A viscous layer model for a shallow water free surface flow
Résumé
Shallow Water equations are widely used at several scales for liquid flows when the depth is smaller than the longitudinal scale. They are based on assumptions on the velocity profile which allow to estimate its shape factor and the shear stress at the wall to close the system of equations. We present here a two layers decompositon between an ideal fluid and a viscous layer, in the spirit of IBL (Interactive Boundary Layer) or IVI (Inviscid Viscous Interaction) introduced in aeronautics. It means that the two layers interact. The displacement thickness of the viscous layer and the order of magnitude of the shear stress at the wall are supposed small and of same order of magnitude. We use this small parameter for expansion and obtain a coupled system of two interacting layers, the viscous layer being then a kind of apparent topography due to the displacement thickness. We show the link with classical Shallow Water equations. The assumption on the velocity profile shape is rejected in the viscous layer, which makes assumptions on profiles (shape factor and wall shear stress) more precise. We test the final system on some classical cases like the starting flow, the flow over bump at several Froude regimes. We finally focus on the flow over a bump in subcritical flows. The computed wall shear stress in this configuration presents some characteristic features of the influence of the boundary layer, it depends on the topography, and moreover its maximum is reached before the top of the bump, which is impossible in classical Shallow Water equations. Also, an additional term with the same magnitude as the shear stress appears in the system, which can be interpreted as a correction to the pressure.
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