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HYBRID FINITE-DIFFERENCE THERMAL LATTICE BOLTZMANN EQUATION

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International Journal of Modern Physics B
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PIERRE LALLEMAND
PIERRE LALLEMAND
PIERRE LALLEMAND
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Li-Shi Luo at Old Dominion University
Li-Shi Luo
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Abstract

We analyze the acoustic and thermal properties of athermal and thermal lattice Boltzmann equation (LBE) in 2D and show that the numerical instability in the thermal lattice Boltzmann equation (TLBE) is related to the algebraic coupling among different modes of the linearized evolution operator. We propose a hybrid finite-difference (FD) thermal lattice Boltzmann equation (TLBE). The hybrid FD-TLBE scheme is far superior over the existing thermal LBE schemes in terms of numerical stability. We point out that the lattice BGK equation is incompatible with the multiple relaxation time model.
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January 29, 2003 11:35 WSPC/140-IJMPB 01706
International Journal of Modern Physics B
Vol. 17, Nos. 1 & 2 (2003) 41–47
c
World Scientific Publishing Company
HYBRID FINITE-DIFFERENCE THERMAL LATTICE
BOLTZMANN EQUATION
PIERRE LALLEMAND∗
Laboratoire ASCI, Bˆatiment 506, Universit´e Paris-Sud (Paris XI Orsay)
91405 Orsay Cedex, France
LI-SHI LUO†
ICASE, MS 132C, NASA Langley Research Center
3 West Reid Street, Building 1152, Hampton, Virginia 23681-2199, USA
Received 9 August 2002
We analyze the acoustic and thermal properties of athermal and thermal lattice Boltz-
mann equation (LBE) in 2D and show that the numerical instability in the thermal
lattice Boltzmann equation (TLBE) is related to the algebraic coupling among different
modes of the linearized evolution operator. We propose a hybrid finite-difference (FD)
thermal lattice Boltzmann equation (TLBE). The hybrid FD-TLBE scheme is far supe-
rior over the existing thermal LBE schemes in terms of numerical stability. We point out
that the lattice BGK equation is incompatible with the multiple relaxation time model.
1. Introduction
In spite of its success in solving various challenging problems involving athermal
fluids, the lattice Boltzmann equation (LBE) has not been able to handle realistic
thermal fluids with satisfaction, even though there has been a continuous pursue in
this area for obvious reasons.1–28 The difficulty encountered in the thermal lattice
Boltzmann equation (TLBE) seems to be the numerical instabilities.
The existing thermal lattice Boltzmann models may be classified into three
categories. The first and the simplest one is to use two sets of distributions for the
flow fields and temperature which is treated as a passive scalar.2–6Numerically
this is not very efficient because of too many redundant degrees of freedom, even
though it can be improved somewhat.5The second category of the TLBE models
includes various shock capturing schemes to simulate fully compressible Euler7–9
or Navier-Stokes10–12 equations. The numerical accuracy of these shock capturing
schemes remains mostly unknown. It is not clear what benefit these schemes can
∗email address: lalleman@asci.fr
†email address: luo@icase.edu; homepage: www.icase.edu/˜luo
41
... On the other hand, the hybrid approach, which solves the energy equation using conventional numerical techniques like Finite Differences (FD), offers noticeable advantages. These include lower memory consumption compared to double-population methods [33], good numerical stability [33,34], easy implementation of high-order schemes, and accommodation of variable properties like heat capacity [35]. ...
... These calculations rely on the updated velocity from Step 2. 5. Calculating particle temperature, thermal forcing, and spreading to Eulerian nodes: Utilizing Eqs. (34) and (45), thermal forcing terms are computed, incorporating the updated temperature field in the fluid. Subsequently, these thermal forces are propagated to the Eulerian nodes using Eq. ...
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... There are three main approaches to thermal LBM present in literature: i) hybrid coupling with a macroscopic solver (e.g. finite differences, or finite volume) for evolving the energy equation [19], ii) the double distribution approach, where a second set of populations is used to evolve the temperature field [20] and iii) models based on high order quadrature rules [11,21]. The latter approach provides an elegant and self-consistent kinetic description of thermal compressible flow [22,23] via the high order moments of the particle distribution function. ...
... Treatment of viscous terms. We found the evaluation of viscous terms from Eqns (18) and (19) crucial to ensure numerical stability over a broad range of numerical viscosity . To illustrate this, simulations at various values of have been conducted comparing with a CBC formulation where the Laplacian of temperature and velocity are approximated using second order finite differences. ...
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