Numerical Simulations of Air–Water Flow of a Non-linear Progressive Wave in an Opposing Wind

We present detailed numerical results for two-dimensional viscous air–water flow of a non-linear progressive water wave when the speed of the opposing wind varies from zero to 1.5 times the wave phase speed. It is revealed that at any speed of the opposing wind there exist two rotating airflows, one anti-clockwise above the wave peak and one clockwise above the wave trough. These rotating airflows form a buffer layer between the main stream of the opposing wind and the wave surface. The thickness of the buffer layer decreases and the strength of rotation increases as the wind speed increases. The profile of the average x\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$x$$\end{document}-component of velocity reveals that the water wave behaves as a solid surface producing larger wind stress compared to the following-wind case.