Boundary layer formulations in orthogonal curvilinear coordinates for flow over wind-generated surface waves

The development of the governing equations for fluid flow in a surface following coordinate system is essential to investigate the fluid flow near an interface deformed by propagating waves. In this paper, the governing equations of fluid flow, including conservation of mass, momentum, and energy balance, are derived in an orthogonal curvilinear coordinate system relevant to surface water waves. All equations are further decomposed to extract mean, wave-induced, and turbulent components. The complete transformed equations include explicit extra geometric terms. For example, turbulent stress and production terms include the effects of coordinate curvature on the structure of fluid flow. Furthermore, the governing equations of motion were further simplified by considering the flow over periodic quasi-linear surface waves wherein the wavelength of the disturbance is large compared to the wave amplitude. The quasi-linear analysis is employed to express the boundary layer equations in the orthogonal wave-following curvilinear coordinates with the corresponding decomposed equations for the mean, wave, and turbulent fields. Finally, the vorticity equations are also derived in the orthogonal curvilinear coordinates in order to express the corresponding velocity-vorticity formulations. The equations developed in this paper proved to be useful in the analysis and interpretation of experimental data of fluid flow over wind-generated surface waves. Experimental results are presented in a companion paper.