# Modélisation, analyse mathématique et numérique de divers écoulements compressibles ou incompressibles en couche mince.

Abstract : In the first part, we formally derive the \PFS (\textbf{P}ressurised and \textbf{F}ree \textbf{S}urface) equations for unsteady mixed flows in closed water pipes with variable geometry. We write the numerical approximation of these equations by a VFRoe and a kinetic solver by upwinding the sources terms at the cell interfaces. Particularly, we propose the upwinding of a friction term (given by the Manning-Strickler law) by introducing the notion of \emph{ dynamic slope}. Finally, we construct a well-balanced scheme preserving the still water steady states by defining a stationary matrix especially constructed for the VFRoe scheme. Following this idea, we construct a well-balanced scheme which preserve all steady states. To deal with transition points occurring when the state of the flow changes (i.e. free surface to pressurised and conversely), we extend the \og ghost waves\fg~ method and propose a full kinetic approach. In the second part, we study a simplified version of a compressible primitive equations for the dynamic of the atmosphere. We obtain an existence result for weak solutions global in time for the two-dimensional model. We also state a stability result for weak solutions for the three dimensional version. To this end, we introduce a useful change of variables which transform the initial model in a more simpler one. We present a small introduction to the cavitation phenomena. We recall the different kinds of cavitation and some mathematical models such as the Rayleigh-Plesset equation and a mixed model. As a first step toward the modeling of the cavitation in closed pipes, we propose a bilayer model which take into account the compressibility effect of the air onto the free surface. As pointed out by several authors, such a system, of $4$ equations, is non hyperbolic and generally, eigenvalues cannot be explicitly computed. We propose a numerical approximation by using a kinetic scheme. In the last chapter, we formally derive a sediments transport model based on the Vlasov equation coupled to an anisotropic compressible Navier-Stokes equations. This model is obtained by performing two asymptotic analysis.
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https://tel.archives-ouvertes.fr/tel-00529392
Contributor : Mehmet Ersoy Connect in order to contact the contributor
Submitted on : Monday, October 25, 2010 - 3:06:17 PM
Last modification on : Thursday, October 7, 2021 - 10:53:57 AM
Long-term archiving on: : Friday, October 26, 2012 - 12:20:35 PM

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• HAL Id : tel-00529392, version 1

### Citation

Mehmet Ersoy. Modélisation, analyse mathématique et numérique de divers écoulements compressibles ou incompressibles en couche mince.. Mathématiques [math]. Université de Savoie, 2010. Français. ⟨tel-00529392⟩

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