Hostname: page-component-546b4f848f-fhndm Total loading time: 0 Render date: 2023-06-04T13:50:34.538Z Has data issue: false Feature Flags: { "useRatesEcommerce": true } hasContentIssue false

Segregation, recirculation and deposition of coarse particles near two-dimensional avalanche fronts

Published online by Cambridge University Press:  15 June 2009

J. M. N. T. GRAY*
School of Mathematics and Manchester Centre for Nonlinear Dynamics, University of Manchester, Oxford Road, Manchester M13 9PL, UK
École Polytechnique Fédérale de Lausanne, Ecublens, 1015 Lausanne, Switzerland
Email address for correspondence:


Stratification patterns are formed when a bidisperse mixture of large rough grains and smaller more mobile particles is poured between parallel plates to form a heap. At low flow rates discrete avalanches flow down the free surface and are brought to rest by the propagation of shock waves. Experiments performed in this paper show that the larger particles are segregated to the top of the avalanche, where the velocity is greatest, and are transported to the flow front. Here the particles are overrun but may rise to the free surface again by size segregation to create a recirculating coarse-grained front. Once the front is established composite images show that there is a steady regime in which any additional large grains that reach the front are deposited. This flow is therefore analogous to finger formation in geophysical mass flows, where the larger less mobile particles are shouldered aside to spontaneously form static lateral levees rather than being removed by basal deposition in two dimensions. At the heart of all these phenomena is a dynamic feedback between the bulk flow and the evolving particle-size distribution within the avalanche. A fully coupled theory for such segregation–mobility feedback effects is beyond the scope of this paper. However, it is shown how to derive a simplified uncoupled travelling-wave solution for the avalanche motion and reconstruct the bulk two-dimensional flow field using assumed velocity profiles through the avalanche depth. This allows a simple hyperbolic segregation theory to be used to construct exact solutions for the particle concentration and for the recirculation within the bulk flow. Depending on the material composition and the strength of the segregation and deposition, there are three types of solution. The coarse-particle front grows in length if more large particles arrive than can be deposited. If there are fewer large grains and if the segregation is strong enough, a breaking size-segregation wave forms at a unique position behind the front. It consists of two expansion fans, two shocks and a central ‘eye’ of constant concentration that are arranged in a ‘lens-like’ structure. Coarse grains just behind the front are recirculated, while those reaching the head are overrun and deposited. Upstream of the wave, the size distribution resembles a small-particle ‘sandwich’ with a raft of rapidly flowing large particles on top and a coarse deposited layer at the bottom, consistent with the experimental observations made here. If the segregation is weak, the central eye degenerates, and all the large particles are deposited without recirculation.

Copyright © Cambridge University Press 2009

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)



