Integral methods for shallow free-surface flows with separation

Abstract

We study laminar thin film flows with large distortions of the free surface, using the method of averaging across the flow. Two specific problems are studied: the circular hydraulic jump and the flow down an inclined plane. For the circular hydraulic jump our method is able to handle an internal eddy and separated flow. Assuming a variable radial velocity profile as in Kármán–Pohlhausen's method, we obtain a system of two ordinary differential equations for stationary states that can smoothly go through the jump. Solutions of the system are in good agreement with experiments. For the flow down an inclined plane we take a similar approach and derive a simple model in which the velocity profile is not restricted to a parabolic or self-similar form. Two types of solutions with large surface distortions are found: solitary, kink-like propagating fronts, obtained when the flow rate is suddenly changed, and stationary jumps, obtained, for instance, behind a sluice gate. We then include time dependence in the model to study the stability of these waves. This allows us to distinguish between sub- and supercritical flows by calculating dispersion relations for wavelengths of the order of the width of the layer.