Computers & Fluids 39 (2010) 1542–1548

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Computers & Fluids

journal homepage: www.elsevier .com/locate /compfluid

Multi-relaxation-time lattice Boltzmann model for axisymmetric flows

Liang Wang a, Zhaoli Guo a,b,*, Chuguang Zheng a

a National Laboratory of Coal Combustion, Huazhong University of Science and Technology, Wuhan 430074, PR Chinab State Key Laboratory of Enhanced Oil Recovery, Research Institute of Petroleum Exploration and Development, Beijing 100083, PR China

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

Article history:Received 19 December 2009Received in revised form 6 May 2010Accepted 11 May 2010Available online 15 May 2010

Keywords:Lattice Boltzmann equationMultiple-relaxation-timeAxisymmetric flow

0045-7930/$ - see front matter � 2010 Elsevier Ltd. Adoi:10.1016/j.compfluid.2010.05.007

* Corresponding author at: National Laboratory ofUniversity of Science and Technology, Wuhan 430074

E-mail address: zlguo@hust.edu.cn (Z. Guo).

In this paper, a lattice-Boltzmann equation (LBE) with multi relaxation times (MRT) is presented foraxisymmetric flows. The model is an extension of a recent model with single-relaxation-time [Guoet al., Phys. Rev. E 79, 046708 (2009)], which was developed based on the axisymmetric Boltzmannequation. Due to the use of the MRT collision model, the present model can achieve better numericalstability. The model is validated by some numerical tests including the Hagen-Poiseuille flow, the pul-satile Womersley flow, and the external flow over a sphere. Numerical results are in excellent agree-ment with analytical solutions or other available data, and the improvement in numerical stability isalso confirmed.

� 2010 Elsevier Ltd. All rights reserved.

1. Introduction

Axisymmetric flows are frequently encountered in many prac-tical and engineering problems. Many numerical methods, such asfinite element method (FEM), finite volume method (FVM), finitedifference method (FDM) and particle strength exchange (PSE)method, have been developed for such flows [1–7]. As a newnumerical method for fluid flows, the lattice Boltzmann equation(LBE) has also been applied to such flows. The most straightfor-ward approach for LBE to handle an axisymmetric flow is to em-ploy certain three-dimensional (3D) LBE models with propercurve boundary treatments [8–11]. On the other hand, severalquasi-two-dimensional LBE models were also developed for axi-symmetric flows based on the fact that an axisymmetric flow isa quasi-two-dimension (2D) problem in the meridian plane. Thefirst axisymmetric LBE model was proposed by Halliday et al.[12], in which some source terms were introduced in order to re-cover the cylindrical Navier–Stokes (NS) equations through theChapman–Enskog expansion. The model was further developedto simulate multiphase flows by incorporating some interfacialphysics [13,14]. Later, Lee et al. [15] pointed out that some impor-tant terms are missing in Halliday’s model, and they proposed anew one [15]. Similar models were also developed in Refs.[16,17]. Zhou [18] developed a similar scheme to simplify thesource term. Recently, another version of axisymmetric model

ll rights reserved.

Coal Combustion, Huazhong, PR China.

[19] based on the vorticity-stream equations was constructedfor incompressible axisymmetric flows. These LBE models havebeen applied to different problems [20–28].

The above axisymmetric LBE models are all designed in a top-down fashion starting from the macroscopic hydrodynamicequations. Recently Guo et al. proposed a simple and effectiveaxisymmetric LB model directly based on the axisymmetricBoltzmann equation [29], in which the source terms are muchsimplier.

In all of the existing LBE models the collision operators aremodeled by the Bhatnagar–Gross–Krook (BGK) or single-relaxa-tion-time model. However, it is known that the BGK model maysuffer from numerical instability when simulating fluids with rel-atively low viscosities [30]. One approach suggested to improvenumerical stability is to make use of a multi-relaxation-time(MRT) collision model instead of the BGK model [30–32]. Becausethe relaxation times can be adjusted independently, the numeri-cal stability of the MRT-LBE models can be enhanced. In addition,the LBE with MRT model maintains the simplicity and computa-tional efficiency as the BGK counterparts. In recent years MRT-LBE has received particular attention due to its advantages. How-ever, to the authors’s knowledge, no works have been reported onMRT-LBE for axisymmetric flows. In this paper, we aim to proposesuch an axisymmetric MRT-LBE model based on the BGK modelproposed in Ref. [29].

