7th International Conference on Multiphase Flow
ICMF 2010, Tampa, FL USA, May 30June 4, 2010
An Analysis of High Concentration GasSolid Flows with a Direct Numerical Simulation
Approach
Goodarz Ahmadi,1 Hojjat Nasr,1 and John B. McLaughlin2
1 Department of Mechanical Engineering and Aeronautical, Clarkson University, Potsdam, NY 13699, USA
2Department of Chemical and Biomolecular Engineering, Clarkson University, Potsdam, NY 13699, USA
ahmadi@tclarkson.edu
Keywords: DNS, Interparticle collisions, Twoway coupling, Particle deposition velocity, Direct numerical simulation
Abstract
This study was concerned with the effect of particleparticle collisions and twoway coupling on the particle deposition
velocity in a turbulent channel flow. The time history of the instantaneous turbulent velocity vector was generated by the
twoway coupled direct numerical simulation (DNS) of the NavierStokes equation via a pseudospectral method. The particle
equation of motion included the Stokes drag, the Saffman lift, and the gravitational forces. The effect of particles on the flow
was included in the analysis via a feedback force that acted on the fluid on the computational grid points. Several simulations
for different particle relaxation times and particle mass loading were performed, and the effects of the interparticle collisions
and twoway coupling on the particle deposition velocity, fluid and particle fluctuating velocities, particle normal mean
velocity, and particle concentration were determined. It was found that when particleparticle collisions were included in the
computation, the particle deposition velocity increased. When the particle collision was neglected but the particlefluid
twoway coupling was accounted for, the particle deposition velocity decreased slightly. For the fourway coupling case,
when both interparticle collisions and twoway coupling effects were taken into account, the particle deposition velocity
increased. Comparisons of the present simulation results with the available experimental data and earlier numerical results are
also presented.
Introduction
Study of transport and deposition of aerosols in
particleladen flows is of considerable interest due to its
importance in numerous industrial and environmental
applications. Despite numerous experimental and
computational studies, the interactions of particles with
turbulent eddies is not fully understood due to its
complexity.
Caporaloni et al. (1975) discovered the existence of
turbophoresis. They showed that due to spatial
inhomogeneity of the turbulent intensities, there is a mean
flux of particles towards the wall. The earliest model for
turbulent deposition was reported by Friedlander and
Johnstone (1957). They proposed the socalled freeflight
theory, which implies that particles reaching the stopping
distance from the wall will deposit on the wall. While this
model gives relatively reasonable results for the eddy
impaction regime, some of the assumptions of free flight
model are difficult to justify. McLaughlin (1989) found that
the Saffman lift force had a significant effect on aerosol
deposition although the assumptions of Saffman's theory
were not satisfied by many of the aerosols as they
approached the wall. The simulation results of Wang and
Stock (1993) showed the importance of nonlinear drag for
particles with high settling velocities. Direct numerical
simulations of particle deposition in wallbounded turbulent
flows were performed by McLaughlin (1989). Zhang and
Ahmadi (2000) used DNS to study aerosol particle transport
and deposition in vertical and horizontal turbulent duct
flows. They showed that the wall coherent structure plays an
important role in the particle deposition process. Squires
and Eaton (1991a) simulated a homogeneous isotropic
nondecaying turbulent flow field by imposing an excitation
at low wave numbers, and studied the effects of inertia on
particle dispersion. They also used DNS to study the
preferential microconcentration structure of particles as a
function of Stokes number in turbulent nearwall flows
(Squires and Eaton 1991b). Brooke et al. (1992) employed
DNS to study particle deposition in a channel flow with the
view of evaluating the free flight theory of Friedlander and
Johnstone (1957). Yamamoto et al. (2001) studied the
interaction between turbulence and solid particles in a fully
developed channel flow using large eddy simulation; they
also considered interparticle collisions as an important
mechanism in high mass loading of the particles. They
showed that the shape and scale of particle concentration
calculated considering interparticle collision are in good
agreement with experimental observations of Fessler et al.
(1994). Li et al. (2001) showed that particleparticle
collisions greatly reduce the tendency of particles to
accumulate near the wall.
In this study, the effects of interparticle collisions and
twoway coupling on dispersed and carrier phase
fluctuations were studied using the direct numerical
simulation of the NavierStokes equation via a
pseudospectral method. The particle deposition velocity,
particle fluctuating velocities, particle normal velocity, and
7th International Conference on Multiphase Flow
ICMF 2010, Tampa, FL USA, May 30June 4, 2010
particle concentration profiles were evaluated under
different conditions. The cases of oneway coupling,
twoway coupling, fourway coupling, and interparticle
collisions without twoway coupling were analyzed. The
effects of particleparticle collisions as well as twoway
coupling on the simulation results are discussed.
Nomenclature
CD
d
Fd
F,
H
J
M.L.
N,
Nd
nx x ny x nz
P
Re
S
S P
t
U
U
Ud
u',V', W'
P
z
Greek letters
V
P
7
,w
Drag coefficient
Particle diameter (m)
Drag force (N)
Saffman lift force (N)
Channel width (m)
Particle mass flux
Particle mass loading
Total number of particles
Number of deposited particles
Number of grid points in x, y and z
direction
Pressure (Nm 2)
Reynolds number
Density ratio
Particle feedback force (N)
Time (s)
velocity (ms1)
Friction velocity (ms1)
Particle deposition velocity (ms1)
Streamwise, normal, and spanwise
fluctuating velocities (ms1)
Particle normal velocity (ms1)
Stramwise Coordinate
Distance from the wall (m)
Spanwise Coordinate
Kinematic viscosity ((m2 s')
Density (Kg m3)
Particle relaxation time (s)
Wall shear stress (Nm 2)
'x, Periodic segment in x and z directions
01 Volume fraction
Supersripts
P particle
f fluid
+ Wall unit
Governing equations
Particle phase
The Lagrangian equation of motion of spherical particles
moving in a wallbounded channel flow including the
wallcorrected nonlinear drag and lift forces in wall units is
given as
d ?
=i CDFd + F
dt+
and
=+p
dU
dt+
where
*2
x+ + tu
V V
+ U
U =
U
Here, u + is the nondimensional particle velocity,, CD
is the nonlinear drag correction factor, Fd is the drag
force after including the wall drag corrections, and F1 is
the wall induced and shear induced lift force. In equation
(3), u is the flow shear velocity.
The details of the lift force including the near wall effects
were described in the work of Chen and McLaughlin
(1995) and Zhang and Ahmadi (2000) and therefore are not
repeated here. (Note that only the ycomponent of lift force
is considered in this study.)
Based on a synthesis of available experimental results,
Clift et al. (1978) and Beard et al. (1971) recommended the
following nonlinear drag correction factors:
1.0 +0.1875Rep
CD
1.0 +0.1315Re 82002171n(Re
Rep, 0.01'
0.01< Rep <20
Here, Rep =d+ u+f +p is the particle Reynolds
number. Additional details of the drag and lift forces
were described in detail by Chen and McLaughlin (1995)
and Ounis et al. (1991).
The nondimensional particle relaxation time is defined
as
d2
r+ =S (5)
18
here d+ =du*/v is the nondimensional particle
diameter and S is the particletofluid density ratio.
For evaluating the forces acting on the particles, the fluid
velocities at the locations of particles must be evaluated
using an interpolation technique. An accurate evaluation
of particle velocities is also essential for analysis of
interparticle collisions that depend on the often small
relative particle velocities. In this study, partial Hermite
interpolation method was used for evaluating the fluid
velocities at the locations of particles and for the inverse
action of particle drag on the fluid. To evaluate the particle
deposition velocity, it is assumed that when a particle
reaches to distance of one radius from the wall, it deposits
with no rebound. In order to keep a uniform particle
concentration inside the channel, when a particle is
deposited on the wall, another particle is randomly
introduced in the computational domain.
The hard sphere particleparticle collision model with a
coefficient of restitution equal to 0.95 was used in the
analysis. The procedure for the numerical implementation
of the particleparticle collisions as described by Li et al.
(2001) was implemented in the present analysis. In this
study, the collisions were assumed to be binary since
multiple collisions are extremely rare at the particle
concentrations that were considered.
Gas phase
The instantaneous fluid velocity field in the channel is
evaluated by DNS of the NavierStokes equation using an
additional source term due to the presence of particles. It is
assumed that the flow is incompressible, and a constant
mean pressure gradient in xdirection is imposed. The
corresponding governing equations of motion are:
Continuity Equation: In this study the volume fraction of
particles is very small, ,< <10 3; therefore, the continuity
equation may be expressed as:
V .+f =0 (6)
Momentum Equation: The effect of particles is added to
the NavierStokes equations by an additional source term
using the point force model:
Du +f 1
V+p  i + V +S/VP+ (7)
Dt+ H
where Ef+ =(uf+,vf+,wf+ ) is the fluid velocity vector
in wall units, and p+ is the pressure in wall units. The
coupling between fluid and dispersed phases incorporated
into the momentum equation via a feedback force per unit
mass, which is the negative of the drag and lift forces
acting on the particles exerted by the fluid in a certain
computational cell; the particle feedback force per unit
mass is given by
dNP NP
Si =  (CD Fd +F+ ) (8)
n=l dt n=l
Noslip boundary conditions are assumed on the channel
walls and periodic boundary conditions are imposed in the
x and zdirections as follows:
if+ = 0,
y = +H
f+ (x+ +ml+ ,y+ ,z+ +nA+ ,t+)=if+ (x+ ,y ,z ,t )
(9)
where m and n are integers.
In the present simulations a channel that has halfwidth
H+ inwall units, and a x,+x ,+ periodic segment inx
and z directions is used. A schematic of the flow domain
and the periodic cell are shown in Figure 1. A
nx x ny x nz computational grid in the x, y and
zdirections is employed. The grid spacing in the x and z
directions are constant, while the variation of grid points in
the ydirection is determined by the collocation points of
the C hlib shei series. The distance of the ith grid point in
the ydirection from the centerline is given as:
+ H+
Y, = cos(7ri/AM), O
2
7th International Conference on Multiphase Flow
ICMF 2010, Tampa, FL USA, May 30June 4, 2010
where M =nz1.
The channel flow code used in this study is the one
developed by McLaughlin (1989). The code used a
pseudospectral method for computing the fluid velocity
field. That is, the fluid velocity is expanded in a
threedimensional FourierC hlib s',IK series. The fluid
velocity field is expanded in Fourier series in the x and z
direction, while in the ydirection aC (hbi IsheI series is
used. The code uses an AdamsBashforthCrankNickolson
(ABCN) scheme to compute the nonlinear and viscous
terms in the NavierStokes equation and performs three
fractional time steps to advance the fluid velocity from
time step (n) to time step (n+l). The details of the
numerical techniques were described by McLaughlin
(1989).
Figure 1: Schematics of the channel flow and
computational periodic cell used
Results and discussion
In this section, simulation results for different particle
parameters such as particle fluctuation velocities, particle
streamwise and normal velocities, and particle deposition
velocity are presented. All simulations are performed for
four approaches: 1) oneway coupling, 2) twoway
coupling, 3) fourway coupling, and 4) only interparticle
collisions. The results are compared to show the effect of
the particle feedback force and interparticle collisions on
the aforementioned particle parameters. All variables are
nondimensionalized by the fluid viscosity, v, and the
friction velocity, u* = That is
x= t = u = (11)
V V ,
A temperature of 288 K, v = 1.5 x 105 N.s/m2,
and pf = 1.2kg /m3 for air are used. The friction velocity,
u is assumed to be 0.3 m/s. The channel half width is
H+ = 125 and the streamwise and spanwise periods are
chosen to be A +=630 and A +=630, respectively. The
numbers of grid points in the x, y and zdirections are nx
= 32, ny = 65 and nz = 64. In this case, the Reynolds
number based on the friction velocity, u*, and the half
channel width is 125, while the flow Reynolds number
based on the Hydraulic diameter and the centerline
velocity is about 8000. This condition corresponds to a
7th International Conference on Multiphase Flow
ICMF 2010, Tampa, FL USA, May 30June 4, 2010
channel half width of H = 12.5 mm, the streamwise and
spanwise periods of 31.5 mm. The value for the density
ratio, S = pP /pf, was taken to be 1000. Each simulation
was performed for 2500 time steps, and the
nondimentional time step was chosen to be 0.25.
Simulations were performed for four particle
diameters, d = 20, 25, 30, and 40 um ; this can be
expressed in terms of nondimensional particle relaxation
time, r+ = 9, 14, 20, and 35. Simulations were performed
at two particle mass loadings, M.L.=20% and 40%. It is
important to mention that for particles with + = 8.9 and
at M.L.=40%, approximately 1 200 000 particles were
tracked which required an expensive computation. All
results have been averaged over the simulation time, and
over the streamwise and spanwise directions. Particles
were uniformly distributed in the channel, and the initial
velocity of each particle was set equal to the local fluid
velocity evaluated at the center of the particle. It is
assumed that when a particle hits the wall, it sticks to the
wall and no rebound occurs. When a particle deposits on
the wall, another particle is randomly distributed inside the
computational domain in order to keep the total number of
particles constant.
Figure 2 shows a sample instantaneous velocity
vector plot in the xy plane at t =100 in the case of
oneway coupling. The random deviations from the
expected mean velocity are clearly seen from this figure.
Figures 35 show the velocity field in the yz plane at
t+ =500 in the case of oneway and fourway coupling at
mass loadings of M.L.=20% and M.L.=40%. The near
wall coherent eddies and flow streams toward and away
from the wall can be observed in these figures. Comparing
the results for the oneway coupling in Figure 3 with the
fourway coupling in Figures 4 and 5, it appears that the
presence of solid particles damps the turbulence
fluctuations and also decreases the number of eddies. The
simulated rootmean square (RMS) fluctuation fluid
velocities in the case of one way and fourway coupling at
M.L.=20% and 40% are shown in Figure 6. It is seen that,
the addition of particles with r+ = 20 attenuates the
intensity of the fluctuations, and as particle mass loading
increases, the level of attenuation increases. This trend is in
agreement with earlier experimental data and numerical
results. As was noted before, it has been widely shown
that particles with diameter less than the Kolomogorov
length scale attenuate the turbulence while particles with
diameters larger than the Kolmogorov length scale
augment it.
ioo0h k_%hhhhhh WhSl
0 100 200 300 400 500 600
X
Figure 2: Sample velocity vector plot in xy plane.
100
0 200 400 600
Z+
Figure 3: Velocity vector plot in yz plane in presence of
T+ = 20.0 particles, oneway coupling
100 1 bii
0 200 400 600
z
Z+
Figure 4: Velocity vector plot in yz plane in presence of
T+ = 20.0 particles fourway coupling at M.L.=20%
200 400 600
Figure 5: Velocity vector plot in yz plane in presence of
T+ = 20.0 particles fourway coupling at M.L.=40%
  20% Fowrwmy cooling
 40% Faouw cup tin
2 2
1
10o 10' 10'
y
Figure 6: Effects of r =20.0 particles on the flow
fluctuating velocities
Figure 7 depicts the number of deposited particles versus
time for r = 20.0 for different approaches. Since in each
case the number of tracked particles is different, the
number of deposited particles in this figure is
nondimensionalized by the total number of particles. It is
observed that in the case of twoway coupling, the number
of deposited particles decreases, and as particle mass
loading increases, fewer particles deposit. When the
particle feedback force is eliminated and interparticle
collisions are taken into account, the number of deposited
particles increases; increasing particle mass loading,
enhances the number of deposited particles.
100 200
Figure 7: Normalized number of deposited particles
versus nondimensional time for particles with + =20.0
When the particle feedback force and interparticle
collisions are considered (fourway coupling), the number
of deposited particles increases comparing with oneway
coupling approach and decreases comparing with the case
in which interparticle collisions are only taken into the
calculations. In the case of fourway coupling, as particle
mass loading increases, more particles deposit. It can be
concluded that twoway coupling causes a decrease in the
number of deposited particles, while the interparticle
collisions lead to an increase in the number of deposited
particles. The increase in the number of deposited particles
due to fourway coupling shows that the power of
interparticle collisions in increasing the number of
deposited particles is more than the power of twoway
coupling in decreasing it.
Particle deposition velocity
The dimensionless deposition velocity for particles
released with a uniform concentration CO is defined as
Ud+ =JICOU* (12)
where J is the particle mass flux to the wall per unit time.
In the computer simulation, the particle deposition velocity
may be estimated as
S (Nd /2)/t (13)
(N /2)/(H /2)
where No is the total number of the particles in the
7th International Conference on Multiphase Flow
ICMF 2010, Tampa, FL USA, May 30June 4, 2010
channel, and Nd is the number of deposited particles in
the time interval td+ on the channel walls. Figure 8
shows the nondimensional particle deposition velocity,
ud versus the nondimensional particle relaxation time,
+ To verify the present simulation results, the
experimental data of Papavergos and Headly (1982), and
the numerical results of Li and Ahmadi (1992) and He and
Ahmadi (1999) are shown in this figure. The present
simulation results in the case of oneway coupling
approach are in favorable agreement with the experimental
data and other simulation results. It is observed that as the
particle relaxation time increases, the particle deposition
velocity increases. In the case of four way coupling, the
particle deposition velocity increases, and as mass loading
increases, this value increases.
Figure 8: Nondimensional particle deposition velocity
versus nondimensional particle relaxation time
For particle deposition, the normal component of particle
fluctuating velocity plays a crucial role. Figure 9 shows the
normal fluctuating velocity of the flow and particles with
z+ = 20 for different approaches. Several features are
noteworthy to mention. The normal fluctuating velocity of
particles in the case of oneway coupling is less than the
fluid one in the region where y+ > 10.0 and greater in the
wall region (y+ <10.0). The decrease of the normal
fluctuating velocity of the particles in the region where
y+ > 10.0 can be explained by the fact that inertial
particles are not fully responsive to all turbulent eddies,
and they fluctuate less than the flow. The increase of the
normal fluctuating velocity of the particles in the wall
region ( y+ < 10.0) is because the maximum particle
normal velocity occurs at y+ = 20 and it goes to zero at
the wall; therefore, a wide range of particle normal
velocities is found in the wall region, causing an increase
in the normal particle fluctuating velocity. In the case of
twoway coupling, the particle normal fluctuating velocity
decreases in the entire channel comparing with oneway
coupling approach, and as particle mass loading increases,
the particle normal fluctuating velocity decreases. This can
be explained in terms of turbulence attenuation due to the
presence of the particle feedback force as shown in Figure
6.
When the particle feedback force is eliminated and
interparticle collisions are considered, the particle normal
fluctuating velocity increases in the entire channel
comparing with oneway coupling approach, and as
particle mass loading increases, the level of augmentation
in particle normal fluctuating velocity increases.
In the case of fourway coupling, the particle normal
fluctuating velocity decreases in the region where
y+ > 10.0 and increases in the wall region (y+<10.0)
comparing with oneway coupling approach. It can be
concluded that twoway coupling decreases the particle
normal fluctuating velocity, while interparticle collisions
increase it.
101 102
y
Figure 9: Normal fluctuating velocity versus the distance
from the wall for particles with + =20.0
The particle normal velocity versus the distance from the
wall for r+ = 20.0 particles with and without interparticle
collisions is respectively shown in Figure 10. Despite the
zeromean normal velocity, it is observed that there is a
particle flux toward the wall with a maximum value at
y+ = 20. In the case of fourway coupling and as mass
loading increases, the particle normal velocity decreases.
When twoway coupling is eliminated (only interparticle
collisions), the particle normal velocity increases, but it is
still less than its value in the case of oneway coupling.
Summary and Conclusions
The goal of this paper was to study the effect of
particleparticle collisions and twoway coupling on
particle deposition velocity, fluid and particle fluctuating
velocities, particle normal velocity, and particle
concentration profile in a turbulent channel flow. The time
history of the instantaneous turbulent velocity vector was
generated by the twoway coupled direct numerical
7th International Conference on Multiphase Flow
ICMF 2010, Tampa, FL USA, May 30June 4, 2010
simulation (DNS) of the Navier Stokes equation via a
pseudo spectral method. The particle equation of motion
included the Stokes drag, the Saffman lift, and the
gravitational forces. The effect of particles on the flow is
included in the analysis via a feedback force on the grid
points. The key ideas proposed in this paper may be
summarized as follows:
0
Oneway
 20% colsbons only
 40% coMnlons only
........... 20% twoway ,
0.03 1 .. ... 4 twoway
..   20% fourway .'
\1  44% fourway ,
P+ e
0.12 ' '
10 10' 102
y
Figure 10: Normal particle velocity versus distance from
the wall for particles with r+ =20.0
1) The addition of particles attenuates the intensity of the
fluid fluctuations in x, y, and z directions, and as particle
mass loading increases, the level of attenuation increases.
2) Interparticle collisions increase the particle normal
fluctuating velocity, while twoway coupling decreases it.
In the case of fourway coupling, the particle normal
fluctuating velocity decreases in the region where
y+ > 10.0 and increases in the wall region (y+<10.0);
increasing particle mass loading magnifies this change.
3) Since interparticle collisions and increasing particle
mass loading increase the particle normal fluctuating
velocity in the wall region, it is observed that the particle
deposition velocity increases.
4) In the case of fourway coupling and as mass loading
increases, the particle normal velocity decreases. When
twoway coupling is eliminated, the particle normal
velocity increases, but it is still less than its value in the
case of oneway coupling.
Acknowledgements
The financial support of the Environmental Protection
Agency (EPA) and the NYSTAR Center of Excellence is
gratefully acknowledged.
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