An Eulerian method for transient nonlinear free surface wave problems

Abstract

An Eulerian difference method is developed for the transient potential flow of an incompressible fluid with fully nonlinear free surface conditions. The free surface coordinate y=ν(x, t) and the velocity potential θ(x, y = ν; t) on the free surface are recognized as the primary unknowns to be solved as an initial value problem from the pair of nonlinear partial differential equations representing the dynamic and the kinematic conditions of the free surface. The continuity relation Δ2θ=0 for the velocity potential θ(x, y; t) over the flow field Ω below the free surface is recognized as a subsidiary condition to be enforced at all times. The field of computation is transformed into a time invariant cartesian region with the free surface ν(x, t) represented by a coordinate line (or surface). The iterative solution for θ(x, y; t),p(x; y; t) in this fixed field of computation is facilitated by the use of Fast Fourier Transform (FFT). The iterative process converges rapidly. In terms of this converged θ(x, y; t), the free surface location ν(x, t) and its potential θ(x, y = ν; t) are advanced in time. Results from two planar examples are illustrated. The method is equally applicable to problems in three space dimensions; possibly involving interactive matching with neighboring flow fields. If the initial free surface potential θ(x, y = ν; t = 0) is unknown, difficulties may be encountered in data specification for securing a well posed problem for solution.