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Bilateral Contracts and Grants with Industry
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Section: New Results

Modelling of free surface flows

Participants : Luca Arpaia, Stevan Bellec, Mathieu Colin, Sebastien de Brye, Andrea Filippini, Maria Kazolea, Mario Ricchiuto [Corresponding member] .

We have introduced a new systematic method to obtain discrete numerical models for incompressible free-surface flows. The method consists in first discretizing the Euler equations with respect to the horizontal variables, keeping the vertial z variable and time continuous. We have focused so far on (continuous) Galerkin n finite element discretizations in the horizontal. We then perform an asymptotic analysis on the resulting semi-discrete system. Our initial result, has led to a new dicrete approximation, which we have shown to be consistent with the Boussinesq system known as Peregrine model. We have proven that the method obtained by means of this discrete asymptotic method, has phase and linear shoaling errors far lower that those obtained by discretizing the continuous model directly by means of the Galerkin method. Extensions to other weakly non-linear models have been obtained, and the study of fully nonlinear variants is under way.

We have also investigated the relations between some of the most common weakly nonlinear Boussinesq models. It is known since many years that, for given phase linear shoaling relations, two families of models exist depending on whether the dispersive terms are evaluated using derivatives of the speed, of of the flux (depth times speed). We have shown both analytically and numerically, that, independently on the phase and linear shoaling relations, these two families provide (quanlitatively and quantitatively) only two distinct behaviours when approaching the nonlinear regime. Models based on velocity derivatives, provide taller more asymmetric waves, all models of the same family produce stunningly similar results, even when the linear relations differ considerably.

To extend our initial work on unstructured solvers for dispersive wave models to the fully nonlinear case we have proposed a new framework to approximate the so-called Green-Naghdi equations [99] . 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. In particular, in our approach we fix the method used for the elliptic component (a continuous Galerkin method), and couple it to different hyperbolic shallow water solvers. As long as the hyperbolic step is more than second order accurate, the approach proposed allows accuracies comparable to those of a fourth order finite difference method, with a natural potential for h and p- adaptation on unstructured grids. The two-dimensional extension in in the testing phase.

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 in preparation. Independently on this activity, this year we used our codes to investigate two case studies. The first is the study of the wave conditions for the old Venetian harbour of Chania in Crete [109] . The study compares fully nonlinear-weakly dispersive COULWAVE code, developed at the University of South California, and TUCWave. The models are used to explore the appearance of resonance, eventually determining the resonant frequencies for the entire basin. The second study concerns the conditions for tidal bore formation in convergent alluvial estuaries [69] . 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. Part of this work has been accepted on Estuarine, Coastal and Shelf Science, with a manuscript on the numerical aspects in review on Ocean Modelling.