  Sand ripples are commonly observed in both nature and industry. For example,
they are found on riverbeds and in oil pipelines that transport sand. In both
natural and industrial cases, ripples increase friction between the bed and
fluid and are related to flooding, high pressure drops, and transients. Ripples
appear when sediments are entrained as bed load (a mobile granular layer) and
are usually considered to be the result of initial bedforms that eventually
saturate. Given the small aspect ratio of the initial bedforms, linear analyses
can be used to understand the formation of ripples. This paper presents a
linear stability analysis of a granular bed under a turbulent flow of a liquid.
This analysis takes into consideration all the main mechanisms and parameters
involved in the turbulent liquid case, including some important parameters that
have not yet been considered together such as the bed compactness and the
bed-load threshold shear stress. The results of this analysis are compared with
published experimental results and they show good agreement.
