A stable free-surface boundary solution method for fully nonlinear potential flow models

Abstract

This paper presents a stable method for solving the kinematic boundary condition equation (KBC) in fully nonlinear potential flow (FNPF) models. The method is motivated by a total variation diminishing (TVD) approach, which makes it especially applicable to advection-dominated partial differential equations such as the KBC. It is also simple, and can be easily implemented in existing finite volume-based FNPF models for wave hydrodynamics. The method is systematically assessed through a series of test cases: the propagation of second and fifth-order Stokes waves; focused wave propagation; and wave shoaling in both 2 and 3-D. It was found that the method stabilised the computation in every instance: it successfully eliminated the sawtooth instability, which commonly arises in FNPF models, without a reduction in computational efficiency. Consequently, we avoided the use of undesirable stabilisation techniques that involve artificial dissipation such as low-order smoothing. The method is also accurate: it produced satisfactory numerical solutions that agreed well with experimental, analytical and other published numerical results. It was also found that the method is superior than classical schemes in terms of energy conservation, applicability, and efficiency—all salient features that are essential for large-scale and long-time simulations.