Publication
Steep streams involve shallow, supercritical turbulent flows over a permeable bed made up of coarse particles. They usually exhibit higher flow resistance and stronger mass and momentum exchanges between the stream and subsurface flow than low-gradient streams. Describing their flow dynamics using generalised Manning-Strickler equations has led to empirical relationships with weak predictive power (errors between predictions and data of over one order of magnitude). We studied shallow turbulent flows by employing a mesoscopic approach based on the double-averaged Navier-Stokes equations. More specifically, we were concerned with the possibility of modelling the turbulent and dispersive shear stress equations using simple algebraic equations. To that end, we studied shallow, supercritical turbulent flows over a sloping bed made up of randomly packed spherical particles. Using visualisation techniques based on particle velocimetry imaging and refractive index matched scanning, we were able to reconstruct the velocity field throughout the bed and stream, far from the sidewalls, and estimate the contributions of the dispersive and turbulent shear stresses to the total shear stress. The dispersive shear stress represented less than 20 % of the turbulent shear stress, but because it was concentrated within a thin layer (called the roughness layer) where it outweighed the turbulent shear stress, it had a significant influence on the mean velocity profile. We proposed an algebraic closure equation for dispersive shear stress, based on the mixing-length model used for turbulent shear stress, and we found that it captured closely the mean-velocity and turbulence-intensity profiles of shallow flows over horizontal or sloping permeable beds. Our data suggest that flow dynamics was affected largely by turbulence damping, drag forces and dispersion within the roughness layer, which may explain why steep streams differ from low-gradient streams.