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

Serge Piperno (1) , Marc Bernacki (2)

1 CERMICS - Centre d'Enseignement et de Recherche en Mathématiques et Calcul Scientifique
2 CEMEF - Centre de Mise en Forme des Matériaux

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 P 1 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.

Mots clés: aeroacoustics; acoustic energy; linearized Euler equations; non-uniform steady-state flow; Discontinuous Galerkin method; time domain; energy-conservation

Domaines: Mécanique des fluides [physics.class-ph]; Matériaux

Serge Piperno, Marc Bernacki. Stabilization of Kelvin-Helmholtz instabilities in 3D linearized Euler equations using a non-dissipative discontinuous Galerkin method. European Conference on Computational Fluid Dynamics (ECCOMAS CFD 2006), Sep 2006, Egmond aan Zee, Netherlands. ⟨hal-01858317⟩