Multiple-relaxation-time lattice Boltzmann modeling of. incompressible flows in porous media

Transcription

1 Muliple-relaaion-ime laice Bolzmann modeling of incompressible flows in porous media Qing Liu, Ya-Ling He* Ke Laboraor of Thermo-Fluid Science and Engineering of MOE, School of Energ and Power Engineering, Xi an Jiaoong Universi, Xi an, Shaani, 749, P.R.China (*Corresponding auhor. Absrac In his paper, a wo-dimensional eigh-veloci (DQ8) muliple-relaaion-ime (MRT) laice Bolzmann (LB) model is proposed for incompressible porous flows a he represenaive elemenar volume scale based on he Brinkman-Forchheimer-eended Darc formulaion. In he model, he porosi is included ino he pressure-based equilibrium momens, and he linear and nonlinear drag forces of he porous media are incorporaed ino he model b adding a forcing erm o he MRT-LB equaion in he momen space. Through he Chapman-Enskog analsis, he generalized Navier-Sokes equaions can be recovered eacl wihou arificial compressible errors. Numerical simulaions of several pical wo-dimensional porous flows are carried ou o validae he presen MRT-LB model. The numerical resuls of he presen MRT-LB model are in good agreemen wih he analical soluions and/or oher numerical soluions repored in he lieraure. Kewords: Laice Bolzmann model; Muliple-relaaion-ime; Porous media; incompressible flows; REV scale.. Inroducion

2 Fluid flow in porous media has gained significan research ineres due o is imporance of relaed scienific and indusrial applicaions, which include conaminan ranspor in groundwaer, crude oil eploraion and eracion, radioacive wase managemen, hdrogeolog and so on. Comprehensive lieraure surves of hese applicaions can be found in Refs. [-3]. For incompressible flows in porous media a he represenaive elemenar volume (REV) scale, he Darc model, he Brinkman-eended Darc model and he Forchheimer-eended Darc model have been widel emploed in man sudies. However, he Darc model and he wo eended (Brinkman and Forchheimer) models have some inrinsic limiaions in simulaing porous flows [4]. In order o overcome he shorcomings of he above menioned models, he Brinkman-Forchheimer-eended Darc model (also called he generalized model) has been developed b several research groups [4-7]. In he generalized model, he viscous and inerial forces are incorporaed ino he momenum equaion b using he local volume-averaging echnique. The Darc model and he wo eended models can be regarded as he limiing cases of he generalized model. Moreover, he single phase flow and he ransien flow in porous media can be solved b he generalized model [7]. As poined ou b Vafai and Kim [8], numerical resuls based on he Brinkman-Forchheimer-eended Darc formulaion have been shown o be in good agreemen wih he eperimenal predicions, and a porous medium/free-fluid inerface can be bes deal wih b he Brinkman-Forchheimer-eended Darc formulaion and he coninui of sresses and velociies a he inerface. In he pas several decades, various radiional numerical mehods, such as he finie volume (FV) mehod, he finie difference (FD) mehod, and he finie elemen (FE) mehod, have been emploed o sud porous flows based on he generalized model. The laice Bolzmann (LB) mehod, as a mesoscopic numerical scheme originaes from he laice-gas auomaa (LGA) mehod [9], has achieved significan success in modeling comple fluid

3 flows and relaed ranspor phenomena due o is kineic characerisics [-6]. Owing o he kineic background, he LB mehod has some disincive advanages over he radiional numerical mehods (e.g., see Ref. [7]). Recenl, he LB mehod has been successfull applied o simulae fluid flows in porous media a he REV scale [8-5]. In he REV scale mehod, he deailed geomeric srucure of he media is ignored, and he sandard LB equaion is modified b adding an addiional erm o accoun for he presence of he porous media. Therefore, LB mehod a he REV scale can be used for ssems wih large compuaional domain. I is worh menioning ha he REV LB mehod is ver effecive for simulaing fluid flows in he region which is pariall filled wih a porous medium. As repored in Ref. [], he disconinui of he veloci-gradien a he porous medium/free-fluid inerface can be well capured b he LB mehod wihou including he sress boundar condiion ino he simulaion model. However, o he bes of our knowledge, mos of he eising REV LB models [8-5] for incompressible porous flows emplo he Bhanagar-Gross-Krook (BGK) model (also called he singe-relaaion-ime model) [6] o represen he collision process. Alhough he BGK model has become he mos popular one because of is simplici, here are several well-known criicisms on his model, such as he numerical insabili a low values of viscosi [7-9] and he inaccurac in reaing boundar condiions [3]. On he oher hand, i has been demonsraed ha he deficiencies of he BGK model can be addressed b emploing he muliple-relaaion-ime (MRT) model [3]. Hence, he aim of his paper is o develop a new MRT-LB model for incompressible porous flows a he REV scale based on he generalized model. In he model, a pressure-based MRT-LB equaion wih he eigh-b-eigh collision mari [3] is consruced in he framework of he sandard MRT-LB mehod. The remainder of his paper is organized as follows. In Secion, he MRT-LB model for incompressible porous flows a he REV scale is presened. In Secion 3, numerical ess of he

