Middle-East Journal of Scientific Research 15 (1): 08-13, 2013
ISSN 1990-9233
© IDOSI Publications, 2013
DOI: 10.5829/idosi.mejsr.2013.15.1.2628
Investigation of Flow Around a Confined Elliptical
Cylinder Using the Lattice Boltzmann Method
M. Taeibi-Rahni, 3V. Esfahanian and 2M. Salari
1,2
Department of Aerospace Engineering, Sharif University of Technology, Tehran, Iran
2
Department of Mechanical and Aerospace Engineering,
Science and Research Branch, Islamic Azad University, Tehran, Iran
3
School of Mechanical Engineering, University of Tehran, College of Engineering, Tehran, Iran
1
Abstract: This paper deals with the investigation of the laminar flow past an elliptical cylinder confined in
a channel. In this paper, the Lattice Boltzmann (LB) method is used to simulate flow in two dimensions.
The present LB method with the used boundary conditions is validated in simulations of the
incompressible flow past a circular cylinder. The simulations are carried out in a range of condition, 0
90
(angle of incidence), 5 Re 100, (Reynolds number) for AR=0.25,0.5 (aspect ratio). The effects of those
parameters on the drag and lift coefficients and other flow characteristics of the cylinder are examined in
detail. The results demonstrate that the drag and lift coefficients increase with the increase of . It is seen
that vortex shedding is not occurred at high Reynolds numbers at =0.
Key words: Elliptical cylinder
Laminar flow
Lattice-Boltzmann method
INTRODUCTION
Strouhal number
Drag and lift coefficients
an unconfined elliptic cylinder at various angles of
incidence. They distinguished five regimes for flow
characteristics. Badr [8] solved the oscillating viscous
flow over an elliptical cylinder at Reynolds numbers of
100,500 and 1000. His solution was based on a numerical
integration of the Helmholtz vorticity transport equation
with the stream function equation. Chanada [9]
investigated the unsteady two dimensional flow of
viscous fluid normal to a thin elliptic cylinder using a
vorticity-stream function formulation of the full
Navier-Stocks equations. Choi and Lee [10] Studied
experimentally the flow characteristics around an elliptic
cylinder with axis ratio of AR=2 located near a flat plate
experimentally. They considered various angles of
attack of the cylinder in order to study the interaction
between the cylinder wake and the boundary layer.
Kocabiyik and D’Alessio [11] presented the numerical
calculations of the flow around an inclined elliptic
cylinder oscillating in-line with the incident steady
flow at Reynolds number Re=1000 and axis ratio of 0.5.
They investigated the effects of the angle of inclination
The flow past cylinders has extensive applications
in engineering problems such as cooling towers, heat
exchangers,
chimney stacks, nuclear reactors and
offshore structures. Consequently, a large number of
researchers have studied this kind of flow. Most of
these studies were concerned with the circular cylinders
[1-4]. In contrast, there are few researches that study
the flow over an elliptical cylinder. Imai [5] studied the
uniform flow past an elliptic cylinder at an arbitrary angle
of incidence for the thickness ratio t=0-1 and the
Reynolds numbers Re = 0.1,1. He obtained analytical
expressions for the lift and drag coefficients. Patel [6]
investigated the flow around an impulsively started
elliptic cylinder at various angles of incidence (0, 30, 45
and 90°) at Re = 200 by solving the Navier-Stokes
equations numerically. He observed a Kármán vortex
street at 30 and 45 incidence for Re=60,200. Park et al.
[7] studied the details of the laminar unsteady flows past
Corresponding Author: M. Salari, Department of Mechanical and Aerospace Engineering, Science and Research Branch,
Islamic Azad University, Tehran, Iran.
