Application of OpenFOAM to Simulate Three-Dimensional Flows past a Single and Two Tandem Circular Cylinders

Hongjian Cao and Decheng Wan

State Key Laboratory of Ocean Engineering, School of Naval Architecture, Ocean and Civil Engineering, Shanghai Jiao Tong University Shanghai, China

ABSTRACT Numerical simulations of three-dimensional flows around a single and two circular cylinders in tandem arrangements are presented by applying the open source codes of OpenFOAM. The Reynolds number (Re) range from 100 to 300 is taken into account in the computations. Two tandem cylinders cases are considered to investigate the interference phenomenon and the vortex shedding. The presented results, including Strouhal number (St) and the vortical structures, show that both the Re and the distance between the two tandem cylinders have great influences on the occurrence of three-dimensional structures in the flow fields. KEY WORDS: OpenFOAM; circular cylinders; tandem; transition; interference; vortex shedding.

INTRODUCTION

The investigation of the flow around circular cylinders has important significance in both academic study and ocean engineering application. Many offshore platforms and pipelines made of cylinders may bring about vibration due to the current and wave, and the more complicated flow interference may effects their safety. With simple configuration and complicated vortex shedding, the study of flows around circular cylinders has been drawing more and more attention as the subject of many investigations.

As a popular problem, the uniform flow around a single cylinder has been extensively studied with both experimental and numerical methods. Williamson (1991), Wu et al. (1996) and Norberg (2003) have studied the vortex shedding from the cylinder with experimental methods. Different numerical methods have also been applied to simulate the flow around a circular cylinder and study the vortex shedding, such as discrete vortex method (Meneghini, 1993) and mesh-free least square-based finite difference method (Ding, 2007).

However, most of the above numerical simulations were carried out in two-dimensional domain, which is not sufficient to predict the three-dimensional structures. As is noted in Williamson (1988, 1996) that there is a transition regime from two-dimensional to three-dimensional in the range 150< Re <300, and this transition regime is associated with two discontinuous changes in the wake formation as Re increases. A

critical Reynolds number is given at Re≈190. The three-dimensional structures of the flow occur above this critical value, and there is a laminar vortex shedding from the cylinder below this value. Furthermore, numerical simulations were performed to study the three-dimensional vortex structures in a cylinder wake. The results of three-dimensional numerical simulation done by Thompson et al. (1996) and Zhang & Dalton (1998) were in good agreement with experimental results.

As for the complicated interferences phenomenon in the flow around two tandem circular cylinders with the center-to-center distance T, numerous experimental and numerical investigations have also been carried out over the past several decades. In the earlier work done by Zdravkovich (1977), Igarashi (1981), Chen (1986) and many other researchers, the wake flow behind two tandem cylinders has been extensively investigated. These investigations confirmed that the wake characteristics have important relationship with the Reynolds number (Re) and T/D. As reviewed in Zdravkovich (1977), three regimes of the wake flow patterns behind tandem cylinders pair were found: single bluff body regime (T/D<1.2D-1.8D); asymmetric regime (1.2D-1.8D<T/D <3.4-3.8) and coupled vortex streets regime (T/D>4D). In the last two decades, numerous numerical investigations have also been carried out to simulate the flow around tandem cylinder pair. The two-dimensional simulations were carried out in the work of Slaouti & Stansby (1992), Wu et al. (1994), Mittal (1997, 2001), Meneghini et al. (2001), Farrant et al. (2001), Ding et al. (2007), Mahir (2008), etc. And also many three-dimensional simulations were carried out to investigate the three-dimensional vortical structures, such as Kondo (2005) with a third-order upwind FEM and Carmo & Meneghini (2006) with the spectral element method. Additionally, the numerical simulations also have been done by Deng (2006), Papaionnou (2006), Kitagawa (2008) and Mahjoub (2008) in recent years.

