Progresses in Gas-Kinetic Upwind Euler/Navier-Stokes Solvers

Abstract

This paper summarizes our recent progresses made in the development of gas-kinetic upwind Euler/Navier-Stokes solvers. They are: 1) a simplified Bhatnagar-Gross-Krook (BGK) scheme for the Navier-Stokes equations based on the relaxation between two kinetic flux-vector splitting (KFVS) schemes, 2) the new BGK schemes based on the relaxation between two kinetic wave/particle splitting (KWPS) schemes, 3) the families of generalized KFVS schemes, 4) the families of generalized KWPS schemes, 5) the kinetic flow-variable splitting schemes. It is concluded that although the BGK schemes improve the accuracy over their baseline KFVS/KWPS schemes, they are computationally less efficient. More importantly, a BGK scheme involves a relaxation parameter, an uncertainty similar to the artificial viscosity used in the central discretization approach. So, we further explore the non-relaxation approaches like the generalized KFVS/KWPS schemes and the kinetic flowvariable splitting approach. It is found that although more accurate than the standard KFVS/KWPS schemes, the developed generalized KFVS/KWPS schemes are still not suitable for viscous computations. The kinetic flow-variable splitting approach is the method of choice as the gas-kinetic upwind Euler/Navier-Stokes solvers. Therefore, KFVS should stand for Kinetic Flow-Variable Splitting instead of Kinetic Flux-Vector Splitting.