Surprising Behaviour and Singularity in the Saint Venant Approximation for a Fluid

A research line is reviewed which, over a few years, led to a substantial change of perspective about the simplified models that underlie the description of quasi-onedimensional streams, their instabilities, and their effects upon sandy beds. Even when the flow is assumed to be laminar, the Saint-Venant equation of quasi-onedimensional fluid flow can be formulated in more than one manner; it will be shown that only one of these choices is consistent with the complete three-dimensional Navier- Stokes equations. When the flow is turbulent, an added complication is the presence of a turbulence model, most often of the eddy-viscosity type; it will be shown that such a model can be in strong contrast with a direct numerical simulation of the same phenomenon, even to the point of producing results of opposite sign. In addition, the complete numerical simulation of flow past an undulated bottom exhibits a non-monotonic approach to its long-wave, quasi-onedimensional limit, with a surprising resonance that has no laminar counterpart and must become the subject of future investigations.