We develop well-posedness theory and analytical and numerical solution techniques for Boussinesq-type equations. Firstly, we consider the Cauchy problem for a generalized Boussinesq equation. We show that under suitable conditions, a global solution for this problem exists. In addition, we derive sufficient conditions for solution blow-up in finite time. / Secondly, a generalized Jacobi/exponential expansion method for finding exact solutions of non-linear partial differential equations is discussed. We use the proposed expansion method to construct many new, previously undiscovered exact solutions for the Boussinesq and modified Korteweg-de Vries equations. We also apply it to the shallow water long wave approximate equations. New solutions are deduced for this system of partial differential equations. / Finally, we develop and validate a numerical procedure for solving a class of initial boundary value problems for the improved Boussinesq equation. The finite element method with linear B-spline basis functions is used to discretize the equation in space and derive a second order system involving only ordinary derivatives. It is shown that the coefficient matrix for the second order term in this system is invertible. Consequently, for the first time, the initial boundary value problem can be reduced to an explicit initial value problem, which can be solved using many accurate numerical methods. Various examples are presented to validate this technique and demonstrate its capacity to simulate wave splitting, wave interaction and blow-up behavior.

Nous nous étudions ici le problème d'Euler avec surface libre sur un fond non plat et dans un régime fortement non linéaire où l'hypothèse de faible amplitude de l'équation de KdV n'est pas vérifiée. On sait que, pour un tel régime, une généralisation de l'équation de KdV peut être dérivée et justifiée lorsque le fond est plat. Nous généralisons ici ces résultats en proposant une nouvelle classe d'équations prenant en compte des topographies variables. Nous démontrons également que ces nouveaux modèles sont bien posés. Nous les étudions aussi numériquement. Ensuite, nous améliorons quelques résultats sur l'existence des équations de Green-Naghdi (GN) dans le cas 1D. Dans le cas de 2D, nous dérivons et étudions un nouveau système de la même précision que les équations de GN usuelles, mais avec un meilleur comportement mathématique. / We study here the water-waves problem for uneven bottoms in a highly nonlinear regime where the small amplitude assumption of the KdV equation is enforced. It is known, that for such regimes, a generalization of the KdV equation can be derived and justified when the bottom is flat. We generalize here this result with a new class of equations taking into account variable bottom topographies. We also demonstrate that these new models are well-posed. We then proceed to study them numerically and compare their behavior with the Boussinesq equations over uneven bottoms. Regimes with stronger nonlinearities than the KdV/Boussinesq regime are then investigated. In particular, a variable coefficient generalization of a Camassa-Holm type equation is derived and justified. Wealso study the Green-Naghdi equations that are commonly used in coastal oceanography todescribe the propagation of large amplitude surface waves. We improve previous results on the well posedness of these equations in the case of one dimensional surface waves. In the $2D$ case, we derive and study a new system of the same accuracy as the standard $2D $ Green-Naghdi equations, but with better mathematical behavior.

It is often desirable to study simulated turbulent flows at steady state even if the flow has no inherent source of turbulence kinetic energy. Doing so requires a numerical forcing scheme and various methods have been studied extensively for turbulence that is isotropic and homogeneous in three dimensions. A review of these existing schemes is used to form a framework for more general forcing methods. In this framework, the problem of developing a forcing scheme in Fourier space is abstracted into the two problems of (1) prescribing the spectrum of the input power and (2) specifying a force that has the desired characteristics and that adds energy to the flow with the correct spectrum. The framework is used to construct three forcing schemes for horizontally homogeneous and isotropic, vertically stratified turbulence. These schemes are implemented in large-eddy simulations and their characteristics analyzed. Which method is “best” depends on the purpose of the simulations, but the framework for specifying forcing schemes enables a systematic approach for identifying a method appropriate for a particular application.

No description available.

A significant proportion of the world's population and physical assets are located in low lying coastal zones. Accurate prediction of wave induced run-up and overtopping of sea defences are important in defining the extent and severity of wave action, and in assessing risk to people and property from severe storms and tsunamis. This thesis describes a one-dimensional numerical model based on the Boussinesq equations of Madsen and Sorensen (1992) and the non-linear shallow water equations. The model is suitable for simulating propagation of weakly non-linear and weakly dispersive waves from intermediate to zero depth, such that any inundation and/or overtopping caused by the incoming waves is also calculated as part of the simulation. Wave breaking is approximated by locally switching to the non-linear shallow water equations, which can model broken waves as bores. A piston paddle wavemaker is incorporated into the model for complete reproduction of laboratory experiments. A domain mapping technique is used in the vicinity of the paddle to transform a time-varying domain into a fixed domain, so that the governing equations can be more readily solved. First, various aspects of the numerical model are verified against known analytical and newly derived semi-analytical solutions. The complete model is then validated with laboratory measurements of run-up and overtopping involving solitary waves. NewWave focused wave groups, which give the expected shape of extreme wave events in a linear random sea, are used for further validation. Simulations of experiments of wave group run-up on a plane beach yield very good agreement with the measured run-up distances and free surface time series. Wave-by-wave overtopping induced by focused wave groups is also successfully simulated with the model, with satisfactory agreement between the experimental and the predicted overtopping volumes. Repeated simulations, now driven by second order paddle displacement signals, give insight into second order error waves spuriously generated by using paddle signals derived from linear theory. Separation of harmonics reveals that the long error wave is significantly affecting the wave group shape and leading to enhanced runu-up distances and overtopping volumes. An extensive parameter study is carried out using the numerical model investigating the influence on wave group run-up of linear wave amplitude at focus, linear focus location, and wave group phase at focus. For a given amplitude, both the phase and the focus location significantly affect the wave group run-up. It is also found that the peak optimised run-up increases with the wave amplitude, but wave breaking becomes an inhibiting factor for larger waves. This methodology is proposed for extreme storm wave induced run-up analysis.

No trabalho desenvolvido estudaram-se as forças que atuam em uma partícula quando esta se movimenta em escoamentos, com intuito de obter uma metodologia capaz de representar o movimento de uma partícula em um escoamento. A equação do movimento da partícula foi integrada numericamente considerando os termos de massa aparente, arrasto estacionário, arrasto não estacionário (forças de Boussinesq/Basset) e forças de sustentação; efeito Magnus e efeito Saffman. O método dos volumes finitos foi utilizado para simulação do escoamento. Na análise das forças utilizamos tanto experimentos quanto simulações numéricas (FLUENT) para avaliar e aumentar a validade dos modelos apresentados na revisão bibliográfica. O FLUENT foi validado para obtenção do coeficiente de arrasto estacionário e sustentação devido ao efeito Magnus. Palavras-chaves: Efeito Magnus, efeito Saffman, força de Bousinesq/Basset, movimento de partículas e solução numérica. / In the developed work was studied the forces which act on a particle when these is a moving inside of a flow, in order to find out a methodology which is able to represent the particle dynamics on a flow. The equation of particle motion was integrated with a numerical approach taking in account the apparent mass, static drag, dynamic drag (history term; Boussinesq/Basset force) and lift force; Magnus effect and Saffman effect. The finite volume method was used to simulate the flow. In the force analyses we used experimental and numerical simulation (FLUENT) to evaluate and extend the models shown on the review. FLUENT was validated to determine the static drag coefficient and lift coefficient due to Magnus effect.

No trabalho desenvolvido estudaram-se as forças que atuam em uma partícula quando esta se movimenta em escoamentos, com intuito de obter uma metodologia capaz de representar o movimento de uma partícula em um escoamento. A equação do movimento da partícula foi integrada numericamente considerando os termos de massa aparente, arrasto estacionário, arrasto não estacionário (forças de Boussinesq/Basset) e forças de sustentação; efeito Magnus e efeito Saffman. O método dos volumes finitos foi utilizado para simulação do escoamento. Na análise das forças utilizamos tanto experimentos quanto simulações numéricas (FLUENT) para avaliar e aumentar a validade dos modelos apresentados na revisão bibliográfica. O FLUENT foi validado para obtenção do coeficiente de arrasto estacionário e sustentação devido ao efeito Magnus. Palavras-chaves: Efeito Magnus, efeito Saffman, força de Bousinesq/Basset, movimento de partículas e solução numérica. / In the developed work was studied the forces which act on a particle when these is a moving inside of a flow, in order to find out a methodology which is able to represent the particle dynamics on a flow. The equation of particle motion was integrated with a numerical approach taking in account the apparent mass, static drag, dynamic drag (history term; Boussinesq/Basset force) and lift force; Magnus effect and Saffman effect. The finite volume method was used to simulate the flow. In the force analyses we used experimental and numerical simulation (FLUENT) to evaluate and extend the models shown on the review. FLUENT was validated to determine the static drag coefficient and lift coefficient due to Magnus effect.

The prediction of coastal groundwater movement is necessary in coastal management. However, the study in this field is still a great challenge due to the involvement of tidal-groundwater interactions and the phenomena of hydrodynamic dispersion between salt-fresh water in the coastal region. To date, numerous theories for groundwater dynamic have been made available in analytical, numerical and also experimental forms. Nevertheless, most of them are based on the zeroth-order shallow flow, i.e. Boussinesq approximation. Two main components for coastal unconfined aquifer have been completed in this Thesis: the vertical beach model and the sloping beach model. Both solutions are solved in closed-form up to higher order with shallow water parameter ([epsilon]) and tidal amplitude parameter ([alpha]). The vertical beach solution contributes to the higher-order tidal fluctuations while the sloping beach model overcomes the shortcomings in the existing solutions. From this study, higher-order components are found to be significant especially for larger value of [alpha] and [epsilon]. Other parameters such as hydraulic conductivity (K) and the thickness of aquifer (D) also affect the water table fluctuations. The new sloping solution demonstrated the significant influence of beach slope ([beta]) on the water table fluctuations. A comprehensive comparison between previous solution and the present sloping solution have been performed mathematically and numerically and the present solution has been demonstrated to provide a better prediction

Le sujet principal de cette thèse est l'étude de la contrôlabilité exacte de deux équations dispersives, l'équation de Korteweg-de Vries et la "bonne" équation de Boussinesq. En ce qui concerne l'équation de Korteweg-de Vrie, on étend un résultat de Rosier en montrant la contrôlabilité exacte en tout temps de l'équation non linéaire autour d'une solution stationnaire proche de zéro mais non nulle, ce pour des longueurs de domaine spatial critiques. Cette démonstration utilise en particulier la méthode d'unicité hilbertienne couplée avec la méthode des multiplicateurs et un théorème de point fixe. Ensuite, nous étudions le problème de la contrôlabilité exacte de l'équation de Boussinesq pour deux contrôles différents. On utilise également la méthode d'unicité hilbertienne pour ces problèmes en appliquant une inégalité de Ingham. On obtient ainsi un résultat de contrôlabilité exacte pour des temps arbitrairement petits. Nous implémentons ensuite cette méthode de facon numérique pour l'équation de Boussinesq avec un contrôle portant sur la dérivée seconde a droite, tant sur le problème linéaire que non linéaire.

No description available.