INTERNATIONAL JOURNAL OF APPLIED ENGINEERING RESEARCH, DINDIGUL Volume 1, No 3, 2010

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1 CFD analysis of 2D unsteady flow around a square cylinder Gera.B, Pavan K. Sharma, Singh R.K Reactor Safety Division, Bhabha Atomic Research Centre, Trombay, Mumbai, India pa1.sharma@gmail.com ABSTRACT Unsteady flow around bluff bodies is an area of great research for scientist for several years. Flows around buildings, chimneys are examples where the fluid is in motion. Atmospheric dispersion of pollutants around bluff bodies has intensified the need to understand wake behavior. The vortex shedding frequency depends on different aspects of the flow field such as the end conditions, blockage ratio of the flow passage. In the present work a numerical simulation was carried out for flow past a square cylinder to see the wake behaviour. A twodimensional unsteady flow past a square cylinder has been investigated numerically for the Reynold number (Re) considered in the range so that flow is laminar. The main objectives of this study were to capture the features of flows past a square cylinder in a domain with the use of CFD. Finite volume method has been used with staggered grid arrangement. The incompressible SIMPLE algorithm was used for the velocity pressure coupling. The second order discretisation was used both for space and time. Power law scheme was used on a nonuniform grid and for time discretisation Crank Nicholson was used. A high resolution grid has been used to avoid spurious oscillations and to keep the errors within limits. The lift coefficient and velocity component in the wake region were monitored for calculation of Strouhal number. The variation of Strouhal number with Reynold number was found from the analysis. Keywords: Vortex Shedding, CFD, Strauhal number, SIMPLE, drag coefficient 1. Introduction Over the last 100 years, the flow around slender cylindrical bluff bodies has been the subject of intense research, mainly owing to the engineering significance of structural design, flowinduced vibration, and acoustic emissions. In recent years, such studies have received a great deal of attention as a result of increasing computer capabilities, improvements in experimental measurement techniques. The vast majority of these investigations have been carried out for the flow around a circular cylinder, whereas, from an engineering point of view, it is also necessary to study flow around other bluff body shapes, such as sharp edged rectangular cross sectional cylinders. Structures that typically have rectangular or near rectangular cross sections include architectural features on buildings, the buildings themselves, beams, fences and occasionally stays and supports in internal and external flow geometries. When these structures are exposed to cross flow the separation takes places from the upper and lower portion of the body. Due to instability the phenomenon of vortex shedding develops known as von Karman Vortex Street. When scaled with the cross stream bluff body dimension and incoming velocity magnitude, the critical Reynold number where the vortex shedding start is of the order of 50 for a zero angle of incidence. In the present work analysis was limited to Reynold number less than 250, below Re less than 250 the flow remains laminar and wake region is two dimensional in nature. In this range the vortex shedding is characterised by a very well defined frequency. The vortex shedding frequency 602

2 and more generally the wake behavior depend on different aspects of the flow field such as the side and end conditions, blockage ratio of the flow passage, upstream velocity and the aspect ratio of the structures. A significant amount of work has been published in many literatures for experimental and numerical study of flow past a square cylinder. Experimental results for Reynolds number <200 for zero angle of incidence are available in Okajima [1], Davis and Moore [2]. Extensive literature is available for numerical investigation on 2D flow around square cylinder is available. Investigation has been carried out for various parameters i.e. effect of Reynolds number, effect of outlet boundary condition, effect of domain extent, effect of grid size, effect of time step, effect of blockage ratio etc. In each of those studies, only some of these aspects are investigated and no investigation has been found which extensively covers all these aspects. The distance from the computational inlet to the cylinder, Xu ranges from 4 (Kelkar and Patankar [3]) to 125 (Stegell and Rockliff [4]). As shown by Sohankar et. al. [5], when using a free stream condition at the inlet, the necessary distance for obtaining results independent of this inlet location is about 10 units. When Xu was increased from 7.5 to 11.1 units, there was a 9.3% decrease in RMS lift (the RMS lift is perhaps the best overall indicator when comparing results in vortex shedding flows). A further increase up to Xu= 18 gave negligible changes in the global results (less than 1%). For all references, a free stream condition U=1 and V=0 is prescribed at the inlet. The effect of blockage was investigated numerically by Stansby and Slaouti [6], Anagnostopoulos et. al. [7], Behr et. al. [8] and Turki et. al. [9]. It is shown that with increasing blockage parameter the Strouhal number and drag coefficient increase, while the base suction and stagnation pressure coefficients increase. At high Reynolds numbers this is also observed experimentally for rectangular cylinders, circular cylinders and flat plates. The influence of domain size, especially the location of the outflow boundary, Xd is investigated by Sohankar et. al. [5] and Behr et. al. [10]. It is shown that if Xd is selected less than 2.5 from the body, then the temporal periodicity of the solution is lost. The minimum value of Xd is found to be 6.5. It was also concluded that reliable results for both types of boundary conditions are obtainable with Xd Sohankar et. al. [5] have investigated the influence of Xd between 3 and 26 using the standard Neumann condition at the outlet (hereafter referred to as the NBC) and Convective condition at the outlet (hereafter referred to as the CBC). The results indicate that in order to obtain results independent of the outlet, Xd must be around 26 for NBC at the outlet. Some refinement studies are carried out by Sohankar et. al. [5] [11] and Franke et al. [12]. Some limited studies with different time steps and near wall resolutions were performed. They conclude that the distance of the first grid point (δ) away from the body has a strong influence on the results. For flow around square cylinders they used δ= A grid refinement study for flow around a square cylinder at Re = 500 was presented by Arnal et al. [13]. They employ three grids ( , and 40645) and report some significant grid dependences. For example, the RMS lift was decreased three times when going from the finest to the coarsest grid. For a square cylinder at zero incidence the effects of time step, distribution of grid points, size of cells adjacent to the body, upstream and downstream 603

3 extents of the calculation domain and blockage were thoroughly investigated by Sohankar et al. [11]. The influence of Reynolds number from 45 to 250 at blockage 5% was also presented. In that study, when using a highly non uniform grid, some recommendations for the required size of the domain, grid distribution, time step and spatial resolution in the nearbody region are provided. These recommendations have directly been used in the present study. The present work is aimed at understanding the wake characteristics in laminar wake of a square cylinder. A numerical simulation approach has been adopted in the present work. The flow configuration comprises of a square cylinder placed normal to free stream in an infinite medium. The Reynolds number based on the cylinder height and the average velocity considered in the numerical study is in the range Governing Equations and Boundary Conditions Flow past a square cylinder has been simulated by solving numerically the unsteady Navier Stokes equations for an incompressible fluid in a two dimensional geometry. The equations for continuity and momentum may be expressed in the dimensionless form as follows: Continuity u v + x y = 0 (1) X momentum 2 2 u p 1 u u + ( uu ) + ( vu ) = + ( + ) t x y x Re 2 2 x y (2) Y momentum 2 2 v p 1 v v + ( uv ) + ( vv ) = + ( + ) t x y y Re 2 2 x y (3) In the above equations, the velocities are non dimensionalised with the average velocity U in at the inlet, all lengths with the obstacle height B and the pressure by ρu in 2. The geometry considered for analysis is schematically shown in Figure 1. The dimensions related to the geometry are height of computational domain (H) =18.0 thus giving a blockage of 5.5%, extent of domain upstream of body (Xu) =8.5; extent of domain downstream of body (Xd) =17. The boundary conditions employed for the present investigation are (Figure 1) at the inlet, a uniform velocity has been prescribed U in =1.0 and V in =0.0. At outflow condition was used which does not affect the flow in the upstream. To this end, the convective boundary conditions have been used. The confining boundaries (top and bottom) were modeled as the slip boundaries. No slip (u=v=0) boundary conditions were used for the velocities on the obstacle surface. 604

4 3. Numerical Details The flow was assumed to be two dimensional and unsteady. An incompressible SIMPLE finite volume code was used with a staggered grid arrangement. Crank Nicholson scheme of second order was used in time and a second order scheme has been used for convective and diffusive terms; the pressure was treated fully implicitly. The time marching calculations were started with the fluid at rest. A constant time step t =0.025 was used for all calculations. During the iterative sequence, convergence was assessed at the end of each iteration on the basis of the residual source criterion, which compares the sum of the absolute residual sources over all the control volumes in the computational field, for each finite volume equation. Figure 1: Computational domain for the flow around a square cylinder Figure 2: Grid used for computation Outside a region from the body which extends 2 units upstream, downstream and sideways, the grid distribution was made uniform with a constant cell size. Downstream of the body, 605

5 was set to In other parts, =0.25 was used. The distance from the cylinder surface to the nearest grid point defines δ. For this study, δ = on the upstream surface, on side surfaces, and on the downstream surface was used. The hyperbolic tangent function was used for stretching the cell sizes between these limits δ and.the number of nodes distributed over one unit length of the cylinder surface was set equal to 25 for all sides of the body. All the computations have been carried out using the grid size of 211X137. The grid used for the computation is shown in figure 2. One of the main difficulties encountered in numerical solution of the Navier Stokes equations is that of boundary conditions, especially the outflow boundaries. This difficulty is due to the fact that the computational domain is bounded whereas the physical domain is unbounded. Thus the computational domain should be truncated from the real domain by using artificial open boundary conditions such as Neumann (NBC) or convective (CBC) boundary conditions. For high accuracy the computational domain must sometimes be very large and this increases CPU times and the cost of computation. Thus proper boundary conditions can reduce the size of the computational domain and decrease the cost. In most numerical studies, especially those which involve vortex shedding, the outlet boundary condition is a very important issue. A suitable outflow boundary should permit the flow to exit the domain with a smooth discharge of vortices, without affecting the flow in the domain near the outlet and near body flow. Incorrect location of outflow boundaries and non suitable boundary conditions might seriously affect the whole flow structure, especially near the body. For finite difference and finite volume discretization the NBC and CBC are the two most popular outlet boundary conditions. As per the recommendation of Sohankar et al. (1998) CBC has been used in present computation. In general the CBC can be written for both U and V as U i t + U c U i x = 0 (4) Where U c is the convective velocity/phase speed (U 1 =U; U 2 =V). The CBC has been tested with both variable and constant (uniform) velocity, with negligible differences in the statistical results. Thus in this study, as recommended by Sohankar et al. (1998), the value of U c was set to U in. The discretized form of this equation was implemented as n 1 n n n U + t = U U ( U U ) N N ( x ) c N N 1 N (5) Where the superscript denotes the time level and the index N refers to the stream wise grid number at the outlet. 4. Results and Discussion Investigation was carried out for a range of Reynolds number from based on the parameters recommended in literature. Strouhal Number (St), total drag coefficient (C D ), pressure drag coefficient (C DP ), RMS lift coefficient (C L ), surface averaged frontal side pressure coefficient (C Pf ), surface averaged top/bottom side pressure coefficient (C Ptb ), base pressure coefficient at centerline (CP bc ) and stagnation pressure coefficient at centerline (C Ps ) were determined from numerical calculation. The results at various Reynolds number are 606

6 given in Table 1. Up to Reynolds Number 50, the flow is steady. Between Reynolds numbers 50 to 55, instability occurs and vortex shedding appears and flow becomes unsteady (Figure 3). Thus our prediction is in good agreement for zero angle of incidence the critical Reynolds number is 52 as reported by Sohankar et al. (1998). Table 1: Effect of Reynolds number Re St C D C DP C L C Pf C Ptb C Pbc C Ps Lift Coefficient Lift Coefficient Lift Coefficient Lift Coefficient Figure 3: Lift Coefficient with time for (a) Re=50 (b) Re=55 (c) Re=100 (d) Re=

7 Figure 4 shows the variation of total drag coefficient with time for Re=55. Initially there is a sharp drop in the drag coefficient then it remains constant at low value followed by a transitional phase leading to fully developed vortex shedding phase. At high Reynolds number this constant low value phase remains for a very short time. Initially there is a drop in the drag coefficient leads to a minimum value of drag coefficient immediately followed by a transition to fully developed phase at Re=100 (Figure 4). Figure 5 depicts the values of C D and St Vs Re for the present work and comparison with other published results. There is a good agreement between the results qualitatively. Quantitatively there is slight difference in absolute values of the parameters that may be attributed to strong sensitivity to various numerical parameters, blockage ratio at this low value of Re. Figure 6 shows the instantaneous vorticity and stream function contour for Re=75. The same for Re=175 has been shown in Figure Conclusion CFD analysis was carried out for unsteady, incompressible 2D flow around a square cylinder at zero angle of incidence for Re ranging from 50 to 250. The vortex shedding starts between Re 50 and 55 if angle of incidence is zero. The vortex shedding is exhibited by a single dominated frequency for Re>55. The study also predicted the influence of Re on quantities such as Strouhal number and lift, drag, and base suction coefficients. The predicted results show good trends with other reported results. Experimental studies on this flow for these low Reynolds numbers are very scarce. Nevertheless, when considering the effects of different blockages, experimental uncertainties and numerical inaccuracies, the agreement seems satisfactory. Accurate measurements, especially at low Re are still needed. In particular, the question of the transitional Reynolds number, i.e. the critical Re above which the flow ceases to be laminar and cannot be made two dimensional, needs further investigation Drag Coefficient Drag Coefficient Figure 4: Drag Coefficient with time for (a) Re=55 (b) Re=

8 C D Present Sohankar et. al Sharma & Eswaran 2004 Davis & Moore 1982 Sohankar et. al Saha et. al Sohankar et. al Re St Present Sohankar et. al Okajima 1982 Davis & Moore 1982 Sohankar et. al Saha et. al Sohankar et. al Re Figure 5: CD and St Vs Re (Present Work and comparison with published results) Figure 6: Instantaneous vorticity contour (a) Re=75 (b) Re= References Figure 7: Instantaneous Stream Function contour (a) Re=75 (b) Re= Okajima, A., Strouhal numbers of rectangular cylinders, Journal Fluid Mechanics vol (123), 1982, pp Davis, R. W.; and Moore, E. F., A numerical study of vortex shedding from rectangles, Journal Fluid Mechanics vol (116), 1982, pp

9 3. Kelkar, K. M.; and Patankar, S. V., Numerical prediction of vortex shedding behind a square cylinder, International Journal Numerical Methods in Fluids vol (14), 1992, p Stegell, N.; and Rockliff, N., Simulation of the effects of body shape on lock in characteristics in pulsating flow by the discrete vortex method, Proc. 3rd Int. Colloq. on Bluff Body Aerodynamics and Applications, VA., Sohankar, A.; Norberg, C.; and Davidson, L., Low Reynolds Number Flow around a Square Cylinder at Incidence: Study of Blockage, Onset of Vortex Shedding and Outlet Boundary Condition, International Journal for Numerical Methods in Fluids vol (26), 1998, pp Stansby, P. K.; Slaouti, A., Simulation of vortex shedding including blockage by the random vortex and other methods, International Journal Numerical Methods in Fluids vol (17), 1993, pp Anagnostopoulos, P.; Illiadis, G.; and Richardson, S., Numerical study of the blockage effects on viscous flow past a circular cylinder, International Journal Numerical Methods in Fluids vol (22), 1996, pp Behr, M.; Hastretier, D.; Mittal, S.; and Tezduyar, T. E., Incompressible Flow past a circular cylinder: dependence of the computed flow field on the location of the lateral boundaries, Comput. Meth. Appl. Mech. Engg., vol (123), 1995, pp Turki, S.; Abbassi, H.; and Nasrallah, S.B., Effect of the blockage ratio on the flow in a channel with a built in square cylinder, Computational Mechanics, vol (33), 2003, pp Behr, M.; Liou, J.; Shih, R.; and Tezduyar, T. E., Vorticity stream function formulation of unsteady incompressible flow past a cylinder: sensitivity of the computed flow field to the location of the outflow boundary, International Journal Numerical Methods in Fluids vol (12), 1991, pp Sohankar, A.; Davidson, L.; and Norberg, C., Numerical simulation of unsteady flow around a square two dimensional cylinder, Proceeding Twelfth Australasian Fluid Mechanics Conference, Sydney, 1995, pp Franke, R.; Rodi, W.; and Scho Ènung, B., Numerical calculation of laminar vortexshedding flow past cylinders, Journal Wind Engineering and Industrial Aerodynamics, vol (35), 1990, pp Arnal, M. P.; Goering, D. J.; and Humphrey, J. A. C., Vortex shedding from a bluff body adjacent to a plane sliding wall, Journal of Fluids Engineering vol (113), 1991, pp