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Rahman2007.pdf
Journal of Naval Architecture and Marine Engineering
June, 2007
http://jname.8m.net
NUMERICAL INVESTIGATION OF UNSTEADY FLOW PAST A
CIRCULAR CYLINDER USING 2-D FINITE VOLUME METHOD
Md. Mahbubar Rahman1, Md. Mashud Karim2 and Md. Abdul Alim3
1
Department of Natural Science, Stamford University Bangladesh, Dhaka-1209, 2Dept. of Naval Architecture and
Marine Engineering, BUET, Dhaka-1000,3Department of Mathematics, BUET, Dhaka-1000, Bangladesh.
E-mail: [email protected], [email protected], [email protected]
Abstract
The dynamic characteristics of the pressure and velocity fields of unsteady
incompressible laminar and turbulent wakes behind a circular cylinder are
investigated numerically and analyzed physically. The governing equations, written
in the velocity pressure formulation are solved using 2-D finite volume method. The
initial mechanism for vortex shedding is demonstrated and unsteady body forces are
evaluated. The turbulent flow for Re = 1000 & 3900 are simulated using k-ε
standard, k-ε Realizable and k-ω SST turbulence models. The capabilities of these
turbulence models to compute lift and drag coefficients are also verified. The
frequencies of the drag and lift oscillations obtained theoretically agree well with the
experimental results. The pressure and drag coefficients for different Reynolds
numbers were also computed and compared with experimental and other numerical
results. Due to faster convergence, 2-D finite volume method is found very much
prospective for turbulent flow as well as laminar flow.
Keywords: Viscous unsteady flow, laminar & turbulent flow, finite volume method, circular cylinder.
NOMENCLATURE:
a
D
ρ
µ
t
u,v
→
u
U∞
l
k
ε
St
1.
Radius of the circular cylinder
Diameter of the circular cylinder
Density of the fluid
Coefficient of viscosity
Time
Velocity components
Velocity vector
Cp
ωw
CD
CDP
CDV
Pw
Re
Pressure coefficient
Wall vorticity
Drag coefficient
Drag coefficient due to pressure
Drag coefficient due to viscosity
Dimensionless wall pressure
Reynolds number
Free stream velocity
T
The period of vortex shedding.
Turbulence length
Turbulence kinetic energy
Turbulence dissipation
Strouhal number
f
I
Cµ
ω
Frequency of the vortex shedding
Turbulence Intensity
Empirical constant
Specified dissipation rate
Introduction
Flow around a circular cylinder is a fundamental fluid mechanics problem of practical importance. It has
potential relevance to a large number of practical applications such as submarines, off shore structures,
bridge piers, pipelines etc. The laminar and turbulent unsteady viscous flow behind a circular cylinder has
been the subject of numerous experimental and numerical studies, especially from the hydrodynamics point
of view. According to the observation of Sumer (1997); the flow field over the circular cylinder is
symmetric at low values of Reynolds number. As the Reynolds number increases, flow begins to separate
1813-8535 © 2007 ANAME Publication. All rights reserved.
M. M. Rahman, M. M. Karim and M. A. Alim / Journal of Naval Architecture and Marine Engineering 4(2007) 27-42
behind the cylinder causing vortex shedding which is an unsteady phenomenon. For the 40 < Re < 200,
there is a laminar vortex shedding in the wake of the cylinder. The laminar wake transient to turbulence in
the region of Re = 200 to 300. In the subcritical region 300 < Re < 3×105 the wake behind the circular
cylinder becomes completely turbulent and a laminar boundary layer separation occurs. The unsteady flow
was first studied by Payne (1958) for Reynolds number equal to 40 & 100. The numerical study and
physical analysis of the pressure and velocity fields in the near wake of a circular cylinder has been
investigated by Braza et al (1986). Tuann & Olson (1978), Martinez & Minh (1978), Loc (1980),
Coutanceau & Defaye (1991) and recently Lakshmipathy (2004) and Reichl et al. (2005) have investigated
this problem for different Reynolds numbers. Braza et al (1990) predicted the large-scale transition features
in the wake of a circular cylinder for 2000 < Re < 10000. The turbulent flow over cylinder was also
investigated by Rai & Moin (1993) and Mittal (1995) for high Reynolds number. The common points of
interest of this work are the development of the primary unsteady wake behind the circular cylinder and the
evaluation of the drag coefficient and the separation angle with time.
