SIMPLIFIED FINITE DIFFERENCE THERMAL LATTICE ...azwadi/pdf_publication/ISI3.pdfSeptember 3, 2008 10:0 WSPC/140-IJMPB 04861 International Journal of Modern Physics B Vol. 22, No. 22

September 3, 2008 10:0 WSPC/140-IJMPB 04861

International Journal of Modern Physics BVol. 22, No. 22 (2008) 3865–3876c© World Scientific Publishing Company

SIMPLIFIED FINITE DIFFERENCE THERMAL LATTICE

BOLTZMANN METHOD

C. S. NOR AZWADI

Faculty of Mechanical Engineering, Universiti Teknologi Malaysia,

81310 UTM Skudai, Johor, Malaysia

[email protected]

T. TANAHASHI

Department of Mechanical Engineering, Keio University,

Yokohama, Kanagawa, Japan

[email protected]

Received 5 November 2007

In this paper, a well-known finite difference technique is combined with thermal latticeBoltzmann method to solve 2-dimensional incompressible thermal fluid flow problems. Asmall number of microvelocity components are applied for the calculation of temperaturefield. The combination of finite difference with lattice Boltzmann method is found to bean efficient and stable approach for the simulation at high Rayleigh number of naturalconvection in a square cavity.

Keywords: Thermal lattice Boltzmann; finite difference; FDLBM; natural convectionflow.

1. Introduction

The lattice Boltzmann method (LBM) is considered as an alternative approach to

the well-known finite difference, finite element, and finite volume techniques for

solving the Navier-Stokes equations. LBM evolved from Lattice Gas Automata,1

simulates fluid flows by tracking the evolution of the single-particle distribution.

Although as a newcomer in numerical scheme, the lattice Boltzmann approach has

found recent successes in a host of fluid dynamical problems, including flows in

porous media, magnetohydrodynamics, immiscible fluids, and turbulence. How-

ever, the simulation of flows with heat transfer turned out to be much more

difficult.

In general, the current thermal lattice Boltzmann models fall into three cate-

gories: the multi-speed approach,2 the passive scalar approach,3 and the thermal

energy distribution model proposed by He et al.4 The multi-speed approach uses

3865


3866 C. S. Nor Azwadi & T. Tanahashi

the same distribution function in defining the macroscopic velocity, pressure, and

temperature. In addition to mass and momentum, in order to preserve the kinetic

energy in the collision on each lattice point, this model requires more variations

of speed than those of the isothermal model and equilibrium distribution function

usually include higher order velocity terms. However, this model is reported to

suffer severe numerical instability, and is not computationally efficient.5

In the passive scalar model, the flow fields (velocity and density) and the tem-

perature are represented by two different distribution functions. The macroscopic

temperature is assumed to satisfy the same evolution equation as a passive scale,

which is advected by the flow velocity, but does not affect the flow field. It has

been shown that the passive scalar model has better numerical stability than the

multi-speed model.5

He et al.4 in their model introduce the internal energy density distribution func-

tion, which can be derived from the Boltzmann equation. This model is shown to

be a suitable model for simulating real thermal problems. However, the complicated

gradient operator term appears in the evolution equation and thus the simplicity

property of the lattice Boltzmann scheme has been lost.6

The conventional isothermal and thermal lattice Boltzmann models, however,

only give second-order accuracy in space and time. Since Luo and He7,8 and Abe9

demonstrated that the lattice Boltzmann equation is a discretized form of the con-

tinuous Boltzmann equation and the discretization of physical space is not coupled

with the discretization of momentum space, any standard numerical technique can

serve the purpose of solving the discrete Boltzmann equation. It is not surprising

that the well-known finite difference has being introduced in order to improve the

accuracy of isothermal and thermal LBM.

The first finite difference LBM (FDLBM) was due to Reider and Sterling,10

and was examined by Cao et al.11 in more detail. FDLBM was further extended

to curvilinear coordinates with non-uniform grids by Mei and Shyy.12 The study of

FDLBM is still in progress.13–15 However, there are still no evidence of combination

of finite difference with thermal lattice Boltzmann.

The purpose of this paper is to improve the earlier proposed double-distribution

function thermal lattice Boltzmann method (TLBM).16 Although this model has

successfully simulated the natural convection problem to a certain degree with

low computational cost, it is limited for the simulation at low Rayleigh numbers.

However, for real thermal engineering applications, the value of Rayleigh numbers

could be achieved up to 106. In this paper, we apply the finite difference technique to

solve the advection term in the governing equations of double-distribution function

TLBM. The combination of finite difference with TLBM (FDTLBM) contributes

in allowing us to increase the accuracy both in time and space where the high

order accuracy is crucial for the simulation of natural convection at high Rayleigh

numbers.


Finite Difference Thermal Lattice Boltzmann Method 3867

2. Double-Distribution Function TLBM

Following the double-distribution function approach proposed by He4 and Nor

Azwadi and Takahaski,16 the governing equations for these two functions are

∂f

∂t+ c

∂f

∂x= − 1

τv(f − f eq) + Ff , (1)

∂g

∂t+ c

∂g

∂x= − 1

τc(g − geq) , (2)

where the density distribution function f = f(x, c, t) is used to simulate the

density and velocity fields, and the internal energy density distribution function

g = g(x, c, t) is used to simulate the macroscopic temperature field. The macro-

scopic variables, such as the density ρ, velocity u, and temperature T can be eval-

uated as the moment to the distribution function

ρ =

∫

fdc , ρu =

∫

cfdc , ρT =

∫

gdc . (3)

f eq and geq in Eqs. (1) and (2) are the equilibrium distribution function for density

and internal energy, respectively, and is given by

f eq = ρ(1

2πRT)D/2 exp

{

− c2

2RT

} [

1 +c · uRT

+(c · u)22(RT )2

− u2

2RT

]

, (4)

geq = ρT

(

1

2πRT

)D/2

exp

{

− c2

2RT

}

[

1 +c · uRT

]

. (5)

Equation (5) is obtained by assuming that at low Mach number flow (incompressible

flow), the higher order of macroscopic velocity and viscous heat dissipation can be

neglected.6 It has also been proved17 that the above simplification does not alter

the corresponding macroscopic equation of energy. The only change is the value

of the constant parameter in the thermal conductivity, which can be absorbed by

manipulating the parameter τc.

We have also recently shown that the discretized equilibrium distribution func-

tion for both density and internal energy density distribution function can be ob-

tained by applying the Gauss–Hermite quadrature procedure for the calculation of

f eq and geq velocity moments. As a result, a 2-dimensional 9-velocity, D2Q9 lat-

tice model, as shown in Fig. 1(left), is obtained, and the corresponding discretized

equilibrium density distribution function is given by

f eqi = ρωi

[

1 + 3c · uc2

+9(c · u)2

2c4− 3u

2

2c2

]

, (6)

where c =√

3RT and the weights are ω1 = 4/9, ω2 = ω3 = ω4 = ω5 = 1/9,

and, ω6 = ω7 = ω8 = ω9 = 1/36. While the lattice type for the energy model

is 2-dimensional 4-velocity, D2Q4 lattice model shown in Fig. 1(right), and the



5c

1c

8c

6c

7c

4c

3c

2c

0c

1c

4c

3c

2c

Fig. 1. Lattice structure for D2Q9 (left) and D2Q4 (right).

corresponding discretized internal energy density equilibrium distribution function

is given by16

geq1,2,3,4 =1

4ρT

[

1 +c · uc2

]

. (7)

Through a multiscaling expansion, the mass and momentum equation can be

derived from D2Q9 and temperature equation from D2Q4 as below

∇ · u = 0 , (8)

∂u

∂t+ u∇ · u = −1

ρ∇p + ν∇2u , (9)

∂T

∂t+ ∇ · (uT ) = χ∇2T . (10)

The viscosity and thermal diffusivity in these models are related to the time

relaxations as below

ν =1

3τv , (11)

χ = τc . (12)

3. Finite Difference Thermal Lattice Boltzmann Method

(FDTLBM)

The temporal discretization is obtained using second-order Runge–Kutta (modified)

Euler method. The time evolution of particle distributions is then derived by

fn+ 1

2

i = fni +

∆t

2

[

−ci · ∇fni −1

τv(fni − f eq,ni )

]

, (13)

fn+1i = fni + ∆t

[

−ci · ∇fn+1

2

i −1

τv

(

fn+ 1

2

i − feq,n+ 1

2

i

)

]

. (14)



The third-order upwind scheme (UTOPIA) was applied to calculate the advec-

tion term in Eq. (1) as below

cix∂xfi = cixfi(x + 2∆x, y) − 2fi(x + ∆x, y) + 9fi(x, y)

6∆x

+ cix−10fi(x − ∆x, y) + 2fi(x − 2∆x, y)

6∆x, cix > 0 , (15)

cix∂xfi = cix−fi(x − 2∆x, y) + 2fi(x − ∆x, y) − 9fi(x, y)

6∆x

+ cix10fi(x + ∆x, y) − 2fi(x + 2∆x, y)

6∆x, cix < 0 , (16)

ciy∂yfi = ciyfi(x, y + 2∆y) − 2fi(x, y + ∆y) + 9fi(x, y)

6∆y

+ ciy−10fi(x, y − ∆y) + 2fi(x, y − 2∆y)

6∆y, ciy > 0 , (17)

ciy∂yfi = ciy−fi(x, y − 2∆y) + 2fi(x, y − ∆y) − 9fi(x, y)

6∆y

+ ciy10fi(x, y + ∆y) − 2fi(x, y + 2∆y)

6∆y, ciy < 0 . (18)

The same procedures were carried out for the evolution of temperature equation.

