Nonlinear Evolution of a Steep, Focusing Wave Group in Deep Water Simulated With oceanwave3d

Abstract

Steep, focusing waves can experience fast and local nonlinear evolution of the spectrum due to wave–wave interactions resulting in energy transfer to both higher and lower wavenumber components. The shape and kinematics of a steep wave may, thus, differ substantially from the predictions of linear theory. We have investigated the role of nonlinear interactions on group shape for a steep, narrow-banded, directionally spread wave group focusing in deep water using the fully nonlinear potential flow solver, oceanwave3d. Exact second-order correction of the initial conditions has been implemented together with a novel third-order approximate correction based on a Stokes-type formulation for surface elevation combined with a scaling argument for the third-order velocity potential. Four-phase separation reveals that the third-order scheme provides a good estimate for the third-order superharmonics. A quantitative assessment of numerical error has also been performed for the spatial and temporal discretization, including energy conservation, a reversibility check, and validation against previous simulations performed with a higher-order spectral (HOS) code. The initially narrow-banded amplitude spectrum exhibits the formation of “sidelobes” at angles of approximately ±35deg to the spectral peak during the simulated extreme wave event, occurring in approximately ten wave periods, with a preferential energy transfer to high-wavenumber components. The directional energy transfer is attributed to resonant third-order interactions with a discussion of the engineering implications.