Numerical assessments of high-order accurate shock capturing schemes: Kelvin–Helmholtz type vortical structures in high-resolutions

Abstract

This paper investigates performance of extensions of the state-of-the-art high-resolution shock capturing schemes by solving hyperbolic conservation laws in gas dynamics. Such numerical schemes used for the integration of compressible flow simulations should provide accurate solutions for the long time integrations these flows require. To this end, several joint solvers are developed within the framework of the reconstruction and flux-splitting approaches using the underlying MUSCL and WENO frameworks. The numerical assessments include testing and evaluation of various interpolation procedures, flux-limiters, Riemann solvers, flux-splitting schemes as well as their formal order of accuracy. A three-stage optimal TVD Runge–Kutta time stepping is employed for temporal integration. The modular development of these joint solvers provides an ease in characterizing the solution procedures. The performances of these high-resolution solvers are compared for several carefully selected two-dimensional Riemann problems including shock and rarefaction waves as well as joint discontinuities. Based on solutions obtained by all forms of five-point stencil schemes, we demonstrate that the reconstruction based WENO scheme with Roe solver is more accurate than all the versions of the flux-splitting WENO solvers tested in this study. We also show that results are highly dependent on the choice of the flux limiter. Performing benchmark quality high-resolution computations, it is shown that the Euler equations discretized by the fifth-order WENO scheme produce solutions which convect vorticity and create small-scale vortical flow structures which are usually associated with the high Reynolds number viscous flows. Surprisingly, it is found that these Kelvin–Helmholtz instability like vortical structures are not captured in any form of the third-order five-point stencil schemes.