The remaining part of the paper is organized as follows. In Sec-tion 2, a MRT-LBE model is first proposed for axisymmetric flows.Section 3 is devoted to the boundary conditions and method forforce evaluation. Numerical tests are then carried out in Section 4,and the paper is closed by a brief summary in Section 5.

L. Wang et al. / Computers & Fluids 39 (2010) 1542–1548 1543

2. Axisymmetric lattice Boltzmann model with multiplerelaxation times

The axisymmetric LBE model proposed in Ref. [29] is a two-dimension-nine-velocity (D2Q9) model in the cylindrical coordi-nate system. Without considering the azimuthal velocity for axi-symmetric flow, the LBE can be described as

giðxþ cidt ; t þ dtÞ � giðx; tÞ ¼ �x½giðx; tÞ � gðeqÞi ðx; tÞ�

þ dt 1�x2

� �Giðx; tÞ; ð1Þ

where x = (x,r), and x and r are the axial and radial coordinates inthe cylindrical coordinate system, respectively. Here x = 1/s, withs being the dimensionless relaxation time.

The discrete velocities {ci = (cix,cir): i = 0,1, . . . ,8} are given by

ci ¼ð0;0Þ; i¼0;ðcos½ði�1Þp=2�;sin½ði�1Þp=2�Þc; i¼1�4;

ðcos½ði�1Þp=2þp=4�;sin½ði�1Þp=2þp=4�Þffiffiffi2p

c; i¼5�8;

8><>:ð2Þ

where c = dx/dt is the lattice speed, and dx and dt are the lattice con-stant and the time step, respectively.

The discrete equilibrium distribution function (EDF) is definedas

gðeqÞi ¼ rqwi 1þ ci � u

c2sþ ðci � uÞ2

2c4s� juj

2

2c2s

" #ð3Þ

In the above equation, the weighting coefficients wi are given byw0 = 4/9, w1�4 = 1/9 and w5�8 = 1/36, and cs ¼

ffiffiffiffiffiffiRTp

with R and Tbeing the gas constant and temperature, respectively.

The forcing term in Eq. (1) is defined as

Gi ¼ðci � uÞ � ~a

RTgðeqÞ

i ð4Þ

where ~a ¼ ð~ax; ~arÞ is given by

~ax ¼ ax; ~ar ¼ ar þRTr

1þ 2sur

r

� �ð5Þ

where u = (ux,ur) and a = (ax,ar) are the corresponding fluid velocityand force acceleration in the meridian plane, respectively.

The fluid density q and velocity u are defined as

q ¼ 1r

Xi

gi ð6Þ

qux ¼1r

Xi

cixgi þdt

2rqax

" #ð7Þ

qur ¼r

r2 þ sdtRT

Xi

cirgi þdt

2rqar þ

dt

2rqRT

" #ð8Þ

The above axisymmetric LBE model is based on the BGK approx-imation. It is known that numerical instability may appear as fluidviscosity decreases for BGK model in the cartesian coordinate sys-tem. Such defect may also exist in the model described by Eq. (1).One way [31,30,32,33] to overcome this drawback is to employ amultiple-relaxation-time (MRT) model in lieu of the BGK modelto approximate the collision operator.

Our proposed MRT-LBE with a forcing term is expressed as:

giðxþ cidt ; t þ dtÞ � giðx; tÞ ¼ �Sia½gaðx; tÞ � gðeqÞa ðx; tÞ�

þ dt dia �Sia

2

� �Gaðx; tÞ; ð9Þ

where Sia is the component of the collision matrix S, and dia is theKronecker delta function. Note that the MRT model (9) reduces tothe BGK model when setting S = xI.

The evolutionary process Eq. (9) can be decomposed into twosubsteps, i.e., the collision process,

gþi ðx; tÞ ¼ giðx; tÞ � Sia½gaðx; tÞ � gðeqÞa ðx; tÞ� þ dt dia �

Sia

2

� �Gaðx; tÞ;

ð10Þ

and the streaming process,

giðxþ cidt; t þ dtÞ ¼ gþi ðx; tÞ: ð11Þ

As noted in Refs. [31,30], the moment representation provides anatural and convenient way to describe the collision process. Inthe velocity space, the matrix S is usually a full matrix which leadsto inconvenience when performing the collision step. However, inthe moment space the collision matrix can be transformed into adiagonal matrix, which brings much convenience to the implemen-tation of the collision process. We will also adopt this method inthis work. To this end, by multiplying by a transformation matrixM, Eq. (9) can be mapped onto the moment space and expressedas:

gðxþ cidt; t þ dtÞ � gðx; tÞ ¼ �M�1eS ½mðx; tÞ �mðeqÞðx; tÞ�

þ dtM�1 I �

eS2

!bGðx; tÞ; ð12Þ

where g is the 9-tuple vector of the distribution functions, m = Mg isthe moment vector, eS ¼MSM�1 is a diagonal matrix whose compo-nents represent the relaxation rates of the moments. m(eq) denotesthe equilibria in moment space, and bG ¼MG is the correspondingforcing term in moment space.

As shown in Refs. [31,30,34], the transformation matrix M canbe constructed via the Gram–Schmidt orthogonalization procedurefrom some polynomials of the components of the discrete veloci-ties. Here we use that given in Ref. [30]:

M ¼

1 1 1 1 1 1 1 1 1�4 �1 �1 �1 �1 2 2 2 24 �2 �2 �2 �2 1 1 1 10 1 0 �1 0 1 �1 �1 10 �2 0 2 0 1 �1 �1 10 0 1 0 �1 1 1 �1 �10 0 �2 0 2 1 1 �1 �10 1 �1 1 �1 0 0 0 00 0 0 0 0 1 �1 1 �1

0BBBBBBBBBBBBBBBB@

1CCCCCCCCCCCCCCCCAð13Þ

With this transformation matrix, we can define the moments as

m ¼Mg ¼ rðq; e; e; jx; qx; jr ; qr; pxx; pxrÞT ð14Þ

where q is the mass density, e is the total energy, e is related to theenergy square, jx and jr correspond to x and r components of themomentum, i.e., jx = qux, jr = qur, qx and qr are the x and r compo-nents of the energy flux, and pxx and pxr relate to the diagonal andoff-diagonal components of the stress tensor, respectively. The cor-responding equilibria in moment space are given by:

mðeqÞ ¼MgðeqÞ ¼ rðq; eðeqÞ; eðeqÞ; jx; qðeqÞx ; jr; q

ðeqÞr ;pðeqÞ

xx ; pðeqÞxr Þ

T ð15Þ

The density q and momentum j := qu := (jx, jr) are conserved quan-tities. The equilibria of other non-conserved moments, which arefunctions of the conserved moments [31,30,35], are given by

eðeqÞ ¼ qð�2þ 3u2x þ 3u2

r Þ; ð16ÞeðeqÞ ¼ �qð�1þ 3u2

x þ 3u2r Þ; ð17Þ

qðeqÞx ¼ �qux; ð18Þ

qðeqÞr ¼ �qux; ð19Þ

Fig. 1. Illustration of the nonequilibrium extrapolation method for a curvedboundary. Solid circle: wall node, open circle: fluid node, solid square: physicalboundary node in the link of fluid node and wall node. Dashed-dotted linerepresents the symmetry axis and the dashed line represents the ghost boundaryabout the symmetry axis.

1544 L. Wang et al. / Computers & Fluids 39 (2010) 1542–1548

pðeqÞxx ¼ qðu2

x � u2r Þ; ð20Þ

pðeqÞxr ¼ quxur : ð21Þ

The expression of the forcing term bG in moment space can beexplicitly expressed as:

bG ¼MG ¼ r

0�3qðux~ax þ ur~arÞð�2þ 3u2

x þ 3u2r Þ

3qðux~ax þ ur~arÞð�2þ 3u2x þ 3u2

r Þq~ax

qð3~axu2r þ 6uxur~ar � ~axÞ

q~ar

qð3~aru2x þ 6uxur~ax � ~arÞ

�q 3ðux~ax þ ur~arÞðu2x � u2

r Þ � 2ðux~ax � ur~arÞ� ��q 3uxurðux~ax þ ur~arÞ � ðux~ar þ ur~axÞ½ �

0BBBBBBBBBBBBBBBB@

1CCCCCCCCCCCCCCCCA:

ð22Þ

The relaxation matrix eS in moment space is given byeS ¼ diagðs0; s1; . . . ; s8Þ; si P 0 ð23Þ

where si is the relaxation rate for moment mi. Considering that thedensity and momentum are conserved during the collision process,the corresponding relaxation rates, s0, s3, and s5, can take any val-ues. The relaxation parameters for these collision invariants areset to be zero in this work. So, eS can be expressed aseS ¼ diagð0; s1; s2;0; s4;0; s6; s7; s8Þ; ð24Þ

With the equilibria given by Eqs. (16)–(21), the following axi-symmetric hydrodynamic equations can be recovered throughthe Champan–Enskog expansion procedure (see Appendix A fordetails):

@tqþ @aðquaÞ þqur

r¼ 0 ð25Þ

@tðquaÞ þ @bðquaubÞ ¼ �@apþ @b½lð@aub þ @buaÞ�

þ lrð@rua þ @aurÞ �

quaur

r� 2l

r2 urdar ð26Þ

where l = qm is the dynamic viscosity with

m ¼ c2s s� 1

2

� �dt ð27Þ

being the kinematic viscosity, cs ¼ c=ffiffiffi3p

is the sound speed, and s isrelated to the relaxation rates s7 and s8 through s ¼ s�1

7 ¼ s�18 .

In the numerical implementation of this axisymmetric MRTmodel, the collision step is operated in the moment space,

mþ ¼ m� eS ½m�mðeqÞ� þ dtðI �eS2ÞbG; ð28Þ

whereas the streaming step is still executed in the velocity space,

gþ ¼M�1mþ; giðxþ cidt; t þ dtÞ ¼ gþi ðx; tÞ ð29Þ

Following the above computational procedures, the distributionfunctions can be used to obtain the hydrodynamic variables asshown in Eqs. (6)–(8).

3. Boundary conditions and force evaluation

Treatment of boundary conditions is one of the key points forLBE. In this work, we will extend the nonequilibrium extrapolationscheme (NEES) proposed in Refs. [29,36], which was deviced forBGK model, to the current MRT-LBE model. Following the basicidea of the NEES, we decompose the distribution function gi atthe wall node xw (see Fig. 1) into its equilibrium part and nonequi-librium part, giðxwÞ ¼ gðeqÞ

i ðxwÞ þ gðneqÞi ðxwÞ. The equilibrium part is

then approximated as [29] gðeqÞi ðxwÞ � gðeqÞ

i ðqw;uwÞ, where qw =q(x1), and

uw ¼ð3�DÞuðxbÞþðD2�1Þuðx1Þ�ð1�DÞ2uðx2Þ

1þD ; D < 0:75;uðxbÞþðD�1Þuðx1Þ

D ; D P 0:75:

(ð30Þ

where D = jx1 � xbj/jx1 � xwj is the fraction of an intersected link inthe fluid region along ci (x1 and x2 are the two nearest nodes alongci of the wall node xw). For the nonequilibrium part, it is approxi-mated by extrapolation [29],

gðneqÞi ðxwÞ ¼

DgðneqÞi ðx1Þ þ ð1� DÞgðneqÞ

i ðx2Þ; D < 0:75;

gðneqÞi ðx1Þ; D P 0:75:

(ð31Þ

where the nonequilibrium distribution function at x1 or x2 is givenby gðneqÞ

i ¼ gi � gðeqÞi . Thus, the post-collision distribution function at

xw can be determined as

g0ðxwÞ ¼ gðeqÞðqw;uwÞ þM�1ðI � eSÞmðneqÞðxwÞ

þ dtM�1 I �

eS2

!bGðxwÞ; ð32Þ

where m(neq)(xw) = Mg(neq)(xw), and bGðxwÞ ¼MGðxwÞ.The above boundary condition can be implemented in two steps

in an analogous manner to Eqs. (28) and (29). The first step is exe-cuted in moment space as

mþw ¼ ðI � eSÞmðneqÞðxwÞ þ dt I �

eS2

!bGðxwÞ; ð33Þ

Then we can determine the post-collision distribution function g0i atthe wall node xw as:

gþðxwÞ ¼ M�1mþw; g0iðxwÞ ¼ gðeqÞ

i ðqw;uwÞ þ gþi ðxwÞ ð34Þ

The NEES presented above remains the simplicity, second-orderaccuracy, and good robustness [36] as its original version for theBGK model [29]. Noticing that the singularity emerging at r = 0,we employ the symmetry boundary condition [29] in this workto overcome the difficulty in dealing with the axis of symmetry.

