international journal of heat and mass transferfaculty.missouri.edu/zhangyu/pubs/235_li_3d...

International Journal of Heat and Mass Transfer 94 (2016) 222–238

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International Journal of Heat and Mass Transfer

journal homepage: www.elsevier .com/locate / i jhmt

Lattice Boltzmann method simulation of 3-D natural convectionwith double MRT model

http://dx.doi.org/10.1016/j.ijheatmasstransfer.2015.11.0420017-9310/� 2015 Elsevier Ltd. All rights reserved.

⇑ Corresponding author. Tel.: +1 573 884 6936; fax: +1 573 884 5090.E-mail address: [email protected] (Y. Zhang).

Zheng Li a,b, Mo Yang a, Yuwen Zhang b,⇑aCollege of Energy and Power Engineering, University of Shanghai for Science and Technology, Shanghai 200093, ChinabDepartment of Mechanical and Aerospace Engineering, University of Missouri, Columbia, MO 65211, USA

a r t i c l e i n f o a b s t r a c t

Article history:Received 15 October 2015Received in revised form 13 November 2015Accepted 16 November 2015Available online 8 December 2015

Keywords:Cubic cavity natural convectionLattice Boltzmann methodMultiple relaxation time model

Multiple-relaxation-time model (MRT) has more advantages than the many others approaches in thelattice Boltzmann method (LBM). Three-dimensional double MRT model is proposed for the first timefor fluid flow and heat transfer simulation. Three types of cubic natural convection problems are solvedwith proposed method at various Rayleigh numbers. Two opposite vertical walls on the left and right arekept at different temperatures for all three types, while the remained four walls are either adiabatic orhave linear temperature variations. For the first two types of cubic natural convections that four wallsare either adiabatic or vary linearly, the present results agreed very well with the benchmark solutionsor experimental results in the literature. For the third type of cubic natural convection, the front and backsurfaces has linearly variable temperature while the bottom and top surface are adiabatic. The resultsfrom the third type exhibited more general three-dimensional characters.

� 2015 Elsevier Ltd. All rights reserved.

1. Introduction

Lattice Boltzmann method (LBM) has become an increasinglypopular numerical method in the last three decades. It has beenused to simulate various hydrodynamic systems, such as incom-pressible fluid flow [1,2], porous media flow [3], and melting prob-lem [4,5]. Different from traditional computational fluid dynamics(CFD) approaches, LBM is based on the discrete Boltzmann equa-tion in statistical physics and it has two basic steps: collision stepand streaming steps [6]. LBM has several models, which differsfrom each other with the method to handle the collision step.The most popular one is the Lattice Bhatnagar–Gross–Krook(LBGK) [7,8] that simplifies collision step with a single relaxationtime term. Some shortcomings of LBGK are also apparent thatPrandtl number must be unity when the model is applied to ther-mal fluids and it suffers numerical instability [9,10]. To overcomethis limitation, entropy LBM (ELBM) [11,12], two-relaxation-timemodel (TRT) [13,14] and multiple relaxation time model (MRT)[11,15,16] have been proposed. Luo et al. [17] compared variousLBM models and concluded that MRT was preferred due to itsadvantages in accuracy and numerical stability. This article usesMRT model to simplify the collision step.

Lattice Boltzmann method was only valid to solve fluid flowwhen it was proposed [7]. Fluid flow involved with heat transferproblem is important due to its numerous applications in industryfields. Multispeed approach (MS), hybrid method and doubledistribution functions (DDF) are three common thermal LBMmodels. MS obtains the temperature field by adding more discretevelocities to density distribution [18]; it is limited by numericalinstability and narrow range of temperature variation [19]. Inhybrid method, the velocity field is solved with LBM while temper-ature field is obtained using other numerical methods, such asfinite volume method (FVM) and Monte Carlo method. Li et al.[20,21] solved natural convection and melting problems with ahybrid LBM and FVM method. Hybrid LBM and MCM methodwas proved to be suitable for convection simulation [22]. Besidesdensity distribution, DDF includes an addition distribution to ana-lyze the temperature field. Huber et al. [23] proposed a DDF modelfor coupled diffusions based on MRT. This article employs DDFwith MRT to simulate the heat transfer process.

