Modelling the vertical structure of turbulent free-surface flows over rough permeable beds

Understanding diffusive processes across the sediment-water interface is important for quantifying hyporheic exchanges and related biogeochemical processes (e.g., denitrification, biomass growth). Viscous, turbulent and dispersive effects all contribute to the hydrodynamics of flows over rough permeable beds, although their specific role across the interface remains poorly understood. In order to model mixing processes from the permeable bed to the free surface, we have run experiments on low submergence flows over sloping porous beds made of randomly packed glass beads. Fluid velocities were measured using Particle Image Velocimetry (PIV) and a technique called Refractive Index-Matched Scanning (RIMS) allowing the interior of the bed to be examined. By applying the double averaging methodology, we determined the following mesoscale profiles: porosity, mean velocity, turbulent and dispersive stresses from the subsurface to the free surface. We observed a turbulent boundary layer over the rough bed for at intermediate Reynolds numbers, i.e. Re = O(1000). Under these flow conditions, viscosity played a non-negligible role through the van Driest damping effect. Based on the Prandtl mixing length theory and mechanistic considerations, we propose a new theoretical model to reproduce turbulent and dispersive stress profiles. In contrast to existing boundary layer models, our model takes into account the continuity of the porosity profile and the associated damping effects on flow momentum in the roughness layer. A good agreement is found between the model and classic flow resistance laws employed for low-land and gravel-bed rivers (e.g. Chézy, Manning-Strickler, Keulegan). A 1D vertical effective diffusion model is derived from the hydrodynamic model and should contribute improving our understanding of depth dependent diffusive processes across the sediment-water interface and thus hyporheic exchanges in various settings.