Skip to main content Accessibility help
×
Home

Multiple solutions for granular flow over a smooth two-dimensional bump

  • S. Viroulet (a1), J. L. Baker (a1), A. N. Edwards (a1), C. G. Johnson (a1), C. Gjaltema (a1), P. Clavel (a1) and J. M. N. T. Gray (a1)...

Abstract

Geophysical granular flows, such as avalanches, debris flows, lahars and pyroclastic flows, are always strongly influenced by the basal topography that they flow over. In particular, localised bumps or obstacles can generate rapid changes in the flow thickness and velocity, or shock waves, which dissipate significant amounts of energy. Understanding how a granular material is affected by the underlying topography is therefore crucial for hazard mitigation purposes, for example to improve the design of deflecting or catching dams for snow avalanches. Moreover, the interactions with solid boundaries can also have important applications in industrial processes. In this paper, small-scale experiments are performed to investigate the flow of a granular avalanche over a two-dimensional smooth symmetrical bump. The experiments show that, depending on the initial conditions, two different steady-state regimes can be observed: either the formation of a detached jet downstream of the bump, or a shock upstream of it. The transition between the two cases can be controlled by adding varying amounts of erodible particles in front of the obstacle. A depth-averaged terrain-following avalanche theory that is formulated in curvilinear coordinates is used to model the system. The results show good agreement with the experiments for both regimes. For the case of a shock, time-dependent numerical simulations of the full system show the evolution to the equilibrium state, as well as the deposition of particles upstream of the bump when the inflow ceases. The terrain-following theory is compared to a standard depth-averaged avalanche model in an aligned Cartesian coordinate system. For this very sensitive problem, it is shown that the steady-shock regime is captured significantly better by the terrain-following avalanche model, and that the standard theory is unable to predict the take-off point of the jet. To retain the practical simplicity of using Cartesian coordinates, but have the improved predictive power of the terrain-following model, a coordinate mapping is used to transform the terrain-following equations from curvilinear to Cartesian coordinates. The terrain-following model, in Cartesian coordinates, makes identical predictions to the original curvilinear formulation, but is much simpler to implement.

  • View HTML
    • Send article to Kindle

      To send this article to your Kindle, first ensure no-reply@cambridge.org is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about sending to your Kindle. Find out more about sending to your Kindle.

      Note you can select to send to either the @free.kindle.com or @kindle.com variations. ‘@free.kindle.com’ emails are free but can only be sent to your device when it is connected to wi-fi. ‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.

      Find out more about the Kindle Personal Document Service.

      Multiple solutions for granular flow over a smooth two-dimensional bump
      Available formats
      ×

      Send article to Dropbox

      To send this article to your Dropbox account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your <service> account. Find out more about sending content to Dropbox.

      Multiple solutions for granular flow over a smooth two-dimensional bump
      Available formats
      ×

      Send article to Google Drive

      To send this article to your Google Drive account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your <service> account. Find out more about sending content to Google Drive.

      Multiple solutions for granular flow over a smooth two-dimensional bump
      Available formats
      ×

Copyright

This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited.

Corresponding author

Email address for correspondence: nico.gray@manchester.ac.uk

References

Hide All
Aker, B. & Bokhove, O. 2008 Hydraulic flow through a channel contraction: multiple steady states. Phys. Fluids 20, 056601.
Ames Research & Staff 1953 Equations, tables and charts for compressible flow. NACA Tech. Rep. 1135.
Andreotti, B., Claudin, P., Devauchelle, O., Durán, O. & Fourrière, A. 2012 Bedforms in a turbulent stream: ripples, chevrons and antidunes. J. Fluid Mech. 690, 94128.
Baines, P. G. & Whitehead, J. A. 2003 On multiple states in single-layer flows. Phys. Fluids 15 (2), 298307.
Baker, J. L., Barker, T. & Gray, J. M. N. T. 2016 A two-dimensional depth-averaged 𝜇(I)-rheology for dense granular avalanches. J. Fluid Mech. 787, 367395.
Bouchut, F., Mangeney-Castelnau, A., Perthame, B. & Vilotte, J. P. 2003 A new model of Saint-Venant and Savage-Hutter type for gravity driven shallow water flows. C. R. Acad. Sci. Paris 336, 531536.
Bouchut, F. & Westdickenberg, M. 2004 Gravity driven shallow water models for arbitrary topography. Commun. Math. Sci. 2, 359389.
Branney, M. J. & Kokelaar, B. P. 1992 A reappraisal of ignimbrite emplacement: progressive aggradation and changes from particulate to non-particulate flow during emplacement of high-grade ignimbrite. Bull. Volcanol. 54, 504520.
Brodsky, E. E., Evgenii Gordeev, E. & Kanamori, H. 2003 Landslide basal friction as measured by seismic waves. Geophys. Res. Lett. 30 (1–5), 2236.
Brodu, N., Richard, P. & Delannay, R. 2013 Shallow granular flows down flat frictional channels: steady flows and longitudinal vortices. Phys. Rev. E 87 (2), 191210.
Chadwick, P. 1976 Continuum mechanics. In Concise Theory and Problems. George Allen & Unwin (republished Dover 1999).
Chanson, H. 2009 Current knowledge in hydraulic jumps and related phenomena. A survey of experimental results. Eur. J. Mech. (B/Fluids) 28 (2), 191210.
Chanut, B., Faug, T. & Naaim, M. 2010 Time-varying force from dense granular avalanches on a wall. Phys. Rev. E 82, 041302.
Cui, X. & Gray, J. M. N. T. 2013 Gravity-driven granular free-surface flow around a circular cylinder. J. Fluid Mech. 720, 314337.
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.
Defina, A. & Susin, F. M. 2003 Stability of a stationary hydraulic jump in an upward sloping channel. Phys. Fluids 15 (12), 38833885.
Delannay, R., Valance, A., Mangeney, A., Roche, O. & Richard, P. 2017 Granular and particle-laden flows: from laboratory experiments to field observations. J. Phys. D 50, 053001.
Doyle, E. E., Hogg, A. J. & Mader, H. 2011 A two-layer to modelling the transformation of dilute pyroclastic currents into dense pyroclastic flows. Proc. R. Soc. Lond. A 467 (2129), 13481371.
Edwards, A. N. & Gray, J. M. N. T. 2015 Erosion-deposition waves in shallow granular free-surface flows. J. Fluid Mech. 762, 3567.
Farin, M., Mangeney, A. & Roche, O. 2014 Fundamental changes of granular flow dynamics, deposition, and erosion processes at high slope angles: insights from laboratory experiments. J. Geophys. Res.-Earth Surf. 119, 504532.
Faug, T. 2015 Depth-averaged analytic solutions for free-surface granular flows impacting rigid walls down inclines. Phys. Rev. E 92, 062310.
Faug, T., Beguin, R. & Chanut, B. 2009 Mean steady granular force on a wall overflowed by free-surface gravity-driven dense flows. Phys. Rev. E 80, 021305.
Faug, T., Childs, P., Wyburn, E. & Einav, I. 2015 Standing jumps in shallow granular flows down smooth inclines. Phys. Fluids 27, 073304.
Favreau, P., Mangeney, A., Lucas, A., Crosta, G. & Bouchut, F. 2010 Numerical modeling of landquakes. Geophys. Res. Lett. 37, L15305.
Fourriere, A., Claudin, P. & Andreotti, B. 2010 Bedforms in a turbulent stream: formation of ripples by primary linear instability and of dunes by nonlinear pattern coarsening. J. Fluid Mech. 649, 287328.
GDR-MiDi 2004 On dense granular flows. Eur. Phys. J. E 14, 341365.
Gray, J. M. N. T. 2001 Granular flow in partially filled slowly rotating drums. J. Fluid Mech. 441, 129.
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.
Gray, J. M. N. T. & Edwards, A. N. 2014 A depth-averaged 𝜇(I)-rheology for shallow granular free-surface flows. J. Fluid Mech. 755, 503534.
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.
Gray, J. M. N. T., Wieland, M. & Hutter, K. 1999 Gravity-driven free surface flow of granular avalanches over complex basal topography. Proc. R. Soc. Lond. A 455, 18411874.
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. Lond. A 342, 573600.
Grigorian, S. S., Eglit, M. E. & Iakimov, I. L. 1967 New state and solution of the problem of the motion of snow avalance. Snow Avalanches Glaciers. Tr. Vysokogornogo Geofizich Inst. 12, 104113.
Hakonardottir, K. M. & Hogg, A. J. 2005 Oblique shocks in rapid granular flows. Phys. Fluids 17, 0077101.
Hakonardottir, K. M., Hogg, A. J., Batey, J. & Woods, A. W. 2003 Flying avalanches. Geophys. Res. Lett. 30, 2191.
Ippen, A. T. 1949 Mechanics of supercritical flow. ASCE 116, 268295.
Iverson, R. M. & Denlinger, R. P. 2001 Flow of variably fluidized granular masses across three-dimensional terrain 1. Coulomb mixture theory. J. Geophys. Res. 106 (B1), 553566.
Johannesson, T., Gauer, P., Issler, P. & Lied, K.2009 The design of avalanche protection dams: recent practical and theoretical developments. Tech. Rep. 112, EUR 23339. European Commission.
Johnson, C. G., Kokelaar, B. P., Iverson, R. M., Logan, M., LaHusen, R. G. & Gray, J. M. N. T. 2012 Grain-size segregation and levee formation in geophysical mass flows. J. Geophys. Res. 117, F01032.
Jop, P., Forterre, Y. & Pouliquen, O. 2006 A constitutive relation for dense granular flows. Nature 44, 727730.
Kurganov, A. & Tadmor, E. 2000 New high-resolution central schemes for nonlinear conservation laws and convection-diffusion equations. J. Comput. Phys. 160, 241282.
Laney, C. B. 1998 Computational Gas Dynamics. Cambridge University Press.
Lawrence, G. A. 1987 Steady flow over an obstacle. J. Hydraul. Engng 113 (8), 981991.
Lax, P. D. 1957 Hyperbolic systems of conservation laws 2. Commun. Pure Appl. Maths 10 (4), 537566.
Levy, C., Mangeney, A., Bonilla, F., Hibert, C., Calder, E. & Smith, P. 2015 Friction weakening in granular flows deduced from seismic records at the Soufrière hills volcano, Montserrat. J. Geophys. Res. 120 (11), 75367557.
Mangeney, A., Bouchut, F., Thomas, N., Vilotte, J. P. & Bristeau, M. O. 2007 Numerical modeling of self-channeling granular flows and of their levee-channel deposits. J. Geophys. Res. 112, F02017.
Mangeney, A., Roche, O., Hungr, O., Magnold, N., Faccanoni, G. & Lucas, A. 2010 Erosion and mobility in granular collapse over sloping beds. J. Geophys. Res. 115, F03040.
Mangeney-Castelnau, A., Vilotte, J. P., Bristeau, M. O., Perthame, B., Bouchut, F., Simeoni, C. & Yerneni, S. 2003 Numerical modeling of avalanches based on Saint-Venant equations using a kinetic scheme. J. Geophys. Res. 108, 2527.
Moretti, L., Mangeney, A., Capdeville, Y., Stutzman, E., Huggel, C., Schneider, D. & Bouchut, F. 2012 Numerical modeling of the Mount Steller landslide flow history and of the generated long period seismic waves. Geophys. Res. Lett. 39, L16402.
Ockendon, J., Howison, S., Lacey, A. & Movchan, A. 2004 Applied Partial Differential Equations. Oxford University Press (revised edition).
Pouliquen, O. 1999 Scaling laws in granular flows down rough inclined planes. Phys. Fluids 11 (3), 542548.
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.
Rouse, H. 1938 Fluid Mechanics for Hydraulic Engineers. McGraw-Hill.
Savage, S. B. & Hutter, K. 1989 The motion of a finite mass of granular material down a rough incline. J. Fluid Mech. 199, 177215.
Savage, S. B. & Hutter, K. 1991 The dynamics of avalanches of granular materials from initiation to run-out. I. Analysis. Acta Mechanica 86, 201223.
Shapiro, A. H. 1977 Steady flow in collapsible tubes. Trans. ASME J. Biomech. Engng 99 (3), 126147.
Stoker, J. J. 1949 The breaking of waves in shallow water. Ann. N.Y. Acad. Sci. 51, 345572.
Taberlet, N., Richard, P., Valance, A., Losert, J.-M., Jenkins, J. T. & Delannay, R. 2003 Superstable granular heap in a thin channel. Phys. Rev. Lett. 91 (26), 264301.
Tai, Y. C., Wang, Y. Q., Gray, J. M. N. T. & Hutter, K. 1999 Methods of similitude in granular avalanche flows. In Advances in Cold-Region Thermal Engineering and Sciences: Technological, Environmental and Climatological Impact (ed. Hutter, K., Wang, Y. Q. & Beer, H.), Lecture Notes in Physics, vol. 533, pp. 415428. Springer.
Thielicke, W. & Stamhuis, E. 2014 PIVlab – toward user-friendly, affordable and accurate digital particle image velocimetry in MATLAB. J. Open Res. Softw. 2, e30.
Weiyan, T. 1992 Shallow Water Hydrodynamics. Elsevier.
Whitham, G. B. 1974 Linear and Nonlinear Waves. John Wiley.
Wiederseiner, S., Andreini, N., Epely-Chauvin, G., Moser, G., Monnereau, M., Gray, J. M. N. T. & Ancey, C. 2011 Experimental investigation into segregating granular flows down chutes. Phys. Fluids 23, 013301.
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.
Wierschem, A. & Aksel, N. 2004 Hydraulic jumps and standing waves in gravity-driven flows of viscous liquids in wavy open channels. Phys. Fluids. 16 (11), 38683877.
MathJax
MathJax is a JavaScript display engine for mathematics. For more information see http://www.mathjax.org.

