The Harten-Lax-van Leer with contact (HLLC) scheme is known to be plagued by various forms of numerical shock instabilities. In this paper, we propose a new framework for developing shock stable, contact and shear preserving approximate Riemann solvers based on the HLLC scheme for the Euler system of equations. The proposed framework termed as HLLC-SWM (Selective Wave Modified) scheme identifies and increases the magnitude of the inherent diffusive HLL component within the HLLC scheme in the vicinity of a shock wave while leaving its antidiffusive component unmodified to retain accuracy on linearly degenerate wavefields. We present two strategies to compute the requisite supplementary dissipation which results in HLLC-SWM-E and HLLC-SWM-P variants. Through a linear perturbation analysis of the HLLC-SWM framework, we clarify how the additional dissipation introduced in this way helps in damping of unphysical perturbations in primitive quantities under a derived CFL constraint. A matrix based stability analysis of a steady two-dimensional normal shock is used to show that both variants of the HLLC-SWM scheme are shock stable over a wide range of inlet Mach numbers. Results from standard test cases demonstrate that the HLLC-SWM schemes are capable of computing shock stable solutions on a variety of problems while retaining positivity and exact inviscid contact ability. On viscous flows, while the HLLC-SWM-P variant * Corresponding author is quite accurate, the HLLC-SWM-E variant introduces slight inaccuracy which can be corrected through a simple Mach number based switching function.
Various forms of numerical shock instabilities are known to plague many contact and shear preserving approximate Riemann solvers, including the popular Harten-Lax-van Leer with Contact (HLLC) scheme, during high speed flow simulations. In this paper we propose a simple and inexpensive novel strategy to prevent the HLLC scheme from developing such spurious solutions without compromising on its linear wave resolution ability. The cure is primarily based on a reinterpretation of the HLLC scheme as a combination of its well-known diffusive counterpart, the HLL scheme, and an antidiffusive term responsible for its accuracy on linear wavefields. In our study, a linear analysis of this alternate form indicates that shock instability in the HLLC scheme could be triggered due to the unwanted activation of the antidiffusive terms of its mass and interface-normal flux components on interfaces that are not aligned with the shock front. This inadvertent activation results in weakening of the favourable dissipation provided by its inherent HLL scheme and causes unphysical mass flux variations along the shock front. To mitigate this, we propose a modified HLLC scheme that employs a simple differentiable pressure based multidimensional shock sensor to achieve smooth control of these critical antidiffusive terms near shocks. Using a linear perturbation analysis and a matrix based stability analysis, we establish that the resulting scheme, called HLLC-ADC (Anti-Diffusion Control), is shock stable over a wide range of freestream Mach numbers. Results from standard numerical test cases demonstrate that the HLLC-ADC scheme is indeed free from the most common manifestations of shock instability including the Carbuncle phenomenon without significant loss of accuracy on shear dominated viscous flows.
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