A fully nonlinear numerical method for modeling wave–current interactions

Abstract

The presence of current in the ocean can significantly modify the characteristics of ocean waves, and it is considered as an important factor responsible for the occurrence of extreme waves, e.g., rogue waves, which are well known as great threats to ocean engineering practices. The magnitude and direction of ocean current normally vary spatially and ocean waves can become very large and steep. Accurate and efficient phase-resolved numerical methods of fully nonlinear wave–current interactions on a large scale in three dimensions (3D) are required to understand their properties, but the existing phase-resolved methods are all based on the assumption of linear or weakly nonlinear interactions. This paper will address the issues and present a fully nonlinear numerical method to model the 3D interactions between waves and varying current on a large scale using a phase-resolved formulation. A new set of equations describing the three-dimensional, fully nonlinear interactions between waves and horizontally shearing current is proposed. They are derived by making no assumption on wave steepness or the order of wave–current interactions. The resulting new equations correctly describe the free surface boundary conditions by representing the fully nonlinear wave–current interactions, removing the limitation to the small wave steepness of the existing formulations in literature.On this basis, the recently developed Enhanced Spectral Boundary Integral (short as ESBI) method is further enhanced to be able to model the wave–current interactions using the new equations, by developing the appropriate procedure for dealing with the extra terms related to nonlinear wave–current interactions. The new equations are used as the prognostic equations for updating the free surface in time domain, and a fast converging iterative technique is employed to solve them. The robustness of the newly developed method is demonstrated through comparing with experimental data available in literature and good agreements are observed in the several different cases, including the 3D fully nonlinear interactions between ocean waves and horizontally varying current. A comparison with the Higher-Order Spectrum (HOS) method based on weak-nonlinear formulations of wave–current interactions is also made to confirm larger error does appear if the wave steepness is high by using the HOS. The method presented in the paper can be employed to simulate the real evolution of ocean waves on current in a phase-resolved way to give deep insights to the dynamics of wave–current interactions, which may not be done correctly by the existing methods so far.