Time-harmonic acoustic scattering in a complex flow

We are interested in the numerical simulation of time harmonic acoustic scattering in presence of a complex flow on a unstructured mesh. Galbrun's equation, whose unknown is the perturbation of displacement, is attractive, compared to the linearized Euler's equations, because it is close to a wave equation which allows the use of classical Lagrange Finite Element, and it is well adapted to take into account boundary conditions, like impedance or interface with an elastic structure. However, a direct discretization of Galbrun's equation with Lagrange Finite Element leads to numerical troubles. We propose a method that allows both to obtain a stable numerical scheme and non-reflecting artificial boundary conditions. This method requires to introduce a new quantity related to hydrodynamic vortices which satisfies a convection equation. A hybrid numerical method is proposed, coupling finite elements for Galbrun's equation and a Discontinuous Galerkin scheme for the convection equation. Several 2D numerical results are presented to show the efficiency of the method. In the 3D case, an attractive alternative to Galbrun's equation is Goldstein's equations: here the vorticity vanishes where the flow is potential which reduces the cost of the Discontinuous Galerkin scheme.