Modélisation, analyse mathématique et numérique de divers ...

Modeling, mathematical and numerical analysis of various compressible
or incompressible flows in thin layer
Abstract
In the first part, we formally derive the PFS (Pressurised and Free Surface) 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 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 « ghost waves » 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.
Keywords: Unsteady mixed flows in closed water pipes, Shallow water equations (Saint-Venant),
Pressurised equations, Finite Volume, VFRoe solver, Kinetic solver, Well-balanced scheme, Compressible
primitive equations, Existence and stability results, Anisotropic compressible Navier-
Stokes equations, Cavitation, Bilayer model, Saint-Venant-Exner equations.
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