A dynamic boundary model for implementation of boundary conditions in lattice-boltzmann method

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1 Journal of Mchanical Scinc and Tchnology Journal of Mchanical Scinc and Tchnology 22 (28) 92~2.springrlink.com/contnt/ x A dynamic boundary modl for implmntation of boundary conditions in lattic-boltzmann mthod Jinfn Kang, Sangmo Kang and Yong Kon Suh * Dpartmnt of Mchanical Enginring, Dong-A nivrsity, 84, Hadan-dong, Saha-gu, Busan, 64-74, Kora (Manuscript Rcivd August 6, 27; Rvisd March 4, 28; Accptd March, 28) Abstract Mthodology of implmntation of th no-slip boundary condition in th lattic-boltzmann mthod affcts ovrall accuracy of th numrical solutions as ll as th stability of th solution procdur. W propos a n algorithm, i.., th mthod of using a dynamic quation for stablishing no-slip boundary conditions on alls. Th distribution functions on th all along ach of th links across th physical boundary ar assumd to b composd of quilibrium and non-quilibrium parts hich inhrit th ida of Guo s xtrapolation mthod (22). In th proposd algorithm, mployd a dynamic quation to corrct th vlocity rror occurring on th physical boundary. Numrical rsults sho that th dynamic boundary modl is faturd ith improvd accuracy and simplicity. Th proposd mthod is postulatd to b usful spcially in th study on microfluidic mixing. Kyords: Dynamic-quation mthod; Lattic-Boltzmann mthod; No-slip boundary condition Introduction Lattic-Boltzmann mthod (LBM) has bcom a promising altrnativ fluid-dynamic computational platform asid from th traditional CFD mthod du to its simplicity for implmntation and as in handling complx boundary conditions. Hovr, in th dvlopmnt of LBM, thr ar still svral problms opn to furthr improvmnt. Th corrct implmntation of th no-slip boundary condition, among othrs, is on of th crucial tchniqus to improv, as it plays an important rol in th ovrall accuracy of th numrical solutions as ll as th stability of th solution procdur. Thr ar various approximat mthods for th tratmnt of no-slip boundary condition. Among thm, th most rprsntativ mthods ar th bounc-back mthod [-4], Yu s intrpolation mthod [5] and Guo s xtrapolation mthod [6]. Th bouncback schm is particularly simpl and has playd a * Corrsponding author. Tl.: , Fax.: addrss: yksuh@dau.ac.kr DOI.7/s y major rol in making LBM popular among CFD rsarchrs, particularly as applid to th porous-mdia flo. Hovr, th bounc-back mthod producs rsults ith only first-ordr spatial accuracy [7] unlss a propr spatial arrangmnt of th boundary location (i.., placing th boundary all just halfay btn th grid nods) is stablishd [3]. By applying Yu's and Guo's mthod, can gt mor accurat rsults, but th stability of solution is not alays satisfying, spcially hn applid to microfluidic flos that ar faturd ith lo Rynolds numbrs and lo vlocity valus; in LBM simulation, such paramtr sttings ncssitat larg valu of rlaxation tim τ, hich usually causs a stability problm. In ordr to nhanc th accuracy and numrical stability of LBM, particularly for daling ith microfluidic flos, dvlopd a n modl, i.., dynamic boundary modl. Th prsnt study is dvotd to th fasibility of this dynamic tratmnt mthod. Accuracy and stability studis hav bn carrid out on 2-D Poisuill flo, oscillating Coutt flo btn to paralll plans, Coutt flo btn to circular cylindrs and lid-drivn cavity flo.

