Simulating compositional convection in the presence of rotation by lattice Boltzmann model

lable at ScienceDirect

International Journal of Thermal Sciences 49 (2010) 2093e2107

Contents lists avai

International Journal of Thermal Sciences

journal homepage: www.elsevier .com/locate/ i j ts

Simulating compositional convection in the presence of rotationby lattice Boltzmann model

Sheng Chen a,b,*

a State Key Laboratory of Coal Combustion, Huazhong University of Science and Technology, Wuhan 430074, ChinabNational Technology Center, WISCO, Wuhan 430083, China

a r t i c l e i n f o

Article history:Received 13 December 2009Received in revised form22 June 2010Accepted 22 June 2010Available online 23 July 2010

Keywords:Compositional convectionRotationLattice Boltzmann method

* State Key Laboratory of Coal Combustion, HuazhoTechnology, Wuhan 430074, China.

E-mail address: [email protected].

1290-0729/$ e see front matter � 2010 Elsevier Masdoi:10.1016/j.ijthermalsci.2010.06.016

a b s t r a c t

In most practical situations, compositional convection accompanies rotation due to the nature or arti-ficial controls. For simplicity, usually cylindrical coordinates are used for simulating such axisymmetricconvectional flows. However, to date the investigation on this field is quite sparse. In this paper a simpleLB model is employed to reveal the characteristics of such flows. In the present model, the governingequations for velocity, temperature and concentration fields all are solved in the same fashion. Even witha coarse grid resolution, the present model still works well for rotationally compositional convectionwith high Rayleigh number and relatively high Lewis number. With the aid of this model, two differentkinds of nontrivial compositional convection accompanying rotation, namely the compositionalconvection in a rotating annulus with horizontal temperature and vertical solutal gradients and that withopposing temperature and concentration gradients, are investigated. Especially, the latter is studied indetailed for the first time. It is found that due to the presence of rotation, the convective motion will besuppressed, for both situations. However, the former is more sensitive to the effect of rotation. For thelatter, secondary flowwill emerge when the Taylor number is big than 5.68 � 107 and the oscillatory flowis easier to emerge than its irrotational counterpart when the ratio of buoyancy forces approaches tounity. Moreover, the Galilean invariance and its influence on the present model are discussed.

� 2010 Elsevier Masson SAS. All rights reserved.

1. Introduction

The study of compositional convective motions has receivedconsiderable attention for a long time because of its relevance tomany industrial and geophysical applications [1e8]. In most prac-tical situations, rotation accompanies compositional convectiondue to the nature or artificial controls, for example, salt diffusing inoceans [6,7] andmaterial processing in rotatingmachines [3,4]. Theeffect of rotation plays very important role in these convectivemotions. By the introduction of rotation effect, the system entails,in addition to the usual forces, the Coriolis force, the centrifugalforce, and the curvature effects in the overall force balance, whichsignificantly enhance the complicated degrees of the problems.Generally, axisymmetric flows in a vertically-mounted rotatingcircular annulus have been studied as benchmark models for thesetypes of convections [2e4,7]. For simplicity, cylindrical coordinates

ng University of Science and

son SAS. All rights reserved.

were employed for numerical simulations [3,7]. However, even todate, the open literature on this field is quite sparse.

In the last two decades, the lattice Boltzmann (LB) models havematured for simulating and modeling complicated physical,chemical and social systems [9e27]. The implementation of an LBprocedure is quite easy. Parallelization of an LB model is naturalsince the relaxation is local and the performance increases nearlylinearly with the number of CPUs. Moreover, the LB models havebeen compared favourably with spectral methods [28], artificialcompressibility methods [29], finite volume methods [30,31], finitedifference methods [32e36], projection methods [37,38] andmultigrid method [39,40], all quantitative results further validateexcellent performance of the LB method not only in computationalefficiency but also in numerical accuracy. Due to these advantages,the LB method has been successfully used to simulate manyproblems, from laminar single phase flows to turbulent multiphaseflows [11,12]. Recently, an excellent work to combine the LB modelproposed by Luo and Girimaji with appropriate boundary condi-tions designed by Verhaeghe et al. [41] to study double-diffusivenatural convection in cubical and square enclosures was reportedby Verhaeghe and his cooperators [42]. What should be empha-sized is that the LB method has advantages over traditional

mailto:[email protected]

www.sciencedirect.com/science/journal/12900729

http://www.elsevier.com/locate/ijts

http://dx.doi.org/10.1016/j.ijthermalsci.2010.06.016



Nomenclature

c fluid particle speedRi,Ro radius of inner/outer wallTa Taylor numberu! fluid velocity vectore!k discrete velocity vectorK radius ratiog, f distribution functionsA aspect ratioNu Nusselt numberSh Sherwood numberRe Reynolds numberC mass fractionRa Rayleigh numberPr Prandtl numberLe Lewis numberN BrunteVaisala frequencySt frequency ratioS Svanberg vorticityT temperaturep pressureRp ratio of buoyancy forces

x! phase space

Greek symbols6 weight coefficientDt time step3 Ekman numbern kinematic viscositys, sJ relaxation timer densityu vorticityj streamfunctionF swirlY source termx, z weight coefficientsd sum of distributions in Eq. (35)c parameter to determine relaxation time in Eq. (36)2 pseudo time stepU collision term in Eq. (37)

Subscripts and superscriptseq equilibriumk discrete direction0 reference value

z

Ri

Ro

top wall

bottom wall

(rotational axis)

gravity

outer wall

inner wall

Fig. 1. The configuration of computational domain.

S. Chen / International Journal of Thermal Sciences 49 (2010) 2093e21072094

numerical methods for simulation of rotating flow: in conventionalcomputational fluid dynamics methods, high-order (for examplesecond-order) discretized scheme for the convectional term willdiverge [43].

But surprisingly, to the best knowledge of the present author, todate there does not exist any open literature on using LB models tostudy compositional convection in the presence of rotation. Untilnow there have been only several publications [43e45] discussinghow to simulate rotating flows with temperature differences by theLB models in cylindrical coordinates. In order to bridge the gap,a simple LB model for compositional convection in the presence ofrotation, which is an extension of our previous work [45e47], isemployed to investigate compositional convection in the presenceof rotation in this paper. The presentmodel possesses three obviousadvantages inheriting from our previous models proposed in Refs.[45e47]: The first, the present model keeps the simplicity of code,in which the governing equations for velocity, concentration andtemperature fields all are solved by the two-dimensional five-speeds (D2Q5) LB model [9]. The second, even with a coarse gridresolution, the present model can still work efficiently and stablyfor flow with high Rayleigh numbers accompanying rotation.Consequently the computational cost is significantly reduced. Thethird, the derivation of the present model is quite straightforward.The numerical experiments for several benchmark tests validatethe excellent performance of the present model. In addition, thepresent model can work well for simulations with relatively highLewis number (Le ¼ k/D, where k is the thermal conductivity and Dis the species diffusivity) around the order of ten. It is well knownthat the high Lewis number is a challenge for a numerical schemedesigned for compositional convection [48]. In most previous LBmodels, the Lewis number is limited about the order of unity[10,11,43,44]. With the aid of this model, two different kinds ofnontrivial compositional convection accompanying rotation,namely the compositional convection in a rotating annulus withhorizontal temperature and vertical solutal gradients and that withopposing temperature and concentration gradients [4,49], areinvestigated. Especially, the latter is studied in detailed for the firsttime. For the present study, it is the main purpose to deepen our

knowledge of the effect of rotation on compositional convectionthrough the present model.

The rest of the paper is organized as follows. The governingequations for compositional convection in a rotating annulus arepresented in Section 2. In Section 3, a simple LB model for suchconvectional phenomena accompanying rotation is introduced. InSection 4 numerical experiments are performed to validate thepresent model. The investigation for compositional convection inrotating annuli and corresponding new findings are presented inSection 5. Conclusion is made in the last section.

2. Governing equations for compositional convectionin the presence of rotation

The configuration of the vertical annulus is illustrated in Fig. 1,which is popularly used in previous studies [4]. The inner wall with

S. Chen / International Journal of Thermal Sciences 49 (2010) 2093e2107 2095

the radius Ri and the outer wall with Ro. K¼ Ro/Ri is the radius ratio.The aspect ratio A¼H/(Ro� Ri).H is the height of the annular cavity.The entire setup rotates steadily at angular frequency U0 about thecentral axis z.

With the Boussinesq assumption, the primitive-variables-basedgoverning equations for compositional convection accompanyingrotation in the cylindrical coordinates read [3,4,7]:

1rvðruÞvr

þ vwvz

¼ 0; (1)

vuvt

þ uvuvr

þwvuvz

� v2

r¼ �1

r

vpvr

þ n

�V2u� u

r2

�; (2)

vv

vtþ u

vv

vrþw

vv

vzþ uv

r¼ n

�V2v� v

r2

�; (3)

vwvt

þ uvwvr

þwvwvz

¼ �1r

vpvz

þ nV2wþ gaTDT � gacDC; (4)

vTvt

þ uvTvr

þwvTvz

¼ kV2T ; (5)

vCvt

þ uvCvr

þwvCvz

¼ DV2C: (6)

where

V2 ¼ 1r

v

vr

�rv

vr

�þ v2

vz2:

and u, v andw are radial, azimuthal and axial velocity components;p is the pressure; T is the temperature; C is the concentration; n isthe kinetic viscosity; g is the gravitational acceleration along thenegative z-axis; k is the thermal conductivity; r is the density; DT isthe temperature difference; DC is the concentration difference; D isthe species diffusivity; aT and ac are the coefficients of thermalexpansion and compositional expansion respectively.

Because the coupling term uv/r in the discretized governingequation leads to a false production of angular momentum that isdifficult to eliminate, usually a physically conserved quantity theswirl F ¼ rv, viz the angular momentum per unit mass, is intro-duced to overcome this difficulty [47,56]. Then Eqs. (2) and (3)become

vuvt

þ uvuvr

þwvuvz

� F2

r3¼ �1

r

vpvr

þ n

�V2u� u

r2

�; (7)

vF

vtþ u

vF

vrþw

vF

vz¼ n

rv

vr

�r3

v

vr

�F

r2

��þ n

v2F

vz2: (8)

For axisymmetric flows in the cylindrical coordinate system,computation time can be reduced if the problem is reformulated sothat the three variables u, w, p are eliminated in favor of thevorticity u and Stokes streamfunction j [46,47], which are definedas

u ¼ vwvr

� vuvz

; (9)

u ¼ 1rvj

vz; (10)

w ¼ �1rvj

vr: (11)

Taking H;1=ð2U0Þ; ðC � C0Þ=DC and ðT � T0Þ=DT as the char-acteristic length, time, concentration and temperature scales, thedimensionless vorticityestreamfunction-based governing equa-tions read

v~Sv~t

þ ~uv~Sv~r

þ ~wv~Sv~z

þ v

v~z

0@ ~F2

~r4

1A ¼ bT~r

v~Tv~r

� bc~r

v~Cv~r

þ 3

(1~r

v

v~r

�1~r

v

v~r

�~r2~S��

þ v2~S

v~z2

);

(12)

v~F

v~tþ u

v~F

v~rþw

v~F

v~z¼ 3

(1~r

v

v~r

"~r3

v

v~r

~F

~r2

!#þ v2 ~F

v~z2

); (13)

v~Tv~t

þ ~uv~Tv~r

þwv~Tv~z

¼ 3

Pr

"1~r

v

v~r

~rv~Tv~r

!þ v2~T

v~z2

#; (14)

v~Cv~t

þ ~uv~Cv~r

þwv~Cv~z

¼ 3

PrLe

"1~r

v

v~r

~rv~Cv~r

!þ v2~C

v~z2

#; (15)

v

v~r

1~r

v~j

v~r

!þ 1

~rv2~j

v~z2¼ �~r~S: (16)

in the above equations the parameters with tildes represent thedimensionless counterparts. We omit the tildes from this pointforward for clarity. S ¼ u=r is the Svanberg vorticity for numeri-cally stable modeling of physically unstable flows [47]. Pr ¼ n=k isthe Prandtl number and Le ¼ k=D is the Lewis number.3 ¼ n=ð2U0H2Þ is the Ekman number. bT ¼ aTgDT=ð4U2

0HÞ is thethermal Rossby number and bc ¼ acgDC=ð4U2

0HÞ is the solutalRossby number. N ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiacgDC=H

pis the BrunteVaisala frequency

and St ¼ ðN=U0Þ2, which denotes the ratio of buoyancy frequencyto Coriolis frequency, is the stratification parameter. Rp ¼ Rac=RaTis the ratio of buoyancy forces, where RaC ¼ aCgDCH3=ðKVÞ is thesolutal Rayleigh number and RaT ¼ aTgDTH3=ðDVÞ is the thermalRayleigh number. When Rac is fixed, smaller St represents rotationwith higher frequency.

3. Lattice Boltzmann model for compositional convectionin the presence of rotation

By performing the following coordinate transformation[43,44,46,47]:

ðr; zÞ1ðx; yÞ; (17)

ðu;wÞ1ðu; vÞ: (18)

Eqs. (10)e(16) can be written in the pseudo-Cartesiancoordinates:

u ¼ 1xvj

vy; (19)

v ¼ �1xvj

vx; (20)

vSvt

þ uvSvx

þ vvSvy

¼ 3

v2Svx2

þ v2Svy2

!þ So; (21)


vFþ uvFþ v

vF ¼ 3v2F

2 þ v2F2 þ Fo; (22)

vt vx vy

vx vy

!

vTvt

þ uvTvx

þ vvTvy

¼ 3

Pr

v2Tvx2

þ v2Tvy2

!þ To; (23)

vCvt

þ uvCvx

þ vvCvy

¼ 3

PrLe

v2Cvx2

þ v2Cvy2

!þ Co; (24)

v2j

vy2þ v2j

vx2¼ Q: (25)

In Eqs. (21)e(25), the source terms caused by the coordinatetransformation and the buoyant forcing due to temperature andconcentration differences read

So ¼ 33x

vSvx

þ bTx

vTvx

� bcx

vCvx

; (26)

Fo ¼ �3vF

vx; (27)

To ¼ 3

Pr1xvTvx

; (28)

Co ¼ 3

Pr Le1xvCvx

; (29)

Q ¼ ��x2Sþ v

�: (30)

Eqs. (21) and (22) (the governing equations for the rotating flowfield), Eq. (23) (the governing equation for the temperature field)and Eq. (24) (the governing equation for the concentration field),which have the same formulation except different coefficients, arenothing but advectionediffusion equations with source terms.There are many matured efficient lattice Boltzmann models for thistype of equation [46,47]. In this paper a D2Q5model is employed tosolve these equations because in the D2Q5 model the forcing-liketerm can be treated more easily [46]. It reads:

gkð x!þ c e!kDt; t þ DtÞ � gkð x!; tÞ¼ �s�1

hgkð x!; tÞ � gðeqÞk ð x!; tÞ

iþ DtYo;k (31)

In Eq. (31) e!k (k ¼ 0.4) are the discrete velocity directions:

e!k ¼ ð0;0Þ : k ¼ 0ðcosðk� 1Þp=2; sinðk� 1Þp=2Þ : k ¼ 1;2;3;4

and c ¼ Dx/Dt is the fluid particle speed. Dx, Dt and s are the latticegrid spacing, the time step and the dimensionless relaxation timerespectively. Yo;k is the discrete form of the source term Yo [15,46],Yo ¼ So;Fo; To;Co for Eqs. (21)e(24) respectively. Yo;k satisfies:Xk�0

Yo;k ¼ Yo (32)

In the present study, the spatial derivatives in the source termsare discretized by usual central difference scheme.

Junk gave out an excellent analysis between the LB method andthe finite difference scheme for fluid flow [50]. Later, van der Smanet al. carried out outstanding works on the relationship betweenthe LB method and the finite difference scheme for con-vectionediffusion flow [51e53]. In addition, the Galilean invariance

is discussed in detail in Refs. [51,52], which gives us insight todevelop LB models with high-order accuracy. The possible choicessatisfying the constraint Eq. (32) are

Yo;k ¼ 6kYo (33)

The equilibrium distribution gk(eq) is defined by

gðeqÞk ¼ 6kd

�1þ e!k, u

!c2s

�(34)

where cs is the speed of sound. d ¼ S;F; T;C for Eqs. (21)e(24)respectively and is obtained by

d ¼Xk�0

gk (35)

According to van der Sman’s analysis [51,52], the optimum are60 ¼ 1=2 and 6k ¼ 1=8 for k ¼ 1.4.

The dimensionless relaxation time s is determined by

c ¼ c2s ðs� 0:5Þ (36)

c ¼ 3; 3; 3=Pr; e=PrLe for Eqs. (21)e(24) respectively. For Sman’schoice c2s ¼ 1=4c2.

Eq. (25) is just the Poisson equation, which also can be solved bythe LB method efficiently. In the present study, the D2Q5 modeldesigned by Chai and Shi [54] is employed. By theway, Chai’s modelalso can be derived by the method proposed by Tölke [55]. Theevolution equation for Eq. (25) reads

fkð x!þ c e!kD2; 2þ D2Þ � fkð x!; 2Þ ¼ Uk þU0k (37)

where Uk ¼ �s�1j ½fkð x!; 2Þ � f ðeqÞk ð x!; 2Þ�, U0

k ¼ D2zQD andD ¼ ðc2=2Þð0:5� sjÞ. sj > 0:5 is the dimensionless relaxation time[46]. 2 and D2 are the pseudo time and time step for solving thePoisson equation. fk(eq) is the equilibrium distribution function, anddefined by

f ðeqÞk ¼ ðx0 � 1:0Þj : k ¼ 0xkj : k ¼ 1;2;3;4

(38)

xk and zk are weight parameters given as x0 ¼ z0 ¼ 0; xk ¼zk ¼ ð1=4Þðk ¼ 1.4Þ. j is obtained by

j ¼Xk�1

fk (39)

The detailed derivation from Eqs. (31) and (38) to Eqs. (21)e(25)can be found in the Appendix.

4. Numerical validation

Firstly the present model is validated by TayloreCouette flowwithin an annular cavity which is a benchmark test for rotatingflow [44]. The annular cavity consists of a rotating inner solid walland a rest outer solid wall, as illustrated in detail in Fig. 1 inRef. [44]. Fig. 2 shows the streamfunction for A ¼ 3.8, K ¼ 2, Rp ¼ 0,RaT ¼ 0, St ¼ 0 and Re ¼ Ri(Ro � Ri)U0/n ¼ 150. From this figure, wecan see the four cell secondary modes. These contours and flowpattern agree well with results in Ref. [44].

Then we validated the present model by setting K ¼ 1, St/N

and A ¼ 2.0 which means the infinite curvature and representsa rectangular cavity without rotation (in our simulation it isimplemented by setting Ri ¼ 105 � (Ro � Ri) and St ¼ 1010). Figs. 3and 4 illustrate the isothermal lines, the isoconcentration lines andthe stream lines for Rp¼ 0.8 and Rp¼ 1.3 respectively, with Pr¼ 1.0,Le ¼ 2.0, RaT ¼ 105. When Rp < 1.0, the flow is primarily dominated

0 0.5 10

1

2

3

z

r

Fig. 2. Streamfunction for TayloreCouette flow within an annular cavity: Re ¼ 150.


by thermal buoyancy effects, and a large central clockwise thermalrecirculation is predicted with the isotherms not being horizontallyuniform in the core region within the enclosure. Furthermore, theconcentration contours are distorted in the core of the enclosurewith a stable stratification in the vertical direction except near theinsulated walls of the enclosure. A stagnant zone in the corners ofthe enclosure is also observed. Whereas for Rp > 1.0 the flow ismainly dominated by compositional buoyancy effects. In the caseRp ¼ 1.3, a counterclockwise compositional recirculation exists inthe core region of the enclosure along with two clockwise thermalrecirculations occurring near the top right and bottom left cornersof the enclosure. The contours for temperature and concentrationare almost parallel to each other within the center of the enclosureaway from the walls. In both cases, the isothermal lines, the iso-concentration lines and the stream lines all are point symmetricwith respect to the geometric center of the enclosure. The resultsobtained by the present model agree well with that in Refs. [49,57].

To quantify the results, the average Nusselt number Nu and theaverage Sherwood number Sh at the inner wall obtained bythe present model are listed in Table 1, with that obtained by thefinite difference method in Ref. [49]. The excellent agreementbetween them demonstrates the capability of the present modelagain. In the table, the numbers in the brackets indicate the gridresolution used.

0 0.2 0.4 0.6 0.8 10

0.5

1

1.5

2

0 0.2 0.40

0.5

1

1.5

2

Fig. 3. Isothermal lines, isoconcentration lines and stream lines for Pr ¼

The average Nusselt number Nu is calculated by

Nu ¼ �ZH0

�vTvr

�dz (40)

The average Sherwood number Sh is calculated by

Sh ¼ �ZH0

�vCvr

�dz (41)

Two different uniform grid resolutions 50 � 100 and 100 � 200are used for above simulation.We found that the present model canwork well even with very coarse grid resolution 50 � 100. In orderto demonstrate the numerical stability and efficiency of the presentmodel, the grid Peclet and Courant numbers defined as

Pe ¼ j u!jDxk

(42)

Cr ¼ j u!jDtDx

(43)

are computed. In the present study Dt ¼ Dx ¼ c ¼ 1, so the gridMach number Mag z Cr.

Figs. 5 and 6 illustrate grid Peclet Pe and Courant Cr numbers forthe two different grid resolutions. One can see that Pe< 2 and Cr �1 even for the coarse grid resolution. The differences of Pe and Cr

between the two different grid resolutions are very slight. It is wellknown that for numerical stability, Cr � 1 must be kept for the LBmethod [12] and significant numerical dispersion would emergewhen Pe[2 [58]. But as van der Sman [51] demonstrated that thecriterion that for stability Pe < 2, does not hold for LB schemes ingeneral, especially if the D2Q9 lattice is used.

In order to demonstrate the capability of the present model forconvection with high Rayleigh number, we also simulated thebenchmark tests used in Refs. [42,60], namely double-diffusiveconvection in an irrotational square cavity with RaT¼ 107, Pr¼ 0.71,Le ¼ 1.0, and Rp ¼ 0.8. In our simulation it is also implemented bysetting Ri ¼ 105 � (Ro � Ri) and St ¼ 1010. Two different grid reso-lutions 100 � 100 and 200 � 200 are used. The computational costof the latter (CPU Time: 33800 s) is much more expensive than theformer (CPU Time: 8441 s) although the numerical differencesbetween them are quite small, as shown in Table 2.

Fig. 7 shows the isotherm, isoconcentration and streamfunction.Nu and Sh obtained by the present model are listed in Table 2, with

0.6 0.8 1 0 0.2 0.4 0.6 0.8 10

0.5

1

1.5

2

1.0, Le ¼ 2.0, RaT ¼ 105, Rp ¼ 0.8, A ¼ 2.0, St ¼ 1010 and K ¼ 1.0.

0 0.2 0.4 0.6 0.8 10

0.5

1

1.5

2

0 0.2 0.4 0.6 0.8 10

0.5

1

1.5

2

0 0.2 0.4 0.6 0.8 10

0.5

1

1.5

2

Fig. 4. Isothermal lines, isoconcentration lines and stream lines for Pr ¼ 1.0, Le ¼ 2.0, RaT ¼ 105, Rp ¼ 1.3, A ¼ 2.0, St ¼ 1010 and K ¼ 1.0.


that published in Ref. [60]. The present results are in good agree-ment with previous data again.

In order to demonstrate the computational efficiency of thepresent model, the natural convection in square cavity with105 � Ra � 107 was also simulated. When Ra � 106 the flow islaminar whereas it becomes transitional as Ra � 107 [61]. Tables 3and 4 list the numerical results, the number of iterative step and thecomputational time (Unit second) of the present model whena stable average Nu has been obtained, comparing with that of thecomputer source code used in Ref. [13] for reactive flow simulation.The computer code in Ref. [13] solves the flow field using the D2Q9lattice model as well as solving the temperature/concentration fieldby the D2Q5 lattice model. In the present model, it is implementedby setting A¼ 1.0, Rp ¼ 0, Ri ¼ 105 � (Ro � Ri) and St¼ 1010. The gridresolution is 150 � 150. All simulations were performed on a Pen-tium 4 (3.0 G CPU). It can be seen that for vortex dominated flow,the present model based on vorticityestreamfunction equations isfaster than the primitive-variables-based one, which agrees withprevious studies [37,56,59].

5. Results and discussion

Two different kinds of nontrivial compositional convectionaccompanying rotation are simulated in the present study. Theyshare the same physical configuration illustrated in Fig. 1 but withdifferent boundary conditions.

The first case is compositional convection in a rotating annuluswith horizontal temperature and vertical solutal gradients, which isimportant in material processing technologies and has beensimulated using the time splitting technique in Ref. [4]. Theboundary conditions are: j ¼ u ¼ w ¼ 0 and F ¼ r2U0 at all walls;T ¼ 1.0 and vC/vr ¼ 0 at the inner wall; T ¼ 0 and vC/vr ¼ 0 at theouter wall; C ¼ 0 and vT/vz ¼ 0 at the top; C ¼ 1.0 and vT/vz ¼ 0 atthe bottom. The initial conditions are j ¼ u ¼ w ¼ 0, T ¼ 0 and vC/vz ¼ DC/H. The boundary conditions and the initial conditions usedin the present study all are same as that in Ref. [4]. The value of

Table 1Average Nusselt number Nu and average Sherwood number Sh.

Nu Sh

Ref.[49]

Present(50 � 100)

Present(100 � 200)

Ref.[49]

Present(50 � 100)

Present(100 � 200)

Rp ¼ 0.8 3.67 3.6897 3.6871 4.89 4.9156 4.8995Rp ¼ 1.3 2.10 2.1255 2.1130 3.15 3.1615 3.1599

Svanberg vorticity S at walls is calculated using the methodproposed in our previous work [46,47].

The second case is compositional convection in a rotatingannulus with opposing temperature and concentration gradients,which is of great fundamental and practice interest to understandconvectional phenomena in geology and chemical processes [49].However, to the best knowledge of the present authors, almost allprevious studies (please see Refs. [49,57] and references therein)only focus on the convectional flows in planar rectangular enclo-sures without rotation. The present study perhaps is the firstexploration on the rotational counterpart. The boundary conditionsfor this case are: j¼ u¼w¼ 0 at all walls; T¼ 0.5 and C¼ 0.5 at theinner wall; T ¼ �0.5 and C ¼ �0.5 at the outer wall; vT/vz ¼ vC/vz ¼ 0 at the top and bottom walls. The initial conditions arej¼ u¼w¼ 0, T¼�0.5 and C¼�0.5. The boundary conditions andthe initial conditions used in the present study all are same as theirirrotationally planar counterparts [49,57]. The value of Svanbergvorticity S at walls is also calculated using the method proposed inour previous work [46,47].

As mentioned above, in the present study Dt¼ Dx¼ c¼ 1, so thecorresponding dimensionless relaxation time varies from 1.610 to1.985 according to different Pr, Le, RaT, St and Rp. In the rest part, theaspect ratio A is fixed at 1.0 and the radius ratio K ¼ 2.0. The Lewisnumber is relatively high Le ¼ 10. It is well known that the highLewis number is a challenge for a numerical scheme designed forcompositional convection [48]. In most previous LB models, theLewis number is limited around the order of unity [10,11,43,44].

5.1. Compositional convection in a rotating annulus withhorizontal temperature and vertical solutal gradients

For the first case, we validate the present model by settingPr ¼ 1.0, RaT ¼ 105, Le ¼ 10.0 and Rp ¼ 0.1 with different St(St ¼ 0.1 and 10.0 respectively), which all are same as that inRef. [4]. Other dimensionless numbers can be determined by theabove five parameters. Figs. 8 and 9 illustrate the numericalresults. When Rp < 1.0, the thermal buoyancy effect outweighs thesolutal buoyancy effect. It is recalled that, in the present flowsetup, the horizontally directed thermal gradient is the primarycause for convective motions. The stable solutal gradient, which isaligned in the vertical, tends to inhibit vertical flows. Fig. 8exemplifies the case with small St ¼ 0.1. The comparative influ-ence of rotation effect is substantial. The meridional circulationflows and the associated azimuthal flow are moderate in magni-tude. The isotherms in the interior are generally tilted from thelower side of the inner sidewall to the upper side of the outersidewall. In the interior region closer to the outer sidewall, the

0 0.2 0.4 0.6 0.8 10

0.5

1

1.5

2

1.6

1.5

1.4

1.3

1.2

1.1

1

0.9

0.8

0.7

0.6

0.5

0.4

0.3

0.2

0.1

z

r

0 0.2 0.4 0.6 0.8 10

0.5

1

1.5

2

0.038

0.036

0.034

0.032

0.03

0.028

0.026

0.024

0.022

0.02

0.018

0.016

0.014

0.012

0.01

0.008

0.006

0.004

0.002

z

r

a b

Fig. 5. (a) Grid Peclet and (b) Courant numbers for Pr ¼ 1.0, Le ¼ 2.0, RaT ¼ 105, Rp ¼ 0.8, A ¼ 2.0, St ¼ 1010 and K ¼ 1.0: grid resolution 50 � 100.


dominant balance is between the vertical shear of azimuthalvelocity and the horizontal gradient of temperature. This isa manifestation of the well known thermal wind relation. Thestratification of the solutal field is mostly confined to the zonesadjacent to the horizontal endwalls. In the middle portion of thecavity, due to the convective activities, the solutal field has beenfairly homogeneous. As the stratification number St increases, therelative importance of the rotation decreases. Consequently, theconvective activities, driven by the horizontally applied temper-ature gradient, intensify. The isotherms in the interior core exhibitthe tendency of a horizontal alignment, suggesting an approach toa linear temperature stratification. When St is large (see Fig. 9 for

0 0.2 0.4 0.6 0.8 10

0.5

1

1.5

21.05

0.95

0.85

0.75

0.65

0.55

0.45

0.35

0.25

0.15

0.05

r

z

a

Fig. 6. (a) Grid Peclet and (b) Courant numbers for Pr ¼ 1.0, Le ¼ 2.0, RaT ¼

St ¼ 10.0), the rotation effect is minor, and the global flowstructure resembles that of a sidewall-heated thermal convectionin a non-rotating environment. There exists only one vortexcenter close to the outer wall. It should be mentioned here that, inthe present formulation, the value of e increases with increasingSt. Therefore, for the example treated in Fig. 9, appreciable viscouseffects are also felt throughout the flow field. The conclusionsobtained by the present model agree qualitatively with that inprevious work [4]. The deviations between the present plots andthat in Ref. [4] result from that the cylindrical coordinates used inRef. [4] rotate with the annulus but in the present study thecylindrical coordinates are rest.

0 0.2 0.4 0.6 0.8 10

0.5

1

1.5

20.038

0.034

0.03

0.026

0.022

0.018

0.014

0.01

0.006

0.002

r

z

b

105, Rp ¼ 0.8, A ¼ 2.0, St ¼ 1010 and K ¼ 1.0: grid resolution 100 � 200.


Nu Sh

Ref.[60]

Present(100 � 100)

Present(200 � 200)

Ref.[60]

Present(100 � 100)

Present(200 � 200)

RaT ¼ 107 10.6 10.5680 10.5968 10.6 10.5693 10.5471

Table 3Comparison of Nu between different methods for natural convection within squarecavity.

Ra Ref. [61] Ref. [13] Present

105 4.5226 4.488 4.509106 8.805 8.683 8.871107 16.79 15.967 16.574


Table 5 lists the average Nusselt number at the inner wall. In thetable, the numbers in the brackets indicate the grid resolutionsused and we find that the coarse one (100 � 100) is fine enough toproduce grid-independent solutions.

0

0.2

0.4

0.6

0.8

1

r

z

a

0

0.2

0.4

0.6

0.8

1

r

z

b

0 0.2 0.4 0.6 0.8 1

0 0.2 0.4 0.6 0.8 1

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

r

z

c

Fig. 7. Isotherm (a), isoconcentration (b) and streamfunction (c) at RaT ¼ 107, Pr ¼ 0.71,Le ¼ 1, A ¼ 1.0, St ¼ 1010 and Rp ¼ 0.8.

5.2. Compositional convection in a rotating annulus withopposing temperature and concentration gradients

In the second case we fix the Prandtl number Pr ¼ 1.0 and theLewis number Le ¼ 10.0 but set variable thermal Rayleigh number105 � RaT � 106, stratification parameter 0.01� St� 0.1 and ratio ofbuoyancy forces 0.1� Rp � 1.3. The grid resolution 100� 100 is fineenough for simulations with above parameters. However, forhigher RaT, a finer grid resolution is required to guarantee numer-ical stability, for example grid size 150� 150 for RaT ¼ 107. To revealthe effects of St, Rp and RaT on the behaviors of convectional flows,we assume one of these parameters varying while the others arefixed. To the best knowledge of the present authors, perhaps thepresent study is the first exploration on this case. Because there arenot open published data for this case, we use the results obtainedby the explicit finite volume method (FVM) for comparison.

5.2.1. Effect of StFigs. 10 and 11 illustrate the effect of St on the isothermal, iso-

concentration and streamfunction lines. Different from the firstcase (cf. Fig. 8), even with very small stratification parameterSt ¼ 0.1, the rotation effect is still minor for compositionalconvection with opposing temperature and concentration gradi-ents. When Rp < 1.0, the flow is primarily dominated by thermalbuoyancy effects, and a large clockwise thermal recirculation ispredicted with the isotherms nearly horizontally uniform in thecore region within the annulus. Furthermore, the concentrationcontours are stably stratified in the vertical direction except nearthe insulated walls of the annulus. The global flow structureresembles its irrotational planar counterpart [49]. However, whileSt decreases successively, the comparative significance of rotationincreases obviously. When St ¼ 0.01, the isotherms in the interiorare generally tilted from the lower side of the inner sidewall to theupper side of the outer sidewall, similar with that in Fig. 8. There isonly one vortex center, either for St ¼ 0.1 or for St ¼ 0.01. But forlower St, the vortex center moves towards the outer sidewall.

To quantify the results, the average Nusselt number Nu and theaverage Sherwood number Sh at the inner wall obtained by thepresent model are listed in Table 6, with that obtained by the FVM.The excellent agreement between them demonstrates the capa-bility of the present model again. From this table, we can find thatNu and Sh decreasewith St, same as that in the first case (cf. Table 5).

Table 4Comparison of computational efficiency between different methods for naturalconvection within square cavity.

Ra Ref. [13] Present

Unit: Second Unit: Second

105 Iterations 205000 55400CPU time 8200 5263



0

0.2

0.4

0.6

0.8

1

r

z

a

0

0.2

0.4

0.6

0.8

1

r

z

b

0 0.2 0.4 0.6 0.8 1

0 0.2 0.4 0.6 0.8 1

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

r

z

c

Fig. 8. Isotherm (a), isoconcentration (b) and streamfunction (c) at RaT ¼ 105, Pr ¼ 1.0,Le ¼ 10.0, St ¼ 0.1 and Rp ¼ 0.1.

0

0.2

0.4

0.6

0.8

1

z

r

a

0

0.2

0.4

0.6

0.8

1

r

z

b

0 0.2 0.4 0.6 0.8 1

0 0.2 0.4 0.6 0.8 1

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

r

z

c


Table 5Average Nusselt number at the inner wall.

Present (100 � 100) Present (200 � 200)

Nu (St ¼ 0.1) 3.5533 3.5620Nu (St ¼ 10.0) 5.6574 5.6601


5.2.2. Effect of RaTTo show the effect of RaT on the flow structure along with to

validate the capability of the present model for high Rayleighnumbers and high frequency rotation, in this subsection we setRaT ¼ 106.

Fig. 12 illustrates the isotherms, isoconcentrations and stream-function contours when St ¼ 0.1, which all are similar with theircounterparts with lower thermal Rayleigh number RaT ¼ 105 (cf.Fig. 10), except the clockwise vortex center moves towards thedown outer corner and the isotherms within the core region aremore acclivitous. Table 7 lists the average Nusselt number Nu andthe average Sherwood number Sh at the inner wall obtained by thepresent model together with that obtained by the FVM. From this

table, it is obvious that Nu and Sh increase with RaT. This conclusionis straightforward and has been proven in previous analogousstudies [49,57]. In this scenario, though the thermal Rayleighnumber is very high, the flow field still can achieve steady status.

0

0.2

0.4

0.6

0.8

1

r

z

a

0

0.2

0.4

0.6

0.8

1

r

z

b

0 0.2 0.4 0.6 0.8 1

0 0.2 0.4 0.6 0.8 1

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

r

z

c


0

0.2

0.4

0.6

0.8

1

r

z

a

0

0.2

0.4

0.6

0.8

1

r

z

b

0 0.2 0.4 0.6 0.8 1

0 0.2 0.4 0.6 0.8 1

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

r

z

c



Nu Sh

FVM Present FVM Present

St ¼ 0.1 4.6473 4.6061 11.8891 11.2078St ¼ 0.01 3.0787 2.9144 8.9457 8.7008


When the rotation frequency increases, the flow field willbecome unsteady, for example, when St ¼ 0.01. Fig. 13 shows theisotherms, isoconcentrations and streamfunction contours at iter-ative step 5 � 105. At that time the unsteady flow has been fullydeveloped. There is a large clockwise thermal recirculation exists inthe core region of the annulus with several small counterclockwisecompositional recirculations occurring near the corners of theenclosure. The flow is still dominated by thermal buoyancy effects,but the concentration and temperature contours are distorted overthe whole domain. Only near the sidewalls, there exist verticalstratifications in the very thin layers. In this study, we find thesecondary flow will emerge when Ta � 5.68 � 107, whereTa ¼ 4U2

0R4i =n

2½1� ðRi=RoÞ2�2 is the Taylor number.

The time average Nusselt number Nut and the time averageSherwood number Sht at the inner wall obtained by the presentmodel are listed in Table 8, with that obtained by the FVM. The timeaverage Nusselt number Nut is calculated by

0

0.2

0.4

0.6

0.8

1

z

r

a

0

0.2

0.4

0.6

0.8

1

z

r

b

0 0.2 0.4 0.6 0.8 1

0 0.2 0.4 0.6 0.8 1

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

z

r

c


0

0.2

0.4

0.6

0.8

1

r

z

a

0

0.2

0.4

0.6

0.8

1

r

z

b

0 0.2 0.4 0.6 0.8 1

0 0.2 0.4 0.6 0.8 1

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

r

z

c



Nut ¼ �1t

Zt0

ZH0

�vTvr

�dzdt (44)

and the time average Sherwood number Sht is calculated by


Nu Sh


St ¼ 0.1 8.6521 8.4799 21.7322 20.5443

Sht ¼ �1t

Zt0

ZH0

�vCvr

�dzdt (45)

Compared with Table 7, from Table 8 we can find that theconvectional heat transfer and the convective mass transport aresignificantly inhibited against increasing rotating frequency.

Table 8Time average Nusselt number Nut and time average Sherwood number Sht.

Nut Sht


St ¼ 0.01 5.6806 5.2524 16.0049 15.1016

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

r

z

a

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

r

z

b

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

r

z

c


0

0.2

0.4

0.6

0.8

1

r

z

a

0

0.2

0.4

0.6

0.8

1

r

z

b

0 0.2 0.4 0.6 0.8 1

0 0.2 0.4 0.6 0.8 1

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

r

z

c



5.2.3. Effect of RpIt is well known that unsteady oscillatory flow due to the vari-

ation of Rp in double-diffusive convection is of fundamental interestand practical importance in engineering [62]. But, existing worksall are restricted in planar irrotational situations (please seeRef. [62] and references therein for detail). In previous analogousstudies [49,57] usually the cases with Rp ¼ 0.8 and Rp ¼ 1.3 werechosen for the representatives to illustrate the effect of Rp, so in thispart we choose the same values. Other parameters are RaT ¼ 106,

Pr ¼ 1.0, Le ¼ 10.0 and St ¼ 0.1, which almost all are same as that inRef. [57] except St and Le. In previouswork [49,57], it shows that theirrotationally compositional convection in a planar cavity withopposing temperature and concentration gradients becomeunsteady only when Rp ¼ 1.0. However, different from their irro-tational planar counterparts [57], in the present study we find theflows will become unsteady evenwhen Rp a little far away from 1.0(for example, Rp ¼ 0.8 and 1.3), due to the presence of rotation.

Figs. 14 and 15 illustrate the effect of Rp on the isothermal, iso-concentration and streamfunction lines at iterative step 5 � 105. Asabove mentioned, at that time the unsteady flow has been fullydeveloped.

Table 9Time average Nusselt number Nut and time average Sherwood number Sht.

Nut Sht


Rp ¼ 0.8 4.2688 4.2306 11.0501 10.7229Rp ¼ 1.3 2.9590 2.8499 8.5951 8.3502


When Rp < 1.0, the flow is primarily dominated by thermalbuoyancy effects, and a large clockwise thermal recirculation existswithin the annulus, similar with its irrotational planar counterpart[57]. However, different from its irrotational planar counterpart[57], the concentration and temperature contours are significantlydistorted in the core region, with vertical stratifications in thevicinity of the sidewalls of the annulus. Whereas for Rp > 1.0 theflow is mainly dominated by compositional buoyancy effects. In thecase Rp ¼ 1.3, a large counterclockwise compositional recirculationexists in the core region of the annulus along with several smallclockwise thermal recirculations occurring near the corners of theenclosure, which also similar with its irrotational planar counter-part [57]. There exist vertical thermal and solutal stratifications inthe very narrow zones near the sidewalls.

To quantify the results, the time average Nusselt number Nut

and the time average Sherwood number Sht at the inner wallobtained by the present model are listed in Table 9, with thatobtained by the FVM. From this table, we can find Nut and Sht forRp ¼ 0.8 are bigger than that for Rp ¼ 1.3, which agrees with theconclusion abstracted from their irrotational planar counterparts[49,57].

6. Conclusion

Compositional convection in the presence of rotation is offundamental interest and practical importance. But the work todesign a suitable and efficient LB model on it is rare, which inspiritsthe present study. In this paper we design a simple LB model tobridge the gap, following the line proposed in our previous work.The present model possesses three obvious advantages inheritingfrom our previous models: The first, the present model keeps thesimplicity of code, in which the governing equations for velocity,concentration and temperature fields all are solved by the D2Q5 LBmodel. The second, with a coarse grid resolution, the present modelcan still work efficiently and stably for rotationally convectionalflow with high Rayleigh number and relatively high Lewis number.The third, the derivation of the present model is quite straightfor-ward. Moreover, the Galilean invariance and its influence on thepresent model are discussed.

We first validated the model by simulating TayloreCouette flowand double-diffusive convection without rotation. The resultsobtained by the present model agree well with previous data. Weinvestigated the grid Peclet and Courant numbers which demon-strate the excellent performance of the present model even witha relatively coarse grid resolution. And comparisons of thecomputation efficiency between the present model and someothers LBmodels are made. It shows that the present model is moresuitable for vortex dominated flow simulation, for example thecompositional convection considered in this study.

Then we simulated the compositional convection in a rotatingannulus with horizontal temperature and vertical solutal gradients,which is important in engineering. When St varies, the patterns ofconvectional flows change significantly. The present modelcaptures the features very well. We also applied our model tocompositional convection in a rotating annulus with opposingtemperature and concentration gradients, to date onwhich there is

seldom open literature. The present study perhaps is the firstdetailed exploration on this case. We find again the numericalresults obtained by the presentmodel are excellent agreementwiththat obtained by the explicit finite volume method. The influencesof changes of RaT, St and Rp on the streamfunction, isoconcentrationand isothermal lines are significant. When RaT ¼ 105, the flow willkeep steady even with very high frequency rotation St ¼ 0.01. Butfor RaT ¼ 106, the flow will become unsteady either when Rpapproaches to unity or when rotation frequency increases toSt ¼ 0.01. The secondary flow will emerge when Ta � 5.68 � 107.

Because compositional convection plays an important role ina gulf stream frontal intrusion [6,63]. Such kind of frontal intrusioncan be described clearly by the present model (please see Figs.13e15). So it is possible to extend the present model to investi-gate these societal important problems, whichwill be considered inour future study.

Acknowledgments

This work was partially supported by the Alexander von Hum-boldt Foundation during the stay of S. Chen in the group led by Prof.M. Krafczyk in Germany. The present author should like to thankF. Verhaeghe for his kind help.

Appendix A. Recovery of the governing equations

The recovery of the governing equations is aided by the Chap-maneEnskog expansion. Expanding the distribution functions andthe time and space derivatives in terms of a small quantity 3

vt ¼ 3v1t þ 32v2t ; va ¼ 3v1a;Yo;k ¼ 3Yo;1k; gk

¼ gð0Þk þ 3gð1Þk þ 32gð2Þk þ. (A.1)

To perform the ChapmaneEnskog expansion we must firstlyTaylor expand Eq. (31):

Dkgk þ12D2kgk ¼ �1

s

�gk � gðeqÞk

�þ Yo;k (A.2)

where Dk ¼ vt þ cek;ava. Please bear in mind Dt ¼ c ¼ 1 in thepresent model, so we omit them from now on in the derivation forclarity. Substituting Eq. (A.1) into Eq. (A.2), we get

3D1k

�gð0Þk þ 3gð1Þk

�þ 32v2tg

ð0Þk þ 32

12D21kg

ð0Þk

¼ �1s

�gð0Þk þ 3gð1Þk þ 32gð2Þk � gðeqÞk

�þ 3Yo;1k (A.3)

where D1k ¼ v1t þ ek;av1a. And then, we can obtain the followingequations in consecutive order of the parameter 3:

O�30�: gð0Þk ¼ gðeqÞk (A.4)

O�31�: D1kg

ð0Þk ¼ �1

sgð1Þk þ Yo;1k (A.5)

O�32�: v2tg

ð0Þk þ 1

2D21kg

ð0Þk þ D1kg

ð1Þk ¼ �1

sgð2Þk (A.6)

Eq. (A.6) can be simplified by Eq. (A.5):

v2tgð0Þk þ

�1� 1

2s

�D1kg

ð1Þk ¼ �1

sgð2Þk � 1

2v1tYo;1k þ ek;av1aYo;1k

�(A.7)


With the assumption of weakly compressible flow, we can replacethe convective time-scale operator with a spatial derivative, asdemonstrated in Ref. [52,64]

v1tw� uav1a (A.8)

then Eq. (A.7) becomes

v2tgð0Þk þ

�1� 1

2s

�D1kg

ð1Þk ¼ �1

sgð2Þk � 1

2v1a�ek;a � ua

�Yo;1k

(A.9)

BecauseXk

gðiÞk ¼ 0 ði � 1Þ (A.10)

andXk

Yo;1k ¼ Yð1Þo ;

Xk

ek;aYo;1k ¼ 0 (A.11)

but mention thatXk

ek;agð1Þk wvads0 (A.12)

Taking summation over above equations we can obtain:

vt1dþ uav1ad ¼ Yo (A.13)

and

v2tdþ v1apð1Þa ¼ �1

2v1auaYo;1k (A.14)

where

pð1Þa ¼

�1� 1

2s

�Xk

ek;agð1Þk (A.15)

In Eq. (A.14) there is a velocity dependent term, which can beremoved by a redefinition of the flux, as Ladd et al. proposed [65]

epð1Þa ¼ pð1Þ

a þ 12Youa (A.16)

Easy implementation is like Ladd handles the body force: half ofthe forcing (source) before the collision, other half after the colli-sion [52,65].

Combining Eqs. (A.13), (A.14) and (A.16), we can obtain Eqs.(21)e(24) if d ¼ S, F, T and C respectively.

Now, we discuss the Galilean invariant about the lattice D2Q5model. With the aid of Eqs. (A.6), (A.8) and (A.11), Eq. (A.15) can betransformed into [51,52]

pð1Þa ¼ �

�s� 1

2

�v1bXk

ek;a�ek;b � ub

�gð0Þk (A.17)

For both D2Q5 and D2Q9 models, we have:

Xk

ek;aubgð0Þk ¼ uaubd (A.18)

and for the second-order moment in D2Q5:Xk

ek;aek;bgð0Þk ¼ dc2s dab (A.19)

where cs is the speed of sound and dab is the Kronecker delta.

However, for the D2Q9 it contains a velocity dependent terms,which exactly compensates for the above velocity dependent termEq. (A18)Xk

ek;aek;bgð0Þk ¼ dc2s dab þ uaubd (A.20)

Hence, D2Q9 has no velocity dependent terms in the dissipativeflux and is thus Galilean invariant, while the D2Q5 retains velocitydependent terms and violates Galilean invariance. In order todecrease this error, Cr and/or Pe must be small enough (namelyOðuaÞwOðeÞ) to guarantee

pð1Þa ¼ � c2s ðs� 0:5Þv1adþ O

�e2�

(A.21)

The detailed derivation from the evolution equation Eq. (37) tothe governing equation Eq. (25) can be found in our previous work[46] and Ref. [54].

References

[1] M.G. Worster, Instabilities of the liquid and mushy regions during solidifica-tion of alloys, J. Fluid Mech. 237 (1992) 649e669.

[2] I.A. Eltayeb, E.A. Hamza, Compositional convection in the presence of rotation,J. Fluid Mech. 354 (1998) 277e299.

[3] J. Lee, S.H. Kang, M.T. Hyun, Double-diffusive convection in a rotating cylinderwith heating from below, Int. J. Heat Mass Transfer 40 (1997) 3387e3394.

[4] H.J. Sung, W.K. Cho, J.M. Hyun, Double-diffusive convection in a rotatingannulus with horizontal temperature and vertical solutal gradients, Int. J. HeatMass Transfer 36 (1993) 3773e3782.

[5] J.S. Turner, Multicomponent convection, Annu. Rev. Fluid Mech. 17 (1985)11e44.

[6] R.W. Schmitt, Double diffusion in oceanography, Annu. Rev. Fluid Mech. 26(1994) 255e285.

[7] J.M. Hyun, H.S. Kwak, Flow of a double-diffusive stratified fluid in a differen-tially-rotating cylinder, Geophys. Astrophys. Fluid Dyn. 46 (1989) 203e219.

[8] R.T. Pappalardo, A.C. Barr, The origin of domes on Europa: the role of ther-mally induced compositional diapirism, Geophys. Res. Lett. 31 (2004) L01701/1eL01701/4.

[9] Y. Qian, D. d’Humieres, P. Lallemand, Lattice BGK models for Navier-Stokesequation, Europhys. Lett. 17 (1992) 479e484.

[10] R. Benzi, S. Succi, M. Vergassola, The lattice Boltzmann equation: theory andapplications, Phys. Rep. 222 (1992) 145e197.

[11] S. Chen, G.D. Doolen, Lattice Boltzmann method for fluid flows, Annu. Rev.Fluid Mech. 30 (1998) 329e364.

[12] S. Succi, The Lattice Boltzmann Equation for Fluid Dynamics and Beyond.Oxford University Press, Oxford, 2001.

[13] S. Chen, Z. Liu, C. Zhang, et al., A novel coupled lattice Boltzmann model forlow Mach number combustion simulation, Appl. Math. Comput. 193 (2007)266e284.

[14] A.J.C. Ladd, Numerical simulations of particulate suspensions via a discretizedBoltzmann equation. Part 2. Numerical results, J. Fluid Mech. 271 (1994)311e339.

[15] S. Chen, S.P. Dawson, G.D. Doolen, et al., Lattice methods and their applica-tions to reacting systems, Comput. Chem. Eng. 19 (1995) 617e646.

[16] Y. Qian, S. Succi, S. Orszag, Recent advances in lattice Boltzmann computing,Annu. Rev. Comput. Phys. 3 (1995) 195e242.

[17] G. Hazi, A.R. Imre, G. Mayer, et al., Lattice Boltzmann methods for two-phaseflow modeling, Ann. Nucl. Energy 29 (2002) 1421e1453.

[18] D. Yu, R. Mei, L.S. Luo, et al., Viscous flow computations with the method oflattice Boltzmann equation, Prog. Aerospace Sci. 39 (2003) 329e367.

[19] M.C. Sukop, D.T. Thorne Jr., Lattice Boltzmann Modeling: an Introduction forGeoscientists and Engineers. Springer, Heidelberg, Berlin, New York, 2006.

[20] S. Arcidiacono, J. Mantzaras, I.V. Karlin, Lattice Boltzmann simulation ofcatalytic reactions, Phys. Rev. E 78 (2008) 046711.

[21] D. Qi, Direct simulations of flexible cylindrical fiber suspensions in finiteReynolds number flows, J. Chem. Phys. 125 (2006) 114901.

[22] B. Shi, Z. Guo, Lattice Boltzmann model for nonlinear convection-diffusionequations, Phys. Rev. E 79 (2009) 016701.

[23] S. Chen, Z. Liu, B. Shi, et al., Computation of gas-solid flows by finite differenceBoltzmann equation, Appl. Math. Comput. 173 (2006) 33e49.

[24] D. Raabe, Overview of the lattice Boltzmann method for nano- and microscalefluid dynamics in materials science and engineering, Model. Simul. Mater. Sci.Eng. 12 (2004) 13e46.

[25] J. Meng, Y. Qian, X. Li, et al., Lattice Boltzmann model for traffic flow, Phys.Rev. E 77 (2008) 036108.

[26] S. Chen, Analysis of entropy generation in counter-flow premixed hydrogen-air combustion, Int. J. Hydrogen Energy 35 (2010) 1401e1411.


[27] S. Chen, J. Li, H.F. Han, Z.H. Liu, C.G. Zheng, Effects of hydrogen addition onentropy generation in ultra-lean counter-flow methane-air premixedcombustion, Int. J. Hydrogen Energy 35 (2010) 3891e3902.

[28] D.O. Martinez, W.H. Matthaeus, S. Chen, et al., Comparison of spectral methodand lattice Boltzmann simulations of two-dimensional hydrodynamics, Phys.Fluids 6 (1994) 1285e1298.

[29] X. He, G.D. Doolen, T. Clark, Comparison of the lattice Boltzmann method andthe artificial compressibility method for Navier-Stokes equations, J. Comput.Phys. 179 (2002) 439e451.

[30] S. Chen, M. Krafczyk, Entropy generation in turbulent natural convection dueto internal heat generation, Int. J. Thermal Sci. 48 (2009) 1978e1987.

[31] S. Chen, J. Tolke, M. Krafczyk, Numerical investigation of double-diffusive(natural) convection in vertical annuluses with opposing temperature andconcentration gradients, Int. J. Heat Fluid Flow 31 (2010) 217e226.

[32] S. Chen, B. Shi, Z. Liu, Z. He, Z. Guo, C. Zheng, Lattice-Boltzmann simulation ofparticle-laden flow over a backward-facing step, Chin. Phys. 13 (2004)1657e1664.

[33] S. Chen, Z. Liu, B. Shi, Z. He, C. Zheng, A novel incompressible finite-differencelattice Boltzmann equation for particle-laden flow, Acta Mech. Sin. 21 (2005)574e581.

[34] S. Chen, Z.W. Tian, Simulation of microchannel flow using the lattice Boltz-mann method, Phys. A 388 (2009) 4803e4810.

[35] S. Chen, Z.W. Tian, Simulation of thermal micro-flow using lattice Boltzmannmethod with Langmuir slip model, Int. J. Heat Fluid Flow 31 (2010)227e235.

[36] S. Chen, Z. Liu, Z. Tian, B. Shi, C. Zheng, A simple lattice Boltzmann scheme forcombustion simulation, Comput. Math. Appl. 55 (2008) 1424e1432.

[37] S. Chen, J. Tolke, M. Krafczyk, A new method for the numerical solution ofvorticity-streamfunction formulations, Comput. Methods Appl. Mech. Eng.198 (2008) 367e376.

[38] S. Chen, J. Tolke, M. Krafczyk, Simple lattice Boltzmann subgrid-scale modelfor convectional flows with high Rayleigh numbers within an enclosedcircular annular cavity, Phys. Rev. E 80 (2009) 026702.

[39] S. Chen, Z. Liu, Z. He, C. Zhang, Z. Tian, B. Shi, C. Zheng, A novel latticeBoltzmann model for reactive flows with fast chemistry, Chin. Phys. Lett. 23(2006) 656e659.

[40] S. Chen, Z. Liu, Z. He, et al., A new numerical approach for fire simulation, Int. J.Mod. Phys. C 18 (2007) 187e202.

[41] F. Verhaeghe, S. Arnout, B. Blanpain, P. Wollants, Lattice-Boltzmann modelingof dissolution phenomena, Phys. Rev. E 73 (2006) 036316.

[42] F. Verhaeghe, B. Blanpain, P. Wollants, Lattice Boltzmann method for double-diffusive natural convection, Phys. Rev. E 75 (2007) 046705.

[43] Y. Peng, C. Shu, Y.T. Chew, J. Qiu, Numerical investigation of flows in Czo-chralski crystal growth by an axisymmetric lattice Boltzmann method, J.Comput. Phys. 186 (2003) 295e307.

[44] H. Huang, T.S. Lee, C. Shu, Hybrid lattice Boltzmann finite-difference simula-tion of axisymmetric swirling and rotating flows, Int. J. Numer. Methods Fluids53 (2007) 1707e1726.

[45] S. Chen, J. Tolke, M. Krafczyk, Numerical simulation of fluid flow and heattransfer inside a rotating disk-cylinder configuration, Phys. Rev. E 80 (2009)016702.

[46] S. Chen, J. Tolke, S. Geller, M. Krafczyk, Lattice Boltzmann model for incom-pressible axisymmetric flows, Phys. Rev. E 78 (2008) 046703.

[47] S. Chen, J. Tolke, M. Krafczyk, Simulation of buoyancy-driven flows ina vertical cylinder using a simple lattice Boltzmann model, Phys. Rev. E 79(2009) 016704.

[48] P.W. Shipp, M. Shoukri, M.B. Carver, Effect of thermal Rayleigh and Lewisnumbers on double diffusive natural convection in closed annulus, Numer.Heat Transfer A 24 (1993) 451e465.

[49] A.J. Chamkha, H. Al-Naser, Hydromagnetic double-diffusive convection ina rectangular enclosure with opposing temperature and concentrationgradients, Int. J. Heat Mass Transfer 45 (2002) 2465e2483.

[50] M. Junk, A finite difference interpretation of the lattice Boltzmann method,Numer. Methods Partial Differential Equations 17 (2001) 383e402.

[51] R.G.M. van der Sman, Galilean invariant lattice Boltzmann scheme for naturalconvection on square and rectangular lattices, Phys. Rev. E 74 (2006) 026705.

[52] R.G.M. van der Sman, Finite Boltzmann schemes, Comput. Fluids 35 (2006)849e854.

[53] R.G.M. van der Sman, M.H. Ernst, Convection-diffusion lattice Boltzmannscheme for irregular lattices, J. Comput. Phys. 160 (2000) 766e782.

[54] Z. Chai, B. Shi, A novel lattice Boltzmann model for the Poisson equation, Appl.Math. Modell 32 (2008) 2050e2058.

[55] J. Tolke, A thermal model based on the lattice Boltzmann method for lowMachnumber compressible flows, J. Comput. Theo. Nanosci. 3 (2006) 579e587.

[56] W.E. Langlois, Buoyancy-driven flows in crystal-growth melts, Annu. Rev.Fluid Mech. 17 (1985) 191e215.

[57] C. Ma, Lattice BGK simulations of double diffusive natural convection ina rectangular enclosure in the presences of magnetic field and heat source,Nonlinear Anal. Real World Appl. 10 (2009) 2666e2678.

[58] T.J. Chung, Computational Fluid Dynamics. Cambridge University Press,Cambridge, 2002.

[59] S. Chen, A large-eddy-based lattice Boltzmann model for turbulent flowsimulation, Appl. Math. Comput. 215 (2009) 591e598.

[60] C. Beghein, F. Haghighat, F. Allard, Numerical study of double-diffusive naturalconvection in a square cavity, Int. J. Heat Mass Transfer 35 (1992) 833e846.

[61] H.N. Dixit, V. Babu, Simulation of high Rayleigh number natural convection ina square cavity using the lattice Boltzmann method, Int. J. Heat Mass Transfer49 (2006) 727e739.

[62] K. Ghorayeb, H. Khallouf, A. Mojtabic, Onset of oscillatory flows in double-diffusive convection, Int. J. Heat Mass Transfer 42 (1999) 629e643.

[63] J. Williams, The role of double diffusion in a Gulf Stream frontal intrusion, J.Geophys. Res. 86 (1981) 1917e1928.

[64] E.G. Flekkoy, Lattice Bhatnagar-Gross-Krook models for miscible fluids, Phys.Rev. E 47 (1993) 4247e4257.

[65] J.C. Ladd, R. Verberg, Lattice-Boltzmann simulations of particle-fluid suspen-sions, J. Stat. Phys. 104 (2001) 1191e1251.

Documents

Simulating compositional convection in the presence of rotation by lattice Boltzmann model