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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: [email protected]
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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

REFERENCES

Abramowitz, M. & Stegun, I. 1970 Handbook of Mathematical Functions, 9th edn, 3.3.7. Dover.Google Scholar
Aranson, I. S., Malloggi, F. & Clement, E. 2006 Transverse instability in granular flows down an incline. Phys. Rev. E 73, 050302(R).CrossRefGoogle ScholarPubMed
Bartelt, P. & McArdell, B. W. 2009 Granulometric investigations of snow avalanches. J. Glaciol. 55, (193), 829–833.CrossRefGoogle Scholar
Baxter, J., Tüzün, U., Heyes, D., Hayati, I. & Fredlund, P. 1998 Stratification in poured granular heaps. Nature 391, 136.CrossRefGoogle Scholar
Bertran, P. 2003 The rock-avalanche of February 1995 at Claix (French Alps). Geomorphology 54, 339–346.CrossRefGoogle Scholar
Calder, E. S., Sparks, R. S. J. & Gardeweg, M. C. 2000 Erosion, transport and segregation of pumice and lithic clasts in pyroclastic flows inferred from ignimbrite at Lascar Volcano, Chile. J. Volcanol. Geotherm. Res. 104, 201–235.CrossRefGoogle Scholar
Chadwick, P. 1999 Continuum Mechanics: Concise Theory and Problems. Dover.Google Scholar
Costa, J. E. & Williams, G. 1984 Debris flow dynamics. Open-File Rep., 84-606 (videotape). US Geological SurveyCrossRefGoogle Scholar
Cui, X., Gray, J. M. N. T. & Johannesson, T. 2007 Deflecting dams and the formation of oblique shocks in snow avalanches at Flateyri, Iceland. J. Geophys. Res. 112, F04012.Google Scholar
Dolgunin, V. N. & Ukolov, A. A. 1995 Segregation modelling of particle rapid gravity flow. Powder Technol. 83, 95–103.CrossRefGoogle Scholar
Doyle, E. E., Huppert, H. E., Lube, G., Mader, H. M. & Sparks, R. S. 2007 Static and flowing regions in granular collapses down channels: insights from a sedimenting shallow water model. Phys. Fluids 19, 106601.CrossRefGoogle Scholar
Félix, G. & Thomas, N. 2004 Relation between dry granular flow regimes and morphology of deposits: formation of levées in pyroclastic deposits. Earth Planet. Sci. Lett. 221, 197–213.CrossRefGoogle Scholar
Golick, L. A. & Daniels, K. E. 2009 Mixing and segregation rates in sheared granular materials. Phys. Rev. E 80, 042301.CrossRefGoogle ScholarPubMed
Goujon, C., Dalloz-Dubrujeaud, B. & Thomas, N. 2007 Bidisperse granular avalanches on inclined planes: a rich variety of behaviours. Eur. J. Phys. E 23, 199–215.CrossRefGoogle Scholar
Grasselli, Y. & Herrmann, H. J. 1997 On the angles of dry granular heaps. Physica A 246, 301–312.CrossRefGoogle Scholar
Gray, J. M. N. T. 2001 Granular flow in partially filled slowly rotating drums. J. Fluid Mech. 441, 1–29.CrossRefGoogle Scholar
Gray, J. M. N. T. & Ancey, C. 2009 Segregation, recirculation and deposition of coarse particles near two-dimensional avalanche fronts. J. Fluid Mech. 629, 387–423.CrossRefGoogle Scholar
Gray, J. M. N. T. & Chugunov, V. A. 2006 Particle-size segregation and diffusive remixing in shallow granular avalanches. J. Fluid Mech. 569, 365–398.CrossRefGoogle Scholar
Gray, J. M. N. T. & Cui, X. 2007 Weak, strong and detached oblique shocks in gravity-driven granular free-surface flows. J. Fluid Mech. 579, 113–136.CrossRefGoogle Scholar
Gray, J. M. N. T. & Hutter, K. 1997 Pattern formation in granular avalanches. Contin. Mech. Thermodyn. 9, 341–345.CrossRefGoogle Scholar
Gray, J. M. N. T., Shearer, M. & Thornton, A. R. 2006 Time-dependent solutions for particle-size segregation in shallow granular avalanches. Proc. R. Soc. A 462, 947–972.CrossRefGoogle Scholar
Gray, J. M. N. T., Tai, Y. C. & Noelle, S. 2003 Shock waves, dead-zones and particle-free regions in rapid granular free-surface flows. J. Fluid Mech. 491, 161–181.CrossRefGoogle Scholar
Gray, J. M. N. T. & Thornton, A. R. 2005 A theory for particle size segregation in shallow granular free-surface flows. Proc. R. Soc. A 461, 1447–1473.CrossRefGoogle Scholar
