Numerically stable formulations of convective terms for turbulent compressible flows

TL;DR

This paper systematically analyzes and characterizes energy-preserving formulations of the convective terms in the compressible Navier-Stokes equations, introducing novel split forms and demonstrating their effectiveness through numerical tests.

Contribution

It introduces a two-parameter family of split forms for convective terms, including new formulations, and analyzes their conservation properties for improved numerical stability.

Findings

New split forms enhance conservation properties.

Adaptive energy formulation improves robustness.

Numerical tests confirm theoretical advantages.

Abstract

A systematic analysis of the discrete conservation properties of non-dissipative, central-difference approximations of the compressible Navier-Stokes equations is reported. A general triple splitting of the nonlinear convective terms is considered, and energy-preserving formulations are fully characterized by deriving a two-parameter family of split forms. Previously developed formulations reported in literature are shown to be particular members of this family; novel splittings are introduced and discussed as well. Furthermore, the conservation properties yielded by different choices for the energy equation (i.e. total and internal energy, entropy) are analyzed thoroughly. It is shown that additional preserved quantities can be obtained through a suitable adaptive selection of the split form within the derived family. Local conservation of primary invariants, which is a fundamental property to build high-fidelity shock-capturing methods, is also discussed in the paper. Numerical tests performed for the Taylor-Green Vortex at zero viscosity fully confirm the theoretical findings, and show that a careful choice of both the splitting and the energy formulation can provide remarkably robust and accurate results.