Skip to main content

The motion of a finite mass of granular material down a rough incline

  • S. B. Savage (a1) (a2) and K. Hutter (a1) (a3)

Rock, snow and ice masses are often dislodged on steep slopes of mountainous regions. The masses, which typically are in the form of innumerable discrete blocks or granules, initially accelerate down the slope until the angle of inclination of the bed approaches the horizontal and bed friction eventually brings them to rest. The present paper describes an initial investigation which considers the idealized problem of a finite mass of material released from rest on a rough inclined plane. The granular mass is treated as a frictional Coulomb-like continuum with a Coulomb-like basal friction law. Depth-averaged equations of motion are derived; they bear a superficial resemblance to the nonlinear shallow-water wave equations. Two similarity solutions are found for the motion. They both are of surprisingly simple analytical form and show a rather unanticipated behaviour. One has the form of a pile of granular material in the shape of a parabolic cap and the other has the form of an M-wave with vertical faces at the leading and trailing edges. The linear stability of the similarity solutions is studied. A restricted stability analysis, in which the spread is left unperturbed shows them to be stable, suggesting that mathematically both are possible asymptotic wave forms. Two numerical finite-difference schemes, one of Lagrangian, the other of Eulerian type, are presented. While the Eulerian technique is able to reproduce the M-wave similarity solution, it appears to give spurious results for more general initial conditions and the Lagrangian technique is best suited for the present problem. The numerical predictions are compared with laboratory experiments of Huber (1980) involving the motion of gravel released from rest on a rough inclined plane. Although in these experiments the continuum approximation breaks down at large times when the gravel layer is only a few particle diameters thick, the general features of the development of the gravel mass are well predicted by the numerical solutions.

Hide All
Abramowitz, M. & Stegun, I. A., 1964 Handbook of Mathematical Functions, with Formulas, Graphs and Mathematical Tables. Dover.
Alean, J.: 1984 Untersuchungen über Entstehungsbedingungen und Reichweiten von Eislawinen. Mitteilung No. 74 der Versuchsanstalt für Wasserbau, Hydrologie und Glaziologie an der ETH, pp. 1217.
Alean, J.: 1985 Ice avalanche activity and mass balance of a high altitude hanging glacier in the Swiss Alps. Ann. Glaciol. 6, 248249.
Augenstein, D. A. & Hogg, R., 1978 An experimental study of the flow of dry powders over inclined surfaces. Powder Technol. 19, 205215.
Bailard, J.: 1978 An experimental study of granular-fluid flow. Ph.D. dissertation, University of California, San Diego.
Bridgwater, J.: 1972 Stress–velocity relationships for particulate solids. ASME Paper 72-MH-21. 7pp.
Buggisch, H. & Stadler, R., 1986 On the relation between shear rate and stresses in one-dimensional steady flow of moist bulk solids. Proc. World Congress Particle Technology, Part III. Mechanics of Pneumatic and Hydraulic Conveying and Mixing, Nürnberg, 16–18 April 1986, pp. 187202.
Campbell, C. S. & Gong, A., 1986 The stress tensor in a two-dimensional granular shear flow. J. Fluid Mech. 164, 107125.
Davies, T. R. H.: 1982 Spreading of rock avalanche debris by mechanical fluidization. Rock Mech. 15, 929.
Dent, J. D.: 1986 Flow properties of granular materials large overburden loads. Acta Mech. 64, 111122.
Dwight, H. B.: 1968 Tables of Integrals and Other Mathematical Data. Macmillan.
Erismann, T.: 1986 Flowing, rolling, bouncing, sliding: Synopsis of basic mechanisms. Acta Mech. 64, 101110.
Fowler, A. C.: 1980 Waves on glaciers. J. Fluid Mech. 140, 283321.
Goguel, J.: 1978 Scale dependent rockslide mechanisms. In Rockslides and Avalanches, vol. 1 (ed. B. Voight), pp. 167180. Elsevier.
Gubler, H.-U.: 1987 Measurements and modelling of snow avalanche speeds. In Avalanche Formation, Movement and Effects, IAHS Publ. 126 (ed. B. Salm & H.-U. Gubler), pp. 405420.
Haff, P. K.: 1983 Grain flow as a fluid-mechanical phenomenon. J. Fluid Mech. 134, 401430.
Hanes, D. M. & Inman, D. L., 1985 Observations of rapidly flowing granular-fluid mixtures. J. Fluid Mech. 150, 357380.
Heim, A.: 1882 Der Bergsturz von Elm. Deutsch Geol. Gesell. Zeitschrift 34, 74115.
Heim, A.: 1932 Bergsturz und Menscheleben. Beiblatt zur Vierteljahresschrift der Natf. Ges. Zürich 20, 1218.
Hsü, K.: 1975 On sturzstorms – catastrophic debris streams generated by rockfalls. Geol. Soc. Am. Bull. 86, 129140.
Hsü, K.: 1978 Albert Heim: Observations on landslides and relevance to modern interpretations. In Rockslides and Avalanches, vol. 1 (ed. B. Voight), pp. 6993. Elsevier.
Huber, A.: 1980 Schwallwellen in Seen als Folge von Felsstürzen. Mitteilung No. 47 der Versuchsanstalt für Wasserbau, Hydrologie und Glaziologie an der ETH, pp. 1122.
Hungr, O. & Morgenstern, N. R., 1984a Experiments on the flow behaviour of granular materials at high velocity in an open channel flow. Géotechnique 34, 405413.
Hungr, O. & Morgenstern, N. R., 1984b High velocity ring shear tests on sand. Géotechnique 34, 415421.
Hutter, K., Plüss, Ch. & Maeno, N. 1988 Some implications deduced from laboratory experiments on granular avalanches. Mitteilung No. 94 der Versuchsanstalt für Wasserbau, Hydrologie und Glaziologie an der ETH, pp. 323344.
Hutter, K. & Savage, S. B., 1988 Avalanche dynamics: The motion of a finite mass of gravel down a mountain side. Proc. 5th Intl Symp. on Landslides, July 7–9, Lausanne, Switzerland. 7pp.
Hutter, K., Szidarovszky, F. & Yakowitz, S., 1986 Plane steady shear flow of a cohesionless granular material down an inclined plane. A model for flow avalanches. Part II: Numerical results. Acta Mech. 63, 87112.
Hyman, J. M.: 1976 The method of lines solution of partial differential equations. ERDA Mathematics and Computing Laboratory Rep. C00-3077-139. Courant Institute of Mathematical Sciences, N.Y.U.
Ishida, M., Hatano, H. & Shirai, T., 1980 The flow of solid particles in an aerated inclined channel. Powder Technol. 27, 712.
Jenkins, J. T. & Richman, M. W., 1985 Grad's 13-moment system for a dense gas of inelastic spheres. Arch. Rat. Mech. Anal. 87, 355377.
Jenkins, J. T. & Savage, S. B., 1983 A theory for the rapid flow of identical, smooth, nearly elastic particles. J. Fluid Mech. 130, 186202.
Johnson, P. C. & Jackson, R., 1987 Frictional–collisional constitutive relations for granular materials, with application to plane shearing. J. Fluid Mech. 176, 6793.
Kent, P. E.: 1965 The transport mechanism in catastrophic rockfalls. J. Geol. 74, 7983.
Knight, P. C.: 1983 The role of particle collisions in determining high strain rate flow behaviour. Proc. Intl. Symp. on the Role of Particle Interactions in Powder Mech. Eindhoven, pp. 172182.
