INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN FLUIDSInt. J. Numer. Meth. Fluids 2012; 70:200–221Published online 10 October 2011 in Wiley Online Library (wileyonlinelibrary.com/journal/nmf). DOI: 10.1002/fld.2685

Coupling of finite volume method and thermal lattice Boltzmannmethod and its application to natural convection

H.B. Luan, H. Xu, L. Chen, Y.L. Feng, Y.L. He and W.Q. Tao*,†

Key Laboratory of Thermo-Fluid Science and Engineering of MOE, School of Energy and Power Engineering, Xi’anJiaotong University, Xi’an, Shaanxi 710049, China

SUMMARY

On the basis of the existing density distribution function reconstruction operator, the temperature distribu-tion operator was derived to calculate heat transfer by coupling the lattice Boltzmann method (LBM) withthe finite volume method. The present coupling model was validated by two-dimensional natural convectionflows with and without an isolated internal vertical plate. The results from the coupling model agree wellwith those from the pure-finite volume method, pure-LBM and references, and all the physical quantitiescross the coupled interface smoothly. On the basis of residual history curves, it is likely that the convergenceproperty and the numerical stability of the present model are better than those of the pure-LBM at fine gridnumbers and high Rayleigh numbers. Copyright © 2011 John Wiley & Sons, Ltd.

Received 22 January 2011; Revised 9 August 2011; Accepted 16 August 2011

KEY WORDS: lattice Boltzmann method; natural convection; reconstruction operator; finite volume;Navier–Stokes; viscous flows; stability; coupling

1. INTRODUCTION

In recent years, the lattice Boltzmann method (LBM) [1–4] has developed into a viable alterna-tive promising tool in modeling and simulating physics in fluids. It was found to be as stable andas accurate as conventional numerical methods in simulations of the single-component, isothermalfluid flow [5–7]. Apart from its applications for simulating fluid flows, modeling of convective heattransfer problems is also developed quickly. In general, the existing thermal lattice Boltzmann mod-els fall into three categories: (i) the multispeed (MS) approach [8–10]; (ii) the double distributionfunction (DDF) approach [11–13]; and (iii) the hybrid approach [14, 15]. In the MS approach, onlythe density distribution function is used. Additional speeds and higher-order velocity terms must beincluded in the equilibrium distribution function to obtain the temperature evolution equation at themacroscopic level. Researchers have found that the model suffers from severe numerical instabilityand the allowed temperature variation is limited to a narrow range [16]. In the DDF approach, thetemperature [12], internal energy [11] or total energy [13] is simulated using a separate distribu-tion function that is independent of the density distribution function. This approach can enhancethe numerical stability compared with the MS approach. In the hybrid approach [14, 15], the flowsimulation is accomplished by using the lattice Boltzmann equation, while the energy equation issolved by different numerical methods (e.g. finite difference method).

It is worth noting that hybrid models involving LBM are now used quite extensively. Generalmotivations of hybrid LBM are laid down by Succi et al. [17] who reported that as we tackle multi-disciplinary problems of growing complexity, it is increasingly evident that we need to upgrade the

*Correspondence to: W.Q. Tao, Key Laboratory of Thermo-Fluid Science and Engineering of MOE, School of Energyand Power Engineering, Xi’an Jiaotong University, Xi’an, Shaanxi 710049, China.

†E-mail: wqtao@mail.xjtu.edu.cn

Copyright © 2011 John Wiley & Sons, Ltd.

EXTENDING THE CFVLBM TO NATURAL CONVECTION 201

method to extend the range of scales the LBM can access and to couple it downward and upwardwith microscopic and macroscopic methods. In the literature, there are many papers proposingvarious hybrid models involving LBM, such as hybrid LBM and finite difference method (FDM)for nanoscale electroosmosis in microfluidic channels [18], hybrid LBM and FDM for convectiveflows [15], hybrid LBM and molecular dynamics simulation (MD) for simulation of colloidal sys-tems [19], hybrid LBM and finite volume method (FVM) for axisymmetric swirling and rotatingflows [20], hybrid LBM and discrete element modeling for fluid–particle interaction systems [21],hybrid LBM and FVM for compressible flow [22], hybrid LBM and MD for dense fluids [23],hybrid LBM and FEM for fluid–solid interaction with external flow [24], hybrid LBM and FDMfor phonon transport in crystalline [25], and hybrid LBM and FVM for fluid flows around complexgeometries [26]. These hybrid methods can be divided into two categories: (i) for the same compu-tational domain, different physical quantities/fields are solved by different numerical methods andthe results are coupled at their governing equations [14, 15, 18–22] and (ii) for the same physicalquantity/field, different regions are solved by different numerical methods and the results are cou-pled at their interfaces [23–26]. In general, the hybrid models provide three advantages comparedwith pure LBM: first, they extend the application range of LBM for complicated fluid problems[18–26]; second, they improve the computational efficiency of LBM [20,26], which provides a newdirection without considering the parallelization of LBM [27,28]; third, they enhance the numericalproperty of LBM (accuracy and stability) [15, 20–22, 25].

It is usually accepted that different scale problems have different numerical methods that aremost applicable to the corresponding scale. In this paper, we developed a new hybrid thermal LBMbased on domain decomposition and coupling at the interface. The computational domain is decom-posed in two regions in which the LBM and the FVM are used respectively. In the LBM zone,the DDF model is adopted. As indicated in [29–32], the key issue in the exchange of informa-tion between the macroscale results and mesoscale/microscale results is the establishment of theso-called reconstruction operator. In the authors’ previous works [26, 32], an analytic second-orderexpression, a reconstruction operator, is derived for the exchange of information from macrophysicalquantities (velocity and density) to the single-particle distribution function of LBM. The reconstruc-tion operator is successfully applied to the coupling of FVM and LBM (hereafter CFVLBM) intwo-dimensional simple [32] and complex geometries [26]. The CFVLBM can substantiallyimprove the total computational efficiency. However, these computations are mainly conducted forisotherm fluid flows. In engineering, fluid flow is often accompanied by heat transfer and thermalfluid flow problems are widely encountered. For multiscale thermal problems, the coupling betweena numerical solution of energy equation from a macroscopic method and meso/microscale resultsfrom LBM is highly required. Thus, it is a very challenging task and there is an urgent need to extendthe CFVLBM to thermal fluid flows. In this paper, a novel reconstruction operator is proposed toreconstruct the temperature distribution function from a macroscopic temperature. By combiningwith the density–velocity reconstruction operator it can be used to solve the thermal fluid flows.

The rest of the paper is organized as follows. First, the DDF thermal lattice Boltzmann model ispresented in Section 2. Then, the coupling scheme between the FVM and thermal LBM model ispresented in Section 3. Numerical examples on natural convection in a square cavity are illustratedin Section 4. Finally, some conclusions are given in Section 5.

2. THE DDF THERMAL LATTICE BOLTZMANN MODEL

2.1. Density distribution function and temperature distribution function

A popular kinetic model of LBM adopted in the literature is the single relaxation time (SRT) approx-imation, the so-called Bhatnagar–Gross–Krook (BGK) model [33–35]. The completely discretizedequation (also the evolution equation) is

fi .xC ci�t , t C�t/D fi .x, t/�1

�f

hfi .x, t/� f .eq/

i .x, t/i

, (1)

where fi is the density distribution function and �f is the dimensionless relaxation time.

Copyright © 2011 John Wiley & Sons, Ltd. Int. J. Numer. Meth. Fluids 2012; 70:200–221DOI: 10.1002/fld

202 H. B. LUAN ET AL.

The nine-velocity square lattice model D2Q9 [36] has been successfully used for simulatingtwo-dimensional flow. The macroscopic density � and velocity vector u can be evaluated as

�D

8XiD0

fi (2a)

�uD

8XiD0

cifi . (2b)

The DDF model solves the density field and the temperature field by using two independentlattice BGK equations [11, 12]. Corresponding to Equation (1), the evolution of temperature field isgoverned by:

gi .xC ci�t , t C�t/� gi .x, t /D�1

�g

hgi .x, t /� g.eq/

i .x, t /i

, (3)

where gi is the temperature distribution function, and �g is the dimensionless relaxation time.The fluid temperature T is defined as

T D

8XiD0

gi . (4)

2.2. A coupled lattice BGK model with the Boussinesq approximation

In the study of natural convection, the well-known Boussinesq approximation [37] is often used.It is assumed that all fluid properties (density, viscosity, thermal diffusivity) can be considered asconstant except the density � in the body force term, where it is assumed to be a linear function ofthe temperature

�D �0 Œ1� ˇ .T � T0/� , (5)

where �0 and T0 are the reference fluid density and temperature, respectively, and ˇ is the coefficientof thermal expansion.

The natural convection adopted Boussinesq approximation can be simulated by adding an externalforce term Fi to the right-hand side of the evolution Equation (1) [38].

fi .xC ci�t , t C�t/D fi .x, t /�1

�f

hfi .x, t /� f .eq/

i .x, t /iCFi . (6)

Fi is defined as

Fi D !i�

�1�

1

2�f

��3

c2.ci �F /C

9

c4.ci �F /

2 �3

c2F 2�

, (7)

where c is the particle velocity vector, ! is the weights factor and F D�gˇ .T � T0/.

3. THE PRINCIPLE AND PROCEDURE OF CFVLBM

For multiscale simulation, a fast converged algorithm of the continuum method adopted is highlyrequired. In the computational heat transfer community, FVM is the most widely adopted discretiza-tion approach and the SIMPLE-series algorithm [39,40] is the most popular method for dealing withthe coupling between velocity and pressure. To enhance the computational efficiency and robust-ness, our group has developed CLEAR [41, 42], and IDEAL [43, 44]. In both the CLEAR andIDEAL algorithms, the improved pressure and velocity are solved directly, rather than by addinga correction term to the intermediate solution. A further improvement of the solution procedure isdeveloped in the IDEAL algorithm, so its convergence rate and robustness is generally better thanthat of CLEAR. In this article, the two-dimensional IDEAL collocated grid algorithm is adopted andthe Stability Guaranteed Second-Order Difference (SGSD) scheme is used for the discretization ofthe convective term [45].

Copyright © 2011 John Wiley & Sons, Ltd. Int. J. Numer. Meth. Fluids 2012; 70:200–221DOI: 10.1002/fld

EXTENDING THE CFVLBM TO NATURAL CONVECTION 203

3.1. Reconstruction operator for density distribution function

As indicated above, an analytic expression has been derived in [31] and adopted in [26, 32] toreconstruct the density–velocity distribution function fi from macrophysical quantities velocityand pressure. For the readers’ convenience and also for further derivation of the thermal couplingscheme, the derivation process of the density–velocity distribution function is briefly presentedbelow. In the following derivation the fluid is assumed to be incompressible, hence the fluctuationof the density is neglected.

According to the Chapman–Enskog method, we can introduce the following time and space scaleexpansion:

@t D "@.1/t C "

2@.2/t (8a)

@x˛ D "@.1/x˛

. (8b)

The small expansion parameter " can be viewed as the Knudsen number Kn, which is the ratio ofthe molecular mean free path over the characteristic length scale of the flow, and ˛ represents thetwo coordinate directions.

The distribution fi is expanded around the distributions f .0/i as follows:

fi D f.0/i C "f

.1/i C "2f

.2/i (9)

with Xi

f.1/i D 0,

Xi

cif.1/i D 0,

Xi

f.2/i D 0,

Xi

cif.2/i D 0. (10a-d)

Then, the fi .xC ci�t , t C�t/ in Equation (1) is expanded about x and t , which gives

fi .xC ci�t , t C�t/D fi .x, t/C�tDi˛fi .x, t /C.�t/2

2D2i˛fi .x, t /COŒ.�t/3�, (11)

where Di˛ D @t C ci@x˛ for concise expression.Substituting Equation (11) into Equation (1) yields the following equation

�tDi˛fi C.�t/2

2D2i˛fi D�

1

�f.fi � f

.eq/i /COŒ.�t/3�. (12)

Furthermore, substituting Equations (8a), (8b), and (9) into Equation (12) obtains

"D.1/i˛ f

.0/C "2hD.1/i˛ f

.1/i C @

.2/t f

.0/i

i+"2

�t

2

hD.1/i˛

i2f.0/i

D�1

�t�f

�f.0/i C "f

.1/i C "2f

.2/i � f

.eq/i

�COŒ.�t/3�.

(13)

Then by matching the scales of "0, "1 and "2, we have

"0 W f.0/i D f

.eq/i (14)

"1 W f.1/i D��t�fD

.1/i˛f.0/i COŒ.�t/2� (15)

"2 W f.2/i D��t�f

hD.1/i˛ f

.1/i C @

.2/t f

.0/i

i-�f

.�t/2

2

hD.1/i˛

i2f.0/i COŒ.�t/3�. (16)

Considering Equations (2a) and (2b), we can sum Equation (16) over the phase space. Then thefirst order of the continuity equation and momentum equation can be derived.

