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Towards a theoretical picture of dense granular flows down inclines

Abstract

Unlike most fluids, granular materials include coexisting solid, liquid or gaseous regions, which produce a rich variety of complex flows. Dense flows down inclines preserve this complexity but remain simple enough for detailed analysis. In this review we survey recent advances in this rapidly evolving area of granular flow, with the aim of providing an organized, synthetic review of phenomena and a characterization of the state of understanding. The perspective that we adopt is influenced by the hope of obtaining a theory for dense, inclined flows that is based on assumptions that can be tested in physical experiments and numerical simulations, and that uses input parameters that can be independently measured. We focus on dense granular flows over three kinds of inclined surfaces: flat-frictional, bumpy-frictional and erodible. The wealth of information generated by experiments and numerical simulations for these flows has led to meaningful tests of relatively simple existing theories.

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Figure 1: A typical facility for producing dense granular flows down inclines.
Figure 2: Snapshots of periodic MD simulations of 10,000 grains in a domain bounded by frictionless sidewalls separated by W = 20d along z, streamwise length L = 25d, inclination α = 20°, a friction coefficient μ = 0.8 between grains, and other grain interaction parameters found in Bi et al.27.
Figure 3: Profiles along y in the simulations of Fig. 2 for a bumpy base (black lines) and a flat base (red lines)27.
Figure 4: Diagrams of dimensionless flow height against tanα, highlighting similarities between simulations and experiments.
Figure 5: Snapshot of MD simulations with frictional sidewalls (μ = 0.8), 60,000 grains and α = 40° (refs 8, 27).
Figure 6: Profiles along y highlighting similarities between SSH experiments and numerical simulations27.
Figure 7: Predictions of the kinetic theory for stresses on planes parallel to the flow compared with numerical simulations of a two-dimensional inclined flow of identical spheres34.
Figure 8: Ratio of shear and normal stress (effective friction) on a flat, frictional wall versus Froude number Fr based on mean velocity and weight6.

References

  1. Maloney, C. & Lemaître, A. Universal breakdown of elasticity at the onset of material failure. Phys. Rev. Lett. 93, 195501 (2004).

    Google Scholar 

  2. Johnson, W. L. Bulk glass-forming metallic alloys: science and technology. Mater. Res. Soc. Bull. 24, 42–56 (1999).

    CAS  Google Scholar 

  3. Azanza, E., Chevoir, F. & Moucheront, P. Experimental study of collisional granular flows down an inclined plane. J. Fluid Mech. 400, 199–227 (1999).

    Google Scholar 

  4. Forterre, Y. & Pouliquen, O. Longitudinal vortices in granular flow. Phys. Rev. Lett. 86, 5886–5889 (2001).

    CAS  Google Scholar 

  5. Xu, H., Louge, M., & Reeves, A. Solutions of the kinetic theory for bounded collisional granular flows. Continuum Mech. Thermodyn. 15, 321–349 (2003).

    CAS  Google Scholar 

  6. Louge, M. Y. & Keast, S. C. On dense granular flows down flat frictional inclines. Phys. Fluids 13, 1213–1233 (2001).

    CAS  Google Scholar 

  7. Pouliquen, O. Scaling laws in granular flows down a bumpy inclined plane. Phys. Fluids 11, 542–548 (1999).

    CAS  Google Scholar 

  8. Taberlet, N. et al. Super stable granular heap in thin channel. Phys. Rev. Lett. 91, 264301 (2003).

    Google Scholar 

  9. Silbert, L. E. et al. Granular flow down an inclined plane: Bagnold scaling and rheology. Phys. Rev. E 64, 51302 (2001).

    CAS  Google Scholar 

  10. GDR MiDi. On dense granular flows. Eur. Phys. J. E 14, 341–365 (2004).

  11. Pouliquen, O. & Chevoir, F. Dense flows of dry granular materials. C. R. Acad. Sci. Paris, Phys. 3, 163–175 (2002).

    CAS  Google Scholar 

  12. Campbell, C. S. Rapid granular flows. Annu. Rev. Fluid Mech. 22, 57–92 (1990).

    Google Scholar 

  13. Goldhirsch, I. Rapid granular flows. Annu. Rev. Fluid Mech. 35, 267–292 (2003).

    Google Scholar 

  14. Walton, O. R. in Particulate Two-Phase Flows (ed. Roco, M.) Ch. 25, 884–911 (Butterworth-Heinemann, Boston, 1993).

    Google Scholar 

  15. Foerster, S. F., Louge, M. Y., Chang, H. & Allia, K. Measurements of the collision properties of small spheres. Phys. Fluids 6, 1108–1115 (1994).

