Numerical Simulation of Lid-Driven Cavity Flow Using the Lattice Boltzmann Method

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1 Proceedngs of the 3th WSEAS Internatonal Conference on APPLIED MATHEMATICS (MATH'8) Numercal Smulaton of Ld-Drven Cavty Flow Usng the Lattce Boltzmann Method M.A. MUSSA, S. ABDULLAH *, C.S. NOR AZWADI 3, N. MUHAMAD AND K. SOPIAN 2 Department of Mechancal and Materal Engneerng 2 Solar Energy Research Insttute (SERI) Natonal Unversty of Malaysa 436 UKM Bang, Selangor MALAYSIA 3 Faculty of Mechancal Engneerng Technology Unversty of Malaysa 83 Skuda, Johor MALAYSIA * shahrr@eng.ukm.my Abstract: - Ths paper presents the smulaton of ld-drven cavty for deep and shallow flow usng the lattce Boltzmann method. The effect the Reynolds number on the flow pattern at aspect rato of 5,.5,.5, 4 was studed extensvely. The lattce Boltzmann method s one of the most recent smulaton technques based on molecular theory whch s found to be a very effcent numercal tool due to ts capablty to go deeper nto the partcle s doman, smulatng ther nteracton among groups of partcle and relatng the parameters back to macro condton. The comparson of the results was made wth CFD software, verson 6.. Excellent agreement was obtaned even relatvely wth coarse grds appled on lattce the Boltzmann smulaton scheme. Keywords: - Lattce Boltzmann method, dstrbuton functon, mcroscopc velocty, tme relaxaton, ld-drven cavty flow Introducton The lattce Boltzmann method (), a numercal method evolved from mathematcal statstcal approach has been well accepted as an alternatve numercal scheme n computatonal flud dynamcs feld. In comparson wth other numercal schemes, s a bottom up approach, derves the Naver- Stokes uaton from statstcal behavor of partcles dynamcs. The magnary propagaton and collson processes of flud partcles are reconstructed n the formulaton of scheme. These processes are represented by the evoluton of partcle dstrbuton functon, f ( x, t) whch descrbes the statstcal populaton of partcles at locaton x and tme t. The advantages of nclude smple calculaton procedure, sutablty for parallel computaton, ease and robust handlng of multphase flow, complex geometres, nterfacal dynamcs and others []. A few standard benchmark problems have been smulated by and the results are found to agree well wth the correspondng Naver- Stokes solutons [2]. The ld-drven cavty flow s one of the most mportant benchmarks for new numercal method to be developed. It represents the flow of a rectangular or square geometry where the flow s drven by a tangental moton wth constant velocty of a sngle ld, representng the Drchlet boundary condtons. Moreover, the drven-cavty flow exhbts a number of nterestng physcal features [3]. However, on the other hand, the smulaton of ld-drven flow nsde cavty s scarcely performed. Only a few studes have dealt wth the nfluence of the heght of cavty on the flow nsde. In comparson wth the flow nsde a square cavty, one new parameter has to be taken nto account, the aspect rato of the cavty, K = W/H, W beng the cavty wdth and H the depth. As shown by Hou et al. [3] and Shen et al. [4], the physcs of drven flow n a cavty ndcate the exstence of crtcal aspect rato at whch the corner eddes merge and form a prmary eddy. The dependence of the vortex structure on the aspect rato at dfferent Reynolds numbers was nvestgated by Pan and Acrvos [5]. A flow vsualzaton experment was conducted to nvestgate the effect of nerta force n the flow structure n the Reynolds number of range 2 Re 4. Due to expermental lmtaton, the shallow cavty flow should consst essentally of a ISSN: ISBN:

2 Proceedngs of the 3th WSEAS Internatonal Conference on APPLIED MATHEMATICS (MATH'8) sngle nvscd core of unform vortcty, whle the vortex structure below the prmary vortex n the deep cavty could not be captured. Therefore, the objectves of the present artcle are to extend the smulatons for cavty flows and to nterpret the results wth partcular focus on the structure of prmary and secondary eddes. 2 Numercal Method Hstorcally, s the logcal development of lattce gas automata (LGA) method [6]. Lke n LGA, the physcal space s dscretzed nto unform lattce nodes. Every node n the network s then connected wth ts neghbours through a number of lattce veloctes to be determned through the model chosen. The lattce Boltzmann uaton s gven by: f t + c f,α =Ω( f ) () x α where f s the dstrbuton functon for partcles wth velocty c at poston x and tme t. Equaton () conssts of two parts; propagaton (left-hand sde) whch refers to the propagaton of dstrbuton functon to the next node n the drecton of ts Ω (rght-hand sde) whch represent the collson of the partcle dstrbuton functon. In, magntude of c s set up so that n each tme step t, every dstrbuton functon propagates n a dstance of lattce nodes spacng x. Ths wll ensure that dstrbuton functon arrves exactly at the lattce nodes after t and colldes smultaneously. There are a few versons of collson operator probable velocty, and collson ( f) ( f) Ω publshed n lterature. However, the most well accepted verson due to ts smplcty and effcency was the Bhatnagar-Gross-Crook (BGK) collson model [7]. The uaton that represents ths model s gven by: ( x, c, t) f ( x, c, t) f Ω ( f) = (2) τ where f s the ulbrum dstrbuton functon and τ s the tme to reach ulbrum condton durng collson process and s called relaxaton tme. Equaton (2) of the BGK collson model descrbes that τ of non-ulbrum dstrbuton relaxes to ulbrum state wthn tme τ on every collson process. By replacng the BGK collson model nto the Boltzmann uaton, the BGK Boltzmann Equaton s obtaned, f f f f + cα = (3) t xα τ The general form of the lattce velocty model s expressed as DnQm where D represents spatal dmenson and Q s the number of connecton (lattce velocty) at every node. In ths paper, the nne-mcroscopc velocty or nne-bt model (D2Q) s used. The lattce geometry s shown n Fg. c 7 c 4 c 3 c 6 c c 2 c c 8 c 5 Fg. (D2Q) lattce geometry The ulbrum dstrbuton functon of the nne-bt model s [8]: f = ρω + 3( c u) + ( c u) u (4) 2 2 The weghts are: ω 4,, = ω 2~5 = ω 6~ = (5) 36 and the mcroscopc velocty components are: c = (6) The macroscopc quanttes can be calculated from: ρ (7) = = f u cf f (8) Through the Chapman-Enskog procedure, the ncompressble Naver-Stokes uatons derved from the ncompressble []: u = () u 2τ + u u= P+ 2 u () t 6 By comparng the momentum uaton obtaned from the second Newton s law wth uaton (), the relaton between tme relaxaton τ and shear vscosty ν s obtaned as follow: τ = 3 ν + 2 () Equaton () descrbes that the value of τ must be kept hgher than.5 n order to avod negatve value of knematc vscosty. Ths lmts our smulaton to low value of Reynolds number. However ths lmtaton can be partly solved by usng hgh number of nodes but wll lead to long computatonal tme. Ths s to be addressed n our future works. ISSN: ISBN:

3 Proceedngs of the 3th WSEAS Internatonal Conference on APPLIED MATHEMATICS (MATH'8) 3 Results and Dscusson Ths secton appled the lattce Boltzmann scheme dscussed n the preceedng secton to smulate the ld-drven cavty flow at dfferent Reynolds number. 3. Code Valdaton For the purpose of code valdaton, we carred out the smulaton of ld-drven flow n a square cavty, K = and compared wth benchmark result []. Ld-drven flow n a square cavty s well-known as a standard test case for the numercal schemes of flud flows. Supposed that the ld s located at the top boundary y = H, and moves wth a constant speed, U from left to rght. The Reynolds number for the system s gven by: UL Re = (2) ν Smulatons were done at Re= usng grd sze 2 2. dstrbuton are n good agreement wth the prevous work of Gha et. al []. 3.2 Deep Cavty Flow was carred out to study flud flow over a deep cavty wth aspect rato K = 5 and.5. D2Q model wth the BGK collson model used. Shown n Fg. 3 are the streamlnes for cavty of aspect rato.5. The three fgures correspond to Re=,4 and respectvely. These smulaton employed 2 nodes. The smulatons were consdered to have reached steady state when the r.m.s. change n horzontal and vertcal velocty decreased to magntude 6 or less. (a) (b) (c) Fg.3 Streamlne at steady state condton (a) Re = (b) Re = 4 and (c) Re = It may be seen from Fg. 3 that, for low Reynolds number smulaton Re=, a counter-rotatng vortex s formed below the movng ld. As the Reynolds number ncreases ( Re= 4), the center of the prmary eddy begns to move downwards wth respect to the top ld. For the case of hgh Reynolds number ( Re=), the prmary eddy s formed at the center of the geometry. The secondary vortex can also be clearly seen at low Raylegh number and ntally formed at the lower rght of the geometry. As we ncrease the Reynolds number, ths vortex shft to the center left of the geometry test case. Ths s perhaps due to the effect of vscous effect produced by the prmary vortex. Fg.2 (top) Streamlne at steady state condton. (bottom) Velocty profle at md-heght (y-velocty) and md-wdth (x-velocty) of cavty Fg. 2 shows plot of stream functon and velocty profle for the Reynolds numbers consdered. It s apparent that the flow structure and velocty 3.3 Shallow Cavty Flow In ths secton numercal soluton of lattce Boltzmann method for rectangular shallow cavtes s present. Fg.4 shows that for aspect rato D =.5 (a),(b) and (c), the prmary vortex center descend to the ccnter of cavty as the Reynolds number ncreased. Two secondary vortces also can be ISSN: ISBN:

4 Proceedngs of the 3th WSEAS Internatonal Conference on APPLIED MATHEMATICS (MATH'8) clearly seen at the lower corners of the cavty wth left vortex s bgger than the rght vortex. (a) (b) (c) Fg.4 Steamlne at steady state condton for K =.5 (a) Re = (b) Re = 4 and (c) Re = For aspect rato D = 4 (a),(b) and (c), as Reynolds number s ncreased, Fg.5 shows that the prmary vortex began to splt nto two vortces. No any secondary vortces n the corners has been notced. Re = Re = 4 Re = Fg.6 Comparson between and for K = 5 (a) (b Re = Re = 4 (c) Fg.5 Steamlne at steady state condton for K = 4 (a) Re = (b) Re = 4 and (c) Re = Fgs. 6, 7, 8 and show the comparson of centerlne velocty profle obtaned from and. In ths comparson, K = 5,.5,.5 and 4 for Re =, 4 and has been selected. It may be observed that the present results are n excellent agreement wth the results for every tested case. Re = Fg.7 Comparson between and for K = Re = Re = Re = Fg.8 Comparson between and for K =.5 ISSN: ISBN:

5 Proceedngs of the 3th WSEAS Internatonal Conference on APPLIED MATHEMATICS (MATH'8) Re = Re = Re = Fg. Comparson between and for K = 4 4 Concluson The flow structure of a two-dmensonal vscous flow n ld-drven deep cavtes has been studed numercally usng lattce Boltzmann scheme. The dynamc of the prmary and secondary vortces are well captured by these smulatons. Whenever comparson s possble, the present results are found to be n good agreement wth the results obtaned by CFD software, verson 6.. The effects of boundary condtons n the lateral drectons (threedmensonal) effect wll be consdered n our future study. Acknowledgments The authors would lke to acknowledge the Mnstry of Scence, Technology and Innovaton, Malaysa for sponsorng ths work. References: [] S. Chen, and G. Doolen,, Lattce Boltzmann Method for Flud Flows, Annual Revew of Flud Mechancs, Vol. 3, 8, pp [2] G. McNamaea, and B. Alder,, Analyss of Lattce Boltzmann Treatment of Hydrodynamcs., Physca A, Vol. 4, 3, pp [3] S. Hou, Q. Zou, S. Chen, G. Doolen and A. C. Cogley, Smulaton of cavty flow by the lattce Boltzmann Method., Journal of Computatonal Physcs, Vol. 8, 5, pp [4] C. Shen, and J. M. Floryan, Low Reynolds Number Flows over Cavtes, Physcs of Fluds, 85, Vol. 28, pp.-22. [5] F. Pan and A. Acrvos, Steady Flows n Rectangular Cavtes., Journal of Flud Mechancs, 67, Vol. 28, pp [6] U. Frsh, B. Hasslacher and Y. Pomeau, Lattce gas automata for the Naver-Stokes uaton, Physcs Rev. Letters, vol. 56, 86, pp [7] P. L. Bhatnagar, E. P. Gross and M. Krook, A Model for Collson Processes n Gasses, Phys. Rev. Vol. 4, 54, pp [8] C.S. Nor Azwad and T. Tanahash, Threedmensonal thermal lattce-boltzmann smulaton of natural convecton n a cubc cavty, Intl. J. Modern Physcs B, Vol. 2, 27, pp [] L. S. Luo and X. He, Lattce Boltzmann model for the ncompressble Naver-Stokes uaton, J. Stat. Phys. Vol. 88, 7, pp [] U. Gha, K. N. Gha, and C. T. Shn, Hgh Re Soluton for Incompressble Flow usng the Naver-Stokes Equatons and a Multgrd Method, Journal of Computatonal Physcs, Vol. 48, 82, pp [] C. S. Azwad and T. Tanahash, Smplfed Thermal Lattce Boltzmann n Incompressble Lmt., Internatonal Journal of Modern Physcs B, Vol. 2, 26, pp [2] DP. Zegler, Boundary Condtons For Lattce Boltzmann Smulatons. J. Stat Phys. Vol. 7, 3, pp [3] D.V. Patl, K.N. Lakshmsha and B. Rogg. Lattce Boltzmann smulaton of ld-drven flow n deep cavtes, Journal of Computers & Fluds, Vol. 35, 26, No., pp [4] M. Cheng, K.C. Hung, Vortex structure of steady flow n a rectangular cavty, Journal of Computers & Fluds, Vol. 35, 26, No., pp ISSN: ISBN: