Linear and nonlinear instabilities of a granular bed: Determination of the scales of ripples and dunes in rivers

Abstract

Granular media are frequently found in nature and in industry and their transport by a fluid flow is of great importance to human activities. One case of particular interest is the transport of sand in open-channel and river flows. In many instances, the shear stresses exerted by the fluid flow are bounded to certain limits and some grains are entrained as bed-load: a mobile layer which stays in contact with the fixed part of the granular bed. Under these conditions, an initially flat granular bed may be unstable, generating ripples and dunes such as those observed on the bed of rivers. In free-surface water flows, dunes are bedforms that scale with the flow depth, while ripples do not scale with it. This article presents a model for the formation of ripples and dunes based on the proposition that ripples are primary linear instabilities and that dunes are secondary instabilities formed from the competition between the coalescence of ripples and free surface effects. Although simple, the model is able to explain the growth of ripples, their saturation (not explained in previous models) and the evolution from ripples to dunes, presenting a complete picture for the formation of dunes. ✩ c 2016. This manuscript version is made available under the CC-BY-NC-ND 4.0 license http://creativecommons.org/licenses/by-nc-nd/4.0/

Length scale of density waves in the gravitational flow of fine grains in pipes

“…Nonetheless, the behavior of granular flows is not well understood and the rheology of granular media is a matter of debate. Given the importance of granular flow in nature and industry, considerable work has been done to understand the dynamics and instabilities of granular media [2][3][4][5][6].…”

Prediction of the bed-load transport by gas–liquid stratified flows in horizontal ducts

“…In the present model, it is assumed that the flow is turbulent and that h/H ≪ H/D = O(1), therefore an outer region is expected to exist. Near the bed the perturbations are driven by the pressure gradient of the outer region, so that the solutions obtained by [18] are valid and may be adapted to pressure driven flows. This is done next.…”

“…where τ 0 is the shear stress on a flat bed (basic state) and τ pert is the perturbation of the shear stress caused by the bed undulation. The shear stress perturbation τ pert was found by [18] as…”

“…as showed in [18], where k is the wavenumber in the longitudinal direction and the tilde denotes variables in the Fourier space.…”

Formation of sand ripples under a turbulent liquid flow

“…They are usually considered to be a result of initially two-dimensional bedforms that saturate eventually [4,5]. Dunes are bedforms whose wavelengths scale with both the flow field and the flow depth [2,3] and may be considered as the result of coalescence of ripples [6,7,8,9]. Ripples and dunes increase friction between the bed and fluid and are related to flooding, high pressure drops, and transients.…”