    CAS  Google Scholar 

  16. Kharaz, A. H., Gorham, D. A. & Salman, A. D. An experimental study of the elastic rebound of spheres. Powder Technol. 120, 281–291 (2001).

    CAS  Google Scholar 

  17. Louge, M. Y. & Adams, M. E. Anomalous behavior of normal kinematic restitution in the oblique impacts of a hard sphere on an elastoplastic plate. Phys. Rev. E 65, 021303 (2002).

    Google Scholar 

  18. Thornton, C. Coefficient of restitution for collinear collisions of elastic perfectly plastic spheres. J. Appl. Mech. 64, 383–386 (1997).

    CAS  Google Scholar 

  19. Zhang, X. & Vu-Quoc, L. A method to extract the mechanical properties of particles in collision based on a new elasto-plastic normal force–displacement model. Mech. Mater. 34, 779–794 (2002).

    Google Scholar 

  20. Ahn, H., Brennen, C. E. & Sabersky, R. H. Measurements of velocity, velocity fluctuations, density, and stresses in chute flows of granular materials. J. Appl. Mech. 58, 792–803 (1991).

    CAS  Google Scholar 

  21. Drake, T. G. Granular flow: physical experiments and their implications for microstructural theories. J. Fluid Mech. 225, 121–151 (1991).

    CAS  Google Scholar 

  22. Berton, G., Delannay, R., Richard, P., Taberlet, N. & Valance, A. Two-dimensional inclined chute flows: transverse motion and segregation. Phys. Rev. E 68, 051303 (2003).

    Google Scholar 

  23. Hanes, D. M. & Walton, O. R. Simulations and physical measurements of glass spheres flowing down a bumpy incline. Powder Tech. 109, 133–144 (2000).

    CAS  Google Scholar 

  24. Walton, O. R. Numerical simulation of inclined chute flows of monodisperse, inelastic, frictional spheres. Mech. Mater. 16, 239–247 (1993).

    Google Scholar 

  25. Luding, S. in The Physics of Granular Media (eds Hinrichsen, H. & Wolf, D.) 299–324 (Wiley-VCH, Weinheim, 2004).

    Google Scholar 

  26. Moreau, J. J. Some numerical methods in multibody dynamics: applications to granular materials. Eur. J. Mech. A 13, 93–114 (1994).

    Google Scholar 

  27. Bi, W., Delannay, R., Richard, P., Taberlet, N. & Valance, A. 2D and 3D confined granular chute flows: experimental and numerical results. J. Phys. Condens. Matter 17, 1–24 (2005).

    Google Scholar 

  28. Johnson, P. C., Nott, P. & Jackson, R. Frictional–collisional equations of motion for particulate flows and their application to chutes. J. Fluid Mech. 210, 501–535 (1990).

    CAS  Google Scholar 

  29. Baumberger, T. & Caroli, C. Multicontact solid friction: a macroscopic probe of pinning and dissipation on the mesoscopic scale. Mater. Res. Soc. Bull. 23, 41–46 (1998).

    Google Scholar 

  30. Silbert, S. L., Landry, J. W. & Grest, G. S. Granular flow down a rough inclined plane; transition between thin and thick piles. Phys. Fluids 15, 1–10 (2003).

    CAS  Google Scholar 

  31. Louge, M. Y. Model for dense granular flows down bumpy inclines. Phys. Rev. E 67, 061303 (2003).

    Google Scholar 

  32. Daerr, A. & Douady, S. Two types of avalanche behaviour in granular media. Nature 399, 241–243 (1999).

    CAS  Google Scholar 

  33. Silbert, L. E., Grest, G. S. & Plimpton, S. J. Boundary effects and self-organization in dense granular flows. Phys. Fluids 14, 1–10 (2002).

    Google Scholar 

  34. Mitarai, N. & Nakanishi, H. Bagnold scaling, density plateau, and kinetic theory analysis of dense granular flow. Phys. Rev. Lett. 94, 128001 (2005).

    Google Scholar 

  35. Ancey, C. Dry granular flows down an inclined channel: experimental investigation on the frictional–collisional regime. Phys. Rev. E 65, 011304 (2001).

    Google Scholar 

  36. Hungr, O. & Morgenstern, N. R. Experiments on the flow behaviour of granular materials at high velocity in an open channel. Géotechnique 34, 405–413 (1984).

    Google Scholar 

  37. Jop, P., Forterre, Y. & Pouliquen, O. Crucial role of side walls for granular surface flows: consequences for rheology. J. Fluid Mech. 541, 167–192 (2005).

    Google Scholar 

  38. Rajchenbach, J. Granular flows. Adv. Phys. 49, 229–256 (2000).

    Google Scholar 

  39. Khakhar, D. V., Orpe, A. V., Andersen, P. & Ottino, J. M. Surface flows of granular materials: model and experiments in heap formation. J. Fluid Mech. 441, 255–264 (2001).

    CAS  Google Scholar 

  40. Komatsu, T. S., Inagaki, S., Nakagawa, N. & Nasuno, S. Creep motion in a granular pile exhibiting steady surface flow. Phys. Rev. Lett. 86, 1757–1760 (2001).

    CAS  Google Scholar 

  41. Félix, G. & Thomas, N. Relation between dry granular flow regimes and morphology of deposits: formation of levees in pyroclastic deposits. Earth Planet. Sci. Lett. 221, 197–213 (2004).

    Google Scholar 

  42. Jenkins, J. T. in Dynamics: Models and Kinetic Methods for Non-Equilibrium Many Bodied Systems (ed. Karkheck, J.) 313–323 (Kluwer, Dordrecht, 2000).

    Google Scholar 

  43. Chapman, S. & Cowling, T. G. The Mathematical Theory of Nonuniform Gases. (Cambridge Univ. Press, Cambridge, 1970).

    Google Scholar 

  44. Kirkwood, J. G. Selected Topics in Statistical Mechanics (Gordon & Breach, New York, 1967).

    Google Scholar 

  45. Jenkins, J. T. Rapid granular flow down inclines. Appl. Mech. Rev. 47, S240–S244 (1994).

    Google Scholar 

  46. Bocquet, L., Errami, L. J. & Lubensky, T. C. A hydrodynamic model of a jammed-to-flowing transition in gravity driven granular materials. Phys. Rev. Lett. 89, 184301 (2002).

    Google Scholar 

  47. Savage, S. B. in Mechanics of Granular Materials: New Models and Constitutive Relations (eds Jenkins, J. T. & Satake, M.) 261–282 (Elsevier, Amsterdam, 1983).

    Google Scholar 

  48. Mills, P., Tixier, M. & Loggia, D. Model for stationary dense granular flow along an inclined wall. Europhys. Lett. 45, 733–738 (1999).

    CAS  Google Scholar 

  49. Ancey, C. & Evesque, P. Frictional–collisional regime for granular suspension flow down an inclined channel. Phys. Rev. E 62, 8349–8360 (2000).

    CAS  Google Scholar 

  50. Aranson, I. S. & Tsimring, L. S. Continuum description of partially fluidized granular flows. Phys. Rev. E 65, 061303 (2002).

    Google Scholar 

  51. Anderson, K. G. & Jackson, R. A. Comparison of the solutions of some proposed equations of motion of granular materials for fully developed flow down inclined planes. J. Fluid Mech. 241, 145–168 (1992).

    CAS  Google Scholar 

  52. Pouliquen, O., Forterre, Y. & Ledizes, S. Dense granular flow down incline as a self-activated process. Adv. Complex Syst. 4, 441–450 (2001).

    Google Scholar 

  53. Lemaitre, A. Origin of a repose angle: kinetics of rearrangement for granular materials. Phys. Rev. Lett. 89, 064303 (2002).

    Google Scholar 

  54. Pouliquen, O. On the shape of granular fronts down rough inclined planes. Phys. Fluids 11, 1956–1958 (1999).

    CAS  Google Scholar 

  55. Jenkins, J. T. & Savage, S. B. A theory for the rapid flow of identical, smooth, nearly elastic, spherical particles. J. Fluid Mech. 130, 187–202 (1983).

    Google Scholar 

  56. Lun, C. K. K., Savage, S. B., Jeffrey, D. J. & Chepurniy, N. Kinetic theories for granular flow: inelastic particles in Couette flow and slightly inelastic particles in a general flow field. J. Fluid Mech. 140, 223–256 (1984).

    Google Scholar 

  57. Jenkins, J. T. & Richman, M. W. Grad's 13-moment system for a dense gas of inelastic spheres. Arch. Rat. Mech. Anal. 87, 355–377 (1985).

    Google Scholar 

  58. Jenkins, J. T. & Zhang, C. Kinetic theory for nearly elastic, slightly frictional spheres. Phys. Fluids 14, 1228–1235 (2002).

    CAS  Google Scholar 

  59. Lun, C. K. K. Kinetic theory for granular flow of dense, slightly inelastic, slightly rough spheres. J. Fluid Mech. 223, 539–559 (1991).

    Google Scholar 

  60. Chou, C. -S. & Richman, M. W. Constitutive theory for homogeneous granular shear flows of highly elastic spheres. Physica A 259, 430–448 (1998).

    CAS  Google Scholar 

  61. Sela, N. & Goldhirsch, I. Hydrodynamic equations for rapid flows of smooth inelastic spheres, to Burnett order. J. Fluid Mech. 361, 41–74 (1998).

    CAS  Google Scholar 

  62. Montanero, J. M., Garzo, V., Santos, A. & Brey, J. J. Kinetic theory of simple granular shear flows of smooth hard spheres. J. Fluid Mech. 389, 391–411 (1999).

    Google Scholar 

  63. Richman, M. W. Boundary conditions based on a modified Maxwellian velocity distribution for flows of identical, smooth, nearly elastic spheres. Acta Mech. 75, 227–240 (1988).

    Google Scholar 

  64. Richman, M. W. & Marciniec, R. P. Gravity-driven granular flows of smooth, inelastic spheres down bumpy inclines. J. Appl. Mech. 57, 1036–1043 (1990).

    CAS  Google Scholar 

  65. Nishimura, K., Kosugi, K. & Nakagawa, M. Experiments on ice-sphere flows along an inclined chute. Mech. Mater. 16, 205–209 (1993).

    Google Scholar 

  66. Silbert, L. E., Ertas, D., Grest, G. S., Halsey, T. C. & Levine, D. Analogies between granular jamming and the liquid-glass transition. Phys. Rev. E 65, 051307 (2002).

    Google Scholar 

  67. Aranson, I. S. & Tsimring, L. S. Continuum description of avalanches in granular media. Phys. Rev. E 64, 020301 (2001).

    CAS  Google Scholar 

  68. Wolfson, D., Tsimring, L. S. & Aranson, I. S. Partially fluidized shear granular flows: continuum theory and MD simulations. Phys. Rev. Lett. 90, 254301 (2003).

    Google Scholar 

  69. Mills, P., Tixier, M. & Loggia, D. Influence of roughness and dilatancy for dense granular flow along an inclined wall. Eur. Phys. J. E 1, 5–8 (2000).

    CAS  Google Scholar 

  70. Chevoir, F., Prochnow, M., Jenkins, J. T. & Mills, P. in Powders and Grains 01 (ed. Kishino, Y.) 373–376 (Balkema, Lisse, 2001).

    Google Scholar 

  71. Bonamy, D. & Mills, P. Diphasic non-local model for granular surface flows. Europhys. Lett. 63, 42–48 (2003).

    CAS  Google Scholar 

  72. Rajchenbach, J. Dense, rapid flows of inelastic grains under gravity. Phys. Rev. Lett. 90, 144302 (2003).

    Google Scholar 

  73. da Cruz, F., Emam, S., Prochnow, M., Roux, J. -N. & Chevoir, F. Rheophysics of dense granular materials: discrete simulation of plane shear flows. Phys. Rev. E 72, 021309 (2005).