From this combination, the accuracy of the FDTLBM is second order in time and

third order in space. The time step used in the computation is varied between 0.1

and 0.001, depending on the Rayleigh number and mesh size.

4. Natural Convection in a Square Cavity

Numerical simulation for the natural convection flow in a square cavity with a

hot wall on the left side and cool wall on the right side up to Rayleigh number,

Ra = 106 was carried out to test the effectiveness of the FDTLBM. Figure 2 shows

a schematic diagram of the setup in the simulation.

The conventional no-slip boundary conditions1 are imposed on all the walls of

the cavity. The thermal conditions applied on the left and right walls are T (x =

0, y) = TH and T (x = L, y) = TC . The top and bottom walls being adiabatic,

∂T/∂y = 0.

The temperature difference between the left and right walls introduces a tem-

perature gradient in a fluid, and the consequent density difference induces a fluid

motion, that is, convection.

In the simulation, the Boussinesq approximation is applied to the buoyancy

force term.

ρG = ρβg0(T − Tm)j , (19)



0,0 =∂

∂=

y

Tu

x

y

HTT =

= 0u

CTT =

= 0u

0,0 =∂

∂=

y

Tu

0g

L

L

Fig. 2. Schematic geometry for natural convection in a square cavity.

where β is the thermal expansion coefficient, g0 is the acceleration due to gravity,

Tm is the average temperature, and j is the vertical direction opposite to that of

gravity. So the external force in Eq. (1) will be

Ff = 3G(c − u)f eq . (20)

The dynamical similarity depends on two dimensionless parameters: the Prandtl

number, Pr and the Rayleigh number, Ra

Pr =ν

χ, Ra =

g0β∆TL3

νχ. (21)

In all simulations, Pr is set to be 0.71 and through the grid dependence study,

the grid sizes of 101 × 101, 151 × 151, 201 × 201, and 251 × 251 are suitable forRayleigh number 103, 104, 105, and 106, respectively. The convergence criterion for

all the cases tested is

Max|(

(u2 + v2)n+1)

1

2 −(

(u2 + v2)n)

1

2 | ≤ 10−7 , (22)

= Max|T n+1 − T n| ≤ 10−7 , (23)

where the calculation is carried out over the entire system.

5. Numerical Results

Figures 3–6 show the time development of isotherms and their corresponding

streamlines for all the Rayleigh numbers simulations.

At the beginning of the simulation for Ra = 103, a vortex appears at the center

left of the cavity. As time evolves, the vortex is shifted to the center of the cavity.

The isotherms are almost vertically parallel to the wall, indicating that conduction

is the dominant heat transfer mechanism. For Ra = 104, a vertically oval-shaped



Fig. 3. Time development of isotherms and streamlines for Ra = 103.




vortex appears at the center left of the cavity. After that, the vortex is shifted

to the center of the cavity and its shape changes to horizontal oval due to the

convection effect. Isotherms start to be horizontally parallel to the wall at the cavity

center. This indicates that the heat transfer mechanisms are mixed conduction and

convection.

For the simulation at Ra = 105, two vortices appear, where one at the top

left and the other one at the bottom right of the cavity when the system achieved

equilibrium condition. All isotherms are almost horizontally parallel to the wall,

indicating that the convection is the main heat transfer mechanism. The vortices

continue to break up when the Rayleigh number is increased up to 106.

Figure 7 shows the non-dimensional temperature profile given at the mid-height

of the cavity for the laminar flow simulations. The profiles show the rapid change in

the heat transfer mechanisms from conduction to convection. From a 45◦ slope at

low Rayleigh number, the temperature profiles become horizontal lines in the cavity

center and all temperature gradients are located in the interior of the boundary

layer, which has developed near the vertical walls. Near the center of the cavity, the

curves change slope and there is a vortex corresponding to each change. It can be

clearly seen that the steep variation of the temperature near the walls is resolved

quite well.

Figures 8 and 9 present similar profiles for the horizontal and vertical velocity

components, respectively. Both figures show a gradually increasing velocity near

the center, and the development of narrow boundary layers along the walls. The

peak values of the horizontal and vertical velocities increase due to the intensified

convective activities with increase in Rayleigh number. The steep rise in the vertical

velocity gradient at the point on the hot and cold walls also confirms the increased

310=Ra

410=Ra

610=Ra

510=Ra

Fig. 7. Non-dimensional temperature profile at the mid-height of the cavity.



310=Ra

410=Ra

610=Ra

510=Ra

Fig. 8. Non-dimensional horizontal velocity at the mid-width of the cavity.

