7th International Conference on Multiphase Flow
ICMF 2010, Tampa, FL USA, May 30June 4, 2010
LES Simulation of GasSolid TwoPhase 3D Wake of a Square Cylinder Confined in a
Channel
Li Zhang, Hongtao Liu, Qiang Tang, Liangming Pan, Canglei Qin, Lin Ding
Chongqing University, College of Power Engineering,
No. 174 Shazheng St., Shapingba, Chongqing, 400044, P.R China
lizhang~cqu.edu.cn
Keywords: Gassolid twophase flow, Large eddy simulation, Square cylinder
Abstract
In present paper, an EulerianLagrangian model is proposed to study gassolid twophase flow past a square cylinder
confined in a channel at high Reynolds number. In this model, the singlephase flow is investigated by large eddy simulation
(LES) and the particle trajectories are tracked by Lagrangian tracking method. The model is validated by the comparison of the
Strouhal number and drag coefficients across square cylinder with literature. Then, gassolid twophase flow across square
cylinder is studied and particle distribution is obtained under different Stokes number for particle transport cases. The
settlement, entrainment and aggregation of solid particles moving with the largescale coherent vortex structure in the wake of
square cylinder are numerically investigated, and the effects of St number on the distribution of solid particles are obtained.
1. Introduction
The diffused distribution of particles in bluff body
wakes, especially particles in the cylinder wake, is a
common phenomenon in nature. There are many industrial
applications and technical background, such as combustion
and pneumatic conveying of coal powder, fluidized bed,
and dust deposition, etc. Therefore, the research on the
mechanism of particle diffused distribution in the cylinder
wake has important engineering value and practical
significance.
The mechanism of square columns the wake past a
rectangular cylinder is difficult from that of circular
cylinder. In case of the flow past a rectangular cylinder,
however, the location of flow separation is fixed at
upstream corners of the cylinder due to the abrupt
geometrical change. And it becomes the research hot spots
that the features is complex in the flow past the square
cylinder with high Reynolds number, such as flow
separation, reattachment and unsteady flow in the wake
region. There have been some studies on the flow past a
rectangular cylinder. Lyn and Rodi (1994) performed
experiments for the flow past a square cylinder at relatively
high Reynolds number (21,400). In their study, they
focused on the shear layer formed by flow separation from
the upstream corner of a square cylinder rather than the
nearwake region. Okajima (1982) carried out experiments
on the flow past a rectangular cylinder of diverse aspect
ratio (1, 2, 3 and 4) at various Reynolds number
(7020,000), and found the relation between Strouhal
number and Reynolds number. Muralami et al.( 1995) ,
and Yu(1997) investigated the flow past a square cylinder
at Re = 105 by largeeddy simulation. The results show that
the complex features of the flow past a bluff body can
accurately Description by LES model. Kim et al ga ini~i
used large eddy model to simulate turbulent flow past a
square cylinder confined in a channel. They reported the
timeaveraged LES results are in good agreement with the
experiments which were currently available.
The numerical studies of particle dispersion in plane
mixing layers, plan jets, wake gas flows behind a cylinder
or cylinders, and sprays have been made extensively, and it
was found that the large organized vortex structures had a
dominate effect on the dispersion of particles over a range
of Stokes numbers(e.g. Aggarwal 1994; Brandon 2001 :Ling
et al.1998; Longmire 1992: Narayanan 2002 ). In recent years,
LES were widely applied to the study of gassolid
twophase flow (e.g. Geiss et al. 2004; Lakehal 2002).
Although the flow field can be well predicted by the
largeeddy simulation model, the accuracy and the
reliability of LES predictions is affected by the accurate
modeling of the subgridscale (SGS) phase interactions
and the correct representation of the initial/boundary
conditions for all phases. Miller & Bellan (2000)
conducted a thorough analysis of the SGS effects using
DNS results for a transitional mixing layer, and they also
concluded that neglecting the SGS velocity fluctuations in
LES might lead to gross errors in the prediction of the
particle drag force. Armenio et al. (1999) investigated the
effects of the SGS on particle motion. Their work indicated
that using a filtered velocity field alone to advance the
particles can lead to serious inaccuracies; thus the
importance of the SGS closures is emphasized.
Up to now, however, only a few numerical studies on
the particle dispersion in gassolid twophase flows past
square cylinders have been conducted. Brandon &
Aggarwal (2001) combined staggeredgrid control volume
approach and the MarkerandCell (MAC) technique
Discrete Vortex Method with solid particle tracking to
simulate the particle motion in gasparticle 2D flows past a
square cylinder at Reynolds number of 250 and 1000. In
their work, the particle dynamics for particles with various
Stokes numbers from 0.01 to 5 were simulated using the
modified BassetBousinesqOseen (BBO) equation. They
got the characterize particle dispersion and deposition as a
function of the Reynolds number, particle Stokes number
(St), and density ratio.
As motion of the particle is largely dominated by the
response of the particle to the fluid fluctuating velocity in
the wake past the square cylinder, particle distribution
under different Stokes number for particle has to be studied.
In this paper, particle motion across a square cylinder
confined in a channel is numerically simulated by means of
LES for the gas flow and a Lagrangian trajectory method
for the particle phase.
Nomenclature
7th International Conference on Multiphase Flow
ICMF 2010, Tampa, FL USA, May 30June 4, 2010
1
To = 2vt S + 3 rk
3 (4)
with vt is the subgrid scale viscosity:
t,=ca) S (5)
Here C, is the Smagorinsky constant, its value is taken
equal to 0.1 in present work.
S = 2S)S (62
a = (A xar yA z)3
Here, S,3 is the rate of strain tensor of the filtered velocity
field.
CD Particle drag coefficient
Cd drag coefficient
d, particle diameter
f, frequency of vortex shedding
h the diameter of square cylinder
g gravitational constant (ms2)
m, mass of particle (kg)
p pressure (Pa)
Re Reynolds number
ReP particle Reynolds number
Sii rate of strain tensor of the filtered velocity field
St particle Stokes number
Str Strouhal number
t time
T* timestep
u, velocity vector
u, Particle velocity vector
mean streamwise velocity at the cylinder
location.
x, position vector
Greek letters
p constant mass density (air density), kgm3
p, particle density, kgm3
vt, subgrid scale viscosity (m2 S1
r,3 the subgrid scale (SGS) stress tensor(Pa)
Subsripts
p particle
2. Continuous phase equations
Large eddy model is used to simulate the gas phase flow.
The flow field variables are separated into a largescale
component and a subgrid scale (SGS) component by
filtering. In the incompressible and Newtonian fluid,
continuity and NavierStokes equations can be written as
the following:
8 jpu )
= 0 (1)
~x
80 Buu, 1 y 0 8 8W,
at 8x, p 8x, 8x,8x, 8x, 8x,
To = UpUJ quI (3)
Here r11 is the subgrid scale (SGS) stress tensor. The effect
of the subgrid scales on the resolved scales is modelled by
the SGS stress according to the Smagorinsky model. (1963):
du
8x,
3 Particle phase equations
Lagrangian approach is employed to predict the
properties of each particle directly from the equations of
motion. Basic assumptions for particle motion are as
follows :
(1) All particles are rigid spheres with equal diameter and
density.
(2) The density of the particles is much larger compared
with that of the fluid.
(3) Particleparticle interactions are negligible.
(4) Dilute twophase particleladen flow is assumed and the
effect of the particles on the fluid flow is neglected.
(5) Collisions with boundaries are assumed to be perfectly
elastic.
Under the above assumptions, the motion of particles
satisfies the second law of Newton. Magnus force, Saffman
force and Basset force are neglected because density ratio
between particle phase and gas phase is quite large. So, the
particle motion equation can be expressed as:
d u
m,
id2
p f ~,
p~) + m~g
d t 1 g
Here, g is the acceleration of gravity, d, is particle diameter,
CD iS the drag coefficient (Hinds, 1984).
24
Re, Re, < 1 (10)
Re,
0.44Re 400 < Re,
Here, Re, is the particle Reynolds which is defined as
Re~p =l~i (11)
The particle Stokes number defined as:
St = P"d(12)
18pu
3. Physical model and Flow parameters
3.1 Computational domain
Fig.1 shows the physical configuration of the
computational domain. The square cylinder is placed at the
center of the channel. Finite volume method is employed to
discretize the governing equations. Nonuniform staggered
S =' 1 8 J
(b) streamlines
Fig 3 Distribution of vorticity magnitude and the
streamlines of the square cylinder
7th International Conference on Multiphase Flow
ICMF 2010, Tampa, FL USA, May 30June 4, 2010
observe the tail of the vortex can only be detached from the
wake.
(a) VOrticity magnitude
grid is applied in the Cartesian coordinate system as shown
in Fig. 2, in which the grid system can be seen on an xy
plane. The grid spacing in the immediate vicinity of the
cylinder is 0:01h. It turns out that 201x91x51 grids are
adequate in streamwise (x), normal (y) and spanwise (z)
directions, respectively a
Sh// Iwall
3h h
Fig.1 Physical configuration.
