Formation of ripples over a sand bed submitted to a turbulent shear flow

We investigate the process of ripple formation when a sand bed is submitted to a steady and turbulent liquid flow. The sand transport dynamics is described in terms of a simple relaxation law which accounts for the fact that the transport rate does not adapt instantaneously to its equilibrium value. The equilibrium sand flux is evaluated using a standard law based on the estimation of the flow shear stress calculated at the sand bed surface. The latter is estimated from an analytical resolution of the flow over a deformed sand bed which is based on the Jackson and Hunt calculation [ J. C. R. Hunt, Quart. J. R. Met. Soc. 101, 929 ( 1975)]. Within this model, we investigate the stability of the sand bed and are able to derive analytical scaling laws for the wavelength and phase velocity of the most dangerous mode. In the deep flow limit, the model predicts the occurrence of a single mode of instability corresponding to the formation of ripples. Predictions of our model are compared with previous models and available experimental data.