Time‐Domain Solution for Second‐Order Wave Diffraction
Journal of waterway, port, coastal, and ocean engineering
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... without my written permission. Abstract A numerical method based on potential flow theory is developed for simulating transient, second-order interactions of ocean waves with large fixed bodies of arbitrary shape in two and three dimensions. The physical problem is represented by a mathematical model composed of a fluid domain bounded by the body surface, the still water surface, the seabed, and a control surface truncating the infinite fluid region. The nonlinear free surface boundary conditions defined on the instantaneous free surface are expanded about the still water level by a Stokes expansion procedure. The flow potential to second order is thereby defined with respect to a time-independent boundary which includes the still water surface, and its solution involves a numerical discretization of an integral equation. With the potential separated into incident and scattered components, the Sommerfeld radiation condition applied to the scattered potential is modified to incorporate a time-dependent celerity to account for the transient and second-order effects. The free surface boundary conditions and the radiation condition are then satisfied to second order by a numerical integration in time. An alternative second-order solution is derived based on a different expansion procedure in which the nonlinear free surface boundary conditions and an integral equation defined on the instantaneous free surface are both expanded by a Taylor series, and terms up to second order are retained. The two approaches give rise to identical first-order problems, but give rise to second-order problems which are apparendy different. The discrepancy arises from the second-order integral equation in which additional second-order terms are retained. The physical interpretations and limitations of these terms are explored and their effects on the evaluations of wave forces are assessed. Applications of the present method are made to studies of regular wave diffraction around a fully submerged and a semi-submerged circular cylinder in two dimensions, and around a bottom-mounted surface-piercing circular cylinder in three dimensions. The stability and ii numerical accuracy of the proposed solution and the treatment of the radiation condition to second order are examined. Comparisons of computed wave forces and runup are made with previous theoretical and experimental results and these indicate favourable agreement. iii Table of Contents Abstract ii List of Tables vii List of Figures viii List of Symbols xi Acknowledgement xv 1.