Fig.2 Grid system
3.2 boundary conditions and numerical treatment
Finite volume method is employed to discretize the
governing equations. The SIMPLE algorithm (Partankar,
1980) is adopted to couple the pressure and velocity fields.
In this work, Reynolds number based on h and Um is
fixed as 104, and the blockage ratio is 20%. Periodic
boundary condition is applied in the spanwise direction. The
uniform velocity is imposed as an inlet boundary condition.
At outlet, a convective condition is employed:
Bu 8v 8w
(13)
8x 8x 8x
Noslip condition is imposed at all solid walls.
One thousand particles are evenly spaced in the region of
y/h=2.510.5 at the inlet and added at intervals of 5 time
steps.
4 Results and Discussion
4.1 Singlephase flow across square cylinders
In this part, the flow field and local characteristics around
square cylinder are studied and the forces on cylinders are
analyzed. The timestep here is 0.001 s and totally 5000
timesteps are calculated.
The distributions of vorticity magnitude and velocity
vector field at timestep of 3000 are shown in Fig. 3. The
results show that there is a very intense change of velocity
field around square cylinder because the vortex Street I was
formed by stress concentration on four corners of the square
cylinder. It can be observed that the vortices changed
direction in the wake of the wall. But can not clearly
T*"
(a)Cd
(b)C,
Fig.4 Drag coefficient and lift coefficient changes with time
7th International Conference on Multiphase Flow
ICMF 2010, Tampa, FL USA, May 30June 4, 2010
n ....;
4. 
= lilli
0 2 4 8 10 2 14 1 18 20 22 2 25 2 0
am
(a) T*=1500
The Strouhal number (Str) is a dimensionless number to
describe oscillating flow mechanisms In these paper, the
Strouhal number defined as:
Ub
where fB is the frequency of vortex shedding, h is the
diameter of square cylinder and Ub iS the mean streamwise
velocity at the cylinder location. The simulation result
showed the Str number of vortex street break off is 0.123,
which is very similar to 1.2~1.3 for 1042Re2103 TepOrted by
Atsushi (1982). For lift and drag coefficient showed in Fig.
4, we can see there is the fluctuation of drag coefficient with
a mean value of 2.146,. which is similar to the value of
Cd=2.1 for Re2104 by White (1994).
4.2 Gassolid twophase flow across square cylinder
Figure 5 shows the longitudinal distribution of particles
with Stokes number of 0.01 at different times. It can be seen
from Fig. 5.13(a) that the particles were constantly
entrained and dragged by the largescale vortex because of
its advantage. Particles with Stokes number of 0.01 have
smaller aerodynamic response time scale so it can rapidly
respond to fluid changes and showed a strong perfonnance
to follow the fluid. With the development of the largescale
vortex structure in the flow field, the distribution of
particles became unifonn and even some particles were
directly entrained into the vortex core area of the largeeddy
structure. Also, the particles distribution showed the same
quasicoherent structures with flow structures. With the
change of time, particle distribution along the flow direction
became unifonn as shown in Fig5(c) and Fig5(d). The main
reason is that the largescale vortex structure would develop
to break up, and fonn smallscale vortex structures in the
flow field. The directions of these smallscale vortex
structures were different, and even the magnitude of their
characteristic time scales were the same as the air
aerodynamic response time of particle. All these factors
would break the regular distribution of the original particle,
which lead homogenization of the particles in the wake
around the square cylinder.
The effect of Stokes number on distribution of particles
flowing the cylinders is investigated. It can be seen from Fig.
6 that the change of stokes number has a significant impact
on the particle distribution in the flow field. When the
Stokes number is 0.05, the particles distribution is similar to
that of particles with Stokes number of 0.01. With the
particle size increasing, its inertia also increased. The
aerodynamic response time of particles is greater than the
characteristic time scales of some smallsized vortex
structures. And some particles unchanged in the direction of
motion. So the length of particle unifonn distribution
increased along the flow direction.
When stokes number is 0.1, the priority agglomeration
effect began to appear in the regions where largescale
vortex structures existed in the flow field. Particles mainly
distributed in the region near the border of the vortex
structures. The partial disperse of particles was affected by
the vortex structures, so the particle distribution is
nonunifonn in the flow.
(b)T*=2000
(c)T*=2.5000
.:P.