Gray, J. M. N. T., Wieland, M. & Hutter, K. 1999 Free surface flow of cohesionless granular avalanches over complex basal topography. Proc. R. Soc. A 455, 1841–1874.CrossRefGoogle Scholar
Grigorian, S. S., Eglit, M. E. & Iakimov, Iu. L. 1967 New state and solution of the problem of the motion of snow avalanche. Snow Avalanches Glaciers, Trans. Vysokogornogo Geofiz. Inst. 12, 104–113.Google Scholar
Gruber, U. & Bartelt, P. 2007 Snow avalanche hazard modelling of large areas using shallow water numerical methods and GIS. Environ. Model. Softw. 22, (10), 1472–1481.CrossRefGoogle Scholar
Herrmann, H. J. 1998 On the shape of a sandpile. In Physics of Dry Granular Media (ed. Herrmann, H. J., Hovi, J.-P. & Luding, S.), NATO ASI Series, pp. 697–702. Kluwer Academic.CrossRefGoogle Scholar
Hill, K. M., Gioia, G. & Amaravadi, D. 2004 Radial segregation patterns in rotating granular mixtures: waviness selection. Phys. Rev. Lett. 93, (224301), 1–4.CrossRefGoogle ScholarPubMed
Iverson, R. M. 1997 The physics of debris-flows. Rev. Geophys. 35, 245–296.CrossRefGoogle Scholar
Iverson, R. M. 2003 The debris-flow rheology myth. In Debris-Flow Hazards Mitigation: Mechanics, Prediction and Assessment (ed. Rickenmann, D. & Chen, C. L.), pp. 303–314. Millpress.Google Scholar
Iverson, R. M. & Denlinger, R. P. 2001 Flow of variably fluidized granular masses across three-dimensional terrain. Part 1. Coulomb mixture theory. J. Geophys. Res. 106, (B1), 553–566.Google Scholar
Iverson, R. M. & Vallance, J. W. 2001 New views of granular mass flows. Geology 29 (2), 115–118.2.0.CO;2>CrossRefGoogle Scholar
Jesuthasan, N., Baliga, B. R. & Savage, S. B. 2006 Use of particle tracking velocimetry for measurements of granular flows: review and application. KONA 24, 15–26.CrossRefGoogle Scholar
Jiang, G.-S., Levy, D., Lin, C.-T., Osher, S. & Tadmor, E. 1998 High-resolution nonoscillatory central schemes with non-staggerred grids for hyperbolic conservation laws. SIAM J. Numer. Anal. 35, (6), 2147–2168.CrossRefGoogle Scholar
Jiang, G.-S. & Tadmor, E. 1998 Nonoscillatory central schemes for multidimensional hyperbolic conservation laws. SIAM J. Sci. Comput. 19 (6), 1892–1917.CrossRefGoogle Scholar
Johanson, J. R. 1978 Particle segregation . . . and what to do about it. Chem. Engng, 183–188.Google Scholar
Jomelli, V. & Bertran, P. 2001 Wet snow avalanche deposits in the French Alps: structure and sedimentology. Geografis. Annal. A 83 (1/2), 15–28.CrossRefGoogle Scholar
Jop, P., Forterre, Y. & Pouliquen, O. 2006 A constitutive relation for dense granular flows. Nature 44, 727–730.CrossRefGoogle Scholar
Khakhar, D. V., McCarthy, J. J. & Ottino, J. M. 1999 Mixing and segregation of granular materials in chute flows. Chaos 9, 594610.CrossRefGoogle ScholarPubMed
Lajeunesse, E., Mangeney-Castelnau, A. & Vilotte, J. P. 2004 Spreading of a granular mass on a horizontal plane. Phys. Fluids 16, 2371.CrossRefGoogle Scholar
Linares-Guerrero, E., Goujon, C. & Zenit, R. 2007 Increased mobility of bidisperse granular flows. J. Fluid Mech. 593, 475–504.CrossRefGoogle Scholar
Lube, G., Huppert, H. E., Sparks, R. S. J. & Hallworth, M. A. 2004 Axisymmetric collapses of granular columns. J. Fluid Mech. 508, 175–199.CrossRefGoogle Scholar
Makse, H. A., Havlin, S., King, P. R. & Stanley, H. E. 1997 Spontaneous stratification in granular mixtures. Nature 386, 379–382.CrossRefGoogle Scholar
Mangeney, A., Bouchut, F., Thomas, N., Vilotte, J. P. & Bristeau, M. O. 2007 Numerical modelling of self-channeling granular flows and of their levee-channel deposits. J. Geophys. Res. 112, F02017.Google Scholar
McIntyre, M., Rowe, E., Shearer, M., Gray, J. M. N. T. & Thornton, A. R. 2007 Evolution of a mixing zone in granular avalanches. AMRX 2007, abm008.Google Scholar
