The structure of hypersonic shock waves using Navier-Stokes equations modified to include mass diffusion

TL;DR

This paper introduces modified Navier-Stokes equations incorporating mass diffusion to better predict hypersonic shock wave structures, addressing previous model limitations and ensuring physical, stable solutions with simpler numerical implementation.

Contribution

The paper proposes a new modification to Navier-Stokes equations that fully includes molecule mass diffusion, resulting in stable, physical solutions for shock wave structures.

Findings

Modified equations produce shock structures similar to extended hydrodynamic models.

Solutions are stable and physically consistent.

The approach simplifies numerical implementation compared to traditional models.

Abstract

Howard Brenner has recently proposed modifications to the Navier-Stokes equations that relate to a diffusion of fluid volume that would be significant for flows with high density gradients. In a previous paper (Greenshields & Reese, 2007), we found these modifications gave good predictions of the viscous structure of shock waves in argon in the range Mach 1.0-12.0 (while conventional Navier-Stokes equations are known to fail above about Mach 2). However, some areas of concern with this model were a somewhat arbitrary choice of modelling coefficient, and potentially unphysical and unstable solutions. In this paper, we therefore present slightly different modifications to include molecule mass diffusion fully in the Navier-Stokes equations. These modifications are shown to be stable and produce physical solutions to the shock problem of a quality broadly similar to those from the family of extended hydrodynamic models that includes the Burnett equations. The modifications primarily add a diffusion term to the mass conservation equation, so are at least as simple to solve as the Navier-Stokes equations; there are none of the numerical implementation problems of conventional extended hydrodynamics models, particularly in respect of boundary conditions. We recommend further investigation and testing on a number of different benchmark non-equilibrium flow cases.