Most of the experimental studies investigated the steady and unsteady behaviors of the alternating vortices
in the wake. The work of Tritton (1971), Lourenco & Shih (1993),and Anderson (1995) should be
mentioned. Besides these theoretical and numerical investigations, some experimental visualizations have
been described by Honji & Taneda (1969), Beaudan & Moin (1994) and Coutanceau & Bouard (1977).
All of the above numerical studies have some common characteristics: they solve the unsteady NavierStokes equation in two dimensional Helmholdtz (vorticity & stream function) formulation; they described
the relevant flow by the global parameters such as Strouhal number as a main feature of the unsteady wake,
drag and lift coefficients in the wall region; nevertheless, poor analysis is provided for the near wake
characteristics. The main goal of the present study is consequently to (i) visualize the laminar and turbulent
unsteady flow field for different Reynolds numbers; (ii) check the capability of different turbulent models
for the simulation the unsteady flow over circular cylinder. (ii) compute drag coefficients along the surfaces
of the cylinder for different Reynolds numbers; (iii) plot contours of stream function, velocity vectors and
static pressure to visualize vortex shedding at different time for Re=100, 1000 & 3900. Numerical results
presented here are compared with experimental measurements and other numerical results and the
agreements are found satisfactory.
2.
Physical Model
The flow field around the cylinder is modeled in two dimensions with the axes of the cylinder
perpendicular to the direction of flow. The cylinder is modeled as a circle and a square flow domain is
created surrounding the cylinder. The flow from left to right with the cylinder of diameter, d submersed in
an incompressible fluid is considered. The computational domain consists of an upstream 23 times the
radius to downstream 40 times the radius and the width of the domain is 50 times the radius of the cylinder.
The problem setup together with the important dimensions is shown in Fig.1.
A typical rectangular mesh used for simulation of this study. This particular mesh has approximately 15659
nodes, 30924 faces and 15380 quadrilateral cells with considerable mesh concentration both around the
cylinder and in the wake. To facilitate meshing, a square with side length of three times diameter of the
cylinder is created around the cylinder.
3. Mathematical Model
3.1 Governing Equations
The governing equations of the unsteady flow of an incompressible viscous fluid past a circular cylinder
are the classical continuity and Navier- Stokes equations. They are written in the following form:
→
div( u ) = 0
(1)
Numerical Investigation of Unsteady Flow Past a Circular Cylinder using 2-D Finite Volume Method
28
M. M. Rahman, M. M. Karim and M. A. Alim / Journal of Naval Architecture and Marine Engineering 4(2007) 27-42
50 R
R
23 R
Periodic
Pressure Outlet
Velocity Inlet
Periodic
40 R
Fig.1: Schematic diagram of the flow field around circular cylinder
∂ (u )
⎛ →⎞ 1
+ div⎜ u u ⎟ = div(Γ∇u )
∂t
⎝ ⎠ ρ
(2)
Equation (2) has the rate of change term and the convective term in the left hand side and the diffusion term
(Γ= diffusion coefficient) on the right hand side. The equation is used as the starting point for
computational procedure in finite volume method. The key step of the finite volume method (Versteeg &
Malalsekera,1995) is the integration of Equation (2) over a two dimensional control volume CV yielding
∫
CV
∂ (u )
1
⎛ →⎞
dv + ∫ div⎜ u u ⎟dv = ∫ div(Γ∇u )dv
∂t
ρ CV
⎝ ⎠
CV
(3)
The volume integrals in the 2nd term on the left hand side, the convective term and in the right hand side the
diffusion term can be re- written as integrals over the entire boundary surface of the control volume by
using Gauss divergence theorem. Then Equation (3) becomes
→
⎞
∂⎛
1
⎜ ∫ (u )dv ⎟ + ∫ n.⎛⎜ u u ⎞⎟dA = ∫ n.(Γ∇u )dA
⎜
⎟
∂t ⎝ CV
ρA
⎠ A ⎝ ⎠
(4)
In the steady state case, the rate of change term is equal to zero. This leads to the integrate form of the
steady transport equation
∫ n.(uU )dA = ρ ∫ n.(Γ∇u )dA
1
A
(5)
A
In time dependent (unsteady) case, it is also necessary to integrate with respect to time t over a small time
interval ∆t from (say) t to t+ ∆t . This yields the most general integral form of the transport equation.
⎞
∂⎛
1
⎛ →⎞
⎜
⎟
(
)
u
dv
dt
n
.
+
⎜ u u ⎟dAdt = ∫ ∫ n.(Γ∇u )dAdt
∫∆t ∂t ⎜⎝ CV∫
∫
∫
⎟
ρ ∆t A
⎠
∆t A ⎝
⎠
(6)
Numerical Investigation of Unsteady Flow Past a Circular Cylinder using 2-D Finite Volume Method
29
M. M. Rahman, M. M. Karim and M. A. Alim / Journal of Naval Architecture and Marine Engineering 4(2007) 27-42
In finite volume method, flow domain is divided into a number of control volumes or cells. Discritized
form of Eq. (6) must be set up at a nodal point placed within each control volume in order to solve the
problem. For control volumes adjacent to the domain boundaries, the general equation is modified to
incorporate boundary conditions. The resulting system of linear algebraic equations is then solved to obtain
the velocity and pressure distribution at each nodal point. Finally, drag and lift coefficients are computed as
follows:
CD =
D
0.5ρU 2 ∞ d
CL =
L
0.5ρU 2 ∞ d
2π
C DP = ∫ Pw cos dx C DV =
0
2π
C Lp = ∫ Pw sin xdx
0
2 2π
ω w sin xdx
Re ∫0
C LV =
2 2π
ω w cos xxdx
Re ∫0
(7)
(8)
Where, D & L represent drag and lift force.
Also, the pressure coefficient is defined as:
1
C P = (P − P∞ ) / ρU 2 d .
2
(9)
Where, the subscripts P and V respectively represent the contributions of the pressure and viscous force. Pw
is the dimensionless wall pressure and ωw is the dimensionless wall vorticity defined as ω w = ωR / U ∞ ,
(here R = 0.5). The dimensionless Reynolds number is given by Re = dU ∞ ρ / µ
(10)
Also, the dimensionless Strouhal number is expressed as St = fd / U ∞
(11)
Where, f is the frequency of the vortex shedding (=1/T), d is the diameter of the cylinder and U ∞ the free
stream velocity.
In case of turbulent flow, the viscous runs are done using standard k-ε, realizable k-ε and Shear-Stress
Transport (SST) k-ω model (Lakshmipathy, 2004).
The standard k-ε model is a semi-empirical model based on model transport equations for the turbulence
kinetic energy (k) and its dissipation rate (ε). The model transport equation for k is derived from the exact
equation, while the model transport equation for ε was obtained using physical reasoning and bears little
resemblance to its mathematically exact counterpart. In the derivation of the k- ε model, it was assumed
that the flow is fully turbulent, and the effects of molecular viscosity are negligible. The standard k-ε model
is therefore valid only for fully turbulent flows.
On the other hand the term “realizable" means that the model satisfies certain mathematical constraints on
the normal stresses, consistent with the physics of turbulent flows. The most straightforward way to ensure
the realizability is to make Cµ variable by sensitizing it to the mean flow (mean deformation) and the
turbulence (k, ε). The turbulence kinetic energy k is given by
k=
3
(U avg I )2
2
(12)
Where, Uavg is the mean flow velocity and I is the turbulence intensity defined by
I = 0.16(Re )
−
1
8
(13)
Turbulence length can be written as l = 0.07 d and turbulence dissipation rate ε as
Numerical Investigation of Unsteady Flow Past a Circular Cylinder using 2-D Finite Volume Method
30
M. M. Rahman, M. M. Karim and M. A. Alim / Journal of Naval Architecture and Marine Engineering 4(2007) 27-42
3
ε = Cµ
3
4
k2
l
(14)
Where, Cµ is an empirical constant specified in the turbulent model (approximately 0.09).
In this study, another turbulent model is used to simulate the turbulent flow named shear-stress transport
(SST) k-ω model. It is so named because the definition of the turbulent viscosity is modified to account for
the transport of the principal turbulent shear stress. It is such a feature that gives the SST k- ω model an
advantage in terms of performance over both the standard k-ε and realizable k-ε model. Other modification
includes the addition of a cross-diffusion term in the ω-equation and a blending function to ensure that the
model equations behave appropriately in both the near-wall and far-field zones. In SST k-ω model the
specific dissipation rate ω can be found by
1
ω=
k2
1
4
(15)
Cµ l
The control volume technique consists of integrating the governing equations about each control volume,
yielding discrete equations that conserve each quantity on a control volume basis. The coupling of pressure
and velocity is done using PISO algorithm (Versteeg & Malalsekera ,1995).
