Analytical lattice Boltzmann solutions for thermal flow problems

Abstract

Analytical solutions based on two 13-bit (hexagonal and square) and one 17-bit square lattice Boltzmann BGK models have been obtained for the Couette flow, with a temperature gradient at the boundaries. The analytical solutions for the unknown distributions functions are written as polynomials in powers of the space variable and the coefficients of the expansion are estimated in terms of characteristic flow quantities, the single relaxation time and the lattice spacing. The analytical solutions of the two 13-bit models contain some nonlinear deviations from the thermal hydrodynamic constraints and the analytical solutions, while the 17-bit square lattice model results into an exact representation of the nonisothermal Couette flow problem.