C 6 8
10 12 11 16 18 20 22 2d 26 28 30
(d) T*=3000
Fig.5 Distribution characteristics of particles with Stokes
=0.01 in the vertical direction at different times
When Stokes number is 0.5, particle's aerodynamic
response time is close to the characteristic time scale of the
vortex structures which is fonned in the flow around a
square cylinder. Also particle's inertia increased and began
to be thrown out of the vortex core region. There is a
obvious region near the border of vortex structure where
particles are mainly distributed, so the nonunifonn
distribution of the internalempty and extemnalfull is
fonned. The vortex profile has been shown in the vortex
street. The aerodynamic response time of particles is lager
than the characteristic time scales of the smallscale vortex
structures. Particles can unifonnly distribute in the region
where the smallscale vortex structures existed. As a result,
particles distributed nonunifonnly in the whole flow field.
When Stokes number is 1, the inertia of particles were
larger, and particles were thrown out of the vortex core
region easily. Under the action of the stretching and folding
among vortex structures, the local enriclunent phenomenon
of particles at the junction of multiple vortex structures
appeared.
Figure shows the distribution characteristics of particles
with different Stokers number the horizontal direction at
T*=2000. When stokes numbers were 0.01, 0.05 and 0.1, a
certain amount of particles distributed in the back region of
the square cylinder. The reason is that the aerodynamic
response time of particles is small, so the particles can
timely respond to the change of flow field and develop with
the vortex. When stokes numbers were 0.01, 0.05 and 0.1,
p s
(a) St=0.01
~mmn  
x/h
(b) St =0.05
7th International Conference on Multiphase Flow
ICMF 2010, Tampa, FL USA, May 30June 4, 2010
(a) St= 0.05
'~~.~;.i
0
~xth
(b) St= 0.1
........
''
~:C:' ~ 1.
p? .
~ ".e 'Li.'ii.. .*.
lo 15 20 15 30
(c) St= 0.5
(c) St0.1
0 2 4 6 10 1 14 6 18 0 22 24 26 26 3
(d) St= 1
Fig.6 Distribution characteristics of particles with different
Stokes number in the vertical direction at T*=2500
there are almost no particles distributed in the back region
of the square cylinder. This is because the particles with
Stokes number of 0.5 and 1 distributed in external regions
of the vortex structure when the largescale vortex stmectures
rolled up after the fluid flowed around a square cylinder. At
same time, impacted by the largescale transverse vortex
stmectures, most of the particles were thrown out of the
vortex core region and distributed in the external region of
the transverse vortex. So there were few of particles in the
vortex, which resulted blank distribution. Along the flow
direction, the largescale flow direction vortex and
transverse vortex gradually turned into smallscale vortex
structures, and particle concentration phenomenon occurred
in local area.
5 Conclusions
In this study, LES is employed to predict turbulent flow
fields past a square cylinder confined in a channel with a
blockage ratio of 20% at Re=104 and Lagrangian approach
is used to predict the properties of each particle. The
dispersion of particles with different Stokes number has
been studied, particle distribution and time series of vortex
evolution is obtained.
(1) The distribution of the particles was affected by the
vortex structure which fonned in the flow around a square
cylinder. The characteristic time scale of different sizes
vortex structures were various when fonned by largescale
vortex broken. Under the same particle aerodynamic
response time, the distribution of particle exhibited
(d) St =0.5
(e) sSt1
Fig.7 Distribution characteristics of particles in the
horizontal direction at T*=2000
diversity.
(2) In the generation and development process of the
largescale vortex stmectures, particles with different Stokes
number have different dispersion characteristics. When
Stokes number is 1, a larger number particles gathered near
the periphery of large vortex structures and the particles
dispersion stmectures like quasicoherent structures of the
flow were fonned. Also the enriclunent phenomenon of
particles occurred in the part. So the particle distribution is
most uneven. The particles with Stokes number of 0.01 had
the similar coherent structures to the flow field that the
distribution of particle concentration is more evenly.
(3) The dispersion of particles in the horizontal direction
mainly affected by the flow direction vortex and transverse
vortex which were fonned due to flow across a square
cylinder. When the largescale flow direction vortex and
j:lj ~3 e~p~
7th International Conference on Multiphase Flow
ICMF 2010, Tampa, FL USA, May 30June 4, 2010
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the dispersion of particles changes in the horizontal
direction.
Acknowledgements
The author would like to acknowledge the National nature
Science Foundation of Chongqing China ( CSTC,
2009BA6067).
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