies lengthwise through it because
in this time the trail may be bent by atmo-
spheric disturbances. It is helpful to lay the
gas trail in a stable layer of air, i.e., where
there is a temperature inversion.
One has been used in wind tunnel tests and
consists of feeding a very small quantity of
active gas into the suspected region of break-
away. This discolors the surface areas of the
breakaway, the region having been sensitized
as described in section 9:3.
The second method, and there is some
flight evidence that it will work, is to dis-
charge smoke into the breakaway region in
adequate quantity to mark the area involved.
Either method can be used without affecting
the actual flow breakaway.
(b) For Boundary Layer Observations
Tests under transient conditions have not
so far been required at high altitudes and the
chemical method has not been used over 7000
feet. Drier air conditions at high altitude
might call for a modification of the chemical
action described under "Chemical Methods"
in section 9:3, and such a modification is
referred to in par. 9:5(b).
A chemical method (that has been kept in
reserve in case chemical technique tests
should be needed in very dry air conditions)
has been devised and tested under laboratory
conditions. The moisture content needed in
the air is only a trace, instead of the 50
relative humidity required for the chemical
action described in section 9:3.
(c) Shock Wave Observations
In carrying out flight tests with the optical
chniques described in section 9:3,"Optical
Techniques," it is not necessary to do the
flying under dusk conditions although the
initial development was done under such
The active gas used is again chlorine and
the sensitive coating on the test surface is
of a nature that requires much more effort
9:10
to remove and renew for each test as
opposed to washing the starch coating away
with a mild scouring powder.
On a swept wing the inboard end of a cover
must be protected by an overlapping strip to
prevent it lifting and, in fact, end strips are
always useful in preventing a cross-wind re-
moving a cover while the aircraft is taxying
out to take off.
(c) Protection of Surfaces Before
Transition Tests
This is sometimes a problem in itself
especially with regard to insect contamina-
tion near the leading edge of a wing.
9:6 CALIBRATION OF AN ANGLE OF
ATTACK MEASURING SYSTEM
(a) Description of the Method
Dust and grit may be wiped from surfaces
treated for dry or wet evaporation tests and
this is best done at the take-off point on the
runway. Surfaces chemically sprayed may
have the forward part of the wing left un-
treated if the flow there is known to be
laminar and so reduce the risk of turbulence
wedges due to grit. This also permits a
covering to be used to keep flies off.
Because angle of attack is custom arily
measured by a vane-type of pressure pickup
type sensing device, the instrument readings
are influenced by local flow conditions.
Thus, the relation between the angle sensed
by the instrument and the actual angle of
attack of the airplane must be determined by
test calibration.
Flies collected near the leading edge
during take-off and early climb can com-
pletely spoil a test. Testing within two hours
after sunrise is usually a sufficient safe-
guard in summer in a temperate climate but
tests in the heat of the day call for a cover
that can be shed at altitude.
The calibration procedure usually em-
ployed consists of first placing the airplane
in stabilized horizontal : flight under calm
air conditions. In these circumstances the
relative wind velocity is horizontal and obvi-
ously the weight acts vertically. Therefore,
the angle between the airplane's reference
axis and the axis of a vertical level in the
airplane provides a measure of the actual
angle of attack (the angle of attack of the
reference line being 90° minus the above-
mentioned angle).
For small test panels, paper has been
used which rips in flight along the leading
edge.
A more satisfactory method has been to
cover the forward surface with two sheets of
thin waterproof cloth held on after being
wetted with water. The upper cover extends
around the leading edge to a point just behind
the stagnation point of the wing section at
take-off and is shed in flight by increasing
the angle of attack to near the stall, when
the stagnation point moves back and the
forward moving air lifts the edge. The
lower cover has its front edge protected from
the airstream by the top cover overlapping
it slightly. These "fly away" covers usually
.
depart together. The lower cover should
extend to about 20% wing chord if possible
but 10 or 12% is sufficient for the top cover
to give complete fly protection,
The instrument utilized for calibration in
France is the vertical level SF IM (J32),
attached to one of the standard recorders,
All or A20. The inclination of the recorder
axis with respect to the longitudinal airplane
reference axis is measured on the ground by
means of an artillery level. An auxiliary
instrument, the adjustable collimeter (SF IM-
U-60), is often provided to aid the pilot to
maintain stable flight.
It is important that the speed be held
constant during tests and that vertical atmo-
spheric motions be negligible; however, it
is possible to correct for deviation of the
test flight path from the horizontal, since
9:11
(d) Conduct of Tests and Data
Reduction*
this flight path angle may be computed from
readings of speed and altitude. The angle of
attack is then given as the difference of the
measured angle and the flight path angle.
For cach stabilized run deter-
minc:
(b) Summary of Purpose of the Tests
The airplane attitude from the
vertical level
The pressure difference dpi
Because angle of attack is generally
measured in terms of some related parameter,
the purpose of the calibration is to relate
the actual angle of attack to the measured
parameter.
The dynamic pressure (compres-
sible)
Generally, in France, the parameter
measured is the pressure difference dpi
(furnished by properly placed pickups) divided
by the compressible dynamic pressure, qc.
This case is the only one considered here
since procedures for others may easily be
determined.
From these data, determine the angle of
attack "a". If the vertical speed is zero,
the angle is at once determined from the
level reading; otherwise,
where a:
adoY
actual angle of attack of reference
line
(c) Test Equipment
Qo = observed angle
(1) Required Equipment
y: flight path angle
Twin pickup angle of attack head
Differential pressure gaged for
measuring dpi
Differential pressure gage for
measuring 9c:
and tan y: w/V, where w : vertical speed
and V : true flight path speed.
Knowing a , a curve of dp:/9c: f (a) is
prepared for each configuration considered.
From the foregoing, curves of a= f (Vcal)
or Ci= f(a) may be deduced for each con-
figuration at given weights and altitudes.
A vertical level
A sensitive altimeter
9:7 MEASUREMENT OF SIDESLIP ANGLE
An artillery level
(2) Desired Equipment
Ambient temperature pickup
The sideslip angle is defined as the angle
between the longitudinal fuselage axis and the
flight path as viewed from above the flight
path. As pointed out in section 9:1, this
angle, like the angle of attack, must be
measured in connection with the investigation
of stability characteristics and aircraft
armament effectiveness.
Equipment for determining en-
gine operating conditions
Thrust meter
Manifold pressure
torquemeter.
gage
or
Tachometer or revolution
counter
* Editor's note: Sections on pilot technique
and on the duties of the observer have been
deleted for the sake of brevity.
9:12
1
Theoretically, the simplest way to meas-
ure sideslip angle would be first to calibrate
a sideslip detector in a wind tunnel and then
mount this same detector on a boom long
enough to place it outside the region of
disturbed flow created by the airplane. It is,
however, impracticable to use a boom long
enough to accomplish this purpose and
accordingly, other means must be employed
than described above.
mally located on a boom; however, the
differential pressure type may just as well
be built into the nose of the airplane. Since
the calibration depends on the pickup loca-
tion, the location must be specified precisely
if usable results are to be obtained. In
particular, the location of the support axis
with respect to the plane of symmetry is of
prime importance.
For low speed, light aircraft, it is possible
to calibrate the sideslip indicator in flight
using a straight road or other similar ground
reference to establish a compass heading
with the pilot flying along the road at a con-
stant sideslip angle. The numerous and ob-
vious practical difficulties associated with
this method render it useless for tests of
high-speed airplanes.
In flight, the sideslip angle is measured
using a sighting collimeter previously aligned
to the aircraft's plane of symmetry. Using
this instrument, the angle between the smoke
trace and the airplane's longitudinal plane
of symmetry may be directly measured and
the results recorded using a camera or
other means.
For high-speed and other aircraft, a
theoretically and practically satisfactory
method of calibrating the sideslip indicated is
to lay a straight smoke trail in the atmo-
sphere as a base reference. An airplane
equipped with smoke generators follows a
straight course and the pilot of the airplane
to be calibrated follows the smoke trail at
an altitude approximately ten feet below the
smoke at a stabilized sideslip angle.
As an aid to the pilot in establishing
the steady sideslip angle, a thread may be
affixed to the windshield (if it has a flat
section) and several marks may be made
on either side of the thread to provide
angular reference. If desired (as in con-
nection with stability analysis), a position
gyro may be installed to record the angle
of bank (if any) or in multiplace airplanes,
this same information may be recorded by
an observer reading the position of the
"ball" of the ball-bank indicator. Similarly,
control position indicators may be installed,
again to provide information on stability
characteristics.
(a) Trace Plane Equipment
The trace airplane must have performance
comparable to the test vehicle and must be
equipped with a smoke-laying device. It is
not necessary that the smoke used remain
visible over great distances; however, the
trace should remain distinct for a distance
of about 1500 feet aft of the trace airplane.
No other special equipment is required for
the trace airplane.
The normal instruments required for
speed and altitude measurement should be
installed and, of course, a means provided
for recording the indications of the side-
slip detector.
I
(c) Test Configuration and Range of Tests
(b) Test Plane Equipment
The test configurations to be investigated
are determined by the reason for conducting
sideslip investigations in the first place. If
the tests are conducted in conjunction with
stability investigations, the configuration,
as well as range of sideslip angles, speeds
and altitudes is determined by the extent of
Sideslip detectors in common use are
of either the differential static pressure or
vane type. The vane type detector is nor-
9:13
the desired stability checks. On the other
hand, if the sideslip detector is ultimately to
furnish information to a gun-sight computer,
it may be possible to limit the tests to, say,
the "cruise configuration."
In conducting the data runs, both the
trace airplane and the test airplane are
stabilized at the required speed and altitude
with the test airplane about 600 to 1200 feet
behind the trace plane and at an altitude of
about ten feet below the smoke trace. The
trace plane must maintain a rigorously
constant heading during the tests to provide
a straight smoke trail.
Normally, the sideslip calibration should
not be sensitive to the c.g. location. More-
over, if a differential pressure type pickup
is employed, the indication should be a
function only of the sideslip angle B at low
speed.
The pilot of the test airplane, using
rudder control, adjusts the heading of his
airplane to the desired angle aligning the
collimeter with the smoke emission point
on the trace airplane and, crossing controls,
banks the airplane to follow the smoke trail.
At higher speeds, Mach number effects
will alter the pressure coefficient distribu-
tion. For high-speed airplanes, calibrations
should first be conducted at low altitude over
the range of speeds which the airplane is
capable of attaining. High altitude checks
should then be made to determine the effect
of Mach number.
When stabilized flight is achieved, data
may be recorded. This operation is repeated
for each desired sideslip angle, speed and
altitude.
The range of sideslip angles to be in-
vestigated depends on the purpose of the
calibration. Frequently calibrations extend-
ing over 6° of sideslip will be sufficient.
(e) Test Limitation - Zero Sideslip
Angle Checks
(d) Conduct of Tests
The principal practical difficulty encoun-
tered in conducting these tests is that after
the pilot of the test airplane has succeeded
in nullifying his drift with respect to the
smoke, the sideslip angle as read on the
collimeter is no longer exactly the desired
value. This difficulty can be overcome by
using photo recording and then plotting the
data to obtain the calibration for the desired
sideslip angles.
Prior to obtain ing actual data, certain
preliminary checks should be made:
(1) The system operation should be
ground checked.
(2) It should be determined that the
nose boom if used does not vibrate exces-
sively in the speed range to be checked. If
vibration is encountered, the natural fre-
quency may be lowered by adding ballast
along the boom.
(3) The range of instrumentation
should be investigated to make sure it is
adequate to cover the range of pressures
and/or angular travels encountered.
It is apparent that tests must be conducted
in calm air to insure that the smoke trail is
not distorted or bent by wind currents.
9:8 AIRBRAKE EVALUATION
(4) The required engine speeds at
the stabilized flight speeds and altitudes to
be tested should be determined to avoid
excessive time being devoted to achieving
stabilized flight.
Airbrakes are the mechanism which allows
the pilot to increase the airplane drag and
are particularly important on low drag air-
planes.
9:14
If v is the descent velocity, then v: V siny,
so that the preceding becomes
This discussion considers the nature of
tests required for the evaluation of airbrake
characteristics from the performance stand-
point (variation of flight path angle at con-
stant airspeed, deceleration value, etc.) and
from the handling qualities standpoint (mo-
ments due to operation of the brake, resulting
vibrations, etc.).
Av: ACD
=
?
2 mg
9:4
The increased Aco produced by the air-
brakes permits the pilot to use them as a
control" of deceleration or of flight path
angle during a descent. We shall see that
the classic flight equations permit us to
relate the drag change ACO, the deceleration
and the flight path inclination one to the
other.
From this equation we see that, provided
ACO is not a function of Mach number or
angle of attack, it is possible (for a given
configuration) to determine ACo by merely
measuring the change in descent rate due to
the airbrakes at one airspeed.
If, on the other hand, it is desired to meas-
ure the influence of some parameter (for
example M) on ACO , measurement of the
descent rates at the same angle of attack but
at various Mach numbers permits us to
obtain the desired results easily.
(a) Variation of Descent Angle at
Constant Airspeed
(b) Deceleration in Level Constant
Altitude Flight
Let F be the net thrust of the engine
(insufficient in this case to maintain altitude
at the speed V), and let YI
and
Y2
be the
descent angles with dive brakes retracted
and extended respectively. Then with brakes
retracted*, summation of forces along the
flight path gives
If we assume that prior to opening the
airbrakes, the speed is stabilized at the
value Vi , then
F + mg-sin ya co sve
F: { vi?cos.
9:5
9:1
and with brakes extended
If we presume that the airbrakes open
instantaneously and if the pilot has kept
the airplane at constant altitude, the instan-
taneous deceleration due to the airbrakes
may be computed from comparison of Eq.9:5
with
F+mg.sin ya 2 (Co+Aco) sve
9:2
and since the flight speeds are the same
in both cases,
2
dv
F-moto Eco + ACO) SVI
dt
9:6
mg (sin 72-sin y) = cosv?
,
y2
whence
9:3
-
OV
dt
2
р
2m
ACo sve 4C09.
*Editor's note: Assuming the thrust line to
be aligned with the flight path.
9:7
9:15
Let us consider Fig. 9:5, which presents
curves of airplane drag with airbrakes re-
tracted (CI), airbrakes open (C2) and thrust
available (C3) as functions of airspeed V for
one given altitude.
Therefore, for a given airplane, the in-
stantaneous deceleration will be inversely
proportional to the mass; proportional to
ACO, which is determined essentially by the
shape and arrangement of the airbrakes
(but sometimes also by the angle of the
attack and Mach number); proportional to
the dynamic pressure; or at constant altitude,
proportional to the square of the speed and
at constant speed proportional to the density p.
If the airplane is in steady level flight at
point Aj and the airbrakes are extended,
the deceleration is given by
Since Eq. 9:7 may be written
dvi
dt
- 4,8
AB
.
9:9
dy YPM?
dt 2
dV
YpMSACD
9:8
If the pilot maintains level flight (constant
altitude) following extension of the airbrakes,
the deceleration at the speed V is
where y: specific heat ratio
p : ambient pressure
dv
dt
.
AB
9:10
it follows that a deceleration started at a
given Mach number (due to the opening of
the dive brakes) decreases with increasing
altitude,
where A and B are the points on the curves
C3 (not CI) and C2 where the abscissa has a
value of V.
These conclusions permit us, among other
things, to calculate the effectiveness of air-
brakes under conditions different from those
of the tests. However, it is well to exercise
caution for the following reasons:
(1) the opening of the airbrakes is
not instantaneous and therefore, the speed
is no longer V when the drag coefficient
achieves a value Co + ACD.
Since AB is a function only of Vi , the
differential equation (9:10) can be solved
by graphical methods and it is possible to
determine the decrease in level flight speed
at the end of a given time,
We conclude these remarks on level
flight deceleration by pointing out that in
certain cases the deceleration in finite time
may vary in the opposite fashion from the
instantaneous deceleration (obtained when
the airbrakes are first opened).
(2) Co may depend on Mach number
and angle of attack.
(3) Finally, the instantaneous decel-
eration is not necessarily the best criterion
for airbrake effectiveness. It will often be
more informative to determine the decrease
in speed over a finite time interval (for
example, 5 or 10 seconds) and this deter-
mination, as we shall see, is not as simple
as that of dv/dt.
Consider Fig. 9:6, which illustrates the
case of an airplane where Co increases rapid-
ly above a certain speed (Mach number), and
where Aco due to airbrakes is a constant
(curves C, and C2). We shall compare the
decelerations obtained starting at
cruising speed V below the drag rise to
those obtained starting at the maximum level
speed vi, which is in the drag rise region.
some
9:16
THRUST
AND
DRAG
C2
А
B
A2
C3
V2 v V V
AIR SPEED
Fig. 9:5
In the first case,
dV
-
d = 4; Bi
dV
A, B,
where Ai Bi > A, B,
> A, BI since for constant
ACO
and at the end of a given time (say 10
seconds), the speed will have fallen to V3;
but in the second case,
A1 B; - AB
6).
9:17
Now, from Fig. 9:6, the speed cannot
decrease below Vi regardless of the length
of time the brakes are used; thus, if (V'
V ) < (V V, ), although we have an
initially greater deceleration at V,' than at
V , the time taken to effect a given speed
change may be larger at Vi than at V.
To put this another way, reasoning
analogous to that which we have just
considered can help the test engineer
to establish a test program and to in-
terpret the various results obtained from
it. However, care should be used in extra-
polation less fallacious conclusions be ob-
tained. For this reason, in the following
paragraphs we shall consider the testing of
airbrakes under the most varied conditions
of utilization.
In these circumstances, should we con-
clude that the airbrake effectiveness in-
creases or decreases with speed?
THRUST
ه ای
AND
DRAG
C
AL
c's
A
B
C3
A
1
|
V₂
Vz
Y
V.
Vå vi
빌
​AIR SPEED, V
Fig. 9:6
9:18
(c) Detailed Tests Quantities to be
Measured and Instruments Required
case of a single seat plane, a ground ob-
server may assist the pilot using radio
communication, for example, by counting
the seconds following airbrake deflection,
the pilot being responsible for reading the
airspeed indicator at 5 and 10 seconds.
(1) Altitude should be measured to
+1 mb using conventional altimeters or
barographs.
(1) Loading
(2) Airspeed should be measured
with a conventional airspeed indicator.
(3) Longitudinal deceleration com-
puted either from the slope of the curve of
true speed versus time or read from a longi-
tudinal accelerometer accurate to +0.02g
(difficult to obtain) range + 0.2g to -0.8g.
The loading of the airplane influences
the decelerations and the descent angles
and, in practice, an average weight is used
for each test condition.
It is assumed a priori, that the airplane
center of gravity location is unimportant
to the tests.
(4) Load factor n measured by an
accelerometer with a range of -lg to +4g.
(2) General Test Conditions*
(5) Surface load(s) should be meas-
ured within 1 kg.
(6) Control position should be meas-
ured to an accuracy of 1%- It is important
that precise control of the airbrakes be
available at maximum deflection to avoid
yaw at certain speeds.
Since most experience with airbrakes has
been gained on high performance combat
type airplanes, the comments here apply
principally to these. The tests required for
a transport plane would certainly be of more
limited scope.
If unusual characteristics are found, it
may be necessary to install longitudinal
position measuring gyros, vibration pickups
and flow visualization systems such as de-
scribed earlier in the chapter.
To be of maximum utility, airbrakes should
be capable of safe extension throughout the
entire airplane operating range of speeds
and Mach numbers. A sequence of tests
demonstrating this might comprise the
following:
(7) Instruments Required by the Pilot
to Conduct the Tests. For proper per-
formance of the tests, the normal panel
instruments are generally sufficient provided
they are correctly calibrated. However, if
not already installed, a pilot's airbrake
position indicator with markings at the ex-
treme positions is very useful as an auxiliary
item.
a. Clean configuration - low alti-
tude at maximum allowable calibrated air-
speed.
b. Clean configuration high
altitude at maximum allowable Mach number.
c. Clean configuration - low alti-
tude at (1) calibrated airspeed corresponding
to maximum level flight airspeed and (2) to
calibrated airspeed corresponding to the
speed for best energy climb.
