7th International Conference on Multiphase Flow
ICMF 2010, Tampa, FL USA, May 30June 4, 2010
Numerical Investigation of ParticleLaden Turbulent Channel Flow with Either
Longitudinal or Transverse Roughness Elements on the Lower Wall
Christos Dritselis
Department of Mechanical Engineering, University of Thessaly
Athens Avenue, 38334 Volos, Greece
dritseli@mie.uth.gr
Keywords: LES, Lagrangian particle tracking, immersed boundary method
Abstract
The effect of twodimensional roughness elements on the particle transfer in confined gasparticle turbulent flows are
investigated by using large eddy simulation combined with discrete particle simulation. Square bars separated by a rectangular
cavity are placed transversely or longitudinally on the lower wall of the channel. Numerical results are obtained for several
values of the cavity width to the roughness height ratio. The trajectories of noncolliding particles with three different response
times are determined by the drag force. The influence of wall roughness on the gasparticle flow is assessed by examining the
mean streamwise and the rootmeansquare velocities of the fluid and the particulate phases. The differences between the two
geometries are demonstrated through comparisons with the results of the corresponding case with two smooth walls.
Introduction
Several aspects of confined gasparticle turbulent
flows have been investigated using combinations of either
Direct Numerical Simulation (DNS) or Large Eddy
Simulation (LES) approaches combined with Discrete
Particle Simulation (DPS) [see, for example, McLaughlin,
1989; Pan & Banerjee, 1997; Rouson & Eaton, 2001;
Yamamoto et al., 2001; Vance et al., 2006; Dritselis &
Vlachos, 2008(a); 2008(b)]. Despite the relative large
amount of work, the effect of wall roughness on the
mechanisms of particle transport by turbulence has not been
adequately explored.
Experiments and numerical simulations have shown
that the statistics of both the gas and the particulate phases
might be significantly affected by the interaction of particles
with a rough wall (see, for example, Sommerfeld, 1992;
Sommerfeld & Huber, 1999; Sommerfeld & Kussin, 2004;
Squires & Simonin, 2006; Konan et al., 2009). In the present
study, the effect of wall roughness on the dynamic behavior
of particles is numerically studied by using LES of the
gaseous flow coupled with DPS of the solid spherical
particles in a fully developed turbulent channel flow. The
lower channel wall consists of square bars placed either
transversely or longitudinally, separated by a rectangular
cavity (see figure 1). Simulations are performed for several
values of the cavity width w to the roughness height ratio k,
while the height of the square bars is 10 % of the wall
normal distance of the outer flow 2h, i.e., k = 0.2h.
The trajectories of particles are determined only by the
drag force. The influence of the subgrid fluid turbulence is
ignored. Interparticle collisions and the momentum exchange
between the two phases are neglected in the present oneway
treatment. The main objective of the present study is to
examine the overall effect the roughness has on the outer
gasparticle flow. Although it is more appropriate to speak of
a turbulent flow over obstacles, such configurations can be
considered as simple models for the investigation of the
influence of wall roughness on the turbulent transport of
particles.
Governing Equations
In the LES/DPS approach, the instantaneous turbulent
fluid velocity is decomposed into a component of the large
flow scales and a second one that corresponds to the small
scales of the turbulent flow. The LES equations for an
incompressible fluid flow are obtained by filtering the
continuity and NavierStokes equations
du ,
0,
ax
+
at ax
1 Dp Du r,'
p, x x x Ox ox
where iu, and j are the large scale components of the fluid
velocity and pressure, respectively, pf, v are the density and
the kinematic viscosity of the carrier phase, respectively, H is
the extra pressure gradient required to keep the mass flow
rate constant, and 6, is the Kronecker delta. The extra term
r,' represents the influence of the subgridscale motions on
the resolved gridscale fluid velocities. It is the traceless part
of the subgridscale stress tensor, defined as
1 
r = r r 32 with = u u u u.
3
The subgrid stress r,' is accounted for by using the
dynamic Smagorinsky subgridscale turbulence model. It is
7th International Conference on Multiphase Flow
ICMF 2010, Tampa, FL USA, May 30June 4, 2010
Y
4h
2.2hl .
