Computations of nonlinear gravity waves by a desingularized boundary integral method.

Abstract

A desingularized boundary integral equation method combined with an Eulerian-Lagrangian time-stepping technique is developed for nonlinear gravity wave problems. The desingularization distance between the boundary and the sources is related to the local mesh size to ensure convergence. Tests for some simple problems show that desingularization significantly reduces the computer time required to compute the influence matrix of the resulting algebraic system. The algebraic system is still adequately well-conditioned to allow fast iterative solutions. Accurate solutions can be obtained for a large range of desingularization distances on the order of the mesh size. Several nonlinear water wave problems are then investigated. The first problem considers upstream runaway solitons due to a disturbance moving near critical speed in two-dimensional shallow water. Results from the desingularized method with the fully nonlinear free surface boundary condition agree well to those using the fKdV model for weak disturbances. The fully nonlinear model predicts larger solitons than the fKdV model for strong disturbances and also predicts the breaking of waves for some stronger disturbances. Next, the problem of three-dimensional waves due to a submerged moving spheroid show good comparison to those from other algorithms. Finally, the generation of inner-angle wavepackets in the wake of a ship is investigated. The three most probable causes of the wavepackets are examined: interference of the wave systems by the bow and stern; free-surface nonlinear effects; and wake unsteadiness due to translation and oscillation of the disturbance. The wake is studied with nonlinear calculations using the desingularized method and with linear calculations using a time-domain Green function and the stationary phase method. It is shown that nonlinear effects are not essential to the generation and persistence of inner-angle wavepackets; the phenomenon can be explained by unsteady linear theory.