A multiple-relaxation-time lattice Boltzmann model for simulating. incompressible axisymmetric thermal flows in porous media

Transcription

1 A mulple-elaxaon-me lace Bolmann model fo smulang ncompessble axsymmec hemal flows n poous meda Qng Lu a, Ya-Lng He a, Qng L b a Key Laboaoy of Themo-Flud Scence and Engneeng of Mnsy of Educaon, School of Enegy and Powe Engneeng, X an Jaoong Unvesy, X an, Shaanx, 70049, Chna b School of Enegy Scence and Engneeng, Cenal Souh Unvesy, Changsha 40083, Chna Absac In hs pape, a mulple-elaxaon-me (MRT) lace Bolmann (LB) model s developed fo smulang ncompessble axsymmec hemal flows n poous meda a he epesenave elemenay volume (REV) scale. In he model, a DQ9 MRT-LB equaon s poposed o solve he flow feld n addon o he DQ5 LB equaon fo he empeaue feld. The souce ems of he model ae smple and conan no velocy and empeaue gaden ems. The genealed axsymmec Nave-Sokes equaons fo axsymmec flows n poous meda ae coecly ecoveed fom he MRT-LB model hough he Chapman-Enskog analyss n he momen space. The pesen model s valdaed by numecal smulaons of seveal ypcal axsymmec hemal poblems n poous meda. The numecal esuls agee well wh he daa epoed n he leaue, demonsang he effecveness and accuacy of he pesen MRT-LB model fo smulang axsymmec hemal flows n poous meda. Keywod: lace Bolmann model; axsymmec hemal flows; mulple-elaxaon-me; poous meda; hea ansfe.. Inoducon The analyss of axsymmec convecve anspo pocesses n poous meda has aaced

2 consdeable aenon due o s mpoance n many aeas such as geohemal enegy sysems, conamnan anspo n goundwae, undegound eamen of nuclea wase maeals, fbous nsulaon, cude ol exacon, chemcal caalyc eacos, eleconc devce coolng, and poous hea exchanges, o name only a few applcaons. Compehensve leaue suveys concenng hs opc have been gven n he books by Neld and Bejan [], and Vafa []. In he pas seveal decades, axsymmec flud flow and hea ansfe poblems n poous meda a he epesenave elemenay volume (REV) scale have been numecally nvesgaed by many eseaches usng some adonal numecal echnques, such as he fne dffeence mehod [3-5], fne volume mehod [6, 7], and fne elemen mehod [8]. The lace Bolmann (LB) mehod, as a mesoscopc numecal echnque ognaes fom he lace-gas auomaa (LGA) mehod [9], has become an effecve compuaonal mehod fo smulang complex flud flows and modelng complex physcs n fluds owng o s knec backgound [0-8]. In he pas decade, he LB mehod fo axsymmec flud flow and hea ansfe poblems has also aaced much aenon. Geneally speakng, hee-dmensonal axsymmec flud flow and hea ansfe poblems can be educed o quas-wo-dmensonal ones n cylndcal coodnae sysem, whch gealy enhances he compuaonal effcency. To make full use of hs feaue, Hallday e al. [9] poposed he fs axsymmec LB model fo ncompessble axsymmec flows by noducng he velocy and spaal dependen souce ems no he LB equaon o accoun fo he addonal ems asng fom he cylndcal coodnae sysem. Snce hen, many axsymmec LB models have been developed o sudy axsymmec sohemal flows [0-9] and hemal flows [30-33]. The above menoned axsymmec LB models ae lmed o axsymmec flud flow and hea ansfe poblems n he absence of poous meda. Recenly, Rong e al. [34] poposed an axsymmec LB model fo

3 axsymmec convecon hea ansfe n poous meda a he REV scale. The axsymmec LB model [34] employs he Bhanaga-Goss-Kook (BGK) collson opeao and can be used o smulae axsymmec flows n a medum wh consan o vaable poosy. Howeve, o he bes of he auhos knowledge, no epos have been found n he leaue abou he applcaons of he mulple-elaxaon-me (MRT) LB mehod o axsymmec flows n poous meda a he REV scale. In he LB communy, has been wdely acceped ha he MRT-LB model [35, 36] has bee numecal sably and accuacy han he BGK-LB model [37]. As epoed n Refs. [5, 6], he numecal sably of he axsymmec MRT-LB model s much bee han he axsymmec BGK-LB model a low vscoses. Ths movaes he pesen wok, whee he man pupose s o develop an axsymmec MRT-LB model fo smulang ncompessble axsymmec hemal flows n poous meda based on some pevous sudes [4, 5, 3, 38]. In he model, a DQ9 MRT-LB equaon s poposed o solve he flow feld of he axsymmec hemal flows n poous meda n he famewok of he sandad MRT-LB mehod. The poosy s ncluded no he equlbum momens, and he lnea and nonlnea dag foces of he poous max ae ncopoaed no he model by addng a souce em no he MRT-LB equaon n he momen space. The es of hs pape s oganed as follows. In Secon, he macoscopc govenng equaons fo axsymmec hemal flows n poous meda ae befly noduced. In Secon 3, he MRT-LB model fo axsymmec hemal flows n poous meda s pesened n deal. In Secon 4, numecal smulaons of seveal ypcal axsymmec hemal flows n poous meda ae caed ou o valdae he effecveness and accuacy of he pesen MRT-LB model. Fnally, Secon 5 concludes he pape.. Macoscopc govenng equaons The flud flow s assumed o be lamna, ncompessble, and Newonan whou vscous hea

4 dsspaon and pessue wok. The Bnkman-Fochheme-exended Dacy model (also called he genealed model) [39] and he local hemal equlbum model [] ae employed o descbe he momenum and enegy ansfe n poous meda, especvely. In cylndcal coodnae sysem, he macoscopc govenng equaons fo ncompessble axsymmec hemal flows n homogeneous and soopc poous meda a he REV scale ae gven as follows [39]: whee u u 0, () p u uu uu u e u F, () u uu uu p e u F, (3) T T T u u e T, (4), (5) u and u ae he componens of he velocy u n he - (adal) and - (axal) decons, especvely, s he flud densy, p s he pessue, T s he empeaue, s he poosy of he poous meda, e and meda, especvely. The oal body foce F F, F e ae he effecve knec vscosy and hemal dffusvy of he poous, whch s nduced by he sold max and ohe exenal foces, can be expessed as [40, 4] F F u u u G, (6) K K whee s he knec vscosy of he flud, and u u u. Based on he Boussnesq appoxmaon, G s gven by 0 G = g T T j + a, (7) whee g s he gavaonal acceleaon, s he hemal expanson coeffcen, T 0 s he efeence empeaue, j s he un veco n he -decon, and a s he acceleaon nduced by

5 ohe exenal foces. Accodng o Egun s expemenal elaon [4], he geomec funcon F and he pemeably K of he poous meda can be esmaed n ems of he poosy and he pacle damee d p as [43] F , d p K (8) The flow govened by Eqs. ()-(4) s chaaceed by and seveal dmensonless paamees: he Raylegh numbe Ra, he Dacy numbe Da, he Pandl numbe P, he Reynolds numbe Re, he vscosy ao J, and he hemal dffusvy ao λ, whch ae defned as follows 3 g TL K Ra, Da L, P, LU Re =, e J, λ e, (9) whee T s he empeaue dffeence (chaacesc empeaue), L s he chaacesc lengh, U s he chaacesc velocy, and s he hemal dffusvy of he flud. The fs em and he second em on he gh-hand-sde of Eq. (6) ae he Dacy (lnea) and Fochheme (nonlnea) dag foces due o he sold max, especvely. The nonlnea dag foce em can be negleced fo a small Dacy numbe Da o Reynolds numbe Re. Howeve, fo lage Dacy numbe Da o Reynolds numbe Re, he nfluence of he nonlnea dag foce mus be consdeed [4]. As and Da, he genealed axsymmec Nave-Sokes equaons ()-(3) educe o he axsymmec Nave-Sokes equaons fo fee flud flows (whou poous meda). In addon, when F 0 (he nonlnea dag foce s negleced), Eqs. () and (3) educe o he Bnkman-exended Dacy equaon. 3. MRT-LB model fo axsymmec hemal flows n poous meda The macoscopc govenng equaons ()-(4) fo ncompessble axsymmec hemal flows n poous meda can be ewen as follows: ju j u, (0)

6 u u u u u j k j j j 0 u j k j p k ku j juk u j ju F j, () ut T j u jt j e jt e T, () whee j, k ndcae he o componen, s he effecve dynamc vscosy, and e j s he Konecke funcon wh wo ndces. The undelne ems n Eqs. () and () ase fom he cylndcal coodnae sysem. In he pesen wok, he mehods epoed n Refs. [4, 3] ae adoped o ecove hese ems n he macoscopc equaons. In wha follows, he axsymmec MRT-LB model fo axsymmec hemal flows n poous meda s pesened n deal. 3. DQ9 MRT-LB equaon fo he flow feld I has been wdely acceped ha he MRT collson model can enhance he numecal sably and accuacy of he LB scheme as compaed wh he BGK collson model. The dmensonless elaxaon mes of he conseved (hydodynamc) and nonconseved (knec) momens can be adjused ndependenly when usng he MRT collson model [5, 6, 35, 36, 44-46]. Accodng o Refs. [6, 44], we noduce he followng DQ9 MRT-LB equaon o solve he flow feld: eq,, f,, x e f x Λ f f, S x S x e x, (3) whee f x, s he volume-aveaged densy dsbuon funcon wh velocy e, e e a poson =, x and me, f eq, x s he equlbum dsbuon funcon, S s he souce em accounng fo he oal body foce and he addonal ems asng fom he cylndcal coodnae sysem, and max and Λ M ΛM s he collson max, n whch M s a 9 9 Λ MΛM dag( s0, s,, s8 ) s a 99 ohogonal ansfomaon dagonal elaxaon max ( s 0 s ae dmensonless elaxaon aes). The nne dscee veloces e 0,,,8 of he DQ9 model ae gven by [37]

7 0,0, 0 e cos,sn c, ~ 4. (4) cos 4,sn 4 c, 5 ~ 8 whee c s he lace speed wh and especvely. In he pesen LB model, c s se o be (.e., ). epesenng he lace spacng and me sep, The mplcness of Eq. (3) can be elmnaed by usng a new dsbuon funcon f f 0.5 S, fom whch he followng explc DQ9 MRT-LB equaon can be obaned: eq,,,, Λ f x e f x M Λ m x m x M I S, (5) whee I s he un max, and he boldface symbols, f, m, m eq, and S, ae 9-dmensonal column vecos, e.g.,, f0,, f,,, f8, T f x x x x, n whch T s he anspose opeao, and S = MS = MS0, S,, S8 T s he souce em n he momen space. The ansfomaon max lnealy ansfoms he dscee dsbuon funcons f 9 (velocy space) o he velocy momens 9 m (momen space): m Mf, f = M m. (6) The nne velocy momens ae gven by m m0, m, m, m3, m4, m5, m6, m7, m8 T, e,, j, q, j, q, p, p, (7) whee m 0 s he flud densy, m e s elaed o he enegy, m s elaed o he enegy squae, 3,5, m j ae componens of he momenum, T J j j u, 4,6, m q ae elaed o he enegy flux, and m7,8 p, ae elaed o he dagonal and off-dagonal componens of he sess enso [36]. As epoed n Ref. [5], he adal coodnae has been ncopoaed no he velocy momens (7) based on Guo e al. s axsymmec LB model [4]. Wh he odeng of he above specfed velocy momens, he ansfomaon max s gven by ( c ) [36]

8 Μ (8) Among he nne velocy momens m 0,,, 8, only he densy m0 and momenum m3,5 j, ae conseved (hydodynamc) quanes, he ohe velocy momens ae nonconseved (knec) quanes. The equlbum velocy momens eq m fo he velocy momens m ae gven by T eq eq eq eq eq eq eq, e,, j, q, j, q, p, p m. (9) whee [38] e 3 eq u u, 3 eq u u, q eq u, q eq u, p eq u u, p uu. (0) eq The evoluon pocess of he MRT-LB equaon (5) consss of wo seps,.e., he collson sep and seamng sep. Usually, he collson sep s caed ou n he momen space as eq m m m m I S, () whle he seamng sep s sll mplemened n he velocy space,, f x e f x, f M m. () The dagonal elaxaon max Λ s gven by Λ =dag, s, s, s, s, s, s, s, s. (3) e j q j q The souce em S n he momen space s gven by [38]

9 whee F, F F s gven by 0 6u F u F 6u F u F F S F, (4) F F u F u F u F uf F F a, F F, (5) n whch a s defned as 0.5 a cs u [4], s he dmensonless elaxaon me fo he flow feld, and c c 3 s he sound speed of he DQ9 model. The flud densy and velocy = u, u s u ae defned as [4] 8 f, (6) 0. (7) 8 u j 0.5 ej f Fj cs j cs j 0 Noe ha he oal body foce F, F F also conans he macoscopc velocy u. Accodng o Eqs. (6) and (7), he macoscopc velocy u can be calculaed explcly by u j v j, (8) l l l v j0 j0 whee = v, v v, j0 l and l ae gven by l j0 v e f G c, (9) 8 j j j s j 0 c 0.5 s j K, F l. (30) K The pessue p s defned as p c s, and he effecve knec vscosy s defned as e cs (3) wh s s7,8.

10 Though he Chapman-Enskog analyss n he momen space (see Appendx A fo deals), he genealed axsymmec Nave-Sokes equaons (0) and () can be ecoveed fom he pesen MRT-LB model. I should be noed ha, as and Da, he pesen MRT-LB model educes o he MRT-LB model [5] fo axsymmec flows n he absence of poous meda. When he nne elaxaon aes s 0 s ae se o be a sngle value,.e., Λ I, hen he pesen MRT-LB model s equvalen o he BGK-LB model wh he followng equlba [34]: f eq e u e u whee 0 49, ~4 9, and 5~ cs cs cs u, (3) 3. DQ5 LB equaon fo he empeaue feld The empeaue feld govened by Eq. () s solved by usng he hemal axsymmec LB model of L e al. [3]. The evoluon equaon fo he empeaue feld s gven by eq g x e, g x, g g x, g x, x,, (33) whee g, x s he empeaue dsbuon funcon, g eq, x s he equlbum empeaue dsbuon funcon, and, e x s he souce em. The elaxaon paamee g s gven by g T / T 0.5 [3], n whch T s he dmensonless elaxaon me elaed o e. In he pesen sudy, he DQ5 model s employed, of whch he fve dscee veloces ae 0,0, 0 e. (34) cos,sn c, ~ 4 The equlbum empeaue dsbuon funcon eq g s defned as g eq T e u, (35) cst whee c s he sound speed of he DQ5 model, and 0,,, 4 st ae wegh coeffcens gven by

11 0, 0. (36) 0 4, ~ 4 Fo he DQ5 model,, and 0 c e e st 0. In he pesen wok, 0 s se o be 35 wh c c 5 5. st The souce em can be chosen as [3] u g The macoscopc empeaue T s defned as eq. (37) 4 T g. (38) 0 Though he Chapman-Enskog analyss [3] of he LB equaon (33), he macoscopc empeaue equaon () can be ecoveed n he ncompessble lm. The effecve hemal dffusvy e s gven by c. The empeaue feld govened by he axsymmec convecon-dffuson e st T equaon () can also be solved by he MRT-LB model. Deals of he MRT-LB model fo axsymmec convecon-dffuson equaon can be found n Ref. [33]. 4. Numecal valdaon In hs secon, numecal smulaons of seveal ypcal axsymmec hemal flows n poous meda ae caed ou o valdae he pesen MRT-LB model. The es poblems nclude he hemally developng flow n a ppe flled wh poous meda, naual convecon flow n a vecal annulus, naual convecon flow n a vecal annulus flled wh poous meda, and naual convecon flow n a vecal poous annulus wh dscee heang. In smulaons, we se 0,, c, J =, and λ. Unless ohewse saed, he nonequlbum exapolaon scheme [47] s adoped o ea he velocy and hemal bounday condons of f and g. The elaxaon aes s 0 s ae chosen as follows: s s j, s., sq., and. Fo e s s naual convecon poblems n poous annulus, he dmensonless elaxaon me ( s ) can

12 be fully deemned n ems of P, Ra and Ma [5, 48], whee Ma U c 3U s he Mach s numbe ( U g TL s he chaacesc velocy). The dmensonless elaxaon mes and T can be deemned as MaJL P 0.5, c Ra s T st cs λ 0.5, (39) c JP whee c 3, s c 5, and Ma s se o be 0. n he pesen wok. st 4. Themally developng flow n a ppe flled wh poous meda In hs subsecon, we apply he pesen LB model o sudy he hemally developng flow n a ppe flled wh flud-sauaed poous meda, whch has been numecally nvesgaed n Ref. [6]. The compuaonal doman and bounday condons of hs poblem ae skeched n Fg.. H and D ae he lengh and damee of he ppe, especvely. A he nle ( 0 ), u 0, u Un, and T T n. A he oule ( H ), he gadens of he velocy and empeaue n he decon ae se o eo (fully developed flow). Fo 0 H and ( D s he nne adus of he ppe), no slp condon s mposed wh T T. The local Nussel numbes along he ppe wall s defned as w [6] Nu D T T T, (40) w w b whee T b s he bulk empeaue b D D T utd ud. (4) 0 0 In smulaons, we se P 0.7,, F 0, and Re= LU 00 (he chaacesc n lengh L D ). A unfom gd N N s employed, coespondng o an aspec ao A H 6. The local Nussel numbes along he ppe wall fo dffeen Dacy numbes ae shown n Fg.. Fom he fgue we can obseve ha he Nussel numbe nceases as he Dacy numbe deceases. The hea ansfe effec can be enhanced by nseng poous maeal no he ppe.

13 On he ohe hand, he nfluence of he Dacy numbe on he pessue dop s sgnfcan when he ppe s fully flled wh poous maeal. As he Dacy numbe deceases, he flow essance nceases. Fo moe deals on hs opc, eades ae efeed o Ref. [6]. The Nussel numbes fo dffeen Dacy numbes n he fully developed flow egon ae measued and ncluded n Table. The numecal esuls gven by Mohamad [6] usng he fne volume mehod ae also ncluded n Table fo compason. As shown n he able, he pesen esuls ae n good ageemen wh hose epoed n he leaue. 4. Naual convecon flow n a vecal annulus I s noed ha as and Da, he pesen LB model educes o a LB model fo axsymmec hemal flows whou poous meda. In hs subsecon, we wll es he pesen LB model by smulang he naual convecon flow n a vecal annulus beween wo coaxal vecal cylndes whou poous meda, whch has been numecally suded by many eseaches [3, 33, 49, 50]. The confguaon of he poblem s skeched n Fg. 3. The nne cylnde wall (locaed a adus ) and oue cylnde wall (locaed a adus o ) ae mananed a consan bu dffeen empeaues T and T ( T T o ), especvely, whle he hoonal walls ae adabac. The adus o ao R = o and aspec ao A H L ae boh se o be. The aveage Nussel numbes on he nne cylnde wall ( Nu ) and oue cylnde wall ( Nu o ) ae defned as [3, 50] Nu,, 0, H o o T d, (4) o H T whee T T T s he empeaue dffeence, T T T o s he efeence empeaue, 0 o and L o s he chaacesc lengh. In smulaons, we se P 0.7,, and 8 Da 0. Numecal smulaons ae caed ou fo 3 Ra 0, 4 0, and 5 0 based on a N N unfom gd. The seamlnes and

14 sohems fo dffeen Ra ae llusaed n Fg. 4. The pesen esuls ae n good ageemen wh hose epoed n pevous sudes [3, 33, 49, 50]. To quanfy he esuls, he aveage Nussel numbes on he nne cylnde wall ( Nu ) pedced by he pesen LB model ae ncluded n Table. The publshed daa obaned by dffeen numecal mehods n Refs. [3, 33, 49, 50] ae also lsed n Table fo compason. As shown n Table, he pesen numecal esuls agee well wh he esuls epoed n pevous sudes. 4.3 Naual convecon flow n a vecal annulus flled wh poous meda Naual convecon hea ansfe n a vecal poous annulus beween wo coaxal vecal cylndes has been nvesgaed boh expemenally and numecally by many eseaches [3, 4, 8, 34]. In hs subsecon, we wll valdae he pesen LB model by smulang such flows. The confguaon of he poblem s he same as ha skeched n Fg. 3, bu he annulus s flled wh flud-sauaed poous meda. The numecal esuls pedced by he pesen LB model ae valdaed agans he numecal soluons of Pasad e al. [3, 4]. The aspec ao A and adus ao R of he annulus ae se o be and 5.338, especvely. In smulaons, we se P, , and Da Numecal smulaons ae caed ou fo Ra 00, 500, and 000 based on a N N unfom gd. Hee, Ra s he Dacy-Raylegh numbe defned as Ra RaDa. The seamlnes and sohems pedced by he pesen LB model ae shown n Fg. 5. Fom he fgue can be seen ha as Ra nceases, he coe of he flow feld shfs owads he op wall of he annulus. Fo a lage Ra, he empeaue gaden nea he lef ho wall s lage and he sohems shf owads he lef vecal cylnde wall. These obsevaons agee well wh hose epoed n Ref. [3]. In ode o examne he behavo of he hea ansfe nsde he poous annulus, Fg. 6 llusaes he empeaue pofles a fou dffeen

15 heghs ( H 0.5, 0.5, 0.75, ) of he annulus fo dffeen Dacy-Raylegh numbes. The empeaue pofles n he fgue clealy ndcae ha, as he Dacy-Raylegh numbe nceases, he empeaue a he cene of he annulus deceases. The numecal esuls gven by Pasad e al. [3, 4] ae also ncluded n Fg. 6 fo compason. As shown, he numecal esuls obaned by he pesen LB model agee well wh he avalable esuls n pevous sudes. 4.4 Naual convecon flow n a vecal poous annulus wh dscee heang In hs subsecon, he numecal esuls pedced by he pesen LB model fo naual convecon flow n a vecal poous annulus wh an soflux dscee heae ae pesened. The compuaonal doman and bounday condons of he poblem ae skeched n Fg. 7. An soflux dscee heae of lengh l ( lh 0.4 ) and sengh q ( q k T, n whch k s he hemal conducvy) s placed on he nne cylnde wall of he annulus, and he unheaed pas of he nne cylnde wall ae hemally nsulaed. The oue cylnde wall s mananed a a consan empeaue T o, whle he boom and op walls ae hemally nsulaed. The dsance beween he boom wall and he cene of he heae s h ( hh 0.5 ). The aspec ao A and adus ao R of he annulus ae se o be and, especvely. The aveage Nussel numbe Nu along he dscee heae s defned as [5] whee Nu() s he local Nussel numbe h0.5l Nu Nu() d l, (43) h0.5l ql Nu() k T T h o, (44) n whch T h s he local empeaue of he dscee heae, and L o s he chaacesc lengh. The empeaue dffeence T ql k, and he efeence empeaue T0 To. In smulaons, we se 7 Ra 0, P 0.7, 0.9, F 0, T 0, and T. Numecal o smulaons ae caed ou fo Da 0, 4 0, and 6 0 based on a N N unfom

16 gd. The seamlnes and sohems fo dffeen Dacy numbes ae ploed n Fg. 8, fom whch we can obseve ha he flud flow and hea ansfe nsde he annulus songly depend on Da. A low Dacy numbe Da = 0 6, he sengh of he flow feld s weak and a weak-convecon sucue can be obseved fom he sohems. As Da nceases o 0, moe convecve mxng occus nsde he annulus, and he man voex moves owads he cold vecal cylnde wall. Moeove, can be obseved ha he local empeaue nsde he annulus deceases apdly as he Dacy numbe nceases o 0. To quanfy he esuls, he aveage Nussel numbes along he dscee heae and he maxmum values of he seam funcon ae measued and lsed n Table 3 ogehe wh he esuls fom Ref. [5]. As shown, ou esuls agee well wh hose epoed n pevous sudes. 5. Conclusons In hs pape, we have pesened an axsymmec MRT-LB model fo smulang ncompessble axsymmec hemal flows n poous meda a he REV scale. In he model, a DQ9 MRT-LB equaon s poposed o solve he flow feld n addon o he DQ5 LB equaon fo he empeaue feld. The souce ems of he pesen MRT-LB model ae smple and conan no velocy and empeaue gaden ems. The equlbum momens ae modfed o accoun fo he poosy of he poous meda, and he lnea and nonlnea dag foces of he poous max ae ncopoaed no he model by addng a souce em no he MRT-LB equaon n he momen space. Though he Chapman-Enskog analyss of he MRT-LB equaon n he momen space, he genealed axsymmec momenum equaon can be coecly deved n he ncompessble lm. The effecveness and accuacy of he pesen LB model s demonsaed by numecal smulaons of seveal ypcal axsymmec hemal poblems n poous meda. I s found ha he pesen esuls agee well wh he daa epoed n pevous sudes.

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18 998;30: [] Succ S. The Lace Bolmann Equaon fo Flud Dynamcs and Beyond. Oxfod: Claendon Pess; 00. [] Kang Q, Zhang D, Chen S. Unfed lace Bolmann mehod fo flow n mulscale poous meda. Phys Rev E 00;66: [3] L Q, He YL, Wang Y, Tao WQ. Coupled double-dsbuon-funcon lace Bolmann mehod fo he compessble Nave-Sokes equaons. Phys Rev E 007;76: [4] Succ S. Lace Bolmann acoss scales: fom ubulence o DNA anslocaon. Eu Phys J B 008;64: [5] He YL, Wang Y, L Q. Lace Bolmann Mehod: Theoy and Applcaons. Bejng: Scence Pess; 009. [6] Feng Y, Sagau P, Tao W. A hee dmensonal lace model fo hemal compessble flow on sandad laces. J Compu Phys 05; 303: [7] He YL, Lu Q, L Q. Thee-dmensonal fne-dffeence lace Bolmann model and s applcaon o nvscd compessble flows wh shock waves. Physca A 03;39: [8] Wang J, Wang D, Lallemand P, Luo LS. Lace Bolmann smulaons of hemal convecve flows n wo dmensons. Compu Mah Appl 03;66:6-86. [9] Hallday I, Hammond LA, Cae CM, Good K, Sevens A. Lace Bolmann equaon hydodynamcs. Phys Rev E 00;64:008. [0] Lee TS, Huang H, Shu C. An axsymmec ncompessble lace Bolmann model fo ppe flow. In J Mod Phys C 006;7: [] Res T, Phllps TN. Modfed lace Bolmann model fo axsymmec flows. Phys Rev E

19 007;75: [] Res T, Phllps TN. Numecal valdaon of a conssen axsymmec lace Bolmann model. Phys Rev E 008;77: [3] Zhou JG. Axsymmec lace Bolmann mehod. Phys Rev E 008;78: [4] Guo Z, Han H, Sh B, Zheng C. Theoy of he lace Bolmann equaon: lace Bolmann model fo axsymmec flows. Phys Rev E 009;79: [5] Wang L, Guo Z, Zheng C. Mul-elaxaon-me lace Bolmann model fo axsymmec flows. Compu Fluds 00;39: [6] L Q, He YL, Tang GH, Tao WQ. Impoved axsymmec lace Bolmann scheme. Phys Rev E 00;8: [7] Tang GH, L XF, Tao WQ. Mcoannula eleco-osmoc flow wh he axsymmec lace Bolmann mehod. J Appl Phys 00;08:4903. [8] Zhou JG. Axsymmec lace Bolmann mehod evsed. Phys Rev E 0;84: [9] Wang Y, Shu C, Teo CJ. A faconal sep axsymmec lace Bolmann flux solve fo ncompessble swlng and oang flows. Compu Fluds 04;96:04-4. [30] Peng Y, Shu C, Chew YT, Qu J. Numecal nvesgaon of flows n Cochalsk cysal gowh by an axsymmec lace Bolmann mehod. J Compu Phys 003;86: [3] Huang H, Lee TS, Shu C. Hybd lace Bolmann fne-dffeence smulaon of axsymmec swlng and oang flows. In J Nume Meh Fluds 007; 53: [3] L Q, He YL, Tang GH, Tao WQ. Lace Bolmann model fo axsymmec hemal flows. Phys Rev E 009;80: [33] L L, Me R, Klausne JF. Mulple-elaxaon-me lace Bolmann model fo he axsymmec

20 convecon dffuson equaon. In J Hea Mass Tansfe 03;67: [34] Rong F, Guo Z, Cha Z, Sh B. A lace Bolmann model fo axsymmec hemal flows hough poous meda. In J Hea Mass Tansfe 00;53: [35] d Humèes D. Genealed lace-bolmann equaons, n aefed gas dynamcs: heoy and smulaons. Pog Asonau Aeonau 99;59: [36] Lallemand P, Luo LS. Theoy of he lace Bolmann mehod: Dspeson, dsspaon, soopy, Gallean nvaance, and sably. Phys Rev E 000;6: [37] Qan YH, d Humèes D, Lallemand P. Lace BGK models fo Nave-Sokes equaon. Euophys Le 99;7: [38] Lu Q, He YL, L Q, Tao WQ. A mulple-elaxaon-me lace Bolmann model fo convecon hea ansfe n poous meda. In J Hea Mass Tansfe 04;73: [39] Nhaasu P, Seehaamu KN, Sundaaajan T. Non-Dacy double-dffusve naual convecon n axsymmec flud sauaed poous caves. Hea Mass Tansfe 997;3: [40] Hsu CT, Cheng P. Themal dspeson n a poous medum. In J Hea Mass Tansfe 990;33: [4] Guo Z, Zhao TS. Lace Bolmann model fo ncompessble flows hough poous meda. Phys Rev E 00;66: [4] Egun S. Flud flow hough packed columns. Chem Eng Pog 95;48: [43] Vafa K. Convecve flow and hea ansfe n vaable-poosy meda. J Flud Mech 984;47: [44] McCacken ME, Abaham J. Mulple-elaxaon-me lace-bolmann model fo mulphase flow. Phys Rev E 005;7:03670.

21 [45] L Q, Luo KH, He YL, Gao YJ, Tao WQ. Couplng lace Bolmann model fo smulaon of hemal flows on sandad laces. Phys Rev E 0;85:0670. [46] L L, Me R, Klausne JF. Bounday condons fo hemal lace Bolmann equaon mehod. J Compu Phys 03;37: [47] Guo ZL, Zheng CG, Sh BC. Non-equlbum exapolaon mehod fo velocy and pessue bounday condons n he lace Bolmann mehod. Chn Phys 00;: [48] L Q, He YL, Wang Y, Tang GH. An mpoved hemal lace Bolmann model fo flows whou vscous hea dsspaon and compesson wok. In J Mod Phys C 008;9:5-50. [49] Kuma R, Kalam MA. Lamna hemal convecon beween vecal coaxal sohemal cylndes. In J Hea Mass Tansfe 99;34: [50] Venkaachalappa M, Sanka M, Naaajan AA. Naual convecon n an annulus beween wo oang vecal cylndes. Aca Mechanca 00;47: [5] Chapman S, Cowlng TG. The Mahemacal Theoy of Non-Unfom Gases. London: Cambdge Unvesy Pess; 970. [5] Cha Z, Zhao TS. Effec of he focng em n he mulple-elaxaon-me lace Bolmann equaon on he shea sess o he san ae enso. Phys Rev E 0;86: Appendx A: Chapman-Enskog analyss of he axsymmec DQ9 MRT-LB model The Chapman-Enskog expanson mehod [35, 44, 5, 5] s adoped o deve he genealed axsymmec Nave-Sokes equaons (0) and () fom he pesen MRT-LB model. To hs end, he followng expansons n me and space ae noduced [5]: n,, 0 n! n f x e e f x, (A.a) 0 () () f f f f, (A.b)

22 , j j, S S, F F (A.c) whee s a small expanson paamee, S S, S,, S 0 8 T, F F, F. Wh he above expansons, we can deve he followng equaons as consecuve odes of he paamee : 0 : 0 eq f f, (A.a) : 0 Λ D f M ΛM f M I M S, (A.b) : f 0 D f 0 D f M ΛM f, (A.c) whee D e e j j ( j, S S, S,, S ), 0 8 M S. In he momen space, he above equaons can be ewen as follows: 0 : 0 eq m m, (A.3a) : : 0 Λ Dm Λ m I S, (A.3b) 0 Λ Λ m D I m D I S Λ m, (A.3c) whee D MD M I C, j j C j can be gven explcly by C j M eji M, Λ Λ, and T 0, e,, F, q, F, q, p, p m. (A.4) C , C Fom Eq. (A.3b), he followng equaons n he momen space a he me scale can be obaned: u u, (A.5a) 0

23 3 u s s e S, (A.5b) 3 u s u u s S, (A.5c) u uu s3 u s3 F 3 3 S, (A.5d) u u u u s u s q S , (A.5e) u u u s5 u s5 F 5 3 S, (A.5f) u u u u s6 u s6 q 6 3 S, (A.5g) u u u u s s p S u u u u s s p S , (A.5h). (A.5) Fom Eq. (A.3c), he followng equaons a he me scale coespondng o he consevave vaables, u, and u can be obaned: 0, (A.6a) s s s s u F 6 e p p s3 s s7 s8 S 3 S S7 S8 0 6, (A.6b) s s s s u F p 6 e p s5 s 8 s s 7 S 5 S 8 S S 70 6, (A.6c) Noe ha e, p and p n Eqs. (A.6b) and (A.6c) ae unknowns o be deemned. Wh he ad of Eqs. (A.5b), (A.5h) and (A.5), we can ge:

24 3 u u s se S u u s s p u u 3 3 S uu s s p u u 3 3 S 8 8 8, (A.7a), (A.7b). (A.7c) Neglecng he ems of ode 3 O( u ) and hghe-ode ems of he fom u ( u u ) j k k j, usng Eqs. (A.5a), (A.5d) and (A.5f), we can oban: u u u, (A.8a) u u F, (A.8b) whee u u u uf, (A.8c) uu u F u F. Wh he above equaons, we can oban: u u F uf 3 s ee u u se, (A.8d), (A.9a) u F uf s p u u s, (A.9b) 3 s u F uf s p u u. (A.9c) 3 Subsung Eq. (A.9) no Eq. (A.6), he followng equaons a he me scale can be deved: 0, (A.0a) u u e u u B u u e u u, (A.0b)

25 whee e and u u e u u B u u e u u, (A.0c) B ae he effecve knec vscosy and bulk vscosy e cs, s B cs. (A.) se Combnng Eq. (A.0) ( me scale) wh Eq. (A.5) ( me scale) ( ), he genealed axsymmec Nave-Sokes equaons (0) and () can be obaned n he ncompessble lm ( 0 0, whee 0 s he mean flud densy, s he densy flucuaon and s of he ode O( u ) ).

26 Fgue Capons Fg.. Compuaonal doman and bounday condons of he hemally developng flow n a ppe flled wh poous meda. Fg.. Local Nussel numbes along he ppe wall fo dffeen Dacy numbes. Fg. 3. Compuaonal doman and bounday condons of he naual convecon flow n a vecal annulus. Fg. 4. Seamlnes (a) and sohems (b) fo 3 Ra 0 (lef), 4 Ra 0 (mddle), and 5 Ra 0 (gh). Fg. 5. Seamlnes (lef) and sohems (gh) fo dffeen Dacy-Raylegh numbes: (a) Ra 00 ; (b) Ra 500 ; (c) Ra 000. Fg. 6. Tempeaue pofles a fou dffeen heghs of he annulus fo dffeen Dacy-Raylegh numbes: (a) Ra 00 ; (b) Ra 500 ; (c) Ra 000. Fg. 7. Compuaonal doman and bounday condons of he naual convecon flow n a vecal poous annulus wh dscee heang. Fg. 8. Seamlnes (lef) and sohems (gh) fo dffeen Dacy numbes: (a) Da 0 6 ; (b) Da 0 4 ; (c) Da 0.

27 Nu T=Tw u=u=0 T n U n D T=Tw u=u=0 H Fg.. Compuaonal doman and bounday condons of he hemally developng flow n a ppe flled wh poous meda Da=0, 0, Whou poous /L Fg.. Local Nussel numbes along he ppe wall fo dffeen Dacy numbes.

28 Adabac g H T To o Adabac Fg. 3. Compuaonal doman and bounday condons of he naual convecon flow n a vecal annulus. (a) Seamlnes (b) Isohems Fg. 4. Seamlnes (a) and sohems (b) fo 3 Ra 0 (lef), 4 Ra 0 (mddle), and 5 Ra 0 (gh).

29 (a) Ra (b) Ra (c) Ra 000 Fg. 5. Seamlnes (lef) and sohems (gh) fo dffeen Dacy-Raylegh numbes: (a) Ra 00 ; (b) Ra 500 ; (c) Ra 000.

30 (TT o ) /T (TT o ) /T (TT o ) /T /H= Pasad e al. [4] Pesen 0.6 /H= /H=0.5 /H= ( ) /L (a) Ra /H= Pasad e al. [4] Pesen /H= /H=0.5 /H= ( )/L (b) Ra /H= Pasad e al. [3] Pesen /H= /H=0.5 /H= ( )/L (c) Ra 000 Fg. 6. Tempeaue pofles a fou dffeen heghs of he annulus fo dffeen Dacy-Raylegh numbes: (a) Ra 00 ; (b) Ra 500 ; (c) Ra 000.

31 Adabac g q' l To H q' To h o Adabac Fg. 7. Compuaonal doman and bounday condons of he naual convecon flow n a vecal poous annulus wh dscee heang.

32 (a) Da (b) Da (c) Da 0 Fg. 8. Seamlnes (lef) and sohems (gh) fo dffeen Dacy numbes: (a) Da 0 6 ; (b) Da 0 4 ; (c) Da 0.

33 Table Capons Table. Compasons of he Nussel numbes fo dffeen Dacy numbes. Table. Compasons of he aveage Nussel numbes n he pesen sudy wh hose epoed n pevous sudes. Table 3. Compasons of he aveage Nussel numbes and he maxmum values of he seam funcon n he pesen sudy wh he esuls n Ref. [5]. Table. Compasons of he Nussel numbes fo dffeen Dacy numbes. Da Ref. [6] Pesen Whou poous

34 Table. Compasons of he aveage Nussel numbes n he pesen sudy wh hose epoed n pevous sudes. Ra Ref. [3] Ref. [33] Ref. [49] Ref. [50] Pesen Table 3. Compasons of he aveage Nussel numbes and he maxmum values of he seam funcon n he pesen sudy wh he esuls n Ref. [5]. Da Nu max Ref. [5] Pesen Ref. [5] Pesen