Z 8h
Figure 1: Channel geometry with longitudinal (a) and transverse (b) square bars located on the lower wall.
modeled as
r=2CASS, (4)
where S, and SI = (2SSj S,)1/2 are the resolved strainrate
tensor and its magnitude, respectively. The characteristic
length is A = (Ax Ay ,' ', where Ax, Ay, and Az are the
grid spacings in the x, y, and z directions, respectively. The
model parameter C is determined by the dynamic procedure
proposed by Germano et al. (1991) based on Lilly's (1991)
modification
C(x2) = LM (5)
' 0 ili ini.1 ,, given as
L = uu uu (6)
M = 2A2S 2A S 1, (7)
where (A) denotes variables calculated on the test filter and
( ) averaging over the planes in the homogeneous
streamwise x and spanwise z directions. The present large
eddy simulations are performed by using a box filter in the
physical space based on the trapezoidal rule. No filtering is
applied in the wallnormal direction, while the width of the
test filter is twice the size of the grid spacing in the
homogeneous directions. The velocity of the carrier phase
satisfies periodic boundary conditions in the x and z
directions, and nonslip conditions at the walls.
The trajectories of the particles are calculated in a
Lagrangian reference frame considering only the drag force.
The particle equations of motion are
dx,
= v, (8)
dt
dv: fl 3 P, C
dt m 4 p, d
where v,", if@p, are the velocity of the n particle and the
undisturbed fluid velocity at the particle position xp,"
respectively. In equation (9), fD," is the drag force, mp(=
ppd,3/6) is the spherical particle mass, pp, d, are the particle
density and diameter, respectively, and bold symbols
indicate vectors. Equation (9) is appropriate for particles
with diameters of the same order or smaller than the
smallest length scales of the fluid motion. The coefficient
CD corrects the Stokes drag force for inertial effects at
nonnegligible particle Reynolds numbers (see Clift et al.,
1978)
CD 4 (1+0.15Re 687)
Re
where Re. = Id/v .
Other forces, such as Basset, added mass, and
pressure gradient forces are discarded from the particle
equation of motion. This is justified by the relative high
mass density ratio between the phases considered in the
present study (Armenio & Fiorotto, 2001). The lift force has
also been neglected in order to study the effects of wall
roughness on the particle transport within a manageable
parameter range.
The particles exiting the channel through the planes
normal to the streamwise (x) and spanwise (z) periodic
directions were reintroduced in the flow domain from the
corresponding opposite boundary plane with their exiting
velocities. Particlewall collisions were considered to be
perfectly elastic.
Numerical Methods
The time evolution of the present flows is obtained by
numerically solving the filtered equations (1) and (2) using
a semiimplicit, fractional step method (see Orlandi, 2000).
A secondorder central finite differencing scheme on a
staggered grid is used for the spatial discretization. The time
integration of the discretized equations is done by a
combination of an explicit secondorder AdamsBashforth
method for the nonlinear convection terms and the extra
subgrid stress and an implicit secondorder CrankNicolson
method for the linear diffusion terms. The resulting system
of algebraic equations is inverted by an approximate
factorization technique. The fluid momentum equations are
advanced in time by using the pressure gradient at the
previous time step, yielding an intermediate nonsolenoidal
fluid velocity field. The latter velocity field is projected by a
scalar quantity onto a solenoidal one.
The roughness elements are treated by an immersed
boundary technique (see, for details, Orlandi & Leonardi
2006), which consists of imposing zero values to all fluid
velocity components on the stationary boundary surface,
k I w
k w
.v ;, v n),
which does not necessarily coincide with the computational
grid. This approach allows the solution of flows over
complex geometries without the need of computationally
intensive bodyfitted grids. At the first grid point outside
each square bar, all the viscous derivates in the filtered
NavierStokes equations are discretized by using the
distance between the fluid velocity components and the
boundary of the roughness element and not the actual mesh
size. This is done in order to avoid describing the geometry
in a stepwise way and to perform numerical simulations by
maintaining a constant fluid mass flow rate in the channel.
The positions and velocities of the particles are
computed simultaneously with the momentum equations of
the carrier phase by time integration of equations (8) and (9)
with a secondorder explicit AdamsBashforth method.
Particle wall collisions are identified by geometric criteria.
The fluid velocity at the particle position is computed by
interpolation with Lagrange polynomials. The interpolation
scheme switches to one side near the channel walls. It
should be noted that loworder, timeefficient interpolation
schemes (e.g., threedimensional linear interpolation and
Lagrange polynomials) are found to be accurate enough and
they have been widely used in wallbounded turbulent flows
laden with particles (see, for example, Rouson & Eaton,
2001; Yamamoto et al., 2001).