3.2 Boundary Condition
The wall boundary conditions used in this steady are those of impermeability and non-slip condition, i.e., u
= 0, v = 0
In the physical domain the flow is not confined. Nevertheless, a fictitious external rectangle boundary is
needed at a large distance from the cylinder (Fig.1) in order to solve the governing equations numerically.
Uniform free stream condition with velocity U ∞ =1.0m/s are applied at the inlet boundary. The periodic
condition is considered at the lateral boundaries. Also the flow exit is treated as a pressure outlet.
4. Results and Discussion
4.1. The vortex shedding:
4.1.1 The Laminar Flow (Re = 100)
Experimental result shows that, for Reynolds number less than 40 the flow over the circular cylinder
reaches the steady state within 15 second and two symmetric vortices are formed behind the cylinder (
Braza et al, 1986). As Reynolds number becomes higher than 40 the flow reports a loss of symmetry in the
wake and alternating eddies are formed and convected in the wake. This generates the alternating
separation of vortices, which are convected and diffused away from the cylinder, forming the well-known
Karman vortex streets. Such destabilizing effects always occur during any physical experiment on the flow
around a circular cylinder. The Strouhal number (St) is found to be 0.164 for Re=100. This result agrees
well with the experimental value (0.164~0.165) reported by Tritton (1959).
The results of unsteady laminar flow for Re = 100 are presented in Figs. 2-4. The instantaneous stream
lines for Re = 100 are shown in Fig.2 at the four phases during one cycle of vortex shedding. In Fig. 3 &
Fig.4 the directions of the velocity vectors and corresponding contours of pressure are plotted respectively
at those phases for the same Reynolds number. The alternating formation, convection and diffusion of the
vortices are clearly shown.
Numerical Investigation of Unsteady Flow Past a Circular Cylinder using 2-D Finite Volume Method
31
M. M. Rahman, M. M. Karim and M. A. Alim / Journal of Naval Architecture and Marine Engineering 4(2007) 27-42
(a)
(b)
(c)
(d)
Fig. 2: Contours of Stream function at (a) t = 48.2s, (b) t = 51.2s,
(c) t = 54.2s & (d) t = 57.2s for Re=100
Numerical Investigation of Unsteady Flow Past a Circular Cylinder using 2-D Finite Volume Method
32
M. M. Rahman, M. M. Karim and M. A. Alim / Journal of Naval Architecture and Marine Engineering 4(2007) 27-42
(a)
(b)
(c)
(d)
Fig. 3: Contours of Velocity vectors at (a) t = 48.2s, (b) t = 51.2s, (c) t = 54.2s & (d) t = 57.2s
for Re=100
Numerical Investigation of Unsteady Flow Past a Circular Cylinder using 2-D Finite Volume Method
33
M. M. Rahman, M. M. Karim and M. A. Alim / Journal of Naval Architecture and Marine Engineering 4(2007) 27-42
(a)
(b)
(c)
(d)
Fig. 4: Contours of Static Pressure at (a) t = 48.2s, (b) t = 51.2s, (c) t = 54.2s & (d) t = 57.2s for
Re=100
Numerical Investigation of Unsteady Flow Past a Circular Cylinder using 2-D Finite Volume Method
34
M. M. Rahman, M. M. Karim and M. A. Alim / Journal of Naval Architecture and Marine Engineering 4(2007) 27-42
4.1.2 The Turbulent Flow (Re = 1000 and 3900)
The experimental work of Roshko (1954) locates the beginning of the laminar-to-turbulent transition at
Reynolds numbers 200 – 300. Beyond this Reynolds number but less than 3×105 the wake of the cylinder is
completely turbulent and the boundary layer separation is laminar (Sumer, 1997). It is known that the flow
around the cylinder is two dimensional only when Re < 200. For Larger Reynolds number, the vortex
shedding occurs in cells and therefore the flow is generally simulated in 3-D. But the 3-D simulation is very
much complicated and treated as a numerically costly procedure. Therefore, the turbulent flow over a
circular cylinder for Reynolds numbers 1000 & 3900 is investigated here using 2-D finite volume method.
