Bhanuman Barmanand S. Bhattacharyya

1. Department of Mathematics, University of Gour Banga, Malda 732103, India

2. Department of Mathematics, Indian Institute of Technology Kharagpur, Kharagpur 721302, India

Control of Vortex Shedding and Drag Reduction through Dual Splitter Plates Attached to a Square Cylinder

Bhanuman Barman1and S. Bhattacharyya2*

1. Department of Mathematics, University of Gour Banga, Malda 732103, India

2. Department of Mathematics, Indian Institute of Technology Kharagpur, Kharagpur 721302, India

In this paper we have made a numerical study on the control of vortex shedding and drag reduction of a cylinder by attaching thin splitter plates. The wake structure of the cylinder of square cross-section with attached splitter plates is analyzed for a range of Reynolds number, based on the incident stream and height of the cylinder, in the laminar range. The Navier-Stokes equations governing the flow are solved by the control volume method over a staggered grid arrangement. We have used the semi-implicit method for pressure-linked equation (SIMPLE) algorithm for computation. Our results show that the presence of a splitter plate upstream of the cylinder reduces the drag, but it has a small impact on the vortex shedding frequency when the plate length is beyond 1.5 time the height of the cylinder. The presence of a downstream splitter plate dampens the vortex shedding frequency. The entrainment of fluid into the inner side of the separated shear layers is obstructed by the downstream splitter plate. Our results suggest that by attaching in-line splitter plates both upstream and downstream of the cylinder, the vortex shedding can be suppressed, as well as a reduction in drag be obtained. We made a parametric study to determine the optimal length of these splitter plates so as to achieve low drag and low vortex shedding frequency.

square cylinder; splitter plate; vortex shedding; drag reduction; semi-implicit method; pressure-linked equation

1 Introduction1

The vortex shedding and formation of a Karman vortex street behind a cylinder has been the object of numerous experimental and numerical studies because of the fundamental mechanisms that this flow exhibits and also because of its several practical relevance. The classical view of a vortex street in cross section consists of regions of concentrated vorticity shed into the downstream flow from alternate sides of the body (and with alternate sense of rotation), giving the appearance of an upper row of negative vortices and lower row of positive vortices. This alternate shedding of vortices in the near wake leads to large fluctuating pressure forces in a direction transverse to the flow and may cause structural vibrations, acoustic noise or resonance.

There have been numerous investigations in the past aiming to alter or suppress the pattern of vortex shedding. The flow separation from the cylinder surface causes significant pressure drop in the rear part of the cylinder, resulting in considerable drag increase. Reducing the drag is critically important in certain engineering applications and both active and passive control techniques have been proposed to achieve that goal. In general, flow control techniques for reducing the aerodynamic drag exerted on a bluff body are classified into two types： active and passive control techniques. Active control methods control the flow by supplying external energy through several means such as rotational oscillatory motion of the bluff body or jet blowing. Because of this, active control techniques require complex mechanical devices that supply external power to the flow. Passive control techniques control the vortex shedding by modifying the shape of the bluff body or by attaching additional devices in the flow stream. Therefore, the passive control technique is energy free and often easier to implement. Among the passive control techniques, the splitter plate has been considered as one of the most successful devices to control the vortex shedding behind a cylinder (Ali et al., 2011). Another approach of controlling the flow behind a bluff body is to place a smaller bluff body in tandem with the main body (Bhattacharyya and Dhinakaran, 2008). The reattachment of shear layers emerging from the front bluff body with the main body causes a substantial drag reduction and damping in oscillation.

