Inviscid flow separation at the crest of an erodible dune

The question of the place where a fluid flow separates on a curved obstacle is among the most difficult in fluid mechanics. A criterion based on the cancellation, at that place, of the inviscid flow singularity was proposed by Brillouin [Annal. Chimie Phys. 23, 145 (1911)] and Villat [J. Math. Pures Appl. 10, 231 (1914)], which is at the starting point of the triple-deck theory. A similar question arises for the flow over an erodible sand dune with sharp crest, where the a priori unknown dune profile and location of the crest result from the coupling of the fluid flow with the sand motion at the dune surface. We show, by computing the potential flow with Levi-Civita's conformal transformation and using a standard closure law for the shear stress driving the sand flux, that the Brillouin-Villat condition here provides the appropriate criterion. We emphasize that within the present model where the bed shear stress is in phase with the flow velocity, so that the flat bed is linearly stable, it is the separation which allows the existence of a self-preserving dune shape. For a dune traveling without deformation on a nonerodible ground, the Brillouin-Villat condition eventually selects its velocity. A parallel is drawn with the Kutta-Joukovski condition for the flow around an airfoil.