With the development of the CFD techniques, lots of numerical methods have been involved in the commercial software such as Fluent, CFX, Phoneics, Star-CD, etc. However, for commercial interest, the source codes of these commercial packages are not opened to the users, which has restricted the development of CFD methods. In Recent years, the Open source Field Operation and Manipulation (OpenFOAM) C++ libraries provide users the open source codes for developing new CFD methods. In the OpenFOAM, numerous solvers and utilities covering a wide range of problem are provided with different mesh including polyhedral mesh for handling complex geometry. Large-scale parallel computing can also be implemented with OpenFOAM. The OpenFOAM is also supplied with pre- and post-processing environments. The interface to the pre- and post-processing are themselves OpenFOAM

Proceedings of the Twentieth (2010) International Offshore and Polar Engineering Conference Beijing, China, June 2025, 2010 Copyright © 2010 by The International Society of Offshore and Polar Engineers (ISOPE) ISBN 978-1-880653-77-7 (Set); ISSN 1098-6189 (Set); www.isope.org

702

utilities, thereby ensuring consistent data handling across all environments. The users can not only use OpenFOAM as a software, but also can modify all the codes of OpenFOAM, even creative new solvers and numerical schemes for particular problems. The object-oriented C++ programming language lays a good basis for the development of OpenFOAM, as well as the development of the CFD. The main objective of this paper is to take the advantages of OpenFOAM and present the numerical simulations of three-dimensional flow around a single and two circular cylinders in tandem arrangements. The numerical simulations are implemented by solving the Navier-Stokes equations for incompressible viscous flow with the Reynolds number (Re) range from 100 to 300, covering the critical value for the flow transition from two- to three-dimension.

This paper is organized as follows: a brief description of the numerical method developed in OpenFoam is presented firstly. Then the reliability and efficiency of the numerical method with a benchmark case of flow around a circular cylinder in a channel is shown. After that, the simulations of uniform flow around a single and two tandem circular cylinders are carried out. The three-dimensional results and discussions including the Strouhal number, drag and lift coefficients and iso-surfaces of vorticity are presented. Finally, a brief conclusion is drawn. NUMERICAL METHOD Governing Equations

In this paper, the fluid is assumed to be incompressible and has constant density ρ and constant dynamic viscosity μ . The Navier–Stokes equations governing incompressible fluid flow are as follows:

0=∂∂

i

ixu

(1)

ij

i

jj

ijixp

xu

xxuu

tu

∂∂

−=∂∂

∂∂

−∂

∂+

∂∂

)()(

νρρρ , (2)

where i, j=1, 2, 3; u, p, ρ and ν are the velocity，pressure, density and kinematics viscosity respectively.

Eq. 2 can be written in the vector form:

pUUUtU

−∇=∇⋅∇−⋅∇+∂∂ )()( μρρ , (3)

where ∇ represents the Hamilton operator and μ is the dynamic viscosity.

The codes in the solver of OpenFOAM are programmed with the C++ language. Each term in Eq. 3 was represented by the corresponding codes as follows:

solve (

fvm::ddt(rho,U) + fvm::div(phi,U) - fvm::laplacian(mu,U) == -fvc::grad(p) ); in which, rho is ρ , mu is μ , phi is Uρϕ = ; The fvm and fvc are classes of the C++ language, which represent the finite volume calculator and finite volume method; The ddt( ) is the time derivative, div( ), laplacian( ) and grad( ) are the discrete functions corresponding to the convection term, laplacian term and gradient term in the momentum equation, respectively. Numerical Schemes

The OpenFOAM provides the users with numerous different numerical schemes. It is convenient to choose the numerical schemes what we want to discrete the momentum equation.

In this paper, FVM and PISO algorithm are used. The numerical discrete schemes are chosen as follows: the convection term with the Gauss cubic scheme, the laplacian term with the Gauss linear corrected scheme, the gradient of pressure term with the Gauss linear scheme and the derivative of time is with the Euler implicit scheme. VALIDATION OF THE METHOD

In order to validate the computational codes provided by OpenFOAM, a benchmark case of flow around a circular cylinder in a channel is chosen as the test case. The three-dimensional simulations have been carried out with Re = 20 and 100.

We choose the computational model provided by Schafer (1996), as illustrated in Fig.1. The height and width of the channel are equal to 0.41m and the length is 2.5m. The diameter of the circular cylinder is D = 0.1m and the length is L = 0.41m. As described in a coordinate system, the span of cylinder is along the z-axis, and the inflow is aligned with x-axis.

