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We study the time-harmonic acoustic scattering in a duct in presence of a flow and of a discontinuous impedance boundary condition. Unlike a continuous impedance, a discontinuous one leads to still open modeling questions, as in particular the singularity of the solution at the abrupt transition and the choice of the right unknown to formulate the scattering problem. To address these questions we propose a mathematical approach based on variational formulations set in weighted Sobolev spaces. Considering the discontinuous impedance as the limit of a continuous boundary condition, we prove that only the problem formulated in terms of the velocity potential converges to a well-posed problem. Moreover we identify the limit problem and determine some Kutta-like condition satisfied by the velocity: its convective derivative must vanish at the ends of the impedance area. Finally we justify why it is not possible to define limit problems for the pressure and the displacement. Numerical examples illustrate the convergence process.
@article{M2AN_2014__48_5_1529_0,
author = {Luneville, Eric and Mercier, Jean-Francois},
title = {Mathematical modeling of time-harmonic aeroacoustics with a generalized impedance boundary condition},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis },
pages = {1529--1555},
publisher = {EDP-Sciences},
volume = {48},
number = {5},
year = {2014},
doi = {10.1051/m2an/2014008},
mrnumber = {3264364},
zbl = {1301.35079},
language = {en},
url = {http://geodesic.mathdoc.fr/articles/10.1051/m2an/2014008/}
}
TY - JOUR AU - Luneville, Eric AU - Mercier, Jean-Francois TI - Mathematical modeling of time-harmonic aeroacoustics with a generalized impedance boundary condition JO - ESAIM: Mathematical Modelling and Numerical Analysis PY - 2014 SP - 1529 EP - 1555 VL - 48 IS - 5 PB - EDP-Sciences UR - http://geodesic.mathdoc.fr/articles/10.1051/m2an/2014008/ DO - 10.1051/m2an/2014008 LA - en ID - M2AN_2014__48_5_1529_0 ER -
%0 Journal Article %A Luneville, Eric %A Mercier, Jean-Francois %T Mathematical modeling of time-harmonic aeroacoustics with a generalized impedance boundary condition %J ESAIM: Mathematical Modelling and Numerical Analysis %D 2014 %P 1529-1555 %V 48 %N 5 %I EDP-Sciences %U http://geodesic.mathdoc.fr/articles/10.1051/m2an/2014008/ %R 10.1051/m2an/2014008 %G en %F M2AN_2014__48_5_1529_0
Luneville, Eric; Mercier, Jean-Francois. Mathematical modeling of time-harmonic aeroacoustics with a generalized impedance boundary condition. ESAIM: Mathematical Modelling and Numerical Analysis , Tome 48 (2014) no. 5, pp. 1529-1555. doi: 10.1051/m2an/2014008
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