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Large particle segregation, transport and accumulation in granular free-surface flows

Published online by Cambridge University Press:  19 May 2010

J. M. N. T. GRAY  and
B. P. KOKELAAR
J. M. N. T. GRAY*
Affiliation:
School of Mathematics and Manchester Centre for Nonlinear Dynamics, University of Manchester, Manchester M13 9PL, UK
B. P. KOKELAAR
Affiliation:
Earth and Ocean Sciences Department, University of Liverpool, Liverpool L69 3GP, UK
*
†Email address for correspondence: nico.gray@manchester.ac.uk
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Abstract

Particle size segregation can have a significant feedback on the motion of many hazardous geophysical mass flows such as debris flows, dense pyroclastic flows and snow avalanches. This paper develops a new depth-averaged theory for segregation that can easily be incorporated into the existing depth-averaged structure of typical models of geophysical mass flows. The theory is derived by depth-averaging the segregation-remixing equation for a bi-disperse mixture of large and small particles and assuming that (i) the avalanche is always inversely graded and (ii) there is a linear downslope velocity profile through the avalanche depth. Remarkably, the resulting ‘large particle transport equation’ is very closely related to the segregation equation from which it is derived. Large particles are preferentially transported towards the avalanche front and then accumulate there. This is important, because when this is combined with mobility feedback effects, the larger less mobile particles at the front can be continuously shouldered aside to spontaneously form lateral levees that channelize the flow and enhance run-out. The theory provides a general framework that will enable segregation-mobility feedback effects to be studied in detail for the first time. While the large particle transport equation has a very simple representation of the particle size distribution, it does a surprisingly good job of capturing solutions to the full theory once the grains have segregated into inversely graded layers. In particular, we show that provided the inversely graded interface does not break it has precisely the same solution as the full theory. When the interface does break, a concentration shock forms instead of a breaking size segregation wave, but the net transport of large particles towards the flow front is exactly the same. The theory can also model more complex effects in small-scale stratification experiments, where particles may either be brought to rest by basal deposition or by the upslope propagation of a granular bore. In the former case the resulting deposit is normally graded, while in the latter case it is inversely graded. These completely opposite gradings in the deposit arise from a parent flow that is inversely graded, which raises many questions about how to interpret geological deposits.

JFM classification

Granular media: Granular media Waves/Free-surface Flows: Wave breaking
Type
Papers
Information
Journal of Fluid Mechanics , Volume 652 , 10 June 2010 , pp. 105 - 137
Copyright
Copyright © Cambridge University Press 2010

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References

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Gray and Kokelaar supplementary movie

Movie 1. An animation of the wave steepening problem shown in figure 5. In each panel a vertical slice through the avalanche is shown with the x axis along the horizontal and the z axis along the vertical. The white region corresponds to large particles, the dark grey region contains small particles and intermediate shades of grey represent the particle concentration in mixed regions. In the top panel an exact solution for the position of the inversely graded interface is shown for the one-dimensional depth averaged transport theory, while in the lower panel a shock capturing numerical solution is shown for the full two-dimensional hyperbolic theory with Sr=0 α=0. Both solutions are identical prior to the interface breaking at t=1. At subsequent times the breaking size segregation wave in the full solution is represented by a shock in the depth-averaged theory.

Download Gray and Kokelaar supplementary movie(Video)
Video 188.3 KB

Gray and Kokelaar supplementary movie

Movie 1. An animation of the wave steepening problem shown in figure 5. In each panel a vertical slice through the avalanche is shown with the x axis along the horizontal and the z axis along the vertical. The white region corresponds to large particles, the dark grey region contains small particles and intermediate shades of grey represent the particle concentration in mixed regions. In the top panel an exact solution for the position of the inversely graded interface is shown for the one-dimensional depth averaged transport theory, while in the lower panel a shock capturing numerical solution is shown for the full two-dimensional hyperbolic theory with Sr=0 α=0. Both solutions are identical prior to the interface breaking at t=1. At subsequent times the breaking size segregation wave in the full solution is represented by a shock in the depth-averaged theory.

Download Gray and Kokelaar supplementary movie(Video)
Video 603.7 KB

Gray and Kokelaar supplementary movie

Movie 2. An animation of the wave merging problem shown in figure 7. In each panel a vertical slice through the avalanche is shown with the x axis along the horizontal and the z axis along the vertical. The white region corresponds to large particles, the dark grey region contains small particles and intermediate shades of grey represent the particle concentration in mixed regions. In the top panel an exact solution for the position of the inversely graded interface is shown for the one-dimensional depth averaged transport theory, while in the lower panel a shock capturing numerical solution is shown for the full two-dimensional hyperbolic theory with Sr=0 α=0. Two breaking size segregation waves form at the outset and propagate downslope. The upper one moves faster than the lower one and they merge at t=3.25 to form a single breaking size segregation wave. In the exact depth averaged solution in the top panel the breaking waves are represented as shocks in the interface height that move at the same speed as the breaking waves.

