7th International Conference on Multiphase Flow,
ICMF 2010, Tampa, FL, May 30 June 4, 2010
Numerical simulation of fully resolved particles
in roughwall turbulent open channel flow
C. ChanBraun, M. GarciaVillalba and M. Uhlmann
Institute for Hydromechanics, Karlsruhe Institute of Technology, 76128 Karlsruhe, Germany
chanbraun@ifh.unikarlsruhe.de
Keywords: particulate flow, sediment transport, turbulence, resolved particles, DNS, immersed boundary method
Abstract
We have performed an interfaceresolved direct numerical simulation of particleladen, horizontal openchannel flow.
The channel bottom boundary was roughened with a fixed layer of spheres and about 9000 particles were allowed to
move within the computational domain. The density ratio of the solid and fluid phase was 1.7 and the bulk Reynolds
number of the flow was 2880. In the present configuration, the particles tend to accumulate near the bed because
of gravity, but due to the turbulent motions, a cycle of resuspension and deposition is produced. This leads to a
particle concentration profile which decreases with the distance from the bed. It is found that the presence of particles
strongly modifies the mean fluid velocity and turbulent fluctuation profiles. The dispersed phase lags the carrier
phase on average across the whole channel height. Both observations confirm previous experimental evidence. The
different observations suggest that particle inertia, finitesize and finiteReynolds effects together with gravity play an
important role in this flow configuration. Several potential mechanisms of turbulenceparticle interaction are discussed.
Introduction
The erosion, transport and deposition of sediment by
turbulent openchannel flow are important mechanisms
for fluvial engineering applications. These phenomena
are far from being completely understood due to the
complex interactions between the turbulent flow and the
sediment particles. Turbulent flow in an openchannel
is statistically inhomogeneous in the wallnormal direc
tion, and the Reynolds numbers of interest are typically
high, which leads to a wide range of velocity and length
scales in the flow. In addition, the presence of a range of
sediment sizes, shapes and compositions further com
plicates the description. Sediment particles tend to ac
cumulate near the bed of the channel due to the effect
of gravity. However, under certain conditions, the turbu
lent motions might erode the bed and entrain sediments
which can then be transported by the carrier fluid. On
average this resuspension and deposition cycle leads to a
concentration profile which decreases with the distance
from the bed.
In order to improve the understanding of such flows,
it is necessary to simplify the problem under consider
ation, while at the same time retaining the fundamental
physics of the original problem. In this study, we con
sider fullydeveloped, particleladen, horizontal, open
channel flow. The particles are spherical, all of them
with the same diameter. For this configuration, we have
performed direct numerical simulations (DNS) with in
terface resolution.
In spite of the difficulty of performing experiments in
multiphase flow, similar configurations as the one stud
ied here have been investigated experimentally over the
last two decades (Kaftori et al. 1995; Tanibre et al. 1997;
Kiger and Pan 2002; Righetti and Romano 2004; Muste
et al. 2009). Most of these experiments were performed
using dilute suspensions, with a volumetric concentra
tion of order 0(10 3), and various kinds of sediments.
For example, in the study of Tanibre et al. (1997) the
density ratio between the solid and fluid phase was very
high since they employed glass and PVC particles in air.
The density ratio was lower in the rest of the studies,
ranging from values as low as 1.05 in the experiments of
Kaftori et al. (1995) (polystyrene particles in water) to
2.6 in Kiger and Pan (2002) and Righetti and Romano
t(2' 14) (both used glass particles in water), while Muste
et al. (2009) covered the range 1.032.65 using natural
sand and neutrallybuoyant sand (crushed nylon) in wa
ter. These experiments have shown the existence of a
velocity lag (in the mean) between particle and fluid ve
locities, and modifications to the turbulence character
istics of the flow due to the presence of the suspended
particles.
7th International Conference on Multiphase Flow,
ICMF 2010, Tampa, FL, May 30 June 4, 2010
Computational studies are only now beginning to ap
pear, with the exception of the pioneering work of Pan
and Banerjee (1997), who conducted resolved DNS of
turbulent particulate flow in a horizontal channel using
160 stationary and mobile particles.
