Hostname: page-component-cd9895bd7-gxg78 Total loading time: 0 Render date: 2024-12-24T05:20:02.725Z Has data issue: false hasContentIssue false
  • English
  • Français

Steady inclined flows of granular-fluid mixtures

Published online by Cambridge University Press:  16 November 2009

D. BERZI  and
J. T. JENKINS
D. BERZI*
Affiliation:
Department of Environmental, Hydraulic, Infrastructures and Surveying Engineering, Politecnico di Milano, Milan 20133, Italy
J. T. JENKINS
Affiliation:
Department of Theoretical and Applied Mechanics, Cornell University, Ithaca, NY 14853, USA
*
†Email address for correspondence: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

We extend a recent theory for steady uniform gravity-driven flow of a highly concentrated granular-fluid mixture over an erodible bed between frictional sidewalls. We first include angles of inclination greater than the angle of repose of the particles; then, we introduce a boundary condition for flow over a rigid bumpy bed. We compare the predictions of the resulting theory with the volume flow rates, depths and angles of inclination measured in the experiments on dry and variously saturated flows over rigid and erodible boundaries. Finally, we employ the resulting theory, with the assumption that the flow is shallow, to solve, in an approximate way, for the variation of height and average velocities along a steady non-uniform inclined flow of a granular-fluid mixture that moves over a rigid bumpy bed. The solutions exhibit features of the flow seen in the experiments – for example, a dry bulbous snout in advance of the fluid, whose length increases with increasing number of the particles and that disappears with increasing velocity – for which satisfactory explanations were lacking.

JFM classification

Granular media: Granular media Geophysical and Geological Flows: Shallow water flows
Type
Papers
Information
Journal of Fluid Mechanics , Volume 641 , 25 December 2009 , pp. 359 - 387
Copyright
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.)

References

REFERENCES

Ancey, C. 2007 Plasticity and geophysical flows: a review. J. Non-Newton. Fluid 14, 4–35.CrossRefGoogle Scholar
Armanini, A., Capart, H., Fraccarollo, L. & Larcher, M. 2005 Rheological stratification in experimental free-surface flows of granular-liquid mixtures. J. Fluid Mech. 532, 269–319.CrossRefGoogle Scholar
Bagnold, R. A. 1954 Experiments on a gravity-free dispersion of large solid spheres in a Newtonian fluid under shear. Proc. R. Soc. Lond. A 225, 49–63.Google Scholar
Berzi, D. & Jenkins, J. T. 2008 a Approximate analytical solutions in a model for highly concentrated granular–liquid flows. Phys. Rev. E 78, 011304.CrossRefGoogle Scholar
Berzi, D. & Jenkins, J. T. 2008 b A theoretical analysis of free-surface flows of saturated granular–liquid mixtures. J. Fluid Mech. 608, 393–410.CrossRefGoogle Scholar
Cassar, C., Nicolas, M. & Pouliquen, O. 2005 Submarine granular flows down inclined planes. Phys. Fluids 17, 103301.CrossRefGoogle Scholar
Chen, C. L. & Ling, C. H. 1998 Rheological equations in asymptotic regimes of granular flow. J. Engng. Mech. 124 (3), 301–310.Google Scholar
Chow, V. T. 1959 Open Channel Hydraulics. McGraw-Hill.Google Scholar
da Cruz, F., Sacha, E., Prochnow, M., Roux, J. & Chevoir, F. 2005 Rheophysics of dense granular materials: discrete simulation of plane shear flows. Phys. Rev. E 72, 021309.CrossRefGoogle ScholarPubMed
Dallavalle, J. 1943 Micromeritics. Pitman.CrossRefGoogle Scholar
Davies, T. R. H. 1988 Debris flow surges – a laboratory investigation. In Mitteilungen der Versuchsanstalt für Wasserbau, Hydrologie und Glaziologie 96. Technischen Hochschule Zürich.Google Scholar
Davies, T. R. H. 1990 Debris flow surges – experimental simulation. J. Hydrol. (New Zealand) 29, 18–46.Google Scholar
Drew, D. A. & Passman, S. L. 1999 Theory of Multicomponent Fluids. Springer.Google Scholar
Fraccarollo, L., Larcher, M. & Armanini, A. 2007 Depth-averaged relations for granular–liquid uniform flows over mobile bed in a wide range of slope values. Granul. Matter 9, 145–157.CrossRefGoogle Scholar
GDR MiDi 2004 On dense granular flows. Eur. Phys. J. E 14, 341–365.CrossRefGoogle Scholar
Goldhirsch, I. 2003 Rapid granular flows. Annu. Rev. Fluid Mech. 35, 267–293.Google Scholar
Hungr, O. 2000 Analysis of debris flow surges using the theory of uniformly progressive flow. Earth Surf. Process. Landf. 25, 483–495.Google Scholar
Hunt, B. 1984 Perturbation solution for dam-break floods. J. Hydraul. Engng 110 (8), 1058–1071.Google Scholar
Hunt, B. 1994 Newtonian fluid mechanics treatment of debris flows and avalanches. J. Hydraul. Engng 120 (12), 1350–1363.CrossRefGoogle Scholar
Iverson, R. M. 1997 The physics of debris flows. Rev. Geophys. 35 (3), 245–296.CrossRefGoogle Scholar
Jenkins, J. T. 2001 Boundary conditions for collisional grain flows at bumpy, frictional boundaries. In Granular Gases (ed. Poschel, T. & Luding, S.), pp. 125–139. Springer.CrossRefGoogle Scholar
Jenkins, J. T. 2006 Dense shearing flows of inelastic disks. Phys. Fluids 18, 103307.CrossRefGoogle Scholar
Jenkins, J. T. 2007 Dense inclined flows of inelastic spheres. Granul. Matter 10, 47–52.CrossRefGoogle Scholar
Jenkins, J. T. & Askari, E. 1991 Boundary conditions for granular flows: phase interfaces. J. Fluid Mech. 223, 497–508.CrossRefGoogle Scholar
Jenkins, J. T. & Askari, E. 1999 Hydraulic theory for a debris flow supported on a collisional shear layer. Chaos 9, 654–658.Google Scholar
Johnson, P. C. & Jackson, R. 1987 Frictional–collisional constitutive relations for granular materials, with application to plane shearing. J. Fluid Mech. 176, 67–93.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, 167–192.CrossRefGoogle Scholar
Komatsu, T. S., Inagaki, S., Nakagawa, N. & Nasuno, S. 2001 Creep motion in a granular pile exhibiting steady surface flow. Phys. Rev. Lett. 86, 1757–1760.CrossRefGoogle Scholar
Larcher, M., Fraccarollo, L., Armanini, A. & Capart, H. 2007 Set of measurement data from flume experiments on steady, uniform debris flows. J. Hydraul. Res. 45, 59–71.CrossRefGoogle Scholar
de Marsily, G. 1981 Quantitative Hydrogeology: Groundwater Hydrology for Engineers. Academic Press.Google Scholar
Mitarai, N. & Nakanishi, H. 2005 Bagnold scaling, density plateau, and kinetic theory analysis of dense granular flow. Phys. Rev. Lett. 94, 128001.Google Scholar
Mitarai, N. & Nakanishi, H. 2007 Velocity correlations in dense granular shear flows: effects on energy dissipation and normal stress. Phys. Rev. E 75, 031305.Google Scholar
Pouliquen, O. 1999 a On the shape of granular fronts down rough inclined planes. Phys. Fluids 11 (7), 1956–1958.CrossRefGoogle Scholar
Pouliquen, O. 1999 b Scaling laws in granular flows down rough inclined planes. Phys. Fluids 11 (3), 542–548.CrossRefGoogle Scholar
Richard, P., Valance, A., Mtayer, J.-F., Sanchez, P., Crassous, J., Louge, M. & Delannay, R. 2009 Rheology of confined granular flows: scale invariance, glass transition, and friction weakening. Phys. Rev. Lett. 101, 248002.CrossRefGoogle Scholar
Richardson, J. F. & Zaki, W. N. 1954 Sedimentation and fluidization: part 1. Trans. Inst. Chem. Engrs 32, 35–53.Google Scholar
Richman, M. W. 1988 Boundary conditions based on a modified Maxwellian velocity distribution function for flows of identical, smooth, nearly elastic spheres. Acta Mech. 75, 227–240.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.Google Scholar
Silbert, L. E., Ertas, D., Grest, G. S., Halsey, T. C., Levine, D. & Plimpton, S. J. 2001 Granular flow down an inclined plane: Bagnold scaling and rheology. Phys. Rev. E 64, 51302.CrossRefGoogle ScholarPubMed
Taberlet, N., Richard, P., Valance, A., Losert, W., Pasini, J. M., Jenkins, J. T. & Delannay, R. 2003 Superstable granular heap in a thin channel. Phys. Rev. Lett. 91, 264301.CrossRefGoogle Scholar
Takahashi, T. 1991 Debris Flow. IAHR Monograph Series. Balkema.Google Scholar
Tubino, M. & Lanzoni, S. 1993 Rheology of debris flows: experimental observations and modelling problems. Excerpta Ital. Contrib. Field Hydraul. Engng 7, 201–236.Google Scholar