HYBRID LBM-FVM AND LBM-MCM METHODS FOR FLUID FLOW AND HEAT TRANSFER SIMULATION

HYBRID LBM-FVM AND LBM-MCM METHODS FOR FLUID FLOW AND HEAT TRANSFER SIMULATION Zheng L a,b, Mo Yang b and Yuwen Zhang a* a Department of Mechancal and Aerospace Engneerng, Unversty of Mssour, Columba, MO 65211, USA b College of Energy and Power Engneerng, Unversty of Shangha for Scence and Technology, Shangha 200093, Chna ABSTRACT The flud flow and heat transfer problems encountered n ndustry applcatons span nto dfferent scales and there are dfferent numercal methods for dfferent scales problems. It s not possble to use sngle scale method to solve problems nvolvng multple scales. Multscale methods are needed to solve problems nvolvng multple scales. In ths chapter, meso-macro-multscale methods are developed by combnng varous sngle scale numercal methods, ncludng lattce Boltzmann method (LBM), fnte volume method (FVM) and Monte Carlo method (MCM). Macroscale methods nclude FVM, whle LBM and MCM belongs to mesoscale methods. Two strateges exst n combng these numercal methods. For the frst one, the whole doman s dvded nto multple subdomans and dfferent domans use varous numercal methods. Message passng among subdomans decdes the accuracy of ths type of multscale numercal method. For the second one, varous parameters are solved wth dfferent numercal methods. These two types of multscale methods are both dscussed n ths chapter. Keywords: Lattce Boltzmann method, Fnte volume method, Monte Carlo method, Multscale method, Natural convecton 1. INTRODUCTION Development of CFD nvolves very wde varaton of scales [1] rangng from, nano-/mcro-, meso- and macroscales. Molecular dynamcs (MD) [2] s applcable to nano- and mcroscale problems and LBM s a typcal mesoscopc scale method [3]; Fnte dfference method (FDM) [4] and FVM [5] on the other hand, falls nto the category of macroscale approach [6]. Realstc problems may nvolve more than one scale and they are so-called multscale problems. Multscale transport phenomena exsts n many ndustry areas, such as: fuel cell, laser materal nteracton and electroncs coolng. It s mpossble to solve a multscale problem usng any sngle-scale method. For example, MD smulaton cannot be used n the entre smulaton doman, and FVM s not sutable for the mcroscopc regon; LBM costs several tmes more computatonal tme than the FVM to obtan the same accuracy n the macroscopc problem [7]. It s necessary to buld multscale method to solve these multscale problems. In ths chapter, meso-macro-multscale methods are developed by combnng varous sngle scale numercal methods, ncludng LBM, FVM and MCM whle two strateges exst n combng these numercal methods. For the frst one, the whole doman s dvded nto multple subdomans and dfferent domans use varous numercal method. Message passng among subdomans decdes the accuracy of ths type of multscale numercal method. For the second one, varous parameters are solved wth dfferent numercal methods. For the frst strategy of multscale methods, there are some exstng results reported n the lterature: MD- FVM [8-10] hybrd methods were developed to combne mcroscale and macroscale numercal methods. In references [11-13], LBM-MD was proposed to combne mesoscale and mcroscale numercal methods. Ths chapter focus on multscale methods combnng mesoscale and macroscale numercal methods. LBM-FDM [14- * Correspondng Author. Emal: zhangyu@mssour.edu

2 Zheng L, Mo Yang and Yuwen Zhang 16] were advanced to solve varous multscale problems. However, the FDM tself has the lmtaton when solvng problems wth complex computatonal doman [6]. Ths shortfall restrcts the development of LBM- FDM because the one of the most attractve advantages of LBM s ts sutablty to solve the problems n complex computatonal doman. LBM-FVM [17-19] were developed to solve conducton-radaton heat transfer and compressble flud flow problems. Luan et al. [20] solved natural convecton usng the LBM-FVM wth the general reconstructon operator [21] reachng persuasve results. Chen et al. [22] used ths method to fulfll LBM- FVM wth varous grds settngs. However general reconstructon operator s newly proposed to fulfll the combne method whch means more valdatons are needed for the general reconstructon operator tself. In ths chapter, now exstng boundary condtons nonequlbrum extrapolaton scheme [23] and fnte-dfference velocty gradent method [16] are employed to fulfll ths type of multscale method. Second strategy of multscale method s a new choce for LBM to solve heat transfer problems. Ths s a promsng numercal method because the man advantage of the LBM les n obtanng a velocty feld. Several researchers [24, 25] obtaned some good results usng the LBM-FDM hybrd method. However, the FDM tself has the lmtaton when solvng problems wth complex computatonal doman [6]. Ths shortfall restrcts the development of LBM-FDM because the one of the most attractve advantages of LBM s ts sutablty to solve the problems n complex computatonal doman. There are lmted efforts on the hybrd lattce Boltzmann wth fnte volume method (LBM-FVM) [26, 27] for conducton-radaton problems. Meltng problems are solved wth LBM-FVM hybrd method [28]. Applcaton of LBM-FVM to natural convecton problems wll be dscussed n ths chapter. Monte Carlo method s a wdely used numercal method for heat transfer problem [29]. It can solve conducton, convecton and radaton heat transfer problems. In addton, both MCM and LBM belong to mesoscopc scale method. It s possble to combne LBM and MCM to solve convecton heat transfer problems, whch wll also be dscussed n ths chapter. 2. LATTICE BOLTZMANN METHOD Lattce Boltzmann method s a promsng mesoscale method for flud flow and heat transfer smulaton. Instead of solvng mass, velocty and energy conservaton equatons as tradtonal CFD methods, LBM reaches macroscale parameter usng statstcal behavors of partcles as shown n Fg. 1, whch represent large amounts of flud molecules. Fgure 1 Partcle behavor These partcles can steam and collson to each other n the computatonal doman. It can be employed to solve varous flud flow and heat transfer problems [30-32]. In ths secton, LBM for flud flow and heat transfer smulaton s ncluded.

Hybrd LBM-FVM and LBM-MCM Methods for Flud Flow and Heat Transfer Smulaton 3 2.1 Boltzmann equaton Statstcal behavors of partcles that are not n thermodynamc equlbrum can be descrbed by the Boltzmann equaton: f f f + + a = f collson t r (1) where f s the densty dstrbuton, and Ω s the collson operator that s dctated by the collson rules., r and a are the partcle s velocty, locaton and acceleraton, respectvely. To solve ths Boltzmann equaton, we need to smplfy the collson term frst. In ths chapter, we employed the wdely used Bhatnagar-Gross-Krook model to fulfll ths process. 2.2 Lattce Bhatnagar-Gross-Krook model The Bhatnagar-Gross-Krook model (BGK) that uses the Maxwell equlbrum dstrbuton, f eq, wll be used n ths chapter: f eq 1 = n exp m/2 2 RT g u 2 2RT g (2) where m s the dmenson of the problem. Equaton (2) descrbes the stuaton that the system has reached to the fnal equlbrum. The BGK model assumes that the collson term s the tme relaxaton from densty dstrbuton to the Maxwell equlbrum dstrbuton. Assumng the relaxaton tme s v, the Boltzmann equaton under the BGK model can be expressed as: f f f 1 eq + + a =- f f (3) t r v The LBM used n ths chapter s a specal scheme of LBGK. Only lmted numbers of drectonal dervatves are appled to Eq. (3) and there must be enough nformaton to obtan the macroscopc governng equaton. Fgure 2 Nne drectons n D2Q9 model The D2Q9 model s used and nne drectons are selected n the 2-D problem shown n the Fg. 2. The velocty n every drecton s: 3