Due to computational limitations, our present study
does not exactly correspond to any of the experiments
mentioned above. We have chosen parameter values
that are comparable to those of the study of Kiger and
Pan (2002), with somewhat lower Reynolds number and
density ratio, and in particular a substantially larger solid
volume fraction. Details of our configuration are given
below.
The objective of the present study is to investigate the
effects observed in previous experiments with the aid of
numerical simulations. In particular we are interested
in the lag between average particle and fluid velocities,
the modification of fluid turbulence properties due to the
dispersed phase, the particle concentration profile and
the particle velocity fluctuations.
Numerical method
The present simulations have been carried out with the
aid of a variant of the immersed boundary technique (Pe
skin 1972, 2002) proposed by Uhlmann '111 '5,,1 This
method employs a direct forcing approach, where a lo
calized volume force term is added to the momentum
equations. The additional forcing term is explicitly com
puted at each time step as a function of the desired parti
cle positions and velocities, without recurring to a feed
back procedure; thereby, the stability characteristics of
the underlying NavierStokes solver are maintained in
the presence of particles, allowing for relatively large
time steps. The necessary interpolation of variable val
ues from Eulerian grid positions to particlerelated La
grangian positions (and the inverse operation of spread
ing the computed force terms back to the Eulerian grid)
are performed by means of the regularized delta function
approach of Peskin (1972, 2002). This procedure yields
a smooth temporal variation of the hydrodynamic forces
acting on individual particles while these are in arbitrary
motion with respect to the fixed grid.
Since particles are free to visit any point in the com
putational domain and in order to ensure that the regu
larized delta function verifies important identities (such
as the conservation of the total force and torque dur
ing interpolation and spreading), a Cartesian grid with
uniform isotropic mesh width Ax Ay Az is em
ployed. For reasons of efficiency, forcing is only applied
to the surface of the spheres, leaving the flow field inside
the particles to develop freely.
The immersed boundary technique is implemented
in a standard fractionalstep method for incompress
y/H
z/H
0.25 0
Figure 1: Closeup of the computational domain. Both
fixed and moving particles are shown at a
given instant.
ible flow. The temporal discretization is semiimplicit,
based on the CrankNicholson scheme for the viscous
terms and a lowstorage threestep RungeKutta proce
dure for the nonlinear part (Verzicco and Orlandi 1996).
The spatial operators are evaluated by central finite
differences on a staggered grid. The temporal and spatial
accuracy of this scheme are of second order.
The particle motion is determined by the Runge
Kuttadiscretized Newton equations for translational and
rotational rigidbody motion, which are explicitly cou
pled to the fluid equations. The hydrodynamic forces
acting upon a particle are readily obtained by summing
the additional volume forcing term over all discrete forc
ing points. Thereby, the exchange of momentum be
tween the two phases cancels out identically and no spu
rious contributions are generated. The analogue proce
dure is applied for the computation of the hydrodynamic
torque driving the angular particle motion.
During the course of a simulation, particles can ap
proach each other closely. However, very thin inter
particle films cannot be resolved by a typical grid and
therefore the correct buildup of repulsive pressure is not
captured which in turn can lead to possible partial 'over
lap' of the particle positions in the numerical computa
tion. In practice, we use the artificial repulsion potential
of Glowinski et al. (1999), relying upon a shortrange re
pulsion force (with a range of 2Ax), in order to prevent
such nonphysical situations.
The present numerical method has been submitted
to exhaustive validation tests (Uhlmann 2003, 2005a,b,
2006b), as well as grid convergence studies (Uhlmann
2006a). The computational code has been applied to the
case of vertical plane channel flow (Uhlmann 2008).
7th International Conference on Multiphase Flow,
ICMF 2010, Tampa, FL, May 30 June 4, 2010
Reb = 2880 ReT = 216
Ubh/uT = 13.9 D+ = 10
pplpf = 1.70 H/D = 21
St = 9.7 Stb = 0.64
t, = 2.2% gh/Ubh = 0.70
D/a A = 6 Ax+ = 1.7
Table 1: Parameters of the direct numerical simulation.
* V
0.6
0.4
><
Setup of the simulation
The flow configuration consists ot turbulent open
channel flow with suspended freelymoving spherical
particles over a rough wall. The rough wall is formed
by a layer of fixed spheres which are tightly packed in
a cubic arrangement (see Figure 1). The layer of fixed
spheres consists of Npx 192 particles in streamwise
and Npz 48 particles in spanwise direction. The spac
ing between the particle centres is D + 2Ax, where D
is the sphere diameter and Ax is the grid spacing. Be
low this layer of spheres a rigid wall is located at y = 0.
As can be seen in Figure 1 this rigid wall is roughened
by spherical caps that can be defined by the part above
y 0 of spheres located at y D /2 2(D/2 Ax)
in a cubical arrangement, staggered to the fixed layer of
spheres above.
Periodic boundary conditions are applied in stream
wise and spanwise directions. At the upper boundary a
freeslip condition is employed. At the bottom boundary
and on the surface of the fixed spheres a noslip condi
tion is applied.
Table 1 summarises the numerical and physical flow
parameters. The computational domain dimensions are
Lx/H x Ly/H x Lz/H 12 x 1 x 3, in streamwise,
wallnormal and spanwise directions, respectively. An
equidistant Cartesian grid was used throughout the entire
domain using 1536 x 129 x 384 grid points. The fixed
as well as the moving particles are of spherical shape
and have a diameter of D/H 4.69 10 2. The global
solid volume fraction, s,, was set to s = 2._" which
corresponds to 9216 particles.
In order to scale the results, two quantities need to
be specified: the friction velocity u, and the virtual
wall location yo, since for a rough wall, the real posi
tion of the wall cannot be unambiguosly defined. In the
present work we define the position of the virtual wall as
yo = 0.8D, measured from the bottom of the computa
tional domain. This value has been used often in the past
(Nezu and Nakagawa 1993) and the results are not very
sensitive to small variations. Once yo was specified, u,
was obtained using a linear fit of the total shearstress
profile. Knowing the position of the virtual wall the ef
..
*
I
T (
" T 
vo
v,
,v ,
,      
1 0.8 0.6 0.4
Su' v') / u
0.2 0
Figure 2: Wallnormal profile of the streamwise
and wallnormal particle velocity cross
correlation (u'v)/u_. Averaging was
performed over different time intervals: dots,
8 flow through times; squares, 4 flow through
times; triangles, 2 flow through times.
fective flow depth, h, can be defined as the distance from
the virtual wall to the top boundary, h H yo. The
bulk velocity based on the domain height H is defined
as UbH 1/H fJo(U)dy. Hereby, U is the time aver
aged streamwise velocity component and angle brackets
denote spatial averaging along the homogeneous direc
tions. The bulk Reynolds number, Reb UbHH/V,
was kept constant at a value of 2880. This corresponds
to a friction Reynolds number, Re = uh/v 180
in case of a smooth wall. In the present twophase flow,
the friction Reynolds number obtained Re = u.h/v is
216. Further quantities that can be derived are the par
ticle diameter in wallunits D+ uD/v 10, the
Stokes number based on the particle time scale and the
friction time scale, St 9.7, and the Stokes number
based on the particle time scale and the outer time scale
tb h/Ubh, Stb 0.64. The nondimensional gravity
acceleration was set to gh/Uh = 0.79.
The grid resolution of the simulation in terms of wall
units was Ax+ 1.7. This is a very high resolution
away from walls, however it is marginal very close to
the wall and very close to the surface of the spheres.
Thus the present simulation should be considered as
marginally resolved and a finer resolution is desirable.
The time step was chosen such that the CFL number re
mains below 0.3.