(d) Abbreviated Tests
It is possible to conduct a brief evaluation
of the airbrakes without employing recording
equipment using a knee pad, the panel instru-
ments and the aid of an observer. In the
*Editor's note: This section has been abbre-
viated from the original in that specific
numerical values have been deleted.
9:19
.
d. Clean configuration high
altitude at calibrated airspeed corresponding
to level flight Vmax.
b. It is of utmost importance in
the case of the flights of tests c., d. and e.,
that the pilot establish and maintain level
flight as closely as possible. From the tests
we may then establish curves of true and
calibrated airspeed and longitudinal decel-
eration as functions of time.
- ap-
e. Clean
configuration
proach speed at reduced thrust.
f. Clean configuration - combat
dive at reduced thrust at various speeds and
flight path angles (angle held constant during
descent).
(3) Handling Qualities
Q
g. Clean configuration - descent
from high to low altitude at reduced thrust -
comparison of descents with and without
airbrakes,
The measurement of the control forces (or
force change) due to opening of the airbrakes
should be investigated to determine that the
brakes produce no unsatisfactory changes in
"feel"'*, which limit their use. In cases
where no force change is involved, the air-
brakes should also be deflected with the
primary controls free to permit measurement
of load factor variation and change in trim
altitude.
.
h. Clean configuration let
down
part flaps - thrust adjusted to hold a
given speed or glide path angle.
i. Landing configuration -other-
wise same as h.
A sharp diving moment at the instant of
brake extension is generally dangerous,
whereas a nose up moment is troublesome if
it is of any magnitude,
(e) Suitability Tests
(1) Usable Limits
During extension of the airbrakes, note
should be made of:
In general it should be established initially
that:
a. Vibration
a. The airbrakes are usable at the
maximum Mach number and maximum cali-
brated airspeed without the appearance of
dangerous phenomena.
b. Assymetric effects due to
unequal opening time or interaction with a
primary control surface.
c. Destabilizing effects.
b. The airbrakes give the re-
quired deceleration or glide path character-
istics.
(4) Utilization
The pilot should report on:
(2) Performance Characteristics
a.
The advantages of using part
flap.
b. Time of operation
a. Increase of flight path angle -
The increase in flight path angle due to
airbrakes for a given initial speed should be
investigated for conditions f., 8., h. and i.,
listed above. In some instances it is also
necessary to investigate the variation, due to
airbrakes, of airspeed for a fixed flight path
angle.
*We are here considering the case of air-
brakes, not pull out flaps" which were used
toward the end of the last war to produce
stalling moments in a dive.
9:20
c. The requirement for a positive
position indicator.
e. The possible uses during
operational flight.
d. The control location.
f. The use during a let down.
REFERENCES
1. Gray, W. e., "A Chemical Method of Indicating Transition in The Boundary Layer,"
R.A.E. Technical Note, No, Aero 1466, July, 1944.
2. Pringle, G. E. and Main-Smith, J. D., "Boundary Layer Transition Indicated by Sublima-
tion,” R.A.E. Technical Note No. Aero 1652, June, 1945.
3. Richards, E. J. and Burstall, F.H., “The 'China Clay' Method of Indicating Transition,"
R & M 2126, August, 1945.
4.
Lamplough, F. E., “Shock-wave Shadow Photography in Tunnel and in Flight,” Aircraft
Engineering, April, 1951.
9:21
AGARD FLIGHT TEST MANUAL
VOLUME I
APPENDICES
AGARD FLIGHT TEST MANUAL
VOLUME I, CHAPTER 10
PERFORMANCE TESTING OF HELICOPTERS
Ву
Ralph B. Lightfoot
Sikorsky Aircraft Division
of United Aircraft Corporation
VOLUME I, CHAPTER 10
CHAPTER CONTENTS
Page
SUMMARY
TERMINOLOGY
10:1
INTRODUCTORY COMMENTS
10:1
10:2
FLIGHT TEST PROGRAM
10:2
10:3
GROUND RESONANCE
10:4
10:4
CONTROL AND CENTER OF GRAVITY LIMITS
10:4
10:5
AIRSPEED CALIBRATION
10:6
10:6
LEVEL FLIGHT PERFORMANCE MEASUREMENT
10:7
10:7
CLIMB PERFORMANCE DETERMINATION
10:12
10:8
DESCENT PERFORMANCE
10:17
10:9
TAKE-OFF AND LANDING PERFORMANCE
10:18
10:10
MISCELLANEOUS TESTS
10:23
10:11
CALCULATION METHODS
10:26
10:12
BLADE STALL LIMITATIONS
10:30
CONCLUDING REMARKS
10:13
10:31
REFERENCES
10:32
SUMMARY
The performance of a helicopter may be determined by methods
similar to those used on conventional aircraft. Particular differences
exist in evaluating vertical climb performance and establishing values
of profile and parasite drag. Methods have been established to cor-
relate power, gross weight, rotor speed, forward speed, altitude,
and temperature by accounting for transmission efficiency, changes
in profile drag with mean lift and Mach number, and parasite drag
with speed. The resultant agreement with the original aerodynamic
analysis falls within the accuracy of flight test measurement. Ab-
solute values of efficiency, parasite and profile drags must be deter-
mined with torquemeters and full scale wind tunnel tests. Otherwise,
relative values may be assigned each parameter to render satisfac-
tory performance analysis.

TERMINOLOGY
Mechanical Efficiency
Tip Speed Ratio
V
Flight Path Velocity
VNE
Never Exceed Speed
VH
Horizontal Velocity
Vy
Vertical or Climb Velocity
VRB
Retreating Blade Velocity
BHP
Engine Brake Horsepower for Power Curves
HAV
Power Available to Main Rotor
HP
Shaft Power to Main Rotor
Pi
Induced Power
PO
Profile Power
Hp
Parasite Power
Pc
Excess Power for Climb
W
Gross Weight
R
Blade Radius
с
Blade Chord
B
Tip Loss Factor
b
Number of Blades
P
Air Density
V
Induced Velocity
vo
Induced Velocity in Hovering
Density Ratio
oe
Rotor Solidity Ratio
N
Rotor Speed
TERMINOLOGY
(Continued)
12
Rotor Angular Velocity
CLm
Mean Rotor Lift Coefficient
CT
Rotor Thrust Coefficient
CDE
Parasite Drag Coefficient
Inflow Factor
u
Velocity of Airflow through Rotor
ca
Torque Coefficient
с
Damping of the Oleo Struts
M
Effective Mass of Helicopter Minus Blades
n
Number of Blades or Maneuver Load Factor
E
Mass of One Blade
C&
Blade Damping Rate
1
Mass Moment of Inertia about Drag Hinge
E
Фр
Natural Frequency of Fuselage on Landing Gear
A3
1/2 [n.m/(M + nm)] · ml/1
.
1
Distance from Drag Hinge to Center of Mass of Blade
р
s/wp
C/(M + nm) W
P
р
λφ
Co/lwp
K
Empirical Factor
v
Azimuth Angle in Rotor Plane
83
Offset Angle of Pitch Control
A
Ground Effect Parameter
h
Altitude
10:1
INTRODUCTORY COMMENTS
inconsistent data and provides a sound basis
for establishing performance under any con-
dition desired.
Numerous methods have been proposed to
evaluate helicopter performance. Frequently
flight tests have not borne out performance
predictions, and therefore some doubt has
been cast on the validity of basic helicopter
theories. Flight tests have been conducted
over a period of years, constantly striving
to improve observation and piloting technique,
as well as vibration characteristics, instru-
mentation, procedures, and analysis.
To obtain good flight test results, several
cardinal rules apply. Data should be re-
corded only during stabilized flight condi-
tions. Pilots must be sufficiently trained
in performance testing and helicopter opera-
tion to hold desired conditions long enough
to establish steady state conditions. The
vibration levels of the aircraft and the con-
trols must not hamper operation. Airspeed
indicator installation errors should be mini-
mized. Zero airspeed must be carefully
obtained for reliable hovering and ground
effect data. Rotor speed should produce
optimum performance considering vibration,
stall, Mach number, and profile drag char-
acteristics. As already pointed out, experi-
ence indicates that energy methods with
proper factors offer the best method of
analyzing performance tests.
The principal items of performance to be
tested are as follows:
It has been found that all of these factors
have been responsible for the discrepancies
noted. Observers have recorded data during
unstabilized flight conditions. Pilots have
not been sufficiently trained as regards per-
formance testing methods and helicopter
operation to hold desired conditions long
enough to establish steady state conditions.
The vibration levels of the aircraft and the
controls have hampered operation. Airspeed
indicator and altimeter installations have
demonstrated extreme calibration variations
arising from both dynamic and static pressure
errors. Zero airspeed has been difficult to
attain for reliable hovering and ground effect
data. Rotor speed has not always agreed
with design values for the production of op-
timum performance because of vibration,
stall, Mach number, or profile drag charac-
teristics.
Attempts to resolve test data into force
components have become unwieldy, whereas
energy methods required empirical factors
for the determination of efficiency, induced,
and profile powers. Experience now indi-
cates that energy methods used with proper
correction factors offer the best method of
analyzing performance tests.
This system
of analysis of flight test data has been used
extensively to demonstrate the performance
and airworthiness of all Sikorsky helicopters
to the certifying and procuring agencies of
the U. S. Government.
By reversing the procedures of design
performance analysis, it is possible to re-
duce all test data to basic parameters
(Ref. 1). Fairing these data erases the
(1) Ground resonance
(2) Control and c. g. limits
(3) Airspeed calibration
(4) Level flight
(5) Best rate-of-climb
(6) Vertical climb
(7) Best rate-of-descent
(8) Take-off distance
(9) Landing distance
(10) Allowable crosswind
(11) Fuel consumption
(12) Cooling
(13) Critical altitude
(14) Carburetor heat rise
(15) Carbon monoxide
(16) Fuel system operation
(17) Torque distribution
(18) Efficiency
(19) Profile drag
(20) Parasite drag
(21) Stability and control
(22) Miscellaneous equipment
operation
(23) Blade stall and Mach number
effects
10:1
These items may be tested in a program
similar to that outlined below.
b. Ground resonance.
c. Flight measurements
of
magnitude of various exciting
frequencies.
d. Stress and motion survey in
flight.
10:2
FLIGHT TEST PROGRAM
(a) Personnel Required
(2) Cooling: Engine, accessories,
transmission
(1) Pilots
(2) Mechanics
(3) Test engineers
(4) Weight engineers
(5) Photographers
(6) Government inspector
(as needed)
(b) Apparatus
(1) Temperature recorder
(2) Pressure recorder
(3) Photopanel
(4) Oscillograph
(5) Theodolitic camera
(6) Standard movie camera
(7) Speed course
(8) Pitot static trailing bomb
(9) Carbon monoxide indicator
(10) Weighted cord with yarn tufts
a. Hovering, max and min RPM
at rated power at sea level
(overload to hold
within
ground effect).
b. Hovering at rated power at
hovering
ceiling, normal
weight.
C.
Vertical climb, rated power,
normal weight.
d. Climb at best rate-of-climb
speed to service ceiling.
e. High speed, max RPM, 30
minutes.
f. Cruising speed, max and min
RPM, 30 minutes.
(3) Airspeed calibration
(c) Weight and Balance
(1) Basic aircraft without test equip-
ment.
a. Longitudinal c. g. location.
b. Vertical c. g. location.
C. Lateral c. g. location.
Pressure survey with pitot
static bomb.
b. Speed course 30 mph to max
speed.
c. Pitot static bomb O to max
speed, level flight, autorota-
tion, and climb.
(4) Climb, sawtooth
(2) Test aircraft with equipment in-
stalled.
Weight of each item of test
equipment.
b. Arm of each item of test
equipment.
a. Rated power; max and min
RPM normal weight; 0, 50,
100 mph.
b. Rated power; optimum RPM;
normal weight; 0, 10, 20, 30,
40, 50, 60, 70, 80, 90, 100
mph.
c. Rated power, optimum RPM,
minimum weight, О and best
speed.
d. Rated power, optimum RPM,
max overload weight, 0 and
best speed.
(d) Flight Tests
(1) Vibration
a. Survey of vibration charac-
teristics (aircraft suspended
to simulate flight).
10:2
C.
Maneuvers
a.
(5) Autorotation, sawtooth
Descent with no power at
speeds indicated in (4) above.
b. Descent with no power at
five rotor speeds from max
to min.
(6) Vertical climb
1. 360 degree turn left and
right at 0 and 20 mph.
2. Left and right turns at
50%, 75%, and 100% max
level flight speed.
3. Acceleration from min to
max speed.
4. Deceleration from max
to min speed.
5. Pullout from dive at 0,
50%, 75%, and 100% max
speed.
6. Pullout from dive and
turn at 75% max speed.
7. Max pitching, rolling,
and yawing angular ac-
celeration at 0, 50%,
75%, max speed.
a. Hover in ground effect: wheel
clearance 0, 5 ft., 10 ft.,
D-10 ft., D-5 ft., (D = rotor
diameter), rated
power,
RPM, normal weight.
b. Hover out of ground effect:
rated power and RPM, nor-
mal weight, at ten altitudes
up to hovering ceiling 100,
500, 1000, 1500, 2000, 2500,
3000, 3500, 4000, 4500, 5000
ft.
C. Hover in ground effect wheel
clearance: repeat a at mini-
mum weight; 0, 5, 10, 15,
D-10, D-5, D ft.
d. Hover out of ground effect:
repeat b at minimum weight.
e. Vertical climb at normal and
minimum weight.
.
(9) Airspeed and RPM limitations,
max forward c. g.
a. Max rotor speed in autoro-
tation, max speed of flare
(reduce pitch below min to
meet design value).
b. Rated power (full throttle)
2000, 5000, 8000 ft. at five
rotor speeds from max to
min RPM, Note forward
speed for greatest tolerable
roughness. Correlate with
structural flight test sub-
stantiation.
(7) Level flight
a. Power required at sea level,
0 to Vmax at max and min
RPM.
b. Power required at critical
altitude, min to max speed,
max and min RPM.
C. Power required at 75% of
service ceiling, min to max
speed, max to min RPM.
(8) Controllability
(10) Take-off and landing, crosswind
tests with theodolitic camera
a.
Rated power, optimum RPM,
accelerate to and climb at
20, 35, 50 mph.
b. Repeat a at 95%, 90%, 85%,
80%, 75% power.
At best sinking speed in ap-
proach, simulate power fail-
ure and subsequent landing
from various altitudes to de-
termine lowest for safety.
d. Repeat at 60 mph, 30 mph,
20 mph, 0 mph.
C.
a. Max aft c. g., max and min
RPM; stick position, min to
max speed.
b. Max forward c. g., max and
min RPM; stick position, 20
mph (backward) to max
speed.
10:3
e.
In ground effect, fly sideways
into and with wind right and
left, up to max controllable
speed.
head is desirable to prevent excessive chord-
wise stresses at the root of the blade. How-
ever, a soft acting gear is usually required
to absorb the shock loads of landing. These
limitations should be explored and demon-
strated to be of such magnitude that all
characteristics are achieved, i.e., no ground
resonance, low blade stresses, adequate
strength of landing gear.
(11) Miscellaneous
a.
Carbon monoxide.
b. Heating and ventilating ratio,
C. Maximum or critical inclina-
tion of fuselage for fuel sys-
tem.
d. Carburetor heat rise.
10:3 GROUND RESONANCE
Ground resonance is a violent unstable
motion of the aircraft while in contact with
the ground, usually when partially airborne.
It is caused by a motion of the blade in the
plane of rotation coupled with a rocking
motion of the aircraft as a whole. The tires,
landing gear, and pylon structure act as a
spring with a rate which is incompatible with
that of the natural frequency of the blade
about a real or effective drag hinge in the
plane of rotation.
The vibration characteristics may be
safely examined if the aircraft is first re-
strained by at least four cables attached
to the highest stationary point on the main
rotor pylon. Two cables on each side may
be looped around steel stakes driven in the
ground at considerable distance from the
helicopter to provide maximum horizontal
restraint to the rotor head. A ground crew
should stand ready to snub out any vibration
which may develop by pulling the loose ends
of the cable very quickly if any evidence of
resonance appears. Resonance should be
checked throughout the complete range of
RPM and power conditions before the re-
straining cables are released. Excitation
may be introduced by periodical application
of the control column in circular motions in
the direction of rotation at progressive rates
from approximately 0.25 to 1.5 times rotor
speed,
Ground resonance is prevented by rigid
blades of proper natural frequency or by a
combination of dampers on the blade acting
in the plane of rotation and on the landing
gear usually serving also as oleo struts.
Proper damping is determined as a function
of the product of the two damper rates.
In Ref. 3 it is noted that for most heli-
copters the ground resonance will be sta-
bilized when the following inequality holds
In operation, the resonance character-
istics should be checked during take-off and
landing at zero speed, as well as during
power-off landings at high forward speeds,
as indicated on Flight Test Plan (d)(1)b and
(d)(10)c and d. Under all conditions, any
oscillation which may be introduced should
be damped or immediately made avoidable
by a change of power conditions. However,
no resonance should occur at any operating
condition such as during a rev-up to take-off
speed.
Ap:λφ >
13
Ź (A-3)
10:1
10:4 CONTROL AND CENTER
OF GRAVITY LIMITS
Initial tests may be made to check the
limits experimentally. Adjustable orifices in
the landing gear and the dampers may be
set at various positions, and tests conducted
for resonance. Low damping in the rotor
Mechanical and aerodynamic character-
istics of the rotorcraft will impose limitations
10:4
stalling speed of the retreating blade where
the actual velocity at the blade element is
equivalent to the rotational speed diminished
by the forward speed
SR-V = VRB:
-:
on the forward speed, rotor speed, gross
weight, altitude, power, and the location of
the center of gravity (Ref. 5). Mechanically,
the limitations will be imposed because of
the inclination of the aircraft and the amount
of control required will be limited by the
allowable displacement of the control system.
Thus, with an aft location of the center of
gravity, more forward displacement of the
control system is required to attain a given
speed. At high power or low rotor speed
conditions, more control is required.
In general
LE C ¿ PSV?
and
24
V =
✓
CLPS
Thus, the stalling speed
If a stalled condition is encountered on
the retreating blade, more pitch and control
displacement will be required to attain a given
speed, The stalled condition may cause
undesirable vibration before maximum con-
trol or power is attained. The situation may
be relieved by decreased pitch and power,
by increased rotor speed or a decrease of
forward speed, gross weight, or altitude.
These relations should be established as
indicated in Fig. 10:1 to insure a satis-
factory vibration level,
VRS
2L
va
: NR-V
CLmax PS
and the maximum forward speed in knots for
blade stall is
2L
V : NR-
CL maxes
Detailed analysis of these results may
lead to consideration of Mach number in-
fluence on maximum lift coefficient. How-
ever, a practical correlation of data may be
made by empirical determination of the
NR
(30) (1.69)
KnW
Rcb
/
leo
140
NR
16.13
KnW
Rcb
Por
120
10:2
100
2000 FT
, M.P.H.
5000 FT
Ve
8000 FT.
BO
where 1.69 converts ft./sec. to knots, Kis
the correlation factor determined from flight
tests for the particular helicopter, and n is
the load factor. As a matter of interest, it
has been noted that the magnitude of Vrs or
the stalling speed of the blade section, is of
the order of 190 knots for a disc loading of
three pounds per square foot and a solidity
ratio of 0.05.
60
40
160
170
180
190
200
210
ROTOR R.P.M.
Test data are obtained under conditions
outlined in the Flight Test Program, (d)(9)b.
Starting at a relatively low forward speed,
Fig. 10:1
Rotor and Airspeed Limitation
10:5
require the least correction of observed data.
Since the dynamic pressure at low speeds
is given by
- į ove,
pitch and throttle should be adjusted to maxi-
mum positions consistent with the pitch-
throttle synchronization and the power rating
of the engine to obtain the desired rotor
speed. Under the initial condition, a climb
will ensue.
More forward speed should be
achieved by control stick displacement, noting
maximum level flight speed. The control
stick should be displaced more forward to
obtain as high a speed as possible consistent
with the structural limitations imposed by
the stress analysis or the maximum allow-
able stresses revealed by fatigue studies
based on the stress survey in Flight Plan
item (d)(1)d.
we know that
2 (0,-)
V
and
Vcal :
Ve
2 2 (07-D) :
Po
Frequently before these limits are ob-
tained, the vibration level of the control stick
or the aircraft will become intolerable for
pilot comfort or even uncontrollable. This
speed does not usually exceed 115% of the
maximum level flight speed. The allowable
never exceed speed may be taken as 90% of
the maximum speed demonstrated (to provide
a margin of safety and comfort). Detail
study of blade stall should also consider the
influence of local Mach number. Within the
limits of full throttle operation, a preliminary
but practical determination of the never
exceed speed appears to be defined by the
relation in Eq. 10:2 with K determined ex-
perimentally as discussed above.
It is seen that proper registration of both
static p and total p, pressures is needed for
correct airspeed indication. It is also noted
that some error may be tolerated in both
pressures provided one error compensates
the other.
A series of total head tubes may be lo-
cated in likely positions, and coupled with
the static reading of a trailing pitot static
bomb to register airspeed on corresponding
indicators. Similarly,- several static pres-
sures may be picked up by tubes mounted
flush to the skin of the fuselage, or other
assumed good locations. These pressures
.
coupled with the total head pressure of the
trailing bomb may be used to actuate several
airspeed indicators.
10:5
AIRSPEED CALIBRATION
The conditions for proper airspeed indica-
tion are that the pitot head register the total
head due to speed alone and that the static
head register a pressure equal to the static
pressure infinitely far away from the air-
craft. In a rotary wing aircraft, the whole
ship is in a slipstream generated by the
rotors so that the areas affected only by
forward speed are limited. Also, the pres-
sure produced by the downwash varies be-
neath the disc. Exact locations for the pitot
head and static heads are peculiar to each
type of ship, and should be determined by
extensive survey to arrive at locations that
The several indicator readings may be
compared with the master indicator (the one
connected to the pitot static bomb only) at
all airspeeds and altitudes, including climb,
autorotation, glide, and yawed flight. While
it may not be possible to find locations where
separate total head and static pressure are
accurately determined, it is usually possible
to find a static source location which will
compensate for the error of total head regis-
tration. Similarly yawed flight with a single
static tube may produce errors which can
10:6
where BHP
= engine power
be corrected by dual static openings on
either side of the fuselage.
η
: efficiency of power transmis-
sion accounting for losses to
cooling fans, gear
fans, gear boxes,
torque compensators, etc.
Ho
profile power to overcome
profile drag of rotor blades
Tests in level flight at speeds over 30
mph may be conducted over a speed course.
Calibration for low speeds, in climb and in
autorotation may be determined using a trail-
ing pitot static bomb. The altimeter or
static pressure error caused by compensa-
tion should be checked with and without the
rotor turning, in the ground effect and by
flying at various speeds at known elevations
such as in line with a radio tower and the
horizon. The rate-of-climb indication may
be checked by noting the sense of indication
when changing from level flight to climb or
glide. The errors should not exceed those
allowed for conventional aircraft.
HI
: induced power to sunport hel-
icopter
HPp
-
parasite power to overcome
resistance to forward flight
HPc
-
excess power for climb
Pacc : accelerating power for turns
and maneuvers
10:6
LEVEL FLIGHT PERFORMANCE
MEASUREMENT
For unaccelerated level flight
-
Pacc - Hc=0
and
781P =
Hot Pi + Po
Reliable power-required data can only be
obtained under extremely smooth air con-
ditions with exceptionally good piloting tech-
nique and with test conditions held long
enough to insure stabilized flight (Ref. 8).
Data should be obtained at extreme gross
weight conditions, several widely separated
altitudes, throughout the allowable RPM
range, and with the center of gravity located
at the extreme and intermediate positions.
Tests may also be conducted to determine
the optimum yaw angle, using a calibrated
yaw head mounted on a boom beyond the
rotor slipstream and fuselage interference
effects.
Profile, induced, and parasite power are
discussed in the following paragraphs.
(a) Profile Power, HP.
)
For a blade element of span dr with re-
sultant velocity normal to the blade span
axis and in the plane of the rotor disc,
Ut
dPo = { cod dr vi
ui
)
2
.
Ut
Although several ways of analyzing per-
formance data and correlating the various
items of performance are theoretically avail-
able, numerous measuring difficulties have
led to our use of an energy system of analysis.
In this system, the power developed by the
engine is accounted for by:
where cy is the profile drag coefficient and
U, is
Up : Nr + x R siny.
BHP :
HO+HP+ 10 + 1Pc+ Hacc
η
10:3
The average profile power is obtained by
integrating over the radius and around the
10:7
azimuth circle and dividing by 2n; thus, for
b blade
For the ideal rotor the induced velocity
in hovering is
27
R
Pore("/" { cod or nex
.
dr 12
2
2 TPR2
3
+ 3 urR sin y
+ 34? VR? siny Jardy.
3 R' sin? y
Because of the high rotor tip pressure
gradient losses exist which may be accounted
for by inserting a tip loss factor B in the
above equation, thus
+
TR
Now, bc : 0 + R, and for average values
e
of
де
and
T
2 TpRB?
ਭਾ
Cd, we have
2 TT
Po
eR Pools
1"
Х
In forward flight the induced velocity is
reduced due to the greater mass flow rate.
This problem is considered in various ref-
erences and Fig. 10:2 is reproduced from
Ref. 18, showing a plot of v/vo against
V/vo where V is the net rotorcraft speed.
R
+ KR* sin v
+ Žu? Rʻsin?
+ H'R*siny
dy
Fig. 10:2 shows the induced flow ratios
for several values of flight path inclination
y, where
or
се Рcd
R$ 12
4
Rºn
-1 Vy
Y = tan
VH
.
+
[12
t +37]
)
Ce 1
e Pcd
"RRS
TT
These data apply to cases where the tip
speed ratio 4 is less than 0.15.
01
.ر3 + 1)
(
*)
8
Bennett, in Ref. 19, suggests that includ-
ing the effects of radial flow changes the
constant 3 to a value of about 4.65; thus,
converting to horsepower
If the helicopter configuration includes a
fixed airfoil section intended to produce lift
and unload the rotor in forward flight, the
gross weight, W, must be reduced by the
amount of lift received from the lifting sur-
face, when substituting W for T in the above
relations.
ce PCG RØR
(1 + 4.654?).
4400
10:4
For the case of no auxiliary lift
(b) Induced Power, Pi
HP; - 550
W
W
V 2R2 B2p
Vz
From the simple momentum theory, the
induced power is
10:5
Ty
-
550
where v/vo may be obtained from Fig. 10:2
or empirical data.
where T = rotor thrust
v : induced velocity.
We shall now consider the nature of the
tip loss factor B for use in Eq. 10:5. Sissigh,
10:8

in Ref. 27, suggests that
B - 1
chord at 70% R
1.5 R
For a good blade design flight tests have
shown that values of B = 0.98 are attain-
able; however, rectangular untwisted blades
may have values of B as low as 0.90.
rotor hub, and all nonlifting components of
the helicopter. It may be determined basi-
cally from drag estimates of all components
of the helicopter and from wind tunnel tests
of the fuselage without the rotor blades.
Flight test data may then be used to adjust
the relationship empirically, also serving to
consider scale effect and such other losses
not properly accounted for in the theoretical
treatment.
(c) Parasite Power, Hp
The parasite power, Pp, is the power re-
quired to overcome the drag of the fuselage,
From hovering test data, the induced
power, Hi and profile power, Ho may be
determined. (See section 10:6(d).) Then the
1.0
0.8
0.6
v/ V.
Y=0°
0.4
Y = 2,0°
Y=4.0°
0.2
o
4
8
10
2
6
V/v.
Fig. 10:2 Induced Velocity in Forward Flight (from Ref. 18)
)
10:9
parasite power is Pp : 7 BIP -
H-HR
Under test conditions, the value measured
is an "equivalent" value; i.e.,
This is accomplished as follows:
From hovering test data we may compute
the induced power IP, and at the same time
we may directly measure the total power
nBIP
n BH. Thus, the profile power in hover is
IP p = IP pe
repe
vero
IPO = 7BHP - HP
and since 4 : 0 during hover, we have
From several level flight performance
runs at various speeds, gross weights, rotor
speeds, and altitudes, the parasite power
may be determined and plotted versus for-
ward speed. It may be convenient to express
the drag in coefficient form, as
Ho
CeCOPTR$123
.
4400
201
CDF
whence
Da R? V?
cd
440012
CEPTORS
ρπΩ
-
and since Pp: (D4V)/550
11001PV 용
​PUR? V3
10:6
Cof
-
Typical Cor data are presented in Fig.
10:3.
Now, co is a function of the rotor thrust
coefficient or alternatively, mean rotor lift
coefficient Clm, so that tests should be
conducted throughout a range of gross weight,
engine speeds, and altitudes to obtain cd as
a function of CT or Clm:
In Fig. 10:4 are shown typical test data
vs Cum Cum is defined below.
·
.
(d) Data Correlation
It was mentioned in section 10:6(c) that
hovering tests could be used as a basis for
establishing profile power characteristics.
of co
016
.05
con
.014
.04
012
.03
PARASITE DRAG COEFFICIENT, D
DO
010
02
008
.01
.006
O
20
60
80
100
2
LO
Vy , M.P.H.
5
CLME
MEAN
Fig. 10:3 Parasite Drag in Level Flight
Fig. 10:4 Rotor Profile Drag Coefficient
as a Function of Mean Lift Coefficient
10:10
and
A mean rotor lift coefficient may be
defined by the equation
2T .BR
U drdy
ST
Re (278° + oH BROD
a
Te vi drove (2pp
BR
TT
2
2
3
T:
- .
Dr Slang) cup cum dr Oy
(
с
dy
also
627 R sin v
"**;
) kaina
vi drd y = VPRO
hence:
where B is the tip loss factor.
In forward flight no thrust is provided by
the reverse flow region of the retreating
blade, that is over a circular region of radius
O to - R sin y ; thus, with azimuth 0 degrees
being forward, we may rewrite the above
integral as
BR
T:
27
CLm. TB'R
BR
2CTORY
ce
TTKBR)
2
R?
27
3
븘 ​6 % 99 ficuCum drony
10:7
Let
27 - R sin y
uCumcardy.
K:
"Olm
ر4
B
B
2
블
​2
H
97
For an average chord c, we have
then
2 TT
BR
ܗܬܘ
KI?CT).
de
T:
pbcc
21
115.7*03 drev
-"";ara
10:8
27.-HR sin y
u
Fig. 10:5 presents calculations for K at
various tip speed ratios for B: 0.98.
.
T may be expressed as
3.3
TECT PTSR *
and since
3.1
bc
1
• TR
2.
we have
FACTOR "K"
2016 (AR)
2.7
CLm
2 BR
is""Dierov- ."S*** sin parou
y
U drdy
2
2.5
Now
2.3
.
.2
N
.3
.5
Ur-Br + H NR sin v
u$ =(p2 + 2HPR sin y + ki? R? sin?u)
Fig. 10:5 Lift Coefficient Factor
vs Tip Speed Ratio
3
10:11
As a final check and means of fairing
data, all performance information may be
reduced to profile drag form, i.e., since
rate of turning may affect climb performance,
but the usual reduction methods and rec-
tilinear climbs appear to provide reasonable
data.
H: 781P - HP - HP - HPC
-
and
(a) Climbs at Best Rate-of-Climb Speed
cd
.
4400 (nBIP-HP: - PPP-HPC)
σε πρ Ω' R(1 + 4.65μ2)
10:9
With this value plotted against the cor-
responding value of
The airspeeds for best rate-of-climb at
various altitudes are established by the con-
ventional sawtooth procedure described ear-
lier in this volume. Following the establish-
ment of the climb schedule or schedules,
continuous climbs to altitude are made (at
least two at near maximum gross weight and
two at near minimum weight).
Readings of altitude, time, and manifold
pressure should be recorded every thirty
seconds while the airspeed, outside and car-
buretor air temperatures, and engine RPM
should be recorded periodically.
CLM
: K
2 Ct
e
all test data may be well faired to yield
basic parameters from which rational per-
formance characteristics may be determined
for any gross weight, altitude, forward speed,
rotor speed, temperature, humidity, etc.
While most subsequent calculations will
conveniently utilize this cd vs CLm relation-
ship, it may be expedient to express the drag
in terms of Cų, i.e.
The flight test data, corrected for instru-
ment calibration, should be entered on the
calculation tables. These data are reduced
as indicated below. The rate-of-climb is
taken as the rate of change of pressure alti-
tude and used as the rate-of-climb at the
intermediate altitude. It is corrected to
true rate-of-climb at standard altitude by
the multiplying factor,
Co = 8,+8, C++ 8, c
The actual techniques of data reduction
are presented in section 10:11, which serves
to clarify specific procedures.
Absolute Outside Air Temperature
Absolute Standard Air Temperature
10:7 CLIMB PERFORMANCE
DETERMINATION
Horsepower is derived from the engine
manufacturer's power curve and corrected
to actual power at standard altitude by the
multiplying factor,
Absolute Standard Air Temperature
Climb tests should be conducted both
under forward speed conditions and in pure
vertical flight.
JAbs.Carb.Air Temp. x Abs.Out.Air Temp.
Tests are conducted at extreme gross
weight conditions and with variations in
speed for sawtooth climbs as indicated in
Refs. 9, 10, and 11. The effect of rotor speed
should be studied as well as the effect of
c. g. location. Temperature, humidity, and
All data are corrected to actual power
using power curves, and these values are
plotted for each climb.
Due to variations of weight and power
10:12
found as follows:
throughout the tests, all data must be re-
duced to the same gross weight, and rated
power at the density altitude of the test.
The flight test data corrected to standard
altitude are plotted for each climb as func-
tions of altitude. The true rate-of-climb
values versus altitude are faired using the
altitude versus time curve to determine the
slope. The rate-of-climb at each altitude
where data are recorded should be converted
into observed excess thrust horsepower by
the formula:
Di = Co; s v?
Coi is the induced drag coefficient, P is the
density of the air in lb. sec. 2/ft. , S is the
area of the rotor disc = TR? sq. ft. and V
is the true airspeed in ft./sec.
The induced drag coefficient is deter-
mined by the analogical relation
2
CO;
Cis
7D²B2
Obs.
Excess :
Test Weight x True Rate-of-Climb
THP
33,000
where D is the rotor diameter = 2BR, B : tip
loss factor : 0.98, and CL is the lift coeffi-
cient of the rotor disc defined again by
analogy by
L
where the test weight is determined by com-
puting average fuel consumptions and actual
fuel consumed during the flight to correct
the initial gross weight.
q
P
2
2
을
​S v
The power developed during the climbs
may be somewhat below or above that avail-
able at standard altitude. The rotor effi-
ciency may be assumed to be 80%. There-
fore, the thrust horsepower not used that
could have been available for climb is found
by:
with L being the lift in pounds corresponding
to the gross weight, and V again being the
true airspeed in ft./sec.
By substituting in the above equation for
Di, we find that
2 w?
D; =
TP O v2 82
Die rede
TIP
Not Used
0.80 (Rated Installed LP -
Actual Test Hp ).
Multiplying this by Po/ Po the equation
becomes
2 W2
To find the available excess thrust horse-
power for climb under standard conditions,
the thrust horsepower not used is added to
the observed excess thrust horsepower.
0, "
;
" oʻgʻv?
)
The rate-of-climb at the desired gross
weight is obtained as follows. The induced
horsepower is determined by a formula
based on analogy with fixed wing aircraft,
i.e.
where P Po is the ratio of the density of
air at any altitude to the density of the air
at sea level, subsequently designated by o.
Simplifying this by substituting known
values and changing V from ft./sec, to mph,
the induced drag becomes
.
Di
2.97
Di x V
375
HP:
124.5 · W2
o².v².o
B?
B
10:10
where V is the true airspeed in miles per
hour, and D; is the induced drag in pounds
(Note that o and oe are not the same quan-
tity.)
10:13
Substituting this expression for induced
drag, the induced horsepower equation for
B: 0.98 is:
approximation and then adjusted by level
flight test data. Hovering test data deter-
mine an average value of mean drag co-
efficient at a mean lift coefficient. Level
flight at the same lift coefficient allows
evaluation of parasite drag from the relation
124.5 · w?
0.353 · W2
PP
.
375.02.V.o.8?
D2.V.0
10:11
Copa
1100
PT RP v3
(78HP - HP-HO!
10:12
where W is the desired gross weight, D is
the rotor diameter, V is the true speed in
mph, and o is the density ratio,
From the energy equation,
The change in excess power is:
Profile Power Po,
ก
BHP
- (HP; + P + PC)
A Excess Power :
HP; (corr.'vt.) - HP
(test wt.)
and 7
where BHP is the rated power at the altitude,
is 0.80 to allow for losses due to
cooling, transmission, torque compensation,
and non-uniformity of airflow.
Excess Power : Avail. Ex. TIP (test
wt.)- A Excess Power if the test weight is
less than the correct weight; ( A Excess
Power is added if the test weight is greater
than the corrected weight.)
cd
Solving for in Eq. 10:4,
4400 PO
OPTNR (I + 4.65 ?)
cd
.
The standard rate-of-climb for the cor-
rected gross weight can now be found by:
Ex. Power x 33,000
SR-O-C:
W
This equation yields values of cd which
are plotted against mean lift coefficient
Clm, the mean lift coefficient again being
determined by the relation
The SR-O-C curves are faired for the
high and low weights and from these faired
curves are obtained approximate horsepower
required for climb by means of the
formula
CLm
2ÇI
.
к
१८.
where the factor K is presented in Fig. 10:5.
IPC,
२
SR-O-CW
33,000
C
As before
Induced power is found by
CT
:
W
ρπΩ?R
: Thrust Coefficient,
124.5 W2
Pi
LO
375 D' B? Vo
o
Solidity
Ř po dr 7,0607 :
Ratio
Parasite power is found by
s'RS
10:13
CD, в°рт
1100
CDf • parasite drag coefficient, and may be
determined from wind tunnel tests as a first
As noted before, values of cd are also
calculated from hovering performance and
a faired cd vs Cum curve obtained as in
Fig. 10:4.
10:14
(b) Vertical Climbs and Hovering
Using A, we find that
Hi
PP i
WA
W
550 V2 RPBP
ſ
10:15
Hovering is a special case of the vertical
climb wherein the vertical speed is zero.
From the simple momentum theory, the
thrust produced by a rotor in hovering or
vertical flight is T : Mv, when Mis the mass
flow rate of air through the rotor and v, is
the total change in velocity imparted to the
air moving from rest above the rotor.
To determine profile power, tests must
be conducted out of ground effect, and then
we have
IP = RIPA -Hico
.
The mass flow rate M is
wheren
M = "R? Bº up;
is a factor accounting for trans-
mission losses, cooling losses, etc.; n has
an average value of 0.80, thus
where u is the average axial flow through
the rotor.
Po: 8HX-HICO
Hrea
Since V, • 2 v and v: U -V, we have
and Ireq is the engine power requirement
of the test as measured by a torquemeter
or determined from engine charts.
T = 2TR 8 u (u-VIP;
U
.
in hovering, V= 0 and therefore
Knowing Po, we find cg from Eq. 10:6, i.e.,
4400 120
Cd
:
u?
-
ਵਲੋਂ
T
27 R2 BPP
e PT NRS
се
The induced power in hover is simply
Tu, therefore
Thus, cd can be determined as a function
of Clm from hovering tests out of ground
effect.
P
R = Tv
T
2 RBP
10:14
Hovering tests may now be conducted in
ground effect and 1 = 1; Hic determined
as a function of altitude. Normally, ground
effect is measurably present to an altitude
equal to the rotor diameter. In Ref. 21, it
is suggested that
and
IP i
-
W
W
550 V2R? BP
A =
)
where T: W since we are in equilibrium.
where D = rotor diameter and h= height of
3/4 radius station of coned rotor above
ground.
Induced power is less in ground effect
than in pure hover, and ground effect may be
expressed in terms of a factor A defined as
HP:
A:
HP.
i
which is less than or equal to one. l; is
the induced power far from the ground.
This factor should be checked by test on
each helicopter. Practically speaking, zero
speed performance of the helicopter is dif-
ficult to measure directly because of large
effects introduced by small deviations from
the trim hover condition.
