Analytical Lattice Boltzmann Solutions for Thermal Flow Problems

TRANSPORT THEORY AND STATISTICAL PHYSICS

Vol. 32, Nos. 5–7, pp. 639–650, 2003

Analytical Lattice Boltzmann Solutions for

Thermal Flow Problems

Dimitris Valougeorgis* and Stergios Naris

Department of Mechanical and Industrial Engineering,

University of Thessaly, Volos, Greece

ABSTRACT

Analytical solutions based on two 13-bit (hexagonal and square) and

one 17-bit square lattice Boltzmann BGK models have been obtained

for the Couette flow, with a temperature gradient at the boundaries.

The analytical solutions for the unknown distributions functions are

written as polynomials in powers of the space variable and the

coefficients of the expansion are estimated in terms of characteristic

flow quantities, the single relaxation time and the lattice spacing. The

analytical solutions of the two 13-bit models contain some nonlinear

deviations from the thermal hydrodynamic constraints and the

analytical solutions, while the 17-bit square lattice model results

into an exact representation of the nonisothermal Couette flow

problem.

*Correspondence: Dimitris Valougeorgis, Department of Mechanical and

Industrial Engineering, University of Thessaly, Volos, 38334, Greece; E-mail:

[email protected].

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639

DOI: 10.1081/TT-120025070 0041-1450 (Print); 1532-2424 (Online)

Copyright & 2003 by Marcel Dekker, Inc. www.dekker.com

+ [1.8.2003–10:43am] [639–650] [Page No. 639] f:/Mdi/Tt/32(5-7)/120025070_TT_032_005-007_R1.3d Transport Theory and Statistical Physics (TT)

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I. INTRODUCTION

In the last few years the lattice Boltzmann method (LBM) has beendeveloped into an effective computational scheme for a broad variety ofthermo-fluid physical systems (Chen and Doolen, 1998). Since the appear-ance of the LBM and its predecessor, the lattice gas automata (LGA) someanalytical solutions have been obtained for these methods for two andthree dimensional flows (Cornubert et al., 1991; Henon, 1987; Luo,1997; Luo et al., 1991). More recently, analytical solutions of the distribu-tion functions for Poiseuille and Couette flows are found for the triangularand square LBMmodels (He et al., 1997; Zou et al., 1995). All these resultsallow one to calculate the viscosity from given collision rules, to improvethe implemented boundary conditions and to justify the accuracy to expectfrom the method. Overall analytical LB approaches are enhancing ourunderstanding of the method. Nevertheless no analytical solution hasbeen previously reported for thermal flow problems. It is well knownthat one of the major shortcomings of the LBM is the lack of a satisfactorythermal model for heat transfer problems. One of the methodologies todevelop thermal lattice Boltzmann models is the so-called ‘‘multi-speedapproach’’ (Alexander et al., 1993; Chen et al., 1994). Although thisapproach has been shown to suffer from numerical instability(McNamara et al., 1995), some recent work has provided new alternativesand potential in this approach (Pavlo et al., 1998a; 1986b). Some of thevelocity discretization models studied in previous work include 13-bitmodels for either the hexagonal (Alexander et al., 1993) or the square(Qian, 1993) grids, as well as typical 17-bit square lattices.

In the present work an investigation on the accuracy to expect fromthe aforementioned multi-speed models is attempted. The nonisothermalCouette flow is chosen as a typical thermal flow model problem and ananalytical LBM formulation approach developed earlier (He et al., 1997;Zou et al., 1995) is extended to include heat transfer effects. Analyticalexpressions of the distribution functions are obtained and some guidanceis given for thermal flow applications.

