key: cord-0224789-nu1013k1
authors: Xu, Ao; Tao, Shi; Shi, Le; Xi, Heng-Dong
title: Transport and deposition of dilute microparticles in turbulent thermal convection
date: 2020-07-11
journal: nan
DOI: nan
sha: 85467e9077dce7bab8a2d775d2a71315fd6bff05
doc_id: 224789
cord_uid: nu1013k1

We analyze the transport and deposition behavior of dilute microparticles in turbulent Rayleigh-B'enard convection. Two-dimensional direct numerical simulations were carried out for the Rayleigh number ($Ra$) of $10^{8}$ and the Prandtl number ($Pr$) of 0.71 (corresponding to the working fluids of air). The Lagrangian point particle model was used to describe the motion of microparticles in the turbulence. Our results show that the suspended particles are homogeneously distributed in the turbulence for Stokes number ($St$) less than $10^{-3}$, and they tend to cluster into bands for $10^{-3} lesssim St lesssim 10^{-2}$. At even larger $St$, the microparticles will quickly sediment in the convection. We also calculate the mean-square displacement (MSD) of the particle's trajectories. At short time intervals, the MSD exhibits a ballistic regime, and it is isotropic in vertical and lateral directions; at longer time intervals, the MSD reflects a confined motion for the particles, and it is anisotropic in different directions. We further obtained a phase diagram of the particle deposition positions on the wall, and three deposition states depending on the particle's density and diameter were identified. An interesting finding is that the dispersed particles preferred to deposit on the vertical wall where the hot plumes arise, which is verified by tilting the cell and altering the rotation direction of the large-scale circulation.

ticles are homogeneously distributed in the turbulence for Stokes number (St) less than 10 −3 , and they tend to cluster into bands for 10 −3 St 10 −2 . At even larger St, the microparticles will quickly sediment in the convection. We also calculate the mean-square displacement (MSD) of the particle's trajectories. At short time intervals, the MSD exhibits a ballistic regime, and it is isotropic in vertical and lateral directions; at longer time intervals, the MSD reflects a confined motion for the particles, and it is anisotropic in different directions. We further obtained a phase diagram of the particle deposition positions on the wall, and three deposition states depending on the particle's density and diameter were identified. An interesting finding is that the dispersed particles preferred to deposit on the vertical wall where the hot plumes arise, which is verified by tilting the cell and altering the rotation direction of the large-scale circulation. a a) The following article has been submitted to Physics of Fluids. After it is published, it will be found at Link (https://publishing.aip.org/resources/librarians/products/journals/). a) Electronic mail: axu@nwpu.edu.cn

Transport and deposition of solid particles (or liquid droplets) in turbulent thermal convection occur ubiquitously in environmental science [1] [2] [3] [4] . For example, suspended atmospheric pollutant particles (PM10, PM2.5) that originated from dust and smoke will severely influence the air quality 5, 6 . Another example is that pathogen laden droplets in confined indoors, which will cause viral and bacterial infectious diseases (SARS, spreading in hospitals, schools, and airplanes [7] [8] [9] [10] . In such dispersed multiphase flow, the evolution of the phase interface may not be a primary concern 11 . From the aspect of particle kinematics, important control parameters include the density ratio of the particle to its surrounding fluid Γ = ρ p /ρ f , and the size ratio Ξ = d p /l f . Here, ρ p and d p are the particle density and particle size, respectively. ρ f is the fluid density, l f is the characteristic fluid length. When Ξ ≪ 1, the Lagrangian particle model can be used to track the dispersed phase. Moreover, when the volume fraction of the dispersed phase is small, the dominant effect is that of the carrier flow on the dynamics of the dispersed phase, but not vice versa. Thus, a one-way interphase coupling approach can be adopted to track the motions of particles 12, 13 . Previous studies have shown that even in homogeneous isotropic turbulence, dispersed particles may not distribute homogeneously but exhibit preferential concentration [14] [15] [16] [17] . For light particles with a density ratio of Γ ≪ 1, they concentrate in regions of high vorticity; for heavy particles with a density ratio of Γ ≫ 1, they are expelled from rotating regions.