10:15
Care must be exercised to avoid small
sinking speeds which include entry into the
vortex ring flow state (Ref. 12). Under
these conditions, large amounts of power may
be expended without regaining a true hover-
ing attitude. Several procedures may cir-
cumvent this dilemma. Operation in the
early morning, about sunrise, frequently
provides still air conditions, Anemometers
on top of buildings turn very slowly and
wind socks droop, indicating wind speeds
less than five mph up to altitudes of several
hundred feet. Under these conditions ground
and building references serve very well to
indicate zero airspeed.
eliminates the need for exact deterinination
of zero airspeed. The test conditions are
intended to simulate the worst conditions
anticipated and to give substantial variation
of thrust coefficient. The standard correc-
tions for horsepower and temperature should
be made as shown previously in section
10:7(a).
In vertical flight with nonzero airspeed
T: 2TRB ulu-ve
and
ст
:
T
PZP - R
2 ulu-V) Bº
R?
Letting
3
At altitudes reference may be made to a
cloud of smoke or to supersensitive indica-
tions of a swiveling trailing bomb. A long
light plumb line with yarns attached and with
a light weight on the end will indicate any
slight relative wind by movement of the
yarns in a direction opposite to the flight
path. Practically, it is sufficiently reliable
to use the service airspeed indicator pending
accurate results of the calibration done on
Flight Plan (d)(3)c.
.
12R
CT: 282,2- 21182
NR
or
CT
V=NR
-
Rate-of-climb in fps.
B?
Experience has indicated that true hover-
ing may be attained starting from a level
flight attitude of about 40 mph. Power is
added gradually, followed by a reduction of
airspeed to retain constant altitude. The
process is repeated with small increments of
power and decrements of speed until zero
airspeed is obtained.
V = 6012 Ria
et
-) :
Rate-of-climb in fpm.
In Ref. 29 it is shown that approximately
(translating to our notation)
co: CTA +
ст. + че са
8
and therefore
At several weight conditions, measure-
ments should be made of the power required
to hover with the wheels several inches, 5 ft.,
10 ft., 15 ft., 20 ft., 25 ft., and 30 ft., etc.,
off the ground under practically zero wind
conditions. These flights are considered as
being in the ground effect. Flights should
also be made to determine the maximum
power required to hover at substantially zero
airspeed at several altitudes out of the
ground effect. All powers should be taken
using several engine speeds.
[co-ed] ¢ *
Ć
Ceca
8
=
so that
V = 60 N R X
CT
2 X B4
ਵਿo - 4) - ]
<z«Co-se
·
This method yields conservative data and
10:16
Now
we have
I2R
550 (
HR) + Şakit av henge
2
V :
है-
W
2 Rp B2
TT
and
or
CT
2 ulu-V/B?
R?
*
ਏ . () (1)
550 (IPA-HP)
w
V:
w
W
2 RB/(550(PAPP.)
ਓਗੈਸ
(tt./sec.)
Therefore,
10:17
СТ
21B?
: U - V :v
.
where, as noted before, IPA: 80% (meas-
ured power) and Po is based on hovering
test values of cd.
and
601 R
V :
cd ,-
de cd
(co
CT
8
The use of these relations is illustrated
in section 10:11.
10:16
From Eq. 10:16
10:8 DESCENT PERFORMANCE
5501P
(a) Forward Flight Descents
e ca
8
.
PT 123R5
Also, by definition,
co
Q
R' 12R2
P
HP.550
RSPI'R?
is the available rotor horse-
where IPA
power.
The best sinking speed and rotor RPM
should be determined for various gross
weights, as indicated in Fig. 10:6. It is also
necessary to study partial power descents to
determine the region of roughness which lies
between the helicopter and autogiro state
so that operation in this vortex ring region
may be avoided in normal operation, as
indicated by Fig. 10:7. The descent region,
varying from approximately 500 to 700 ft./
min. at zero speed to level flight at 60 mph,
usually exhibits roughness and partial loss
of control, especially as vertical descent is
approached.
W
Crਨਾਕ ਹੋ ਜਾਂਦਾ
T
CTTRp
and Eq. 10:16 becomes
(Ita-tP!-v.
v : 550
-
W
)
Since
W
27RPUB?
W
2 RPP(V+v) B2
For several helicopters tested at disc
loadings between 2 and 4 lbs./sq. ft., it
appears that this condition develops when
only 85% of the power required for level
flight is delivered by the engine. The con-
dition is aggravating at low speeds, but is
of little consequence as the speed for mini-
mum power required is approached.
TT
or
2
vzore
है -
W
27 RPP B?
(b) Vertical Descent
The optimum pitch or rotor RPM should
)
10:17
be determined to insure minimum rate of
descent and to establish a safe margin below
the stall of the blades. Changes in delta-3
or alpha-1 hinge effects have considerable
bearing on the control characteristics in
autorotation,
the restraint in the plane of flapping may
introduce excessive normal stresses at the
root of the blade.
The flapping and drag hinges are fre-
quently inclined to the principal coordinate
planes. The effect of inclination may often
be accomplished in the cyclic pitch control
system. The inclination of either or both
hinges is an effective means of automatically
reducing pitch in the event of a power failure
to one allowing autorotation. A reduction of
power produces an immediate decrease of
rotor speed 12
speed , and of the accelerating
torque factor, and a consequent lower value
of inflow, a.
Variations of disc loading which occur with
power changes, accelerations, maneuvers,
etc., will also alter the coning angle and
pitch, introducing changes of rotor speed to
the chagrin of the pilot. This change in
speed is most disconcerting when flaring
out for a landing. The rotor speed increases,
as is desired to store up energy for the
final pull-up, but the unexpected reduction
of pitch with pull-up limits additional sup-
port, and the machine “squashes through."
Angles up to 8 : 20 degrees have been
63
found satisfactory.
10:9 TAKE-OFF AND LANDING
PERFORMANCE
(a) Take-Off Distance
Since ^ is usually a negative quantity
for the power-on condition, it causes an
increase of coning when the pitch is fixed.
But if pitch, BR reduces with increased
coning, the coning angle and rotor speed
may be maintained at a safe value if the
hinge inclination, 8, is properly chosen. For
high power conditions, the inflow and pitch
setting may be relatively high. This pro-
hibits sole reliance on 8, hinge action since
Take-off distance can best be determined
using theodolitic camera equipment. With
this equipment accurate measurement of
helicopter motion is made by a camera which
records time and horizontal and vertical
1800
-200
1600
1400
+200
R.O.D., FT./MIN.
R.O.D., FT./MIN.
.
1200
+400
REGION OF ROUGHN
1000
+ 600
800
+800
160
170
180
190
200
210
O
O
20
40
80
100
60
Ve , M.P.H.
ROTOR R.P.M.
.
Fig. 10:6 Limitation of Rotor
Speed in Autorotation
Fig. 10:7 Region of Roughness
in Partial Power Descent
10:18
displacement, provided the take-off and land-
ing is made in a plane perpendicular to the
line of camera vision. The camera is nor-
mally located 1500 ft. from the centerline
of a runway. A stripe marked by paint or
tape locates the runway and may be approxi-
mately 1000 ft. long. The height of the heli-
copter may be determined accurately up to
approximately 100 ft. Tests should be con-
ducted with virtually no wind.
of climb will occur with greater forward
speed, considerable reduction in take-off
distance to clear a 50-ft. obstacle may be
obtained by selecting a take-off speed slightly
in excess of the speed required for best
angle of climb. Obviously zero distance
would be required to clear a 50-ft. obstacle
if sufficient vertical performance is avail-
able.
By noting time and distance, both velocity
and acceleration can be determined by dif-
ferentiation. For example; plot distance
versus time, and measure slopes at corre-
sponding time intervals. Acceleration can
be measured by plotting velocity against
time and again noting slope at certain time
intervals. The effects of altitude, speed,
gross weight, temperature, and humidity
should be determined. Although higher rates
Above the altitude where vertical take-off
over the 50-ft, obstacle may be accomplished,
determination of the actual technique to be
used in achieving best take-off distance is
a matter of adding the distance required to
accelerate to a given airspeed. The effects
of altitude, temperature, gross weight, and
humidity must be considered. As indicated
on the Flight Plan (d)(10)a and b, the effect
of forward speed and altitude are simulated
for the take-off ground run. The air run in

Z
ALPHA HINGE
PARALLEL TO Z AXIS
NUMBER ONE
PLANE
NORMAL TO PLANE
OF ROTATION
X
NUMBER TWO
PLANE
NORMAL TO
BLADE RADIUS
-DELTA
THREE
NUMBER THREE
PLANE
PARALLEL TO Y AXIS
DELTA HINGE
Y
Alpha Hinge: Hinge axis about which the blade hunts, perpendicular to the plane of rotation
of the blades.
Delta Hinge: Hinge axis about which the blade flaps, in plane of rotation perpendicular to
the blade span.
Number One Plane: Plane perpendicular to the plane of rotation and passing through the
pitch change axis.
Number Two Plane: Plane perpendicular to the number one plane and to the plane of rotation.
Number Three Plane: Plane of rotation perpendicular to the axis of rotation.
An alpha one hinge is an alpha hinge that is rotated in a plane parallel to the number
one plane, etc.
Fig. 10:8 Hinge Nomenclature
10:19
Swash Plate
Arm
Pitch Control Arm
Hinge Line (Horiz.)
Flapping
Blade Arm
[ Rotation
Blade Feathering Axis ( Pitch Change)
s
at
Lag Hinge Line (Vert.)
Z Axis
Fig. 10:9 83 Hinging
10:20
the subsequent climb is determined from the
sawtooth climbs of Flight Plan (d)(4)b. Com-
bining the two distances and accounting for
temperature, take-off distance may be pre-
sented as shown in Fig. 10:10.
as though following engine failure. These
landings may be made from any speed, but
at low altitudes, consideration must be given
to the particular helicopter characteristics.
Contact with the ground has been made at
speeds as low as zero airspeed, although the
speed is usually about 20 mph. The amount
of wind determines the actual ground contact
speed. At low speeds, the vertical velocity
increases with the extreme case occurring
with vertical descent power off. For ex-
ample; a particular helicopter at 40-50 mph
has a rate of descent which is approximately
1200 fpm. This increases to approximately
2400 fpm at true zero airspeed.
(b) Landing Distance
The usual landing procedure employs
power so that a controlled vertical descent
may be made requiring no approach or
rolling distance after passing a 50-ft. barrier.
The critical case is landing with power off,
10,000

STAND.
9,000
-50°F
O°F
50°F
8,000
35
100°F
7,000
30
40 M.P.H.I.A.S.
6,000
ALTITUDE, FT.
25
20
5,000
10
35 M.P.H. 1.A.S.
4,000
3,000
30 M.P.H. 1. A.S.
2,000
1,000
25 M.P.H. 1.A.S.
20 M.P.H. 1.A.S.
10 M.P.H. 1.A.S.
OM.P.H. 1.A.S.
200 400 600 800 1000 1200 1400 1600 1800 2000 2200
DISTANCE, FT
Fig. 10:10 50-Ft. Obstacle Take-Off Characteristics
10:21
or so is attained a normal flare-out landing
may be accomplished; with less speed it may
be best not to use a flare, but increase pitch
to maximum shortly before ground contact.
>
In zero airspeed autorotation, the sinking
speed may approach 40 fps, but it will require
several feet loss of altitude to attain. Up to
approximately 10 ft., the sinking speed will
not cause structural failure of the aircraft.
From 50 to 75 ft., vertical landings may be
effected without injury to personnel and only
minor damage to the aircraft.
Continuing the example, from altitudes
between 100 and 300 ft., it would be difficult
to make a safe vertical landing, power off.
The aircraft may be placarded against hover-
ing in this region. At the higher altitudes,
some forward speed can be attained and a
safe landing effected. Between 300 and 400
ft. altitude, quick pilot reaction is required,
while above 400 ft., the landing can be made
quite easily.
The distance required to land over a 50-ft.
obstacle can also be measured satisfactorily
with the theodolitic camera equipment. It is
desirable to determine the effects of gross
weight, speed, altitude, and temperature.
As noted above, for some helicopters there
appears to be an altitude interval from which
a safe autorotative landing cannot be made
from a hovering attitude in the event of
power failure. Some vertical drop is re-
quired to attain control and forward speed.
This minimum altitude may be determined
as shown on Flight Plan (d)(10)c and d by a
succession of flights leaving it to the dis-
cretion of an experienced test pilot as to
whether he is able to nose down, attain a
reasonable glide speed, flare out and roll
the ship to a stop without striking a tail
rotor blade, tail skid, or tail cone on the
ground,
For the case cited above, landing proce-
dures have been established for power fail-
ures at the very low altitudes and at the
higher altitudes. When close to the ground,
although there is very little time available,
the main rotor pitch should be increased as
rapidly as possible. At the higher altitudes,
pitch should be decreased quickly to increase
the rotor speed. The stick should be moved
forward to drop the nose and a forward
speed of approximately 40 mph acquired to
hold the desired rate of descent. When the
ground clearance is approximately 35 ft., a
flare should be executed by pulling back on
the stick to decrease speed to approximately
35 mph. The stick may then be moved for-
ward to lower the nose to a level position
and to eliminate the possibility of dragging
the tail. The succeeding descent to the ground
may be controlled by pulling up on the main
pitch control to permit a normal landing at
a contact speed of approximately 20 mph.
The 50 mph approach is usually obtained
first. An altitude such as 500 ft. may be
selected as the starting point. If the landing
is successful and the pilot does not have to
apply power at the last instant to avoid a
crash, a lower altitude may be attempted,
until the lowest reasonable altitude is
reached. This program is repeated at vari-
ous forward speeds. With large departures
from the 50 mph approach speed, more alti-
tude is usually required to effect a safe
landing following a simulated power failure.
The landing gear characteristics will also
limit the maximum height from which the
helicopter may be dropped following a power
failure. Correspondingly, these minimum
and maximum altitudes will vary with speed,
tending to converge at a minimum approach
speed as indicated in Fig. 10:11.
For hovering at intermediate altitudes,
i.e., 10 to 400 ft. (covered by aircraft plac-
ard), a satisfactory procedure is difficult
to recommend. At the lower altitudes it
might be wise to decrease pitch slightly and
then increase pitch rapidly to a maximum.
At higher altitudes, it would be helpful to
gain forward speed; if an airspeed of 40 mph
The maximum rotor speed desired for a
satisfactory flare should be determined, one
which will not exceed design criteria. The
accelerations experienced by various com-
ponents of the aircraft should be demonstrated
10:22
to fall below design values for all landings.
With multi-engine helicopters, many of the
above problems are avoided.
while the aircraft is tied down, it is possible
to reproduce the numerous power and air-
flow conditions experienced in flight, so that
prolonged tests are unnecessary. See Chap-
ter 6 for data reduction procedures.
10:10 MISCELLANEOUS TESTS
(a) Allowable Crosswind
(c) Cooling
The auxiliary rotor capacity and fuselage
characteristics will allow hovering in a
considerable crosswind. The magnitude of
this wind should be determined so that opera-
tion will not be attempted under dangerous
conditions. It has been found satisfactory to
determine crosswind limits by flying side-
ways in view of the theodolite camera equip-
ment and noting the speed at which directional
control is lost.
Experience has indicated that critical
cooling usually occurs at zero airspeed and
maximum power. Tests may be conducted
at hovering ceiling, but the same result may
be obtained more easily by overloading the
aircraft to lower the hovering ceiling into
the ground effect. Basic cooling problems
may be studied and corrected under these
conditions. However, the adequate cooling
of the engine and transmission should be
demonstrated in hovering, climb, cruising,
and high speed conditions with maximum and
minimum RPM (Ref. 15).
(b) Fuel Consumption
(d) Critical Altitude
The variable pitch and RPM possible on
a helicopter facilitates the determination of
specific fuel consumption. With accurate
fuel flow meters installed, it is possible to
determine the fuel consumption with a series
of ground, hovering, and overload hovering
flight tests. Under these conditions, or
500
Critical altitude (Ref. 11) is the highest
altitude at which the engine manufacturer's
rated power at rated RPM may be obtained
under standard conditions of temperature
and pressure on a particular installation.
Forward facing carburetor airscoops, pres-
sure rise of cooling fan, carburetor duct
friction losses, temperature rise, etc., may
alter the supercharging effect on critical
altitude. Forward speed will increase the
desirable ram effect while heat transfer
through the ducts will decrease the critical
altitude,

oo
AL TITUDE AT POWER FAILURE
200
AVOID
100
The net effect of these factors may be
determined by noting the manifold pressure
in a climb at full throttle above the critical
altitude indicated by the engine manufacturer.
The effects of forward speed may be checked
by a series of level flight runs at maximum
and minimum speeds using full throttle at
several altitudes, approximately 1000 ft.
apart near the indicated critical altitude. The
manifold pressure is corrected to standard
temperature conditions at pressure altitude
20
40
60
80
100
Vo , M.P.H.
Fig. 10:11 Maximum and Minimum Altitude
vs. Speed for Safe Power-Off Landing
10:23
by an impeller chart as presented in Chapter
6, considering the unavoidable carburetor
heat rise. The corrected manifold pressure
is then converted to power available. It is
this power that should be used in correcting
performance data to full power conditions,
the chart will determine the CO concentra-
tion. Such a kit is available from the Mines
Safety Appliance Company, Pittsburgh, Penn-
sylvania, U. S. A.
(g) Fuel System Operation
(e) Carburetor Heat Rise
Carburetor heat rise is the temperature
difference between carburetor air tempera-
ture and outside air temperature. The amount
of carburetor heat rise available should be
determined with variations in outside air
temperature and power. The value should be
checked in level flight at various speeds to
include the effects of ram and in climb.
The full cold air value should be deter-
mined as well as the ram produced by the
cooling fan. The ram developed by a cooling
fan may be sufficient to counteract the car-
buretor heat rise normally encountered in the
full cold position. This effect will be ac-
counted for in the critical altitude study
described above.
Depending on the particular fuel system
installation and the quantity of fuel in the
tanks at a given instant, it is possible to
interrupt the flow of fuel to the carburetor
before the tanks are exhausted if the ship
is put into some unconventional attitude. It
is necessary to study all possible combina-
tions of ship attitude and fuel quantity to
determine the amount of fuel that may be
left in the tanks before engine stoppage will
occur.
Critical attitudes may be determined in
flight; for example, with a forward c. g. the
fuselage will nose down the most and possibly
uncover the sump of a flat fuel tank. The
aircraft is correspondingly oriented on the
ground, and the engine run until the fuel
pressure warning light comes on or the
engine stops, The fuel remaining in the
system is then drained and measured. This
residual fuel is deducted from the nominal
tank capacity and classified as part of the
empty weight.
(1) Carbon Monoxide
Depending upon the configuration of the
aircraft, tests should be conducted while
hovering in little or no wind, in forward,
backward, and sideward motion, having the
doors and windows open and closed. Tests
should also be conducted in cruise, high
speed, and autorotation conditions unless it
is apparent that any condition is not critical.
(h) Torque Distribution
The CO content should not exceed 0.005%
(Ref. 14). The CO content is measured by
using a small glass tube filled with two
chemicals that react and change color when
exposed to a CO concentration. The glass
tube is open to the atmosphere at one end,
and the other end is inserted into a rubber
squeeze bulb.
A sample of air is drawn
into the bulb over the chemicals. The tube
is then compared to a standard color chart
which is part of the kit. Proper matching of
the colors of the chemicals in the tubes with
For academic purposes it is desirable to
know the distribution of power to the various
components of the aircraft. For satisfactory
performance analysis, however, it is also
necessary to know where the power developed
by the engine goes. It is desirable to have
torque meters to determine the power de-
livered by the engine and that absorbed by
the main rotor, auxiliary rotor, transmission,
and cooling fan.
Torque may be measured with wire strain
gages indicating directly on a galvanometer
calibrated to read torque. A permanent
record may also be obtained on an oscillo-
graph, but at a slight loss in accuracy of
10:24
no load registration. The strain gages should
be mounted in pairs on opposite sides of any
shaft in question with the axes of the gages
at 45 degrees to the axis of the shaft to form
a complete Wheatstone bridge for accuracy
and temperature compensation. The im-
pressed voltage on the gages and the re-
sponse to load passes through a slip ring
arrangement.
meters are provided to determine the power
delivered to the main rotor. Without the
torque meter, it will be necessary to assume
a transmission efficiency, obtain a first
approximation of profile drag from hovering
tests, determine parasite drag on the basis
of the initial profile drag values, and com-
plete the lift drag polar from climb, and high
altitude test data.
