A lattice Boltzmann approach for solving scalar transport equations

Transcription

1 Phl. Trans. R. Soc. A (2011) 369, do: /rsta A lattce Boltzmann approach for solvng scalar transport equatons BY RAOYANG ZHANG*, HONGLI FAN AND HUDONG CHEN Exa Corporaton, 55 Network Drve, Burlngton, MA 01803, USA A lattce Boltzmann (LB) approach s presented for solvng scalar transport equatons. In addton to the standard LB for flud flow, a second set of dstrbuton functons s ntroduced for transport scalars. Ths LB approach fully recovers the macroscopc scalar transport equaton satsfyng an exact conservaton law. It s numercally stable and scalar dffusvty does not have a Courant Fredrchs Lewy-lke stablty upper lmt. Wth a suffcent lattce sotropy, numercal solutons are ndependent of grd orentatons. A generalzed boundary condton for scalars on arbtrary geometry s also realzed by a precse control of surface scalar flux. Numercal results of varous benchmarks are presented to demonstrate the accuracy, effcency and robustness of the approach. Keywords: lattce Boltzmann; scalar transport; knetc theory 1. Introducton The lattce Boltzmann method (LBM) has now become a promsng alternatve numercal method to conventonal computatonal flud dynamcs (CFD) for smulatng varous flud flows. Unlke conventonal CFD based on drectly solvng macroscopc contnuum equatons, the LBM starts from a mesoscopc knetc (.e. Boltzmann) equaton to determne macroscopc flud dynamcs [1]. Its knetc nature brngs many advantages over the conventonal methods, such as easy handlng of complex geometres, parallel computaton, low numercal dsspaton and effcent smulatons of mult-phase flows wth thermodynamc phase transtons (cf. [1 4]). It has been appled extensvely n research and real engneerng applcatons. In many flud flow smulatons, addtonal scalar quanttes usually need to be solved; for example, temperature varaton n heat transfer problems. Because of lack of a global H-theorem [5], most LB approaches wth full energy conservaton encounter challengng numercal nstablty ssues unless hghorder LB models are appled [6]. Smultaneous realzaton of mass, momentum and temperature boundary condtons (BCs) s also extremely challengng. To overcome these dffcultes, one soluton s to solve an addtonal scalar equaton for temperature [4]. *Author for correspondence ([email protected]). One contrbuton of 26 to a Theme Issue Dscrete smulaton of flud dynamcs: methods Ths journal s 2011 The Royal Socety

2 A lattce Boltzmann scalar solver 2265 Generally, there are two types of methods to solve scalar transport equatons along wth a basc LB flud solver, one s the fnte dfference (FD) method and the other s an LB scalar solver. Because of the Lagrangan nature of the background LB flud solver, explct FD schemes are commonly appled on the same cubc lattce grds for convenence to solve the scalar equaton. There are many explct FD schemes n the lterature. Nonetheless, owng to the very strct Courant Fredrchs Lewy (CFL) constrant, there exsts a stablty upper bound for the normalzed value of dffusvty, and also a mxture of upwnd and central schemes for scalar advecton s usually necessary [7]. A drect consequence of such a treatment s the ncreased numercal dffuson and loss of exact Gallean nvarance. Symmetry of the schemes could also be sacrfced, whch causes grd orentaton dependence of numercal solutons. Furthermore, t s dffcult to accurately calculate scalar gradent and scalar advecton n regons contanng arbtrary geometres due to mssng neghbour nformaton. Thus, near wall numercal accuracy could be sgnfcantly compromsed. There are not many works on LB scalar solvers n the lterature. In 1987, the concept of usng mult-component formulaton was proposed n lattce gas framework [8]. In 1997, Shan [9] used a separate dstrbuton functon to solve the temperature equaton and conducted Raylegh Benard smulatons n two and three dmensons. Later on He et al. [10] proposed a smlar approach to solve nternal energy dstrbuton functon. Peng et al. [11] further extended the nternal energy model for three-dmensonal ncompressble thermal smulatons. One of the advantages of these LB scalar solvers s numercal stablty. Unlke explct FD, LB scalar solvers naturally satsfy the CFL condton. The scalar advecton s exact and there s no need to calculate scalar gradent explctly. Scalar dffusvty does not have an upper lmt. Although the exstng LB scalar solvers possess many advantages, they all fal on a fundamental nvarance condton. That s, f there s no external scalar source, a unform scalar dstrbuton should be mantaned at all tmes no matter the background flud flow. In ths paper, a new LB scalar approach s proposed that addresses the above problem. In addton, a generalzed BC for the scalar on arbtrary geometry, whch s largely mssng n prevous studes, s also presented. Theoretcal detals are descrbed n 2 followed by results for smple but representatve benchmark problems to demonstrate the accuracy, stablty and symmetry of the new LB scalar solver. Conclusons are gven n New lattce Boltzmann scalar solver (a) The flud solver Varous types of LBM could be appled for solvng flud flows, whch serve as the background carrer for scalar transport. For smplcty, the standard D3Q19 Bhatnagar Gross Krook (BGK) model [1] s used n ths study: f (x + c, t + 1) = f (x, t) 1 t (f (x, t) f eq (x, t)) + g (x, t). (2.1) Phl. Trans. R. Soc. A (2011)

