Double MRT Thermal Lattice Boltzmann Method for Simulating Natural Convection of Low Prandtl Number Fluids
📝 Abstract
The purposes of this paper are testing an efficiency algorithm based on LBM and using it to analyze two-dimensional natural convection with low Prandtl number. Steady state or oscillatory results are obtained using double multiple-relaxation-time thermal lattice Boltzmann method. The velocity and temperature fields are solved using D2Q9 and D2Q5 models, respectively. With different Rayleigh number, the tested natural convection can either achieve to steady state or oscillatory. With fixed Rayleigh number, lower Prandtl number leads to a weaker convection effect, longer oscillation period and higher oscillation amplitude for the cases reaching oscillatory solutions. At fixed Prandtl number, higher Rayleigh number leads to a more notable convection effect and longer oscillation period. Double multiple-relaxation-time thermal lattice Boltzmann method is applied to simulate the low Prandtl number fluid natural convection. Rayleigh number and Prandtl number effects are also investigated when the natural convection results oscillate.
💡 Analysis
The purposes of this paper are testing an efficiency algorithm based on LBM and using it to analyze two-dimensional natural convection with low Prandtl number. Steady state or oscillatory results are obtained using double multiple-relaxation-time thermal lattice Boltzmann method. The velocity and temperature fields are solved using D2Q9 and D2Q5 models, respectively. With different Rayleigh number, the tested natural convection can either achieve to steady state or oscillatory. With fixed Rayleigh number, lower Prandtl number leads to a weaker convection effect, longer oscillation period and higher oscillation amplitude for the cases reaching oscillatory solutions. At fixed Prandtl number, higher Rayleigh number leads to a more notable convection effect and longer oscillation period. Double multiple-relaxation-time thermal lattice Boltzmann method is applied to simulate the low Prandtl number fluid natural convection. Rayleigh number and Prandtl number effects are also investigated when the natural convection results oscillate.
📄 Content
1
Double MRT Thermal Lattice Boltzmann Method for Simulating Natural Convection of Low Prandtl Number Fluids
Zheng Lia,b, Mo Yanga and Yuwen Zhangb,1 a School of Energy and Power Engineering, University of Shanghai for Science and Technology, Shanghai 200093, China b Department of Mechanical and Aerospace Engineering, University of Missouri, Columbia, MO 65211, USA
ABSTRACT
Purpose: The purposes of this paper are testing an efficiency algorithm based on LBM
and using it to analyze two-dimensional natural convection with low Prandtl number.
Design/methodology/approach: Steady state or oscillatory results are obtained using
double multiple-relaxation-time thermal lattice Boltzmann method. The velocity and
temperature fields are solved using D2Q9 and D2Q5 models, respectively.
Findings: With different Rayleigh number, the tested natural convection can either
achieve to steady state or oscillatory. With fixed Rayleigh number, lower Prandtl
number leads to a weaker convection effect, longer oscillation period and higher
oscillation amplitude for the cases reaching oscillatory solutions. At fixed Prandtl
number, higher Rayleigh number leads to a more notable convection effect and longer
oscillation period.
Originality/value: Double multiple-relaxation-time thermal lattice Boltzmann method
is applied to simulate the low Prandtl number (0.001 – 0.01) fluid natural convection.
Rayleigh number and Prandtl number effects are also investigated when the natural
convection results oscillate.
Keywords: lattice Boltzmann method, multiple-relaxation-time model, natural
convection, low Prandtl number
NOMENCLATURE c lattice speed p c specific heat (J/kgK) sc sound speed ie particle speed if density distribution iF body force Fo Fourier number g gravity acceleration 2 / m s
G effective gravitational acceleration 2 / m s
gi energy distribution k thermal conductivity (W/m k) M transform matrix for density distribution i m moment function for density distribution Ma Mach number
1 Corresponding author. Email: zhangyu@missouri.edu. Tel: 001-573-884-6936. Fax: 001-573-884- 5090 2
N transform matrix for density distribution
in moment function for energy distribution
p pressure (Pa)
P non-dimensional pressure
Pr Prandtl number
Q collision matrix for energy distribution
Ra Rayleigh number
s relaxation time in density distribution
S collision matrix for density distribution
t time (s)
T temperature
K
u velocity in x-direction (m/s)
iu particle speed in energy distribution
U non-dimensional velocity in x-direction
v velocity in y-direction (m/s)
V non-dimensional velocity in y-direction
V velocity
thermal diffusivity
2 /
m
s
thermal expansion (K-1)
t
time step (s)
non-dimensional temperature
viscosity (Kg/ms)
Density (kg/m3)
relaxation time in energy distribution
non-dimensional time
kinematic viscosity
2 /
m
s
- Introduction Lattice Boltzmann method (LBM) has been developed into a promising numerical method in the last two decades. It can be used to solve different fluid flow problems, such as incompressible fluid flow (Guo and Zhao, 2002), compressible fluid flow (Kataoka and Tsutahara, 2004) and multiphase fluid flow (Luo, 2000). Instead of solving the macroscopic continuum and momentum equations as the traditional computational fluid dynamics (CFD), the LBM is based on solving the discrete Boltzmann equation in statistical physics via two basic steps: collision step and streaming step (Succi, 2001). There are different LBM models for fluid flow problems. Lattice Bhatnagar-Gross-Krook (LBGK) simplifies the collision term with one relaxation time (Chen and Chen, 1991; Chen and Doolen, 1998). Based on LBGK model, Li et al. (2014a) and Li et al. (2014b) use combined LBM and finite volume method to solve lid driven flow and natural convection, respectively. Although it is widely used, LBGK is limited by the numerical instability (Lallemand and Luo, 2000). To overcome this limitation, entropy LBM (ELBM) (Chikatamarla et al., 2006; Chikatamarla and Karlin, 2006), two-relaxation-time model (TRT) (Ginzburg, 2005; Ginzburg and d’Humieres, 2007) and multiple relaxation time model (MRT) (Lallemand and Luo, 2000; Lallemand and Luo, 2003) have been proposed. The difference among these models lies in the ways to simplify the collision term while their streaming steps are the same. Luo et al. (2011) compared these models by using them 3
to solve the lid driven flow problem. It was concluded that the MRT was preferred due to its advantages in accuracy and numerical stability. The fluid flow problem with heat transfer also can be solved using LBM. Multispeed approach (MS), hybrid method and double distribution functions (DDF) are the common thermal LBM models. The MS approach obtains the temperature field by addi
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