Other torque meters which might be em-
ployed are specially designed shafts for
magnetic strain gages or hydraulic restraint
of plane pinions in planetary transmissions.
The hydraulic pressure should be converted
to terms of torque.
Approximate values
are:
Wind tunnel tests of an actual sample of
the rotor blade at flight Reynolds number
and Mach number will further serve to es-
tablish the relationship of section lift and
drag coefficients (Ref. 16). Reduction of
data to mean rotor blade lift and profile
drag coefficients offers a very convenient
method of fairing out test data obtained at
various altitudes, weights, RPM, speeds,
Main rotor
.80 - .90
and powers.
Auxiliary rotor
.07-0
(j) Parasite Drag
Transmission
.02 - .05
Cooling fan
.05 - .10
The most satisfactory information ob-
tained on parasite drag has been determined
by full scale wind tunnel tests on the air-
craft with the rotor removed. With sufficient
flight test data, a relative value may be
determined which renders satisfactory per-
formance analysis. Hovering and level
flight tests are conducted at the same mean
rotor blade lift coefficient.
The torque information makes it possible
to determine the efficiency of the transmis-
sion system so that a factor (7) may be de-
termined for various tip speed ratios (ulex-
pressing the percent of power developed by
the engine which is delivered to the main
rotor. With little or no knowledge of the
power distribution, an efficiency of approxi-
mately 80% may be assumed. Any discrep-
ancy will then manifest itself as profile or
parasite drag, but has little effect on the
final performance data.
The first approximation of profile power
obtained in hovering tests, when parasite
drag is obviously zero, together with induced
power, may be deducted from forward flight
data to yield a parasite drag value. It will
usually be found that parasite drag coefficient
varies with speed and is often higher than
anticipated at high speed (see Fig. 10:13).
(i) Profile Drag
(k) Stability and Control
With sufficiently reliable flight test data,
it is possible to ascribe relative values to
profile drag which, although not absolute,
render fairly accurate means of adjusting
flight test data for variations of gross weight,
altitude, etc. More nearly absolute values
of profile drag may be obtained if torque
Present stability and control require-
ments for conventional aircraft have been
relaxed for helicopter application due to the
high degree of controllability present and the
special pilot training required. For the
present, certain elements of stability and
10:25
devices, it is desirable to investigate the
stability characteristics and rate them for
flying quality. Subsequently it may then
be possible to incorporate an automatic
pilot.
control may be checked. At high speed,
forward stick motion should increase speed
and back stick should decrease speed. If
lateral motion is also required, no reversal
should be present. During the high speed
dives there should be no tendency for the
nose to drop. It is desirable for the rotor-
craft to remain level laterally and for the
nose to rise with up-gusts when extreme
forward or fixed control is applied. During
pullouts from dives, only a moderate roll
should be allowed, and rotor speed should
not exceed prescribed limits during entry or
pullout from dive.
(1) Miscellaneous Equipment
1
The optional equipment provided for a
helicopter should also be subject to examina-
tion if it involves the safety and operation
of the aircraft or personnel. Thus, a hoist
should be checked for capacity, operating
characteristics, and effect on performance
and control characteristics while in opera-
tion. Dual and servo controls should comply
with the usual requirements. Litters for
rescue work should not endanger the occu-
pants by the possibility of carbon monoxide
fumes nor require extraordinary technique
if they are to be loaded while in flight. Floats
should be checked for ground resonance
characteristics and consideration given to
their power-off landing characteristics.
Insufficient experience with stable heli-
copters is available to permit dictating the
type or degree of stability desired. Static
stick position stability as noted above is
desirable, i.e., forward stick for forward
motion. It may be desirable to introduce a
stable stick force gradient, which returns
the stick to the trimmed position if left free.
However, slight instability here is preferred
to any large sacrifice of maneuverability.
For instance, since the pilot must direct
all his attention to operation when rapidly
changing from hovering to high speed, etc.,
it is detrimental to fight large stick forces
while doing so. The opposing inertia forces
inherently resist enough without adding more
load. For a long range cruising operation,
this may not be true, and a small stable
stick force gradient is desirable, probably of
the order of 0.1 lb./mph,
10:11 CALCULATION METHODS
Lateral stick forces may be of similar
magnitude. The time of oscillation for the
stick-fixed condition should be of sufficiently
long period to be readily controllable.
20-second period is acceptable. With an un-
stable stick force gradient, the period of
the stick-free dynamic instability may be as
low as 10 seconds, because the pilot is
always in control.
Flight test data are reduced to standard
conditions by methods outlined in earlier
chapters of this volume to the extent of
determining gross weight, actual power,
true speed, true rate-of-climb, and density
altitude. For changes in gross weight of
less than 2%, the test data may be scaled up
to a desired weight by inverse proportion.
For changes in gross weight in excess of 5%,
induced power, must be corrected using the
ratio (W/W, X
(Ref. 17). For changes of
gross weight in excess of 5%, the variation
of mean lift coefficient will require consid-
eration of a new profile drag. For climb
test data less than 500 ft./min., all factors
usually must be considered. This, then,
requires a fairly reliable evaluation of the
relation between mean lift coefficient and
mean profile drag coefficient.
Test data and performance results tend
to substantiate the assumptions of uniform
induced flow over the effective radius, al-
lowing the use of momentum analysis. Twist
If stable stick forces of sufficiently high
degree are introduced, the difficulty of dy-
namic stick-free instability will largely dis-
appear. Until continued operating experience
is gained with the several possible stabilizing
10:26
where the factor K is given in Fig. 10:5
derived from Ref. 20.
and taper may be accounted for in the ex-
pressions for tip loss and effective solidity
and by use of an equivalent blade angle at
3/4 radius. The variation of induced power
with speed and angle of climb may be ex-
pressed by the ratio of Fig. 10:2, taken from
Ref. 10. The profile power is effective over
the whole radius and includes radial losses
as suggested by Ref. 19. The parasite power
is determined from level flight or full scale
wind tunnel tests. The climbing power is
that excess power which produces climb. The
total power dispersed may then be accounted
for by Eq. 10:3.
A polar curve may be constructed by
plotting Co vs Clm as in Fig. 10:4. Fairing
a curve through the scattered points elimi-
nates the vagaries of test data accumulated
under a variety of conditions of weight, speed,
altitude, power, temperature, and piloting
techniques. It may be helpful in fairing data
for CDF to first fair the data of equivalent
parasite power HP pe plotted against Ve,
equivalent airspeed.
n 7 BIP = 'Po + HP; + Ppt PC
+ +
+ c
where
The empirical values of cd and Cop, Figs.
10:3 and 10:4, when used in the basic power
equation, Eq. 10:3, will give properly cor-
related performance results.
Fig. 10:12
indicates the agreement of original test data
to be expected with empirical calculations
in climb tests, while Fig. 10:13 shows the
agreement in level flight.
n : transmission efficiency
:
BIP: actual brake horsepower delivered
by engine
Hi : induced
W
power
VOTRBF
W induced velocity in
VoVZTRAB? hovering (see Eq.10:14)
ਨ
V
W
550 = to V
C
Special consideration must be given to
vertical climb performance. Although
Pp: 0 and IP : ,
7 BHP - HP - Ho, the in-
duced power in vertical climb is reduced
because of the improved operating conditions.
In ground effect which exists up to a rotor
diameter, the induced power is also, as ex-
plained earlier, reduced approximately but
should be determined for each helicopter.
For take-off without a prepared runway
surface at altitude or overload conditions,
the value of 1 is usually taken for a 10 ft.
wheel clearance.
.
B :
.92 for straight untwisted blade
.98 for tapered twisted blade
Wy : variable with flight path speed V
and y from Fig. 10:2.
angle
- tan
- ROC below
of climb 60 V 30 mph
IPO profile de pod
R'n'
-(1 + 4.6543
power
4400
PP parasite
power
1100
Pc climbing, ROC · W
power 33,000
ROC = rate-of-climb in ft./min.
The
4400(BHP-IP: LPP IPC)
profile
ce TPR (1+4.65427
Co Parevo
Then corrected vertical climb is ex-
pressed in fpm as:
.
zippe
3
550(Hayitpe!
AW
W
2TAR B2)
ROC = 60 Vc = 60
:
drag, cd
• ( ) {550
$) ਦਾ ਸ
W
(Pav-
The mean lift
coefficient, Cum
: K
oe
from Eq. 10:17.
10:27
= .013 at 300 ft. (.016 at 5700 ft.)
'd
.
sea
(.014 at 2600 ft.)
For example, W : 5000 lbs., R = 25.25 ft.,
RPM = 189, BHP = 500 HP at level
(400 HP at 5700 ft.), B = .97, de = .05,
coning angle (B.) = 6 degrees = sin' 0.1,
height of helicopter = 14.1 ft., disc area
TR? : 2000 sq. ft., and tip speed S R = 500
ft./sec.
.
from Fig. 10:4
=
oe PT RP (NR)
Рп
HO
с
d'
4400
Find hovering ceiling for 10 ft. wheel
clearance above ground. Assume two alti -
tude's, -300 ft. (P = .0024) and 5700 ft.
(P = .002).
(.05)(.013)(.0024)(2000)(125,000,000)
4400
89 HP at -300 ft. (91 HP at 5700 ft.)
h :
10 + 14.1 + (.75)(25.2)(0.1) = 26 ft.
=
P
-
av
.8 (500) = 400 HP at -300 ft.
(320 HP at 5700 ft.)
D
50.4 ft.
:
h/D : .516
D.
(h/b)} , 0.8
550
(400 - 89
5000
-.8
(40
5000
(40096.97.2 .0024
л
w
5000
-
-
ROC = 60
1.0024
0024)
CT-
PT R’ (12 R)? (2000)6.0024)(25,000)
)
5000
550(400-89)
.00417 at -300 ft. (.005 at 5700 ft.)
1270 ft./min. at -300 ft.
-
(240 ft./min. at 5700 ft.)
(3.29](2) CT
.
C
(3.29)(2)(.00417),
.05
m
The rate of change of climb
.55 at -300 ft.
1270 - 240
5700 + 300
(.66 at 5700 ft., .6 at 2600 ft.)
172 ft./min./1000 ft.
450
20000
o TEST POINTS, SEA LEVEL
• TEST POINTS, 10,000 FT.
O TEST POINTS
400
16000
SEA LEVEL
4500 LB
12000
350
ALTITUDE, FT.
B.H.P.
O
B000
300
5000 LB.
4000
250
-10,000 FT
o
400
O
1600
200
2000
O
20
40
60
80
100
800
1200
ROC, FT./MIN.
V , M.P.H.
1
Fig. 10:12 Climb Performance
at Best ROC Speed
Fig. 10:13 Power Required
in Level Flight
10:28
By extrapolation
Δh :
240
172
x 1000 = 1400 ft.
expected. For example and continuing the
case quoted above:
= .008 from Fig. 10:3
at 60 mph (88 ft./sec.)
Ср
The altitude at which the helicopter may
hover with its wheels ten feet off the ground
is 5700 + 1400 = 7100 ft.
CD, por v
PTR?
f
1100
P
Po
:
3
Effects of Temperature and Humidity
(.008).0024)(2000)(88)
1100
24 HP at -300 ft.
и е
V
12R
88
500
: .175; 4.654?: .143
It is realized that increased temperature
reduces the power developed by the engine
and increases the effective altitude of opera-
tion for power required. The effect of
humidity has been established by the engine
manufacturers. Test data obtained at car-
buretor temperatures above 75°F and rela-
tive humidity above 75% should be used with
caution. Every effort should be made to
conduct performance tests under atmospheric
conditions which avoid these limits.
о с РТ
2
R” (SR)". (1+4.65 m?)
HO
e d
4400
:
89 (1.143) = 102 HP at -300 ft.
:
ſa
W
2TR BP
=
ve
5000
(4000)(.97 8.0024)
: 23.5 ft./sec.
To evaluate climb performance at tem-
peratures other than that established for
standard conditions, it is satisfactory to
work from a density basis (Ref. 22). Climb
performance is reduced to standard con-
ditions of density altitude (Ref. 23). There-
fore, for any pressure and temperature the
aircraft will experience a certain density
altitude, and develops the climb corres-
ponding to that density altitude if the engine
experiences a temperature normally encoun-
tered at that density altitude.
V!.
88
23.5
.
= 3.75; v/.
: .25 from Fig. 10:12.
W
.
Pi
N
v/v
550 OV2TR BP
5000
-
550 (.25)(23.5) = 53 HP
Pc
Hav - HOPPp Pi
3
Since this is not the case, the engine power
available is subject to a correction (Ref. 24).
At that pressure and temperature, a certain
power may be developed. The difference
between this power and that which would be
developed at that density altitude under
standard conditions is corrected for trans-
mission efficiency and subtracted from the
excess power which would have produced
the climb at that density altitude under
standard conditions.
:
(.8)(500) -102 -24 -53 = 221 HP
ROC
221 x 23,000
5000
.
= 1460 ft./min.
at -300 ft. (approx. sea level)
The corrected excess power converted
into climb yields the performance to be
At sea level pressure altitude with 100°F
10:29
Hc = (.8) (285) = 102 (.0015) (.028)
7.0024)(.013)
:
outside air temperature and a 20°F car-
buretor heat rise, the density altitude is
2600 ft. where P: .0022. At 2600 ft.
under standard conditions, the temperature
is 50°F and the engine would develop
450 HP. The excess power for climb may
be obtained from the above values accounting
for density changes and for increased pro-
file drag coefficient noted above.
-246.0015)
-53
7.0024)
0024
= 9 HP.
.0015
ROC
9 x 33,000
5000
: 59 ft./min.
-
He
7.0022 014
(.8) (450) - 102
1.0024/ 1013
.
1200 - 59
15,000 - 2600
1141
12,400
: 92.2 ft./min./1000 ft.
-
-24
.0022
-53
,0024
.0024
.0022
: 182 HP
Δh
.
50
x 1000 = 640 ft.
92.2
ROC
.
182 x 33,000
5000
: 1200 ft./min. at 2600 ft.
:
By extrapolation ceiling = 15,640 ft. under
standard conditions.
10:12 BLADE STALL LIMITATIONS
8
This rate-of-climb can be obtained only
when standard conditions prevail at the
carburetor corresponding to a pressure al-
titude of 2600 ft. and 50°F. But at sea level
and 100° + 20°: 120°F the engine can develop
actual HP : indicated HP ASAT/ACAT :
500 (460 + 59)/(460 120) = 473 BHP or
23 BHP more than that required for the same
performance density altitude under
standard conditions.
at
Unlike a conventional aircraft, rotor blade
stall occurs at high forward speed, but
actually it is due to the relatively low speed
and high angle of attack of the retreating
blade (Ref. 25). From this conception, the
same factors which affect stalling on a con-
ventional aircraft hold true on a helicopter.
With a given rotor speed, as forward speed
is increased, the retreating blade operates
at lower relative speeds, and higher lift
coefficients to maintain equilibrium, until
the airflow is stalled over the tip (Ref. 26).
If transmission efficiency is taken as
80%, the excess power developed for climb is
: HP
НР.
CT
+ .80BHP: 182+(.8)(23)=
182 + 18 = 200 hp.
The climb at 100°F at sea level becomes:
ROCI
200 x 33,000
5000
For a given forward speed, as rotor
speed is reduced, the same effect is pro-
duced. If the gross weight is increased,
generally higher lift coefficients will be
required precipitating an earlier stall. As
altitude is increased, the lower air density
will require operation at higher lift coeffi-
cients making the stall appear earlier. The
first appearance of a stall may be incon-
sequential, nor does it spread rapidly to
cause trouble as is the case with the con-
ventional aircraft.
• 1320 ft./min.
instead of the 1460 ft./min, obtained under
standard conditions at sea level. Continuing
the example for standard conditions, at
15,000 ft. P = 0015, Clm =[(55)6.0024))/(.0015)-
88, cd = .028, and engine power : 285 HP.
The profile drag increases as the stall
is approached, requiring more profile and
10:30
total power.
Depending upon the aerody-
namic characteristics of the airfoil section
in the stalled conditions, the aircraft or the
controls will become rough due to the periodic
loss of lift and change of aerodynamic
twisting moment on the blade. With flexible
blades of constant pitching moment, the
roughness signs may not be apparent, but
an unusually large increment of power and
control will be required for increased forward
speed.
altitude, and center of gravity location are
dependent variables and their relationship
should be established, At some sacrifice
of performance, the minimum RPM, and
extreme aft c. g. may be selected and the
maximum speed determined at several alti-
tudes.
Further increase in speed beyond these
conditions is not warranted, but safe opera-
tion at design gross weight should be possible:
Climb performance may be checked up
to 75% of the absolute ceiling. However,
the blades may stall at 85% without indications
from the low altitude test. The test data
should be checked up to service ceiling to
detect any decrease in slope of the rate of
climb vs. altitude curve, or by calculation,
using Bailey's methods of Ref. 20, to insure
maximum lift coefficient of the tip of the
retreating blade does not exceed Clmax
1.6 for smooth airfoil contours and CLmax
1.2 for poor airfoil contours.
(a) Up to the “Dive Test Speed,”
(b) Down to minimum rotor speed,
(c) Up to operating ceiling, and
10:13 CONCLUDING REMARKS
(d) With extreme aft location of the center
gravity.
During (a) the helicopter should be flown
at maximum rated rotor speed in level flight
up to maximum speed, then nosed over to
attain "Dive Test Speed". The test may be
repeated at 50% rated BHP, and again in
autorotation.
The foregoing test and analysis procedure
provides a means of establishing the heli-
copter possibilities and limitations. While
the evaluation of efficiency, induced, para-
site, and profile power values may not be
absolute, the parameters serve as a medium
of correlation for test data obtained under
conditions other than standard. The prin-
cipal items of performance may be presented
on a chart similar to those shown in this
chapter.
With power on, the minimum rotor speed,
9
10:31
REFERENCES
1.
Lightfoot, R. B., "CAA Testing of Helicopter," presented at the Annual Meeting of
the Institute of the Aeronautical Sciences, January, 1947.
2.
CAA Flight Engineering Report No. 17, “The Apparent Effect of Pilot Technique
and Atmospheric Disturbances upon the measured Rate of Climb of an Airplane."
3.
Deutsch, M. I., "Ground Vibrations of Helicopters,” Journal of the Aeronautical
Sciences, May, 1946, Vol. 13, No. 5.
4.
Horvay, G., “Vibrations of Helicopters on the Ground,” Journal of the Aeronautical
Sciences, November, 1946.
5.
Meyers, Gary, C. Jr., “Flight Measurements of Helicopter Blade Motion with a Com-
parison Between Theoretical and Experimental Results,” NACA Technical Note 265,
April, 1947.
6.
Specification MIL-1-6115 A.
7.
Civil Air Regulations, Part 03, “Airplane Airworthiness-Normal, Utility, Acrobatic,
and Restricted Purpose Categories."
8.
CAA Flight Engineering Report 11, “Methods for Determining Airplane Cruising
Range,” August 29, 1944.
9.
CAA Flight Engineering Report No. 3, “Airplane Climb Performance."
10.
Beerer, J. G., “The Reduction of Flight Test Performance Data to Standard Air
Conditions by the Temperature Altitude Method,” Journal of the Aeronautical Sciences,
October, 1946.
11.
Hamlin, B., “Flight Testing - Conventional and Jet Propelled Airplanes,” The Mac-
millan Company, New York, 1946.
12.
Glauert, H., “The Elements of Airfoil and Airscrew Theory," The Macmillan Company,
New York, 1943.
13.
CAA Flight Engineering Report 4, "CAA Equipment for Recording Airplane Take-Off
and Landing Characteristics."
14.
Civil Air Regulations, Part 06, “Rotorcraft Airworthiness."
15.
Specification AA-T-62.
16.
Tetervin, N., “Airfoil Section Data from Tests of Ten Practical Construction Sections
of Helicopter Rotor Blades,” submitted by Sikorsky Aircraft Division of United Aircraft
Corporation, NACA M. R. for AAF, September 6, 1944.
17.
CAA Flight Engineering Report 10, “Effect of Airplane Weight Upon Rate of Climb,"
September 15, 1943.
10:32
18.
NACA Advance Restricted Report 15E1D dated June 1945 by Coleman, R. P., Feingold,
A. V., and Stenyen, C. W., "Evalutation of the Induced Velocity Field of an Idealized
Helicopter Rotor."
19.
Bennett, J. A. J., "Rotary Wing Theory," Aircraft Engineering article taken from
“Aircraft Engineering," January through August, 1940.
20.
Bailey, F. J., "A Simplified Theoretical Method of Determining the Characteristics
of a lifting Rotor in Forward Flight," NACA T.R. No. 716.
21.
Knight, M. and Hefner, R. A., "Analysis of Ground Effect on the Lifting Airscrew,'
NACA T. N. No. 835, December, 1941.
22.
CAA Flight Engineering Report 12, "The Effect of Air Temperature Upon the Rate
of Climb of an Airplane Equipped with a Constant Speed Propeller," December 1, 1943.
23.
CAA Flight Engineering Report 13, “Altitude and Its Effect on Airplane Performance,”
June 13, 1944.
24.
CAA Flight Engineering Report 17, “The Effect of Power Upon the Calculated Airplane
Climb Performance," November 1, 1945,
25.
Gustafson, F. B., and Gessow, A., “Effect of Blade Stalling on the Efficiency of a
Helicopter Rotor as Measured in Flight,” Proceedings of Third Annual Forum of
American Helicopter Society, March 1947, or NACA T. N. No. 1250, April, 1947.
26.
>
Gustafson, F. B., and Gessow, A., “The Effect of Rotor-Tip Speed on Helicopter
Hovering Performance and Maximum Forward Speed,” NACA Wartime Report ARR
No. 16A16, March, 1946.
27.
Sissigh, G., "Contribution to the Aerodynamics of Rotating-Wing Aircraft," Technical
Memorandum 921, NACA, December, 1939.
28.
Wheatley, John B., "Aerodynamic Analysis of the Autogiro Rotor with a Comparison
Between Calculated and Experimental Results,"Report No. 487, NACA, January 16, 1934.
29.
Dommasch, D. O., "Elements of Propeller and Helicopter Aerodynamics," Pitman,
1953.
$
10:33
1
1
i
1
AGARD FLIGHT TEST MANUAL
VOLUME I, CHAPTER 11
THE EFFECT OF THE GROUND ON THE PERFORMANCE OF A HELICOPTER
By
I. C. Cheeseman
and
J. D. L. Gregory
Aeroplane & Armament Experimental Establishment
United Kingdom
C
VOLUME I, CHAPTER 11
CHAPTER CONTENTS
Page
TERMINOLOGY
11:1
INTRODUCTORY COMMENTS
11:1
11:2
THEORETICAL ANALYSIS FOR A SINGLE ROTOR HELICOPTER
11:1
(a) General
11:1
(b) Ground Effect at Zero Airspeed
11:1
(c) Ground Effect in Forward Flight
11:2
(d) Power Required in the Ground Cushion in Hovering
and in Forward Flight at Constant Weight
11:2
(e) The Effect of a Change of Blade Loading on the Ground Effect
at Constant Power
11:3
11:3
THEORETICAL ANALYSIS FOR TANDEM ROTOR HELICOPTERS
11:5
11:4
FLIGHT TESTS
11:6
(a) General
11:6
(b) Tests at Constant Weight
11:7
(c) Tests at Constant Power
11:7
COMPARISON OF THEORY AND EXPERIMENT
11:5
11:7
REFERENCES
11:9
TERMINOLOGY
A
Disc Area
b
Number of Rotor Blades
с
Chord at 0.7 Radius
CT
T/TPR (12R)?, Thrust Coefficient
Ст/с
Blade Loading Coefficient
E
Ratio of Effective Power at the Rotor to Total Engine Power
G
Vertical Separation of Rotors on Tandem Rotor Helicopter
k
Constant Defined in Section 11:2
P
Engine Power
PR
Power Required to Rotate the Rotor
R
Rotor Radius
S
Horizontal Separation of Rotor Centers on Tandem Rotor Helicopters
T
Rotor Thrust
T.
Rotor Thrust away from the Ground
Tg
Rotor Thrust in the Ground Cushion
u
Total Flow Normal to the Rotor Disc
UCO
Value of u away from the Ground
ug
Value of u in Ground Cushion
u
u'oo
Ratio of Normal Velocities through Rear and Front Rotors of Tandem Rotor
Helicopter
८
Velocity of Flow Induced by the Rotor Normal to the Rotor Disc
Voo
Value of v away from the Ground
vg
g
Value of v on Ground Cushion
T
VT
2 TOR?
TERMINOLOGY
(Continued)
Aircraft Speed
> N 6
Height of Rotor

General Angle Measured from Mean Direction of Flow when in Forward Flight
Defined in Section 11:2(C)
η
Nondimensional Constant Defined in Section 11:2
Tan" V/v = Mean Direction of Flow under the Rotor
0.
Mean Blade Collective Pitch Angle
Inflow Ratio : u/R
N
Р
Air Density
o
bc/1.4R+ Solidity Taken at 0.7 Radius
12
Rotor Angular Velocity
1
11:1
INTRODUCTORY COMMENTS
more nearly resembles the flow from a
source, and so the method of Betz in which
the sink is replaced by a source is used in
the zero airspeed case. This model is not
sufficiently detailed to show the effect of
forward speed and a development has been
made which is discussed below.
(b) Ground Effect at Zero Airspeed
The rotor energy equation at zero air-
speed may be written
The performance of a helicopter is al-
tered by its proximity to the ground. In
general the handling qualities of single rotor
helicopters are not altered, but some effects
have been experienced on tandem rotor
machines. The ability to operate close to
the ground at zero ground speed in different
wind speeds is important in certain opera-
tional roles. An approximate theoretical
method of estimating the performance of
helicopters at any height and airspeed has
been devised and compared with experimental
results. The theory has also been extended
to cover the case of tandem rotor helicopters
and this extension is now being checked by
flight tests.
Because the effect of the ground is de-
pendent on the induced flow at the rotor,
disc loading is an important parameter when
comparing helicopters in the same flight
condition, A secondary parameter is blade
loading.
EP - PR: Tv
11:1
where EP is the effective power at the rotor,
PR is the power required to rotate the rotor,
T the thrust, and v the induced velocity.
Keeping EP - PR constant, the thrust in-
side the ground cushion Tg is related to the
thrust outside the ground cushion To , by
O
Tg
Too
Voo
Vg
11:2
11:2 THEORETICAL ANALYSIS
FOR A SINGLE ROTOR
HELICOPTER
(a) General
where vg and Vos are the induced velocities
inside and outside the ground cushion re-
spectively.
If 8 Voo is the change in induced velocity
due to the presence of the ground, then
vg : VO
8 vo and hence
2
To
Top
1
SVO
VOO
1 -
11:3
Proximity to the ground changes the
induced flow through the rotor and this may
be mathematically represented by the method
of images. To apply this method, a model
of the flow beneath the lifting rotor has to
be developed. For various purposes several
such models have been devised (Refs. 1, 2,
and 3) but in all cases the induced flow builds
up to the theoretical value of twice the in-
duced flow at infinity. If these models are
used the value of the ground effect so ob-
tained is large when compared to experi-
mental values (Ref. 4).
A report by Betz (Ref. 5) suggested that
the flow around a rotor could be approxi-
mated by a three-dimensional hydrodynamic
sink. This indicated that the ground effect
for a hovering helicopter decreased with de-
crease in height. The flow beneath the rotor
To calculate 8 Vo , a three-dimensional
source having a strength of Avco/4 is
assumed, A being the rotor disc area. This
value is chosen so that the source has the
same flow per unit time as the induced flow
of the rotor. Hence the image source pro-
.
duces an upflow at the rotor of
AVDO
.
SVO
ž
16 TT
11:4
€
11:1
Thus by analogy to Eq. 11:2 at constant
effective power,
where Z is the height of the rotor above the
ground.
Substituting from Eq. 11:4 into Eq. 11:3,
the ground effect at zero airspeed is then
expressed by
Tg
um
ug
ܢ4-,
Too
δυ
-
11:7
Too
2
R
(2)
Hence
16
11:5
To
1
Too
From this equation the rapid fall-off in
ground effect with height is easily seen.
1- ito (
A
'+
(c) Ground Effect in Forward Flight
11:8
The simple model used above is not
sufficiently exact to show any change of
ground effect with forward speed. This is
because the directional properties of the
flow beneath a lifting rotor have a consid-
erable bearing on the ground effect.
This expression is plotted in Fig. 11:1
from which it is seen that the ground effect
as well as decreasing with height falls off
very rapidly with forward speed. Since it
is assumed that the disc inclination to the
horizontal is zero, a simple relationship
exists between V/V, and V/V, Using the
experimental data of Ref. 7, the figures are
presented in the more usual form with V/VT,
instead of Viv, as the nondimensional velocity
parameter.
a
An attempt to allow for this effect has
been made by arbitrarily assuming a source
with a strength of the form (k Au/47)cos
where a is the polar angle measured from
the line of symmetry, n? 1 and k is an
arbitrary constant, The line of symmetry
is assumed to be along the resultant flow
direction beneath the rotor.
(d) Power Required in the Ground Cush-
ion in Hovering and in Forward Flight at
Constant Weight
In many applications data on the power
1.2
To obtain the strength of the source, the
flow over a unit sphere is determined. From
this it is found that n must be an even in-
teger, the lowest possible value, namely
n = 2, being assumed. The total flow is then
adjusted by means of the constant k to give
equivalence in the case of V = 0 with Eq. 11:4.
To/T.1
O
Assuming that in low speed level flight
the rotor disc inclination to the horizontal
is small, then the change in induced velocity
su due to the image source can be written
VY
=0.5
VG
sc
1.0
= 2.0
SV
.
δυ :
Au cos20
16 z
T
1.0
0.5
1.0
1.5
2.0
2.5
3.0
Z/R
11:6
where 8 = tan (V/v).
Fig. 11:1 Variation of Thrust
with Height at a Given Airspeed
11:2
required to maintain a helicopter at constant
weight in the ground cushion is needed.
This is calculated from Eq. 11:8.
is therefore made by starting from the
blade element thrust formula
Too • Ź pobc SPR* (013-12)
.
2
11:9
Consider a helicopter of weight W flying
at speed V outside the ground cushion with
an effective rotor power of (EP-PR). If
this helicopter is now moved into the ground
cushion the same power will maintain a
weight of (Tg/T.) W where Tg/Tc is
given by Eq. 11:8. Let (EP-PRO be the
effective rotor power required to maintain
a weight of (Tg/TO) W at a speed V
outside the
ground cushion. Then
(EP-PR)g/(EP-PR), represents the effect
of the ground cushion on power at constant
weight.
where a is the lift curve slope of the blade
section, b the number of blades with mean
chord c, the rotor angular velocity,
o the mean collective pitch, and the inflow
ratio. Since only slow speeds are being
considered the terms in have been neg-
lected in Eq. 11:9.
Typical curves of (EP-PR),/(EP-PRIO
are shown in Fig. 11:2.
Taking the change in induced flow from
Eq. 11:6 an expression for the ground effect
is deduced as follows. We assume To
is given by the same expression as to
except that dg replaces 1. Then
(e) The Effect of a Change of Blade
Loading on the Ground Effect at Constant
Power
-
To-To-
pabc n°RY (1-1)
?.
From Eq. 11:8
In Ref. 6 a variation of ground effect
with the blade loading parameter CTO
is found. An estimate of the effect of vari-
ation of the ground effect with blade loading
2
ug
TO
R
::- (2) * (1-4
ok
*
U DO
Thus
V.0.6
| Pabc R3
X
2
: 1+
$ 0.8
Nie
(EP-PR'S
(EP-PRICE
-048
Too
16 () * (1+() 3)
ਚ)
1.0
We may express ^ (Ref. 7) as
$
+ 2.0
Too
WA
1 TV2 TPR (NR)?
:
AVI
NR
1.0
0.5
1.0
1.5
2.0
VINT
Fig. 11:2 Variation of Power
with Forward Speed at a Given Height
where is a nondimensional parameter
depending on V.
11:3
Then with
For any
Ст =
TO
TT PR?(2R)2
G :
bc
This expression indicates that the ground
effect varies with (CT) /o rather than
Co as suggested in Ref. 6.
particular helicoptero is a constant, and
the blade loading parameter is varied at
constant disc loading by varying the RPM,
Typical curves have been plotted in Fig. 11:3.
From these it is seen that an increase in
CT/
Co leads to a decrease in the ground
effect. In general however the blade loadings
of the helicopters have been in the neighbor -
hood of 0.1 and it has been found, for per-
formance estimates, that blade loading can
be neglected and Eq. 11:8 can be used.
OTR
Tg
.it
Too
0.25 navo
vom
16((71)
71+
SYCAMORE
Copy
0.0589
0.0692
0.0777
0.0912
W+
4600
5400
287
4600
250
5400
250
2
287
0
al

1.3
= 0.0589
0.0692
00777
0.0912
1.2
11
0.5
R
To/TO
1.1
Z
= 1.0
R
FOR ALL
얼
​VALUES AT
= 2.0
1.0
0.4
0.8
1.2
1.6
2.0
VIVA
Fig. 11:3 Variation of Tg/T with Speed and Blade Loading at a Given Height
11:4

11:3 THEORETICAL ANALYSIS
FOR TANDEM ROTOR
HELICOPTERS
case is replaced by a source and an image
source placed symmetrically beneath itself.
In this way any combination of gap and
stagger between the two rotors can
be
treated. Following through the analysis for
constant power the expressions for the change
of thrust on front and rear rotors deduced
are,
The simple model developed in 11:2(c)
can be used in connection with the tandem
rotor configuration. Each rotor in this
ola
= 0.2 , Ş= 2.0
= 0
Ř=
0.5 ,
= 1.8
1.25
1.25
NIE
0.5
-1.0
5
U OD
UM
1.20
1.20
1.15
1.15
U OD
=0
Z
R
0.5,
= 1.0
To/TC
To/TC
uoo
2 -0.5,
R
름
​che
.ها
هنا
JO-1.8
$
-1.0,
= 1.8
1.10
R
1.10
U...
근
​R
1.0,
= 1.0
uc
Z
R
= 1.0,
-1.0,
1.05
.
-2.00
름
​1.05
11
O: 1.0
U.
U - 1.8
& unge = 1.8
-2.0,
Uge: 1.0
<
-2.0,4
Nulac
u : 1.0
u'.
- 2.0, "=1.8
VOO
1.00
O
2
1
2
R
1.00
o
1
v
FRONT ROTOR
VT
REAR ROTOR
>>
Fig. 11:4 The Ground Effect on a Tandem Rotor Helicopter
11:5
uce
{Tool front
= 1 +
16
cos 20
(Z/R)2
u'on
cos2(6+Y)cos y
Z
{ + <
*****
2R)
In these expressions the value of
u uw is the ratio of the normal induced
velocity at the front to that at the rear
rotor and is introduced to allow for the
interference of one rotor with the flow
through the other. From Fig. 11:4 in
which u". /um takes the value of 1, and
1.8 it is seen that this effect is small and
in general this ratio can be taken to be
unity.
cos?
+
u'on
c.
u"
2.
NICE
+
GR
sto
s
Tool
=1+
reor
係​)
k
(+ (
cos? (0-4)Cosy
G12
ZR) 2R)
+
For tandem rotor helicopters, since the
ground effect on the two rotors is different,
there will also be a pitching moment asso-
ciated with the change of thrust, For a
helicopter with zero gap this results in a
nose-down pitching moment.
where
cos?0 = 1+ (WUZcos? (8 +4):
I
>
0
V/)?
V/
i +
v + [s/122+6)]
*{v^v}{22
S
2Z + G
This effect has been noticed by the pilots
during a cushion take-off as a sudden in-
crease in nose-down attitude necessitating
a backward cyclic control movement. This
effect is modified by fuselage attitude, Figs.
11:4 and 11:5 being drawn for G/R = 0.2
and 0 respectively. The method of cal-
culation given here, however, enables
estimates of the change of trim due to the
ground effect, with any combination of gap
and stagger and fuselage attitude, to be
determined.
22 + G
cosy
{s* + (22+ G)2
}
GPS
1.125
REAR ROTOR THRUST
11:4 FLIGHT TESTS
(a) General
1.100
FRONT ROTOR
THRUST
T/T 1.075
There are two principal ways of con-
ducting flight tests, namely, at either constant
weight or constant power.
The former
test method is preferred if a torquemeter
or some other accurate power measuring
device is available. Care should be taken
not to operate in turbulent conditions and
in particular, regions of thermal activity
should be avoided.
= 0
1.050
pero
-
름
​: 2.0
0.8
R
1.025
O
0.4
0.8
1.2
1.6
V/V,
The test techniques described are ap-
plicable to both tandem and single rotor
helicopters. For tandem rotor machines the
fuselage attitude should be maintained con-
stant during each test by use of the dif-
ferential collective pitch trimmer. The
Fig. 11:5 The Ground Effect
on a Tandem Rotor Helicopter
11:6
experiments should be repeated for a range
of fuselage attitudes.
(b) Tests at Constant Weight
in the ground cushion. The helicopter is
then hovered as near to the ground as
possible. While the helicopter is main-
tained at constant power the ballast is
progessively jettisoned and the new height
measured photographically. The ballast
is loaded in a manner that keeps the heli-
copter center of gravity as constant as
possible. By plotting all up weight of the
helicopter against height, the value of To
is giveh by the asymptote to the curve.
The helicopter is hovered at zero ground
speed as near to the ground as possible.
By increasing the power in small steps
the helicopter is hovered at different heights
which are measured photographically with
a ground mounted camera.
The power
required at each height is obtained either
from torquemeter readings or, if a cali-
brated engine is installed, from boost and
RPM readings. In this case the normal
allowances for tail rotor and gear box
power losses have to be made.
For some helicopters the disposable load
is insufficient to cover the height of the
ground cushion, in which case, after the
initial test, the helicopter is reballasted
and hovered at a new power setting at
height just below that reached in the pre-
vious test. In this case the power required
is adjusted to the value used in the final
trip by an appropriate weight correction.
The experiment is then repeated with the
helicopter in forward flight, the airspeed
and wind gradient being obtained as pre-
viously suggested.
Unless the tests are made in still wind
conditions the wind speed at various heights
above the ground must be determined. This
can be conveniently done by flying the heli-
copter, with and against the wind at a
constant indicated airspeed, past the
ground mounted camera. From the photo-
graphic record the ground speed is deter-
mined and knowing the aircraft instrument
position error the true wind speed at any
height is determined. An anemometer is
also mounted about six feet above the ground
to provide surface wind data.
!
In this test the blade loading of the
helicopter varies, but in general this
variation is small and can be neglected.
An estimate of the variation is obtained
from 11:2(e).
11:5 COMPARISON OF THEORY
AND EXPERIMENT
In order to cover the range of forward
speeds the helicopter is flown past the ground
camera and the photographic record used
to determine both height and ground speed.
The results obtained are plotted as
(EP-PR)g/(EP-PR). against V and Z/R.
If the effect of the blade loading coeffi-
cient is
required the tests should be
repeated at the same weight but different
rotor RPM.
In Fig. 11:6 the theoretical curves ob-
tained from 11:2(c) have been plotted with
a series of experimental points obtained
by the methods described in 11:4 on a
variety of single rotor helicopters. It is
seen that
agreement is good for Z/R
greater than 0.6 but that the theory over-
estimates the ground effect for lower values
of ZR.
(c) Tests at Constant Power
For this test the helicopter is loaded
with jettisonable ballast to the maximum
gross weight at which it will just hover
Insufficient tests have as yet been made
on tandem rotor helicopters to enable a
comparison between theory and experiment
to be obtained.
11:7
1.2
EXPERIMENTAL
RESULTS
0.4
V
-=0
V,
:
0.5
V,
To/Too 1.1
> is is
: 1.0
>1>
!
0.5
V
: 2.0
V,
>>
1.