"1 W @.1/t �C @.1/x˛ .�u˛/COŒ.�t/2�D 0 (17a)

Copyright © 2011 John Wiley & Sons, Ltd. Int. J. Numer. Meth. Fluids 2012; 70:200–221DOI: 10.1002/fld

204 H. B. LUAN ET AL.

@.1/t .�u˛/C @

.1/xˇ

��u˛uˇ C pı˛ˇ

COŒ.�t/2�D 0. (17b)

By the same way, we can obtain the second order of the continuity equation and momentumequation according to Equation (15):

"2 W @.2/t �COŒ.�t/3�D 0 (18a)

@.2/t .�u˛/� �@

.1/xˇ

°�h@.1/x˛ uˇ C @

.1/xˇu˛

i±COŒ.�t/3�D 0. (18b)

The formulas according to the chain rule of derivatives read

@tf.eq/i D @�f

.eq/i @t�C @uˇf

.eq/i @tuˇ (19a)

@x˛f.eq/i D @�f

.eq/i @x˛�C @uˇf

.eq/i @x˛uˇ . (19b)

From the density equilibrium distribution function, we can get that

@uˇf.eq/i D !i�

�1

c2s.ciˇ � uˇ /C

1

c4sci˛ciˇu˛

�(20)

@�f.eq/i D

1

�f.eq/i . (21)

Furthermore, substituting Equations (17a)–(21) into Equation (15), we can derive the first orderexpression of distribution function fi

f.1/i D��f�t

h@.1/t f

.0/i C ci@

.1/x˛f.0/i

i

D��f�th@�f

.0/i @.1/

t�C @uˇf

.0/i @

.1/t uˇ C ci

�@�f

.0/i @.1/x˛ �C @uˇf

.0/i @.1/x˛ uˇ

�i

D��f�t

��@�f

.0/i @.1/x˛ .�u˛/�

1

�@uˇf

.0/i @.1/x˛

��u˛uˇ C pı˛ˇ

Cci

�@�f

.0/i @.1/x˛ �C @uˇf

.0/i @.1/x˛ uˇ

�i

D��f�t

�Ui˛f

.0/i

1

�@.1/x˛ �CUi˛!i�

�1

c2sUiˇ C

1

c4sciˇci�u�

�@.1/x˛ uˇ � f

.0/i @.1/x˛ u˛

�!i

�1

c2sUi˛ C

1

c4sci˛ci�u�

�@.1/x˛p

�, (22)

where Ui˛ D ci˛ � u˛ .The second-order expression of fi in Equation (16) is calculated as follows:

f.2/i D��t�f

hD.1/i˛ f

.1/i C @

.2/t f

.0/i

i�.�t/2�f

2

hD.1/i˛

i2f.0/i

D��t�f

hD.1/i˛

����tD

.1/i˛ f

.0/i

�C @

.2/t f

.0/i

i�.�t/2�f

2

hD.1/i˛

i2f.0/i

D��t�f @.2/t f

.0/i C .�t/2�f

��f �

1

2

�hD.1/i˛

i2f.0/i

. (23)

We can ignore the second-order derivative of f .2/i , then

f.2/i D��t�f @

.2/t f

.0/i . (24)

By the chain rule of derivatives, it gives

@.2/t f

.0/i D @�f

.0/i @

.2/t �C @uˇf

.0/i @

.2/t uˇ D @uˇf

.0/i @

.2/t uˇ . (25)

Copyright © 2011 John Wiley & Sons, Ltd. Int. J. Numer. Meth. Fluids 2012; 70:200–221DOI: 10.1002/fld

EXTENDING THE CFVLBM TO NATURAL CONVECTION 205

Using Equations (18b) and (20), we get

@.2/t f

.0/i D @uˇf

.0/i @

.2/t uˇ

D1

�@uˇf

.0/i @

.2/t

��uˇ

D �!i

�1

c2s

�ciˇ � uˇ

C1

c4sci˛ciˇu˛

�@.1/x˛

���@.1/x˛ uˇ C @

.1/xˇu˛

��

D �!i�

�1

c2s

�ciˇ � uˇ

C1

c4sci˛ciˇu˛

� �1

�@.1/x˛ �

�@.1/x˛ uˇ C @

.1/xˇu˛

�

C@.1/x˛

�@.1/x˛ uˇ C @

.1/xˇu˛

�i. (26)

Therefore, the following expression is obtained

f.2/i D��t�f �!i�

�1

c2sUiˇ C

1

c4sci˛ciˇu˛

��1

�@.1/x˛ �

�@.1/x˛ uˇ C @

.1/xˇu˛

�

C@.1/x˛

��@.1/x˛ uˇ C @

.1/xˇu˛

��i. (27)

Here, we introduce an approximation of @uˇf.0/i by dropping terms of a higher order than u2 as

follows:

@uˇf.0/i D !i�

�1

c2sUiˇ C

1

c4sciˇci�u�

��Uiˇ

c2sf.0/i . (28)

Assuming that the velocity field is divergence-free, written as

@x˛u˛ D 0 . (29)

According to Equations (28) and (29), we can rewrite the expressions of f .1/i and f .2/i as,

f.1/i D��f�t

�Ui˛f

.0/i

1

�@.1/x˛ �CUi˛Uiˇf

.0/i

1

c2s@.1/x˛ uˇ �Ui˛f

.0/i

1

�c2s@.1/x˛p

�

D��f�tUi˛Uiˇf.0/i c�2s @.1/x˛ uˇ

(30)

f.2/i D��t�f �Uiˇf

.0/i c�2s

�1

�@.1/x˛ �

�@.1/x˛ uˇ C @

.1/xˇu˛

�C�@.1/x˛

�2uˇ

�

D��t�f �Uiˇf.0/i c�2s

�1

�S.1/

˛ˇ@.1/x˛ �C

�@.1/x˛

�2uˇ

�,

(31)

where S˛ˇ D @xˇu˛ C @x˛uˇ .Finally, we can derive the expression of fi

fi D f.0/i C "f

.1/i C "2f

.2/i

D f.0/i � ��tUi˛Uiˇf

.0/i c�2s @x˛uˇ � ��t�Uiˇf

.0/i c�2s

�1

�S˛ˇ@x˛�C @

2x˛uˇ

�

D f.eq/i

1� ��tUiˇc

�2s

�Ui˛@x˛uˇ C �@

2x˛uˇ C ��

�1S˛ˇ@x˛��

. (32)

Equation (32) is an analytic expression for reconstructing the density distribution function (DF) fromthe macro variables. Hereafter, we call it as velocity density function (DF) reconstruction operator.