3 2266 R. Zhang et al. Here, f (x, t)( = 1,..., 19) s the partcle dstrbuton functon, t s the sngle relaxaton tme and f eq (x, t) s the equlbrum dstrbuton functon wth a thrd-order expanson n flud velocty, ( f eq (x, t) = rw 1 + c u + (c u) 2 T 0 2T0 2 u2 + (c u) 3 2T 0 6T0 3 c u 2T0 2 u ), 2 (2.2) where T 0 = 1/3. The dscrete lattce veloctes c are 0 c = (±1, 0, 0), (0, ±1, 0), (0, 0, ±1) (±1, ±1, 0), (±1, 0, ±1), (0, ±1, ±1) (2.3) wth w 0 = 1/3 for rest partcle, w = 1/18 for states of Cartesan drectons and w = 1/36 for states of b-dagonal drectons. g (x, t) s the external body force term [12]. The hydrodynamc quanttes r and u are moments of the partcle dstrbuton functon r(x, t) = f (x, t) and r(x, t)u(x, t) = c f (x, t). (2.4) (b) The scalar solver In addton to the flud solver, a separate set of dstrbuton functons, T,s ntroduced for scalar transport: ( T (x + c, t + 1) = T(x, t) ) F (x, t), (2.5) t T F (x, t) = c u(x, t) c j f j (x, t)(t j (x, t) T(x, t)) (2.6) r(x, t)t 0 j and T(x, t) = f (x, t)t (x, t). (2.7) r(x, t) T s the scalar dstrbuton functon and T s the scalar beng solved for. t T s the relaxaton tme correspondng to scalar dffusvty. The f, r, T 0 and u are defned n equatons (2.1), (2.2) and (2.4), respectvely. In general, the lattce speed set for scalar dstrbuton does not need to be the same as the lattce speed set for flud dstrbuton because the current scalar solver s an addtonal system attached to the basc flud solver. Dfferent lattce speed set for scalar could be appled as long as the scalar lattce speed set s a subset of the flud lattce speed set. For example, a sx-speed LB model may be used for scalar smulaton when the 19-speed LB model s used for flud smulaton. Snce the 19-speed LB model has a hgher order lattce symmetry than the sx-speed LB model, the same 19-speed lattce model for scalar s used here. Unlke the regular BGK, the so-called BGK regularzed/fltered collson operator form s used here [13,14]. Such a form ensures that only the frst-order non-equlbrum moment contrbutes to scalar dffuson n the hydrodynamc range. All non-equlbrum moments of hgher orders are fltered out by ths collson process. Its drect benefts nclude elmnaton of numercal nose exhbted n BGK and mproved robustness. The scalar T serves as Phl. Trans. R. Soc. A (2011)