Roshko (1955) was among the first to study the control of vortex shedding behind a circular cylinder through attaching a splitter plate along the downstream. This was examined numerically by Kwon and Choi (1996) for various plate lengths and at lower range of Reynolds numbers (Re) i.e., Re between 80 to 160. They found that the vortex shedding frequency due to a circular cylinder decreases with the increase of splitter plate length and the vortex shedding completely disappears when the length of the splitter plate is larger than a critical length, which depends on the Reynolds number. Vortex shedding from bluff bodies with splitter plates was experimentally investigated by Nakamura (1996). The free shear layers emerging from either sides of a bluff body roll up to form vortices, and these vortices are shed alternately from each side of the body. These studies showthat by varying the length of the splitter plate the interaction of the shear layer can be modified. Flow visualization study due to Kawai (1990) shows that a splitter plate reduces the three-dimensionality in the formation region by stabilizing the transverse ‘flapping’ of the shear layers. Lin and Wu (1994) found that a splitter plate attached to the cylinder having a length double the diameter of the cylinder could suppress the vortex shedding. Tiwari, et al. (2005) carried out a numerical study to analyze the effect on the flow characteristics of the splitter plate mounted on the back of the circular tube. Their results indicate that the splitter plate changed the wake characteristics and the size of the wake was decreased due to the presence of the splitter plate. A recent study by Ali et al. (2011) shows that a splitter plate can fundamentally change the flow structure of the square cylinder wake via a variety of hydrodynamic interaction mechanisms. The control of wake behind a circular cylinder by attaching a freely rotatable splitter plates was studied experimentally for a higher range of Reynolds number i.e., Re varies between 3×104to 6×104by Gu et al. (2012). The effect of a downstream detached splitter plate on the flow around a surface mounted circular cylinder of finite height was studied experimentally by Igbalajobi et al. (2013) for Reynolds number 7.4×104.

Reducing the drag is critically important in certain engineering applications, and both active and passive control techniques have been proposed to achieve that goal. When a geometric modification is done either through an attached/detached splitter plate or by placing an in-line bluff body, the flow approaching the bluff body is subjected to a momentum loss. This causes a significant reduction in pressure along the upstream face of the bluff body. Thus, the drag experienced by the body gets modified. In presence of the splitter plate upstream of the cylinder, a bifurcated flow impinges on the cylinder leading to a lower forward stagnation pressure in front of the cylinder than in absence of a plate. Several studies on drag reduction and control of flow by placing another cylinder in tandem have been reported e.g., Bhattacharyya and Dhinakaran (2008). The control of flow-induced forces in a laminar regime on a circular cylinder using a detached splitter plate placed horizontally in the wake region is studied by Hwang et al. (2003). Zhou et al. (2005) studied numerically the effect of the presence of a control rod placed upstream of a square cylinder for fixed a value of Reynolds number, i.e., Re = 250. They observed that in the presence of the control rod, the drag on the cylinder gets significantly reduced and even the lift gets suppressed. Hwang and Yang (2007) found that substantial drag reduction of a circular cylinder is possible through introduction of dual detached splitter plates. They examined the flow for Reynolds number up to 160 for plates length equal to the diameter of the cylinder. The control of vortex shedding and drag reduction of a circular cylinder by attaching splitter plates upstream and/or downstream of the cylinder for Reynolds number in the turbulent range was made by Qiu et al. (2014). Their experimental study shows that a substantial drag reduction is achieved due to the presence of the front splitter plate.

The study on the control of wake due to a square cylinder by introducing splitter plate is rather limited. Flows over square cylinders are important in many engineering applications, especially in flows around bridges, buildings, marine risers and in the context of augmentation of heat transfer from PCBs. The basic difference between the flow field past a square cylinder and a circular cylinder is that the points of separation for the former one are fixed at the upstream corners, whereas the points of separation for the latter case move back and forth depending on the oncoming fluid velocity. The fluid travels downstream at a large trajectory angle from each of these points and a comparatively larger recirculation zone is generated (Ali et al., 2011). Ozono (1999) studied the effects on vortex shedding frequency by placing a splitter plate along the centreline of the square cylinder. Turki (2008) made a numerical study on passive control of wake due to a square cylinder by introducing an attached splitter plate downstream of the cylinder. Ali et al. (2011) investigated the effect of the length of a downstream attached splitter plate on the wake of a square cylinder at Reynolds number 150. Based on the length of the splitter plate, they have identified three flow regimes.

In this paper, the control of vortex shedding as well as drag reduction by attached splitter plates both upstream and downstream of a square cylinder is investigated. The splitter plates are placed along the horizontal centreline of the cylinder. Most of the experimental studies considered the Reynolds number in the subcritical region i.e., Re is O(104), in which the flow is relatively little influenced by the variation of Reynolds number. We have restricted our attention for the range of Reynolds number for which the flow can be considered laminar and two-dimensional. This range of Reynolds number may correspond to flow around an offshore installation of characteristic length 10 cm in an approaching stream in the order of 1 m/s. The laminar range of Reynolds number also appear in analyzing flow around an electronics device where the characteristic length is in the millimeter range. In the following section we have described the mathematical formulation. We have developed the present computer code to compute the unsteady Navier-Stokes equations through the algorithm as described in section 3. The results are discussed in section 4, which is followed by a concluding section.

2 Problem formulation

A long square cylinder of side D placed in a uniform flow (from left to right) with velocity U∞ is considered. Splitter plates of fixed length L1 and L2 are attached in the upstream and the downstream of the cylinder, respectively. The length of the splitter plates are varied. A schematic diagram of the flow configuration is presented in the Fig. 1. The non-dimensional Navier-Stokes equations governing the fluid flow are given

where the non-dimensional variables are defined as

Fig. 1 Sketch of the test case and annotations

The variables with asterisks denote dimensional variables. Here u and v denotes the Cartesian components of velocity, ν is the kinematic viscosity, ρ is the fluid density.

The governing equations (1)-(3) are subjected to the following boundary conditions：

A no-slip condition is imposed on the surface of the cylinder and plates. The drag and lift coefficients (CD, CL) are obtained by considering the viscous and pressure forces acting on the cylinder. They are determined as

where FDand FLare the drag and lift forces experienced by the cylinder, respectively.

The non-dimensional surface pressure coefficient (Cp) is defined by

The instantaneous lift, drag and pressure coefficients of the cylinder are averaged over a period of shedding cycle to obtain,, andwhich are given by

3 Numerical method and validation

We developed a computer code to compute the governing equations based on the numerical algorithm as described below. The governing equations (1)-(3) are solved in a coupled manner through the finite volume method over a staggered grid system. In the staggered grid arrangement, the velocity components are stored at the midpoints of the cell sides to which they are normal. The scalar quantity, such as pressure is stored at the center of the cell. The discretized form of the governing equations are obtained by integrating over an elemental rectangular cells using the finite volume method. A pressure correction-based iterative algorithm, SIMPLE (Patankar, 1980) is used for solving the governing equation with the specified boundary conditions. A first-order implicit scheme is used to discretize the time derivatives. The pressure link between continuity and momentum is accomplished by transforming the continuity equation into Poisson equation for pressure. The Poisson equation implements a pressure correction for a divergent velocity field. At every iteration step, the resulting block tri-diagonal system of algebraic equations are solved through a block elimination method. A single iteration consists of the following sequential steps：

1) The Poisson equation for pressure correction is solved using the successive under relaxation method. In this case the under-relaxation parameter is chosen as 0.8.

2) The velocity field at each cell is updated using the pressure correction.

The iteration at each time step is continued until the divergence-free velocity field is obtained. However for this purpose, the divergence in each cell is towed below a pre-assigned small quantity (ε). In the present case ε is 0.5×10-4. A time-dependent numerical solution is achieved by advancing the flow field variables through a sequence of short time steps of duration δt. At the initial stage of motion, the time step δt is taken to be 0.001 which has been subsequently increased to 0.005 after the transient state.

A non-uniform grid distribution is incorporated in the computational domain (Fig. 2). The grid is finer near the surface of the square cylinder and plates. The first grid point is placed at a distance from each wall is 0.01D. The computation is performed on 471×352 grids with the first and second number being the number of mesh points in x-direction and y-direction, respectively.

Fig. 2 The arrangement of the computational grid in the computational domain

In order to assess the accuracy of our numerical method, we have compared our results for the attached downstream splitter plate i.e., L1/D=0 with the results due to Ali et al. (2011). Fig. 3(a) shows the comparison of time-averaged drag coefficientwith (Ali et al., 2011) at different values of the downstream plate length (L2/D) for Reynolds number 150. The comparison of Strouhal number with (Ali et al., 2011) at different values of L2/D at Reynolds number 150 is shown in Fig. 3(b). Fig. 3(a), and (b) shows that our computed results are in good agreement with those of Ali etal.(2011). A grid independency test has also been made in Fig. 3(a). It shows that the maximum percentage difference inis 3% when the grid size is increased from 420×310 to 471×352.