The local undisturbed fluid velocity i@p," in equation
(9) is approximated by the fluid velocity that results from
the numerical solution of equations (1) and (2), calculated at
the particle positions. The effect of subgridscales on the
particle motion is not taken into account. Thus, the particle
trajectories are calculated by using only the resolved fluid
velocity in the LES computations. This is a reasonable
assumption given the filtering due to the particle inertia and
the moderate Reynolds number of the flow, for which there
is a relatively weak effect of the unresolved on the resolved
scales (see, for example, Armenio et al. 1999; Yamamoto et
al., 2001; Vance et al., 2006).
Numerical Details
All simulations are performed at the same Reynolds
number ofReb(= ub2h / v) = 5600 based on the bulk velocity
Ub(= 2 / 3ul) and the distance between the smooth upper wall
and the crest of the lower rough wall 2h, where ul is the
centreline velocity of the initially laminar profile. A fixed
fluid mass flow rate is considered and, thus, for all cases the
velocity ub and ul are constants. For this reason, a mean
pressure gradient II was imposed in the streamwise fluid
momentum equation. This quantity was properly adjusted
during the simulations by integrating over the whole
channel domain the right hand side of the discretized ua
momentum equation. The turbulent centerline velocity uc
and the wall friction velocity u, = (tw / v)1/2 vary, where r, is
the wall shear stress.
Two computational domains are used in the numerical
simulations. For transverse elements, the dimensions of the
channel in the streamwise and spanwise directions are L, =
8h and L, = ih, respectively. The computational mesh is
201 x94x33. Each roughness element is discretized by using
30 almost equidistant grid points over the range of 1.2 <
y/h < 1, while 5 grid points are used in the x direction. The
rest 64 of the grid points are unevenly distributed in 1
7th International Conference on Multiphase Flow
ICMF 2010, Tampa, FL USA, May 30June 4, 2010
< 1. For roughness elements aligned in the flow direction,
the computational mesh is 33x94x101 and the channel
dimensions are L, = 8h and L, = 4h in the streamwise and
spanwise directions, respectively, This is done in order to
have an integer number of elements along the z direction.
The number of grid points per element is kept constant,
equal to 30 and 5 in the y and z directions, respectively. For
all cases, the CFL number was fixed at a constant value of
0.5, ensuring a small enough time step and, consequently, a
good accuracy for both the fluid and particle velocities.
At first, numerical simulations are performed for
various w/k values without considering the particulate phase,
using as initial conditions the fully developed turbulent
channel flow with two smooth walls at Reb = 5600. For the
latter case, the lower smooth wall is also described by the
immersed boundary method. For transverse elements, . I, =
0, 1, 3, and 7 are studied, while for longitudinal elements,
. I = 0, 1, 3, and 9 are studied. Next, the trajectories of 105
copper particles were calculated in fully developed
turbulent channel flows with one rough wall. Initially, the
particles were uniformly distributed in the region of 1 <
y/h < 1, with velocities equal to those of the fluid at their
positions. Results are obtained for three samples of particles
with a density ratio of S = p, / pf = 1000 and diameters
ranging from 1 to 16 pm, respectively. For the sake of
brevity, only the results for particles with d, = 16 pm are
presented here.
After an initially transient time (typically a few
particle time constants), the particulate phase reaches a
stationary state, where its statistics do not significantly
change with time. All the results shown in the study have
been calculated by postprocessing 100 instantaneous flow
fields saved every 5 h/u1 time units after the initial transient
period. Convergence of the statistics was ensured by
checking that the results would not change with increasing
the number of the processed flow fields.
The mean streamwise velocity and the rootmean
square velocity fluctuations of the carrier phase predicted
using the dynamic Smagorinsky turbulence model agree
reasonably well with corresponding DNS and other
published LES results for the case of a fully developed
turbulent channel flow at Reb = 5600 [see Dritselis &
Vlachos, 2008(a); 2008(b)]. Finally, the accuracy of the
present numerical treatment for the roughness elements was
verified through extensive comparisons with the results of
the Furuya et al. (1976) experiment for a boundary layer
over twodimensional transverse circular rods and with the
DNS results of Leonardi et al. (2003) for the turbulent
channel flows with square bars placed on one wall.
