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Large eddy simulation of particle-laden turbulent channel flow

Qunzhen Wanga) and Kyle D. SquiresDepartment of Mechanical Engineering, 209 Votey Building, University of Vermont,Burlington, Vermont 05405

Received 25 July 1995; accepted 26 January 1996

Particle transport in fully-developed turbulent channel flow has been investigated using large eddy

simulation LES of the incompressible NavierStokes equations. Calculations were performed at

channel flow Reynolds numbers, Re , of 180 and 644 based on friction velocity and channel half

width ; subgrid-scale stresses were parametrized using the Lagrangian dynamic eddy viscositymodel. Particle motion was governed by both drag and gravitational forces and the volume fraction

of the dispersed phase was small enough such that particle collisions were negligible and properties

of the carrier flow were not modified. Material properties of the particles used in the simulations

were identical to those in the DNS calculations of Rouson and Eaton Proceedings of the 7th

Workshop on Two-Phase Flow Predictions 1994 and experimental measurements of Kulicket al.

J. Fluid Mech. 277, 109 1994 . Statistical properties of the dispersed phase in the channel flow at

Re180 are in good agreement with the DNS; reasonable agreement is obtained between the LES

at Re644 and experimental measurements. It is shown that the LES correctly predicts the greater

streamwise particle fluctuation level relative to the fluid and increasing anisotropy of velocity

fluctuations in the dispersed phase with increasing values of the particle time constant. Analysis of

particle fluctuation levels demonstrates the importance of production by mean gradients in the

particle velocity as well as the fluid-particle velocity correlation. Preferential concentration of

particles by turbulence is also investigated. Visualizations of the particle number density field nearthe wall and along the channel centerline are similar to those observed in DNS and the experiments

of Fessler et al. Phys. Fluids 6, 3742 1994 . Quantitative measures of preferential concentration

are also in good agreement with Fessler et al. Phys. Fluids 6, 3742 1994 . 1996 American

Institute of Physics. S1070-6631 96 00905-4

I. INTRODUCTION

Gas-phase turbulent flow fields laden with small heavy

particles occur in a wide range of engineering and scientific

disciplines. Examples are as diverse as pollutant dispersion

in the atmosphere and contaminant transport in industrialapplications. In these as well as many other instances the

primary interactions are from fluid to particles only, i.e., in

the dilute regime in which particle collisions as well as the

effect of particles on fluid mass and momentum transport is

negligible. Even with the restriction to dilute flows, accurate

prediction of particle-laden turbulence is important in order

to gain a better understanding of particle transport by turbu-

lence as well as ultimately improve the design of the engi-

neering devices in which two-phase flows occur.

The principal difficulty with the prediction of particle-

laden turbulent flows is that traditional approaches model

particle transport using gradient transport hypotheses and do

not accurately account for the complex interactions betweenparticles and turbulence. Traditional methods are usually

based on the Reynolds-averaged NavierStokes RANS

equations in which the entire spectrum of velocity fluctua-

tions is represented indirectly using various parametrizations,

e.g., K- models. A primary shortcoming of RANS methods

for the prediction of particle-laden turbulent flows is related

to deficiencies associated with the model used to predict

properties of the Eulerian turbulence field. Accurate predic-

tion of particle transport is strongly dependent upon provid-

ing a realistic description of the time-dependent, three-

dimensional velocity field encountered along particle

trajectories e.g., see Berlemont et al.,1 Simonin et al.2 . De-

ficiencies in the prediction of turbulence properties in RANScalculations will in turn adversely impact prediction of dis-

persed phase transport. Thus, predictive techniques which

are generally applicable to a wide class of turbulent two-

phase flows and accurately represent the underlying structure

of turbulence are needed.

The most sophisticated approach to representing the un-

derlying structure of turbulence and, hence, particle transport

in turbulent flows is direct numerical simulation DNS . In

DNS the NavierStokes equations are solved without ap-

proximation other than those associated with the numerical

method . DNS has been successfully employed in a number

of studies which have increased our fundamental understand-

ing of many of the mechanisms governing particle interac-

tions with turbulence. Much of this work has been performed

in isotropic turbulence and has shown, for example, that par-

ticles with time constants of the order of the Kolmogorov

timescale are preferentially concentrated into regions of low

vorticity and high strain rate e.g., see Squires and Eaton,3

Wang and Maxey4 . Preferential concentration results from

an inertial bias in the particle trajectory and also leads to

substantial increases in particle settling velocities in homo-

geneous turbulence Wang and Maxey4 . DNS has also been

used to examine particle transport in wall-bounded shear

a Phone: 802 656-1940; FAX: 802 656-1929; electronic mail:

[email protected], [email protected]

1207Phys. Fluids 8 (5), May 1996 1070-6631/96/8(5)/1207/17/$10.00 1996 American Insti tute of Physics

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flows e.g., see Rashidi et al.,5 Pedinotti et al.,6 Rouson and

Eaton7 . Rashidi et al.5 and Pedinotti et al.6 have shown that

particles can become preferentially concentrated in the low-

speed streaks which characterize the near-wall region and are

subsequently resuspended by ejections from the wall. Rou-

son and Eaton7 found that particles possessing similar mate-

rial properties to those used in the experiments of Kulick

et al.8 see also Fessler et al.9 demonstrated effects of pref-

erential concentration. Application of DNS to the study of

particle deposition in turbulent boundary layers by

McLaughlin,10 Brooke et al.,11,12 and Chen et al.13 has

clearly shown that particles accumulate in the near wall re-

gion.

Aside from the interactions between particles and turbu-

lence resulting in preferential concentration in wall-bounded

shear flows, mean gradients in the fluid and particle veloci-

ties have complex effects on particle velocity fluctuations.

Both DNS and experiments have demonstrated that particle

fluctuations consistently lead those of the fluid near the wall

and, contrary to measurements in isotropic turbulence, that

the streamwise velocity variance increases with increasing

particle response time e.g., see Rogers and Eaton,14 Kulick

et al.,8

Rouson and Eaton7

. Analyses by Liljegren15

andReeks16 are consistent with these findings and have demon-

strated the important effect of mean shear in both the fluid

and particle velocity field. Some Eulerian-based models have

also recognized the importance of accounting for the effect

of mean shear on particle fluctuation levels e.g., see Kataoka

and Serizawa,17 Abou-Arab and Roco,18 Reeks,19 Hwang and

Shen,20 Simonin et al.2 .