4 Zheng L, Mo Yang and Yuwen Zhang (0,0) 1 a a e c( cos, sn ) 2,3,4,5 2 2 (2a1) (2a1) 2 c( cos, sn ) 6,7,8,9 4 4 (4) where c s a constant n the lattce unt. The densty dstrbuton f n the fxed drecton can be obtaned by ntegratng Eq. (3): 1 f,, eq r e t t t f r t f r, t fr, ttfr, t (5) where t s the magntude of the tme step whle s low, eq f can be smplfed as: v F s the body force n the fxed drecton. When the velocty e u ( e u) u R T R T RT 2 2 eq 1 2 2 g 2 g g f m 2 e 2 2RT g exp 2 RgT (6) (7) where eq f can be further smplfed wth regardng c s that s the speed of sound n the lattce unt: e u ( e u) u 2 2 eq 1 2 4 2 cs 2cs 2cs f (8) where 4 1 9 1 2,3, 4,5 9 1 6,7,8,9 36 2.3 Chapman-Enskog Expanson Applyng the followng Chapman-Enskog expanson equatons K r r 2 K K t t t 1 1 2 (9) (10) (11)

Hybrd LBM-FVM and LBM-MCM Methods for Flud Flow and Heat Transfer Smulaton 5 f f Kf K f (12) 0 1 2 2 a a a a to Eq. (5), the macroscopc governng equatons can be obtaned from LBM: t 0 V (13) V T VV p V V 2 VVV (14) t cs c t 2 2 1 s v (15) Equaton (14) dffers from the macroscopc momentum conservaton from due to presence of the term 2 VVV. Fortunately, t can be neglected when Mach number s low, whch s the case n c s consderaton. Thus, the LBM satsfes the same macroscopc governng equatons as the macroscopc method, whch meets the requrement to combne LBM wth FVM. To obtan the macroscopc parameter, the followng two addtonal equatons are needed. 9 f (16) 1 9 V ef (17) 1 where e s the vector representng the velocty n every dscrete drecton and V s the vector of the macroscopc velocty. 2.4 Thermal LBM model Two dstrbuton functons are selected for the flud flow and heat transfer n LBM. The densty and energy dstrbutons are represented by f and g, whch are related by the buoyancy force. D2Q9 model for f has been ncluded n Secton 2.1. The buoyancy force can be obtaned as: where the pressure, p, equals e V eq F tg f (18) p 2 c s. The D2Q5 model [33] s used for the temperature feld. There are fve dscrete velocty at each computng node shown s Fg. 2-3. Smlar to the densty dstrbuton, the energy dstrbuton can be obtaned by 1 eq gret, ttgr, t g r, tg r, t, 1,2,...5 (19) The macroscopc energy equaton can be obtaned from by Eq. (19) usng Chapman-Enskog expanson. T 5

6 Zheng L, Mo Yang and Yuwen Zhang Fgure 3 Fve drectons n D2Q5 model Then the relaxaton tme T s related to the thermal dffusvty as c t 2 2 1 s T (20) The equlbrum energy dstrbuton n Eq. (19) s g T e 1 V eq T 2 cs (21) where: T 1 3 1 6 1 2,3,4,5 (22) The temperature at each computng node can be obtaned as: 5 T g (23) 1 Two dmensonal double dstrbutons LBM model for flud flow and heat transfer smulaton s ntroduced n ths Secton 2. 3. TWO SCHEMES FOR HYBRID LATTICE BOLTZMANN and FINITE VOLUME METHODS Two schemes for hybrd LBM-FVM method are proposed n ths Secton 3. The key pont of the hybrd method s to pass the nformaton on the nterface between LBM and FVM. It s dffcult to transfer velocty obtaned from FVM nto node populaton that s needed n LBM. Nonequlbrum extrapolaton scheme [23] and fnte-dfference velocty gradent method [16] are boundary condtons for LBM. They wll be used n ths secton to pass nformaton between the FVM and LBM zones. The ld-drven flow problem s solved to test the proposed methods.

Hybrd LBM-FVM and LBM-MCM Methods for Flud Flow and Heat Transfer Smulaton 7 3.1 Fnte volume method Lattce Boltzmann method for 2-D flud flow has been ncluded n Secton 2. SIMPLE s a very popular FVM algorthm [34] that solves the general equatons n macroscopc scale based on the control volume shown n Fg. 4. Ths algorthm s employed n ths Chapter as FVM. Fgure 4 Control volume n 2-D FVM For a 2-D problem n Cartesan coordnate system, the general equatons can be expressed as: u v S t x y x x y y (24) The SIMPLE algorthm s employed to solve Eq. (24) n ths chapter. QUICK scheme [6] s selected to have an accuracy good enough. 3. 2 Combned LBM and FVM method for pure flud flow problems 3.2.1 Problem Statement Ld drven flow s used to test the method to combne LBM and FVM. No-slp boundary condtons are appled to ths 2-D problem, and the flow s drven by a constant ld velocty u 0 on the top of the square cavty whle the veloctes on all other boundares are zero shown n Fg. 5. Ths problem can be descrbed by the followng governng equatons: u v 0 t x y 2 2 u uu vu p u u 2 2 t x y x x y 2 2 v uv vv p v v 2 2 t x y y x y (25) (26) (27) whch are subject to the followng boundary condtons: x 0 u 0 v 0 (28) x H u 0 v 0 (29) y 0 u 0 v 0 (30) 7

8 Zheng L, Mo Yang and Yuwen Zhang yh uu0 v 0 (31) In addton, the Reynolds number s defned wth the constant velocty on the top. Re 0 uh (32) Fgure 5 Ld-drven flow 3.2.2 Descrpton of the combng method Lattce unt s appled to LBM whle SIMPLE algorthm uses non-dmensonal procedure. As dscussed above, e has dfferent values n dfferent drectons n the lattce unt. In order to combne these two methods wth dfferent unts together, they must be used to descrbe the same stuaton n the actual unt. In the LBM unt converson process, whch changes all the propertes nto lattce unt, the speed of sound c and tme step t are fxed so that the densty dstrbutons are all on the computatonal nodes. s cs 1/ 3, t 1 (33) n, the Assumng that the real speed of sound s u sound and the number of nodes n the y-drecton s 1 dmensonless veloctes n LBM are: U L u (34) 3u sound V L and the tme step s t v 3u H L n 3c s sound 1 (35) (36) Thus the coordnates n lattce unt become X L nx H (37)

