A discourse on Galilean invariance, SUPG stabilization, and the variational multiscale framework

Galilean invariance is one of the key requirements of many physical models adopted in theoretical and computational mechanics. Spurred by recent research developments in shock hydrodynamics computations [G. Scovazzi, Stabilized shock hydrodynamics: II. Design and physical interpretation of the SUPG operator for Lagrangian computations. Comput. Methods Appl. Mech. Engrg., in press, doi:10.1016/j.cma.2006.08.009], a detailed analysis on the principle of Galilean invariance in the context of SUPG operators is presented. It was observed in G. Scovazzi (in press) that lack of Galilean invariance can yield catastrophic instabilities in Lagrangian computations. Here, the analysis develops at a more general level, and an arbitrary Lagrangian–Eulerian (ALE) formulation is used to explain how to consistently derive Galilean invariant SUPG operators. Stabilization operators for Lagrangian and Eulerian mesh computations are obtained as limits of the stabilization operator for the underlying ALE formulation. In the case of Eulerian meshes, it is shown that most of the SUPG operators designed for compressible flow computations to date are not consistent with Galilean invariance. It is stressed that Galilean invariant SUPG formulations can provide consistent advantages in the context of complex engineering applications, due to the simple modifications needed for their implementation.