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Mathematical modeling of time-harmonic aeroacoustics with a generalized impedance boundary condition. (English) Zbl 1301.35079
Summary: 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.
MSC:
35P25 Scattering theory for PDEs
76Q05 Hydro- and aero-acoustics
35J20 Variational methods for second-order elliptic equations
35J05 Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation
65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
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