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Dickson, Ronald. "Algebro-geometric solutions of the Boussinesq hierarchy /." free to MU campus, to others for purchase, 1998. http://wwwlib.umi.com/cr/mo/fullcit?p9904841.
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Lin, Qun. "The well-posedness and solutions of Boussinesq-type equations." Curtin University of Technology, Department of Mathematics and Statistics, 2009. http://espace.library.curtin.edu.au:80/R/?func=dbin-jump-full&object_id=129030.
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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.<br>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.<br>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.
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Moore, Kieron R. "Coupled Boussinesq equations and nonlinear waves in layered waveguides." Thesis, Loughborough University, 2013. https://dspace.lboro.ac.uk/2134/13636.
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There exists substantial applications motivating the study of nonlinear longitudinal wave propagation in layered (or laminated) elastic waveguides, in particular within areas related to non-destructive testing, where there is a demand to understand, reinforce, and improve deformation properties of such structures. It has been shown [76] that long longitudinal waves in such structures can be accurately modelled by coupled regularised Boussinesq (cRB) equations, provided the bonding between layers is sufficiently soft. The work in this thesis firstly examines the initial-value problem (IVP) for the system of cRB equations in [76] on the infinite line, for localised or sufficiently rapidly decaying initial conditions. Using asymptotic multiple-scales expansions, a nonsecular weakly nonlinear solution of the IVP is constructed, up to the accuracy of the problem formulation. The asymptotic theory is supported with numerical simulations of the cRB equations. The weakly nonlinear solution for the equivalent IVP for a single regularised Boussinesq equation is then constructed; constituting an extension of the classical d'Alembert's formula for the leading order wave equation. The initial conditions are also extended to allow one to separately specify an O(1) and O(ε) part. Large classes of solutions are derived and several particular examples are explicitly analysed with numerical simulations. The weakly nonlinear solution is then improved by considering the IVP for a single regularised Boussinesq-type equation, in order to further develop the higher order terms in the solution. More specifically, it enables one to now correctly specify the higher order term's time dependence. Numerical simulations of the IVP are compared with several examples to justify the improvement of the solution. Finally an asymptotic procedure is developed to describe the class of radiating solitary wave solutions which exist as solutions to cRB equations under particular regimes of the parameters. The validity of the analytical solution is examined with numerical simulations of the cRB equations. Numerical simulations throughout this work are derived and implemented via developments of several finite difference schemes and pseudo-spectral methods, explained in detail in the appendices.
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Li, Shenghao. "Non-homogeneous Boundary Value Problems for Boussinesq-type Equations." University of Cincinnati / OhioLINK, 2016. http://rave.ohiolink.edu/etdc/view?acc_num=ucin1468512590.
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Walkley, Mark Andrew. "A numerical method for extended Boussinesq shallow-water wave equations." Thesis, University of Leeds, 1999. http://etheses.whiterose.ac.uk/1291/.
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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.
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Rivas, Ivonne. "Analysis and Control of the Boussinesq and Korteweg-de Vries Equations." University of Cincinnati / OhioLINK, 2011. http://rave.ohiolink.edu/etdc/view?acc_num=ucin1321371582.
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Nascimento, Clair do. "Efeito de localização para as equações estacionarias classicas de Boussinesq em um canal." [s.n.], 2009. http://repositorio.unicamp.br/jspui/handle/REPOSIP/307389.
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Orientador: Jose Luiz Boldrini<br>Tese (doutorado) - Universidade Estadual de Campinas, Instituto de Matemática, Estatística e Computação Científica<br>Made available in DSpace on 2018-08-13T20:05:11Z (GMT). No. of bitstreams: 1
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Previous issue date: 2009<br>Resumo: Consideramos o fluxo de um fluido viscoso e incompressível em um canal bidimensional
semi-infinito, dadas velocidade e temperatura possivelmente nao nulas na entrada deste
canal. Assumindo que este fluido e governado pelas equações estacionarias classicas de
Boussinesq, sob hipoteses adequadas sobre as condições de fronteira, mostramos que pela
aplicação de certas forças sublineares (que dependendem da velocidade e da temperatura
do fluido) é possíivel parar o fluxo a uma distancia finita da entrada do canal. Mais
especificamente, a uma distancia finita da entrada do canal a velocidade e a temperatura
do fluido se anulam e assim temos o chamado efeito de localização (ou que a solução e
localizada). Este trabalho e feito em duas etapas. Primeiramente, usando um argumento de ponto
fixo com o auxilio do teorema de Leray-Schauder, mostramos a existencia de uma solução
fraca. Na segunda etapa provamos que tal solução é localizada usando estimativas do tipo
energia adequadas similares aquelas utilizadas por Bernis. Devido ao fato de que o nosso
dominio (o canal) é ilimitado, por razões tecnica, as etapas anteriores são feitas primeiramente
considerando soluções aproximadas em dominios limitados obtidos pelo truncamento
do canal; o resultado desejado 'e então obtido tomando o limite destas soluções
aproximadas usando cuidadosamente que certas estimativas são uniformes com respeito
a tais dominios truncados.<br>Abstract: We consider the flow of an incompressible viscous fluid in a bidimensional semi-infinity
strip, given possible non-zero velocities and temperatures at the strip entrance. Assuming
that flow is governed by the Boussinesq classic stationary equations, under suitable hypotheses
on the boundary conditions, we show that by applying certain sub-linear forces
(depending of velocity and temperature) it is possible to stop the flow at a finite distance
of the strip entrance. More specifically, at finite distance of the strip entrance, the velocity
and temperature become zero, and thus we have what is called the localization effect (or
that the solution is localized).
This work is done in two stages. First, by using a fixed point argument with help
of Leray-Schauder theorem, we show the existence of a weak solution of the system of
equations describing the flow. Second, we proof that such solution is localized by using
suitable energy estimates similar to those used by Bernis. Due the fact our domain, the
strip, is unbounded, for technical reasons the previous stages are firstly done by considering
associated approximate solutions on bounded domains, obtained by truncation of the
strip; the desired result is obtained by taking the limit of these approximate solutions by
using carefully that some estimates are uniform with respect to such truncated strips.<br>Doutorado<br>Equações Diferenciais Parciais<br>Doutor em Matemática Aplicada
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McQuarrie, Shane Alexander. "Data Assimilation in the Boussinesq Approximation for Mantle Convection." BYU ScholarsArchive, 2018. https://scholarsarchive.byu.edu/etd/6951.
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Many highly developed physical models poorly approximate actual physical systems due to natural random noise. For example, convection in the earth's mantle—a fundamental process for understanding the geochemical makeup of the earth's crust and the geologic history of the earth—exhibits chaotic behavior, so it is difficult to model accurately. In addition, it is impossible to directly measure temperature and fluid viscosity in the mantle, and any indirect measurements are not guaranteed to be highly accurate. Over the last 50 years, mathematicians have developed a rigorous framework for reconciling noisy observations with reasonable physical models, a technique called data assimilation. We apply data assimilation to the problem of mantle convection with the infinite-Prandtl Boussinesq approximation to the Navier-Stokes equations as the model, providing rigorous conditions that guarantee synchronization between the observational system and the model. We validate these rigorous results through numerical simulations powered by a flexible new Python package, Dedalus. This methodology, including the simulation and post-processing code, may be generalized to many other systems. The numerical simulations show that the rigorous synchronization conditions are not sharp; that is, synchronization may occur even when the conditions are not met. These simulations also cast some light on the true relationships between the system parameters that are required in order to achieve synchronization. To conclude, we conduct experiments for two closely related data assimilation problems to further demonstrate the limitations of the rigorous results and to test the flexibility of data assimilation for mantle-like systems.
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Hu, Weiwei. "Approximation and Control of the Boussinesq Equations with Application to Control of Energy Efficient Building Systems." Diss., Virginia Tech, 2012. http://hdl.handle.net/10919/38664.
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In this thesis we present theoretical and numerical results for a feedback control problem defined by a thermal fluid. The problem is motivated by recent interest in designing and controlling energy efficient building systems. In particular, we show that it is possible to locally exponentially stabilize the nonlinear Boussinesq Equations by applying Neumann/Robin type boundary control on a bounded and connected domain. The feedback controller is obtained by solving a Linear Quadratic Regulator problem for the linearized Boussinesq equations. Applying classical results for semilinear equations where the linear term generates an analytic semigroup, we establish that this Riccati-based optimal boundary feedback control provides a local stabilizing controller for the full nonlinear Boussinesq equations. In addition, we present a finite element Galerkin approximation. Finally, we provide numerical results based on standard Taylor-Hood elements to illustrate the theory.<br>Ph. D.
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Haas, Tobias [Verfasser], and Guido [Akademischer Betreuer] Schneider. "Amplitude equations for Boussinesq and Ginzburg-Landau-like models / Tobias Haas ; Betreuer: Guido Schneider." Stuttgart : Universitätsbibliothek der Universität Stuttgart, 2019. http://d-nb.info/1211649709/34.
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Rosenblatt, Heather Leah. "Asymptotics and Borel Summability: Applications to MHD, Boussinesq equations and Rigorous Stokes Constant Calculations." The Ohio State University, 2013. http://rave.ohiolink.edu/etdc/view?acc_num=osu1373987953.
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Paiva, Lynnyngs Kelly Arruda Saraiva de. "Estabilidade de soluções ondas viajantes periodicas para as equações de Boussinesq e de Korteweg - de Vries." [s.n.], 2005. http://repositorio.unicamp.br/jspui/handle/REPOSIP/307223.
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Orientadores: Jaime Angulo Paiva, Marcia A. G. Scialom<br>Tese (doutorado) - Universidade Estadual de Campinas, Instituto de Matematica, Estatistica e Computação Cientifica<br>Made available in DSpace on 2018-08-04T09:39:02Z (GMT). No. of bitstreams: 1
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Previous issue date: 2005<br>Doutorado<br>Matematica<br>Doutor em Matemática
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Guesmia, Mustapha. "Etude numerique des equations de boussinesq. Application a la simulation des raz-de-maree d'origine sismique." Paris 11, 1996. http://www.theses.fr/1996PA112472.
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Les tsunamis sont des ondes marines d'origine sismique, consecutifs a un seisme sous-marin. Ils sont generes par un deplacement vertical du fond de l'ocean. Au cours de ces dix dernieres annees, plus d'une dizaine de tsunamis majeurs ont ete observes dans l'ocean pacifique, tuant plus de 2000 personnes et causant des degats considerables. La frequence des raz-de-maree dans l'atlantique n'est pas aussi importante. Le tsunami consecutif au seisme du 1#e#r novembre 1755 fut l'un des plus violents que l'on observa dans cette zone. L'epicentre de ce seisme a ete localise a 300 km, au sud-ouest de st vincente. De magnitude 8. 75, ce tsunami fut l'un des plus destructeurs en europe, ou l'on observa des hauteurs de vagues de plusieurs metres, notamment le long du littoral portugais et marocain. L'objet de cette these a ete de construire des modeles numeriques bases sur les equations des ondes longues, et de les appliquer a la reproduction de tsunamis pour lesquels des etudes hydrauliques ou sismologiques ont ete effectuees. Cette etude s'inscrit dans le programme gitec, acronyme pour genese et impact des tsunamis sur les cotes europeennes, qui est un projet europeen de cooperation entre plusieurs institutions et laboratoires. Il fait partie de projets supportes par la commission des communautes europeennes. Il concerne la detection, l'alerte, la modelisation et la prevention des tsunamis
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Mccabe, Maurice Vincent. "Modelling nearshore waves, runup and overtopping." Thesis, University of Manchester, 2011. https://www.research.manchester.ac.uk/portal/en/theses/modelling-nearshore-waves-runup-and-overtopping(16ee1ecf-542c-4e3d-a150-fcb4d3981f6d).html.
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Coastal flooding from wave overtopping causes considerable damage. Presently, to model wave overtopping one can either make use of physical model tests or empirical tools such as those described in the EurOtop manual. Both these methods have limitations; therefore, a quick and reliable numerical model for wave overtopping would be a very useful tool for a coastal engineer.This research aims to test and develop a numerical model (in one horizontal dimension) for nearshore waves, runup and overtopping. The Shallow Water And Boussinesq (SWAB) model solves the Boussinesq-type equations of Madsen and Sorensen (1992) for non breaking waves and the nonlinear shallow water equations for breaking waves. Through testing against a range of physical model data using regular and random waves, the SWAB model's transfer from non-breaking to breaking waves was optimised. It was found that a wave height to water depth ratio worked consistently well as a breaking criterion.A set of physical model tests were carried out, based on previous field testing of wave overtopping that had previously taken place at Anchorsholme, Blackpool. The SWAB model was used to simulate some of these physical model tests, giving good results for mean overtopping rates. SWAB models the force imposed by steep walls and recurve walls on the incident flow; this force was found to have a significant effect on overtopping rates. A comparison was made between mean overtopping rates from the SWAB model, the physical model tests, empirically-based software (PC-Overtopping) and the field data. The physical model and SWAB results compared well with the field data, though the empirical software gave large overestimates.The SWAB model was applied to the analysis of overtopping at Walcott, Norfolk. It was found that beach levels affected overtopping rates, but not as much as different randomly phased wave trains. A simulation of a recent storm event was performed, with overtopping rates being slightly lower than those reported by local residents. A joint probability analysis showed that the predicted frequency of such an event was in line with these reports.An alternative modelling technique was also tested, where a spectral energy model was coupled with a nonlinear shallow water solver. Results for wave runup parameters were very accurate, when the coupling location is at the seaward edge of the surf zone. Extension of this modelling technique into two horizontal dimensions would be more straightforward than with the SWAB model.
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Lima, Fabiana Goulart de. "Soluções fracas para um sistema de equações de Oberbeck-Boussinesq." reponame:Biblioteca Digital de Teses e Dissertações da UFRGS, 2002. http://hdl.handle.net/10183/118194.
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Neste trabalho, utilizando o método espectral de Galerkin, provamos a existência de soluções fracas (quando a dimensão n é maior que 2) e existência e unicidade de soluções fracas (quando a dimensão é 2) para um sistema de equações diferenciais parciais que descrevem o movimento de um fluido quimicamente ativo em um domínio limitado em Rn, n 2≥2.<br>In this work, by using the spectral Galerkin method, we prove the existence of weak solutions (when the dimension n is great than 2) and existence and uniqueness of weak solutions (when the dimension is 2) for a system of partial differential equations that describes the motion of a chemical active fluid in a bounded domain in Rn, n≥2.
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Isa, Mukheta Bin. "A study of the soliton solutions of the Boussinesq and other nonlinear evolution equations of fluid mechanics." Thesis, University of Newcastle Upon Tyne, 1988. http://hdl.handle.net/10443/723.
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After introducing the nonlinear evolution equations of interest: the finite depth fluid (FDF), the Kadomtsev-Petviashvili (KP), the Classical and the ordinary Boussinesq equations, formal asymptotic derivations of the KP and the FDF equations are given for the description of surface and interfacial waves. The N-soliton solution of the FDF equation is reconstructed as a finite sum of Wronskian type determinants. This solution is then shown to reduce to the solutions of the KdV and the Benjamin - Ono equations under specific limiting conditions. Interactions between two solitons of the FDF equation are studied and their interaction properties are shown to reduce to those of the KdV and the Benjamin - Ono equations. Computer plots of the interactions of two-soliton solutions of the FDF and the Benjamin - Ono equations are given. Resonance phenomena in solitons are studied with reference to the KP equation. After discussion of the basic concepts of these phenomena, the N-soliton solution is shown to reduce to the Wronskian of N/2 functions (N-even), each of which represents a triad of solitons when the solitons resonate in pairs. Asymptotic behaviour of the interactions between a triad and a soliton and between two triads are examined and the phase shifts of the triads are obtained directly from the Wronskian representation. The interactions are analysed in detail with reference to numerical computations of the full solutions. After showing that the Classical Boussinesq equations are obtained from Whitham's shallow water wave equations, the basic concept of Hirota's pq=c reduction of the first modified KP hierarchy is outlined. The Classical Boussinesq equations are shown as the pq=O reduction of the same hierarchy. The solution of the hierarchy is manipulated to incorporate the pq=O reduction. As a result of these limiting procedures applied to the problem, Wronskian solutions of the Classical Boussinesq equations in terms of rational functions are produced. Finally the pq=c reduction of the KP hierarchy is applied to the ordinary Boussinesq equation. Using this, the N-soliton solution is expressed as a finite sum of Wronskian type determinants. Analytic verification made for the two-soliton solution shows that a number of Wronskian identities are needed for this purpose. The reason for this behaviour is examined.
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Akmel, Dé Godefroy. "Études sur les équations de Boussineq." Paris 11, 1996. http://www.theses.fr/1996PA112337.
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Dans ce travail, nous présentons quelques propriétés de certaines équations de Boussinesq généralisées. Dans la première partie, nous étudions le comportement en grand temps de la solution (n,v) des équations n_t + n_x1 + 3/2_npnx1 - 1/6_βnx1x1t + 1/2_γvx2 - 1/2εv = 0, γv_t + εn + γn_x2 = 0, (x_1, x_2) ∈ R, avec p ≥ 1 entier, β,ε,γ > 0 réels. En particulier, nous montrons que pour p > 6, et pour de petites données initiales, la norme infinie de n decroît vers zéro comme t^(-1/3), alors que celle de v reste bornée. La preuve est basée sur l'analyse du problème linéaire associé aux équations ci-dessus. L'absence de terme régularisant en x_2 dans ces équations nous empêche de faire l'étude de l'existence locale via les méthodes classiques d'estimations d'énergie. C'est pour contourner cette difficulté que nous travaillons avec des transformées de Fourier et des intégrales oscillantes. De même, à cause la forme particulière de ces équations, l'étude du comportement asymptotique a été faite dans les espaces de Sobolev à poids. Dans la seconde partie de cette thèse, nous étudions l'explosion en temps fini des solutions d'équations de Boussinesq. Après avoir traité le problème d'existence locale, nous montrons que sous certaines conditions, ces solutions explosent en temps fini.
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Bosi, Umberto. "A unified spectral/hp element depth-integrated Boussinesq model for nonlinear wave-floating body interaction." Thesis, Bordeaux, 2019. http://www.theses.fr/2019BORD0084/document.
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Le secteur de l’énergie houlomotrice s’appuie fortement sur la modélisation mathématique et la simulation d’expériences physiques mettant en jeu les interactions entre les ondes et les corps. Dans ce travail, nous avons développé un modèle d’interaction de fidélité moyenne vague-corps pour la simulation de structures tronquées flottantes fonctionnant en mouvement vertical. Ce travail concerne l’ingénierie de l’énergie marine, pour des applications telles que les convertisseurs d’énergie de vague (WEC) à absorption ponctuelle, même si ses applications peuvent aussi être utilisées en ingénierie maritime et navale. Les motivations de ce travail reposent sur les méthodes standard actuelles pour décrire l’interaction corps-vague. Cellesci sont basées sur des modèles résolvant le flux de potentiel linéaire (LPF), du fait de leur grande efficacité. Cependant, les modèles LPF sont basés sur l’hypothèse de faible amplitude et ne peuvent pas répresenter les effets hydrodynamiques non linéaires, importants pour le WEC opérant dans la région de résonance ou dans les régions proches du rivage. En effet, il a été démontré que les modèles LFP prédisent de manière excessive la production de puissance, sauf si des coefficients de traînée sont calibrés. Plus récemment, des simulations Reynolds Averaged Navier-Stokes (RANS) ont été utilisées pour les WEC. RANS est un modèle complet et précis, mais très coûteux en calcul. Il n’est ni adapté à l’optimisation d’appareils uniques ni aux parcs énergétiques. Nous avons donc proposé un modèle de fidélité moyenne basé sur des équations de type Boussinesq, afin d’améliorer le compromis entre précision et efficacité. Les équations de type Boussinesq sont des modèles d’ondes intégrées en profondeur et ont été un outil d’ingénierie standard pour la simulation numérique de la propagation d’ondes non linéaires dans les eaux peu profondes et les zones côtières. Grâce à l’élimination de la dimension verticale, le modèle résultant est très efficace et évite la description temporelle de la limite entre la surface libre et l’air. Jiang (2001) a proposé un modèle de Boussinesq unifié, décomposant le problème en deux domaines : surface libre et corps. Dans cette méthode, le domaine du corps est également modélisé par une approche intégrée en profondeur - d’où le terme unifié. Récemment, Lannes (2016) avait analysé de manière rigoureuse une configuration similaire dans une équation non linéaire en eaux peu profondes, en déduisant une solution exacte et semi-analitique pour des corps en mouvement. Suivant la même approche, Godlewski et al. (2018) a élaboré un modèle de flux d’eau peu profonde encombrée. [...] Dans cette thèse, nous développons les résultats présentés par Eskilsson et al. (2016) et Bosi et al. (2019). Le modèle est étendu à deux dimensions horizontales. Le modèle 1D est vérifié à l’aide de solutions fabriquées et validé par rapport aux résultats publiés sur l’interaction vague-corps en 1D pour les pontons fixes et corps en mouvement de soulèvement forcé et libre. Les résultats des preuves de concept de la simulation de plusieurs corps sont présentés. Nous validons et vérifions le modèle 2D en suivant des étapes similaires. Enfin, nous mettons en oeuvre la technique de verrouillage, une méthode de contrôle de mouvement du corps pour améliorer la réponse au mouvement des vagues. Il est démontré que le modèle possède une excellente précision, qu’il est pertinent pour les applications d’ondes en interaction avec des dispositifs à énergie houlomotrice et qu’il peut être étendu pour simuler des cas plus complexes<br>The wave energy sector relies heavily on mathematical modelling and simulation of the interactions between waves and floating bodies. In this work, we have developed a medium-fidelity wave-body interaction model for the simulation of truncated surface piercing structures operating in heave motion, such as point absorbers wave energy converters (WECs). The motivation of the work lies in the present approach to wave-body interaction. The standard approach is to use models based on linear potential flow (LPF). LPF models are based on the small amplitude/ small motion assumption and, while highly computational efficient, cannot account for nonlinear hydrodynamic effects (except for Morison-type drag). Nonlinear effects are particularly important for WEC operating in resonance, or in nearshore regions where wave transformations are expected. More recently, Reynolds Averaged Navier-Stokes (RANS) simulations have been employed for modelling WECs. RANS is a complete and accurate model but computationally very costly. At present RANS models are therefore unsuited for the optimization of single devices, not to mention energy farms. Thus, we propose a numerical model based built on Boussinesq-type equations to include wave-wave interaction as well as finite body motion in a computationally efficient formulation. Boussinesq-type equations are depth-integrated wave models and are standard engineering tool for numerical simulation of propagation of nonlinear wave in shallow water and coastal areas. Thanks to the elimination of the vertical dimension and the avoidance of a time-dependent computational the resulting model is very computational efficient. Jiang (Jiang, 2001) proposed a unified Boussinesq model, decomposing the problem into free surface and body domains. Notably, in Jiang’s methodology also the body domain is modeled by a depth-integrated approach –hence the term unified. As all models based on Boussinesq-type equations, the model is limited to shallow and intermediate depth regimes. We consider the Madsen and Sørensen model, an enhanced Boussinesq model, for the propagation of waves. We employ a spectral/hp finite element method (SEM) to discretize the governing equations. The continuous SEM is used inside each domain and flux-based coupling conditions are derived from the discontinuous Galerkin method. The use of SEM give support for the use of adaptive meshes for geometric flexibility and high-order accurate approximations makes the scheme computationally efficient. In this thesis, we present 1D results for the propagation and interaction of waves with floating structures. The 1D model is verified using manufactured solutions. The model is then validated against published results for wave-body interaction. The hydrostatic cases (forced motion and decay test) are compared to analytical and semi-analytical solutions (Lannes, 2017), while the non-hydrostatic tests (fixed pontoon and freely heaving bodies) are compared to RANS reference solutions. The model is easily extended to handle multiple bodies and a proof-of-concept result is presented. Finally, we implement the latching technique, a method to control the movement of the body such that the response to the wave movement is improved. The model is extended to two horizontal dimensions and verified and validated against manufactured solutions and RANS simulations. The model is found to have a good accuracy both in one and two dimensions and is relevant for applications of waves interacting with wave energy devices. The model can be extended to simulate more complex cases such as WEC farms/arrays or include power generation systems to the device
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Alves, Wolney. "Modelling of wave generation in sewer systems by intermittent discharge devices using the Saint-Venant and Boussinesq equations." Thesis, Heriot-Watt University, 1996. http://hdl.handle.net/10399/708.
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Bellec, Stevan. "Nouvelle approche pour l'obtention de modèles asymptotiques en océanographie." Thesis, Bordeaux, 2016. http://www.theses.fr/2016BORD0182/document.
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Dans ce manuscrit, nous nous inéressons à l'étude du mouvement des vagues soumises uniquement à leur poids par le biais d'équations asymptotiques. Nous commençons par rappeler la dérivation des principaux modèles généralement utilisés (Boussinesq, Green-Naghdi,...). Nous introduisons également un nouveau modèle exprimé en amplitude-flux qui correspond à une variante des équations de Nwogu. Dans le second chapitre, nous démontrons un résultat d'existence en temps long pour ces nouvelles équations et nous étudions l'existence d'ondes solitaires pour les équations de Boussinesq. Ce travail permet notamment de calculer avec une grande précision ces solutions exactes. Le troisième chapitre détaille les différences non linéaires que l'on retrouve entre les différentes équations de Boussinesq (modèles en flux-amplitude comparés aux modèles en vitesse-amplitude). Enfin, les deux derniers chapitres introduisent un nouveau paradigme pour trouver des schémas numériques adaptés aux modèles asymptotiques. L'idée est d'appliquer une analyse asymptotique aux équations d'Euler discrétisées. Ce nouveau paradigme est appliqué aux équations de Peregrine, de Nwogu et de Green-Naghdi. Plusieurs cas tests sont proposés dans ces deux chapitres<br>In this work, we are interested in the evolution of water waves under the gravity force using asymptotics models. We start by recalling the derivation of most used models (Boussinesq, Green-Naghdi,...) and we introduce a new model expressed amplitude-flux, which is an alternative version of the Nwogu equations. In the second chapter, we prove a long time existence result for the new model and we investigate the existence of solitary waves for the Boussinesq models. This work allow us to compute these solutions with a good precision. The third chapter highlights the nonlinear differences between the Boussinesq equations (amplitude-flux models versus amplitude-velocity models). Finally, the two last chapter introduce a new paradigm in order to find numerical schemes adapted to asymptotics models. The idea is to apply an asymptotic analysis to a discretized Euler system. This new paradigm is applied to Peregrine equations, Nwogu equations and Green-Naghdi equations. Test cases are presented in these two chapters
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Sedjro, Marc Mawulom. "On the almost axisymmetric flows with forcing terms." Diss., Georgia Institute of Technology, 2012. http://hdl.handle.net/1853/44879.
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This work is concerned with the Almost Axisymmetric Flows with Forcing Terms which are derived from the inviscid Boussinesq equations. It is our hope that these flows will be useful in Meteorology to describe tropical cyclones. We show that these flows give rise to a collection of Monge-Ampere equations for which we prove an existence and uniqueness result. What makes these equations unusual is the boundary conditions they are expected to satisfy and the fact that the boundary is part of the unknown. Our study allows us to make inferences in a toy Almost Axisymmetric Flows with a forcing term model.
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He, Jiao. "Comportement d’un fluide autour d’un petit obstacle, problèmes de convections et dynamique chaotique des films liquides." Thesis, Lyon, 2019. http://www.theses.fr/2019LYSE1166/document.
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Cette thèse est consacrée à trois différentes équations d’évolution non-linéaires dans le cadre de mécanique des fluides : le système fluide-solide, le système de Boussinesq et un modèle de films liquides. Pour le système fluide-solide, nous étudions l’évolution d’un petit solide en mouvement dans un fluide newtonien incompressible dans le cas où l’obstacle se contracte vers un point. En supposant que la densité du solide tend vers l’infini, nous montrons la convergence des solutions du système fluide-solide vers une solution des équations de Navier-Stokes dans $\mathbb{R}^d$ , avec $d^2$ et 3. Pour le problème de convection, nous travaillons sur l’unicité des solutions ‘mild’ du système de Boussinesq et généralise de plusieurs manières différentes des résultats classiques d’unicité pour les équations de Navier-Stokes. Dans la dernière partie, nous exposons nos contributions à l’étude des interface 2D de films liquides en dimension trois. Nous montrons qu’une variante 2D, non-local, de l’équation de Kuramoto-Sivashinsky admet un attracteur globale compact et obtenons enfin une majoration du nombre d’oscillations spatiales des solutions<br>This thesis is devoted to three different non-linear evolution equations in fluid mechanics : the fluid-solid system, the Boussinesq system and a falling films model. For the fluid-solid system, we study the evolution of a small moving solid in incompressible viscous fluid in the case the obstacle converges to a point. Assuming that the density of the solid tends to infinity, we prove that the rigid body has no influence on the limit equation by showing the convergence of solutions of the fluid-solid system towards to a solution of the Navier-Stokes equations in the full $\mathbb{R}^d$ , avec $d^2$ et 3. For the convection problem, we provide several uniqueness classes on the velocity and the temperature and generalize some classical uniqueness result for ‘mild’ solutions of the Navier-Stokes equations. We then work on a falling films model in three dimensions (2D interface). We show that a non-local variant of the Kuramoto-Sivashinsky equation admits a compact global attractor and we study the number of spatial oscillations of the solutions
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Zhang, Xin. "Étude qualitative des solutions du système de Navier-Stokes incompressible à densité variable." Thesis, Paris Est, 2017. http://www.theses.fr/2017PESC1215/document.
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Dans cette thèse, on s'intéresse à deux problèmes provenant de l'étude mathématique des fluides incompressibles visqueux : la propagation de la régularité tangentielle et le mouvement d'une surface libre.La première question concerne plus particulièrement l'étude qualitative de l'évolution de quantités thermodynamiques telles que la température dans l'équation de Boussinesq sans diffusion et la densité dans le système de Navier-Stokes non homogène. Typiquement, on suppose que ces deux quantités sont, à l'instant initial, discontinues le long d'une interface à régularité h"oldérienne. Comme conséquence de résultats de propagation de régularité tangentielle pour le champ de vitesses, on établit que la régularité des interfaces persiste pour tout temps aussi bien en dimension deux d'espace, qu'en dimension supérieure (avec condition de petitesse). Notre approche suit celle du travail de J.-Y. Chemin dans les années 90 pour le problème des poches de tourbillon dans les fluides incompressiblesparfaits.Dans le cas présent, outre cette hypothèse de régularité tangentielle, nous n'avons besoin que d'une régularité critique sur le champ de vitesses.La démonstration repose sur le calcul para-différentiel et les espaces de multiplicateurs.Dans la dernière partie de la thèse, on considère le problème à frontière libre pour le système de Navier-Stokes incompressible à deux phases. Ce système permet de décrire l'évolution d'un mélange de deux fluides non miscibles tels que l'huile et l'eau par exemple. Différents cas de figure sont étudiés : le cas d'un réservoir borné, d'une goutte ou d'une rivière à profondeur finie.On établit l'existence et l'unicité à temps petit pour ce problème. Notre démonstration repose fortement sur des propriétés de régularité maximale parabolique de type $L_p$-$L_q<br>This thesis is dedicated to two different problems in the mathematical study of the viscous incompressible fluids: the persistence of tangential regularity and the motion of a free surface.The first problem concerns the study of the qualitative properties of some thermodynamical quantities in incompressible fluid models, such as the temperature for Boussinesq system with no diffusion and the density for the non-homogeneous Navier-Stokes system. Typically, we assume those two quantities to be initially piecewise constant along an interface with H"older regularity.As a consequence of stability of certain directional smoothness of the velocity field, we establish that the regularity of the interfaces persist globally with respect to time both in the two dimensional and higher dimensional cases (under some smallness condition). Our strategy is borrowed from the pioneering works by J.-Y.Chemin in 1990s on the vortex patch problem for ideal fluids.Let us emphasize that, apart from the directional regularity, we only impose rough (critical) regularity on the velocity field. The proof requires tools from para-differential calculus and multiplier space theory.In the last part of this thesis, we are concerned with the free boundary value problem for two-phase density-dependent Navier-Stokes system.This model is used to describe the motion of two immiscible liquids, like the oil and the water. Such mixture may occur in different situations, such as in a fixed bounded container, in a moving bounded droplet or in a river with finite depth. We establish the short time well-posedness for this problem. Our result strongly relies on the $L_p$-$L_q$ maximal regularity theoryfor parabolic equations
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Sitanggang, Khairil Irfan. "Boussinesq-equation and rans hybrid wave model." [College Station, Tex. : Texas A&M University, 2008. http://hdl.handle.net/1969.1/ETD-TAMU-2795.
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Liu, Fang-Lan. "Some asymptotic stability results for the Boussinesq equation." Diss., Virginia Tech, 1993. http://hdl.handle.net/10919/40052.
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Sjölander, Filip. "Numerical solutions to the Boussinesq equation and the Korteweg-de Vries equation." Thesis, KTH, Skolan för teknikvetenskap (SCI), 2021. http://urn.kb.se/resolve?urn=urn:nbn:se:kth:diva-297544.
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The aim of the report is to numerically construct solutions to two analytically solvable non-linear differential equations: the Korteweg–De Vries equation and the Boussinesq equation. To accomplish this, a range of numerical methods where implemented, including Galerkin methods. To asses the accuracy of the solutions, analytic solutions were derived for reference. Characteristic of both equations is that they support a certain type of wave-solutions called "soliton solutions", which admit an intuitive physical interpretation as solitary traveling waves. Theses solutions are the ones simulated. The solitons are also qualitatively studied in the report.
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Sun, Weizhou. "LOCAL DISCONTINUOUS GALERKIN METHOD FOR KHOKHLOV-ZABOLOTSKAYA-KUZNETZOV EQUATION AND IMPROVED BOUSSINESQ EQUATION." The Ohio State University, 2016. http://rave.ohiolink.edu/etdc/view?acc_num=osu1480327264817905.
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Filippini, Andrea Gilberto. "Free surface flow simulation in estuarine and coastal environments : numerical development and application on unstructured meshes." Thesis, Bordeaux, 2016. http://www.theses.fr/2016BORD0404/document.
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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<br>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
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Dufresne, Margarita. "Modélisation de la houle par éléments finis." Compiègne, 1997. http://www.theses.fr/1997COMP0986.
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Les modèles bidimensionnels horizontaux de la surface libre dans les cas de la distribution de pression hydrostatique (modèle de Saint-Venant) et non-hydrostatique ont été démontrés. Le développement des équations de Boussinesq et de Serre par la méthode de perturbation et leur analyse détaillée ont été effectués avec précision du domaine de validité de chaque équation. La grande variété de modèles de type Serre-Boussinesq résulte des multiples alternatives qui s'offrent lors du choix de la composante horizontale de la vitesse, ainsi que l'interprétation que l'on donne à cette dernière. La correction de Madsen introduit pour améliorer le caractère dispersif des équations servis comme une base de "fabrication" des différents modèles. Les modèles éléments finis des équations de type Serre-Boussinesq en une et deux dimensions ont été établis, en utilisant pour la discrétisation temporelle des schémas non-diffusifs de type Lax-Wendroff. La validation des modèles monodimensionnels a été effectuée par comparaison avec les solutions analytiques ou bien avec mesures expérimentales pour différents types de conditions aux limites dans le cas du fond plat et du fond quelconque. L'aptitude des modèles bidimensionnels horizontaux éléments finis des équations Serre-Boussinesq à simuler correctement des phénomènes physiques tels que la réfraction, la diffraction et la réflexion a été illustrée par plusieurs tests numériques et la comparaison avec ses analogues expérimentaux. La simulation pour une géométrie bidimensionnelle complexe a été effectuée avec un maillage irrégulier avec raffinement local à l'aide du logiciel I-DEAS. Un modèle de déferlement, basé sur l'intervention d'une diffusion turbulente sous forme d'un terme de dispersion dans l'évaluation de quantité de mouvement a été proposé. Un accord satisfaisant entre les résultats du calcul et les enregistrements expérimentaux a été obtenu<br>The two-dimensional (horizontal plane) models of free surface wave propagation are deduced from the fundamental equations of fluid mechanics. They are based on the non Iinear non-dispersive wave approach described by Saint-Venant equations (hydrostatic pressure), and on the non-linear dispersive wave approach described by Serre and Boussinesq type equations (non-hydrostatic pressure). The Boussinesq and Serre equations are developed using perturbation method with definition of the domain of validity of various approximations. A considerable number of Serre and Boussinesq type models is due to the choice of the kind of horizontal velocity, for which we give unambiguous interpretation. Higher-order terms introduced by Madsen to improve frequency dispersion serve as a base of "product" of different Boussinesq-type modeis. A one-dimensional and a two-dimensional (in plane) finite elements model of Serre and Boussinesq-type equations with improved frequency dispersion are presented. The time discretisation is based on Lax-Wendrofftype non-diffusive scheme. The one-dimensional numerical models are validated comparing with theoretical solutions and results obtained experimentally for horizontal and uneven bottom with various boundary conditions. The two-dimensional (in plane) Serre-Boussinesq finite elements models, capable to predict the refraction, diffraction and reflection are validated with good agreements between numerical and experimental results. The irregular meshs for complex bathymetry are created using I-DEAS code. A new one-dimensional breaking wave propagation model based on the Boussinesq type equations is developed by introduction of turbulent dissipation. Satisfactory agreements between numerical results and experiences are obtained
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Chazel, Florent. "Influence de la topographie sur les ondes de surface." Phd thesis, Université Sciences et Technologies - Bordeaux I, 2007. http://tel.archives-ouvertes.fr/tel-00200419.
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Dans cette thèse, nous considérons le problème d'Euler surface libre sur un domaine à fond non plat, dans le cadre du régime d'ondes longues de faible amplitude. L'objectif est de construire, justifier et comparer de nouveaux modèles asymptotiques pour ce problème, permettant de prendre en compte les effets liés aux variations bathymétriques. En premier lieu, nous construisons rigoureusement deux classes de modèles de Boussinesq symétriques dans le cadre de deux régimes topographiques distincts, celui de faible variations bathymétriques et celui de fortes variations. Dans un second temps, nous retrouvons et discutons dans le cas de faibles variations topographiques l'approximation classique de Korteweg-de Vries, et proposons une nouvelle approximation via l'ajout de termes bathymétriques. Dans une troisième partie, ces deux modèles, ainsi que les modèles de Boussinesq construits dans la première partie, sont simulés numériquement et comparés sur des cas tests de topographie. Enfin, il est présenté une étude numérique des équations de Green-Naghdi, dont le domaine de validité physique est plus étendu, ainsi qu'une comparaison numérique de ce modèle avec les modèles précédents sur des bathymétries spécifiques.
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Attaoui, Abdelatif. "Existence de solutions faibles et faible-renormalisées pour des systèmes non linéaires de Boussinesq." Phd thesis, Université de Rouen, 2007. http://tel.archives-ouvertes.fr/tel-00259252.
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La thèse est consacrée essentiellement à l'étude de systèmes non linéaires d'évolution issus d'un modèle de Boussinesq : couplage entre les équations de Navier-stokes avec un second membre F(µ), où F est une force de gravité proportionnelle à des variations de densité qui dépendent de la température et l'équation de l'énergie.<br />Le premier chapitre nous donne un résultat d'existence d'une solution faible-renormalisée du système de Boussinesq en dimension 2, dans le cas où F est bornée.<br />Dans le chapitre 2, on aborde le cas de fonctions F plus générales : F vérifie une hypothèse de croissance. On démontre l'existence de solutions pour toutes données initiales ou pour des données initiales petites selon la croissance de F.<br />Dans le chapitre 3, nous faisons une généralisation des résultats du chapitre 2 mais sans le terme de convection.<br />Dans le chapitre 4, le manque de stabilité de l'énergie de dissipation dans L1(Q) en dimension 3, nous contraint à transformer de façon formelle le système de Boussinesq. On démontre l'existence d'une solution faible de ce nouveau système en dimension 3.
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Aldbaissy, Rim. "Discrétisation du problème de couplage instationnaire des équations de Navier-Stokes avec l'équation de la chaleur." Thesis, Sorbonne université, 2019. http://www.theses.fr/2019SORUS013.
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Les équations aux dérivées partielles issues de la nature n’ont pas de solutions explicites et ne peuvent de ce fait qu’être résolue de manière approchée. Le travail présenté dans cette thèse porte d’une part sur la résolution du système de Navier-stokes couplé avec l’équation de la température. Ce couplage est connu sous le nom du modèle de Boussinesq. La viscosité et la force extérieure sont non linéaires dépendent de la température. D’autre part, sur la validation numérique des résultats théoriques obtenus dans le cadre académique et industriel. Ce travail porte sur deux parties. Dans la première, nous nous intéressons à l’approximation numérique de la solution des schémas discrets proposés en utilisant la méthode d’Euler semi-implicit pour la dicrétisation en temps et la méthode des éléments finis pour la discrétisation en espace d’ordre un. Dans le but de gagner en temps et en ordre de convergence, nous discrétisons le problème de couplage en ordre deux en temps et en espace, respectivement par la méthode BDF et la méthode des éléments finis d’ordre deux. Nous effectuons ainsi l’analyse de l’erreur a priori des schémas proposés et nous terminons par valider les résultats théoriques déjà obtenus par des simulations numériques en utilisant le logiciel Freefem++. La deuxième partie est dédiée à la modélisation du phénomène bouchon qui apparaît de temps en temps durant l’impression 3D. Dans le but d’améliorer l’algorithme séquentiel 2D et pouvoir passer ensuite à la simulation 3D, nous effectuons des calculs parallèles basés sur la méthode de décomposition de domaine. Les résultats obtenus montrent que cette méthode n’est pas efficace en termes de scalablité. Nous utilisons alors une méthode de préconditionnement à un niveau où les essais numériques décèlent une dépendance de la convergence en fonction du nombre de processeurs et de la physique du modèle. D’où l’idée d’ajouter au préconditionneur un deuxième niveau par la résolution du problème grossier<br>The analytical solutions of the majority of partial differential equations are difficult to calculate, hence, numerical methods are employed. This work is divided into two parts. First, we study the time dependent Navier-Stokes equations coupled with the heat equation with nonlinear viscosity depending on the temperature known as the Boussinesq (buoyancy) model . Then, numerical experiments are presented to confirm the theoretical accuracy of the discretization using the Freefem++ software. In the first part, we propose first order numerical schemes based on the finite element method for the space discretization and the semi-implicit Euler method for the time discretization. In order to gain time and order of convergence, we study a second order scheme in time and space by using respectively the second order BDF method "Backward Differentiation Formula" and the finite element method. An optimal a priori error estimate is then derived for each numerical scheme. Finally, numerical experiments are presented to confirm the theoretical results. The second part is dedicated to the modeling of the thermal instability that appears from time to time while printing using a 3D printer. Our purpose is to build a reliable scheme for the 3D simulation. For this reason, we propose a trivial parallel algorithm based on the domain decomposition method. The numerical results show that this method is not efficient in terms of scalability. Therefore, it is important to use a one-level preconditioning method "ORAS". When using a large number of subdomains, the numerical test shows a slow convergence. In addition, we noticed that the iteration number depends on the physical model. A coarse space correction is required to obtain a better convergence and to be able to model in three dimensions
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Crepeau-Jaisson, Emmanuelle. "Contrôlabilité exacte d'équations dispersives issues de la mécanique." Paris 11, 2002. http://www.theses.fr/2002PA112210.
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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 Vries, 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 façon numérique pour l'équation de Boussinesq avec un contrôle portant sur la dérivée seconde à droite, tant sur le problème linéaire que non linéaire<br>In this thesis, we study the exact controlability of two dispersive equations, the Korteweg-de Vries equation and the "good" Boussinesq equation. First, for the Korteweg-de Vries equation, we extend a result of Rosier. We prove that for critical length, the nonlinear equation is exactly controlable in a neighbourhood of a small non nul stationary solution. This study uses the hilbert uniqueness method with the multiplier theory and a fixed point theorem. Secondly, we study the exact controllability of the "good" Boussinesq equation with two different boundary controls. We use again the hilbert uniqueness method but with Ingham inequality. Lastly, we apply this method for a numerical approach of the controllability of the Boussinesq equation both for linear and nonlinear equations. The control is applied to the second spatial derivative, at the right endpoint
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Santos, Maurício Cardoso. "Controlabilidade exata de sistemas parabólicos, hiperbólicos e dispersivos." Universidade Federal da Paraíba, 2014. http://tede.biblioteca.ufpb.br:8080/handle/tede/7432.
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Previous issue date: 2014-07-25<br>Coordenação de Aperfeiçoamento de Pessoal de Nível Superior<br>In this thesis, we study controllability results of some phenomena modeled by Partial
Differential Equations (PDEs):
Multi objective control problem, for parabolic equations, following the Stackelber-Nash
strategy is considered: for each leader control which impose the null controllability for
the state variable, we find a Nash equilibrium associated to some costs. The leader
control is chosen to be the one of minimal cost.
Null controllability for the linear Schrödinger equation: with a convenient space-time
discretization, we numerically construct boundary controls which lead the solution of
the Schrödinger equation to zero; using some arguments of Fursikov-Imanuvilov (see
[Lecture Notes Series, Vol 34, 1996]) we construct controls with exponential decay at
final time.
Null controllability for a Schrödinger-KdV system: in this work, we combine global
Carleman estimates with energy estimates to obtain an observability inequality. The
controllability result holds by the Hilbert Uniqueness Method (HUM).
Controllability results for a Euler type system, incompressible, inviscid, under the influence
of a temperature are obtained: we mainly use the extension and return methods<br>Nesta tese, estudaremos resultados de controle para alguns problemas da teoria das equações
diferenciais parciais (EDPs):
Problema de controle multi objetivo para um problema parabólico, seguindo estratégias
do tipo Stackelberg-Nash: para cada controle líder, que impõe a controlabilidade nula
para o estado, encontramos seguidores, em equilíbrio de Nash, associados a funcionais
custo. Em seguida, determinamos o líder de menor custo. Controlabilidade nula para a equação de Schrödinger linear: com uma discretização
espaço-tempo adequada, construímos numericamente controles-fronteira que conduzem
a solução de Schrödinger a zero; utilizando técnicas de Fursikov-Imanuvilov (veja [Lecture
Notes Series, Vol 34, 1996]) contruímos controles que decaem exponencialmente no
tempo final.
Controlabilidade nula para um sistema acoplado Schrödinger-KdV: neste trabalho, combinando
estimativas globais de Carleman com estimativas de energia, obtemos uma desigualdade
de observabilidade. O resultado de controlabilidade segue pelo método de
unicicade Hilbert (HUM).
Controlabilidade para um sistema do tipo Euler, incompressível, invíscido, sob influência
de uma temperatura: Utilizamos os métodos de extensão seguido do método do retorno
para provar resultados de controlabilidade para este sistema
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Teo, Hhih-Ting, and h. teo@griffith edu au. "Tidal Dynamics in Coastal Aquifers." Griffith University. School of Engineering, 2003. http://www4.gu.edu.au:8080/adt-root/public/adt-QGU20030729.155028.
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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
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Guibourg, Sandrine. "MModélisations numérique et expérimentale des houles bidimensionnelles en zone cotière." Université Joseph Fourier (Grenoble), 1994. http://www.theses.fr/1994GRE10160.
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Une analyse theorique detaillee des equations de boussinesq et de serre a ete realisee. Les domaines de validite de chaque equation ont ete determines theoriquement. Ces equations d'ondes longues sont discretisees selon un schema aux differences finies pour des ondes de surface libre sur fond plat et fond variable. Par le biais d'une comparaison numerique avec des essais experimentaux d'ondes longues sur fond plat, les modeles numeriques ont ete etendus a la description des ondes courtes. Un terme dispersif correctif a ete introduit pour ameliorer les capacites dispersives des modeles. Des essais numeriques de propagation d'ondes longues sur un talus ont egalement ete compares aux experiences. Une etude de l'interaction d'une houle courte de haute frequence avec une onde solitaire nous a conduit a mesurer le dephasage que subit l'onde courte apres le passage du soliton. Nous nous sommes consacres a la validation experimentale d'une comparaison entre les modeles de boussinesq et de serre sur des plages peu inclinees, ainsi qu'a l'evolution du nombre d'ursell le long de la plage. L'etude experimentale a ensuite ete etendue aux phenomenes de run up, de run down et aux calculs des coefficients de reflexion des plages etudiees. Pour calculer numeriquement les run up, nous avons ameliore le modele de serre par des conditions de trait de cote variable
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Capistrano, Filho Roberto De Almeida. "Contrôle d'équations dispersives pour les ondes de surface." Thesis, Université de Lorraine, 2014. http://www.theses.fr/2014LORR0031/document.
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Dans cette thèse, nous prouvons des résultats concernant le contrôle et la stabilisation d'équations dispersives étudiées sur un intervalle borné. Pour commencer, nous étudions la stabilisation interne du système de Gear-Grimshaw, qui est un système de deux équations de Korteweg-de-Vries (KdV) couplées. Nous obtenons une décroissance exponentielle de l'énergie totale associée au modèle en introduisant une fonction de Lyapunov convenable. Nous prouvons aussi des résultats de contrôlabilité à zéro et exacte pour l'équation de Korteweg-de Vries avec un contrôle distribué à support dans un sous-intervalle du domaine. Pour la contrôlabilité à zéro du système linéarisé, nous utilisons l'approche classique basée sur la dualité qui ramène le problème à l'étude d'une inégalité d'observabilité qui, dans ce travail, est établie à l'aide d'une inégalité de Carleman. Ensuite, utilisant des fonctions plateau, nous prouvons un résultat de contrôlabilité exacte. Dans les deux cas, le résultat concernant le système non linéaire est obtenu à l'aide d'un argument de point fixe. Enfin, dans la lignée du résultat de contrôlabilité au bord obtenu par L. Rosier pour KdV, nous prouvons que le système linéaire de Boussinesq de type KdV-KdV est exactement contrôlable lorsque des contrôles sont appliqués au bord. Notre méthode repose sur l'utilisation de multiplicateurs et l'approche de la dualité mentionnée ci-dessus. Lorsqu'un mécanisme d'amortissement est introduit au bord, nous montrons que le système non linéaire est aussi exactement contrôlable et que l'énergie associée au modèle décroit exponentiellement<br>This work is devoted to prove a series of results concerning the control and stabilization properties of dispersive models posed on a bounded interval. Initially, we study the internal stabilization of a coupled system of two Korteweg-de Vries equations (KdV), the so-called Gear-Grimshaw system. Defining a convenient Lyapunov function we obtain the exponential decay of the total energy associated to the model. We also prove results of null and exact controllability for the Korteweg-de Vries equation with a control acting internally on a subset of the domain. In the case of the null controllability for the linear model, we use a classical duality approach which reduces the problem to the study of an observability inequality that, in this work, is proved by means of a Carleman inequality. Then, making use of cut-off functions, the exact controllability is also investigated. In both cases, the result for the nonlinear system is obtained by means of fixed-point argument. Finally, in view of the result of the boundary controllability obtained by L. Rosier for the KdV equation, we prove that the linear Boussinesq system of KdV-KdV type is exactly controllable when the controls act in the boundary conditions. Our analysis is performed using multipliers and the duality approach mentioned above. Adding a damping mechanism in the boundary, it is proved that the nonlinear system is also exactly controllable and that the energy associated to the model decays exponentially
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Kao, Cyril. "Fonctionnement hydraulique des nappes superficielles de fonds de vallées en interaction avec le réseau hydrographique." Phd thesis, ENGREF (AgroParisTech), 2002. http://tel.archives-ouvertes.fr/tel-00003957.
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L'objectif de cette thèse est de contribuer à une meilleure compréhension du fonctionnement hydraulique d'un système de nappe superficielle de fond de vallée, alimentée par un versant et drainée par un fossé en régime transitoire. Ceci doit aboutir à terme à un outil de modélisation permettant de tester des scénarios de gestion ou d'aménagements de ces zones humides. Le parti (pari ?) choisi a été de fonder les efforts de modélisation sur l'approche " saturée 1D " (équation de Boussinesq), tout en utilisant des modèles plus sophistiqués (Laplace, Richards) afin de servir de référence lors de la discussion et l'élaboration des hypothèses simplificatrices. En particulier, certaines conditions aux limites ont été étudiées : (i) hétérogénéité de la recharge de la nappe ; (ii) déterminisme des phénomènes de surface de suintement ; (iii) l'affleurement ; (ij) une condition aval transitoire. Des expérimentations ont été menées sur modèle physique au laboratoire (maquette MASHyNS) et sur le terrain (bassin versant du Ru de Cétrais, Loire Atlantique). Un modèle (SIDRA 2+), fondé sur une résolution numérique de l'équation de Boussinesq, a été utilisé et adapté à la prise en compte des conditions aux limites particulières évoquées. Le modèle a été calé et validé à partir des données expérimentales et a permis de prédire avec une excellente précision la position de la surface libre de la nappe à différentes distances du ru de Cétrais, en période hivernale.
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Wang, Yunli. "Etude expérimentale et numérique des oscillations hydrodynamiques en milieux poreux partiellement saturés." Thesis, Toulouse, INPT, 2010. http://www.theses.fr/2010INPT0127/document.
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Cette thèse vise à étudier expérimentalement, analytiquement et numériquement, les conséquences de variations et d'oscillations hydrodynamiques à forte variabilité temporelle en milieux poreux partiellement saturés. Les problèmes que nous étudions comportent des surfaces libres tant à l'extérieur qu'à l'intérieur des milieux poreux, celles-ci étant définies comme des isosurfaces de pression d'eau égale à la pression atmosphérique (Pwater = Patm). Les différentes études expérimentales réalisées en laboratoire sont, respectivement : une expérience d'imbibition dans une boite à sable avec effets capillaires importants; la transmission d'oscillations de la surface libre à travers un massif sableux intercalaire dans un petit canal à houle (IMFT, Toulouse); l'étude de la dynamique et de la propagation des oscillations des niveaux d'eau dans un grand canal à houle (HYDRALAB, Barcelone), partiellement recouvert d'un fond sableux incliné, avec mesures de niveaux d'eau en pleine eau et sous le sable, et mesures du fond sableux (érosion/dépôts). Pour les études théoriques, nous avons développés des solutions analytiques linéarisées. Un exemple de problème traité analytiquement est: l'équation linéarisée de Dupuit-Boussinesq (D-B) transitoire à surface libre, en hypothèse d'écoulements plans et vidange/remplissage instantané : oscillations forcées, transmission et dissipation d'ondes à travers une boite à sable rectangulaire. Nous avons aussi développé une solution de l'équation faiblement non linéaire de Dupuit- Boussinesq (D-B) pour étudier le problème d'imbibition avec variation abrupte du niveau d'eau amont (suivi temporel du front de saturation). Nous avons pu étudier les différents types de problèmes transitoires liés aux expériences citées plus haut par simulation numérique. En particulier, nous avons simulé des écoulements partiellement saturés et insaturés, en coupe verticale, à l'aide d'un code de calcul (BIGFLOW 3D) qui résoud l'équation de Richards généralisée en régime transitoire. Nous avons ainsi étudié numériquement en régime non saturé, l'expérience d'imbibition dans un sable initialement sec à frontières verticales (IMFT sandbox), puis l'expérience de propagation d'ondes dans le grand canal à houle de Barcelone (laboratoire HYDRALAB) comportant une plage de sable inclinée, avec un couplage complètement intégré entre les zones micro-poreuse (sable) et “macro-poreuse” (pleine eau). Pour analyser les résultats de cette dernière expérience et les comparer aux simulations, nous avons utilisé plusieurs méthodes de traitement et d'analyse des signaux : analyse de Fourier (spectres de fréquences) ; ondelettes discrètes multi-résolution (Daubechies) ; analyses corrélatoires simple et croisée. Ces méthodes sont combinées avec des méthodes de préfiltrage pour estimer dérives et résidus (moyennes mobiles ; ondelettes multi-résolution). Cette analyse des signaux a permis de comprendre et quantifier la propagation à travers une plage de sable. Au total, les différentes approches de modélisation mis en oeuvre, associé à des procédures de calage en situation de couplage transitoire non linéaire ont permis de reproduire globalement les phénomènes de propagation de teneur en eau et de niveau d'eau dans les différentes configurations étudiées<br>This thesis aims at investigating experimentally, analytically and numerically, the consequences of hydrodynamic variations and oscillations with high temporal variability in partially saturated porous media. The problems investigated in this work involve “free surfaces” both outside and inside the porous media, the free surface being defined as the “atmospheric” water pressure isosurface (Pwater = Patm). The laboratory experiments studied in this work are, respectively: Lateral imbibition in a dry sand box with significant capillary effects; Transmission of oscillations of the free surface through a vertical sand box placed in a small wave canal (IMFT, Toulouse); Dynamics of free surface oscillations and wave propagation in a large wave canal (HYDRALAB, Barcelona), partially covered with sand, with measurements of both open water and groundwater levels, and of sand topography (erosion / deposition). For theoretical studies, we have developed linearized analytical solutions. Here is a sample problem that was treated analytically in this work: The linearized equation of Dupuit-Boussinesq (DB) for transient free surface flow, assuming horizontal flow and instantaneous wetting/drainage of the unsaturated zone: forced oscillations, wave transmission and dissipation through a rectangular sandbox. We also developed a weakly nonlinear solution of the Dupuit-Boussinesq equation to study the sudden imbibition (temporal monitoring of the wetting front). We have studied the different types of transient flow problems related to the experiments cited above by numerical simulation. In particular, we have simulated unsaturated or partially saturated transient flows in vertical cross-section, using a computer code (BIGFLOW 3D) which solves a generalized version of Richards’ equation. Thus, using the Richards / BIGFLOW 3D model, we have studied numerically the experiment of unsaturated imbibition in a dry sand (IMFT sandbox), and then, with the same model, we have also studied the partially saturated wave propagation experiment in the large Barcelona wave canal (HYDRALAB laboratory), focusing on the sloping sandy beach, with coupling between the micro-porous zone (sand) and the “macro-porous” zone (open water). To interpret the results of the latter experiment and compare them to simulations, we use several methods of signal analyzis and signal processing, such as: Fourier analysis, discrete multi-resolution wavelets (Daubechies), auto and cross-correlation functions. These methods are combined with pre-filtering methods to estimate trends and residuals (moving averages; discrete wavelet analyses). This signal analyzis has allowed us to interpret and quantify water propagation phenomena through a sandy beach. To sum up, different modeling approaches, combined with model calibration procedures, were applied to transient nonlinear coupled flow problems. These approaches have allowed us to reproduce globally the water content distributions and water level propagation in the different configurations studied in this work
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Haddad, Adel. "Modèles numériques à faibles nombres de Mach pour l'étude d'écoulements en convection naturelle et mixte." Thesis, Aix-Marseille 1, 2011. http://www.theses.fr/2011AIX10154.
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Le modèle numérique que nous avons développé au cours de cette thèse présente deux caractéristiques principales : un modèle dilatable pour l'eau et la prise en compte de domaines ouverts. Les difficultés associées au premier aspect concernent l'adaptation de la loi d'état de l’eau au modèle dilatable sous l’approximation à faibles nombres de Mach, tandis que celles associées au second sont relatives à la mise en œuvre de conditions aux limites numériques de sortie compatibles avec l'algorithme de projection utilisé. Les résultats de simulations d'écoulement de convection mixte en canal horizontal chauffé par le bas ont été confrontés à celles utilisant l'approximation de Boussinesq et aux expériences<br>The 3D numerical model which we developed in this thesis presents two main features: a Low-Mach-Number approximation for water along with an open boundary condition formulation. Indeed, the difficulties related to the former point stand in a computationally efficient adaptation of the water equation of state in the framework of Low Mach number approximation, whereas the difficulties related to the latter concern the introduction of Open Boundary Conditions in the projection algorithm used. We have computed a mixed convection flow in a horizontal channel uniformly heated from below and compared the results obtained with both the Boussinesq approximation and experimental results
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Glenn-Levin, Jacob Benjamin. "Incompressible Boussinesq equations and spaces of borderline Besov type." Thesis, 2012. http://hdl.handle.net/2152/ETD-UT-2012-05-5143.
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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.<br>text
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Chou, Shih-En, and 周世恩. "Boussinesq Equations for Waves PropagatingOver Artificial Sand Bars." Thesis, 2003. http://ndltd.ncl.edu.tw/handle/50577071054634589500.
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碩士<br>國立成功大學<br>水利及海洋工程學系碩博士班<br>91<br>The one part of this paper is to develop the numerical model based on the 2nd — order fully nonlinear Boussinesq equations of Wei et al. (1995), and the Boussinesq model has been applied to compute wave fields for several cases of wave propagation for the rationality of the model. The another part of this paper is to apply the Boussinesq model to the simulation of the Bragg reflection of monochromatic and random waves due to artificial sand bars, for which experimental data have been presented by Davies and Heathershaw (1984) and Kirby and Anton (1990). The numerical results are compared with the theoretical solutions of Miles (1981) and the corresponding results using the evolution equation for mild slope equation of Hsu et al. (2003). For the monochromatic wave, the Boussinesq model can predict the reflection coefficients of the primary and second-harmonic resonance well. For the random waves, the reflection coefficients of the primary resonance are smaller and the reflection bandwidth is wider than the monochromatic wave, so the Bragg reflection of random waves is different from that of the monochromatic wave.
In addition, the Boussinesq model is applied to study the affecting factors of the Bragg reflection, including the number, the height and the spacing of artificial sand bars. The results are that increasing the number and the height of the sand bars, the reflection coefficients of the primary and second-harmonic resonance raise and increasing the spacing of sand bars, the reflection coefficients of the second-harmonic resonance increase, but that of the primary resonance decrease.
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Wei, Ge. "Simulation of water waves by Boussinesq models." 1997. http://catalog.hathitrust.org/api/volumes/oclc/40868412.html.
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Chang, Kuo-Wei, and 張國緯. "Numerical Calculations of One-Dimensional Wave Fields Using Boussinesq Equations." Thesis, 2001. http://ndltd.ncl.edu.tw/handle/29085108703215105427.
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碩士<br>國立臺灣大學<br>造船及海洋工程學研究所<br>90<br>The purpose of this research is to develop a numerical model based on Boussinesq equations that can predict the wave transformation. By [2,2] Padé approximation, different velocity parameter Boussinesq equations are derived. The effects of different Boussinesq type equations on linear dispersion relations, group velocity and shoaling gradients were discussed. The limitation of on the applications of Boussinesq equations can be extended to 0.5 under suitable choice of the velocity parameter with the requirement of the difference of the phase velocities from calculation and from linear dispersion relation less than 5﹪. The fourth-order Adams-Bashforth-Moulton predictor-corrector scheme with proper absorbing boundary conditions was imposed as the basic numerical scheme. Finally, numerical results were verified with past experimental and theoretical results.
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Tseng, I.-Fan, and 曾以帆. "On the Improvement of Boundary Conditions and Applications of Boussinesq Equations." Thesis, 2005. http://ndltd.ncl.edu.tw/handle/40738647075016694891.
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博士<br>國立成功大學<br>水利及海洋工程學系碩博士班<br>93<br>To improve the weak nonlinearity and weak dispersion of the classical Boussinesq equation, a 2nd-order fully nonlinear Boussinesq model based on Wei and Kirby (1995)’s scheme is established in this study. This model also uses the eddy viscosity technique to model breaking, and a “slotted beach” to simulate run-up phenomena. The damping coefficients of the sponge layer boundary in this model are derived theoretically. The present result differs from former researches in which the free parameters in the damping coefficients are suggested by numerical tests to control the effect of the sponge layer. Numerical experiments show that the proposed damping coefficients work efficiently on reducing the energy of reflected waves from the sponge layer. The numerical tests are performed to verify the applicability and validity of the present model.
The present model is performed to simulate the deformation of waves propagating over the varying topography, including shoaling, breaking, recovery, runup and setup, etc. With different wave conditions and beach slopes, numerical analysis of the surf similarity parameter, runup elevation and reflection coefficient result in extended range of the empirical formulas.
This study is also applied to simulate the Bragg reflection of monochromatic and random waves due to artificial sand ripples. The numerical results are compared with the theoretical solutions of Miles (1981), and with the corresponding results using the evolution equation for mild slope equation of Hsu et al. (2003) and the experimental data. For the monochromatic wave, the present model can predict the reflection coefficients of the primary and second-harmonic resonance well. For the random waves, the reflection coefficients of the primary resonance are smaller and the reflection bandwidth is wider than the monochromatic wave, so the Bragg reflection of random waves is different from that of the monochromatic wave. In addition, present model is applied to study the affecting factors of the Bragg reflection, including the number, the height and the spacing of artificial sand ripples.
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Huang, Yuan-Fang, and 黃遠芳. "A Study in Boussinesq Equations for Wave Transformations on Porous Beds." Thesis, 2003. http://ndltd.ncl.edu.tw/handle/79113456888759653372.
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碩士<br>國立臺灣大學<br>工程科學與海洋工程學系<br>91<br>The purpose of this research is to develop a numerical model for studying the properties of wave transformations over porous beds. The basic equations, called Boussinesq equations, which is a set of vertically-integrated equations for the porous bed. By the analysis of the dispersion relations, the limitation of relative depth can be extended to 0.5 if the maximum relative error for the phase celerity is less than 5%. The fourth-order Adams-Bashforth-Moulton predictor-corrector scheme with proper absorbing boundary conditions is imposed for numerical scheme. Consider a wave passing a flat porous bed, the wave height and velocity would be influenced by the different properties of porous beds. If the porosity is fixed, the wave height and velocity would decrease by the increase of permeability. Moreover, in order to verify the usefulness of the present model, the present of results were compared with results computed by Cruz et al. (1997).
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Chen, Ying-Chih, and 陳盈智. "Comparison of Numerical Solution of Shallow-Water Equations and Boussinesq Equations for Wave Propagation over a Submerged Bar." Thesis, 2012. http://ndltd.ncl.edu.tw/handle/26891633165150445527.
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碩士<br>國立臺灣海洋大學<br>海洋環境資訊學系<br>100<br>Numerical model of shallow water equations (SWE) and Boussinesq equations (BE) based on space-time least-square finite element method (STLSFEM) are developed to study wave deformation and wave-bathymetry interactions. Condition number of the resulting system of equations serves as an indicator for choosing of variables, interpolations, and possible treatment of boundary conditions. Test cases include (1) standing wave in a closed flat basin, (2) propagation of solitary wave in a flat channel, (3) sinusoidal wave propagation in a flat channel, and (4) propagation of sinusoidal wave over a submerged bar (Beji and Battjes, 1993). Computed results are compared with linearized approximation or available experiment data. Results of the SWE and BE model are consistent with exact solution for shallow waters (long waves). However, predictions of SWE model show inaccuracy in deep waters (short waves). In the propagation of sinusoidal wave over a submerged bar simulation, comparison of computed results of non-dispersive SWE model with experiment data shows that the non-dispersive SWE model is not accurate in the deep waters (short waves) and high-harmonics are generated due to non-linear effect. In order to accurately simulate the deep water waves, the dispersive BE or non-hydrostatic pressure model is suggested for future study.
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Chiang, Wen-Ping, and 江文平. "Comparison of Numerical Solution of Shallow-Water Equations and Boussinesq Equations for Standing Wave in a Flat Closed Basin." Thesis, 2012. http://ndltd.ncl.edu.tw/handle/45763866724885134655.
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碩士<br>國立臺灣海洋大學<br>海洋環境資訊學系<br>100<br>Both shallow-water equations (SWE) and Boussinesq equations (BE) are used to simulate standing waves in a flat closed basin with various water depths, ranging from shallow waters to deep waters, where h is the water depth and is the wave length, respectively. Numerical model is based on the space-time least-squares finite-element method (STLSFEM). It is found that computed wave velocity of shallow water waves of both SWE and BE model agrees well with solution of the Airy theory. Computed wave velocity of intermediate water waves and deep water waves of SWE model does not agree with value of Airy theory due to neglecting of vertical acceleration and dispersion. Dispersive BE model gives better prediction of wave amplitude, frequency and conservation property than non-dispersive SWE model does for intermediate and deep water waves.
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Kim, Dae Hong. "Turbulent Flow and Transport Modeling by Long Waves and Currents." 2009. http://hdl.handle.net/1969.1/ETD-TAMU-2009-08-6979.
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This dissertation presents models for turbulent flow and transport by currents
and long waves in large domain.
From the Navier-Stokes equations, a fully nonlinear depth-integrated equation
model for weakly dispersive, turbulent and rotational flow is derived by a perturbation
approach based on long wave scaling. The same perturbation approach is applied
for the derivation of a depth-integrated transport equation. As the results, coherent
structures generated by the turbulence induced by the bottom friction and topography
can be predicted very reasonably.
The three dimensional turbulence effects are incorporated into the flow model by
employing a back scatter model. The back scatter model makes it possible to predict
turbulent transport: It contributes to the energy transport and the lateral turbulent
diffusion through relying on the turbulent intensity, not by relying on an empirical
diffusion constant. The inherent limitation of the depth-integrated transport equation,
that is, the limitation for the near field prediction is recognized in the derivation
and the numerical simulation.
To solve the derived equation set, a highly accurate and stable finite volume
scheme numerical solver is developed. Thus, the numerical solver can predict dispersive
and nonlinear wave propagation with minimal error. Also, good stability is
achieved enough to be applied to the dam-break flows and undular tidal bores. In addition, a robust moving boundary scheme based on simple physical conditions is
presented, which can extend the applicability area of the depth-integrated models.
By the comparison study with experimental data, it is expected that the numerical
model can provide high confidence results for the wave and current transformations
including shocks and undular bores on complex bathymetry and topography. For
the accurate near field transport prediction, a three dimensional transport model in
?-coordinate coupled with the depth-integrated flow model is developed. Like the
other models, this model is also intended for large domain problems, and yet efficient
and accurate in the far field and near field together.
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Panda, Nishant. "Discontinuous Galerkin methods for resolving non linear and dispersive near shore waves." Thesis, 2014. http://hdl.handle.net/2152/26857.
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Near shore hydrodynamics has been an important research area dealing with coastal processes. The nearshore coastal region is the region between the shoreline and a fictive offshore limit which usually is defined as the limit where the depth becomes so large that it no longer influences the waves. This spatially limited but highly energetic zone is where water waves shoal, break and transmit energy to the shoreline and are governed by highly dispersive and non-linear effects. An accurate understanding of this phenomena is extremely useful, especially in emergency situations during hurricanes and storms. While the shallow water assumption is valid in regions where the characteristic wavelength exceeds a typical depth by orders of magnitude, Boussinesq-type equations have been used to model near-shore wave motion. Unfortunately these equations are complex system of coupled non-linear and dispersive differential equations that have made the developement of numerical approximations extremely challenging. In this dissertation, a local discontinuous Galerkin method for Boussinesq-Green Naghdi Equations is presented and validated against experimental results. Currently Green-Naghdi equations have many variants. We develop a numerical method in one horizontal dimension for the Green-Naghdi equations based on rotational characteristics in the velocity field. Stability criterion is also established for the linearized Green-Naghdi equations and a careful proof of linear stability of the numerical method is carried out. Verification is done against a linearized standing wave problem in flat bathymetry and h,p (denoted by K in this thesis) error rates are plotted. The numerical method is validated with experimental data from dispersive and non-linear test cases.<br>text