Hybrid lattice Boltzmann finite-difference simulation of axisymmetric swirling and rotating flows

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1 INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN FLUIDS Int. J. Nume. Meth. Fluds 27; 53: Publshed onlne 1 Octobe 26 n Wley InteScence Hybd lattce Boltzmann fnte-dffeence smulaton of asymmetc swlng and otatng flows Habo Huang,,T.S.LeeandC.Shu Flud Dvson, Depatment of Mechancal Engneeng, Natonal Unvesty of Sngapoe, 1 Kent Rdge Cescent, Sngapoe 11926, Sngapoe SUMMARY The asymmetc flows wth swl o otaton wee solved by a hybd scheme wth lattce Boltzmann method fo the aal and adal veloctes and fnte-dffeence method fo the azmuthal o swl) velocty and the tempeatue. An ncompessble asymmetc lattce Boltzmann D2Q9 model was poposed to solve the aal and adal veloctes though nsetng souce tems nto the two-dmensonal lattce Boltzmann equaton. Pesent hybd scheme was fstly valdated by smulatons of Taylo Couette flows between two concentc cylndes. Then the benchmak poblems of melt flow n Czochalsk cystal gowth wee studed and accuate esults wee obtaned. Numecal epement demonstated that pesent asymmetc D2Q9 model s moe stable than the pevous asymmetc D2Q9 model J. Comp. Phys. 23; 1861):295 37). Hence, compaed wth the pevous model, pesent numecal method povdes a sgnfcant advantage n smulaton melt flow cases wth hgh Reynolds numbe and hgh Gashof numbe. Copyght q 26 John Wley & Sons, Ltd. Receved 3 June 26; Revsed 24 August 26; Accepted 25 August 26 KEY WORDS: lattce Boltzmann; asymmetc; souce tem; Taylo Couette flow; cystal gowth 1. INTRODUCTION Many mpotant engneeng flows nvolve swl o otaton, fo eample, the flows n combuston, tubomachney and mng tanks. Hee, we ae nteested n the asymmetc flows wth swl and otaton. Snce the gadent fo any vaable n the azmuthal decton s zeo, an asymmetc swlng flow s a quas-thee-dmensonal 3D) poblem fo conventonal Nave Stokes NS) solves n the cylndcal coodnate system. In ths pape two typcal asymmetc swlng and otatng flows would be studed. Coespondence to: Habo Huang, Flud Dvson, Depatment of Mechancal Engneeng, Natonal Unvesty of Sngapoe, 1 Kent Rdge Cescent, Sngapoe 11926, Sngapoe. E-mal: g3118@nus.edu.sg Copyght q 26 John Wley & Sons, Ltd.

2 178 H. HUANG, T. S. LEE AND C. SHU One s Taylo Couette flow between two concentc cylndes. At low otatonal speed of the nne cylnde, the flow s steady and the votces ae plana. Thee-dmensonal votces would begn to appea when the speed of otaton eceeds a ctcal value that depends on the ato of adus of two cylndes. Pevously, thee ae some studes on Taylo Couette flow usng the conventonal NS solves [1]. The othe typcal asymmetc swlng flow s the melt flow n Czochalsk CZ) cystal gowth. CZ cystal gowth s one of the majo pototypcal systems fo melt-cystal gowth. It has eceved the most attenton because t can povde lage sngle cystals. In typcal CZ cystal gowth systems, the hgh Reynolds numbe and Gashof numbe of the melt make numecal smulaton dffcult. The conventonal CFD methods such as fnte volume and fnte-dffeence methods have been developed to smulate the CZ cystal gowth flow poblems [2 4]. The second-ode cental dffeence scheme s usually chosen to dscetze the convecton tems n NS equatons. Howeve, fo melt flows wth hgh Reynolds numbe and Gashof numbe whch ae the equement of gowth of lage and pefect cystals, the convecton tems n the NS equatons become domnant and the second-ode cental dffeence scheme may be unsutable due to enhanced numecal nstablty [3]. Howeve, f the low-ode upwnd scheme s appled, we can only obtan accuate solutons by usng vey fne gd [3]. Consdeng the dscetzaton poblem n conventonal CFD method, lattce Boltzmann method LBM) was poposed to smulate the melt flow n CZ cystal flow [5]. It s well known that LBM has been poposed as an altenatve numecal scheme fo solvng the ncompessble NS equatons [6, 7] and have poven to be supeo n accuacy and effcency fo cetan applcatons. One man advantage s that the convecton opeato of LBM n phase space s lnea whch may ovecome the above dscetzaton poblem n conventonal CFD method. Snce the standad two-dmensonal 2D) LBM s based on the Catesan coodnate system, the asymmetc swlng flows cannot be smulated as a quas-3d poblem n cylndcal coodnates usng standad 2D LBM. On the othe hand, of couse, the asymmetc swlng flows can be solved by the 3D LBM [8, 9] whch usng the 3D cubc lattces wth pope cuved wall bounday teatment dectly. Howeve, that means a lage gd sze. It s not effcent to smulate an asymmetc swlng flow poblem n that way. To smulate the asymmetc flows wthout otaton moe effcently, n 21, Hallday et al. [1] poposed an asymmetc D2Q9 model and t seems vey successful fo smulaton steady flow n staght tube. The man dea of the model s nsetng seveal spatal and velocty-dependent souce tems nto the 2D lattce Boltzmann equaton LBE). Followng the dea of Hallday et al. [1], Penget al. [5] used LBM to study the melt flow n CZ cystal gowth as a quas-3d poblem. They poposed an asymmetc D2Q9 LBM to solve the aal and adal velocty n an asymmetc plane and swl velocty and tempeatue wee solved by fnte-dffeence method. Howeve, Peng et al. [5] only smulated test cases of lowe Reynolds numbe and Gashof numbe. It was found that the asymmetc model poposed by Peng et al. [5] s unstable fo smulatons of melt flows wth hgh Reynolds numbe Re = 1 4 ) and hgh Gashof numbe G = 1 6 ) even wth vey fne gd such as 2 2. On the othe hand, snce the model poposed by Peng et al. s deved fom the standad D2Q9 model, the compessble effect of standad D2Q9 model [11, 12] may be nvolved n the smulaton. To mpove the numecal stablty and elmnate the compessblty effect of standad LBM, hee a new ncompessble asymmetc D2Q9 model was poposed. Ths asymmetc D2Q9 model was deved fom the ncompessble D2Q9 model poposed by He and Luo [12]. Copyght q 26 John Wley & Sons, Ltd. Int. J. Nume. Meth. Fluds 27; 53:

3 HYBRID LATTICE BOLTZMANN FINITE-DIFFERENCE SIMULATION 179 In ths pape, the asymmetc swlng flows would be solved by a hybd scheme. The aal and adal veloctes wee solved by LBM and swl velocty and tempeatue wee solved by fnte-dffeence method. Ths hybd scheme was fstly valdated by smulaton of Taylo Couette flows between two concentc cylndes. Then the melt flows n CZ cystal gowth wee studed n detal. Ou numecal esults wee compaed wth avalable data of Raspo et al. [4] and Xu et al. [3]. Though numecal tests, pesent asymmetc D2Q9 model was poved moe stable than the model of Peng et al. [5] fo steady asymmetc swlng and otatng flows. As a esult, ou hybd scheme can gve accuate esults fo melt flow wth hgh Reynolds numbe and hgh Gashof numbe usng smalle gd sze. 2. NUMERICAL METHODS We consde the poblems of the lamna asymmetc swlng flow of an ncompessble lqud wth an as n decton. The followng contnuty equaton 1) and NS momentum equatons 2) n the pseudo-catesan coodnates, ) ae used to descbe the flow n aal and adal dectons β u β = u t u α + β u β u α ) + 1 ρ α p ν β β u α ) = u αu + ν 1) u α u ) δ α + u2 z δ α + S 2) whee u β β =, ) s the two components of velocty and u α s the velocty u o u. S s the addtonal souce tem that may appea n some flow poblems. In the above equatons we adopt the Ensten summaton conventon. Fo the asymmetc swlng flow, thee ae no ccumfeental gadents but thee may stll be non-zeo swl veloctes u z. The momentum equaton of azmuthal velocty s u z t u z + u + u u z = ν 2 u z + 2 u z 2 2 ) + ν uz u ) z u u z Hee we poposed an asymmetc lattce Boltzmann D2Q9 model to ecove above equatons 1) and 2) though the Chapman Enskog epanson efe to Append A). Snce the lattce Bhatnaga Goss Kook LBGK) model s the smplest model among the LBE models n applcaton, ou asymmetc lattce Boltzmann model s deved fom an ncompessble LBGK D2Q9 model [12]. The nne dscete veloctes of ou model ae as follows:, ), = e = cos[ 1)π/2], sn[ 1)π/2])c, = 1, 2, 3, 4 4) 2cos[ 5)π/2 + π/4], sn[ 5)π/2 + π/4])c, = 5, 6, 7, 8 3) whee c = δ /δ t, and n ou studes c = 1. δ and δ t ae the lattce spacng and tme step sze. The evaluaton equaton to descbe 2D flow n, ) pseudo-catesan coodnates s llustated as Equaton 5) whch s smla to the evaluaton equaton fo standad D2Q9 model n 2D, y) Copyght q 26 John Wley & Sons, Ltd. Int. J. Nume. Meth. Fluds 27; 53:

4 171 H. HUANG, T. S. LEE AND C. SHU Catesan coodnates. The dffeence s that a souce tem h,, t) was ncopoated nto the mcoscopc evaluaton equaton f + ce δ t, + ce δ t, t + δ t ) f,, t) = 1 τ [ f eq,, t) f,, t)]+h,, t) 5) In Equaton 5), f,, t) s the dstbuton functon fo patcles wth velocty e at poston, ) and tme t. τ s the ela tme constant. The ela tme constant τ and the flud vscosty satsfes equaton ν = c 2 s δ tτ.5) 6) whee c s = c/ 3. In Equaton 5), the equlbum dstbuton f eq of ncompessble D2Q9 model [12] s defned as [ f eq p e u,, t) = ω cs 2 + ω ρ cs 2 + e u) 2 ] 2cs 4 u2 2cs 2, =, 1, 2,...,8 7) whee p s the pessue and ρ s the densty of flud. In the above epessons, fo D2Q9 model, ω = 4 9, ω = 1 9 = 1, 2, 3, 4), ω = 36 1 = 5, 6, 7, 8). It s notced that the man dffeence between above ncompessble D2Q9 model and the standad D2Q9 model s the fom of Equaton 5). 3. HYBRID SCHEME 3.1. Implementaton of the asymmetc model and fnte dffeence In numecal smulatons, one must ensue that the Mach numbe s low and the densty fluctuaton δρ) s of ode OM 2 ) [12]. The addtonal lmt L /c s T ) 1 s llustated n ou devaton efe to Append A). In ths pat, we manly dscuss the mplementaton of the asymmetc D2Q9 model. In the D2Q9 model, f,, t) s the dstbuton functon. The macoscopc pessue p and momentum ρ u ae defned as 8 = f = p/c 2 s, 8 = f e α = ρ u α 8) The two man steps of lattce BGK model ae collson and steamng. In the collson step, a goup of calculatons 9) and 1) ae mplemented f ne f +,, t) = f eq,, t) + = f,, t) f eq,, t) 9) 1 1 ) f ne + δ t h 1) + δ 2 t τ h2) 1) whee f eq s the equlbum momentum dstbuton functon whch can be obtaned though Equaton 7), f ne s the non-equlbum pat of dstbuton functon, h 1) and h 2) ae the souce tems added nto the collson step, whch can calculated though Equatons A11) and A23), Copyght q 26 John Wley & Sons, Ltd. Int. J. Nume. Meth. Fluds 27; 53:

5 HYBRID LATTICE BOLTZMANN FINITE-DIFFERENCE SIMULATION 1711 espectvely efe to Append A). Fo smplcty, n ou study, δ t = δ f + s the post-collson dstbuton functon. In the steamng step, the new dstbuton functon value obtaned fom Equaton 1) would popagate to neghboung eght lattces. That pocedue can be epesented as follows: f + e δ t, + e δ t, t + δ t ) = f +,, t) 11) Fo the velocty devatons n Equaton A23), the tems u + u, u and u can all be obtaned though Equaton 12) wth α =, β = ; α = β = and α = β =, espectvely ρ ν β u α + α u β ) = 1 1 ) 8 f 1) e α e β = 1 1 ) 8 f ne e α e β + oε 2 ) 12) 2τ = 2τ = Fo the tem u n Equaton A23), t s equal to u + u ) u.snce u + u ) can be easly obtaned by Equaton 12), only value of u s left unknown to detemne u.hee fnte-dffeence method s used to obtan u at lattce node, j), whch can be calculated by u ), j = u ) +1, j u ) 1, j )/2δ ) 13) The values of u + u, u, u, u and u fo the lattce nodes whch just on the wall bounday can also be calculated fom Equatons 12) and 13). Hence, fo the addtonal souce tem n ou model, most velocty gadent tems can be obtaned fom hgh-ode momentum of dstbuton functon, whch s consstent wth the phlosophy of the LBM. Fo the momentum equaton of azmuthal velocty, t was solved by fnte-dffeence method. Equaton 3) can be solved eplctly by usng fst-ode fowad dffeence scheme n tme and the second-ode cental dffeence scheme e.g. Equatons 15) and 16)) fo space dscetzaton as follows: u n+1 z = u n z + δ t [ u n u n z + u n ) 2 un z u n + ν z 2 + ) 2 u n z 2 + ν u n ) ] z un z un un z 14) u n z = un z ) +1, j u n z ) 1, j 2δ 15) 2 u n z 2 = un z ) +1, j + u n z ) 1, j 2u n z ), j δ 2 16) 3.2. Bounday condton Bounday condton s an mpotant ssue when usng LBM to smulate the flud flows [13]. Its well known that bounce back scheme s one of the smplest schemes n LBM. Hee ths scheme was appled fo non-slp wall bounday condton. Fo the asymmetc bounday condton.e. the -as), the specula eflecton scheme was appled to lattce nodes n as [5]. As we know, specula eflecton scheme can be appled to fee-slp bounday condton whee no momentum s to be echanged wth the bounday along the tangental component. Hence, fo the fee suface e.g. = H, R <<R c n Fgue 4) n ou smulated case, the specula bounday condton s also appled. Copyght q 26 John Wley & Sons, Ltd. Int. J. Nume. Meth. Fluds 27; 53:

6 1712 H. HUANG, T. S. LEE AND C. SHU When usng the fnte-dffeence method to solve the equaton fo swl velocty o the heat equaton, we may encounte the Neumann bounday condton. Hee the Neumann bounday condton was tansfeed nto the Dchlet bounday condton. Fo eample, f T/ = was mposed n the bounday =.e. the -as, efe to Fgue 4), the T value n the bounday lattce node 1, j) can be detemned by etapolaton fom the nne lattce nodes as T ) 1, j = 4T ) 2, j T ) 3, j )/3, whee j s the lattce nde n coodnate. 4. RESULTS AND DISCUSSION 4.1. Taylo Couette flows Fgue 1 llustates the geomety of Taylo Couette flow, ou computatonal doman s a plane. The govenng equatons fo the asymmetc swl flow ae Equatons 1) 3) wth S =n Equaton 2). The bounday condtons used n ou smulaton ae also llustated n Fgue 1. The Reynolds numbe s defned as Re = WD/ν, whee W s the azmuthal velocty of nne cylnde, D s the gap of the annulus and ν s the flud vscosty. The adus ato of nne cylnde and out cylnde s set as.5. The aspect ato s set as 3.8. Fstly, the gd ndependence of the esults was eamned and t was found that wth 2 76 unfom gd, pesent numecal method can gve out vey accuate esults. The mamum steam functon values n plane fo cases of Re = 85, 1 and 15 ae lsted n Table I. It seems that even wth gd 2 76, the esults of ou hybd scheme agee well wth those of Lu [1] whch wee obtaned by vey fne gd. The contous of steam functon, pessue and votcty fo case Re = 15 ae llustated n Fgue 2. Fom Fgue 2, we can see the fou cell seconday modes. These contous and flow patten also agee well wth esults of Lu [1]. u = u = u z =W u = u = u z = Fgue 1. Geomety of Taylo Couette flow and bounday condtons. Copyght q 26 John Wley & Sons, Ltd. Int. J. Nume. Meth. Fluds 27; 53:

7 HYBRID LATTICE BOLTZMANN FINITE-DIFFERENCE SIMULATION 1713 Table I. The mamum steam functon value n plane fo Taylo Couette flow gd 2 76). Re ψ ma ψ ma [1] Fgue 2. The contou of steam functon, pessue and votcty fo case Re = 15 wth gd Secondly, the effcency of ou hybd scheme LBM + FD) and eplct fnte volume method FVM) was compaed. The effcency s evaluated by compang the espectve computng tmes equed. To mnmze the nfluence of computes and convegence cteon, n ths study, both ou LBM+FD solve and FVM solve FLUENT) wee eecuted on a supe compute Compaq ES4: total pefomance of 53 Mflops) n the Natonal Unvesty of Sngapoe. In ou smulatons, the zeo veloctes wee ntalzed eveywhee. The esdual used to monto the convegence s defned usng the u z -momentum equaton fo two solves as follows: FVM: u n z t LBM + FD: u n+1 z u n z δt + u n u n z + u n un z + un un z ν 2 u n z 2 + ) 2 u n z 2 ν u n ) z un z 17) 18) Copyght q 26 John Wley & Sons, Ltd. Int. J. Nume. Meth. Fluds 27; 53:

8 1714 H. HUANG, T. S. LEE AND C. SHU 1.1 u z Resdual.1 1E-3 1E-4 FLUENT =.59, LBM+FD =.68, LBM+FD 1E steps Fgue 3. Convegence hstoy fo FLUENT and the hybd scheme LBM + FD). Table II. Compason of CPU tme fo hybd scheme and FVM smulaton of Taylo Couette flow Re = 1, gd 3 114). Steps CPU tme s) ψ ma FLUENT LBM + FD τ =.59) LBM + FD τ =.68) Lu Note that all the computatons ae caed out on a sngle-cpu of the compute Compaq ES4, whch does not take paallel advantage of the LBM. Fo compason pupose, the case of Taylo Couette flow fo Re = 1 usng gd was smulated. In the eplct FVM solve FLUENT), the couant numbe was set as CFL = 1. The convegence fo the hybd scheme and FVM solve s dsplayed n Fgue 3 n tems of elatve esdual eo the esdual epessons wee nomalzed by the ntal esdual). The oveall convegence tend of ou hybd scheme s smla to that of FVM solve. The CPU tmes fo hybd scheme and FVM ae also lsted n Table II. It seems that to each the same convegence cteon, ou LBM + FD solve τ =.59) takes almost same CPU tme as the eplct FVM solve. The calculaton of LBM + FD solve wth ela tme constant τ =.68 s faste than calculaton wth τ =.59. Accodng to ou epeence, fo a 2D flow case wth same gd, usually the eplct FLUENT solve eques about eght tmes lage CPU tme pe teaton than ou 2D LBM solve. It s also obseved that fo asymmetc cases wthout otaton, the FLUENT solve eques about fou Copyght q 26 John Wley & Sons, Ltd. Int. J. Nume. Meth. Fluds 27; 53:

9 HYBRID LATTICE BOLTZMANN FINITE-DIFFERENCE SIMULATION 1715 R =βr c R =βr c H gavty Cystal A u = u = u = u / = u z = Ω u z / = H gavty Cystal A T=T -R T=T + ) T c -T ) R c -R ) u / = u = u z = Melt u = u = u z =ΩcRc T/ = Melt T=T c u = u = u z =Ωc Rc T/ = Rc Fgue 4. The momentum and themal bounday condtons of melt flow n Czochalsk cystal gowth. tmes lage CPU tme pe teaton than asymmetc LBM. That may be due to the fact that compaed wth a 2D case, FLUENT almost does not eque moe CPU tme fo asymmetc case whle asymmetc LBM need moe effot to calculate the souce tems. Fom Table II, t s found that fo the asymmetc flow wth otaton, compaed wth LBM+FD solve, FLUENT eques about 3.35 tmes lage CPU tme pe teaton. It s also obseved fom ou numecal epement that the tme spent fo the solvng of Equaton 14).e. FD) n ou LBM + FD scheme s aound 12% of total CPU tme Melt flows n CZ cystal gowth In the CZ cystal gowth, the melt flow s vey comple because t s a combnaton of natual convecton due to themal gadents and foced convecton due to otaton of the cystal and the cucble. Hee, the Wheele benchmak poblem [14] n CZ cystal gowth s taken as the test eample to valdate ou numecal method. The confguaton and the momentum and themal bounday condtons ae all llustated n Fgue 4. In the poblem, a vetcal cylndcal cucble flled wth a melt to a heght H = R c otates wth an angula velocty Ω c. In the top of the melt, t s bounded by a coaal cystal wth adus R = βr c β =.4) whch otates wth angula velocty Ω. Thee s a phase bounday between the cystal and melt. In the top ght pat of melt R>R ), thee s a fee suface. The u, u, u z ae the aal, adal and azmuthal velocty components, espectvely. The contnuty and momentum equatons fo CZ cystal gowth can also llustated by Equatons 1) 3). In Equaton 2), snce the Boussnesq appomaton s appled to ths buoyancy foce tem, S = gβ T T c )δ α, whee g s the acceleaton due to gavty, β s the themal epanson coeffcent, T c s the tempeatue of cucble. The govenng equaton of tempeatue s T t T + u + u T = ν 2 ) T P T T 19) Copyght q 26 John Wley & Sons, Ltd. Int. J. Nume. Meth. Fluds 27; 53:

10 1716 H. HUANG, T. S. LEE AND C. SHU Ths equaton can be solved eplctly by fnte-dffeence method as Equaton 14). Howeve, n ths pat, to acceleate convegence ate, fnte-dffeence equatons fo Equatons 3) and 19) wee solved by the tdagonal mat algothm TDMA) at each teaton. The dmensonless paametes: Reynolds numbe Re c, Re, Pandtl numbe P and Gashof numbe G ae defned as Re c = R2 c Ω c ν, Re = R2 c Ω, P = ν ν α, G = gβ T c T )Rc 3 ν 2 whee α s the themal dffusvty. In ou smulatons, P =.5. The value of chaactestc velocty U t = β gt c T )R c was chosen as.15 fo G 1 5 and.25 fo G>1 5.When U t s detemned, the knetc vscosty ν can be detemned by G. And then the elaaton tmes τ s detemned by Equaton 6). Anothe chaactestc velocty U h = R c Ω β s also used when G = n ou smulaton and t s usually set as.1. Fo the esults, R c and ν/r c ae used as the chaactestc length, speed scales. The dmensonless tempeatue s defned as T = T T )/T c T ), whee T s the tempeatue of the cystal. In ou smulatons, the zeo veloctes and zeo tempeatue wee ntalzed eveywhee and the convegence cteon n ou smulaton s set as [u, j, t + δ t ) u, j, t)] 2 +[u, j, t + δ t ) u, j, t)] 2 <1 6 2), j [u, j, t + δ t )] 2 +[u, j, t + δ t )] 2 whee, j ae the lattce nodes nde. To compae wth avalable data of Raspo et al. [4], Buckle et al. [2] and Xu et al. [3], all of the pesent numecal esults ae epessed as steam functon. The steam functon ψ s defned as ψ = u, ψ = u 21) wth ψ = on all the boundaes of computng plane. In the followng, the mnmum and mamum values of steam functon denoted by ψ mn and ψ ma wll be used to compae the esults of ou hybd scheme wth avalable data n the lteatue [3, 4]. Fstly, the gd ndependence of the esults was eamned. Case A2, wth G =, Re = 1 3, Re c =, was calculated usng thee gds. The ψ mn and ψ ma ae compaed wth the esult of Table III. Gd ndependence test fo case A2, G =, Re = 1 3, Re c =. Gd ψ mn ψ ma Refeence [4] Copyght q 26 John Wley & Sons, Ltd. Int. J. Nume. Meth. Fluds 27; 53:

11 HYBRID LATTICE BOLTZMANN FINITE-DIFFERENCE SIMULATION 1717 Raspo et al. [4] n Table III, whee we can see that an 1 1 gd s suffcent to obtan accuate esults. Secondly, as many as 11 cases wth dffeence paamete sets wee smulated. The 11 cases wee classfed nto fou goups. In goup A, the cystal otate wth Re vayng fom 1 2 to 1 4, whle the cucble s at est and G s set to zeo. In goup B, the cystal and cucble otate n opposte dectons. Goup A and B ae all foced convecton poblems. Cases n goup C ae natual convecton poblems. Cases n goup D ae moe lke pactcal applcatons because these melt flows combned both the natual convecton and foced convecton wee nvestgated. Table IV shows the compason of computed mnmum and mamum steam functon fo all the above 11 cases. In the table, the numbe n the backet followed the case type ndcates the gd sze used. If not specfed, the gd used n ou smulaton s 1 1. Fo compason, we also pesent the esults of Xu et al. [3] usng the second-ode dffeence scheme wth a gd sze of 8 8. In all cases, the mamum absolute values of steam functon computed by the LBM agee vey well wth those of Xu et al. [3]. Some vey small devatons between the computed mnmum absolute values of steam functon can be neglected snce the mnmum absolute values of steam functon ae so small compaed wth the mamum absolute values. Due to numecal stablty, the smulaton of cases A3, B3, C2 used fne gds. The ssue of numecal stablty wll be dscussed n detal n Secton 4.3. Fgue 5 shows the calculated steamlnes and tempeatue contous of case A2. That s a typcal esult fo goup A. Thee s a pmay vote nduced by otaton of the cystal. Fo the cases of goup A, when the Reynolds numbe of cystal otaton s nceased fom 1 2 to 1 4, the mamum absolute value of the steam functon nceases fom.2272 to 4.47, whch means the ntensty of vote nceases. Fo hghe Reynolds numbe cases n goup A, the cente of the vote moves towads the sde wall of the cucble and the hghest velocty egon moves fom the uppe left cone to the uppe ght cone. Hence, bette qualty cystal can be poduced f Re s hgh. Fgue 6 llustates the steamlnes and tempeatue contous of case B2, whch epesent the flow patten of goup B. Fo cases n goup B, the cystal and cucble otate n opposte dectons. As a esult, thee ae two votces wth opposte dectons appeang n the uppe left cone just below the cystal and the lowe ght cone. Wth the ncease n otaton speeds of the cystal and cucble, the uppe left vote poduced moves towads ght cone and the lowe ght pmay vote nduced by the cucble otaton moves to the left and domnates the flow feld. It s notced that fo cases of foced convecton poblems whee G = cases n goup A and B), the contous of tempeatue ae vey smla. Fgue 7 shows the steamlnes and tempeatue contous of case C2. In ths natual convecton flow case, the cucble and the cystal ae all at est. Thee s a pmay vote nduced by the tempeatue dffeence between the cystal and cucble. Compaed wth tempeatue contous n Fgues 5 and 6, the tempeatue contous of case C2 n Fgue 7 shows the effect of buoyancy foce on the tempeatue feld. Fgue 8 shows the steamlnes and tempeatue contous of case D2. The steamlnes and contous llustated the combned effects of the natual convectve flow and foced convectve flow. It s found that the steamlnes and tempeatue contous of cases n goup D ae vey smla to those of case C1 whose Gashof numbe s also equal to 1 5. Fom Table IV, t s also found that the ψ ma of cases n goup D ae all vey close to that of case C1. That means n cases of goup D, f Re <1 3, the natual convectve flow domnates the melt flow whle the foce convectve flow nduced by the cystal only has mno effect. Copyght q 26 John Wley & Sons, Ltd. Int. J. Nume. Meth. Fluds 27; 53:

12 1718 H. HUANG, T. S. LEE AND C. SHU Table IV. Some esults fo the test cases by the hybd scheme and QUICK [3]. Case G Re Rec ψ mn ψ ma ψ mn Refeence [3]) ψ ma Refeence [3]) A A A3 2) B B B3 25) C C2 15) D D D Copyght q 26 John Wley & Sons, Ltd. Int. J. Nume. Meth. Fluds 27; 53:

13 HYBRID LATTICE BOLTZMANN FINITE-DIFFERENCE SIMULATION Fgue 5. Steamlnes and tempeatue contous of case A2, G =, Re = 1 3, Re c = Fgue 6. Steamlnes and tempeatue contous of case B2, G =, Re = 1 3, Re c = Numecal stablty compason The numecal stablty of LBM depends on the ela tme τ, the Mach numbe of the flow and the sze of mesh. It s well known that n LBM f τ s vey close to.5, numecal nstablty would appea. τ mn s usually case dependent. The Reynolds numbe s usually defned as Re = UD ν = UcD/δ ) cs 2 τ.5) Copyght q 26 John Wley & Sons, Ltd. Int. J. Nume. Meth. Fluds 27; 53:

14 172 H. HUANG, T. S. LEE AND C. SHU Fgue 7. Steamlnes and tempeatue contous of case C2, G = 1 6, Re =, Re c = Fgue 8. Steamlnes and tempeatue contous of case D2, G = 1 5, Re = 1 2, Re c =. the Mach numbe n LBM s U/c s 1. To smulate cases of hgh Reynolds numbe, wth lmtaton of τ mn and Mach numbe, usually we have to ncease the value of D/δ ).e. enlage the gd sze). Geneally speakng, addng comple poston and tme-dependent souce tems nto the LBE would decease the numecal stablty [15]. Compaed wth the pevous asymmetc D2Q9 model [5], the epesson of the souce tems h 1) and h 2) n pesent model ae much smple snce the swl velocty only appeas n the tem h 2). Hence, the pesent smple model s epected to be moe stable. Copyght q 26 John Wley & Sons, Ltd. Int. J. Nume. Meth. Fluds 27; 53:

15 HYBRID LATTICE BOLTZMANN FINITE-DIFFERENCE SIMULATION 1721 Table V. Numecal stablty compason fo case A1. Gd τ mn Pesent model) τ mn Peng s model [5]) To compae the numecal stablty of ou model and pevous model [5], the benchmak case A1 of melt flow n CZ cystal gowth was smulated by the two models wth the same bounday condton teatment. As we know, the numecal stablty can be demonstated by the mnmum τ value at whch numecal nstablty does not appea. Howeve, t s had to fnd out the eact τ mn. But f the value of τ mn s set as τ mn =.5 + k.125, whee k> s an ntege, we may fnd out τ mn oughly by fndng the mnmum k value at whch numecal nstablty does not appea. So the numecal epement was caed out to fnd τ mn.theτ mn fo the two asymmetc D2Q9 models s lsted n the Table V. Fom Table V, we can see that n all cases, τ mn of pesent model ae smalle than that of Peng et al. [5]. It seems ou asymmetc D2Q9 model s moe stable. The numecal stablty s vey mpotant fo smulaton of hgh Reynolds numbe o hgh Gashof numbe cases. Fo eample, f we want to smulate the case of G = 1 7, wth the lmtaton of ncompessble flow n LBM, U t usually should not eceed.25, snce ν = cs 2δ tτ.5) = U t R c / G, f numecal stablty eques τ.6125, then the mesh pont n R c should satsfy the elatonshp 1/ ) R c = c2 s τ.5) G 474 δ cu t That means to smulate the case of G = 1 7, the coasest gd should be , othewse, the numecal nstablty would encounte n the smulaton. Whle fo the case of G = 1 7,f numecal stablty of the Peng s model [5] eques τ.7375, gd as fne as 1 1 s equed. Hence, ou numecal method povdes a sgnfcant advantage n smulaton melt flow cases wth hgh Reynolds numbe and hgh Gashof numbe. 5. CONCLUSION As conventonal CFD solves, pesent hybd scheme combnng the LBMs and fnte-dffeence method can solve the asymmetc swlng flow as a quas-3d poblem. An asymmetc ncompessble lattce Boltzmann D2Q9 model was poposed n ths pape by ntoducng an addtonal souce tem to LBE. Wth lmt of Mach numbe M 1 andl /c s T ) 1, ths asymmetc D2Q9 model successfully ecoveed the contnuty equaton and momentum equatons fo aal and adal veloctes though Chapman Enskog epanson Append A). The equaton fo swl velocty and the heat equaton wee solved by fnte-dffeence methods. Ths hybd scheme was successfully appled to smulate the Taylo Couette flow between two concentc cylndes. It was found that the esdual convegence behavou of ths hybd scheme s smla to that of eplct FVM. It s found that compaed wth LBM + FD solve, FLUENT Copyght q 26 John Wley & Sons, Ltd. Int. J. Nume. Meth. Fluds 27; 53:

16 1722 H. HUANG, T. S. LEE AND C. SHU eques about 3.35 tmes lage CPU tme pe teaton. Howeve, to each the same convegence cteon, the CPU tme taken by ou LBM+FD solve and eplct FVM solve ae of same ode. The hybd scheme was also appled to smulate flows n CZ cystal gowth. Compaed wth esults n othe lteatues [3, 4], the hybdscheme can gve out vey accuateesults fo benchmak poblems. Pesent asymmetc D2Q9 model also seems moe stable than that of Peng et al. [5]. As a esult, ths scheme can gve accuate esults fo hgh Reynolds numbe and hgh Gashof numbe cases wth smalle gd sze. APPENDIX A: BRIEF CHAPMAN ENSKOG DERIVATION OF THE AXISYMMETRIC D2Q9 MODEL We consde the poblems of the lamna asymmetc swlng flow of an ncompessble lqud wth an as n decton. The contnuty equaton 1) and NS momentum equatons 2) n the pseudo-catesan coodnates, ) would be ecoveed fom ou asymmetc D2Q9 model though the Chapman Enskog epanson. Hee an ncompessble D2Q9 model s used to deve ou asymmetc model. The nne dscete veloctes of ou model ae llustated n Equaton 4). In ou devaton we adopt δ = δ t.the evaluaton equaton to descbe 2D flow n, ) pseudo-catesan coodnates s llustated as Equaton 5). In ou devaton, the Ensten summaton conventon s used. As we know, the stategy fo ncopoatng souce tems nto the LBE can be appled fo flud poblems n whch an etenal o ntenal foce s nvolved. Gavty can be modelled by nsetng souce tems wth dffeent foms nto LBE [16]. To smulate the patcle flud suspensons, the focng tem s epanded n a powe sees n the patcle velocty [17]. The fom of souce tem [17] was also used to deve a coect epesentaton of the focng tem [18]. Fo the above devaton [16 18], snce the etenal o ntenal foces ae smple, souce tem h s only epanded to fst-ode and thee s no h 2) tem. Howeve, to model the spatal and tme-dependent foces n RHS of Equatons 1) and 2), souce tem h would be much moe comple and should be epanded to second-ode as that llustated n Equaton A2) [1]. It s notced that the choce of followng foms of h 1) and h 2) o that n Refeence [1] s only one patcula stategy to ncopoate geometcal souce tems nto the LBM. The othe possble stateges to deve asymmetc models ae suggested n detal n Refeence [15]. At the begnnng, we adopt the followng epansons [1, 12]: f + e, + e, t + 1) = ε n n! Dn f,, t) A1) n= f = f ) + ε f 1) + ε 2 f 2) + t = ε 1t + ε 2 2t + β = ε 1β h = εh 1) + ε 2 h 2) + whee ε = δ t and D t + e β β ), β =, and α, β means o. Copyght q 26 John Wley & Sons, Ltd. Int. J. Nume. Meth. Fluds 27; 53: A2)

17 HYBRID LATTICE BOLTZMANN FINITE-DIFFERENCE SIMULATION 1723 Retanng tems up to Oε 2 ) n Equatons A1) and A2) and substtutng nto Equaton 5) esults n the followng Equatons A3) A5) Oε ) : f ) f eq )/τ = A3) Oε 1 ) : 1t + e β 1β ) f ) + f 1) /τ h 1) = A4) Oε 2 ) : 2t f ) ) 1t +e β 1β ) f 1) + 1 2τ 2 1t+e β 1β )h 1) + 1 τ f 2) h 2) = A5) The dstbuton functon f s constaned by the followng elatonshps: 8 = f ) = p cs 2, 8 = e α f ) = ρ u α A6) 8 = f m) =, 8 = e f m) = fo m> A7) Wth the popetes of the tenso E n) = α e α1e α2,...,e αn [19], wehave 8 = e α e β f ) = ρ u α u β + pδ αβ A8) e α e β e k f ) = ρ c 2 s δ jkδ βα + δ jα δ βk + δ jβ δ αk )u j A9) Mass consevaton and h 1) Summng on n Equaton A4), we obtan at Oε) 1t p/c 2 s ) + ρ β u β = h 1) A1) whch motvates the followng selecton of h 1) when compang wth the taget dynamcs of Equatons 1) and 2). Rewtng A1) n a dmensonless fom, we can see that a condton of L /c s T ) 1 should be satsfed to neglect the fst LHS tem [12], whee L s the chaacte length n decton, T s the chaacte tme of unsteady flow. That s an addtonal lmt of ou devaton besdes condton Mach numbe M 1. To ecove the contnuty Equaton 1), because ω = 1, the followng selecton of h 1) s easonable [1]: h 1) = ω ρ u / A11) Then we poceed to Oε 2 ) now. Summng on n Equaton A5) gves 2t p/c 2 s ) t + e β 1β )h 1) h 2) = A12) Copyght q 26 John Wley & Sons, Ltd. Int. J. Nume. Meth. Fluds 27; 53:

18 1724 H. HUANG, T. S. LEE AND C. SHU Snce 2t p/cs 2 ) = [12] and wth ou taget dynamcs n vew, the emanng tems n Equaton A12) should vansh. Hence, we obtan h 2) = 1 1t + e β 1β )h 1) = 1 [ )] ω ρ u 1t = t In the above pocess, we have used the esults of ω e β =. Momentum consevaton and h 2) Multplyng Equaton A4) by e α and summng on, gves ρ 1t u α + 1β Π αβ = ρ u ) A13) h 1) e α = A14) whee Π αβ = 8 = e α e β f ) s the zeoth-ode momentum flu tenso. Wth Π αβ Equaton A8), Equaton A14) gves ρ 1t u = β Πβ = βpδ β + ρ u β u ) gven by A15) Substtutng Equaton A15) nto Equaton A13), we have a condton on the h 2) h 2) = 1 2 βpδ β + ρ u β u ) A16) Multplyng Equaton A5) wth e α and summng ove gves ρ 2t u α ) ) 1β Π 1) 2τ αβ = 1 1t e α h 1) + 1β e α e β h 1) + 2 whee Π 1) αβ = e αe β f 1) A4) and A9), we have Π 1) αβ = h 2) e α A17) s the fst-ode momentum flu tenso. Wth the ad of Equatons + τ [ )] = τ 1t Π ) αβ + k e α e β e k f ) + τ e α e β f 1) = τ e α e β D 1t f ) e α e β h 1) e α e β h 1) = τ[ 1t Π ) αβ + ρ c 2 s δ αβ j u j + β u α + α u β )] ρ τc 2 s δ αβu / A18) Fo the fst tem n Equaton A18), usng Equatons A1) and A14) and the addtonal condton L /c s T ) 1, t s found that 1t Π ) αβ ae of Ou3 ). Hence, ths tem can be neglected [19]. Then, the second tem n LHS of Equaton A17) can be wtten as 1 1 ) 1β Π 1) 2τ αβ 1 = τ 1 ) ρ 2τ cs [ 2 1β δ αβ j u j + u α + u ) ] β + 1β δ αβ u /) β α = νρ 1β β u α + α u β ) A19) Copyght q 26 John Wley & Sons, Ltd. Int. J. Nume. Meth. Fluds 27; 53:

19 HYBRID LATTICE BOLTZMANN FINITE-DIFFERENCE SIMULATION 1725 Substtutng Equaton A19) nto Equaton A17) and usng the esult of e αe β h 1) = c 2 s δ αβρ u /, Equaton A17) can be wtten as u α t + u βu α + p ν 2 u α β ρ α β 2 = 1 2 c2 s α The RHS of A2) can be wtten as c 2 s 1 τ) α u u ) + ) + ν α β u β ) + h 2) ρ e α h 2) ρ e α A2) Compang momentum Equaton A2) wth Equaton 2), to ecove the NS momentum equatons of aal and adal veloctes, Equaton A21) should be satsfed c 2 s 1 τ) α u ) + h 2 e α = u αu ρ + ν u α u ) δ α + u2 z δ α + S A21) Solvng equaton system of Equatons A16) and A21), we can obtan the epesson of h 2) as follows h 2) ρ = ω [ )] pδβ [ ν β + u β u + 3ω e β u β u 2 ρ e u ) u 2 ω 1 τ) β e β + 3ω z e + e ) βs δ αβ ) u βu ] e β A22) Equaton A22) can be ewtten eplctly as h 2) ρ = ω [ ) ] p + u u + u u + 3ω ν u e + u e u e 2 ρ u 3ω u u e + u e ) ω 1 τ) e u 2 e + ) u e ) u 2 + 3ω z e + e S + e S δ α δ α ) A23) The epesson of h 1) [Equaton A11)], h 2) [Equaton A23)] ae successfully deved and the contnuty equaton 1) and NS equaton 2) can be fully ecoveed. REFERENCES 1. Lu Y. Numecal smulaton of flows n Czochalsk cystal gowth and Taylo votces. M. Eng. Thess, Natonal Unvesty of Sngapoe, Buckle U, Schafe M. Benchmak esults fo the numecal smulaton of flow n Czochalsk cystal gowth. Jounal of Cystal Gowth 1993; 1264): Copyght q 26 John Wley & Sons, Ltd. Int. J. Nume. Meth. Fluds 27; 53:

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