Solving Galbrun’s Equation with a Discontinuous galerkin Finite Element Method

Over many years, scientists and engineers have developed a broad variety of mathematical formulations to investigate the propagation and interactions with flow of flow-induced noise in early-stage of product design and development. Beside established theories such as the linearized Euler equations (LEE), the linearized Navier–Stokes equations (LNSE) and the acoustic perturbation equations (APE) which are described in an Eulerian framework, Galbrun utilized a mixed Lagrange–Eulerian framework to reduce the number of unknowns by representing perturbations by means of particle displacement only. Despite the advantages of fewer degrees of freedom and the reduced effort to solve the system equations, a computational approach using standard continuous finite element methods (FEM) suffers from instabilities called spurious modes that pollute the solution. In this work, the authors employ a discontinuous Galerkin approach to overcome the difficulties related to spurious modes while solving Galbrun’s equation in a mixed and pure displacement based formulation. The results achieved with the proposed approach are compared with results from previous attempts to solve Galbrun’s equation. The numerical determination of acoustic modes and the identification of vortical modes is discussed. Furthermore, case studies for a lined-duct and an annulus supporting a rotating shear-flow are investigated.