Valid Physical Processes from Numerical Discontinuities in Computational Fluid Dynamics

Abstract

Due to the limited cell resolution in the representation of flow variables, a piecewise continuous initial reconstruction with discontinuous jump at a cell interface is usually used in modern computational fluid dynamics methods. Starting from the discontinuity, a flux function in a numerical scheme should be based on the real flow physics, or at least mimic what happens from an initial discontinuity, i.e., the non-equilibrium flow behavior. The adaptation of the exact Riemann solver of the Euler equations assumes the underlying equilibrium flow, and this assumption may introduce intrinsically a mechanism to develop instabilities in strong shock simulations. In order to clarify the flow physics from a discontinuity, the unsteady behavior of one-dimensional contact discontinuity and shock wave is studied on a time scale of (0∼10000) times of the particle collision time. For high Mach number flow simulation, inside a numerical shock layer this time scale and the corresponding length scale may have the same order as the time step and cell size used in a numerical scheme. Therefore, the use of equilibrium solution of the Euler equations in these cases may be invalid physically. In the study of the non-equilibrium flow behavior from a discontinuity, the collision-less Boltzmann equation is first used for the time scale within one particle collision time, then the direct simulation Monte Carlo (DSMC) method will be adapted to get the further evolution solution. The transition from the free particle transport to the dissipative Navier-Stokes solutions are obtained as an increasing of time. The exact Riemann solution becomes a limiting solution with infinite number of particle collisions. Unfortunately, the infinite number of particle collisions never achieves for the gas molecules across the whole shock layer. Therefore, the use of the Riemann solution inside the numerical shock layer is fundamentally flawed. For the continuum flow at high Reynolds number, the non-equilibrium scale should be very small in comparison with cell size and time step, and the Riemann solution can be used here to capture the flow evolution from the discontinuity. In order to develop a robust and accurate numerical scheme for all speed flows, the numerical scheme should be able to describe both equilibrium and non-equilibrium flow behavior. Even for the continuum flow computation, the numerical shock must be considered as an enlarged non-equilibrium region, especially in the strong shock case. The non-equilibrium flow physics, which approaches to the equilibrium one with the increasing of particle collisions, is a valid physical process to develop such a numerical flux function. The use of exact Riemann solution, such as the Godunov method, lacks this kind of mechanism. On the other hand, the gas-kinetic scheme (GKS) follows the non-equilibrium flow physics and its evolution to an equilibrium state, which may be the reason for its absence from shock instabilities in high Mach number flow computations. I. INTRODUCTION The Boltzmann equation is generally regarded as the governing equation for the motion of fluid. It describes the time evolution of a large number of particles through binary collisions in statistical physics. This is a seven-dimensional integral-differential equation, which is more fundamental than the Euler and Navier-Stokes equations. This equation, however, can be simplified under some conditions. For the equilibrium flow, the Boltzmann equation leads to the compressible Euler system which is a nonlinear hyperbolic system of conservation laws. The basic wave structure of the hyperbolic system,