Fig. 3 Comparison of our results for Re=150 with Ali et al. (2011) at different values of the downstream splitter plate length (0 ≤ L2/D≤ 4) when L1/D=0. The effect of grid size is also shown

4 Results and discussion

4.1 Upstream splitter plate

We consider the control of flow by attaching splitter plates both upstream and downstream of the cylinder. While the influence of a downstream splitter plate on the wake is well understood, the impact of the upstream attached splitter plate alone has not been addressed in details before. We first discuss the influence of the upstream splitter plate on the flow past a square cylinder in the laminar range of Reynolds number. Subsequently, the impact of dual splitter plates is discussed.

The instantaneous vorticity lines for different length of the splitter plate is shown in Fig. 4(a)-4(e) for Reynolds number 150. The separated shear layers of the plate reattach on the downstream cylinder. The top and bottom shear layers of the plate have similar vorticity distribution and do not disturb the manner of the vortex shedding from the cylinder. The separated shear layer from the thin plate reattaches to the cylinder and rolls up in a quasi-steady manner. The pattern of the vortex shedding behind the cylinder remains unaltered due to the presence of the upstream plate. However, the circulation strength of the vorticity is increased due to the presence of the splitter plate. The approach shear flow from either sides of the plate increases the vortex strength of the separated shear layer emerging from upper and lower faces of the cylinder, which in turn enhances the vortex shedding frequency as demonstrated below.

Fig. 4 Instantaneous vorticity for different front plate length (L1/D) for Re=150 when L2/D=0

The time averaged surface pressure distribution around the cylinder at Reynolds number, Re=150 is presented in Fig. 5 for different values of the upstream splitter plate length i.e., L1/D=0, 0.5, 1, 2. The pressure is positive along the front face AB of the cylinder with the stagnation pressure occuring at the mid-point of AB. The flow separates fromthe upper and lower corners A and B, where the pressure distribution attains minima. The pressure along the other sides of the cylinder is negative as the flow separates from the lower and upper corners and never reattaches with the cylinder. It is also found that the pressure distribution on the front face of the cylinder changes due to the variation of the plate length. However, the surface pressure distribution along the faces other than the front face follow the similar trend as that of the case of a cylinder without an attached plate (L1/D=0). Since the upstream plate decreases the momentum of the fluid impinging on the cylinder, the stagnation pressure is reduced.

Fig. 5 Surface pressure distribution for different values of the front plate length (L1/D=0, 0.5, 1.0, 1.5, 2.0, 2.5) when Re=150 with L2/D=0

Fig. 6 Effect of upstream splitter plate (L1/D) with L2/D=0 at different Reynolds number

The variation of the Strouhal number (St) and the average drag coefficient (CD) due to the variation of Reynolds number at different values of the upstream splitter plate length is presented in Fig. 6 (a) and (b). At lower range of Re, it is found that the upstream splitter plate has a small effect on the vortex shedding frequency. It may be noted that the study of Hwang and Yang (2007) for Reynolds number up to 160 shows that the presence of an upstream detached splitter plate does not alter the vortex shedding frequency of a circular cylinder. However, we find in Fig.6(a) that for higher range of Re, the vortex shedding frequency is enhanced due to the introduction of a splitter plate. The vorticity in the shear layers emerging from either sides of the cylinder become stronger due to the presence of the splitter plate. This results in an enhanced vortex shedding frequency in presence of an upstream splitter plate.

The variation of the time-averaged drag coefficient () with Reynolds number at different values of the upstream splitter plate length shows that the average drag reduces with the introduction of the upstream splitter plate. Hwang and Yang (2007) also found the reduction in drag experienced by a circular cylinder due to the introduction of a upstream detached splitter plate. We find that at a given value of the Reynolds number, the drag experienced by the cylinder reduces as the length of the plate increases. It is evident from the Fig. 6(b) that the variation of () with Re follow the same trend at different L1/D (=0, 0.5, 1.0, 2.0).

Fig. 6(a) and 6(b) shows that an increase in upstream splitter plate length does not influence the vortex shedding mechanism downstream of the cylinder but produces a monotonic reduction in drag experienced by the cylinder. This reduction inis just due to the reduction of pressure at the front face of the cylinder. We find that when L1/D ≥ 1.5, St becomes independent of L1/D for the three values of Reynolds number considered. However, an increment in the plate length produces a monotonic reduction in the drag force experienced by the cylinder.

4.2 Dual Splitter plate

The influence of a downstream splitter plate on the wake of a square cylinder in the laminar range of Reynolds number have been studied at length by Ali et al. (2011). They found that in the presence of the downstream attached plate, the free shear layers which are emerging from the opposite sides of the cylinder convects further downstream before rolling-up compared to a bare cylinder. When the plate length is bigger, a secondary vortex forms around the trailing edge of the plate. This secondary vortex influences the vortex shedding behind the cylinder.

We find in the previous subsection that the upstream plate does not produce a significant effect on vortex shedding frequency when L1/D ≥ 1.5, however, the presence of the upstream plate produces a significant reduction in the drag experienced by the bluff body. In order to investigate themechanism of drag and vortex shedding frequency reduction, we consider the configuration in which the square cylinder is fitted with both upstream and downstream plates along the horizontal center line. Fig. 7(a)-7(d) shows the influence of dual splitter plates on the flow around a square cylinder. We considered the flow at Reynolds number, Re=150 and varied the length of the downstream plate (0≤L2/D ≤4.0) when the length of the upstream splitter plate is L1/D is taken to be 1.5. The vortex shedding behind the cylinder occurs due to the interaction of shear layers of opposite sign emerging from either sides of the cylinder. The attached downstream plate interferes the fluid entrainment in the shear layers and consequently, influences the vortex shedding. Several authors have already reported on the influence of a downstream splitter plate on vortex shedding, a review of which is already provided in the introduction section. It is found that with the introduction of the downstream plate, the shear layers convect further downstream as compared to a bare cylinder before being rolled up. The increase in the length of the plate causes the shear layers to extend further downstream before they entrain into each other. For a lower range of the downstream plate length, a periodic vortex shedding occurs at Re=150. The downstream splitter plate attached along the centreline produces a symmetry in the wake structure. A tip vortex at the trailing edge of the plate is clearly visible at this Reynolds number. Fig. 7(c) and (d) shows that the wake becomes steady at Re=150 when L2/D is bigger than 2. At a steady state, the interaction between the top and bottom shear layers is suppressed. The shear layer emerging from top and bottom faces of the cylinder transports downstream almost horizontally without forming any vortex in the near wake of the cylinder. It is found that at any given value of Reynolds number (Re≤ 200), there exists a critical value of the downstream splitter plate length for which the vortex shedding is suppressed. This critical value of L2/D depends on the length of the upstream splitter plate (L1/D). Later in this section, we have discussed the suppression of vortex shedding by varying L2/D through computation of the lift coefficient and Strouhal number.

The effect on vortex shedding frequency at Re=150 due to the variation of a downstream plate length at different values of the upstream plate is presented in Fig. 8. The variation of Strouhal number in absence of the upstream plate is also shown. Our computed results show that St reduces with the introduction of the downstream plate. A recent article by Ali et al. (2011) found four distinct regions of variation of St with L2/D. Our results also establish the fact that the dependence of St on L2/D is not uniform. We find that St is further affected when dual splitter plates is considered. It is evident from this result that the dual splitter plates produce a substantial damping effect on vortex shedding frequency.

The knowledge of the parameter values for which the vortex shedding is suppressed is important in several practical contexts. In Fig. 9 we present an estimate of the critical length of the upstream and downstream plates for which the vortex shedding is suppressed at Re=150. The critical plate lengths are the minimum values of the length of the plates for which the lift coefficient becomes independent of time at the Reynolds number. At a given value of the upstream plate length (L1/D), we determine the minimum value of the downstream plate length (L2/D) for which the lift coefficient becomes independent of time for Reynolds number 150. We find that the vortex shedding can be suppressed through a downstream plate alone i.e., L1/D=0. The plate length L2/D=4.5 with L1/D=0 leads to a suppression of vortex shedding at Re=150. Ali et al. (2011) also observed a sudden drop in St for L2/D ≥ 4.5. The length of the downstream plate for the vortex shedding suppression can be decreased by introducing an upstream plate. However, an increase of the upstream plate length beyond 1.5 has practically no influence on the critical length of the downstream plate. We have already noted that an increase in L1/D beyond 1.5 have no influence on the wake flow.

Fig. 7 Instantaneous vorticity contours for different length of the downstream splitter plate (L2/D) at the fixed upstream splitter plate of length L1/D=1.5 and Re=150

The time averaged drag coefficientas well as the pressure dragis presented in Fig. 10 as a function of L2/D for a different choice of L1/D when Re=150. It is evident that the introduction of the downstream plate reduces the drag compared to the unbounded case. The drag further reduces for the attached dual plate configuration. At a fixed value of L1/D, the variation of the time averaged drag with the downstream plate length is not uniform. However, a monotonic reduction in drag due to the increase of upstream plate length at a given value of the downstream plate length is evident. We have seen before in Fig. 6(b) that the increase of the length of the upstream plate in a single plate configuration produces a monotonic reduction in drag. Fig. 8 for vortex shedding frequency and Fig. 10 for drag coefficient may provide an optimal choice for the dual plate configuration to achieve low drag with a low vortex shedding frequency.

Fig. 8 Strouhal number for dual splitter plates configuration at Re=150

Fig. 9 Critical length of splitter plates for suppression of vortex shedding at Re=150

The form of the wake during one vortex shedding cycle is presented in Fig. 11(a)-11(d) for Reynolds number 150 with L1/D=1.5 and L2/D=2.0. The form of the wake is similar to the wake behind a bare cylinder. Compared to a bare cylinder, the shear layers stretched further downstream before being rolled up. The street of vortices is composed of well-defined negative vortices shed from the upper side of the cylinder and positive ones shed from lower side of the cylinder. The formation of a tip vortex at the trailing edge of the downstream plate is evident.

Fig. 10 Time-averaged drag coefficient(solid lines) and time-averaged pressure drag(dashed lines) as a function of the downstream plate length for different values of the upstream plate length (L1/D= 0.0, 0.5, 1.0, 1.5, 2.0) in the dual plates configuration for Re=150

Fig. 11 Vorticity contours during one shedding cycle for dual splitter plates configuration when Re=150, L1/D=1.5, L2/D=2.0

5 Conclusions

A parametric study has been carried out in order to assess the effectiveness of dual attached splitter plates on the reduction of drag and vortex shedding frequency of a square cylinder subjected to an uniform stream. The numerical simulations have been carried out for laminar range of Reynolds number. The form of the wake in presence of an upstream plate as well as dual plates is analyzed. The vortex shedding frequency and drag experienced by the cylinder is determined.

Our computed result shows that the presence of the upstream splitter plate produces an enhancement in the Strouhal number. However, the Strouhal number becomes independent of the upstream plate length for a plate length beyond 1.5. The introduction of the upstream plate reduces the drag of the cylinder significantly and the drag reduces monotonically with the increase of the plate length.

The presence of a downstream splitter plate damps the vortex shedding frequency. Vortex shedding suppression occurs beyond a critical length of the downstream plate. This critical length of the downstream plate can be reduced by introducing an in-line upstream attached plate.

The mechanism of vortex shedding and the form of the wake of a cylinder modified with dual splitter plates is similar to a bare cylinder. Our results show that at Reynolds 150, the presence of dual splitter plates can reduce both the drag and vortex shedding frequency.

Acknowledgement

One of the authors (Bhanuman Barman) wishes to acknowledge IIT Kharagpur for providing the resources to carry out this work.

Ali M, Doolan C, Wheatley V (2011). Low reynolds number flow over a square cylinder with a splitter plate. Physics of Fluids, 23, 033602.

