A benchmark study of kinetic models for shock waves

It is well-known that the Boltzmann collision term can be simplified by Bhatnagar-Gross-Krook (BGK) and Fokker-Planck (FP) models. These two kinetic models satisfy the conservation laws; however, their predictions of the Prandtl number are inconsistent with that derived from the Boltzmann equation. A variety of modifications, such as Shakhov BGK (S-BGK), Ellipsoidal Statistical BGK (ES-BGK), Cubic Fokker-Planck (Cubic-FP), and Ellipsoidal Statistical Fokker-Planck (ES-FP), models have been proposed and they can provide correct Prandtl number. However, in the transition regime, the validity of simplified kinetic models is still controversial. In this work, the performance of four kinetic models is investigated through theoretical analysis and numerical comparison of the high-order moments beyond shear stress and heat flux. The simulations of the shock waves show that the results of these modified kinetic models deviate from the Boltzmann equation as Mach number increases. On the whole, the S-BGK model, which gives the most approximated relaxation coefficients for the production terms of high-order moments, provides the best predictions in the transition regime. For the ES-FP and Cubic-FP models, besides the incorrect relaxation coefficients, their deviations at large Mach numbers are further caused by inconstant Prandtl number and overestimated nonlinear terms, respectively.