Abramowitz, M. & Stegun, I. 1970 Handbook of Mathematical Functions, 9th edn. Dover.Google Scholar
Ancey, C. 2002 Dry granular flow down an inclined channel: experimental investigations on the frictional–collisional regime. Phys. Rev. E 65, 011304.CrossRefGoogle ScholarPubMed
Aranson, I. S., Malloggi, F. & Clement, E. 2006 Transverse instability in granular flows down an incline. Phys. Rev. E 73, 050302(R).CrossRefGoogle ScholarPubMed
Azanza, E. 1998 Ecoulements granulaires bidimensionnels sur un plan incliné. Thèse l'École Nationale des Ponts et Chaussées, France, Spécialité: Structures et matériaux, 263 pp.Google Scholar
Bagnold, R. A. 1954 Experiments on gravity-free dispersion of large solid spheres in a Newtonian fluid under shear. Proc. R. Soc. Lond. A 225, 4963.CrossRefGoogle Scholar
Balmforth, N. J. & Kerswell, R. R. 2005 Granular collapse in two dimensions. J. Fluid Mech. 538, 399428.CrossRefGoogle Scholar
Baran, O., Ertas, D. & Halcey, T. C. 2006 Velocity correlations in dense gravity-driven granular chute flow. Phys. Rev. E 74, 051302, 110.CrossRefGoogle ScholarPubMed
Bartelt, P., Buser, O. & Platzer, K. 2007 Starving avalanches: frictional mechanisms at the tails of finite-sized mass movements. Starving avalanches: frictional mechanisms at the tails of finite-sized mass movements 34 (20), L20407.Google 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, 339346.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. Volcan. Geotherm. Res. 104, 201235.CrossRefGoogle Scholar
Costa, J. E. & Williams, G. P. 1984 Debris flow dynamics. US Geological Survey, Open-file Report, pp. 84–606. (22 minute video) Scholar
Chadwick, P. 1999 Continuum mechanics: precise theory and problems (2nd corr. and enl. ed.), Dover 187pp.Google Scholar
Cruden, D. M. & Hungr, O. 1986 The debris of the Frank slide and theories of rockslide avalanche mobility. The debris of the Frank slide and theories of rockslide avalanche mobility 33 (3).Google 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.CrossRefGoogle Scholar
Daerr, A. 2001 Dynamical equilibrium of avalanches on a rough plane. Phys. Fluids 13, 21152124.CrossRefGoogle Scholar
Deboeuf, S., Lajeunesse, E., Dauchot, O. & Andreotti, B. 2006 Flow rule, self channelization, and levees in unconfined granular flows. Phys. Rev. Lett. 97, 158303.CrossRefGoogle ScholarPubMed
Denlinger, R. P. & Iverson, R. M. 2001 Flow of variably fluidized granular masses across three-dimensional terrain. Part 2. Numerical predictions and experimental tests. J. Geophys. Res. B1, 553566.CrossRefGoogle Scholar
Dolgunin, V. N. & Ukolov, A. A. 1995 Segregation modelling of particle rapid gravity flow. Powder Technol. 83, 95103.CrossRefGoogle Scholar
Douady, S., Andreotti, B., Daerr, A. & Cladé, E. 2002 From a grain to avalanches: on the physics of granular surface flows. C. R. Physique 3, 177186.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
Ehrichs, E. E., Jaeger, H. M., Karczmar, G. S., Knight, J. B., Kuperman, V. Y. & Nagel, S. R. 1995 Granular convection observed by magnetic-resonance-imaging. Granular convection observed by magnetic-resonance-imaging 267 (5204), 16321634.Google ScholarPubMed
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, 197213.CrossRefGoogle Scholar
GDR MiDi 2004 On dense granular flows. Eur. Phys. J. E 14, 341365.CrossRefGoogle ScholarPubMed
Goujon, C., Dalloz-Dubrujeaud, B. & Thomas, N. 2007 Bidisperse granular avalanches on inclined planes: a rich variety of behaviours. Eur. Phys. J. E 23, 199215.CrossRefGoogle Scholar
Gray, J. M. N. T. 2001 Granular flow in partially filled slowly rotating drums. J. Fluid Mech. 441, 129.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, 365398.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, 113136.CrossRefGoogle Scholar
Gray, J. M. N. T. & Hutter, K. 1997 Pattern formation in granular avalanches. Continuum Mech. Thermodyn. 9, 341345.CrossRefGoogle Scholar
Gray, J. M. N. T. & Hutter, K. 1998 Physik granularer Lawinen. Physikal. Blatter 54 (1), 3743.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, 947972.CrossRefGoogle Scholar
Gray, J. M. N. T. & Tai, Y. C. 1998 Particle size segregation, granular shocks and stratification patterns. In Physics of Dry Granular Media (ed. Herrmann, H. J., Hovi, J.-P. & Luding, S.), pp. 697702, NATO ASI series. Kluwer Academic.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, 161181.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, 14471473.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, 18411874.CrossRefGoogle Scholar
Grasselli, Y. & Herrmann, H. J. 1997 On the angles of dry granular heaps. Physica A 246, 301312.CrossRefGoogle Scholar
Grigorian, S. S., Eglit, M. E. & Iakimov, Iu. L. 1967 New state and solution of the problem of the motion of snow avalance. Snow Avalanches Glaciers. Tr. Vysokogornogo Geofizich Inst. 12, 104113.Google 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.), pp. 697702, NATO ASI series. 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, 14.CrossRefGoogle ScholarPubMed
Hsiau, S. S. & Hunt, M. L. 1993 Shear-induced particle diffusion and longitudinal velocity fluctuations in a granular-flow mixing layer. J. Fluid Mech. 251, 299313.CrossRefGoogle Scholar
Issler, D. 2003 Experimental information on the dynamics of dry snow avalanches. In Dynamic Response of Granular and Porous Material under Large and Catastrophic Deformations (ed. Hutter, K. & Kirchner, N.), vol. 11, pp. 251261, Lecture Notes in Applied & Computational Mechanics. Springer.Google Scholar
Iverson, R. M. 1997 The physics of debris-flows. Reviews in Geophysics 35, 245296.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. 303314. Millpress.Google Scholar
