A simple HLLE-type scheme for all Mach number flows

TL;DR

This paper introduces a simple HLLE-type scheme capable of accurately resolving all Mach number flows, including low Mach and high-speed shocks, by employing velocity reconstruction and stability-enhancing techniques.

Contribution

The paper presents a novel HLLE-type scheme that resolves shear and contact waves without added wave structures, improving stability and accuracy across all Mach numbers.

Findings

Successfully resolves low Mach flow features via asymptotic analysis.

Demonstrates robustness against shock instabilities at high speeds.

Capable of capturing flow features at very low Mach numbers.

Abstract

A simple HLLE-type scheme is proposed for all Mach number flows. In the proposed scheme, no extra wave structure is added in the HLLE scheme to resolve the shear wave while the contact wave is resolved by adding a wave structure similar to the HLLEM scheme. The resolution of the shear layers and the flow features at low Mach number are achieved by a velocity reconstruction method based on the face normal Mach number. Robustness against the numerical instabilities is achieved by scaling the velocity reconstruction function in the vicinity of shock with a multi-dimensional pressure sensor. The ability of the proposed scheme to resolve low Mach flow features is demonstrated through asymptotic analysis while stability of the proposed scheme for strong shock is demonstrated through linear perturbation and matrix stability analyses. A set of numerical test cases are solved to show that the proposed scheme is free from numerical shock instability problems at high speeds and is capable of resolving the flow features at very low Mach numbers.