Fig.1. The computational model for flow around a circular cylinder Three uniform meshes with different grids number are generated for the computations. The nearest grid to the cylinder surface is less than 0.002m and the detail parameters are shown in Table 1. Table 1. The details of the mesh for the computation

Mesh Number of Cells

Number of Nodes around the circle

Number of Nodes along the spanwise

I 314880 96 40 II 364800 120 40 III 381440 128 40

The inflow condition is parabolic velocity-inlet described by the

function: 4/))((16),,0( HzHyHyzUzyU m −−= , 0V W= = . (4)

where the mU is the maximum velocity, and the characteristic velocity is the mean velocity:

9/),2/,2/,0(4)( tHHUtU = , (5) and the Reynolds number is defined by:

ν/Re DU= . (6) The outflow condition is defined by:

0=∂∂

=∂∂

=∂∂

zU

yU

xU . (7)

Cylinder Surface and the other boundaries are non-slip wall condition with all the velocity components U=V=W=0.

The drag and lift forces are obtained by integrating the pressure and the skin friction contributions. With the force components, the drag and

703

lift coefficient are computed from:

DHU

FC x

d 22

ρ= ,

DHU

FC y

f 2

2

ρ= . (8)

The Strouhal number is defined as UDfSt /= with the frequency of oscillation of the lift coefficient.

The results for Re = 20 and Re = 100 are presented in Table 2, and compared with the benchmark results. The computed results with the three meshes are all in between the lower bound and upper bound of the benchmark results. The good agreements of the computed results with the benchmark results show the applicability and efficiency of the present numerical method. Table 2. Force coefficients for the three-Dimensional Test Case with different mesh and comparison with the benchmark results

Re Mesh Cd max Cl max St Mesh I 6.1424 0.00914 —— Mesh II 6.14113 0.00911 —— Mesh III 6.14025 0.00909 —— 20 lower bound upper bound (benchmark results)

6.0500 6.2500

0.0080 0.0100 ——

Mesh I 3.2973 -0.00995 0.3390 Mesh II 3.3054 -0.01033 0.3397 Mesh III 3.3026 -0.01027 0.3397 100 lower bound upper bound (benchmark results)

3.2900 3.3100

-0.0110 -0.0080

0.2900 0.3500

RESULTS AND DISCUSSION

In this section, the uniform flow around a single and two tandem cylinders are simulated, and their results are presented and discussed.

Uniform Flow Past a Single Cylinder

A rectangular computational domain 30D×20D×10D is selected as shown in Fig. 1(a). And a circular cylinder is chosen with the diameter (D) equal to 0.1m and the length (L) equal to 10D. As described in the coordinates system, the span of cylinder is along the z-axis, and the inflow is aligned with x-axis. The inflow boundary is 10D in the front of the cylinder centerline. The outflow boundary is 20D behind the cylinder centerline, and two sides are 10D from the centerline.

Uniform meshes are generated by the pre-process tool Gambit. Along the direction of the span of the cylinder, 80 grids are used. And 120 grids are used to divide the cylinder circle. The distance from the nearest grid to the cylinder surface is less than 0.002m. The total number of meshes is controlled less than 1 million in order to balance the accuracy and computational time.

The inflow boundary condition is uniform velocity with only velocity component U along the x-axes, and V = W = 0. The outflow boundary condition is defined by:

),(,0 pU==∇ ψψ . (9) In order to avoid the boundaries’ effects to the flow field, symmetry boundary condition is applied to the other sides of the domain which is defined by:

0=∂∂

=∂∂

=∂∂

zU

yU

xU (10)

(a)

(b)

(c) Fig. 2. Computational domain and mesh for the single cylinder cases: (a) computational domain, (b) uniform mesh for the global domain, (c) details of the mesh around the cylinder.

In Table 3, the simulation results are presented including the average drag coefficients Cd, the amplitude of lift coefficients Cl and the Strouhal numbers( St )at different Re numbers. As can be seen from the Table 4, the Strouhal numbers presented at Re = 100 and 200 agree very well with the experimental results of Williamson (1991) and Norberg (2003). Table 3. Results of the average drag coefficients and Strouhal numbers for a single cylinder at different Reynolds numbers.

Re 100 160 200 240 270 300 Cd 1.3932 1.3533 1.4056 1.3047 1.2937 1.3658Cl 0.3347 0.5501 0.7250 0.6951 0.6992 0.7988St 0.168 0.188 0.195 0.190 0.192 0.196

704

Table 4. Comparison of the Strouhal number (St) for a single cylinder at Re = 100 and 200.

Re 100 200 Present study 0.168 0.195 Williamson (1991) 0.164 0.196 Norberg (2003) 0.168 0.18-0.197

According to Williamson (1988), in the Re range from 100 to 300, the

vortex shedding of a cylinder have two regime: laminar vortex shedding regime (Re<190); 3-D wake transition regime (Re>190) with two different vortex shedding modes named mode A and mode B.

In Fig. 3, the vortex structure in the wake of the cylinder with Re = 100 and 160 is illustrated. For the two cases, there is no 3-D structure appeared, and laminar vortex is shedding from the cylinder and forms a vortex street. The vortex shedding at each cross-section of cylinder is the same.

(a) (b) Fig. 3. The laminar vortex shedding mode: (a) Re=100; (b) Re=160. The blue surfaces correspond to xω < 0, and the red ones to xω > 0. The same illustrations are to all the following figures for the iso-surfaces of vorticity.

(b) Fig. 4. Iso-surfaces of z-vorticity ( zω ): (a) Re=200; (b) Re=270.

(a) (b) Fig. 5. Iso-surfaces of x-vorticity ( xω ): (a) Re=200; (b) Re=270.

Fig. 4 and Fig. 5 show the iso-surface of the vorticity components xω and zω , from which we can see that: the 3-D wake occurs with the vortex shedding mode A at Re = 200, and mode B at Re = 270. In Fig. 6, both the comparisons of presented simulation results with the corresponding results from Zhang & Dalton (1996) are illustrated. The similarity of the iso-surfaces of xω and zω behind a cylinder is obvious.

(a)

(b) Fig. 6. Comparison of the iso-surfaces of xω and zω , Re=200: (a) results from Zhang and Dalton (1996); (b) presented simulation results.

Furthermore, another comparison of the wake behind a cylinder at Re = 300 with the experiments results of Williamson (1996) is shown in Fig.7, and good agreement can be observed.

(a)

705

(b) Fig. 7. Comparison of iso-surfaces of xω , Re=300 : (a) aluminum flake visualizations from Williamson (1996); (b) presented simulation results. Uniform Flow Past two Tandem Cylinders

For the uniform flow around two tandem cylinders cases, the diameter and length of the cylinder selected are the same as the single cylinder case, as well as the computational domain. Differently, the length is 30D+T, where T is the center-to-center distance of the two cylinders. In more detail, as illustrated in Fig. 8(a), the inflow is 10D ahead the upstream cylinder centerline and the outflow is 20D behind the downstream cylinder centerline. The two tandem cylinders are placed along the central line that links the cylinder centers.

Also uniform meshes are generated for computation. As shown in Fig. 8(b), the detail of the grid is the same as the single cylinder case. Also the boundary conditions are identical to the single cylinder cases. With the same numerical method, a series of simulations are carried out with T=2D, 3D, 3.5D, 4D, 5D and Re=160, 200, 240, 270, 300.

(a)

(b) Fig. 8. Computational domain and mesh for the two tandem cylinders: (a) computational domain; (b) details of the mesh around two tandem cylinders.

The numerical results are presented in Table 5. From the results, it can be seen that the drag coefficient of the downstream cylinder changes sign from negative to positive as T/D increases from 3.5 to 4. Additionally,

the Strouhal numbers have a sudden rise. As an example for the cases at Re = 200, the Strouhal numbers change from 0.126 (T/D=3.5) to 0.180 (T/D=4). The same results have also been obtained by Menegnihi (2001) in two-dimension and Carmo & Meneghini (2006) in three-dimension.

Table 5. The Strouhal number and the drag and lift coefficients obtained at different T/D and Re. (the subscript 1 represents the upstream cylinder, and 2 represents the downstream cylinder)

Re T/D St Cd1 Cl1 Cd2 Cl2 3.0 0.124 1.0770 0.0240 -0.0995 0.2406 3.5 0.125 1.0616 0.0256 -0.0424 0.3188

160

4.0 0.174 1.1791 0.6824 0.4932 1.7926 3.0 0.130 1.0362 0.0256 -0.1366 0.2656 3.5 0.126 1.0170 0.0256 -0.0844 0.3254

200

4.0 0.180 1.2975 0.7839 0.8020 1.9200 3.0 0.127 1.0081 0.0340 -0.1420 0.3291 3.5 0.127 0.9896 0.0218 -0.0975 0.3461

240

4.0 0.173 1.1894 0.6449 0.6431 1.6496 3.0 0.129 0.9921 0.0437 -0.1315 0.3670 3.5 0.127 0.9697 0.0376 -0.0975 0.3721

270

4.0 0.174 1.2198 0.7148 0.6159 1.7058 3.0 0.130 0.9759 0.0686 -0.1125 0.4033 3.5 0.128 0.9546 0.0413 -0.1075 0.3874

300

4.0 0.128 0.9757 0.0163 -0.1125 0.3896

According to Zdravkovich (1977), the critical value of T for vortex

formation is about 3.5D~3.8D depending on the Reynolds number. In the presented simulation, T = 3.5D is below the critical value, and the separated layer from upstream cylinder forms a vortex behind downstream cylinder. Thus the pressure behind the downstream cylinder is larger than ahead of it, so the negative drag is obtained. For T>4D, the vortex is shedding from both of the cylinders and forms a vortex street, which is called the co-shedding regime. In this regime, both the drag coefficients of two cylinders are positive. Therefore, the presented results agree well with Zdravkovich (1977).

Moreover, there is an interesting phenomenon that the strouhal numbers for T = 4D have a sudden decrease as Re increases from 270 to 300, as is shown in Fig. 9. It can be explained as follow words. For T = 4D, the separated layer from upstream cylinder forms a vortex ahead of the downstream cylinder in all the cases with Re<270, but forms a vortex behind the downstream cylinder with Re=300. Due to the interference of the two cylinders, the Strouhal number is decreased.

Fig. 9. Strouhal number (St) as a function of Reynolds number (Re).

706

As noted in Williamson (1988) for a single cylinder case, the critical

Reynolds number for three-dimensional structures to appear is about Re=190. Below this value, it is the laminar vortex shedding regime. In presented simulations of two tandem cylinder cases, below the critical Re, there is no three-dimensional structure, as the iso-surfaces of vorticity components zω for Re=160 shown in Fig. 10.

(a) (b)

(c) (d) Fig.10. Iso-surfaces of z-vorticity ( zω ), Re=160 : (a) T=2D; (b) T=3D; (c) T=4D; (d) T=5D.

Now we focus on how the T influents the occurrence of the three-dimensional structures on condition that Re is larger than the critical value (Re>190). With the iso-surfaces of vorticity component zω for different cases are shown in Figs. 11~15, we can observe that the three-dimensional structures appear only for T≥3D. For T=3D and T=3.5D, the three-dimensional structures begin to appear as Re increases up to 240, which are shown as Fig. 13(b) and Fig. 14(b). For the cases with T=4D and T=5D, the three-dimensional structures is quite obvious when Re is lager the critical Reynolds number.

(a) (b)

(c) (d) Fig. 11. Iso-surfaces of z-vorticity ( zω ), T=2D: (a) Re=200; (b) Re=240; (c) Re=270; (d) Re=300.

(a) (b)

(c) (d) Fig. 12. Iso-surfaces of z-vorticity ( zω ), T=3D: (a) Re=200; (b) Re=240; (c) Re=270; (d) Re=300.

(a) (b)

(c) (d) Fig. 13. Iso-surfaces of z-vorticity ( zω ), T=3.5D: (a) Re=200; (b) Re=240; (c) Re=270; (d) Re=300.

(a) (b)

(c) (d) Fig. 14. Iso-surfaces of z-vorticity ( zω ), T=4D: (a) Re=200; (b) Re=240; (c) Re=270; (d) Re=300.

707

(a) (b)

(c) (d) Fig. 15. Iso-surfaces of z-vorticity ( zω ), T=5D: (a) Re=200; (b) Re=240; (c) Re=270; (d) Re=300.

As similar as the single cylinder cases, for the two tandem cylinders cases, there are also two modes of the vortex shedding observed while the three-dimensional structures appear. As the iso-surfaces of vorticity component xω shown in Figs. 16~19, for T=3D and 3.5D, only mode A can be observed in the range 240≤Re≤300. And mode B can be observed at T=5D. Specially, for T=4D, both mode A (at Re= 200, 300) and mode B (at Re=240, 270) can be observed.

