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Breaking size segregation waves and particle recirculation in granular avalanches

Published online by Cambridge University Press:  17 January 2008

A. R. THORNTON  and
J. M. N. T. GRAY
A. R. THORNTON
Affiliation:
School of Mathematics and Manchester Centre for Nonlinear Dynamics, University of Manchester, Oxford Road, Manchester, M13 9PL, UK
J. M. N. T. GRAY
Affiliation:
School of Mathematics and Manchester Centre for Nonlinear Dynamics, University of Manchester, Oxford Road, Manchester, M13 9PL, UK
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Abstract

Particle-size segregation is a common feature of dense gravity-driven granular free-surface flows, where sliding and frictional grain–grain interactions dominate. Provided that the diameter ratio of the particles is not too large, the grains segregate by a process called kinetic sieving, which, on average, causes the large particles to rise to the surface and the small grains to sink to the base of the avalanche. When the flowing layer is brought to rest this stratification is often preserved in the deposit and is known by geologists as inverse grading. Idealized experiments with bi-disperse mixtures of differently sized grains have shown that inverse grading can be extremely sharp on rough beds at low inclination angles, and may be modelled as a concentration jump or shock. Several authors have developed hyperbolic conservation laws for segregation that naturally lead to a perfectly inversely graded state, with a pure phase of coarse particles separated from a pure phase of fines below, by a sharp concentration jump. A generic feature of these models is that monotonically decreasing sections of this concentration shock steepen and eventually break when the layer is sheared. In this paper, we investigate the structure of the subsequent breaking, which is important for large-particle recirculation at the bouldery margins of debris flows and for fingering instabilities of dry granular flows. We develop an exact quasi-steady travelling wave solution for the structure of the breaking/recirculation zone, which consists of two shocks and two expansion fans that are arranged in a ‘lens’-like structure. A high-resolution shock-capturing numerical scheme is used to investigate the temporal evolution of a linearly decreasing shock towards a steady-state lens, as well as the interaction of two recirculation zones that travel at different speeds and eventually coalesce to form a single zone. Movies are available with the online version of the paper.

JFM classification

Granular media: Granular media Waves/Free-surface Flows: Wave breaking Mixing: Granular mixing
Type
Papers
Information
Journal of Fluid Mechanics , Volume 596 , 25 January 2008 , pp. 261 - 284
Copyright
Copyright © Cambridge University Press 2008

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Thornton and Gray supplementary movie

Movie 1. Wave Breaking problem. Animation showing the evolution of a breaking size segregation wave in a frame (ξ,z) moving downstream at the same speed as the steady-state lens ulens = 1. A series of stills for this problem are illustrated in figure 6 and the final steady state is shown in figure 4(a). The initial condition consists of a linearly decreasing concentration shock that joins two constant height sections. In response to linear shear through the avalanche depth the shock steepens and breaks at t = 1 to form an oscillating lens. Computations are performed on a 300 x 300 grid and with Sr = 1. Contours of the small particle concentration are illustrated using the colour bar below, with red corresponding to pure fines and blue to pure large. For t > 20 the time-step is increased to speed up the convergence towards the steady-state solution.

Download Thornton and Gray supplementary movie(Video)
Video 1.2 MB

Thornton and Gray supplementary movie

Movie 1. Wave Breaking problem. Animation showing the evolution of a breaking size segregation wave in a frame (ξ,z) moving downstream at the same speed as the steady-state lens ulens = 1. A series of stills for this problem are illustrated in figure 6 and the final steady state is shown in figure 4(a). The initial condition consists of a linearly decreasing concentration shock that joins two constant height sections. In response to linear shear through the avalanche depth the shock steepens and breaks at t = 1 to form an oscillating lens. Computations are performed on a 300 x 300 grid and with Sr = 1. Contours of the small particle concentration are illustrated using the colour bar below, with red corresponding to pure fines and blue to pure large. For t > 20 the time-step is increased to speed up the convergence towards the steady-state solution.

Download Thornton and Gray supplementary movie(Video)
Video 1 MB

Thornton and Gray supplementary movie

Movie 2. Lens interaction. Animation showing the development of the small particle concentration during the interaction of two breaking size segregation waves in a frame (ξ,z) moving downslope with speed unity. A series of stills for this problem are illustrated in figure 8 and the final steady state is shown in figure 4(a). At t = 0 the sharp downward steps in concentration break to form two lenses that propagate in opposite directions with speed 0.4. Just after t = 3 these begin to coalesce to form a single lens between Hup = 0.9 and Hdown = 0.1 that propagates downslope with speed unity. The results are for linear shear and Sr = 1. Computations are performed on a 300 x 300 grid and contours of the small particle concentration are illustrated using the colour bar above, with red corresponding to pure fines and blue to pure large. For t > 20 the time-step is increased to speed up the convergence towards the steady-state solution.

Download Thornton and Gray supplementary movie(Video)
Video 1.2 MB

Thornton and Gray supplementary movie

Movie 2. Lens interaction. Animation showing the development of the small particle concentration during the interaction of two breaking size segregation waves in a frame (ξ,z) moving downslope with speed unity. A series of stills for this problem are illustrated in figure 8 and the final steady state is shown in figure 4(a). At t = 0 the sharp downward steps in concentration break to form two lenses that propagate in opposite directions with speed 0.4. Just after t = 3 these begin to coalesce to form a single lens between Hup = 0.9 and Hdown = 0.1 that propagates downslope with speed unity. The results are for linear shear and Sr = 1. Computations are performed on a 300 x 300 grid and contours of the small particle concentration are illustrated using the colour bar above, with red corresponding to pure fines and blue to pure large. For t > 20 the time-step is increased to speed up the convergence towards the steady-state solution.

Download Thornton and Gray supplementary movie(Video)
Video 1 MB