numerical simulations of flow around square … - numerical simulations of flow around...numerical...

Columbia International Publishing Journal of Advanced Computing (2013) 1: 59-70 doi:10.7726/jac.2013.1005

Research Article

______________________________________________________________________________________________________________________________ *Corresponding e-mail: [email protected] 1 Department of Marine Technology, Amirkabir University of Technology, Tehran, Iran

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Numerical Simulations of Flow around Square Cylinder Using an Iterative High Order Difference Scheme

Mahdi Yousefi Fard 1, Parviz Ghadimi 1*, Rahim Zamanian 1 Received 13 April 2013; Published online 8 June 2013 © The author(s) 2013. Published with open access at www.uscip.org

Abstract A time dependent two-dimensional numerical simulation of flow around a rectangular cylinder is conducted using a particular high order compact finite difference method. The current forth-order compact scheme is implemented for the simulation of flow around bluff bodies for the first time. This formulation offers a semi-explicit method for the solution of Navier-Stokes equations in stream function-vorticity form and based on a nine point difference scheme with uniform grid spacing. The main advantage of the suggested method is its fast convergence and high accuracy. The proposed numerical method combines the enhanced Fourni’e’s forth order scheme with a semi-explicit formulation. By this combination, very accurate results can be obtained with a relatively coarse mesh in a short time. Uniform grids are applied with this high performance algorithm, and the accuracy of the scheme is proved using the benchmark problem of incompressible laminar flow around a rectangular cylinder at medium and high Reynolds numbers. Current results have good agreement with other numerical and experimental values. Generally, it is shown that, using the high order finite difference method demonstrates fairly competitive results for the current problem. Keywords: Steady 2-D Navier-Stokes equations; Finite difference method; rectangular cylinder; Stream function-vorticity formulation

1. Introduction Study of bluff body wakes is important for many applications in the fluid engineering problems. Bluff body cross sections that are often employed are circular and rectangular, especially square. Flow details behind these geometries mainly depend on Reynolds number and blockage ratio. Flow around a square cylinder is similar in many ways to the flow around a circular cylinder. But the separation mechanism and the consequent dependence of lift, drag, and Strouhal number on the Reynolds number are significantly different. The flow field over a cylinder is symmetric at low

Mahdi Yousefi Fard, Parviz Ghadimi, and Rahim Zamanian / Journal of Advanced Computing (2013) 1: 59-70

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values of Reynolds number. As the Reynolds number increases, flow begins to separate behind the cylinder causing vortex shedding which is an unsteady phenomenon. Among the three mentioned bodies, flow around rectangular bodies has a majority of past research. As noted by Roshko (1954), the shedding frequency is related to the width of the wake. Durao et al. (1988) presented a detailed experimental inspection of flow around a square cylinder as quintessential bluff body, using laser doppler velocimetry at a Reynolds number of 14,000. Similar work has been done by Lyn et al. (1995) at higher Reynolds number. As previously mentioned, flow around the square cylinder is symmetric at low Reynolds numbers. Vortex shedding starts taking place when the Reynolds number increases and critical Reynolds number ( ) presented by many researchers. Okajima (1982) abserved vortex motion at low Reynolds number, and proposed that is below 70. Also, this critical value is suggested by Klekar and Patankar (1992). They determined the value of to be 54. The effect of blockage was investigated numerically by Stansby and Slaouti (1993), Anagnostopoulos et al. (1996), Behr et al. (1995) and Turki et al. (2003). It is shown that, increasing the blockage factor leads to high Strouhal number, drag coefficient and stagnation pressure. A significant amount of research has been published for numerical study of flow past a rectangular cylinder at high Reynolds numbers. Murakami et al. (1990) investigated the performance of the model on a square section. Similarly, Lubcke et al. (2001) tested both RANS an LES method in simulation of flow around rectangular cylinder at Reynolds number Re=22,000. Ghadimi et al. (2013) presented a new forth order finite difference scheme to solve the steady Navier-Stokes equation in the form of vorticity-streamfunction. They extended Fourni’e’s (2006) formulation and applied it to solve Navier-Stokes equation in the benchmark problem of flow in the driven cavity up to . This advanced method has been used in the current study to simulate steady and unsteady flow around square cylinder to examine the accuracy and performance of this scheme. In order to solve unsteady problems, the transient term has been inserted into the ordinary vorticity-stream function formulation.

