LATTICE BOLTZMANN SIMULATION OF FLOW OVER A CIRCULAR CYLINDER AT MODERATE REYNOLDS NUMBERS

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1 THERMAL SCIENCE: Year 014, Vol. 18, No. 4, pp LATTICE BOLTZMANN SIMULATION OF FLOW OVER A CIRCULAR CYLINDER AT MODERATE REYNOLDS NUMBERS by Dharmaraj ARUMUGA PERUMAL a*, Gundavarapu V. S. KUMAR b, and Anoop K. DASS b a Automotve Research Centre, SMBS, VIT Unversty, Vellore, Tamlnadu, Inda, b Department of Mechancal Engneerng, Indan Insttute of Technology Guwahat, Guwahat, Inda Orgnal scentfc paper DOI: 10.98/TSCI A Ths work s concerned wth Lattce Boltzmann computaton of -D ncompressble vscous flow past a crcular cylnder confned n a channel. Computatons are carred out both for steady and unsteady flows and the crtcal Reynolds number at whch symmetry breaks and unsteadness sets n s predcted. Effects of Reynolds number, blockage rato and channel length are studed n some detals. All the results compare qute well wth those computed wth contnuum-based methods, demonstratng the ablty and usefulness of the Lattce Boltzmann method n capturng the flow features of ths nterestng and flud-mechancally rch problem. Key words: -D, crcular cylnder, confned channel, Lattce Boltzmann method, DQ9 model Introducton The classcal problem of vscous ncompressble flow over a crcular cylnder confned n a channel s one of the most wdely studed problems n computatonal flud dynamcs (CFD) [1]. Ths flow, at lower Reynolds numbers (Re), s steady whereas at hgher Reynolds numbers unsteadness develops and the flow becomes perodc. The perodc flow s characterzed by the so called von Karman vortex street generated by alternate and perodc sheddng of vortces from the top and bottom. Ths type of flow problems frequently arse n varous engneerng felds whch offer tough challenges partcularly at hgh Reynolds numbers []. Because of ts popularty, a plethora of numercal, theoretcal and expermental results are avalable for ths problem n the lterature [3-10]. Some of the notable works on flow past a crcular cylnder by the conventonal numercal methods nclude those of Fornberg [4], Braza [5] and Franke et al., [6]. Calhoun [7] descrbed a fnte-dfference/fnte-volume algorthm for solvng the streamfuncton-vortcty equatons n an rregular geometry such as flow past a crcular cylnder. Sahn and Owens [8] used a fnte volume method to solve the flow feld around a crcular cylnder confned n a channel. They nvestgated lateral wall proxmty effects on stablty, Strouhal number, hydrodynamc forces and wake structure behnd the cylnder for a wde range of blockage ratos. Zovatto and Pedrzzett [9] presented solutons for the flow past a crcular cylnder placed nsde a plane channel at varous dstances * Correspondng author; e-mal: perumal.d@vt.ac.n

2 136 THERMAL SCIENCE: Year 014, Vol. 18, No. 4, pp from the walls for dfferent Reynolds number from steady flows to the ntaton of the vortex sheddng regme. Anagnostopoulos and Ilads [10] used the fnte element technque for the soluton of steady and unsteady flow around a crcular cylnder at Re = 10 6 for dfferent blockage ratos. More recently, Sngha and Snhamahapatra [11] studed the flow over a confned cylnder problem and confrmed that transton to vortex sheddng regme s delayed when the channel walls are close to the cylnder because of the nteracton between the vortces from the channel wall and cylnder wake. In the last two decades lattce Boltzmann method (LBM), a non-conventonal numercal method for applcatons n CFD has ganed popularty [1]. Dstngushed from conventonal smulaton methods lke fnte dfference method (FDM), fnte