Asymptotic derivation of the section averaged shallow water equations for natural river hydraulics

The section-averaged shallow water model usually applied in river and open channel hydraulics is derived by an asymptotic analysis that accounts for terms up to second order in the vertical/longitudinal length ratio, starting from the three-dimensional Reynolds-averaged Navier–Stokes equations for incompressible free surface flows. The derivation is carried out under quite general assumptions on the geometry of the channel, thus allowing for the application of the resulting equations to natural rivers with arbitrarily shaped cross sections. As a result of the derivation, a generalized friction term is obtained, that does not rely on local uniformity assumptions and that can be computed directly from three-dimensional turbulence models, without need for local uniformity assumptions. The modified equations including the novel friction term are compared to the classical Saint Venant equations in the case of steady state open channel flows, where analytic solutions are available, showing that the solutions resulting from the modified equation set are much closer to the three-dimensional solutions than those of the classical equation set. Furthermore, it is shown that the proposed formulation yields results that are very similar to those obtained with empirical friction closures widely applied in computational hydraulics. The generalized friction term derived therefore justifies a posteriori these empirical closures, while allowing to avoid the assumptions on local flow uniformity on which these closures rely.