Improved axisymmetric lattice Boltzmann scheme

Transcription

1 Impoved axisymmeic laice Bolzmann scheme Q. Li, Y. L. He, G. H. Tang, and W. Q. Tao Naional Key Laboaoy of Muliphase Flow in Powe Engineeing, School of Enegy and Powe Engineeing, Xi an Jiaoong Univesiy, Xi an, Shaanxi 7149, China This pape poposes an impoved laice Bolzmann scheme fo incompessible axisymmeic flows. The scheme has he following feaues. Fis, i is sill wihin he famewok of he sandad laice Bolzmann mehod using he single-paicle densiy disibuion funcion and consisen wih he philosophy of he laice Bolzmann mehod. Second, he souce em of he scheme is simple and conains no velociy gadien ems. Owing o his feaue, he scheme is easy o implemen. In addiion, he singulaiy poblem a he axis can be appopiaely handled wihou affecing an impoan advanage of he laice Bolzmann mehod: he easy eamen of bounday condiions. The scheme is esed by simulaing Hagen-Poiseuille flow, hee-dimensional Womesley flow, Wheele benchmak poblem in cysal gowh, and lid-diven oaional flow in cylindical caviies. I is found ha he numeical esuls agee well wih he analyical soluions and/o he esuls epoed in pevious sudies. PACS: j Ⅰ. INTRODUCTION Because of is kineic naue and disincive compuaional feaues, he laice-bolzmann (LB mehod, which oiginaes fom he laice-gas auomaa (LGA mehod [1], has been developed ino a 1

2 vey aacive alenaive o convenional numeical mehods. In he LB mehod, insead of solving he macoscopic govening uaions, he discee Bolzmann uaion wih ceain collision models, such as he maix model [, 3], Bhanaga-Goss-Kook (BGK model [4-7], muliple-elaxaion-ime (MRT model [8-13], and he wo-elaxaion-ime (TRT model [14-16], is solved o simulae fluid flows and model physics in fluids. In he lieaue, he main advanages of he LB mehod ae summaized as follows [17]: (i non-lineaiy (collision pocess is local and non-localiy (seaming pocess is linea, while in he Navie-Sokes uaion he convecive em u u is non-linea and non-local a a ime; (ii seaming is exac; (iii complex bounday condiions can be easily fomulaed in ems of elemenay mechanics ules; (iv fluid pessue and he sain enso ae available locally; (v nealy ideal amenabiliy o paallel compuing (low communicaion/compuaion aio. Owing o hese advanages, in he pas wo decades he LB mehod has been successfully applied o vaious flow poblems in science and engineeing [18-4] In ecen yeas, he LB mehod fo axisymmeic flows has aaced much aenion. I is known ha LB simulaions of axisymmeic flows can be handled wih a sandad hee-dimensional (3D LB model. Howeve, such a eamen does no ake he advanage of he axisymmeic popey of he flow: 3D axisymmeic flows ae wo-dimensional (D poblems in a cylindical coodinae sysem. To make use of his popey, much eseach has been conduced. The fis aemp was made by Halliday e al. [5]. The basic idea of Halliday e al. s mehod is o incopoae spaial and velociy dependen souce ems ino he micoscopic evoluion uaion o mimic he addiional axisymmeic conibuions in cylindical coodinaes. Following Halliday e al. s wok, Peng e al. [6] poposed a hybid LB model fo incompessible axisymmeic hemal flows by solving he azimuhal velociy and

3 he empeaue wih a second-ode cene-diffeence scheme. Neveheless, i was lae found ha Halliday e al. s model fails o epoduce he coec hydodynamic momenum uaion due o some missing ems. Afe consideing hese ems, Lee e al. [7] developed a moe accuae axisymmeic LB model. Reis and Phillips [8] have also pesened a modified vesion of Halliday e al. s model by deiving he souce ems in a diffeen manne. The modified model was subsuenly validaed wih seveal numeical ess [9]. In he above models, some complex diffeenial ems wee inoduced ino he second-ode souce em due o he discee effecs on he fis-ode souce em (idenical o a focing em. These complex ems may inoduce some addiional eos and do ham o he numeical sabiliy. He e al. [3, 31] have poined ou he apezium ule is necessay fo he inegaion of a focing em o avoid he spuious effecs in he ecoveed macoscopic uaions. By using a new disibuion funcion o eliminae he impliciness esuling fom he apezium ule, i can be found ha a faco dependen on he elaxaion ime will be included in he focing em and he macoscopic vaiables should be edefined [3]. Following his saegy, Pemnah and Abaham [33] devised a LB scheme fo axisymmeic muliphase flows. The scheme was exended o axisymmeic wo-phase flows wih lage densiy aio in Ref. [34]. Similaly, Zhou [35] ecenly poposed a simplified axisymmeic LB model by adoping a ceneed scheme o simplify he souce em. Besides he above-menioned models, an axisymmeic LB mehod based on he voiciy-seam-funcion uaions of incompessible axisymmeic flows has also been developed [36, 37]. In his mehod, disibuion funcions f ω and f ψ fo ω and ψ ae adoped, whee ω and ψ ae he voiciy and he seam funcion, especively, and is he coodinae in he adial diecion. The adial velociy u and axial velociy u ae obained fom u ( ψ z = and z 3

4 u z ( ψ = wih a second-ode cene-diffeence scheme, whee z is he coodinae in he axial diecion. The auhos poined ou ha he main dawback of he mehod is he difficuly in eaing bounday voiciy. In his egad, an impoan advanage of he LB mehod, he easy eamen of bounday condiions, may be los. In a ecen pape [38], an axisymmeic kineic BGK model of single-paicle densiy disibuion funcion f has been deived fom he coninuous Bolzmann uaion in cylindical coodinaes. Due o a em in he kineic model, a new disibuion funcion f was employed o eplace f when devising an axisymmeic LB model. The souce em of he devised model conains no gadien ems and is much simple han hose in pevious models. Alhough pevious models wee ciicized fo he inclusion of velociy gadien ems ino he souce em, i doesn mean hei benefis can be ignoed. Since hese models ae wihin he famewok of he sandad LB mehod, he geneal philosophy of he LB mehod is eained. When he disibuion funcion involves he coodinae, people may be bewildeed by poblems ha ae seemingly inconsisen wih he philosophy of he LB mehod, such as: why he (adial coodinae of one node can be popagaed o is neighboing node wih he paicle? In addiion, he singulaiy poblem a he axis ( = canno be solved. In numeical applicaions, his poblem is found o cause inconvenience while eaing bounday condiions. On he conay, his poblem can be appopiaely handled in pevious models wihou affecing he easy eamen of bounday condiions [6-9, 33, 34]. I is geneally expeced ha a moe consisen axisymmeic LB scheme can be esablished if he poblems ha plague pevious schemes ae ovecome. Howeve, fom he available cuen lieaue on he axisymmeic LB mehod, people may conclude ha i is impossible o have such an axisymmeic LB scheme. Hence, in his pape, we aim o develop an impoved axisymmeic LB scheme based on 4

5 pevious sudies and o show ha consucing a simple axisymmeic LB scheme wihin he famewok of he sandad LB mehod is possible. The es of he pape is oganized as follows. The macoscopic govening uaions fo incompessible axisymmeic flows and an oiginal axisymmeic LB scheme ae descibed in Sec. Ⅱ. The impoved scheme is poposed in Sec. Ⅲ. Wihou loss of genealiy, boh he BGK and MRT collision models will be consideed. In Sec. Ⅳ, he numeical validaion is pesened. Finally, Secion Ⅴ concludes he pape. Ⅱ. MACROSCOPIC EQUATIONS AND AN ORIGINAL AXISYMMETRIC LB SCHEME The poblem of lamina axisymmeic flows of an incompessible fluid wih an axis in he z diecion is consideed. The macoscopic uaions fo incompessible axisymmeic flows in cylindical coodinaes ae given as follows [7, 35, 39]: juj = u (1 μ ρuu i μui ρ ui j ( uu i j + = ip+ μ jui + ui δ i, ( whee i, j indicae he o z componen, μ is he dynamic viscosiy, and δ i is he Konecke dela wih wo indices. Beaing in mind ha, in he sandad LB mehod he ecoveed macoscopic momenum uaion is ( ρu ( ρuu p μ( u u i + j i j = i + j i j + j i, (3 heefoe we need o ewie Eq. ( as μ ρuu i μui ρ ui j ( uu i j + = ip+ μ j ( jui + iuj + ( ui + iu δ i. (4 The esul μ ( u μ ( u μ( u uδ = = has been used in he above deivaion. j i j i j j i i i In Ref. [33], Pemnah and Abaham adoped he following evoluion uaion fo axisymmeic flows by inegaing he collision and souce ems wih he apezium ule: 5

6 whee ( f f τ 1 δ + G G, (, ( +, + x x e δ + δ ( x+ e δ, + δ ( x, = Ω +Ω ( x, ( x+ e δ, + δ f f Ω =, f is he discee single-paicle densiy disibuion funcion, x is he ( spaial veco, i.e., x= z,, e = ( e, e is he velociy veco of a paicle in he link, δ z (5 is he ime sep, τ is he dimensionless elaxaion ime, and f is he uilibium disibuion, which can be given by ( ( e u e u u f = wρ 1 + +, (6 4 cs cs c s fo he wo-dimensional nine-velociy (DQ9 laice [6], whee cs = c 3 ( c = δ x δ is he sound speed and he weighs w ae given by w = 49, w1 4= 19, and w5 8= 136. If he axisymmeic conibuions of suface ension and phase segegaion effecs ae no consideed, he souce em G is ( ρu e u μ μu ρuu G = w + f ( u + u δ. (7 i i i i i i i ρc s The Chapman-Enskog analysis [4, 41] of Eq. (5 can be found in Ref. [33]. The impliciness of Eq. (5 is eliminaed wih f% = f.5ω.5δ G [31]: f% (, (, (, + δ + δ f% = ω f% f (, + ( 1.5ω δ G (, whee ω 1 ( τ.5 x e x x x x, (8 = +. The macoscopic densiy and velociies ae calculaed by δ ρu ρ = f %, (9 δ μ ρuu μu ρui = e i f + ui iu + % i i ( δ. (1 i Ⅲ. IMPROVED LB SCHEME FOR INCOMPRESSIBLE AXISYMMETRIC FLOWS A. BGK collision model In his secion, an impoved axisymmeic LB scheme will be developed based on he above 6