` = 3.0
1.0
-0-0-0
: 2.0
BI V
1.0
0.5
1.0
1.5
2.0
2.5
3.0
Z/R
Fig. 11:6 Variation of Thrust with Height at a Given Airspeed
11:8
REFERENCES
1.
Castles and DeLeeuw, "'The Normal Component of the Induced Velocity in the Vicinity
of a Lifting Rotor and Some Examples of its Application," NACA TN 2912, March 1953.
2.
.
Knight and Hefner, "Static Thrust Analysis of the Lifting Airscrew," NACA TN 626,
December 1937.
3.
Mangler, “Calculation of the Induced Velocity Field of a Rotor," Unpublished British
M.O.S. Report.
4.
Knight and Hefner, "Analysis of the Ground Effect on the Lifting Airscrew," NACA
TN 835, December 1941.
5.
Betz, “The Ground Effect on Lifting Propellers," NACA TN 836, April 1937.
6.
Zbrozek, "Ground Effect on the Lifting Rotor," British R&M 2347, July 1947.
7.
Oliver, “The Low Speed Performance of a Helicopter," British A. R. C. Current
Paper 122, May 1952.
11:9
AGARD FLIGHT TEST MANUAL
VOLUME I, CHAPTER 12
THE TRANSITION PERFORMANCE OF A HELICOPTER
FOLLOWING A SUDDEN LOSS OF POWER
Ву
I, C. Cheeseman
and
G. F. Langdon
Aeroplane and Armament Experimental Establishment
United Kingdom
VOLUME I, CHAPTER 12
CHAPTER CONTENTS
Page
TERMINOLOGY
12:1
INTRODUCTORY COMMENTS
12:1
12:2
THEORETICAL ANALYSIS FOR SINGLE-ENGINE HELICOPTERS
12:1
(a) Engine Failure during Vertical Flight outside the Ground Cushion
12:3
(b) Engine Failure in Vertical Flight inside the Ground Cushion
12:3
(c) Engine Failure in Forward Flight
12:5
12:3
MULTI-ENGINE HELICOPTERS
12:6
12:4
TEST TECHNIQUES
12:7
(a) Single-Engine Helicopters
12:7
(b) Multi-Engine Helicopters
12:8
REFERENCES
12:10
TERMINOLOGY
Slope of Blade Lift Coefficient Curve
a
Forward Acceleration
A
Power Required to Hover
b
Number of Blades
B
Power Available with One Engine Inoperative
с
Rotor Blade Chord at 0.7 Radius
Ср
Blade Profile Drag Coefficient at the Mean Effective Lift Coefficient
Fuselage Drag
DI
V2
f
T
2 TP R? v?
f
مها
IVE
Vertical Flight Coefficients
F
T
2 TP RU?
F,
+ F
Acceleration Due to Gravity
60
h
CD 8
H
Height Loss
Hi
á Co DCPR
i
Rotor Disc Incidence to Flight Path Positive Upward (rotor angle of attack)
J
Rotor Moment of Inertia
k
uf
AR
K
号​+》
)
F
2
m
Helicopter Mass
TERMINOLOGY
(Continued)
R
Rotor Radius
t
Time
T
Rotor Thrust
u
Resultant Airflow Normal to the Rotor
V
Induced Flow through the Rotor
V
Flight Speed
C
V V
x' z
Components of the Flight Speed in Direction Ox, OZ
V.
(
}
mg
127 PR2
X,Z,
Space Coordinates; x, Horizontal Forward; z, Vertical Downward
P.
Engine Power at Sea Level
n
Number of Engines
E
Efficiency Factor
a
Rotor Disc Attitude to the Horizontal, Positive Upward
х
A Constant
Blade Pitch
do
0
d.
mR"
H
外
​NR
P
Air Density
PO
Air Density at Sea Level
91
T
Ω
Rotor Speed
12:1 INTRODUCTORY COMMENTS
For safe operation of a helicopter it is
important to know the height and speed
bands in which an engine failure may result
in a crash landing. For single-engine heli-
copters engine failure is automatically fol-
lowed by a landing, but for multi-engine
machines, the ability to continue flight in
various situations, such as during take-off
and landing from restricted sites, must be
assessed. Methods of estimating the per-
formance of a helicopter following sudden
loss of power have been developed in order
to gain some preliminary knowledge of the
performance of the helicopter prior to flight
trials and also to enable a comparison to
be
drawn between different helicopter
designs.
pilot can utilize the forward speed to im-
prove the gliding angle, and to maintain
the rotor speed. However, during the flare
out and landing the forward speed should
be not more than 10 knots for a wheeled
undercarriage in order to safeguard the
crew when operating over rough terrain.
There are thus regions in the height-speed
plane in which an engine failure may easily
result in a crash landing, a typical curve
being shown in Fig. 12:1.
In attempting to estimate the transition
performance of a helicopter it is difficult
to allow for the wide variety of possible
pilot's control movements and the resulting
response of the helicopter. A simplifying
assumption has been made, namely that for
each stage of the maneuver the pilot main-
tains a constant disc attitude to the horizontal
and constant collective pitch.
12:2 THEORETICAL ANALYSIS
FOR SINGLE-ENGINE
HELICOPTERS
For axes fixed in space with OX hori-
zontal forward and OZ vertical downward
and assuming the fuselage drag to be
D, ve and the rotor drag to have the ap-
proximate form H, VR, the equations of
motion are
dVx
m
: - Tsina -D, VVx-H, V S R cos a
dt
Following total power failure there are
three main cases to be considered. If
power failure occurs near the ground in
vertical flight, the pilot of existing types
of helicopter has insufficient time to make
any major control changes.
. Hence, the
helicopter drops back to the ground, the
rate of descent being restricted by the util-
ization of some of the kinetic energy stored
in the rotor. The maximum safe height
from which a landing can be made is deter-
mined by the energy absorbing properties
of the undercarriage.
12:1
m
?
dVz
dt
mg - T cos Q - D, VV2 + H, VSR sina
12:2
d12
J
.
Tu
Ω
πρσ R5
8
con?
dt
12:3
and in
If engine failure occurs in vertical flight
when well clear of the ground, the pilot is
assumed to reduce the collective pitch in
an attempt to maintain the rotor speed at
the expense of increasing the rate of descent
and then to make a flare out and landing in
which the rotor energy is used to reduce
the rate of descent to a value which can be
absorbed by the undercarriage.
It can be shown as in Ref. 1, that
T/(2TP R? V?) and u/V are functions of i
thus enabling the equations to be
solved for any compatible set of initial
conditions. Since the relationship for
T/(2TPR? V?) and u/V in terms of i and
is known only graphically in the low speed
region (Ref. 2) the solution is obtained by
a step-by-step integration process.
In forward flight there are cases cor-
responding to the above limits but here the
Normally the rotor drag terms in Eqs.
12:1 and 12:2 can be neglected.
12:1
500

400
AVOID FLIGHT IN
SHADED AREAS
300
HEIGHT, FEET
200
100
o
50
100
150
AIR SPEED, KNOTS
Fig. 12:1 Typical Unsafe Height Band
2.0
VORTEX RING
PROPELLER
STATE
1.0
WORKING
STATE
O
WINDMILL
BRAKE
-1.0
STATE
4:v (240R-++
&•v/(
2 }
2 πρR
τ
Este
-2.0
-2.0
-1.0
O
1.0
Fig. 12:2 Vertical Flight Coefficients
12:2
(a) Engine Failure during Vertical Flight
outside the Ground Cushion
a step increase in the collective pitch setting.
A point is finally reached at which the rate
of descent is reduced to a figure which can
be absorbed by the undercarriage but still
keeping the rotor RPM inside its lower limit.
The equations of motion for this case
take the simpler form since a and Vy are
zero. The equations can be written:
%E
.
mg-T
dt
12:4
.
d12
J
C
Tu
2
PORS
con.
8
By adding together the height lost in the
three stages of the calculation an estimate
of the safety height is found. So far no
allowance has been made for the ground
effect. This is a permissible assumption
during the first two stages of the calculation,
but must be considered during the flare
out. Curves showing the variation of thrust
with height are given elsewhere in this
manual and are also given in Ref. 3. The
modified thrust value is fed into the step-
by-step calculation.
12:5
In vertical flight there is a unique re-
lationship between T/(2*2 R? V?) and u/V
since i is a constant and this is normally
expressed in terms of the rotor coeffi-
cients f, and F. The empirical connection
between these coefficients is shown in
Fig. 12:2. Using the blade element formula
the following relationship is derived:
(b) Engine Failure in Vertical Flight
inside the Ground Crishion
*****
ह 을
​8
3
]
12:6
where y = (a,/8), and is used with Eqs. 12:4
and 12:5 and Fig. 12:2 to determine the
solution for a collective pitch setting 8.
The calculation of this height boundary
is most important because of the difficulty
of obtaining the results experimentally
without severely damaging the test heli-
copter. Some experiments were carried
out in Great Britain using obsolete heli-
copters (Ref. 4) and these confirmed the
rapid increase in rate of descent with height
at which engine failure occurred.
These equations are not exact since no
allowance is made for the rotor operating
in its own downwash. While it is mathe-
matically possible to make such an allowance
there is little information on the value of
this variable. The solution of these equations
is still of use since it represents the case of
a near vertical descent which is what is nor.
mally achieved in practice.
On existing types of helicopters it has
been found that the time available between
engine failure and touch down is probably
too short for the pilot to make any control
movements. The motion is therefore cal-
culated using Eqs. 12:4, 12:5, and 12:6,
assuming that the collective pitch remains
at its initial value. Allowance is made
for the ground effect as indicated above,
Following engine failure it is generally
assumed that no control action is taken for
a period corresponding to the pilot's re-
action time and for the adjustment of the
controls to become effective. Following
this time (assumed to be of the order of
two seconds) the collective pitch is assumed
to have its autorotative value. Flare outs
are calculated at different parts of the
transition to steady autorotation by assuming
On the evidence of the Hoverfly tests
this method overestimates the safe height,
An approximate method has been given in
Ref. 4 but this underestimates the height,
and therefore is not recommended for final
estimation but is useful for comparative
performance estimates. To simplify the
equations it is assumed that pf, is a constant
12:3
2
2
30
250
NORMAL MAX.
200
NORMAL MIN.
20
OBSERVED
R.P.M.
BLADE PITCH, ROTOR SPEED,
150
O
2
3
15
ROTOR SPEED, RADIANS/SEC.
ACCURATE NUMERICAL
SOLUTION
10
10
5
O
2
3
APPROXIMATE NUMERICAL
SOLUTION
30
20
ACCELERATION,
FT./SEC.?
1
TIME FROM ENGINE CUT, SECONDS
10
O
Fig. 12:4 Rotor Speed at Constant Pitch
o
2
3
20
15
10
RATE OF DESCENT,
FT./SEC.
5
15
O
2
APPROXIMATE NUMERICAL
SOLUTION
3
15
RATE OF DESCENT, FT./SEC.
10
OBSERVED
10
HEIGHT LOST,
FT.
5
5
5
3
2
TIME FROM ENGINE CUT, SECONDS
ACCURATE
NUMERICAL SOLUTION
Fig. 12:3 Performance of a Single Rotor Helicopter with
Pitch Maintained at Hovering Value until Touch Down
(engine failure near the ground)
1
TIME FROM ENGINE CUT, SECONDS
Fig. 12:5 Rate of Descent at Constant Pitch
12:4
k, determined from the hovering conditions.
The equations can then be solved to give
so
-k Novor?
12=
, V:
К по +1
KNOT+1
The ground effect may be allowed for
by estimating a mean value from the curves
of Ref. 3 and using this to modify the value
of k. Comparison of these results with
the flight tests of Ref. 4 is shown in
Figs. 12:3, 12:4, and 12:5.
T
12:7
where
(c) Engine Failure in Forward Flight
Kamp
к
(ht
2
Cases similar to those considered in
earlier paragraphs of this section exist in
forward flight. There is now the added
complication that the pilot's control move-
ments are not uniquely defined. In Ref. 1
an investigation into the effect of control
movements on the height lost was made.
and the other symbols are as defined in the
accompanying table.
RATE OF DESCENT,
FEET/SEC.
FORWARD SPEED,
FEET/SEC.
ܘ̄ ܘ̄ ܘ̄ ܘ
o
RATE OF DESCENT,
FEET/SEC.
FORWARD SPEED,
FEET/SEC.
2 4
FOR DISC ATTITUDE Q: -5°
TIME, SEC.
8
1
9
O
01
20
30
130
o
o
o ā ģ
26
27
28
29
a: +7°
TIME , SEC.
2 4 6
HEIGHT LOST,
FEET
ROTOR SPEED,
RADIANS/SEC.
50
100
150
200
26
27
28
29
o
o
8
o
50
001
100
200
26
27
28
29
ROTOR SPEED,
HEIGHT LOST,
FEET
2
TIME, SEC.
4 6
RADIANS/SEC.
8
o
영
​ở
RATE OF DESCENT,
FEETI SEC.
FORWARD SPEED,
FEET/SEC.
Fig. 12:6 Estimated Transition Performance, Effect of Control Movements
12:5
Estimates were made for a helicopter as-
sumed to be flying initially at 112.6 ft./sec.
Three different disc attitudes (a : -5 de-
grees, 0 degrees, and +7 degrees)were as-
sumed and the performance calculated from
Eqs. 12:1, 12:2, and 12:3.
If an engine fails while the helicopter is
flying outside this speed range the speed
must be increased or decreased if the flight
is to continue and while this is done some
loss of height is inevitable, This height
loss determines the take-off and landing
technique to be used when operating from
restricted sites and the minimum size of
the landing ground required for safety if
an engine fails at any point on the flight path.
These results are shown in Fig. 12:6.
From these it is seen that the steady con-
ditions are achieved more rapidly and with
a smaller height loss by adjusting the control
so that the helicopter approaches its best
steady gliding speed. To complete the cal-
culation it is necessary to calculate the
height lost in making a flare out, the end
condition being a rate of descent which can
be absorbed by the undercarriage and a
forward speed less than 10 knots.
Fig. 12:7, due to Hafner (Ref. 6), shows
how the single engine performance affects
the take-off technique.
The behavior of the helicopter after one
engine has failed can be calculated by in-
tegration of the equations of motion if the
control actions are specified.
12:3 MULTI-ENGINE HELICOPTERS
These equations
equations can
written, (Ref. 5),
conveniently be
The case of a helicopter with more than
one engine is rather different from that of
a single-engine machine as complete power
failure is unlikely to occur. Even in this
case, however, it may still be possible to
make an autorotative landing.
dV
m og -H
A:T sin 2 - D, VVx-H, VSR cos a
d
dVz
= (mg - Tcos a)-0, VV2 + H, VSR sin a a
+
dt
In some cases the aircraft will not be
able to hover with one engine inoperative
although it may be able to climb within
some restricted range of forward speed.
.J
d2
dt
EPon
по
[] - - PCOSRºs?
P
peces
NORMAL
TAKE-OFF PATH
TAKE-OFF PATH IF
ONE ENGINE FAILS
ABOVE CRITICAL
HEIGHT.
ܠܕ€
२०
RETURN PATH IF ONE ENGINE
FAILS BELOW CRITICAL HEIGHT.
Fig. 12:7 Take-Off Technique for Restricted Sites
12:6
The work needed to accelerate the heli-
copter is
The last two terms, representing the
induced and parasitic torque and blade profile
torque respectively, are taken as the sum
of their values for each lifting rotor in the
case of multi-rotor machines.
2
mys. mx (-2)
mv
' .
B
A
The work done by gravity is mgH, and
the work done by the engine is
B
XB (-)
87,
.
These equations must be integrated by
some step-by-step method which is labo-
rious. The accuracy of prediction depends
on the degree of approximation to which
the control movements are estimated. As
in the single-engine case, it is usual to
assume that after an arbitrary pilot reaction
time of two seconds the collective pitch is
reduced to the value appropriate to steady
flight with one engine failed and that during
the transition to the final forward speed the
disc attitude a is maintained constant.
Since the sum of these various work
items must be zero
X
H:
29
(-2) (4 m2 +1)
.
A
B
А.
ܘܬܐ)
An alternative approach based on energy
balance has been made in Ref. 6.
Consider the case of a helicopter which
can climb with one engine failed but which
has insufficient power to hover in this con-
dition. Let the power required to hover be
A and the power available with one engine
out be B. When one engine fails while the
aircraft is hovering the pilot is assumed to
take action to accelerate the helicopter to
climb-away speed. Then assuming that the
power required varies linearly with speed
up to the speed at which height can be main-
tained on one engine, power required
= A ( 1 - V/X) where X is a constant which
can be found from the power curve for the
aircraft.
A test program carried out
out using a
single-engine helicopter fitted with a device
which enabled power to be reduced suddenly
(Ref. 7) showed that the energy method is
simple and quick and gives a fair estimate
of the order of height loss to be expected.
Step-by-step integration of the equations
of motion entails considerably more work
but makes possible the comparison of dif-
ferent control techniques.
12:4
TEST TECHNIQUES
(a) Single-Engine Helicopters
Thus
B
:
1
- Å ; t.
(%) (-2).
If the forward acceleration is a constant,
a, then the total work required for sustenta-
tion during the time t, in which the helicopter
reaches the speed V, is
(1) Simulated engine failure near
the ground. An estimate of the height from
above which a safe landing using rotor energy
is not possible can be made as part of the
handling assessment of the aircraft. It is
obviously not possible to determine accu-
rately this height experimentally for each
new type because of the risk of damaging
the helicopter, however a series of tests
using obsolete aircraft has been carried
out (Ref. 4) with the results indicated in
section 12:2 (b).
B2
S'ali-*) at
dt :
АХ
20
(-).
12:7
(2) Simulated engine failure at alti-
tude. A fair idea of the height required to
establish steady autorotation can be obtained
from visual observations of the rotor speed
indicator and altimeter during sudden transi-
tions to autorotation from powered flight at
various speeds. To make a complete in-
vestigation an auto-observer or continuous
trace recorder must be fitted to record
control positions, accelerations, speeds, al-
titude, and attitude. Pilot reaction time may
have a critical effect on the behavior of
helicopters with low inertia rotor systems
and the effect of a delay before corrective
action is taken should be investigated. Tran-
sitions in which the horizontal speed is
both increased and decreased from the initial
flight speed should be made and the optimum
technique at each speed determined.
The first stage is to establish the steady
rate of climb or descent with one engine
throttled back and disengaged from the rotors.
To do this, partial climbs or descents are
carried out over as much of the forward
speed range as possible, using standard per-
formance methods. It is useful to carry
out this test with the operative engines
running at both the five minute power and
maximum continuous ratings. Although the
.
performance should not depend on which
engine it is decided to disengage, it may
be advisable to carry out tests with each
engine throttled back in turn to check on
the cooling of the engines in use.
C
C
Typical experimental results obtained on
Hoverfly I aircraft (Ref. 1) are shown in
Figs. 12:8, 12:9, and 12:10.
When the steady performance is known,
the loss of height following an engine failure
during level flight can be found by disen-
gaging one engine as quickly as possible
and using an auto-observer to record control
movements, rotor speed, altitude, airspeed
and accelerations. At this stage it should
be possible to judge the effect on fall-off
of rator speed of a delay before taking
control action after failure of one engine
and to decide whether an engine failure
warning indicator is necessary.

(b) Multi-Engine Helicopters
The test program is designed to discover
the performance following engine failure in
various conditions of flight.
There remains the question of behavior
after an engine failure during take-off or
landing. If the earlier tests show that the
helicopter is capable of hovering with one
engine out and the remainder at emergency"
1
K TIME AT WHICH ENGINE WAS CUT
STICK POSITION
ba
2.5
FWD
o
AFT
2.5
STICK
POSITION,
INCHES FROM
CENTRAL
300
COLLECTIVE
PITCH,
DEGREES
COLLECTIVE PITCH
0 0
60
50
200
AIRSPEED
AIR SPEED,
KNOTS
40
HEIGHT LOST IN REATTAINING
INITIAL ROTOR SPEED (236 R.PM)
250
30
ROTOR SPEED
HEIGHT LOST, FEET
SPEED,
ROTOR
R.P.M.
