Cure for shock instability - Development of an improved Roe scheme

Abstract

This paper deals with the development of an improved Roe scheme that is free from the shock instability and still preserves the accuracy and efficiency of the original Roe's Flux Difference Splitting (EDS). Roe's FDS is known to possess good accuracy but to suffer from the shock instability, such as the carbuncle phenomenon. As the first step towards a shockstable scheme, Roe's FDS is compared with the HLLE scheme to identify the source of the shock instability. Through a linear perturbation analysis on the odd-even decoupling problem, damping characteristic is examined and control functions / and g are introduced into Roe's FDS to cure the shock instability. In order to satisfy the conservation of total enthalpy, which is crucial in predicting surface heat transfer rate in high speed steady flows, an analysis of dissipation mechanism in the energy equation is carried out to find out the error source and to make the proposed scheme preserve total enthalpy. By modifying the maximum-minimum wave speed, the problem of expansion shock and numerical instability in the expansion region is also remedied without sacrificing the exact capturing of contact discontinuity. Various numerical tests concerned with the shock instability are performed to validate the robustness of the proposed scheme. Then, viscous flow test cases ranging from transonic to hypersonic regime are calculated to demonstrate the accuracy, robustness, and other essential features of the proposed scheme. ITRODUCTION It is essential that a numerical representation of inviscid fluxes, namely a numerical flux function, should guarantee the high level of accuracy, efficiency and robustness in computational fluid dynamics (CFD). In the last three decades, numerous numerical flux functions have been developed and much progress has been achieved [2,5,11,14,19]. The Flux Difference Splitting (FDS) framework is one of the most successful groups among the various approaches to design numerical schemes and is Graduate Student f Assistant Professor, Member AIAA, * Professor, Senior Member AIAA 55 Senior Researcher, Member AIAA Copyright © 2002 by the American Institute of Aeronautics and Astronautics Inc. All rights reserved widely used and studied. FDS schemes are generally based on the idea due to Godunov [1] and the Riemann problem is utilized locally. Godunov showed that after preparing piecewise constant initial data from cell-averaged flow values, numerical flux at a cell interface can be calculated through the exact solution of the Riemann problem. Although this strategy provides a way to obtain a good shock-capturing scheme, the Riemann problem is highly nonlinear and has no closed form solution. In order to overcome this deficiency, many have tried to simplify the step of numerical flux calculation, which leads to the family of Godunov-type schemes or approximate Riemann solvers, such as Roe's FDS [2], Osher's FDS [3] and HLLEM [5], and etc. These FDS schemes can capture contact discontinuity accurately and give good resolution for the boundary layer in viscous flow calculation. Despite these advantages and the good shock capturing property, some disastrous failings are also found in certain problems. This pathological behavior, usually represented as the 'carbuncle phenomenon', was first observed by Peery and Imlay [6] for blunt body computations with Roe's FDS. The carbuncle phenomenon refers to a protuberant shock profile obtained when a supersonic flow over a blunt body is calculated. Quirk [7] reported that approximate Riemann solvers generally suffer from such failings. After the carbuncle phenomenon was observed, many attempts were made to unveil the cause and to cure these failings. The attempts to cure the shock instability can be generally categorized into two groups. One is to use an alternative dissipative scheme in a hybrid manner and the other to employ an entropy fix. Quirk [7] noticed that some schemes possessing the property of the good capturing of contact discontinuity show carbuncle phenomena while others free from carbuncle phenomena do not capture contact discontinuity accurately. Thus, it is suggested that a dissipative scheme, such as HLLE, should be used in shock region while a less dissipative scheme, such as Roe's FDS, should be used elsewhere. In order to flag the cell interface where a dissipative scheme is needed, a pressure gradient sensor is used. Wada and Liou [8], by the same philosophy, suggest a similar flagging procedure but they use a sonic point. For a less dissipative scheme, AUSMDV is used and Hanel's FVS is used for a dissipative scheme. This cure turns out to be very efficient, as shown by the American Institute of Aeronautics and Astronautics (c)2002 American Institute of Aeronautics & Astronautics or Published with Permission of Author(s) and/or Author(s)' Sponsoring Organization. results reported in [7,8]. However, this approach always needs a proper counterpart which complements defects of the original flux function, and the selection of a proper numerical scheme itself is a critical problem. An inadequate counterpart may contaminate the accuracy of numerical solutions, especially in cases of high speed flows. An entropy fix to the linear wave field is a method to limit the minimum value of the wave speed, which is equivalent to the addition of extra numerical dissipation to damp out spurious oscillation. Peery and Imlay [6] propose an anisotropic function for an entropy fix, and Lin [9] designs an isotropic correction function using a pressure gradient sensor. Although this approach may successfully cure the carbuncle phenomenon, its performance always depends on the location and/or the amount of numerical dissipation added. Improper entropy fix may easily broaden shock wave profile and/or deteriorate boundary layer resolution. Thus, the development of a proper sensor which determines the location and the amount of numerical dissipation is crucial. The two approaches to cure the shock instability problem, i.e. the use of dissipative numerical schemes and the employment of an entropy fix, are fundamentally the same in the sense that extra numerical dissipation is added to the original scheme in a way or another, and both need a detection procedure which usually involves a tuning coefficient. So far, it is generally believed that a scheme that can capture contact discontinuity exactly, i.e. a scheme that has vanishing dissipation in stationary contact discontinuity, cannot avoid the shock instability, and the only way to prevent it is to add enough dissipation to damp out oscillation. However, Liou [10] observes that all the tested numerical functions that suffer from the shock instability have a term depending on pressure difference in the mass flux while those free from the shock instability are independent of pressure difference in the mass flux. Based on the numerical analysis and experiment, he suggests the following conjecture: 'The condition D ^ 0, VM , in the mass flux is necessary for a scheme to develop, as t increases, the shock instability as manifested by the odd-even decoupling and carbuncle phenomena. On the other hand, the condition D =0,VM, is sufficient for a scheme to prevent the shock instability from occurring. Here, D stands for the dissipation term depending on pressure difference. This result indicates that it is possible to devise a FDS flux function free from the shock instability with vanishing dissipation in capturing stationary contact discontinuity. Xu [11] explains the shock instability using the Bernoulli equation and a convergent-divergent nozzle concept. According to his explanation, vanishing dissipation in the direction parallel to the shock and the contribution of pressure fluctuation to the mass flux cause the shock instability. The analysis in Ref. [11] is in a qualitative agreement with Liou's conjecture in the sense that the pressure term in the mass flux triggers the shock instability. The present study aims at the design of a new Roebased FDS flux function that is free from shock instability. Following Liou's conjecture and Xu's explanation, we focus on the pressure term in the mass flux of Roe's FDS. In order to maintain the high level of robustness and accuracy, we impose that the newly developed flux function should satisfy the following criteria: • The new flux function should not have any tunable parameter. • The new flux function should capture contact discontinuity for the accurate resolution of boundary layer. • Total enthalpy should be conserved for the accurate prediction of surface heat transfer rate in high speed steady flows. • Robustness in the expansion region should be substantially improved and entropy-violating expansion shock should be removed. The present paper is organized as follows. After introduction, a list of the shock instability is briefly reviewed with some examples. Then, we present the analysis procedure of Roe's flux function and propose Roe with Mach number-based function (RoeM) schemes. And, we present extensive numerical results and discuss properties of the proposed schemes. In order to demonstrate various properties of the flux functions, we apply the proposed schemes to steady and unsteady problems. Finally, concluding remarks are given.