The well-known lattice Boltzmann evolution equation is given by

f�, iðxþ e�, i�t, tþ �tÞ � f�, iðx, tÞ ¼ �1

�½ f�, iðx, tÞ � f

ð0Þ�, i ðx, tÞ� ð1Þ

where f�, iðx, tÞ is the distribution function of the particle of type ð�, iÞ atposition x and time t, f

ð0Þ�, i ðx, tÞ is the corresponding equilibrium function

of the particle, e�, i are the unit velocity vectors along the specified direc-tions and � is the single relaxation time, which controls the rate at whichthe system relaxes to the local equilibrium. All quantities in Eq. (1) are in

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640 Valougeorgis and Naris

nondimensional form (Sterling and Chen, 1996). The choice of fð0Þ�, i ðx, tÞ is

critical in thermal lattice Boltzmann models. To accurately simulatehydrodynamic phenomena Eulerian and Navier–Stokesian descriptionsof real fluids must be fully recovered. In two dimensions this may beachieved by requiring that the moments of f

ð0Þ�, i ðx, tÞ satisfy the relations

(McNamara and Alder, 1993)X�, i

fð0Þ�, i ¼ n ð2aÞ

X�, i

e�, �, i fð0Þ�, i ¼ nu� ð2bÞ

X�, i

e�, �, ie�, �, i fð0Þ�, i ¼ nu�u� þ n"�� ð2cÞ

X�, i

e�, �, ie�, �, ie�, �, i fð0Þ�, i ¼ nu�u�u� þ n"ðu�� þ u�� þ u��Þ ð2dÞ

X�, i

e2�, ie�, �, ie�, �, i fð0Þ�, i ¼ ðu2 þ 6"Þnu�u� þ ðu2 þ 4"Þn"�� ð2eÞ

where Greek subscripts indicate Cartesian components. For a thermalfluid taking into account the symmetry of the moments under exchangeof any pairs of indices there are 13 such constrains in two dimensions(26 constraints in three dimensions). This suggests the need for at least13 different particle velocities in order to guarantee the linear indepen-dence of the left hand side of Eq. (2) and the full recovery of the thermalNavier–Stokes equations up to the fourth order terms. Typicalequilibrium function has the polynomial form (Pavlo et al., 1998b)

fð0Þ�, i ¼ n½A� þ B�ðe�, i � uÞ þ C�ðe�, i � uÞ

2þD�u

2þ E�ðe�, i � uÞ

3

þ F�ðe�, i � uÞu2þ G�u

4þH�ðe�, i � uÞ

2u2 þ I�ðe�, i � uÞ4� ð3Þ

where the coefficients are functions of the local density n ¼P

�, i f�, i andthe internal energy 2n" ¼

P�, i f�, iðe�, i � uÞ

2, while the bulk velocity isdefined by nu ¼

P�, i f�, ie�, i. The form of expression (3) is based on a

Taylor expansion of the Maxwellian equilibrium distribution in the localvelocity u keeping terms up to the fourth power. The coefficients of therelaxation distribution (3) are obtained in such a manner to remove dis-crete lattice effects and consequently the resulting relaxation distributionis not the Maxwellian.

The Couette thermal-flow problem under investigation consists of afluid contained between two plates, the upper one moving with velocity u0

AQ2

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Analytical LBM for Thermal Flow Problems 641

and the lower one is stationary, while a temperature difference existsbetween the boundaries. The x and y components of the velocityu ¼ ðux, uyÞ and normalized energy profiles must satisfy

uxð yÞ ¼ u0y ð4aÞ

uyð yÞ ¼ 0 ð4bÞ

and

"ð yÞ ¼ yþBr

2u20yð1� yÞ

� �ð4cÞ

respectively, where y is the normalized distance from the lower plateand the Brinkman number Br is the product of the Prandtl and Eckertnumbers.

II. ANALYTICAL SOLUTIONS OF THE

13-BIT MODELS

First the 13-bit hexagonal lattice is considered. This model isconsisting of one rest particle,

e�i ¼ 0, ð5aÞ

for � ¼ 0, and two nonzero speeds for which

e�i ¼ � cos�ði � 1Þ

3, sin

�ði � 1Þ

3

� �, ð5bÞ

for � ¼ 1, 2 and i ¼ 1, 2, 3, 4, 5, 6. Taking into account the constrainsmentioned above one can easily solve for the unknown coefficients ofthe equillibrium distribution function. One possible solution is(Alexander et al., 1993):