2 J. Kang t al. / Journal of Mchanical Scinc and Tchnology 22 (28) 92~ Background 2. Lattic-Boltzmann mthod Th origin of LBM tracs back to th ida that on can construct th fluid flos from a group of fictitious particls having idntical mass and spd, but diffrnt vlocity dirctions. Thy hav motions lik suspndd pollns in a cup of atr. Thy can mov, chang thir positions, collid ith ach othr and xchang thir momntum. Th initial modl to raliz this ida is th lattic gas cllular automata (LGCA) hich has bn gradually dvlopd to LBM. In LBM, th statistical concpt, i.., particl mass distribution function, f, is introducd to rplac th ral particl in LGCA; th momntum, dnsity, vlocity and prssur, tc., can b obtaind from th volution of f [4]. During mor than to dcads, LBM has bn dvlopd to svral modl classs adaptd to diffrnt fluid applications. In th prsnt study, mployd th D2Q9 (to-dimnsional nin-vlocity lattic typ shon in Fig. ) LBGK modl [8], and applid th modifid quilibrium distribution function hich as proposd by H and Luo [9] for incomprssibl flos. Th distribution function is dtrmind by th folloing volution quation. f + tt+ t f t = f t τ q f ( x, t) ( x δ, δ ) ( x, ) ( x, ) () In th abov quation, x is th coordinat of th lattic nod of intrst, t th tim, δ t th tim stp (in th prsnt simulation, δ t = ), τ th dimnsionlss rlaxation tim, =,, 2,, 8, th link numbr around a nodal point in th lattic systm, and th discrt vlocity vctor, hich for D2Q9 lattic spac is (,) = = (,),(,),(,),(, ) =,2,3,4 (2) (,),(,),(, ),(, ) = 5,6,7,8 Furthr, f is th particl mass distribution function, q and f th quilibrium distribution function along th link numbr at a nod of intrst givn by f = ρ + ρ u+ ( u ) q 2 4 cs 2cs 2 ( u u) 2cs 2 (3) hr, u is th fluid vlocity, ρ th fluid dnsity, ρ th rfrnc dnsity (constant), c s th sound spd, and th ighting factor; = 4/9 for =, = /9 for =, 2,3, 4, and = /36 for = 5,6,7,8. Th fluid dnsity and mass flux can b valuatd by th folloing formula: 8 8 q f f (4) = = ρ = = 8 8 q f f = = ρ u = = (5) During on-tim-stp computation, Eq. () can b solvd by to sub-stps, i.., collision and straming; (i) collision stp: q f ( x, t) = f ( x, t) f(, t) f (, t) τ x x (6) (ii) straming stp: ( x+ δ, + δ ) = ( x, ) f tt t f t (7) hr ~ dnots th post-collision stat of th distribution function. It is notd that th collision stp is local and th straming stp involvs no spcial computation. 2.2 Boundary condition mthods in litratur Fig.. Discrt vlocity vctors of D2Q9 lattic. In gnral, to finish th straming stp, th distribution functions at th solid nods nar th physical boundaris, f ( x, t) (s Fig. 2), nd to b spci- s

3 94 J. Kang t al. / Journal of Mchanical Scinc and Tchnology 22 (28) 92~2 fid aftr th collision stp. W brifly dscrib hr th thr typical mthods that ar usd in valuating ths functional valus for th purpos to comparativly valuat th dynamic boundary mthod; bounc-back mthod [4], Yu s intrpolation mthod [5], and Guo s xtrapolation mthod [6]. Bounc-back is th most popular and simplst schm. If a nod is includd in boundary solid, th normal collision computation is omittd and th distribution functions ar bouncd back. Th subsqunt straming stp brings f back into th fluid domain. For a stationary all, it is quivalnt to stting f ( xs, t) = f ( x, t), hr dnots th vlocity dirction toard th physical boundary of th fluid domain, and th opposit dirction (Fig. 2). For a moving all, a crtain amount of momntum should b addd to th bouncd particl distribution function: f ( x, t) = f ( x, t) + 6 ρ u (8) s hr ρ is th fluid dnsity at th all, u th dsird xact vlocity spcifid on th boundary all nod W. In Yu s intrpolation schm [5], on obtains f ( x, t) on nod aftr straming stp hich is f ( xs, t δt) brought from th nod S during th straming stp. Yu s schm is constructd basd on th principl of momntum balanc to nsur th noslip condition ( u = ) on th all, considring th momntum balanc in th dirction of : f ( x, t) = f ( x, t) (9) A simpl intrpolation is thn usd to obtain th distribution functions on th boundary all nod f (, t) x givn as f(,) t = f(,) t + [ f( s,) t f(,) t ] x x x x () Hr, is th fraction of an intrsctd link in th fluid rgion givn by x x, = x xs () sing f ( x, t) and f( x + δ t, t), on obtains f ( x, t) using a linar intrpolation: f( x, t) = f( x, t) + [ f( x+ δ t, t) + f ( x, t) ] (2) If th boundary is drivn by non-zro all vlocity, u, thn an xtra momntum trm is addd to f ( x, t) in Eq. (9): f ( x, t) = f ( x, t) + 6 ρ u (3) For Guo s xtrapolation mthod [6], th distribution function at a solid nod nar th all f ( x s, t) is dcomposd into to parts, th quilibrium part and th non-quilibrium on. q n f ( x, t) = f ( x, t) + f ( x, t) (4) s s s n Th non-quilibrium part f ( x s, t) is approximatd from thos of th nighboring fluid nods along n n th link by using f ( xs, t) = f ( x, t) for n n n.75 and f ( xs, t) = f ( x, t) + ( ) f ( x 2, t) for <.75. Th quilibrium part is dtrmind by a fictitious quilibrium distribution q f ( xs, t) = ρs + ρ 2 us cs 2 ( ) ( ) + 4 us u 2 s us 2cs 2cs (5) Fig. 2. Layout of th lattics nar a curvd boundary. hr ρ s is th fluid dnsity on th solid nod S, bing approximatd to b qual to ρ (ρ at th nod in Fig. 2), and u s th fictitious vlocity on th solid nod nar th boundary, is dtrmind by a linar xtrapolation using

4 J. Kang t al. / Journal of Mchanical Scinc and Tchnology 22 (28) 92~2 95 ( ( ) )/ ( b ( ) ) u = u + u (6a) s u = 2u + u /( + ) (6b) s2 2 It is proposd to us us = u s for.75, and us a linar intrpolation from u s and u s2 for <.75, i.., us = us+ ( ) u s2. Th boundarycondition nforcmnt is compltd by th valuation of th post-collision distribution function as follos: f t f t f t (6) q n ( xs,) = ( xs,) + ( τ ) ( xs,) 3. Dynamic boundary modl Similarly to Guo s xtrapolation mthod, in our n mthod also dcomposd th distribution function at a solid all nod along th links across th physical boundary f ( x s, t) into quilibrium and non-quilibrium parts lik Eq. (4). Th quilibrium part is approximatd by Eq. (5). No, in ordr to obtain th fictitious vlocity on th solid nods nar th physical boundary, u s, mployd a dynamic quation givn by du dt s o = r( u u ) (8) hr u is th dsird xact vlocity spcifid on o th boundary all nod W and u th calculatd boundary-all vlocity. Eq. (8) can also b rittn in a discrt form as follos: u = u o + rδ t( u u o ) (9) s s Hr, r is a rlaxation factor, hich affcts th convrgnc proprty of th computation. o According to this dynamic quation, if u is largr than th dsird all vlocity u, th solid nod vlocity u s ill b dcrasd, and vic vrsa. Thn, th valu of fictitious vlocity on th solid nods u s can b automatically corrctd vry tim stp until th numrical all vlocity u matchs th xact vlocity u. For cod implmntation, th prsnt all vlocity o u can b computd by using an xtrapolation mthod ith th givn vlocitis at nods, 2 and 3 (illustratd in Fig. 2). W us on of th folloing thr algorithms for th xtrapolation dpnding on th vlocity data availabl on th narby nods: th-ordr : o o u = u linar xtrapolation: o o o u = ( + ) u u (2) 2 quadraticxtrapolation : o o o o u = ( + )(2 + ) u (2 + ) u2+ ( + ) u3 2 2 In most cass, sinc both u 3 and u 2 ar availabl, us th quadratic xtrapolation algorithm. n Th non-quilibrium part f ( x s, t) is approximatd by using a first-ordr xtrapolation schm basd on th non-quilibrium distribution functions on th nighboring fluid nods: f ( x, t) = β f ( x, t) + ( β) f ( x, t) (2) n n n s 2 hr β is a paramtr to b dtrmind. 4. Computational assssmnt For assssmnt of th proposd dynamic boundary modl, studid svral 2-D fluid flo problms, including Poisuill flo, oscillating Coutt flo btn to planar alls and rotating Coutt flo btn to circular cylindrs. W applid th dynamic mthod as ll as othr mthods to ths flos, compard th rsults and chckd th spatial, tmporal accuracy and th numrical stability. Lid-drivn cavity flo is also studid to tst th ability of handling fluid problms ith gomtric singularity. To valuat th computational accuracy of th rsults, mad us of to quantitis,: th all-slip vlocity rror S and th rror in th vlocity profil ε. Th all-slip vlocity rror S is givn by S = u u (22) Hr, u is th numrical vlocity on th boundary all hich can b obtaind by th quadratic xtrapolation (Eq. (2)). As u is initially knon and fixd, th diffrnc btn u and u provids a masur of th accuracy of th mthod. Th rror ε is calculatd by ε N = N 2 2 u ( j) u( j) / u (23) max j =

