Development and Optimization of Non-Hydrostatic Models for Water Waves and Fluid-Vegetation Interaction

The primary objective of this study is twofold: 1) to develop an efficient and accurate non-hydrostatic wave model for fully dispersive highly nonlinear waves, and 2) to investigate the interaction between waves and submerged flexible vegetation using a fully coupled wave-vegetation model. This research consists of three parts. Firstly, an analytical dispersion relationship is derived for waves simulated by models utilizing Keller-box scheme and central differencing for vertical discretization. The phase speed can be expressed as a rational polynomial function of the dimensionless water depth, $kh$, and the layer distribution in water column becomes an optimizable parameter in this function. For a given tolerance dispersion error, the range of $kh$ is extended and the layer thicknesses are optimally selected. The derived theoretical dispersion relationship is tested with linear and nonlinear standing waves generated by an Euler model. The optimization method is applicable to other non-hydrostatic models for water waves. Secondly, an efficient and accurate approach is developed to solve Euler equations for fully dispersive and highly nonlinear water waves. Discontinuous Galerkin, finite difference, and spectral element formulations are used for horizontal discretization, vertical discretization, and the Poisson equation, respectively. The Keller-box scheme is adopted for its capability of resolving frequency dispersion accurately with low vertical resolution (two or three layers). A three-stage optimal Strong Stability-Preserving Runge-Kutta (SSP-RK) scheme is employed for time integration. Thirdly, a fully coupled wave-vegetation model for simulating the interaction between water waves and submerged flexible plants is presented. The complete governing equation for vegetation motion is solved with a high-order finite element method and an implicit time differencing scheme. The vegetation model is fully coupled with a wave model to explore the relationship between displacement of water particle and plant stem, as well as the effect of vegetation flexibility on wave attenuation. This vegetation deformation model can be coupled with other wave models to simulate wave-vegetation interactions

Dynamic stability and vibrations of slender marine structures at low tension

This thesis describes an analytical and numerical investigation into the dynamic behaviour of long slender structures under forcing, parametric and combined periodic excitations as well as under pulse loading. In particular, this work is directed at obtaining an understanding of the dynamic stability and vibrations of slender vertical marine structures at low tension with applications to the tethers of tension leg platforms (TLPs) and to vertical marine risers. A closed form solution for exact natural frequencies and corresponding mode shapes of slender vertical cylinders is obtained analytically by using the Bessel function. This is followed by analysis to obtain the response of such structures when subjected to lateral forces (forcing excitation). This is done by reducing their governing partial differential equation to a non-linear differential form by applying Galerkin's method and the method of separation of variables. This non-linear equation is then solved analytically and semi-analytically to obtain forced vibrations of the structure. When slender marine structures are subjected to a time-varying axial force, they can exhibit parametric vibrations described by the Mathieu equation. The Mathieu stability chart over such a wide range of parameters is obtained. In addition, a non-linear Mathieu equation for the lower-order instability regions is analytically solved by using a perturbation method. In order to solve the equation of the higher-order instability regions, a fourth-order Runge-Kutta method is employed. In reality, most slender marine structures such as tethers of TLPs are subjected to both time-varying axial forces and lateral forces giving rise to combined parametric and forcing excitation. This combined excitation is solved to demonstrate that the effect of combined excitation on the response of such structures becomes significant compared to forcing or parametric excitation, especially in the even numbers of instability regions for the Mathieu chart. For the same excitation strength, the response amplitude of the structures under combined excitation is found to be the largest in the second instability region. In order to deal with the dynamic stability of slender vertical structures at low tension in the higher-order instability regions, dynamic pulse buckling is also investigated. It is known that if the load duration is short enough, long slender marine structures can survive axial loads much larger than the static Euler load value. In this work, the theory of dynamic pulse buckling is applied to rather short TLP tethers in order to obtain confirmatory criteria for allowable compressive axial load and its duration time. A finite element method with a time-varying stiffness matrix is used to verify analytical results. Results from the finite element method are in good agreement with analytical results. The results of the above theoretical developments are illustrated by application studies on a marine riser and the tethers of three conventional TLPs - on the Hutton, Jolliet and Snorre fields. It is found that the effect of combined excitation is significant on the dynamic behaviour of the tethers at low tension and that the conventional high pretension of their tethers can be reduced to a certain extent

Unconditionally Energy Stable Linear Schemes for a Two-Phase Diffuse Interface Model with Peng-Robinson Equation of State

Many problems in the fields of science and engineering, particularly in materials science and fluid dynamic, involve flows with multiple phases and components. From mathematical modeling point of view, it is a challenge to perform numerical simulations of multiphase flows and study interfaces between phases, due to the topological changes, inherent nonlinearities and complexities of dealing with moving interfaces. In this work, we investigate numerical solutions of a diffuse interface model with Peng-Robinson equation of state. Based on the invariant energy quadratization approach, we develop first and second order time stepping schemes to solve the liquid-gas diffuse interface problems for both pure substances and their mixtures. The resulting temporal semi-discretizations from both schemes lead to linear systems that are symmetric and positive definite at each time step, therefore they can be numerically solved by many efficient linear solvers. The unconditional energy stabilities in the discrete sense are rigorously proven, and various numerical simulations in two and three dimensional spaces are presented to validate the accuracies and stabilities of the proposed linear schemes

A Computational Model of Protein Induced Membrane Morphology with Geodesic Curvature Driven Protein-Membrane Interface

Continuum or hybrid modeling of bilayer membrane morphological dynamics induced by embedded proteins necessitates the identification of protein-membrane interfaces and coupling of deformations of two surfaces. In this article we developed (i) a minimal total geodesic curvature model to describe these interfaces, and (ii) a numerical one-one mapping between two surface through a conformal mapping of each surface to the common middle annulus. Our work provides the first computational tractable approach for determining the interfaces between bilayer and embedded proteins. The one-one mapping allows a convenient coupling of the morphology of two surfaces. We integrated these two new developments into the energetic model of protein-membrane interactions, and developed the full set of numerical methods for the coupled system. Numerical examples are presented to demonstrate (1) the efficiency and robustness of our methods in locating the curves with minimal total geodesic curvature on highly complicated protein surfaces, (2) the usefulness of these interfaces as interior boundaries for membrane deformation, and (3) the rich morphology of bilayer surfaces for different protein-membrane interfaces