京都大学 / 0048 / 新制・課程博士 / 博士(工学) / 甲第22756号 / 工博第4755号 / 新制||工||1744(附属図書館) / 京都大学大学院工学研究科都市社会工学専攻 / (主査)教授 細田 尚, 教授 戸田 圭一, 准教授 音田 慎一郎 / 学位規則第4条第1項該当 / Doctor of Philosophy (Engineering) / Kyoto University / DFAM

Mathematical modeling of ocean waves is based on the formulation and solution of the appropriate equations of continuity, momentum and the choice of proper initial and boundary conditions. Under the influence of gravity, many free surface water waves can be modeled by the shallow water equations (SWE) with the assumption that the horizontal length scale of the wave is much greater than the depth scale and the wave height is much less than the fluid's mean depth. Furthermore, to describe three dimensional flows in the hydrostatic and Boussinesq limits, the multilayer SWE model is used, where the fluid is discretized horizontally into a set of vertical layers, each having its own height, density, horizontal velocity and geopotential. In this study, we used an explicit staggered finite volume method to solve single and multilayer SWE, with and without density stratification and bathymetry, to understand the dynamic of surface waves and internal waves. We implemented a two-dimensional version of the incompressible DYNAMICO method and compare it with a one-dimensional SWE. For multilayer SWE, we considered both two layer and a linear stratification of density, with very small density gradient, consistent with Boussinesq approximation. We used Lagrangian vertical coordinate which doesn't allow mass to flow across vertical layers. Numerical examples are presented to verify multilayer SWE model against single layer SWE, in terms of the phase speed and the steepness criteria of wave profile. In addition, the phase speed of the barotropic and baroclinic mode of two-layer SWE also verified our multilayer SWE model. We found that, for multilayer SWE, waves move slower than single layer SWE and get steeper than normal when they flow across bathymetry. A series of numerical experiment were carried out to compare 1-D shallow water solutions to 2-D solutions with and without density as well as to explain the dynamics of surface wave and internal wave. We found that, a positive fluctuations on free surface causes water to rise above surface level, gravity pulls it back and the forces that acquired during the falling movement causes the water to penetrate beneath it's equilibrium level, influences the generation of internal waves. Internal waves travel considerably more slowly than surface waves. On the other hand, a bumpy or a slicky formation of surface waves is associated with the propagation of internal waves. The interaction between these two waves is therefore demonstrated and discussed. / Thesis / Master of Science (MSc) / In the modelling of ocean wave, the formulation and solution of appropriate equations and proper initial and boundary conditions are required. The shallow water equations (SWE) are derived from the conservation of mass and momentum equations, in the case where the horizontal length scale of the wave is much greater than the depth scale and the wave height is much less than the fluid's mean depth. In multilayer SWE, the fluid is discretized horizontally into a set of vertical layers, each having its own height, density, horizontal velocity and geopotential. In this study, we used an explicit staggered finite volume method to solve single and multilayer SWE, with and without density stratification and bathymetry, to understand the dynamic of surface waves and internal waves. A series of numerical experiments were carried out to validate our multilayer model. It is found that, in the presence of density differences, surface waves for the multilayer SWE move slowly and get more steep than normal when they flow across bathymetry. Also, a positive fluctuations on free surface generates internal waves at the interior of ocean which propagate along the line of density gradient.

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.

L’enjeu de la simulation de la dynamique atmosphérique et océanographique a pris ces dernières années une importance accrue avec la question du réchauffement climatique. Le modèle à simuler est complexe. Il combine les équations de la mécanique des fluides avec celles de la thermodynamique. Au 19ème siècle, le mathématicien Adhémar Barré de Saint-Venant formule un système d’équations aux dérivées partielles décrivant les mouvements d’un fluide soumis à la gravité et de faible épaisseur. Il s’agit des équations Shallow Water. L’objectif de cette thèse est de développer et d’analyser un algorithme de résolution des équations Shallow Water sur une sphère en rotation. Dans un premier temps, j’étudie différents aspects mathématiques des opérateurs aux différences finis utilisés par la suite en géométrie sphérique. Les schémas aux différences obtenus sont utilisés pour résoudre l’équation de transport, l’équation des ondes et l’équation de Burgers. Les propriétés de stabilité précision et conservation sont analysées. Dans un second temps, la grille Cubed-Sphere est introduite et analysée. La structure de ce maillage est analogue à celle d’un cube. L'interprétation de la Cubed-Sphere à l’aide de grands cercles permet de construire des opérateurs sphériques discrets gradient, divergence et vorticité d'ordre au moins égal à 3 (en pratique d'ordre 4). La troisième partie de la thèse est dédiée à différents tests pour le système d’équations Shallow Water ainsi que pour l’équation d’advection. Les résultats démontrent une précision proche de celle obtenue par les algorithmes conservatifs d'ordre 4 les plus récents / The problem to obtain accurate simulations of the atmospheric and oceanic equations has become essential in recent years for a proper understanding of the climate change. The full mathematical model to simulate is rather complex. It consists of the coupling of several equations involving fluid dynamics and thermodynamics. In the 19th century, Adhémar Barré de Saint-Venant first formulated the equations describing the dynamic of a fluid subject to gravity and bottom topography. This system is Shallow Water equations. The goal of this thesis is to develop and analyze a numerical scheme to solve the shallow water equation on a rotating sphere. First, a mathematical analsysis of finite difference operators that will be used on the sphere is presented. These schemes are then used to solve various equations in a spehreical setting, in particular the advection equation, the wave equation and the Burgers equation. Stability, accuracy and conservation properties are studied. In a second part, I consider in detail the Cubed-Sphere grid. This particular spherical grid has the mesh topology of a cube. Another interpretation makes use of great circles, this allows to obtain spherical discret operators gradient, divergence and curl of a preved third order. These operators are numercially of 4th order. Numerial results are show in particular for the SW equations an acurracy similar to the one of conservative schemes of 4th order published recently

With the advances in the computing world, computational fluid dynamics (CFD) is becoming more and more critical tool in the field of fluid dynamics. In the past few decades, a huge number of CFD models have been developed with ever improved performance. In this research a robust CFD model, called Riemann2D, is extended to model flow over a mobile bed and applied to a full scale dam break problem. Riemann2D, an object oriented hyperbolic solver that solves shallow water equations with an unstructured triangular mesh and using high resolution shock capturing methods, provides a generic framework for the solution of hyperbolic problems. The object-oriented design of Riemann2D has the flexibility to apply the model to any type of hyperbolic problem with the addition of new information and inheriting the common components from the generic part of the model. In a part of this work, this feature of Riemann2D is exploited to enhance the model capabilities to compute flow over mobile beds. This is achieved by incorporating the two dimensional version of the one dimensional non-capacity model for erodible bed hydraulics by Cao et al. (2004). A few novel and simple algorithms are included, to track the wet/dry and dry/wet fronts over abruptly varying topography and stabilize the solution while using high resolution shock capturing methods. The negative depths computed from the surface gradient by the limiters are algebraically adjusted to ensure depth positivity. The friction term contribution in the source term, that creates unphysical values near the wet/dry fronts, are resolved by the introduction of a limiting value for the friction term. The model is validated using an extensive variety of tests both on fixed and mobile beds. The results are compared with the analytical, numerical and experimental results available in the literature. The model is also tested against the actual field data of 1957 Malpasset dam break. Finally, the model is applied to simulate dam break flow of Warsak Dam in Pakistan. Remotely sensed topographic data of Warsak dam is used to improve the accuracy of the solution. The study reveals from the thorough testing and application of the model that the simulated results are in close agreement with the available analytical, numerical and experimental results. The high resolution shock capturing methods give far better results than the traditional numerical schemes. It is also concluded that the object oriented CFD model is very easy to adapt and extend without changing the generic part of the model.