Hybrd LBM-FVM and LBM-MCM Methods for Flud Flow and Heat Transfer Smulaton 9 nx YL (38) H In order to allow the boundary velocty n lattce unt n LBM to be same as the dmensonless velocty n SIMPLE, t s assumng that the ld velocty and the speed of sound have the followng relatonshp: 10u0 3u sound (39) whch requres that 10u 0 to be used n the non-dmenson process. The dmensonless ld velocty n the LBM becomes U0 0.1 (40) At ths pont, the only unknown parameter n lattce unt s the knematc vscosty L and t can be obtaned from the Reynolds number: UH 0 L L ReL To satsfy Eq. (15), the relaxaton tme v can be obtaned to fulfll LBM (41) 0 3 L 0.5 (42) As dscussed above, to obtan the same boundary dmensonless velocty as that n lattce unt, the followng non-dmensonal varables are defned: u v x y Us, Vs, Xs, Ys 10u0 10u0 H H t p uh 0 F, P, Re 2 s H /(10 u0 ) 10u0 (43) The governng equatons (24) to (30) can be nondmenonalzed as: U X s s V Y s s 0 (44) 2 2 UU VU 1 U s s s s s P Us Us 2 2 F Xs Ys Xs 10 Res Xs Ys 2 2 UV VV 1 V s s s s s P Vs Vs 2 2 F Xs Ys Ys 10 Res Xs Ys (45) (46) X 0 U 0 V 0 (47) s s s X 1 U 0 V 0 (48) s s s Y 0 U 0 V 0 (49) s s s 9

10 Zheng L, Mo Yang and Yuwen Zhang Y 1 U 0.1 V 0 s s s In order to meet the requrement for descrbng the same stuaton n the actual unt n LBM and SIMPLE, the followng two addtonal equatons are needed. Re s Re (51) F t n L where n 1 s the number of nodes n the y-drecton. By followng the above nondmensonalzaton procedures for LBM and SIMPLE, the non-dmensonal ld velocty n both methods are 0.1. So the same nondmensonal veloctes are reached at the same real locaton for the same Reynolds number n every tme step. Therefore, the non-dmensonal veloctes can be transferred drectly. In order to combne LBM and SIMPLE n the same problem, the computatonal doman s dvded nto two zones as shown n Fg. 6. (50) (52) Fgure 6 Computatonal doman for LBM and SIMPLE A 160 160 unform grd s appled to the entre computatonal doman. In most cases, the wder the message passng zone s, the better the accuracy s. However, enlarged message passng zone also ncreases the computatonal tme n every tme step. Meanwhle, although the grd s unform n the whole doman, the locatons of velocty on the grd are dfferent for LBM and SIMPLE. So addtonal nterpolaton, whch may requre the nformaton on the nearby nodes, s needed n the nformaton sharng process. Three shared grds are selected after testng and the grd n LBM s 80 160 whle that n SIMPLE s 83 160. It s necessary to pont out that t s almost mpossble to transfer the pressure n LBM to that n SIMPLE because they have dfferent ways to obtan the pressure. In LBM, the deal gas law s used to obtan the pressure, whle SIMPLE solves the pressure correcton equaton based on the conservaton of mass. The example n ths secton does not need to transfer the pressure nformaton n the message passng zone shown n Fg. 7 due to the nature of method used to combne LBM and SIMPLE. The nterface between LBM and SIMPLE zone s treated as a fxed velocty boundary at every tme step. Staggered grds are used n SIMPLE and LBM, and the locatons of macroscopc parameters n the computatonal doman are shown n Fgs. 7 and 8. It can be seen that the locatons of macroscopc parameters are dfferent n SIMPLE and LBM, even the grds are the same. Fgure 9 gves a more clear vew about that n one control volume n the message passng zone. The boundares of the LBM and SIMPLE zones are 1 l and 2 l as shown n Fg. 6. Meanwhle, they are the nner nodes n both SIMPLE and LBM zones. In addton, t can also be seen that there s no dfference between

Hybrd LBM-FVM and LBM-MCM Methods for Flud Flow and Heat Transfer Smulaton 11 boundares and nner nodes for LBM as can be seen from Fg. 8. Therefore, transferrng nformaton from SIMPLE to LBM only needs the nformaton on l 1 from the nearby nodes n the SIMPLE zone: u, j, L v, j, L u 1, j1, S u 1, j, S (53) 2 v1, j1, S v, j1, S (54) 2 Fgure 7 Varable locatons n SIMPLE Fgure 8 Computatonal nodes n LBM Fgure 9 Detals on one control volume n the message passng zone 11

12 Zheng L, Mo Yang and Yuwen Zhang The procedure to transfer nformaton from LBM to SIMPLE s smlar except the locatons of nodes are dfferent for nner nodes and boundary nodes as shown n Fg. 6. The followng equatons can be used n ths transfer process: v u, j, S u 1, j1, L u 1, j, L (55) 2, j, S v, j 1, L (56) After transferrng the nformaton between the two zones, the hybrd method turns to fxed velocty problem on the nterfaces l 1 and l 2 shown n Fg. 6 n LBM and SIMPLE. There s no need for any specal treatment to the SIMPLE zone rather than settng the boundary velocty n the program. On the other hand, t s dfferent for the LBM zone because ts orgnal varable s densty dstrbuton on the computatonal nodes. Fgure 10 Boundary condton n LBM at the nterface The three densty dstrbutons f 4, f 8, f 9 and densty are unknown as shown n Fg. 10, whle there are only three equatons: f f f f f f f f f (57) 4 8 9 1 2 3 5 6 7 f f f u f f f (58) 4 8 9 L 2 6 7 f f v f f f f (59) 8 9 L 5 6 3 7 There are several methods to solve ths problem on the boundary [16]. Nonequlbrum extrapolaton scheme and fnte-dfference velocty gradent method are used n ths secton to replace all the densty dstrbutons on the boundary. In the Nonequlbrum extrapolaton scheme (hybrd method 1), we have the followng assumpton: eq eq f f f f 1, 2 9 (60) boundary nner Ths scheme has second order accuracy and can meet the requrement of the combnng method. One the other hand, fnte-dfference velocty gradent method (hybrd method 2) s used to replace all the densty dstrbutons on the boundary. Equaton (12) can be rewrtten as:

Hybrd LBM-FVM and LBM-MCM Methods for Flud Flow and Heat Transfer Smulaton 13 f f Kf O K (61) (0) (1) 2 = + + ( ) The followng equatons can be obtaned dependng on dfferent order of K. K : f f 0 (0) eq 1 eq 1 (1) K : e 1 f f 0 t1 t eq 2 f 1 (1) 1 (2) K : e 1 1 f f 0 t2 t1 2 t (62) (63) (64) Then (1) f can be obtaned from Eq. (63): (1) eq f =-t a1 f t e (65) 1 The hydrodynamc varables also have the relaton wth densty dstrbutons: 9 ee f (66) 1 where s the moment of order 2. Assume that: Q ee c I (67) 2 s where I s the dentty tensor, one obtans: Kf 1 = Q : ue: uu e Q : uu (68) (1) 2 2 cs 2cs (1) Substtutng Eq. (62) to Eqs. (17) and (66) and approxmatng Kf by the frst term only [16], the densty dstrbuton can be expressed as: eq f f Q 2 : u (69) cs So the densty dstrbuton can be related to stran rate tensor S due to the symmetry of Q : eq f f Q 2 : S (70) cs where 1 T S= u+ u (71) 2 13

