Three-dimensional sand ripples as the product of vortex instability
  Three-dimensional sand ripples can be observed under steady liquid flows in both nature and industry. Some examples are the ripples observed on the bed of rivers and in petroleum pipelines conveying sand. Although of importance, the formation of these patterns is not completely understood. There are theoretical and experimental evidence that aquatic ripples grow from two-dimensional bed instabilities, so that a straight vortex is formed just downstream of their crests. The proposition of Raudkivi (2006), that three-dimensionality has its origin in a vortex instability, is employed here. This paper presents a linear stability analysis of the downstream vortex in order to obtain the transverse scales of three-dimensional ripples. The obtained wavelength is compared with experimentally observed ripples. 
Erick de Moraes Franklin
physics.flu-dyn