Numerical solutions of 2-D steady incompressible flow in a driven skewed cavity.

Transcription

1 Published i : ZAMM - Joural of Applied Mathematics ad Mechaics (007 ZAMM - Z. Agew. Math. Mech. 007; Vol 87: pp Numerical solutios of -D steady icompressible flow i a drive skewed cavity. Erca Erturk 1, Bahtiyar Dursu Gebze Istitute of Techology, Eergy Systems Egieerig Departmet, Gebze, Kocaeli 41400, Turkey Key words Drive skewed cavity flow, steady icompressible N-S equatios, geeral curviliear coordiates, fiite differece, o-orthogoal grid mesh Abstract The bechmark test case for o-orthogoal grid mesh, the drive skewed cavity flow, first itroduced by Demirdži et al. [5] for skew agles of 30 ad 45, is reitroduced with a more variety of skew agles. The bechmark problem has o-orthogoal, skewed grid mesh with skew agle (. The goverig -D steady icompressible Navier- Stokes equatios i geeral curviliear coordiates are solved for the solutio of drive skewed cavity flow with oorthogoal grid mesh usig a umerical method which is efficiet ad stable eve at extreme skew agles. Highly accurate umerical solutios of the drive skewed cavity flow, solved usig a fie grid (5151 mesh, are preseted for yolds umber of 100 ad 1000 for skew agles ragig betwee Itroductio I the literature, it is possible to fid may umerical methods proposed for the solutio of the steady icompressible N-S equatios. These umerical methods are ofte tested o several bechmark test cases i terms of their stability, accuracy as well as efficiecy. Amog several bechmark test cases for steady icompressible flow solvers, the drive cavity flow is a very well kow ad commoly used bechmark problem. The reaso why the drive cavity flow is so popular may be the simplicity of the geometry. I this flow problem, whe the flow variables are odimesioalized with the cavity legth ad the velocity of the lid, yolds umber appears i the equatios as a importat flow parameter. Eve though the geometry is simple ad easy to apply i programmig poit of view, the cavity flow has all essetial flow physics with couter rotatig recirculatig regios at the corers of the cavity. Amog umerous papers foud i the literature, Erturk et al. [6], Botella ad Peyret [4], Schreiber ad Keller [1], Li et al. [1], Wright ad Gaskel [30], Erturk ad Gokcol [7], Bejami ad Dey [] ad Nishida ad Satofuka [16] are examples of umerical studies o the drive cavity flow. Due to its simple geometry, the cavity flow is best solved i Cartesia coordiates with Cartesia grid mesh. Most of the bechmark test cases foud i the literature have orthogoal geometries therefore they are best solved with orthogoal grid mesh. However ofte times the real life flow problems have much more complex geometries tha that of the drive cavity flow. I most cases, researchers have to deal with oorthogoal geometries with o-orthogoal grid mesh. I a o-orthogoal grid mesh, whe the goverig equatios are formulated i geeral curviliear coordiates, cross derivative terms appear i the equatios. Depedig o the skewess of the grid mesh, these cross derivative terms ca be very sigificat ad ca affect the umerical stability as well as the accuracy of the umerical method used for the solutio. Eve though, the drive cavity flow bechmark problem serves for compariso betwee umerical methods, the flow is far from simulatig the real life fluid problems with complex geometries with o-orthogoal grid mesh. The umerical performaces of umerical methods o orthogoal grids may or may ot be the same o o-orthogoal grids. Ufortuately, there are ot much bechmark problems with o-orthogoal grids for umerical methods to compare solutios with each other. Demirdži et al. [5] have itroduced the drive skewed cavity flow as a test case for o-orthogoal grids. The test case is similar to drive cavity flow but the geometry is a parallelogram rather tha a square. I this test case, the skewess of the geometry ca be easily chaged by 1 Correspodig author, ercaerturk@gyte.edu.tr, URL:

2 chagig the skew agle (. The skewed cavity problem is a perfect test case for body fitted o-orthogoal grids ad yet it is as simple as the cavity flow i terms of programmig poit of view. Later Oosterlee et al. [17], Louaked et al. [13], Roychowdhury et al. [0], Xu ad Zhag [31], Wag ad Komori [8], Xu ad Zhag [3], Tucker ad Pa [7], Brakkee et al. [3], Pacheco ad Peck [18], Teiglad ad Eliasse [5], Lai ad Ya [11] ad Shklyar ad Arbel [] have solved the same bechmark problem. I all these studies, the solutio of the drive skewed cavity flow is preseted for yolds umbers of 100 ad 1000 for oly two differet skew agles which are 30 ad 45 ad also the maximum umber of grids used i these studies is Peri [19] cosidered the -D flow i a skewed cavity ad he stated that the goverig equatios fail to coverge for 30. The mai motivatio of this study is the to reitroduce the skewed cavity flow problem with a wide rage of skew agle ( ad preset detailed tabulated results obtaied usig a fie grid mesh with 5151 poits for future refereces. Erturk et al. [6] have itroduced a efficiet, fast ad stable umerical formulatio for the steady icompressible Navier-Stokes equatios. Their methods solve the streamfuctio ad vorticity equatios separately, ad the umerical solutio of each equatio requires the solutio of two tridiagoal systems. Solvig tridiagoal systems are computatioally efficiet ad therefore they were able to use very fie grid mesh i their solutio. Usig this umerical formulatio, they have solved the very well kow bechmark problem, the steady flow i a square drive cavity, up to yolds umber of 1000 usig a fie grid mesh. Their formulatio proved to be stable ad effective at very high yolds umbers ([6], [7], [8]. I this study, the umerical formulatio itroduced by Erturk et al. [6] will be applied to Navier-Stokes equatios i geeral curviliear coordiates ad the umerical solutios of the drive skewed cavity flow problem with body fitted o-orthogoal skewed grid mesh will be preseted. By cosiderig a wide rage of skew agles, the efficiecy of the umerical method will be tested for grid skewess especially at extreme skew agles. The umerical solutios of the flow i a skewed cavity will be preseted for yolds umber of 100 ad 1000 for a wide variety of skew agles ragig betwee 15 ad 165 with 15 icremets.. Numerical Formulatio For two-dimesioal ad axi-symmetric flows it is coveiet to use the streamfuctio ( ad vorticity ( formulatio of the Navier-Stokes equatios. I o-dimesioal form, they are give as (1 xx yy 1 yx x y ( xx yy ( where, is the yolds umber, ad x ad y are the Cartesia coordiates. We cosider the goverig Navier-Stokes equatios i geeral curviliear coordiates as the followig ( ( ( ( ( (3 x y x y xx yy xx yy x x y y 1 ( x y ( x y (( x y ( x y ( ( ( (4 xx yy xx yy x x y y

3 Followig Erturk et al. [6], first pseudo time derivatives are assiged to streamfuctio ad vorticity equatios ad usig a implicit Euler time step for these pseudo time derivatives, the fiite differece formulatios i operator otatio become the followig (1 t( t( t( t( 1 x y x y xx yy xx yy t( t (5 x x y y t t t t ( 1 ( x y ( x y ( xx yy ( xx yy t( t( 1 x y x y t ( x x y y (6 Where ad deote the secod order fiite differece operators, ad similarly ad deote the first order fiite differece operators i - ad -directio respectively. The equatios above are i implicit form ad require the solutio of a large matrix at every pseudo time iteratio which is computatioally iefficiet. Istead these equatios are spatially factorized such that (1 t( t( (1 t( t( 1 x y xx yy x y xx yy t( t (7 x x y y t t ( 1 ( x y ( xx yy t( x y t t ( ( x y ( xx yy t( x y 1 1 t ( x x y y (8 The advatage of these equatios are that each equatio require the solutio of a tridiagoal systems that ca be solved very efficietly usig the Thomas algorithm. It ca be show that approximate factorizatio itroduces additioal secod order terms (O( t i these equatios. I order for the equatios to have the correct physical represetatio, to cacel out the secod order terms due to factorizatio the same terms are added to the right had side of the equatios. The reader is referred to Erturk et al. [6] for more details of the umerical method. The fial form of the equatios take the followig form (1 t( t( (1 t( t( 1 x y xx yy x y xx yy t( t x x y y ( t( x y t( xx yy ( t( x y t( xx yy (9 t t ( 1 ( x y ( xx yy t( x y t t ( 1 ( x y ( xx yy t( x y t ( x x y y 1

4 t t ( ( x y ( xx yy t( x y t t ( ( x y ( xx yy t( x y (10 3. Drive Skewed Cavity Flow Fig. 1 illustrates the schematic view of the bechmark problem, the drive skewed cavity flow. We will cosider the most geeral case where the skew agle ca be 90 or 90. I order to calculate the metrics, the grids i the physical domai are mapped oto orthogoal grids i the computatioal grids as show i Fig.. The iverse trasformatio metrics are calculated usig cetral 1 differeces, as a example xi 1, j xi 1, j N 1. Similarly, iverse trasformatio metrics are u N calculated as the followig 1 cos si x, x, y0, y (11 N N N where N is the umber of grid poits. We cosider a (NN grid mesh. The determiat of the Jacobia matrix is foud as si J x y x y N (1 The trasformatio metrics are defied as y, x, y, x x J y J x J y J (13 Substitutig Equatios (11 ad (1 ito (13, the trasformatio metrics are obtaied as the followig, Ncos N x N y, x0, y (14 si si Note that sice we use equal grid spacig, the secod order trasformatio metrics will be all equal to zero such that for example ( x ( x ( x xx x x 0 (15 x Hece xx yy xx yy 0 (16 These calculated metrics are substituted ito Equatios (9 ad (10 ad the fial form of the umerical equatios become as the followig

5 N N 1 ( 1t( ( 1t( si si N cos N N t( t ( t( ( t( (17 si si si t N N t N N 1 ( 1 ( t( ( 1 ( t( si si si si t N cos ( si t N N t N N ( ( t( ( ( t( (18 si si si si The solutio methodology of each of the above two equatios, Equatios (17 ad (18, ivolves a two-stage time-level updatig. First the streamfuctio equatio (17 is solved, ad for this, the variable f is itroduced such that N 1 ( 1 t( f (19 si where N N cos ( 1t( f t( t si si N N ( t( ( t( (0 si si I Equatio (0 f is the oly ukow variable. First, this Equatio (0 is solved for f at each grid poit. Followig this, the streamfuctio ( variable is advaced ito the ew time level usig Equatio (19. The the vorticity equatio (18 is solved, ad i a similar fashio, the variable g is itroduced such that t N N 1 ( 1 ( t( g (1 si si where t N N t N cos ( 1 ( t( g ( si si si t N N t N N ( ( t( ( ( t( ( si si si si As with f, first the variable g is determied at every grid poit usig Equatio (, the vorticity ( variable is advaced ito the ext time level usig Equatio (1.

6 3.1 Boudary Coditios I the computatioal domai the velocity compoets are defied as the followig Ncos N u y y y (3 si si v N (4 x x x O the left wall boudary we have 0, 0, 0 (5 0, j 0, j 0, j where the subscripts 0 ad j are the grid idexes. Also o the left wall, the velocity is zero (u0 ad v0. Usig Equatios (3 ad (4 we obtai 0 0 (6, j ad also (, j 0 0 (7 Therefore, substitutig these ito the streamfuctio Equatio (3 ad usig Thom s formula [6], o the left wall boudary the vorticity is calculated as the followig N si 0, j 1, j (8 Similarly the vorticity o the right wall ( ad the vorticity o the bottom wall ( N, j are defied i,0 as the followig N N N1, j i, 1 N, j, i, 0 si si (9 O the top wall the u-velocity is equal to u1. Followig the same procedure, the vorticity o the top wall is foud as in, N in, 1 N si si (30 We ote that, it is well uderstood ([10], [15], [3], [9] that, eve though Thom s method is locally first order accurate, the global solutio obtaied usig Thom s method preserves secod order accuracy. Therefore i this study, sice three poit secod order cetral differece is used iside the skewed cavity ad Thom s method is used at the wall boudary coditios, the preseted solutios are secod order accurate. I the skewed drive cavity flow, the corer poits are sigular poits for vorticity. We ote that due to the skew agle, the goverig equatios have cross derivative terms ad because of these cross derivative terms

7 the computatioal stecil icludes 33 grid poits. Therefore, the solutio at the first diagoal grid poits ear the corers of the cavity require the vorticity values at the corer poits. For square drive cavity flow Gupta et al. [9] have itroduced a explicit asymptotic solutio i the eighborhood of sharp corers. Similarly, Störtkuhl et al. [4] have preseted a aalytical asymptotic solutios ear the corers of cavity ad usig fiite elemet biliear shape fuctios they also have preseted a sigularity removed boudary coditio for vorticity at the corer poits as well as at the wall poits. I this study we follow Störtkuhl et al. [4] ad use the followig expressio for calculatig vorticity values at the corers of the skewed cavity N 3si VN 1 9 si (31 where V is the speed of the wall which is equal to 1 for the upper two corers ad it is equal to 0 for the bottom two corers. The reader is referred to Störtkuhl et al. [4] for details. 4. sults The steady icompressible flow i a drive skewed cavity is umerically solved usig the described umerical formulatio ad boudary coditios. We have cosidered two yolds umbers, =100 ad =1000. For these two yolds umbers we have varied the skew-agle ( from with 15 icremets. We have solved the itroduced problem with a 5151 grid mesh, for the two yolds umber ad for all the skew agles cosidered. Durig the iteratios as a measure of the covergece to the steady state solutio, we moitored three error parameters. The first error parameter, ERR1, is defied as the maximum absolute residual of the fiite differece equatios of the steady streamfuctio ad vorticity equatios i geeral curviliear coordiates, Equatios (3 ad (4. These are respectively give as N N N cos ERR1maxabs si a si a si a ERR1w maxabs a a a a a 1 N 1 N N N N cos si si si si si i, j i, j (3 The magitude of ERR1 is a idicatio of the degree to which the solutio has coverged to steady state. I the limit ERR1 would be zero. The secod error parameter, ERR, is defied as the maximum absolute differece betwee a iteratio time step i the streamfuctio ad vorticity variables. These are respectively give as ERR max abs i, j i, j i, j i, j ERR max abs (33

8 ERR gives a idicatio of the sigificat digit of the streamfuctio ad vorticity variables are chagig betwee two time levels. The third error parameter, ERR3, is similar to ERR, except that it is ormalized by the represetative value at the previous time step. This the provides a idicatio of the maximum percet chage i ad at each iteratio step. ERR3 is defied as 1 i, j i, j ERR3 maxabs i, j 1 i, j i, j ERR3 maxabs (34 i, j I our computatios, for every yolds umbers ad for every skew agles, we cosidered that covergece 10 was achieved whe both ERR1 ad ERR1 were less tha10. Such a low value was chose to esure the accuracy of the solutio. At these residual levels, the maximum absolute differece i streamfuctio value 17 betwee two time steps, ERR, was i the order of 10 ad for vorticity, ERR, it was i the order of Ad also at these covergece levels, betwee two time steps the maximum absolute ormalized 14 differece i streamfuctio, ERR3, ad i vorticity, ERR3, was i the order of 10, ad10 13 respectively. We ote that at extreme skew agles, covergece to such low residuals is ecessary. For example, at skew agle 15 at the bottom left corer, ad at skew agle 165 at the bottom right corer, there appears progressively smaller couter rotatig recirculatig regios i accordace with Moffatt [14]. I these recirculatig regios cofied i the sharp corer, the value of streamfuctio variable is gettig extremely smaller as the size of the recirculatig regio gets smaller towards the corer. Therefore, it is crucial to have covergece to such low residuals especially at extreme skew agles. Before solvig the skewed drive cavity flow at differet skew agles first we have solved the square drive cavity flow to test the accuracy of the solutio. The square cavity is actually a special case for skewed cavity ad obtaied whe the skew agle is chose as 90. For the square drive cavity flow, the streamfuctio ad the vorticity values at the ceter of the primary vortex ad the locatio of this ceter are tabulated i Table 1 for yolds umbers of =1000, together with results foud i the literature. The preset results are almost exactly the same with that of Erturk et al. [6]. This was expected sice i both studies the same umber of grid poits were used ad also the spatial accuracy of both the boudary coditio approximatios ad the solutios were the same. Furthermore the preseted results are i very good agreemet with that of highly accurate spectral solutios of Botella ad Peyret [4] ad extrapolated solutios of Schreiber ad Keller [1] ad also fourth order solutios of Erturk ad Gokcol [7] with approximately less tha 0.18% ad 0.14% differece i streamfuctio ad vorticity variables respectively. For all the skew agles cosidered i this study ( we expect to have the same level of accuracy we achieved forc. With Li et al. [1], Wright ad Gaskel [30], Bejami ad Dey [] ad Nishida ad Satofuka [16] agai our solutios compare good. After validatig our solutio for 90, we decided to validate our solutios at differet skew agles. I order to do this we compare our results with the results foud i the literature. At this poit, we would like to ote that i the literature amog the studies that have solved the skewed cavity flow ([5], [17], [13], [0], [31], [8], [3], [7], [3], [18], [5], [11] ad [], oly Demirdži et al. [5], Oosterlee et al. [17], Shklyar ad Arbel [] ad Louaked et al. [13] have preseted tabulated results therefore we will maily compare our results with those studies.