The effects of preferential concentration on particle

transport as well as other features such as the effect of mean

velocity gradients on particle fluctuation levels are very dif-

ficult to represent using conventional predictive methods.

DNS, while an extremely powerful tool for supplying infor-

mation which cannot be obtained from experiment, is notpractical for use as a predictive tool because it remains re-

stricted to relatively low Reynolds number turbulent flows.

An approach which is not as severely restricted in the range

of Reynolds numbers as DNS is large eddy simulation

LES . In LES the large, energy containing scales of motion

are calculated directly while only the effect of the smallest

subgrid scales of motion are modeled. Thus, LES predic-

tions are less sensitive to modeling errors than in RANS

calculations and, since the subgrid scales are more universal

than the large scales, it is also possible to represent the effect

of the subgrid scales using relatively simple models. A sig-

nificant advantage of LES over RANS methods is that it

permits a much more accurate accounting of particle-

turbulence interactions.

The primary drawback to the application of LES for the

prediction of complex turbulent flows has traditionally been

much the same as that which currently limits RANS meth-

ods, i.e., an inability of the subgrid-scale SGS model to

account for changes in the spectral content of the turbulence

under a variety of conditions, e.g., changes in the Reynolds

number, type of flow, etc., without ad hoc tuning. The de-

velopment of dynamic SGS modeling Germano et al.21 ,

however, has considerably improved the viability of LES as

a tool for prediction of complex flows since the eddy viscos-

ity is calculated during the course of the computation and in

turn reflects local properties of the flow e.g., see Squires and

Piomelli22 . SGS models which reflect local properties of the

turbulence are especially attractive for the computation of

particle-laden flows since predictions of particle transport

should be expected to be significantly improved by more

accurate SGS models.

Thus, the principal objective of this work is application

of large eddy simulation to computation of a well-defined

turbulent shear flow, fully-developed channel flow, for which

DNS results and experimental measurements exist for com-

parison and evaluation of LES predictions. As discussed

above, particle interactions with turbulence in wall-bounded

shear flows result in both complex statistical behavior of the

particle velocity field as well as complicated structural fea-

tures, i.e., preferential concentration. Therefore, a primary

interest of this study is to determine the utility of LES for

prediction of these effects. Contained in Sec. II is an over-

view of the simulations. Comparison of LES predictions to

DNS results as well as experimental data is presented in

Secs. III and IV. In addition to statistical properties such as

mean and fluctuating particle velocities, instantaneous par-ticle distributions near the channel wall and centerline are

analyzed in detail, both qualitatively and quantitatively, to

investigate the degree to which LES represents preferential

concentration. A summary of the work may be found in Sec.

V.

II. SIMULATION OVERVIEW

A. LES of turbulent channel flow

The turbulent flow between plane channels driven by a

uniform pressure gradient was calculated using LES of the

incompressible NavierStokes equations. The equations

governing transport of the large eddies obtained by filteringthe NavierStokes equations are

ui

x i0, 1

ui

t

xj uiuj

p

x i

1

Re

2ui

xj xj

i j

xji1 , 2

where u i is the fluid velocity, p is the pressure, i j is the

Kronecker delta, and i1,2,3 refers to the x streamwise ,

y normal , and z spanwise directions, respectively the

usual summation notation applies and an overbar, , denotes

application of the filtering operation

. The governing equa-

tions 1 and 2 have been made dimensionless using the

channel half-width and friction velocity u , yielding a

mean pressure gradient of1 corresponding to i1 in Eq.

2 . The Reynolds number in 2 is Reu/, where is

the kinematic viscosity. For fully-developed channel flow

periodic boundary conditions for the dependent variables are

applied in the streamwise and spanwise directions, whereas

the no-slip condition is applied on the channel walls.

The effect of the subgrid scales on the resolved eddies in

2 is represented by the SGS stress, i ju iujuiuj . In this

worki j is parametrized using an eddy viscosity hypothesis,

1208 Phys. Fluids, Vol. 8, No. 5, May 1996 Q. Wang and K. D. Squires



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i j1

3i jkk2TS

i j , 3

where the eddy viscosity is

TC2 S , 4

the resolved-scale strain rate tensor is defined as

Si j1

2

ui

xj

uj

x i, 5

and S 2S

i jS

i j is the magnitude of S

i j . The filter width

is defined as (123)1/3 where 1 , 2 , and 3 are

the grid spacings in the x , y , and z directions, respectively.

The model coefficient C in 4 requires specification in order

to close the system 1 and 2 .

The model coefficient C is determined dynamically us-

ing the resolved velocity field. Following Germano et al.,21 a

second filter, the test filter denoted using , is introduced to

derive an expression for C. Germano23 showed that the re-

solved turbulent stress,

L i juiuju

iu

j , 6

the SGS stress i j , and the subtest-scale stress Ti ju iuj

u

iu

j , are related by

L i jTi ji j. 7

Assuming that a closure similar to 3 may be used to model

the test-field stress Ti j , it is possible to use 7 to derive an

expression for C. The model coefficient C was calculated

following the approach developed by Meneveau et al.24 in

which the error in 7 is minimized along fluid particle tra-

jectories, resulting in an expression for the model coefficient

of the form

C x,t ILMIMM

. 8

In principle, ILM and IMM are obtained from the solution of

separate transport equations. However, to reduce the compu-

tational cost associated with calculation of C, the numerator

and denominator of 8 are obtained using a simple time

discretization, resulting in

ILMn1

x H L i jn1

Mi jn1

x 1 ILMn xui

nt , 9

IMMn1

x Mi jn1Mi j

n1 x 1 IMM

n xui

nt , 10

where H x max(x,0) is the ramp function, the timescale in 10 is defined as T2ILM

1/4 , and

t/Tn

1t/Tn. 11

The quantity Mi j is dependent upon the closure approxima-

tion and for an eddy viscosity model is

Mi j2 S Si j

2 S Si j . 12

The values of ILMn (xuit) and IMM

n (x uit) are obtained

through linear interpolation see Meneveau et al.24 for a fur-

ther discussion . The test filter width in the streamwise and

spanwise directions was twice the grid filter width also see

Germano et al.21 . Test filtering in the streamwise and span-

wise directions was carried out in Fourier space using a sharp

cut-off filter.

The governing equations 1 and 2 were solved nu-

merically using the fractional step method on a staggered

grid e.g., see Kim and Moin,25 Perot,26 Wu et al.27 .

Second-order AdamsBashforth was used for advancement

of the convective terms and part of the SGS term while the

CrankNicholson method was applied for an update of the

viscous terms and a portion of the SGS stress. The Poisson

equation for pressure was solved using Fourier series expan-

sions in the streamwise and spanwise directions together

with tridiagonal matrix inversion.

Large eddy simulations were performed at Reynolds

numbers based on friction velocity and channel half-width of

180 and 644 corresponding to Reynolds numbers of 3,200

and 13,800 based on centerline velocity and channel half-

width . At both Reynolds numbers the flow was resolved

using 646564 grid points in the x , y , and z directions,

respectively. The channel domain for the calculation at

Re

180 was 4

2

4

/3 and 5

/2

2

/2 atRe644. Previous computations of turbulent channel flow

have shown these domain sizes are large enough to avoid

contamination of the flow by periodic boundary conditions

Piomelli28 . The grid spacing in wall coordinates in the x

and z directions was x35 and z12 at Re180

and x79 and z16 at Re644. A stretched grid

was used in the wall-normal direction and for both Reynolds

numbers the first grid point was at y1.

B. Calculation of particle trajectories

The particle equation of motion used in the simulations

describes the motion of particles with densities substantiallylarger than that of the surrounding fluid and diameters small

compared to the Kolmogorov scale:

dv i

dt

f

p

3

4

CD

d vu v iu i gi1 , 13

where v i is the velocity of the particle, u i is the velocity of

the fluid at the particle position, and g is the acceleration of

gravity. The body force acts along the positive streamwise

direction corresponding to flow in a vertical channel. The

fluid and particle densities in 13 are denoted f and p ,

respectively, and d is the particle diameter. Previous compu-

tations of particle-laden turbulent channel flow have shown

that the particle Reynolds number, Rep vu d/, does not

necessarily remain small e.g., see Rouson and Eaton,7 Wang

and Squires29 . Therefore, an empirical relation for CD from

Clift et al.30 valid for particle Reynolds numbers up to about

40 was employed,

CD24

Rep 10.15Rep

0.687 . 14

It should also be noted that 13 is appropriate for describing

the motion of smooth rigid spheres and neglects the influence

of virtual mass, buoyancy, and the Basset history force on

1209Phys. Fluids, Vol. 8, No. 5, May 1996 Q. Wang and K. D. Squires



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particle motion. For particles with material densities largecompared to the fluid these forces are negligible compared to

the drag. Previous investigations have shown that the effect

of the lift force, while relevant to problems of particle depo-

sition, is less significant to this work and therefore the effect

of shear-induced lift in the equation of motion has been ne-

glected see Wang and Squires29 . Finally, the volume frac-

tion of particles is assumed small enough such that particle-

particle interactions are negligible.

From computation of an Eulerian velocity field, 13 was

integrated in time using second-order Adams Bashforth.

Since it is only by chance that a particle is located at a grid

point where the Eulerian velocity field is available, fourth-

order Lagrange polynomials were used to interpolate the

fluid velocity to the particle position see Wang et al.31 for

further discussion . Particle displacements were also inte-

grated using the second-order AdamsBashforth method.

For particles that moved out of the channel in the streamwise

or spanwise directions periodic boundary conditions were

used to reintroduce it in the computational domain. The

channel walls are perfectly smooth and a particle was as-

sumed to contact the wall when their center was one radius

from the wall. Elastic collisions were assumed for particles

contacting the wall.

Properties of the dispersed phase were obtained by fol-

lowing the trajectories of 250,000 particles. The trajectoriesof a large ensemble of particles are required in order to

present statistics for the dispersed phase in the same manner

as for the fluid, i.e., by averaging over homogeneous planes.

Numerical experiments demonstrated adequate statistical

convergence was obtained using this sample size see the

Appendix for further discussion . The particles used in the

simulations possess material characteristics identical to those

in the DNS study of Rouson and Eaton 7 and experiments of

Kulick et al.8 and Fessler et al.9: 28 m diameter Lycopo-

dium particles, 25, 50, and 90 m diameter glass beads, and

70 m copper particles. The particle response time and ra-

dius a expressed in terms of both channel variables ( and

u) and in wall units for the simulations at Re 180 andRe644 are shown in Tables I and II, respectively the

response time ppd2/( 18f) shown in the table is the

value corresponding to Stokes flow around the particle . The

particle parameters at Re644 have been normalized by the

experimental values of channel half-width 20 mm and

friction velocity u0.49 m/s see Kulick et al.8 . The par-

ticle parameters in the DNS at Re 180 were scaled by

Rouson and Eaton7 to match the Stokes numbers defined in

terms of the Kolmogorov timescale in the experimental mea-

surements of Kulick et al..8 For both Reynolds numbers the

particle radius is much smaller than the filter width except

very near the wall where the particle radius can be compa-

rable to the wall-normal grid spacing the first grid point was

at y0.45 at Re180 and at y0.84 at Re644).

Thus, the effect of a nonuniform fluid velocity field on par-

ticle motion near the wall may be less accurately represented

for the larger particles.

III. PARTICLE VELOCITY STATISTICS

From an arbitrary initial condition the Eulerian velocity

field was time advanced to a statistically stationary state. The

particles were then assigned random locations throughout the

channel. The initial particle velocity was assumed to be the

same as the fluid velocity at the particle location. Particles

were then time advanced in the flow field to a new equilib-

rium condition in which particle motion was independent of

initial conditions. Similar to the fluid flow, statistics of the

particle velocity were averaged over the two homogeneous

directions, both channel halves, and time. Fluid statistics

were averaged for 3.5/u at Re180 and for 4/u at

Re 644. The development time, i.e., the time required for

particles to become independent of their initial conditions

was larger for increasing values of the particle response time,

e.g., 0.5/u for the Lycopodium particles and 6/u for the

copper particles at Re180. After an equilibrium condition

had been reached, particle statistics were accumulated for6/u at Re180 and 4.5/u at Re644.

The mean streamwise velocity profile obtained from the

LES calculations is shown in Figure 1 for each particle type

considered in the DNS Fig. 1 a and experiments Fig.

