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“RECEIVED BY DTIE AUG 29 1967
cscf ngaCES
#993.09 ww.65
HYDROMAGNETIC GORILER INSTABILITY IN A BOUNDARY
LAYER ON A CONCAVE WALL
MASTER
T. S. Chang
W. K. Sartory
Submitted for presentation at the Fourth Southeastern
Conference on Theoretical and Applied Mechanics to be
February 29 - March 1, 1968, in New Orleans, Louisiana.
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"
HYDROMAGNETIC GÖRTLER INSTABILITY IN A BOUNDARY
LAYER ON A CONCAVE WALL
T. S. Chang
Oak Ridge National Laboratory
and
North Carolina State University
W. K. Sartory
Oak Ridge National Laboratory
ABSTRACT
The effect of an applied magnetic l'ield on the Görtler instability
of a Blasius boundary-layer flow over a concave wall is investigated
analytically for the case where the fluid is a finite electrical con-
ductor and the wall is perfectly conducting. The magnetic field, which
C
is aligned parallel to the wall and perpendicular to the flow direction,
is found to have a stabilizing effect just as with Taylor instability
considered by earlier investigators. The effect on stability of the
component of velocity normal to the wall in a growing boundary layer is
also considered by making calculations both with and without the normal
flow terms. The normal flow is found to have a destabilizing effect,
especially with small values of the Hartmann modulus. At large values
of the Hartmann modulus, the normal flow appears to suppress the forma-
tion of oscillatory modes of instability.
Consultant and Professor, respectively.
si
NOTATION
Axially (2) applied uniform magnetic induction field
Convective wave number
Б., Бөг ъ,
magnetic induction vector
= a/ar
= D + 1/2
E, F, I, J, M, N Functions of my, mg, , T, , Vo
Dimensionless amplitude functions of (bre bg, Bb, )/(BRB)
= (võ/v) ✓ ēr, Görtler modulus* based on momentum
thickness
= 8/R
f, g, h
Grom
* *
= 87(5./6)
= Volu B ro, Hartmann modulus
= H8/
= 7(7/6)
de idi
+ --- -- - B2
dx2 x dx
Tangential distance downstream of the leading edge
of the plate at which the stability is determined
Roots with negative real parts of (ma - 62,2
- Ūamma - 82) + 2 = 0
Undisturbed fluid pressure
6.
"This parameter was first used by Görtler and for convenience we refer
to it as the "Görtler" modulus, although the name is not in common use.
.:
21
..
.
.
..
..
.
.
.
.
.
.
.
..
P
L
,
... !:"IT
ILI
2 po
* ONOV, magnetic Prandt) modulus
Perturbation of p*
- DV,/uReynolds modulus
- VF
, magnetic Reynolds modulus
0,2
U, V, W
Cylindrical coordinates
Radius of curvature of the conceve wall
- 2R2 (6/7), Taylor modulus
Cylindrical components of the velocity vector
= -4/(.6)
= VN
= _lin (V, U)
10. To
Tangential velocity outside of the boundary layer
m', v', w
= (U, VINCE
Perturbations of the cylindrical components of the
velocity vector
amplitudes of (u', v', Bw') No
uz, Vgs Wid
= 4,
= (f. - )/6
= 3/5
- 7,0/4
Greek Letters
781
= bro
l
= 36/.
iv
KY
mcm
= (0/0)
. 4/ 17
X/ T
, boundary layer thickness parameter
Momentum thickness of the boundary: layer
ao pot < F ois a
Coefficient of viscosity
Magnetic permeability
Kinematic viscosity
Amplitude of p*r/
Fluid density
Electrical conductivity
dir)
Ampiltude of the perturbations
Other Symbols
va
* DMD +
*
regge
+ -
22
.
"..
.
.
Web
..
..
.
.
.
..
..
.
..
.
*.
.
.
.
-
-
-
-
-:
....
."
*!
!
.
'
.
"..
+..
LIST OF FIGURES
Fig. 1. Critical Görtler Modulus Versus Hartmann Modulus.