The initial turbulent flow field was taken from a sim
ulation of openchannel flow over a rough wall without
moving particles. Then, particles were added at random
positions and assigning them an initial traslational ve
locity equal to the local fluid velocity. Subsequently, the
7th International Conference on Multiphase Flow,
ICMF 2010, Tampa, FL, May 30 June 4, 2010
10
.... .... ,,.,
3 0
Figure 3: Entire computational domain, moving particles clustered in the near wall region due to gravity, highspeed
and low speed streaks in blue and red colour visualised by isocontours of constant velocity fluctuation.
simulation was run until the flow reached a statistically
stationary state. The simulation was continued for 120
H/UbH, which corresponds to 10 flowthrough times,
during which flow field statistics were sampled at every
second time step and particle statistics were stored every
tenth time step. This leads to 9000 samples of instanta
neous flow and 4.9 107 samples of particle data from
1850 instantaneous fields. Based on this data the statis
tics were computed that are discussed in the following.
Due to the gravity effect, once the steady state is
reached the concentration of particles is much higher
close to the wall than in the upper part of the chan
nel (see Figures 1 and 3). To provide a measure for
the convergence of the disperse flow statistics, Figure 2
displays the wallnormal profile of the crosscorrelation
between the streamwise and wallnormal particle ve
locities, (u'v') computed over different time intervals.
The statistics of the disperse phase below y/h z 0.3
can be regarded as steady and well converged. Above
y/h z 0.3 the scatter of the data increases. The latter
region is visited by few particles and therefore to obtain
wellconverged particle statistics would require signifi
cantly longer integration times.
Results and Discussion
Figure 3 shows a snapshot of the flow field illustrating
the complexity of the phenomena involved. Particles are
coloured with the distance to the wall and the flow is vi
sualized using isosurfaces of positive (negative) stream
wise velocity fluctuations in red (blue). Particles accu
mulate close to the wall but occasionally, due to the large
scale turbulent motion, particles might reach the upper
part of the domain.
In the following discussion the results of the two
phase flow are compared with the results of a single
phase open channel over a smooth bed at the same
bulk Reynolds number Reb = 2880 which corresponds
to a friction Reynolds number of Re, 182. This
simulation has been performed with the inhouse code
LESOCC2 (Breuer and Rodi 1996; Hinterberger 2004).
The mean streamwise velocity profile of the fluid
phase and the disperse phase are shown in Figures 4 and
5 in comparison with the singlephase smoothwall sim
ulation. The first observation when comparing the pro
files of the fluid phase is that the effect of the disperse
phase is similar to the effect of roughness, i.e. the ve
locity profile in Figure 5 is shifted towards lower values.
However, in addition to this, the gradient of the velocity
profile decreases near the wall in the present situation,
which is untypical for flow over roughness. A change
of the gradient is also in contrast to the observations in
previous experiments of twophase flow in a horizontal
channel (Kaftori et al. 1995; Kiger and Pan 2002). This
effect might stem from the low Reynolds number con
sidered, the lower ratio h/D used in the present simu
lation or the large concentration of particles in the near
wall region. Other explanations are also possible. Both
Figures show that there is a mean velocity lag between
solid and fluid phase over the whole flow depth. A lower
velocity of the disperse phase has been observed by sev
eral previous experiments over a range of Stokes num
bers from Stb of order 0.1 in Kiger and Pan (2002) up
to Stokes numbers of order 1 in Tanibre et al. (1997).
It was also found to occur in flows with different den
sity ratios; from nearly neutrally buoyant in the case of
7th International Conference on Multiphase Flow,
ICMF 2010, Tampa, FL, May 30 June 4, 2010
20
15
10
0 0.5 1
U / Ubh
Figure 4: Wallnormal profile of the mean streamwise
velocity distribution normalised by Ubh of the
fluid phase (solid line) and the solid phase
(dots) in comparison with a single phase open
channel flow over a smooth wall (solid line
markers +).
Kaftori et al. (1995) to high density ratios in Tanibre
et al. (1997). Furthermore, it has been observed also
over a range of different solid volume fraction from as
little as 0.023% in Kaftori et al. (1995) up to 2.2% in
the present case. There could be several explanations
for this phenomenon. It might be explained by the fact
that particles close to the wall, and therefore with low
momentum, are ejected by bursts to the outer region but
retaining their low momentum for some time. Another
possibility is that the driving pressuregradient might act
differently on fluid and particles.