Bhattacharyya S, Dhinakaran S (2008). Vortex shedding in shear flow past tandem square cylinders in the vicinity of a plane wall. Journal of Fluids and Structures, 24(3), 400-417.

DOI： 10.1016/j.jfluidstructs.2007.09.002

Gu F, Wang J, Qiao X, Huang Z (2012). Pressure distribution, fluctuating forces and vortex shedding behavior of circular cylinder with rotatable splitter. Journal of Fluids and Structures, 28, 263-278.

DOI： 10.1016/j.jfluidstructs.2011.11.005

Hwang J, Yang K (2007). Drag reduction on a circular cylinder using dual detached splitter plates. Journal of Wind Engineering and Industrial Aerodynamics, 95(7), 551-564.

DOI： 10.1016/j.jweia.2006.11.003

Hwang J, Yang K, Sun S (2003). Reduction of flow-induced forces on a circular cylinder using a detached splitter plate. Physics of Fluids, 15, 2433.

DOI： http：//dx.doi.org/10.1063/1.1583733

Igbalajobi A, McClean J, Sumner D, Bergstrom D (2013). The effect of a wake-mounted splitter plate on the flow around a surface-mounted finite-height circular cylinder. Journal of Fluids and Structures, 37, 185-200.

DOI： 10.1016/j.jfluidstructs.2012.10.001

Kawai H (1990). Discrete vortex simulation for flow around a circular cylinder with a splitter plate. Journal of Wind Engineering and Industrial Aerodynamics, 33(1), 153-160.

DOI： 10.1016/0167-6105(90)90031-7

Kwon K, Choi H (1996). Control of laminar vortex shedding behind a circular cylinder using splitter plates. Physics of Fluids, 8, 479.

DOI： 10.1063/1.868801

Lin S, Wu T (1994). Flow control simulations around a circular cylinder by a finite volume scheme. Numerical Heat Transfer, Part A Applications, 26(3), 301-319.

DOI： 10.1080/10407789408955994

Nakamura Y (1996). Vortex shedding from bluff bodies with splitter plates. Journal of Fluids and Structures, 10(2), 147-158.

DOI： 10.1006/jfls.1996.0010

Ozono S (1999). Flow control of vortex shedding by a short splitter plate asymmetrically arranged downstream of a cylinder. Physics of Fluids, 11, 2928.

DOI： 10.1063/1.870151

Patankar S (1980). Numerical heat transfer and fluid flow. Hemisphere Publication, USA.

Qiu Y, Sun Y, Wu Y, Tamura Y (2014). Effects of splitter plates and reynolds number on the aerodynamic loads acting on a circular cylinder. Journal of Wind Engineering and Industrial Aerodynamics, 127, 40-50.

DOI： 10.1016/j.jweia.2014.02.003

Roshko A (1955). On the wake and drag of bluff bodies. Journal of Aeronautical Science, 22(2), l955.

DOI： 10.2514/8.3286

Tiwari S, Chakraborty D, Biswas G, Panigrahi P (2005). Numerical prediction of flow and heat transfer in a channel in the presence of a built-in circular tube with and without an integral wake splitter. International Journal of Heat and Mass Transfer, 48(2), 439-453.

DOI： 10.1016/j.ijheatmasstransfer.2004.09.003

Turki S (2008). Numerical simulation of passive control on vortex shedding behind square cylinder using splitter plate. Engineering Applications of Computational Fluid Mechanics, 2(4), 514-524.

DOI： 10.1080/19942060.2008.11015248

Zhou L, Cheng M, Hung K (2005). Suppression of fluid force on a square cylinder by flow control. Journal of Fluids and Structures, 21(2), 151-167.

DOI： 10.1016/j.jfluidstructs.2005.07.002

1671-9433(2015)02-0138-08

10.1063/1.3563619

DOI： 10.1007/s11804-015-1299-5

Received date： 2014-08-19.

Accepted date： 2014-12-22.

*Corresponding author Email： somnath@maths.iitkgp.ernet.in

? Harbin Engineering University and Springer-Verlag Berlin Heidelberg 2015