9 As metioed earlier, Demirdži et al. [5] have preseted solutios for skewed cavity for yolds umber of 100 ad 1000 for skewed agles of 45 ad 30. Figure 3 compares our results of u-velocity alog lie A-B ad v-velocity alog lie C-D with that of Demirdži et al. [5] for =100 ad 1000 for 45, ad also Figure 4 compares the same for 30. Our results agree excellet with results of Demirdži et al. [5]. Table compares our results of the miimum ad also maximum streamfuctio value ad also their locatio for yolds umbers of 100 ad 1000 for skew agles of 30 ad 45 with results of Demirdži et al. [5], Oosterlee et al. [17], Louaked et al. [13] ad Shklyar ad Arbel []. The results of this study ad the results of Demirdži et al. [5] ad also those of Oosterlee et al. [17], Shklyar ad Arbel [] ad Louaked et al. [13] agree well with each other, although we believe that our results are more accurate sice i this study a very fie grid mesh is used. Figure 5 to Figure 8 show the streamlie ad also vorticity cotours for =100 ad =1000 for skew agles from 15 to 165 with 15 icremets. As it is see from these cotour figures of streamfuctio ad vorticity, the solutios obtaied are very smooth without ay wiggles i the cotours eve at extreme skew agles. We have solved the icompressible flow i a skewed drive cavity umerically ad compared our umerical solutio with the solutios foud i the literature for 90, 30 ad 45, ad good agreemet is foud. We, the, have preseted solutios for Sice we could ot fid solutios i the literature to compare with our preseted solutios other tha 90, 30 ad 45, i order to demostrate the accuracy of the umerical solutios we preseted, a good mathematical check would be to check the cotiuity of the fluid, as suggested by Aydi ad Feer [1]. We have itegrated the u-velocity ad v-velocity profiles alog lie A-B ad lie C-D, passig through the geometric ceter of the cavity show by the red dotted lie i Figure 1, i order to obtai the et volumetric flow rate through these sectios. Through sectio A-B, the volumetric flow rate isq udy vdx vdx. Sice the flow is AB ad through sectio C-D it is icompressible, the et volumetric flow rate passig through these sectios should be equal to zero, Q 0. Usig Simpso s rule for the itegratio, the volumetric flow ratesq ad AB Q CD are calculated for every skew agle ( ad every yolds umber cosidered. I order to help quatify the errors, the obtaied volumetric flow rate values are ormalized by the absolute total flow rate through the correspodig sectio at the cosidered ad. Hece Q is ormalized by AB udy vdx ad similarly Q CD is ormalized by vdx. Table 3 tabulates the ormalized volumetric flow rates through the cosidered sectios. We ote QCD that i a itegratio process the umerical errors will add up, evertheless, the ormalized volumetric flow rate values tabulated i Table 3 are close to zero, such that they ca be cosidered as QAB QCD 0. This mathematical check o the coservatio of the cotiuity shows that our umerical solutio is ideed very accurate at the cosidered skew agles ad yolds umbers. We ote that, to the authors best kowledge, i the literature there is ot a study that cosidered the skewed cavity flow at the skew agles used i the preset study other tha 30 ad 45. The solutios preseted i this study are uique therefore, for future refereces, i Table 4 we have tabulated the miimum ad also maximum streamfuctio values ad their locatios ad also the vorticity value at these poits for yolds umber of 100 ad 1000 for all the skew agles cosidered, from 15 to 165 with 15 icremets. I this table the iterestig poit is, at yolds umber of 1000, the stregth of vorticity (absolute value of the vorticity at the ceter of the primary vortex decrease as the skew agle icrease from 15 to 90, havig the miimum value at 90. As the skew agle icrease further from 90 to 165, the stregth of vorticity at the ceter of the primary vortex icrease. At this yolds umber, =1000, the streamfuctio value at the ceter of the primary vortex also show the same type of behavior, where the value of the streamfuctio start to icrease as the skew agle icrease util 90, the start to decrease as the skew agle icrease further. However at yolds umber of 100, the miimum value of the stregth of vorticity ad also the maximum value of the streamfuctio at the ceter of Q CD