4 MRT-LB model are performed for he porous Poiseuille flow, porous Couee flow, lid-driven flow in a square porous cavi, and naural convecion flow in a square porous cavi. Finall, a brief conclusion is made in Secion 4.. MRT-LB model for incompressible flows in porous media. Macroscopic governing equaions The fluid flow is assumed o be wo-dimensional, laminar and incompressible. For isohermal incompressible porous flows a he REV scale, he generalized model proposed b Nihiarasu e al. [4, 7] is emploed in he presen sud. The dimensional governing equaions of he generalized model can be wrien as u, () u u F, () u u p e where is he average fluid densi, u and p are he volume-averaged fluid veloci and pressure, respecivel, is he porosi, and e is he effecive kineic viscosi. F = F, F denoes he oal bod force induced b he porous mari and oher eernal forces, which can be epressed as [6, ] F F u u u a, (3) K K where is he kineic viscosi of he fluid, K is he permeabili, F is he geomeric funcion, a is he bod force due o an eernal force, and u u u, in which u and u are he - and -componens of he macroscopic veloci u, respecivel. Based on Ergun s relaion [33], he geomeric funcion F and he permeabili K of he porous media can be epressed as [34] F , d p K 3 5, (4)

5 where d p is he solid paricle diameer. The flow governed b Eqs. () and () are characerized b he porosi and several dimensionless parameers: he Darc number Da, he viscosi raio J, and he Renolds number Re, which are defined as K e Da, J L, Re LU, (5) where L is he characerisic lengh, and U is he characerisic veloci.. MRT-LB model In his subsecion, a wo-dimensional MRT-LB model wih eigh velociies (DQ8 model) is presened o sud incompressible porous flows. According o Refs. [35, 36], he MRT-LB equaion wih an eplici reamen of he forcing erm can be wrien as eq,,,, Λ f e f M Λ m m M I S, (6) where M is he 8 8 orhogonal ransformaion mari, Λ is he non-negaive 8 8 diagonal relaaion mari, and I is he 8 8 uni mari. The boldface smbols, f, m, m eq, and S represen 8-dimensional (column) vecors:, f,, f,,, f8, f, f,,, f8 8, f e e e, m,, m,,, m8, m where T is he ranspose operaor, f, eq eq eq eq 8 T, T, m, m,, m,,, m, i S = S, S,, S 8 T, is he discree disribuion funcion corresponding o he T, T, discree veloci e and ime,, i m and eq, m are he veloci momens of he discree disribuion funcions f and he corresponding equilibrium momens, respecivel, and S i,,, 8 are componens of he forcing erm S. i

6 In he DQ8 model, he eigh discree velociies e i,,, 8 (see Fig. ) are defined as [3, i 37] where c, in which and cos i,sin i c, i ~ 4 e i, (7) cos i 9 4,sin i 9 4 c, i 5 ~ 8 are he ime sep and laice spacing, respecivel. The sound speed of he DQ8 model is c c 3. In he presen work, c is se o be, which leads o s. The ransformaion mari linearl ransforms he discree disribuion funcions f 8 (veloci space) o heir veloci momens 8 m (momen space): m Mf, f = M m. (8) For he DQ8 model, he eigh veloci momens m i,,, 8 corresponding o he i discree velociies are: where m m p is he pressure, m momenum j, j m, m, m3, m4, m5, m6, m7, m8 p, e, j, q, j, q, p, p T T, (9) e is relaed o energ, m3,5 j, are componens of he J, m4,6 q, are relaed o energ flu, and m7,8 p, are relaed o he diagonal and off-diagonal componens of he sress ensor. Wih he ordering of he veloci momens i m given above, he ransformaion mari M can be easil consruced ( c ) [3]: Μ. () The equilibrium momens eq m for he veloci momens m are given as follows:

7 m T eq 5 u eq eq eq eq eq p, e, j, q, j, q, p, p 3 3, () where eq 3 u u e p, 4 eq q c u, eq q cu, eq 3 u u p, eq 3 uu p 3. () To ge he correc generalized Navier-Sokes equaions () and (), he parameers are chosen as follows:, c c,, and 3 3. In he above equilibrium momens, we have emploed he incompressibili approimaion, i.e., he fluid densi and j, j J u. The relaaion mari Λ is given b Λ =diag s, s, s, s, s, s, s, s =diag, se,, sq,, sq, s, s. (3) The evoluion process of he MRT-LB equaion (6) consiss of wo seps: he collision sep and sreaming sep [7]. Usuall, he collision sep is implemened in he momen space: eq m m m m I S, (4) where m represen he veloci momens afer collision, while he sreaming sep is sill carried ou in he veloci space:,, f e f, (5) i i i where f M m.

8 In order o derive he correc equaions of hdrodnamics, he forcing erm S in he momen space should be chosen appropriael. For he DQ8 MRT-LB model, he forcing erm S can be chosen as 4u F 5 F F S F, (6) F uf uf uf u F where F, F F is given b Eq. (3). The fluid veloci u is defined as u e F, (7) 8 i fi i Noe ha he oal bod force F also conains he veloci u, so Eq. (7) is a nonlinear equaion for u. According o Eqs (3) and (7), he veloci u can be calculaed eplicil b v u l l l v, (8) where v, l and l are given b 8 v ei fi a, l i K, F l K. (9) According o Ref. [39], he fluid pressure p can be deermined b where 4 9 c p f s 8 s i i u, () and s u u 3. Here, u 3e u 4.5 e u.5 u wih 4 9, ~4 9 and 5~8 36. si i i i The effecive kineic viscosi e in he model is defined as

9 e s c () wih s7 s8 s. Through he Chapman-Enskog analsis [3, 35, 38] of he MRT-LB equaion (6) in he momen space, he generalized Navier-Sokes equaions () and () can be recovered eacl in he incompressible limi (see Appendi A for deails). If he eigh relaaion raes si i,,,8 are se o be a single value, i.e., Λ I, hen he presen MRT-LB model reduces o a BGK-LB model wih he following equilibrium disribuion funcion: f p e u e u u eq i i i i 4 cs cs cs cs. () I should be noed ha, as and Da, he presen MRT-LB model reduces o he incompressible MRT-LB model [3] for free-fluid flows wihou porous media. When F, he Brinkman-eended Darc equaion can be obained from he presen MRT-LB model. 3. Numerical ess In his secion, numerical simulaions of several pical wo-dimensional porous flows are carried ou o validae he proposed MRT-LB model. The esing problems include he Poiseuille flow in a channel filled wih porous media, he Couee flow beween wo parallel plaes filled wih porous media, he lid-driven flow in a square porous cavi, and he naural convecion flow in a square porous cavi. The presen numerical resuls are compared wih he analical and numerical soluions in previous sudies. In simulaions, we se, c =, and. The relaaion raes si i 8 are chosen as follows: s s3 s 5, s., s4 s 6., and s7 s8 s. Unless oherwise saed, he non-equilibrium erapolaion scheme [4] is emploed o rea he veloci and emperaure boundar condiions in simulaions. 3. Poiseuille flow in a channel filled wih porous media

10 The firs es problem is he Poiseuille flow in a D channel filled wih porous media. The heigh of he channel is L, and he flow is driven b an eernal force =, a along he channel a direcion. When he flow is full developed in he -direcion (along he channel direcion), he governing equaion of he flow can be epressed as wih u u L u e K F u u a K,,, and he -direcion veloci componen u is zero everwhere. The (3) Renolds number Re is defined as Re Lu, where u is he maimum veloci of he porous Poiseuille flow (wihou he nonlinear drag force, i.e., F ) along he cenerline of he channel []: where K. e ak L u cosh, (4) In simulaions, he porosi is se o be., he viscosi raio J is se o be, he Darc number Da changes from 6 o 3, and he Renolds number Re changes from. o. The relaaion rae s is se o be 5 3 (.6 ) wih a N N 8 8 square mesh. The nonequilibrium erapolaion scheme is emploed o he op and boom plaes for no-slip veloci boundar condiion, and he periodic boundar condiions are emploed o he inle and oule of he channel. A, he veloci momens are se o be heir equilibrium values, and he veloci field is iniialized o be wih a consan pressure p. The veloci profiles prediced b he presen MRT-LB model for differen Re and Da are shown in Fig.. The numerical soluions given b Guo and Zhao [] using he FD mehod are also included in he figure for comparison. From Fig. i is clearl seen ha he presen numerical resuls agree well wih he FD soluions repored in he lieraure. 3. Couee flow beween wo parallel plaes filled wih porous media