Lun, C. K. K., Savage, S. B., Jeffrey, D. J. & Chepurniy, N., 1984 Kinetic theories for granular flow: inelastic particles in Couette flow and slightly inelastic particles in a general flow field. J. Fluid Mech. 140, 223256.
MacCormack, R. W.: 1978 An efficient explicit-implicit characteristic method for solving the compressible Navier-Stokes equations. SIAM-AMS Proc. 11, 130155.
McDonald, B. E. & Ambrosiano, J., 1984 High-order upwind flux correction methods for hyperbolic conservation laws. J. Comp. Phys. 56, 448460.
McSaveney, M. J.: 1978 Sherman Glacier rock avalanche, Alaska, U. S. A. In Rockslides and Avalanches, vol. 1 (ed. B. Voight), pp. 197258. Elsevier.
Melosh, J.: 1986 The physics of very large landslides. Acta Mech. 64, 8999.
Norem, H., Irgens, F. & Schieldrop, B., 1987 A continuum model for calculating snow avalanches. In Avalanche Formation, Movement and Effects. IAHS Publ. 126 (ed. B. Salm & H. Gubler), pp. 363379.
Novosad, J.: 1964 Studies on granular materials. II. Apparatus for measuring the dynamic angle of internal and external friction of granular materials. Collection Czechoslov. Chem. Commun. 29, No. 2697.
Perla, I. P. & Martinelli, M., 1978 Avalanche Handbook, p. 489. US Department of Agriculture Forest Service, Agriculture Handbook.
Plüss, Ch. 1987 Experiments on granular avalanches. Diplomarbeit, Abt X, Eidg. Techn. Hochschule, Zürich. 113 pp.
Roethlisberger, H.: 1974 Möglichkeiten und Grenzen der Gletscherüberwachung. Neue Zürcher Zeitung (Beilage: Aus Forschung und Technik) 196, 29.
Roethlishberger, H.: 1978 Eislawinen und Ausbrüche von Gletscherseen. Jahrbuch der Schweiz. Natf. Ges. Wissenschaftl. Teil. 170–212.
Roethlisberger, H.: 1981 Destructive power of glaciers. In: Switzerland and her Glaciers, from the Ice Age to the Present, pp. 128165. Kümmerli & Frey, Publ. Swiss National Trust office.
Roscoe, K. H.: 1970 The influence of strain in soil mechanics. Géotechnique 20, 129170.
Salm, B.: 1966 Contribution to avalanche dynamics. IUGG/IAHS Symposium on Scientific Aspects of Snow Avalanches, Davos, Switzerland. IAHS Publ. 69, pp. 199214.
Salm, B.: 1968 On nonuniform steady flow of avalanching snow. IUGG/IAHS General Assembly Berne, Switzerland. IAHS Publ. 79, pp. 1929.
Savage, S. B.: 1979 Gravity flow of cohesionless granular materials in chutes and channels. J. Fluid Mech. 92, 5396.
Savage, S. B.: 1983 Granular flows down rough inclines – review and extension. In Mechanics of Granular Materials: New Models and Constitutive Relations (ed. J. T. Jenkins & M. Satake), pp. 26182. Elsevier.
Savage, S. B.: 1984 The mechanics of rapid granular flows. Advances in Applied Mechanics, Vol. 24 (ed. T. Y. Wu & J. Hutchinson), pp. 289366. Academic.
Savage, S. B. & Sayed, M., 1984 Stresses developed by dry cohesionless granular materials sheared in an annular shear cell. J. Fluid Mech. 142, 391430.
Scheidegger, A. E.: 1975 Physical Aspects of Natural Catastrophes. Elsevier.
Shreve, R. L.: 1966 Sherman landslide, Alaska. Science 154, 16391643.
Shreve, R. L.: 1968a Leakage and fluidization in air-lubricated avalanches. Geol. Soc. Am. Bull. 79, 653658.
Shreve, R. L.: 1968b The Blackhawk landslide. Geol. Soc. Am., Spec. Paper 108, 47 pp.
Stadler, R.: 1986 Stationäres, schnelles Fliessen von dicht gepackten trockenen und feuchten Schüttgütern. Dr.-Ing. dissertation, University of Karlsruhe, West Germany.
Stadler, R. & Buggisch, H., 1985 Influence of the deformation rate on shear stress in bulk solids, theoretical aspects and experimental results. EFCE Publication Series No. 49, Reliable Flow of Particulate Solids, Proc. Bergen, Norway, pp. 15.
Stoker, J. J.: 1957 Water Waves, the Mathematical Theory with Applications. Interscience.
Szidarovszky, F., Hutter, K. & Yakowitz, S., 1987 A numerical procedure for solving steady plane granular chute flows. Intl. J. Num. Methods in Engng 24, 19932015.
Voellmy, A.: 1955 Über die Zerstörungskraft von Lawinen. Schweizerische Bauzeitung 73, 159162, 212217, 246249, 280285.
Walton, O. R. & Braun, R. L., 1986 Stress calculations for assemblies of inelastic spheres in uniform shear. Acta Mech. 63, 7386.
Recommend this journal

Email your librarian or administrator to recommend adding this journal to your organisation's collection.

Journal of Fluid Mechanics
  • ISSN: 0022-1120
  • EISSN: 1469-7645
  • URL: /core/journals/journal-of-fluid-mechanics
Please enter your name
Please enter a valid email address
Who would you like to send this to? *


Full text views

Total number of HTML views: 0
Total number of PDF views: 354 *
Loading metrics...

Abstract views

Total abstract views: 1398 *
Loading metrics...

* Views captured on Cambridge Core between September 2016 - 16th January 2018. This data will be updated every 24 hours.