Upper curve: Without normal flow terms
Lower Curve: With normal flow terms
Fig. 2. Critical Wave Number Versus Hartmann Modulus.
Upper curve: Without normal flow terms
Lower durve: With normal f'low terms
Fig. 3. Neutral Stability with Normal Flow Terms, H
= 0.
INTRODUCTION
The authors know of no previous work on the hydromagnetic Görtler
stability problem. The nonmagnetic stability problem has been treated
theoretically by several authors.
Görtler (1) first proposed and demonstrated theoretically that
Taylor instability could occur in a boundary layer on a concave surface.
He made the parallel flow approximation (i.e., he neglected the component
of the primary velocity normal to the surface) in deriving the stability
equations.
Meksyn (2.) solved Görtler's stability equations using a different,
asymptotic method of integration. Hämmerlin [3] also re-solved Görtler's
stability equations by different methods and apparently obtained very
accurate solutions. He first established the peculiar result that the
minimum Görtler modulus for neutral stability occurs at an axial wave
CU
VO
number of zero. The resulting disturbances extend far outside of the
boundary layer.
Smith [4] considered the approximations made in deriving the sta-
bility equations in detail and derived a new set of much more complicated
equations. The additions to the stability equations made by Smith were
of three types:
1. Normal flow terms. Certain terms involving the velocity normal
to the surface in a growing boundary layer were found to he of the same
order of magnitude as the tangential velocity terms.
2. Finite curvature terms. Higher order terms approximately
describing the effect of curvature on the disturbances were retained.
The addition of these terms involves the introduction of another parameter,
and the relative importance of the terms depends on the range of values
chosen for the new parameter.
2
.
3. Growth rate terms. Smith permitted the disturbances to grow
"
L
"
-
N
.
in the downstream direction rather than in time. The growth rate terms
."
A,
...'
....
.
i
t
do not appear in the neutral stability equations.
The purpose of the present investigation is to consider the effect
of an applied magnetic field, aligned parallel to the wall and perpen-
dicular to the flow direction, on Görtler instability when the fluid is
an electrical conductor. With regard to the three additions to the sta-
nan
..
....
.
.
..
LP
.
bility equations made by Smith and noted above, we shall consider sta-
bility both with and without the normal flow terms. We shall not consider
finite curvature or rate of growth at all.
THEORY
We consider flow along a wide concave plate with a uniform radius
of curvature, and adopt a cylindrical coordinate system. The two-
dimensional primary velocity profile is taken to be of the form:
U = u(r, e),
V = v(r, e), W=0
(1)
where U, V, W are the cylindrical (r, 0, z) components of the velocity
vector. The fluid is assumed to be incompressible, viscous, and elec-
trically conducting. An external magnetic induction field B. is applied
in the axial (z) direction. Based on the classical theory of magneto-
hydrodynamics, it is not difficult to demonstrate that the partial dif-
ferential equations as well as the boundary conditions at a perfectly
conducting plate admit the velocity given by Egs. (1) and the applied
magnetic field Bo as a solution.
WCU
MULUT
1.
"
Allowing the flow field and the magnetic induction field to be per- .
turbed slightly, and inserting the perturbed functions into the basic
equations of MHD (5), the following sets of equations (correct to the
zeroth and first order of the perturbations) are obtained.
VA
Zeroth Order Equations
au au ve
U – + V ----
or raer
OP*
a2
lo
= -
+ VID,
ar
av
av
UV
p*
OU
Il no
ar
rao
r
page2 ) V +
rae
au
-
or
U QU
+ - + mom = 0
r rae
where
will
e
3.560 ) = 2*=,0)
ol
W
d. I
w
P is the undisturbed fluid pressure, p is the fluid density, and v is
the kinematic viscosity.
First Order Equations
+
V
la * -- + I
at ar
o
rao
, OU 2V
v 02) u' + ( -
'raer
+
2va
pe ai?
v'
ar
5
2.
do*
to
, av v 2 va
( - t --- -
дr r r2 де?
rə
u' +
ə ə av U
+ U — +1.- +- +--uva) v'
ar rat rat R
at
--
---
Boabe ap*
1.
Ono az
rao
OP
+ U
+
V
V
t
=
0
at
-
rao
ar
inain
Pho az
a
OU
- 2
du
dl
U
+ V
-
-
1
-
Ollo
-
rao
-
Do = 0
ar
ar
rse
hora de
B
av' , av v 2 a
-- + -- + -
az 'ar r rpolo ã o
)
N +
A +
see
--
e et
ala
51.6
-
I
-
Ono
q (A
0 =
de
ole ola
ow'
,
l
ə
--
rae
1
-
Oro
+
+ U -- + V
dr
b
= 0
Olor
au' u' av? Ow'
- + - + + -
ar r roooz
= 0
aby brabe ab
- + I +20+ 2=0
ar r ra e az
(4)
where u', v', w' are the perturbations of the cylindrical components of
the velocity vector; bre be, b, are the perturbations of the cylindrical
components of the magnetic induction vector; p* is the perturbation of
P", He is the magnetic permeability, o is the electrical conductivity,
and
✔
- DLD +
+
.
reaga
aza.
In the above equations, the rationalized MKS system of units is used.
"
.
Normal Mode Equations
The first order perturbation equations, Egs. (4), form a set of
voorsien
coupled, linear differential equations governing the perturbations for
a given stationary profile, (U, V). The perturbations are now resolved
into stationary normal modes of the form:
iberiman memastikan alarmet uimich in die
L
· sin bz
i mesinetr
$(r) { cos bz
n
r
hemmin mi
where
(r) is the amplitude and b is the convective wave number.
In terms of the normal modes, Egs. (4) reduce to a set of coupled,
ordinary differential equations governing the amplitudes as follows:
ter som binding hieman turcam
d
du
idu
2V
atly - 0
12
+
u,
+
R
-
BPH f t
=
0
dx
ax'a
v
1
хәөх
*
dx
dy
v
ale
un + i -It R
u av i
- + - + --
dx x xD e'.
v, - BOH® g = 0
dx
x
) * *[*** ( ) * - *#s=0
Go to ) vs + Pfa- $75 -0
d
+
L
+ RP
ди
f - PR -
m
g
=
0
Dry
ах
xao
most common
va - RPM (--
+
L
+ RP
id 1 av 1
(u
(u - ---
dx x x -
g =
дx
x
that it can have a common
df
-
I H
- h = 0
ax
*
du,
un
ther that
+
+ W, = 0
dx
x
the
more interest
For
where x = x/t., B = bro, (u, v) = (U, V)NoV is the tangential velocity
outside of the boundary layer; u,, V,, W,, f, g, h, , are dimensionless
amplitude functions defined by:
1
(u', v', “') = V. (un cos bz, v, cos bz, w, sin bz/B)
(higa bogs bz) = B. B B (f sin bz, & sin bz, h cos bz/B)
p* = (v/R) Ty cos bz
(6)
R = p Vor/u is the Reynolds modulus, R = V ro Hoo is the magnetic
Reynolds modulus, Pm = RM/R is the magnetic Prandtl modulus, H = Vo7u Bor.
is the Hartmann modulus, H is the coefficient of viscosity, r is the
radius of curvature of the concave wall, and the operator I is given by:
da 1 d 1
dxa
x dx
x
If the magnetic Prandtl modulus is assumed to be small and P is
m
4
set equal to zero, Egs. (5) resoive to the following set of two coupled
differential equations:
you
2v
à ou
1 ui - R (u -- + -- ) Lu
+ B2 HP un ti B?R?
dx
Әх
xa
X
tus =R (v ) In, * ffu, to ) ,30
's - r[uce ) - ( 11 )]2+885 - () --
(8)
-
R
u
of
Lg
+ f2172
x 14
where &, = 8/R.
Boundary-Layer Approximation
Some of the earlier investigators, Görtler [1] for example, have
assumed vardous arbitrary primary or zeroth-order velocity profiles for
stability calculations. Görtler concluded that the stability was not
very sensitive to changes in the shape of the primary profile provided
the momentum thickness was not changed. In the present analysis, both
normal and tangential velocity profiles are needed; and to insure that
the two are consistent, it seems appropriate to consider only calculated
boundary-layer profiles. Following Smith [4], we assume no pressure
gradient and use the Blasius solution as the primary flow.
WE
In Egs. (2), we put ap*/8 = 0 and make the substitutions
* = (r. - r)/8 y = (r.0)/4
Ū = -Ul/(V.) = V/.
(9)
where l is the tangential distance downstream of the leading edge of the
plate at which the stability is to be determined, and 8 = l/Foltv is
the boundary-layer thickness parameter.
If we retain only the lowest-order terms in 8, we obtain the usual
boundary-layer equations in dimensionless form:
әї әї агу
U + V -
ažo ar
aū Jū
- + - = 0
(10)
až JY
Blasius' solution of these equations is of the form
ū(7, 7) = 7(n) where m = /
(11)
By definition, Y = 1.0 at the point where stability is to be determined;
so n and ñ can be used interchangeably in the stability calculations.
If we substitute Eqs. (9) into Eqs. (8) and again retain only the
lowest-order terms in 8,* we obtain:
Smith [4] also retained higher-order terms, which we call the "finite
radius of curvature terms."
12. - (0
pag-lo
) ET +FT; - FIDIZ --
) ++)
HT
100
Ion
(12)
where
I
,
B
1 = 18/,
ū = uz
B = B6/
(Taylor modulus) = 2R2(6/7.)3
x = & (r./8)3
(13)
If the wall is nonpermeable and a perfect electrical conductor,
the boundary conditions for tue perturbations are [5]:
dui
dgn
1004
= 0
at
x
= 0
(14)
We require that the perturbations
vanish as X +
It is possible
to derive boundary conditions which can be applied at the outer edge of
the boundary layer rather than at infinity, thus restricting the integra-
tion to a finite interval. The method is a direct extension of that used
by Hämmerlin [3] for his nonmagnetic, parallel flow analysis of boundary-
layer stability. The resulting boundary conditions are:
L
. ALL
n]4- (*) 7 -
[- ) 54]
Le- -lock 20
la
NO
a
-
-
- + m
-
-
mo
-
-
-
-
---
-
-
-
-
-
-
-
(15)
where
More
2FI + FJ(m + m) + EI(my + m2) + 2JE m, my
2 + FE(m + mg) + my my
N =- CEM ma mz - Icm, + m) - 23 m, m.]
B = 4/(m2 + m )2 – m. m) – 472 – 3ūvm, + m )
F = -um, m (m + m2) + Tool 3mm + 69)
j = ē T(m+ m)
I = Z2 T Tol-m, min - %)
(16)
M,,m are the roots with negative real parts of the polynomial equation:
(ma – 32) 2 - 7. m(m2 - 54) + 2 T2 = 0
(17)
and
and
lŪor Tool - lim (ū, ū)
(18)
NUMERICAL METHOD
Equations (12) were rewritten as a set of eight first-order dif:
ferential equations and integrated numerically on a digital computer
using predictor-corrector formulas of fourth-order accuracy. The numerical
technique of matching the boundary conditions and calculating critical
values of the parameters was described by the authors in an earlier
paper [6]. The outer boundary conditions given by Eqs. (15) were applied
at a value of X = 9. The slope of the Blasius tangential velocity profile
is zero in the first six decimal places at i lor m) = 9.
The error of numerical integration was controlled by successively
halvine; the mesh size and repeating the calculations until the critical
W88
10
Görtler modulus and wave number from two successive calculations agreed
within %. The required number of mesh points was found to increase to
several hundred for those calculations involving very small values of
Ő and T. It was also found that the calculations at small values of B
and T became very sensitive to the accuracy of the primary velocity. This
sensitivity occurred only when the primary normal flow terms were retained
in the stability equations, and may be caused by a cancellation of effects
from the normal and tangential components of the primary flow. To insure
accuracy, the primary profile was therefore obtained by numerically in-
tegrating the Blasius equation from 1 = 0 . 10 using 500 intervals with
a fourth-order predictor-corrector method.
