Mathematical modeling of time-harmonic aeroacoustics with a generalized impedance boundary condition

Eric Luneville; Jean-Francois Mercier

ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique (2014)

  • Volume: 48, Issue: 5, page 1529-1555
  • ISSN: 0764-583X

Abstract

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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.

How to cite

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Luneville, Eric, and Mercier, Jean-Francois. "Mathematical modeling of time-harmonic aeroacoustics with a generalized impedance boundary condition." ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique 48.5 (2014): 1529-1555. <http://eudml.org/doc/273229>.

@article{Luneville2014,
abstract = {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.},
author = {Luneville, Eric, Mercier, Jean-Francois},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique},
keywords = {aeroacoustics; scattering of sound in flows; treated boundary; Myers condition; finite elements; variational formulations; scattering},
language = {eng},
number = {5},
pages = {1529-1555},
publisher = {EDP-Sciences},
title = {Mathematical modeling of time-harmonic aeroacoustics with a generalized impedance boundary condition},
url = {http://eudml.org/doc/273229},
volume = {48},
year = {2014},
}

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 - Modélisation Mathématique et Analyse Numérique
PY - 2014
PB - EDP-Sciences
VL - 48
IS - 5
SP - 1529
EP - 1555
AB - 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.
LA - eng
KW - aeroacoustics; scattering of sound in flows; treated boundary; Myers condition; finite elements; variational formulations; scattering
UR - http://eudml.org/doc/273229
ER -

References

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