11 We now appl he MRT-LB model o he Couee flow in a D channel filled wih fluid-sauraed porous media. The flow is driven b he op plae moving along he -direcion wih a uniform veloci u, while he boom plae is fied. The Renolds number Re of his flow is defined b Re Lu. A he sead sae, he flow sill follows Eq. (4) ( a ) wih u, and u, L u. In simulaions, he porosi is se o be., he viscosi raio J is se o be, and he boundar and iniial condiions are he same as hose in he porous Poiseuille flow. For Re =.,, 5, and, he kineic viscosi is se o be.4,.8,.3, and.8 based on a N N 8 8 square mesh, respecivel. In Fig. 3, he veloci profiles prediced b he presen MRT-LB model are presened for various Re and Da. The FD soluions in Ref. [] are also included in Fig. 3 for comparison. Ecellen agreemen can be found beween he presen MRT-LB resuls and he FD soluions. The MRT-LB model is also applied o simulae he modified Couee flow in a D channel pariall filled wih a fluid-sauraed porous medium, which has been invesigaed in Refs. [,, 4]. A porous laer is posiioned in he lower par of he channel such ha here is a gap beween he op plae and he medium. The porosi is se o be. for L (porous region) and for L (free-fluid region). For small Re and Da, he nonlinear drag force can be negleced, and he Brinkman model is applied in he porous region. A he sead sae, he analical soluion of he flow can be epressed as [, 4]: u d d L L, (5) d ep L L where K, d Ku K, and d u K e. In simulaions, F is se o be zero, and he fluid kineic viscosi is se o be.6 wih a N N 8 8 laice. The relaaion rae s is deermined b s.5 J cs (for free-fluid region, J ). The

12 veloci profiles prediced b he presen MRT-LB model for J and 4 a Da. and Re. are ploed in Fig. 4. As shown, he presen numerical resuls agree well wih he analical ones. The disconinui of he veloci-gradien a he porous medium/free-fluid inerface is well capured b he presen MRT-LB model wihou an special reamen for he boundar condiion a he inerface in simulaions. 3.3 Lid-driven flow in a square porous cavi In his subsecion, we appl he MRT-LB model o simulae he incompressible flow in a square cavi filled wih a fluid-sauraed porous medium. The op wall of he cavi moves from lef o righ wih a uniform veloci u, while he boom, righ, and lef walls of he cavi are fied. The Renolds number Re of his flow is defined as Re Lu, where L is he heigh of he cavi. In simulaions, we se., Re, J =, and he kineic viscosi is se o be.56 based on a N N 8 8 square mesh. The nonequilibrium erapolaion scheme is adoped o rea i he veloci boundar condiions of f on he four walls. The veloci momens are se o be heir equilibrium values, and he veloci field is iniialized o be wih a consan pressure p a. The sreamlines prediced b he presen MRT-LB model for differen Darc numbers wih. and Re are illusraed in Fig. 5. From he figure we can see ha, as Da decreases, he vore inside he cavi becomes weaker, and he boundar laer near he op wall becomes hinner. This can be aribued o he lower permeabili ( K = DaL ) of he medium which resuls in lower fluid veloci. To be more informaive, he horizonal veloci componen u in he verical midplane L and he verical veloci componen u in he horizonal midplane L are ploed in Fig. 6. The FD soluions in Ref. [] are also ploed in Fig. 6 for comparison. Obviousl, he presen numerical resuls are in good agreemen wih he FD soluions.

13 As and Da ends o infini, he presen MRT-LB model reduces o he incompressible MRT-LB model for free-fluid flows wihou porous media. We now appl he MRT-LB model o he lid-driven cavi flow wihou porous media. In simulaions, we se.999, 6 Da, and u.. In Fig. 7, he veloci profiles hrough he cener of he cavi are ploed for differen Renolds numbers. The benchmark soluions of Ghia e al. [4] are also presened for comparison. Ecellen agreemen can be found beween he presen MRT-LB resuls and he benchmark soluions. 3.4 Naural convecion flow in a square cavi filled wih porous media Naural convecion flow in a square cavi filled wih a porous medium (see Fig. 8) has been sudied eensivel b man researchers [4, 7, 43, 44] based on he generalized model. For his problem, he veloci field is solved b he presen MRT-LB model and he emperaure field is solved b he DQ5 MRT-LB model [44] (see Appendi B for deails). The op and boom walls of he cavi are hermall insulaed, while he lef and righ verical walls are kep a consan bu differen emperaures T h and T c ( T h T ), respecivel. The Prandl number Pr and Raleigh number Ra c of his flow are defined as Pr and Ra T T T is 3 g TL Pr, respecivel, where h c he emperaure difference, is he hermal diffusivi of he fluid, g is he graviaional acceleraion, is he hermal epansion coefficien, and L is he disance beween he walls. The bod force a is defined as a j, where T T T = g T T is he reference emperaure, h c and j is he uni vecor in he -direcion. In simulaions, we se Pr, J, = (hermal capaci raio), e ( e is he effecive hermal diffusivi), ζ, ζ ζ T, ζ3 ζ4., Th, and T c =. According o Refs. [3, 45], he dimensionless relaaion imes of he veloci and emperaure fields can be deermined as

14 MaJL 3Pr, c Ra T cs cst.5, (6) J Pr respecivel, where Ma U c 3U is he Mach number ( U g TL is he characerisic s veloci, and usuall Ma.3 ), and c c 5 ( c st is he sound speed of he DQ5 model). The st grid sizes of,, and 5 5 are emploed for Da, 4, and 6, respecivel. Fig. 9 illusraes he sreamlines and isoherms for differen Raleigh numbers and Darc numbers (Darc-Raleigh number * Ra RaDa = ) wih Pr and.6. From he figure we can observe ha, as Da decreases, he veloci and hermal boundar laers become hinner near he ho and cold verical walls. As Da increases o, he sreamlines and isoherms are less crowded near he verical walls and more convecive miing occurs inside he cavi. To quanif he resuls, he average Nussel numbers of he lef verical wall are calculaed and lised in Table. The numerical resuls given b Nihiarasu e al. [4, 7] using he finie elemen mehod are also included in Table for comparison. To sum up, he numerical resuls of he presen MRT-LB model agree well wih hose resuls repored in previous sudies. 4. Conclusion In his paper, an MRT-LB model wih he eigh-b-eigh collision mari has been presened for simulaing incompressible flows in porous media a he REV scale. The ke poin of he MRT-LB model is o include he porosi ino he equilibrium momens, and add a forcing erm o he MRT-LB equaion in he momen space o accoun for he Darc (linear) and Forchheimer (nonlinear) drag forces of he solid mari based on he generalized model. Through he Chapman-Enskog analsis in he momen space, he generalized Navier-Sokes equaions can be recovered eacl wihou arificial compressible errors. Numerical simulaions of he porous Poiseuille flow, porous Couee flow,

15 lid-driven flow in square porous cavi, and naural convecion flow in a square porous cavi have been carried ou o demonsrae he presen MRT-LB model. The numerical resuls of he presen MRT-LB model agree well wih he analical soluions and/or oher numerical soluions repored in previous sudies. Acknowledgemens This work was suppored b he Naional Ke Basic Research Program of China (973 Program) (3CB834). References [] K. Vafai, Handbook of Porous Media, Marcel Dekker, New York,. [] I. Pop, D.B. Ingham, Convecive Hea Transfer: Mahemaical and Compuaional Modeling of Viscous Fluids and Porous Media, Pergamon, Oford,. [3] D.A. Nield, A. Bejan, Convecion in Porous Media, hird ed., Springer, New York, 6. [4] P. Nihiarasu, K.N. Seeharamu, T. Sundararajan, Naural convecive hea ransfer in a fluid sauraed variable porosi medium, In. J. Hea Mass Transfer 4 (997) [5] K. Vafai, C.L. Tien, Boundar and ineria effecs on flow and hea ransfer in porous media, In. J. Hea Mass Transfer 4 (98) [6] C.T. Hsu, P. Cheng, Thermal dispersion in a porous medium, In. J. Hea Mass Transfer 33 (99) [7] P. Nihiarasu, K. Ravindran, A new semi-implici ime sepping procedure for buoanc driven flow in a fluid sauraed porous medium, Compu. Meh. Appl. Mech. Eng. 65 (998)

