Shock waves, dead zones and particle-free regions in rapid granular free-surface flows

Journal of Fluid Mechanics

Volumn 491, Issue , 2003, Pages 161-181

  Gray, J M N T ^a   Tai, Y C ^b   Noelle, S ^c  

^a UNIVERSITY OF MANCHESTER   (United Kingdom)

^b DARMSTADT UNIVERSITY OF TECHNOLOGY   (Germany)

^c RWTH AACHEN UNIVERSITY   (Germany)

Author keywords

[No Author keywords available]

Indexed keywords

AVALANCHES (SNOWSLIDES); FLOW OF FLUIDS; GRANULAR MATERIALS; HYDRAULICS; NUMERICAL METHODS; PROTECTION; SHOCK WAVES;

TOPOGRAPHY;

FLUID MECHANICS;

FLUID MECHANICS; FREE SURFACE FLOW; GRANULAR MEDIUM; MATHEMATICAL ANALYSIS; SHOCK WAVE; TWO-DIMENSIONAL FLOW;

EID: 0142027703     PISSN: 00221120     EISSN: None     Source Type: Journal    
DOI: 10.1017/S0022112003005317     Document Type: Article

References (53)

1. ABRAMOWITZ, M. & STEGUN, I. 1970 Handbook of Mathematical Functions, 9th Edn, 3.3.7. Dover.
2. AMMANN, W., BUSER, O. & VOLLENWYDER, U. 1997 Lawinen. Birkhäuser.
3. ARMINJON, P. & VIALLON, M. C. 1995 Généralisation du schéma de Nessyahu-Tadmor pour une équation hyperbolique à deux dimensions d'espace. C. R. Acad. Sci. Paris 320, 85-88.
4. ARMINJON, P. & VIALLON, M. C. 1999 Convergence of a finite volume extension of the Nessyahu-Tadmor scheme on unstructured grids for a two-dimensional linear hyperbolic equation. SIAM J. Numer. Anal. 36, 738-771.
5. BAXTER, J. TÜZÜN, U., HEYES, D., HAYATI, I. & FREDLUND, P. 1998 Stratification in poured granular heaps. Nature 391, 136.
6. CHADWICK, P. 1976. Continuum Mechanics. Concise Theory and Problems. George Allen & Unwin (republished Dover 1999).
7. CHAUDHRY, M. H. 1993 Open-Channel Flow. Prentice-Hall.
8. COURANT, R. & HILBERT, D. 1962 Methods of Mathematical Physics. Part II. Partial Differential Equations. Interscience, John Wiley and Sons.
9. DENT, J. D., BURREL, K. J., SCHMIDT, D. S., LOUGE, M. Y., ADAMS, E. E. & JAZBUTIS, T. G. 1998. Density, velocity and friction measurements in a dry-snow avalanche. Annal. Glac. 26, 247-252.
10. EGLIT, M. E. 1983 Some mathematical models of snow avalanches. In Advances in Mechanics and the Flow of Granular Materials (ed. M. Shahinpoor), vol. 2, pp. 577-588. Clausthal-Zellerfeld and Gulf Publishing Company.
11. EVANS, S. G., HUNGR, O. & ENEGREN, E. G. 1994 The Avalanche Lake rock avalanche, Mackensie Mountains, Northwest Territories, Canada: description, dating and dynamics. Can. Geotech. J. 31, 749-768.
12. GODLEVSKI, E. & RAVIART, P.-A. 1996 Numerical Approximation of Hyperbolic Systems of Conservation laws. Springer.
13. GODUNOV, S. 1959 A finite difference scheme for numerical computation of the discontinuous solutions of equations of fluid dynamics. Math. Sb. 47, 271-306.
14. GREVE, R. & HUTTER, K. 1993 Motion of a granular avalanche in a convex and concave curved chute: experiments and theoretical predictions. Phil. Trans. R. Soc. Load. A 342, 573-600.
15. GREVE, R., KOCH, T. & HUTTER, K. 1994 Unconfined flow of granular avalanches along a partly curved surface. Part I: Theory. Proc. R. Soc. Lond. A 445, 399-413.
16. GRAY, J. M. N. T. 2001 Granular flow in partially filled slowly rotating drums. J. Fluid. Mech. 441, 1-29.
17. GRAY, J. M. N. T. & HUTTER K. 1997 Pattern formation in granular avalanches. Continuum Mech. & Thermodyn. 9, 341-345.
18. GRAY, J. M. N. T. & HUTTER, K. 1998 Physik granularer Lawinen. Physikalische Blätter 54, 37-43.
19. GRAY, J. M. N. T. & TAI, Y. C. 1998 On the inclusion of a velocity dependent basal drag in avalanche models. Annal. Glac. 26, 277-280.
20. GRAY, J. M. N. T., WIELAND, M. & HUTTER K. 1999 Free surface flow of cohesionless granular avalanches over complex basal topography. Proc. R. Soc. Lond. A 455, 1841-1874.
21. GRIGORIAN, S. S., EGLIT, M. E. & IAKIMOV, IU. L. 1967 New statement and solution of the problem of the motion of snow avalanche. In Snow, Avalanches & Glaciers. Tr. Vysokogornogo Geofizich Inst. 12, 104-113. (In Russian.)
22. HARTEN, A. 1983 High resolution schemes for hyperbolic conservation laws. J. Comput. Phys. 49, 357-393.
