Section: New Results

Non-hydrostatic modelling of free surface flows

Participants : Stevan Bellec, Mathieu Colin [Corresponding member] , Andrea Filippini, Maria Kazolea, Mario Ricchiuto.

This year we have made a lot of progress in the understanding of the properties of Boussinesq-type models for near shore applications. In particular, we have performed a systematic analysis of the nonlinear behaviour of these models in the surf-zone, and in particular of their shoaling properties. These properties influence fundamentally the wave breaking process, and thus the impact of the wave on coastal structures. We have clearly identified two families of physical behaviours, associated to a similar formal structure of the equations. This result has been presented in [30] , [45] , and the full study is currently in revision on the Coastal Engineering journal.

In parallel, we have continued the study of the implementation of wave breaking models, comparing several physical criteria for the detection of the beginning and end of the breaking process. So far, we have only tested the so-called hybrid approach in which the hyperbolic Shallow Water equations are used in breaking regions, and the energy dissipation of breaking waves is modelled by the dissipation of mathematical entropy in shock waves. The work performed complements the initial study performed by M. Kazolea in her PhD and also proposes new physical detection criteria [28] , [44] (a full paper is in preparation).

Furthermore, we have began a systematic study on the existence of particular solutions (such as solitary waves for example) to the different Boussinesq-type models in view of having efficient materials to determinate the efficiency of our numerical schemes and to perform preliminary simulations.

The last important theoretical brick we added this year is the study of fully discrete asymptotic models, obtained by pre-discretizing the two-dimensional incompressible free surface Euler equations with a finite element method, and then by performing an asymptotic development (in terms of the classical nonlinearity and dispersion parameters). We have thus obtained a discrete model which, although consistent with a known continuous Boussinesq system, represents a surprisingly improved discret eversion of these equations, hardly obtainable by classical discretisation choices.

Besides the modelling effort, we have also started woking on real applications. In particular, we have worked on case studies involving harbour dynamics and river hydraulics. In the first case, M. Kazolea has performed a systematic study of the contribution of harbour resonance in the excitation of the Venetian harbor basin of Chania, during typical winder storms. Concerning river hydraulics, we have performed a parametric study of the appearance of tidal bores in estuaries, with parameters given by the tide non-linearity (amplitude), and the friction in the river. Both works will be presented at the next world congress of the International Association for Hydro-Environment Engineering.

External contributors. This work has benefitted from the collaboration with the EPOC lab in Bordeaux, and in particular with P. Bonneton.