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18 Rev. E 8 () [3] I. Ginzburg, D. d Humières, Mulireflecion boundar condiions for laice Bolzmann models, Phs. Rev. E 68 (3) [3] D. d Humières, Generalized laice-bolzmann equaions, in: B.D. Shizgal, D.P. Weaver (Eds.), Rarefied Gas Dnamics: Theor and Simulaions, in: Prog. Asronau. Aeronau., Vol. 59, AIAA, Washingon, DC, 99, pp [3] R. Du, B. Shi, Incompressible MRT laice Bolzmann model wih eigh velociies in D space, In. J. Mod. Phs. C (9) [33] S. Ergun, Fluid flow hrough packed columns, Chem. Eng. Prog. 48 (95) [34] K. Vafai, Convecive flow and hea ransfer in variable-porosi media, J. Fluid Mech. 47 (984) [35] M.E. MaCracken, J. Abraham, Muliple-relaaion-ime laice-bolzmann model for muliphase flow, Phs. Rev. E 7 (5) 367. [36] Q. Li, K.H. Luo, X.J. Li, Laice Bolzmann modeling of muliphase flows a large densi raio wih an improved pseudopoenial model, Phs. Rev. E, 87 (3) 533. [37] Y.H. Qian, D. d Humeier, P. Lallemand, Laice BGK models for Navier-Sokes equaion, Europhs. Le. 7 (99) [38] S. Chapman and T. G. Cowling, The Mahemaical Theor of Non-Uniform Gases, Cambridge Universi Press, London, 97. [39] Z. Guo, B. Shi, N. Wang, Laice BGK model for incompressible Navier-Sokes equaion, J. Compu. Phs. 65 () [4] Z.L. Guo, C.G. Zheng, B.C. Shi, Non-equilibrium erapolaion mehod for veloci and pressure

19 boundar condiions in he laice Bolzmann mehod, Chin. Phs. () 366. [4] N. Mars, D.P. Benz, E.J. Garboczi, Compuer simulaion sud of he effecive viscosi in Brinkman s equaion, Phs. Fluids 6 (994) [4] U. Ghia, K.N. Ghia, C.T. Shin, High-Re soluions for incompressible flow using he Navier-Sokes equaions and a muligrid mehod, J. Compu. Phs. 48 (98) [43] Z. Guo, T.S. Zhao, A laice Bolzmann model for convecion hea ransfer in porous media, Numer. Hea Transfer B 47 (5) [44] Q. Liu, Y.L. He, Q. Li, W.Q. Tao, A muliple-relaaion-ime laice Bolzmann model for convecion hea ransfer in porous media, In. J. Hea Mass Transfer 73 (4) [45] Q. Li, Y.L. He, Y. Wang, G.H. Tang, An improved hermal laice Bolzmann model for flows wihou viscous hea dissipaion and compression work, In. J. Mod. Phs. C 9 (8) 5-5. Appendi A: Chapman-Enskog analsis of he DQ8 MRT-LB model The Chapman-Enskog epansion mehod [3, 35, 38] is adoped o derive he generalized Navier-Sokes equaions () and () from he DQ8 MRT-LB model. To his end, he following epansions in ime and space are inroduced: n f f n e, e, i i i i i n! () () i i i i, (A.a) f f f f, (A.b), j j, S S, F F, (A.c) where is a small epansion parameer, S,, S S T 8, F F, F. Wih he above epansions, we can derive he following equaions from Eq. (6) as consecuive orders of he parameer in he momen space as

20 : eq m m, (A.a) where : j j : D MD M I C (, Λ D m Λ m I S, (A.b) Λ Λ m D I m D I S Λ m, (A.c) j ), D I jdiag e j, e j,, e8 j, j eij C M I M, Λ Λ, and m T, e, F, q, F, q, p, p. (A.3) C and C can be given eplicil b C, C 3. (A.4) From Eq. (A.b), he following equaions a he ime scale can be obained: 5 u p u u, 3 3 (A.5a) s p u u s e 3 3 S, (A.5b) u uu s3 u p s3 F 3 S, (A.5c) u u u u s4 u p s4 q 4 S, (A.5d) u u u s5 u p s5 F 5 S, (A.5e) u u u u s6 u p s6 q 6 S, (A.5f) u u 7 s u u s7 p S, (A.5g)

