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International audience; The appearance of shock anomaly is a major unsolved problem for some low diffusion schemes when simulating the hypersonic flow. In this paper, a simple method is proposed to enhance the robustness of the low diffusion schemes to overcome the shock anomaly. The main idea of this method is adding an appropriate extra term to the original low diffusion schemes without influencing the accuracy in aerodynamic heating prediction. This extra term is derived from the difference between the flux splitting scheme (FVS) and the advection upstream splitting method+ (AUSM+). Adding this term to three low diffusion schemes, seven typical numerical tests are conducted to examine the capability of those schemes. Numerical results show that the three new schemes turn out to be carbuncle-free and shock-stable without losing their original accuracy in prediction of aerodynamic heating, validating the feasibility and reliability of the proposed method.

The purpose of this paper is twofold. First, the application of high-order methods in combination with the popular HLLC Riemann solver demonstrates that the grid-aligned shock instability can strongly affect simulation results when the grid resolution is increased. Beyond the well-documented two-dimensional behavior, the problem is particularly troublesome with three-dimensional simulations. Hence, there is a need for shock-stable modifications of HLLC-type solvers for high-speed flow simulations. Second, the paper provides a stabilization of the popular HLLC flux based on a recently proposed mechanism for grid aligned-shock instabilities Fleischmann et al. (2020) [8]. The instability was found to be triggered by an inappropriate scaling of acoustic and advection dissipation for local low Mach numbers. These low Mach numbers occur during the calculation of fluxes in transverse direction of the shock propagation, where the local velocity component vanishes. A centralized formulation of the HLLC flux is provided for this purpose, which allows for a simple reduction of nonlinear signal speeds. In contrast to other shock-stable versions of the HLLC flux, the resulting HLLC-LM flux reduces the inherent numerical dissipation of the scheme. The robustness of the proposed scheme is tested for a comprehensive range of cases involving strong shock waves. Three-dimensional single- and multi-component simulations are performed with high-order methods to demonstrate that the HLLC-LM flux also copes with latest challenges of compressible high-speed computational fluid dynamics. © 2020 The Author(s)

The present work investigates the dynamics and stability of shock waves in granular gases. The problem was modelled for a piston propagating into a system of disks that can undergo inelastic collisions if an impact threshold is exceeded. The model was addressed numerically at the microscopic and macroscopic levels. The molecular dynamics methodology employed the Event-Driven Molecular Dynamics method, and the continuum model was formulated using the Navier-Stokes equations for granular gases with the transport terms of Jenkins and Richman and a modified cooling rate term. The inviscid steady state shock structure was derived and analyzed. The results indicated that a relaxing shock structure is expected for sufficiently strong shock waves. Beyond this limit the structure was shown to be independent of the initial energy, a finding similar to the strong shock approximation in molecular gases. One-dimensional simulations demonstrated that the molecular dynamics and continuum models yield similar evolutions and structures of the shock wave, validating the continuum description of this study. Two-dimensional results showed that sufficiently strong shock waves can exhibit multi-dimensional instability with high density non-uniformities and convective rolls within the structure, with the size of instabilities shown to scale with the relaxation length of the shock structure. Instabilities were observed with the continuum description only with the inclusion of statistical fluctuations to density mimicking the molecular model. The cases that were unstable were shown to be in a regime whereby statistical fluctuations can become important, following the description for this regime by Bird. Based on these findings, it is proposed that unstable shock behaviour can be observed for highly dissipative shock waves that yield short relaxation length scales, where fluctuations become important. The current work may shed light on unstable shock behaviour observed in dissipative gases, having implications for both granular media and molecular gases.

In this paper, a simple Harten, Lax and van Leer (HLL) type Riemann solver, capable of resolving contact discontinuity accurately, is proposed. Here, the ability to capture the contact is brought about in a way different from Harten, Lax and van Leer with Contact (HLLC) scheme of Toro. In the present method, the flux vector of the Euler equations is split into convective and pressure parts before applying the HLL scheme to each part. Several carefully chosen one- and two-dimensional test problems are solved using the present method in order to demonstrate its robustness and efficacy. The new method is found to be free from problems like spurious expansion shock, carbuncle phenomenon, odd even decoupling, kinked Mach stem, which are usually faced by most of the contact resolving methods. (C) 2012 Elsevier Ltd. All rights reserved.

It is still challenging to predict surface heat-transfer rate in hypersonic flow computations. In this paper, we first performed numerical experiments by changing numerical flux functions and meshes for a hypersonic flow around a hemisphere. Results show that AUSM+ flux function by Liou (1996) on a carefully refined mesh can lead to a shock stable solution with an accurate aerodynamic heating on the surface. Then, a numerical simulation on a hypersonic flow around two bodies involving a shock/shock interaction and a boundary-layer separation has been conducted by using the same method. Although this is a rather difficult problem with a complicated flowfield, comparisons show good agreement with the corresponding experimental data, including the surface heat-transfer rate profile. Therefore, we can say that we have established a numerical method to accurately predict surface heat-transfer in hypersonic shock interacting flows. Finally, detailed analysis of the computed flowfield has been made.

The carbuncle phenomenon is a shock instability mechanism which ruins all efforts to compute grid-aligned shock waves using low-dissipative upwind schemes. The present study develops a stability analysis for two-dimensional steady shocks on structured meshes based on the matrix method. The numerical resolution of the corresponding eigenvalue problem confirms the typical odd–even form of the unstable mode and displays a Mach number threshold effect currently observed in computations. Furthermore, the present method indicates that the instability of steady shocks is not only governed by the upstream Mach number but also by the numerical shock structure. Finally, the source of the instability is localized in the upstream region, providing some clues to better understand and control the onset of the carbuncle.

The current study addresses the stability of shock waves propagating through dissipative media, analogous to both granular media and molecular gases undergoing endothermic reactions. In order to investigate the stability, a simple molecular dynamics model was developed to observe shock waves and their structures with the inclusion of energy dissipation. For this, an Event Driven Molecular Dynamics model was implemented in a 2D environment, where a molecule is represented by a disk. The simulations addressed the formation of a shock wave in a gas by the sudden acceleration of a piston. Inelastic collisions were assumed to occur only if an impact velocity threshold is surpassed, representing the activation energy of the dissipative reactions. Parametric studies were conducted for this molecular model, by varying the strength of the shock wave, the activation threshold and the degree of inelasticity in the collisions. The resulting simulations showed that a shock structure does indeed become unstable with the presence of dissipative collisions. This instability manifests itself in the form of distinctive high density non-uniformities behind the shock wave, which take the form of convective rolls. The spacing and size of this ``finger-like" unstable pattern was shown to be dependent on the degree of inelasticity, the activation energy, and the strength of the driving piston. The mechanism responsible for the instability was addressed by studying the time evolution of the material undergoing the shock wave compression and further relaxation. It is found that the gas develops the instability on the same time scales as the clustering instability in homogeneous gases, first observed by Goldhirsch and Zanetti in granular gases. This confirmed that the clustering instability is the dominant mechanism.