Consistent equations for open-channel flows in the smooth turbulent regime with shearing effects

Consistent equations for turbulent open-channel flows on a smooth bottom are derived using a turbulence model of mixing length and an asymptotic expansion in two layers. A shallow-water scaling is used in an upper – or external – layer and a viscous scaling is used in a thin viscous – or internal – layer close to the bottom wall. A matching procedure is used to connect both expansions in an overlap domain. Depth-averaged equations are then obtained in the approximation of weakly sheared flows which is rigorously justified. We show that the Saint-Venant equations with a negligible deviation from a flat velocity profile and with a friction law are a consistent set of equations at a certain level of approximation. The obtained friction law is of the Kármán–Prandtl type and successfully compared to relevant experiments of the literature. At a higher precision level, a consistent three-equation model is obtained with the mathematical structure of the Euler equations of compressible fluids with relaxation source terms. This new set of equations includes shearing effects and adds corrective terms to the Saint-Venant model. At this level of approximation, energy and momentum resistances are clearly distinguished. Several applications of this new model that pertains to the hydraulics of open-channel flows are presented including the computation of backwater curves and the numerical resolution of the growing and breaking of roll waves.