7 oiginal scheme. Acually, fom Eqs. (4, (7, and (1, i can be seen ha, if we wan o devise a simple axisymmeic LB scheme based on he sandad LB mehod, he em ( u u μ + in he i i macoscopic axisymmeic momenum uaion should be ecoveed in such a way ha he difficulies aising fom his em can be avoided. Moivaed by ou ecen wok [4], we popose he following evoluion uaion: 1 δ e + S S (, δ (, ( f f + + δ + δ x x e. (11 ( x+ e δ, + δ ( x, = Ω +Ω ( x, ( x+ e δ, + δ f f Noe ha, in he Chapman-Enskog pocedue, he las em on he igh-hand side of Eq. (11 will exis in he second-ode expansion of he evoluion uaion. Then no discee laice effecs need o be consideed. The souce em S is given by ( S e = u F u f, μu δ F = i i i i i i ρcs. (1 Hee i can be seen ha S is simple and conains no velociy gadien ems. Accoding o He e al. [31], he impliciness of Eq. (11 can be emoved wih a new disibuion funcion f = f.5ω.5δ S, fom which he following LB scheme can be obained: f (, (, (, x+ eδ + δ f x = ω f f f (, x x + ( 1.5ωf δs ( x,, (13 whee ω = 1+ ( τδ e ( τ +.5. The macoscopic vaiables ae defined as f δ ρu ρ = f, (14 δ ρuu μu ρui e f = + δ i i i i. (15 Muliplying Eq. (14 wih u i and hen subsiuing he esul ino Eq. (15, we can obain Fom Eq. (14, he densiy is given by u i e i f + ( δ μ =. (16 f δi 7

8 f ρ =. ( (.5δ u In he incompessible limi [43] (i.e., ρ = ρ + δρ ρ and δρ is of he ode Ma, whee Ma is he Mach numbe, he viscosiy μ used in F i and Eq. (16 is eplaced wih μ. In summay, uaion (13 ogehe wih Eqs. (1, (16, and (17 consiues an impoved axisymmeic LB-BGK scheme. Fo he sake of demonsaing ha he coesponding macoscopic uaions can be coecly ecoveed in he limi of small Mach numbe, we poceed o pefom he Chapman-Enskog analysis of he evoluion uaion. Fis, aking a second-ode Taylo seies expansion o Eq. (11 in ime and space aound poin ( x,, we have δ δ δ τ τ 1 ( + e f + ( + e f = ( f f ( + e ( f f δ e + δs + ( + e S δ ( f f + Ο( δ 3, (18 whee = (, is he spaial gadien opeao. Accoding o he Chapman-Enskog expansion [5, z 4, 41], he ime deivaive, he disibuion funcion, and he souce em can be wien as, ( ( 1 (, ( ( 1 δ 1 f f δ f δ f S S S = + = + + = + δ, (19 whee ( u f S e u f c u 1 i i ρ sτ iδi ρcs = and ( ( S =. Wih hese muli-scale expansions, we can ewie Eq. (18 in he consecuive odes of δ : ( ( : Ο δ f = f, ( 1 ( : ( Ο δ ( 1 (1 ( + e f + f = S, (1 τ ( (1 1 ( 1 ( 1 Ο( δ : 1f + ( + e f + ( + e f + f τ + τ + e 1 ( ( 1 e ( 1 = ( + e S + S f. Using Eq.(1, Eq. ( can be ewien as ( f (1 ( 8

9 ( (1 1 ( e ( 1 ( 1 1f ( f f f S + + e + = + τ. (3 Summaions of Eq. (1 and Eq. (3 lead o, especively ( ρu ρu ρ + j j =, (4 1ρ =. (5 Combining he above wo uaions ( = + δ 1 gives ρu ρ + j ( ρu j =. (6 Taking he fis-ode momen, e i ( Fom Eq. (1, i is obained ha, of Eqs. (1 and (3, especively, we ge ( ρ ( ρ ρuu i ui + j uu i j = ip, (7 ρc τu + = 1 s i 1 i j i j i (1 ( 1 ( ρu ( e e f e e f δ i. (8 (1 ( ( ( eie j f = τ ( ij + kpijk eie js, (9 whee = and ( ( P ijk = ei e jek f. Fo he DQ9 laice model, ( and ij ( ( ij ei e j f ( ijk P ae given by ( ( ij ρuu i j ρcsδ ij, Pijk ρcs ( uiδ jk ujδik ukδij = + = + +. (3 Some sandad algeba will show ha ( 3 c δ ρ c u ρ c u ρ Ο ( = + ij s ij s j i s i j u, (31 ( ( k Pijk csδ ij k ρuk ρcs jui ρcs iuj csui jρ c suj i = ρ. (3 Wih he above esuls, we have ( ( ( ( ij kpijk csδij ρ k ρuk ρcs jui iuj + = (33 Using Eq. (33 wih Eq. (4 and noing ha ( 3 = s + ( e e S c u u i j δ ij ρ Ο, we can simplify Eq. (9 o ( e e f = τρc u + u. (34 (1 i j s j i i j 9

10 Subsiuing Eq. (34 ino Eq. (8 yields 1 ρcsτui 1( ρui = j τρcs ( jui iuj + + τρcs ( ui + iu δ i. (35 Combining Eq. (35 wih Eq. (7 ( = + δ 1, we can obain μ ρuu i μui ( ρui + j ( ρuu i j = ip+ j μ( jui iuj + + ( ui + iu δ i, (36 whee μ = τρ δ. Clealy, in he incompessible limi ( ρ ρ, Eqs. (6 and (36 educe o he c s axisymmeic coninuiy uaion (1 and he momenum uaion (4, especively. Now a bief compaison beween he impoved and oiginal schemes is made. Fis, boh schemes ae wihin he famewok of he sandad LB mehod using he single-paicle densiy disibuion funcion and have a simple sucue so ha he geneal benefis of he sandad LB mehod ae eained. On he ohe hand, in he impoved scheme, he em ( u u μ + is ecoveed in an efficien i i way ha is consisen wih he philosophy of he LB mehod. As a consuence, he souce em and he calculaions of macoscopic vaiables ae gealy simplified. Accodingly, he poblems ha plague he oiginal scheme ae ovecome. B. MRT collision model 1. MRT-LB mehod In Ref. [34], he MRT collision model, which is an impoan exension of he elaxaion LB mehod poposed by Higuea [, 3], has been employed o consuc an axisymmeic MRT-LB scheme based on he above-menioned oiginal scheme. Much eseach has shown ha he MRT collision model can significanly impove he numeical sabiliy of LB schemes by caefully sepaaing he elaxaion imes of hydodynamic and non-hydodynamic momens. A deailed descipion of he MRT-LB mehod can be found in Refs. [8-13]. Accoding o Refs. [1, 13, 34], a DQ9 MRT-LB 1

11 scheme wih a semi-implici eamen of he souce em is given by whee δ f ( x+ e δ, + δ f ( x, = Λ ( f f + S S ( + ( ( + +, (37 β β β,, x x x e, δ δ 1 Λ=Μ ΛΜ is he collision maix, in which = diag ( sρ, se, sε, sj, sq, sj, sq, sv, sv Λ is a diagonal Maix and M is a ohogonal ansfomaion maix (see Ref. [9]. Though he ansfomaion maix, he disibuion funcion f and is uilibium disibuion f can be pojeced ono he momen space wih m = Mf and m = Mf, whee T f = ( f, f1, L, f8 and (,, T f = f L f 8. Fo he DQ9 laice model, m and m ae given by m whee ρ is he densiy; = ( m = ρ, e, ε, j, q, j, q, p, p x x y y xx xy T ( ρ, e, ε, jx, qx, jy, qy, pxx, pxy ρ ( 1, 3 u, 1 3 u, ux, ux, uy, uy, ux uy, uxuy = + T, (38 T, (39 e is he enegy mode; ε is elaed o enegy squae; ( jx, j y ae he momenum componens; ( qx, q y coespond o enegy flux; and ( xx, xy p p ae elaed o he diagonal and off-diagonal componens of he sess ensos [9]. Because of he implici eamen of he souce em, Eq. (37 canno be diecly applied in numeical simulaions. The following explici MRT-LB scheme can be obained wih f = f.5δ S [1, 13]: ( + δ, (, ( + δ = Λ + δ (, (.5Λ x ( x, f x e f x f f S S. (4 β β β β β Usually, as shown in Ref. [9], he collision pocess of MRT-LB schemes is caied ou in he momen space + Λ m = m Λ ( m m + δ I S %, (41 whee m = Mf and S % = MS, in which ( T S = S, S, L, S, while he seaming pocess is 1 8 implemened in he velociy space + ( + δ, + δ = ( f x e f x,, (4 11

12 whee f = M m +. Accoding o Eq. (41, he collision pocess of he momenum componens can be + 1 wien as ( δ ( j + = j s j j + s S%, (43 x x j x x j ( δ ( j + = j s j j + s S%. (44 y y j y y j The macoscopic uaions ecoveed fom MRT-LB schemes can also be deived hough he Chapman-Enskog analysis, which can be implemened in he momen space. Fo he deails of his pocedue, eades ae efeed o Refs. [9, 1, 13, 44]. Seveal elaionships ae given below consideing ha hey will be used in he nex subsecion: ( 1 ( ( 1 ρ (, (, 1 ρ see = ρ xux + yuy svpxx = xux yuy svpxy = ( xuy + yux, ( whee ( 1 e, ( 1 xx p, and p ( 1 ae defined as xy ( 1 δ m m m, coesponding o ( 1 δ f f f. 3 Noe ha ems of ( Ma Ο have been negleced in Eq. (45.. Axisymmeic MRT-LB scheme I is known ha, wih he benefi of using MRT collision model, he collision pocess of each momen can be manipulaed independenly in he momen space. The appoach of modifying he collision pocess o adjus macoscopic uaions has been used in Ref. [45], in which he wo momens elaed o he enegy flux wee modified o achieve a consisen viscosiy in he macoscopic momenum and enegy uaions. In he pesen wok, i is found ha he collision pocess of he momenum componens can be appopiaely manipulaed o ecove he velociy gadien em ( u u momenum uaion. To his end, we need o evaluae ( u u μ + in he axisymmeic i i i i μ + in a way consisen wih he philosophy of he MRT-LB mehod. Noe ha, fom Eq. (45, we have 1

13 Using Eqs. (45 and (46, we can obain 1 1 ρ s e + s p = u + u 6 3 ( 1 ( 1 e v xx ( x x x x μ γ 1 1 ( + = + ( 1 ( u u see svpxx sv 6. (46 1, (47 μ ( 1 ( uz + zu = γ pxy, (48 whee μ = ( 1.5 δρ 3 γ = ( s δ, and (, z coespond o (, s v, 1.5 v x y. Thus we can modify he collision pocess of he momenum componens as follows: ( x x δγ ( e v + + ( 1 ( 1 = new xx j j s s e p, ( ( 1 ( jy = jy δγ xy new p. (5 ( 1 Hee e, p ( 1 xx, and ( 1 xy p ae given by ( 1 ( 1 m = m +.5 S % (ecalling he uaion f = f.5 δ S, in which m ( 1 is appoximaed by ( 1 δ m m m. The souce em in he momen space is ( m S % = u + S % wih S % given as follows [1]: S% =, S% = 6 u F, S% = 6 u F, 1 S% = F, S% = F, S% = F, S% = F, 3 x 4 x 5 y 6 ( x x y y ( x y y x S% = u F u F, S% = u F + u F, ( whee Fx = F and F y = F z ae given in Eq. (1. I can be eadily poved ha, wih such a choice of y he souce em, he elaionships shown in Eq. (45 will no change. Finally, uaions (41 and (4 ogehe wih he modified collision pocess, Eqs. (49 and (5, consiue a consisen axisymmeic MRT-LB scheme fo incompessible axisymmeic flows. The macoscopic vaiables ae calculaed by Eqs. (16 and (17 hough eplacing f wih f. C. Exension o axisymmeic oaional flows By including he effec of azimuhal oaion, he poposed scheme can be applied o axisymmeic 13

14 oaional flows. The macoscopic govening uaion fo azimuhal velociy u θ in cylindical coodinaes is given by [39] μ ρuu θ μuθ ρ u j ( uju θ + θ = μ j ( juθ + uθ. (5 I is seen ha he above uaion is an advecion-diffusion uaion. Usually, a DQ4 o DQ5 laice model is enough fo an advecion-diffusion uaion in ems of he compuaional accuacy as well as he compuaional efficiency [46]. Fuhemoe, as suggesed in Ref. [36], he souce em of an advecion-diffusion uaion can be eaed moe simply han he usual focing saegy. Hence in his sudy he following evoluion uaion wih a DQ4 laice ( e : = 1,,3,4. is adoped o solve he azimuhal velociy: (, (, (, g + δ + δ = ω (, + δ ( g x e g x g g x g x S x,. (53 Hee g is he disibuion funcion fo azimuhal velociy, ωg = 1+ ( τgδe ( τg + ( e u.5, and ρu θ g 1 ν g = 1+, S u = + g 4 c, (54 whee ν = δτ is he kinemaic viscosiy and τ = τ 3. Clealy, he souce em c g g g S is also simple and conains no gadien ems. Noe ha, when he effec of azimuhal oaion is consideed, an ineial foce ρ δ should be included in F i of Eq. (1. Moeove, in he incompessible u θ i limi, he densiy ρ in he uilibium disibuion funcion g can be diecly eplaced by ρ, and hen he macoscopic azimuhal velociy is calculaed by u = g ρ. θ Ⅳ. NUMERICAL VALIDATION A. Hagen-Poiseuille flow and 3D Womesley flow To validae he poposed scheme, numeical simulaions ae caied ou fo some ypical axisymmeic flows. Fis, we conside he Hagen-Poiseuille flow, which is an axisymmeic seady, 14

15 lamina flow of a viscous fluid hough a pipe of unifom cicula coss-secion and diven by a consan exenal foce in he axial diecion. The analyical soluion fo he axial velociy of he Hagen-Poiseuille flow is given by whee U ar ( μ 4 uz ( = U 1 R, (55 = is he maximum axial velociy in he pipe, a is he exenal foce, and R is he adius of he pipe. In he simulaion, we adop a Nz N = 4 laice ( N z and N exclude he exa layes ouside he boundaies wih a line of symmey a = and a solid wall a =. The no-slip bounday condiion is imposed along he solid wall [47], he peiodic bounday condiions ae applied o he inle and oule, and he specula eflecion bounday [6] is employed along he axisymmeic line. The singulaiy a = is eaed following pevious sudies. In Refs. [9, 33], he souce em a = was evaluaed wih he L'Hôpial's ule; afe applying his ule, he souce em was se o be zeo. Similaly, in Ref. [6], all he ems elaed o ( 1 wee only applied a he posiion of. In ohe wods, hese ems wee appoximaely aken as zeo a =. In he pesen pape, a simila eamen is adoped. The maximum velociy U is se o be.5 wih a 4 = 1 and μ =.. The numeical axial velociy is shown in Fig. 1, in which he analyical soluion is also pesened fo compaison. I is obseved ha he numeical esul agees well wih he analyical one. When he consan foce in he Hagen-Poiseuille flow oscillaes wih a peiod T, he flow will become a 3D Womesley flow, which is an unseady axisymmeic flow in a cicula pipe diven by a peiodic foce a = a cos( ω, whee a is he maximum ampliude and he ω = π T is he angula fuency [48]. The Reynolds numbe is defined as Re = ρudμ wih he chaaceisic c lengh D = R and he chaaceisic velociy U = a ( 4ρω. Hee = R ρω μ is he c 15

16 Womesley numbe. The analyical soluion fo he Womesley flow is [7, 48] ( a ( φ ( φ J R iω uz, = Re 1 e iρω J, (56 whee J is he zeoh ode Bessel funcion of he fis ype and i is he imaginay uni, ( i φ = +. Noe ha Re in Eq. (56 denoes he eal pa of a complex numbe ahe han he Reynolds numbe. In his es, he bounday condiions and he gid sysem ae he same as hose used in he simulaion of Hagen-Poiseuille flow. Following pevious sudies, in he compuaion we se Re = 1, T = 1, =, = 7.97, ρ = ρ = 1, and μ =.1 3 ; he simulaions begin 3 a 1 3 wih an iniial condiion of zeo velociy and he numeical esuls a diffeen imes ae obained afe unning en peiods. Figues and 3 compae he axial velociy pediced by he pesen scheme wih he analyical soluions. I can be found ha he numeical esuls ae in excellen ageemen wih he analyical one. To quanify he esuls, a elaive eo defined as ξ = u( i ua( i ua( i i i (he subscip a denoes analyical is used o compae he pesen soluion wih he numeical soluion shown in Ref. [7], which is a bee one among he esuls epoed in pevious sudies. The global eo ξ is he aveage of ξ ove he peiod. The pesen ξ is.33%, which is smalle han he esul 1.3% epoed in Ref. [7]. B. Axisymmeic oaional flows In his subsecion, he Wheele benchmak poblem in he Czochalski cysal gowh [6, 49-51] and he lid-diven oaional flow in cylindical caviies [5-58] ae aken as he es examples o validae he capabiliy of he poposed scheme fo he simulaion of axisymmeic oaional flows. The configuaion of he Wheele poblem is descied in Fig. 4. In he poblem, a veical cylindical 16

17 cucible of adius Rc filled wih a mel o a heigh H = R oaes wih an angula velociy. On c Ω c he op of he mel, i is bounded by a coaxial cysal wih adius Rx = β R ( β =.4, which oaes c wih an angula velociy Ω x. The flow sucue depends on he Reynolds numbe Re = R Ω ν c c c and Re x = Rc Ω x ν. In he compuaions, a N N = 1 1 z laice is adoped and he value of he chaaceisic velociy U = β R Ω x is aken as U.1 so ha he Mach numbe of he flow is sufficienly small. c The elaxaion imes can be deemined wih U and Re x. The non-uilibium exapolaion scheme [59] is employed o ea diffeen bounday condiions of f and g, excep ha he specula eflecion bounday is applied o f along he axisymmeic line. The zeo velociies ae iniialized eveywhee. A seady sae can be eached afe a numbe of ieaions and he convegence cieion is max ( n+ 1 ( z z u + u u + u n ς, (57 whee n and n + 1 epesen he old and new ime levels, especively. ς is se o be 1 8 in his es [6]. The seamlines of ( ( 3 Re c, Rex = 1, 5 and ( 1, 5 ae pesened in Fig. 5, fom which we can see ha wo voices wih opposie diecions appea in he uppe lef cone and he lowe igh cone. Wih he incease of he Reynolds numbe, he uppe lef voex moves owads igh cone and he lowe igh pimay voex moves o lef and dominaes he whole flow field. These behavios ae also found in pevious numeical sudies. To quanify he esuls, he seam funcion defined as ψ = u z, zψ = u is calculaed and Table 1 shows he compaisons of ψ min and ψ max beween he pesen esuls and he esuls epoed in Refs. [6, 5]. Good ageemen can be concluded fom he able. In he Wheele poblem, if we se Rec = Ω c = and Rx = Rc, he flow will become he 17

18 lid-diven oaional flow in cylindical caviies, which is an impoan geneic poblem invesigaed boh expeimenally [5-54] and numeically [55-58]. The cylindical caviy oaional flow is known o depend on wo paamees, he aspec aio A = H R and he Reynolds numbe Re = R Ω ν. In he lieaue, i has been confimed ha, a ceain combinaions of A and Re, a eciculaion egion will fom along he axis of he cylinde. Such a eciculaion egion is called he voex beakdown bubble. In his sudy, he cases Re = 99 and 19 wih A = 1.5 ae consideed following Ref. [58]. The expeimenal esuls of hese wo cases ae available [54]. In ou simulaions, a gid sysem N N = 1 15 z is adoped. The bounday eamens ae simila o hose used in he Wheele poblem. The elaxaion ime τ is chosen as τ =. o ensue he chaaceisic velociy U =ΩR.1. The obained seamlines ae pesened in Fig. 6. Fom he figue i is seen ha a single voex beakdown appeas a Re = 19, wheeas he esul of Re = 99 do no eveal any voex beakdown. Table shows he magniude ( uz,max and he locaion ( h max H of he maximum axial velociy on he axis, in which he expeimen esuls [54] and he numeical esuls obained fom 3D-LB model [58] ae also lised fo compaison. To sum up, he pesen esuls ae well consisen wih he pevious ones. The compaison beween he BGK and MRT collision models is conduced hough simulaing 3 cylindical caviy oaional flow a Re = 19 wih a low viscosiy v = , which coesponds o τ =.5 and s = 1 ( The convegence cieion ς is se o be v 1 1 as he chaaceisic velociy U is gealy deceased. The simulaion esuls ae pesened in Fig. 7. I can be obseved ha he BGK model is numeically unsable when τ =.5, bu he MRT model can give a sable and coec soluion unde he same condiion. The compaison illusaes ha he enhanced numeical sabiliy of he MRT model compaed wih he BGK model in ha he MRT model 18