4 A lattce Boltzmann scalar solver 2267 ts own equlbrum and no complcated expresson of scalar equlbrum dstrbuton functon s needed. The overall calculaton of collson operator F s rather effcent. It can be shown easly that the collson n equaton (2.5) obeys the scalar conservaton law. Multplyng f (x, t) = f (x + c, t + 1) on both sdes of equaton (2.5) and notcng we have (c u)f rt 0 = f T = ru ru rt 0 = 0, (2.8) f T = rt, (2.9) where T (x, t) denotes the rght-hand sde of equaton (2.5). Hence, the scalar collson operator conserves local rt, whch mples realzaton of local energy conservaton f the scalar s consdered as temperature. Snce T propagates along wth f, the energy dstrbuton E = f T s fully mantaned durng advecton. The global conservaton of rt s therefore acheved. Furthermore, and most notably, ths scheme mantans the exact nvarance on unformty of scalar T. It s straghtforward to see that f T (x, t) = T(x, t) = T = const. everywhere, then F (x, t) 0, and T (x, t + 1) = T everywhere at all later tmes, regardless of the background flow feld. Ths fundamental property s not demonstrated n any prevous LB scalar models. Usng Chapman Enskog expanson, t can be shown that equaton (2.5) recovers the followng second-order macroscopc scalar transport equaton: vrt + V (rut) = V rkvt, (2.10) vt wth k = (t T 1/2)T 0. The unformty nvarance condton ensures that r s outsde of VT. (c) Boundary condton One substantal advantage of the LBM s the capablty of dealng wth complex geometry [1]. In 1998, Chen et al. [15] formulated a general volumetrc LB surface algorthm to acheve frctonless/frctonal BCs on arbtrary geometry. In ther scheme, mass s conserved, both tangental and normal momentum fluxes on the boundary are precsely realzed. The local detaled balance s fully satsfed. Many benchmark studes have well demonstrated the accuracy and robustness of ths scheme [2,3,16]. An adabatc (zero scalar flux) BC on arbtrary geometry for scalar can be derved as a drect extenson of ths scheme. Once the adabatc BC s realzed, a prescrbed fnte flux BC can be accomplshed. Unlke other pont-wse LB, BCs are conducted on a dscretzed set of surface elements. These pece-wse flat surface elements together represent a curved geometry. Durng partcle advecton, each surface element collects ncomng partcles from ts neghbourng flud cells. The ncomng dstrbutons f n, T n are weghted by volume overlappng of paralleleppeds from the underlng surface element wth cells n partcle movng drectons (cf. [15]). After recevng the Phl. Trans. R. Soc. A (2011)

5 2268 R. Zhang et al. ncomng quanttes, the followng surface scalar algorthm s appled to determne the outgong dstrbutons from the surface: where T out = T n f n T n f n T n f out, (2.11) T n = c f n n<0 T n P c n<0 f n P. (2.12) Here, n s the surface normal pontng towards the flud doman and c n < 0, c = c. P ( n c A) s the volume of parallelepped n partcle drecton c assocated wth the surface normal n and area A of a gven surface element, and obvously P = P. Fnally, the outgong dstrbutons are propagated back from the surface element to flud cells accordng to the same surface advecton process as n Chen et al. [15]. It s not dffcult to show that the above surface scalar collson acheves exact zero surface scalar flux. Takng summaton over outgong drectons, the outgong scalar flux s T out P = c n<0 f out c n<0 c n<0 f out T n P c n<0 c n<0 (f n T n f n T n )P. (2.13) Note, P = P and the defnton of T n n equaton (2.12), the second summaton term on the rght-hand sde s zero. In addton, because of the mass flux conservaton c f out n<0 P = c f n n<0 P, the total outgong scalar flux s the same as the total ncomng scalar flux f out T out P = f n T n P. (2.14) Therefore, zero net surface flux (adabatc) BC s fully satsfed on arbtrary geometry. If an external scalar source Q(t) s specfed on the surface, a source term can be drectly added to equaton (2.11) Q(t)P A rc p c P. (2.15) n>0 If the BC has a prescrbed scalar quantty T w (for example, surface temperature), surface heat flux can be calculated accordngly: Q(t) = rc p k(t w T n ). (2.16) 3. Numercal verfcaton Four sets of smulaton results are presented to demonstrate the capablty of the LB scalar solver regardng ts numercal accuracy, stablty, Gallean nvarance, grd orentaton ndependence, etc. Results usng two dfferent second-order FD schemes, van Leer type of flux lmter scheme and drect mxng scheme (mxture of central and frst-order upwnd schemes), are also presented as comparsons. Phl. Trans. R. Soc. A (2011)

