Este trabalho aborda a solução numérica das equações lineares de águas rasas. O método dos elementos finitos e utilizado para a discretização espacial das equações que modelam o problema, e para a discretização temporal, o esquema semi-implícito de Crank-Nicolson é empregado. Além de alguns conceitos comuns quando se trabalha com escoamentos geofísicos, são descritas também a formulação das equações de águas rasas, sua linearização e uma solução analítica para um caso onde o parâmetro de Coriolis é nulo. A escolha adequada de pares de elementos finitos é a principal dificuldade quando se trabalha com esse método para a resolução da equação de águas rasas. Assim, é discutido o uso de quatro pares de elementos finitos e técnicas de estabilização para contornar o surgimento de modos espúrios na solução discreta. Os resultados numéricos são realizados com auxílio do software FreeFem++, onde se pode notar a capacidade dos pares de elementos de reproduzirem o escoamento, através da solução discreta, além das propriedades de conservação de massa e energia de cada discretização. / This work is about the numerical solution of the linear shallow water equations. The finite element method is used for spatial discretization of the equations that model the problem and for the time discretization the semi-implicit Crank-Nicolson scheme is used. Besides the concepts related to geophysical flows, the formulation of the shallow water equations, their linearization and an analytical solution for a case where the Coriolis parameter is zero are also described. The appropriate choice of a pair of finite elements is the main difficulty when working with this method for solving the shallow water equations. The use of four pairs of finite elements and stabilization techniques to circumvent the appearance of spurious modes in the discrete solution are discussed. The numerical results are performed using the software FreeFem++, where one can notice the ability of the elements to represent the discrete solution and mass and energy conservation of each discretization.

Tato diplomová práce se zabývá matematickým modelováním vodních vln v blízkosti pobřeží pomocí parciálních diferenciálních rovnic. Cílem této práce je formulace pohybových rovnic a jejich následné numerické řešení s grafickou interpretací dosažených výsledků.

Shallow water equations are widely used to model water flows in the field of hydrodynamics and civil engineering. They are complex, and except for some simplified cases, no analytical solution exists for them. Therefore, the partial differential equations of the shallow water system have been the subject of various numerical analyses and studies in past decades. In this study, we construct a stable and robust finite volume scheme for the shallow water equations over quadtree grids. Quadtree grids are two-dimensional semi-structured Cartesian grids that have different applications in several fields of engineering, such as computational fluid dynamics. Quadtree grids refine or coarsen where it is required in the computational domain, which gives the advantage of reducing the computational cost in some problems. Numerical schemes on quadtree grids have different properties. An accurate and robust numerical scheme is able to provide a balance between the flux and source terms, preserve the positivity of the water height and water surface, and is capable of regenerating the grid with respect to different conditions of the problem and computed solution. The proposed scheme uses a piecewise constant approximation and employs a high-order Runge-Kutta method to be able to make the solution high-order in space and time. Hence, in this thesis, we develop an adaptive well-balanced positivity preserving scheme for the shallow water system over quadtree grids utilizing different techniques. We demonstrate the formulations of the proposed scheme over one of the different configurations of quadtree cells. Six numerical benchmark tests confirm the ability of the scheme to accurately solve the problems and to capture small perturbations. Furthermore, we extend the proposed scheme to the coupled variable density shallow water flows and establish an extended method where we focus on eliminating nonphysical oscillations, as well as well-balanced, positivity preserving, and adaptivity properties of the scheme. Four different numerical benchmark tests show that the proposed extension of the scheme is accurate, stable, and robust.

Title: Barotropic ocean tide model Author: David Einšpigel Department: Department of Geophysics Supervisor: prof. RNDr. Zdeněk Martinec, DrSc., Department of Geophysics Abstract: The main aim of this thesis is developing of a numerical model of an oceanic circulation forced by the lunisolar tidal potential. The circulation is described by the shallow water equations which are derived from the fundamental balance laws assuming that the ratio of the vertical and horizontal dimension of the investigated problem is small, which leads to the formulation of the 2- D task. Furthermore, programs for solving the shallow water equations were written. Their functionality is demonstrated on several examples. The programs include an ephemeridal tidal modul which computes the complete lunisolar tidal potential. Keywords: oceanic flow, shallow water equations, lunisolar tidal potential

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Yes / A two-dimensional model based on OpenTidalFarm is applied to simulate tidal stream flow around turbines. The model is governed by shallow water equations and is able to optimize the layout of the deployed turbine array in terms of maximizing the energy outputs. Three turbine array layouts including two structured layouts (regular and staggered) and one unstructured layout (optimized) are simulated to investigate the effect of turbine layouts on energy extraction. The present study shows that more energy could be extracted when lateral spacing decreases and longitudinal spacing increases within the same domain, namely the effective turbine layout is to deploy more turbines in the first row to extract energy from undisturbed tidal stream, while larger longitudinal spacing will make it possible for tidal stream to recover more before reaching the next turbines row. Taking the tidal stream turbines array around Zhoushan Islands as a case study, results show that the optimized layout can extract 106.8% energy of that extracted by the regular and staggered layout for a full tide in the same marine area. Additionally, the turbine array has a great influence on tidal stream velocities immediately behind the array and has little effect on far-field wake flow. / National Natural Science Foundation Council of China (51879098), and the Marine Renewable Energy Research Project of State Oceanic Administration (GHME2015GC01).

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.

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.

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.

The two-dimensional depth-averaged shallow water equations have attracted considerable attentions as a practical way to solve flows with free surface. Compared to three-dimensional Navier-Stokes equations, the shallow water equations give essentially the same results at much lower cost. Solving the shallow water equations by the Godunov-type finite volume method is a newly emerging area. The Godunov-type finite volume method is good at capturing the discontinuous fronts in numerical solutions. This makes the method suitable for solving the system of shallow water equations. In this dissertation, both the shallow water equations and the Godunov-type finite volume method are described in detail. A new surface flow routing method is proposed in the dissertation. The method does not limit the shallow water equations to open channels but extends the shallow water equations to the whole domain. Results show that the new routing method is a promising method for prediction of watershed runoff. The method is also applied to turbulence modeling of free surface flow. The κ - ε turbulence model is incorporated into the system of shallow water equations. The outcomes prove that the turbulence modeling is necessary for calculation of free surface flow. At last part of the dissertation, a total load sediment transport model is described and the model is tested against 1D and 2D laboratory experiments. In summary, the proposed numerical method shows good potential in solving free surface flow problems. And future development will be focusing on river meandering simulation, non-equilibrium sediment transport and surface flow - subsurface flow interaction.