1 b . Figure 1 a shows that at Re180 there is good

agreement between LES and DNS. As expected, both the

LES and DNS show that the Lycopodium particles closely

track the fluid flow. Near the wall (y10) the Lycopodium

velocity profile from the DNS slightly lags that of the fluid

while the mean velocity from the LES is nearly equal to the

fluid velocity. Rouson and Eaton7 attribute the lag in the

Lycopodium profile to preferential concentration of Lycopo-

dium particles in the low-speed streaks. The discrepancy be-tween LES and DNS results may be related to the fact that

TABLE I. Particle parameters in turbulent channel flow, Re 180.

28 m

Lycopodium

50 m

glass

70 m

copper

p /( /u) 0.048 0.65 4.50

p 9 117 810

a/ 0.00139 0.00277 0.00388

a 0.25 0.50 0.70

TABLE II. Particle parameters in turbulent channel flow, Re 644.

7 m

Lycopodium

14 m

Lycopodium

28 m

Lycopodium

25 m

glass

50 m

glass

90 m

glass

70 m

copper

p /( /u) 0.0025 0.01 0.04 0.12 0.44 1.54 3.10

p 2 6 26 77 283 992 1996

a/ 0.000175 0.00035 0.0007 0.000625 0.00125 0.00225 0.00175

a 0.113 0.225 0.450 0.403 0.805 1.449 1.127




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the near-wall structures are less well resolved in the LES

and, consequently, preferential concentration of particles in

low-speed regions is less significant than in the DNS, result-

ing in an over-prediction of the mean velocity of the Lyco-

podium particles see Sec. IV A for further discussion . For

the glass beads and copper particles the profiles in Figure

1 a become increasingly larger than the fluid velocity for

increases in the Stokes number. LES predictions of the mean

velocity of 50 m glass beads are in good agreement with

the DNS; the copper velocity profile from the LES calcula-

tions is also in good agreement with DNS results for

y10 but it may be observed from the figure that the DNS

results are larger near the wall. Note also that both LES and

DNS profiles of the copper particles exhibit a slight plateau

near the wall. The plateau may result from the transport of

high velocity particles from the outer region of the channel

to the near-wall region and this feature appears to be some-

what more pronounced in the DNS. However, it is also im-

portant to note that the difference between LES and DNS is

relatively small and confined to a very thin region near the

wall.

Particle mean velocity profiles at Re644 from the

LES calculations are shown in Figure 1 b together with the

experimental measurements for 50 m glass beads and cop-

per particles from Kulick et al.8 Similar to the results at the

lower Reynolds number, LES predictions show greater dif-

ferences in the mean particle velocity relative to the fluid

with increasing Stokes number; the largest differences occur-

ring near the wall. As also observed at Re180, near the

wall the Lycopodium particles slightly lead the fluid and are

then nearly identical to the fluid velocity in the outer region.

Comparison of the profiles for the 50 m glass beads dem-

onstrates relatively good agreement for y greater than about

20. However, for y20 the results in Figure 1 b show that

the experimental measurements increase towards the wall

while LES predictions do not. It may also be observed that

there is relatively poor agreement between the LES and Ku-

lick et al.8 for the copper particles. For y greater than

roughly 10, the mean profile of the copper particles in the

experiment is nearly equal to that of the fluid while in the

LES the copper particles lead the fluid throughout the chan-

nel. As also apparent in Figure 1 b , Kulick et al.8 found that

the mean velocity of the copper particles increased substan-

tially near the wall, an effect not observed in the LES at

either Reynolds number. It should also be noted that the

mass loading corresponding to the experimental measure-

ments shown in the figure is 2%. Kulick et al.8 also mea-

sured particle velocity statistics at higher mass loading where

the effect of particles on fluid turbulence becomes signifi-cant.

Thus, it is clear there are substantial differences between

the LES and experimental measurements of the copper ve-

locities at Re644. Kulick et al.8 attributed the increase in

near-wall copper velocity to the possibility that high-speed

particles rebounding from the wall maintain a significant

fraction of their streamwise momentum. The plateau in the

copper velocity profile near the wall is consistent with notion

of elastic collisions of high-speed particles contacting the

wall and is also in agreement with recent examinations of

particle deposition which have shown that particles are

brought to the near wall region by events with normal veloc-

ity much larger than the local turbulence intensity Brookeet al.12 . Unlike the experimental measurements, however,

the resulting copper mean velocity in the LES does not in-

crease in the near wall region.

Comparison of rms particle velocity fluctuations from

the LES to both DNS results and experimental measurements

is shown in Figure 2 for Re180) and Figure 3 for

Re644). As may be observed in Figure 2, there is in gen-

eral good agreement between LES predictions and the DNS

results of Rouson and Eaton.7 It is clear from the figure that

near the wall the streamwise fluctuation levels increase with

increasing values of the Stokes number while the wall-

normal and spanwise fluctuations are reduced. Comparison

of the LES and DNS results also shows that the peak values

in the LES profiles occur at slightly larger y than in the

DNS. It is further interesting to note that the streamwise rms

velocities of the Lycopodium and glass beads from the LES

calculations are smaller than the DNS values while the wall-

normal and spanwise rms velocities in the LES are slightly

greater than the corresponding values in the DNS.

Rms intensities from the channel at Re644 are shown

in Figure 3. Velocity profiles for the Lycopodium particles

are not available from the experiments; LES predictions are

shown in Figure 3 a for comparison and are consistent with

FIG. 1. Mean streamwise velocity in turbulent channel flow. a

Re180; b Re644. LES:, fluid elements; ---, Lycopodium; , 50

m glass; , copper; Rouson and Eaton7 in a and Kulick et al.8 in b :

; Lycopodium; , 50 m glass; *, copper.




6/17

the data at the lower Reynolds number, i.e., the Lycopodium

fluctuations are nearly equal to the fluid values but lead in

the streamwise direction while lagging in the wall-normal

and spanwise directions. Experimental measurements of par-

ticle velocities are available for the 50 m glass beads and

copper particles and it is evident in Figure 3 b that there is

good agreement between LES predictions and the measured

streamwise intensities of the glass beads for y10. The

wall-normal fluctuations in the experiment are greater than

the LES values but the location of the peak in the wall-

normal fluctuations is reasonably well predicted. The great-

est discrepancy between LES and experimental measure-

ments occurs for the copper particles. Figure 3 c shows that

the streamwise intensities in the experiment are larger than

the LES predictions, and unlike the LES and DNS values at

Re180, the streamwise intensities in the experiment peak

at around y12. Kulick et al.8 showed that the probability

FIG. 2. Root-mean-square velocity fluctuations in turbulent channel flow,

Re180. a Lycopodium; b 50 m glass; c copper. LES fluid: ,

i1; , i2; i3; LES particle:, i1; i2; ---, i3; particle

fluctuations from Rouson and Eaton:7

, i1; * i2; i3.

