Numerical solutions of 2-D steady incompressible driven cavity ow at high Reynolds numbers

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1 INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN FLUIDS Int. J. Numer. Meth. Fluids 25; 48: Published online March 25 in Wiley InterScience ( DOI:.2/d.953 Numerical solutions of 2-D steady incompressible driven cavity ow at high ynolds numbers E. Erturk ; ;, T. C. Corke 2 and C. Gokcol Energy Systems Engineering Department; Gebze Institute of Technology; Gebze; Kocaeli 44; Turkey 2 Department of Aerospace and Mechanical Engineering; University of Notre Dame; IN 46556; U.S.A. SUMMARY Numerical calculations of the 2-D steady incompressible driven cavity ow are presented. The Navier Stokes equations in streamfunction and vorticity formulation are solved numerically using a ne uniform grid mesh of 6 6. The steady driven cavity ow solutions are computed for 6 2 with a maximum absolute residuals of the governing equations that were less than. A new quaternary vortex at the bottom left corner and a new tertiary vortex at the top left corner of the cavity are observed in the ow eld as the ynolds number increases. Detailed results are presented and comparisons are made with benchmark solutions found in the literature. Copyright? 25 John Wiley & Sons, Ltd. KEY WORDS: steady 2-D incompressible N S equations; driven cavity ow; ne grid solutions; high ynolds numbers. INTRODUCTION Numerical methods for 2-D steady incompressible Navier Stokes (N S) equations are often tested for code validation, on a very well known benchmark problem; the lid-driven cavity ow. Due to the simplicity of the cavity geometry, applying a numerical method on this ow problem in terms of coding is quite easy and straight forward. Despite its simple geometry, the driven cavity ow retains a rich uid ow physics manifested by multiple counter rotating recirculating regions on the corners of the cavity depending on the ynolds number. In the literature, it is possible to nd dierent numerical approaches which have been applied to the driven cavity ow problem [ 34]. Though this ow problem has been numerically studied extensively, still there are some points which are not agreed upon. For example; Correspondence to: E. Erturk, Energy Systems Engineering Department, Gebze Institute of Technology, Gebze, Kocaeli 44, Turkey. ercanerturk@gyte.edu.tr, Contract=grant sponsor: Gebze Institute of Technology; contract=grant number: BAP-23-A-22 ceived 9 October 23 vised 26 July 24 Copyright? 25 John Wiley & Sons, Ltd. Accepted 3 January 25

2 748 E. ERTURK, T. C. CORKE AND C. G OKC OL () an interesting point among many studies is that dierent numerical solutions of cavity ow yield about the same results for 6 however, start to deviate from each other for larger, (2) also another interesting point is that while some studies predict a periodic ow at a high ynolds number, some others present steady solutions for even a higher ynolds number. The objective of this work is then to investigate these two points and present accurate very ne grid numerical solutions of steady 2-D driven cavity ow. Among the numerous studies found in the literature, we will give a brief survey about some of the signicant studies that uses dierent types of numerical methods on the driven cavity ow. In doing so the emphasis will be given on three points; on the numerical method used, on the spatial order of the numerical solution and on the largest ynolds number achieved. cently, Barragy and Carey [2] have used a p-type nite element scheme on a strongly graded and rened element mesh. They have obtained a highly accurate (h 8 order) solutions for steady cavity ow solutions up to ynolds numbers of = 2 5. Although Barragy and Carey [2] have presented qualitative solution for = 6 they concluded that their solution for = 6 was under-resolved and needed a greater mesh size. Botella and Peyret [6] have used a Chebyshev collocation method for the solution of the liddriven cavity ow. They have used a subtraction method of the leading terms of the asymptotic expansion of the solution of the N S equations in the vicinity of the corners, where velocity is discontinuous, and obtained a highly accurate spectral solutions for the cavity ow with a maximum of grid mesh of N = 6 (polynomial degree) for ynolds numbers 6 9. They stated that their numerical solutions exhibit a periodic behaviour beyond this. Schreiber and Keller [26] have introduced an ecient numerical technique for steady viscous incompressible ows. The non-linear dierential equations are solved by a sequence of Newton and chord iterations. The linear systems associated with the Newton iteration is solved by LU-factorization with partial pivoting. Applying repeated Richardson extrapolation using the solutions obtained on dierent grid mesh sizes (maximum being 8 8), they have presented high-order accurate (h 8 order in theory) solutions for ynolds numbers 6. Benjamin and Denny [5] have used a method of relaxing the algebraic equations by means of the ADI method, with a non-uniform iteration parameter. They have solved the cavity ow for 6 with three dierent grid mesh sizes (maximum being ) and used a h n extrapolation to obtain the values when h. Wright and Gaskell [34] have applied the block implicit multigrid method (BIMM) to the SMART and QUICK discretizations. They have presented cavity ow results obtained on a grid mesh for 6. Nishida and Satofuka [22] have presented a new higher order method for simulation of the driven cavity ow. They have discretized the spatial derivatives of the N S equations using a modied dierential quadrature (MDQ) method. They have integrated the resulting system of ODEs in time with fourth order Runge Kutta Gill (RKG) scheme. With this they have presented spatially h order accurate solutions with grid size of for Liao [9], and Liao and Zhu [2], have used a higher order streamfunction vorticity boundary element method (BEM) formulation for the solution of N S equations. With this they have presented solutions up to = with a grid mesh of Hou et al. [7] have used lattice Boltzmann method for simulation of the cavity ow. They have used grid points and presented solutions up to ynolds numbers of = 75. Copyright? 25 John Wiley & Sons, Ltd. Int. J. Numer. Meth. Fluids 25; 48:

