Considering bottom curvature in depth-averaged open-channel flow modelling, based on curvilinear coordinates

Accurately modelling open-channel flows on strongly vertically curved bottoms, such as for instance over a spillway, is a challenge for any depth-averaged flow model. This type of computation requires the use of axes properly inclined along the mean flow direction in the vertical plane and a modelling of curvature effects. The present generalized model performs such computations by means of curvilinear coordinates in the vertical plane, enabling for instance one to simulate within one single computation domain the flows in the upstream reservoir, over the spillway, in the stilling basin and in the river reach downstream of a dam. The frame of reference is chosen in such a way that one of the two curvilinear axes follows the local bottom curvature. Hence the set of generalized shallow water equations involves explicitly not only the channel bottom slope, but also the channel vertical curvature and its derivative. The velocity profile is generalized in comparison with the uniform one usually assumed in the conventional shallow-water equations. The pressure distribution is also modified as a function of the bottom curvature and is thus not purely hydrostatic but accounts for effects of centrifugal forces. This enhanced mathematical modelling framework has been implemented in a 2D finite volume model. A specific flux vector splitting technique has been developed and demonstrated to be stable for any flow regime and any bottom curvature. The scheme offers the advantage of being dependent only on the sign of the bottom curvature. For a vanishing bottom curvature, the new model converges smoothly towards the conventional shallow-water equations. Finally, two test cases are detailed and lead to satisfactory validation results for the new model.