A hybrid method for modelling two dimensional non-breaking and breaking waves

This is the first paper to present a hybrid method coupling a Improved Meshless Local Petrov Galerkin method with Rankine source solution (IMLPG_R) based on the Navier Stokes (NS) equations, with a finite element method (FEM) based on the fully nonlinear potential flow theory (FNPT) in order to efficiently simulate the violent waves and their interaction with marine structures. The two models are strongly coupled in space and time domains using a moving overlapping zone, wherein the information from both the solvers is exchanged. In the time domain, the Runge-Kutta 2nd order method is nested with a predictor-corrector scheme. In the space domain, numerical techniques including ‘Feeding Particles’ and two-layer particle interpolation with relaxation coefficients are introduced to achieve the robust coupling of the two models. The properties and behaviours of the new hybrid model are tested by modelling a regular wave, solitary wave and Cnoidal wave including breaking and overtopping. It is validated by comparing the results of the method with analytical solutions, results from other methods and experimental data. The paper demonstrates that the method can produce satisfactory results but uses much less computational time compared with a method based on the full NS model

Contravariant Boussinesq equations for the simulation of wave transformation, breaking and run-up

We propose an integral form of the fully non-linear Boussinesq equations in contravariant formulation, in which Christoffel symbols are avoided, in order to simulate wave transformation phenomena, wave breaking and near shore currents in computational domains representing the complex morphology of real coastal regions. The motion equations retain the term related to the approximation to the second order of the vertical vorticity. A new Upwind Weighted Essentially Non-Oscillatory scheme for the solution of the fully non- linear Boussinesq equations on generalised curvilinear coordinate systems is proposed. The equations are rearranged in order to solve them by a high resolution hybrid finite volume–finite difference scheme. The conservative part of the above-mentioned equations, consisting of the convective terms and the terms related to the free surface elevation, is discretised by a high-order shock- capturing finite volume scheme; dispersive terms and the term related to the approximation to the second order of the vertical vorticity are discretised by a cell-centred finite difference scheme. The shock-capturing method makes it possible to intrinsically model the wave breaking, therefore no additional terms are needed to take into account the breaking related energy dissipation in the surf zone. The model is applied on a real case regarding the simulation of wave fields and nearshore currents in the coastal region opposite Pescara harbour (Italy)

Finite volume schemes for dispersive wave propagation and runup

Finite volume schemes are commonly used to construct approximate solutions to conservation laws. In this study we extend the framework of the finite volume methods to dispersive water wave models, in particular to Boussinesq type systems. We focus mainly on the application of the method to bidirectional nonlinear, dispersive wave propagation in one space dimension. Special emphasis is given to important nonlinear phenomena such as solitary waves interactions, dispersive shock wave formation and the runup of breaking and non-breaking long waves.Comment: 41 pafes, 20 figures. Other author's papers can be downloaded at http://www.lama.univ-savoie.fr/~dutykh

A discontinuous Galerkin method for a new class of Green-Naghdi equations on simplicial unstructured meshes

In this paper, we introduce a discontinuous Finite Element formulation on simplicial unstructured meshes for the study of free surface flows based on the fully nonlinear and weakly dispersive Green-Naghdi equations. Working with a new class of asymptotically equivalent equations, which have a simplified analytical structure, we consider a decoupling strategy: we approximate the solutions of the classical shallow water equations supplemented with a source term globally accounting for the non-hydrostatic effects and we show that this source term can be computed through the resolution of scalar elliptic second-order sub-problems. The assets of the proposed discrete formulation are: (i) the handling of arbitrary unstructured simplicial meshes, (ii) an arbitrary order of approximation in space, (iii) the exact preservation of the motionless steady states, (iv) the preservation of the water height positivity, (v) a simple way to enhance any numerical code based on the nonlinear shallow water equations. The resulting numerical model is validated through several benchmarks involving nonlinear wave transformations and run-up over complex topographies

Bottom changes in coastal areas with complex shorelines

A model for the sea-bottom change simulations in coastal areas with complex shorelines is proposed. In deep and intermediate water depths, the hydrodynamic quantities are calculated by numerically integrating the contravariant Boussinesq equations, devoid of Christoffel symbols. In the surf zone, the propagation of the breaking waves is simulated by the nonlinear shallow water equations. The momentum equation is solved inside the turbulent boundary layer in order to calculate intrawave hydrodynamic quantities. An integral formulation for the contravariant suspended sediment advection-diffusion equation is proposed and used for the sea-bottom dynamic simulations. The proposed model is applied to the real case study of Pescara harbor (in Italy)

A splitting approach for the fully nonlinear and weakly dispersive Green-Naghdi model

The fully nonlinear and weakly dispersive Green-Naghdi model for shallow water waves of large amplitude is studied. The original model is first recast under a new formulation more suitable for numerical resolution. An hybrid finite volume and finite difference splitting approach is then proposed. The hyperbolic part of the equations is handled with a high-order finite volume scheme allowing for breaking waves and dry areas. The dispersive part is treated with a classical finite difference approach. Extensive numerical validations are then performed in one horizontal dimension, relying both on analytical solutions and experimental data. The results show that our approach gives a good account of all the processes of wave transformation in coastal areas: shoaling, wave breaking and run-up

Upwind Stabilized Finite Element Modelling of Non-hydrostatic Wave Breaking and Run-up

In the following report a new methodology is presented to model the propagation, wave breaking and run-up of waves in coastal zones. We represent the different coastal phenomena through the coupling of non-linear shallow water equations with the extended Boussinesq equations of Madsen and Sørensen. Each of the involved equations has a major role in describing a particular physical behaviour of the wave: the latter equations permit to model the propagation, while the non-linear shallow water ones lead waves to locally converge into discontinuities. We start from the third-order stabilized finite element scheme for the Boussinesq equations, developed in a previous scientific work (Ricchiuto and Filippini, J.Comput.Phys. 2014) and develop a non-linear variant, and detach the dispersive from the shallow water terms. A shock-capturing technique based on local non-linear mass lumping that permits in the shallow water regions to degrade locally the scheme to a first-order one across bores (shocks) and dry fronts is proposed. As for the detection of the breaking fronts, the shallow water areas, this involves physics based breaking criteria. We present different definitions of the breaking criterion, including a local implementation of the convective criterion of (Bjørkavåg and H. Kalisch, Phys.Letters A 2011), and the hybrid models of (Kazolea et. al, J.Comput.Phys. 2014), and (Tonelli and Petti, J.Hydr.Res. 2011). The behavior of different breaking criteria is investigated on several cases for which experimental data are available.On décrit une approche pour la simulation de la propagation et déferlement des vagues en proche cote basée sur la couplage entre les équations de Boussinesq améliorées de Madsen and Sorensen, pour la propagation, et les équations Shallow Water, pour le déferlement et le runup. La contruction de ce modele hybride passe d'abord la proposition une variante non-linéaire du schéma élément finis stabilisé de (Ricchiuto and Filippini, J.Comput.Phys. 2014) capable de résoudre les chocs de maniere monotone. Cela est obtenu par un operateur locale de condensation de la matrice de masse qui réduit le schéma de (Ricchiuto and Filippini, J.Comput.Phys. 2014) au schéma de Roe classique. Le couplage entre le modèle Boussinesq et Shallow Water est en suite étudié. On considere différents critères physiques de détection de fronts déferlants. En particulier, on présente une implémentation numérique locale du critère convectif de (Bjorkavag and H. Kalisch, Phys.Letters A, 2011), qui est comparée au critères proposés dans (Kazolea et. al, J.Comput.Phys., 2014) et (Tonelli and Petti, J.Hydr.Res. 2011). Le modèle obtenu est validé sur des nombreux benchmarks avec données expérimentales