Turbulent Free Surface Dynamics in High-speed Flows: A Sensitivity Analysis of the Governing Equations

High-speed flows are naturally occurring in some of the most energy-containing hydraulic structures and steep streams. With flow acceleration, turbulence quantities increase, hence distorting the free surface and eventually leading to self-aeration when the free surface perturbations become unstable. Self-aeration has only been mathematically formulated as such a problem recently, but several empirical evidences have already been disclosed. These equations can be joined into a second-order non-linear differential equation for the perturbations amplitude. However, despite the governing equations for a turbulent free surface are currently known, little has been revealed about their nature and costly, advanced experimental modelling is still to be required to determine the real physical properties of the coefficients of these equations. In this study, the physical interpretation of these coefficients, the most restricting hypotheses and relevant experimental evidences are introduced first. Subsequently, a numerical solution procedure is presented – step by step – in a clear and concise manner, therefore assisting future interested researchers. The governing equations are finally solved numerically while addressing a sensitivity analysis of their parameters, thus providing insight on their nature and revealing some critical aspects. Only upon procurement of the physical parameters of these equations, accurate estimations of the inception point will become feasible for complex three-dimensional flows through simplified general models that could be implemented within Computational Fluid Dynamics codes.