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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
1210 Phys. Fluids, Vol. 8, No. 5, May 1996 Q. Wang and K. D. Squires
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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.
1211Phys. Fluids, Vol. 8, No. 5, May 1996 Q. Wang and K. D. Squires
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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.
1212 Phys. Fluids, Vol. 8, No. 5, May 1996 Q. Wang and K. D. Squires
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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.
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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 .
1214 Phys. Fluids, Vol. 8, No. 5, May 1996 Q. Wang and K. D. Squires
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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 .
1215Phys. Fluids, Vol. 8, No. 5, May 1996 Q. Wang and K. D. Squires
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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 .
1216 Phys. Fluids, Vol. 8, No. 5, May 1996 Q. Wang and K. D. Squires
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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 .
1217Phys. Fluids, Vol. 8, No. 5, May 1996 Q. Wang and K. D. Squires
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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.
1218 Phys. Fluids, Vol. 8, No. 5, May 1996 Q. Wang and K. D. Squires
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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
m glass; d 70 m copper.
1219Phys. Fluids, Vol. 8, No. 5, May 1996 Q. Wang and K. D. Squires
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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
m glass; d 70 m copper.
1220 Phys. Fluids, Vol. 8, No. 5, May 1996 Q. Wang and K. D. Squires
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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.
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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.
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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.
1A. Berlemont, P. Desjonqueres, and G. Gouesbet, Particle Lagrangian
simulation in turbulent flows, Int. J. Multiphase Flow 16, 19 1990 .2O. Simonin, E. Deutsch, and M. Boivin, Large eddy simulation and
second-moment closure model of particle fluctuating motion in two-phaseturbulent shear flows, in Turbulent Shear Flow 9, edited by F. Durst, N.
Kasagi, B. E. Launder, F. W. Schmidt, and J. H. Whitelaw Springer-
Verlag, Heidelberg, 1995 , p. 85.3K. D. Squires and J. K. Eaton, Preferential concentration of particles by
turbulence, Phys. Fluid A 3, 1169 1991 .4L.-P. Wang and M. R. Maxey, Settling velocity and concentration dis-
tribution of heavy particles in homogeneous isotropic turbulence, J. Fluid
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