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Segregation-induced finger formation in granular free-surface flows

  • J. L. Baker (a1), C. G. Johnson (a1) and J. M. N. T. Gray (a1)

Abstract

Geophysical granular flows, such as landslides, pyroclastic flows and snow avalanches, consist of particles with varying surface roughnesses or shapes that have a tendency to segregate during flow due to size differences. Such segregation leads to the formation of regions with different frictional properties, which in turn can feed back on the bulk flow. This paper introduces a well-posed depth-averaged model for these segregation-mobility feedback effects. The full segregation equation for dense granular flows is integrated through the avalanche thickness by assuming inversely graded layers with large particles above fines, and a Bagnold shear profile. The resulting large particle transport equation is then coupled to depth-averaged equations for conservation of mass and momentum, with the feedback arising through a basal friction law that is composition dependent, implying greater friction where there are more large particles. The new system of equations includes viscous terms in the momentum balance, which are derived from the $\unicode[STIX]{x1D707}(I)$ -rheology for dense granular flows and represent a singular perturbation to previous models. Linear stability calculations of the steady uniform base state demonstrate the significance of these higher-order terms, which ensure that, unlike the inviscid equations, the growth rates remain bounded everywhere. The new system is therefore mathematically well posed. Two-dimensional simulations of bidisperse material propagating down an inclined plane show the development of an unstable large-rich flow front, which subsequently breaks into a series of finger-like structures, each bounded by coarse-grained lateral levees. The key properties of the fingers are independent of the grid resolution and are controlled by the physical viscosity. This process of segregation-induced finger formation is observed in laboratory experiments, and numerical computations are in qualitative agreement.

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Copyright

This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited.

Corresponding author

Email addresses for correspondence: james.baker@alumni.manchester.ac.uk, nico.gray@manchester.ac.uk

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JFM classification

Type Description Title
VIDEO
Movie 1

Baker et al. Movie 1
An overhead view showing a laboratory experiment in which a bidisperse mixture of 80% ballotini (white, 75-150μm), 20% carborundum (brown, 305-355μm) is released from rest through a double gate system of inflow thickness 2mm. The initially empty chute is inclined at 27° and roughened with turquoise ballotini (750-1000μm). As material flows down the slope, the large particles are segregated to the surface and preferentially sheared to the front. This becomes unstable due to greater frictional forces and splits into a number of different channels, or fingers, that are bounded by coarse-grained lateral levees.

 Video (18.6 MB)
18.6 MB
VIDEO
Movie 1

Baker et al. Movie 1
An overhead view showing a laboratory experiment in which a bidisperse mixture of 80% ballotini (white, 75-150μm), 20% carborundum (brown, 305-355μm) is released from rest through a double gate system of inflow thickness 2mm. The initially empty chute is inclined at 27° and roughened with turquoise ballotini (750-1000μm). As material flows down the slope, the large particles are segregated to the surface and preferentially sheared to the front. This becomes unstable due to greater frictional forces and splits into a number of different channels, or fingers, that are bounded by coarse-grained lateral levees.

 Video (5.6 MB)
5.6 MB
VIDEO
Movie 2

Baker et al. Movie 2
An overhead view showing a laboratory experiment in which a monodisperse granular material of 100% white ballotini (75-150μm) is released from rest through a double gate system of inflow thickness 2mm. The initially empty chute is inclined at 27° and roughened with turquoise ballotini (750-1000μm). In contrast to the bidisperse case (supplementary movie 1), material propagates approximately uniformly downslope with only slight irregularities due to bed and inflow imperfections.

 Video (5.2 MB)
5.2 MB
VIDEO
Movie 2

Baker et al. Movie 2
An overhead view showing a laboratory experiment in which a monodisperse granular material of 100% white ballotini (75-150μm) is released from rest through a double gate system of inflow thickness 2mm. The initially empty chute is inclined at 27° and roughened with turquoise ballotini (750-1000μm). In contrast to the bidisperse case (supplementary movie 1), material propagates approximately uniformly downslope with only slight irregularities due to bed and inflow imperfections.

 Video (2.2 MB)
2.2 MB
VIDEO
Movie 3

Baker et al. Movie 3
Time-dependent numerical solutions of the system of equations representing a continuous uniform inflow of bidisperse granular material propagating down an empty inclined plane. Panels show contours of the flow thickness (top), depth-averaged concentration of small particles (centre), and depth-averaged flow speed (bottom). A large-rich region quickly develops at the flow front, starts to become unstable and develops into finger-like structures, which elongate and coarsen over time. The slow moving lateral levees bounding the fingers consist predominantly of large particles and channelise the more mobile interior. Note that the fines in the centre of the channel speed up as the fingers form.

 Video (9.6 MB)
9.6 MB
VIDEO
Movie 3

Baker et al. Movie 3
Time-dependent numerical solutions of the system of equations representing a continuous uniform inflow of bidisperse granular material propagating down an empty inclined plane. Panels show contours of the flow thickness (top), depth-averaged concentration of small particles (centre), and depth-averaged flow speed (bottom). A large-rich region quickly develops at the flow front, starts to become unstable and develops into finger-like structures, which elongate and coarsen over time. The slow moving lateral levees bounding the fingers consist predominantly of large particles and channelise the more mobile interior. Note that the fines in the centre of the channel speed up as the fingers form.

 Video (1.5 MB)
1.5 MB

Segregation-induced finger formation in granular free-surface flows

  • J. L. Baker (a1), C. G. Johnson (a1) and J. M. N. T. Gray (a1)

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