A0 ¼ 1�5

2"þ 2"2, A1 ¼

4

9"�

4

9"2, A2 ¼ �

1

36"þ

1

9"2,

B1 ¼4

9�4

9", B2 ¼ �

1

36þ1

9",

C1 ¼8

9�4

3", C2 ¼ �

1

72þ

1

12",

D0 ¼ �5

4þ 2", D1 ¼ �

2

9þ2

9", D2 ¼

1

72�

1

18",

E1 ¼ �4

27, E2 ¼

1

108, F1 ¼ F2 ¼ 0:

ð6Þ

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At this point we note that using the above set of estimates for thecoefficients only the zeroth, first, and second moments of the imposedconstraints corresponding to Eqs. (2a), (2b), and (2c) respectively aresatisfied, while for the third moments, given by Eq. (2d), cubic deviationsare present. Actually none of the possible solutions satisfy exactly therequired constraints.

Now suppose that there is a solution f�, iðx, tÞ of Eq. (1) that exactlyrepresents the Couette flow with a temperature gradient between theboundaries. The solution must contain the following properties:

f�, iðx, tÞ is time independent denoted by f�, iðxÞ ð7aÞ

f�, iðxÞ is a function of one space variable denoted by f�, ið yÞ ð7bÞ

X�, i

f�, ið yÞ ¼ n ð7cÞ

X�, i

f�, ið yÞex, �, i ¼ nuxð yÞ ¼ nu0y ð7dÞ

X�, i

f�, ið yÞey, �, i ¼ 0 ð7eÞ

X�, i

f�, ið yÞðe�, i � u

2Þ

2¼ n"ð yÞ ¼ n yþ

Br

2u20yð1� yÞ

� �ð7fÞ

Equations (7a) and (7b) are due to the fact that the particular flow underinvestigation is steady and fully developed. Equations (7c–7f ) are derivedusing the definitions of the local density, the x and y componentsof velocity and the internal energy respectively supplemented by thewell-known analytical velocity and temperature profiles given in Eqs. (4).

Using properties (7a) and (7b) for � ¼ 1, 2 and i ¼ 0, 1, 4, whichcorrespond to the rest particle and the two particles with motion alongthe x-axis, Eq. (1) may be written as

f�, ið yÞ ¼ f�, ið yÞ �1

�ð f�, ið yÞ � f

ð0Þ�, i ð yÞÞ: ð8Þ

Hence for � ¼ 1, 2 and i ¼ 0, 1, 4 we obtain f�, ið yÞ ¼ fð0Þ�, i ð yÞ. To find the

remaining distribution functions we note that the equilibrium distribu-tions are functions of powers of y up to y3 through linear dependenceon the x-component of the velocity. Thus, the following form of theremaining unknown distribution functions is suggested:

f�, ið yÞ ¼ nða�, i þ b�, i yþ c�, i y2þ d�, i y

3Þ, ð9Þ

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for � ¼ 1, 2 and i ¼ 2, 3, 5, 6. The 32 unknown coefficients areestimated by implementing evolution Eq. (1) in all eight velocitydirections accordingly. For example for � ¼ 1 and i ¼ 2 we have

f1, 2ð yþ �Þ ¼ f1, 2ð yÞ �1

�ð f1, 2ð yÞ � f

ð0Þ1, 2ð yÞÞ, ð10Þ

where � is the vertical spacing between the lattice rows. The expressionsfor f1, 2ð yÞ and f

ð0Þ1, 2ð yÞ given by Eqs. (3) and (9) respectively, are sub-

stituted in Eq. (10) and then the left hand side of Eq. (8) is expandedusing Taylor series. Equating terms of equal powers in y in the resultingequation leads to an algebraic system of linear equations to be solved forthe unknown coefficients. Applying the same procedure to all directionsfor which the distribution function is unknown and solving the systemssymbolically yields