2. Governing Equations The governing equations of viscous fluid flows in stream function (ψ) and vorticity (ω) formulation can be written as (Liggett and Caughey, 1994):

(1)

(2)

where x and y are the Cartesian coordinates and the parameter is the so-called dimensionless Reynolds number. The velocity components ( ) are defined as (3) (4) The nonlinear system is closed with suitable boundary conditions. The general form of these relations, could be written as an elliptic equation

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(

)

( ) ( )

(5)

which is accompanied by Dirichlet boundary conditions on , where is a smooth convex domain in consisting of a series of rectangular shapes. Parameters c, d, and β are assumed to be constant coefficients, while β satisfies the ellipticity condition . The forcing term ( ) as well as the solution ( ) of the problem are assumed to be sufficiently smooth in the computational domain. Discretization of Eq. (5) using the advanced nine point difference scheme (as shown in Fig.1) leads to accurate numerical results with reasonable computational cost. The base idea of forth order compact scheme has been demonstrated by Gupta et al. (1984). The truncation error of the scheme is of order ( ).

Fig 1. Labeling of the nine grid points

The fourth-order compact finite difference formula for the central point ( ) involves the nearest eight neighboring mesh points with mesh spacing h which is given by Fourni’e (2006) as in

∑

∑

(6)

where the coefficients , i=0,1,…,8, are given as (

),

( ),

( ),

( ),

( ),

( ),

( ),

( ),

( ).


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The coefficients , , are

,

( ),

( )

( ),

( ),

,

,

.

Eq. (1) is a Poisson type equation, and changes to fourth-order approximation with and . The vorticity Eq. (2) is a special case of the convection-diffusion equation, and the fourth-order approximation in this case may be obtained with , , and , in Eq. (5). Time marching algorithm is applied by a simple difference method. The velocity components u and v at a grid point ( )are computed from the discrete approximation of Eq. (5) with fourth-order approximation as follows (Gupta, 1991):

(7)

(8)

Stortkuhl et al. (1994) have also presented the following expression for computing the vorticity values on the wall:

[

]

[

]

(9)

where V is the speed of the wall which is equal to 1 for the moving top wall and is equal to 0 for the three stationary walls.

3. Solution Procedure An iterative scheme is used in the solution of the Navier-Stokes. Continuity and momentum equations are solved simultaneously. At the beginning, the initial data except the boundary values of the known functions is set equal to zero. Subsequently, an iterative procedure is implemented, as shown in Fig.2.


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Fig 2. Solution procedure

4. Flow past a square cylinder The capabilities of the proposed method are demonstrated by simulating the laminar steady and unsteady flow over a square cylinder. The flow around a square cylinder at different Reynolds

numbers (

) is computed to study the influence of several parameters, where is the free-

stream velocity, is the cylinder height, and is the kinematic viscosity.

Time=0

Apply boundary conditions

Calculate streamfunction values

Eq. (6)

Assign vorticity boundary condition

Eq. (9)

Calculate vorticity values

Eq. (6)

t=t+Δt

Check the

differences

of variables

between

steps

Relax

the

values

Set initial values as 0

Write the results


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The domain geometry is indicated in Fig.3. To minimize the effects of the upstream and downstream boundaries on the flow, these boundaries are located at distances sufficiently away from the cylinder. In this case, inlet boundary condition is specified at 10 square heights upstream of the body as a uniform velocity, and a simple second order finite difference scheme between two neighboring points is applied. Downstream boundary is located at 50 cylinder heights from the step and direction and uniform velocity profile are specified to achieve a developed flow at the outlet. In the present study, the domain has a width and the blockage( ) is . These dimensions were chosen in order to minimize the boundary effects on the flow development.

