INTRODUCTION
The flow past a row of circular cylinders is an area of considerable interest
to numerical analyst as well as theoreticians. A vast body of knowledge encompassing
analytical, numerical and experimental studies is now available on the flow
past a circular cylinder (Zdravkovich, 2003). This study
will shed light between the drag, blockage ratio (distances between centers
of cylinders), Reynolds number and the Strouhal numbers for the twodimensional
flow past a row of circular cylinder. Systematically studied for large distances
between centers of the cylinders was conducted by Fornberg
(1991).
Over the past few decades, much theoretical and numerical effort has been made
to investigate the distances between centers of cylinders (Boppana
and Gajjar, 2007; Gajjar and Nabila, 2004; Fornberg,
1998; Natarajan et al., 1993; Smith,
1992). Gajjar investigated instability of flow past a cascade of circular
cylinders (Gajjar, 2006).
The interface effect between the cylinders in the row is a very remarkable
at low Reynolds numbers (Tamada and Fujikawa, 1957).
A steady flow through a uniform cascade of normal flat plates numerically and
experimentally studies (Ingham et al., 1990).
However, the studies, especially on numerical simulations of flow past a row
of circular cylinders are still relatively partial. Therefore, the further effort
is made in this study to study numerically the flow past a row of circular cylinders.
In the present study, the Lattice Boltzmann Method (LBM) is employed to simulate
such flows past a row of circular cylinders. LBM is originated from the Lattice
Gas Cellular Automata (LGCA) (Frisch et al., 1986,
1987 ). In LGCA, the fictitious particles move along
lattices with streaming and collision. However, LGCA suffers some drawbacks
such as large statistical noise, unphysical velocitydependent pressure, nonGalilean
invariance and large numerical viscosities. It is the lattice Boltzmann method
that overcomes the drawbacks of LGCA. In LBM, the BhatnagarGrossKrook (BGK)
(Bhatnagar et al., 1954) collision model used
in the standard Boltzmann equation is adopted.
In recent years, the Lattice Boltzmann Method (LBM) has received considerable
attention as an alternative numerical scheme for simulating a variety of fluid
dynamic problems (Chopard and Droz, 1998; WolfGladrow,
2000; Succi, 2001; Benzi et
al., 1992; Chen and Doolen, 1998). A twodimensional
ninevelocity (D2Q9) LBM model (Xiaoyi and LiShi, 1997)
has been used in this study.
The purpose of the present study is an attempt to examine the effect of distances
between centres of cylinders up to Reynolds number of 200. A little information
is available about the effect of distances between centers of cylinders on drag
and wake characteristics. On the other hand, much information is available on
the drag, transition from one regime to another, vortex shedding, etc. The numerical
study has been carried out using the Lattice Boltzmann Method (LBM). This study
shows the capability of the proposed model and the interface effect between
the cylinders in the row is very remarkable at low Reynolds numbers.
LATTICE BOLTZMANN METHOD
A square lattice D2Q9 with unit spacing is used in which each node has eight
nearest neighbors connected by eight links (Fig. 1). Since,
the details of D2Q9 model are given in detail there (Xiaoyi
and LiShi, 1997) for incompressible NavierStokes (NS) equations independent
of the density variation. Some basic features of the proposed LBM alone are
presented here. Hence, there are two types of moving particles. Particles move
along the axis with speed e_{i} = 1, i = 1,2,3,4, particles move along
the diagonal directions with speed
Rest particles also allowed at each node with speed zero. The velocity directions
of these particles are defined as:
where, c = δx/δt, δx and δt are the lattice constant and the time step size, respectively.
The particle distribution function satisfies the following evaluation equation
of the system is:
P_{i}(x+e_{i}δt,t+δt)
= P_{i}(x,t)+Ω_{i} 
_{}(2) 
where, Ω_{i} is the collision operator representing the rate of
change of the particle distribution function due to collision and P_{i}(x,t)
is the density distribution function. The lattice Boltzmann BGK equation (in
lattice units) after BhatnagarGroos Krook (BGK) (Bhatnagar
et al., 1954), with singletime relaxation approximation is:

Fig. 1: 
A twodimensional ninevelocity lattice (D2Q9) model 
where, P_{i}^{(e,q)}_{(x,t)} is the equilibrium pressure
distribution at x, t and τ is the relaxation time which controls the rate
of approach to equilibrium. The pressure p and the macroscopic velocity u, are
defined in terms of the particle distribution function by:
and
The corresponding equilibrium pressure distribution
defined by:
where,
is the constant average density. The weighting factor w_{i} is given
by:
The speed of sound in this model is:
In order to derive the NavierStokes (NS) equations from LBE, under the incompressible flow limit i.e., the Mach number
the ChapmanEnskog expansion (Chapman et al., 1999)
is used, in essence with a standard multiscale expansion proposed by Frisch
et al. (1987), the mass and momentum equations can be derived from
the D_{2}Q_{9} model as follows:
where, the kinematic viscosity, υ, is:
Since, the details of this model are given by Xiaoyi and
LiShi (1997).
PROBLEM STATEMENT AND FORMULATION
The computational domain is drawn in Fig. 2. A twodimensional
flow past a row of circular cylinders with radius R is investigated,
where five different values of B= W[R], i.e., 8R,10R,15R,20R and 25R are examined.
A rectangular computational domain of Bx68R (widthxlength) is selected with
one cylinder lying on the horizontal central line and symmetric boundary condition
is applied at the top and bottom boundaries of the computational domain to represent
a row of cylinders. Five different lattices 60x504,74x504,112x504,148x504 and
168x504 are used for the five values of B, respectively. The relaxation parameter
τ is set to be 0.5173, 0.5086 and 0.5043, respectively.

Fig. 2: 
Schematic flow configuration 
BOUNDARY CONDITIONS
Initial condition: Potential flow solution is adopted as an initial condition.
Inlet boundary: A constant pressure gradient along the xdirection is
imposed (Luo, 1998):
where,
and
are postcollision of distribution function and unit vector along the x axis,
respectively.
Outlet boundary: A simple extrapolation is adopted for exit.
f_{i}(N_{x},j) = 2f_{i}(N_{x}i,j)f_{i}(N_{x}2,j) 
(12) 
where, N_{x}, is the number of lattices in the xdirection.
Surface of the cylinder: Noslip wall boundary condition is applied
to the surface:
and this is realized with a bounceback boundary treatment.
Top and bottom boundaries: On the top and bottom boundaries symmetry
boundary condition is applied by Renwei et al., (1999).
Force evaluation: The momentum exchange method proposed by Dazhi
et al. (2003) was used in the present study for simulating the flow
around the circular cylinder. Since, the details of this model are given here.
Reynolds number R_{e} is defined by:
where, D is the cylinder diameter. Other important dimensionless numbers are
the drag coefficient C_{d}, the lift coefficient C_{l}, the
Strouhal number S_{t} and blockage ratio B. They are defined by the
following formulas:
where, f_{S} is the vortex shedding frequency from the cylinder, F_{d} and F_{l} are the force components in the inline and transverse directions, respectively.
Computation is terminated when the following convergence criteria is satisfied:
The proposed LBM model (Xiaoyi and LiShi, 1997) is
independent of density variation. Instead of Eq. 17 and 18
we used Eq. 22 and 23 for calculating drag
and lift coefficients (because the density, ρ, is here the average density
and is set to be 1.0):
RESULTS
The graphical representation of different compositions and comparison with
literature data (Fornberg, 1991) shown here. Figure
3 shows how the drag coefficients vary with different blockage ratios.
Figure 4 gives similar behavior for drag coefficient with different Reynolds number (R_{e}).
The structures of the vorticity filed are shown in Fig. 5a
and b.
Figure 6a and b show the vortex formation
for different blockage ratios.
The spectrum analysis for the lift coefficient displayed graphically in Fig.
7af.
Figure 811 show the variation of Strouhal
numbers for different blockage ratios, Strouhal number variation for different
Reynolds number R_{e} and similarly examine the effect of drag coefficients
for different Reynolds number R_{e} and blockage ratios.

Fig. 3: 
Variation of drag coefficient with blockage ratios for R_{e}
= 100; line with square box from Fornberg (1991) 

Fig. 4: 
Variation of drag coefficient with Reynolds number for B =
10; line with square box from Fornberg (1991) 


Fig. 5: 
Vorticity contours of vortex wake of the cylinder for R_{e}
= 50, (a) B = 8 and (b) B = 20 


Fig. 6: 
Vorticity contours of vortex wake of the cylinder for
different blockage ratio with (a) R_{e} = 100 and (b) 200 




Fig. 7: 
Spectrum analysis of lift force time history, (a) B = 10,
R_{e} = 50, (b) B = 10, R_{e} =100, (c) B = 10, R_{e}
= 200, (d) B = 25, R_{e} = 50, (e) B = 25, R_{e} = 100 and
(f) B = 25, R_{e} = 200 

Fig. 8: 
Variation of S_{t} with B for different R_{e} 

Fig. 9: 
Variation of S_{t} with R_{e} for different
B 

Fig. 10: 
Variation of C_{d }with R_{e} for different
B 

Fig. 11: 
Variation of C_{d} with B for different R_{e} 
DISCUSSION
In order to study the effect of the blockage ratio, five blockage ratios, 8,
10, 15, 20 and 25 are investigated for R_{e} = 50, 100 and 200. The
results are compared with some data in literature (Fornberg,
1991). It is seen that the drag coefficient increases as the distance between
the cylinders becomes smaller and approaches a steady value as B increases which
corresponds to an unbounded flow past an isolated cylinder in Fig.
3. The drag coefficient decreases as R_{e} increases as expected
for this range of the Reynolds number in Fig. 4. For comparison,
the numerical results of (Fornberg, 1991) using finite
difference method for the same problem are presented in the Fig.
3 and 4 and a good agreement is observed in both figures.
Vortex shedding appears at around R_{e} = 50. However, vortex shedding
starts to appear earlier as the blockage ratio decreases (i.e., as the space
between the cylinders becomes smaller) for a fixed Reynolds number. This is
shown in Fig. 5. The vortex shedding starts at very early
stage during the computation for B = 8 (Fig. 5a), while the
wake still maintains a two steady bubbles for a long calculation time for
B = 20 (Fig. 5b). This means we not find any disturbance
during the whole period of simulations.
Figure 6 the vorticity contours of vortex wake for different
blockage ratios are graphically represented. It is seen that both the longitudinal
and lateral spacing increase as the blockage ratio increases which is consistent
with the observation by Anagnostopoulos and Iliadis (1996).
To study the effect of the blockage on the vortex shedding frequency, the lift force time histories are analyzed using FFT technique for the five blockage ratios. The frequency spectrum analysis results are plotted in Fig. 7 for different blockage ratios.
The results of the spectrum analysis are then shown in Fig. 8
and 9. It can be seen that the vortex shedding frequency decreases
as the blockage ratios increases. In other words, the vortices shed with a higher
frequency when the distance between the cylinders becomes smaller. The shedding
frequency goes up as Reynolds number increases, which shows the general behavior
of S_{t} with R_{e} for this Reynolds number range. This is
similar to an isolated cylinder in a uniform flow as expected. Figure
10 and 11 show that the drag coefficient goes up as Reynolds
number decreases and drag coefficient decreases when the blockage ratios increases
with fixed Reynolds number.
In practice, the flows past a row of circular cylinders are usually more important
and are also hard to incorporate. Therefore, we put more emphasis on them in
this study to numerically investigate such types of flows. There is a slight
difference with the present and literature data (Fornberg,
1991). Many possibilities are here: different numerical method, limitations
of the present method, inflow boundary conditions, side wall boundary conditions
and so on.
CONCLUSION
A numerical investigation on a uniform flow past a row of circular cylinders
is conducted using the lattice Boltzmann method. The effects of the blockage
ratio on the drag coefficient, the vortex wake and vortex shedding frequency
are examined. The results show that both the drag coefficient and the vortex
shedding frequency increase as the cylinders becomes closer. The present results
show good agreement with some data in literature (Fornberg,
1991) and indicate the capability of the LBM in simulating the flow field
with complex geometries. This study also sheds the light on the effect of the
numerical blockage effect on a numerical solution for an unbounded circular
cylinder. When the grids with high blockage are used for reducing the computational
time required, the values of the various hydrodynamic parameters may be considerably
different from what they should be. When R_{e} = 50, as B increases,
the wake flow becomes steady. However, as B decreases the alternate vortex shedding
from the cylinder occurs. The current simulation shows that the proposed method
is capable of handling such types of flows problems.
ACKNOWLEDGMENT
The study described in this research was supported by a grant from National Natural Science Foundation of China (Project No. 90715031). The financial support is gratefully acknowledged.