5 96 J. Kang t al. / Journal of Mchanical Scinc and Tchnology 22 (28) 92~2 It rflcts th root-man-squar of th diffrnc btn th normalizd vlocity obtaind form th numrical computation and that from th analytic solution. Implmnting th no-slip boundary conditions on th solid alls and valuating th prformanc of diffrnt mthods is our main purpos in this study. W applid th proposd dynamic mthod to svral flo modls and compard th accuracy as ll as th numrical stability ith othr thr mthods: bounc-back schm, Yu s intrpolation mthod and Guo s xtrapolation mthod D poisuill flo Th grid systm for 2-D Poisuill flo is spcifid as illustratd in Fig. 3. Th boundary conditions for this flo ar st as follos. For th lft inlt, applid a parabolic profil Fig. 4 shos th variation of S ith rspct to diffrnt numbr of grids N y obtaind by applying four diffrnt boundary-tratmnt mthods. Th valu of S givn by th prsnt mthod is sn to b much smallr than thos by othr schms. In fact, it 5 is on th ordr of hich can b considrd as th computr s machin rror. So suppos that thr is almost no numrical all-slip vlocity rror by applying th prsnt mthod to th Poisuill flo. Fig. 5 shos ε obtaind by applying four diffrnt mthods. W can s that th magnitud of ε by th prsnt mthod is smallr than th othrs by svral ordrs of magnitud. In Fig. 6, ε is also chckd ith diffrnt positions of physical all on th lattic. Th variation of ε by th prsnt mthod not only shos small rror valus but also provids much lss dpndnc on. u ( ) 4 ( )/ in y y H y H 2 = (24) hich corrsponds to th xact vlocity profil of th fully-dvlopd Poisuill flo. Hr, y is th vrtical coordinat ith its origin at th bottom all, its discrt valu on th nod j bing givn by yj = j +, H = N y + 2 th hight of th fluid domain, N y th grid numbr along y dirction, and th givn maximum inlt vlocity hich as st at =. in th lattic unit. Each of th boundary alls is locatd at xactly halfay btn to grid lins, i.., =.5. As th outlt boundary condition, zro gradint as imposd on th main stram vlocity. W applid no-slip boundary condition at th top and bottom solid alls by using diffrnt tratmnt mthods and th sam rlaxation tim τ = 5. In applying th prsnt mthod, hav spcial paramtrs to b dtrmind, rlaxation factor r in (8) and cofficint β in (2). Ths to paramtrs affct th convrgnc proprty and accuracy of th hol numrical computation. For 2-D Poisuill flo, chos r =. and β =. Fig. 4. Variation of all-slip vlocity rror givn by th dynamic mthod in comparison ith othr thr mthods in Poisuill flo ( =.5, τ = 5, =., r =., β = ). Fig. 3. Lattic systm for 2-D Poisuill flo. Fig. 5. Variation of ε rror givn by th dynamic mthod in comparison ith othr thr mthods in Poisuill flo ( =.5, τ = 5, =., r =., β = ).

J. Kang t al. / Journal of Mchanical Scinc and Tchnology 22 (28) 92~2 97 Fig. 6. Dpndnc of ε on rprsnting th boundary all position in Poisuill flo ( τ = 5, =., N = 3, r =., β = ). y Fig. 8.

In this figur, th gray color contours rprsnt th lvl of rror ε ; th dark color dnots a high numrical accuracy and light/hit color a lo accuracy including vn th numrical ovrflo.

2 Coutt flo btn to circular cylindrs (b) In Coutt flo btn to circular cylindrs, th innr circl of radius R is rotating ith linar vlocity =. and th outr circl of radius R 2 is stationary.

