Simulation of Incompressible Flows in Two-Sided Lid-Driven Square Cavities. Part II - LBM

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1 Perumal & Dass CFD Letters Vol. 2(1) Vol. 2 (1) March 2010 Smulaton of Incompressble Flows n Two-Sded Ld-Drven Square Cavtes. Part II - LBM D. Arumuga Perumal 1c and Anoop K. Dass 1 1 Department of Mechancal Engneerng, Indan Insttute of Technology Guwahat, Guwahat , INDIA Receved: 15/10/2009 Revsed 15/01/2010 Accepted 04/03/2010 Abstract The present paper computes the flow n a two-sded ld-drven square cavty by the Lattce Boltzmann Method (LBM). For some aspect ratos there exsts a multplcty of steady solutons, but the square cavty problem gves only a sngle steady soluton for both the parallel and antparallel moton of the walls. It s found that for parallel moton of the walls, there appears a par of counter-rotatng secondary vortces of equal sze near the centre of a wall. Because of symmetry, ths par of counter-rotatng vortces has smlar shapes and ther detaled study as to how they grow wth ncreasng Reynolds number has not yet been made by lattce Boltzmann Method. Such a study s attempted n ths paper through the LBM, as the results of the problem have the potental of beng used for testng varous soluton methods for ncompressble vscous flows. The results for the antparallel moton of the walls are also presented n some detal. To lend credblty to the LBM results they are further compared wth those obtaned from a fnte dfference method (FDM). Keywords: two-sded ld-drven square cavty; lattce Boltzmann method; D2Q9 model; bounce-back boundary condton; fnte dfference method. 1. Introducton Smulaton of ncompressble flows n two-sded ld-drven square cavty flow by FDM s dscussed n some detal n the frst part of our paper [1]. The present work focuses on the computaton and valdaton of two-sded ld-drven square cavty flows by the Lattce Boltzmann Method. The LBM s a relatvely novel technque that has become an alternatve to the tradtonal numercal methods for computng flud flow problems. The method s dscussed n suffcent detals n the books by Wolf-Gladrow [2] and Succ [3]. Chen and Doolen [4] have wrtten an excellent revew paper on the subject. Hstorcally, LBM orgnated from the method of Lattce Gas Cellular automata (LGCA), whch was frst ntroduced n 1973 by Hardy, Pomeau and de Pazzs (HPP) [5]. In 1986, Frsch, Hasslacher and Pomeau (FHP) obtaned the correct Naver-Stokes equatons usng a hexagonal lattce. Lattce Boltzmann Equatons has been used at the cradle of Lattce Gas Automata (LGA) by Frsch et al. [6] to calculate vscosty. To elmnate statstcal nose n 1988 c Correspondng Author: D. Arumuga Perumal Emal: d.perumal@tg.ernet.n Telephone: Fax: All rghts reserved. ISSR Journals 25

2 Smulaton of Incompressble Flows n Two-Sded Ld-Drven Square Cavtes. Part II - LBM McNamara and Zanett [7] dd away wth the Boolean operaton of LGA nvolvng the partcle occupaton varables by neglectng partcle correlatons and ntroducng averaged dstrbuton functons gvng rse to the LBM. Hguera and Jmenez [8] brought about an mportant smplfcaton n LBM by presentng an lattce Boltzmann Equaton (LBE) wth a lnearzed collson operator that assumes that the dstrbuton s close to the local equlbrum state. A partcularly smple verson of lnearzed collson operator based on the Bhatnagar-Gross-Krook (BGK) [9] collson model was ndependently ntroduced by several authors ncludng Koelman [10] and Chen et al. [11]. The lattce BGK (LBGK) model [12, 13] utlzes the local equlbrum dstrbuton functon to recover the macroscopc Naver-Stokes equatons. A revew of computatonal and expermental studes on ld-drven cavty flow can be found n Shankar & Deshpande [14]. They have studed and analyzed corner eddes, nonunqueness, transton and turbulence n the ld-drven cavty. The two-sded ld-drven rectangular cavty problem has been nvestgated n some detals by Kuhlmann and coworkers [15]. Many researchers [17-20] carred out smulatons of one-sded ld-drven cavty flow by the lattce Boltzmann method. Yong G La et al. [19] compared the lattce Boltzmann method and the fnte volume Naver-Stokes solver and concluded that bounce-back boundary condton has better than frst order accuracy. The present work uses Lattce Boltzmann BGK model (LBGK) wth sngle tme relaxaton and bounceback boundary condton to nvestgate the flow drven by parallel and antparallel moton of two facng walls n a square cavty for Reynolds number up to A nne-velocty ncompressble LB model n 2D space has been used n the present work snce t s known to gve more accurate results compared to seven-velocty ncompressble LB model. Ths paper s organzed n four sectons. In Secton 2 numercal methods ncludng LBGK wth sngle tme relaxaton scheme and two-dmensonal nne-velocty lattce model s descrbed. In Secton 3 the two-sded ld-drven cavty problem s descrbed and the results wth parallel and antparallel moton of the walls are presented and valdated. Concludng remarks are made n Secton Numercal Methods 2.1. Lattce Boltzmann Method As has already been mentoned the Lattce Boltzmann method (LBM) represents an alternatve possblty for the drect smulaton of the ncompressble flow. It s seen that the accuracy of the Lattce Boltzmann method s of second order both n space and tme [19]. The Lattce Boltzmann equaton whch can be lnked to the Boltzmann equaton n knetc theory s formulated as [17] f ( x c t, t t) f ( x, t) = (1) where f s the partcle dstrbuton functon, c s the partcle velocty along the th drecton and s the collson term. The so-called LBGK model wth sngle tme relaxaton, whch s a commonly used lattce Boltzmann method s gven by [17] 1 (, ) (, ) = - (, ) (0) f x c t t t f x t f x t f ( x, t) (2) where f (0) ( x, t) s the equlbrum dstrbuton functon at x,t and s the tme relaxaton parameter Two-Dmensonal Nne-Velocty Square Lattce Model The D2Q9 square lattce used here has nne dscrete veloctes. A square lattce s used, each node of whch has eght neghbours connected by eght lnks as shown n Fgure 1. Partcles resdng on a node move to ther nearest neghbours along these lnks n unt tme 26

3 Perumal & Dass CFD Letters Vol. 2(1) 2010 step. The occupaton of the rest partcle s desgnated as f 0. The occupaton of the partcles movng along the x- and y-axes are desgnated as dagonally movng partcles are desgnated as f, 5 f, 1 f, 6 f, 2 f, 7 f, 3 defned as c = 0, = 0 c = (cos( / 4( 1)),sn( / 4( 1))), = 1, 2, 3, 4 c = 2(cos( / 4( 1)),sn( / 4( 1))), = 5, 6, 7, 8. f, whle the occupaton of 4 f. The partcle veloctes are 8 (3) Fgure 1. Two-Dmensonal Nne-Velocty Square Lattce (D2Q9) Model. The macroscopc quanttes such as densty and momentum densty u are defned as velocty moments of the dstrbuton functon f as follows: N ρ= f, =0 (4) N ρ u= f =0 c (5) The densty (whch s drectly related to the pressure) s determned from the partcle dstrbuton functon. The densty and the veloctes satsfy the Naver-Stokes equatons n the low-mach number lmt. Ths can be demonstrated by usng the Chapman-Enskog expanson. In the nne-speed square lattce, a sutable equlbrum dstrbuton functon that has been proposed s [17] (0) 3 2 f = w 1 u, = 0 2 f (0) = 1 3( ) ( ) w c.u c.u u, = 1, 2, 3, 4 (6) f (0) = w 1 3( ) ( ) c.u c.u u, = 5, 6, 7, 8 where the lattce weghts are gven by w = 4/9, w = w = w = w = 1/9 and w = w = w = w = 1/36. The relaxaton tme s related to the vscosty by [19] 6 1 = (7) 2 where s the knematc vscosty measured n lattce unts. It s seen that = 0.5 s the crtcal value for ensurng a non-negatve knematc vscosty. Numercal nstablty can occur for a close to ths crtcal value. Ths stuaton takes place at hgh Reynolds numbers. In ths work Reynolds numbers up to 2000 n a lattce sze of have been nvestgated. 4 27

