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The present work extends the semi-integrated method proposed by Antuono and Brocchini [“Beyond Boussinesq-type equations: Semi-integrated models for coastal dynamics,” Phys. Fluids 25(1), 016603 (2013)], which comprises a subset of depth-averaged equations (similar to Boussinesq-like models) and a Poisson equation that accounts for vertical dynamics. Here, the subset of depth-averaged equations has been reshaped in a conservative-like form and both the Poisson equation formulations proposed by Antuono and Brocchini [“Beyond Boussinesq-type equations: Semi-integrated models for coastal dynamics,” Phys. Fluids 25(1), 016603 (2013)] are investigated: the former uses the vertical velocity component (formulation A) and the latter a specific depth semi-averaged variable, ϒ (formulation B). Our analyses reveal that formulation A is prone to instabilities as wave nonlinearity increases. On the contrary, formulation B allows an accurate, robust numerical implementation. Test cases derived from the scientific literature on Boussinesq-type models—i.e., solitary and Stokes wave analytical solutions for linear dispersion and nonlinear evolution and experimental data for shoaling properties—are used to assess the proposed solution strategy. It is found that the present method gives reliable predictions of wave propagation in shallow to intermediate waters, in terms of both semi-averaged variables and conservation properties. This is the authors' submitted pre-refereed manuscript to the article. The following article appeared in Physics of Fluids and may be found at http://aip.scitation.org/doi/10.1063/1.4984079

Ces dernières décennies, une attention particulière a été portée sur la modélisation mathématique et la simulation numérique de la propagation de vagues en environnements côtiers. Une description physiquement correcte des phénomènes à grande échelle, qui apparaissent dans les régions d'eau peu profonde, doit prendre en compte de forts effets non-linéaires et dispersifs, ainsi que l'interaction avec des bathymétries complexes. Dans un premier temps, une étude du comportement en régime non linéaire de différents modèles de type Boussinesq est proposée, démontrant l'avantage d'utiliser des modèles fortement non-linéaires par rapport à des modèles faiblement non-linéaires et faiblement dispersifs (couramment utilisés). Ensuite, une nouvelle approche flexible pour résoudre les équations fortement non-linéaires et faiblement dispersives de Green-Naghdi est présentée. Cette stratégie permet d'améliorer un code "shallow water" existant par le simple ajout d'un terme algébrique dans l'équation du moment et est particulièrement adapté à l'utilisation de techniques hybrides pour le déferlement des vagues. De plus, la première discrétisation des équations de Green-Naghdi sur maillage non structuré est proposée via des schémas hybrides Volume Fini/Élément Fini. Finalement, les modèles et méthodes développés dans la thèse sont appliqués à l'étude du problème physique de la formation du mascaret dans des estuaires convergents et alluviaux. Cela a amené à la première caractérisation d'estuaire naturel en terme d'apparition de mascaret.; Over the last decades, there has been considerable attention in the accurate mathematical modeling and numerical simulations of free surface wave propagation in near-shore environments. A physical correct description of the large scale phenomena, which take place in the shallow water region, must account for strong nonlinear and dispersive effects, along with the interaction with complex topographies. First, a study on the behavior in nonlinear regime of different Boussinesq-type models is proposed, showing the advantage of using fully-nonlinear models with respect to weakly-nonlinear and weakly dispersive models (commonly employed). Secondly, a new flexible strategy for solving the fully-nonlinear and weakly-dispersive Green-Naghdi equations is presented, which allows to enhance an existing shallow water code by simply adding an algebraic term to the momentum balance and is particularly adapted for the use of hybrid techniques for wave breaking. Moreover, the first discretization of the Green-Naghdi equations on unstructured meshes is proposed via hybrid finite volume/ finite element schemes. Finally, the models and the methods developed in the thesis are deployed to study the physical problem of bore formation in convergent alluvial estuary, providing the first characterization of natural estuaries in terms of bore inception.

The present work completes the validation of the depth-semi-averaged model defined in Antuono et al. [“A depth semi-averaged model for coastal dynamics,” Phys. Fluids 29, 056603 (2017)] providing a numerical implementation in three spatial dimensions. In particular, the model is aimed at describing a wide variety of phenomena typical of the coastal region, as for example nonlinear interaction between waves and the bottom bathymetry (diffraction/refraction phenomena), dispersion effects, and beach inundation. To this purpose, the proposed scheme is tested against benchmark experimental data available in the literature for wave propagation problems in both intermediate and shallow water conditions. A final simulation is carried out by considering a tsunami wave running over a prototype bathymetry resembling a bay-promontory coastline. Different widths of the bay have been considered, describing their influence on the run-up dynamics. This is the authors’ accepted and refereed manuscript to the article. This article may be downloaded for personal use only. Any other use requires prior permission of the author and AIP Publishing. The following article appeared in Physics of Fluids 31, 026601 (2019) and may be found at https://doi.org/10.1063/1.5080307.