Dissertationen zum Thema „Boussinesq-Type“

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.

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.

An investigation into the numerical and physical behaviour of a hybrid finite-volume finite-difference Boussinesq-type model, using a rotational surface roller approach in the surf-zone is presented. The relevant theory for the required development of a numerical model implementing this technique is outlined. The proposed method looks to achieve a more physically realistic description of the hydrodynamics by considering the rotational nature of the highly turbulent flow found during wave breaking. This involves a semi-analytical solution to the vorticity transport equation and provides a mechanism by which energy is dissipated. Resolving vorticity within the flow also allows vertical profiles of the horizontal velocity to be constructed, offering valuable detail that is otherwise unavailable when using equivalent irrotational Boussinesq-type models. By obtaining additional information about the structure of the flow, other quantities can be determined, such as the undertow, which has a key role in morphodynamic processes occurring in this region. These benefits are combined with a finite-volume finite-difference scheme, which yields improvements in stability and possesses inherent shock-capturing capabilities. The ability of the model to replicate laboratory observations is verified, and identified shortcomings are explained in the context of the numerical procedure and the assumptions made during the derivation of the governing equations. Although the weak nonlinearity of the Boussinesq-type equations means the shoaling characteristics of the model do not accurately reflect those found experimentally, the adopted formulation of the finite-volume scheme is shown to prevent the inclusion of the necessary higher-order derivatives which exist in a fully-nonlinear formulation. In order to establish a realistic dissipation mechanism, it is vital that the extent of any misleading numerical artefacts are recognised and their effects alleviated. This study explores a range of physical attributes predicted by the present model and discusses the numerical features of the scheme, evaluating how these influence the results.

Cette thèse s'intéresse au couplage de modèles hydrauliques en zone côtière, à phase résolue, couramment utilisés pour l'étude de la propagation des vagues près du rivage. Malgré de nombreux modèles et des exemples de couplage existants, il y a eu un manque significatif de consensus concernant les artefacts et les problèmes induits par ces stratégies, ainsi qu'une compréhension vague de la façon de les analyser et de les comparer. Pour aborder ce problème, cette recherche adopte une approche de décomposition de domaine, ancrée dans le principe que les modèles de vagues 3D (par exemple, Euler ou Navier-Stokes) servent de solution de référence.Structurée en deux parties, la thèse propose d'abord de nouveaux modèles et les évalue à travers des expériences numériques, identifiant des hypothèses spécifiques sur leur précision et leurs limites. Par la suite, un cadre théorique est développé pour élucider ces découvertes, en utilisant le modèle couplé unidirectionnel comme une référence intermédiaire pour distinguer les effets attendus et inattendus et catégoriser les erreurs par rapport à la solution 3D.L'erreur totale est divisée en trois parties : l'erreur de couplage, l'erreur du modèle de Cauchy, et l'erreur du modèle de demi-droite, et ces concepts sont appliqués au couplage linéaire des modèles de Saint-Venant et de Boussinesq en utilisant le modèle dit 'hybride'. L'analyse confirme que l'erreur de couplage prend en compte les réflexions aux interfaces et varie selon la direction de la propagation. De plus, grâce au choix du modèle unidirectionnel comme référence intermédiaire, cette analyse prouve plusieurs propriétés importantes telles que le caractère bien posé et la taille asymptotique des réflexions. En outre, la thèse aborde également le caractère faiblement bien posé du problème de Cauchy pour le modèle B et ses implications pour les solutions dépendantes du maillage qui ont été signalées. Comme produit dérivé, un nouveau résultat pour le problème de demi-droite du modèle linéaire B est obtenu, pour une classe plus générale de données aux limites, incluant une description de la couche limite dispersive, qui n'avait pas encore été abordée dans la littérature.La définition pragmatique proposée de l'erreur de couplage s'aligne avec et étend les notions existantes de la littérature. Elle peut être facilement appliquée à d'autres modèles BT, équations discrètes, cas linéaires et non linéaires (au moins numériquement), ainsi qu'à d'autres techniques de couplage, tous discutés dans le travail en perspective
This thesis investigates the coupling of coastal phase-resolving water wave models, commonly employed in the study of nearshore wave propagation. Despite numerous models and the existing coupling examples, there has been a significant lack of consensus concerning the artifacts and issues induced by these strategies, as well as a vague understanding of how to analyze and compare them. To tackle this problem, this research adopts a domain decomposition approach, anchored in the principle that 3D water wave models (e.g., Euler or Navier-Stokes) serve as the ideal reference solution.Structured in two parts, the thesis first proposes new models and evaluates them through numerical experiments, identifying specific hypotheses about their accuracy and limitations. Subsequently, a theoretical framework is developed to prove these hypotheses mathematically, utilizing the one-way coupled model as an intermediate reference to distinguish between expected and unexpected effects and categorize errors relative to the 3D solution.The total error is split in three parts—coupling error, Cauchy-model error, and half-line-model error—and these concepts are applied to the linear coupling of Saint-Venant and Boussinesq models using the so called 'hybrid' model. The analysis confirms that the coupling error accounts for wave reflections at the interfaces, and varies with the direction of propagation. Moreover, thanks to the choice of the one-way model as the intermediate reference solution, this analysis proves several important properties such as the well-posedness and the asymptotic size of the reflections. Additionally, the thesis also addresses the weak-wellposedness of the Cauchy problem for the B model and its implications for mesh-dependent solutions that have been reported. As a byproduct, a new result for the half-line problem of the linear B model is obtained for a more general class of boundary data, including a description of the dispersive boundary layer, which had not been addressed in the literature yet.The proposed pragmatic definition of coupling error aligns with and extends existing notions from the literature. It can be readily applied to other BT models, discrete equations, linear and nonlinear cases (at least numerically), as well as other coupling techniques, all of which are discussed in the perspective work

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Coordenação de Aperfeiçoamento de Pessoal de Nível Superior - CAPES
The aim of this thesis is to present some controllability results for some fluid mechanic models. More precisely, we will prove the existence of controls that steer the solution of our system from a prescribed initial state to a desired final state at a given positive time. The two first Chapters deal with the controllability of the Burgers-α and Leray-α models. The Leray-α model is a regularized variant of the Navier-Stokes system (α is a small positive parameter), that can also be viewed as a model for turbulent flows; the Burgers-α model can be viewed as a related toy model of Leray-α. We prove that the Leray-α and Burgers-α models are locally null controllable, with controls uniformly bounded in α. We also prove that, if the initial data are sufficiently small, the pair state-control (that steers the solution to zero) for the Leray-α system (resp. the Burgers-α system) converges as α → 0+ to a pair state-control(that steers the solution to zero) for the Navier-Stokes equations (resp. the Burgers equation). The third Chapter is devoted to the boundary controllability of inviscid incompressible fluids for which thermal effects are important. They will be modeled through the so called Boussinesq approximation. In the zero heat diffusion case, by adapting and extending some ideas from J.-M. Coron [14] and O. Glass [45], we establish the simultaneous global exact controllability of the velocity field and the temperature for 2D and 3D flows. When the heat diffusion coefficient is positive, we present some additional results concerning exact controllability for the velocity field and local null controllability of the temperature. In the last Chapter, we prove the local exact controllability to the trajectories for a coupled system of the Boussinesq kind, with a reduced number of controls. In the state system, the unknowns are: the velocity field and pressure of the fluid (y, p), the temperature θ and an additional variable c that can be viewed as the concentration of a contaminant solute. We prove several results, that essentially show that it is sufficient to act locally in space on the equations satisfied by θ and c.
O objetivo desta tese é apresentar alguns resultados controlabilidade para alguns modelos da mecânica dos fluidos. Mais precisamente, provaremos a existência de controles que conduzem a solução do nosso sistema de um estado inicial prescrito à um estado final desejado em um tempo positivo dado. Os dois primeiros Capítulos preocupam-se com a controlabilidade dos modelos de Burgers-α e Leray-α. O modelo de Leray-α é uma variante regularizada do sistema de Navier-Stokes (α é umparâmetro positivo pequeno), que pode também ser visto como um modelo de fluxos turbulentos; já o modelo Burgers-α pode ser visto como um modelo simplificado de Leray-α. Provamos que os modelos de Leray-α e Burgers-α são localmente controláveis a zero, com controles limitados uniformemente em α. Também provamos que, se os dados iniciais são suficientemente pequenos, o par estado-controle (que conduz a solução a zero) para o sistema de Leray-α (resp. para o sistema de Burgers-α) converge quando α → 0+ a um par estado-controle (que conduz a solução a zero) para as equações de Navier-Stokes (resp. para a equação de Burgers). O terceiro Capítulo é dedicado à controlabilidade de fluidos incompressíveis invíscidos nos quais os efeitos térmicos são importantes. Estes fluidos são modelados através da então chamada Aproximação de Boussinesq. No caso emque não há difusão de calor, adaptando e estendendo algumas idéias de J.-M. Coron [14] e O. Glass [45], estabelecemos a controlabilidade exata global simultaneamente do campo velocidade e da temperatura para fluxos em 2D e 3D. Quando o coeficiente de difusão do calor é positivo, apresentamos alguns resultados sobre a controlabilidade exata global para o campo velocidade e controlabilidade nula local para a temperatura. No último Capítulo, provamos a controlabilidade exata local à trajetórias de um sistema acoplado do tipo Boussinesq, com um número reduzido de controles. Nesse sistema, as incógnitas são: o campo velocidade e a pressão do fluido (y, p), a temperatura θ e uma variável adicional c que pode ser vista como a concentração de um soluto contaminante. Provamos vários resultados, que essencialmente mostram que é suficiente atuar localmente no espaço sobre as equações satisfeitas por θ e c.

Les énergies marines renouvelables (EMR) sont soumises aux vagues générées par le vent. Une caractérisation précise de ces vagues est nécessaire dans les zones côtières et littorales où les vagues interagissent fortement avec le fond, générant de la réfraction et du déferlement parmi d’autres processus.Une étude approfondie sur l’initiation du déferlement est développée. La validité du critère de déferlement conventionnel uc/c (rapport entre la vitesse orbitale horizontale à la crête et la vitesse de phase) est examinée numériquement. Cette étude nous mène à définir un nouveau critère cinématique basé sur le rapport entre la vitesse orbitale maximale ||um|| et c. Ce nouveau critère améliore la détection de l’initiation du déferlement, car la position d’où s’initie l’instabilité conduisant au déferlement est mieux capturée à partir de ||um||. La variabilité spatiale du champ de vagues en zone côtière est majoritairement étudiée à partir de modèles spectraux. La capacité d’un modèle à phase-résolue (type Boussinesq BT) à fournir des informations complémentaires pour les EMR est étudiée. Les modèles spectraux et BT produisent des résultats très différents en termes de hauteur de vagues et de puissance en présence d’une forte réfraction causée par la variabilité de la bathymétrie. On définit une méthode innovante pour extraire des informations liées aux vagues à partir d’images satellites, issues d’un radar à synthèse d’ouverture (SAR), et les comparer aux sorties des modèles. Nos résultats montrent des similitudes encourageantes entre le modèle BT et les données SAR
Since Marine Renewable Energy (MRE) systems are submitted to wind generated waves. Accurate wave characterization is required in the coastal and nearshore environment where the waves are strongly modified by their interaction with the sea bottom, inducing refraction and wave breaking among other processes.A comprehensive study regarding the wave breaking initiation process is developed. The conventional kinematic criterion uc/c (ratio between the horizontal orbital velocity at the crest and the phase velocity) validity is numerically investigated. Our study leads us to a new kinematic wave breaking criterion based on the ratio between the maximum fluid velocity ||um|| near the wave crest and c. This new criterion improves the detection of the breaking initiation, since ||um|| accurately captures the location of the fluid instability leading to breaking.The wave field spatial variability in coastal areas is mostly studied with spectral wave models. We explore the ability of a phase-resolving model (Boussinesq-type, BT) to provide additional wave information for MRE applications.Spectral and BT models lead to significantly different spatial wave height and power patterns in the presence of strong bottom-induced refraction. We define an innovative methodology to extract wave information from satellite Synthetic Aperture Radar (SAR) images for comparison with models’ outputs. Our results highlight encouraging similarities between the BT model and SAR data

In this study, the evolution of the numerical water wave models with the theoretical background and the governing equations are briefly discussed and a numerical model MIKE21 BW which can be applied to wave problems in nearshore zone is presented. The numerical model is based on the numerical solution of the Boussinesq type equations formulated on time domain. Nonlinearity and frequency dispersion is included in the model. In order to make comparison between the results of nonlinear shallow water equations with Boussinesq terms, MIKE21 BW and NAMIDANCE are applied to the problem of wave propagation in the long distances and runup on simple and composite slopes. The numerical experiments are applied to Datç
a Marina and the results are compared to the results of the physical experiments on wave disturbance in Datç
a Marina. In these comparisons the reflection characteristics of different coastal boundaries in the harbor area are tested and the internal parameters in the model are calibrated accordingly. The numerical model MIKE21 BW is applied to skenderun harbor as a case study. The input wave parameters are selected from the wave climate study for Iskenderun Harbor. The model is set up and the agitation inside the harbor is computed according to four different incoming wave scenarios. The disturbance maps inside the harbor for different incoming wave scenarios are obtained. The critical regions v of the harbor according to disturbance under different wave conditions are presented and discussed.

Dans cette thèse, nous introduisons un nouveau modèle instationnaire de vagues valable de la zone de levée à la zone de jet de rive adapté à l'étude de la submersion. Le modèle est basé sur les équations de Serre Green-Naghdi (S-GN), dont l'application à la zone de surf reste un domaine de recherche ouvert. Nous proposons une nouvelle approche pour gérer le déferlement dans ce type de modèle, basée sur la représentation des fronts déferlés par des chocs. Cette approche a été utilisée avec succès pour les modèles basés sur les équations de Saint-Venant (SV) et permet une description simple et efficace du déferlement et des mouvements de la ligne d'eau. Dans ces travaux, nous cherchons à étendre le domaine de validité du modèle SV SURF-WB (Marche et al. 2007) vers la zone de levée en incluant les termes dispersifs propres aux équations de S-GN. Des basculements locaux vers les équations de SV au niveau des fronts permettent alors aux vagues de déferler et dissiper leur énergie. Le modèle obtenu, appelé SURF-GN, est validé à l'aide de données de laboratoire correspondant à différents types de vagues incidentes et de plages. Il est ensuite utilisé pour analyser la dynamique des fronts d'ondes longues de type tsunami en zone littorale. Nous montrons que SURF-GN peut décrire les différents types de fronts, d'ondulé non-déferlé à purement déferlé. Les conséquences de la transformation d'une onde de type tsunami en train d'ondulations lors de la propagation sur une plage sont ensuite considérées. Nous présentons finalement une étude de la célérité des vagues déferlées, basée sur les données de la campagne de mesure in-situ ECORS Truc-Vert 2008. L'influence des non-linéarités est en particulier quantifiée
In this thesis, we introduce a new numerical model able to describe wave transformation from the shoaling to the swash zones, including overtopping. This model is based on Serre Green-Naghdi equations, which are the basic fully nonlinear Boussinesq-type equations. These equations can accurately describe wave dynamics prior to breaking, but their application to the surf zone usually requires the use of complex parameterizations. We propose a new approach to describe wave breaking in S-GN models, based on the representation of breaking wave fronts as shocks. This method has been successfully applied to the Nonlinear Shallow Water (NSW) equations, and allows for an easy treatment of wave breaking and shoreline motions. However, the NSW equations can only be applied after breaking. In this thesis, we aim at extending the validity domain of the NSW model SURF-WB (Marche et al. 2007) to the shoaling zone by adding the S-GN dispersive terms to the governing equations. Local switches to NSW equations are then performed in the vicinity of the breaking fronts, allowing for the waves to break and dissipate their energy. Extensive validations using laboratory data are presented. The new model, called SURF-GN, is then applied to study tsunami-like undular bore dynamics in the nearshore. The model ability to describe bore dynamics for a large range of Froude number is first demonstrated, and the effects of the bore transformation on wave run-up over a sloping beach are considered. We finally present an in-situ study of broken wave celerity, based on the ECORS-Truc Vert 2008 field experiment. In particular, we quantify the effects of non-linearities and evaluate the predictive ability of several non-linear celerity models

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
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

Ce travail est consacré à l'étude de quelques problèmes de contrôlabilité concernant plusieurs modèles issues de la mécanique des fluides. Dans le Chapitre 2, on obtient la contrôlabilité locale à zéro du système de Navier-Stokes avec contrôles distribués ayant une composante nulle. La nouveauté la plus importante est l'absence de conditions géometriques sur le domaine de contrôle. Le Chapitre 3 étend ce résultat pour le système de Boussinesq, où le couplage avec l'équation de la chaleur permet d'avoir jusqu'à deux composantes nulles dans le contrôle agissant sur l'équation du fluide. Le Chapitre 4 traite l'existence de contrôles insensibilisants pour le système de Boussinesq. En particulier, on montre la contrôlabilité à zéro d'un système en cascade issu du problème d'insensibilisation où le contrôle dans l'équation du fluide possède deux composantes nulles. Pour ces problèmes, on suit une approche classique. On établit la contrôlabilité à zéro du système linéalisé autour de zéro par une inégalité de Carleman pour le système adjoint avec des termes source. Puis, on obtient le résultat pour le système non linéaire par un argument d'inversion locale.Dans le Chapitre 5, on étudie quelques aspects de la contrôlabilité à zéro d'une équation de KdV linéaire avec conditions au bord de type Colin-Ghidaglia. On obtient une estimation du coût de la contrôlabilité à zéro qui est optimal par rapport au coefficient de dispersion. Sa preuve repose sur une inégalité de Carleman avec un comportement optimal en temps. Puis, on montre que le coût de la contrôlabilité à zéro explose exponentiellement par rapport au coefficient de dispersion lorsque le temps final est suffisamment petit
This work is devoted to the study of some controllability problems concerning some models from fluid mechanics. First, in Chapter 2, we obtain the local null controllability of the Navier-Stokes system with distributed controls having one vanishing component. The main novelty is that no geometric condition is imposed on the control domain. In Chapter 3, we extend this result for the Boussinesq system, where the coupling with the temperature equation allows us to have up to two vanishing components in the control acting on the fluid equation. Chapter 4 deals with the existence of insensitizing controls for the Boussinesq system. In particular, we prove the null controllability of the cascade system arising from the reformulation of the insensitizing problem, where the control on the fluid equation has two vanishing components. For these problems, we follow a classical approach. We establish the null controllability of the linearized system around the origin by means of a suitable Carleman inequality for the adjoint system with source terms. Then, we obtain the result for the nonlinear system by a local inversion argument.In Chapter 5, we study some null controllability aspects of a linear KdV equation with Colin-Ghidaglia boundary conditions. First, we obtain an estimation of the cost of null controllability, which is optimal with respect to the dispersion coefficient. This improves previous results on this matter. Its proof relies on a Carleman estimate with an optimal behavior in time. Finally, we prove that the cost of null controllability blows up exponentially with respect to the dispersion coefficient provided that the final time is small enough

The Boussinesq approximation is a set of fluids equations utilized in the atmospheric and oceanographic sciences. They may be thought of as inhomogeneous, incompressible Euler or Navier-Stokes equations, where the inhomogeneous term is a scalar quantity, typically representing density or temperature, governed by a convection-diffusion equation. In this thesis, we prove local-in-time existence and uniqueness of an inviscid Boussinesq system. Furthermore, we show that under stronger assumptions, the local-in-time results can be extended to global-in-time existence and uniqueness as well. We assume the density equation contains nonzero diffusion and that our initial vorticity and density belong to a space of borderline Besov-type. We use paradifferential calculus and properties of the Besov-type spaces to control the growth of vorticity via an a priori estimate on the growth of density. This result is motivated by work of M. Vishik demonstrating local-in-time existence and uniqueness for 2D Euler equations in borderline Besov-type spaces, and by work of R. Danchin and M. Paicu showing the global well-posedness of the 2D Boussinesq system with initial data in critical Besov and Lp-spaces.
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