Approche asymptotique pour l'étude mathématique et la simulation numérique de la propagation du son en présence d'un écoulement fortement cisaillé

Lauris Joubert 1
1 POEMS - Propagation des Ondes : Étude Mathématique et Simulation
Inria Saclay - Ile de France, UMA - Unité de Mathématiques Appliquées, CNRS - Centre National de la Recherche Scientifique : UMR7231
Abstract : This thesis deals with the modelling and the numerical simulation of acoustic propagation in a flow. The aim of this work is to derive approximate models that easily take into account thin shear layers of the reference flow (boundary layer, mixing shear layer...). The model retained for these studies is the Galbrun's equation. The first part is devoted to the study of the acoustic propagation in a thin bidimensional duct. By an asymptotic analysis, which can be seen as a low frequency approach, we derive an approximate model. It has a non-local integral term in the transverse coordinate. Because of its non-standard structure, the stability analysis is difficult and an ad hoc analysis is needed. Using this original approach we find well-known results of hydrodynamic instabilities in laminar flows (in the incompressible case), but we also get some new results. We finally give a numerical method based on a quasi-explicit expression of the solution. The second part focuses on the consideration of boundary layers. We consider a bidimensional problem with a flat wall. The cases of a rigid wall and of an impedant wall are investigated. In both cases, using an asymptotic analyis, we replace the boundary layer by an approximate limit condition. These two generalized impedance conditions involve the resolution of the limit equation of the thin duct, and their stability analysis relie on the results obtained in the first part. Then, we explore the physical and mathematical properties of these approximate problems.
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Submitted on : Thursday, January 6, 2011 - 2:50:07 PM
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Lauris Joubert. Approche asymptotique pour l'étude mathématique et la simulation numérique de la propagation du son en présence d'un écoulement fortement cisaillé. Equations aux dérivées partielles [math.AP]. Ecole Polytechnique X, 2010. Français. ⟨pastel-00553081⟩

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