A Mathematical and Numerical Examination of Wave-Current Interaction and Wave-Driven Hydrodynamics
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A new derivation of an elliptic extended mild-slope wave equation, including the effects of energy dissipation and current, has been accomplished. A Galerkin-Eigenfunction method was used for this derivation and the final equation has been used to create a Finite Element Wave-Current Interaction Model (NM-WCIM). The NM-WCIM solves for the complex value of velocity potential, from which all other wave properties can be obtained. An iterative solution scheme based on the gradient of wave phase is implemented to solve for wave-current interaction. A novel post-processing technique for the NM-WCIM has been developed to obtain wave energy rays and hence breaking wave heights and eddy viscosity values. The model has been calibrated and tested against measured data and published results of similar models. The NM-WCIM was used to examine scenarios with complex wave-current interaction and bathymetry, including a case study of Casheen Bay on the west coast of Ireland. The NM-WCIM has proven itself to be an advancement over previous similar models in terms of efficiency and accuracy. Equations for the conservation of mass and momentum have been derived to examine wave-driven hydrodynamics in and around the surf-zone. These equations use a radiation stress driving force obtained, using a unique formula, from the velocity potential results of the NM-WCIM. The conservation equations also include turbulent diffusion terms based on eddy viscosity and a general bottom friction term for flow in any direction. The conservation of mass and momentum formulae have been used to develop a depth-integrated Finite Element Wave-Driven Hydrodynamic Model (NM-WDHM) which iterates to a converged solution using a finite difference time-stepping procedure. This model was calibrated against measured data and published results of similar models and has been used in a coupled system with the NM-WCIM to examine many scenarios with complex bathymetric and wave conditions, including those in Casheen Bay. The NM-WDHM has proven itself to be both accurate and computationally efficient.