Generalized local equilibrium in the cascaded lattice Boltzmann method.

By realizing the insufficient degree of Galilean invariance of the traditional multiple-relaxation-time collision operators, Geier [Phys. Rev. E 73, 066705 (2006)] proposed to relax differently the moments shifted by the macroscopic velocity, leading to the so-called cascaded lattice Boltzmann method (LBM). This paper points out that (a) the cascaded LBM essentially consists in adopting a generalized local equilibrium in the frame at rest; (b) this new equilibrium does not affect the consistency of LBM; and finally (c) if the raw moments are relaxed in the frame at rest as usual and the number of relaxation frequencies is reduced, the proposed derivation leads to the two-relaxation-time collisional operator with proper polynomial equilibrium.