19 is able o achieve sable esuls a lowe viscosiies [1]. Finally, an impoan issue should be poined ou. Sicly speaking, in he axisymmeic LB-MRT mehod he bulk viscosiy should be ual o he dynamic viscosiy, which uies s e = s v. This is because, fo an axisymmeic LB scheme in cylindical coodinaes, he pseudo divegence of velociy u = u is nonzeo. While in he Caesian coodinae sysem, he eal divegence of velociy is i i zeo fo incompessible flows. In simulaions, an appopiae diffeence beween s v and s e may be pemied if needed. Howeve, when s e significanly diffes fom s v, lage eos will be inoduced. The sable soluion pesened in Fig. 7 is obained based on se = sv. In fac, a MRT collision model wih se = sv is simila o a wo-elaxaion-ime (TRT collision model [14-16], which is an impoan and naual simplificaion of MRT collision model [6]. In TRT models, he momens of even ( s e sε s v = = and odd odes ae elaxed a diffeen aes. Ⅳ. CONCLUSIONS In his pape, an impoved axisymmeic LB scheme has been developed fo incompessible axisymmeic flows. I has been shown ha consucing a simple axisymmeic LB scheme wihin he famewok of he sandad LB mehod using he single-paicle densiy disibuion funcion is possible. The main saegy is o ecove he em ( u u μ + in he macoscopic momenum uaion in i i an efficien way ha is consisen wih he philosophy of he LB mehod. The Chapman-Enskog analysis has been employed o demonsae ha he macoscopic uaions can be coecly ecoveed in he limi of small Mach numbe. As a esul of he change, he souce em becomes simple and conains no velociy gadien ems. Fuhemoe, he calculaions of macoscopic vaiables ae simplified. The singulaiy poblem a he axis is appopiaely eaed following pevious sudies, 19

20 eaining he easy eamen of bounday condiions. In he poposed scheme, boh he BGK and MRT collision models have been consideed. In addiion, is exension o axisymmeic oaional flows is also pesened. Numeical simulaions have been caied ou fo some ypical axisymmeic flows. The numeical expeimens show ha he esuls pediced by he pesen scheme ae in good ageemen wih he analyical soluions and he esuls epoed in pevious sudies. The compaison beween he BGK and MRT collision models has also been made. I is shown ha he MRT collision model exhibis an excellen numeical sabiliy compaed wih he BGK model when he viscosiy appoaches zeo. This feaue makes MRT-LB schemes moe useful in pacical applicaions. ACKNOWLEDGMENTS This wok was suppoed by he Key Pojec of Naional Naual Science Foundaion of China (No and he Naional Basic Reseach Pogam of China (973 Pogam (No. 7CB69. Refeences [1] U. Fisch, B. Hasslache, and Y. Pomeau, Phys. Rev. Le., 56, 155 (1986. [] F. Higuea and J. Jiménez, Euophys. Le. 9, 663 (1989. [3] F. Higuea, S. Succi, and R. Benzi, Euophys. Le. 9, 345 (1989. [4] J. M. V. A. Koelman, Euophys. Le., 15, 63 (1991. [5] S. Chen, H. Chen, D. Maínez, and W. Mahaeus, Phys. Rev. Le. 67, 3776 (1991. [6] Y. H. Qian, D. d Humièes, and P. Lallemand, Euophys. Le. 17, 479 (199. [7] X. He and L.-S. Luo, Phys. Rev. E 56, 6811 (1997.

21 [8] D. d Humièes, in Raefied Gas Dynamics: Theoy and Simulaions, Pog. Asonau. Aeonau. Vol. 159, edied by B. D. Shizgal and D. P. Weave (AIAA, Washingon, D.C., 199. [9] P. Lallemand and L.-S. Luo, Phys. Rev. E 61, 6546 (. [1] D. d'humièes, I. Ginzbug, M. Kafczyk, P. Lallemand, and L.-S. Luo, Phil. Tans. R. Soc. Lond. A 36, 437 (. [11] P. J. Della, J. Compu. Phys. 19, 351 (3. [1] M. E. MaCacken and J. Abaham, Phys. Rev. E 71, 3671 (5. [13] K. N. Pemnah and J. Abaham, J. Compu. Phys. 4, 539 (7. [14] I. Ginzbug, Adv. Wae Resou. 8, 1171 (5. [15] I. Ginzbug, Adv. Wae Resou. 8, 1196 (5. [16] I. Ginzbug, Comm. Comp. Phys. 3, 47 (8. [17] S. Succi, Eu. Phys. J. B 64, 471 (8. [18] R. Benzi, S. Succi, and M. Vegassola, Phys. Rep., 145 (199. [19] S. Chen and G. D. Doolen, Annu. Rev. Fluid Mech., 3, 39 (1998. [] A J. C. Ladd and R. Vebeg, J. Sa. Phys. 14, 1191 (1. [1] D. Yu, R. Mei, L.-S. Luo, and W. Shyy, Pog. Aeosp. Sci. 39, 39 (3. [] B. Dünweg and A. J. C. Ladd, Adv. Polym. Sci. 1, 89 (9. [3] S. Succi, The Laice Bolzmann Equaion fo Fluid Dynamics and Beyond (Claendon Pess, Oxfod, 1. [4] M. C. Sukop and D. T. J Thone, Laice Bolzmann modeling: An inoducion fo geoscieniss and enginees (Spinge, Belin, 5. [5] I. Halliday, L. A. Hammond, C. M. Cae, K. Good, and A. Sevens, Phys. Rev. E 64, 118 1

22 (1. [6] Y. Peng, C. Shu, Y. T. Chew, and J. Qiu, J. Compu. Phys. 186, 95 (3. [7] T. S. Lee, H. Huang, and C. Shu, In. J. Mod. Phys. C 17, 645 (6. [8] T. Reis and T. N. Phillips, Phys. Rev. E 75, 5673 (7. [9] T. Reis and T. N. Phillips, Phys. Rev. E 77, 673 (8. [3] X. He, X. Shan, and G. D. Doolen, Phys. Rev. E 57, R13 (1998. [31] X. He, S. Chen, and G. D. Doolen, J. Compu. Phys. 146, 8 (1998. [3] Z. Guo, C. Zheng, and B. Shi, Phys. Rev. E 65, 4638 (. [33] K. N. Pemnah and J. Abaham, Phys. Rev. E 71, 5676 (5. [34] S. Mukhejee and J. Abaham, Phys. Rev. E 75, 671 (7. [35] J. G. Zhou, Phys. Rev. E 78, 3671 (8. [36] S. Chen, J. Tölke, S. Gelle, and M. Kafczyk, Phys. Rev. E 78, 4673 (8. [37] S. Chen, J. Tölke, and M. Kafczyk, Phys. Rev. E 79, 1674 (9. [38] Z. Guo, H. Han, B. Shi, and C. Zheng, Phys. Rev. E 79, 4678 (9. [39] F. M. Whie, Fluid Mechanics (5h ed., McGaw-Hill, New Yok, 3. [4] S. Chapman and T. G. Cowling, The Mahemaical Theoy of Non-Unifom Gases, 3d ed. (Cambidge Univesiy Pess, Cambidge, UK, 197. [41] S. Hou, Q. Zou, S. Chen, G. Doolen, and A. C. Cogley, J. Compu. Phys. 118, 39 (1995. [4] Q. Li, Y. L. He, G. H. Tang, and W. Q. Tao, Phys. Rev. E 8, 377 (9. [43] X. He and L.-S. Luo, J. Sa. Phys. 88, 97 (1997. [44] R. Du, B. Shi, and X. Chen, Phys. Le. A 359, 564 (6. [45] L. Zheng, B. Shi, and Z. Guo, Phys. Rev. E 78, 675 (9.

23 [46] S. Chen e al., Appl. Mah. Compu. 193, 66 (7. [47] Q. Zou and X. He, Phys. Fluids 9, 1591 (1997. [48] A. M. Aoli, A. G. Hoeksa and P. M. A. Sloo, In. J. Mod. Phys. C 13, 1119 (. [49] A. A. Wheele, J. Cysal Gowh 1, 691 (199. [5] D. Xu, C. Shu, and B. C. Khoo, J. Cysal Gowh 173, 13 (1997. [51] H. Huang, T. S. Lee, and C. Shu, In. J. Nume. Mehods Fluids 53, 177 (7. [5] K. Houigan, L.J.W. Gaham, and M.C. Thompson, Phys. Fluid 7, 316 (1995. [53] K. Fujimua, H.S. Koyama, and J.M. Hyun, Tans ASME: J. Fluids Eng. 119, 45 (1997. [54] K. Fujimua, H. Yoshizawa, R. Iwasu, and H.S. Koyama, J. Fluids Eng. 13, 64 (1. [55] A.Y. Gelfga, J.M. Ba-Yoseph, and A. Solan, J. Fluid Mech. 311, 1 (1996. [56] H.M. Blackbun and J.M. Lopez, Phys. Fluids 1, 698 (. [57] E. See and P. Bonoux, J. Fluid Mech. 459, 347 (. [58] S. K. Bhaumik and K. N. Lakshmisha, Compu. Fluids 36, 1163 (7. [59] Z. L. Guo, C. Zheng, and B. Shi, Chin. Phys. 11, 366 (. [6] K. N. Pemnah and S. Banejee, Phys. Rev. E 8, 367 (9. 3

24 1..8 u z (/U /R FIG. 1. Analyical (solid line and numeical (symbol esuls of Hagen-Poiseuille flow. 4

25 u z (,.8 n = 3.6 n =.4 n = 1. n =. -. n = n = 14 n = n = /R FIG.. Analyical (solid line and numeical (symbol esuls of Womesley flow a diffeen ime = nt 16 wih n =,1,, 3,1,13,14,15. 5

26 u z (,.8 n = 4.6 n = 5.4 n = 6. n = 7. n = n = n = 1 n = /R FIG. 3. Analyical (solid line and numeical (symbol esuls of Womesley flow a diffeen ime = nt 16 wih n = 4, 5, 6, 7,8, 9,1,11. 6

27 z cysal fee suface R x Ω x mel Ω c H Ω c R c FIG. 4. Configuaion of he Wheele poblem. 7

28 (a (b FIG. 5. Seamlines of he Wheele poblem: (a 3 = = ; (b Re = 1, Re = 5. Rex 1, Rec 5 x c 8

29 (a (b FIG. 6. Seamlines of cylindical caviy oaional flow: (a Re = 99 ; (b Re = 19. 9

30 = = seady-sae FIG. 7. Simulaions of cylindical caviy oaional flow a Re = 19 wih a low viscosiy ν = using he BGK (lef and MRT (igh collision models. 3

31 Table 1. Compaisons of minimum and maximum seam funcion fo he Wheele poblem. Refeence = = Rex 1, Rec 5 = = 3 Rex 1, Rec 5 ψ min ψ max ψ min ψ max Pesen Ref. [6] Ref. [5] Table. Compaisons of magniude and locaion of he maximum axial velociy on he axis fo cylindical caviy oaional flow. Refeence Re = 99 Re = 19 u z,max max h H z,max u hmax H Pesen Expeimenal [54] D LB model [58]