15.1 Introduction

For time-dependent and nonlinear free-surface motions, analytical solutions are rarely available, and theoretical studies and practical applications rely very much on numerical simulations. Numerical methods are developed according to the specific applications of interest, based on assumptions regarding the dominant physical processes involved.

For instance, for marine structures and ships with characteristic dimensions larger than those of the surface waves, viscous effects and surface tension can often be neglected. On the other hand, nonlinear free-surface effects become important under conditions of large motions and extreme loads, for which high performance, safety and ultimate survivability are of concern.

Significant advances in the simulation of nonlinear wave dynamics have been made in the past 30 years. A number of articles give comprehensive reviews of this progress. These include the excellent reviews by Mei (1978) for ideal, linear free-surface flows using integral-equation and finite-element methods; Schwartz and Fenton (1982) for ideal, nonlinear flows with emphasis on theoretical methods; Yeung (1982) for both linear and nonlinear ideal flows; Floryan and Rasmussen (1989) for viscous flows with moving interfaces; Tsai and Yue (1996) for incompressible, nonlinear free-surface flow; and Scardovelli and Zaleski (1999) for viscous free-surface and inter-facial flows focusing on fixed-grid volume-of-fluid methods.

Numerical methods in nonlinear free-surface flow computations can be generally categorized into volume-discretization and boundary-discretization approaches. Volume-discretization methods are applicable to both inviscid and viscous flows, while boundary-discretization approaches are mainly used when the flow is inviscid and irrotational.

Volume discretization methods can be broadly classified in terms of...