Due to the injected buoyancy and the effect of the domain boundaries, turbulent thermal convection is generally inhomogeneous and anisotropic. A simple paradigm system to study thermal convection is the Rayleigh-Bénard (RB) cell, where a fluid layer is heated from the bottom and cooled from the top [18] [19] [20] [21] [22] [23] [24] . The control parameters of the RB system include the Rayleigh number Ra = β g∆ T H 3 /(ν f κ f ) and the Prandtl number Pr = ν f /κ f . The Ra describes the strength of buoyancy relative to thermal and viscous dissipative effects. The Pr describes thermophysical fluid properties. Here, β , κ f , and ν f are the thermal expansion coefficient, thermal diffusivity, and kinematic viscosity of the fluid, respectively. g is the gravitational acceleration. ∆ T is the imposed temperature difference between the top and bottom fluid layers of height H. In the RB convection, ubiquitous coherent structures include thermal plumes and large-scale circulation (LSC) 25, 26 .

Specifically, sheet-like plumes that detached from boundary layers transform into mushroom-like ones via mixing, merging, and clustering 27 . Due to plume-vortex and plume-plume interactions, thermal plumes further self-organize into the LSC that spans the size of the convection cell 28 .

Although the dynamics of single-phase turbulent thermal convection has been thoroughly investigated, the complex interactions between dispersed immiscible phase and its surrounding fluid in turbulent thermal convection remain less explored. One of the few studies by Puragliesi et al. 29 focused on particle deposition in side-heated convection cell (i.e., heated from one vertical side and cooled from the other vertical side). They found that a strong recirculating zone contributes to the decreased gravitational settling, thus resulting in particles suspending with longer time. Because the driven force, namely the temperature gradient, in side-heated convection cell is perpendicular to that in the RB convection cell, the fluid and particle dynamics are expected to be different in these two cells. Lappa 30 analyzed the pattern produced by inertial particles dispersed in the localized rising thermal plume. He identified the average behavior of particles by revealing the mean evolution. It should be noted that although thermal plumes are the building blocks of turbulent thermal convection, the LSC, which is another essential feature of the turbulent thermal convection, is missing in such analysis. In addition to the one-way coupling between the dispersed phase and the carrier flow, Park et al. 31 further investigated the RB turbulence modified by inertial and thermal particles. Changes of integrated turbulent kinetic energy and heat transfer efficiency were quantified. Results showed that particles with Stokes number (to be defined in Sec. II C) of order unity maximize the heat transfer efficiency. However, particles with such high Stokes number (either heavy density or large size) will sediment quickly in the air, which may be of limited interest for studying suspended atmospheric pollutant particles or pathogen laden droplets.

In this work, our objective is to shed light on the dynamics of atmospheric pollutant particles or pathogen laden droplets. We simulate transport and deposition of dilute microparticles in an RB convection cell with air as the working fluid (i.e., Pr = 0.71) at high Ra number (i.e., Ra = 10 8 ), such that ubiquitous features of the turbulent thermal convection (including thermal plumes and the LSC) naturally arise. We choose the typical particle parameters as 10 µm ≤ d p ≤ 100 µm and 400 kg/m 3 ≤ ρ p ≤ 4000 kg/m 3 , and the corresponding particle Stokes number (i.e., 3.67 × 10 −4 ≤ St ≤ 0.37) is much lower than that by Park et al. 31 (i.e., 0.1 < St < 15). The rest of this paper is organized as follows: In Sec. II, we present numerical details for the simulations, including direct numerical simulation of thermal turbulence and Lagrangian point particle model.

In Sec. III, we analyzed the particle transport behavior via flow visualization and particle meansquare displacement calculation; followed by statistics of particle deposition behavior, such as time history of the particle deposition ratio and phase diagram of particle deposition location. In Sec. IV, the main findings of the present work are summarized.

In incompressible thermal flows, temperature variation will cause density variation, thus resulting in a buoyancy effect. Following the Boussinesq approximation, the temperature can be treated as an active scalar, and its influence on the velocity field is realized through the buoyancy term.