Natural convection plays an important role in many industryfields. Numerous two-dimensional results were reported to discussnatural convection effect in various processes [4,5,19–22,24–28].The research on three-dimensional cases are scant although theyare more general. Leong et al. [29] provided experimental Nusseltnumbers for a cubical-cavity benchmark problem in three-dimensional natural convection. Tric et al. [30] solved three-dimensional natural convention at different Rayleigh numbers

Nomenclature

c lattice speedcp specific heat (J/kg K)cs sound speedei particle speedfi density distributionFi body forceg gravitational acceleration (m/s2)G effective gravitational acceleration (m/s2)k thermal conductivity (W/m k)M transform matrix for density distributionmi moment function for density distributionMa Mach numberN transform matrix for density distributionni moment function for energy distributionp pressure (Pa)P non-dimensional pressurePr Prandtl numberQ collision matrix for energy distributionRa Rayleigh number

S collision matrix for density distributiont time (s)T temperature (K)U non-dimensional velocity in x-directionv velocity in y-direction (m/s)V non-dimensional velocity in y-directionw velocity in y-direction (m/s)W non-dimensional velocity in y-directionV velocitya thermal diffusivity (m2/s)b thermal expansion (K�1)Dt time step (s)h non-dimensional temperaturel viscosity (Kg/ms)q density (kg/m3)m kinematic viscosity (m2/s)xi value factor for velocityxT,i value factor for temperature

Z. Li et al. / International Journal of Heat and Mass Transfer 94 (2016) 222–238 223

numerically. Wakashima and Saitoh [31] obtained Benchmarksolutions for natural convection in a cubic cavity using the high-order time-space method. Salat et al. [32] did experimental andnumerical investigation of turbulent natural convection in a largeair-filled cavity. This problem was also investigated using LBM[33]. Azwadi and Syahrullall [34] employed a double LBGK modelto simulate natural convection in a cubic cavity. LBM models inDDF can be different from each other. MRT and LBGK were appliedto solve velocity and temperature fields respectively in a cubic cav-ity mix convection flow [35,36]. This article proposed a doubleMRT model, which has not been reported by our knowledge, forthree-dimensional fluid flow and heat transfer simulation. Theobjective of this paper is to discuss three-dimensional naturalconvections with proposed double MRT model.

2. Natural convection in a cubic cavity

Physical model of natural convection in a cubic cavity is shownin Fig. 1. The cubic cavity with an edge length of H is filled withworking fluid of air. The Prandtl number is fixed at 0.71. Boussinesqassumption is employed. Then the governing equations are

@u@x

þ @v@y

þ @w@z

¼ 0 ð1Þ

q@u@t

þ u@u@x

þ v @u@y

þw@u@z

� �¼ � @p

@xþ l @2u

@x2þ @2u

@y2þ @2u

@z2

!ð2Þ

q@v@t

þ u@v@x

þ v @v@y

þw@v@z

� �¼ � @p

@yþ l @2v

@x2þ @2v

@y2þ @2v

@z2

!ð3Þ

q@w@t

þ u@w@x

þ v @w@y

þw@w@z

� �¼ � @p

@z

þ l @2w@x2

þ @2w@y2

þ @2w@z2

!

þ qgbðT � TlÞ ð4Þ

q@T@t

þ u@T@x

þ v @T@y

þw@T@z

� �¼ k

@2T@x2

þ @2T@y2

þ @2T@z2

!ð5Þ

Three types of natural convections are in consideration in thisarticle. They differ from each other with the thermal boundary set-tings. In all these cubic cavity natural convections, the verticalwalls (y = 0 and y = H) are kept at Th and Tc respectively. u, v andw are velocities in the x-, y- and z-directions. Non-slip boundaryconditions are employed.

y ¼ 0; T ¼ Th ð6Þ

y ¼ H; T ¼ Tc ð7Þ

u ¼ v ¼ w ¼ 0 for all boundaries ð8ÞIn the first type cubic cavity natural convection, the remained

four walls are all adiabatic. Natural convection with this settingis solved as benchmark problem in Ref. [30–33]. Leong et al. [29]argued that this setting was not physically-realizable becauseadiabatic boundary condition was hard to be reached.

Fig. 1. Cubic natural convection.

Fig. 2. D3Q19 model.

224 Z. Li et al. / International Journal of Heat and Mass Transfer 94 (2016) 222–238

The second type of cubic cavity natural convection is aphysically-realizable benchmark problem. Its remained four wallshave a linear temperature variation from the cold surface to hotsurface. Experimental results are reported in Refs. [29,37]. In thisarticle, we use:

x ¼ 0; T ¼ Th þ ðTc � ThÞ � y=Hx ¼ H; T ¼ Th þ ðTc � ThÞ � y=Hz ¼ 0; T ¼ Th þ ðTc � ThÞ � y=Hz ¼ H; T ¼ Th þ ðTc � ThÞ � y=H

8>>><>>>:

ð9Þ

The following boundary conditions are applied to the third typecubic natural convection.

x ¼ 0; T ¼ Th þ ðTc � ThÞ � y=Hx ¼ H; T ¼ Th þ ðTc � ThÞ � y=Hz ¼ 0; @T=@z ¼ 0z ¼ H; @T=@z ¼ 0

8>>><>>>:

ð10Þ

In the remained parts of this article, these three types ofproblems are referred to as Type 1, Type 2 and Type 3.

