Stabilized Perfectly Matched Layer for Advective Acoustics

Abstract

In this paper we present a stabilized Perfectly Matched Layer (PML) for the subsonic aeroacoustics equations. The PML method was introduced by Bérenger [2] to compute the solution of Maxwell’s equations in unbounded domain. This method was adapted by Hu [4] to the aeroacoustics equations. We will first recall his model, considering an horizontal uniform mean flow M. We have to rewrite the acoustics equations whose unknowns are the pressure and the velocity (p, U = (u, v)),

$$ \left\{ \begin{gathered} \partial _t p + M\partial _x p + \partial _x u + \partial _y v = 0, \hfill \\ \partial _t u + M\partial _x u + \partial _x p = 0, \hfill \\ \partial _t v + M\partial _x v + \partial _y p = 0, \hfill \\ \end{gathered} \right. $$

((1))

in a so called “split form” in the unknowns (p _x , p _y , u,v _x,v _y), where p _x and p _y (resp. v _x and v _y ) are non-physical variables, whose sum gives the pressure (resp. the y-component of the velocity) p = p _x + p _y (resp. v = v _x + v _y ). One then obtains the “PML model” by adding a zero-order absorption term proportional to some absorption coefficient σ. Finally, an absorbing layer is obtained by replacing the aeroacoustics equations by the PML model inside a layer of finite width that surrounds the bounded domain of interest for the computations. This absorbing layer has the property to be “perfectly matched”. This means that a wave propagating in the domain of interest does not produce any reflection when it meets the interface with the absorbing layer, whatever its frequency and its angle of incidence are.