a1, 2 ¼ a1, 5 ¼ A1 � b1, 2�� c1, 2�2� � d1, 2�

3�,

a1, 3 ¼ a1, 6 ¼ A1 � b1, 3�� c1, 3�2� � d1, 3�

3�,

a2, 2 ¼ a2, 5 ¼ A2 � 2b2, 2�� 4c2, 2�2� � 8d2, 2�

3�,

a2, 3 ¼ a2, 6 ¼ A2 � 2b2, 3�� 4c2, 3�2� � 8d2, 3�

3�,

b1, 2 ¼ �b1, 5 ¼B1

2u0 � 2c1, 2�� 3d1, 2�

2�,

b1, 3 ¼ �b1, 6 ¼ �B1

2u0 � 2c1, 3�� 3d1, 3�

2�,

b2, 2 ¼ �b2, 5 ¼ B2u0 � 4c2, 2�� 12d2, 2�2�,

b2, 3 ¼ �b2, 6 ¼ �B2u0 � 4c2, 3�� 12d2, 3�2�,

c1, 2 ¼ c1, 5 ¼C1 þ 4D1

4u20 � 3d1, 2��,

c1, 3 ¼ c1, 6 ¼C1 þ 4D1

4u20 � 3d1, 3��,

c2, 2 ¼ c2, 5 ¼ ðC2 þD2Þu20 � 6d2, 2��,

c2, 3 ¼ c2, 6 ¼ ðC2 þD2Þu20 � 6d2, 3��,

d1, 2 ¼ �d1, 5 ¼E1

8u30, d1, 3 ¼ �d1, 6 ¼ �

E1

8u30,

d2, 2 ¼ �d2, 5 ¼ E2u30, d2, 3 ¼ �d2, 6 ¼ �E2u

30:

ð11Þ

Putting these results into Eq. (9) we obtain analytical expressions forall 13 distribution functions. These analytical expressions are substitutedfinally into Eqs. (7c–7f ) to findX

�, i

f�, i ¼ n ð12aÞ

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X�, i

ex, �, i f�, i ¼ nyþ1

3ny�ð2� � 1Þ�2 ð12bÞ

X�, i

ey, �, i f�, i ¼ 0 ð12cÞ

X�, i

ðe�, i � uÞ2

2f�, i ¼ n"�

1

3ny2�ð2� � 1Þ �

2

3n"�ð2� � 1Þ

� ��2 ð12dÞ

It is noticed that the conservation of mass and y-momentum equationsare satisfied exactly, while there is a second order discrepancy in thex-momentum and energy equations. It is seen that the derived analyticalsolutions although represent an exact solution of the LB model, Eq. (1)do not represent an exact solution of the nonisothermal Couette flowproblem. The analytical solution depends on � and �. It is seen that as �goes to zero or � approaches 1/2 the exact solution is recovered.

Next the 13-bit square lattice is considered. This model has one restparticle

e�, i ¼ 0 ð13aÞ

for � ¼ 0 and three nonzero speeds for which

e�, i ¼ e�ðcos�i, sin�iÞ ð13aÞ

for � ¼ 1, 3 and i ¼ 1, 2, 3, 4 where e1 ¼ 1, e3 ¼ 2, �i ¼ ði � 1Þ�=2 and

e�, i ¼ e�ðcos�i, sin�iÞ ð13aÞ

for � ¼ 2 and i ¼ 1, 2, 3, 4 where e2 ¼ffiffiffi2

p, �i ¼ ði � 1=2Þ�=2. Using the

constraints given by Eqs. (2) and the above set of discrete velocities wefind the coefficients of the equilibrium function (3) to be

A0 ¼1

2ð2� 5"þ 4"2Þ, A1 ¼

2

3ð"� "2Þ, A3 ¼

1

24ð�"þ 4"2Þ,

B1 ¼1

3ð2� 3"Þ, B2 ¼

"

4, B3 ¼

1

24ð�1þ 3"Þ,

C1 ¼1

3ð2� 3"Þ, C2 ¼

1

8, C3 ¼

1

96ð�1þ 6"Þ,

D0 ¼ �5

4þ ", D1 ¼

1

3", D2 ¼

1

8ð�1� 2"Þ, D3 ¼

1

24",

E1 ¼1

3, E2 ¼

1

8, E3 ¼

1

96, F1 ¼ �

1

2, F2 ¼ �

1

8,

G0 ¼1

4, H1 ¼ �

1

6, H3 ¼

1

96:

ð14Þ

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The coefficients, which are not included in the above equations, aretaken equal to zero. Higher order terms have been added to the expres-sion of the equilibrium function and as a result in this case the first fourmoments of the equilibrium, given by Eqs. (7a–7d), are recovered.However, still it is not possible to find a solution to satisfy the fourthorder constraints given by Eq. (2e).