6 J. Kang t al. / Journal of Mchanical Scinc and Tchnology 22 (28) 92~2 97 Fig. 6. Dpndnc of ε on rprsnting th boundary all position in Poisuill flo ( τ = 5, =., N = 3, r =., β = ). y Fig. 8. Lattic systm for Coutt flo btn to circular cylindrs. (a) W valuatd th numrical stability of th prsnt and othr mthods in t of th ε -contour in th paramtr spac (, τ ) as shon in Fig. 7. In this figur, th gray color contours rprsnt th lvl of rror ε ; th dark color dnots a high numrical accuracy and light/hit color a lo accuracy including vn th numrical ovrflo. Th prsnt mthod shos supriority as th dark color dominats larg rang of th valu of τ and. 4.2 Coutt flo btn to circular cylindrs (b) In Coutt flo btn to circular cylindrs, th innr circl of radius R is rotating ith linar vlocity =. and th outr circl of radius R 2 is stationary. Th grid systm is fixd at For such a Coutt flo, hav an analytical solution givn by κ R R uθ ( R) = 2 2 κ R R2 (25) (c) Fig. 7. Stability rprsntd by ε dpnding on τ and in Poisuill flo; (a) Guo s mthod, (b) Yu s mthod, and (c) dynamic mthod ( =.5, N = 3, r =., β = ). y hr uθ ( R) is th vlocity componnt along th tangntial dirction, R th radial coordinat and κ th ratio of innr and outr circl radius ( κ = R/ R2). W fixd R 2 = 42.5, and adjustd κ to control th gomtry. Figs. 9 and sho ε obtaind by applying diffrnt boundary-tratmnt mthods at τ =.6 (Fig. 9) and τ =.5 (Fig. ). Th paramtrs r and β for th dynamic mthod ar st at r =. and β =. Ths rsults indicat that th advantag of th dynamic boundary modl

7 98 J. Kang t al. / Journal of Mchanical Scinc and Tchnology 22 (28) 92~2 Fig. 9. Comparison of ε among diffrnt mthods in Coutt flo at τ =.6 (R=, r =., β = ). (a) Fig.. Comparison of ε among diffrnt mthods in Coutt flo at τ =.5 (R=, r =., β = ). appars mor distinctiv for larg τ valus. Fig. shos th distribution of ε in th paramtr spac ( τ, ). Considring that th numrical rsults ith ε lss than. ar accptabl, can s that th dynamic mthod is mor robust than Yu s and Guo s mthod in trating th boundary conditions of Coutt flo btn to circular cylindrs. (b) 4.3 Oscillating coutt flo For an oscillating Coutt flo btn to paralll plans, th channl gomtry is st at L = 4H + (Fig. 2). Th boundary all is locatd halfay btn to nighboring horizontal grid lins, i.., st =.5. Th bottom all of th channl oscillats ith vlocity u(, t) = sin( ωt) (26) (c) Fig.. Stability rprsntd by ε dpnding on τ and in Coutt flo; (a) Guo s mthod, (b) Yu s mthod, and (c) dynamic mthod ( κ =.2, r =., β = ).

8 J. Kang t al. / Journal of Mchanical Scinc and Tchnology 22 (28) 92~2 99 Fig. 2. Sktch of oscillating Coutt flo. Fig. 4. Lattic systm for 2-D lid-drivn cavity flo. Fig. 3. Comparison of ε among diffrnt mthods in oscillating Coutt flo ( =., τ = 5, r =, β = ). ndr such an arrangmnt, th flo vlocity has an xact solution givn by λ y u = sin( ωt λy) (27) xa hr λ = ω/2υ and υ is th kintic viscosity in lattic unit. In th study, st ω =., =. and τ = 5. For th dynamic mthod, slctd r = and β =. Th numrical rsults shon by ε in Fig. 3 indicat that th prsnt mthod is mor accurat in trating this fluid problm too than othrs. 4.4 Lid-drivn cavity flo In ordr to chck th flxibility of th prsnt mthod in solving th standard flo problm but having cornr singularitis, mployd it to th liddrivn cavity flo (Fig. 4). W applid Guo s and prsnt mthods in implmnting th no-slip boundary conditions for stationary sid and bottom alls and =. for th sliding top all. In simulation, th grid systm is fixd at 5 5, th paramtrs at =.5, τ =.8 and R = 5 in both mthods and Fig. 5. Comparison of u and v vlocitis along vrtical and horizontal cntrlin of lid-drivn cavity flo btn Guo s and dynamic mthods ( =., τ =.8, R = 5, =.5, r =., β = ). r =. and β = in th dynamic mthod. Fig. 5 shos rsults bing in a vry good agrmnt ith Guo s. In fact, it is an laborat task to apply th prsnt mthod to th cornr of lid-drivn flo. Spcial considration and car nd to b givn, bcaus diffrnt choic of th xtrapolation schm to obtain o u lads to a diffrnt ffct, and vn influncs th solution stability. 4.5 Convrgnc study and furthr discussions W hav dmonstratd that by using th dynamic boundary modl, on can gt mor accurat and stabl rsults. But th convrgnc is anothr proprty to b chckd. W found that th rlaxation tim τ is th main factor affcting th convrgnc proprty of th dynamic boundary modl. W studid th convrgnc proprty for th 2-D Poisuill flo at =. and =.5. For th dynamic mthod,