Results and Discussion
Results are presented to address the effect of
roughness elements on the mean streamwise and the root
meansquare fluid and particle velocities. Figure 2 shows
the mean streamlines averaged in time and in the spanwise
direction for various values of . I, for transverse roughness
elements. The characteristics of these distributions have
been extensively described previously (see, for example,
Leonardi et al., 2003; Cui et al., 2003), and here only the
main characteristics are summarized. For . I, = 1, the whole
cavity is occupied by a single recirculation region. The size
7th International Conference on Multiphase Flow
ICMF 2010, Tampa, FL USA, May 30June 4, 2010
03 04
Figure 4: Mean fluid velocity profiles
elements.
for longitudinal
04
02
o
o 
06
1
(c) 5
Figure 2: Mean streamlines
elements with , = 1 (a), , h
flow is from left to right.
for transverse roughness
: 3 (b), and wlk = 7 (c). The
of the vortex is increased for , = 3 and a smaller
secondary vortex with opposite circulation is observed close
to the left vertical wall of the cavity. For these two cases of
, = 1 and 3, the separation of the flow occurs at the edge
of each element, while the flow reattaches at the vertical
wall of the following roughness element. For wlk = 7, a
large recirculation zone is located downstream of each
element, which is accompanied by a smaller vortex. For this
case, the flow reattaches on the bottom wall of the cavity
and separates again as the vertical wall of the next element
is approached. These results are in good agreement with
those of Leonardi et al. (2003) who used a similar immerse
boundary technique and with the results of Cui et al. (2003)
who used a bodyfitted numerical code to study the
turbulent flow in similar geometries.
Figures 3 and 4 show the wallnormal distributions of
the mean fluid velocity U in the streamwise direction for
transverse and longitudinal roughness elements, respectively.
Averages have been performed with respect to the x and z
directions, since our primary interest is to assess the effect
of wall roughness on the overlying flow and not to account
E U.'
A 0.3
v 0.3
Figure 3: Mean fluid velocity profiles for transverse
elements.
1 0.5
0.5 1
Figure 5: Turbulent intensity profiles for transverse
elements: solid lines, , = 0, dashed lines = 1 (a), , /=
3 (b), and , = 7 (c). Red lines: (u121/2, green lines:
( 22 ) 1/2, blue lines: ( u32 1/2. (b) and (c) are shifted
upwards by 0.25 and 0.5.
(b)
. (a) 
n LA:*2 , _l . . i .
o
 



7th International Conference on Multiphase Flow
ICMF 2010, Tampa, FL USA, May 30June 4, 2010
_ U.'4
A
v 0.3
0.2
0.1
Figure 6: Turbulent intensity profiles for longitudinal
elements: solid lines, , = 0, dashed lines , = 1 (a), /. =
3 (b), and i = 9 (c). Red lines: (2) 1 2, green lines:
( 22) 1/2, blue lines: ( u32) 1/2. (b) and (c) are shifted
upwards by 0.25 and 0.5.
for variations within one roughness wavelength. Figure 3
shows that the maximum value of U is shifted towards
smooth wall. This indicates that flow disturbances can
propagate from the rough wall the smooth one. This is more
pronounced with increasing . / On the other hand, large
changes are observed near the crest of the roughness
elements as revealed by the U distributions in figure 4. The
present results of U agree reasonably well with the DNS
results of Orlandi et al. (2006), for the flow cases of both
transverse and longitudinal square roughness elements.
(b)
(a)
0.5
0 9 . .I .
0.5 0 0.5
ylh
Figure 7: Profiles of the mean streamwise velocities of
particles (lines) and the fluid along the particle trajectory
(symbols) for transverse elements: (a) i = 0, (b) i /. = 1,
(c) I, = 3, and (d) wlk = 9. (b), (c), and (d) are shifted
upwards by 0.25, 0.5, and 0.75, respectively.
Figure 8: Profiles of the mean streamwise velocities of
particles (lines) and the fluid along the particle trajectory
(symbols) for longitudinal elements: (a) I /. = 0, (b) w/k = 1,
(c) I, = 3, and (d) , I = 9. (b), (c), and (d) are shifted
upwards by 0.25, 0.5, and 0.75, respectively.
Figures 5 and 6 show the wallnormal distributions of the
fluid turbulent intensities ( ,2 )1/2 for the cases with
transverse and longitudinal roughness elements, respectively.
For both types of geometry, these profiles are increased near
the crest of the roughness elements. For the transverse
elements, the increase in ( ,2) 1/2 is observed for larger y/h
values. The present results of (u,2 1/2 agree also well with
the corresponding DNS results of Orlandi et al. (2006) for
both the transverse and longitudinal roughness elements.
Figure 9: (v,2) /2 (symbols) and ( 1@p,/2 ) (lines) for
transverse elements: (a). i.=0, (b). i= L, (c). 1.=3, and (d)
. =7. Red color: (x2 1/2, ( @,x2 ) /2 green color:
vy2 ) 1/2 ( U@,y) 1/2; blue color: (v,2 ( @p,2 ) 1. (b),
(c), and (d) are shifted upwards by 0.25, 0.5, and 0.75.
1.2
1
(d)
0.8
v 0.6 (c)
> 0.4 
0.2
(a)
S 0.5 0 0.5 1
ylh
Figure 10: ( v,2 1/2 (symbols) and ( U@,2) 1/2 (lines) for
longitudinal elements: (a) I. .=0, (b) w/k=, (c) . ,=3, and
(d) /.=7. Red color: (v2)1 /2, (u ,x2)/2; green color:
( y2 )1/2, ( p,y2 )1/2; blue color: (Vz2 1/2, ( @p,2 1/2. (b),
(c), and (d) are shifted upwards by 0.25, 0.5, and 0.75.
Figures 7 and 8 show the distributions of the mean
streamwise velocity of particles V and of the fluid velocity
"seen" by the particles Up, for the cases with transverse and
longitudinal roughness elements, respectively. It is seen that
the maxima of V and U@, are located closer to the smooth
wall for increasing . I for the transverse elements. On the
contrary, the profiles of V are modified mostly in the region
of the longitudinal roughness elements. For all . cases, V
is increased near the crest of the elements. This result is
qualitatively similar to that generated by the interparticle
collisions on V [see, for example, Yamamoto et al., 2001;
Vance et al., 2006; Dritselis & Vlachos, 2008(a)].
Figures 9 and 10 show the wallnormal distributions
of the turbulent intensities (v,2) 1/2 and (u@p,2) 1/2 for the
cases with roughness elements placed on the lower wall
transversely and longitudinally, respectively. It is revealed
that these quantities are increased in the region near the
crest of the elements. This behavior is also observed when
interparticle collisions are important and they are taken into
account in the simulations. For the transverse elements, the
changes in (v,2) 1/2 and (up,2) 1/2 occupy a larger region
ofy/h values, as compared to the distributions f. i ,. /. = 0.
In the past, wall roughness has been indirectly
accounted for in numerical simulations based on virtual wall
models (Sommerfeld, 1992; Squires & Simonin, 2006;
Konan et al., 2009). The present study provides original
results and confirms previous findings, showing that the
roughness effect can be considered to be similar to that
produced by interparticle collisions, i.e., a considerable
enhancement of the wallnormal particle velocity, which
consequently leads to a very different overall particle
behavior.
7th International Conference on Multiphase Flow
ICMF 2010, Tampa, FL USA, May 30June 4, 2010
Conclusions
Results from LES of the gaseous flow and DPS of the
particles were presented in order to investigate the effects of
twodimensional roughness elements on the particle
transport in turbulent channel flow. Square bars separated
by a rectangular cavity were placed transversely or
longitudinally on the lower wall of the channel for several
values of the cavity width to the roughness height ratio.
The profiles of the mean streamwise and the root
meansquare velocities of the particulate phase are modified
by the roughness elements. The changes in the above
quantities are qualitatively similar to those produced by
interparticle collisions. The longitudinal elements influence
the overlying gasparticle flow mainly in the region close to
their crests, while the changes due to the rough wall
propagate to the outer flow for the transverse ones.
Nomenclature
C parameter of dynamic Smagorinsky model
CD drag force coefficient
d, particle diameter
fD,n drag force
h wallnormal distance of the outer flow
L,,M, Stress tensors of the dynamic model
mp particle mass
p fluid pressure
Re Reynolds number
S Density ratio between the two phases
S,, (S) strainrate tensor (magnitude)
t time
igp,n Component of fluid velocity seen by the particles
uc turbulent centerline fluid velocity
u, fluid velocity component
ul laminar centerline fluid velocity
u, wall friction velocity
v, particle velocity component
w, k cavity width and height of roughness element
x,y,z streamwise, wallnormal, and spanwise
coordinates
Xp,I particle position
Ax, Ay,Az Grid spacing in the x, y, z directions
II extra pressure gradient
pf fluid density
pp Particle density
ry Subgridscale stress tensor
Greek letters
1, Kronecker delta
A characteristic length scale
v fluid kinematic viscosity
Superscripts
n index of particle
r reduced quantities
A variables calculated on the test filter
(variables averaged in the homogeneous planes
and time
( filtered quantities
Subsripts
P
F
D
B
particles
fluid
drag
fluid properties along particle trajectory
bulk
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