Méthode asymptotique numérique pour le calcul des branches solutions et des instabilités dans les fluides et pour les problèmes d'interaction fluide-structure

Abstract : This work deals with the computing of solution branches of the stationary Navier-Stokes equations with an asymptotic numerical method (anm). Some bifurcating indicators are introduced on solution branches to detect stationary and Hopf bifurcation points. Anm is the combination of a perturbation technique and finite element method. Perturbation technique transforms the initial non linear problem into a succession of linear ones which have the same tangent operator. So these linear problems are solved numerically by finite element method and only one triangulation of the tangent stiffness matrix is needed to determine analytically a part of the solution branch. The first chapter of this thesis is an overview of the anm and continuation technique to compute solution branches of stationary Navier-Stokes equations. The second chapter deals with the application of the anm when a Petrov-Galerkin weighting is used. The main difficulty of such a problem is the non-quadric form of the equations. Initial relations must then be modified to introduce asymptotic expansions and continuation method defined in chapter one. The two following chapters concern the detection of singular points on stationary solution branches. Two kinds of instabilities are investigated, stationary and Hopf bifurcations. Bifurcation indicators permit us to evaluate numerically these singular points. Several numerical examples show the efficiency and the reliability of such indicators. In the last chapter we apply the numerical procedures previously defined to a problem of fluid/structure interaction : a cylinder in a cross flow. Equations of motion of the fluid and the cylinder are described in a arbitrary Lagrangian-Eulerian frame
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Jean-Marc Cadou. Méthode asymptotique numérique pour le calcul des branches solutions et des instabilités dans les fluides et pour les problèmes d'interaction fluide-structure. Autre [cond-mat.other]. Université Paul Verlaine - Metz, 1997. Français. ⟨NNT : 1997METZ022S⟩. ⟨tel-01777208⟩

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