A Third-order Compact Gas-kinetic Scheme on Unstructured Meshes for Compressible Navier-Stokes Solutions

TL;DR

This paper introduces a novel third-order compact gas-kinetic scheme for unstructured meshes, enabling accurate and stable simulation of compressible viscous flows, including boundary layers, vortices, and shocks, without complex flux integration or multi-stage methods.

Contribution

The paper presents the first third-order compact gas-kinetic scheme on unstructured meshes, utilizing a high-order gas evolution model for improved flow variable reconstruction.

Findings

Achieves third-order accuracy with a compact stencil.

Stable under CFL condition above 0.5.

Effectively captures boundary layers, vortices, and shocks.

Abstract

In this paper, for the first time a compact third-order gas-kinetic scheme is proposed on unstructured meshes for the compressible viscous flow computations. The possibility to de sign such a third-order compact scheme is due to the high-order gas evolution model, where a time-dependent gas distribution function at a cell interface not only provides the fluxes across a cell interface, but also the time evolution of the flow variables at the cell interface as well. As a result, both cell averaged and cell interface flow variables can be used for the initial data reconstruction at the beginning of next time step. A weighted least-square reconstruction has been used for the construction of a third-order initial condition. Therefore, a compact third-order gas-kinetic scheme with the involvement of neighboring cells only can be developed on unstructured meshes. In comparison with other conventional high-order schemes, the current method avoids the use of Gaussian points for the flux integration along a cell interface and the multi-stage Runge-Kutta time stepping technique. The third-order compact scheme is numerically stable under CFL condition above 0.5. Due to the multidimensional gas-kinetic formulation and the coupling of inviscid and viscous terms, even with unstructured meshes the boundary layer solution and the vortex structure can be accurately captured in the current scheme. At the same time, the compact scheme can capture strong shocks as well.