310=Ra

410=Ra

610=Ra

510=Ra

310=Ra

410=Ra

610=Ra

510=Ra

Fig. 9. Non-dimensional vertical velocity at the mid-width of the cavity.

convective activity at higher Rayleigh number values observed in Fig. 9. The changes

in the velocity direction correspond to slope changes of the temperature profile and

lead to vortex development.

In order to validate the present numerical algorithm, the predicted results

are compared with the results obtained by the Navier–Stokes equation approach.

Among the characteristic numerical values of the flow, the comparisons concern the

average Nusselt number at the mid-plane wall, Nuave the maximum value for hor-

izontal and vertical velocity components, umax and vmax with the positions where

they occur.



Table 1. Comparison between the present results (FDTLBM) and aNavier–Stokes solver.18

Ra

103 104 105 106

umax FDTLBM 3.638 16.127 34.700 65.827

N–S Solver18 3.634 16.182 34.810 65.330

y FDTLBM 0.810 0.820 0.855 0.852

N–S Solver18 0.813 0.823 0.855 0.848

vmax FDTLBM 3.691 19.584 68.319 221.071

N–S Solver18 3.679 19.509 68.220 216.750

x FDTLBM 0.180 0.120 0.065 0.040

N–S Solver18 0.179 0.120 0.066 0.038

Nuave FDTLBM 1.117 2.233 4.483 8.723

N–S Solver18 1.116 2.234 4.510 8.798

As shown in Table 1, for Ra = 103 and Ra = 104, the present results are in

close agreement with the Navier–Stokes solution obtained by Davis.18 However,

small discrepancies can be seen for higher values of Rayleigh numbers, Ra = 105

and Ra = 106. For all values of Rayleigh number considered in the present analysis,

the average Nusselt number for the system have been predicted with less than 3%

error and can be accepted for real engineering applications.

6. Conclusion

The natural convection in a differentially heated square cavity has been studied

using the double-distribution approach thermal lattice Boltzmann method with

small microvelocity components applied in the internal energy distribution func-

tion. The evolution of lattice Boltzmann equations have been discretized using the

third-order accuracy finite difference upwind, UTOPIA scheme. From Figs. 7–9,

the boundary layers for the velocities and temperature can be observed clearly. As

expected, the thermal boundary layer is thicker than the velocity boundary layer

for every Rayleigh number simulations. The flow patterns including the boundary

layers and vortices can be seen clearly. The results obtained demonstrate that this

new approach in the double-distribution function thermal lattice Boltzmann model

is a very efficient procedure to study flow and heat transfer in a differentially heated

square enclosure.

Acknowledgments

The authors wish to acknowledge Universiti Teknologi Malaysia, Keio University

and the Malaysian government for supporting these research activities.

References

1. J. Hardy, Y. Pomeau and D. Pazzis, J. Math. Phys. 14, 1746 (1973).



2. G. McNamara and B. Alder, Phys. A. 194, 218 (1993).3. X. Shan, Phys. Rev. E. 55, 2780 (1997).4. X. He, S. Shan and G. D. Doolen, J. Comp. Phys. 146, 282 (1998).5. H. Chen and C. Teixeira, Comp. Phys. Comm. 129, 21 (2000).6. Y. Peng, C. Shu and Y. T. Chew, Phys. Rev. E 68, 020671 (2003).7. L. S. Luo and X. He, Phys. Rev. E 55, R6333 (1997).8. X. He and L. S. Luo, Phys. Rev. E 56, 6811282 (1997).9. T. Abe, J. Comp. Phys. 131, 241 (1997).

10. M. B. Reider and J. D. Sterling, Comp. Fluids 24, 459 (1995).11. N. Cao, S. Chen, S. Jin and D. Martinez, Phys. Rev. E 55, R21 (1997).12. Z. R. Mei and W. Shyy, J. Comp. Phys. 143, 426 (1998).13. J. Tolke, M. Krafczvk, M. Schulz, E. Rank and R. Berrios, Int. J. Mod. Phys. C 9,

1143 (1998).14. G. Hazi, Int. J. Mod. C 13, 67 (1998).15. T. Seta and R. Takahashi, J. Stat. Phys. 107, 557 (2002).16. C. S. Nor Azwadi and T. Takahashi, Int. J. Mod. Phys. B 20, 2437 (2006).17. Z. Guo, Y. Shi and T. S. Zhao, Phys. Rev. E 70, 066310 (2004).18. D. V. Davis, Int. J. Numer. Meth. Fluids 3, 249 (1983).

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SIMPLIFIED FINITE DIFFERENCE THERMAL LATTICE ...azwadi/pdf_publication/ISI3.pdfSeptember 3, 2008 10:0 WSPC/140-IJMPB 04861 International Journal of Modern Physics B Vol. 22, No. 22