8
Middle-East J. Sci. Res., 15 (1): 08-13, 2013
and velocity ratio and observed two distinct forms of
vortex street were dependent on the values of the angle
of inclination. Faruquee et al. [12] examined the effects
of axis ratio (AR) on the fluid flow around an unbounded
elliptical cylinder using FLUENT software at a Reynolds
number of 40 based on hydraulic diameter. They implied
the wake size and the drag coefficient increase with the
increase of AR. Sivakumar et al. [13] studied the flow
characteristics of the two dimensional incompressible
flow of non-Newtonian power-law fluids over an
elliptical cylinder. Stack and Bravo [14] studied
numerically the flow separation behind two-dimensional
ellipses with the aspect ratios ranging from 0, a flat
plate, to 1, a circular cylinder, for Reynolds numbers less
than 10 using both a cellular automata model and
FLUENT software. They reported for Reynolds numbers
greater than 1, the relationship between the critical
aspect ratio for flow separation and Reynolds number is
linear. Nejat et al. [15] studied two dimensional
incompressible flow of non-Newtonian power-law fluid
over a confined pair of elliptical tandem cylinders.
Also, they investigated the heat transfer characteristics
of this problem. Their studied range of condition was ,
0.2 n 1.8 (power-law index), 1 Pr 100 (Prandtl number),
0.25 E 2 (aspect ratio) and 1.25 L 20 (distance between
cylinders) for 1 Re 40. They illustrated that in the heat
exchangers an elliptical cylinder with aspect ratio AR=0.5
can be more efficient than circular cylinders for moderate
Reynolds numbers. Most of the mentioned works
studied flow over an unconfined elliptical cylinder.
Hence, the goal of this study is to investigate the
hydrodynamic characteristics of fluid flows over an
elliptic cylinder using the lattice Boltzmann method
(LBM). The simulations are carried out in the range of
5 Re 100 and for range of inclinations (0
90) and for
two values of the aspect ratio AR=0.2 and AR =0.5.
A brief review of the Lattice Boltzmann method will
be presented in the next section. Then, we explain the
problem and boundary conditions. In section 3, we
validate our approach by solving the flow around a
single circular cylinder. After that, the numerical results
for simulation of flow over an elliptic cylinder are
presented. Finally, the conclusions are presented.
Where f (xi , t )
direction at position x and time t.
relaxation time and f (eq ) (x, t )
1
− [f (x, t ) − f (eq ) (x, t )],
t , t + t ) − f (x, t ) =
is the dimensionless
is the equilibrium
distribution function. We use D2Q9 scheme in this
work in which D refers to space dimensions and Q to the
number of particles at a computational node. In D2Q9
Lattice, the discrete velocities e are presented as:
e0=0,
e =
c (cos(( − 1)
4
),sin(( − 1)
e = 2c (cos(( − 1)
4
4
), for
),sin(( − 1)
4
), for
1,3,5,7
=
= 2,4,6,8
(2)
Where c =∆x ∆t is the lattice speed and x and t
are the lattice spacing and the time step, respectively.
The equilibrium distribution functions for D2Q9 models
are:
3
9
3
f ( eq ) = w [1 + 2 ea .u + 4 (ea .u )2 − 2 u . u ],
(3)
2c
c
c
Where w is the weighting factors and given by: w0
= 4/9, wi = 1/9 for i = 1 – 4 and wi = 1/36 for i = 5 – 8.
The fluid density
=
8
∑f
and velocity u are defined as:
(4.a)
,
=0
u=
8
∑e
f .
(4.b)
=1
In addition, the pressure and
viscosity are calculated as p = cs2 and
where cs = c 3 is the speed of sound.
the kinematic
=
( − 1 2 )c s 2 ∆ t ,
Problem Definition and Boundary Conditions: The
geometry of fluid and boundary conditions are
illustrated in Figure 1.The aspect ratio of the elliptical
cylinder is defined by AR = b/a, where 2a and 2a are the
diameters of the elliptical cylinder along the flow and
normal to the flow respectively. The length of channel
L, is 44a. The distance of cylinder from the channel inlet
L1, is 3a. The distance of cylinder from the upper wall of
the channel h1 and the distance of cylinder from the lower
wall of the channel h2 are equal to 3.2a and 3a,
respectively. The Reynolds number is defined by
Re = 2Ua
with the average inflow velocity of
U = 2U (0, H 2, t ) / 3 . The drag and lift coefficients are defined
as:
Lattice Boltzmann Method: The Lattice Boltzmann
equation
with
Bhatnagar-Gross-Krook
(BGK)
approximation [16] can be expressed as:
f (x + e
is the distribution function in
(1)
9
Middle-East J. Sci. Res., 15 (1): 08-13, 2013
inlet
L1
Table1: Verification of mesh independency for a singlecylinder at Re = 20
h1
H
(no-slip boundary)
h2
Y
X
D/
outlet
L2
Fig. 1: The geometric configuration for the flow past an
elliptical cylinder confined in a channel
CD
CL
Diverged
5.615
5.611
5.609
5.609
Diverged
0.011
0.01
0.01
0.01
x
10
20
30
40
50
Table 2: The drag coefficient, the lift coefficient and the length of the
recirculating zone for the flow past a single cylinder with the
results of Ref. [20]
This work
CFD lower bound in Ref. [20]
CFD upper bound in Ref. [20]
FD
=
,
CL
U 2a
FL
.
U 2a
CL
La
0.01
0.0104
0.011
0.0817
0.0842
0.0852
This table shows that D/ x = 40 is a good
resolution. The results of simulation are presented in
Table 2. According to this table, results are in a good
agreement with the results of Ref [20].
Fig. 2: The geometric configuration for the flow past a
cylinder asymmetrically placed in the channel [20]
=
CD
CD
5.609
5.57
5.59
RESULTS AND DISCUSSIONS
(5)
In this work, The flow characteristics of fluid flows
passing over an elliptical cylinder in channel have
been
investigated
numerically using the lattice
Boltzmann method in different values of Reynolds number
(5 Re 100), for different attack angles (0
90) and
for two values of the aspect ratio AR=0.2 and AR=0.5.
To show the independency of results from the grid
size, mesh independency is verified through different
grids at Re = 20, for = 90 and AR = 0.5 (Table 3). As it is
obvious, for grids finer than 881 × 165, no considerable
change is seen. So we used this resolution in our
simulations.
The Strouhal number is defined as St = 2af U , where f
is the frequency of vortex shedding.
At the channel inlet, a parabolic velocity profile is
imposed. At the inlet and the channel walls we use Zou
and He boundary condition [17] to simulate these
boundary conditions. At the outlet, we use an
extrapolation boundary
condition proposed by
chen et al. [18]. At the cylinder boundary, we use the
second order curved boundary
condition which
presented by Filippova and Hänel (FH) [19].
Validation of the Code: To validate our code, the steady
flow past a cylinder placed in the channel in 2D at Re=20
is simulated [20]. This flow has been validated by many
researchers numerically and experimentally, so it can be
used as a benchmark problem. The Schematics of
computational domain and the boundary conditions are
indicated in Figure 2. The channel height is H = 0.41m
and D = 0.1m is the cylinder diameter. The length of
recirculation is calculated as La = xr – xe, where x e is the
x-coordinate of the end of the cylinder and xr is the
x-coordinate of the end of the recirculation area.
Other parameters are defined as for the circular cylinder
except using D in place of 2a.
To show the independency of results from the
grid size, mesh independency is verified through
different grids at Re = 20 (Table 1). In Table 1, we use a
number of meshes with different resolutions in terms of
D/ x.
Table 3: Grid independence study for elliptical cylinder
Mesh spacing ( x= y)
Total mesh size
441×83
0.005
CD
5.676
661×124
0.0033
5.699
881×165
0.0025
5.704
1101×206
0.002
5.707
1761×329
0.00125
5.709
Table 4: Drag coefficient, lift coefficient and Strouhal number for an
elliptical cylinder at
= 40
AR = 0.2
10
Re
CD
CL
CDmax
CLmax
Dt
5
8.673
–3.609
-
-
-
20
3.742
–2.516
-
-
-
100
-
-
2.618
–1.412
0.251
Middle-East J. Sci. Res., 15 (1): 08-13, 2013
According to this table as the Reynolds
number increases, drag and lift coefficients decrease.
Time histories of lift and drag coefficients for Re =100
are shown in Figure 4, which is followed by velocity
and pressure contours for the mentioned Reynolds
number in Figure 5. As it is obvious, vortex shedding
occurred at this Reynolds number. Because of periodic
vortex shedding the lift and drag coefficients diagrams
have oscillating properties.
To investigate the effects of attack angle on
the flow characteristics, we study unsteady flow
around an
elliptical
cylinder
at
different
inclinations. At Re =100, for an aspect ratio of AR=0.5
the pressure contours and streamlines for different
attack angles is plotted in Figure 6 and Figure 7
respectively. It can be seen from these figures that vortex
shedding is not occurred at = 0. The maximum drag
coefficient CDmax, the maximum lift coefficient CLmax and
Strouhal number at Re =100 for different inclinations are
presented in Table 5. It can be seen from this table that
the drag and lift coefficients are minimum at = 0 and as
increases they increase.
Table 5: The maximum drag coefficient, the maximum lift coefficient and
Strouhal number for an elliptical cylinder at = 40
0
45
90
CDmax
CLmax
St
1.377
2.823
5.472
–0.012
–0.4
2.07
0.228
0.217
Streamline patterns for an aspect ratio of AR=0.2
and inclination of = 40 at Reynolds numbers Re = 5,
20 and 100 are shown in Figure 3. At low Reynolds
number (Re = 5) the steady flow past the elliptical cylinder
occurred without separation as shown in Figure 3(a).
As Reynolds number increases the separation of the
steady flow is clearly observed in Figure 3(b). At Re = 100,
the flow becomes unsteady and a periodic vortex
shedding takes place as shown in Figure 3(c).
The simulation results of flow past an inclined elliptical
cylinder are presented for drag coefficients, lift
coefficients and Strouhal numbers in Table 4. It should
be noted that for unsteady flow both the drag and lift
coefficients are periodic functions in time. So we
measure the maximum drag coefficient CDmax and the
maximum lift coefficient CLmax.
Fig. 3: Streamlines for an elliptical cylinder at different Reynolds numbers for an aspect ratio AR=0.2: (a) Re =5 (b)
Re =20 (c) Re =100
Fig. 4: Time histories of lift and drag coefficients for the flow past an elliptical cylinder at Re =100
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Middle-East J. Sci. Res., 15 (1): 08-13, 2013
Fig. 5: Flow characteristics for an inclined elliptical cylinder at Re =100
Fig. 6: Pressure contours for an elliptical cylinder at Re =100 for different inclinations: (a)
(c) = 90
Fig. 7: Streamlines for an elliptical cylinder at Re =100 for different inclinations: (a)
CONCLUSION
= 0, (b)
= 0, (b)
= 45 and
= 45 and (c)
= 90
condition for the no-slip boundary conditions at curved
boundaries. For boundary conditions at channel walls
and the inflow and outflow boundary conditions we
use Zou and He boundary condition. Validation of
code shows that the Lattice Boltzmann method with the
In this paper we use the lattice Boltzmann method
to simulate laminar flow over an elliptical cylinder in
channel in two dimensions. We apply FH boundary
12
Middle-East J. Sci. Res., 15 (1): 08-13, 2013
mentioned boundary conditions can be used very
effectively and produce accurate results in agreement
with published results. Different Reynolds numbers and
inclinations for two aspect ratios have been used in
order to investigate the effect of various parameters on
the flow characteristics. The results show that at lower
Reynolds number the flow past the cylinder with has no
separation. The separation is occurred as the Reynolds
number increases. Much increase in the Reynolds
number results in the increased size of the vortices.
Finally, at a critical Reynolds number, vortex shedding
is occurred.
It is seen that at Re =100 in different inclinations
vortex shedding is occurred except at = 0 in which the
drag and lift coefficients are the minimum. Also it is
observed that the drag and lift coefficients are increased
as the inclination is increased.
9.
10.
11.
12.
13.
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