Abstract
The stability of a uniform flow above an erodible bed composed of a bimodal mixture of sediments is investigated by means of linear analysis. Results show that, for any given set of the flow and sediment parameters, two distinct modes of instability arise, each one characterized by its own wave speed, growth rate and longitudinal wavelength, each one involving spatial variations of both grain size density and bed elevation. Although at a linear level no information on the amplitude of the perturbations is gathered, the analysis of the eigenvectors associated with the two modes of instability allows for an easy classification in terms of the relative amplitudes of the perturbations of bed elevation and size density. One eigenvalue is shown to be associated with the modifications of bed forms induced by the presence of the heterogeneous mixture, such as the local accumulation of finer and coarser material along the unit wavelength, the other with the formation of the low-amplitude sorting waves known as bedload sheets. In the present unidirectional shallow-water framework, only the sorting wave is found to be unstable, since dunes and antidunes, the relevant bed forms for this case, require a more refined rotational flow model in order to become unstable. On the other hand, the simple flow model adopted allows for the formulation of an algebraic eigenvalue problem that can be solved analytically, allowing for a deep insight into the mechanisms that drive both instabilities.
Highlights
Sediments in nature can seldom be considered as homogeneous
One of the most popular simplifications adopted in the study of morphodynamic problems is the quasi-steady approximation, whereby times derivatives are dropped in the flow equations
A linear stability analysis of a uniform flow over a bed composed of an even mixture of two grain sizes is presented
Summary
Sediments in nature can seldom be considered as homogeneous. Despite this, among the many simplifying assumptions usually adopted in the modelling of sediment transport, the one of well-sorted material is probably the most common. From a theoretical point of view, linear stability analyses are known to be the ideal tool to shed light on the mechanisms responsible for the formation of a wide variety of patterns, so it is not surprising that in the nineties several attempts were made to include sorting into existing studies on bed forms, initially developed for a well-sorted sediment (see Colombini & Parker (1995), Seminara, Colombini & Parker (1996) and Lanzoni & Tubino (1999), among others) These analyses were mainly focused on the formulation of coupled sediment transport and flow models valid for heterogeneous mixtures, including the effect of hiding and of the local surface roughness. The absence of a cutoff in the short-wave range is quite often the symptom of the lack of an essential, stabilizing ingredient in the analysis, as is the case of the interfacial tension in Kelvin–Helmholtz and Rayleigh–Taylor instabilities, which transform an ill-posed (and irregular) problem into a well-posed one (Truzzolillo & Cipelletti 2017)
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