DISCUSSION OF RESULTS
Stability results for a boundary layer on a concave wall are usually
reported in terms of the Görtler modulus based on momentum thickness
91/2
Com
V7ft. - [5Jue
as! ol
(29)
mom
where ő is the momentum thickness of the boundary layer (= 0.664 8 for the
Blasius profile).
We also use the corresponding wave number
Brnom = B(7/6)
(20)
and Hartmann modulus
Hom = H (7/8).
(21)
mom
Critical results have been obtained both with and without the primary
normal flow terms. Results without primary normal flow were obtained by
11
setting Ū = 0 in Egs. (12) and (15).
Figure 1 shows the critical Görtler modulus, Gmome as a function of
the Hartmann modulus, Hom; and Fig. 2 shows the critical wave number
mom
Berm as a function of the Hartman modulus. The upper curves apply to the
mom
case of no normal flow and may be considered a direct hydromagnetic exten-
sion of the work of Görtler. In the limit of Hmom * O the critical Görtler
modulus approaches 0.3 and the critical wave number approaches zero, in
good agreement with the results of Hämmerlin [3] for the Blasius profile.
As Hmom approaches 2, the critical wave number curves upward sharply and
mom
the critical Görtler modulus appears to be approaching an asymptote of the
form Gmom o Home This behavior is an indication that the normal modes
MOI
mom
for neutral stability are becoming oscillatory rather than stationary.
whiie it is possible to carry the critical stationary mode calculations
beyond the point where oscillatory modes begin to form, very complicated
results of no particular physical significance usually result. The prob-
lem of oscillatory mode formation in hydromagnetic Taylor instability has
been discussed in detail by the authors in an earlier paper [7]. In the
present work, only stationary modes have been considered and no attempt
has been made to extend the calculations further,
The lower curves of Figs. 1 and 2 apply to the case where the normal
flow terms of Egs. (12) and (15) are retained. They may be considered a
hydromagnetic extension of the critical calculations of Smith, with the
reservations noted in the introduction. The figures show that in the
VA
limit of Hom + 0, both the critical wave number and the critical. Görtler
modulus approach zero. This conclusion is in conflict with the results
of Smith [4], who found a critical wave number, Pmome of about 0.07 and a
12
t ers ti
i a Prin
103

.
. Wedstr
... ... .
: -.
G MOM
10°3 102 101 100 101 102
H MOM
Fig. I. Critical Görtler Modulus vs Hartmann Modulus
Upper Curve: Without Normal Flow Terms
Lower Curve: With Normal Flow Terms
13

BETA MOM
10-3 10-2 10-1 100 101 102
H MOM
Fig. 2. Critical Wave Number vs Hartmann Modulus
Upper Curve. Without Normal Flow Terms
Lower Curve: With Normal Flow Terms
.
14
mom
critical Görtler modulus of about 0.32, To investigate further, a neutral
stability (Gmom versus Bmom) diagram has been calculated for Hmom = 0,
with the normal flow terms in Egs, (12) and (15) retained. The results
are shown in Fig. 3. The curve is in good agreement with the results of
Smith in the range Bmom > 0.4, and in particular at the point Brom = 0.664
where Smith made particularly careful calculations. As the wave number
decreases further, however, the neutral curve of Smith passes through a
minimum and rises again, while the present curve extrapolates tomom:=1:0
at Brom = 0.
The difference in the neutral curve can probably be attributed to
i mom
two sources: the additional finite radius of curvature terms retained
by Smith but not considered in the present analysis, and the numerical
methods used.