(a) (b) Fig. 16. Iso-surfaces of z-vorticity ( xω ), T=3D: (a) Re=240; (b) Re=300.

(a) (b) Fig. 17. Iso-surfaces of z-vorticity ( xω ), T=3.5D: (a) Re=240; (b) Re=300.

(a) (b)

(c) (d) Fig. 18. Iso-surfaces of x-vorticity ( xω ), T=4D: (a) Re=200; (b) Re=240; (c) Re=270; (d) Re=300.

(a) (b)

(c) (d) Fig. 19. Iso-surfaces of x-vorticity ( xω ), T=5D: (a) Re=200; (b) Re=240; (c) Re=270; (d) Re=300.

708

CONCLUSIONS The numerical simulations of three-dimensional incompressible

viscous flow around a single and two tandem cylinders by means of the CFD tools provided by OpenFOAM have been presented. In the single cylinder case, three-dimensional structures with two vortex shedding mode A and mode B were observed at Re≥200. Good agreements between the computed results of the Strouhal number, the drag and lift coefficients and the iso-surfaces of the vorticity and the corresponding results in available literature have been shown. In the two tandem cylinders case, the flow is complicated due to the interferences of the two cylinders. It is found that both the drag inversion and the sudden increase of Strouhal number happen as T changes from 3.5D to 4D, which agrees well with other researchers’ results. The three-dimensional structures appearing at Re≥200 and T ≥3D are observed. The critical value of T for three-dimensional structures appearing is in the range 2D＜T≤3D. Moreover, there is also a transition of the vortex shedding modes. But it is different for different Re and T. Specially for T=4D, there is an obvious transition of the vortex shedding from mode B to mode A between Re=270 and 300, which is due to the interference of the two cylinders.

From these results, it is found that the complex three-dimensional structures of flow around one and two cylinders can be accurately captured by solving the Navier-Stokes equations based on the tools of OpenFOAM. Since the OpenFOAM provides open source codes, it is convenient to choose and change, even modify the numerical schemes. In addition, with the utilities of OpenFOAM as interfaces to other pre- and post-processing, a quick and fine mesh generation and data post-processing become easier. The OpenFOAM is not only a convenient and efficient CFD tool, but also lays a good basis for constructing new numerical methods and schemes. Obviously, OpenFOAM will achieve more success and make greater contributions to the development of CFD. ACKNOWLEDGEMENTS

The support of National Natural Science Foundation of China (Grant No. 50739004), National 863 Plan Project of Ministry of Science and Technology of China (Grant No. 2009AA09Z301, 2006AA09A107), PhD Program Foundation of Ministry of Education of China (Grant No. 20060248039), Program for New Century Excellent Talents in University (Grant No. NCET-06-0404) and the Program for Professor of Special Appointment (Eastern Scholar) at Shanghai Institutions of Higher Learning for this work is gratefully acknowledged. REFERENCES Braza, M, Chassaing, P, and Ha Minh, H (1986). “Numerical study and

physical analysis of the pressure and velocity fields in the near wake of a circular cylinder,” J Fluid Mech, Vol 165, pp 79-130.

Carmo, BS, and Meneghini, JR (2006). “Numerical investigation of the flow around two circular cylinders in tandem,” J Fluids Struct, Vol 22, Nos 6-7, pp 979-988.

Chen, SS (1986). “Review of flow-induced vibration of two circular cylinders in crossflow,” Journal of Pressure Vessel Technology, Transaction of the ASME, Vol 108, No 4, pp. 382-393.

Deng, J, Ren, AL, Zou, JF, and Shao, XM (2006). "Three-dimensional flow around two circular cylinders in tandem arrangement," Fluid Dyn Res, Vol 38, No 6, pp 386-404.

Ding, H, Shu, C, Yeo, YO, and Xu, D (2007). “Numerical simulation of flows around two circular cylinders by mesh-free least square-based finite difference methods,” Int J Numer Methods Fluids, Vol 53, pp 305-332.

