On the Efficient Numerical Simulation of Directionally Spread Surface Water Waves

• Published in: Journal of computational physics Vol. 174; no. 1; pp. 277 - 305
• Main Authors: Bateman, W.J.D., Swan, C., Taylor, P.H.
• Format: Journal Article
• Language: English
• Published: Elsevier Inc 20.11.2001
• Subjects: numerical wave modeling; 3-D wave modelling; nonlinear wave–wave interactions; directional water waves; extreme ocean waves
• ISSN: 0021-9991; 1090-2716
• DOI: 10.1006/jcph.2001.6906

This paper concerns the description of transient and highly nonlinear, near-breaking, surface water waves that are characterized by a spread of wave energy in both frequency and direction. A new spectral wave model is described that allows both the unsteadiness and the directionality of a wave field to be described in a fully nonlinear sense. The methodology underlying the scheme is similar to the unidirectional model developed previously by Craig and Sulem [13]. An approximation of the Dirichlet–Neumann operator is made that transforms the boundary values of the velocity potential, φ, at the water surface into values of φz. This allows an initial spatial representation of the water surface elevation and the velocity potential on this surface to be time marched using fast Fourier transforms. The advantages of this technique lie in both its efficiency and its robustness. These are of fundamental importance when seeking to model extreme ocean waves, involving broad-banded frequency spectra and realistic directional spreads, since they incorporate a large range of horizontal length scales. In its present form, the model is appropriate to waves propagating on water of constant depth; it runs on a PC and is sufficiently stable to predict the evolution of near-breaking waves. Indeed, the only significant restriction arises due to the Fourier series representation. This requires the water surface elevation to be a single-valued function of the horizontal coordinates and therefore limits the model to non-overturning waves. The new numerical scheme is validated against a fifth-order Stokes solution for regular waves and the recent experimental observations provided by Johannessen and Swan [3]. These latter comparisons are particularly important, confirming that the model is able to describe the rapid and highly significant energy transfers that occur across the wavenumber spectrum in the vicinity of an extreme event. These are strongly dependent upon the directionality of the wavefield and critically important when seeking to define the characteristics of an extreme, near-breaking, wave. The paper concludes with an example of the formation of a realistic, fully nonlinear and directionally spread wave group in the open ocean.