9 2 J. Kang t al. / Journal of Mchanical Scinc and Tchnology 22 (28) 92~2 (a) (b) (c) (d) Fig. 6. Convrgnc proprtis of diffrnt mthods usd for implmnting th no-slip boundary condition in LBM for th Poisuill flo at =., =.5, and β = ; (a) τ =.6, (b) τ =, (c) τ = 3, (d) τ = 5. chckd th prformanc at th β valu fixd at β = but ith th r valu in th rang. r. Fig. 6 displays th numrical rsults at four diffrnt τ valus. Th comparison shos that at small valu of τ, applying th dynamic boundary modl can caus rathr slo convrgnc and can rval fluctuation at th initial transint stat. Hovr, incras of τ brings fastr convrgnc as ll as highr accuracy ith th dynamic boundary modl. W hav shon that applying th dynamic boundary modl to th implmntation of no-slip boundary conditions in LBM is suitabl and robust spcially hn τ is larg. Th Rynolds numbr, R = Lυ /, can b rittn as R = 3 L /( τ.5) from th rlation τ = 3υ +.5. So, to mak th Rynolds numbr small, must st ithr small or τ larg. On th othr hand in studying th fluid mixing, th valu of must b st as larg as possibl bcaus th mixing phnomnon is usually govrnd by particl advction, and small ould lad to longr CP tim. Thrfor, in ordr to apply LBM to th fluid mixing spcially in microfluidics, hich is charactrizd by lo Rynolds numbrs, must find a suitabl schm such as th prsnt mthod that is robust vn at high τ valus. Evn so, still nd to invstigat th ffct of r on th convrgnc proprty mor laboratly and hopfully find som ay to improv th convrgnc proprty.

10 J. Kang t al. / Journal of Mchanical Scinc and Tchnology 22 (28) 92~ Conclusions In this study, hav prsntd a n LBM boundary-condition-tratmnt mthod, namly, dynamic boundary modl. W applid th prsnt mthod togthr ith othr xisting typical mthods to svral flo problms. Th comparison study dmonstratd that th proposd dynamic boundary modl posssss th ability to produc high accuracy. W also drivd a conclusion that our dynamic boundary modl is particularly suitabl in simulating microfluidic mixing problm as it is mor robust and accurat at lo Rynolds numbrs than othr mthods. Acknoldgmnt This ork as supportd by th Kora Scinc and Enginring Foundation (KOSEF) through th National Rsarch Laboratory Program fundd by th Ministry of Scinc and Tchnology (No. 25-9). Rfrncs [] D. P. Ziglr, Boundary Conditions for Lattic Boltzmann Simulations, J. Stat. Phys., 7 (993) [2] A. J. C. Ladd, Numrical Simulation of Particular Suspnsions via a Discrtizd Boltzmann Mthod, J. Flui Mch., 27 (994) [3] X. H, Q. Zou, L. S. Luo and M. Dmbo, Analytic Solutions and Analysis on Non-slip Boundary Condition for th Lattic Boltzmann BGK Modl, J. Stat. Phys., 87 (997) [4] S. Succi, Th Lattic Boltzmann Equation for Fluid Dynamics and Byond, Oxford, Clarndon Prss, (2) [5] D. Yu, R. Mi and W. Shyy, A nifid Boundary Tratmnt in Lattic Boltzmann Mthod, st Arospac Scincs Mting and Exhibit, AIAA, (23) [6] Z. Guo, C. Zhng and B. Shi, An Extrapolation Mthod for Boundary Conditions in Lattic Boltzmann Mthod, Phys. Fluids, 4 (6) (22) [7] I. Ginzbourg and P. M. Aldr, Boundary Flo Condition Analysis for th Thr-dimnsional Lattic Boltzmann Modl, J. Phys. Ⅱ Franc, (994) [8] Y. H. Qian, D. D Humirs and P. Lallmand, Lattic BGK Modls for Navir-Stoks Equation, Europhys. Ltt., 7 (992) [9] X. H and L. Luo, Lattic Boltzmann Modl for th Incomprssibl Navir-stoks Equation, J. Stat. Phys., 88 (997)

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