Implementation and Evaluation of Breaking Detection Criteria for a Hybrid Boussinesq Model
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Implementation and Evaluation of Breaking Detection Criteria for a Hybrid Boussinesq Model Paola Bacigaluppi1
· Mario Ricchiuto2
· Philippe Bonneton3
Received: 2 February 2019 / Accepted: 12 November 2019 © Springer Nature Switzerland AG 2019
Abstract The aim of the present work has been to develop a model able to represent the propagation and transformation of waves in nearshore areas. The focus is on the phenomena of wave breaking, shoaling, and run-up. These different phenomena are represented through a hybrid approach obtained by the coupling of non-linear Shallow Water equations with the extended Boussinesq equations of Madsen and Sørensen. The novelty is the switch tool between the two modelling equations: a critical free surface Froude criterion. This is based on a physically meaningful new approach to detect wave breaking, which corresponds to the steepening of the wave’s crest which turns into a roller. To allow for an appropriate discretization of both types of equations, we consider a finite element Upwind Petrov Galerkin method with a novel limiting strategy that guarantees the preservation of smooth waves as well as the monotonicity of the results in presence of discontinuities. We provide a detailed discussion of the implementation of the newly proposed detection method, as well as of two other well-known criteria which are used for comparison. An extensive benchmarking on several problems involving different wave phenomena and breaking conditions allows to show the robustness of the numerical method proposed, as well as to assess the advantages and limitations of the different detection methods.
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Paola Bacigaluppi [email protected] Mario Ricchiuto [email protected] Philippe Bonneton [email protected]
1
Institute of Mathematics, University of Zurich, Winterthurerstrasse 190, 8057 Zürich, Switzerland
2
Team CARDAMOM, Inria Bordeaux Sud-Ouest, 200 avenue de la Vieille Tour, 33405 Talence Cedex, France
3
University of Bordeaux, CNRS, UMR 5805 EPOC, Allée Geoffroy Saint-Hilaire, 33615 Pessac, France
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P. Bacigaluppi et al.
Keywords Wave breaking detection · Convective criterion · Critical free surface froude criterion · Hybrid model · Boussinesq and Non-linear Shallow Water equations · Non-linear SUPG FEM
1 Introduction Wave breaking is a fundamental phenomenon, especially in the nearshore region. It drives wave set-up and set-down, wave run-up, nearshore circulations (e.g. longshore and rip currents), sediment transport, forces on structures, and many other effects. At large scales, an important effect of wave breaking is the energy dissipation, which is the result of many local hydrodynamic phenomena, ultimately leading to dissipation due to small-scale viscous effects. In this paper we focus on the wave breaking closure when wave propagation is performed by means of depth averaged Boussinesq models. Among the main families of closures found in literature we mention the eddy viscosity formulations [14,25,38,42,52], the breaking roller methods [8,50],
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