FIG. 3. Root-mean-square velocity fluctuations in turbulent channel flow,

Re644. a Lycopodium; b 50 m glass; c copper. LES fluid: ,

i1;, i2; i3; LES particle:,i1; , i2; ---, i3; particle fluc-

tuationsfrom Kulicket al.:8 , i 1; *, i 2.




7/17

distribution function pdf of the streamwise copper velocity

was bimodal at y12, demonstrating that the streamwise

copper intensities should not be interpreted as the width of a

Gaussian velocity distribution. Pdfs of the copper velocities

in the LES around y12 were examined and not found to

exhibit a similar bi-modal structure as in the experiment.

Finally, comparison of the spanwise fluctuation levels for all

the particles show a reduction with increasing Stokes number

and greater similarity to the fluid for the smaller Stokes num-

bers.

It is difficult to speculate as to the precise cause of the

differences between LES predictions and both DNS results

and experimental measurements. Errors in the SGS model

used in LES contribute to the differences as well as other

factors such as the different particle sample sizes and the

interpolation scheme used to obtain fluid velocities at par-

ticle positions in the computations. Another source of error

in LES predictions of the particle velocity statistics is due to

the neglect of particle transport by SGS velocities. In LES

the smallest scales of motion are not resolved by the compu-

tational grid, only their effect on the large eddies is repre-

sented via the SGS model. Thus, only the large-scale veloc-

ity field is directly available in a LES computation fordetermining particle motion; for the results presented above

the effect of subgrid-scale velocity fluctuations on particle

transport were neglected. One measure of the error is the

value of the particle relaxation time relative to the smallest

resolved timescale in the LES, T/ ( S )1/ S . AtRe180 the timescale T increases from about 0.011 near

the wall to roughly 0.11 near the channel centerline while at

Re644 increases from 0.002 to 0.08. Thus, some of the

particle relaxation times considered in the LES are compa-

rable to the smallest resolved timescale see Tables I and II .

The effect of the SGS velocity field on particle transport

was investigated by adding SGS fluctuations to the fluid ve-

locity used in the particle equation of motion 13 . Calcula-tions were performed in the channel flow at Re644 in

which the fluid velocity in the particle equation of motion

was the resolved component ui , directly available in the

LES, plus a subgrid contribution u i. The magnitude of the

SGS fluctuation u i was determined by solving a transport

equation for SGS kinetic energy, q 2. The transport equation

used for determination of q2 is that proposed by

Schumann,32 i.e.,

q 2

tuj

q 2

xj2T S

2xj

1

3lq

2/2q 2

xj

2q 2

xj xj12cq

3

l, 15

where

c2

3k0

3/2

,lmin ,cy , 16

with the Kolmogorov constant k01.6. Shown in Figure 4 is

the profile of the SGS fluctuation q 2/3 along with the re-solved components. SGS fluctuations are larger than the re-

solved wall-normal velocity near the wall, and the SGS en-

ergy peaks at about the same location as the resolved

streamwise fluctuations.

In the simulations SGS intensities were obtained from

q2

and specified to have the same relative magnitudes as theresolved-scale intensities. The component fluctuations u i

were then scaled by random numbers sampled from a Gauss-

ian distribution and added to ui at the particle location. The

complete velocity, i.e., uiu i, was subsequently used in

13 to determine the particle velocity. Figure 5 shows the

wall-normal fluctuations for the 28 m Lycopodium par-

ticles at Re644 both with and without SGS velocity fluc-

tuations included in 13 . As is evident from the figure there

is a negligible effect of the SGS fluid velocity on particle

fluctuations. The relative difference is greatest near the wall

where SGS fluctuations are large compared to the resolved

components, but the change in the wall-normal particle ve-

locities due to the addition of the SGS fluid velocity is lessthan 1%. Although not shown here, the difference in particle

fluctuations in the other directions is similar and there is

essentially no effect on the mean streamwise particle veloc-

ity. Figure 5 demonstrates the strong filtering effect of par-

FIG. 4. Rms velocity fluctuations in turbulent channel flow, Re 644.

q 2/3; ---, u1,rms ; , u2,rms ; , u3,rms .

FIG. 5. Wall-normal velocity fluctuations for 28 m Lycopodium particles,

Re644. , without SGS velocity in the particle equation of motion; ---,

with SGS velocity in the particle equation of motion.




8/17

ticle inertia on the fluid velocity spectrum. For particles with

material densities large compared to the carrier flow the re-

sponse of particles to the frequency spectrum of the turbu-

lence can be shown to be proportional to 1/( p)2 ( is the

frequency . Thus, for increasing values of the relaxation time

and/or frequency the filtering of high frequency motions by

particle inertia is significant, consistent with the results in

Figure 5.

Figures 2 and 3 show that at both Reynolds numbers the

streamwise fluctuations of the particle velocities are signifi-

cantly larger than the wall-normal or spanwise values and are

also larger than the streamwise fluctuations in the fluid. Fur-

thermore, the wall-normal and spanwise fluctuations are re-

duced for increasing particle inertia while the streamwise

values increase see also Rogers and Eaton,14 Kulicket al.8 .

The relative strength of the streamwise intensities relative to

the other components is clearly illustrated through examina-

tion of the diagonal components of

bu,v

f2

fifi, 17

where f is either the fluid or particle velocity fluctuation. Ifthe fluctuations fi are isotropic each component of bshould be 1/3 and the deviation from this value is a measure

of the anisotropy in fi . Shown in Figures 6 and 7 is a com-

parison of the diagonal components of the anisotropy tensor

for each particle type to those in the fluid. It is evident that

for all particles considered in the calculations the anisotropy

of the particle fluctuations is larger than the fluid. The fluc-

tuation levels of the copper particles exhibit the greatest an-

isotropy, e.g., the wall-normal and spanwise components are

never greater than 0.1.

Perhaps more significant than the increased anisotropy

of the particle fluctuations is that the streamwise intensities

of the particles exhibit significant increases relative to the

fluid in the near-wall region and the difference becomes

larger with increases in the Stokes number. The enhanced

fluctuation levels of the particle intensities and larger anisot-

ropy, while counter-intuitive, demonstrates the significant ef-

fect of the mean-velocity gradient on particle fluctuations.