Figure 6 shows the velocity fluctuations of the two
phases in comparison with the fluctuations in the smooth
wall simulation. For the statistics of the fluid phase the
strongest effect of the dispersed phase can be seen in the
decrease of the nearwall peak of the streamwise com
ponent m,,,. The profiles of the fluctuations in wall
normal and spanwise direction differ only marginally
from the profiles of the smooth wall simulation. This
is in agreements with the findings of Kiger and Pan
(2002) who, however, observed a less pronounced drop
of the nearwall peak in the streamwise fluctuations.
This might be due to the smaller volume fraction in their
experiment. Above y/h = 0.4 the single and twophase
fluid statistics overlap except for the region close to the
upper boundary. Here, the streamwise and spanwise
fluctuations are smaller compared to the single phase
case while the wallnormal component again overlaps
with the single phase case. This could indicate that the
difference between the singlephase flow and the two
phase flow in this region can be found in the largescale
Figure 5: Wallnormal profile of the mean streamwise
velocity distribution normalised by u, of the
fluid phase (solid line) and the solid phase
(dots) in comparison with a single phase open
channel flow over a smooth wall (solid line
markers +).
turbulent structures rather than in the small scale struc
tures.
We turn now our attention to the statistics of the
disperse phase. Near the wall, the fluctuations of the
streamwise and spanwise components are smaller than
the fluid counterparts, while the wallnormal compo
nent is larger, i.e. the particle fluctuation energy is more
isotropic than the fluid turbulent kinetic energy. As men
tioned before, in this region there is a higher concentra
tion of particles, and therefore the collision rate is in
creased. As a consequence a more isotropic distribu
tion is expected. Kaftori et al. (1995) have also observed
that the fluctuations of the wallnormal particle velocity
are larger than the fluid fluctuations near the wall, while
Kiger and Pan (2002) did not observe this effect. Con
versely, in the outer region (above y/h > 0.4) all par
ticle velocity fluctuation components are larger than the
fluid counterparts and exhibit smaller wallnormal gra
dients. This can possibly be explained by the increased
inertia of the disperse phase.
In Figure 7 the profiles of the crosscorrelation be
tween streamwise and wallnormal velocity fluctuations
(Reynolds stress) for both solid and fluid phases is
shown. The statistics for the fluid phase above y/h
0.3 collapse with the singlephase data. In this region
the disperse phase seems to have only little effect on the
distributionof the Reynolds stress. Below y/h = 0.3 the
Reynolds stress of the twophase flow is smaller. This
could be due to some sort of interference between the
particles and the buffer layer structures, the latter could
be somewhat damped by the presence of the former lead
. 0.6
0.4
0.2
7th International Conference on Multiphase Flow,
ICMF 2010, Tampa, FL, May 30 June 4, 2010
u' /u
rms T
Figure 6: Wallnormal profile of rootmean square ve
locity fluctuations normalised by u, of the
fluid phase (solid line) and the solid phase
(solid markers) in comparison with a single
phase open channel flow over a smooth wall
(solid line markers +); from left to right:
wallnormal, spanwise and streamwise fluctu
ations.
ing to a lower Reynolds stress. Comparing the disperse
phase crosscorrelation and the fluid Reynolds stress, it
is clear that a strong deviation can be found between
(u'v') and (uv') below y/h ~ 0.4. Visibly, the par
ticle motion in the streamwise and wallnormal direc
tions is less correlated than the fluid motion. Different
explanations are possible, in particular the wellknown
crossingtrajectories effect of heavy particles in turbu
lent flow (Yudine 1959). This observation deserves fur
ther consideration.
