Valid Physical Processes from Numerical Discontinuities in Computational Fluid Dynamics

TL;DR

This paper investigates the physical validity of numerical discontinuities in CFD by analyzing non-equilibrium flow behavior from a discontinuity using kinetic theory and DSMC, revealing limitations of Riemann solutions in shock layers.

Contribution

It introduces a kinetic-theory-based analysis of flow discontinuities in CFD, highlighting the non-equilibrium physics and limitations of traditional Riemann solutions in high Mach number flows.

Findings

Discontinuities can represent non-equilibrium shock layers.

Riemann solutions are limiting cases of particle collisions.

High Mach flows involve highly non-equilibrium shock regions.

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 Riemann problem in the Godunov method is solved for the flux evaluation across the cell interface in a finite volume scheme. With the increasing of Mach number in the CFD simulations, the adaptation of the Riemann solver seems introduce intrinsically a mechanism to develop instabilities in strong shock regions. Theoretically, the Riemann solution of the Euler equations are based on the equilibrium assumption, which may not be valid in the non-equilibrium shock layer. In order to clarify the flow physics from a discontinuity, the unsteady flow behavior of one-dimensional contact and shock wave is studied on a time scale of (0~10000) times of the particle collision time. 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 (NS) solutions are obtained as an increasing of time. The exact Riemann solution becomes a limiting solution with infinite number of particle collisions. For the high Mach number flow simulations, the points in the shock transition region, even though the region is enlarged numerically to the mesh size, should be considered as the points inside a highly non-equilibrium shock layer.