Experimental results are presented in a companion paper. The equations developed in this paper proved to be useful in the analysis and interpretation of experimental data of fluid flow over more » Finally, the vorticity equations are also derived in the orthogonal curvilinear coordinates in order to express the corresponding velocity–vorticity formulations. 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. 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. For example, turbulent stress and production terms include the effects of coordinate curvature on the structure of fluid flow. The complete transformed equations include explicit extra geometric terms. All equations are further decomposed to extract mean, wave-induced and turbulent components. 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.
![orcaflex coordinate system wind and wave coordinates orcaflex coordinate system wind and wave coordinates](https://www.open-ocean.org/wp-content/uploads/2021/04/image-5.png)
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.