Lattice Boltzmann Shakhov kinetic models for variable Prandtl number on Cartesian lattices

Two-dimensional lattice Boltzmann (LB) models for the Shakhov kinetic equation are developed. In contrast to several previous thermal LB models with variable Prandtl number, the present approach deals with the models on Cartesian lattices. This allows the standard collide-and-stream implementation. The discrete velocity local equilibrium is evaluated as a projection of the Shakhov local equilibrium on the orthonormal basis spanned by the Hermite polynomials. The local equilibrium distribution explicitly depends on the Prandtl number, and the Shakhov LB models can be established on well-known lattices having 25 and 37 velocities. Next, in order to reproduce the rarefied effects better, a semiautomatic approach for construction of high-order lattices which minimizes the difference between the half moments evaluated from the discrete velocity equilibrium and the Maxwell distribution is proposed. This problem is equivalent to solving a system of linear equations with constraint. Two high-order lattices having 37 velocities are deduced. Several test problems are considered: the thermal wave decay and the thermal Couette flow in the hydrodynamic limit for various Prandtl and Mach numbers, rarefied effects in the athermal Couette and Poiseuille flows, and rarefied effects in the Fourier flow. It is demonstrated that the new models show a good accuracy while considering rarefied flows in slip and transition regimes and some particular problems in the ballistic regime (the Knudsen paradox).