Section: New Results

Geophysical flows

A numerical scheme for the Saint-Venant equations

Participants : Emmanuel Audusse, Christophe Chalons [Univ. Versailles] , Philippe Ung.

In order to improve the numerical simulations of the shallow-water equations, one has to face three important issues related to the well-balanced, positivity and entropy-preserving properties, as well as the ability to handle vacuum states. In that purpose, we propose a Godunov-type method based on the design of a three-wave Approximate Riemann Solver (ARS) which satisfies all aformentioned properties.

Two-phase flows

Participants : Frédéric Coquel [CNRS] , Jean-Marc Hérard [EDF] , Khaled Saleh [IRSN] , Nicolas Seguin.

After having developed numerical schemes for models of compressible two-phase flows [17] , [19] , we have proven some fundamental properties of these systems: symmetrizability and (non strict) convexity of the entropy [18] . This enables us now to address the well-posedness of these models when the relaxation terms are included.

Non-hydrostatic models

Participants : Marie-Odile Bristeau, 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. We have already contributed to this subject but the obtained results extend previous ones [29] in several directions:

• the derivation of the model is more rigorous and follows the entropy-based moment closures proposed in [28] ,

• the properties of the model and especially its connections with Green-Nagdhi model have been investigated,

• a family of analytical solutions for the proposed model have been obtained.

These analytical solutions emphasize the non-hydrostatic effects appearing for large slope variations.

Fluids with complex rheology

Participants : Anne Mangeney, Jacques Sainte-Marie.

We have been able

• to develop detection, characterization and localisation methods applicable to the seismic signals generated by rockfalls and thus to analyse ths spatio-temporal change of rockfall localisation and properties during several years, making it possible to show how rockfalls can be used as a precursor of volcanic activity,

• to propose an empirical “universal” law describing friction weakening in landslides over a broad range of volumes and geological contexts,

• to propose a new debris flow model with an energy balance,

• show the existence of a slow propagation phase in granular flows, playing a key role in their dynamics and in erosion processes.

Dynamics of sedimentary river beds with stochastic fluctuations

Participants : Emmanuel Audusse, Philippe Ung.

The Exner equation is a coarse model for the dynamics of sedimentary river beds, derived using both many heuristics and empirism. Though, it is also quite practical for hydraulic engineering applications, and efficient enough in numerous situations. Our goal in this work is to improve the model by including some effects that have been neglected so far in the heuristics. In particular, inline with other current research directions in the field, we study the possibility of introducing some stochasticity in the model. To this end, we suggest to numerically experiment some recently proposed variations of the Exner equation based on the introduction of stochastic fluctuations within the standard formulation.

This project has been the subject of a study during the 2013 session of the CEMRACS.