3 DRIVEN CAVITY FLOW AT HIGH REYNOLDS NUMBERS 749 Goyon [3] have solved the streamfunction and vorticity equations using Incremental Unknowns. They have presented steady solutions for 6 75 on a maximum grid size of Rubin and Khosla [25] have used the strongly implicit numerical method with 2 2 coupled streamfunction vorticity form of the N S equations. They have obtained solutions for ynolds numbers in the range of 6 3 with a grid mesh of 7 7. Ghia et al. [2] later have applied a multi-grid strategy to the coupled strongly implicit method developed by Rubin and Khosla [25]. They have presented solutions for ynolds numbers as high as = with meshes consisting of as many as grid points. Gupta [5] has used a fourth (h 4 ) order compact scheme for the numerical solution of the driven cavity ow. He has used a 9 point (3 3) stencil in which the streamfunction and vorticity equations are approximated to fourth order accuracy. He has used point-sor type of iteration and presented steady cavity ow solutions for 6 2 with a maximum of 4 4 grid mesh. Li et al. [8] have used a fourth (h 4 ) order compact scheme which had a faster convergence than that of Gupta [5]. They have solved the cavity ow with a grid size of for Bruneau and Jouron [7] have solved the N S equations in primitive variables using a full multigrid-full approximation storage (FMG-FAS) method. With a grid size , they have obtained steady solutions for 6 5. Grigoriev and Dargush [4] have presented a BEM solution with improved penalty function technique using hexagonal subregions and they have discretized the integral equation for each subregion as in FEM. They have used a non-uniform mesh of 54 quadrilateral cells. With this they were able to solve driven cavity ow up to = 5. Aydin and Fenner [] have used BEM formulation with using central and upwind nite dierence scheme for the convective terms. They have stated that their formulation lost its reliability for ynolds numbers greater than. To the authors best knowledge, in all these studies among with other numerous papers found in the literature, the maximum ynolds number achieved for the 2-D steady incompressible ow in a lid-driven cavity is = 2 5 and is reported by Barragy and Carey [2]. Although Barragy and Carey [2] have presented solutions for = 6, their solution for this ynolds number display oscillations related to boundary layer resolution issues due to a coarse mesh. Nallasamy and Prasad [2] have presented steady solutions for ynolds numbers up to 6 5, however, their solutions are believed to be inaccurate as a result of excessive numerical dissipation caused by their rst order upwind dierence scheme. A very brief discussion on computational as well as experimental studies on the lid-driven cavity ow can be found in Shankar and Deshpande [27]. Many factors aect the accuracy of a numerical solution, such as, the number of grids in the computational mesh (h), and the spatial discretization order of the nite dierence equations, and also the boundary conditions used in the solution. It is an obvious statement that as the number of grids in a mesh is increased (smaller h) a numerical solution gets more accurate. In this study the eect of number of grid points in a mesh on the accuracy of the numerical solution of driven cavity ow is investigated, especially as the ynolds number increases. For this, the governing N S equations are solved on progressively increasing number of grid points (from to 6 6) and the solutions are compared with the highly accurate benchmark solutions found in the literature. Copyright? 25 John Wiley & Sons, Ltd. Int. J. Numer. Meth. Fluids 25; 48:

4 75 E. ERTURK, T. C. CORKE AND C. G OKC OL Order of spatial discretization is another factor on the accuracy of a numerical solution. Finite element and spectral methods provide high spatial accuracy. This study also investigates the eect of spacial accuracy on the numerical solution of driven cavity ow. For this, following [26], repeated Richardson extrapolation is used and then the extrapolated results are compared with the benchmark solutions. Another factor on the accuracy of a numerical solution is the type and the order of the numerical boundary conditions used in computation. For driven cavity ow, this subject was briey discussed by Weinan and Jian-Guo [32], Spotz [28], Napolitano et al. [35] and also by Gupta and Manohar [6]. Until Barragy and Carey [2], many studies have presented steady solutions of driven cavity ow for = [5, 2, 2, 26]. They [2] have presented solutions for = 2 5. In all these studies that have presented high ynolds number numerical solutions, the largest grid size used was In this study, the grid size will be increased up to 6 6 and the eect of the grid size on the largest computable ynolds number solution of the driven cavity ow will be investigated. This paper presents accurate, very ne grid (6 6) numerical solutions of 2-D steady incompressible ow in a lid-driven cavity for ynolds numbers up to = 2. A detailed comparisons of our results with mainly the studies mentioned above as well as with other studies not mentioned here, will be done. 2. NUMERICAL METHOD For two-dimensional and axi-symmetric ows it is convenient to use the streamfunction ( ) and vorticity (!) formulation of the Navier Stokes equations. In non-dimensional form, they are given ) =! where, is the ynolds number, and x and y are the Cartesian coordinates. The N S equations are nonlinear, and therefore need to be solved in an iterative manner. In order to have an iterative numerical algorithm, pseudo time derivatives are assigned to these Equations () and (2) Since we are seeking a steady solution, the order of the pseudotime derivatives will not aect the nal solution. Therefore, an implicit Euler time step is used for these pseudotime derivatives, which is rst order (t) accurate. Also for the non-linear terms in the vorticity Copyright? 25 John Wiley & Sons, Ltd. Int. J. Numer. Meth. Fluids 25; (4)