Smith actually concludes from his numerical results that
the stability is not very sensitive to the finite radius of curvature
terms, at least in the range covered by his calculations. Nevertheless,
it is likely that at very small wave numbers, where the disturbances are
known to extend far outside of the boundary layer, an appreciable effect
of the radius of curvature could occur and could account for the difference
in results. The numerical method used by Smith was Galerkin's method, and
the solution was assumed to have the form of a polynomial times an expo-
nential factor which decayed far from the wall. The polynomial coeffi.
cients were calculated in solving the problem, but the exponential decay
rate was assumed, mainly on the basis of numerical experimentation. Our
results indicate that, at Bom = 0.1, the final exponential rate of decay
of the perturbations far from the wall is over an order of magnitude lower
mom
than the lowest rate of decay assumed by Smith, This could account for a
progressive loss of accuracy in Smith's results at small wave numbers,
15

MOM
010-1
10-2
10•34
104
10°3
102
BETA MOM
101
100
· Fig. 3. Neutral Stability with Normal Flow Terms, H MOMO
16
At large values of Hm, the lower curves of Figs. 1 and 2 indicate
that mom mom. The form of these asymptotes is consistent with the
results of Chandrasekhar (5) for the hydromagnetic Taylor stability prob-
lem. While no investigation of oscillatory modes has been made for the
present case, the occurrence of the usual asymptotic behavior at large
Ho suggests that oscillatory modes are not affecting the critical point
in this case.
mom
The conclusion that when the normal flow terms are included and
Hmom = 0, then mom and smom are both zero is certainly peculiar; and
while we believe that it is the correct solution of the particular mathe-
matical formulation of the stability problem adopted in this paper, it
probably requires some further discussion of physical significance.
The result that the critical wave number is zero was obtained by
Hämmerlin (3), who neglected the normal flow terms. This result implies
that the size of the critical disturbance is not controlled or limited by
the finite boundary-layer thickness. Since a disturbance of infinite size
is hardly acceptable, some factor other than the boundary-layer thickness
must limit the size. The additional conclusion of this work is that,
when the normal flow terms are included, some additional factor must be
introduced even to obtain a nonzero critical Görlter modulus. Depending
on the physical situation, the limiting factor could be the finite span
of the wall, or it could be the finite radius of curvature or the need to
attain a finite rate of disturbance growth. The latter two factors have
been considered by Smith. In the hydromagnetic case, magnetic damping
serves to limit the growth of a disturbance beyond the boundary layer,
and a finite critical Görtler modulus and wave number ere obtained without
introducing any additional factors.
17
ACKNOWLEDGMENT
This research was sponsored by the U. S. Atomic Energy Commission
under contract with the Union Carbide Corporation. The authors wish
to thank Dr. J. J. Keyes, Jr., for his comments.
REFERENCES
1. Görtler, H., "über eine dreidimensionale Instabilität laminarer
Grenzschichten an konkaven Wänden," Gesellshaft der Wissenschaften
zu Göttingen, Nachrichten, Mathematik, Vol. 2, No. 1, 1940.
2.
Meksyn, D., "Stability of Viscous Flow over Concave Cylindrical
Surfaces," Royal Society of London, Proceedings, Vol. 203, Series A,
1950, pp. 253-265.
3,
Hämmerlin, G., "über das Eigenwertproblem der dreidimensionalen
Instabilitat laminarer Grenzschichten an konkaven Wänden,"
of Rational Mechanics and Analysis, Vol. 4, 1955, pp. 279-321.
Wänden," Journal
4.
Smith, A. M. o., "On the Growth of Taylor-Görtler Vortices along
Highly Concave Walls," Quarterly of Applied Mathematics, Vol. 13,
1955, pp. 223-262.
5. Chandrasekhar, S., Hydrodynamic and Hydromagnetic Stability,
Clarendon Press, Oxford, 1961.
6. Chang, T. S., and Sartory, W. K., "Hydromagnetic Stability of
Dissipative Vortex Flow, " The Physics of Fluids, Vol. 8, No. 2,
1965, pp. 235-241.
7.
Chang, 1. S., and Sartory, W. K., "On the onset of Instability by
Oscillatory Modes in Hydromagnetic Couette Flow," (to be published);
also see "Transition from Stationary to Oscillatory Modes of Insta-
bility in Magnetohydrodynamic Couette Flow," USAEC Report ORNL-TM-1404,
Oak Ridge National Laboratory, 1966.
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