Galilean invariance and stabilized methods for compressible flows

摘要

In a recent work (Comput. Methods Appl. Mech. Eng. 2007; 196(4-6):966-978), it was observed that lack of Galilean invariance led to catastrophic instabilities when stabilized methods were used in Lagrangian shock hydrodynamics computations. By means of an arbitrary Lagrangian-Eulerian (ALE) formulation, Galilean invariant SUPG operators were consistently derived in (Comput. Methods Appl. Mech. Eng. 2007; 196(4-6):1108-1132), and their Lagrangian and Eulerian limits were compared to the most commonly used stabilized formulations. In the particular case of Eulerian meshes, it was shown that most of the SUPG operators designed to date for compressible flow computations are not invariant. However, due to the significant overhead of algebraic manipulations, the use in (Comput. Methods Appl. Mech. Eng. 2007; 196(4-6):1108-1132) of the referential form of the ALE equations made the presentation of the main ideas quite involved. The present paper addresses this particular issue, since the invariance analysis is presented with the aid of the intuitive current configuration reference frame, more familiar to computational fluid dynamicists.