Almost extreme waves

TL;DR

This paper numerically investigates the detailed structure of almost extreme periodic traveling waves with vorticity, revealing boundary layer behaviors, angle variations, and oscillations near the wave crest and trough, independent of vorticity as the wave approaches the extreme form.

Contribution

It provides high-accuracy numerical analysis of wave surface angles and boundary layers in flows with vorticity, highlighting universal features near the extreme wave limit.

Findings

Boundary layer angle rises sharply to ~30.38° at the crest.

Outer region angle varies with vorticity, reaching >30° for positive vorticity.

Oscillations in the transition region resemble Gibbs phenomenon, becoming vorticity-independent.

Abstract

Numerically computed with high accuracy are periodic traveling waves at the free surface of a two dimensional, infinitely deep, and constant vorticity flow of an incompressible inviscid fluid, under gravity, without the effects of surface tension. Of particular interest is the angle the fluid surface of an almost extreme wave makes with the horizontal. Numerically found are: (i) a boundary layer where the angle rises sharply from $0^\circ$ at the crest to a local maximum, which converges to $30.3787\dots^\circ$ as the amplitude increases toward that of the extreme wave, independently of the vorticity, (ii) an outer region where the angle descends to $0^\circ$ at the trough for negative vorticity, while it rises to a maximum, greater than $30^\circ$, and then falls sharply to $0^\circ$ at the trough for large positive vorticity, and (iii) a transition region where the angle oscillates about $30^\circ$, resembling the Gibbs phenomenon. Numerical evidence suggests that the amplitude and frequency of the oscillations become independent of the vorticity as the wave profile approaches the extreme form.