6 A lattce Boltzmann scalar solver DT/T x/l Fgure 1. Temperature profles at t = 81. Crcles, LB; trangles, flux lmter; crosses, mxng scheme; sold lne, theory. (a) Shearwave decay The frst test case s a temperature shearwave decay carred by a constant unform flud flow. The ntal temperature dstrbuton s a unform one plus a spatal snusodal varaton wth lattce wavelength L = 16 and magntude d = 6.67% : T(x) = T A (1 + d sn(2px/l)). T A s a constant. The velocty of background mean flow s 0.2 and the thermal dffusvty k s Wth such a low resoluton and k, the numercal stablty and accuracy can be well valdated. For temperature decay wthout background flow, both the LB scalar solver and the FD methods show excellent agreements wth theory. Wth non-zero background mean flow, the LB scalar solver s stll able to accurately compare wth theory. However, the FD results show notceable numercal errors. The temperature profles at lattce tme step 81 are plotted n fgure 1. Numercal dffuson s seen clearly for the flux-lmter FD scheme, whle nether the correct temperature profle nor ts locaton can be mantaned by the mxng FD scheme. (b) Inclned channel wth volume heat source The second test case s a smulaton of temperature dstrbuton n a channel flow wth dfferent lattce orentatons. The channel walls are free-slp, and the flud flow stays unform wth U 0 = 0.2 as a result. The channel wdth s 50 (lattce spacng) and the flow Re s The thermal dffusvty k s The temperature on the wall s fxed at T w = 1/3. A constant volume heat source q = s appled n the bulk flud doman. The flow has perodc BC n the streamwse drecton, whch s easy to realze n lattce algned stuaton. When the channel (lght colour) s tlted as shown n fgure 2, the n and out channel boundares are matched perfectly n coordnate drectons so that the perodc Phl. Trans. R. Soc. A (2011)

7 2270 R. Zhang et al T/T y/h Fgure 2. Temperature dstrbutons across tlted channel wth streamwse perodc BC. Sold lne, LB; dashed lne, flux lmter; dashed-dotted lne, mxng scheme; small crcles, non-nclned. BC s once agan realzed n the streamwse drecton. In order to demonstrate lattce ndependence, we choose the tlted angle to be Lke the frst test case, the temperature dstrbutons usng the LB scalar solver and the two FD schemes match the analytc soluton very well when the channel s lattce algned. However, the results from the FD schemes depart sgnfcantly from theory when the channel s tlted. The smulaton results of temperature dstrbuton across the tlted channel are shown n fgure 2. The LB results are clearly shown to be ndependent of lattce orentatons. The errors from the FD methods are also orgnated from ther fundamental dffculty n dealng wth gradent calculaton on a tlted boundary orentaton, so that addtonal numercal artefacts are ntroduced. Snce the LB scalar partcle advecton s exact wth the BC presented here, t s thus able to acheve a lattce orentaton ndependent scalar evoluton. (c) Temperature propagaton n an nclned channel Owng to lack of neghbour nformaton n a non-lattce algned near wall regon on Cartesan grds, t s extremely dffcult to get accurate estmaton of local gradents, whch s essental for FD based methods. Furthermore, because of strong dependency on upwnd nformaton, the calculaton of scalar advecton can be further compromsed for FD methods. In contrast, the boundary treatment of the LB scalar solver acheves exact local scalar flux conservaton as dscussed above. The scalar advecton n such a near wall regon can be computed accurately. Hgh temperature convecton n a channel tlted by 30 s conducted as a demonstraton. The free-slp and adabatc BCs are enforced at sold walls and flud velocty s constant U 0 = along the channel. The thermal dffusvty k s Intally, the temperature s 1/3 everywhere except for T = 4/9 at the nlet. Then ths temperature front should be convected by the unform background flud flow to downstream locatons at later tmes wthout dstorton. The computed temperature front dstrbutons across the channel at lattce tme Phl. Trans. R. Soc. A (2011)

