A fractional step method for unsteady free-surface flow with applications to non-linear wave dynamics

Abstract

SUMMARY A fractional step method is developed for solving the time dependent two-dimensional Euler equations with full non-linear free-surface boundary conditions. The geometry of the free surface is described by a height function, and its evolution is tracked by integrating in time the kinematic boundary conditions based on the free-surface volume flux. The fluid domain is discretised by adapting a time-varying curvilinear grid to all boundaries, including the free surface. Mass and momentum equations are discretised by a conservative finite volume formulation, taking into account the time dependency of the grid. A fractional step type method is developed for integrating the fluid motion in time. The method is applied to a non-linear standing wave in a square container, testing for compliance with mass and energy conservation and comparing computed wave period with other results. Non-linear travelling waves are simulated in channels with either constant depth or varying depth and non-linear wave processes involving both triad interactions and quartet interactions are studied. Results are compared with both experimental data and theoretical results and excellent agreement is found. Interaction of waves and currents is studied. The blocking of waves in an opposing current is simulated and found to show good agreement with theoretical results. The method is intended to be a first step towards a full description of wave dynamics interacting with structures and currents. © 1998 John Wiley & Sons, Ltd.