Diffraction of Linear Water Waves on a Parabolic Wall: Small Parameter Approach

摘要

The wave energy conversion system developed by Energetech features a vertical parabolic wall that focuses energy onto an OWC chamber. Here we present a simplified computationally-efficient analytical model to investigate the wave capture dynamics and calculate wave forces on the parabolic wall. At low order, the diffraction of plane linear water waves on a vertically-slotted bottom-mounted surface-piercing cylindrical wall is taken as an approximate model for a parabolic wall. Calculation of the free surface elevation inside the wall, and wave loading on the wall, is investigated. The vertically-slotted cylindrical wall represents part of cylinder, with the opening directed against incident waves. The fluid domain is divided into two regions; an interior fluid cylinder matching to the cylindrical wall, and an exterior region extending to infinity. The solution for linear water waves is sought in both regions satisfying the matching pressure and normal velocity conditions between the internal and external fluid regions, and zero normal velocity conditions on the rigid boundary. The method of solution is based on perturbation theory, using a perturbation parameter defined in terms of the surface geometry of the cylinder. The analysis includes terms up to the first-order in this parameter, where the zeroth-order solution corresponds to a circular cylinder. The velocity potentials at the zeroth- and first-orders are expressed as eigenfunction expansions involving unknown coefficients that are determined through the cylinder boundary conditions. The mixed boundary-value problem is transformed into dual series relations and solutions are obtained by the least-square approximation method. Numerical results are presented to illustrate the influence of the magnitude of shape perturbations on the resulting hydrodynamic force and the wave amplitude inside parabolic wall.