Stabilization of Kelvin-Helmholtz instabilities in 3D linearized Euler equations using a non-dissipative discontinuous Galerkin method

Abstract

We present in this paper a time-domain Discontinuous Galerkin dissipation-free method for the transient solution of the three-dimensional linearized Euler equations around a steady-state solution. In the general context of a non-uniform supporting flow, we prove, using the well-known symmetrization of Euler equations, that some aeroacoustic energy satisfies a balance equation with source term at the continuous level, and that our numerical framework satisfies an equivalent balance equation at the discrete level and is genuinely dissipation-free. Moreover, there exists a correction term in aeroacoustic variables such that the aeroacoustic energy is exactly preserved, and therefore the stability of the scheme can be proved. This leads to a new filtering of Kelvin-Helmholtz instabilities. In the case of P1 Lagrange basis functions and tetrahedral unstructured meshes, a parallel implementation of the method has been developed, based on message passing and mesh partitioning. Three-dimensional numerical results confirm the theoretical properties of the method. They include test-cases where Kelvin-Helmholtz instabilities appear and can be eliminated by addition of the source term.