The effect of the mean velocity gradient has been considered

in analyses which are predicated upon an accurate prescrip-

tion of the Lagrangian correlation of fluid elements measured

along the trajectory of a particle e.g., see Liljegren,15

Reeks16 . The effect of mean shear on particle intensities

may also be considered through direct examination of the

particle equation of motion

13

. As shown by Simonin

et al.,2 the equation governing the transport of the particle

intensities can be written as

D

Dt vv PD , 18

where

D

Dt vv t

Vm x m vv , 19

v is the particle velocity fluctuation, and Vm is the particle

mean velocity no summation on repeated Greek indices .

The first term on the right-hand side represents production by

the mean gradient of the particle velocity:

P2 vvmV

x m. 20

FIG. 6. Diagonal components of the anisotropy tensor, Re180. a Lyco-podium; b 50 m glass; c copper. , b11

u ; --- b22u ; , b33

u ; , b11v ;

*, b22v ; , b33

v .




9/17

The second term on the right-hand side of 18 represents

transport by the particle velocity fluctuations,

D1

x m vvvm , 21

where is the volume fraction of the dispersed phase. The

last term represents turbulent momentum transfer and is ex-

pressed as

fp

3

2

CD

d vu vu v . 22

As shown by Simonin et al.,2 can be approximated as

d

p , 23

where d

and p

are destruction and production of par-ticle fluctuations, respectively, and may be written as

d

2

A vv ,

p

2

A uv , 24

where A is the mean particle relaxation time,

Ap

f

4

3

d

CD

1

vu, 25

( CD denotes the drag coefficient averaged over the par-

ticles .

In canonical wall-bounded flows such as a two-

dimensional boundary layer or turbulent channel flow theproduction term P is non-zero only in the streamwise di-

rection. A representative profile of the terms in 18 is shown

in Figure 8 for the 50 m glass beads at Re180. As can

be observed in Figure 8 a streamwise fluctuations are pro-

duced through both interaction with mean gradients, P 11 , as

well as through the fluid-particle covariance represented by

11p . It is important to note that the production by the fluid-

particle covariance is of the same order of magnitude as

P11 and provides a means by which greater anisotropy can

occur in the particle intensities. In fact, though not shown

here, the production term 11p for the Lycopodium particles

is substantially larger than that due to mean gradients. As can

also be observed in each of Figure 8, production of stream-wise intensities is balanced primarily by the contribution

from 11d . It is interesting to note, however, that close to the

wall the attenuation of particle intensities is balanced mostly

by the turbulent transport term D was calculated as the

difference between the other terms in Eq. 18 . Similar be-

havior has also been observed by Simonin et al.2 Compari-

son of the figures also shows that the terms in 18 for the

wall-normal and spanwise components are substantially

smaller than for the streamwise component. Peaks in both

p and

d for 2 and 3 are also collocated with the

peaks in the intensities shown in Figure 2.

The results in Figure 8 clearly show that turbulent mo-

mentum transfer acts as both a source and sink for

particle intensities. In particular, the fluid-particle covariance

has a substantial effect on particle fluctuation levels through

p . Shown in Figure 9 are the non-zero terms of the fluid-

particle covariance at Re180. As is evident in Figure 9 a

the covariance u 1v1 is the largest for the Lycopodium par-

ticles and decreases with increasing Stokes number. Com-

parison of the figures also shows that streamwise component

of the covariance is substantially larger than the other com-

ponents. It should then be expected that the production term

p will become increasingly important at smaller Stokes

FIG. 7. Diagonal components of the anisotropy tensor, Re 644. a Lyco-

podium; b 50 m glass; c copper. , b11u ; ---, b22

u ;; , b33u ; b11

v ; *,

b22v ; ; b33

v .




10/17

numbers. It is also interesting to note that the off-diagonal

components peak at approximately the same y. For the

Lycopodium particles the covariance tensor is nearly sym-

metric with u 2v1 u 1v2 . For the glass beads and copper

particles, however, the asymmetry of the covariance be-

comes more apparent with u 2v1 becoming increasinglylarger than u1v2 . Thus, lower-level closure models forthese quantities should necessarily reflect this asymmetry

see also Reeks,19 Liljegren33 . Simonin et al.2 derived

second-order closures for these quantities and show that pro-

duction of u2v1 is proportional to the mean gradient of the

particle velocity while u 1v2 is proportional to the meangradient of the fluid velocity and that the non-symmetrical

form of the closure is related to the modeling of the pressure

correlations, specifically the rapid pressure term. The other

diagonal components of the fluid-particle covariance in Fig-

ure 9 show that the peak of the wall-normal component,

u2

v2

, occurs at slightly larger y than the spanwise value,

u 3v3 . This behavior correlates both with the location ofthe peak in 22

p and 33p .

IV. PREFERENTIAL CONCENTRATION

A. Near-wall region

Both experimental measurements and numerical simula-

tions have shown that inertial bias in particle trajectories re-

sults in a preferential concentration of particles in regions of

low vorticity or high strain rate see Eaton and Fessler34 for

a general review . In wall-bounded shear flows it is reason-

ably well known that particle concentration fields near the

wall are non-uniform with the largest number densities oc-curring in the low-speed streaks Pedinotti et al.,6 Rouson

and Eaton7 . It is also important to point out, however, that

recent investigations have shown that preferential concentra-

tion occurs throughout the channel Fessler et al.9 . Further-

more, Wang and Maxey4 have shown that preferential con-

centration obeys Kolmogorov scaling, i.e., particles with

time constants and settling velocities close to the Kolmog-

orov scales will exhibit the largest effects of preferential con-

centration. Since the smallest scales of motion in LES are not

resolved, it is of interest to examine the degree to which

preferential concentration is reproduced in the present LES.

Shown in Figure 10 a are streamwise velocity contours

in an x-z plane at y3.6 at Re180. As evident in the

figure, the streaky structure of the near-wall region is repre-

sented in the LES. The instantaneous particle distribution in

the same plane and at the same time is also shown in Figure

10. A streaky structure in the number density of the Lycopo-

dium particles similar to that observed in the velocity field is

apparent in Figure 10 b . It is also clear from the figure that

the number density is less well organized for the 50 m

glass beads and the copper particle distribution in Figure

10 d appears random. Similar behavior was also observed

by Rouson and Eaton7 using DNS and demonstrates that at

Re180 the LES does represent, at least qualitatively, pref-

erential concentration of particles by turbulence. The stream-

wise velocity contours in an x-z plane at y4.8 and

Re644 are shown in Figure 11 a . As can be seen from

the figure, the streaky structure of the streamwise velocity is

also evident at the higher Reynolds number. Number density

distributions for the 28 m Lycopodium, 50 m glass beads,

and 70 m copper particles are shown in Figures 11 b ,

11 c , and 11 d , respectively, and are similar to those ob-

tained at the lower Reynolds number. Lycopodium particles

again exhibit a structure somewhat similar to the velocity

field and appear more organized than the glass beads and

copper particles.