Finally, we discuss the vertical distribution of parti
cles in the channel. It is customary in the literature
to study equilibrium suspensions by using an equation
for the concentration of the dispersed phase. This can
be derived from a convectiondiffusion equation of the
concentration (Garcia 2008). Further assumptions are
the use of a logarithmic law of the wall for the mean
streamwise velocity distribution of the fluid phase and an
isotropic eddy viscosity model for the turbulent concen
tration flux. This later assumption is particularly ques
tionable in light of the higher particle inertia. These hy
potheses lead to the following law for the concentration
of the dispersed phase
(C) (h y) /y ()
Ca ha)/a
Hereby, = vl/Ku,, where v, is the 'nominal' set
tling velocity of the particle given by a balance between
0 "         
1 0.8 0.6 0.4 0.2 0
(u'v' /u
Figure 7: Wallnormal profile of crosscorrelation
of streamwise and wallnormal velocities
(Reynolds stress) normalised by u2 of the
fluid phase (solid line) and the solid phase
(solid markers) in comparison with a single
phase open channel flow over a smooth wall
(solid line with markers).
drag and immersed weight, using the standard drag for
mula (Clift et al. 1978), K z 0.41 the von Kirmin con
stant and u the friction velocity. In equation (1) the
quantity C, is a nearbed reference concentration mea
sured at a distance y yo a, where a cannot be
zero because the expression becomes singular. Here we
have selected a value of a which corresponds roughly
to three particle diameters (a/h 0.15), leading to
Ca 0.0533. Equation (1) is often called the Rousean
distribution for suspended sediment.
Figure 8 shows the concentration of particles versus
the vertical coordinate normalized by the concentration
Ca, together with the evaluation of the Rousean distri
bution for two different B values. The first one as given
by its definition, = vl/uu,, does not provide a good
approximation for the present particle distribution. This
might be due to the high degree of simplification inher
ent in the formula, in particular the modeling of the ver
tical turbulent concentration flux by means of a simple
eddy viscosity. A further evaluation with an optimized
exponent (40 % lower value of /) leads to a remarkable
fit with the present DNS data for y/h > 0.12.
Conclusions
We have performed interfaceresolved direct numeri
cal simulation of sediment transport in horizontal open
channel flow. In our configuration the wall bound
ary was roughened by means of fixed spherical parti
cles. The simulated parameter values were comparable
104 102 10
C/C
a
Figure 8: Wallnormal profile of particle concentration
(solid line with markers) in comparison with
Rouse formula, eqn. (1), using the original
definition of B (dashdotted line) and using a
40% lower value of B (dashed line).
to those of the study of Kiger and Pan (2002), with the
main exception of a substantially larger solid volume
fraction.
Our results show that the presence of particles
strongly modifies the mean fluid velocity profile, lead
ing to lower gradients in the immediate vicinity of the
wall and higher gradients somewhat outside of the re
gion exhibiting the highest particle concentration. The
dispersed phase, on the other hand, lags the carrier phase
on average across the whole channel height (here we
refer to the difference in the mean of the streamwise
component of the velocities corresponding to the two
phases). The fluid velocity fluctuations are similar to
the singlephase reference data, except for the stream
wise component, which is strongly damped in the buffer
layer. Particle velocity fluctuations are smaller than the
fluid counterpart near the wall and larger in the outer
flow. The fluid Reynolds stress is found to be damped
near the wall and essentially unchanged in the outer flow.
The particle "Reynolds stress" is much lower than the
fluid counterpart in the nearwall region, and both are
similar outside.
Finally, we have compared the concentration profile
with the Rousean distribution. Although the original for
mula already involves a freetochoose parameter (the
location of the reference point), we need to adapt the ex
ponent (given as a function of the settling velocity, the
friction velocity and the Karman constant) in order to
obtain a good fit of the DNS data.
The different observations suggest that particle iner
tia, finitesize and finiteReynolds effects together with
gravity play an important role in this flow configuration.
7th International Conference on Multiphase Flow,
ICMF 2010, Tampa, FL, May 30 June 4, 2010
Several potential mechanisms of turbulenceparticle in
teraction have been discussed in the text. It is our in
tention to gather the additional statistical data which
is necessary to verify those hypotheses in the future.
More urgently, however, we need to repeat the cur
rent simulation at increased spatial resolution, which is
slightly marginal. Furthermore, it needs to be estab
lished whether the details of the numerical treatment of
particle collisions affect the results significantly.
Acknowledgements
This work was supported by the German Research Foun
dation (DFG) under project JI 18/191. The computa
tions have been carried out at the Steinbuch Centre for
Computing (SCC) of Karlsruhe Institute of Technology.
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