Derivation of lattice Boltzmann equation via analytical characteristic integral

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.

On-node lattices construction using partial Gauss–Hermite quadrature for the lattice Boltzmann method

A concise theoretical framework,the partial Gauss–Hermite quadrature(pGHQ),is established to construct on-node lattices of the lattice Boltzmann(LB)method under a Cartesian coordinate system.Compared with the existing approaches,the pGHQ scheme has the following advantages:extremely concise algorithm,unifies the constructing procedure for symmetric and asymmetric on-node lattices,and covers a full-range quadrature degree of a given discrete velocity set.We employ the pGHQ scheme to search the local optimal and asymmetric lattices for{n=3,4,5,6,7}moment degree equilibrium distribution discretization on the range[-10,10].The search reveals a surprising abundance of available lattices.Through a brief analysis,the discrete velocity set shows a significant influence on the positivity of equilibrium distributions,which is considered as one of the major impacts of the numerical stability of the LB method.Hence,the results of the p GHQ scheme lay a foundation for further investigations to improve the numerical stability of the LB method by modifying the discrete velocity set.It is also worth noting that pGHQ can be extended into the entropic LB model,even though it was proposed for the Hermite polynomial expansion LB theory.