Fig 3. Configuration of the test case.

5. Results and Discussion An iterative scheme is used in the solution of the Navier-Stokes. Continuity and momentum equations are solved simultaneously. At the beginning, the initial data except the boundary values of the known functions is set zero. For every Reynolds number, iterations are continued until both the maximum residual of mass and momentum equations are less than 10-5 in the computational domain. Figure 4 shows the streamlines contours along the domain for Re=10, 20 and 50. At these Reynolds numbers, the flow is laminar and steady. Also, reattachment length is presented for each of the Reynolds numbers and compared with Yoon et al. (2010) results. Here, is defined as the distance between the cylinder center and the near-wake saddle point as mentioned by Perry et al. (1982).

Re=10 Re=20


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Re=50

Fig 4. Streamline contour for Re=10, 20 and 50.

Fig 5. Variation of the length of recirculation region against Re for Re=10, 20 and 50.

At moderate Reynolds numbers, the flow is laminar and unsteady. In the current study, numerical simulation is presented at a series of extra Reynolds numbers above 100 with unsteady flow pattern. A time marching approach is performed to simulate the unsteady flow in every time step. Figures 6 and 7 show the time-dependent streamline contour for both Re=100 and 200. The periodicity of unsteady flow field has been displayed in both contours.

T=100 T=101

1

1.5

2

2.5

3

3.5

4

0 10 20 30 40 50 60

L r

Re

Yoon et al.

Current Study


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T=102 T=103

T=104 T=105

T=106 T=107

T=108 T=109

T=110

Fig 6. Streamline contour for Re=100


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Downstream flow oscillated and streamline contours were found in 10 seconds period of time. This period is about 7.0 seconds for Re=200.

T=100 T=101

T=102 T=103

T=104 T=105

T=106 T=107

T=108 T=109


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T=110

Fig 7. Streamline contour for Re=200.

Reference Strouhal numbers (

) for Re=100, 200 were derived by Robichaux et al. (1999),

where is the period of vortex shedding. Table 1 shows Strouhal numbers for Re=100, 200 and the current results are compared against those calculated by Robichaux et al. (1999). Comparison indicates good agreement. Table 1 Comparison of Strouhal numbers

Reynolds Number Current study Strouhal number Robichaux et al. (1999) Strouhal number

100 0.145 0.150

150 0.155 0.165

200 0.160 0.168

6. Conclusion

High order finite difference simulation has been carried out to study the vortex shedding downstream of two-dimensional square cylinder. In this method, a nine point finite difference scheme with uniform grid spacing has been implemented to solve Navier-Stokes equations in vorticity-streamfunction formulation. According to this scheme, a time marching method is used to simulate unsteady flow and find the Strouhal number for medium Reynolds numbers. Main advantage of the present numerical method is simplicity and fast convergence. Different flow simulations were performed at various Reynolds numbers. Results indicate that the suggested numerical scheme is proved to work for Reynolds numbers up to 200 for this problem. The robustness and accuracy of the present solution is proved by comparison of the pressure distribution over the cylinder for three low Reynolds numbers. Stream function pattern for these low Reynolds numbers have been presented and the reattachment length was compared with experimental data. Furthermore, for the medium Reynolds number, unsteady flow has been reviewed to find Strouhal number. Strouhal numbers are the same as experimental and other numerical data. This means that the proposed time dependent high order scheme is comparable with other solutions in efficiency, simplicity and accuracy and could be used for unsteady flow and high Reynolds numbers.


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http://dx.doi.org/10.1108/EUM0000000004030

http://dx.doi.org/10.1007/s00466-003-0496-2

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