Turbulent diffusion near a free surface

We study numerically and analytically the turbulent diffusion characteristi cs in a low-Froude-number turbulent shear flow beneath a free surface. In t he numerical study, the Navier-Stokes equations are solved directly subject to viscous boundary conditions at the free surface. From an ensemble of su ch simulations, we find that a boundary layer develops at the free surface characterized by a fast reduction in the value of the eddy viscosity. As th e free surface is approached, the magnitude of the mean shear initially inc reases over the boundary (outer) layer, reaches a maximum and then drops to zero inside a much thinner inner layer. To understand and model this behav iour, we derive an analytical similarity solution for the mean flow. This s olution predicts well the shape and the time-scaling behaviour of the mean flow obtained in the direct simulations. The theoretical solution is then u sed to derive scaling relations for the thickness of the inner and outer la yers. Based on this similarity solution, we propose a free-surface function model for large-eddy simulations of free-surface turbulence. This new mode l correctly accounts for the variations of the Smagorinsky coefficient over the free-surface boundary layer and is validated in both a priori and a po steriori tests.