A Stabilised High-Order Finite Element Model for the Linearised Euler Equations

Abstract

The propagation of sound in complex flows is a critical issue for many industries. When modeling turbomachinery noise radiating from engine exhausts, the jet shear layer induces a strong refraction of the sound waves. This can be described by the Linearised Euler Equations (LEE). Most of the difficulties associated with time-domain solutions of the LEE can be avoided by working in the frequency domain. Standard finite elements suffer from large dispersion errors and to improve the computational efficiency we resort here to highorder FEM. The FEM is also known to encounter stability issues for advection-diffusion problems that can be corrected by adding artificial diffusion terms in the formulation. In this paper, we aim at investigating dedicated high-order stabilisation schemes for the time-harmonic LEE. A dispersion analysis of the one-dimensional time-harmonic transport equation is provided. The optimal stabilisation parameter is derived so as to cancel the dispersion error, for each polynomial order of the shape functions. The performance of the resulting stabilised formulation is investigated on a two-dimensional test case with unstructured meshes. The steady parameter used in the literature for the LEE performs well in the high-resolution regime, as attested by the results of the sound propagation from a semi-infinite circular duct with non-uniform mean flow. The sound propagation and radiation are accurately described, as well as the interactions between the acoustic waves and the hydrodynamic field resulting in the vorticity shedding from the duct lip.