Effect of Mesh Quality on Flux Reconstruction in Multi-Dimensions

TL;DR

This paper analyzes how non-uniform grid quality affects Flux Reconstruction stability and dissipation in multi-dimensional simulations, revealing that mesh expansion causes instability while contraction leads to excess dissipation.

Contribution

It develops theoretical methods to understand multi-dimensional mesh effects on Flux Reconstruction and compares its performance to finite volume methods under mesh deformation.

Findings

FR is more resilient to mesh deformation than FV.

Mesh expansion causes instability; contraction causes dissipation.

Kinetic energy preserving schemes improve resilience to mesh issues.

Abstract

Theoretical methods are developed to understand the effect of non-uniform grids on Flux Reconstruction (FR) in multi-dimensions. The analysis reveals that the same effect of expanding and contracting grids is seen in two dimensions as in one dimension. Namely, that expansions cause instability and contractions cause excess dissipation. Subsequent numerical experiments on the Taylor-Green Vortex with jittered elements show the effect of localised regions of expansion and contraction, with an initial increase in the kinetic energy observed on non-uniform meshes. Some comparison is made between second-order FR and second-order finite volume (FV). FR is found to be more resilient to mesh deformation, however, FV is found to be more resolved when operated at second order on the same mesh. In both cases, it is recommended that a kinetic energy preserving/conservation formulation should be used as this can greatly increase resilience to mesh deformation.