FIG. 8. Terms in 18 at Re

180 for 50 m glass beads. a

1; b2; c 3. , D ; ---,

d ; , p ; , P .




11/17

A quantitative measure of preferential concentration in

the near-wall region is the ratio of the streamwise fluid ve-

locity at the particle location to the average fluid velocity in

the same plane. This measure was first used by Pedinotti

et al.6 and should be unity for a random distribution of par-

ticles

i.e., no preferential concentration

and smaller than

unity if particles are preferentially concentrated in low-speed

regions. Figure 12 shows the pdf of this quantity at both

Reynolds numbers. Consistent with the instantaneous distri-

butions in Figures 10 and 11, the pdf of the Lycopodium

particles exhibits the greatest difference as compared to a

random distribution while the glass beads and copper par-

ticles possess distributions closer to random. In particular,

the higher probability of small values of the streamwise ve-

locity being measured in the vicinity of Lycopodium par-

ticles indicates a preferential concentration in low-speed re-

gions.

B. Channel centerline

Fessler et al.9 have recently demonstrated that preferen-

tial concentration also occurs along the centerline of turbu-

lent channel flow. In addition to the five types of particles

examined in the experiments two sets of Lycopodium par-

ticles with smaller diameters were used in the LES atRe644 to further examine the effect of response time on

preferential concentration see Table II . Both visualizations

and quantitative measures were obtained in the experiments

and therefore provide a suitable benchmark for comparison

to LES predictions.

The instantaneous particle distributions along the chan-

nel centerline from the LES calculation at Re644 are

shown in Figure 13. Similar to the behavior observed by

Fessler et al.,9 the figure show that the copper particles are

randomly distributed whereas the Lycopodium particles and

FIG. 9. Fluid-particle covariance in turbulent channel flow, Re 180. a u1v1 . , Lycopodium; ---, glass; , copper. b Lycopodium; c 50 m glass;

d copper., u1v2 ; ---, u 2v1 ; , u2v2 ; , u3v3 .




12/17

glass beads exhibit varying degrees of preferential concen-

tration. Of the three types of particles shown in Figure

13 a c , the preferential concentration of the 25 m glass

beads appears more significant than for the Lycopodium par-

ticles, which is in turn stronger than for the 50 m glass

beads. Thus, results in the figures again demonstrate that the

LES reflects preferential concentration of particles by turbu-

lence and exhibits the same qualitative features as in the

experiments.

One approach to quantifying preferential concentration

is through calculation of the pdf of the particle number den-

sity. For a random distribution of particles the pdf is Poisson

distributed,

Fp k ek

k!, 26

where is the average number of particles per cell. Shown

in Figure 14 are pdfs of the particle number density at

Re644 together with the Poisson distribution. The pdf was

FIG. 10. Velocity contours and particle distributions from LES at t6/u , y3.6, Re180. a Streamwise velocity; b 28 m Lycopodium; c 50

m glass; d 70 m copper.




13/17

calculated by subdividing the region 0.975y/1.025 into

a network of cells of cross-sectional area 2.6 mm2 in the

x-z plane . The pdf of copper particles is very similar to the

Poisson distribution whereas the other particles differ signifi-

cantly from a random distribution; the greatest difference

appearing to occur for the 25 m glass beads. Thus, consis-

tent with Figure 13 as well as Fessler et al.,9 Figure 14

shows that preferential concentration is not a monotonic

function of the Stokes number.

Fessler et al.9 also defined as a measure of preferential

concentration the deviation from a Poisson distribution,

Dp

, 27

where and p are the standard deviations for the measured

particle distribution and the Poisson distribution, respec-

tively. For particles exhibiting preferential concentration

some cells have large number densities, whereas other cells

substantially lower number densities relative to the mean,

resulting in large positive values of D . Since number densi-

ties are obtained by defining a network of cells in an x-z

plane, it is therefore important to consider the effect of the

FIG. 11. Velocity contours and particle distributions from LES at t3/u , y4.8, Re644. a Streamwise velocity; b 28 m Lycopodium; c 50





14/17

cell size when calculating D . For very small cell sizes, the

dimension of a cell is smaller than the Kolmogorov length-

scale and the particle distribution will appear random, result-

ing in D being zero. For very large cells, regions of high and

low particle number density will be contained within the

same cell and the resulting value of D will also be close to

zero. Between these two extremes there is a cell size where

D is maximized see Fessler et al.9 for a further discussion .

As shown in Fessler et al.,9 the cell size at which 27 is

maximum is a function of the particle type. Therefore, pdfs

of the number density field along the channel centerline were

calculated using several cell sizes. The maximum value from

27 , D ma x , in the LES is compared to Fessler et al.9 in

Figure 15 a . Both the LES and the experiments indicate that

Dma x

exhibits a peak and is largest for the 25 m glass

beads. Wang and Maxey4 used DNS of isotropic turbulence

and showed that preferential concentration of heavy particles

obeys Kolmogorov scaling. Fessler et al.9 estimated the Kol-

mogorov timescale and found that the ratio of the response

time for 25 m glass beads to the Kolmogorov timescale is

2.2. However, the Stokes number defined in terms of the

Kolmogorov timescale for the 28 m Lycopodium is 0.74

and thus the results in Figure 15 a seem to contradict Wang

and Maxey.4 Fessler et al.9 attributed a possible cause of the

discrepancy to the wider range of lengthscales in the experi-

ment as compared to DNS. In this regard it is interesting to

note that while the LES is performed at the same Reynolds

number as the experiment, the range of scales in the compu-

tation is smaller since the subgrid-scale motions are not re-

solved. Thus, it is unlikely that the 25 m glass beads ex-

hibit slightly stronger effects of preferential concentration

because of an extended range of scales. A more likely cause

of the discrepancy, which is discussed by Fessler et al.,9 is

the difference between the cell size used for computation of

the number density field relative to the Kolmogorov length-

scale in the experiment and DNS. Fessler et al.9 found that

for smaller cell sizes the Lycopodium particles exhibited

slightly larger values of D as compared to that for the 25

m glass beads. Similar behavior was observed when ana-

lyzing the LES results.