23. HILL, K. M., KHARKAR, D. V., GILCHRIST, J. F., MCCARTHY, J. J. & OTTINO, J. M. 1999 Segregation-driven organization in chaotic granular flows. Proc. Natl Acad. Sci. USA 96 (21), 11 689-12 210.
24. HUNGR, O. 1995 A model for the runout analysis of rapid flow slides. debris flows and avalanches. Can. Geotech. J. 32, 610-623.
25. HUTTER, K., KOCH, T., PLÜSS C. & SAVAGE, S. B. 1995 The dynamics of avalanches of granular materials from initiation to run-out. Acta Mech. 109, 127-165.
26. HUTTER, K., SIEGEL, M., SAVAGE, S. B. & NOHGUCHI, Y. 1993 Two-dimensional spreading of a granular avalanche down an inclined plane. Part I Theory. Acta Mech. 100, 37-68.
27. JIANG, G. & TADMOR, E. 1997 Non-oscillatory central schemes for multidimensional hyperbolic conservation laws. SIAM J. Sci. Comput. 19, 1892-1917.
28. JÓHANNESSON, T. 2001 Run-up of two avalanches on the deflecting dams at Flateyri, northwestern Iceland. Annal. Glac. 32, 350-354.
29. KELLER, S., ITO, Y. & NISHIMURA, K. 1998 Measurements of the vertical velocity distribution in ping pong ball avalanches. Annal. Glac. 26, 259-264.
30. KOCH, T., GREVE, R. & HUTTER, K. 1994 Unconfined flow of granular avalanches along a partly curved chute. II. Experiments and numerical computations. Proc. R. Soc. Lond. A 445, 415-435.
31. KRÖNER, D. 1997 Numerical Schemes for Conservation Laws. Wiley-Teubner.
32. KULIKOVSKII, A. G. & EGLIT, M. E. 1973 Two-dimensional problem of the motion of a snow avalanche along a slope with smoothly changing properties. Prikl. Mat. Mekh. 37 (5), 837-848 (in Russian: English transl. J. Appl. Maths Mech. 37 (5), 792-803.
33. LE VEQUE, R. J. 1990 Numerical Methods for Conservation Laws. Birkhäuser.
34. LIE, K. A. & NOELLE, S. 2003 An improved quadrature rule for the flux-computation in staggered central difference schemes in multidimensions. J. Sci. Comput. 18, 69-81.
35. MCCARTHY, J. J., WOLF, J. E., SHINBROT, T. & METCALFE, G. 1996 Mixing of granular materials in slowly rotating containers. AIChE J. 42, 3351-3363.
36. MAKSE, H. A., HAVLIN, S., KING, P. R. & STANLEY, H. E. 1997 Spontaneous stratification in granular mixtures. Nature 386, 379-382.
37. METCALFE, G. SHINBROT, T, MCCARTHY, J. J. & OTTINO, J. M. 1995 Avalanche mixing of granular solids. Nature 374, 39-41.
38. NESSYAHU, H. & TADMOR, E. 1990 Non-oscillatory central differencing for hyperbolic conservation laws. J. Comput. Phys. 87, 408-463.
39. POULIQUEN, O. 1999 Scaling laws in granular flows down rough inclined planes. Phys. Fluids 11, 542-548.
40. 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.
41. REYNOLDS, O. 1885 On the dilatency of media composed of rigid particles in contact. Phil. Mag. 20, 469-481.
42. SAVAGE, S. B. 1979 Gravity flow of cohesionless granular materials in chutes and channels. J. Fluid Mech. 92, 53-96.
43. 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.
44. SAVAGE, S. B. & HUTTER, K. 1991 The dynamics of avalanches of granular materials from initiation to run-out. Part I: Analysis. Acta Mechanica 86, 201-223.
45. SHINBROT, T., ALEXANDER, A. & MUZZIO, F. J. 1999 Spontaneous chaotic granular mixing. Nature 397, 675.
46. SHINBROT, T. & MUZZIO, F. J. 2000 Nonequilibrium patterns in granular mixing and segregation. Physics Today March 2000, 25-30.
47. STOKER, J. J. 1957 Water Waves. Interscience.
48. 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.
49. TORO, E. F. 1997 Riemann Solvers and Numerical Methods for Fluid Dynamics. Springer.
50. VAN LEER, B. 1979 Towards the ultimate conservative difference scheme V. J. Comput. Phys. 32, 101-136.
51. WHITHAM, G. B. 1973 Linear and Non-linear Waves. John Wiley & Sons.
52. WIELAND, M., GRAY, J. M. N. T. & HUTTER, K. 1999 Channelized free-surface flow of cohesionless granular avalanches in a chute with shallow lateral curvature. J. Fluid Mech. 392, 73-100.
53. YEE, H. C. 1987 Construction of explicit and implicit symmetric TVD schemes and their applications. J. Comput. Phys. 68, 151-179.