A fully non-linear model for three-dimensional overturning waves over an arbitrary bottom

An accurate three-dimensional numerical model, applicable to strongly non-l inear waves, is proposed. The model solves fully non-linear potential flow equations with a free surface using a higher-order three-dimensional bounda ry element method (BEM) and a mixed Eulerian-Lagrangian time updating, base d on second-order explicit Taylor series expansions with adaptive time step s. The model is applicable to non-linear 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 reflective or absorbin g boundary conditions specified on lateral boundaries. In the BEM, boundary geometry and field variables are represented by 16-node cubic 'sliding' qu adrilateral elements, providing local inter-element continuity of the first and second derivatives. Accurate and efficient numerical integrations are developed for these. elements. Discretized boundary conditions at intersect ions (corner/edges) between the free surface or the bottom and lateral boun daries are well-posed in all cases of mixed boundary conditions. Higher-ord er tangential derivatives, required for the time updating, are calculated i n a local curvilinear co-ordinate system, using 25-node 'sliding' fourth-or der quadrilateral elements. Very high accuracy is achieved in the model for mass and energy conservation. No smoothing of the solution is required, bu t regridding to a higher resolution can be specified at any time over selec ted areas of the free surface. Applications are presented for the propagati on of numerically exact solitary waves. Model properties of accuracy and co nvergence with a refined spatio-temporal discretization are assessed by pro pagating 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 sho w good agreement with a two-dimensional solution proposed earlier. Finally, three-dimensional overturning waves are generated over a 1:15 sloping bott om having a ridge in the middle, thus focusing wave energy. The node regrid ding method is used to refine the discretization around the overturning wav e. Convergence of the solution with grid size is also verified for this cas e. Copyright (C) 2001 John Wiley & Sons, Ltd.