225
100
200
NORMAL ACCELERATION
200
9
NORMAL
ACCELERATION,
O
HEIGHT
LOST,
FEET
HEIGHT LOST IN ATTAINING
APPROX. STEADY RATE OF DESCENT
100
HEIGHT_LOST
o
2
8
10
6
TIME, SECS.
o
20
30
40
50
60
INITIAL AIR SPEED IN LEVEL FLIGHT, KNOTS
Fig. 12:8 Transition to Autorotation
after Engine Cut in Level Flight
Fig. 12:9 Height Lost in Transition
C
12:8
240
INITIAL ROTOR SPEED IN LEVEL FLIGHT
230
0_
ROTOR SPEED AT TIME OF ATTAINING
STEADY RATE OF DESCENT
investigate the effect on the control of the
aircraft of failure of one engine and the
effect of opening the others to full power.
If, as will probably be the case with a
twin-engine machine, the helicopter cannot
hover with one engine failed some such
flight path as that shown in Fig. 12:7 will
have to be adopted in the event of an engine
failure during take-off. Since the shape
of this flight path will determine the size
of the site and the obstruction line for safe
operation it is important that it should be
measured accurately.
220
ROTOR SPEED, R.P.M.
0
MINIMUM ROTOR SPEED
DURING TRANSITION
210
NORMAL MINIMUM PERMISSIBLE
ROTOR SPEED
200
190
20
30
40
50
60
INITIAL AIRSPEED IN LEVEL FLIGHT, KNOTS
Flight tests have been carried out in
Great Britain in which one engine of a
twin-engine helicopter was throttled back
suddenly during vertical and backward take-
offs and the flight path during the subsequent
transition to forward flight recorded by
kine-theodolite. An auto-observer and
Hussenot continuous trace recorders were
fitted in the aircraft. Full results of these
tests are not
not yet available, however it
appears that auto-observer records of altim-
eter and A.S.I. readings may yield a suf-
ficiently accurate flight path shape, as shown
by the comparison of one actual flight path
measured by theodolite with that deduced
from auto-observer instruments (Fig. 12:11).
Fig. 12:10
Variation of Rotor Speed
in Transition
or "take-off" rating, it will be possible to
make a safe landing under these conditions
from any height and it is only necessary to
400
0
HEIGHT, FEET
200
O FROM AUTO-OBSERVER
FROM KINE THEODOLITES
400
800
1200
HORIZONTAL DISTANCE, FEET
Fig. 12:11 Flight Path after Single Engine Failure
12:9
REFERENCES
1.
O'Hara and Mather, “The Performance after Power Failure of a Helicopter with
Blade Pitch Control, Parts I and II," British R. & M. 2797.
2.
Oliver, “The Low Speed Performance of a Helicopter," British A.R.C. C.P. 122.
3.
Cheeseman and Bennett, “The Effect of the Ground on a Lifting Rotor,'' Unpublished
British M. O. S. Report.
4.
Wilkinson, “The Performance of a Single Rotor Helicopter Following Power Cut
whilst Hovering. Flight Tests with a Hoverfly I near the Ground," Unpublished British
M. U. S. Report.
5.
.
Oliver, “The Performance of a Multi-Engine Helicopter Following Failure of One
Engine during Take-Off or Landing," Unpublished British M. O, S, Report.
6.
Hafner, "The Domain of the Helicopter," Seventh Bleriot Lecture, Journal of the
Royal Aeronautical Society, October 1954.
7.
Langdon, “An Experimental Investigation into the Performance of a Helicopter Fol-
lowing Sudden Reduction in Power," Unpublished British M. O. S. Report.
را
C
Air Force-USAFE, Wsbn, Ger- 56-2628
12:10
APPENDIX I, PARTI
ICAN STANDARD ATMOSPHERE
Temperatures
Pressures*
PP
Altitude
(feet)
ICAN Standard
ос
°C Abs.
psi
Absolute
Speed**
of sound
ft.per sec.
Millibars
p/Po
0
1,000
2,000
3,000
4,000
5,000
1.000
0.985
0.971
0.957
0.942
0.928
15.0
13.0
11.0
9.1
7.1
5.1
288.0
286.0
284.0
282.1
280.1
278.1
1,013.2
977.1
942.1
908.1
875.1
843.0
14.70
14.17
13.66
13.17
12.69
12.23
1.000
0.964
0.930
0.896
0.864
0.832
1,117
1,113
1,109
1,105
1,102
1,098
6,000
7,000
8,000
9,000
10,000
0.914
0.900
0.887
0.873
0.859
3.1
1.1
-0.8
-2.8
-4.8
276.1
274.1
272.2
270.2
268.2
812.0
781.8
752.6
724.3
696.8
11.78
11.34
10.92
10.51
10.11
0.801
0.772
0.743
0.715
0.688
1,094
1,090
1,086
1,082
1,078
11,000
12,000
13,000
14,000
15,000
0.846
0.833
0.819
0.806
0.793
-6.8
-8.8
-10.7
-12.7
-14.7
266.2
264.2
262.3
260.3
258.3
670.2
644.4
619.4
595.2
571.8
9.72
9.35
8.98
8.63
8.29
0.661
0.636
0.611
0.588
0.564
1,074
1,070
1,066
1,062
1,058
16,000
17,000
18,000
19,000
20,000
0.780
0.768
0.755
0.742
0.730
-16.7
-18.7
-20.6
-22.6
-24.6
256.3
254.3
252.4
250.4
248.4
549.1
527.2
506.0
485,5
465.6
7.97
7.65
7.34
7.04
6.75
0.542
0.520
0.499
0.479
0.460
1,054
1,050
1,046
1,041
1,037
21,000
22,000
23,000
24,000
25,000
0.718
0.705
0.693
0.681
0.669
-26.6
-28.6
-30.5
-32.5
-34.5
246.4
244,4
242.5
240.5
238.5
446.4
427.9
410.0
392,7
376.0
6.48
6.21
5.95
5.70
5.45
0.441
0.422
0.405
0.388
0.371
1,033
1,029
1,025
1,021
1,017
ICAN standard conditions
** ICAN standard conditions.
tional to T.
For conditions other than ICAN, the speed of sound is propor-
A-1
APPENDIX I, PART I
ICAN STANDARD ATMOSPHERE
Temperatures
Pressures*
Speed**
Altitude
(feet)
volpo
ICAN Standard
OC
OC Abs.
psi
Absolute
Millibars
of sound
ft.per sec.
p/PO
26,000
27,000
28,000
29,000
30,000
0.658
0.616
0.635
0.623
0.612
- 36.5
-38.5
-40.4
-42.4
-44.4
236.5
234.5
232.6
230.6
228.6
359.9
344.3
329.3
314.9
300.9
5.22
4.99
4.78
4.57
4.36
0.355
0.340
0.325
0.311
0.297
1,012
1,003
1,004
999
995
31,000
32,000
33,000
34,000
35,000
0.601
0.589
0.578
0,568
0.557
-46.4
-18.4
-50.3
-52.3
-54.3
226.6
224.6
222.7
220.7
218.7
287.5
274.5
262.0
250.0
238.4
4.17
3.98
3.80
3.63
3.46
0.284
0.271
0.259
0.247
0.235
991
987
982
973
973
-56.3
-56.5
216.7
216.5
969
968
36,000
37,000
38,000
39,000
40,000
0.546
0.533
0.521
0.508
0.496
227.3
216.7
206.5
196.8
187.6
3.30
3.14
2.99
2.85
2.72
0.224
0.214
0.204
0.194
0.185
41,000
42,000
43,000
44,000
45,000
0.485
0.473
0.462
0,451
0.440
178.8
170.4
162.4
154.8
147.5
2.60
2.47
2.35
2.24
2.14
0.176
0.168
0.160
0.153
0.146
46,000
47,000
48,0000
49.000
50,000
0.430
0.419
0.409
0.400
0.390
140.6
134.0
127.7
121.7
116.0
2.04
1.94
1.85
1.76
1.68
0.139
0.132
0.126
0.120
0.115
ICAN standard conditions
** ICAN standard conditions.
tional to ✓T.
For conditions other than ICAN, the speed of sound is propor-
A-2
APPENDIX I, PART II
PROPERTIES OF THE NACA
STANDARD ATMOSPHERE
h,
T,
o
Vc,
v X 104
(1000 ft.) | °F, Abs.
P,
P.
in. Hg. slugs per cu.ft.
mph
ft. per sec.
0
ܘܫ ܘܬ ܚ ܛ
2
3
4
5
518.4
514.8
511.3
507.7
504.1
500.6
29.92
28.86
27.82
26.81
25.84
24.89
0.002378
.002309
.002242
.002176
.002112
.002049
1.0000
0.9710
.9428
.9151
.8881
.8616
1.0000
0.9854
.9710
9566
.9424
.9282
760.9
758.3
755.7
753.0
750.4
747.7
1.567
1.604
1.644
1.684
1.725
1.768
Onda
497.0
493.4
489.9
486.3
482.7
23.98
23.09
22.22
21.38
20.58
,001988
.001928
.001869
,001812
.001756
.8358
.8106
.7859
7619
.7384
9142
.9003
.8865
.8729
.8593
745.1
742.3
739.7
737.0
734.3
1.812
1.857
1.905
1.954
2.004
10
11
12
13
14
15
479,2
475.6
472.0
468.5
464.9
19.79
19.03
18.29
17.57
16.88
.001702
.001648
.001596
.001545
.001496
.7154
.6931
.6712
.6499
.6291
.8458
.8325
.8193
.8062
.7931
731.6
728.8
726.1
723.4
720.6
2.055
2,109
2.165
2.223
2.281
16
17
18
19
20
416.3
457.8
454.2
450.6
447.1
16.21
15.56
14.94
14.33
13.75
.001448
.001401
.001355
.001311
.001267
.6088
.5891
.5698
.5509
.5327
.7803
.7675
.7548
.7422
.7299
717.8
715.0
712.2
709.4
706.6
2.342
2.405
2.471
2.537
2.608
21
22
23
24
25
443.5
439.9
436.4
432.8
492.2
13.18
12.63
12.10
11.59
11.10
.001225
,001183
.001143
.001103
.001065
.5148
.4974
.4805
.4640
.4480
.7175
.7053
.6932
.6812
.6693
703.8
701.0
698.1
695.3
692.4
2.680
2.756
2.834
2.916
3.000
26
27
28
29
30
425.7
422.1
418.5
415.0
411.4
10.62
10.16
9.720
9.293
8.880
.001028
.000992
.000957
.000922
.000889
.4323
.4171
.4023
.3879
.3740
.6575
.6458
.6343
.6228
.6116
689.5
686.6
683.7
680.8
677.9
3.086
3.175
3.268
3.369
3.468
* ICAN standard conditions
** ICAN standard conditions. For conditions other than ICAN, the speed of sound is propor-
tional to T.
A-3
APPENDIX I, PART II
PROPERTIES OF THE NACA
STANDARD ATMOSPHERE
I,
h,
(1000 ft.)
v X 104
°F, Abs.
P,
P,
in. Hg. slugs per cu.ft.
Vc,
mph
ft. per sec
31
32
33
34
35
407.8
404.3
400.7
397.2
393.6
8.483
8.101
7.732
7.377
7.036
.000857
.000826
.000795
.000765
.000736
.3603
.3472
.3343
.3218
.3098
.6003
.5892
.5781
.5673
.5566
674.9
672.0
669.0
666.0
663.0
3.570
3.678
3.792
3.911
4.034
35.332
36
37
38
39
40
392.4
392.4
392.4
392.4
392.4
392.4
6.926
6.711
6.397
6.098
5.813
5.544
.000727
.000705
.000672
.000640
.000610
.000582
.3058
.2963
.2824
2692
.2567
.2448
.5530
.5443
.5314
.5188
.5067
.4948
662.0
662.0
662.0
662.0
662.0
662.0
4.073
4.204
4.410
4.625
4.852
5.089
41
42
43
44
45
392.4
392.4
392.4
392,4
392.4
5.284
5.038
4.802
4.578
4.365
.000555
.000529
.000504
.000480
.000458
.2333
.2225
.2120
2021
.1927
.4830
.4717
4604
.4496
.4390
662.0
662.0
662.0
662.0
662.0
5.338
5.599
5.873
6.161
6.462
46
47
48
49
50
392.4
392,4
392.4
392.4
392.4
4.162
3.967
3.782
3.605
3.438
.000437
.000417
.000397
.000379
.000361
.1838
.1752
.1670
.1592
1518
.4287
.4186
4086
.3990
.3896
662.0
662.0
662.0
662.0
662.0
6.778
7.110
7.459
7.824
8.206
51
52
53
54
55
392.4
392,4
392.4
392.4
392.4
3.276
3.124
2.979
2.840
2.707
.000344
.000328
.000313
.000298
.000284
.1447
.1379
.1315
.1254
.1195
.3804
.3714
.3626
.3541
.3457
662.0
662.0
662.0
662.0
662.0
8.607
9.028
9.470
9.933
10.42
56
57
58
59
60
392.4
392.4
392.4
392.4
392,4
2.581
2.461
2.346
2.236
2.132
.000271
.000258
.000246
.000235
.000224
.1140
,1087
.1036
.0987
.0941
.3376
.3296
.3219
.3142
.3068
662.0
662.0
662.0
662.0
662,0
10.93
11.46
12.02
12.61
13.23
* ICAN standard conditions
** ICAN standard conditions. For conditions other than ICAN, the speed of sound is propor-
tional to VT.
A-4
.50
45
40
35
.45
30
25
20
15
10
.40!
5
SEA LEVEL
.35
.30
.25
PARAMETERS OF 1,000 FEET Hp
.20
.15
100
120
140
160 180
200
220
240
260
280
300
320
340
VCAL IN KNOTS
APPENDIX II (1) Graphical Relations Between Calibrated Airspeed, Mach Number and Pressure Altitude for Subsonic Speeds
(NACA Standard Atmosphere for Altimeter and Airspeed Indicator Calibration)
A-5
.88
PARAMETERS OF Hp
1,000 FEET
.85
.80
80
55
.75
40
35
.70
30
25
.65
20
15
.60
10
.33
SEA LEVEL
.50
100
120
140
160
180
200
280
300
320
340
360
VCAL IN
220 240 260
IN KNOTS
APPENDIX II (2) Graphical Relations Between Calibrated Airspeed, Mach Number and Pressure Altitude for Subsonic Speeds
(NACA Standard Atmosphere for Altimeter and Airspeed Indicator Calibration)
A-6
1.00
.95
Os
.90
.85
20
.80
.75
710
.70
.65
SEA LEVEL
.60
PARAMETERS OF Hp
2,000 FEET
.55
.50
320
340
360
380
400
420
440
460
480
500
VCAL IN KNOTS
APPENDIX II (3) Graphical Relations Between Calibrated Airspeed, Mach Number
and Pressure Altitude for Subsonic Speeds
(NACA Standard Atmosphere for Altimeter and Airspeed Indicator Calibration)
A-7
440
420
50,000 FT.
400
45,000 FT.
40,000 FT.
380
35,000 FT.
360
30,000 FT.
X
340
,25,000 FT.
20,000 FT.
320
15,000 FT.
10,000 FT.
V, IN KNOTS
300F
5,000 FT.
1,000 FT.
280
SEA LEVEL
260
240
220
200
STANDARD DAY CONDITIONS
180
160
160
180
200
220
240
260
280
300
320
340
VCAL IN KNOTS
APPENDIX III Graphical Relation Between True Airspeed, Altitude
and Calibrated Airspeed Under Standard Conditions
(NACA Standard Atmosphere)
A-8
!
Too
I
M
7
1+ € m2(2), 8-1.40
То
T..-T.
E : RECOVERY FACTOR :
Ty-Ta
-
1.20
E = 1.0
E =
E = 0.9
1.16
E = 0.8
E = 0.7
1.12
Te = 0.6
1 € =0.5
ܘܘܟ
Ta
1.08
1.04
1.00
.10
.20
.30
.40
.50
.60
.70
.80
.90
1.00

ma
M
APPENDIX IV Mach Number Temperature Relations for Various Recovery Factors
A-9

Zone
150.000
15
50004.h
250
300
350
400
450
500
550
kom
11.
RELATIONS BETWEEN TRUE
SPEED V, CALIBRATED AIR-
SPEED Vcal, MACH NUMBER
M, AND PRESSURE ALTI-
(SUBSONIC)
VALID
FOR
M€ 1.0
14
Veal = 100 kt
TUDE hp
13-1
M-0.60
M=0.50
M=0.40
M10.70
M0.80
40,000)
T, CONSTANT. 56.5°C
M1.00
M0.90
140,00d
1
12
NOTATION: Di impact pressure. D, (MCO)
Distatic pressure
cspeed of sound
1) BASIC RELATIONS
PD
..(M)
(for isentropic
150 kt
flow
10
Neal
30,000
9
Vcal: 200 kr
8
• 250 ky
Vcol
7
Vcal
Vcal. 300 km
+50°C
(2) M. Vic
T.14, R.227 (m2,211°C
30,000
2) CALIBRATED AIR SPEED
(1) At eoch altitude Rwas computed
40°C or a function of and M from (1)
(2) At eoch value of 0, -, the value
of calibrated airspeed is determin
ed from:
+30°C
Pro
Subscript, designatos *
level conditions
popold
3) Under non standard conditions
Vis given by the abscind values
+20°C corresponding to the intersec-
tion of the ordinate with
the constant Mach number
line which DOIS through
the junction of the
+10°c curves Ycal and no
20,000
Ncal. 350k1
6
cal*400**
NCC 450kt
Cal 500k
10,000
3
ON
Ncal. 550kt
goood qi-
toºc
2
Vcal f600 ki
Foc
M0.30
M: 0.40
M +0.50
M=0.60
M+0.70
M-0.80
M0.90
M=1.00
+20°C
250
300
350
200
400
400
600
450
850
500
950
550
1,000
650 knots
1,200
km/h
450
500
550
600
650
700
750
800
900
1,050
1,100
1,150
APPENDIX V (1) Relations Between True Airspeed V, Calibrated Airspeed Vcal ,
Mach Number M and Pressure Altitude hp (Supersonic)
A-10
1

ܪ
200000
250
300
350
90008 Ap
400
450
550
500
15
kan
11.
Veal = 100 kt
RELATIONS BETWEEN TRUE
SPEED V, CALIBRATED AIR-
SPEED Vcal , MACH NUMBER
M, AND PRESSURE ALTI-
(SUBSONIC)
VALID
FOR
M€ 1.0
14
TUDE hp
13-
M = 0.40
M = 0.60
M=0.50
M = 10.70
M = 0.80
40,000
12-
M1.00
M +0.90
for Tr. CONSTANT:56.5"
40,00
NOTATION: Di impoct pressure D, (M
, static pressure
cspeed of sound
1) BASIC RELATIONS
11
10
cal" 150 k1
30,000
9
Vcal: 200 ml
8
Vcal.- 250 ky
D.
. (M)
(for isentropic
flow)
+50°C
(2)
M. Vic
7:14R.227 ( m2,21/C
30.00
2) CALIBRATED AIR SPEED
(1) At eoch altitude D-, - computed
+40°C os a function of Boy and M from (1)
(2) Al och value of D-Ps the value
of calibrated airspeed is determin
ed from:
Vcoi
-30°C
Subscript,o, designates 10
level conditions
20.00
3) Under non standard conditions
Vis given by the abscisso values
+20°c corresponding to the internec-
tion of the ordinate with
the constant Moch number
line which pane through
the junction of the
Floºc curvas Vcal and
7
Vcol. 300 kt
20,000
Ncal. 350kt/
6
09
Ncal 400k/
5
Ncore 450 kt
4
•500kr
10,000
3
10.000
Neol
Near"550kt
to°c
2
Vcai 600 ki
.
1
FIOC
O
M=0.30
M = 0.40
M-0.50
M=0.60
M0.70
M20.80
M-0.90
001.no
(20°C
300
200
400
250
450
350
650
400
750
450
850
500
950
700
550
1,000
600
1,100
650 knots
1,200
km/h
500
550
600
BOO
900
1,050
1,150
APPENDIX V (1) Relations Between True Airspeed V, Calibrated Airspeed Vcal,
Mach Number M and Pressure Altitude hp (Supersonic)
A-10
.
1
UNIVERSITY OF MICHIGAN
3 9015 06435 3850
The HF Group
Indiana Plant
TO