14 Zheng L, Mo Yang and Yuwen Zhang Thus, the nformaton of the stran rate tensor on l 2 s needed to be transferred to LBM zone from the FVM zone. In addton, more nformaton needs to be transferred from the SIMPLE zone to LBM zone besdes the velocty. u x, u y, v x and v on 1 y l are also needed n order to get the stran rate tensor. u y approxmated by a centered dfference. The conservaton of mass requres that: v u y x v, and x The followng steps should be taken to transfer nformaton between the two zones n the method to combne LBM and SIMPLE (see Fg. 6): 1. Assume the velocty on x = l 2. 2. Use SIMPLE to solve the velocty n the SIMPLE zone. 3. Transfer the nformaton on x = l 1 from the SIMPLE zone to the LBM zone. 4. Use the LBM to solve the velocty n the LBM zone. 5. Transfer the nformaton on l 2 from LBM zone to SIMPLE zone. 6. Go back to step 2 untl the velocty s converged. 3.3 Two schemes results comparson u x be (72) (a) SIMPLE (b) LBM (c) Hybrd method 1 (d) Hybrd method 2 Fgure 11 Streamlnes at Re = 100 [36]

Hybrd LBM-FVM and LBM-MCM Methods for Flud Flow and Heat Transfer Smulaton 15 The ld drven flow s wdely used as a benchmark soluton to test the accuracy of a numercal method. To assess the hybrd methods, the results of Ref. [35] are used for comparson. The ld-drven flow s solved for three dfferent Reynolds numbers at 100, 400, and 1000, respectvely. Pure SIMPLE wth QUICK scheme, LBM wth nonequlbrum extrapolaton boundary method, and the two methods combnng SIMPLE and LBM are appled to solve ths problem. As dscussed above, all these four methods use the same grd of 160 160, whle the grd n the message passng zone n the hybrd method s 3 160. Fgures 11 to 13 show the streamlnes at three Reynolds numbers obtaned from the three methods, whle Fg. 14 and 15 are the horzontal velocty profles n the mddle of the x-drecton and vertcal velocty profles n the mddle of the y-drecton comparng wth that n Ref. [35]. It should be ponted out that the veloctes n the reference need to multply by 0.1 because the non-dmensonal processes are dfferent. (a) SIMPLE (b) LBM (c) Hybrd method 1 (d) Hybrd method 2 Fgure 12 Streamlnes at Re = 400 [36] The streamlnes obtaned from the combned method are hghly smlar to that obtaned from pure SIMPLE and pure LBM as shown n the Fgs. 11 to 13. The postons of the centers of the prmary vortces are (0.6125, 0.7375) by SIMPLE, (0.61875, 0.74375) by LBM, and (0.6172, 0.7344) n Ref. [35] at Re=100. And these three locatons from these three sources are (0.5500, 0.60625), (0.55625, 0.6125), and (0.5547, 0.6055) for Re=400, and (0.5250, 0.55625), (0.53125, 0.56875), and (0.5313, 0.5625) for Re=1000. The dfferences of the locatons the centers of the prmary vortces are nsgnfcant n three cases from SIMPLE and LBM. And the streamlnes got from the two methods are hghly smlar In addton to that, the horzontal velocty profles n the mddle of the x-drecton and vertcal velocty profles n the mddle of the y-drecton obtaned by SIMPLE and LBM are very close to that n reference as shown n Fgs. 14 and 15. In other words, the SIMPLE and LBM used n ths secton are relable. Due to ths, the stream lnes obtaned from SIMPLE can be treated as standard results. Therefore, t can be concluded that the SIMPLE and LBM used n ths secton are relable and the accuracy of the combne method only depends on the soluton of the nterface tself. 15

16 Zheng L, Mo Yang and Yuwen Zhang (a) SIMPLE (b) LBM (c) Hybrd method 1 (d) Hybrd method 2 Fgure 13 Streamlnes at Re = 1000 [36] (a) Re = 100 (b) Re = 400 (c) Re = 1000 Fgure 14 Horzontal velocty profles (a) Re = 100 (b) Re = 400 (c) Re = 1000 Fgure 15 Vertcal velocty profles

Hybrd LBM-FVM and LBM-MCM Methods for Flud Flow and Heat Transfer Smulaton 17 On the other hand, the streamlnes obtaned from the hybrd method 1 and hybrd method 2 for Reynolds numbers at 100, 400 and 1000 shown n the Fgs. 11 to 13 are almost the same as those obtaned from SIMPLE and LBM. The postons of the centers of the prmary vortces obtaned from the hybrd method for three dfferent Reynolds numbers are (0.6125, 07375), (0.5547, 0.6055) and (0.5313, 0.5625) for hybrd method 1 and (0.6000, 071875), (0.5750, 0.60625) and (0.54375, 0.56875) for hybrd method 2 respectvely. Thus, the dfferences from the reference are stll at the same level as that from pure SIMPLE and LBM. The hybrd method can reach the same flow pattern as the other two methods. Especally the results at the message passng zone dd not show any nstablty. When regardng the detals of the flud feld, these two hybrd methods show dfferent accuraces n dfferent Reynolds numbers. The horzontal velocty profles n the mddle of the x-drecton and vertcal velocty profles n the mddle of the y-drecton obtaned by hybrd method 1 satsfy the result from Ref. [35] for Reynolds number equals 100 and 400. Meanwhle, the horzontal velocty profles n the mddle of the x-drecton and vertcal velocty profles n the mddle of the y-drecton obtaned by hybrd method 2 satsfy the result from Ref. [35] for Reynolds number equals 400 and 1000. Nonequlbrum extrapolaton scheme n hybrd method 1 obtan the unknown densty dstrbuton by assumng the nonequlbrum part of the densty dstrbuton equals to that on the nearby nner nodes. So the bgger the velocty gradent s, the worse the accuracy s. By contrast, the fnte-dfference velocty gradent method n hybrd method 2 obtans the he unknown densty dstrbuton by relatng the nonequlbrum part of the densty dstrbuton wth the velocty dstrbuton. So the bgger the velocty gradent s, the worse the accuracy s. Snce the velocty gradent on the nterface wll become more vald wth the ncreasng of Reynolds number. Hybrd method 1 s sutable for the case when Reynolds number s low whle hybrd method 2 s sutable for the hgh Reynolds number case. For the ncompressble problem, LBM needs several tmes more computatonal tme than that of the SIMPLE whle LBM saves a great deal of computatonal tme n the complex flud flow problems [37]. The total computatonal tme of any hybrd method always depends on the computatonal tme of the slower one. The extra tme consumpton of message passng can be neglected comparng wth the total computatonal tme. Thus, the computatonal tme n the ld-drven flow depends on that n the LBM zone when these two zones have the same grds. The computatonal effcency of the hybrd methods are between those of SIMPLE and LBM. The man purpose to solve the ld-drven flow wth the hybrd methods whch reach the same accuracy wth more tme consumng comparng wth SIMPLE s to certfy these two hybrd methods can buld a relaton between LBM and FVM to solve the flud flow problem together. These hybrd methods have the further to save tme wth the same accuracy for the problem ncludng several parts that can take advantages of both SIMPLE and LBM n ther subdomans. 4. A COUPLED LATTICE BOLTZMANN AND FINITE VOLUME METHOD In Secton 3, frst strategy to fulfll LBM-FVM multscale method for flud flow smulaton s dscussed. Heat transfer plays an mportant role n many multscale problems. A coupled LBM-FVM method for flud flow and heat transfer s proposed n ths secton. After takng heat transfer n consderaton, three more settngs are needed n LBM-FVM hybrd method comparng wth that n last chapter: temperature nformaton transfer between subdomans; temperature nterpolaton due to the dfferences between FVM and LBM n computatonal nodes locatons; densty nformaton transfer between subdomans regardng temperature felds are hghly related wth densty. Nonequlbrum extrapolaton scheme has been proved to be vald n combng LBM and FVM for the low speed flud flow smulaton n last chapter. Nature convecton n ths chapter satsfes ths speed requrement. So nonequlbrum extrapolaton scheme s employed to transfer the velocty and temperature nformaton n ths chapter. Natural convectons n a squared enclosure wth dfferent Raylegh numbers are solved usng the coupled method and the results are compared wth those obtaned from pure LBM and pure FVM for valdaton of the combned method. 17