Copyright © 2011 John Wiley & Sons, Ltd. Int. J. Numer. Meth. Fluids 2012; 70:200–221DOI: 10.1002/fld

206 H. B. LUAN ET AL.

3.2. Reconstruction operator for temperature distribution function

Following Equations (8a)–(13), we can obtain gi in the scales of "0, "1 and "2,

"0 W g.0/i D g

.eq/i (33)

"1 W g.1/i D��t�gD

.1/i˛g.0/i CO.�t/2� (34)

"2 W g.2/i D��t�g

hD.1/i˛ g

.1/i C @

.2/t g

.0/i

i�.�t/2�g

2

hD.1/i˛

i2g.0/i COŒ.�t/3�. (35)

Therefore, we can derive the macroscopic equations at the t1 D "t and t2 D "2t time scales:

@.1/t T C @.1/x˛ .u˛T /D 0 (36)

@.2/t T C @.1/x˛

��1�

1

2�g

�q.1/˛

�D 0, (37)

where q.1/˛ D8PiD0

ci˛g.1/i .

After some standard algebraic manipulations, we have

q.1/˛ D��g�th@.1/t .u˛T /C c

2s @.1/x˛Ti

. (38)

For flows that vary slowly in time, the first term in the bracket on the right-hand side of Equation(38) is indeed much smaller than the second term by the analysis of orders of magnitude and, hence,can be neglected [46].

Substituting Equation (38) into Equation (37), we get

@.2/t T C @.1/x˛

��1�

1

2�g

����g�tc

2s @.1/x˛T��D 0. (39)

Supposing ˛T D c2s��g �

12

�t , we can rewrite Equation (39) as

@.2/t T D ˛T

�@.1/x˛

�2T . (40)

Figure 1. Interface structure between two regions of CFVLBM for natural convection in a square cavity.

Copyright © 2011 John Wiley & Sons, Ltd. Int. J. Numer. Meth. Fluids 2012; 70:200–221DOI: 10.1002/fld

EXTENDING THE CFVLBM TO NATURAL CONVECTION 207

Introducing the formulas according to the chain rule of derivatives

@tg.eq/i D @T g

.eq/i @tgC @uˇg

.eq/i @tuˇ (41)

@x˛g.eq/i D @T g

.eq/i @x˛T C @uˇg

.eq/i @x˛uˇ , (42)

and from the temperature equilibrium distribution function, we can get the following expression:

@uˇg.eq/i D @uˇ

!iT

�1C c�2s ci�u�

�D c�2s !iTciˇ (43)

@T g.0/i D @T

!iT

�1C c�2s ci�u�

�D T �1g

.eq/i . (44)

(a) Ra=103

(b) Ra=104

(c) Ra=105

(d) Ra=106

Figure 2. The contour lines of u�velocity (the results of FVM, LBM, and CFVLBM, shown fromleft to right).

Copyright © 2011 John Wiley & Sons, Ltd. Int. J. Numer. Meth. Fluids 2012; 70:200–221DOI: 10.1002/fld

208 H. B. LUAN ET AL.

The first-order expression of the distribution function gi can be derived as

g.1/i D��g�tD

.1/i˛ g

.0/i

D��g�t�@.1/t g

.0/i C ci˛@

.1/x˛g.0/i

�

D��g�th@T g

.0/i @

.1/t T C @uˇg

.0/i @

.1/t uˇ C ci˛

�@T g

.0/i @.1/x˛ T C @uˇg

.0/i @.1/x˛ uˇ

�i

D��g�th�@T g

.0/i @.1/x˛ .u˛T /� �

�1@uˇg.0/i @.1/x˛

��u˛uˇ C pı˛ˇ

Cci˛

�@T g

.0/i @.1/x˛ T C @uˇg

.0/i @.1/x˛ uˇ

�i

D��g�th�u˛@T g

.0/i @.1/x˛ T � u˛@uˇg

.0/i @.1/x˛ uˇ � �

�1@uˇg.0/i @.1/x˛

�pı˛ˇ

Cci˛@T g

.0/i @.1/x˛ T C ci˛@uˇg

.0/i @.1/x˛ uˇ

i

D��g�thUi˛@T g

.0/i @.1/x˛ T CUi˛@uˇg

.0/i @.1/x˛ uˇ � �

�1@uˇg.0/i @.1/xˇp

i

D��g�thUi˛T

�1g.eq/i @.1/x˛ T CUi˛!iTciˇc

�2s @.1/x˛ uˇ � �

�1!iTciˇ@.1/xˇ�i

. (45)

(a) Ra=103

(b) Ra=104

(c) Ra=105

(d) Ra=106

Figure 3. The contour lines of v-velocity (the results of FVM, LBM, and CFVLBM, shown fromleft to right).

Copyright © 2011 John Wiley & Sons, Ltd. Int. J. Numer. Meth. Fluids 2012; 70:200–221DOI: 10.1002/fld

EXTENDING THE CFVLBM TO NATURAL CONVECTION 209

The second-order expression of the distribution function gi can be derived as

g.2/i D��g�t

²@.2/t g

.0/i C

�1�

1

2�g

�D.1/i˛ g

.1/i

³

D��g�t

²@.2/t g

.0/i �

��g �

1

2

��tD

.1/i˛

�D.1/i˛ g

.0/i

�³

D��g�t

�@.2/t g

.0/i � ˛T c

�2s

�D.1/i˛

�2g.0/i

�. (46)

The second-order derivative of g.0/i can be ignored, then

g.2/i D��g�t@

.2/t g

.0/i

D��g�th@T g

.0/i @

.2/t T C @uˇg

.0/i @

.2/t uˇ

i

D��g�t

�˛T @T g

.0/i

�@.1/x˛

�2T C!iciˇc

�2s ��1T @t2

��uˇ

�

(a) Ra=103

(b) Ra=104

(c) Ra=105

(d) Ra=106

Figure 4. The streamlines (the results of FVM, LBM, and CFVLBM, shown from left to right).

Copyright © 2011 John Wiley & Sons, Ltd. Int. J. Numer. Meth. Fluids 2012; 70:200–221DOI: 10.1002/fld

210 H. B. LUAN ET AL.

D��g�t

�˛T T

�1g.0/i

�@.1/x˛

�2T C!iciˇc

�2s ��1T �

h@.1/x˛

���@.1/x˛ uˇ C @

.1/xˇu˛

��i�

D��g�t

²˛T T

�1g.0/i

�@.1/x˛

�2T C!iciˇc

�2s ��1T �

h�@.1/x˛

�@.1/x˛ uˇ C @

.1/xˇu˛

�

C�@.1/x˛ uˇ C @

.1/xˇu˛

�@.1/x˛ �

i±

D��g�t

²˛T T

�1g.0/i

�@.1/x˛

�2T C!iciˇc

�2s ��1T �

���@.1/x˛

�2uˇ C S

.1/

˛ˇ@.1/x˛ �

�³. (47)

Finally, the expression of gi is derived as

gi D g.0/i C "g

.1/i C "

2g.2/i C � � �

D g.eq/ � �g�t�Ui˛T

�1g.eq/1 @x˛T CUi˛!iTciˇc

�2s @x˛uˇ � �

�1!iTciˇ@x˛��

� �g�t°˛T T

�1g.eq/i .@x˛ /

2 T C!iciˇc�2s ��1T �

h� .@x˛ /

2 uˇ C S.1/

˛ˇ@x˛�

i±

D g.eq/°1� �g�tT

�1hUi˛@x˛T C ˛T .@x˛ /

2 Ti±

(a) Ra=103

(b) Ra=104

(c) Ra=105

(d) Ra=106

Figure 5. The contour lines of temperature (the results of FVM, LBM, and CFVLBM, shown fromleft to right).

Copyright © 2011 John Wiley & Sons, Ltd. Int. J. Numer. Meth. Fluids 2012; 70:200–221DOI: 10.1002/fld

EXTENDING THE CFVLBM TO NATURAL CONVECTION 211

� �g�t!iciˇTc�2s

°Ui˛@x˛uˇ C � .@x˛ /

2 uˇ C �S˛ˇ��1@xˇ�

±C �g�t!iciˇT�

�1@xˇ�

D g.eq/°1� �g�tT

�1hUi˛@x˛T C ˛T .@x˛ /

2 Ti±

C�g!iTciˇ

�f Uiˇ

�fi � f

.eq/i

�

f.eq/i

C �g�t!iciˇT��1@xˇ� (48)

Equation (48) is an analytic expression for the reconstruction of the temperature distributionfunction gi from the macro variables. It will be called the temperature DF reconstruction operator.

Equation (32) combined with Equation (48) can be applied to the thermal fluid flows. The recon-struction operations are essential to establish an effective information exchange scheme from themacrosolver to the micro/mesosolver and to construct a reasonable initial field for accelerating themicroscopic computation.

3.3. Computational procedure

To illustrate the basic idea of the CFVLBM, the natural convection in a square cavity is adopted asan example. The computational domain is decomposed in two regions in which the LBM and theFVM are used, respectively. In Figure 1 the decomposition of the computational domain is schemat-ically shown. We can easily control the coarseness and fineness of the grids according to the zone

(a) Ra=103

(b) Ra=104

(c) Ra=105

(d) Ra=106

Figure 6. The contour lines of pressure (the results of FVM, LBM, and CFVLBM, shown from left to right).

Copyright © 2011 John Wiley & Sons, Ltd. Int. J. Numer. Meth. Fluids 2012; 70:200–221DOI: 10.1002/fld

212 H. B. LUAN ET AL.

spatial scale in each region. When the grid systems at the interface of the two subregions are notidentical, space interpolation at the interface is required when transferring the information at theinterface. In this paper, we choose the FVM grid size equal to lattice size to avoid the spatial inter-polation. LineMN is the FVM region boundary located in the LBM subregion, while lineAB is theLBM region boundary located in the FVM subregion. Hence, the subdomain between the two linesis the overlapped region (often called hand-shaking region) where both LBM and FVM methods areadopted. This arrangement of the interface is convenient for the information exchange between thetwo neighboring regions [47].

The CFVLBM computation procedures are conducted as follows:

Step 1. With some assumed initial boundary conditions at the lineMN , the FVM simulation in theleft region is performed.

Step 2. After a temporary solution is obtained, the information at the line AB is transformed intothe density–velocity distribution function by Equation (32) and the temperature distributionfunction by Equation (48).

Step 3. The LBM simulation is carried out in the right region. In detail, this step can be divided intofour substeps: (1) streaming; (2) collision for density distribution function; (3) streaming;and (4) collision for temperature distribution function.

Step 4. The temporary solution of LBM at the line MN is transported into the macro variables andthe FVM simulation is repeated.

Step 5. Such computation is repeated until the results are within an allowed tolerance.

(a) Ra=103

(b) Ra=104

(c) Ra=105

(d) Ra=106

Figure 7. The contour lines of vorticity (the results of FVM, LBM, and CFVLBM, shown from left to right).

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EXTENDING THE CFVLBM TO NATURAL CONVECTION 213

4. RESULTS AND DISCUSSION

4.1. Natural convection in a square cavity

The natural convection in a square cavity is a benchmark problem in the computational heat transfercommunity and a lot of numerical studies have been conducted [48–50]. The physical model forthe problem is shown in Figure 1. The height of the square cavity H D1.0 m. The computationaldomain is divided into two subregions. The left part is the FVM region with length scale 0.525 � 1.0and grid number 105 � 200. The right part is the LBM region with length scale 0.5 � 1.0 and gridnumber 100 � 200. The length scale of the overlapped region is 0.025 � 1.0 with grid number 5 �200. The Prandtl number (P r D �=a) is fixed at 0.71. All surrounding walls are rigid and imperme-able. The vertical walls located at x D0 m and x D1 m are retained to be isothermal but at differenttemperatures of Th and Tc, respectively. The horizontal sidewalls are taken as adiabatic. The buoy-ancy force because of gravity works downwards. Fluid properties are defined as density � D 1.0kg/m2, kinetic viscosity � D7.5E–4 m2/s and gravity acceleration g D10. The computations areconducted for four Rayleigh numbers (Ra D gˇH 3�T=a� D 103, 104,105, and 106). The resultsof CFVLBM simulation are compared carefully with the results of pure-FVM and pure-LBM. Therelaxation times in DDF model are �f D 0.95 and �g D 1.134.

Figures 2(a–d) show contour lines of u_velocity at four different Ra numbers. The agreementbetween the results of FVM, LBM, and CFVLBM is quite good. The contour lines of CFVLBM arevery smooth across the hand-shaking region.

Figures 3(a)–(d) show contour lines of v_velocity at four different Ra numbers. For Ra D 103,104, and 105, the contour lines of CFVLBM are almost identical to the results of the other threemethods and smooth across the interface. For Ra D 106, the flow pattern is more complicated andnearly reach to transition zone [49]. Little unsmoothness of the contour lines occurs at the coupledinterface in Figure 3(d).

Figures 4(a)–(d) show streamlines at four differentRa numbers. From these figures, all of the fourmethods obtain similar results, and the streamlines in CFVLBM are smooth across the interface. Forlow value of Ra, a central vortex appears as the typical feature of the flow. As Ra increases, thevortex tends to become elliptic and finally breaks up into two vortices at Ra D 105. When Rareaches to 106, the two vortices move towards the walls, giving space for a third vortex to develop.

Figures 5(a)–(d) show the change of isotherms as the Ra number increases. Satisfactory consis-tence can be found among the results of the four methods and the contour lines of CFVLBM aresmooth across the coupled interface. For small Ra, almost vertical isotherms appear, implying thatheat is mainly transferred by conduction between hot and cold walls. AsRa increases, the dominant

Figure 8. Comparisons of u-velocity along the vertical central line and v-velocity along the horizontalcentral line (RaD 103).

Copyright © 2011 John Wiley & Sons, Ltd. Int. J. Numer. Meth. Fluids 2012; 70:200–221DOI: 10.1002/fld

214 H. B. LUAN ET AL.

heat transfer mechanism changes from conduction to convection, and the isotherms at the center ofthe cavity are horizontal. The higher the Ra, the wider the horizontal part and the thinner the ther-mal boundary layer along the vertical wall. All of our results, including those observations shownin Figure 2–5, are qualitatively in good agreement with the results reported in previous studies.

Attention is now turned to the pressure contour line, which is a more appropriate parameter toensure the quality of good smoothness and consistence at the coupling interface [32]. As can beobserved in Figures 6(a)–(d), there is a little dislocation in the coupling interface of CFVLBM. Thisis because the density of the FVM zone is constant, while the density of LBM zone is changinginherently related to the weak compressibility of the lattice Boltzmann model. In the LBM region,the velocity and temperature of boundary AB (Figure 1) can be directly obtained from that of theFVM zone, but the density cannot be obtained from the FVM zone. In the paper, the density isobtained through the linear extrapolation from the inner neighboring values of the LBM zone.

To be more informative, the contours of vorticity distribution are examined. The results are pre-sented in Figures 7(a)–(d). It can be observed that the predicted CFVLBM results are agreeablewith other methods. However, there is some unsmoothness at the interface in the coupled model.The dislocation becomes visible near the center. This is because the vorticity in the center part is

Figure 9. Comparisons of u-velocity along the vertical central line and v-velocity along the horizontalcentral line (RaD 104).

Figure 10. Comparisons of u-velocity along the vertical central line and v-velocity along the horizontalcentral line (RaD 105).

Copyright © 2011 John Wiley & Sons, Ltd. Int. J. Numer. Meth. Fluids 2012; 70:200–221DOI: 10.1002/fld

EXTENDING THE CFVLBM TO NATURAL CONVECTION 215

less intensive than that near the confining wall. Thus, a relatively large discrepancy can be observedin the center region. When we couple numerical solutions at the interface, the quality of the vorticitycontour plot is more sensible to that of velocity. This is because vorticity is the first derivative ofvelocity; its smoothness requirement is more strict than that of velocity.

To further quantify the results of CFVLBM, the comparisons of u-velocity along the verticalline and v-velocity along the horizontal line through the geometric center are conducted. For con-venience of presentation, nondimensionalized velocities are presented. The nondimensionalizedu-velocity and v-velocity are based on the umax and vmax extracted from the two center lines and thereference length for coordinate x and y is the wall heightH . From Figures 8–11, it can be observedthat the predicted results from four methods are all in good agreement.

The variation of nondimensional temperature along the horizontal centerline of the cavity isshown in Figure 12. It can be clearly seen that the results of the four methods confirm each othervery well. At Ra D 103, the temperature drops linearly from the conduction mechanism. Withthe increase of Ra the temperature gradient at the two vertical wall increases and the center partbecomes flat. At Ra D 106, the temperature reverse can be observed, which is characterized by avalley near the hot wall and a summit near the cold wall.

Figure 11. Comparisons of u-velocity along the vertical central line and v-velocity along the horizontalcentral line (RaD 106).

Figure 12. Comparisons of temperature profile along the horizontal central line (Ra from 103 to 106/.

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216 H. B. LUAN ET AL.

Furthermore, local and average Nusselt numbers are calculated for the hot walls. The local (Nu)and mean (Num) Nusselt numbers at the hot wall are determined by

NuD�H

�T

�@T

@x

�w

(49)

Num D

Z H

0

Nu dy. (50)

The five characteristic parameters predicted by CFVLMB are in good agreement with the resultsof a previous study [50] for each Ra as shown in Table I.

Table I. Comparison of CFVLBM solution with previous works for different Ra values.

RaD 103 RaD 104 RaD 105 RaD 106

Present Ref. [50] Present Ref. [50] Present Ref. [50] Present Ref. [50]

Num 1.110 1.114 2.230 2.245 4.507 4.510 8.807 8.806Numax 1.561 1.581 3.539 3.539 7.738 7.637 17.71 17.442(y=H/max 0.099 0.099 0.145 0.143 0.083 0.085 0.375 0.0368Numin 0.690 0.670 0.584 0.583 0.746 0.773 0.978 1.001(y=H/min 0.995 0.994 0.995 0.994 0.997 0.999 0.997 0.999

0.0 2.0x104 4.0x104 6.0x104 8.0x104 1.0x105 1.2x10510-9

10-8

10-7

10-6

10-5

10-4

10-3

10-2

Res

idua

l

Grid number 200x200 300x300 400x400

CPU Time (s)(a) different grid number at Ra=106

0.0 2.0x104 4.0x104 6.0x104 8.0x104 1.0x105 1.2x105 1.4x10510-11

10-10

10-9

10-8

10-7

10-6

10-5

10-4

10-3

10-2

RaRa=104

Ra=105

Ra=106

Res

idua

l

CPU Time (s)(b) different Ra with grid number 400 ×400

Figure 13. Residual curves of DDF model.

Copyright © 2011 John Wiley & Sons, Ltd. Int. J. Numer. Meth. Fluids 2012; 70:200–221DOI: 10.1002/fld

EXTENDING THE CFVLBM TO NATURAL CONVECTION 217

Finally, residual history is discussed. The iteration convergence criterion for reaching the steadystate is defined by Equation (51).

ResidualD

PPju .i , j , t C�t/� u .i , j , t /j C

PPjv .i , j , t C�t/� v.i , j , t /jPP

.ju .i , j , t /j C jv .i , j , t /j/. (51)

For pure-LBM, residual curves show that the oscillation frequency increases as the grid numberincreases (Figure 13(a)) and the Ra increases (Figure 13(b)). Figure 14 shows the comparison ofresidual curves between CFVLBM and pure-LBM at Ra D 106 with grid number 400 � 400. Itcan be seen that the curve of the coupled model is more stable than pure LBM because there are nothermal fluctuations in the FVM regions.

4.2. Natural convection in a square cavity with an internal isolated vertical plate

As pointed in [26], the CFVLBM is more efficient than pure-LBM and more accurate than pure-FVM for complex flows such as flow in porous media. In this examples, we extend the CFVLBM tothermal flows in a square cavity with an internal isolated vertical plate [51, 52] shown in Figure 15.The height of the cavity is H D1 with grid number 200 � 200. The plate exists in the LBM zones.The collision between fluid particle and the solid sites of the plate is handled by using a bounceback boundary condition. The height and width of the plate are m D41/80 and n D 13/160. Thedistances between the walls are a D33/160 and b D3/40. All the fluid property parameters follow

5.0x104 1.0x105 1.5x105 2.0x10510-9

10-8

10-7

10-6

10-5

10-4

10-3

10-2

Res

idua

l

CPU Times (s)

LBM

CFVLBM

Figure 14. Comparison of couple model and pure LBM at RaD 106 with grid number 400 � 400.

Figure 15. An enclosure cavity with an internal isolated vertical plate.

Copyright © 2011 John Wiley & Sons, Ltd. Int. J. Numer. Meth. Fluids 2012; 70:200–221DOI: 10.1002/fld

218 H. B. LUAN ET AL.

Subsection 4.1. The Rayleigh number equals to 1.1� 105. The relaxation times in DDF models are�f D 0.95 and �g D 1.134.

The streamlines are shown in Figure 16. The result of CFVLBM (Figure 16(a)) is in good agree-ment with the result in [52] (Figures 16(b) and (c)). The minor difference between the numericalresults and the experiment results may be caused by the measuring technique [52]. Figure 17 showsthe isotherms of CFVLBM and LBM. Satisfactory consistence can be found among the resultsand the contour lines of CFVLBM are smooth across the coupled interface. Furthermore, the com-parisons of v-velocity along the horizontal line through the geometric center are conducted. InFigure 18, it can be observed that the predicted results are in good agreement between CFVLBMand LBM.

5. CONCLUSIONS

In this paper, the so-called temperature reconstruction operator is derived for the DDF thermalLBM model, seemingly first in the literature. Combined with the velocity reconstruction opera-tor, a CFVLBM is proposed to solve a two-dimensional convective heat transfer problem. Thetwo reconstruction operators can directly lift the macroscopic velocity, density, and temperaturefields to mesoscopic velocity distribution function and temperature distribution function of LBM,respectively.

(a) CFVLBM

(b) Numerical results in ref. [52]

(c) Experiment results in ref. [52]

Figure 16. The streamlines at RaD 1.1� 105.

Copyright © 2011 John Wiley & Sons, Ltd. Int. J. Numer. Meth. Fluids 2012; 70:200–221DOI: 10.1002/fld

EXTENDING THE CFVLBM TO NATURAL CONVECTION 219

(a) CFVLBM

(b) LBM

Figure 17. The isotherms at RaD 1.1� 105.

0.0 0.2 0.4 0.6 0.8 1.0-0.10

-0.08

-0.06

-0.04

-0.02

0.00

0.02

0.04

0.06

0.08

0.10

v

x

CFVLBM LBM

Figure 18. Comparison of v-velocity along the horizontal central line.

The natural convection in a square cavity is used to validate the feasibility and reliability of thederived reconstruction operator and the proposed CFVLBM scheme. The residual curve shows thatthe coupled model is more stable than pure LBM. Taking all things considered, we can come to theconclusion that the CFVLBM is reliable and accurate for the present simulations till RaD 106.

Natural convection in cavity with a vertical plate is another sample problem. The good agreementindicates that the CFVLBM is reliable and accurate.

ACKNOWLEDGEMENT

This work was supported by the Key Project of National Natural Science Foundation of China. (No.50636050, 51136004).

Copyright © 2011 John Wiley & Sons, Ltd. Int. J. Numer. Meth. Fluids 2012; 70:200–221DOI: 10.1002/fld

220 H. B. LUAN ET AL.

REFERENCES

1. Succi S. Lattice Boltzmann Equation for Fluid Flow and Beyond. Clarendon Press: Oxford, 2001.2. Chen SY, Doolen GD. Lattice Boltzmann method for fluid flows. Annual Review of Fluid Mechanics 1998;

30:329–364.3. He YL, Wang Y, Li Q. Lattice Boltzmann Method: Theory and Applications. Science Press: Beijing, 2008.4. Guo ZL, Zheng CG. Theory and Applications of Lattice Boltzmann Method. Science Press: Beijing, 2008.5. Martinez DO, Matthaeus WH, Chen S. Comparison of spectral method and lattice Boltzmann simulations of

two-dimension hydrodynamics. Physics of Fluids 1994; 6:1285–1298.6. Noble DR, Georgiadis JG, Buckius RO. Comparison of accuracy and performance for lattice Boltzmann and

finite difference simulations of steady viscous flow. International Journal for Numerical Methods in Fluids 1996;23(1):1–18.

7. Yoshino M, Matsuda Y, Shao C. Comparison of accuracy and efficiency between the lattice boltzmann method andthe finite difference method in viscous/thermal fluid flows. International Journal of Computational Fluid Dynamics2004; 18(4):333–345.

8. Alexander FJ, Chen S, Sterling JD. Lattice Boltzmann thermo-hydrodynamics. Physical Review E 1993;47(4):R2249–R2252.

9. Qian YH. Simulating thermohydrodynamics with lattice BGK models. Journal of Scientific Computing 1993;8(3):231–241.

10. Chen Y, Ohashi H, Akiyama M. Thermal lattice Bhatnagar–Gross–Krook model without nonlinear deviations inmacrodynamic equations. Physical Review E 1994; 50(4):2776–2783.

11. He X, Chen S, Doolen GD. A novel thermal model for the lattice Boltzmann method in incompressible limit. Journalof Computational Physics 1998; 146(1):282–300.

12. Guo ZL, Shi BC, Zheng CG. A coupled lattice BGK model for the Boussinesq equations. International Journal forNumerical Methods in Fluids 2002; 39(4):325–342.

13. Guo ZL, Zheng CG, Shi BC, Zhao TS. Thermal lattice Boltzmann equation for low Mach number flows: decouplingmodel. Physicla Review E 2007; 75(036704):1–15.

14. Lallemand P, Luo LS. Theory of the lattice Boltzmann method: acoustic and thermal properties in two and threedimensions. Physical Review E 2003; 68(036706):1–25.

15. Mezrhab A, Bouzidi M, Lallemand P. Hybrid lattice-Boltzmann finite-difference simulation of convective flows.Computers and Fluids 2004; 33(4):623–641.

16. McNamara G, Garcia AL, Alder BJ. Stabilization of thermal lattice Boltzmann models. Journal of Statistical Physics1995; 81(1–2):395–408.

17. Succi S, Filippova O, Smith G, Kaxiras E. Applying the lattice boltzmann equation to multiscale fluid problems.Computing in Science and Engineering 2001; 3(6):26–37.

18. Hlushkou D, Kandhai D, Tallarek U. Coupled lattice-Boltzmann and finite-difference simulation of electroosmosisin microfluidic channels. International Journal for Numerical Methods in Fluids 2004; 46(5):507–532.

19. Chatterji A, Horbach J. Combining molecular dynamics with lattice Boltzmann: a hybrid method for the simulationof (charged) colloidal systems. Journal of Chemical Physics 2005; 122(18):18490318.

20. Huang HB, Lee TS, Shu C. Hybrid lattice Boltzmann finite-difference simulation of axisymmetric swirling androtating flows. International Journal for Numerical Methods in Fluids 2007; 53(11):1707–1726.

21. Han K, Feng YT, Owen D. Coupled lattice Boltzmann and discrete element modeling of fluid-particle interactionproblems. Computers and Structures 2007; 85(11-14Sp. Iss. SI):1080–1088.

22. Joshi H, Agarwal A, Puranik B, Shu C, Agrawal A. A hybrid FVM-LBM method for single and multi-fluidcompressible flow problems. International Journal for Numerical Methods in Fluids 2010; 62(4):403–427.

23. Dupuis A, Kotsalis EM, Koumoutsakos P. Coupling lattice Boltzmann and molecular dynamics models for densefluids. Physical Review E 2007; 75:0467044.

24. Kwon YW, Jo JC. 3D modeling of fluid-structure interaction with external flow using coupled LBM and FEM.Journal of Pressure Vessel Technology 2008; 130(2):0213012.

25. Christensen A, Graham S. Multiscale lattice Boltzmann modeling of phonon transport in crystalline semiconductormaterials. Numerical Heat Transfer Part B-Fundamentals 2010; 57(2):89–109.

26. Luan HB, Xu H, Chen L, He YL, Tao WQ. Evaluation of the coupling scheme of FVM and LBM for fluid flowsaround complex geometries. International Journal of Heat and Mass Transfer 2011; 54:1975–1985.

27. Desplat JC, Pagonabarraga I, Bladon P. Ludwig: a parallel lattice-Boltzmann code for complex fluids. ComputerPhysics Communications 2001; 134(3):273–290.

28. Pan CX, Prins JF, Miller CT. A high-performance lattice Boltzmann implementation to model flow in porous media.Computer Physics Communications 2004; 158(2):89–105.

29. Tao WQ, He YL. Recent advances in multiscale simulation of heat transfer and fluid flow problems. Progress inComputational Fluid Dynamics 2009; 9:151–157.

30. Tao W Q, He YL. Multiscale simulations of heat transfer and fluid flow problems. Proceedings of the 14thInternational Heat Transfer Conference IHTC-14, Washington D.C., USA, IHTC14-2340, August 8.

31. Xu H. Lattice Boltzmann model for turbulent fluid flow simulation and its application in multiscale analysis,Dissertation of Xi’an Jiaotong University, 2009.

32. Luan HB, Xu H, Chen L, Sun DL, Tao WQ. Numerical illustrations of the coupling between lattice Boltzmannmethod and finite-type macro-numerical methods. Numerical Heat Transfer B 2010; 57(2):147–171.

Copyright © 2011 John Wiley & Sons, Ltd. Int. J. Numer. Meth. Fluids 2012; 70:200–221DOI: 10.1002/fld

EXTENDING THE CFVLBM TO NATURAL CONVECTION 221

33. Bhatnagar PL, Gross EP, Krook M. A model for collision processes in gases, part I. small amplitude processes incharged and neutral one-component system. Physical Review 1954; 94:511–525.

34. He X, Luo LS. A priori derivation of the lattice Boltzmann equation. Physical Review E 1997; 55:R6333–6.35. He X, Luo LS. Theory of the lattice Boltzmann equation: from Boltzmann equation to lattice Boltzmann equation.

Physical Review E 1997; 56:6811–7.36. Qian YH, d’Humiéres D, Lallemand P. Lattice BGK models for Navier-Stokes equation. Europhysics Letters 1991;

15:603–607.37. Gray DD, Giorgin A. The validity of the Boussinesq approximation for liquids and gases. International Journal of

Heat Mass Transfer 1976; 19:545–551.38. Guo ZL, Zhao TS. Lattice Boltzmann model for incompressible flows through porous media. Physical Review E

2002; 66:036304.1–036304.9.39. Patankar SV. Numerical Fluid Flow and Heat Transfer. McGraw-Hill: New York, 1980.40. Tao WQ. Numerical Heat Transfer, 2nd ed. Xi’an: Xi’an Jiaotong University Press: Xi’an, 2000.41. Tao WQ, Qu ZG, He YL. A novel segregated algorithm for incompressible fluid flow and heat transfer prob-

lems - CLEAR (Coupled and Linked Equations Algorithm Revised) part I: mathematical formulation and solutionprocedure. Numerical Heat Transfer B 2004; 45:1–17.

42. Tao WQ, Qu ZG, He YL. A novel segregated algorithm for incompressible fluid flow and heat transfer problems -CLEAR (Coupled and Linked Equations Algorithm Revised) part II: application examples. Numerical Heat TransferB 2004; 45:19–48.

43. Sun DL, Qu ZG, He YL, Tao WQ. An efficient segregated algorithm for incompressible fluid flow and heat trans-fer problems - IDEAL (Inner Doubly Iterative Efficient Algorithm for Linked Equations) part I: mathematicalformulation and solution procedure. Numerical Heat Transfer B 2008; 53:1–17.

44. Sun DL, Qu ZG, He YL, Tao WQ. An efficient segregated algorithm for incompressible fluid flow and heat transferproblems - IDEAL (Inner Doubly Iterative Efficient Algorithm for Linked Equations) part II: application examples.Numerical Heat Transfer B 2008; 53:18–38.

45. Li ZY, Tao WQ. A new stability-guaranteed second-order difference scheme. Numerical Heat Transfer B 2002;42:349–365.

46. Guo ZL, Zhao TS. A lattice Boltzmann model for convection heat transfer in porous media. Numerical Heat TransferB 2005; 47:157–177.

47. Tao WQ. Recent Advances in Computational Heat Transfer. Science Press: Beijing, 2005.48. Davis G. Natural convection of air in a square cavity: a benchmark numerical solution. International Journal for

Numerical Methods in Fluids 1983; 3:249–264.49. Saitoh T, Hirose K. High-accuarcy bench mark solutions to natural convection in a square cavity. Computational

Mechanics 1989; 4:417–427.50. Kuznik F, Vareilles J, Rusaouen G, Krauss G. A double-population lattice Boltzmann method with non-uniform

mesh for the simulation of natural convection in a square cavity. International Journal of Heat and Fluid Flow 2007;28:862–870.

51. Yang M, Tao WQ, Wang QW, Lue SS. On identical problems of natural convection in enclosures and applications ofthe identity character. Journal of Thermal Science 1993; 2:116–125.

52. Wang QW, Yang M, Tao WQ. Natural convection in a square enclosure with an internal isolated vertical plate. Heatand mass transfer 1994; 29:161–169.

Copyright © 2011 John Wiley & Sons, Ltd. Int. J. Numer. Meth. Fluids 2012; 70:200–221DOI: 10.1002/fld