Extreme two-dimensional water waves: an assessment of potential design solutions

Abstract

This paper concerns the description of extreme 2-D water waves arising due to the focusing of wave components in a random or irregular sea. Exact numerical calculations have been undertaken in a range of uniform water depths using the time-stepping solution outlined by Dold and Peregrine (Proc. 19th Int. Conf. on Costal Engng. 1 (1984) 955). This procedure is first validated against laboratory data (Phil. Trans. R. Soc. Lond. Ser. A. 354 (1996) 1) and, secondly, used to provide bench-mark data against which both existing design solutions and two recently developed wave models are assessed. Six laboratory-scale test cases are established, involving both broad- and narrow-banded frequency spectra in deep, intermediate and shallow water. With the predicted water surface elevations used as input to the various wave kinematics models, the vertical distribution of the maximum horizontal velocity arising beneath the largest wave crest provides a yard-stick with which to judge their relative success. These comparisons clearly indicate that the commonly applied design solutions provide a poor description of the water particle kinematics. This arises because such solutions are unable to model both the nonlinearity and the unsteadiness that characterises extreme waves. In contrast, the local Fourier series solution (Appl. Ocean Res. 14 (1992) 93) and the double Fourier series solution (Appl. Ocean Res. 16 (1994) 101) provide improved representations. However, as an extreme wave group evolves, the nature of the nonlinear wave–wave interactions and, in particular, the local (and rapid) re-distribution of energy within the frequency domain, is shown to have important implications for the accuracy of these solutions. For example, the local Fourier series solution is shown to be unable to model the frequency-difference terms, responsible for the set-down beneath the wave group, and therefore provides a poor description of the fluid velocities beneath the still water level. In contrast, the double Fourier series solution explicitly incorporates both the nonlinearity and the unsteadiness of the wave, but is limited in terms of the total number of Fourier components that can practically be included. As a result, the largest velocities arising close to the water surface are typically under-predicted in the very largest wave events. Nevertheless, both these kinematic models provide a significant improvement over the existing design solutions. Having demonstrated the importance of these energy shifts in relation to the prediction of the water particle kinematics, the final section of the paper addresses the wider, practical, implications. In particular, questions are raised concerning the statistical NewWave model (Proc. 1st Int. Offshore and Polar Eng. Conf. 3 (1991) 64) that predicts the most probable shape of an extreme wave based on the underlying linear wave spectrum, S ηη ( ω). If a wave is highly nonlinear such an approach will under-estimate both the crest-trough asymmetry and the maximum crest elevation. Indeed, it is argued that the observed energy shifts provide a possible explanation for the occurrence of so-called “freak” waves, or those that are larger or occur more often than is statistically predicted. The near-surface water particle kinematics associated with such waves are difficult to predict, but highly relevant to the onset of dynamic response (or “ringing”), the occurrence of wave slamming, and the incidence of green–water inundation.