Depth-averaged model for open-channel flows in the smooth turbulent case and reconstruction of the 3D-fields

A two-dimensional depth-averaged model is derived for open-channel flows in the smooth turbulent case. The derivation is consistently obtained with a method of matched asymptotic expansions in the outer and inner layers using a mixing length model of turbulence including the free-surface reduction of the eddy viscosity. The shear effects are taken into account by an extra tensor variable of the model called enstrophy tensor. The friction coefficient is an explicit expression of the water depth. The three-dimensional (3D) velocity field and the friction velocity can be reconstructed from the values of the depthaveraged quantities. Numerical simulations show that the enstrophy can be used to evaluate the development of the turbulent boundary layer. In the case of subcritical unsteady flows, the reconstructed velocity can be described with a logarithmic law modified by Coles' wake function with apparent von Kármán constant, integration constant and wake-strength parameter, which differ from their values in steady flows. In the viscous sublayer, the steady-state relation between the velocity and the vertical coordinate, in the inner scaling, is not valid for unsteady flows. Large errors on the calculation of the von Kármán constant can be made if the validity of the steady-state relation is assumed for unsteady flows