Derivation of lattice Boltzmann equation via analytical characteristic integral**Project supported by the National Science and Technology Major Project, China (Grant No. 2017ZX06002002).

Abstract

A lattice Boltzmann (LB) theory, the analytical characteristic integral (ACI) LB theory, is proposed in this paper. ACI LB theory takes the Bhatnagar–Gross–Krook (BGK)-Boltzmann equation as the exact kinetic equation behind Navier–Stokes continuum and momentum equations and constructs an LB equation by rigorously integrating the BGK-Boltzmann equation along characteristics. It is a general theory, supporting most existing LB equations including the standard lattice BGK (LBGK) equation inherited from lattice-gas automata, whose theoretical foundation had been questioned. ACI LB theory also indicates that the characteristic parameter of an LB equation is collision number, depicting the particle-interaction intensity in the time span of the LB equation, instead of the traditionally assumed relaxation time, and the over-relaxation time problem is merely a manifestation of the temporal evolution of equilibrium distribution along characteristics under high collision number, irrelevant to particle kinetics. In ACI LB theory, the temporal evolution of equilibrium distribution along characteristics is the determinant of LB method accuracy and we numerically prove this.