Lattice-Boltzmann model for axisymmetric thermal flows

Transcription

1 Lace-Bolzmann model fo asymmec hemal flows Q. L, Y. L. He, G. H. Tan, and W. Q. Tao Naonal Key Laboaoy of Mulphase Flow n Powe Enneen, School of Eney and Powe Enneen, X an Jaoon Unvesy, X an, Shaan 71009, Chna In hs bef epo, a hemal lace-bolzmann (LB) model s pesened fo asymmec hemal flows n he ncompessble lm. The model s based on he double-dsbuon-funcon LB mehod, whch has aaced much aenon snce s emeence fo s ecellen numecal sably. Compaed wh he esn asymmec hemal LB models, he pesen model s smple and eans he nheen feaues of he sandad LB mehod. Numecal smulaons ae caed ou fo he hemally developn lamna flows n ccula ducs and he naual convecon n an annulus beween wo coaal vecal cylndes. The Nussel numbe obaned fom he smulaons aees well wh he analycal soluons and/o he esuls epoed n pevous sudes. PACS: j;.0.+e In ecen yeas, he lace-bolzmann (LB) mehod fo smulan asymmec flows has aaced much aenon [1-10]. In fac, he LB smulaon of asymmec flows can be decly handled wh a hee-dmensonal (3D) LB model. Howeve, such a eamen does no ake he advanae of he asymmec popey of he flow: a 3D asymmec flow can be educed o a quas-d poblem. To make use of hs feaue, Hallday e al. [1] fs suded he D LB mehod fo

2 asymmec flows n 001. Some souce ems conann densy and velocy adens wee noduced no he mcoscopc evoluon uaon. Howeve, hs mehod fals o epoduce he coec hydodynamc momenum uaon due o some mssn ems: he em ρ uu s mssn n he ecoveed momenum uaon and some addonal ems nvolvn he fs-ode souce em ae mssn n he second-ode epanson of he mcoscopc evoluon uaon. These mssn ems wee noced by Lee e al. [] and Res e al. [3, ]. By addn hese ems, Lee e al. developed a moe accuae asymmec LB model. Res e al. edeved Hallday e al. s model and hen pesened a modfed veson. Zhou [] ecenly poposed a smplfed asymmec sohemal model, n whch he souce ems ae smple, ye sll conan a velocy aden em whch should be deemned wh a fne-dffeence scheme. Mos ecenly, Guo e al. [6] developed a smple and conssen LB model fo asymmec sohemal flows based on he connuous Bolzmann uaon. The souce ems n he model conan no adens and ae ease o mplemen. Thee ae also seveal aemps fo consucn asymmec hemal LB models. The fs aemp was made by Pen e al. [7] houh he hybd LB appoach. In he model, he azmuhal velocy and he empeaue feld ae solved by he second-ode cene dffeence scheme. Lae, Huan e al. [8] found ha, fo flows wh hh Reynolds numbe and Rayleh numbe, he convecon ems n he Nave-Sokes uaons become domnan and he second-ode cene dffeence scheme s unsuable due o he enhanced numecal nsably. Then hey poposed an mpoved veson of Pen e al. s model. Recenly, Chen e al. [9, 10] poned ou ha, alhouh Huan e al. s hybd LB model s moe numecally sable han Pen e al. s model, oo many complcaed souce ems es n he model and a ea deal of lace ds ae sll ued fo numecal sably. Nocn hs poblem, hey devsed a hemal LB model fo asymmec hemal flows based on he

3 vocy-seam-funcon (VSF) uaons [10]. The souce ems ae smplfed by nvokn he VSF fomulaon bu sll conan seveal aden ems. Guo e al. ecenly aued ha Chen e al. s model wll become vey neffcen fo unseady flows because a Posson uaon mus be solved a evey me sep [6]. Meanwhle, he bounday condon s no easy o mplemen n he VSF-based numecal mehods. In he leaue, Lallemand and Luo [11] have poned ou ha he hybd LB appoach snfcanly devaes fom he sandad LB mehod, whch means loses some nheen feaues of he sandad LB mehod, and only povdes a compomsed soluon. Alenavely, he double-dsbuon-funcon (DDF) LB appoach [1-17], whch ulzes wo dffeen dsbuon funcons, one fo he velocy feld and he ohe fo he empeaue o eney feld, has aaced much aenon snce s emeence fo s ecellen numecal sably as well as he eann of nheen feaues of he sandad LB mehod. The am of hs sudy s o develop a hemal LB model fo smulan asymmec hemal flows based on he DDF LB appoach. The velocy feld of he ncompessble asymmec hemal flows can be solved wh sohemal asymmec LB models. In wha follows we focus on dscussn he mcoscopc evoluon uaon fo solvn he empeaue feld. The macoscopc empeaue uaon of ncompessble asymmec hemal flows n a cylndcal coodnae sysem can be wen as 1 T + u T =( χ T) + χ T, (1) whee T s he empeaue; ndcaes he o componen, hee and ae he coodnaes n adal and aal decons, especvely; u s he componen of velocy n he decon; and χ s he hemal dffusvy. Wh he connuy uaon u = u, we can

4 ewen Eq. (1) as 1 ut T + ( ut ) =( χ T) + χ T. () The undelned ems ase fom he cylndcal pola coodnae. In ode o ecove hese ems, we noduce he follown empeaue evoluon uaon: 1 τ ( + e δ, + δ ) (, ) = ( + e δ, + δ ) ( + e δ, + δ ) 1 e (, ) (, ) δ (, ) (, ) τ δ + S ( + δ, + δ) + S (, ), e (3) whee s he empeaue dsbuon funcon; τ s non-dmensonal elaaon me fo he empeaue feld; S s he souce em; and he dscee veloces { ( e e ) e =, : = 0,1, K, 8} ae specfed by he sandad DQ9 lace. The undelned em n Eq. (3) s used o ecove he second em on he R.H.S of Eq. (). Acually, we can pove ha, when a smla eamen s combned wh Zhou s sohemal asymmec model [], he souce ems of he eaaned model wll conan no aden ems. s chosen as = Tf = Tw 1+ cs + 0. cs 0.u cs ρ e u e u, whee cs = c 3 ( c = δ δ s he lace speed) s he sound speed and he wehs w ae ven by w 0 = 9, w1 = 19, and w 8= 136. I can be found ha sasfes = ρt, e = ρtu, () e e Tuu pt j = j + j. () ρ δ Thouh he second-ode Taylo sees epanson, he evoluon uaon (3) can be educed o δ δ 1 τ τ ( + e ) + ( + e ) = ( ) ( + e )( ) ) e δ δ ( ) δ S ( e ) S Ο( δ ) δ, (6) whee = (, s he spaal aden opeao. By noducn he follown epansons [18]

5 , ( 0) ( 1) ( ) 0 δ 1 δ = + = + + δ, (7) we can ewe Eq. (6) n he consecuve odes of : Ο δ δ as ( 0) Ο 1: =, (8) : Ο δ e + = S, (9) ( 0 ) (1) τ (1) ( 0) () 1 0 e 0 e 0 e τ τ e 1 = + + () 1 0 e S. Usn Eq. (9), we can ewe Eq. (10) as ( 0 ) (1) 1 () e () 1 1 ( 0 ) τ (1) (10) + + e + =. (11) Takn he summaons of Eqs. (9) and (11), we can oban, especvely ( ρ ) ( ρ ) 0 T + j ujt = S, (1) 1 + = (1) () 1 ( ρt) ( e ) e. (13) 1 To ecove he ae macoscopc empeaue uaon, S should be ven by S = ρtu. Fom Eq. (9), we have (1) ( 0) ( 0) = τ ( 0 + j j ) + τ e e e e e S. (1) Fom Eqs. (), (), and (8), we can oban ( ) 0 0 e = u 0 ρt + ρt 0u, (1) ( 0) j ( j ) j ( ρ j) ρ j j e e = u Tu + Tu u + T p + pt, (16) whee u 0 s evaluaed as 0u = 0 ρu u 0ρ ρ = uj ju p ρ. (17) Accodn o Eqs. (1), (16), and (17), we can ewe Eq. (1) as (1) e = τ u 0 ρt + uj ρtuj es + pt. (18) If we caefully choose es = ρtuu, hen Eq. (18) can be educed o

6 (1) e = τ pt. Theefoe n hs sudy he souce em S s chosen as In he pesence of a body foce ( S u =. (19) F = ρa ), acually a focn em F = ρtw ( e a ) c s should also be consdeed n Eq. (3). Bu hs em seemnly can be neleced n mos cases [1-1]. Subsun he uaon e (1) = τ pt no Eq. (13) and hen combnn Eq. (1) wh Eq. (13) ( 0 δ 1 = + ), we can oban he follown macoscopc empeaue uaon: 1 ρut ( ρt) + ( ρut ) = ( ρχ T) + ρχ T, (0) whee he hemal dffusvy χ s ven by χ = δτ 3. In he ncompessble lm wh ρ ρ0, c Eq. (0) s jus he ae macoscopc empeaue uaon. To hs end, we can smply modfy as = ρ0 f ρ. Fo small Mach-numbe flows, can be fuhe smplfed by nelecn he ems of Ο( u ) [1]. In hs suaon, based on a DQ lace wh fou decons e1, e, e3, and can also be used: = T 1+ e u c oehe wh e χ = δτ. c To elmnae he mplcness of Eq. (3), follown He e al. [13], a new dsbuon funcon ( ) = + 0. τ 0.δ S can be noduced. Thouh some sandad aleba, he evoluon uaon fo can be obaned (, ) (, ) (, ) + δ + δ = ω (, ) + ( 1 0.ω) δ (, ) e S, (1) whee ω s ven by ω = 1+ ( e τδ ) ( τ + ) 0.. The macoscopc empeaue can be calculaed fom he new dsbuon funcon as T = 0 ( u ) ρ δ. The velocy feld s solved by usn sohemal asymmec LB models. Hee Guo e al. s sohemal asymmec LB model s adoped, whch can be befly summazed as follows [6]: he evoluon uaon fo velocy feld s ( + δ, + δ ) (, ) = ω (, ) (, ) + δ ( 1 0.ω ) f% e f% f f% f% f G,, ()

7 wh ωf 1 ( τ f 0. ) G = e u a % f% c s, a% = a, = +, a % = a + c s 1 δτ fu, and f % = f, whee f τ s he non-dmensonal elaaon me fo he velocy feld; a and a ae he componens of he eenal foce acceleaon n he and decons, especvely. The macoscopc densy and velocy ae calculaed by ρ = f % and u = e f + 0.δρa + 0.δρcsδ ρ + τ fδc sδ %. The knemac vscosy s ven by ν = δτ 3. Equaons (1) and () oehe wh he coespondn ulbum dsbuons and c f he souce ems consue he pesen hemal DDF LB model fo asymmec hemal flows. Two numecal ess ae consdeed o valdae he poposed model. The fs es s he hemally developn flow n a ccula duc, whch s a classcal poblem descbed n many hea ansfe ebooks. A unfom empeaue pofle T n = 10 and a hemally fully developed flow ae especvely mposed a he nle and oule. Two dffeen hemal bounday condons (BC), he consan wall empeaue BC (Type 1) and he consan wall hea flu BC (Type ), ae consdeed a he wall. In smulaons, he elaaon me τ f = 0.6, he Pandl numbe P = ν χ = 0.7, and a d sze of N N = 69 8 s adoped, coespondn o an aspec ao LD= 68 / 81 = 8, whch s suffcen o descbe a hemally developn flow. The peodc BC s appled n he aal decon fo he velocy BCs wh a body foce ρa = ρ 10, whle he non-ulbum eapolaon BC s appled a he nle and he wall fo he hemal BCs. Meanwhle, he ouflow s supposed o be fully developed and obeys he Neumann ule. The conous of he local Nussel numbe, whch s defned as Nu = D T T T b, whee D s he damee, T = 1, and w w w D D T πutd πud b = 0 0 s he bulk empeaue, ae ploed n F. 1 alon he aal decon. The Nussel numbes ae 3.67 and.376 especvely n he hemal fully developed eon fo he wo dffeen BCs. Compaed wh he coespondn analycal soluons [19], 3.66 and.36, he elave eos ae 0.38% and 0.37%,

8 especvely. The second es s he naual convecon n an annulus beween wo coaal vecal cylndes [0, 1], whch s a smplfed epesenaon of many paccal poblems and has been nvesaed boh numecally and epemenally by a numbe of eseaches. The poblem s skeched n F., whee s he avaon acceleaon, T and T ae he consan empeaues of he nne and oue o cylndes, especvely, and T > T o. The adus ao o and he aspec ao h ( ) o ae boh se o be.0. The naual convecon s chaacezed by he Pandl numbe P = ν χ and he Rayleh numbe 3 Ra = β T T P ν, whee β s he hemal epanson coeffcen. o o The buoyancy foce s ven by ρa = ρ( T T ), whee T ( T T ) = +. Numecal smulaons o ae caed ou fo Ra = 10 and 10. A d sze of N N = s adoped. The seamlnes and sohems a he seady sae ae shown n F. 3 and, especvely. A a modeae Ra numbe ( 10 ), a smple cculaon s obseved. When he Ra numbe nceases, he buoyancy foce acceleaes he cculaon of flud flow and he naual convecon s snfcanly enhanced. As a esul, he sohemals ae ealy defomed. To quanfy he esuls, n Table 1, he Nussel numbes defned as h ( 1 ) Nu = h T T T d ae compaed wh he aveae Nussel numbe, o, o 0, o o ( Nu ( Nu Nu ) = + ) epoed n Refs. [0, 1]. Meanwhle, he aveae Nussel numbes obaned o fom he DQ lace ae 3.19 and.78 especvely fo Ra = 10 and 10, whch ae n ood aeemen wh he esuls obaned fom he DQ9 lace wh = ρ f ρ. 0 In summay, we have pesened a hemal LB model fo asymmec hemal flows based on he DDF LB appoach. The souce ems of he model conan no aden ems. Compaed wh he esn asymmec hemal LB models, he pesen model s smple and eans he nheen feaues of he sandad LB mehod. Two numecal ess have been consdeed o valdae he poposed

9 model. Numecal esuls ae compaed wh he analycal soluons and/o he esuls epoed n pevous sudes. The compasons show he capably and elably of he model. Ths wok was suppoed by he Key Pojec of Naonal Naual Scence Foundaon of Chna (No ). [1] I. Hallday, L. A. Hammond, C. M. Cae, K. Good, and A. Sevens, Phys. Rev. E 6, (001). [] T. S. Lee, H. Huan, and C. Shu, In. J. Mod. Phys. C 17, 6 (006). [3] T. Res and T. N. Phllps, Phys. Rev. E 7, (007). [] T. Res and T. N. Phllps, Phys. Rev. E 77, (008). [] J. G. Zhou, Phys. Rev. E 78, (008). [6] Z. Guo, H. Han, B. Sh, and C. Zhen, Phys. Rev. E 79, (009) [7] Y. Pen, C. Shu, Y. T. Chew, and J. Qu, J. Compu. Phys. 186, 9 (003). [8] H. Huan, T. S. Lee, and C. Shu, In. J. Nume. Mehods Fluds 3, 1707 (007). [9] S. Chen, J. Tölke, S. Gelle, and M. Kafczyk, Phys. Rev. E 78, (008). [10] S. Chen, J. Tölke, S. Gelle, and M. Kafczyk, Phys. Rev. E 79, (009). [11] P. Lallemand and L.-S. Luo, Phys. Rev. E 68, (003). [1] X. Shan, Phys. Rev. E, 780 (1997). [13] X. He, S. Chen, and G. D. Doolen, J. Compu. Phys. 16, 8 (1998). [1] Y. Sh, T. S. Zhao, and Z. L. Guo, Phys. Rev. E 70, (003). [1] Z. Guo, C. Zhen, B. Sh, and T. S. Zhao, Phys. Rev. E 7, (007). [16] G. H. Tan, W. Q. Tao, and Y. L. He, Phys. Rev. E 7, (00). [17] Q. L, Y. L. He, Y. Wan, and W. Q. Tao, Phys. Rev. E 76, 0670 (007). [18] S. Hou, Q. Zou, S. Chen, G. Doolen, and A. C. Coley, J. Compu. Phys. 118, 39 (199).

10 nd [19] Y. A. Cenel, Hea ansfe: A paccal appoach ( ed, McGaw-Hll, Boson, 003). [0] R. Kuma and M. A. Kalam, In. J. Hea Mass Tansfe 3, 13 (1991). [1] M. Venkaachalappa, M. Sanka, and A. A. Naaajan, Aca Mechanca 17, 173 (001). 0 1 Type 1 Type Nu E (/D)/(ReP) FIG. 1. Local Nussel numbe dsbuon alon he aal decon fo he hemally developn flow. = 0 T T T o h o = 0 T FIG.. Naual convecon beween coaal vecal cylndes.

11 a b FIG. 3. Seamlnes fo Ra = 10 (a) and 10 (b). a b FIG.. Isohems fo Ra = 10 (a) and 10 (b). Table. 1. Compason of he Nussel numbe. Ra Ref. [0] Ref. [1] Nu Nu o