Multiple-relaxation-time model for the correct thermohydrodynamic equations
Lin Zheng, Baochang Shi, and Zhaoli Guo
*
National Laboratory of Coal Combustion, Huazhong University of Science and Technology,
Wuhan 430074, People’s Republic of China
Received 27 December 2007; revised manuscript received 12 May 2008; published 12 August 2008
A coupling lattice Boltzmann equation LBE model with multiple relaxation times is proposed for thermal
flows with viscous heat dissipation and compression work. In this model the fixed Prandtl number and the
viscous dissipation problems in the energy equation, which exist in most of the LBE models, are successfully
overcome. The model is validated by simulating the two-dimensional Couette flow, thermal Poiseuille flow,
and the natural convection flow in a square cavity. It is found that the numerical results agree well with the
analytical solutions and/or other numerical results.
DOI: 10.1103/PhysRevE.78.026705 PACS numbers: 47.11.-j, 44.05.+e
I. INTRODUCTION
The lattice Boltzmann equation LBE method has been
considered as a powerful numerical tool for simulating com-
plex athermal or isothermal fluid flows and associated trans-
port phenomena 1,2. LBE was first proposed as a numerical
scheme by McNamara and Zanetti 3, which obtained a
smoother macroscopic behavior than the lattice gas. Unlike
conventional numerical schemes based on discretization of
the macroscopic continuum equation, the LBE is based on
the microscopic or mesoscopic kinetic equation. The main
ideas for LBE can be manifested by two aspects: One is that
it treats the fluid on a statistical level and the particle density
distribution function is solved by a discrete Boltzmann equa-
tion; another is to construct a simplified kinetic model that
can represent the essential physics of microscopic or mesos-
copic processes so that the macroscopic averaged properties
can obey the desired macroscopic equations.
Because of the advantages of the LBE and easy boundary
treatment, LBE has been applied to thermal flows. To clearly
understand the mechanism of the thermal flows, recently,
three categories of thermal LBE models have been proposed,
i.e., the multispeed MS approach, the double-distribution-
function DDF approach, and the hybrid approach. The MS
approach is a straightforward extension of the athermal LBE
for isothermal flows 4 –15, which obeys the Boltzmann
equation; the DDF approach utilizes two different distribu-
tion functions, one for the velocity field and the other for the
temperature or internal energy field 16–24; the hybrid ap-
proach can be considered as another version of the DDF
approach 25. Nevertheless, the energy equation in the hy-
brid approach is solved by different numerical methods
rather than by LBE.
However, the application of MS, DDF, and hybrid ap-
proaches for thermal flows still has many challenges. The
fundamental problem with most of the MS models is the
insufficient truncation in the equilibrium distribution func-
tion and the lack of isotropy in the lattice model 26,27.
Furthermore, the fixed Prandtl number exists in most of the
MS models, although this problem has been overcome by
some modified models. Nevertheless, in most of the existing
improved MS models except for Ref. 28, the viscous co-
efficient in the energy equation is not consistent with that in
the momentum equation 29–31 as Pr 1: The transport co-
efficient of the viscous term appearing in the energy equation
is the thermal conductivity rather than the shear viscosity. On
the other hand, although the DDF models have improved
numerical stability and can overcome the fixed Prandtl num-
ber problem, the DDF based approaches rely on the conve-
nient mathematical artifacts whose purpose is merely to re-
cover the three conservation equations at the continuum
level, and the viscous heat dissipation and compression work
done by the pressure were ignored by most of the models
except for those proposed in Refs. 18–20. Furthermore, all
of the existing DDF models are “decoupling models, i.e.,
the momentum equation adopts an equation of state with a
constant temperature. This decoupling between the energy
and momentum equations may result in a large error when
applied to problems where the temperatures have significant
influences on the velocity field. In the hybrid approach, the
energy equation is solved by other numerical methods and
the flow simulation was not only decoupled from the energy
equation, but also usually ignored the viscous dissipation
term and compression work.
It should be pointed out that, to the authors’ knowledge,
the improved MS model in Ref. 28 has firstly removed the
inconsistent viscous problem between the momentum equa-
tion and the energy equation. However, to obtain the correct
thermohydrodynamic equations, the third-order moments—
the eigenvector of the collision matrix, of which the corre-
sponding eigenvalue is related to thermal conductivity—are
not in the frame of the lattice but the moving fluid 28. This
local varying eigenvector may require the collision operator
to be varied locally and may be the reason for its few appli-
cations. The detailed analysis of this model can be found in
Ref. 28.
Up to date, most of the LBE models adopt the BGK col-
lision model 32 which is approximated by a relaxation pro-
cess with a single relaxation time. What should be mentioned
is that the LBE models with multiple relaxation times MRT
have also attracted much attention in recent years. Due to
their apparent advantages over the BGK model 32, the
MRT-LBE models have been successfully applied to a vari-
ety of isothermal flows 33–36 and recently formulated in a
more general fashion 37.
Compared with the BGK model, the additional freedoms
in the MRT model, i.e., different physical modes can be ma-
*
Corresponding author; zlguo@hust.edu.cn
PHYSICAL REVIEW E 78, 026705 2008
1539-3755/2008/782/02670510 ©2008 The American Physical Society 026705-1