volume method (FVM), and fnte element method (FEM), the lattce Boltzmann method provdes stable and effcent numercal calculatons for the macroscopc behavour of fluds, although descrbng the flud n a mcroscopc way [13-1]. It s observed that a large number of attempts have been made to study flow over a cylnder usng contnuum-based methods [-5] and a very few attempts have been made usng lattce Boltzmann method. Ths paper ams at brngng out the merts of the lattce Boltzmann method n capturng varous flow features of ths nterestng and flud-mechancally rch problem. Besdes presentng steady and unsteady results for varous Reynolds numbers, the present work also predcts, through LBM, the crtcal Reynolds number at whch unsteadness sets n. Lattce Boltzmann method In the last one and a half decade or so LBM has emerged as a new and effectve approach of computatonal flud dynamcs (CFD) and t has acheved consderable success n smulatng flud flows and heat transfer [14]. The justfcaton for ths approach s the fact that the collectve behavour of many mcroscopc partcles s behnd the macroscopc dynamcs of a flud and ths dynamcs s not partcularly dependent on the detals of the mcroscopc phenomena exhbted by the ndvdual molecules. The LBM therefore uses a smplfed knetc model to work wth. In spte of ts knetc orgn LBM however, s manly used to capture the macroscopc behavour represented by varables descrbng flud flow and heat transfer. The LBGK model wth the sngle tme relaxaton, whch s a commonly used lattce Boltzmann method, s gven by [15]: eq f( t, t t) f( 1 x e D D x, t) [ f( x, t) f ( x, t) ] (1) t where f s the partcle dstrbuton functon, f eq ( x, t ) the equlbrum dstrbuton functon at x, t, e the partcle velocty along the th drecton, and t the tme relaxaton parameter. The DQ9 square lattce (fg. 1) used here has nne dscrete veloctes. The partcle veloctes are defned as [15]: e 00,, 0 p p e cos ( 1),sn ( 1), 1,,3,4 () 4 4 Fgure 1. -D nne-velocty square lattce model p p e ( 1), sn ( 1), 5, 6, 7, 8 4 4

3 THERMAL SCIENCE: Year 014, Vol. 18, No. 4, pp The macroscopc quanttes such as densty r and momentum densty r u are defned as velocty moments of the dstrbuton functon f as follows: r N f 0 (3) ru f e (4) 0 where N = 8. The densty s determned from the partcle dstrbuton functon. The densty and the veloctes satsfy the Naver-Stokes equatons n the low-mach number lmt [1]. In the DQ9 square lattce, a sutable equlbrum dstrbuton functon that has been proposed s [15]: eq f rw 1 3 u, 0 f f eq eq N rw[ 13( eu ) 45. (eu) 15. u], 1,, 3, 4 rw [ 13( eu) 45. ( e u) 15. u ], 5678,,, where the lattce weghts are gven by w 0 = 4/9, w 1 = w = w 3 = w 4 = 1/9, w 5 = w 6 = w 7 = w 8 = 1/36. The relaxaton tme whch fxes the rate of approach to equlbrum s related to the vscosty by [13]: 6n 1 t (6) where n s the knematc vscosty measured n lattce unts. Problem and assocated features of LBM The schematc dagram of the flow past a crcular cylnder confned n a channel s shown n fg.. The present problem deals wth a cylnder wth crcular cross-secton wth dameter D mounted centrally nsde a plane channel of heght H wth blockage rato B = H/D. The channel length L s fxed at L/D =50 to reduce the nfluence of outflow boundary condtons on accuracy. An upstream length of l = L/4 or 1.5D has been chosen. In ths problem, the nlet boundary condtons are gven by a unform velocty profle. The LBM smulaton studes are carred out for dfferent values of Reynolds numbers Re = UD/g, where U s the velocty at the channel entry and D s the cylnder dameter. It s known that, n LBM smulaton the boundary condton for the partcle dstrbuton functon on a sold wall can be gven by a popular approach known as bounce-back scheme [13]. Therefore, n the present work, we use bounce-back boundary condton on the top and bottom walls, whch ndcates no-slp. We use bounce-back wth addtonal momentum to the nlet boundary wth known velocty to provde the requred nlet condton for the partcle dstrbuton functon f, as proposed by Yu et al. [14]. It creates a mechansm for the pressure wave from (5) Fgure. Schematc dagram of the flow past a crcular cylnder confned n a channel

4 138 THERMAL SCIENCE: Year 014, Vol. 18, No. 4, pp the nteror of the computatonal doman to nteract wth the nlet boundary and t can be reflected back. At the outlet of the computatonal doman, f can usually be approxmated by a smple extrapolaton boundary condton proposed by Guo et al. [1]. On the sold curved boundary,. e., on the surface of the cylnder, a robust second order accurate boundary treatment proposed by Bouzd et al. [16] s used for the partcle dstrbuton functon. In the present LBM smulaton, we use the momentum exchange method to compute the flud force on the crcular cylnder. The total force actng on a sold body by flud can be wrtten as [14]: N d x F d e f x t f x e t t w x e a a [ ~ ~ a( b, ) a ( b ad, )][ 1 ( b a )] allx b 1 dt (7) where N d s the number of non-zero lattce velocty vectors and w(x b + e a ) s an ndcator, whch s 0 at x f and 1 at x b. The nner summaton calculates the momentum exchange between a sold node at x b, and all possble neghbourng flud nodes around that sold node. The outer summaton calculates the force contrbuted by all boundary nodes x b. The two most mportant characterstc quanttes of flow around a cylnder are the coeffcent of drag and coeffcent of lft. The coeffcents are defned as: coeffcent of drag C D F x 1 rau coeffcent of lft F y CL 1 rau a where F x and F y are the x- and y-components of the total flud force actng on the cylnder, A the projected area. Themes of present studes are: for a fxed blockage rato to study the crcular cylnder flow feld characterstcs, for dfferent blockage ratos to study the crcular cylnder flow feld characterstcs. for dfferent outlet boundary locatons to study the cylnder flow feld characterstcs, and for dfferent lengthwse cylnder locatons to study the flow feld characterstcs. Results and dscussons Fxed blockage rato For a blockage rato B = H/D = 8 computatons are carred out at dfferent Reynolds numbers wth the help of lattce Boltzmann method wth sngle-relaxaton-tme collson model. The lattce sze of s used for the present confguraton. Steady flow Streamlne patterns for a blockage rato B = 8 at Reynolds numbers Re = 4, 5, 10, 0, 30, and 40 are shown n fg. 3. At low Reynolds number (Re = 4) the steady flow past the crcular cylnder perssts wthout separaton as shown n fg. 3(a). As Reynolds number ncreases the separaton of the steady flow lamnar boundary layer s clearly observed n fgs. 3(b) and 3(c). At Re = 5, separaton of steady flow just begns and as Reynolds number ncreases a par of vortces develops behnd the crcular cylnder and they grow n sze wth Reynolds number. It s seen, fgs. 3(c)-3(f), that two vortces downstream of the cylnder, symmetrcally placed about the a

5 THERMAL SCIENCE: Year 014, Vol. 18, No. 4, pp Fgure 3. Streamlne patterns for steady flows past a crcular cylnder at dfferent Reynolds numbers for a blockage rato B = 8: (a) Re = 4, (b) Re = 5, (c) Re = 10, (d) Re = 0, (e) Re = 30, and (f) Re = 40; lattce sze: channel centrelne, develop and reman attached to the cylnder. From fgs. 3(a)-3(f), t s agan seen that for steady flows the sze of the vortces ncreases wth Reynolds number. In tab. 1, we present the coeffcent of drag for dfferent steady-flow Reynolds numbers. Expectedly as the Reynolds number ncreases coeffcent of drag (CD) decreases. Table 1. Coeffcent of drag for dfferent steady-flow Reynolds numbers Authors Re = 10 Re = 0 Re = 30 Re = 40 Trtton (expermental) [] Fornberg (numercal) [4] Calhoun (numercal) [7] Present LBM Unsteady flow In our LBM smulaton the perodc flow s computed at Re = 60, 100 and 150 for whch the streamlne patterns at a certan nstant are shown n fgs. 4(a)-(c). At Re = 60, many expermental and numercal results have shown conclusvely that eventually asymmetry sets n

6 140 THERMAL SCIENCE: Year 014, Vol. 18, No. 4, pp Fgure 4. Instantaneous streamlne lne patterns for flows past a crcular cylnder at dfferent Reynolds numbers for a blockage rato B = 8: (a) Re = 60 (b) Re = 100, and (c) Re = 150; lattce sze: and the flow becomes unsteady. From fgs. 4(a)-(c), t s seen that as the Reynolds number ncreases the streamlne patterns becomes wavy and snuous on the rear-sde of the crcular cylnder. Fgure 5 depcts the vortcty contours for flow past a crcular cylnder at varous Reynolds numbers (Re = 4, 5, 40, 60, 100, and 150). Interestngly for the steady flow, on the lne of symmetry downstream of the cylnder flow s rrotatonal. Steady flow vortces seem to be more and more elongated as the Reynolds number ncreases. From the vortcty contours we clearly observe that vortces wth negatve and postve vortces are alternatvely shed at Re = 60, 100, and 150. Ths s known as the famous von Karman vortex street. The pont of separaton s also clearly observed from the vortcty contours. From fgs. 5(d)-(f) t s seen that for a fxed length of the confnng channel, number of perodc vortctes shed ncreases wth Reynolds number. Thus wthout quantfyng the assocated frequences for the Reynolds numbers, t s possble to say that frequency of vortex sheddng ncreases as Reynolds number ncreases. Fgure 5. Vortcty contours for flows past a crcular cylnder at dfferent Reynolds numbers for a blockage rato B = 8: (a) Re = 4, (b) Re = 5, (c) Re = 40, (d) Re = 60, (e) Re = 100, and (f) Re = 150; lattce sze: In order to study the flow feld characterstcs, velocty varaton along the channel centrelne are presented for Re = 45. Fgure 6 shows the dstrbutons of x- and y-velocty com-

7 THERMAL SCIENCE: Year 014, Vol. 18, No. 4, pp Fgure 6. Instantaneous results at a certan moment for the flow over a crcular cylnder: (a) stream wse u, and (b) cross stream, v, veloctes along the centerlne (y = 0); Re = 45, lattce sze: ponents along the channel centrelne. It s observed that fluctuatons of velocty at the ext boundary are small. At the surface of the cylnder velocty s zero and at upstream locatons flow velocty vares smoothly wthout much oscllaton. Fgure 7 shows smlar plots when the channel length s ncreased from 500 to 600 lattce unts. It s observed that the fluctuaton at the outflow boundary s smaller for ths ncreased channel length. Ths s the justfcaton for applyng the outflow boundary condton at a large dstance from the cylnder. Fgure 7. Instantaneous results at a certan moment for the flow over a crcular cylnder: (a) stream wse u, and (b) cross stream v, veloctes along the centerlne (y = 0); Re = 45, lattce sze: Fgure 8 shows the perodc tme varaton of the lft coeffcent (C L ) and drag coeffcent (C D ) for the flow past a crcular cylnder at Re = 60 and 100. It s seen that for each tme perod, C D has two crests and two troughs of unequal ampltudes, whch are a consequence of the perodc vortex sheddng from the top and bottom surfaces. The same fgure shows that C L for each tme perod has just one trough and one crest; ths s because C L s not nfluenced by the

8 14 THERMAL SCIENCE: Year 014, Vol. 18, No. 4, pp Fgure 8. Flow over a crcular cylnder: Tme dependent lft and drag coeffcents (C L and C D )at (a) Re = 60 and (b) Re = 100 on a lattce sze of Table. Mean drag coeffcent and coeffcent of lft for the flow over a cylnder at Re = 100 Authors C Dmean C L Braza et al. [5] ±0.5 Calhoun [7] ±0.98 Present LBM 1.90 ±0.89 Fgure 9. Onset of unsteadness for a blockage rato B = 8.0 (A for u, A for v as a functon of Re) pressure dstrbuton on the rear sde of the cylnder, whch s strongly nfluenced by the vortex sheddng at the top and bottom sdes of the cylnder. The value of the lft force fluctuaton s drectly connected to the formaton and sheddng of the eddy and, therefore, ts value vares between a postve maxmum and a negatve maxmum of equal magntude. In tab., we present the mean drag coeffcent and coeffcent of lft for Re = 100. All the results computed wth present LBM compare well wth exstng numercal results [17-19]. Fgure 9 shows how a crtcal Reynolds number can be roughly predcted [0]. Here Au ndcates, at a pont n the wake regon, the ampltude of fluctuatons of u when perodcty sets n and Av ndcates the same for v. For fve dfferent Reynolds numbers, close to the crtcal, A u and A u are plotted aganst Re. Lnear fts through these ponts ntersects at a pont, whch approxmately ndcates the Reynolds number at whch vortex sheddng starts. Here t s observed that for a blockage rato B = 8 crtcal Reynolds number s between 43 and 44. Dfferent blockage ratos To study the effect of dfferent blockage ratos Re = 44 s chosen n the present work. Cylnder locaton of l = L/4 and a channel length of L/D = 50 s consdered here. Fgure10 shows the vortcty contours for confned flow over a crcular cylnder at dfferent blockage ratos. Sx dfferent blockage ratos B = 4, 7, 8, 10, 15, 0, 30 and 40 are studed n the present work. From

9 THERMAL SCIENCE: Year 014, Vol. 18, No. 4, pp Fgure 10. Instantaneous vortcty contours for flows past a crcular cylnder at dfferent blockage ratos (a) B =4,(b)B =7,(c)B =8,(d)B = 10, (e) B = 15, (f) B = 0, (g) B = 30, and (h) B =40 our LBM smulatons we observe that the moderate blockage rato B = 8 onwards unsteadness develops n the flow feld. The flow happens to be steady for lower values of blockage rato B (4 and 7 here). It can be clearly observed that at low blockage ratos wall boundary layer has strong nfluence over the flow around the cylnder so that t nhbts vortex sheddng. Flow becomes unsteady at hgher values of blockage rato B. As the blockage rato ncreases computaton tme requred for obtanng unsteadness ncreases. Dfferent outlet boundary condtons To study the effect of dfferent outlet boundary locatons Re = 44 s taken. In ths case, blockage rato B = 8 and upstream length l = L/4 s chosen. In the present case L/D = 30, 40, 50, and 60 are consdered. Fgure 11 shows the vortcty contours for confned flow over a crcular cylnder at dfferent outlet boundary locatons. It s clearly observed that the outlet boundary locaton s also playng mportant role n the vortex sheddng process for a partcular Reynolds number. For the extrapolaton boundary condton on the partcle dstrbuton functon at the outflow boundary to hold accurately, the boundary should be far enough downstream of the cyl-

10 144 THERMAL SCIENCE: Year 014, Vol. 18, No. 4, pp Fgure 11. Vortcty contours for flows past a crcular cylnder at dfferent outlet boundary locatons for Re = 44, l = 1.5D: (a) L/D = 30, (b) L/D = 40, (c) L/D = 50, and (d) L/D =60 nder. In our present study, smulaton results reveal the fact that L/D = 50 onwards perodc solutons are obtaned for Re = 44. Dfferent cylnder locatons To study the effect of dfferent cylnder locatons Re = 44 s chosen n the present work. As usual a blockage rato of B = 8 and L/D = 50 are consdered. All the cases studed so far take l = 1.5D. In the present case, l = 5.0D, 8.5D, 1.5D, 16.0D and 5.0D are consdered. Fgure 1 shows the nstantaneous vortcty contours for confned flow over a crcular cylnder for varous cylnder locatons. It s seen that only for the cylnder locaton of 1.5D and beyond the vortex sheddng appears. Also dfferences n the sze and strength of the vortces are observed as the cylnder locaton changes. Ths can be attrbuted to the fact that at a dfferent cylnder locatons the wall boundary layer produces dfferent effect. Also as l ncreases, the ext boundary may come too near for the prescrbed outflow boundary condton to hold accurately. Fgure 1. Instantaneous vortcty contours for flows past a crcular cylnder at dfferent cylnder locatons for B =8,Re=44:(a)l = 5.0D (b) l = 8.5D, (c) l = 1.5D, (d) l = 16.0D, and (e) l = 5.0D Conclusons In ths paper, LBM computaton s carred out for the flow past a crcular cylnder confned n a channel. Three dfferent aspects of the problem are studed. Frst, for a fxed blockage

11 THERMAL SCIENCE: Year 014, Vol. 18, No. 4, pp rato computatons are carred out to study the effect of Reynolds number on the flow pattern. At lower Reynolds numbers the flow s steady and a par of statonary vortces symmetrc about the channel centrelne develops and remans attached to the rear sde of the cylnder. As the Reynolds number ncreases the vortces grow n sze and fnally at a certan Reynolds number, the symmetry breaks and perodcty of flow sets n downstream of the cylnder. Usng a plot of ampltudes of fluctuaton n the x- and y-component of veloctes at a downstream pont, the crtcal Reynolds number at whch unsteadness sets n s found to be between 43 and 44, a result substantated by experments. Plots of u and v along the channel centerlne downstream of the cylnder s also gven to show that the oscllatons are small at the outflow boundary where extrapolaton seems approprate to be used to determne the boundary partcle dstrbuton functon. Perodc varatons of the lft and drag coeffcents at two Reynolds number are plotted and explanatons are gven for ther typcal behavour. Then the effect of blockage rato, defned n ths work as the rato of the channel wdth to the cylnder dameter, on the flow characterstcs s studed. It s seen that lower blockage rato tends to nhbt unsteadness. The effect of the outlet boundary locaton s also studed and expectedly t s seen that keepng the boundary at a large dstance downstream tends to favourably nfluence the accuracy of the computatons. Overall t s seen that the lattce Boltzmann method can be used very effectvely to produce accurate results and capture all the flow features that conventonal contnuum-based methods lke fnte dfference (F D ) and fnte volume (F V ) methods are capable of capturng. References [1] Patel, V. A., Karman Vortex Street behnd a Crcular Cylnder by the Seres Truncaton Method, Journal of Computatonal Physcs, 8 (1978), 1, pp [] Trtton, D. J., Experments on the Flow Past a Crcular Cylnder at Low Reynolds Numbers, Journal of Flud Mechancs, 6 (1959), 4, pp [3] Kalta, J. C., Ray, R. K., A Transformaton-Free HOC Scheme for Incompressble Vscous Flows Past an Impulsvely Started Crcular Cylnder, Journal of Computatonal Physcs, 8 (009), 14, pp [4] Fornberg, B., A Numercal Study of Steady Vscous Flow Past a Crcular Cylnder, Journal of Flud Mechancs, 98 (1980), 4, pp [5] Braza, M., et al., Numercal Study and Physcal Analyss of the Pressure and Velocty Felds n the Near Wake of a Crcular Cylnder, Journal of Flud Mechancs, 165 (1986), Apr., pp [6] Franke, R., et al., Numercal Calculaton of Lamnar Vortex Sheddng Flow Past Cylnders, Journal of wnd Engneerng and Industral Aerodynamcs, 35 (1990), 1-3, pp [7] Calhoun, D., A Cartesan Grd Method for Solvng the Two-Dmensonal Streamfuncton-Vortcty Equatons n Irregular Regons, Journal of Computatonal Physcs, 176 (00),, pp [8] Sahn, M., Owens, R. G., A Numercal Investgaton of Wall Effects up to Hgh Blockage Ratos on Two-Dmensonal Flow Past a Confned Crcular Cylnder, Physcs of Fluds, 16 (004), 5, pp. 5-0 [9] Zovatto, L., Pedrzzett, G., Flow about a Crcular Cylnder between Parallel Walls, Journal of Flud Mechancs, 440 (001), Aug., pp. 1-5 [10] Anagnostopoulos, P., Ilads, G., Numercal Study of the Blockage Effects on Vscous Flow Past a Crcular Cylnder, Internatonal Journal for Numercal Methods n Fluds, (1996), 11, pp [11] Sngha, S., Snhamahapatra, K. P., Flow Past a Crcular Cylnder between Parallel Walls at Low Reynolds Numbers, Journal of Ocean Engneerng, 37 (010), 8-9, pp [1] Perumal, D. A., Dass, A. K., Multplcty of Steady Solutons n Two-Dmensonal Ld-Drven Cavty Flows by the Lattce Boltzmann Method, Computers & Mathematcs wth Applcatons, 61 (011), 1, pp [13] Perumal, D. A., Dass, A. K., Smulaton of Incompressble Flows n Two-Sded Ld-Drven Square Cavtes, Part II-LBM, CFD Letters, (010), 1, pp [14] Yu, D., et al., Vscous Flow Computatons wth the Method of Lattce Boltzmann Equaton, Progress n Aerospace Scences, 39 (003), 5, pp

12 146 THERMAL SCIENCE: Year 014, Vol. 18, No. 4, pp [15] Hou, S., et al., Smulaton of Cavty Flow by the Lattce Boltzmann Method, Journal of Computatonal Physcs, 118 (1995),, pp [16] Bouzd, M., et al., Momentum Transfer of a Lattce Boltzmann Flud wth Boundares, Physcs of Fluds, 13 (001), 11, pp [17] Islam, S. U., Zhou, C. Y., Numercal Smulaton of Flow around a Row of Cylnders Usng the Lattce Boltzmann Method, Informaton Technology Journal, 8 (009), 4, pp [18] Shu, C., et al., Taylor Seres Expanson- and Least Square-Based Lattce Boltzmann Method: An Effcent Approach for Smulaton of Incompressble Vscous Flows, Progress n Computatonal Flud Dynamcs, 5 (005), 1/, pp [19] Ganaou, M. E., Semma, E. A., A Lattce Boltzmann Coupled to Fnte Volumes Method for Solvng Phase Change Problems, Thermal Scence, 13 (009),, pp [0] Nemat,H.,et al., Mult-Relaxaton-Tme Lattce Boltzmann Model for Unform Shear Flow over a Rotatng Crcular Cylnder, Thermal Scence, 15 (011), 3, pp [1] Guo, Z., et al., An Extrapolaton Method for Boundary Condtons n Lattce Boltzmann Method, Physcs of Fluds, 14 (00), 6, pp [] Coutanceau, M., Bouard, R., Expermental Determnaton of the Man Features of the Vscous Flow n the Wake of a Crcular Cylnder n Unform Translaton. Part 1: Steady Flow, Journal of Flud Mechancs, 79 (1977),, pp [3] De, A. K., Dalal, A., Numercal Smulaton of Unconfned Flow Past a Trangular Cylnder, Internatonal Journal for Numercal Methods n Fluds, 5 (006), 7, pp [4] Sharma, V., Dhman, A. K., Heat Transfer from a Rotatng Crcular Cylnder n the Steady Regme: Effects of Prandtl Number, Thermal Scence, 16 (01), 1, pp [5] Mat, D. K., Dependence of Flow Characterstcs of Rectangular Cylnders Near a Wall on the Incdent Velocty, Acta Mechanca, (011), 3, pp Paper submtted: September 18, 011 Paper revsed: May 9, 01 Paper accepted: May 9, 01