Mass Flux Schemes and Connection to Shock Instability.pdf

Journal of Computational Physics 160, 623–648 (2000) doi:10.1006/jcph.2000.6478, available online at on Mass Flux Schemes and Connection to Shock Instability Meng-Sing Liou Turbomachinery and Propulsion Systems Division, NASA Glenn Research Center at Lewis Field, Cleveland, Ohio 44135 Received November 24, 1998; revised February 10, 2000 We analyze numerical mass fluxes with an emphasis on their capability for ac- curately capturing shock and contact discontinuities. The study of mass flux is use- ful because it is the term common to all conservation equations and the numerical diffusivity introduced in it bears a direct consequence to the prediction of contact (stationary and moving) discontinuities, which are considered to be the limiting case of the boundary layer. We examine several prominent numerical flux schemes and analyze the structure of numerical diffusivity. This leads to a detailed investigation into the cause of certain catastrophic breakdowns by some numerical flux schemes. In particular, we identify the dissipative terms that are responsible for shock instabil- ities, such as the odd–even decoupling and the so-called “carbuncle phenomenon.” As a result, we propose a conjecture stating the connection of the pressure differ- ence term to these multidimensional shock instabilities and hence a cure to those difficulties. The validity of this conjecture has been confirmed by examining a wide class of upwind schemes. The conjecture is useful to the flux function development, for it indicates whether the flux scheme under consideration will be afflicted with these kinds of failings. Thus, a class of shock-stable schemes can be identified. In- terestingly, a shock-stable scheme’s self-correcting capability is demonstrated with respect to carbuncle-contaminated profiles for flows at both low supersonic and high Mach numbers. Key Words: upwind methods; mass flux; shock (carbuncle) instability; AUSM schemes. 1. INTRODUCTION Numerical representation of inviscid