3. Lattice Boltzmann method

Lattice Boltzmann method has been employed to solve manyfluid flow and heat transfer problems [19,21,36]. MS, hybridmethod and DDF are the common thermal LBM models. The DDFin LBM will be used to solve velocity and temperature fields,respectively.

Streaming and collision are the basic processes in LBM. ManyLBM models exist and they differ from each other by the ways tosimplify the collision term. MRT is selected due to its advantagesin accuracy and numerical stability [17]. To the best of the authors’knowledge, double MRT model for three-dimensional fluid flowand heat transfer simulation has not been reported by now. In thisarticle, 3-D double MRT model is proposed and three types ofnatural convections are solved with it for validation.

3.1. D3Q19-MRT–LBM model for fluid flow

Lattice Boltzmann equation can describe the statistical behaviorof a fluid flow.

f ðr þ eDt; t þ DtÞ � f ðr; tÞ ¼ Xþ F ð11Þwhere f is the density distribution, Dt is the time step, X is thecollision term and F is the body force. To simplify this equation, itis assumed that each computing nodes has 19 directions as shownin Fig. 2 and these velocities are given by:

ei ¼ c

0 1 �1 0 0 0 0 1 �1 1 �1 1 �1 1 �1 0 0 0 00 0 0 1 �1 0 0 1 1 �1 �1 0 0 0 0 1 �1 1 �10 0 0 0 0 1 �1 0 0 0 0 1 1 �1 �1 1 1 �1 �1

264

375 ð12Þ

where c is the lattice speed and it relates to the sound speed cs as:

c2 ¼ 3c2s ð13ÞIn this D3Q19 model, Eq. (11) can be expressed as:

f iðr þ eiDt; t þ DtÞ � f iðr; tÞ ¼ Xi þ Fi; i ¼ 1;2; . . .19 ð14Þwhere the force term in the equation can be obtained as:

Fi ¼ DtG � ðei � VÞp

f eqi ðr; tÞ ð15Þ

where G is the effective gravitational force:

G ¼ �bðT � TlÞg ð16ÞThe equilibrium distribution function f eqi ðr; tÞ is expressed as:

f eqi ðr; tÞ ¼ qxi 1þ ei � Vc2s

þ ðei � VÞ22c4s

� V � V2c2s

" #; i ¼ 1;2; . . .19 ð17Þ

The density weighting factors xi are:

xi ¼1=3; i ¼ 11=18; i ¼ 2;3; . . . ;71=36; i ¼ 8;9; . . . ;19

8><>: ð18Þ

To satisfy the continuum and momentum conservations, thecollision term in MRT is:

Xi ¼ �M�1 � S � ½miðr; tÞ �meqi ðr; tÞ�; i ¼ 1;2; . . .19 ð19Þ

where mi(r, t) and meqi ðr; tÞ are moments and their equilibrium

functions; M and S are the transform matrix and collision matrix

respectively. d’Humieres et al. [10] introduced the detailedparameter settings in D3Q19-MRT model.

Macroscopic parameters are relate to the density distributionsas:

q ¼X19i¼1

f iðr; tÞ; qV ¼X19i¼1

f iðr; tÞ

m ¼ M � ½ f 1; f 2; � � � f 19 �T

8>><>>: ð20Þ


In this model, transform matrix M is:

Fig. 3. D3Q7 model.

Table 1Type 1, Nu3D comparison.

Ref.[30]

Ref.[31]

Ref.[33]

Ref.[41]

50 � 50 � 50 60 � 60 � 60

Ra = 1 � 104 2.05 2.06 2.08 2.10 2.07 2.08Ra = 1 � 105 4.34 4.37 4.38 4.36 4.42 4.39

Table 2Type 1, Numax comparison.

Ref. [30] Ref. [34] 50 � 50 � 50 60 � 60 � 60

Ra = 1 � 104 2.25 2.30 2.27 2.27Ra = 1 � 105 4.61 4.67 4.72 4.69


Ref. [33] 50 � 50 � 50 60 � 60 � 60

Ra = 1 � 104 3.72 3.66 3.68Ra = 1 � 105 7.88 7.98 7.98

Table 4Type 1, velocity comparison, Ra = 1 � 104.

U V W

50 � 50 � 50 2.20 16.88 19.2360 � 60 � 60 2.19 16.83 19.20Ref. [30] 2.16 16.72 18.98

Table 5Type 1, velocity comparison, Ra = 1 � 105.

U V W

50 � 50 � 50 10.09 45.00 72.5060 � 60 � 60 9.98 44.81 72.72Ref. [30] 9.70 43.91 71.11

M ¼

1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1�30 �11 �11 �11 �11 �11 �11 8 8 8 8 8 8 8 8 8 8 8 812 �4 �4 �4 �4 �4 �4 1 1 1 1 1 1 1 1 1 1 1 10 1 �1 0 0 0 0 1 �1 1 �1 1 �1 1 �1 0 0 0 00 �4 4 0 0 0 0 1 �1 1 �1 1 �1 1 �1 0 0 0 00 0 0 1 �1 0 0 1 1 �1 �1 0 0 0 0 1 �1 1 �10 0 0 �4 4 0 0 1 1 �1 �1 0 0 0 0 1 �1 1 �10 0 0 0 0 1 �1 0 0 0 0 1 1 �1 �1 1 1 �1 �10 0 0 0 0 �4 4 0 0 0 0 1 1 �1 �1 1 1 �1 �10 2 2 �1 �1 �1 �1 1 1 1 1 1 1 1 1 �2 �2 �2 �20 �4 �4 2 2 2 2 1 1 1 1 1 1 1 1 �2 �2 �2 �20 0 0 1 1 �1 �1 1 1 1 1 �1 �1 �1 �1 0 0 0 00 0 0 �2 �2 2 2 1 1 1 1 �1 �1 �1 �1 0 0 0 00 0 0 0 0 0 0 1 �1 1 �1 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 �1 �1 10 0 0 0 0 0 0 0 0 0 0 1 �1 �1 1 0 0 0 00 0 0 0 0 0 0 1 �1 1 �1 �1 1 �1 1 0 0 0 00 0 0 0 0 0 0 �1 �1 1 1 0 0 0 0 1 �1 1 �10 0 0 0 0 0 0 0 0 0 0 1 1 �1 �1 �1 �1 1 1

2666666666666666666666666666666666666666664

3777777777777777777777777777777777777777775

ð21Þ

(a) Surface temperature distribution (b)Temperature isosurfaces

(c) X=0.5 (d) Y=0.5

(e) Z=0.5 (f) Nusselt number distribution Fig. 4. Type 1, temperature results, Ra ¼ 1� 104.



The corresponding macroscopic moments are:

m ¼ ðq; e; e; jx; qx; jy; qy; jz; qz; 3pxx; 3pxx

pww; pww; pxy; pyz; pxz; mx; my; mzÞT ð22ÞThe collision matrix S in moment space is the diagonal matrix

S ¼ diagðs1; s2; s3; s4; s5; s6; s7; s8; s9; s10; s11; s12; s13; s14; s15;

s16; s17; s18; s19Þ ð23ÞWith s9 equaling to s13, the equilibrium moments are:

meq ¼ q; �11qþ19 j2xþj2yþj2zq0

; 3q� 112

j2xþj2yþj2zq0

; jx; �23 jx; jy; �2

3 jy; jz; �23 jz;

�

2j2x�ðj2yþj2z Þq0

; �122j2x�ðj2yþj2z Þ

q0;j2y�j2zq0

; �12j2y�j2zq0

;jxjyq0;jy jzq0; jx jzq0; 0; 0; 0

�T

ð24Þ

where

jx ¼ qux; jy ¼ quy; jz ¼ quz ð25ÞThe constant q0 in Eq. (24) is the mean density in the system

and it is usually set to be unity. Taking q0 in to account can reducecompressible effect in the model [38]. Then the unknown parame-ters in collision matrix S are:

s1 ¼ s4 ¼ s6 ¼ s8 ¼ 1:0; s2 ¼ 1:19; s3 ¼ s11 ¼ s13 ¼ 1:4;s5 ¼ s7 ¼ s9 ¼ 1:2s17 ¼ s18 ¼ s19 ¼ 1:98; s10 ¼ s12 ¼ s14 ¼ s15 ¼ s16 ¼ 1=ð3mþ 0:5Þ

8><>:

ð26ÞThe velocity field is solved using this D3Q19-MRT model.

Non-slip boundary conditions in this article are fulfilled usingbounce-back boundary conditions [10].

Fig. 5. Type 1, streamtrace

3.2. D3Q7-MRT–LBM model for heat transfer

Yoshida and Nagaoka [39] proposed a Multiple-relaxation-timelattice Boltzmann model for convection and anisotropic diffusionequation. In this D3Q7 model, seven discrete velocities are neededfor a three-dimensional problem. Li et al. [40] discussed boundaryconditions for this thermal LBM model. This model has not beeninvolved by any DDF model in LBM. In this article, we propose adouble MRT model for fluid flow and heat transfer problemsimulation. Velocity and temperature fields are solved withD3Q19-MRT and D3Q7-MRT, respectively.

D3Q7-MRT model is valid to solve the following standardconvection–diffusion equation.

@/@t

þ @

@xjðv j/Þ ¼ @

@xiDij

@/@xj

� �ð27Þ

where / is a scalar variable and Dij is the diffusion coefficient.Energy equation for the benchmark problem in this article isshown in Eq. (5), which is a special case of Eq. (27). Therefore,D3Q7 model can be used to solve the energy equation in thisarticle.

