The Role of Momentum Interpolation Mechanism of the Roe Scheme in the Shock Instability

TL;DR

This paper investigates the role of the momentum interpolation mechanism in the Roe scheme and proposes an improved method to suppress shock instability by adaptively controlling MIM based on local flow conditions.

Contribution

The study reveals the critical influence of MIM on shock instability and introduces an adaptive Roe scheme that effectively reduces instability across different flow regimes.

Findings

MIM significantly impacts shock instability.

The improved scheme suppresses odd-even decoupling.

Adaptive coefficients based on Mach number enhance stability.

Abstract

The shock instability phenomenon is a famous problem for the shock-capturing scheme. By subdividing the numerical dissipation of the Roe scheme, the term of pressure-difference-driven modification for the cell face velocity is regarded as a version of the momentum interpolation method (MIM), which is necessary for incompressible flows to suppress the pressure checkerboard problem. Through the analysis and odd-even decoupling test, it is discovered that MIM plays the most important role on the shock instability. In fact, for non-linear flows MIM should be completely removed, but unexpected MIM is activated on the cell face nearly parallel to the flow for high Mach number flows or low Mach number flows in shock. For such conditions, two coefficients are designed based on local Mach number and a shock detector, respectively, and then the improved Roe scheme is proposed, which gives consideration to requirement of MIM for incompressible and compressible flows and is validated for good performance of avoiding odd-even decoupling. Therefore, the aim of decreasing rather than increasing numerical dissipation to cure the shock instability can be achieved.