10 the primary vortex occur at 105. I order to explai this behavior, we decided to look at the locatio of the ceter of the primary vortex at differet skew agles. The stregth of vorticity at the ceter of the primary vortex is proportioal with the vertical distace betwee the ceter of the primary vortex ad the top movig lid. As the ceter of the primary vortex move away from the top movig wall, the stregth of vorticity at the ceter should decrease. Figure 9 shows the vertical distace betwee the ceter of the primary vortex ad the top movig lid ad as a fuctio of the skew agle for both =100 ad As this figure show at =100 the eye of the primary vortex is at its farthest positio from the top movig lid at 105 where as at =1000 the same occurs at 90, explaiig the miimum vorticity stregth we obtai for the primary vortex at 105 for =100 ad also at 90 for =1000. For future refereces, i Table 5 ad 6 we have tabulated the u-velocity profiles alog lie A-B (show i Figure 1 for yolds umber of 100 ad 1000 respectively, ad similarly, i Table 7 ad 8 we have tabulated the v-velocity profiles alog lie C-D (also show i Figure 1 for yolds umber of 100 ad 1000 respectively. 5. Coclusios I this study the bechmark test case for o-orthogoal grid mesh, the skewed cavity flow itroduced by Demirdži et al. [5] for skew agles of 30 ad 45, is reitroduced with a variety of skew agles. The skewed cavity flow is cosidered for skew agles ragig betwee with 15 icremets, for =100 ad =1000. The goverig Navier-Stokes equatios are cosidered i most geeral form, i geeral curviliear coordiates. The o-orthogoal grids are mapped oto a computatioal domai. Usig the umerical formulatio itroduced by Erturk et al. [6], fie grid solutios of streamfuctio ad vorticity equatios are obtaied with very low residuals. The umerical formulatio of Erturk et al. [6] have proved to be very effective o o-orthogoal problems with o-orthogoal grid mesh eve at extreme skew agles. The drive skewed cavity flow problem is a challegig problem ad it ca be a perfect bechmark test case for umerical methods to test performaces o o-orthogoal grid meshes. For future refereces detailed results are tabulated. fereces [1] M. Aydi ad R. T. Feer, Boudary Elemet Aalysis of Drive Cavity Flow for Low ad Moderate yolds Numbers, Iteratioal Joural for Numerical Methods i Fluids, 37 (001, [] A. S. Bejami ad V. E. Dey, O the Covergece of Numerical Solutios for -D Flows i a Cavity at Large, Joural of Computatioal Physics 33 (1979, [3] E. Brakkee, P. Wesselig ad C. G. M. Kassels, Schwarz Domai Decompositio for the Icompressible Navier Stokes Equatios i Geeral Co-ordiates, Iteratioal Joural for Numerical Methods i Fluids 3, ( [4] O. Botella ad R. Peyret, Bechmark Spectral sults o the Lid-Drive Cavity Flow, Computers ad Fluids 7, ( [5] I. Demirdži, Z. Lilek ad M. Peri, Fluid Flow ad Heat Trasfer Test Problems for No-orthogoal Grids: Bech-mark Solutios, Iteratioal Joural for Numerical Methods i Fluids 15, ( [6] E. Erturk, T. C. Corke ad C. Gokcol, Numerical Solutios of -D Steady Icompressible Drive Cavity Flow at High yolds Numbers, Iteratioal Joural for Numerical Methods i Fluids 48, ( [7] E. Erturk ad C. Gokcol, Fourth Order Compact Formulatio of Navier-Stokes Equatios ad Drive Cavity Flow at High yolds Numbers, Iteratioal Joural for Numerical Methods i Fluids 50, ( [8] E. Erturk, O. M. Haddad ad T. C. Corke, Numerical Solutios of Lamiar Icompressible Flow Past Parabolic Bodies at Agles of Attack, AIAA Joural 4, (

11 [9] M. M. Gupta, R. P. Maohar ad B. Noble, Nature of Viscous Flows Near Sharp Corers, Computers ad Fluids 9, ( [10] H. Huag ad B. R. Wetto, Discrete Compatibility i Fiite Differece Methods for Viscous Icompressible Fluid Flow, Joural of Computatioal Physics 16, ( [11] H. Lai ad Y. Y. Ya, The Effect of Choosig Depedet Variables ad Cellface Velocities o Covergece of the SIMPLE Algorithm Usig No-Orthogoal Grids, Iteratioal Joural of Numerical Methods for Heat & Fluid Flow 11, ( [1] M. Li, T. Tag ad B. Forberg, A Compact Forth-Order Fiite Differece Scheme for the Steady Icompressible Navier-Stokes Equatios Iteratioal Joural for Numerical Methods i Fluids 0, ( [13] M. Louaked, L. Haich ad K. D. Nguye, A Efficiet Fiite Differece Techique For Computig Icompressible Viscous Flows, Iteratioal Joural for Numerical Methods i Fluids 5, ( [14] H. K. Moffatt, Viscous ad resistive eddies ear a sharp corer, Joural of Fluid Mechaics 18, ( [15] M. Napolitao, G. Pascazio ad L. Quartapelle, A view of Vorticity Coditios i the Numerical Solutio of the - Equatios, Computers ad Fluids 8, ( [16] H. Nishida ad N. Satofuka, Higher-Order Solutios of Square Drive Cavity Flow Usig a Variable-Order Multi-Grid Method, Iteratioal Joural for Numerical Methods i Fluids 34, ( [17] C. W. Oosterlee, P. Wesselig, A. Segal ad E. Brakkee, Bechmark Solutios for the Icompressible Navier- Stokes Equatios i Geeral Co-ordiates o Staggered Grids, Iteratioal Joural for Numerical Methods i Fluids 17, ( [18] J. R. Pacheco ad R. E. Peck, Nostaggered Boudary-Fitted Coordiate Method For Free Surface Flows, Numerical Heat Trasfer Part B 37, ( [19] M. Peri, Aalysis of Pressure-Velocity Couplig o No-orthogoal Grids, Numerical Heat Trasfer Part B 17, ( [0] D. G. Roychowdhury, S. K. Das ad T. Sudararaja, A Efficiet Solutio Method for Icompressible N-S Equatios Usig No-Orthogoal Collocated Grid, Iteratioal Joural for Numerical Methods i Egieerig 45, ( [1] R. Schreiber ad H. B. Keller, Drive Cavity Flows by Efficiet Numerical Techiques, Joural of Computatioal Physics 49, ( [] A. Shklyar ad A. Arbel, Numerical Method for Calculatio of the Icompressible Flow i Geeral Curviliear Co-ordiates With Double Staggered Grid, Iteratioal Joural for Numerical Methods i Fluids 41, ( [3] W. F. Spotz, Accuracy ad Performace of Numerical Wall Boudary Coditios for Steady D Icompressible Streamfuctio Vorticity, Iteratioal Joural for Numerical Methods i Fluids 8, ( [4] T. Stortkuhl, C. Zeger ad S. Zimmer, A Asymptotic Solutio for the Sigularity at the Agular Poit of the Lid Drive Cavity, Iteratioal Joural of Numerical Methods for Heat & Fluid Flow 4, ( [5] R. Teiglad ad I. K. Eliasse, A Multiblock/Multilevel Mesh fiemet Procedure for CFD Computatios, Iteratioal Joural for Numerical Methods i Fluids 36, ( [6] A. Thom, The Flow Past Circular Cyliders at Low Speed, Proceedigs of the Royal Society of Lodo Series A 141, ( [7] P. G. Tucker ad Z. Pa, A Cartesia Cut Cell Method for Icompressible Viscous Flow, Applied Mathematical Modellig 4, ( [8] Y. Wag ad S. Komori, O the Improvemet of the SIMPLE-Like method for Flows with Complex Geometry, Heat ad Mass Trasfer 36, ( [9] E. Weia ad L. Jia-Guo, Vorticity Boudary Coditio ad lated Issues for Fiite Differece Schemes, Joural of Computatioal Physics 14, ( [30] N. G. Wright ad P. H. Gaskell, A Efficiet Multigrid Approach to Solvig Highly circulatig Flows, Computers ad Fluids 4, (

12 [31] H. Xu ad C. Zhag, Study Of The Effect Of The No-Orthogoality For No-Staggered Grids The sults, Iteratioal Joural for Numerical Methods i Fluids 9, ( [3] H. Xu ad C. Zhag, Numerical Calculatio of Lamiar Flows Usig Cotravariat Velocity Fluxes, Computers ad Fluids 9, (

13 U = 1 A C d = 1 D d = 1 B a skewed cavity with U = 1 A d = 1 d = 1 C B D b skewed cavity with Fig. 1. Schematic view of drive skewed cavity flow

14 Fig.. Trasformatio of the physical domai to computatioal domai -1 x y i+1,j i,j y y x x y x y x y x y x Computatioal domai Physical domai i-1,j i,j-1 i,j+1 x = N 1 N 1 y = N ( si =1 =1

15 =100 α=45 ο =100 α=45 ο grid idex alog A-B Computed Demirdzic et al. (199 ^ v-velocity Computed Demirdzic et al. (199 ^ u-velocity grid idex alog C-D =1000 α=45 ο =1000 α=45 ο grid idex alog A-B Computed Demirdzic et al. ( u-velocity ^ v-velocity Computed Demirdzic et al. ( grid idex alog C-D ^ Fig. 3. Compariso of u-velocity alog lie A-B ad v-velocity alog lie C-D, for =100 ad 1000, for skew agle α=45 ο

16 =100 α=30 ο =100 α=30 ο grid idex alog A-B Computed Demirdzic et al. (199 ^ v-velocity Computed Demirdzic et al. (199 ^ u-velocity grid idex alog C-D =1000 α=30 ο =1000 α=30 ο grid idex alog A-B Computed Demirdzic et al. ( u-velocity ^ v-velocity Computed Demirdzic et al. ( grid idex alog C-D ^ Fig. 4. Compariso of u-velocity alog lie A-B ad v-velocity alog lie C-D, for =100 ad 1000, for skew agle α=30 ο

17

18

19

20

21 Vertical distace α α a =100 b =1000 Fig. 9. The vertical distace betwee the ceter of the primary vortex ad the top movig lid versus the skew agle Vertical distace

22 ferece Grid Accuracy x y Preset Study 513x513 h Erturk et al. [6] 513x513 h Botella & Peyret [4] N=160 N= Schreiber & Keller [1] Extrapolated 6 h Li et al. [1] 19x19 4 h Wright & Gaskell [30] 104x104 h Erturk & Gokcol [7] 601x601 4 h Bejami & Dey [] Extrapolated high order Nishida & Satofuka [16] 19x19 h Table 1 Compariso of the properties of the primary vortex for square drive cavity flow; the maximum streamfuctio value, the vorticity value ad the locatio of the ceter, for =1000

23 Skew =100 =1000 Agle mi max mi max Preset study E E E E-03 ( (x,y (1.1680, (0.56, (1.456, (0.904, Demirdži et al. [5] E E E E-03 (3030 (x,y (1.1664, (0.569, (1.4583, (0.9039, Oosterlee et al. [17] E E E E-03 (5656 (x,y (1.1680, (0.591, (1.4565, (0.9036, Shklyar ad Arbel [] E E E E-03 (3030 (x,y (1.1674, (0.511, (1.4583, (0.8901, Louaked et al. [13] E E-03 (1010 (x,y - - (1.4540, (0.8980, Preset study -7.03E E E E-0 ( (x,y (1.1119, (0.3395, 0.14 (1.3148, (0.7780, Demirdži et al. [5] -7.06E E E E-0 (3030 (x,y (1.1100, (0.3387, (1.3130, (0.7766, Oosterlee et al. [17] E E E E-0 (5656 (x,y (1.1100, (0.3390, (1.318, (0.7775, Shklyar ad Arbel [] E E E E-0 (3030 (x,y (1.1146, (0.308, (1.310, (0.7766, Louaked et al. [13] E E-0 (1010 (x,y - - (1.3100, (0.7760, Table Compariso of the miimum ad maximum streamfuctio value ad the locatio of these poits, for yolds umber of 100 ad 1000, for skew agles of 30 ad 45

24 Q AB udyvdx udy vdx Q CD vdx vdx Skew Agle =100 =1000 =100 =1000 = E-05.75E E E-05 =30.089E E-05.69E E-06 =45.056E E E E-05 =60.054E E-05.5E-05.11E-06 =75.17E E E E-05 =90.40E E E E-06 = E E E E-05 =10.399E E E E-06 =135.56E E E E-06 = E E E E-07 = E E E E-06 Table 3 Normalized volumetric flow rates through sectios A-B ad C-D

25 =100 =1000 Skew Agle mi max mi max = E E E E E E-01 = 30 = 45 = 60 (x, y (1.1393, (0.7447, (1.4009, (0.8705, E E E E E E-01 (x, y (1.1680, (0.56, (1.456, (0.904, E E E E E E-01 (x, y (1.1119, (0.3395, 0.14 (1.3148, (0.7780, E E E E E E-01 (x, y (0.9844, (0.1914, (1.0703, (0.5879, = E E E E E (x, y (0.8089, (0.0890, (0.6888, (0.9095, = E E E E E (x, y (0.615, (0.9434, (0.5313, (0.8633, = E E E E E (x, y (0.467, (0.8844, (0.385, (0.7869, = E E E E E (x, y (0.539, (0.7949, 0.15 (0.50, (0.6836, = E E E E E (x, y (0.1055, (0.6708, (0.1390, 0.39 (0.5554, = E E E E E (x, y ( , (0.5003, (0.0478, (0.3994, = E E E E E (x, y ( , (0.77, ( , (0.140, Table 4 Tabulated miimum ad maximum streamfuctio values, the vorticity values ad the locatios, for yolds umber of 100 ad 1000, for various skew agles

26 Grid idex = 15 = 30 = 45 = 60 = 75 = 90 = 105 = 10 = 135 = 150 = E E E E E E E E E E E E E E E E E E E E E E E E E-0-4.6E E E E E E-0-1.7E E E E E-0-6.8E E E E E E E E E E E E-0-1.9E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E-0.086E E E E E E E E E E E E E E E E E E E E E-01.84E E E E E E E E E E E E E E E Table 5 Tabulated u-velocity profiles alog lie A-B, for various skew agles, for =100

27 Grid idex = 15 = 30 = 45 = 60 = 75 = 90 = 105 = 10 = 135 = 150 = E E E E E E E E E-0.430E E E E E-0.177E E E E E E E E E E E-0.909E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E-0.539E E E E E E E E E E-0.477E E E E E E E E E E-0.305E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E-0.016E E E E E E E E E E E-0.883E E E E E-01.51E E E E E-03.58E E E E E E E E E E E E E E E E E E E Table 6 Tabulated u-velocity profiles alog lie A-B, for various skew agles, for =1000

28 Grid idex = 15 = 30 = 45 = 60 = 75 = 90 = 105 = 10 = 135 = 150 = E E E E E E E E E E-0 4.3E E E E E E E E E E E E E E E E E E E-01.01E E E E E-04.31E E E E E-01.07E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E-0.468E E E E E E E E E-0.06E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E Table 7 Tabulated v-velocity profiles alog lie C-D, for various skew agles, for =100

29 Grid idex = 15 = 30 = 45 = 60 = 75 = 90 = 105 = 10 = 135 = 150 = E E E E E E E E E E-04.87E E E E-0-4.4E-0.846E E E E E E-0.674E E E E E E E E E E E E E E E E E E E E E E E E E E E-0.530E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E-0.581E E E E E E E E E-0 5.6E E E E E E E E E E E E E E E E E E E E E-03.37E E E E E E E E E E E-0.136E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E-0 1.9E E E Table 8 Tabulated v-velocity profiles alog lie C-D, for various skew agles, for =1000