Each computing nodes have seven discrete velocities shown inFig. 3:

ui ¼ c

0 1 �1 0 0 0 00 0 0 1 �1 0 00 0 0 0 0 1 �1

264

375 ð28Þ

Similar to the density distribution, energy distribution gi can beobtained by:

giðr þ uiDt; t þ DtÞ � giðr; tÞ ¼ �N�1 � Q � ½niðr; tÞ � neqi ðr; tÞ�;

i ¼ 1;2; . . .7 ð29Þ

results, Ra ¼ 1� 104.

(a) Surface temperature distribution (b) Temperature isosurfaces

(c) X=0.5 (d) Y=0.5

(e) Z=0.5 (f) Nusselt number distribution Fig. 6. Type 1, temperature results, Ra ¼ 1� 105.


where neqi ðr; tÞ is the equilibrium function for niðr; tÞ, N and Q are the

transform matrix and collision matrix for the energy distribution[39].

Macroscopic parameters relate to the energy distributionsas:

T ¼X7i¼1

giðr; tÞ;

n ¼ N � ½ g1; g2; � � � g7 �T

8><>: ð30Þ

The energy weight factors xT;i are:

(a) 3D results (b) X=0.5

(c) Y=0.5 (d) Z=0.5

Fig. 7. Type 1, streamtrace results, Ra = 1 � 105.


Ref. [29] LBM

Ra = 1 � 104 1.52 1.49Ra = 1 � 105 3.10 3.06


xT;i ¼1=4; ði ¼ 1Þ;1=8; ði ¼ 2;3; . . . ;7Þ

�ð31Þ

Transform matrix in D3Q7 model is defined as:

N ¼

1 1 1 1 1 1 10 1 �1 0 0 0 00 0 0 1 �1 0 00 0 0 0 0 1 �16 �1 �1 �1 �1 �1 �10 2 2 �1 �1 �1 �10 0 0 1 1 �1 �1

2666666666664

3777777777775

ð32Þ

and its corresponding equilibrium moments are:

neq ¼ ½ T; uT; vT; wT; aT; 0; 0 �T ð33Þ

where a is a constant relating to the coefficient xT,1 by:

a ¼ ð7xT;1 � 1Þ ¼ 3=4 ð34ÞThe definition of collision matrix Q is:

Q�1 ¼

s1 0 0 0 0 0 00 sxx sxy sxz 0 0 00 sxy syy syz 0 0 00 sxz syz szz 0 0 00 0 0 0 s5 0 00 0 0 0 0 s6 00 0 0 0 0 0 s7

2666666666664

3777777777775

ð35Þ

The off-diagonal components correspond to the rotation of prin-cipal axis of anisotropic diffusion [39]. The relaxation coefficientssij are related to the diffusion coefficient matrix by:

sij ¼ 12dij þ Dt

eDx2Dij ð36Þ

where e is a constant 0.25 in a three-dimensional problem, and dij isthe Kronecker’s delta:

dij ¼0 if i–j;

1 if i ¼ j:

�ð37Þ

The relaxation coefficient s1 for the conserved quantity does notaffect the numerical solution, and s5, s6 and s7 only affect errorterms. After testing, this article use 1 for these four coefficients.

Temperature field is solved using this D3Q7-MRT model andthermal boundary conditions are solved based on the settings inRef. [40].

4. Results and discussions

Lattice velocity c is always set as unity in LBM. Therefore,parameters in any lattice unit are also non-dimensional. For a nat-ural convection problem, Ref. [27] describes detailed settings forparameters in lattice unit.

X¼ xH ; Y ¼ y

H ; Z¼ zH ; uc ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffigbðTh�TcÞH

p; Ma¼ uc

cs; U¼ uffiffi

3p

cs;V ¼ vffiffi

3p

cs;

W ¼ wH ; s¼ t�

ffiffi3

pcs

H ; h¼ T�TcTh�Tc

; P¼ p3qc2s

; Pr¼ ma ;Ra¼ gbðTh�TcÞH3Pr

m2

8<:

ð38Þ


(c) X=0.5 (d) Y=0.5

(e) Z=0.5 (f) Nusselt number distribution Fig. 8. Type 2, temperature results, Ra = 1 � 104.


Mach number Ma, Prandtl number Pr and Rayleigh number Raare the character parameters.

Natural convection problem is fully defined with Pr and Ra. LBMincludes the speed of sound, cs. So we have to include Ma to fulfillthis non-dimensional process for lattice unit. Wang et al. [25]

demonstrated that Ma has little effect on accuracy of MRT simula-tion. Incompressible air is the working fluid. Ma is 0.1 while Pr is0.71 for all the cases in this article. Three types of natural convec-tions in Section 2 are solved for various Rayleigh numbers rangedfrom 1 � 104 to 1 � 105.


(c) Y=0.5 (d) Z=0.5



The two vertical walls (Y = 0 and Y = 1) are kept at Th and Tc forall the cases in consideration. The local Nusselt number is definedas

Nu ¼ hk=H

��Y¼0

¼ � @h@Y

��Y¼0

ð39Þ

The average Nusselt number Nu3D:

Nu3D ¼Z 1

0

Z 1

0ðNuÞdXdZ ð40Þ

and the maximum Nusselt number Numax on the heat wall areimportant parameters to discuss three-dimensional naturalconvection problems. Due to the symmetry of the cubic natural con-vection, the mid-plane of the cubic (X = 0.5) plays an important rolein this problem and the average Nusselt number Nu2D at mid-planeis also in consideration.

Nu2D ¼Z 1

0ðNuX¼0:5ÞdZ ð41Þ

Besides Nu2D, Nu3D and Numax, the maximum velocities in alldirections are also discussed.

4.1. Type 1 natural convection

For type 1 natural convection, two different sets of grids(50 � 50 � 50 and 60 � 60 � 60) are employed. Refs. [30–32,41]reported Nu3D for Ra = 1 � 104 and Ra = 1 � 105. Table 1 showsthe comparison between present results and that in references.These references results agree with each other well and their aver-ages (2.07 for Ra = 1 � 104 and 4.36 for Ra = 1 � 105) can be viewed

as standard results. For the case of Ra = 1 � 104, the presentnumerical results in different grids are both close to the standardone. For the case of Ra = 1 � 105, however, the results obtainedusing grid of 60 � 60 � 60 agreed better with the results in the lit-erature. Refs. [31,34] reported Nu2D for this type of natural convec-tion. Table 2 is the comparison between results obtained from thepresent LBM and those from the references. The mean values fromthe references (2.28 for Ra = 1 � 104 and 4.64 for Ra = 1 � 105) aretaken as standard ones. For the case of Ra = 1 � 104, the resultsfrom the two grid number are the same and their differencesbetween standard one are negligible. For the case of Ra = 1 � 105,the differences between two the results from the two grid numbersand standard one are within 2% and the result from grid number of60 � 60 � 60 grids is closer to the standard one. Refs. [30,33]reported the Numax and maximum velocities, respectively. Non-dimensional process in these references is different from that inarticle. We can get the velocity Us in reference unit with thenumerical results Ul using the following equation:

Us ¼ Ul

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi3Pr � Ra

p=Ma ð42Þ

Tables 3–5 are the comparisons between numerical and refer-ence results. They indicate that the results from different gridnumbers agreed well with those from the references. The abovecomparisons indicated that Nu2D, Nu3D, Numax and maximum veloc-ities in different directions results all agreed well with referenceones; thus the proposed double MRT model is valid for the Type1 cubic natural convection simulation. Considering the computa-tional efficiency and accuracy, the grid number of 50 � 50 � 50 issuitable for the case of Ra = 1 � 104 while 60 � 60 � 60 more


(c) X=0.5 (d) Y=0.5



appropriate for thee case of Ra = 1 � 105. The other two types ofnatural convection simulations for various Rayleigh numbers alsohave the same grid settings.

Temperature and velocity fields for the cubic natural convectionare very important. But few references include three-dimensionalvisual results. Fig. 4 shows the temperature field for Type 1problem at Ra = 1 � 104. Surface temperature distribution and

temperature isosurfaces are included in Fig. 4(a) and (b). Thetemperature turns to be higher with increasing z in the cubiccavity. Convection has dominated the heat transfer process andtemperature isosurfaces does not change a lot in the x-direction.Fig. 4(c)–(e) show the temperature distributions on differentlocations for Type 1 problem. Regarding the boundary settings, itis common to argue the working condition on the mid-plane of


(c) Y=0.5 (d) Z=0.5


Table 7Type 2 benchmark solutions.

Umax Vmax Wmax Nu2D Nu3D Numax

Ra = 1 � 104 3.33 21.18 22.42 1.75 1.49 2.68Ra = 1 � 105 30.30 68.97 94.20 3.67 3.05 5.91


the cubic (X = 0.5) can be viewed as a two-dimensional problem[30]. Temperature field (X = 0.5) agrees well with the two-dimensional results in Ref. [27]. In the other two locations(Y = 0.5 and Z = 0.5), isothermal lines are almost parallel to the X-axis. It supports the two-dimensional assumption. Fig. 4(f) showsthe Nusselt number distribution on the hot surface (Y = 0). Nusseltnumbers at the mid-plane of the cubic (X = 0.5) are higher thanthat in the other regions due to the side wall effect to thesethree-dimensional problem. Non-slip condition is applied to allthe boundaries and it slows down the convection flow near theboundaries. Consequently, the convection effect to heat transferis also lowered.

The velocity field in the cubic cavity is also discussed. Fig. 5shows the streamtraces result for Ra = 1 � 104. Its main tendencyis a two-dimensional flow in the YZ plane. It also has a tendencyto flow to the center of the cavity in the X-direction. Fig. 5(b)–(d)show the streamtraces on different locations. The results on themid-plane of the cubic (X = 0.5) agreed well with two-dimensional ones in Ref. [27]. The results on Y = 0.5 show thatthe fluid have a tendency to flow to the cavity center in theX-direction. Meanwhile, two intersections exist for the stream-traces on the surface Z = 0.5. It indicates fluid flowing to the cavitycenter in the X-direction joins the two-dimensional flow in the Yand Z directions on the mid-plane of the cubic (X = 0.5).

Type 1 problem is also discussed for Ra = 1 � 105. Fig. 6 showsits temperature results. Convection effect is stronger comparingwith the case at lower Rayleigh number. And temperature differ-ence between top and bottom of the cavity turns to be greater.The temperature isosurfaces’ changes in the X-direction are lim-ited. Fig. 6(c)–(e) are temperature distributions on different loca-tions. Because of the boundary settings and symmetry of thisproblem, the working condition on mid-plane of the cubic(X = 0.5) is still close to a two-dimensional one. The temperaturefield for X = 0.5 agreed well with the two dimensional result inRef. [27]. Temperature distributions for Y = 0.5 and Z = 0.5 alsoprove that the two-dimensional assumption on the mid-plane ofthe cubic (X = 0.5) is reasonable. Nusselt number distribution onthe hot surface is shown in Fig. 6(f). It decreases with increasingZ. Its isolines are almost parallel to the X axis except the bottomregion. In that region, Nusselt number turns to be higher whenclosing to the mid-plane of the cubic (X = 0.5). As shown in Fig. 7(a), the velocity field is more complicated and flow is stronger thanthat of the case of Ra = 1 � 104. The two-dimensional flow in the Y-and Z-directions is still the main tendency. Fig. 7(b)–(d) include thesurface streamtraces on different locations. Results on the mid-plane of the cubic (X = 0.5) agree well with the two-dimensionalones in Ref. [27]. On the surface of Y = 0.5, fluid flows to the centerin the X-direction. Four streamtrace intersections exist on the sur-face (Z = 0.5). It indicates a stronger three-dimensional effect to thefluid flow.

Type 1 cubic natural convection is widely used as a benchmarkproblem to test numerical methods for three-dimensional fluidflow and heat transfer simulations. The above results show thatthe proposed double MRT model is reliable for this kind of prob-lem. Meanwhile, few references include three-dimensional visual


(c) X=0.5 (d) Y=0.5



results. Since type 1 problem is not physically-realizable [29], wewill continue to discuss type 2 cubic natural convection.


It is physically-realizable regarding its boundary conditionsettings in Section 2. Leong et al. [29] obtained the experimental

results of Nu3D for this type of natural convection. ForRa = 1 � 104 and Ra = 1 � 105, the present Nu3D results agreewith Ref. [29] ones well shown in Table 6. It also proves that theproposed double MRT model is valid for three-dimensional fluidflow and heat transfer simulation. More detailed results areincluded for type 2 cubic natural convection as benchmarksolutions.


(c) Y=0.5 (d) Z=0.5



Fig. 8 shows the temperature results for Type 2 problem atRa = 1 � 104. Hot and cold surfaces (Y = 0 and Y = 1) are kept at con-stant temperatures. The remained four side walls have linear tem-perature distributions in the Y-direction. The temperatureisosurfaces show that convection dominates the heat transferprocess. Temperature differences in the X-direction are significant.It indicates that type 2 problem has clear 3-D characteristics.Temperature distributions on different locations are shown inFig. 8(c)–(e). Results on the mid-plane of the cubic (X = 0.5) showthe convection effect. Results on the surfaces (Y = 0.5 and Z = 0.5)indicate that temperature differences at the center of the cavityare not significant. Fig. 8(f) shows the Nusselt number distributionon the hot surface for this working condition. The location at whichmaximum Nusselt number is reached is higher in the cavity thanthat in Type 1 problem with the same Rayleigh number. Nusseltnumber at the mid-plane of the cubic (X = 0.5) can be lower thanthat in the other locations at the same height. From Fig. 9(a), wecan find two-dimensional flow in the Y- and Z-directions. Andthe flow in the X-direction is also strong. Streamtraces on differentlocations are shown in Fig. 9(b)–(d). One vortex locates on themid-plane of the cubic (X = 0.5). It is quite similar to that in Type1 problem for Ra = 1 � 104. The surface (Y = 0.5) has four symmetryvortexes. Streamtraces on the surface (Z = 0.5) have two intersec-tions. They show the flow tendency in all directions.

Convection effect is more valid in Type 2 cubic natural convec-tion when Rayleigh number is 1 � 105 as shown in Fig. 10. Nusseltnumber isolines in Fig. 10(f) have similar tendency as that in Fig. 8(f). The difference is that Nusselt numbers are higher due to thestronger convection effects. All three cases discussed above (Type1 problem for Ra = 1 � 104 and Ra = 1 � 105; Type 2 problem forRa = 1 � 104) all have strong two-dimensional flows in the Y- andZ-directions. Cubic cavity streamtraces in Fig. 11 shows this

two-dimensional flow is not as strong as that in the other cases.Moreover, Fig. 11(b)–(d) include streamtraces on differentsurfaces. Mid-plane of the cubic (X = 0.5) have two vortexes whiletwo and four streamtraces intersections exists on the surfaces(Y = 0.5 and Z = 0.5), respectively. Character factors for type 2problem with different Rayleigh numbers are shown in Table 7,which can be used as benchmark solutions.

Comparing with Type 1 problem, Type 2 cubic natural convec-tion has three advantages to be a benchmark problem to testnumerical method for a three-dimensional fluid flow and heattransfer: (1) it is physically-realizable, (2) it has experimentalresults that agree well with the present numerical ones, and (3)three-dimensional effect is more valid in type 2 problem. On theother hand, type 2 temperature isosurfaces do not change a lot inthe X-direction at the region close to the cubic cavity top as canbe seen Figs. 8 and 10. To discuss a real three-dimensional prob-lem, we propose Type 3 cubic natural convections.


In type 3 cubic natural convection, surfaces (Y = 0 and Y = 1)have constant temperatures, and side walls (X = 0 and X = 1) havelinear temperature distributions in the Y-direction while the topand bottom of the cubic are kept adiabatic. We discuss the type3 problem for Ra = 1 � 104 first. Fig. 12(a)–(e) show the tempera-ture field and temperature distribution on different locations forthis case and three-dimensional features are clearly shown. Nus-selt number distribution in Fig. 12(f) is similar to that in Type 1problem shown in Fig. 4(f). For the same Rayleigh number, Nusseltnumbers are lower than that in Type 1 problem and higher thanthat in Type 2 problem. Boundary with linear temperature distri-bution lowers the convection effect, comparing with the adiabatic


(c) X=0.5 (d) Y=0.5



condition. Streamtraces shown in Fig. 13 indicate the flow in the X-,Y- and Z-directions are all strong. Fig. 13(b)–(d) include the stream-traces on different locations. Mid-plane of the cubic (X = 0.5) has onevortex, four vortexes exist on the surface (Y = 0.5), and streamtraceson the surface (Z = 0.5) have two intersections. These results aresimilar to that in Type 2 problem for Ra = 1� 104.

Type 3 problem is then discussed for Ra = 1 � 105 and the resultsare shown in Figs. 14 and 15. They showmore complicated velocityand temperature fields, which have clear three-dimensionalfeatures. Temperature isosurfaces changes significantly in the X-,Y- and Z-directions. For the same Rayleigh number, Nusseltnumbers for Type 3 problem are still lower than that in Type 1


(c) Y=0.5 (d) Z=0.5


Table 8Type 3 benchmark solutions.

Umax Vmax Wmax Nu2D Nu3D Numax

Ra = 1 � 104 3.80 18.85 20.84 2.21 1.80 3.59Ra = 1 � 105 21.73 59.19 86.26 4.66 3.94 7.96


problem and higher than that in Type 2 problem. Streamtracesresults indicate flow in all directions are strong. It shows clear3-D characteristics in type 3 problem. Table 8 records the character-istic quantities for various Rayleigh numbers. They can be used asbenchmark solutions for Type 3 problem.

5. Conclusions

Three-dimensional double MRT model is proposed for LBM forfluid flow and heat simulation. Three types of cubic natural convec-tion problems with various Rayleigh numbers are solved with theproposed method. Temperature field, hot surface Nusselt numberdistribution, velocity field, Nu2D, Nu3D, Numax and maximum veloc-ities in different directions are discussed. The results of Type 1problem agreed well with the reference ones, and the results fromType 2 problem fit the reported experimental results well. There-fore, the proposed double MRT is valid for three-dimensional sim-ulation. Type 2 problems are more physically-realizable comparingwith the type 1 problems. Their numerical results are reported forthe first time. Type 3 problems are also investigated because theirresults have more general three-dimensional features. All thesethree types’ 3-D natural convection results can be used as bench-mark solutions for further researches.

Acknowledgments

Support for this work by Chinese National Natural ScienceFoundations under Grants 51129602 and 51476103, and Innova-tion Program of Shanghai Municipal Education Commission underGrant 14ZZ134 are gratefully acknowledged.

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