Nonlinear instabilities on a granular bed sheared by a turbulent liquid flow

Abstract

The granular media is of great importance in our quotidian, and their transport 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 grains stay in contact with the fixed part of the granular bed. Under these conditions, a flat granular bed may be unstable, generating ripples and dunes. In a recent article (Franklin, 2010), the mechanisms of this instability were explained and a linear stability analysis was presented, in which a scaling between the fluid flow conditions and the typical length of the initial bed-forms was proposed. The present paper proposes a nonlinear stability analysis (weakly nonlinear approach) applicable to sheared granular beds, shedding light on the evolution of the bed-forms after their initial phase. The scope of the nonlinear analysis is the same as that of Franklin (2010): granular beds under turbulent liquid flows and in the presence of bed-load. It is shown here that, in this case, the initial instabilities saturate (supercritical bifurcation). Also, a discussion is made on some published experimental data

The feedback effect caused by bed load on a turbulent liquid flow

“…The tests nominal flow rates were of 5, 5.5, 6, 6.5 and 7m 3 /h, corresponding to cross-section mean velocities in the range 0.19m/s ≤ U ≤ 0.29m/s and to Reynolds number Re = U 2H gap /ν in the range 16000 < Re < 27000, where H gap is the distance from the granular bed to the top wall. The flow rates' lower and upper bounds were fixed to, respectively, correspond to the bed-load threshold [1,3,18] and to avoid the formation of ripples on the granular bed within the time scale of the tests [19,20,21,22].…”

The perturbation of a turbulent boundary layer by a two-dimensional hill

“…The longitudinal evolution of the perturbed flow is of importance to understand the formation of sand ripples. For the growth of ripples, the fluid flow must be an unstable mechanism, which means that the surface shear stress caused by the fluid must reach its maximum upstream of the bedform crests [1,2,3,11]. Therefore, in the following we focus our attention on the region upstream of the ripple crest.…”

“…The new velocity and stress distributions are of importance for many engineering applications. For instance, when in the presence of sediment transport the stress distribution along the hill is an essential parameter to understand the bed instabilities [1,2,3].…”

“…In the case of loose granular beds, the longitudinal evolution of the shear stress determines if the fluid flow is an unstable mechanism, so that Eqs. 2 to 4 are of importance for stability analyses of sand ripples and dunes [1,2,3,11]. Formally, asymptotic methods are applied to hills with aspect ratios of h max (4L) −1 < 0.05 [5,6,12], where the total length of the bedform is approximately 4L and h max corresponds to its maximum height.…”

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

“…This means that the modes resonating with the dominant one will grow much faster than the others, which can be neglected, so that the analysis is then made on a bounded number of modes. It was showed by [10] that, after the initial exponential growth (linear phase), the instabilities saturate, i.e., they attenuate their growth rate and maintain the same wavelength.…”