The transport of granular matter by a fluid flow is frequently found in nature and in industry. When the shear stresses exerted by the fluid flow on a granular bed are bounded to some limits, a mobile granular layer known as bed-load takes place, in which the grainsstay in contact with the fixed part of the granular bed. Under these conditions, an initiallyflat granular bed may be unstable, generating ripples and dunes, such as those observed indeserts, but also in pipelines conveying sand. There are evidences that these forms have atypical length correlated to their initial wavelength. So, the length-scale of the initial linear instabilities is a key point to understand the typical structures observed. This paper presents a theoretical study of the initial instabilities on a granular bed sheared by aturbulent liquid flow without free-surface effects, when bed-load is present. This studyconsists of a linear stability analysis, taking into consideration fluid flow, relaxation and gravity effects, and it is compared to published experimental data. It is proposed here,differently from many previous studies, that the initial wavelength of bed-forms varies with flow conditions when the fluid is a liquid.
two-phase flow; granular bed; bed-load; instabilities; pattern formation
