Numerical Solutions of 2D Riemann Problems of Gas Dynamics Using a Hybrid PVU-M+ Scheme

Abstract

The present work consists of numerical solution of two-dimensional Riemann problems for gas dynamics using PVU-M+ scheme (Hasan et al. Comput Fluid 119:58–86 2015 [1]). The test cases and their initial data are taken from Lax and Liu (SIAM J Sci Comput 19:319–340, 1998 [2]). The flow domain consists of a square geometry divided in 4 quadrants such that there exists only one planar wave (shock, rarefaction or slip line) between each two quadrants. The solutions are analyzed through density contours and line plots at a suitable location in X–Y plane. Although the PVU-M+ scheme resolved most flow features, it showed difficulty in capturing pressure field in a weak velocity field behind a discontinuity (rarefaction/shock). To overcome this, pressure field at cell interface is calculated based on acoustic speeds and a pressure gradient at cell center is obtained using this pressure field which is then blended with forward/backward approximation of pressure gradient using a mean hybrid weight function, ${\overline{W} }_{h}=\frac{1}{3}{\sum }_{k=1}^{3}{\left({W}_{h}\right)}_{i+2-k}$, where the hybrid weight function defined as, ${W}_{h}=A{e}^{-{\left(\frac{M}{B}\right)}^{2}}$ has two adjustable constants (“A” and “B”) which can be adjusted according to the flow problems and ‘M’ being the local Mach number based on convective flux velocity (u or v). This modified PVU-M+ scheme is named “PVU-M+H” where “H” stands for hybrid. Recommended values of “A” and “B” are 0.2 and 0.12, respectively, for all type of flow problems. All test cases were solved keeping these values of “A” and “B”.