4 Smulaton of Incompressble Flows n Two-Sded Ld-Drven Square Cavtes. Part II - LBM Boundary Condtons In LBM several boundary condtons have been proposed [17-20]. The bounce-back scheme was used n these smulatons to copy the velocty no-slp condton on walls. In ths scheme, the partcle dstrbuton functon at the wall lattce node s assgned to be the partcle dstrbuton functon of ts opposte drecton. The basc argument for the use of ongrd bounce-back model' s that t s both mathematcally applcable and qute relevant for LBE smulatons of flud flows n smple bounded domans. For ths reason, ths boundary condton has been employed here on the two statonary walls. However for the movng walls, the equlbrum boundary condton s appled [17]. At the lattce nodes on the movng walls, flow-varables are re-set to ther pre-assumed values at the end of every streamngstep. A ld-velocty of U = 0.1 has been consdered n ths work. Snce Mach number s U/c s, where c s equals 1/ 3 a Mach number of , well wthn the ncompressble lmt, s obtaned Numercal Procedure The veloctes are assumed to be zero at the tme of startng the smulatons for all nodes. Intally, the equlbrum dstrbuton functon that corresponds to the flow-varables s assumed as the unknown dstrbuton functon for all node at t = 0. Also a unform flud densty =1.0 s mposed ntally. The boundary condtons for the parallel and antparallel wall moton are shown n Fgures 2 (a) and 2 (b). The soluton procedure of the above LBM at each tme step comprse the streamng and collson step, applcaton of boundary condtons, calculaton of partcle dstrbuton functon followed by calculaton of macroscopc varables. The LBE s solved n the soluton doman subjected to the above ntal and boundary condtons on a unform 2D square lattce structure. It s seen that the numercal algorthm of the lattce Boltzmann method s relatvely smpler compared wth conventonal Naver-Stokes methods that use technques lke fnte dfference, fnte volume or fnte element to dscretze the equatons at the macroscopc level. Another beneft of the present approach s the easness of programmng. Fgure 2. Two-Sded Ld-Drven Cavty for (a) parallel wall moton (b) antparallel wall moton wth LBM boundary condtons for the movng walls Fnte Dfference Results for Comparson As the LBM method s ntended to be used to compute the flow n a relatvely unexplored problem, Ref [1] that uses FDM to compute the flow n the same geometry serves as an effectve bass for comparson of most of the results presented n ths paper. The FDM code 28

5 Perumal & Dass CFD Letters Vol. 2(1) 2010 numercally solves the 2D Naver-Stokes equatons n the stream functon-vortcty form gven by = x y u + v = +. (9) t x y Re x y Ths form of the Naver-Stokes equaton s known to be amenable for hghly accurate numercal soluton. The code, whch s valdated aganst establshed results, uses ADI technque and second-order accurate central dfferencng for space dscretzaton. Consequently these results, whch are the only ones avalable for comparson, are hghly relable. Thus, favourable comparson of the present LBM results wth these accurate FDM results wll grant legtmacy to them. Ths n turn would lend added credblty to the results for ths relatvely unexplored problem, so that the flow confguraton could be used as a good test case for algorthm valdaton Valdaton of the LBM Code The developed LBM code s used to compute the sngle ld-drven square cavty flow for Re = 1000 on a lattce structure. Well-establshed results computed by Gha et al. [16] exst for the same problem whch s used for the present code-valdaton exercse. Fgures 3(a) and 3(b) shows the steady-state x-component of the velocty along the vertcal centrelne and the y-component of the velocty along the horzontal centrelne of the cavty at Re = Here the top ld moves from left to rght and t s observed that the agreement between our LBM results and those of Gha et al. [16] s excellent. The close agreement gves credblty to the result of our LBM code and t stands valdated. The same fgure also dsplays the FDM results [1] of the present authors, whch agan are n excellent agreement wth these results. (8) Fgure 3. Code valdaton: (a) u-velocty along vertcal centrelne and (b) v-velocty along horzontal centrelne for sngle ld-drven square-cavty (Re = 1000). 3. Two-Sded Ld-Drven Cavty Flow An ncompressble vscous flow n a square cavty whose top and bottom walls move n the same (parallel moton) or opposte (antparallel moton) drecton wth a unform velocty s the problem nvestgated n the present work. In the case of parallel wall moton, a free shear layer exsts mdway between the top and bottom walls apart from the wall bounded shear layers, whereas n the case of antparallel wall moton, only wall bounded shear layers exst. 29

6 Smulaton of Incompressble Flows n Two-Sded Ld-Drven Square Cavtes. Part II - LBM 3.1. Parallel Wall Moton In the parallel moton we consder, both the upper and lower plates move from left to rght n the x drecton wth the same velocty. Fgure 4 shows the streamlne patterns for varous Reynolds numbers on a lattce structure. Expectedly, the streamlnes are found to be symmetrcal wth respect to the horzontal centerlne. Fgure 4(a) shows the streamlne pattern for Re = 100. Two counter-rotatng prmary vortces symmetrcal to each other are seen to form wth a `free shear layer n between. At ths Reynolds number the prmary vortex cores are seen to be somewhat away from the centres of the top and the bottom halves of the cavty towards the rghthand top and rghthand bottom corners respectvely. Fgure 4(b) shows the streamlne pattern for Re = 400. At ths Reynolds number s seen also a par of counter-rotatng secondary vortces symmetrcally placed about the horzontal centrelne near the centre of the rght wall. Fgures 4(c) and 4(d) show the streamlne patterns for Re = 1000 and Re = 2000 respectvely. It s seen that wth the ncrease n Reynolds number the prmary vortex cores move towards the centres of the top and bottom halves of the cavty and the secondary vortex par grow n sze. At all the Reynolds numbers the counter-rotatng pars of prmary and secondary vortces mantan ther symmetry about the horzontal centerlne. (a) Re = 100 (b) Re = 400 (c) Re = 1000 (d) Re = 2000 Fgure 4. Streamlne pattern for parallel wall moton at (a) Re = 100 (b) Re = 400 (c) Re = 1000 and (d) Re = 2000 on a lattce. 30

7 Perumal & Dass CFD Letters Vol. 2(1) 2010 (a) Re = 100 (b) Re = 400 (c) Re = 1000 (d) Re = 2000 Fgure 5. Vortcty contours for parallel wall moton at (a) Re = 100 (b) Re = 400 (c) Re = 1000 and (d) Re = 2000 on a lattce. Fgure 6. A magnfed vew of secondary vortces for parallel wall moton at Re = 1000 (a) LBM, (b) FDM [1]. 31

8 Smulaton of Incompressble Flows n Two-Sded Ld-Drven Square Cavtes. Part II - LBM Fgure 5 shows the vortcty contours. In Fgure 6 a magnfed vew of the secondary vortces at Re = 1000 s compared wth those obtaned by FDM [1] to show that the vortces have been captured accurately. The other results are now authentcated by comparson wth the same reference. Fgures 7-10 show the comparson for horzontal velocty profles along vertcal lnes and vertcal velocty profles along horzontal lnes passng through dfferent ponts of the cavty for varous Reynolds numbers. Agreement of the velocty profles gven by LBM and FDM s once agan excellent. Table 1 gves the locatons of the vortces gven by the LBM and FDM for Re = 100, 400, 1000, 1500 and All these results show that the agreement s very good, whch further substantates the accuracy of the present LBM computatons. We thus accurately lst varous flow detals for ths relatvely unexplored confguraton. Fgure 7. Parallel wall moton, Re = 100: (a) horzontal velocty u along vertcal lnes (x=0.25, 0.50, 0.75), (b) vertcal velocty v along horzontal lnes (y=0.25, 0.50, 0.75). Fgure 8. Parallel wall moton, Re = 400: (a) horzontal velocty u along vertcal lnes (x=0.25, 0.50, 0.75), (b) vertcal velocty v along horzontal lnes (y=0.25, 0.50, 0.75). Fgure 9. Parallel wall moton, Re = 1000: (a) horzontal velocty u along vertcal lnes (x=0.25, 0.50, 0.75), (b) vertcal velocty v along horzontal lnes (y=0.25, 0.50, 0.75). 32

9 Perumal & Dass CFD Letters Vol. 2(1) 2010 Fgure 10. Parallel wall moton, Re = 2000: (a) horzontal velocty u along vertcal lnes (x=0.25, 0.50, 0.75), (b) vertcal velocty v along horzontal lnes (y=0.25, 0.50, 0.75). TABLE 1: LOCATIONS OF THE VORTICES FOR PARALLEL WALL MOTION: a. FDM [1], b. LBM. Prmary vortex centres Secondary vortex centres Re 100 Bottom Top Bottom Top x y x y x y x y a b a b a b a b a b Antparallel Wall Moton In the antparallel wall moton the upper and lower walls move n opposte drectons along the x-axs wth the same velocty. Fgure 11 shows the streamlne patterns on a lattce sze of for the antparallel wall moton. A sngle prmary vortex centred at the geometrc centre of the cavty s formed at low Reynolds numbers as shown n Fgure 11(a) for Re = 100 and Fgure 11(b) for Re = 400. The streamlne patterns for Re = 1000 and 2000 are shown n Fgures 11(c) and 11(d) respectvely. The ncreased Reynolds number results n the appearance of two secondary vortces near the top left and the bottom rght corners of the cavty and a very small shft of the prmary vortex centre from the geometrc centre of the cavty. It wll not be out of place to menton here that for the much-examned sngle ld-drven cavty flow the secondary vortex near the tralng edge of the movng wall does not appear at a Reynolds number as low as 1000 but much beyond that (at some value hgher than 2000). It s also seen that wth the ncrease n Reynolds number the shft of the prmary vortex centre from the geometrc centre of the cavty s small so that t remans very close to the geometrc centre even 33

10 Smulaton of Incompressble Flows n Two-Sded Ld-Drven Square Cavtes. Part II - LBM for these hgher values of Re = 1000 and However, n lne wth parallel wall moton, between Re = 1000 and 2000 the secondary vortces are seen to grow n sze. The vortcty contours for varous Reynolds numbers are shown n Fgure 12. A magnfed vew of the secondary vortces (the ones at the top left corner) at Re = 1000 gven by the present LBM and FDM [1] s shown n Fgure 13. The agreement s found to be very good. Fgures show the comparsons for horzontal velocty profles along vertcal lnes and vertcal velocty profles along horzontal lnes passng through dfferent ponts of the cavty for the varous Reynolds numbers and the agreement s excellent once agan. Table 2 gves the locatons of the vortces gven by the present LBM and FDM [1] for Re = 100, 400, 1000, 1500 and Clearly the results gven by the LBM agree very well wth those of the FDM. We thus see that the LBM results gven by all the fgures and tables are n excellent agreement wth those gven by the FDM [1]. Ths lends credblty to the current LBM results for ths problem. (a) Re = 100 (b) Re = 400 (c) Re = 1000 (d) Re = 2000 Fgure 11. Streamlne pattern for antparallel wall moton at (a) Re = 100 (b) Re = 400 (c) Re = 1000 and (d) Re = 2000 on a lattce. 34

11 Perumal & Dass CFD Letters Vol. 2(1) 2010 (a) Re = 100 (b) Re = 400 (c) Re = 1000 (d) Re = 2000 Fgure 12. Vortcty contours for antparallel wall moton at (a) Re = 100 (b) Re = 400 (c) Re = 1000 and (d) Re = 2000 on a lattce. Fgure 13. A magnfed vew of secondary vortces for antparallel wall moton at Re = 1000 (a) LBM, (b) FDM. 35

12 Smulaton of Incompressble Flows n Two-Sded Ld-Drven Square Cavtes. Part II - LBM Fgure 14. Antparallel wall moton, Re = 100: (a) horzontal velocty u along vertcal lnes (x=0.25, 0.50 and 0.75), (b) vertcal velocty v along horzontal lnes (y=0.25, 0.50 and 0.75). Fgure 15. Antparallel wall moton, Re = 400: (a) horzontal velocty u along vertcal lnes (x=0.25, 0.50 and 0.75), (b) vertcal velocty v along horzontal lnes (y=0.25, 0.50 and 0.75). Fgure 16. Antparallel wall moton, Re = 1000: (a) horzontal velocty u along vertcal lnes (x=0.25, 0.50 and 0.75), (b) vertcal velocty v along horzontal lnes (y=0.25, 0.50 and 0.75). Fgure 17. Antparallel wall moton, Re = 2000: (a) horzontal velocty u along vertcal lnes (x=0.25, 0.50 and 0.75), (b) vertcal velocty v along horzontal lnes (y=0.25, 0.50 and 0.75). 36

13 Perumal & Dass CFD Letters Vol. 2(1) 2010 TABLE 2: LOCATIONS OF THE VORTICES FOR ANTIPARALLEL WALL MOTION: (a) FDM [1], (b) LBM. Re Prmary Vortex (PV) x 100 a b a b a b a b a b y Secondary Vortces (SV) Bottom Rght Top Left x y x y Concluson In ths work a relatvely unexplored flow confguraton n a two-sded ld-drven square cavty s computed wth the LBM. The flow s nvestgated for both the parallel and antparallel moton of two of the facng walls. In the case of parallel wall moton, besdes two prmary vortces, there also appears a par of counter-rotatng secondary vortces symmetrcally placed about the centerlne parallel to the moton of the walls. About ths centerlne also appears a free shear layer wth the ncrease n Reynolds number. In the case of ant-parallel wall moton, besdes a sngle prmary vortex, there appears two secondary vortces near the tralng edges of the movng walls. Each of these near-tarlng-edge vortces are seen to appear at a much lower Reynolds number compared wth that n the sngle ld-drven cavty flow. The results of nvestgaton of ths relatvely unexplored problem comprse fgures and tables detalng streamlnes and vortcty contours, varous velocty profles and features of vortces lke ther centres and shapes. All these LBM results compare very well wth the only exstng (and accurate) set of FDM results. Ths not only lends credblty to, but also benchmarks these results for the present flow confguraton. Consequently these results, lke those of the sngle ld-drven cavty flow, may be used for valdatng the algorthms for computng steady flows governed by the 2D ncompressble Naver- Stokes equatons. References [1] Perumal, D.A, and A.K. Dass, Smulaton of Incompressble flows n two-sded ld-drven square cavtes. Part I FDM, CFD Letters, 2(1), pp [2] Wolf-Gladrow, D.A., Lattce-Gas Cellular Automata and Lattce Boltzmann Models: An Introducton. 2000: Sprnger-Verlag Berln-Hedelberg. [3] Succ, S., The Lattce Boltzmann Method for Flud Dynamcs and Beyond. 2000: Oxford Unversty Press. [4] Chen, S., and G.D. Doolen, Lattce Boltzmann method for flud flows. Annual Revew of Flud Mechancs, : p [5] Hardy, J., Y. Pomeau, and O. de Pazzs, Tme Evoluton of a Two-Dmensonal classcal lattce system. Physcal Revew Letters, : p [6] Frsch, U., B. Hasslacher, and Y. Pomeau, Lattce-Gas automata for the Naver-Stokes equatons. Physcal Revew Letters, : p [7] McNamara, G.R., and G. Zanett, Use of the Boltzmann Equaton to Smulate Lattce-Gas Automata. Physcal Revew Letters, : p

14 Smulaton of Incompressble Flows n Two-Sded Ld-Drven Square Cavtes. Part II - LBM [8] Hguera., and Jmenez, Boltzmann approach to Lattce Gas Smulatons. Europhyscs Letters, (7): p [9] Bhatnagar P.L., E.P. Gross and M. Krook, A Model for Collson Processes n Gases. I. Small Ampltude Processes n Charged and Neutral One-Component Systems. Physcal Revew, : p [10] Koelman J.M.V.A., A Smple Lattce Boltzmann Scheme for Naver-Stokes Flud Flow. Europhyscs Letters, (6): p [11] Chen S., H. Chen, Martnez and Matthaeus, Lattce Boltzmann Model for Smulaton of Magnetohydrodynamcs. Physcal Revew Letters, (27): p [12] Qan Y.H., D. d'humeres and P. Lallamand, Lattce BGK Models for Naver-Stokes Equaton. Europhyscs Letters, (6): p [13] Chen H., S. Chen, and Matthaeus, Recovery of the Naver-Stokes Equaton usng a lattcegas Boltzmann method. Physcal Revew A, (8): p. R [14] Shankar, P.N, and M.D. Deshpande, Flud Mechancs n the Drven Cavty. Annual Revew of Flud Mechancs, : p [15] Kuhlmann H.C., M. Wanschura, and H.J. Rath, Flow n two-sded ld-drven cavtes: nonunqueness, nstablty, and cellular structures. Journal of Computatonal Physcs, : p [16] Gha U., K.N. Gha, and C.T. Shn, Hgh-Re solutons for ncompressble flow usng Naver- Stokes equatons and a multgrd method. Journal of Computatonal Physcs, : p [17] Hou, S., Q. Zou, S. Chen, G. Doolen, and A.C. Cogley, Smulaton of Cavty Flow by the Lattce Boltzmann Method. Journal of Computatonal Physcs, : p [18] Perumal, D.A, and A.K. Dass, Lattce Boltzmann Smulaton of Two and Three Dmensonal Flow n a Ld-Drven Cavty. n 2008 Proceedngs of the Int. Conference on Advances n Mechancal Engg, IC-ICAME IISc Bangalore, INDIA. [19] La Y.G., C.L. Ln, and J. Huang, Accuracy and Effcency study of a Lattce Boltzmann method for steady-state flow smulatons. Numercal Heat Transfer, Part B, : p [20] Yu D., R. Me, L.S. Luo, and W. Shyy, Vscous flow computatons wth the method of lattce Boltzmann equaton. Progress n Aerospace Scences, : p