Iverson, R. M. 2005 Debris-flow mechanics. In Debris Flow Hazards and Related Phenomena (ed. Jakob, M. and Hungr, O.), pp. 105134. Springer–Praxis.CrossRefGoogle Scholar
Iverson, R. M. & Denlinger, 2001 Flow of variably fluidized granular masses across three-dimensional terrain. Part 1. Coulomb mixture theory. J. Geophys. Res. 106 (B1), 553566.CrossRefGoogle Scholar
Iverson, R. M. & Vallance, J. W. 2001 New views of granular mass flows. New views of granular mass flows 29 (2), 115118.Google Scholar
Jenkins, J. T. & Mancini, F. 1987 Balance laws and constitutive relations for plane flows of a dense, binary mixture of smooth nearly elastic, circular disks. J. Appl. Mech. 54, 2734.CrossRefGoogle Scholar
Jenkins, J. T. & Yoon, D. 2001 Segregation in binary mixtures under gravity. Segregation in binary mixtures under gravity 88 (19), 194301194304.Google ScholarPubMed
Jesuthasan, N., Baliga, B. R. & Savage, S. B. 2006 Use of particle tracking velocimetry for measurements of granular flows: review and application. KONA 24, 1526.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), 1528.CrossRefGoogle Scholar
Jop, P., Forterre, Y. & Pouliquen, O. 2005 Crucial role of sidewalls in granular surface flows: consequences for the rheology. J. Fluid Mech. 541, 167192.CrossRefGoogle Scholar
Jop, P., Forterre, Y. & Pouliquen, O. 2006 A constitutive relation for dense granular flows. Nature 44, 727730.CrossRefGoogle Scholar
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, 475504.CrossRefGoogle Scholar
Lube, G, Huppert, H. E., Sparks, R. S. J. & Hallworth, M. A. 2004 Axisymmetric collapses of granular columns. J. Fluid Mech. 508, 175199.CrossRefGoogle Scholar
Makse, H. A., Havlin, S., King, P. R. & Stanley, H. E. 1997 Spontaneous stratification in granular mixtures. Nature 386, 379382.CrossRefGoogle Scholar
McIntyre, M., Rowe, E., Shearer, M., Gray, J. M. N. T. & Thornton, A. R. 2008 Evolution of a mixing zone in granular avalanches. AMRX 2008, abm008, 112.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. 197218. Wiley.Google Scholar
Möbius, M. E., Lauderdale, B. E., Nagel, S. R. & Jaeger, H. M. 2001 Size segregation of granular particles. Nature 414, 270.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, 466480.CrossRefGoogle Scholar
Pierson, T. C. 1986 Flow behavior of channelized debris flows, Mount St. Helens, Washington. In Hillslope Processes (ed. Abrahams, A. D.), pp. 269296. Allen & Unwin.Google Scholar
Pouliquen, O. 1999 a Scaling laws in granular flows down rough inclined planes. Phys. Fluids 11 (3), 542548.CrossRefGoogle Scholar
Pouliquen, O. 1999 b On the shape of granular fronts down rough inclined planes. Phys. Fluids 11 (7), 19561958.CrossRefGoogle Scholar
Pouliquen, O., Delour, J. & Savage, S. B. 1997 Fingering in granular flows. Nature 386, 816817.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, 133151.CrossRefGoogle Scholar
Pouliquen, O. & Vallance, J. W. 1999 Segregation induced instabilities of granular fronts. Chaos 9 (3), 621630.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. 2008 Free-surface granular flows down heaps. J. Engng Math. 60, 221240.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, 177215.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, 311335.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, 6186.CrossRefGoogle Scholar
Silbert, L. E., Ertas, D., Grest, G. S., Halsey, T. C., Levine, D. & Plimpton, S. J. 2001 Granular flow doen an inclined plane: Bagnold scaling and rheology. Phys. Rev. E 64, 051302.Google Scholar
Silbert, L. E., Landry, J. W. & Grest, G. S. 2003 Granular flow down a rough inclined plane: transition between thin and thick piles. Granular flow down a rough inclined plane: transition between thin and thick piles 15 (1), 110.Google Scholar
Taberlet, N., Richard, P., Henry, E. & Delannay, R. 2004 The growth of a super stable heap: an experimental and numerical study. The growth of a super stable heap: an experimental and numerical study 68 (4), 515521.Google Scholar
Taberlet, N., Richard, P., Valance, A., Losert, W., Pasini, J. M., Jenkins, J. T. & Delannay, R. 2003 Superstable granular heap in a thin channel. Superstable granular heap in a thin channel 91 (26), 264301, 14.Google Scholar
Thomas, N. 2000 Reverse and intermediate segregation of large beads in dry granular media. Phys. Rev. E 62 (1), 961974.Google ScholarPubMed
Thornton, A. R. & Gray, J. M. N. T. 2008 Breaking size-segregation waves and particle recirculation in granular avalanches. J. Fluid Mech. 596, 261284.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, 125.CrossRefGoogle Scholar
Vallance, J. W. 1994 Experimental and field studies related to the behavior of granular mass flows and the characteristics of their deposits. PhD thesis, Michigan Technological University.Google Scholar
Vallance, J. W. 2000 Lahars. In Encyclopedia of Volcanoes (ed. Sigurdsson, H.), pp. 601616. Academic.Google 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. 3151. Kluwer.CrossRefGoogle 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, 73100.CrossRefGoogle Scholar
Williams, S. C. 1968 The mixing of dry powders. Powder Technol. 2, 1320.CrossRefGoogle Scholar
Zhang, Y. H. & Reese, J. M. 2000 The influence of the drag force due to the interstitial gas on granular flows down a chute. Intl J. Multiphase Flow 26, 20492072.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, 14.CrossRefGoogle Scholar

Gray and Ancey supplementary movie

Movie 1. An animation showing how the stratification pattern shown in figure 1 is built up by the passage of two avalanches. 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. The avalanches are brought to rest by a normal shock (Gray & Hutter 1997; Gray, Tai & Noelle 2003) and 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 Ancey supplementary movie(Video)
Video 252 KB