21 uu 8 s u u s8 p S. (A.5h) From Eq. (A.c), he following equaions a he ime scale can be obained: u p, (A.6a) s s s u F e p s p 4 s3 s s7 s8 S 3 S S7 S8 4, (A.6b) s s s u F p e s p 4 s5 s8 s s7 S 5 S8 S S7 4, (A.6c) Following he approach in Ref. [3], we can obain: u p, (A.7) equaion Combining Eqs. (A.5a) and (A.6a) wih Eq. (A.7) leads o he following incompressible coninui u u, (A.8) Noe ha e, p and p in Eqs. (A.6b) and (A.6c) are unknowns o be deermined. According o Eqs. (A.5b), (A.5g) and (A.5h), we can ge: s s e p u u 3 3 S, (A.9a) u u s7 s 7 p u u S, (A.9b) uu s8 s 8 p u u S. (A.9c) Neglecing he erms of order 3 O( u ) and higher-order erms of he form u ( u u ) j k k j, using Eqs. (A.5c) and (A.5e), we can obain:

22 u uf, (A.a) u uf, (A.b) u u uf uf. (A.c) Wih he above equaions, we can obain: 4 s uf u F s e u u, (A.a) 5 5 u F u F s 7 p u u s 7, (A.b) 3 s u 8 F u F s 8 p u u. (A.c) 3 Subsiuing Eq. (A.) ino Eq. (A.6), he following equaions a he ime scale can be derived: u u u u u 5 s 3 s7 u u, (A.a) 3 s8 u u u u u 3 s8 5 s u u. (A.b) 3 s7 Combining Eq. (A.) ( ime scale) wih Eqs. (A.5c) and (A.5e) ( ime scale), he following equaions can be derived ( ): u u u u p 5 s u u

23 u u u u F, (A.3a) 3 3 s7 s8 u u u u p u u 3 s8 u u u u F. (A.3b) 5 s 3 s7 Wih he aid of he incompressible coninui equaion u u, he incompressible generalized Navier-Sokes equaions () and () can be obained from Eq. (A.3). The effecive kineic viscosi is defined as c.5 wih s7 s8 s. e s Appendi B: DQ5 MRT-LB model The macroscopic emperaure equaion of naural convecion flow in porous media can be wrien as T u T e T, (B.) where is he hermal capaci raio, and e is he effecive hermal diffusivi. For he emperaure field, he DQ5 MRT-LB equaion is defined as eq g e, g, N,, n n, (B.) where N is a 5 5 orhogonal ransformaion mari, and is a diagonal relaaion mari. The boldface smbols, g, n, and eq n are 5-dimensional (column) vecors:, g,, g,,, g4, g, g,,, g4 4, g e e e, n,, n,,, n4, n eq eq eq eq 4 T, T, n, n,, n,,, n, T, T,

24 where g, is he emperaure disribuion funcion,, i n and eq, n are he veloci momens of he discree disribuion funcions g and he corresponding equilibrium momens, respecivel. In he DQ5 model, he five discree velociies e i,,, 4 are given b i,, i e i. (B.3) cos i,sin i c, i ~ 4 For he DQ5 model, he ransformaion mari N is given b N. (B.4) 4 The ransformaion mari N linearl ransforms he discree disribuion funcions g 5 o heir veloci momens 5 n : In he ssem of g equilibrium momens as [44] i i n Ng,, onl he emperaure g = N n. (B.5) 4 T n g is conserved quani. The i i n eq i,,, 4 for he veloci momens n i,,, 4 are defined n eq T, ut n, eq ut n, eq n eq 3 T, eq 4 n, (B.6) where is a consan ( 4 ). In he presen work, is se o be. The diagonal relaaion mari is given b: 3 4 =diag, ζ, ζ, ζ, ζ. (B.7) The effecive hermal diffusivi is defined as c.5 e e st T wih ζ ζ T and c 4 c. st