Download Gray and Kokelaar supplementary movie(Video)
Video 146.1 KB

Gray and Kokelaar supplementary movie

Movie 2. An animation of the wave merging problem shown in figure 7. In each panel a vertical slice through the avalanche is shown with the x axis along the horizontal and the z axis along the vertical. The white region corresponds to large particles, the dark grey region contains small particles and intermediate shades of grey represent the particle concentration in mixed regions. In the top panel an exact solution for the position of the inversely graded interface is shown for the one-dimensional depth averaged transport theory, while in the lower panel a shock capturing numerical solution is shown for the full two-dimensional hyperbolic theory with Sr=0 α=0. Two breaking size segregation waves form at the outset and propagate downslope. The upper one moves faster than the lower one and they merge at t=3.25 to form a single breaking size segregation wave. In the exact depth averaged solution in the top panel the breaking waves are represented as shocks in the interface height that move at the same speed as the breaking waves.

Download Gray and Kokelaar supplementary movie(Video)
Video 471.6 KB

Gray and Kokelaar supplementary movie

Movie 3. An animation showing how the stratification pattern in figure 9 is built up by the passage of two avalanches that are brought to rest by a combination of deposition and the upslope propagation of a granular bore (Gray & Ancey 2009). Each avalanche has a coarse rich flow front and is strongly inversely graded behind, with large white sugar crystals on top of smaller more mobile iron spheres. Once the entire avalanche has come to rest, the stationary free-surface forms the new slope for the next avalanche to flow down. By placing a ruler along the initial slope of the pile it is possible to visualize the deposition of large particles as the coarse rich front flows past.

Download Gray and Kokelaar supplementary movie(Video)
Video 415.5 KB

Gray and Kokelaar supplementary movie

Movie 3. An animation showing how the stratification pattern in figure 9 is built up by the passage of two avalanches that are brought to rest by a combination of deposition and the upslope propagation of a granular bore (Gray & Ancey 2009). Each avalanche has a coarse rich flow front and is strongly inversely graded behind, with large white sugar crystals on top of smaller more mobile iron spheres. Once the entire avalanche has come to rest, the stationary free-surface forms the new slope for the next avalanche to flow down. By placing a ruler along the initial slope of the pile it is possible to visualize the deposition of large particles as the coarse rich front flows past.

Download Gray and Kokelaar supplementary movie(Video)
Video 252.4 KB

Gray and Kokelaar supplementary movie

Movie 4. An animation of the large particle transport and accumulation problem shown in figure 14. A vertical slice through the avalanche front is shown in a frame moving with speed uF. The ξ axis lies along the horizontal and the z axis is along the vertical coordinate. In the moving frame the front is fixed at ξ=0 and the free-surface of the avalanche lies along the solid line. The white region below the free-surface contains large particles and the dark grey region contains fines. Initially the avalanche front is composed of all small particles, and at subsequent times large particles are advected towards the flow front, reaching it at τ=10. The inversely graded interface (dot dash line) then becomes multiple valued and a shock fitting procedure is used to locate the position of the discontinuity that divides the bouldery flow front from the inversely graded avalanche behind. This exact solution for the depth-averaged transport model is for parameters α=0 and β=0.3. Note that large particles are sheared towards the flow front and then accumulate there to create a bouldery margin.

Download Gray and Kokelaar supplementary movie(Video)
Video 876.5 KB

Gray and Kokelaar supplementary movie

Movie 4. An animation of the large particle transport and accumulation problem shown in figure 14. A vertical slice through the avalanche front is shown in a frame moving with speed uF. The ξ axis lies along the horizontal and the z axis is along the vertical coordinate. In the moving frame the front is fixed at ξ=0 and the free-surface of the avalanche lies along the solid line. The white region below the free-surface contains large particles and the dark grey region contains fines. Initially the avalanche front is composed of all small particles, and at subsequent times large particles are advected towards the flow front, reaching it at τ=10. The inversely graded interface (dot dash line) then becomes multiple valued and a shock fitting procedure is used to locate the position of the discontinuity that divides the bouldery flow front from the inversely graded avalanche behind. This exact solution for the depth-averaged transport model is for parameters α=0 and β=0.3. Note that large particles are sheared towards the flow front and then accumulate there to create a bouldery margin.

Download Gray and Kokelaar supplementary movie(Video)
Video 2.2 MB