Kinetic theory-based force analysis in lattice Boltzmann equation

In the gas kinetic theory, it is shown that the zeroth order of the density distribution function [Formula: see text] and local equilibrium density distribution function were the Maxwellian distribution [Formula: see text] with an external force term, where [Formula: see text] is the fluid density, [Formula: see text] is the physical velocity and [Formula: see text] is the temperature, while in the lattice Boltzmann equation (LBE) method, numerous force treatments were proposed with a discrete density distribution function [Formula: see text] apparently relaxed to a given state [Formula: see text], where the given velocity [Formula: see text] could be different with [Formula: see text], and the Chapman–Enskog (CE) analysis showed that [Formula: see text] and local equilibrium density distribution function should be [Formula: see text] in the literature. In this paper, we start from the kinetic theory and show that the [Formula: see text] and local equilibrium density distribution function in LBE should obey the Maxwellian distribution [Formula: see text] with [Formula: see text] relaxed to [Formula: see text], which are consistent with kinetic theory. Then the general requirements for the force term are derived, by which the correct hydrodynamic equations could be recovered at Navier–Stokes (NS) level, and numerical results confirm our theoretical analysis.