Surface gravity waves from direct numerical simulations of the Euler equations: A comparison with second-order theory

When the wave spectrum is sufficiently narrow-banded and the wave steepness is sufficiently high, the modulational instability can take place and waves can be higher than expected from second-order wave theory. In order to investigate these effects on the statistical distribution of long-crested, deep water waves, direct numerical simulations of the Euler equations have been performed. Results show that, for a typical design spectral shape, both the upper and lower tails of the probability density function for the surface elevation significantly deviate from the commonly used second-order wave theory. In this respect, the crest elevation is observed to increase up to 18% at low probability levels. It would furthermore be expected that wave troughs become shallower due to nonlinear effects. Nonetheless, the numerical simulations show that the trough depressions tend to be deeper than in second-order theory.