Section: New Results
Modelling of complex flows
Dynamics of sedimentary river beds with stochastic fluctuations
Participants : Emmanuel Audusse, Philippe Ung.
We studied in  the behaviour of the solution of the Saint-Venant–Exner equations when a stochastic term is introduced in the model through the sediment flux. A first investigation was done considering periodic boundary conditions and the next part of this study is devoted to the case when physical ones are imposed. Our goal is to investigate the possibility to bring out a characteristic long time behaviour and to establish a relation between the injected noise and the physical parameters involved in the model. This work was achieved in collaboration with Sébastien Boyaval from Lab. Hydraulique Saint-Venant.
Participants : Nora Aïssiouene, Marie-Odile Bristeau, Edwige Godlewski, Dena Kazerani, Anne Mangeney, Jacques Sainte-Marie, Nicolas Seguin.
The objective is to derive a model corresponding to a depth averaged version of the incompressible Euler equations with free surface and to develop a robust numerical method for the resolution of the model.
Concerning the modelling aspect, a non-hydrostatic shallow water-type model approximating the incompressible Euler and Navier-Stokes sytems with free surface was developped and published in  . The closure relations are obtained by a minimal energy constraint instead of an asymptotic expansion. The model slightly differs from the well-known Green-Naghdi model and is confronted with stationary and analytical solutions of the Euler system corresponding to rotational flows.
The numerical approximation relies on a projection-correction type scheme. The hyperbolic part of the system is approximated using a kinetic finite volume solver and the correction step implies to solve an elliptic problem involving the non-hydrostatic part of the pressure.
In one dimension, the resolution of the incompressibility problem leads to solve a mixed problem where the pressure and the velocity are defined in compatible approximation spaces. This step uses a variationnal formulation of the shallow water version of the incompressibility condition.
This numerical scheme satisfies classical properties (positivity, well-balancing and consistency) and a discrete entropy inequality. Several numerical experiments are performed to confirm the relevance of our approach.
This approach will allow us to extend the numerical method in higher dimensions and to treat particular difficult cases occuring in specific geophysical situations (dry/wet interfaces).
Plasticity in Shallow Water equations
Participant : Nicolas Seguin.
In collaboration with Bruno Després and Clément Mifsud from Univ. Pierre et Marie Curie, we proposed in  a new definition of solutions for hyperbolic Friedrichs' systems in bounded domains, which follows the idea of Lions' dissipative solutions and Otto's boundary formulation for conservation laws. We proved in the classical settings existence and uniqueness. The goal of this project is to be able to incorporate nonlinear effects of plasticity in models of elasticity or overflowing in channels for shallow water flows, by adding entropy compatible constraints.
Management of marine energies
Participants : Cindy Guichard, Martin Parisot, Jacques Sainte-Marie, Julien Salomon.
The purpose of this project is to model floating devices (like buoys) in the context of recovering energy from water resources (seas and oceans). If the free surface flow can be handled by means of the Saint-Venant equations, the area under the buoys requires a different modelling (for example equivalence with springs) as the surface is constrained. The Archimedes' principle is also involved. Some preliminary numerical results were obtained thanks to the FreshKiss3D code.
To go further, the optimisation of the overall process is also under consideration. Indeed, to maximise the amount of recovered energy, the bathymetry, the shape of the buoy, the number of buoys are critical parameters which must be modelled in view of industrial applications. Optimal control methods are applied to determine the best configuration depending on the devices: optimisation of the kinetic energy for water-turbines or of the potential energy for buoys.