25 Figure Capions Fig.. Discree velociies of he DQ8 model. Fig.. Veloci profiles of he porous Poiseuille flow for differen Re and Da wih. and J. Solid lines represen he FD soluions [] and smbols represen he presen MRT-LB resuls. Fig. 3. Veloci profiles of he porous Couee flow for differen Re and Da wih. and J. Solid lines represen he FD soluions [] and smbols represen he presen MRT-LB resuls. Fig. 4. Veloci profiles of he porous Couee flow for differen viscosi raio J wih Re =. and Da =.. Solid lines represen he analical soluions from Eq. (5) and smbols represen he presen MRT-LB resuls. Fig. 5. Sreamlines of he lid-driven flow in a porous cavi for differen Da wih. and Re : (a) Da ; (b) Da 3 ; (c) Da 4. Fig. 6. Veloci profiles hrough he cener of he cavi: (a) horizonal veloci componen u in he verical midplane L ; (b) verical veloci componen u in he horizonal midplane L. Solid lines represen he FD soluions [] and smbols represen he presen MRT-LB resuls. Fig. 7. Veloci profiles hrough he cener of he cavi: (a) horizonal veloci componen u in he verical midplane L ; (b) verical veloci componen u in he horizonal midplane L. Solid lines represen he benchmark soluions [4] and smbols represen he presen MRT-LB resuls. Fig. 8. Naural convecion in a square cavi filled wih a porous medium. Fig. 9. Sreamlines and isoherms for differen Ra and Da wih Pr and.6 : (a) Da, 5 Ra ; (b) Da 4, 7 Ra ; (c) Da 6, 9 Ra.

26 Fig.. Discree velociies of he DQ8 model.

27 ..8 u/u.6.4. Da= 5, =. Re=. Re= Re=5 Re= Lines: FD soluions /L (a)..8 u/u.6.4. Re=., =. Da= 6 Da= 4 Da= 3 Lines: FD soluions /L (b) Fig.. Veloci profiles of he porous Poiseuille flow for differen Re and Da wih. and J. Solid lines represen he FD soluions [] and smbols represen he presen MRT-LB resuls.

28 ..8 u/u.6.4. Da=., =. Re=. Re=5 Re= Lines: FD soluions /L (a)..8.6 /L.4. Re=, =. Da= Da= 3 Da= 5 Lines: FD soluions....4 u/u.6.8. (b) Fig. 3. Veloci profiles of he porous Couee flow for differen Re and Da wih. and J. Solid lines represen he FD soluions [] and smbols represen he presen MRT-LB resuls.

29 ..8 /L.6.4. Re=., Da=. J= J=4 Lines: Analical soluions u/u Fig. 4. Veloci profiles of he porous Couee flow for differen viscosi raio J wih Re =. and Da =.. Solid lines represen he analical soluions from Eq. (5) and smbols represen he presen MRT-LB resuls.

30 (a) Da (b) Da (c) Da 4 Fig. 5. Sreamlines of he lid-driven flow in a porous cavi for differen Da wih. and Re : (a) Da ; (b) Da 3 ; (c) Da 4.

31 ..8 /L.6.4. Re=, =. Da= Da= 3 Da= 4 Lines: FD soluions u(l/, )/u (a).5..5 v(,l/)/u Re=, =. Da= Da= 3 Da= 4 Lines: FD soluions /L (b) Fig. 6. Veloci profiles hrough he cener of he cavi: (a) horizonal veloci componen u in he verical midplane L ; (b) verical veloci componen u in he horizonal midplane L. Solid lines represen he FD soluions [] and smbols represen he presen MRT-LB resuls.

32 ..8.6 /L.4. Da= 6, =.999 Re= Re=4 Re= Lines: Ghia e al. [4] u(l/, )/u (a).4. v(,l/)/u Da= 6, =.999 Re= Re=4 Re= Lines: Ghia e al. [4] /L (b) Fig. 7. Veloci profiles hrough he cener of he cavi: (a) horizonal veloci componen u in he verical midplane L ; (b) verical veloci componen u in he horizonal midplane L. Solid lines represen he benchmark soluions [4] and smbols represen he presen MRT-LB resuls.

33 Adiabaic Porous Medium Th g Tc Adiabaic Fig. 8. Naural convecion in a square cavi filled wih a porous medium.

34 (a) Da, 5 Ra (b) Da 4, 7 Ra (c) Da 6, 9 Ra Fig. 9. Sreamlines and isoherms for differen Ra and Da wih Pr and.6 : (a) Da, 5 Ra ; (b) Da 4, 7 Ra ; (c) Da 6, 9 Ra.

35 Table Capion Table. Comparisons of he average Nussel numbers for differen Ra, Da and wih Pr.. Da Table. Comparisons of he average Nussel numbers for differen Ra, Da and wih Pr.. Ra Ref.[4] Ref.[7] Presen Ref.[4] Ref.[7] Presen Ref.[4] Ref.[7] Presen