JFM classification

Type Description Title
VIDEO
Movies

Viroulet et al. supplementary movie
Time evolution of the jet to the steady state. As the oncoming material flows over the top of the bump it is able to detach from the base and follow ballistic trajectories, before landing and coming into contact with the chute once again. The experiment is performed at a constant slope angle θ = 39◦

 Video (2.3 MB)
2.3 MB
VIDEO
Movies

Viroulet et al. supplementary movie
Time-dependent evolution of the shock towards steady state. As the oncoming material from the inflow collides with the layer of static particles upstream of the bump there is a sharp decrease in bulk velocity and associated increase in flow thickness. This shock propagates upstream until it reaches an equilibrium position. The experiment is performed at a constant slope angle θ = 39◦

 Video (4.6 MB)
4.6 MB
VIDEO
Movies

Viroulet et al. supplementary movie
An initially empty chute leads to the formation of a jet, and a shock is then generated by temporarily placing a rigid obstacle into the path of the flow. After it has settled down to an equilibrium state, the flow is again obstructed downstream of the shock. This momentarily causes the shock to migrate upstream, but as soon as the obstacle is removed the shock relaxes back to its steady-state position. Similarly, sweeping away small amounts of the slower moving material in the shock causes it to temporarily move downstream before returning to its original position.

 Video (8.3 MB)
8.3 MB
VIDEO
Movies

Viroulet et al. supplementary movie
Numerical simulation showing the full time-dependent development of the solution from the impingement of the avalanche onto the static deposit to the formation of the static deposit at the end.

 Video (7.8 MB)
7.8 MB

Metrics

Full text views

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

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed