A Fully Nonlinear Model for Three-dimensional Overturning Waves over Arbitrary Bottom 1

An accurate three-dimensional (3D) numerical model, applicable to strongly nonlinear waves, is proposed. The model solves fully nonlinear potential ow equations, with a free surface, using a higher-order 3D Boundary Element Method (BEM) and a mixed Eulerian-Lagrangian time updating, based on second-order explicit Taylor series expansions, with adaptive time steps. The model is applicable to nonlinear wave transformations from deep to shallow water over complex bottom topography, up to overturning and breaking. Arbitrary waves can be generated in the model, and re ective or absorbing boundary conditions speci ed on lateral boundaries. In the BEM, boundary geometry and eld variables are represented by 16-node cubic \sliding" quadrilateral elements, providing local inter-element continuity of the rst and second derivatives. Accurate and e cient numerical integrations are developed for these elements. Discretized boundary conditions at intersections (corner/edges) between the free surface or the bottom, and lateral boundaries are well-posed in all cases of mixed boundary conditions. Higher-order tangential derivatives, required for the time updating, are calculated in a local curvilinear coordinate system, using 25-node \sliding" 4th-order quadrilateral elements. Very high accuracy is achieved in the model for mass and energy conservation. No smoothing of the solution is required, but regridding to a higher resolution can be speci ed at any time over selected areas of the free surface. Applications are presented for the propagation of numerically exact solitary waves. Model properties of accuracy and convergence with a re ned spatio-temporal discretization are assessed by propagating such a wave over constant depth. The shoaling of solitary waves up to overturning is then calculated over a 1:15 plane slope, and results show good agreement with a twodimensional solution proposed earlier. Finally, 3D overturning waves are generated over a 1:15 sloping bottom having a ridge in the middle, thus focusing wave energy. The node regridding method is used to re ne the discretization around the overturning wave. Convergence of the solution with grid size is also veri ed for this

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