Section: New Results

Modelling of complex flows

A new shallow water type model involving non-hydrostatic effects is derived in [37] . Under the assumption that the horizontal velocity is close to its vertical mean value, the model enables to recover the energy from the Euler system before integration. Link with the non-hydrostatic published in [18] is identified. Compared to the aforementioned models, the new system consists of more equations (6). However, the numerical strategy presented in the paper does not induce extra computational time.

Estimating the energy loss in elastic waves during an impact is an important problem in seismology and in industry. Three complementary methods to estimate the elastic energy radiated by bead impacts on thin plates and thick blocks from the generated vibration are proposed in [30] . The first two methods are based on the direct wave front and are shown to be equivalent. The third method makes use of the diffuse regime. These methods are shown to be relevant to establish the energy budget of an impact. The radiated elastic energy estimated with the presented methods is quantitatively validated by Hertz's model of elastic impact.

In [25] we propose a strategy to approximate incompressible free surface Euler and Navier-Stokes models. The main advantage of the proposed models is that the water depth is a dynamical variable of the system and hence the model is formulated over a fixed domain.

The proposed strategy extends previous works approximating the Euler and Navier-Stokes systems using a multilayer description. Here, the needed 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.