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Segregation-induced fingering instabilities in granular free-surface flows

Published online by Cambridge University Press:  21 August 2012

M. J. Woodhouse ,
A. R. Thornton ,
C. G. Johnson ,
B. P. Kokelaar  and
J. M. N. T. Gray
M. J. Woodhouse*
Affiliation:
School of Mathematics and Manchester Centre for Nonlinear Dynamics, University of Manchester, Oxford Road, Manchester M13 9PL, UK
A. R. Thornton
Affiliation:
Department of Applied Mathematics and Department of Mechanical Engineering, University of Twente, 7500 AE Enschede, The Netherlands
C. G. Johnson
Affiliation:
School of Mathematics and Manchester Centre for Nonlinear Dynamics, University of Manchester, Oxford Road, Manchester M13 9PL, UK
B. P. Kokelaar
Affiliation:
Department of Geology and Geophysics, University of Liverpool, 4 Brownlow Street, Liverpool L69 3GP, 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
*
†Email address for correspondence: mark.woodhouse@bristol.ac.uk
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Abstract

Particle-size segregation can have a significant feedback on the bulk motion of granular avalanches when the larger grains experience greater resistance to motion than the fine grains. When such segregation-mobility feedback effects occur the flow may form digitate lobate fingers or spontaneously self-channelize to form lateral levees that enhance run-out distance. This is particularly important in geophysical mass flows, such as pyroclastic currents, snow avalanches and debris flows, where run-out distance is of crucial importance in hazards assessment. A model for finger formation in a bidisperse granular avalanche is developed by coupling a depth-averaged description of the preferential transport of large particles towards the front with an established avalanche model. The coupling is achieved through a concentration-dependent friction coefficient, which results in a system of non-strictly hyperbolic equations. We compute numerical solutions to the flow of a bidisperse mixture of small mobile particles and larger more resistive grains down an inclined chute. The numerical results demonstrate that our model is able to describe the formation of a front rich in large particles, the instability of this front and the subsequent evolution of elongated fingers bounded by large-rich lateral levees, as observed in small-scale laboratory experiments. However, our numerical results are grid dependent, with the number of fingers increasing as the numerical resolution is increased. We investigate this pathology by examining the linear stability of a steady uniform flow, which shows that arbitrarily small wavelength perturbations grow exponentially quickly. Furthermore, we find that on a curve in parameter space the growth rate is unbounded above as the wavelength of perturbations is decreased and so the system of equations on this curve is ill-posed. This indicates that the model captures the physical mechanisms that drive the instability, but additional dissipation mechanisms, such as those considered in the realm of flow rheology, are required to set the length scale of the fingers that develop.

JFM classification

Granular media: Granular media Instability: Instability Geophysical and Geological Flows: Shallow water flows
Type
Papers
Information
Journal of Fluid Mechanics , Volume 709 , 25 October 2012 , pp. 543 - 580
Copyright
Copyright © Cambridge University Press 2012

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Footnotes

‡

Current address: School of Mathematics, University of Bristol, University Walk, Bristol BS8 1TW, UK.

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Woodhouse et al. supplementary movie

An overhead view showing the time-dependent evolution of the laboratory experiment shown in figure 1 in which a bi-disperse mixture of spherical (white) ballotini (75-150 μm) and irregular (brown) carborundum grains (315-350 μm) is moving down an inclined plane covered with (turquoise) glass ballotini (750-1000 μm). Kinetic sieving and squeeze expulsion cause particle-size segregation, with the large grains rising towards the free-surface while the small particles fall towards the base. Velocity shear through the avalanche results in a preferential transport of large grains towards the flow front where they may be overrun, but rise to the surface again by size segregation. Therefore the larger, more resistive particles accumulate at the flow front which leads to a lateral instability. The front then degenerates into a series of distinct fingers bounded by large-rich lateral levees. As the fingers evolve, narrow fingers are cannibalized by wider neighbours and the tips of fingers are seen to split.

Download Woodhouse et al. supplementary movie(Video)
Video 5 MB

Woodhouse et al. supplementary movie

An overhead view showing the time-dependent evolution of the laboratory experiment shown in figure 1 in which a bi-disperse mixture of spherical (white) ballotini (75-150 μm) and irregular (brown) carborundum grains (315-350 μm) is moving down an inclined plane covered with (turquoise) glass ballotini (750-1000 μm). Kinetic sieving and squeeze expulsion cause particle-size segregation, with the large grains rising towards the free-surface while the small particles fall towards the base. Velocity shear through the avalanche results in a preferential transport of large grains towards the flow front where they may be overrun, but rise to the surface again by size segregation. Therefore the larger, more resistive particles accumulate at the flow front which leads to a lateral instability. The front then degenerates into a series of distinct fingers bounded by large-rich lateral levees. As the fingers evolve, narrow fingers are cannibalized by wider neighbours and the tips of fingers are seen to split.

Download Woodhouse et al. supplementary movie(Video)
Video 8.1 MB

Woodhouse et al. supplementary movie

An oblique head on view showing the time-dependent evolution of the laboratory experiment shown in figure 2 in which a bi-disperse mixture of spherical (white) ballotini and irregular (brown) carborundum grains moves down an inclined plane towards the camera. Large grains gather at the flow margins and form lateral levees that confine more mobile fines-rich material in a channel. The accumulation of static regions of large material can cause deviation in the flow direction, as seen here. In addition, as the flow wanes, material can be seen draining off the raised lateral levees, exposing fines-rich material lining the interior of the channels

Download Woodhouse et al. supplementary movie(Video)
Video 12.3 MB

Woodhouse et al. supplementary movie

An oblique head on view showing the time-dependent evolution of the laboratory experiment shown in figure 2 in which a bi-disperse mixture of spherical (white) ballotini and irregular (brown) carborundum grains moves down an inclined plane towards the camera. Large grains gather at the flow margins and form lateral levees that confine more mobile fines-rich material in a channel. The accumulation of static regions of large material can cause deviation in the flow direction, as seen here. In addition, as the flow wanes, material can be seen draining off the raised lateral levees, exposing fines-rich material lining the interior of the channels

Download Woodhouse et al. supplementary movie(Video)
Video 9.9 MB

Woodhouse et al. supplementary movie

Results of a numerical computation in which solutions of the governing equations representing the flow of a bi-disperse granular mixture from a hopper with a fixed height onto an inclined plane. Contours of the concentration of small particles (top panel), depth-averaged flow speed (middle panel) and flow depth (bottom panel) are shown. At early times the material propagates down the plane maintaining a uniform front, with large particles accumulating at the front. Perturbations to the front grow and numerous small wavelength lobes and clefts appear at the front and on the interface between the large-rich front and the material behind. The perturbations develop into distinct fingers separated by slow-moving large-rich material (levees). During the further evolution of the flow, finger splitting and cannibalization events are observed.

Download Woodhouse et al. supplementary movie(Video)
Video 2.2 MB

Woodhouse et al. supplementary movie

Results of a numerical computation in which solutions of the governing equations representing the flow of a bi-disperse granular mixture from a hopper with a fixed height onto an inclined plane. Contours of the concentration of small particles (top panel), depth-averaged flow speed (middle panel) and flow depth (bottom panel) are shown. At early times the material propagates down the plane maintaining a uniform front, with large particles accumulating at the front. Perturbations to the front grow and numerous small wavelength lobes and clefts appear at the front and on the interface between the large-rich front and the material behind. The perturbations develop into distinct fingers separated by slow-moving large-rich material (levees). During the further evolution of the flow, finger splitting and cannibalization events are observed.

Download Woodhouse et al. supplementary movie(Video)
Video 7.2 MB