The governing equations can be written as

where u f , p and T are the fluid velocity, pressure and temperature, respectively. ρ 0 and T 0 are the reference density and temperature, respectively.ŷ is the unit vector in the vertical direction. In the above equations, all the transport coefficients are assumed to be constants.

We adopt the lattice Boltzmann (LB) method [32] [33] [34] [35] as the numerical tool for direct numerical simulation of turbulent thermal convection. The advantages of the LB method include easy implementation and parallelization, as well as low numerical dissipation 36 . In the LB method, to solve

Eqs. 1a and 1b, the evolution equation of the density distribution function is written as 32,33

To solve Eq. 1c, the evolution equation of temperature distribution function is written as 32,33

Here, f i and g i are the density and temperature distribution function, respectively. x is the fluid parcel position, t is the time, and δ t is the time step. e i is the discrete velocity along the ith direction. M is a 9 × 9 orthogonal transformation matrix based on the D2Q9 discrete velocity model; N is a 5 × 5 orthogonal transformation matrix based on the D2Q5 discrete velocity model.

The equilibrium moments m (eq) in Eq. 2 are

The equilibrium moments n (eq) in Eq. 3 are

a T is a constant determined by thermal diffusivity as a T = 20 √ 3κ − 6. The relaxation matrix S is S = diag(s ρ , s e , s ε , s j , s q , s j , s q , s ν , s ν ), and the kinematic viscosity of the fluid is calculated

The macroscopic fluid variables of density ρ, velocity u f and temperature T are calculated as

More numerical details on the LB method and validation of the in-house code can be found in our previous work 37- 39 .

We consider small particles such that their presences do not modify the turbulence structure, namely one-way coupling between the multiphase. Here, 'small' means the diameter of the particle is smaller than the Kolmogorov length scale of the turbulence; however, the diameter of the particle should still be much larger than the molecular mean free path, such that the effect of Brownian motion can be neglected. In addition, the particles are assumed to be isotropic, such that we only consider the motion of the particle and neglect the rotation of the particle 40, 41 . Specifically, the particles' motions are described by Newton's second law as

The total force F total exerted on the particle include the net gravitational force F G and the drag force F D . Specifically, particles experience a gravitational force in the direction of gravitational acceleration, as well as buoyancy in the opposite direction. The net gravitational force F G is given by

where ρ p and V p are the density and volume of the particle, respectively. Meanwhile, the particle experiences a drag force that acts to catch up with the changing velocity of the surrounding fluid.

The drag force F D is given by

where m p and u p are the mass and velocity of the particle, respectively. τ p = ρ p d 2 p /(18µ f ) is the particle response time and d p is the particle diameter. The particle Reynolds number

. When Re p is much less one, namely a Stokes drag law is valid, we have f (Re p ) ≈ 1. In general, Clift et al. 42 give the relationship f (Re p ) = 1 + 0.15Re 0.687 p for Re p < 40.

We consider the particle motions in a 2D convection cell with size H × H. The top and bottom walls of the cell are kept at a constant cold and hot temperatures, respectively; the other two vertical walls are adiabatic. All four walls impose no-slip velocity boundary conditions. Our simulation protocol is as follows. We start the simulation of single-phase turbulent thermal convection, namely without considering the particles' motion. The particles are released in the turbulence after statistically stationary state has reached, which takes 500 t f . Here, t f = H/(gβ ∆ T ) denotes free-fall time units. We then advance the fluid flows and the motion of the particles simultaneously. A total number of 10,000 particles are initially placed at the cell central region (see Fig.   1 for the illustration, the 10,000 particles are initially grouped into a 100 × 100 array, and each particle is placed half grid spacing away from the other). The initial velocities of the particles are equal to that of the local fluid. The initial particle configuration approximates the transport of pollutant particles emitted from a source, and the dilute particles may mimic the particle-laden fluid in a cough 43 . We average 2,000 t f to obtain statistics for the turbulent flows and the particles.

When a particle hits the wall, we assume it will deposit on the wall and no longer transport in the convection cell.

We provide simulation results for a fixed Rayleigh number of Ra = 10 8 and Prandtl number of Pr = 0.71 (corresponding to the working fluids of air at 300 K). Other detailed simulation parameters are listed in Table 1 . The mesh size is 513 × 513, such that the grid spacing ∆ g and time interval ∆ t is properly resolved to compare with the Kolmogorov and Batchelor scales.

