Linearized Navier-Stokes and Euler Equations for the Determination of the Acoustic Scattering Behaviour of an Area Expansion

Abstract

In this paper, we investigate the scattering behaviour of plane acoustic waves at an area expansion. Therein, the mean flow separates at the trailing edge and expands into a jet in the downstream duct. This configuration supports complex interaction effects between acoustic waves and mean flow field. The methodology involves a two step hybrid approach. First, a highly resolved large eddy simulation (LES) is performed to extract the mean flow field. Then, the acoustic field is computed in frequency space, yielding the unknown scattering matrix coefficients. Two different sets of governing equations are used for this task: the linearized Navier-Stokes equations (LNSEs) and linearized Euler equations (LEEs). By definition, these sets of equations are convection dominated and are therefore susceptible to numerical instabilities. In this paper a consistent finite element Galerkin/least-squares (GLS) approach is used to stabilize the different sets of equations. Unlike the introduction of artificial viscosity, this technique partially preserves the accuracy of the discretization order. The results show that both equation sets, viz. LNSEs and LEEs capture the complex interaction between acoustic waves and the free shear layer in detail. It is shown that acoustic diffusion effects of the LNSEs are of small order and may be neglected for the acoustic determination of the scattering behaviour of sudden area expansions.