Wang and Maxey4 used a somewhat analogous measure

as Fessler et al.9 in quantifying preferential concentration,

D kk0

F k Fp k 2, 28

where F(k) and Fp(k) are the pdfs for the actual and random

distributions, respectively. LES results show that, similar to

D , D k also exhibits a maximum which is dependent upon thecell size over which the number density field is defined. The

maximum value is shown in Figure 15 b and again shows a

peak for the 25 m glass beads, confirming that particles

with relaxation times close to the Kolmogorov timescale ex-

hibit the strongest effects of preferential concentration.

V. SUMMARY

Large eddy simulations were carried out and particle

transport was studied in turbulent channel flows at Reynolds

numbers Reu/180 and Re644, corresponding to

FIG. 12. Probability distribution function of the difference between the fluid

velocity at the particle location and average fluid velocity. a Re180, pdf

measured for 5.4y13.4; b Re644, pdf measured for

5.7y13.7. ,random distribution; ---, Lycopodium; , 50 m glass;copper.

FIG. 13. Particle distribution for the region 0.975y/

1.025 att2/u , Re644. a 28 m Lycopodium; b 25 m glass; c 50





15/17

RecUc/3,200 and Rec13,800, respectively. The La-

grangian dynamic eddy viscosity model of Meneveau et al.24

was used to parametrize subgrid-scale stresses in the calcu-

lations. Several statistical measures of the particle velocity

and concentration field together with instantaneous number

density distributions were obtained from the calculations and

compared to DNS results and experimental measurements.

Both the particle mean velocity and rms fluctuations from the

LES are in good agreement with the DNS results of Rouson

and Eaton7 at Re 180 and demonstrates that LES is nearly

as accurate as the DNS. Reasonable agreement is obtained

with the experimental measurements of Kulick et al.9 at the

higher Reynolds number, except in the near-wall region.

DNS results also do not agree particularly well with the ex-

periments near the wall, indicating that the discrepancy in

this region is probably related to the modeling assumptions

used for the particle phase which is not treated exactly, even

in the DNS . Consistent with previous analyses and measure-

ments, particle fluctuation levels in the streamwise direction

are greater than those in the fluid and increase with increas-

ing particle inertia. LES results show the importance of pro-

duction by both the mean velocity gradient as well as the

fluid-particle covariance term. Preferential concentration was

found to be reasonably well reproduced in the LES both near

the wall and along the channel centerline. Visualizations and

statistical measures are in good agreement with those ob-

tained by Rouson and Eaton7 and Fessler et al.9

Discrepancies between the numerical simulations,

whether LES or DNS, and experimental measurements may

be due to factors such as electrostatic effects and particle

collisions present in the experiment but currently not incor-

porated into either DNS and LES. Sommerfeld35 has shown

that small changes in the particle-wall collision model can

have a relatively large effect on statistics of the particle ve-

locity, especially for particles with large time constants. In-

corporation of electrostatic effects, different wall collisions

FIG. 14. Probability distribution function of particle number density in 0.975y/1.025 at t2/u , Re644. Cell size used for calculation of pdf is 2.6

mm. a 28 m Lycopodium; b 25 m glass; c 50 m glass; d 70 m copper.




16/17

models, as well as representations of particle-particle colli-

sions should be expected to shed light on the differences

between the LES and DNS and experimental measure-

ments. Accounting for particle collisions may be quite im-portant since their effect is thought to reduce the anisotropy

of particle fluctuation levels Simonin et al.2 . Incorporation

of a model for SGS velocity fluctuations yielded a very little

effect on particle velocity statistics. Given the relatively ac-

curate predictions of particle statistics at the moderate Rey-

nolds numbers considered in this study, higher Reynolds

number calculations i.e., coarser numerical resolution or

statistics more sensitive to the small-scale velocity field are

needed to examine the effect of the SGS velocity on particle

transport.

APPENDIX A: SAMPLE SIZE

Sample sizes necessary for obtaining a continuum repre-

sentation of particle statistics were determined using velocity

FIG. 16. Effect of sample size on the mean velocity in turbulent channel

flow, Re180. , Eulerian; ---, 100 000 particles; , 250 000 particles;

, 500 000 particles.

FIG. 17. Effect of sample size on the rms fluctuating velocity in turbulent

channel flow, Re 180. a streamwise; b wall-normal; c spanwise. ,

Eulerian; ---, 100 000 particles; , 250 000 particles; , 500 000 par-

ticles.

FIG. 15. Maximum values ofD defined in 27 and Dk defined in 28 for

particles in the region 0.975y/1.025, Re644. a D max ; b

Dk,max . , LES; *, Fessler et al.9.




17/17

fields from LES of the channel flow at Re180. Varying

numbers of particles were randomly distributed throughout

the channel. Statistics were then calculated by interpolating

fluid velocities to the particle position and averaging over

x-z planes. Statistics obtained in this manner were then com-

pared to the mean velocity and rms intensities obtained from

the Eulerian grid. For a sufficiently large sample size, the

statistics of the velocity v i obtained from the particles should

be the same as that obtained from the Eulerian field ui .

Shown in Figure 16 is a comparison of the Eulerian mean

velocity to the mean profiles obtained using 100 000,

250 000 and 500 000 particles. It is clear that the mean pro-

file is adequately resolved using 100 000 particles. However,

Figure 17 suggests that this is not the case for the root-mean-

square rms fluctuating velocity. The rms fluctuations ob-

tained using 100 000 particles differ from the value obtained

from the grid while the profiles obtained using 250 000 and

500 000 particles are quite close to the Eulerian values.

Based upon the results in Figures 16 and 17, a sample size of

250 000 particles is adequate for resolution of the mean and

rms profiles and was used for the simulations reported in this

work.

ACKNOWLEDGMENTS

The authors are grateful to Professor John Eaton and Mr.

Damian Rouson for supplying the DNS results and experi-

mental measurements as well as providing helpful comments

on the manuscript. This work is supported by the National

Institute of Occupational Safety and Health Grant No.

OH03052-02 . Computer time for the simulations was sup-

plied by the Cornell Theory Center.

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