18 Zheng L, Mo Yang and Yuwen Zhang 4.1 Combned LBM-FVM method for flud flow and heat transfer problem 4.1.1 Problem statement Natural convecton of ncompressble flud n a squared enclosure as shown n Fg. 16 s used to test the coupled method. For the velocty feld, non-slp condton s appled to all boundares. The left boundary s kept at a constant temperature T h whle the rght boundary has a lower constant temperature of T l. The top and bottom boundares are adabatc. Applyng Boussnesq assumpton, the problem can be descrbed by the followng governng equatons: Fgure 16 Physcal model of the natural convecton problem u v 0 x y 2 2 u u u p u u u v 2 2 t x y x x y t x y y x y 2 2 v v v p v v u v g 2 2 T Tl 2 2 T T T T T cp u v k 2 2 t x y x y (73) (74) (75) (76) Equatons (73) (76) are subject to the followng boundary and ntal condtons: x 0, u0, v 0, T Th (77) x H, u 0, v 0, T Tl (78) y 0, u0, v 0, T / y 0 (79) y H, u 0, v 0, T / y 0 (80) Regardng Eq. (18) n the vew of Chapman-Enskog expanson, Eq. (76) can be obtaned from the D2Q9 model n LBM when effectve gravty acceleraton G s defned as: G T T l g (81) where s the volume expanson coeffcent of the flud.

Hybrd LBM-FVM and LBM-MCM Methods for Flud Flow and Heat Transfer Smulaton 19 The SIMPLE algorthm wth QUICK scheme [6] s employed to solve Eqs. (73) - (76). Prandtl number, Pr, and Raylegh number, Ra, are the two non-dmensonal parameters governng the natural convecton. Pr= (82) 3 g T T H Pr For LBM Mach number, Ma, s needed: Ma u h l Ra = (83) 2 c (84) cs where c g Th Tl H. Snce the natural convecton n consderaton s ncompressble, Ma can be any number n the ncompressble regon. u s the speed of sound that equals Applyng the followng non-dmensonal varables x y u v X, Y, U, V H H 3c 3c t 3 cs T T p,, P H T T 3 c l 2 h l s s s (85) to Eqs. (73)-(80), the dmensonless governng equatons are obtaned: U V X Y 0 (86) Pr U V Ma X Y X 3Ra X Y 2 2 U U U P U U 2 2 2 2 2 V V V P Pr V V Ma U V Ma 2 2 X Y Y 3Ra X Y 3 U V Ma X Y 3RaPr X Y 2 2 1 2 2 (87) (88) (89) For the heat transfer at the left boundary, Nusselt number, Nu, can be obtaned by the nondmensonal temperature gradent at the surface: Nu (90) X X 0 whch reflects the rato of convecton to the conducton heat transfer across the wall. 19

20 Zheng L, Mo Yang and Yuwen Zhang 4.1.2 Descrpton of the Coupled Method (a) Dvded vertcally (b) Dvded horzontally Fgure 17 Computatonal domans for LBM and FVM Ths coupled method s desgned to solve a sngle problem wth FVM and LBM smultaneously. The computatonal doman s dvded nto LBM and FVM zones, and there s a publc area between these two zones. The artfcal boundary of FVM zone s the nner nodes of LBM zone whle the LBM artfcal boundary s nsde the FVM zone. Two knds of geometry settngs are appled to test ths coupled method shown n Fg. 17. To fulfll the coupled method, the nformaton on the artfcal boundary needs to be obtaned from the other subdoman. For the FVM zone, the velocty and temperature on the artfcal boundary are needed from LBM zone. Densty s not needed because FVM s solvng the ncompressble flow. The pressure on that boundary can be obtaned drectly from the FVM zone tself [6]. On the other hand, LBM needs the velocty, temperature and the densty nformaton of the artfcal boundary from the FVM zone. It s not straghtforward to transfer the densty from an ncompressble FVM zone to the compressble LBM zone. The average densty n the message passng zone, ρ 0, s calculated n the message LBM zone, and the FVM zone provdes the average pressure n L the message passng zone, p. Meanwhle, the pressure on the LBM artfcal boundary, p, can be obtaned S L S from the FVM zone pressure, p. It s shown that there s very small dfference between p and p n the message passng zone. But ths small dfference leads evdent error when I tred to fulfll ths combne method. It s also found that the pressure gradents dfferences from the two methods n the message passng zone s not evdent ether, Then the dfference between p L and p s smlar as that between p L S L and p where p s average pressure n the message passng zone calculated by the LBM zone results. Therefore t s reasonable to assume that L L S p p p p (91) Regardng the relaton between pressure and densty n LBM, t can be obtaned that Then the unknown L 2 2 S cs 0cs p p (92) L can be obtaned that L p S 0 2 0cs p 1 (93) The densty nformaton on the artfcal boundary of LBM can be obtaned by the result of the FVM result. Staggered grd s appled to SIMPLE algorthm.

Hybrd LBM-FVM and LBM-MCM Methods for Flud Flow and Heat Transfer Smulaton 21 Fgure 18 Varables locatons Fgure 18 shows the locatons of varables n a control volume for LBM and FVM. The varables n LBM all locates on the corner of the control volume whle velocty, pressure and temperature n FVM have dfferent locatons n the control volume. Central dfference s appled to message passng processes due to the varable locaton dfferences. After transferrng the nformaton from each other, the FVM and LBM zones need to be solved ndependently for each tme step. For the FVM zone, temperatures and veloctes on the four boundares are known so that the soluton procedure s straghtforward. There are several choces for the LBM boundary condtons. Nonequlbrum extrapolaton scheme s appled to both velocty and temperature felds n the solvng process. Assumng x b s the boundary node and x f s ts nearby nner mode, the densty and energy dstrbuton at the artfcal boundary are: eq eq f ( x,) t f ( x,) t f ( x,) t f ( x,) t (94) b b f f eq eq g ( x,) t g ( x,) t g ( x,) t g ( x,) t (95) b b f f The temperatures on the boundares are known n every tme step, Eq. (95) can be appled to the thermal boundary condtons for Eq. (93). For the velocty feld, the boundary densty s only known on the artfcal boundary. It s common to approxmate the densty on the fxed boundary by ( x, t) ( x, t) (96) b f 4.2 Two geometry results comparson Natural convecton n a squared enclosure s solved for three dfferent Raylegh numbers at 10 4, 10 5 and 10 6 whle the Prantl number s kept at 0.71. Pure LBM and pure FVM are reported n the lterature to be sutable for the natural convecton n a cavty. Thus, these two methods are appled to solve the test cases. If the LBM results agree wth the FVM results, t can be concluded that these results can be used as standard results for comparson. It can also verfy that the codes for the two subdomans are relable n the coupled method. Then only the message passng method between the two subdomans affects the results from the coupled methods. Two coupled methods wth dfferent geometry settngs are appled. When the doman s dvded vertcally, t s referred to as Coupled Method 1. And the Coupled Method 2 dvdes the doman horzontally as shown n Fg. 17. Temperature feld, streamlne and Nusselt number on the left wall obtaned from these four methods are compared for the three cases. Nondmensonal varables defned n Eq. (85) are appled n the comparsons. Fgures 19 and 20 show comparsons of temperature feld and streamlnes obtaned by dfferent methods for the case that Raylegh number s 10 4. It s obvous that natural convecton has domnated the heat transfer process and there s a stream lne vertex near the center of the cavty. Temperature felds and streamlnes obtaned from pure LBM and pure FVM agree wth each other well as shown n Fg. 19 and 20. In addton, there s not any notceable dfference between the results obtaned from coupled methods 1 and 2 and the results of coupled methods agreed wth that from the pure FVM and pure LBM very well. 21

22 Zheng L, Mo Yang and Yuwen Zhang a. FVM b. LBM c. Coupled method 1 d. Coupled method 2 Fgure 19 Temperature felds at Ra=10 4 [38] a. FVM b. LBM c. Coupled method 1 d. Coupled method 2 Fgure 20 Nusselt numbers at Ra=10 4 [38] Fgure 21 shows the Nusselt number at heated wall along the vertcal drecton of the enclosure obtaned from dfferent methods. There s a lttle dfference between the Nusselt numbers obtaned from pure LBM and pure FVM. The cause of ths dfference s that FVM s based on the ncompressble flud assumpton whle LBM

Hybrd LBM-FVM and LBM-MCM Methods for Flud Flow and Heat Transfer Smulaton 23 s based on compressble flud assumpton. Meanwhle the Nusselt number tendences of the two coupled methods are very close to that of the two pure methods. Fgure 21 Nusselt numbers at Ra=10 4 [38] a. FVM b. LBM c. Coupled method 1 d. Coupled method 2 Fgure 22 Temperature felds at Ra=10 5 [38] 23

24 Zheng L, Mo Yang and Yuwen Zhang a. FVM b. LBM c. Coupled method 1 d. Coupled method 2 Fgure 23 Streamlnes at Ra=10 5 [38] Fgure 24 Nusselt numbers at Ra=10 5 [38] Fgures 22 and 23 show ther temperature feld and streamlnes obtaned by dfferent methods for the case that Raylegh number s 10 5. It can be see that the pure FVM and pure LBM reach smlar temperature felds and streamlnes. The convecton effect becomes more pronounced as the Raylegh number ncreases. Two vertexes appear and the temperature gradents near the vertcal boundary ncrease. The two coupled method results stll agree very well wth that n the pure methods as shown n Fgs. 22 and 23. Meanwhle, Fg. 24 shows that the dfference between Nusselt numbers obtaned from pure FVM and pure LBM s larger than that n Fg. 21; but the largest dfference s stll round 2%. The Nusselt numbers from the two coupled methods are closer to the results of pure LBM than that of the pure FVM. Snce both coupled methods 1 and 2 have half regons wth LBM that do not have ncompressble flud assumpton, the flud n the entre computatonal doman of the coupled methods can be consdered as compressble. The Nusselt number dfferences between the two coupled methods are not larger than that between the two pure methods.

Hybrd LBM-FVM and LBM-MCM Methods for Flud Flow and Heat Transfer Smulaton 25 a. FVM b. LBM c. Coupled method 1 d. Coupled method 2 Fgure 25 Temperature felds at Ra=10 6 [38] a. FVM b. LBM c. Coupled method 1 d. Coupled method 2 Fgure 26 Streamlnes at Ra=10 6 [38] Convecton contnues to become stronger when Raylegh number s ncreased to 10 6. Fgures 25 and 26 show that all four methods yeld the smlar temperature felds and streamlnes. There are stll two ndependent stream lne vertexes that are closer to the vertcal boundares; ths ndcates a stronger convecton effect comparng wth the results when Raylegh number s 10 5. As for the Nusselt number, Fg. 27 shows that the results from the coupled methods 1 and 2 are very close to that from the pure LBM. And the dfference between the two coupled methods and pure FVM s stll acceptable. 25

26 Zheng L, Mo Yang and Yuwen Zhang Fgure 27 Nusselt numbers at Ra=10 6 [38] Therefore, the results obtaned from the four methods agreed wth each other very well at dfferent Raylegh numbers of 10 4, 10 5 and 10 6. The geometrc settngs do not affect the accuracy of the coupled method. The results of ths work demonstrated that the coupled method s relable to solve natural convecton problems. 5. A HYBRID LATTICE BOLTZMANN AND FINITE VOLUME METHOD In Sectons 3 and 4, LBM-FVM multscale method was proposed to solve flud flow and heat transfer problems. The whole doman s dvded nto two subdomans and these subdoman are solved wth LBM and FVM respectvely. Ths strategy of hybrd method s sutable for smulatng problems nvolved wth multple length scales. In another strategy of hybrd LBM-FVM method, velocty, pressure and densty are obtaned from the densty dstrbuton solved by LBM, whle the temperature feld can be obtaned drectly usng FVM based on the other solved macroscopc varables from LBM. Ths s a promsng numercal method snce the man advantages of LBM les n obtanng velocty feld. Two addton settngs are needed to fulfll ths LBM-FVM hybrd method: the varable locatons n LBM and FVM are dfferent and proper dfference s needed n the smulaton process; FVM and LBM are dfferent tme scale methods and tme steps are set dfferent n LBM and FVM. Pure thermal LBM, pure FVM, and hybrd LBM-FVM are employed to solve the natural convecton n a fxed-boundary cavty wth dfferent Raylegh numbers. The results are compared wth reference ones for valdaton. 5.1 Coupled LBM-FVM approach The velocty feld s obtaned by the LBM whle FVM solves the temperature feld n the hybrd method. They have been ncluded n Secton 2. Dfferent from the multscale method n Chapters 3 and 4, the LBM-FVM hybrd method n ths chapter apply LBM and FVM to the whole computng doman. The locatons for the temperature and velocty n the control volume are shown n Fg. 28. A corrected method to solve convectve-dffuson equaton based on SIMPLE was proposed n reference [39]. Based on ths result, a LBM-FVM hybrd method s proposed n ths chapter. The temperature feld can be solved by energy balance n the control volume after the velocty feld s obtaned from the LBM. Assume the grd n LBM s 2NX 2NY after mesh ndependent testng for the LBM. In the natural convecton cases solved n ths secton, 160 160 s the selected mesh. It s qute straghtforward to assume that the grd for the temperature feld s the same as that n the velocty feld shown n Fg. 29. The veloctes on the faces of control volume that are needed n the FVM are obtaned by the nterpolatng of the LBM results. Fgure 30 shows another method to set the grd for FVM that NX NY grd s appled to FVM for the temperature feld.

Hybrd LBM-FVM and LBM-MCM Methods for Flud Flow and Heat Transfer Smulaton 27 The dashed lnes show the new control volume for the temperature feld. The obtaned velocty results can be appled to FVM drectly wthout nterpolaton. Then the temperature on the computng nodes of can be obtaned by the nterpolaton of the FVM results for these two grds settng method, and they can reach the same results after comparng the results. Snce the computatonal tme for the temperature feld n the second method s much shorter than that of the frst one, the second method s preferred n the hybrd method. Fgure 28 Control volume n FVM Fgure 29 Grd settng 1 Fgure 30 Grd settng 2 27

28 Zheng L, Mo Yang and Yuwen Zhang Non-dmensonal process n FVM s the same as that n Secton 4 and 2NX 2NY s employed n LBM. Correspondently, tme steps n FVM and LBM are H / 3c s and H / 2NX 3c s respectvely. Therefore, n every FVM tme step, LBM runs 2NX tme steps. 5.2 Results and dscussons Natural convecton n squared enclosure s wdely used as a benchmark problem for valdaton of the numercal methods. Pure LBM, pure FVM and the hybrd method are appled to smulate the natural convecton ndependently for three dfferent Raylegh numbers at 10 4, 10 5 and 10 6, whle the Prantl number s 0.71. It s reported by several groups ndependently that pure LBM and pure FVM are both sutable for the natural convecton n ths range [40, 41]. So when the results of these two methods agree wth each other well, they can be treated as the benchmark results for comparson. Under the Boussnesq assumpton, varaton of densty wth temperature s consdered n the buoyancy force. In the gravtatonal feld, there s a tendency that lghter flud goes upward and the heaver one goes downward. The boundary layer of a vertcal wall wth hgh constant temperature turns thcker wth the ncreasng of the heght, whle the boundary layer of a vertcal wall wth low constant temperature becomes thcker wth the decreasng of the heght. So the flud near the left wall moves up but the flud near the rght wall moves down. The closer to the left boundary, the hotter the flud s snce the left boundary has the hghest temperature. Therefore the upward movng flud near the left moves towards rght near the top of the cavty and the downward movng flud moves towards left near the bottom of the cavty. And the two flows combne together as a vortex. For the case that Raylegh number s equal to 10 4, the streamlnes and temperature felds obtaned by the pure FVM and pure LBM agreed wth each other well as shown n Fgs. 31 and 32, respectvely. It s shown that the natural convecton has governed the heat transfer process. There s a streamlne vertex n the center of the cavty and the temperature feld also shows the character of the convecton. The streamlnes and temperature feld obtaned from the hybrd method matches the results from the other two methods very well. a. FVM b. LBM c. Hybrd method Fgure 31 Streamlnes comparson at Ra=10 4 [42] a. FVM b. LBM c. Hybrd method Fgure 32 Temperature feld comparson at Ra=10 4 [42]

Hybrd LBM-FVM and LBM-MCM Methods for Flud Flow and Heat Transfer Smulaton 29 The natural convecton effect becomes more obvous when the Raylegh number s ncreased to 10 5. Fgures 33 and 34 show that the streamlnes and temperature felds from the three methods agree wth each other very well. There are two vertexes n the flow feld and the temperature gradent at the left and rght boundares s larger than that for the case of Ra = 10 4.The dfferences of the streamlnes and temperature felds among the three methods are stll not notceable when the Raylegh number grows to 10 6 as shown n Fgs. 35 and 36. The two streamlne vertexes locate farther from the cavty center and the temperature changes turn closer to the left and rght boundares. So the streamlnes and temperature felds of the hybrd method agree wth those obtaned by the other two methods for the three Raylegh numbers. a. FVM b. LBM c. Hybrd method Fgure 33 Streamlnes comparson at Ra=10 5 [42] a. FVM b. LBM c. Hybrd method Fgure 34 Temperature feld comparson at Ra=10 5 [42] a. FVM b. LBM c. Hybrd method Fgure 35 Streamlnes comparson at Ra=10 6 [42] Comparson of Nusselt number obtaned from the three methods s shown n the Fg. 5-10 for dfferent cases. It can be seen that the results agree wth each other very well n the three cases. The man dfference locates at the maxmum Nusselt number part, whch s more obvous as the Raylegh number ncreases. Meanwhle Table 29

30 Zheng L, Mo Yang and Yuwen Zhang 1 and 2 gve the comparsons of the maxmum Nusselt number and ts locaton for the three methods and the results from Ref. [41]. For the maxmum Nusselt number, the hybrd result s closest to the reference result n the frst two cases whle the pure LBM has the closest result when Raylegh number equals 10 6. The hybrd method result s closest to the reference result n all the three cases. The largest dfference between all the three methods results and the reference result s stll under 3% whch s acceptable. a. FVM b. LBM c. Hybrd method Fgure 36 Temperature feld comparson at Ra=10 6 [42] Table 1 Comparson of the maxmum Nusselt numbers [42] Ra=10 4 Ra=10 5 Ra=10 6 Hybrd method 3.5324 7.6970 17.3354 FVM 3.5486 7.8382 17.8399 LBM 3.5481 7.7907 17.5133 Reference [41] 3.5309 7.7201 17.5360 a. Ra=10 4 b. Ra=10 5 c. Ra=10 6 Fgure 37 Comparson of Nusselt numbers [42] Table 2 Comparson of the locatons of maxmum Nusselt numbers Ra=10 4 Ra=10 5 Ra=10 6 Hybrd method 0.1437 0.0812 0.0437 FVM 0.1375 0.0813 0.0334 LBM 0.1375 0.0750 0.0313 Reference [41] 0.1439 0.0820 0.0392 The streamlnes and temperature felds obtaned by the three methods agree wth each other very well for dfferent Raylegh numbers at 10 4, 10 5 and 10 6. The hybrd method also has a good accuracy for the Nusselt number when comparng wth the reference and the two pure methods results. Thus, the hybrd LBM-FVM s relable for the natural convecton smulaton.

