Section: New Results

Modelling of free surface flows

• Participants: Luca Arpaia , Stevan Bellec , Mathieu Collin , Sebastien De Brye , Andrea Filippini , Maria Kazolea, Luc Mieussens, and Mario Ricchiuto

• Corresponding member: Mario Ricchiuto

We have introduced a new systematic method to obtain discrete numerical models for incompressible free-surface flows. our approach allows to recover discrete asymptotic equations from a semi-discretized form (keeping the vertical $z$ variable and time continuous) of the incompressible Euler equations with free surface. In particular, starting from a (continuous) Galerkin finite element discretization in the horizontal direction, we perform an asymptotic analysis of the resulting semi-discrete system. This has allowed to obtain new discrete equivalents of the Peregrine equations [5], as well as enhanced variants in the spirit of [115]. This has been done in the PhD of S. Bellec. We have demonstrated that the resulting discrete equations present dispersion characteristics much improved w.r.t. those obtained by directly discretizing the asymptotic Boussinesq equations with continuous finite elements. This has been confirmed by numerical experiments on long wave propagation benchmarks. Concerning more classical continuous Boussinesq models, additional work has been done to characterize some of their exact solutions. This has provided some improved solutions to benchmark our codes, as well as some additional knownledge on these models [4].

This year we extended our work on fully non-linear weakly dispersive wave models in two dimensional horizontal coordinates. The proposed framework in [11], to approximate the so-called Green-Naghdi equations is followed. The method proposed, while remaining unsplit in time, is based on a separation of the elliptic and hyperbolic components of the equations. This separation is designed to recover the standard shallow water equations in the hyperbolic step, so that the method can be written as an algebraic correction to an existing shallow water code. More precisely, we re-write the standard form of the equations by splitting the original system in its elliptic and hyperbolic parts, through the definition of a new variable, accounting for the dispersive effects and having the role of a non-hydrostatic pressure gradient in the shallow water equations. We consider a two-step solution procedure. In the first step we compute a source term by inverting the elliptic coercive operator associated to the dispersive effects; then in a hyperbolic step we evolve the flow variables by using the non-linear shallow water equations, with all nonhydrostatic effects accounted by the source computed in the elliptic phase. The advantages of this procedure are firstly that the GN equations are used for propagation and shoaling, while locally reverting to the non-linear shallow water equations to model energy dissipation in breaking regions. Secondly and from the numerical point of view, this strategy allows each step to be solved with an appropriate numerical method on arbitrary unstructured meshes. We propose a hybrid finite element (FE) finite volume (FV) scheme, where the elliptic part of the system is discretized by means of the continuous Galerkin FE method and the hyperbolic part is discretized using a third-order node-centered finite volume (FV) technique. This work was a part of Andrea Filippini's PhD and a research paper is under preparation.

We also continue our study on wave breaking techniques on unstructured meshes [55]. In particular, we evaluate the coupling of both a weakly and a fully non-linear Boussinesq system with a turbulence model. We reformulate an evolution model for the turbulent kinetic energy, initially proposed by Nwogu [115], and evaluate its capabilities to provide sufficient dissipation in breaking regions. We also compare this dissipation to the one introduced by the numerical discretization. A research paper on the topic, is under preparation. Further more we studied and tested the application and validation of TUCWave code on the transformation breaking and run-up of irregular waves. Its is the first time that an unstructured high-resolution FV numerical solver for the 2D extended BT equations of Nwogu is tested on the generation and propagation of irregular waves. A research paper is under preparation.

The tools developed have been also used intensively in funded research programs. Within the TANDEM project, several benchmarks relevant to tsunami modelling have been performed and several common publications with the project partners are submitted and/or in preparation [54], [45]. We also our code SLOWS, to study the conditions for tidal bore formation in convergent alluvial estuaries [7]. A new set of dimensionless parameters has been introduced to describe the problem, and the code SLOWS has been used to explore the space of these parameters allowing to determine a critical curve allowing to characterize an estuary as "bore forming" or not. Surprising physical behaviours, in terms of dissipation and nonlinearity of the tides, have been highlighted.

Finally, in collaboration with F. Veron (University of Delaware at Newark, USA), L. Mieussens has developed a model to describe the effect of rain falling on water waves [20]. This model is based on a kinetic description of rain droplets that is used to compute the induced pression on a water wave. This allows to estimate the dissipation (or amplification) of the wave due to rainy conditions.