Section: New Results

Modelling of complex flows

The Shallow Water model with Roof: derivation and simulation

Participants : Edwige Godlewski, Cindy Guichard, Martin Parisot, Jacques Sainte-Marie, Fabien Wahl.

In view of taking into account interactions with floating structures, a shallow water type model is derived. In a first step a constraint corresponding to a static roof is considered and a relaxation approach is proposed in order to solve the model numerically. A particular attention is paid to the energy law as an application to marine energy devices is planned. The CPR scheme proposed in [17] is adapted to our case and implemented in one space dimension. Finally the numerical results are tested on analytical solutions, as well stationary as non-stationary ones [26].

Modelling of Sediment Transport

Participants : Emmanuel Audusse, Léa Boittin, Martin Parisot, Jacques Sainte-Marie.

A new model for sediment transport in river context is proposed. The model is derived from the Navier-Stokes equations by performing simultaneously the thin layer approximation and the diffusive limit. The well-posedness of the model is studied in a simplified case.

Layer-averaged Euler and Navier-Stokes equation

Participants : Marie-Odile Bristeau, Bernard Di Martino, Cindy Guichard, Jacques Sainte-Marie.

In [3], we propose a strategy to approximate incompressible hydrostatic free surface Euler and Navier-Stokes models. The proposed strategy extends previous works approximating the Euler and Navier-Stokes systems using a multilayer description. Here, the required closure relations are obtained using an energy-based optimality criterion instead of an asymptotic expansion. Moreover, the layer-averaged description is successfully applied to the Navier-Stokes system with a general form of the Cauchy stress tensor.

Layerwise Discretization for Non-Hydrostatic flows

Participants : Martin Parisot, Yohan Penel, Jacques Sainte-Marie.

In collaboration with Enrique Fernández-Nieto (Sevilla).

The work presented in [25] aims at deriving a new semi-discretisation with respect to the vertical variable of the Euler equations. It results in a hierarchy of multilayer model involving both hydrostatic and non-hydrostatic parts of the pressure field. All models are proven to satisfy an energy inequality. Moreover, the linear dispersion relation is given for each one with an explicit formula which converges to the exact Airy formula when the number of layers goes to infinity.

Two-phase (grains/fluid) model for geophysical debris flows

Participant : Anne Mangeney.

We developped a thin-layer depth-averaged model describing the two-phase flow made of granular material saturated by a fluid and include compression/dilatation effects. We solved numerically these equations and were able to accurately reproduce laboratory experiments.

Multi-layer model for viscoplastic granular flows

Participant : Anne Mangeney.

In collaboration with Enrique Fernández-Nieto and Gladys Narbona-Reina (Sevilla).

A multi-layer model was developped to simulate granular flow dynamics and deposit based on viscoplastic behaviour ($\mu \left(I\right)$-rheology). The numerical model made it possible to reproduce for the first time the increase of runout distance of granular material when flowing on erodible beds.