As noted in Ref. [29], momentum exchange will come about as aresult of the interactions between the fluid and the wall. The

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

Analytical

τ=1.25τ=0.75

Fig. 2. Velocity profile of the Poiseuille flow at Re = 40, dx = R/16.

Table 1Comparisons of mmin(smin) against different grid resolution.

1 R/dx mmin (smin) mMRT/mBGKRef. [29] Present

0.0005 5 0.3333 � 10�3(0.505) 0.3333 � 10�4(0.5005) 0.100.0005 10 0.0667 � 10�3(0.502) 0.1667 � 10�4(0.5005) 0.250.0001 16 0.0042 � 10�3(0.5002) 0.0208 � 10�4(0.5001) 0.500.00005 32 0.0010 � 10�3(0.5001) 0.0052 � 10�4(0.50005) 0.50

L. Wang et al. / Computers & Fluids 39 (2010) 1542–1548 1545

momentum exchange at a wall node also leads to a force exertingon the wall,

Fðxb; tÞ ¼ �ci

dt

g0iðxw; tÞrw

þg0i0 ðx1; tÞ

r1

; ð35Þ

where xb is a boundary node, x1 denotes the lattice point at whichthe particles move toward xb with velocity ci0 ¼ �ci. Therefore, thetotal drag force on the wall exerted by the fluid can be computed as

Fx ¼ 2pZ

CdxrFxðxÞ dl � �2pd2

x

dt

Xcix g0iðxw; tÞ þ g0i0 ðx1; tÞ� �

; ð36Þ

where C is the body surface in the meridian half-plane.

4. Numerical tests

In this section, some numerical tests are conducted to validatethe proposed MRT-LBE model. The test problems include the stea-dy and unsteady flows through a circular tube, and the flow over asphere. These flows have been well studied by many researchersand can serve as good benchmark problems for the present LBEmodel.

In our simulations, the symmetry boundary condition is appliedto the symmetric axis, and the NEES is used to treat other bound-aries. The relaxation rates s7 and s8 are determined by Eq. (27)according to the value of Reynolds number, while others are cho-sen to be s1 = 1.1, s2 = 1.0, s4 = s6 = 1.2. Different values of s1, s2,and s4 (=s6) are also used and it is found that the difference isnegligible.

4.1. Steady poiseuille flow

We first consider the Hagen-Poiseuille flow through a straightpipe of radius R driven by a constant force a = (ax,0). Here, the ra-dius R is set to be 1, and the computational domain is restrictedwithin 0 6 x 6 0.1 and 0 6 r 6 R. The symmetry axis lies at r = 0and the solid wall is located at r = R. The boundary conditions forthe fluid variables are as follows:

r ¼ 0 :@u@r¼ 0; 8u;

r ¼ R : ux ¼ ur ¼ 0:

In the streamwise (x) direction, periodic boundary conditions areapplied to the inlet and outlet. With these conditions, the axialvelocity can be described as

uxðrÞ ¼ u0 1� r2

R2

� �; ð37Þ

where u0 = axR2/4m is the maximum velocity.

In our simulation, the Reynolds number is defined as Re = 2Ru0/m, and the grid resolution is dx = R/16. The density and velocity ofthe fluid are initialized to be q = 1.0 and ux = ur = 0.0 in the wholedomain. After a number of iterations a steady state is reached,where the criterion for steady state isP

i;jjuxðxi; rj; tÞ � uxðxi; rj; t � 100dtÞjPi;jjuxðxi; rj; tÞj

< 10�6; ð38Þ

where the summation is taken over all grid points.Fig. 2 presents the velocity profiles at Re = 40 with two different

values of the relaxation time s (0.75 and 1.25) together with theanalytical solution. It is seen that the numerical results are inexcellent agreement with the analytical solutions.

We now compare the numerical stability of the present MRTmodel with that of the BGK model proposed in Ref. [29]. SinceLBE will encounter numerical instability for small viscosity or asthe relaxation time s approaches to 0.5, we can assert the stability

properties of the MRT model and the BGK model by measuring theminimum value of s (i.e., smin) at which the computation is still sta-ble. However, it is hard to determine the exact smin. As proposed byHuang et al. [24], we can roughly obtain smin by selecting s with asmall increment 1 from 0.5, and recognize smin as the value atwhich numerical instability does not appear. Following this idea,we measured smin for both the MRT and BGK models with differentlattice sizes. After having the value of smin, the corresponding min-imum viscosity, mmin, can be obtained subsequently from Eq. (27).The results are listed in Table 1.

It is observed from the Table that the values of smin of both mod-els decrease with increasing grid resolution. In particular, smin ofthe present MRT model is considerably closer to 0.5 than that ofthe BGK model on the same mesh. Also, the ratios of mmin of thepresent model to that of the BGK model are always less than unity,which means lower viscosity can be attained in the current model.These results clearly demonstrate the improvement in numericalstability by making use of the MRT model in LBE.

4.2. Unsteady womersley flow

To further test the present MRT model, we now apply it to theunsteady axisymmetric Womersley flow in a pipe. The geometryof this flow is the same as that of the Hagen-Poiseuille flow, exceptthat the driven force is oscillating with a frequency X, i.e., ax = G-cos(Xt). Here G is the maximum amplitude of the cosinusoidallyvarying force. The analytical solution of the Womersley flow is

uxðr; tÞ ¼ RealGiX

1� J0ðrs=RÞJ0ðsÞ

eiXt

� �; ð39Þ

where R is the radius of the tube, i stands for the imaginary unit, s isdefined as s ¼ aði� 1Þ=

ffiffiffi2p

; J0 is the zero-order Bessel function, anda is the Womersley number defined by a ¼ R

ffiffiffiffiffiffiffiffiffiX=m

p, and Real de-

notes the real part of a complex number.In our simulation, boundary conditions are the same as those

used in the Poiseuille flow. The velocity is initialized to be zeroeverywhere in the system, and all numerical measurements aredone after running 10T, where T stands for the period of the driving

Fig. 4. The smin dependence of different mesh size.

1546 L. Wang et al. / Computers & Fluids 39 (2010) 1542–1548

force (T = 2p/X). Fig. 3 shows the velocity profiles at four differenttimes of the flow at two Womersley numbers, i.e., a = 8 and a = 16,for Re = 1200. Here the Reynolds number is defined as Re = 2Ru0/mwith u0 = GR2/4m. As shown, the agreement between the numericalsolutions and the analytical solutions given by Eq. (39) is quitegood. In the simulations, the lattice spacing dx and the relaxationtime s are taken to be R/20 and 0.6 respectively in the case ofa = 8. In the case of a = 16, dx is set to be R/80 so that the Machnumber of the flow is sufficiently small. The numerical stabilityof the MRT model is also tested. The values of smin are plottedagainst the grid size in Fig. 4, together with the results of theBGK model. It is clearly seen that smin of the MRT model is insensi-tive to grid resolution, whereas that of the BGK model is stronglygrid-dependent. The ratios of the achievable minimum viscositiesof the two models, mMRT/mBGK, are 0.04, 0.11, 0.25, 0.50 and 0.20for R/dx = 5, 10, 20, 32, and 64, respectively. These observations fur-ther demonstrate the improvement of the numerical stability ofthe MRT model.

Fig. 5. Drag coefficient with respect to Reynolds numbers for axisymmetric flowover a sphere (dx = R/16).

4.3. Flow over a sphere

The present MRT model is further tested by the flow over asphere. This flow has been a long-lasting subject of experimentaland numerical investigation [39,37,38,41–44,40,45] for its wide-spread applications as well as complicated fluid dynamics.

We simulate the flow at moderate Reynolds numbers (Re) rang-ing from 5 to 500, where Re is based on the free-stream velocity u0

and the sphere diameter 2R. The computational domain is 0 6 x 640R and 0 6 r 6 10R, where the sphere is positioned at (20R,0).No-slip boundary condition is applied to the surface of the sphere.The free-stream boundary conditions (ux = u0,ur = 0) and theconvection boundary conditions (@tu + u0@xu = 0, for u = q,ux,ur)are applied to the inlet and outlet of the flow, respectively.The boundary condition at the side boundary is set to be @ru = 0.The NEES is used to realize these boundary conditions.

Simulations are conducted on a mesh of Nx � Nr = 641 � 162size. The momentum-exchange method described in Section 3 isused to compute the fluid force acting on the sphere. To reachthe steady state, a number of time steps are performed. The work-ing criterion for steady state is given by Eq. (38). In Fig. 5 the dragcoefficient, Cd ¼ 4Fx=qu2

0R2, is plotted together with previous liter-ature data. Good agreement is observed between the LBE resultand other numerical and experimental data. Flows with higherRe are also simulated to investigate the numerical stability of theproposed MRT-LBE. It is observed that the computation blows upfor the BGK model [29] as Re P 330, whereas the present MRT

(a) (

Fig. 3. Comparsion of the simulated velocity profile (circle) with the analytical Womedx = R/80.

model is much more robust. For instance, the computation is stillstable even as Re = 1800. Of course at such a high Reynolds num-ber, the flow should be unsteady and non-axisymmetric, here thiscase is used only to demonstrate the good numerical stability androbustness of the present MRT model.

We also tested the minimum values of s (i.e., smin) at which thecomputation is stable for this flow at Re = 100 with different meshsizes, which are shown in Fig. 6. It is found that the correspondingviscosity ratio, mMRT/mBGK, varies from 0.100 to 0.125. These results

b)

rsley solution (solid line) at Re = 1200 and s = 0.6. (a) a = 8, dx = R/20; (b) a = 16,

Fig. 6. Comparison of smin of the flow past a sphere with different lattice spacing(Re = 100).

L. Wang et al. / Computers & Fluids 39 (2010) 1542–1548 1547

again demonstrated the improved numerical stability of the pres-ent MRT model in comparison with the BGK model [29].

5. Summary

In this paper we have proposed a LBE model with multiple relax-ation times for axisymmetric flows. The model is analyzed by theChapman–Enskog method and the axisymmetric hydrodynamicequations in the cylindrical coordinate system are recovered.

Numerical simulations of several axisymmetric flows have beenconducted to validate the proposed axisymmetric MRT-LBE model.Numerical accuracy are confirmed by comparing the LBE resultswith the analytical or existing numerical or experimental data inthe literature. Numerical stability is also tested, and the resultssuggest that the present MRT model is much stable than that ofthe corresponding BGK model [29]. These findings indicate thepresent MRT-LBE can serve as a powerful method for computationof axisymmetric flows.

Acknowledgements

L. Wang would like to thank Dr. L. Zheng for helpful discussions.This work is supported by the National Natural Science Foundationof China (10972087) and the National Basic Research Program ofChina (2006CB705804). Z. Guo is also supported by an open grantfrom the State Key Laboratory of EOR.

Appendix A. Champan–Enskog analysis of the axisymmetricMultiple-relaxation-time LBE model

In this appendix, we will analyze the proposed axisymmetricMRT model using the Champan–Enskog expansion method [46].First, we introduce the following expansions:

gi ¼X1n¼0

engðnÞi ; @t ¼X1n¼0

en@tn ð40Þ

giðxþ cidt ; t þ dtÞ ¼X1n¼0

en

n!Dn

i giðx; tÞ ð41Þ

where e = dt is a small expansion parameter, and Di = @t + ci �r = @t + cia@a in Eq. (41).

Through the above expansions, we can obtain the followingrelationship for Eq. (9) in the consecutive orders of the parametere as:

e0 : gð0Þi ¼ gðeqÞi ; ð42Þ

e1 : D0igð0Þi ¼ �Siagð1Þa þ Gi ð43Þ

e2 : @t1 gð0Þi þ D0i dia �Sia

2

� �gð1Þa ¼ �Siagð2Þa ð44Þ

where D0i ¼ @t0 þ ci � r. Multiplying M on both sides of Eqs. (42)–(44), we can obtain the following moment equations:

e0 : mð0Þ ¼ mðeqÞ ð45Þe1 : bD0mð0Þ ¼ �eSmð1Þ þ bG ð46Þ

e2 : @t1 mð0Þ þ bD0 I �eS2

!mð1Þ ¼ �eSmð2Þ ð47Þ

where bD0 ¼MD0M�1;D0 ¼ diagðD00;D01; . . . ;D08Þ, and

mð1Þ ¼ rð0; eð1Þ; eð1Þ;0; qð1Þx ;0; qð1Þr ;pð1Þxx ;pð1Þxr Þ

T:

From Eq. (46), the equations in moment space at the t0 time scalecan be explicitly written as:

r@t0

qqð�2þ 3u2Þqð�1þ 3u2Þ

qux

�qux

qur

�qur

qðu2x � u2

r Þquxur

0BBBBBBBBBBBBBBBB@

1CCCCCCCCCCCCCCCCAþ r@x

qux

0�qux

pþ qu2x

qð�1þ 3u2r þ u2

x Þquxur

quxur

23 qux

13 qur

0BBBBBBBBBBBBBBBB@

1CCCCCCCCCCCCCCCCA

þ @r

rqur

0�rqur

rquxur

rquxur

rðpþ qu2r Þ

rqð�1þ 3u2x þ u2

r Þ� 2

3 rqur

13 rqux

0BBBBBBBBBBBBBBBB@

1CCCCCCCCCCCCCCCCA¼ r

0�s1eð1Þ

�s2eð1Þ

0�s4qð1Þx

0�s6qð1Þr

�s7pð1Þxx

�s8pð1Þxr

0BBBBBBBBBBBBBBBB@

1CCCCCCCCCCCCCCCCA

þ r

0�3qðux~ax þ ur~arÞð�2þ 3u2Þ3qðux~ax þ ur~arÞð�2þ 3u2Þ

q~ax

qð3~axu2r þ 6uxur~ar � ~axÞ

q~ar

qð3~aru2x þ 6uxur~ax � ~arÞ

�q 3ðux~ax þ ur~arÞðu2x � u2

r Þ � 2ðux~ax � ur~arÞ� ��q 3uxurðux~ax þ ur~arÞ � ðux~ar þ ur~axÞ½ �

0BBBBBBBBBBBBBBBB@

1CCCCCCCCCCCCCCCCA: ð48Þ

where p ¼ c2s q; u2 ¼ u2

x þ u2r . Similarly, the equations of the con-

served moments at the t1 time scale can be obtained from Eq.(47) as:

@t1 ðqÞ ¼ 0; ð49Þ

r@t1 ðquxÞ þ r@x16

1� s1

2

� �eð1Þ þ 1

21� s7

2

� �pð1Þxx

þ @r r 1� s8

2

� �pð1Þxr

h i¼ 0; ð50Þ

r@t1 ðqurÞ þ r@x 1� s8

2

� �pð1Þxr

h iþ @r r

16

1� s1

2

� �eð1Þ � 1

21� s7

2

� �pð1Þxx

� � ¼ 0: ð51Þ

1548 L. Wang et al. / Computers & Fluids 39 (2010) 1542–1548

Notice that eð1Þ; pð1Þxx and pð1Þxr are unknown to be determined. UsingEq. (48) and neglecting the terms of order O(Ma)3, where Ma = u/cs

is the Mach number, we can obtain:

@t0q ¼ �q @xux þ @rur þur

r

� �ð52Þ

@t0 ðqu2x Þ ¼ 2qux~ax ð53Þ

@t0 ðqu2r Þ ¼ 2qur~ar ð54Þ

@t0 ðquxurÞ ¼ qux~ar þ qur~ax þ quxur @xux þ @rur þur

r

� �ð55Þ

With the above results, we can get:

� s1eð1Þ ¼ 6p @xux þ @rur þur

r

� �ð56Þ

� s7pð1Þxx ¼ 2pð@xux � @rurÞ ð57Þ

� s8pð1Þxr ¼13qð@xur þ @ruxÞ þ quxur @xux þ @rur þ

ur

r

� �ð58Þ

By substituting the above results into Eqs. (50) and (51), we canobtain the following two equations at the t1 time scale:

r@t1 ðquxÞ ¼ @xsð1Þxx þ @rsð1Þxr ð59Þr@t1 ðqurÞ ¼ @xsð1Þrx þ @rsð1Þrr ð60Þ

where

sð1Þxx ¼ r qf0 @xux þ @rur þur

r

� �þ qm0ð@xux � @rurÞ

h i;

sð1Þxr ¼ sð1Þrx ¼ r qm0ð@xur þ @ruxÞ þ 3qm0uxur @xux þ @rur þur

r

� �h i;

sð1Þrr ¼ r qf0 @xux þ @rur þur

r

� �þ qm0ð@rur � @xuxÞ

h i:

with m0 ¼ c2s ðs� 0:5Þ and f0 ¼ c2

s ðs�11 � 0:5Þ, where s = 1/s7 = 1/s8.

Based on the first-order Eq. (48) for the conserved moments andsecond-order results, i.e., Eqs. (49), (59) and (60), we can obtain theaxisymmetric hydrodynamic Eqs. (25) and (26), where the kine-matic shear and bulk viscosities are respectively given as:

m ¼ 13

1s7� 1

2

� �dt ; f ¼ 1

31s1� 1

2

� �dt :

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