Introducing Eqs. (7a) and (7b) we obtain f�, ið yÞ ¼ fð0Þ�, i ð yÞ for � ¼ 1, 3

and i ¼ 0, 1, 3. In this case the remaining unknown distribution functionstake the form

f�, ið yÞ ¼ nða�, i þ b�, i yþ c�, i y2þ d�, i y

3þ e�, i y

4Þ ð15Þ

for � ¼ 1, 2, 3 and i ¼ 2, 3, 4, 6, 7, 8. Following the same procedure asbefore we find

a1, 2 ¼ a1, 4 ¼ A1 þ c1, 2�2�ð�1þ 2�Þ þ e1, 2�

4�ð�1þ 8� � 12�2Þ,

a2, 1 ¼ a2, 3 ¼ A2 � b2, 1�� c2, 1�2� � d2, 1�

3� � e2, 1�4�,

a2, 2 ¼ a2, 4 ¼ A2 � b2, 2�� c2, 2�2� � d2, 2�

3� � e2, 2�4�,

a3, 2 ¼ a3, 4 ¼ A3 þ 4c3, 2�2�ð�1þ 2�Þ þ 16e3, 2�

4�ð�1þ 8� � 12�2Þ,

b1, 2 ¼ �b1, 4 ¼ �2c1, 2�� þ 4e1, 2�3�ð�1þ 3�Þ,

b2, 2 ¼ �b2, 4 ¼ �B2u0 � 2c2, 2�� 3d2, 2�2� � 4e2, 2�

3�,

b2, 1 ¼ �b2, 3 ¼ B2u0 � 2c2, 1�� 3d2, 1�2� � 4e2, 1�

3�,

b3, 2 ¼ �b3, 4 ¼ �4c3, 2�� þ 32e3, 2�3�ð�1þ 3�Þ,

c1, 2 ¼ c1, 4 ¼ D1u20 þ 6e1, 2�

2�ð�1þ 2�Þ,

c2, 1 ¼ c2, 3 ¼ ðC2 þD2Þu20 � 3d2, 1�� 6e2, 1�

2�,

c2, 2 ¼ c2, 4 ¼ ðC2 þD2Þu20 � 3d2, 2�� 6e2, 2�

2�,

c3, 2 ¼ c3, 4 ¼ D3u20 þ 24e3, 2�

2�ð�1þ 2�Þ,

d2, 1 ¼ �d2, 3 ¼ ðE2 þ F2Þu30 � 4e2, 1��,

d2, 2 ¼ �d2, 4 ¼ �ðE2 þ F2Þu30 � 4e2, 2��,

d1, 2 ¼ �d1, 4 ¼ �4e1, 2��,

d3, 2 ¼ �d3, 4 ¼ �8e3, 2��,

e2, 1 ¼ e2, 3 ¼ ðG2 þH2 þ I2Þu40,

e3, 2 ¼ e3, 4 ¼ G3u40,

e2, 2 ¼ e2, 4 ¼ ðG2 þH2 þ I2Þu40,

e1, 2 ¼ e1, 4 ¼ G1u40:

ð16Þ

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Once the 13 distribution functions are estimated they are substitutedback into Eqs. (7) to find that all properties are fully satisfied. As a resultin the analytical solutions, based on the 13-bit square lattice model, allflow characteristics are recovered and the LB evolution equation issatisfied. Hence the solution is an exact representation of the thermalCouette flow problem and it is valid for any relaxation time � and latticespacing. The only pitfall is that the Navier–Stokes equations are notfully recovered since the fourth order constraints are not satisfied.This drawback is circumvented in the next session by proposing a 17discrete velocity model.

The obtained results of the two 13-bit models are indicative for theaccuracy to expect implementing the 13-bit hexagonal and square lattice.It is seen that the accuracy of the 13-bit square lattice is improvedcompared with the accuracy of the 13-bit hexagonal lattice. No remarkshowever, can be made regarding stability issues of the two models.Actually previous stability analysis performed on the two models(Pavlo et al., 1998b) has shown that the 13-bit hexagonal model ismore stable than the 13-bit square model.

III. ANALYTICAL SOLUTIONS OF

THE 17-BIT MODEL

The 17-bit model introduced here is a straightforward extension ofthe 9-bit model used in isothermal problems. It is consisting of one restparticle,

e�, i ¼ 0, ð17aÞ

for � ¼ 0, and four nonzero speeds for which

e�, i ¼ e�ðcos�i, sin�iÞ, ð17bÞ

for � ¼ 1, 3 and i ¼ 1, 2, 3, 4 where e1 ¼ 1, e3 ¼ 2, �i ¼ ði � 1Þ�=2 and

e�, i ¼ e�ðcos�i, sin�iÞ, ð17cÞ

for � ¼ 2, 4 and i ¼ 1, 2, 3, 4 where e2 ¼ffiffiffi2

p, e4 ¼ 2

ffiffiffi2

p, �i ¼ ði � 1=2Þ�.

The analytical formulation has been described extensively in theprevious section and so we will be brief here presenting only the new

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material. The coefficients of the 17-bit equilibrium distribution functions,expressed by Eq. (3) are

A0 ¼1

2ð2� 5"þ 4"2Þ, A1 ¼

2

3ð"� "2Þ, A3 ¼

1

24ð�"þ 4"2Þ

B1 ¼1

3ð2� 3"Þ, B2 ¼

1

4", B3 ¼

1

24ð�1þ 3"Þ

C1 ¼1

3ð2� 3"Þ, C2 ¼

1

12ð2� 3"Þ,

C3 ¼1

96ð�1þ 6"Þ, C4 ¼

1

384ð�1þ 6"Þ

D0 ¼ �1

8ð5þ 7"Þ, D1 ¼

1

3ð�1þ 4"Þ,

D2 ¼ �1

2", D3 ¼

1

48ð1� "Þ, D4 ¼ �

1

32"

E1 ¼ �1

3, E2 ¼ �

1

24, E3 ¼

1

48, E4 ¼

1

384,

F1 ¼1

2, F2 ¼ �

1

8, F3 ¼ �

1

16,

G0 ¼1

4, G2 ¼

1

24, G4 ¼ �

1

96,

H1 ¼ �1

6, H2 ¼ �

1

24, H3 ¼

1

96, H4 ¼

1

384

ð18Þ

Again the coefficients, which are not included in the above expres-sions, are taken equal to zero. The set of equilibrium functions resultingfrom the above constants, unlike the ones obtained by the two 13-bitmodels, satisfy all 13 constraints given in Eq. (2).

The distribution functions for � ¼ 1, 3 and i ¼ 0, 1, 3 are equal to thecorresponding equilibrium distributions. The unknown distributionfunctions for the remaining directions have a polynomial form identicalto Eq. (15). The unknown coefficients for � ¼ 1, 3 with i ¼ 2, 4 and � ¼ 2with i ¼ 1, 2, 3, 4 are given by Eq. (16), while the coefficients for � ¼ 4and i ¼ 1, 2, 3, 4 are given by

a4, 1 ¼ a4, 3 ¼ A4 � 2b4, 1�� 4c4, 1�2� � 8d4, 1�

3� � 16e4, 1�4�,

a4, 2 ¼ a4, 4 ¼ A4 � 2b4, 2�� 4c4, 2�2� � 8d4, 2�

3� � 16e4, 2�4�,

b4, 1 ¼ �b4, 3 ¼ 2B4u0 � 4c4, 1�� 12d4, 1�2� � 32e4, 1�

3�,

b4, 2 ¼ �b4, 4 ¼ 2B4u0 � 4c4, 2�� 12d4, 2�2� � 32e4, 2�

3�,

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c4, 1 ¼ c4, 3 ¼ ð4C4 þD4Þu20 � 6d4, 1�� 24e4, 1�

2�,

c4, 2 ¼ c4, 4 ¼ ð4C4 þD4Þu20 � 6d4, 2�� 24e4, 2�

2�,

d4, 1 ¼ �d4, 3 ¼ 2ð4E4 þ F4Þu30 � 8e4, 1��,

d4, 2 ¼ �d4, 4 ¼ �2ð4E4 þ F4Þu30 � 8e4, 2��,

e4, 1 ¼ e4, 3 ¼ ðG4 þ 4H4 þ 16I4Þu40,

e4, 2 ¼ e4, 4 ¼ ðG4 þ 4H4 þ 16I4Þu40: ð19Þ

The resulting analytical distribution functions satisfy exactly allproperties described by Eqs. (7) and the LB Eq. (1) and can be consideredas an exact representation of the thermal Couette flow. The analyticalsolution is independent of � and �.

IV. CONCLUDING REMARKS

In conclusion we have developed analytical solutions of the thermalCouette flow with a temperature difference at the boundaries usingthree different discrete velocity models. The analytical solutions for thediscretized distribution functions must satisfy all 13 thermal hydrodynam-ic constraints in order to ensure full recovery of the Navier–Stokesequations, the flow characteristics and the profiles of the macroscopicsolutions. It is seen that the solutions based on a 17-bit square lattice fulfilsall above requirements. The solution based on the 13-bit square latticealthough represents exactly the thermal flow problem does not recoverthe fourth order constraint. Finally the solution based on the 13-bithexagonal model posses a second order error in the thermal flow solutionsand a third order deviation in the third order constraints.

ACKNOWLEDGMENT

This work has been partially supported by the AssociationEURATOM-Hellenic Republic (Contract ERB 5005 CT 990100).

REFERENCES

Alexander, F. J., Chen, S., Sterling, J. D. (1993). Lattice Boltzmannthermohydrodynamics. Phys. Rev. E 47(4):R2249–R2252.

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Chen, S., Doolen, G. D. (1998). Lattice Boltzmann method for fluidflows. Annu. Rev. Fluid Mech. 30:329–364.

Chen, Y., Ohachi, H., Akiyama, M. (1994). Thermal lattice Bhatnagar–Gross–Krook model without non-linear deviations in macro-dynamic equations. Phys. Rev. E 50(3):2776–2783.

Cornubert, R., d’Humieres, D., Levermore, D. (1991). A Knudsen layertheory for lattice gases. Physica D 47(6):241–259.

He, X., Zou, Q., Luo, L. S., Dembo, M. (1997). Analytical solutions ofsimple flows and analysis of nonslip boundary conditions for thelattice Boltzmann BGK model. J. Stat. Phys. 87:115–136.

Henon, M. (1987). Viscosity of a lattice gas. Complex Systems 1:763–789.Luo, L. S. (1997). Analytic solutions of linearized lattice Boltzmann

equation. J. Stat. Phys. 88:913–926.Luo, L. S., Chen, H., Chen, S., Doolen, G. D., Lee, Y. C. (1991).

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McNamara, G., Alder, B. (1993). Analysis of the lattice Boltzmanntreatment of hydrodynamics. Physica A 194:218–228.

McNamara, G., Garcia, A. L., Alder, B. J. (1995). Stabilization ofthermal lattice Boltzmann models. J. Stat. Phys. 81:395–408.

Pavlo, P., Vahala, G., Vahala, L. (1998a). Higher order isotropic velocitygrids in lattice methods. Phys. Rev. Lett. 80(18):3960–3963.

Pavlo, P., Vahala, G., Vahala, L., Soe, M. (1998b). Linear stabilityanalysis of thermo-lattice Boltzmann models. J. Comput. Phys.139:79–91.

Qian, Y. H. (1993). Simulating thermohydrodynamics with lattice BGKmodels. J. Sci. Comput. 8(3):231–242.

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Zou, Q., Hou, S., Doolen, G. D. (1995). Analytical solutions of the latticeBoltzmann BGK model. J. Stat. Phys. 81:319–339.

Received September 15, 2001Revised June 25, 2002Accepted June 25, 2002

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Analytical Lattice Boltzmann Solutions for Thermal Flow Problems