https://hal.inria.fr/hal-01698300/document

On wave breaking for Boussinesq-type models

A flexible genuinely nonlinear approach for nonlinear wave propagation, breaking and run-up

The accurate numerical simulation of wave disturbance within harbours requires consideration of both nonlinear and dispersive wave processes in order to capture such physical effects as wave refraction and diffraction, and nonlinear wave interactions such as the generation of harmonic waves. The Boussinesq equations are the simplest class of mathematical model that contain all these effects in a variable depth, shallow water environment. There are a variety of Boussinesq-type mathematical models and it is necessary to compare and contrast them both for their limitations with respect to the physical parameters of the problem and also for their ease of application as part of a suitable numerical model. It is decided here to consider a set of extended Boussinesq equations which provide an accurate model of the wave processes over a greater range of depths than the classical Boussinesq mathematical model.

A method-of-lines numerical algorithm is proposed for these problems, combining a finite element spatial discretisation with existing, adaptive order, adaptive step size time integration software. Two simpler one-dimensional, nonlinear, dispersive wave models; the Korteweg-de Vries equation and Regularised Long Wave equation, are used in the initial development of the numerical methods. It is shown that within the shallow water framework a linear finite element method is sufficiently accurate for these problems.

This numerical method is then applied to the one-dimensional extended Boussinesq equations. It is shown how the previously developed method can be directly used and that it is of similar accuracy to a previously published finite difference method. Initial conditions and boundary conditions are described in detail taking into account physical, mathematical and computational considerations. A new formulation of internal wave generation is developed which allows reflected waves to pass through the wave generation region. The performance of the numerical model is demonstrated by comparison against theoretical results, a previously published finite difference model and experimental results.

The two-dimensional extended Boussinesq equation system is rewritten in a form suitable for the application of a linear triangular finite element spatial discretisation. The formulation of appropriate initial and boundary conditions in combination with the application of the time integration software to this equation system is considered in detail. The performance of the numerical method is tested by comparison with experimental data and the suitability of the model for harbour design is investigated by simulation of a realistic harbour geometry and wave conditions.

A Numerical Method forExtended BoussinesqShallow-Water Wave Equations

https://hal.inria.fr/hal-01053036/document

On the nonlinear behaviour of Boussinesq type models: Amplitude-velocity vs amplitude-flux forms

Modelling the impact of climate change on harbour operability: The Barcelona port case study

In this paper, we present a long time existence theory for a new enhanced Boussinesq-type system with constant bathymetry written in a conservative form. We also prove the existence of solitary wave for a large class of asymptotic models, including Beji-Nadaoka, Madsen-Sorensen and Nwogu equations. Furthermore, we give a procedure to calculate numerically these particular solutions and we present some effective computations.

On the existence of solitary waves for Boussinesq type equations and a new conservative model.

In this paper, we investigate the nonlinear properties of Boussinesq models. In particular, we consider the wave shoaling obtained in physical regimes which go from linear to weakly nonlinear, to the wave breaking limit. For a given asymptotic accuracy in terms of dispersion and nonlinearity, we consider two families of models: the first depending on derivatives of the velocity, the second on derivatives of the volume flux. We show that, while linear dispersion and linear shoaling characteristics are strongly dependent on the type of dispersive terms introduced, when approaching the nonlinear regime the only influencing factor is whether the model is in amplitude-velocity of amplitude-flux form. We investigate these two alternative formulations of several known models, and propose a new model with a compact differential form, and the same linear characteristics of the model of Nwogu. The nonlinear shoaling properties of the models are investigated numerically showing that inside one given family, all the models have almost identical behaviour.

On Nonlinear Shoaling Properties of Enhanced Boussinesq Models

In this paper, we discuss a new systematic method to obtain discrete asymptotic numerical models for incompressible free-surface flows. The method consists of first discretizing the Euler equations in the horizontal direction, keeping both the vertical and time derivatives continuous, and then performing an asymptotic analysis on the resulting system. The asymptotics involve the ratios wave amplitude over depth, denoted by $\varepsilon$, and depth over wavelength, denoted by $\sigma$. For simplicity, in this paper we only consider the weakly nonlinear scaling in which both $\sigma^4$ and $\varepsilon\sigma^2$ are very small and of the same order. We investigate the properties of the fully discrete Boussinesq model obtained by neglecting terms proportional to these quantities. Our study reveals that if the interaction between terms arising from the discretization and from the PDE is properly accounted for, the resulting discrete system has improved linear dispersion and shoaling approximations w.r.t. the d...

/pdf/discrete-asymptotic-equations-for-long-wave-propagation-389wya507x.pdf

Discrete Asymptotic Equations for Long Wave Propagation

Ce document se concentre sur la modelisation de l'ecoulement des vagues sur le littoral, ainsi que la derivation des equations. Pour cela, on part des equations d'Euler incompressibles et irrotationnelles qui regissent cet ecoulement. On souhaite les simplifier suffisamment pour y adapter un schema numerique precis et performant, tout en preservant les caracteristiques de dispersion lineaire du modele de base (modele d'Airy, detaille au debut) pour des profondeurs de l'ordre de la moitie de la longueur d'onde des vagues. Apres les premieres simplifications, on obtient le modele de "Shallow-water", dont les caracteristiques de dispersion ne conviennent pas. On cherche donc a affiner la derivation des equations, on obtient alors les equations de Peregrine et de Green-Naghdi. Celles-ci ameliorent la profondeur limite de validite. On cherche alors a les ameliorer en introduisant un parametre libre, ce qui nous amene aux equations de Beji-Nadaoka (ou Madsen et Sorensen) puis aux equations de Nwogu. La fin de ce document se concentre sur une caracteristique de dispersion lineaire plus restrictive : le gradient de shoaling.

https://hal.inria.fr/hal-00860438/document

Stevan Bellec

Papers

On the nonlinear behaviour of Boussinesq type models: Amplitude-velocity vs amplitude-flux forms

On the existence of solitary waves for Boussinesq type equations and a new conservative model.

On Nonlinear Shoaling Properties of Enhanced Boussinesq Models

Discrete Asymptotic Equations for Long Wave Propagation

Sur des Modèles Asymptotiques en Océanographie