5 DRIVEN CAVITY FLOW AT HIGH REYNOLDS NUMBERS 75 equation, a rst order (t) accurate approximation is used and the governing Equations (3) and (4) are written as follows: n+ 2 n+ = n +t! n 2! n n n =! n In operator notation the above equations look like the following. t 2 2 ) n+ = n +t! n 2 t 2 2 ( ) n ( ) n+ =! The above equations are in fully implicit form where each equation requires the solution of a large banded matrix which is not computationally ecient. Instead of solving the Equations (7) and (8) in fully implicit form, more eciently, we spatially factorize the equations, such as ( t ( 2 ( 2 )( 2 )( t ) n+ = n +t! n 2 ( ) 2 n+ =! Here, we split the single LHS operators into two operators, where each only contains derivatives in one direction. The only dierence between Equations (7) and (8) and (9) and () are the t 2 terms produced by the factorization. The advantage of this process is that each equation now requires the solution of a tridiagonal system, which is numerically more ecient. As the solution converges to the steady state solution, the t 2 terms due to the factorization will not become zero and will remain in the solution. To illustrate this, Equation (9) can be written in explicit form as n+ n+ 2 t = n +t! n () 2 As the solution converges to a steady state, n+ becomes equal to n and they cancel each other from the above equation and at convergence, Equation () therefore appears as the +! = In Equation (2), there is no guarantee that the RHS, which is the consequence of the factorization, will be small. Only a very small time step (t) will ensure that the RHS of Equation (2) will be close to zero. This, however, will slow down the convergence. In addition, the product may not be small 4 =@x 2 can be O[=t] or higher in many ow situations. Copyright? 25 John Wiley & Sons, Ltd. Int. J. Numer. Meth. Fluids 25; 48:

6 752 E. ERTURK, T. C. CORKE AND C. G OKC OL In order to overcome this problem, we add a similar term at the previous time level to the RHS of Equation () to give the following nite dierence form: n+ t n+ @x 2 t = n +t! n +t 4 n (3) 2 As a result, at steady state, n+ converges to n 4 n+ =@x 2 converges 4 n =@x 2, so that they cancel each out, and the nal solution converges to the correct physical 2 +! = (4) We would like to emphasize that the added t 2 term on the RHS is not an articial numerical diusion term. Such terms are often used to stabilize the numerical schemes so that convergence can be achieved. Here, the added t 2 term on the RHS is not meant to stabilize the numerical scheme. The only reason for this approach is to cancel out the terms introduced by factorization so that the equations are the correct physical representation at the steady state. One of the most popular spatially factorized schemes is undoubtedly the Beam and Warming [4] method. In that method, the equations are formulated in delta () form. The main advantage of a delta formulation in a steady problem is that there will be no second order (t 2 ) terms due to the factorization in the solution at the steady state. However in our formulation the second order (t 2 ) terms due to factorization will remain even at the steady state. Adding a second order (t 2 ) term to the RHS of the equations to cancel out the terms produced by factorization, easily overcomes this problem. One of the main dierences between the presented formulation and a delta formulation is in the numerical treatment of the vorticity equation. In a delta formulation of the vorticity equation, there appears t order convection and dissipation terms on the RHS (explicit side) of the numerical equations. In the presented formulation, on the RHS of the equation there are no t order terms. Instead cross derivative terms (@ 4 =@x =@x@y 2, 2 =@x@y) appear on the RHS. Note that these terms are t 2 order. Since t is small, these t 2 terms are even smaller. Our extensive numerical tests showed that this approach has better numerical stability characteristics and is more eective especially at high ynolds numbers compared to a delta formulation of the steady streamfunction and vorticity equations. Applying the approach of removing second order terms due to factorization on the RHS, to both Equations (9) and (), the nal form of the numerical formulation we used for the solution of streamfunction and vorticity equation becomes ( t ( 2 2 t ( =! n ( 2 ( n+ = n +t! n + t )( 2 ( ) )( ( @y n (5) )! 2 ( ) n Copyright? 25 John Wiley & Sons, Ltd. Int. J. Numer. Meth. Fluids 25; 48:

7 DRIVEN CAVITY FLOW AT HIGH REYNOLDS NUMBERS 753 The solution methodology for the two equations involves a two-level updating. First the streamfunction equation is solved. For Equation (5), the variable f is introduced such that ) ( n+ = f 2 and t 2 ) ( f = n +t! + t 2 t 2 ) n 2 In Equation (8) f is the only unknown. First, this equation is solved for f at each grid point. Following this, the streamfunction variable ( ) is advanced into the new time level using Equation (7). Next, the vorticity equation is solved. In a similar fashion for Equation (6), the variable g is introduced such that ( t and ( 2 @ 2 ( ) n g =! n + ) n n+ = g ( 2 ( ) 2 ( ) n As with f, g is determined at every grid point using Equation (2), then vorticity variable (!) is advanced into the next time level using Equation (9). We note that, in this numerical approach the streamfunction and vorticity equations are solved separately. Each equation is advanced into a next time level by solving two tridiagonal systems, which allows the use of very large grid sizes easily. The method proved to be very eective on ow problems that require high accuracy on very ne grid meshes [9, ]. 3. RESULTS AND DISCUSSION The boundary conditions and a schematics of the vortices generated in a driven cavity ow are shown in Figure. In this gure, the abbreviations BR, BL and TL refer to bottom right, bottom left and top left corners of the cavity, respectively. The number following these abbreviations refer to the vortices that appear in the ow, which are numbered according to size. We have used the well-known Thom s formula [3] for the wall boundary condition, such that =! = 2 h 2 2U h (2) Copyright? 25 John Wiley & Sons, Ltd. Int. J. Numer. Meth. Fluids 25; 48:

8 754 E. ERTURK, T. C. CORKE AND C. G OKC OL u= v= TL2 TL u= v= Primary Vortex u= v= BL3 BL BL2 u= v= BR BR2 BR3 Figure. Schematic view of driven cavity ow. where subscript refers to points on the wall and refers to points adjacent to the wall, h refers to grid spacing and U refers to the velocity of the wall with being equal to on the moving wall and on the stationary walls. We note that, it is well understood [28, 32, 35, 36] that, even though Thom s method is locally rst order accurate, the global solution obtained using Thom s method preserves second order accuracy. Therefore in this study, since three point second order central dierence is used inside the cavity and Thom s method is used at the wall boundary conditions, the presented solutions are second order accurate. During our computations we monitored the residual of the steady streamfunction and vorticity Equations () and (2) as a measure of the convergence to the steady state solution, where the residual of each equation is given as R = n+ i ;j 2 i; n+ j + i+;j n+ + x 2 n+ i; j 2 i; n+ j + i; n+ j+ +! n+ y 2 i; j (22) R! =! i ;j n+ 2!n+ i; j +! i+;j n+ +! n+ x 2 i; j 2!n+ i; j +! n+ y 2 i; j+ n+ i; j+ i; n+ j 2y! i+;j n+!n+ i ;j + 2x n+ i+;j i ;j n+ 2x! n+ i; j+!n+ i; j 2y (23) The magnitude of these residuals is an indication of the degree to which the solution has converged to steady state. In the limit these residuals would be zero. In our computations, for all ynolds numbers, we considered that convergence was achieved when for each Equations (22) and (23) the maximum of the absolute residual in the computational domain (max( R ) Copyright? 25 John Wiley & Sons, Ltd. Int. J. Numer. Meth. Fluids 25; 48:

9 DRIVEN CAVITY FLOW AT HIGH REYNOLDS NUMBERS 755 Table I. Comparison of the properties of the primary vortex; the maximum streamfunction value, the vorticity value and the location of the centre, for =. Spatial ference No Grid accuracy! x y Present 4 4 h Present h Present 6 6 h Present Extrapolated h Barragy and Carey ference [2] p = Botella and Peyret ference [6] N = 28 N = Botella and Peyret ference [6] N = 6 N = Schreiber and Keller ference [26] h Schreiber and Keller ference [26] 2 2 h Schreiber and Keller ference [26] 4 4 h Schreiber and Keller ference [26] Extrapolated h Wright and Gaskell ference [34] h Nishida and Satofuka ference [22] h Benjamin and Denny ference [5] h Benjamin and Denny ference [5] Extrapolated High order Li et al. ference [8] h Ghia et al. ference [2] h Bruneau and Jouron ference [7] h Goyon ference [3] h Vanka ference [3] h Gupta ference [5] 4 4 h Hou et al. ference [7] h Liao and Zhu ference [2] h Grigoriev and Dargush ference [4] and max( R! )) was less than. Such a low value was chosen to ensure the accuracy of the solution. At these residual levels, the maximum absolute dierence in streamfunction value between two time steps, (max( n+ n )), was in the order of 6 and for vorticity, (max(! n+! n )), it was in the order of 4. And also at these convergence levels, between two time steps the maximum absolute normalized dierence in streamfunction, (max( ( n+ n )= n )), and in vorticity, (max( (! n+! n )=! n )), were in the order of 3, and 2 respectively. We have started our computations by solving the driven cavity ow from R = to = 5 on a grid mesh. With using this many number of grids, we could not get a steady solution for = 75. The rst natural conclusion was that the cavity ow may not be steady at = 75, and therefore steady solution may not be computable. Even though we have used a pseudo-time iteration, the time evolution of the ow parameters, either vorticity or streamfunction values at certain locations, were periodic suggesting that the ow indeed may be periodic, backing up this conclusion. Besides, in the literature many studies claim that the driven cavity ow becomes unstable between a ynolds number of 7 and 8, for example, Peng et al. [23], Fortin et al. [] and Poliashenko and Aidun [24] predict unstable ow beyond of 742, and 7763, respectively. However, on the Copyright? 25 John Wiley & Sons, Ltd. Int. J. Numer. Meth. Fluids 25; 48:

10 756 E. ERTURK, T. C. CORKE AND C. G OKC OL Table II. Richardson extrapolation for streamfunction values at the primary vortex. Grid O(h 2 ) O(h 4 ) O(h 6 ) other hand there were many studies [2, 5, 2, 2, 26] that have presented solutions for a higher ynolds number, =. This was contradictory. We then have tried to solve the same case, = 75, with a larger grid size of This time we were able to obtain a steady solution. In fact, with number of grids, we carry our steady computations up to = 2 5. Barragy and Carey [2] have also presented steady solutions for = 2 5. Interesting point was, when we tried to increase furthermore with the same grid mesh, we have again obtained a periodic behaviour. Once again the natural conclusion was that the steady solutions beyond this may not be computable. Having concluded this, we could not help but ask the question what happens if we increase the number of grids furthermore. In their study Barragy and Carey [2] have presented a steady solution for = 6. Their solution for this ynolds number displayed oscillations. They concluded that this was due to a coarse mesh. This conclusion encouraged us to use larger grid size than in Copyright? 25 John Wiley & Sons, Ltd. Int. J. Numer. Meth. Fluids 25; 48:

11 DRIVEN CAVITY FLOW AT HIGH REYNOLDS NUMBERS 757 Table III. Richardson extrapolation for vorticity values at the primary vortex. Grid!O(h 2 )!O(h 4 )!O(h 6 ) order to obtain solutions for ynolds numbers greater than 2 5. At this point, since the focus was on the number of grids, we realized that among the studies that presented steady solutions at very high ynolds numbers [2, 5, 2, 2, 26], the maximum number of grids used was Larger grid sizes have not been used at these high ynolds numbers. This was also encouraging us to use more grid points. We, then, decided to increase the number of grids and tried to solve for = 5 with 4 4 number of grids. This time we were able to obtain a steady solution. This fact suggests that in order to obtain a numerical solution for 2-D steady incompressible driven cavity ow for 2 5, a grid mesh with more than grid points is needed. With using 4 4 grid points, we continued our steady computations up to = 2. Beyond this our computations again displayed a periodic behaviour. This time the rst natural thing came to mind was to increase the Copyright? 25 John Wiley & Sons, Ltd. Int. J. Numer. Meth. Fluids 25; 48:

12 758 E. ERTURK, T. C. CORKE AND C. G OKC OL.75 Y X.5 Eddies BL,BL Eddies BR, BR Figure 2. Streamline contours of primary and secondary vortices, =. number of grids. We have tried to use grids, however again we could not get a steady solution beyond = 2. Thinking that the increase in number of grids may not be enough, we then again increased the number of grids and have used 6 6 grids. The situation was the same, and the maximum ynolds number that we can obtain a steady solution with using 6 6 grids was = 2. We did not try to increase the number of grids furthermore since the computations became time consuming. Whether or not steady Copyright? 25 John Wiley & Sons, Ltd. Int. J. Numer. Meth. Fluids 25; 48:

13 DRIVEN CAVITY FLOW AT HIGH REYNOLDS NUMBERS Y Eddy TL X Eddies BL, BL Eddies BR, BR2, BR Figure 3. Streamline contours of primary and secondary vortices, = 5. computations of driven cavity ow are possible beyond = 2 with grid numbers larger than 6 6, is still an open question. One of the reasons, why the steady solutions of the driven cavity ow at very high ynolds numbers become computable when ner grids are used, may be the fact that as the number of grids used increases, h gets smaller, then the cell ynolds number or so called Peclet number dened as c = uh= decreases. This improves the numerical stability characteristics Copyright? 25 John Wiley & Sons, Ltd. Int. J. Numer. Meth. Fluids 25; 48:

14 76 E. ERTURK, T. C. CORKE AND C. G OKC OL Eddy TL X Eddies BL, BL2, BL3 Eddies BR, BR2, BR Figure 4. Streamline contours of primary and secondary vortices, =. of the numerical scheme used [29, 32], and allows high cavity ynolds numbered solutions computable. Another reason may be that fact that ner grids would resolve the corner vortices better. This would, then, help decrease any numerical oscillations that might occur at the corners of the cavity during iterations. We note that all the gures and tables present the solution of the nest grid size of 6 6 unless otherwise stated. Copyright? 25 John Wiley & Sons, Ltd. Int. J. Numer. Meth. Fluids 25; 48:

15 DRIVEN CAVITY FLOW AT HIGH REYNOLDS NUMBERS Y Eddies TL, TL X.5 Eddies BL, BL2, BL3.45 Eddies BR, BR2, BR Figure 5. Streamline contours of primary and secondary vortices, = 5. The accuracy of a nite dierence solution is set by the mesh size, and by the spatial order of the nite dierence equations and the boundary approximations. At low ynolds numbers ( = ), dierent numerical method solutions found in the literature agree with each other. However, the solutions at higher ynolds numbers ( ) have noticeable discrepancies. We believe it is mainly due to dierent spatial orders and dierent grid mesh sizes and dierent boundary conditions used in dierent studies. Copyright? 25 John Wiley & Sons, Ltd. Int. J. Numer. Meth. Fluids 25; 48:

16 762 E. ERTURK, T. C. CORKE AND C. G OKC OL Y X Eddies BL, BL2, BL Eddies BR, BR2, BR Figure 6. Streamline contours of primary and secondary vortices, = 2. Table I tabulates the minimum streamfunction value, the vorticity value at the centre of the primary vortex and also the centre location of the primary vortex for = along with similar results found in the literature with the most signicant ones are underlined. In Table I among the most signicant (underlined) results, Schreiber and Keller [26] have used Richardson extrapolation in order to achieve high spatial accuracy. For = their extrapolated solutions are h 6 order accurate and these solutions are obtained by using repeated Richardson extrapolation on three dierent mesh size solutions. Copyright? 25 John Wiley & Sons, Ltd. Int. J. Numer. Meth. Fluids 25; 48:

17 DRIVEN CAVITY FLOW AT HIGH REYNOLDS NUMBERS 763 Table IV. Comparison of the minimum streamfunction value, the vorticity value at the centre of the primary vortex for various ynolds numbers Present ! Barragy and Carey ference [2] ! Schreiber and Keller ference [26].2292!.9263 Li et al. ference [8] ! Benjamin and Denny ference [5].22! Ghia et al. ference [2] ! Liao and Zhu ference [2] ! Hou et al. ference [7].24.27! Grigoriev and Dargush ference [4].229! Following the same approach, we have solved the cavity ow on three dierent grid mesh (4 4, and 6 6) separately for every ynolds number ( = 2 ). Since the numerical solution is second order accurate, the computed values of streamfunction and vorticity have an asymptotic error expansion of the form (x i ;y j )= i; j + c h 2 + c 2 h 4 +!(x i ;y j )=! i; j + d h 2 + d 2 h 4 + (24) We can use Richardson extrapolation in order to obtain high-order accurate approximations [26]. We have used repeated Richardson extrapolations using the three dierent mesh size solutions. The extrapolated values of the streamfunction and the vorticity at the centre of the primary vortex are given in Tables II and III, respectively. The extrapolated results tabulated in Tables II and III are, in theory, h 6 order accurate. Looking back to Table I, for =, our extrapolated results are in very good agreement with the extrapolated results of Schreiber and Keller [26]. In fact our extrapolated streamfunction value at the primary vortex diers only.3% and the vorticity value diers only.95% from the extrapolated results of Schreiber and Keller [26]. With spectral methods, very high spatial accuracy can be obtained with a relatively smaller number of grid points. For this reason, the tabulated results from Botella and Peyret [6] are believed to be very accurate. Our results are also in very good agreement such that our extrapolated results diers.6% in the streamfunction value and.98% in the vorticity value from the results of Botella and Peyret [6]. Copyright? 25 John Wiley & Sons, Ltd. Int. J. Numer. Meth. Fluids 25; 48:

18 764 E. ERTURK, T. C. CORKE AND C. G OKC OL Table V. Properties of primary and secondary vortices; streamfunction and vorticity values, (x; y) locations Primary vortex! (x; y) (:53; :565) (:52; :5433) (:55; :535) (:533; :537) (:57; :53) (:57; :5283) (:5; :5283) (:5; :5267) (:5; :5267) (:5; :5267).728E E 2.364E 2.322E E E E E E E 2 BR! (x; y) (:8633; :7) (:835; :97) (:85; :733) (:79; :65) (:7767; :6) (:767; :55) (:7483; :5) (:7367; :467) (:7267; :45) (:727; :433).2326E E E 2.57E 2.593E E E 2.633E 2.632E E 2 BL! (x; y) (:833; :783) (:85; :) (:733; :367) (:65; :57) (:583; :633) (:55; :683) (:533; :77) (:57; :75) (:483; :87) (:483; :833).54962E E 6.4E E E E E E E E 3 BR2!.7776E E.348E (x; y) (:997; :67) (:99; :) (:9783; :83) (:957; :47) (:935; :667) (:9283; :87) (:9267; :883) (:9283; :967) (:93; :33) (:937; :83).8422E E 7.728E 7.222E 6.5E E 5.272E E E E 4 BL2!.2982E 2.938E E.6862E.398E.68557E (x; y) (:5; :5) (:67; :67) (:83; :83) (:7; :7) (:67; :2) (:25; :37) (:367; :47) (:483; :483) (:567; :533) (:6; :55).34455E 3.446E 2.29E E E 2.32E E 2.365E E 2 TL! (x; y) (:433; :89) (:633; :9) (:667; :933) (:77; :97) (:75; :97) (:767; :97) (:8; :933) (:87; :933) (:87; :933).2562E 9.449E E E E E 7.296E E 7 BR3!.7828E 3.228E E E E E E E 2 (x; y) (:9983; :7) (:9967; :33) (:9967; :5) (:995; :5) (:995; :5) (:995; :67) (:9933; :67) (:9933; :83).4382E 9.593E 9.28E E E E 8 BL3!.78229E E 3.824E E 2.844E 2.728E 2 (x; y) (:7; :7) (:7; :7) (:7; :33) (:33; :33) (:33; :33) (:33; :33).2752E E E E E 4 TL2! (x; y) (:67; :837) (:5; :8267) (:2; :8233) (:233; :82) (:25; :82) Copyright? 25 John Wiley & Sons, Ltd. Int. J. Numer. Meth. Fluids 25; 48:

19 DRIVEN CAVITY FLOW AT HIGH REYNOLDS NUMBERS 765 Barragy and Carey [2] have only tabulated the location of the vortex centres and the streamfunction values at these locations yet they have not presented any vorticity data. Nevertheless, their solutions based on p-type nite element scheme also have very high spatial accuracy (h 8 order accurate). They have presented the maximum ynolds number solutions ( = 2 5) for the cavity ow, to the best of our knowledge, found in the literature. We will show that the agreement between our results and results of Barragy and Carey [2] is excellent through the whole ynolds number range they considered ( ). Comparing solutions for = in Table I, the dierence is.22% in the streamfunction value. Wright and Gaskell [34] have used a very ne grid mesh with grid points in their solutions. With this many grids their solution should be quite accurate. Their streamfunction and vorticity value are very close to our solutions with a dierence of.3 and.4%, respectively. Nishida and Satofuka [22] have presented higher order solutions in which for = their solutions are h 8 order accurate. Comparing with our results we nd a very good agreement such that our extrapolated results diers.4% in streamfunction value and.36% in vorticity value. Grigoriev and Dargush [4] have solved the governing equations with a boundary element method (BEM). They have only tabulated the streamfunction values and the location of the centres of the primary and secondary vortices for driven cavity ow for ynolds numbers up to 5. Their solution for = is quite accurate. Benjamin and Denny [5] have used a h n extrapolation to obtain the values when h using the three dierent grid mesh size solutions. Their extrapolated results are close to other underlined data in Table I. It is quite remarkable that even a simple extrapolation produces such good results. Hou et al. [7] have simulated the cavity ow by the lattice Boltzmann method. They have presented solutions for 6 75 obtained by using lattice. However their results have signicant oscillations in the two upper corners of the cavity. Especially their vorticity contours have wiggles aligned with lattice directions, for. Li et al. [8] have used a compact scheme and presented fourth order accurate (h 4 ) results for a grid mesh. Ghia et al. [2] have used a second order method (h 2 ) with a mesh size of These results, together with that of Goyon [3] and Liao and Zhu [2], might be considered as somewhat under-resolved. Our calculations showed that for a second order (h 2 ) spatial accuracy, even a 4 4 grid mesh solutions can be considered as under-resolved for =. The results of Bruneau and Jouron [7] and Vanka [3] appear over-diusive due to upwind dierencing they have used. From all these comparisons we can conclude that even for = higher order approximations together with the use of ne grids are necessary for accuracy. Figures 2 6 show streamfunction contours of the nest (6 6) grid solutions, for various ynolds numbers. These gures exhibit the formation of the counter-rotating secondary vortices which appear as the ynolds number increases. Table IV compares our computed streamfunction and vorticity values at the centre of the primary vortex with the results found in the literature for various ynolds numbers. The agreement between our Richardson extrapolated results (h 6 order accurate) with results of Barragy and Carey [2] based on p-type nite element scheme (h 8 order accurate) is quite remarkable. One interesting result is that the streamfunction value at the centre of the Copyright? 25 John Wiley & Sons, Ltd. Int. J. Numer. Meth. Fluids 25; 48:

20 766 E. ERTURK, T. C. CORKE AND C. G OKC OL =2 =2 =75 =5 =25 = =75 =5 =25 =.8.6 y.4.2 u-velocity Computed Ghia et. al.f.[2] Figure 7. Computed u-velocity proles along a vertical line passing through the geometric centre of the cavity at various ynolds numbers. primary vortex decreases at rst as the ynolds number increases until =, however then starts to increase as the ynolds number increases further. This behaviour was also documented by Barragy and Carey [2]. In terms of quantitative results, Gupta and Manohar [6], stated that for the numerical solutions of driven cavity ow, the value of the streamfunction ( ) at the primary vortex centre appears to be a reliable indicator of accuracy, where as the location of the vortex centre (x and y) is limited by the mesh size, and the value of the vorticity (!) at the vortex centre is sensitive to the accuracy of the wall boundary conditions. Nevertheless, the locations of the primary and secondary vortices, as well as the values of the streamfunction ( ) and vorticity (!) at these locations are presented in Table V. These are comparable with the solutions of Barragy and Carey [2] and Ghia et al. [2]. It is evident that as the ynolds number increases, the centre of the primary vortex moves towards the geometric centre of the cavity. Its location is almost invariant for 7 5. Our computations indicate the appearance of a quaternary vortex at the bottom left corner (BL3) at ynolds number of. We would like to note that, this quaternary vortex (BL3) did not appear in our solutions for = with grid sizes less than This would suggest that, especially at high ynolds numbers, in order to resolve the small vortices appear at the corners properly, very ne grids have to be used. Moreover, our solutions indicate another tertiary vortex at the top left side of the cavity (TL2) at = 2 5. These vortices (BL3 and TL2) were also observed by Barragy and Carey [2]. Figures 7 and 8 present the u-velocity proles along a vertical line and the v-velocity proles along a horizontal line passing through the geometric centre of the cavity respectively and also Tables VI and VII tabulates u- and v-velocity values at certain locations respectively, for future references. These proles are in good agreement with that of Ghia et al. [2] shown by symbols in Figures 7 and 8. We have solved the incompressible ow in a driven cavity numerically. A very good mathematical check on the accuracy of the numerical solution would be to check the continuity of the uid, as suggested by Aydin and Fenner [], although this is done very rarely in driven cavity ow papers. We have integrated the u-velocity and v-velocity proles, considered in Figures 7 and 8, along a vertical line and horizontal line passing through the geometric centre of the cavity, in order to obtain the net volumetric ow rate, Q, through these sections. Since Copyright? 25 John Wiley & Sons, Ltd. Int. J. Numer. Meth. Fluids 25; 48:

21 DRIVEN CAVITY FLOW AT HIGH REYNOLDS NUMBERS =2 =2 =75 =5 =25 v-velocity = =75 =5 =25 = x Computed Ghia et. al.f. [2] Figure 8. Computed v-velocity proles along a horizontal line passing through the geometric centre of the cavity at various ynolds numbers. Copyright? 25 John Wiley & Sons, Ltd. Int. J. Numer. Meth. Fluids 25; 48:

22 768 E. ERTURK, T. C. CORKE AND C. G OKC OL Table VI. Tabulated u-velocity proles along a vertical line passing through the geometric centre of the cavity at various ynolds numbers. y the ow is incompressible, the net volumetric ow rate passing through these sections should be equal to zero, Q =. The u-velocity and v-velocity proles at every ynolds number considered are integrated using Simpson s rule, then the obtained volumetric ow rate values are divided by a characteristic ow rate, Q c, which is the horizontal rate that would occur in the absence of the side walls (Plane Couette ow), to help quantify the errors, as also suggested by Aydin and Fenner []. In an integration process the numerical errors will add up. Nevertheless, the volumetric ow rate values (Q = u dy =Q c and Q 2 = v dx =Q c) tabulated in Table VIII are close to zero such that, even the largest values of Q =:75 and Q 2 =:2424 in Table VIII can be considered as Q Q 2. This mathematical check on the conservation of the continuity shows that our numerical solution is indeed very accurate. Distinctively than any other paper, Botella and Peyret [6] have tabulated highly accurate Chebyshev collocation spectral vorticity data from inside the cavity, along a vertical line and along a horizontal line passing through the geometric centre of the cavity, for =. Comparing our solutions with theirs [6] in Figures 9 and, we nd that the agreement with Botella and Peyret [6] is remarkable, with the maximum dierence between two solutions being.8%. Between the whole range of = and the maximum found in the literature, = 2 5, our computed results agreed well with the published results. Since there is no pre- Copyright? 25 John Wiley & Sons, Ltd. Int. J. Numer. Meth. Fluids 25; 48:

23 DRIVEN CAVITY FLOW AT HIGH REYNOLDS NUMBERS 769 Table VII. Tabulated v-velocity proles along a horizontal line passing through the geometric centre of the cavity at various ynolds numbers. x viously published results (to the best of our knowledge) of steady driven cavity ow beyond 2 5 in the literature, we decided to use Batchelor s [3] model in order to demonstrate the accuracy of our numerical solutions at higher ynolds numbers. As seen in Figures 7 and 8, the u- and v-velocity proles change almost linearly in the core of the primary vortex as ynolds number increases. This would indicate that in this region the vorticity is uniform. As increases thin boundary layers are developed along the solid walls and the core uid moves as a solid body with a uniform vorticity, in the manner suggested by Batchelor [3]. At high ynolds numbers the vorticity at the core of the eddy is approximately constant and the ow in the core is governed 2 = C (25) where C is the constant vorticity value. With streamfunction value being = on the boundaries, inside the domain (x; y)=([ :5; :5]; [ :5; :5]) the following expression is a solution to Equation (25) ( y 2 = C 2 ) 8 + A n cosh(2n )x cos(2n )y n= (26) Copyright? 25 John Wiley & Sons, Ltd. Int. J. Numer. Meth. Fluids 25; 48:

24 77 E. ERTURK, T. C. CORKE AND C. G OKC OL Table VIII. Volumetric ow rates through a vertical line, Q, and a horizontal line, Q 2, passing through the geometric centre of the cavity. Q = u dy Q 2 = v dx Q c Q c Computed Botella & Peyret f.[6] y ω Figure 9. Computed vorticity values along a vertical line passing through the geometric centre of the cavity, =. where A n = 4( )n+ (2n ) 3 3 (cosh(n 2 )) (27) Copyright? 25 John Wiley & Sons, Ltd. Int. J. Numer. Meth. Fluids 25; 48:

25 DRIVEN CAVITY FLOW AT HIGH REYNOLDS NUMBERS ω -2 Computed Botella & Peyret f. [6] X.6.8 Figure. Computed vorticity values along a horizontal line passing through the geometric centre of the cavity, =. The mean square law proposed by Batchelor [3] relates the core vorticity value with the boundary velocities. From using the relation [( x ) 2 +( y ) 2 ]ds = U 2 ds = (28) where U is the boundary velocity, we can have C = n= (2n ) 4 (cosh(n =2)) 2 ( ( ) n (2n )A n cosh(2n )x) dx (29) n= 2 From the above expression, the numerical value of C (vorticity value at the core) can be found approximately as.8859, in accordance with the theoretical core vorticity value of.886 at innite, calculated analytically by Burggraf [8] in his application of Batchelor s model [3], which consists of an inviscid core with uniform vorticity, coupled to boundary layer ows at the solid surface. Copyright? 25 John Wiley & Sons, Ltd. Int. J. Numer. Meth. Fluids 25; 48: