Ce travail s'inscrit dans l'étude mathématique d'écoulements de fluides environnementaux. Nous en abordons deux aspects, à travers deux contextes distincts d'application.En lien avec la simulation des écoulements en milieux poreux, on s'intéresse dans une première partie à la discrétisation d'opérateurs de diffusion anisotropes hétérogènes par des méthodes de volumes finis sur des maillages généraux. Dans le but d'obtenir des solutions approchées qui respectent les bornes physiques des modèles, notre attention se porte sur la conservation du principe du maximum pour les opérateurs elliptiques. Nous présentons des mécanismes généraux permettant de corriger tout schéma volumes finis afin de garantir un principe du maximum discret tout en préservant certaines de ses propriétés principales. On étudie en particulier les propriétés de coercivité et de convergence des schémas corrigés.La deuxième partie est consacrée à la construction de modèles approchés pour la propagation des vagues en eaux peu profondes et sur des topographies irrégulières. A cet effet, nous proposons tout d'abord une adaptation de la démarche d'étude classique à des écoulements bidimensionnels sur des topographies polygonales. Dans un cadre plus général, nous développons ensuite une démarche formelle qui débouche sur des alternatives non locales à quelques modèles classiques (équations de Saint-Venant, équations de Serre, système de Boussinesq). Ces nouveaux modèles contiennent des termes régularisants pour les contributions du fond. / This work investigates two research questions associated with environmental flows and their mathematical modeling.The first part is devoted to the development of finite volume methods for anisotropic and heterogeneous diffusion operators arising in models of porous media flows. To ensure that the approximate solutions lie within physical bounds, we aim at maintaining a discrete analogous of the maximum principle for elliptic operators. Starting from any given cell-centered finite volume scheme, we present a general approach to devise non-linear corrections providing a discrete maximum principle while retaining some main properties of the scheme. In particular, we study the coercivity and convergence properties of the modified schemes.The second part of this work focuses on the derivation of approximate models for shallow water wave propagation over rough topographies. In the particular case of one-dimensional polygonal bottom profiles, we first propose an adaptation of the usual derivation method using complex analysis tools. We then develop a formal approach to account for more general topographies. We propose nonlocal alternatives to some classical models (namely Saint-Venant equations, Serre equations and Boussinesq system). All these alternative models only involve smoothing contributions of the bottom.

Les courants turbulents de bord ouest sont l'un des phénomènes les plus dominants des océans, il en existe aux faibles latitudes aussi. Ils sont caractérisés par une dynamique très turbulente avec une forte production d'énergie cinétique, et une forte variabilité interne. Plusieurs régions existent où les courants de bord ouest se rétrofléchissent (décollage de la côte) pour former des structures cohérentes: des anticyclones, des bursts (arrachements) et des dipoles. Circulant le long de la côte, les courants de bord ouest interagissent très fortement avec le bord ouest et la bathymétrie et sont donc un problème de couche limite. Cependant aucune étude du point de vue de la théorie de couche limite n'a été jamais été faite. Cette thèse aborde le problème d'un point de vue de couche limite par l'utilisation d'un modèle idéalisé "shallow water" à très haute résolution (2.5km) afin d'isoler et de comprendre les processus. Les résultats sont ensuite appliqués à des sorties de modèle réaliste Drakkar (~10km) basé sur le code NEMO. Le courant de Somali est ensuite pris pour cette application. / Strong western boundary currents are one of dominant features of the world oceans, also at low latitudes. They exhibit a turbulent dynamics and their region is a source of strong kinetic energy production and internal variability of the worlds oceans. Several places exists where the western boundary currents retrofect (i.e separation from the coast) and generate coherent structures as anticyclonic eddies, bursts and dipoles. The dynamics of turbulent western boundary currents has so far not been extensively studied in the viewpoint of turbulent boundary-layer theory. The approach followed in this thesis is to use a fine resolution (2.5km) reduced-gravity shallow water model to understand the turbulent boundary-layer processes and then apply these findings to the Ocean General Circulation Model NEMO in the Drakkar configuration (~10km). The case of the Somali Currentis considered for this application.

Utilising tidal currents as a renewable energy resource is presently under consideration to meet the requirements of increasing worldwide energy demand and the need to reduce carbon emissions. In this respect, in-stream tidal devices are proposed to convert the kinetic energy of currents into useful extractable power. In order to extract a useful amount of energy from tidal currents, the proposed devices need to be deployed in an array or farm-like format. Due to the thrust exerted by the devices within an array, the natural flow regime will inevitably be changed. In light of this, this study aims to estimate the maximum power that can be extracted by tidal turbine arrays and assess the far-field effects of energy extraction in the designated areas around the UK for various array configurations. In this thesis, the ocean tides are modelled using the long wave equations, commonly referred as the shallow water equations (SWEs). A numerical solver based on a Runge-Kutta discontinuous Galerkin finite element method is employed to solve the SWEs. One main advantage of the discontinuous Galerkin method is that it approximates the solution individually at each element, which allows for discontinuities within the solution system while ensuring mass conservation locally and globally. The selected numerical solver has been verified against several benchmark tests. It is then modified to include a line discontinuity to represent the effect of tidal turbine array(s) in a coastal basin. The algorithm implemented in the numerical solver involves a sub-grid model, which is based on Linear Momentum Actuator Disk Theory (LMADT) to approximate the local flow-field in the presence of the turbines. This near-field approach allows the flow velocity at the turbine to be estimated with a greater accuracy. As the power available to the turbines is related to the velocity at the turbine blades, the characterisation of the designated tidal site as a resource using LMADT may be more accurate than previously proposed methods. An additional advantage of using LMADT is that it provides a distinction between the power extracted by the turbines and the total amount of power that is removed from the tidal stream, including the wake mixing losses. The methodology employed in this thesis has been applied to two tidal basins around the UK; the Anglesey Skerries (a headland) and the Bristol Channel (an oscillating bay). A comprehensive unstructured triangular finite element model has been constructed to simulate the naturally occurring tides at these regions. The constructed model has then been validated against field measurement. The validated model is used to conduct parametric studies, which evaluate the importance of tidal array locations, configurations and operating conditions on the available power at the Anglesey Skerries and the Bristol Channel sites. The parametric study aims to evaluate a realistic upper limit of available power at each site considered. This study also provides a unique analysis to examine the potential tidal farm interactions by deploying several tidal arrays at both Anglesey Skerries and the Bristol Channel.

Bed topography identification in open channel and glacier flows is of paramount importance for the study of the respective flows. In the former, the knowledge of the channel bed topography is required for modelling the hydrodynamics of open channel flows, fluvial hydraulics, flood propagation, and river flow monitoring. Indeed, flow models based on the Shallow Water Approximation require prior information on the channel bed topography to accurately capture the flow features. While in the latter, usable bedrock topographic information is very important for glacier flow modellers to accurately predict the flow characteristics. Experimental techniques to infer the bed topography are usually used but are mostly time consuming, costly, and sometimes not possible due to geographical restrictions. However, the measurement of free surface elevation is relatively easy. Alternative to experimental techniques, it is therefore important to develop fast, easy-to-implement, and cost-effective numerical methods. The inverse of the classical hydrodynamic problem corresponds to the determination of hydraulic parameters from measurable quantities. The forward problem uses model parameters to determine measurable quantities. New one-shot and direct pseudo-analytical and numerical approaches for reconstructing the channel bed topography from known free surface elevation data is developed for one-dimensional shallow water flows. It is shown in this work that instead of treating this inverse problem in the traditional partial differential equation (PDE)-constrained optimization framework, the governing equations of the direct problem can be conveniently rearranged to obtain an explicit PDE for the inverse problem. This leads to a direct solution of the inverse problem which is successfully tested on a range of benchmark problems and experimental data for noisy and noiseless free surface data. It was found that this solution approach creates very little amplification of noise. A numerical technique which uses the measured free surface velocity to infer the channel bed topography is also developed. The one-dimensional shallow water equations along with an empirical relationship between the free surface and the depth averaged velocities are used for the inverse problem analysis. It is shown that after a series of algebraic manipulation and integration, the equation governing the inverse problem simplifies to a simple integral equation. The proposed method is tested on a range of analytical and experimental benchmark test cases and the results confirm that, it is possible to reconstruct the channel bed topography from a known free surface velocity distribution of one-dimensional open channel flows. Following the analysis of the case of one-dimensional shallow water flows, a numerical technique for reconstructing the channel bed topography from known free surface elevation data for steep open channel flows is developed using a modified set of equations for which the zero-inertia shallow water approximation holds. In this context, the shallow water equations are modified by neglecting inertia terms while retaining the effects of the bed slope and friction terms. The governing equations are recast into a single first-order partial differential equation which describes the inverse problem. Interestingly, the analysis shows that the inverse problem does not require the knowledge of the bed roughness. The forward problem is solved using MacCormack’s explicit numerical scheme by considering unsteady modified shallow water equations. However, the inverse problem is solved using the method of characteristics. The results of the inverse and the forward problem are successfully tested against each other. In the framework of full two-dimensional shallow water equations, an easy-to-implement and fast to solve direct numerical technique is developed to solve the inverse problem of shallow open channel flows. The main underlying idea is analogous to the idea implemented for the case of one-dimensional reconstruction. The technique described is a “one-shot technique” in the sense that the solution of the partial differential equation provides the solution to the inverse problem directly. The idea is tested on a set of artificial data obtained by first solving the forward problem. Glaciers are very important as an indicator of future climate change or to trace past climate. They respond quickly compared to the Antarctica and Greenland ice sheets which make them ideal to predict climate changes. Glacier bedrock topography is an important parameter in glacier flow modelling to accurately capture its flow dynamics. Thus, a mathematical technique to infer this parameter from measured free surface data is invaluable. Analogous to the approaches implemented for open channel flows, easy-to-implement direct numerical and analytical algorithms are developed to infer the bedrock topography from the knowledge of the free surface elevation in one space dimension. The numerical and analytical methods are both based on the Shallow Ice Approximation and require the time series of the ablation/accumulation rate distribution. Moreover, the analytical method requires the knowledge of a non-zero glacier thickness at an arbitrary location. Numerical benchmark test cases are used to verify the suitability and applicability of the algorithms.

Coarse-grained beaches are particularly prevalent in the UK, composed of accumulations of either gravel, or mixed sand and gravel sediments. The aim of the work presented in this thesis is to improve capabilities for predicting coarse-grained beach 2D profile development. In particular, the effects of infiltration and sediment sorting are considered. In this study, the public domain numerical model, XBeach (v12) is developed further. This model was initially developed for studying sandy environments especially for the case of dune erosion. Here, the model is modified to enhance its capability to predict beach profile change on coarse-grained beaches. Improvements include: use of Lagrangian interpretation of velocity in place of Eulerian for driving sediment movement; introduction of a new morphological module based upon Soulsby’s sediment transport equation for waves and currents; incorporation of Packwood’s infiltration approach in the unsaturated area of the swash region; and implementation of a multiple sediment fraction algorithm for sediment sorting of mixed sediments. These changes are suggested and justified in order to significantly improve the application of this model to gravel and mixed beaches, especially with regard to swash velocity asymmetry which is responsible for development of the steep accretionary phase steep berm above waterline and sediment sorting. A comparison between model simulation and large scale experiments is presented with particular regard to the tendency for onshore transport and profile steepening during calm conditions; offshore transport and profile flattening during storm conditions; and sediment sorting in the swash zone. Data used for this and the model calibration comes from the Large Wave Channel (GWK) of the Coastal Research Centre (FZK) in Hannover, Germany. The results are found to agree well with the measured experimental data on gravel beach profile evolution. This is due to the inclusion of infiltration in the model which weakens the backwash volume and velocity in a more satisfying manner than through the use of asymmetric swash friction and transport coefficient. The model also simulates sediment sorting of a mixed sediment beach. However, the profile comparisons were not satisfactory due to limitations of the numerical model such as the constant permeability rate used throughout the simulation and the non-conservation of the sediment volume in the laboratory data by an order of 50%. From the simulation, it was found that the fine sediment moves offshore and the coarser sediment moves onshore. This is because of infiltration weakens the backwash velocity; the coarser sediment moving onshore barely moves back offshore while the fine sediment remains in motion. This pattern agrees with the pattern obtained from sediment samples analysis in the experiment and provides an explanation for the existence of composite beaches. The model is also shown to be capable of switching from accretionary to erosive conditions as the wave conditions become more storm-like. Again, the model simulations were in a good agreement with the observations from the GWK dataset. Numerical model simulations on the effects of the tidal cycle on coarse-grained beach profile evolution were also carried out. This preliminary investigation showed that the model was able to predict the anticipated profile change associated with a coarse-grained beach under such wave and tidal forcing. Tidally forced accretion and erosion were compared with those predicted under similar beach sediments and wave conditions for constant water level. The main differences are that the affected area is wider and the berm is located on the upper beach during flood for both gravel and mixed beaches. Therefore, the model developed in this study can be seen to be a robust tool with which to investigate cross-shore beach profile change on coarse-grained beaches and sediment sorting on mixed beaches. Further work is also indicated.

Esta dissertação tem por objetivo a implementação de um código para simular problemas hidrodinâmicos, bem como a possibilidade de controlar as elevações de onda resultantes numa determinada região por meio de uma vazão ótima controlada dentro do sistema estudado. O algoritmo implementado é baseado nas equações de águas rasas, as quais são aplicáveis em situações onde a altura d’água é de ordem muito menor do que as dimensões do sistema, que é discretizado espacial e temporalmente pelo Método dos Elementos Finitos e pelo método CBS (Characteristic Based-Split), respectivamente. O método de controle consiste na busca de uma curva de vazão de controle ótima que minimize a função objetivo, a qual compara os valores de altura d’água que se deseja encontrar em uma região especificada com os calculados pela simulação numérica. Para isso, utiliza-se um algoritmo evolutivo SCE-UA (Shuffled Complex Evolution – University of Arizona), que busca otimizar parâmetros de geração das curvas de vazão de controle, podendo estas serem modeladas por NURBS (Non- Uniform Rational B-Splines), que são capazes de encontrar a solução ótima, ou modeladas com curvas de forma triangular (linear) ou parabólica (quadrática) que apresentam uma solução aproximada de fácil implementação. Por fim, várias aplicações são realizadas, tanto para a simples simulação, quanto para o controle de problemas hidrodinâmicos, a fim de validar os algoritmos desenvolvidos e os resultados obtidos mostraram que os objetivos foram alcançados, encontrando uma forma eficiente de se fazer o controle de enchentes. / Implementation of a computational code for the numerical simulation of hydrodynamic problems as well as the ability to control the resulting wave elevations in a specific area, using an optimal flow controlled within the studied system are the aims of this work. The implemented algorithm is based on the shallow waters equations, which are applicable in situations where the water height is much smaller than the system dimensions, and are spatial and temporally discretized by the Finite Element Method and the CBS method (Caractheristic Based-Split), respectively. The control method consists in finding an optimal control flow curve that minimizes the objective function, which compares the objective value of water elevations in a specified region with those calculated by numerical simulation. An evolutionary algorithm called SCE-UA (Shuffled Complex Evolution - University of Arizona), which looks for optimize parameters of control flow curves generation, is used. These curves may be modeled by NURBS (Non-Uniform Rational B-Splines) which are able to find the optimal solution, or by curves of triangular (linear) or parabolic quadratic forms, which are an approximate solution easy to implement. Finally, several applications are performed for both simulation and control of hydrodynamic problems in order to validate the developed algorithms, and the results showed that the aims of this work were reached, finding an efficient way to control floods.

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In shallow water environments, sound propagation experiences multiple interactions with the surface/bottom interfaces, with hydrodynamic disturbances such as internal waves, and with tides and fronts. It is thus very difficult to make satisfactory predictions of sound propagation in shallow water. Given that many of the ocean characteristics can be modeled as stochastic processes, the statistical measure, spatial coherence, is consequently an important quantity. Spatial coherence provides valuable information for array performance predictions. However, for the case of long-range, low frequency propagation, studies of spatial coherence influenced by various environmental parameters are limited insofar as having the appropriate environmental data with which to model and interpret the results. The comprehensive Asian Seas International Experiment 2001 (ASIAEX01) examined acoustic propagation and scattering in shallow water. Environmental oceanographic data were taken simultaneously with the acoustic data. ASIAEX01 provided a unique data set which enabled separate study of the characteristics of the oceanographic features and their influence on long range sound propagation. In this thesis, the environmental descriptors considered include sediment sound speed and attenuation, background internal waves, episodic non-linear internal waves, and air-sea interface conditions. Using this environmental data, the acoustic data are analyzed to show the characteristics of spatial coherence in a shallow water waveguide. It is shown that spatial coherence can be used as an inversion parameter to extract geoacoustic information for the seabed. Environmental phenomena including internal waves and wind-generated surface waves are also studied. The spatial and temporal variations in the sound field induced by them are presented. In addition, a tank experiment is presented which simulates propagation in a shallow water waveguide over a short range. Based on the data model comparison results, the model proposed here is effective in addressing the major environmental effects on sound propagation in shallow water.

This thesis investigates the feasibility of using fluid flow as a deformable model for segmenting vessels in 2D and 3D medical images. Exploiting fluid flow in vessel segmentation is biologically plausible since vessels naturally provide the medium for blood transportation. Fluid flow can be used as a basis for powerful vessel segmentation because streaming fluid regions can merge and split providing topological adaptivity. In addition, the fluid can also flow through small gaps formed by imaging artifacts building connections between disconnected areas. In our study, due to their simplicity, parallelism, and low computational cost compared to other fluid simulation methods, linearized shallow water equations (LSWE) are used. The method developed herein is validated using synthetic data sets, two clinical datasets, and publicly available simulated datasets which contain Magnetic Resonance Angiography (MRA) images, Magnetic Resonance Venography (MRV) images and retinal angiography images. Depending on image size, one to two order of magnitude speed ups are obtained with developed parallel implementation using Nvidia Compute Unified Device Architecture (CUDA) compared to single-core and multicore CPU implementation.

Analytical solutions for the linear and nonlinear shallow-water wave equations are developed for evolution and runup of tsunamis &ndash / long waves&ndash / over one- and two-dimensional bathymetries. In one-dimensional case, the nonlinear equations are solved for a plane beach using the hodograph transformation with eigenfunction expansion or integral transform methods under different initial conditions, i.e., earthquake-generated waves, wind set-down relaxation, and landslide-generated waves. In two-dimensional case, the linear shallow-water wave equation is solved for a flat ocean bottom for initial waves having finite-crest length. Analytical verification of source focusing is presented. The role of focusing in unexpectedly high tsunami runup observations for the 17 July 1998 Papua New Guinea and 17 July 2006 Java Island, Indonesia tsunamis are investigated. Analytical models developed here can serve as benchmark solutions for numerical studies.