    Google Scholar 

  74. Jop, P., Forterre, Y. & Pouliquen, O. A constitutive law for dense granular flows. Nature 441, 727–730 (2006).

    CAS  Google Scholar 

  75. Campbell, C. S, Cleary, P. & Hopkins, M. A. Long run-out landslides: a study by computer simulation. J. Geophys. Res. 100, 8267–8283 (1995).

    Google Scholar 

  76. Campbell, C. S. Granular shear flow at the elastic limit. J. Fluid Mech. 465, 261–291 (2002).

    CAS  Google Scholar 

  77. Babic, M., Shen, H. H. & Shen, H. T. The stress tensor in granular shear flows of uniform, deformable disks at high solids concentrations. J. Fluid Mech. 219, 81–118 (1990).

    CAS  Google Scholar 

  78. Zhang, D. Z. & Rauenzahn, R. M. Stress relaxation in dense and slow flows. J. Rheol. 44, 1019–1041 (2000).

    CAS  Google Scholar 

  79. Ertas, D. & Halsey, T. C. Granular gravitational collapse and chute flow. Europhys. Lett. 60, 931–937 (2002).

    CAS  Google Scholar 

  80. Wildman, R. D., Huntley, J. M., Hansen, J. -P. & Parker, D. J. Granular temperature profiles in three-dimensional vibrofluidized granular beds. Phys. Rev. E 63, 061311 (2001).

    CAS  Google Scholar 

  81. Fukushima, E. Granular flow studies by NMR: a chronology. Adv. Complex Syst. 4, 1–5 (2001).

    Google Scholar 

  82. Dixon, P. K. & Durian, D. J. Speckle visibility spectroscopy and variable granular fluidization. Phys. Rev. Lett. 90 184302 (2003).

    CAS  Google Scholar 

  83. Pouliquen, O., Belzons, M. & Nicolas, M. Fluctuating particle motion during shear induced granular compaction. Phys. Rev. Lett. 91, 014301 (2003).

    CAS  Google Scholar 

  84. Toiya, M., Stambaugh, J. & Losert, W. Transient and oscillatory granular shear flow. Phys. Rev. Lett. 93, 088001 (2004).

    Google Scholar 

  85. Richard, P., Philippe, P., Barbe, F., Bourlè s, T. X. & Bideau, D. Analysis by x-ray microtomography of a granular packing undergoing compaction. Phys. Rev. E 68, 020301 (2003).

    Google Scholar 

  86. Jenkins, J. T. & Richman, M. Kinetic theory for plane flows of a dense gas of identical, rough, inelastic, circular disks. Phys. Fluids 28, 3485–3494 (1985).

    CAS  Google Scholar 

  87. Verlet, L. & Levesque, D. Integral equations for classical fluids. III. The hard discs system. Mol. Phys. 46, 969–980 (1982).

    CAS  Google Scholar 

  88. Luding, S. Global equation of state of two-dimensional hard sphere system. Phys. Rev. E 63, 042201 (2001).

    CAS  Google Scholar 

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Acknowledgements

We thank Daniel Bideau, Gérard Le Caër, Luc Oger, Nathalie Thomas, and our colleagues in the Groupement de Recherche Milieux Divises (GDR MiDi) for valuable discussions. We thank James T. Jenkins for contributing several paragraphs on merits of the kinetic theory, and Namiko Mitarai for providing data shown in Fig. 7. The preparation of this review was assisted by financial support from the GDR MiDi and US–France Cooperative Research grant INT-0233212. Our research in dense, inclined flows is sponsored by the French Ministry of Education and Research (ACI PCN (INSU): Écoulements gravitaires: modélisation des processus), the CNRS (PNRN: Programme National des Risques Naturels, écoulements gravitaires), and NASA grants NCC3-468, NAG3-2705, NCC3-797 and NAG3-2353.

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Delannay, R., Louge, M., Richard, P. et al. Towards a theoretical picture of dense granular flows down inclines. Nature Mater 6, 99–108 (2007). https://doi.org/10.1038/nmat1813

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