A new and generalized lattice Boltzmann model for simulating thermal two-phase flow is described. In this model, the single component multi-phase lattice Boltzmann model proposed by Shan and Chen is used to simulate the fluid dynamics. The temperature field is simulated using the passive-scalar approach, i.e., through modeling the density field of an extra component, which evolves according to the advection-diffusion equation. By coupling the fluid dynamics and temperature field through a suitably defined body force term, the thermal two-phase lattice Boltzmann model is obtained. In this paper, the theoretical foundations of the model and the validity of the thermal lattice Boltzmann equation method are laid out, illustrated by analytical and numerical examples. In a companion paper (P. Yuan and L. Schaefer, 2006, ASME J. Fluids Eng., 128, pp. 151–156), the numerical results of the new model are reported.

1.
Chen
,
S.
, and
Doolen
,
G. D.
, 1998, “
Lattice Boltzmann Method for Fluid Flows
,”
Annu. Rev. Fluid Mech.
0066-4189,
30
, pp.
329
364
.
2.
Yu
,
D.
,
Mei
,
R.
,
Luo
,
L.
, and
Shyy
,
W.
, 2003, “
Viscous Flow Comutations With the Method of Lattice Boltzmann Equation
,”
Prog. Aerosp. Sci.
0376-0421,
39
, pp.
329
367
.
3.
Chen
,
S.
,
Chen
,
H.
,
Martinez
,
D.
, and
Matthaeus
,
W.
, 1991, “
Lattice Boltzmann Model for Simulation of Magnetohydrodynamics
,”
Phys. Rev. Lett.
0031-9007,
67
, pp.
3776
3779
.
4.
Shan
,
X.
, and
Chen
,
H.
, 1993, “
Lattice Boltzmann Model for Simulation Flows With Multiple Phases and Components
,”
Phys. Rev. E
1063-651X,
47
, pp.
1815
1819
.
5.
Shan
,
X.
, and
Chen
,
H.
, 1994, “
Simulation of Nonideal Gases and Liquid-Gas Phase Transitions by the Lattice Boltzmann Equation
,”
Phys. Rev. E
1063-651X,
49
, pp.
2941
2948
.
6.
Shan
,
X.
, and
Doolen
,
G. D.
, 1995, “
Multicomponent Lattice-Boltzmann Model With Interparticle Interaction
,”
J. Stat. Phys.
0022-4715,
81
, pp.
379
393
.
7.
Swift
,
M.
,
Osborn
,
W.
, and
Yeomans
,
J.
, 1995, “
Lattice Boltzmann Simulation of Nonideal Fluids
,”
Phys. Rev. Lett.
0031-9007,
75
, pp.
830
833
.
8.
Swift
,
M.
,
Orlandini
,
S.
,
Osborn
,
W.
, and
Yeomans
,
J.
, 1996, “
Lattice Boltzmann Simulations of Liquid-Gas and Binary-Fluid Systems
,”
Phys. Rev. E
1063-651X,
54
, pp.
5041
5052
.
9.
Nourgaliev
,
R.
,
Dinh
,
T.
,
Theofanous
,
T.
, and
Joesph
,
D.
, 2003, “
The Lattice Boltzmann Equation Method: Theoretical Interpretation, Numerics and Implications
,”
Int. J. Multiphase Flow
0301-9322,
29
, pp.
117
169
.
10.
Alexander
,
F. J.
,
Chen
,
S.
, and
Sterling
,
J. D.
, 1993, “
Lattice Boltzmann Thermohydrodynamics
,”
Phys. Rev. E
1063-651X,
47
, pp.
R2249
R2252
.
11.
Shan
,
X.
, 1997, “
Simulation of Rayleigh—Bénard Convection Using a Lattice Boltzmann Method
,”
Phys. Rev. E
1063-651X,
55
, pp.
2780
2788
.
12.
Yuan
,
P.
, and
Schaefer
,
L.
, 2006, “
A Thermal Lattice Boltzmann Two-Phase Flow Model and Its Application to Heat Transfer Problems—Part 2. Integration and Validation
,”
ASME J. Fluids Eng.
0098-2202,
128
, pp.
151
156
.
13.
Succi
,
S.
, 2001,
The Lattice Boltzmann Equation for Fluid Dynamics and Beyond
,
Oxford University Press
,
Oxford
, UK.
14.
He
,
X.
, and
Luo
,
L.
, 1997, “
Theory of the Lattice Boltzmann Method: From the Boltzmann Equation to the Lattice Boltzmann Equation
,”
Phys. Rev. E
1063-651X,
56
, pp.
6811
6817
.
15.
Chen
,
S.
,
Martinez
,
D.
, and
Mei
,
R.
, 1996, “
On Boundary Conditions in Lattice Boltzmann Method
,”
Phys. Fluids
1070-6631,
8
, pp.
2527
2536
.
16.
Inamuro
,
T.
,
Yoshino
,
M.
, and
Ogino
,
F.
, 1995, “
A Non-Slip Boundary Condition for Lattice Boltzmann Simulations
,”
Phys. Fluids
1070-6631,
7
, pp.
2928
2930
.
17.
He
,
X.
,
Zou
,
Q.
,
Luo
,
L.
, and
Dembo
,
M.
, 1997, “
Analytic Solutions and Analysis on Non-Slip Boundary Condition for the Lattice Boltzmann BGK Model
,”
J. Stat. Phys.
0022-4715,
87
, pp.
115
136
.
18.
Mei
,
R.
,
Luo
,
L.
, and
Shyy
,
W.
, 1999, “
An Accurate Curved Boundary Treatment in the Lattice Boltzmann Method
,”
J. Comput. Phys.
0021-9991,
155
, pp.
307
329
.
19.
Hou
,
S.
, 1995, “
Lattice Boltzmann Method for Incompressible Viscous Flow
,” Ph.D. thesis, Department of Mechanical Engineering, Kansas State Univ., Manhattan, Kansas, USA.
20.
Sukop
,
M.
, and
Or
,
D.
, 2004, “
Lattice Boltzmann Method for Modeling Liquid-Vapor Interface Configurations in Porous Media
,”
Water Resour. Res.
0043-1397,
40
,
W01509
.
21.
Kang
,
Q.
,
Zhang
,
D.
, and
Chen
,
S.
, 2002, “
Displacement of a Two-Dimensional Immiscible Droplet in a Channel
,”
Phys. Fluids
1070-6631,
14
, pp.
3203
3214
.
22.
Buick
,
J.
, and
Greated
,
C.
, 2000, “
Gravity in the Lattice Boltzmann Model
,”
Phys. Rev. E
1063-651X,
61
, pp.
5307
5320
.
23.
Martys
,
N. S.
, and
Chen
,
H.
, 1996, “
Simulation of Multicomponent Fluids in Complex Three-Dimensional Geometries by the Lattice Boltzmann Method
,”
Phys. Rev. E
1063-651X,
53
, pp.
743
750
.
24.
Hou
,
S.
,
Shan
,
X.
,
Zou
,
Q.
,
Doolen
,
G.
, and
Soll
,
W.
, 1997, “
Evaluation of Two Lattice Boltzmann Models for Multiphase Flows
,”
J. Comput. Phys.
0021-9991,
138
, pp.
695
713
.
25.
Yang
,
Z.
,
Dinh
,
T.
,
Nourgaliev
,
R.
, and
Sehgal
,
B.
, 2001, “
Numerical Investigation of Bubble Growth and Detachment by the Lattice-Boltzmann Method
,”
Int. J. Heat Mass Transfer
0017-9310,
44
, pp.
195
206
.
26.
McNamara
,
G.
,
Garcia
,
A. L.
, and
Alder
,
B. J.
, 1995, “
Stabilization of Thermal Lattice Boltzmann Models
,”
J. Stat. Phys.
0022-4715,
81
, pp.
395
408
.
27.
Pavlo
,
P.
,
Vahala
,
G.
, and
Vahala
,
L.
, 1998, “
Higher Order Isotropic Velocity Grids in Lattice Methods
,”
Phys. Rev. Lett.
0031-9007,
80
, pp.
3960
3963
.
28.
Inamuro
,
T.
,
Yoshino
,
M.
,
Inoue
,
H.
,
Mizuno
,
R.
, and
Ogino
,
F.
, 2002, “
A Lattice Boltzmann Method for a Binary Miscible Fluid Mixture and Its Application to a Heat-Transfer Problem
,”
J. Comput. Phys.
0021-9991,
179
, pp.
201
215
.
29.
Shu
,
C.
,
Peng
,
Y.
, and
Chew
,
Y.
, 2002, “
Simulation of Natural Convection in a Square Cavity by Taylor Series Expansion- and Least Squares-Based Lattice Boltzmann Method
,”
Int. J. Mod. Phys. C
0129-1831,
13
, pp.
1399
1414
.
30.
Busse
,
F.
, 1986,
Hydrodynamics Instabilities and the Transition to Turbulence
,
2nd ed.
,
Springer-Verlag
,
Berlin
, Germany.
You do not currently have access to this content.