An analytical model of wave bottom boundary layers incorporating turbulent relaxation and diffusion effects

To calculate the effects of turbulent relaxation on oscillatory turbulent boundary layers, a viscoelastic term is added to an eddy viscosity model. The viscoelastic term parameterizes the lag of turbulent properties in response to imposed oscillatory shear and is proportional to the ratio between the timescales of eddy dissipation and of the oscillating flow. It is found that the turbulent relaxation plays an important role in the phase variations of velocity and shear stress with elevation, and that it decreases the friction factor and the phase lead of bed shear stress over free stream velocity. To assess the effects of turbulent diffusion in this problem, the viscoelastic model is extended by further introducing a turbulent diffusion term in the model. The comparisons between these two models indicate that turbulent diffusion significantly reduces the magnitudes of shear stress and velocity perturbation in the outer region of the boundary layer. It is also found that the effects of turbulent relaxation and diffusion increase with increasing relative roughness. As a result, the analytical solutions demonstrate an overall improvement over the eddy viscosity model in predicting the observed temporal evolution of velocity and shear stress profiles; this improvement is more distinct for rough beds than smooth beds.