Here, the Kolmogorov length scale is estimated by the global criterion In the simulations, the non-dimensional control parameters for the particles include the density ratio of the particle to its surrounding fluid Γ = ρ p /ρ f , and the size ratio Ξ = d p /l f . By combing the Γ and Ξ, we can obtain the particle Stokes number (St) and the Archimedes number (Ar) as

where τ p is particle response time. The St describes the particle inertia relative to that of the fluid, and the Ar describes the ratio of gravity forces to the viscous forces. Because we have fixed the Ra and the Pr in the simulation, namely thermal convection related quantities are fixed, we then have

p . The St and the Ar numbers can be uniquely determined by d p and ρ p , as shown in Fig. 2 . We explore the parameter space of 10 µm ≤ d p ≤ 100 µm and 400 kg/m 3 ≤ ρ p ≤ 4000 kg/m 3 , denoted by the black circles in Fig. 2 . We note that the estimated Kolmogorov length scale is η K = 2.27 mm, and the largest particle volume fraction of all cases is only 0.02%. Thus, for dilute particles with diameters fall in the range mentioned above, the one-way coupling strategy is justified to model their motions. For particles with larger number but still similar size, i.e., particles with higher particle volume fraction, a four-way coupling strategy is necessary to describe the interactions between the particle and its surrounding fluid 46 

A. Particle transport in the convection cell On the other hand, we also notice the differences in the spatial pattern of particles' positions: the particles are more homogeneously distributed in the turbulence at relatively smaller particle density (see Figs. 3(a-c), which corresponds to 3.67 × 10 −4 ≤ St ≤ 9.19 × 10 −4 and 0.115 ≤ Ar ≤ 0.183). In contrast, they tend to cluster into bands at relatively larger particle density (see to those in homogeneous isotropic turbulence [14] [15] [16] [17] . We also notice there are fewer particles in the corner rolls of the convection with the increase of particle density. The previous study by Park et al. 31 indicates that the clustering behavior in thermal turbulence occurs at much larger particle St number (namely St ≈ 1) when the dimensionless particle settling velocity V g /U buoy = ρ p d 2 p g/(18µ f ) / gβ ∆H is fixed as 0.001. However, if we assume the carrier fluid is air, a quantitative estimation shows that simultaneously achieving St ≈ 1 and would result in artificially tiny gravity value (almost seven orders of magnitude smaller than 9.8 m/s 2 ).

The-above visualizations illustrate the preferential distribution of particles in the thermal turbulence. To quantitatively describes the spatial distribution of the particles, we divide the simulation domain into 100 × 100 uniform subcells and calculate the local particle number density as

where N(i, j,t) is the number of suspended particles found inside the (i, j)-th small square subcell (here 1 ≤ i, j ≤ 100), and N total (t) is the number of suspended particle in the whole convection cell at time t. In Fig. 4 , we plot the local particle number density at t = 500 t f , where we can observe homogenous local particle number densities for 400 kg/m 3 ≤ ρ p ≤ 1000 kg/m 3 . The local particle number densities are more inhomogeneous for 2000 kg/m 3 ≤ ρ p ≤ 4000 kg/m 3 , which is due to higher particle inertia and longer particle response time to the carrier flow.

We further calculate the relative standard deviation of the local particle number density, namely, the root-mean-square (r.m.s.) of particle number density normalized by the volume-averaged particle number density, which is defined as

Here,n(t) denotes the volume-averaged particle number density at time t. In Fig. 5 , we plot the time histories of the relative standard deviation for particles with a diameter of 10 µm. We can see that the deviations decrease rapidly during the initial transient state (i.e., t 250 t f ), which is due to the dispersion of the particle group after being released in the turbulence. At t 250 t f , the relative standard deviations nearly reach a plateau, indicating the well dispersion of the particles in the turbulence. We also found that the relative standard deviation of local particle number density depends on the St and Ar, as light density and small size of the particles favor their dispersion.

We then analyze the statistics of particles' trajectories by calculating their mean-square dis-

Here, r(t) is the particle's position at time t, and τ is the lag time between the two positions taken by the particles. The average · · · represents a time-average over t and an ensemble-average over trajectories. When a particle is deposited on the wall, we will stop tracking its trajectory. Fig. 6(a) are isotropic. We decompose the distance vector r into a lateral (r x ) and vertical (r y ) part, and calculate the MSD in the lateral and vertical directions separately as

From Fig. 6(b) , we can see that the MSD is isotropic at short time intervals, while the differences between MSD x and MSD y are apparent at longer time intervals. We can also roughly estimate how the particle is constrained in different directions by calculating the square root of the plateau MSD value. The results in Fig. 6 (b) indicates that the vertical region of constraint is a bit larger than that of the lateral region. The reason is that most of the suspended particles are trapped within the elliptical primary roll, whose major axis has a longer vertical component than the horizontal one. Thus, when the LSC advects the particles, they will 'travel' longer distance in the vertical direction than the lateral one.

The-above analysis focused on relatively small particles that will be dispersed in the turbulence.

In contrast, for relatively large particles (e.g., particles with d p = 100 µm), they will sediment quickly after being released in the turbulence, as shown in Fig. 7 . The carrier flow minorly influences the particles' motions, and the particle group almost remains their initial shape (namely the square shape due to the artificial simulation setting, see Fig. 1 ) during the sedimentation. Because the LSC of the convection is clockwise rotated, the deposition location of the particle group on the bottom wall will be left side offset their initial horizontal position. We also observe that the shape of the lighter particle group will stretch more during the sedimentation. In comparison, heavier influenced with such small tilted angle 50, 51 . In the tilted case, the hot plumes arise along the right vertical wall, and we can see more particles are deposited on the right vertical wall. Thus, a general conclusion is that particles prefer to deposited on the vertical wall where the hot plumes arise.

With the numerical simulations in a wide range of d p and ρ p parameter spaces, we can then obtain the phase diagram for the particle deposition positions on the walls. As shown in Fig. 9 , particles with smaller d p and ρ p are more easily suspended and well dispersed in the flow; thus, the particles have chances to deposit on the left and right vertical walls, and of course, most of the particles will deposit on the bottom wall due to the gravity sedimentation (denoted as 'Three walls'

in the phase diagram). For particles with larger d p and ρ p , the carrier flows minorly influences them, and the particles will only deposit on the bottom wall (denoted as 'One wall' in the phase diagram). The 'One wall' deposition state also corresponds to the initially released particle group not well dispersed in the turbulence. Sandwiched between the 'Three walls' and 'One wall' states is the 'Two walls' state, where particles will deposit on the bottom wall and one vertical wall at medium d p and ρ p (namely, medium St and Ar). This transition state of particle deposition on only one vertical one is due to that particles exhibit cluster behavior and they are not well dispersed in the flow compared to the cases in 'Three walls' state. On the other hand, in the transition state, the particles will still be majorly advected in the convection compared to the cases in 'One wall' state, and if particles deposit, they will only deposit on vertical walls where the hot plumes arise.

For the explored parameter space of d p and ρ p , we confirm that there are no particles deposited on the top wall. From the phase diagram, we can also observe the borders between different states are strongly correlated with the St and the Ar numbers. 

In this work, we have performed numerical simulations of particle motion in turbulent thermal convection. Specifically, we analyzed the statistics of particle transport and deposition in 2D

square RB convection cell. The main findings are summarized as follows:

1. The suspended particles are more homogeneously distributed in the turbulence at St less than 10 −3 , and they tend to cluster into bands for 10 −3 St 10 −2 . At even larger St, the particles' motion will be minorly influenced by the turbulence, and they will sediment quickly and deposit on the boundary walls. 3. We obtained a phase diagram of the particle deposition positions, and three deposition states were identified: particles deposited on three walls, two walls, and one wall. Although most of the particles will deposit on the bottom wall, we found there is still a tiny portion of particles deposited on the vertical wall. Moreover, the particles preferred to deposit on the vertical wall where the hot plumes arise. 

The data that support the findings of this study are available from the corresponding author upon reasonable request.

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