In this research, the turbulent flow for both Reynolds numbers is simulated using three turbulence models
namely, Standard k-ε, Realizable k-ε and Shear-Stress Turbulence (SST) k-ω turbulence model. The
vortex shedding is observed and the drag & lift forces is also calculated using these methods.
For unsteady case, time interval have been chosen small enough to capture the vortex shedding. The easiest
way to set up the right time step size is one that ensures the maximum Courant number very near to unity.
To avoid running the case for very long time, a small perturbation has been introduced in the domain. This
perturbation is generally in the form of very small velocity in perpendicular direction to the flow direction.
This assists the solver to start vortex shedding. Since there is nothing to sustain this perturbation in the
domain for very long time, it dies out and does not affect the calculation of Cl and Cd. The dimensionless
frequency of vortex shedding is the Strouhal numbers found by different methods are shown in Fig.5.
0.35
(Laminar)
(Standard k-epsilon)
(Realizable k-epsilon)
(SST K-omega)
(Experimental)
Strouhal Number
0.30
0.25
0.20
0.15
0.10
0.05
0
1000
2000
3000
4000
Reynolds Number
Fig.-5: Strouhal Number Vs Reynolds Number
The vortex shedding is visualized by means of the contours of stream function, velocity vectors and static
pressure. Fig. 6 show the stream function during one cycle of lift coefficient for Re = 1000. At the same
time the velocity vector and pressure fields are also shown in Fig. 7. The alternating vortex shedding
clearly observed in these figures. Fig. 8 shows the stream function and Fig. 9 shows the velocity vector &
pressure field during one cycle of lift coefficient for Re = 3900. The alternating formation, convection and
diffusion of the vortices are clearly shown again.
Numerical Investigation of Unsteady Flow Past a Circular Cylinder using 2-D Finite Volume Method
35
M. M. Rahman, M. M. Karim and M. A. Alim / Journal of Naval Architecture and Marine Engineering 4(2007) 27-42
The results in this section show that the vortex shedding is an intrinsic phenomenon of the flow, well
predicted by the solution of the Navier- Stokes equations. The vortex shedding is generated by a loss of
symmetry of the two dimensional symmetric structures in the wake of the circular cylinder.
(a)
(b)
(c)
(d)
Fig.-6: Stream Function (a) T=120.12s,(b) T=121.80 (c)T=123.48s, (d) T=125.16s for Re=1000
Numerical Investigation of Unsteady Flow Past a Circular Cylinder using 2-D Finite Volume Method
36
M. M. Rahman, M. M. Karim and M. A. Alim / Journal of Naval Architecture and Marine Engineering 4(2007) 27-42
(a)
(b)
Fig.-7: (a) Velocity Vector (b) Static Pressure at T=120.12s, T=121.80 T=123.48s, T=125.16s for
Re=1000
Numerical Investigation of Unsteady Flow Past a Circular Cylinder using 2-D Finite Volume Method
37
M. M. Rahman, M. M. Karim and M. A. Alim / Journal of Naval Architecture and Marine Engineering 4(2007) 27-42
(a)
(b)
(c)
(d)
Fig.-8: Stream Function (a)T=75s, (b) T=76.23s, (c)T=77.46s and (d)T=78.79s for Re=3900
4.2. The Forces:
The drag force is a result of the convective motion of the cylinder through the fluid. Because of this motion
and of the non-slip condition of the wall, a tangential velocity gradient is created in the direction normal to
the wall. The mean value of the drag coefficient calculated by the present
method for different Reynolds numbers is very close to the experimental results of Tritton (1959), Tritton
(1971) and Anderson (2005). The total drag is also due to two effects: the pressure field (CDp) and the
viscous forces (CDv) calculated by the Eqs. (7-8). For Re = 100 the mean value of the pressure drag, CDp is
0.917, which corresponds to 75% of the total drag. This value is closer to the experimental result (0.995) of
Numerical Investigation of Unsteady Flow Past a Circular Cylinder using 2-D Finite Volume Method
38
M. M. Rahman, M. M. Karim and M. A. Alim / Journal of Naval Architecture and Marine Engineering 4(2007) 27-42
Roshko (1954) than that (1.02) of Braza et al.(1986). Numerical values of the components of total drag
coefficient are shown in Table 1.
