Ripples and dunes in a turbulent stream . Part 1 : Turbulent flow

Using a second order turbulent closure, the turbulent flow over a wavy bottom is derived as a systematic expansion with respect to the amplitude of the corrugation. The article is focused on the analysis of the dynamical mechanisms controlling the different regions of the flow. It aims also at calculating the shear and normal stresses applied to the ground, which constitute key ingredients of the morphodynamics of ripples and dunes. Amongst the non-linear effects, we investigate the relation between the geometry of the bottom and the resulting hydrodynamical roughness. We also show that most of the first non-linearities exert a negative feedback on the basal stress. Finally, we study the influence of the presence of a free surface at distance from the ground comparable to the bottom wavelength. Due to the resonance of gravity surface waves, the phase lag between the shear stress and the bottom profile shifts downstream. This is a crucial point to understand ripples and dunes instabilities, detailed in part 3.