Farrant, T, Tan, M, and Price, WG (2001). “A cell boundary element method applied to laminar vortex shedding from circular cylinders,”

Comput Fluids, Vol 30, pp 211-236. Igarashi, Tamotsu (1984). “Characteristics of the flow around two

circular cylinders arranged in tandem - (2nd Report, Unique phenomenon at small spacing),” Bulletin of the JSME, Vol 27, No 233, pp 2380-2387.

Kitagawa, T, and Ohta, H (2008). ‘Numerical investigation on flow around circular cylinders in tandem arrangement at a subcritical Reynolds number,” J Fluids Struct, Vol 24, No 5, pp 680-699.

Kondo, N, and Matsukuma, D (2005). “Numerical simulation for flow around two circular cylinders in tandem,” Int J Comput Fluid Dyn, Vol 19, No 4, pp 277-288

Mahir, N, and Altac, Z (2008). “Numerical investigation of convective heat transfer in unsteady flowpast two cylinders in tandem arrangements,” Int J Heat Fluid Flow, Vol 29, pp 1309-1318.

Mahjoub Said, N, Mhiri, H, Bournot, H, and Le Palec, G (2008). “Experimental and numerical modelling of the three-dimensional incompressible flow behaviour in the near wake of circular cylinders,” J Wind Eng Ind Aerodyn, Vol 96, No 5, pp. 471-502.

Meneghini, JR, Saltara, F, Siqueira, CLR, and Ferrari Jr, JA (2001). “Numerical simulation of flow interference between two circular cylinders in tandem and side-by-side arrangements,” J Fluids Struct, Vol 15, No 2, pp 327-350.

Mittal, S, Kumar, V, and Raghuvanshi, A (1997). “Unsteady incompressible flows past two cylinders in tandem and staggered arrangements,” Int J Numer Methods Fluids, Vol 25, No 11, pp 1315-1344.

Mittal, S, and Kumar, V (2001). “Flow-induced oscillations of two cylinders in tandem and staggered arrangements,” J Fluids Struct, Vol 15, No 5, pp 717-736.

Norberg, C (2003). “Fluctuating lift on a circular cylinder: review and new measurements,” J Fluids Struct, Vol 17, pp 57-96.

Papaioannou, GV, Yue, DKP, Triantafyllou, MS, and Karniadakis, GE (2006). “Three-dimensionality effects in flow around two tandem cylinders,” J Fluid Mech, Vol 558, pp 387-413.

Schafer, M, and Turek, S (1996). “Notes on Numerical computations of laminar flow around a cylinder,” Notes on Numerical Fluid Mechanics, Vol 52, pp 547-566.

Slaouti, A, and Stansby, PK (1992). “Flow around two circular cylinders by the random-vortex method,” J Fluids Struct, Vol 6, No 6, pp 641-670.

Thompson, M, Hourigan, K, and Sheridan, J (1996). “Three-dimensional instabilities in the wake of a circular cylinder,” Experimental Thermal and Fluid Science, Vol 12, pp 190-196.

Williamson, CHK (1988). “The Existence of Two Stages in the Transition to Three-Dimensionality of a Cylinder Wake,” Phys Fluids, Vol 31, pp 31-65.

Williamson, CHK (1991). “2-D and 3-D Aspects of the Wake of a Cylinder, and Their Relation to Wake Computations,” Lect Appl Math, Vol 28, pp 719-751.

Williamson, CHK (1996). “Vortex dynamics in the cylinder wake,” Annual Review of Fluid Mechanics, Vol 28, pp 477-539.

Wu, J, Welch, LW, Welsh, MC, Sheridan, J, and Walker, GJ (1994). “Spanwise wake structures of a circular cylinder and 2 circular cylinders in tandem,” Experimental Thermal and Fluid Science, Vol 9, No 3, pp 299-308.

Wu, J, Sheridan, J, Welsh, M, and Hourigan, K (1996). Three-dimensional vortex structures in a cylinder wake. J Fluid Mech, Vol 312, pp 201-222.

Zdravkovich, MM, and Pridden, DL (1977). “Interference between two circular cylinders; Series of unexpected discontinuities,” J Industrial Aerodynamics, Vol 2, No 3, pp 255-270.

Zhang, J, and Dalton, C (1998). “A three-dimensional simulation of a steady approach flow past a circular cylinder at low Reynolds number,” Int J Numer Methods Fluids, Vol 26, pp 1003-1022.

709