(a)
(b)
Fig.-9: (a) Velocity Vector (b) Static Pressure at T=75s, T=76.23s, T=77.46s and T=78.79s for Re=3900
Numerical Investigation of Unsteady Flow Past a Circular Cylinder using 2-D Finite Volume Method
39
M. M. Rahman, M. M. Karim and M. A. Alim / Journal of Naval Architecture and Marine Engineering 4(2007) 27-42
The pressure and drag coefficients of the turbulent flow at Re = 1000 & 3900 have also been calculated
using three turbulent models. The predicted Cd values by standard k-epsilon model are quite closely to the
experimental values shown in Table 2. But for the visualization of vortex shedding the Realizable k-epsilon
turbulence model is more effective, as this model captures the separating flow better than standard kepsilon model.
On the other hand, the SST k-omega model is much more recommendable for high Reynolds numbers. The
global periodic character of the flow is found to be essentially the same as the Reynolds number increases.
For the Reynolds numbers of 1000 and 3900 it should be recalled that the unsteady lift and drag coefficient
oscillates periodically. Fig. 10 shows the lift and drag coefficient for Re=1000. The drag coefficients as a
function of different Reynolds numbers are compared with experimental and other numerical results in Fig.
11 and in Table 2. In most of the cases, the computed results show better agreement with experimental data
compared to other predicted values.
Fig.-10: Time History of (a) Lift coefficient and (b) Drag coefficient at Re = 1000
Table 1: Components of Total Drag Coefficient.
Re
Pressure Coefficient
Viscous coefficient
Total Drag Coefficient
100
0.917
0.329
1.245
1000
0.876
0.119
0.995
3900
0.877
0.12
0.997
5. Conclusion
The complex problem of the unsteady laminar and turbulent flow around a circular cylinder at Reynolds
numbers of 100, 1000 and 3900 is studied using 2-D finite volume method. A second-order-accurate
numerical scheme is used, based upon a velocity-pressure formulation and conservative schemes. Three
different turbulent models are used to tackle the complex behavior of turbulent flow. From this study it is
found that the two-dimensional finite volume method computes hydrodynamic forces and captures vortex
shedding very well. Even at high Reynolds number, the method is very much applicable without loss of
Numerical Investigation of Unsteady Flow Past a Circular Cylinder using 2-D Finite Volume Method
40
M. M. Rahman, M. M. Karim and M. A. Alim / Journal of Naval Architecture and Marine Engineering 4(2007) 27-42
accuracy. It is also observed that standard k-epsilon model computes drag coefficients accurately, where
the realizable k-epsilon turbulence model is more effective for visualization of vortex shedding. The SST komega model is much more recommendable for high Reynolds numbers. Most of the results presented in
this research are compared with experimental data and are in better agreement with respect to other
numerical results. In this research, 2-D method is found very prospective one for turbulent flow due to its
faster convergence. However, the method is restricted here only at two Reynolds numbers (1000 & 3900)
to analyze turbulent flow around circular cylinder; more studies are required to make it applicable at higher
Reynolds numbers.
(Experimental)
(Martinez et al)
(Loc)
(Braza et al)
(Jordan & Fromm)
(Mittal)
(Lakshmipathy)
(Breuer)
(Present)
1.40
1.35
1.30
Drag Coefficient Cd
1.25
1.20
1.15
1.10
1.05
1.00
0.95
0.90
0.85
0.80
0
800
1600
2400
3200
4000
Reynolds Number
Fig.-11: Drag Coefficient Vs Reynolds Number
Table2: Drag coefficient for different Reynolds numbers with experimental and some numerical
results.
Re
100
1.24-1.26
(Tritton,1959)
1000
0.9
(Anderson, 2005)
3900
0.98+/-0.05
Lourenco & Shih (1993)
Tuann & Olson (1978)
1.25
--
--
Martinez et al. (1978)
1.1
--
--
Loc (1980)
1.15
--
--
Braza et al. (1986)
1.17
1.15
--
Jordon & Fromm(1972)
--
1.2
--
Beaudan & Moin (1994)
--
--
1.74
Lakshmipathy
--
--
0.87
Breuer (1998)
Present
-1.245
-0.995
1.08
0.997
Experimental
Numerical Investigation of Unsteady Flow Past a Circular Cylinder using 2-D Finite Volume Method
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M. M. Rahman, M. M. Karim and M. A. Alim / Journal of Naval Architecture and Marine Engineering 4(2007) 27-42
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Numerical Investigation of Unsteady Flow Past a Circular Cylinder using 2-D Finite Volume Method
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