Middleton, G. V. 1970 Experimental studies related to problems of flysch sedimentation. In Flysch Sedimentology in North America (ed. Lajoie, J.), pp. 253–272. Business and Economics Science Ltd.Google Scholar
Middleton, G. V. & Hampton, M. A. 1976 Subaqueous sediment transport and deposition by sediment gravity flows. In Marine Sediment Transport and Environmental Management (ed. Stanley, D. J. & Swift, D. J. P.), pp. 197–218. Wiley.Google Scholar
Nessyahu, H. & Tadmor, E. 1990 Non-oscillatory central differencing for hyperbolic conservation laws. J. Comput. Phys. 87, 408–463.CrossRefGoogle Scholar
Phillips, J. C., Hogg, A. J., Kerswell, R. R. & Thomas, N. H. 2006 Enhanced mobility of granular mixtures of fine and coarse particles. Earth Planet. Sci. Lett. 246, 466–480.CrossRefGoogle Scholar
Pierson, T. C. 1986 Flow behaviour of channelized debris flows, Mount St. Helens, Washington. In Hillslope Processes (ed. Abrahams, A. D.), pp. 269–296. Allen and Unwin.Google Scholar
Pitman, E. B., Nichita, C. C., Patra, A., Bauer, A., Sheridan, M. & Bursik, M. 2003 Computing granular avalanches and landslides. Phys. Fluids 15, (12), 3638–3646.CrossRefGoogle Scholar
Pouliquen, O. 1999 a Scaling laws in granular flows down rough inclined planes. Phys. Fluids 11 (3), 542–548.CrossRefGoogle Scholar
Pouliquen, O. 1999 b On the shape of granular fronts down rough inclined planes. Phys. Fluids 11 (7), 1956–1958.CrossRefGoogle Scholar
Pouliquen, O., Delour, J. & Savage, S. B. 1997 Fingering in granular flows. Nature 386, 816–817.CrossRefGoogle Scholar
Pouliquen, O. & Forterre, Y. 2002 Friction law for dense granular flows: application to the motion of a mass down a rough inclined plane. J. Fluid Mech. 453, 133–151.CrossRefGoogle Scholar
Pouliquen, O. & Vallance, J. W. 1999 Segregation induced instabilities of granular fronts. Chaos 9, (3), 621–630.CrossRefGoogle ScholarPubMed
Rognon, P. G., Roux, J. N., Naaim, M. & Chevoir, F. 2007 Dense flows of bidisperse assemblies of disks down an inclined plane. Phys. Fluids 19, 058101.CrossRefGoogle Scholar
Savage, S. B. & Hutter, K. 1989 The motion of a finite mass of granular material down a rough incline. J. Fluid Mech. 199, 177–215.CrossRefGoogle Scholar
Savage, S. B. & Lun, C. K. K. 1988 Particle size segregation in inclined chute flow of dry cohesionless granular solids. J. Fluid Mech. 189, 311–335.CrossRefGoogle Scholar
Shearer, M. Gray, J. M. N. T. & Thornton, A. R. 2008 Stable solutions of a scalar conservation law for particle-size segregation in dense granular avalanches. Eur. J. Appl. Math. 19, 61–86.CrossRefGoogle Scholar
Tai, Y.-C., Noelle, S., Gray, J. M. N. T. & Hutter, K. 2002 Shock-capturing and front-tracking methods for granular avalanches. J. Comput. Phys. 175, 269–301.CrossRefGoogle Scholar
Thornton, A. R., Gray, J. M. N. T. & Hogg, A. J. 2006 A three-phase mixture theory for particle size segregation in shallow granular free-surface flows. J. Fluid Mech. 550, 1–25.CrossRefGoogle Scholar
Thornton, A. R. & Gray, J. M. N. T. 2008 Breaking size-segregation waves and particle recirculation in granular avalanches. J. Fluid Mech. 596, 261–284.CrossRefGoogle Scholar
Vallance, J. W. & Savage, S. B. 2000 Particle segregation in granular flows down chutes. In IUTAM Symposium on Segregation in Granular Materials (ed. Rosato, A. D. & Blackmore, D. L.), pp. 31–51. Kluwer.CrossRefGoogle Scholar
Whitham, G. B. 1974 Linear and Nonlinear Waves. Wiley.Google Scholar
Wieland, M., Gray, J. M. N. T. & Hutter, K. 1999 Channelised free surface flow of cohesionless granular avalanches in a chute with shallow lateral curvature. J. Fluid Mech. 392, 73–100.CrossRefGoogle Scholar
Williams, S. C. 1968 The mixing of dry powders. Powder Technol. 2, 13–20.CrossRefGoogle Scholar
Zuriguel, I., Gray, J. M. N. T., Peixinho, J. & Mullin, T. 2006 Pattern selection by a granular wave in a rotating drum. Phys. Rev. E 73, (061302), 1–4.CrossRefGoogle Scholar

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