A Gas-kinetic BGK Scheme for the Navier-Stokes Equations and Its Connection with Artificial Dissipation and Godunov Method

Kun Xu

Abstract: This paper presents an improved gas-kinetic scheme based on the Bhatnagar-Gross-Krook (BGK) model for the compressible Navier-Stokes equations. The current method extends the previous gas-kinetic Navier-Stokes solver developed by Xu and Prendergast (J. Comput. Phys., {\bf 114}, 9) by (a). implementing a general nonequilibrium state based on the Chapman-Enskog expansion of the BGK model as the initial gas distribution function at each time step, (b). using a different way in the construction of the equilibrium state at a cell interface, and (c). keeping two slopes of the macroscopic variables in the evaluation of spatial variations of the equilibrium state. As a result, the previous requirement of particle collision time $\tau$ being less than the time step $\Delta t$ for the validity of the BGK Navier-Stokes solver is removed, and the scheme becomes more robust than the previous one. The current BGK method approximates the Navier-Stokes solutions accurately regardless of the ratio between the particle collision time and the numerical time step. The Kinetic Flux Vector Splitting Navier-Stokes (KFVS NS) solver developed by Chou and Baganoff comes to be the limiting case of the current method when the particle collision time is much larger than the time step. The Equilibrium Flux Method (EFM) of Pullin for the Euler equations is also a limiting case of the current method when the nonequilibrium part in the initial gas distribution function disappears and the collision time is much larger than the time step. The dissipative mechanism in the KFVS and many other FVS schemes is qualitatively analyzed from their relation with the BGK scheme. Also, in this paper, the appropriate implementation of boundary condition for the kinetic scheme, different limiting solutions, and the Prandtl number fix are presented. The connection among von Neumann $\&$ Richtmyer's artificial dissipation, Godunov method, and the gas-kinetic BGK scheme is discussed, from which two concepts of {\sl dynamical} and {\sl kinematical dissipation} are introduced. The principles for constructing accurate and robust numerical schemes for the compressible flow simulation are proposed. Many numerical tests, which include highly non-equilibrium case, i.e., Mach $10$ shock structure, are included to validate the BGK scheme for the viscous solutions and support the physical and numerical analysis for different schemes.

Paper:
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Author(s):
Kun Xu, <makxu@ust.hk>
Publishing information:
submitted to J. Comput. Phys., August 2000