8 A lattce Boltzmann scalar solver T/T m y/h Fgure 3. Temperature propagaton fronts across tlted channel. Sold lne, LB; dashed lne, flux lmter; dashed-dotted lne, mxng scheme. step 2000 are shown n fgure 3. The temperature front of the LB scalar solver mantans a nearly straght profle. On the other hand, the temperature fronts from the two FD schemes show substantal dstortons n near wall regons. It s also worth mentonng that the LB scalar result shows the thnnest temperature front whch mples that the LB scalar solver has a smaller numercal dffuson. (d) Raylegh Bénard natural convecton Raylegh Bénard (RB) natural convecton s a classcal benchmark for accuracy verfcaton of numercal solvers. It has a smple case set-up but complex physcs phenomena. When Raylegh number Ra exceeds a certan crtcal value, the system experences a transton from no-flow to convecton. Current study s carred out under the Boussnesq approxmaton, n that the buoyancy force actng on the flud s defned as arg(t T m ) where a s the thermal expanson rate, g s gravty, and T m s the average temperature value of the top and bottom boundares. Snce the most unstable wave number s k c = when Ra exceeds the crtcal value Ra c = , the resoluton s used n the study. Pr used here s A set of smulatons wth varous Ra are conducted. Owng to page lmtaton, only Nusselt number Nu versus Ra s presented here. When RB convecton s establshed, the heat transfer between two plates s greatly enhanced. The enhancement of the heat transfer s descrbed by Nu = 1 + u y T H /kdt. Our results are compared wth those of Clever & Busse [17] nfgure 4. As shown, the agreements are very good for Ra less than Our results slghtly underestmate the heat transfer at hgher Ra. Other LB methods also showed a smlar trend [9,10]. Phl. Trans. R. Soc. A (2011)

9 2272 R. Zhang et al. 6 5 Nusselt number Raylegh number Fgure 4. Nusselt number versus Raylegh number. Squares, LB; trangles, Clever & Busse [17]; sold lne, 1.56(Ra/Ra c ) Conclusons A new LB scalar solver ncludng a generalzed BC on arbtrary geometry s presented n ths paper. The algorthm s smple and effcent. The fundamental flaw assocated wth lack of nvarance of unform temperature n the prevous LB scalar models s fully removed. Its numercal stablty, accuracy, symmetry and robustness are demonstrated by some representatve benchmark cases. Fnally we would lke to pont out that ths LB scalar solver can be trvally extended to smulate multple scalars. References 1 Chen, S. & Doolen, G. D Lattce Boltzmann method for flud flows. Annu. Rev. Flud Mech. 30, (do: /annurev.flud ) 2 L, Y., Shock, R., Zhang, R. & Chen, H Numercal study of flow past an mpulsvely started cylnder by lattce Boltzmann method. J. Flud Mech. 519, (do: / S ) 3 Chen, H., Kandasamy, S., Orszag, S., Shock, R., Succ, S. & Yakhot, V Extended Boltzmann knetc equaton for turbulent flows. Scence 301, (do: /scence ) 4 Zhang, R. & Chen, H A lattce Boltzmann method for smulatons of lqud vapor thermal flows. Phys. Rev. E 67, (do: /physreve ) 5 Chen, H. & Texera, C H-theorem and orgns of nstablty n thermal lattce Boltzmann models. Comput. Phys. Commun. 129, (do: /s (00) ) 6 Sebert, D. N., Hegele Jr, L. A. & Phlpp, P. C Lattce Boltzmann equaton lnear stablty analyss: thermal and athermal models. Phys. Rev. E 77, (do: / PhysRevE ) Phl. Trans. R. Soc. A (2011)

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