Hybrd LBM-FVM and LBM-MCM Methods for Flud Flow and Heat Transfer Smulaton 31 6. A COMBINED LATTICE BOLTZMANN AND MONTE CARLO METHOD To take advantages of both LBM and FVM, a LBM-FVM hybrd method was proposed and verfed n Secton 5. Monte Carlo method has vald advantages n solvng some heat transfer problems. LBM-MCM combned method s proposed for flud flow and heat transfer smulaton n ths secton. Velocty feld s solved wth LBM whle temperature s obtaned usng MCM. Natural convecton wth dfferent Raylegh numbers are solved to verfy ths numercal method. 6.1 Monte Carlo Method (MCM) for heat transfer A rectangular grd system of mesh sze x y s selected for the two-dmensonal computng doman shown n Fg. 38. The nner computng node, j relates to ts surroundng nodes by: T P T P T P T P T (97), j x 1, j x 1, j y, j1 y, j1 where the possbltes P x, Px, Py and Py are postve; ther sum has to be 1. Fgure 38 Computng grd For the conducton heat transfer problem, these possbltes are: y/ x Px Px 2 y/ xx/ y x/ y Py Py 2 y/ xx/ y y/ xuy y/ x Px Px D D x/ yvx x/ y Py Py D D (98) (99) where D 2 y/ xx/ y uyv x (100) For the convecton heat transfer problem, the horzontal velocty u and vertcal velocty v have effects on Px, Px, Py and Py. The possbltes should be modfed to the followng equaton: 31

32 Zheng L, Mo Yang and Yuwen Zhang The statstcal procedure MCM uses random walkers to solve the heat transfer problem [43]. A random walker locates at node, jat begnnng and a random number RN s chosen n the unformly dstrbuton set from 0 to 1. Ths random walker wll change ts poston by the followng rules:,, 1, 1, f 0 RN Px from, j to 1, j f Px RN Px Py from j to j f Px Py RN Px PyP x from j to j f 1Py 1 RN 1 from, j to, j 1 Once the random walker has completed ts frst step, the procedure contnues for the second step. Ths process goes on tll that random walker reaches the boundary. Then the boundary condton T w 1 s recorded for the nner node, j. Ths process s repeated N -1 tmes and the recorded boundary condtons are T 2 to w T N. Wth these N results, the MCM estmaton for, n 1 T j can be expressed as: (101) N 1 T, j Tw n (102) N Then all the nner computng nodes can be obtaned by ths method. The treatments to dfferent knd boundares can be found n reference [43]. 6.2 Coupled LBM-MCM method A combned LBM-MCM method s desgned to solve a flud flow and heat transfer problem wth LBM and MCM smultaneously. Same grd system s used n both LBM and MCM. The velocty feld s obtaned by LBM and MCM solves the temperature feld. Fgure 39 shows the flowchart for the combned method. For the natural convecton problem n consderaton, temperature and velocty have effects on each other. Therefore ths method needs to run LBM and MCM n consequence for each step. After settng the ntal condton, LBM solves the velocty feld wth the ntal temperature feld. Then the temperature feld s obtaned by MCM wth the LBM velocty result. Ths temperature can be appled to LBM n next step. Ths process s repeated tll converged results are reached. w Fgure 39 Flowchart for combned LBM-MCM

Hybrd LBM-FVM and LBM-MCM Methods for Flud Flow and Heat Transfer Smulaton 33 6.3 Numercal methods valdaton 6.3.1 Valdaton of MCM In Secton 3, LBM has been proved to be vald n solvng ncompressble flud flow problems. In ths secton, MCM s verfed for heat transfer smulaton. As dscussed n secton 6.1, MCM has the same approach n solvng conducton and convecton heat transfer problems. A pure conducton problem s solved to test the MCM. In Fg. 40, the left, rght and bottom of the two-dmensonal doman are kept at T c and the top boundary has a hgher temperature of T h. It s assumed that the thermal conductvty s ndependent from the temperature. The energy equaton for ths problem s Fgure 40 Two-dmensonal steady state heat conducton 2 2 T T 0, 0 x H, 0 y H 2 2 x y (103) wth the followng boundary condtons: T T, c x 0 or H,0 y H (104) T T, c y 0, 0 x H (105) T T, h y H, 0 x H (106) Defnng the followng non-dmensonal varables x y X, Y, H H T T c Th Tc (107) Equatons. (103)-(106) become 2 2 0, 0 X 1, 0 Y 1 2 2 X Y (108) 33 wth the followng boundary condtons: 0, X 0 or1, 0 Y 1 (109) 0, Y 0, 0 X 1 (110) 1, Y 1, 0 X 1 (111) Ths problem can be solved analytcally by separaton of varable method [44].

34 Zheng L, Mo Yang and Yuwen Zhang n1 n1 2 1 1snh( n Y) sn( nx ) (112) n snh( n) Fgure 41 Conducton temperature feld comparson [45] Fgure 41 shows the comparson of the analytcal and MCM temperature felds. As dscussed n secton 6.2, MCM s a statstcal method to smulate the heat transfer process. Its results are based on a large number of random walkers results. The MCM sothermal lnes are not that smooth due to ts procedure nature. These two methods sothermal lnes agree wth each other well. Therefore, MCM n ths secton s vald for the heat transfer problem. 6.3.2 Natural convecton n rectangular enclosure 4 In case 1, Fgure 42 shows the streamlnes for Case 1 that Ra and Pr are 10 and 0.71, respectvely. There s a vortex n the cavty due to the convecton effect. Fgure 43 s the temperature feld obtaned from the combned method. Both streamlnes and temperature feld agree wth the benchmark solutons very well except the unsmoothness of some sothermal lnes. As dscussed n the pure conducton problem, the unsmoothness s led by the nature of MCM and ts effect to the streamlnes s nsgnfcant. Meanwhle, Table 1 summarze maxmum Nusselt number Nu max, maxmum Nusselt number locaton Y Nu and average Nusselt number Nu max ave obtaned from the LBM-MCM and Reference (Davs, 1983). It can be seen that the combned LBM-MCM method has a good agreement wth benchmark standard solutons for these three parameters for Case 1. Fgure 42 Streamlnes n case 1 [45]

Hybrd LBM-FVM and LBM-MCM Methods for Flud Flow and Heat Transfer Smulaton 35 Fgure 43 Temperature feld n Case 1 [45] 5 Case 2 s also studed when Ra grows to 10 and Pr s kept 0.71. Convecton plays a more mportant rule due to the ncreased Ra. Two vortexes locate n the cavty are shown n Fg. 44. And the temperature feld n Fg. 45 also ndcates a pronounced convecton effect. Table 3 also showed the comparson of Nu max, Y Nu and Nu max ave obtaned from the present LBM-MCM wth the benchmark solutons for ths case. The streamlnes, temperature fled and Nusselt number results n combned LBM-MCM method all agree wth the benchmark solutons well. Smlar to Case 1, the unsmooth sothermal lnes effect s nsgnfcant. Therefore the combned LBM-MCM method can gve good predctons to these two natural convecton cases. Fgure 44 Streamlnes n case 2 [45] Fgure 45 Temperature feld n Case 2 [45] The streamlnes temperature feld and Nusselt number obtaned from the present LBM-MCM approach agree wth that of the benchmark solutons well. Thus, the combned LBM-MCM s relable for the natural convecton smulaton. 35