Efficient numerical methods are devised for unsteady free surface motions of inviscid fluids and viscous fluids. For inviscid fluids, a numerical method which can be easily implemented is developed and for viscous fluids, a general method is proposed.
For inviscid fluids, the governing equation becomes a Laplace``s equation, which is treated here by means of a series expansion of the velocity potential. The free surface is represented with a height function. The present method is applied to surface gravity waves and Faraday resonances for the stability and accuracy of the method.
For an example of versatility of the numerical method, unsteady motions of sink flows having a free surface are studied numerically. First, for a line sink, results on dip formation are obtained corroborating with Tyvand``s series solution[Phys. Fluids A 4, 671 (1992)],except for small disagreements for large t owing to the finiteness of calculational domain of our numerical approach. Next, to investigate the drain size effect in dip formations, free surface flows due to a sink array are considered. With increasing r, a parameter for the drain size, the effective sink strength $q_eff$ is found to decrease, which we argue to be in the scaling law: $q_eff/q_0=r^ε$ where $q_0$ is the total sink strength and \epsilon is a constant. Overall, the drain size effect is small compared with the other effects due to the submerged depth or the total sink strength.
For viscous fluids, marker particles are used for the free surface representation and a particle simulation technique is adopted, where a momentum equation in Lagrangian coordinates is solved. The numerical method is applied to surface gravity waves which is usually very hard to approach with numerical methods for viscous flows. The numerical results was found to be accurate even with a coarse grid system.