Temperature-scaled collision process for the high-order lattice Boltzmann model

We postulate that the relaxations of the distribution function in the lattice Boltzmann model should be self-similar under temperature scaling. Based on this postulation, a multiple-relaxation-time collision model in the relative, temperature-scaled reference frame is devised with Hermite expansion. Resorting to the relation between the Hermite basis with the temperature-scaled relative velocity and the Hermite basis with the raw velocity, the relaxations in the temperature-scaled reference frame can be converted to those in the raw reference frame with some correction terms to eliminate the cross-talk effects among the relaxations of different orders. The highest-order nonequilibrium relative central moment is filtered due to the insufficient discrerization in the velocity space. The highest-order collision term can be recursively obtained from the lower-order collision terms. The improved performance is validated by the double shear layer flow, shock tube flow, and the Taylor-Green vortex flow.