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Section: New Results
Numerical methods for free-surface flows
Godunov schemes for the low Froude regime
Participants : Emmanuel Audusse, Do Minh Hieu, Yohan Penel.
We investigated in [29] the behaviour of collocated Godunov type finite volume schemes when applied to the 1d linear wave equation with Coriolis force in collaboration with S. Dellacherie and P. Omnes (CEA). Accuracy for short time and stability were proven for different versions of the classical Godunov schemes, including some schemes already proposed in the literature (Bouchut et al, [42] ). Next step will be to include linear advection and then to study the fully non linear shallow water model. Then results will be extended to 2d problems for which geometrical constraints should be taken into account.
Numerical method for non-hydrostatic models
Participants : Nora Aïssiouene, Marie-Odile Bristeau, Edwige Godlewski, Jacques Sainte-Marie.
In [1] , a numerical method based on a prediction-correction scheme in one dimension has been developed and compared to experimental data and analytical solutions. The issue is then to extend the method in higher dimensions. We propose a variational framework for the resolution of a non-hydrostatic Saint-Venant type model with bottom topography. This model is a shallow water type approximation of the free surface incompressible Euler system and slightly differs from the Green-Naghdi model. The resolution of the incompressibility constraint leads to an elliptic problem involving the non-hydrostatic part of the pressure. This step uses a variational formulation of a shallow water version of the incompressibility condition. Several numerical experiments are performed to confirm the relevance of our approach. This work is exposed in [18] .
Uncertainties with the topography
Participants : Emmanuel Audusse, Nicole Goutal, Philippe Ung.
We propose to study the uncertainty related to the Saint-Venant system. A perturbation is introduced in the bottom topography such that the topography deviation is characterized by two parameters: its amplitude and its smoothness. In particular, we extend the work previously done with periodic boundary conditions and suggest a treatment of the physical ones. In doing so, we are interested in the influence of the topography deviation on the hydraulic quantities, and in particular, we numerically exhibit a relationship between the spatial correlations of the topography and the water height. Furthermore, we complete the study by a comparison of the outputs between the two flow regimes – fluvial and torrential.
Coupled Stokes-Exner model
Participant : Nora Aïssiouene.
In the framework of the 2015 CEMRACS session (Coupling Multi-Physics Models involving Fluids), we explored an approach to model the sediment transport. In [17] , we consider a coupling between the Exner equation and the Stokes system to model sediments in geophysical flow phenomena. We focus on a model without free surface and used some numerical tests to evaluate the relevance of the method. The fluid structure interaction theory and methods have been applied on the coupled system and the objective is to test the proposed method which can be extend to a free surface model. The library Feel++ and the high computing performance embedded have been used to test the solution method. Therefore, the goal of this project is to understand the impact of the sediment transport on the flow using Navier-Stokes with a free surface system coupled with the standard Exner equation. This work has been done in collaboration with Tarik Amtout, Matthieu Brachet, Emmanuel Frenod, Romain Hild, Christophe Prud'homme, Antoine Rousseau and Stéphanie Salmon.