A short review on lattice Boltzmann method: comprehensive historic resources and theoretical problems to be addressed

Abstract

This short review presents a detailed introduction on lattice Boltzmann method (LBM) involving its theoretical background, originality, fundamental ideas with main characteristics, historic developments, important review papers, recognized books, applications, and available computer codes, which provides comprehensive reference resources for students and researchers intending to engage the related research. Through reading the publications, it is revealed the following theoretical problems to be addressed. a) Maxwell-Boltzmann distribution developed from dilute gases concerning only the pressure-internal energy, but not reflecting the one produced by viscous stresses. b) Three conservation laws cannot be directly derived only by the lattice Boltzmann transport equation, but needing a multi-scaling expansion, and the energy conservation equation has not included the external energy generation source. c) LBM implementation process only updates the two macroscopic parameters: mass density and mean velocity, but not internal energy, and therefore it is not a completed theory, due to the internal energy in complex flows is a function of time and space points. These revealed issues could be the reasons causing current LBM schemes failing to simulate flows with high speeds or high compressibility concerning obvious changes of internal energy in flow fields. Author gives a paper of theoretical investigations on LBM, accepted recently, trying to address these revealed problems.

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