The Self-Organization of Ripples Towards Two-Dimensional Forms

Abstract

Ripples are two-dimensional bed forms of dune-like shape which develop from a flat bed under well defined conditions. Two-dimensional ripples are the result of an interaction of the erodible bed with a three-dimensional turbulent flow field, $\underline u = \overline {\underline u } + \underline {u'}$. The question is then: Why do ripples become two-dimensional and periodic? One may suggest that due to the flow separation on the crest of the ripples, a strong two-dimensional component is superimposed onto the threedimensional turbulent channel flow. Nevertheless this explanation is too simple, since it implies that two-dimensional bed forms are already present. In case of an infinitesimal two-dimensional disturbance, one can analyze the stability of the bed by introducing a sediment transport equation. This approach is problematic for two reasons. The primary periodicity is unknown, and the turbulent flow is represented in the transport equation by the mean wall shear stress, ${\tau _w} \approx {\left. {\overline {u'v'} } \right|_w}$, only. Moreover, film sequences show that the early sediment transport produces a three dimensional bed configuration. It is thus not clear how three-dimensional preforms evolve into two-dimensional ripples. This self-organization needs to be explained. A first hint is the fact that the sand grains have to be smaller than k⁺>12.5, with k⁺ = ku^τ/v and u^τ = sqrt(τ ^ω/p) for ripples to form. This implies that the separation on individual roughness elements are strong enough to disturb the self organization process. In other words, one doesn’t need a hydraulically smooth wall, with k⁺<5 where the grains are embedded in the viscous sublayer.