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Weak, strong and detached oblique shocks in gravity-driven granular free-surface flows

Published online by Cambridge University Press:  02 May 2007

J. M. N. T. GRAY  and
X. CUI
J. M. N. T. GRAY
Affiliation:
School of Mathematics and Manchester Centre for Nonlinear Dynamics, University of Manchester, Manchester M13 9PL, UK
X. CUI
Affiliation:
School of Mathematics and Manchester Centre for Nonlinear Dynamics, University of Manchester, Manchester M13 9PL, UK
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Abstract

Hazardous natural flows such as snow-slab avalanches, debris flows, pyroclastic flows and lahars are part of a much wider class of dense gravity-driven granular free-surface flows that frequently occur in industrial processes as well as in foodstuffs in our kitchens! This paper investigates the formation of oblique granular shocks, when the oncoming flow is deflected by a wall or obstacle in such a way as to cause a rapid change in the flow height and velocity. The theory for non-accelerative slopes is qualitatively similar to that of gasdynamics. For a given deflection angle there are three possibilities: a weak shock may form close to the wall; a strong shock may extend across the chute; or the shock may detach from the tip. Weak shocks have been observed in both dense granular free-surface flows and granular gases. This paper shows how strong shocks can be triggered in chute experiments by careful control of the downstream boundary conditions. The resulting downstream flow height is much thicker than that of weak shocks and there is a marked decrease in the downstream velocity. Strong shocks therefore dissipate much more energy than weak shocks. An exact solution for the angle at which the flow detaches from the wedge is derived and this is shown to be in excellent agreement with experiment. It therefore provides a very useful criterion for determining whether the flow will detach. In experimental, industrial and geophysical flows the avalanche is usually accelerated, or decelerated, by the net effect of the gravitational acceleration and basal sliding friction as the slope inclination angle changes. The presence of these source terms necessarily leads to gradual changes in the flow height and velocity away from the shocks, and this in turn modifies the local Froude number of the flow. A shock-capturing non-oscillating central method is used to compute numerical solutions to the full problem. This shows that the experiments can be matched very closely when the source terms are included and explains the deviations away from the classical oblique-shock theory. We show that weak shocks bend towards the wedge on accelerative slopes and away from it on decelerative slopes. In both cases the presence of the source terms leads to a gradual increase in the downstream flow thickness along the wedge, which suggests that defensive dams should increase in height further down the slope, contrary to current design criteria but in accordance with field observations of snow-avalanche deposits from a defensive dam in Northwestern Iceland. Movies are available with the online version of the paper.

Type
Papers
Information
Journal of Fluid Mechanics , Volume 579 , 25 May 2007 , pp. 113 - 136
Copyright
Copyright © Cambridge University Press 2007

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References

REFERENCES

Abramowitz, M. & Stegun, I. 1970 Handbook of Mathematical Functions, 9th Edn. Dover.Google Scholar
Amarouchene, Y. & Kellay, H. 2006 Speed of sound from shock fronts in granular flows. Phys. Fluids 18, 031707.CrossRefGoogle Scholar
Ames Research Staff 1953 Equations, tables and charts for compressible flow. NACA Rep. 1135.Google Scholar
Anderson, J. D. 1982 Modern Compressible Flow. McGraw-Hill.Google Scholar
Brater, E. F., King, H. F., Lindell, J. E. & Wei, C. Y. 1996 Handbook of Hydraulics, 7th Ed. McGraw-Hill.Google Scholar
Brennan, C. E., Sieck, K. & Paslaski, J. 1983 Hydraulic jumps in granular material flow. Powder Tech. 35 31–37.CrossRefGoogle Scholar
Chadwick, P. 1976 Continuum Mechanics. Concise Theory and Problems. George Allen & Unwin (republished Dover 1999).Google Scholar
Cui, X., Gray, J. M. N. T. & Jóhannesson, T. 2007 Delecting dams and the formation of oblique shocks in snow avalanches at Flateyri, Iceland. J. Geophys. Res. (in press).CrossRefGoogle Scholar
Denlinger, R. P. & Iverson, R. M. 2001 Flow of variably fluidized granular masses across three-dimensional terrain 2. Numerical predictions and experimental tests. J. Geophys. Res. B1, 553–566.CrossRefGoogle Scholar
Gray, J. M. N. T. & Hutter, K. 1997 Pattern formation in granular avalanches. Continuum Mech. Thermodyn. 9, 341–345.CrossRefGoogle Scholar
Gray, J. M. N. T. & Hutter, K. 1998 Physik granularer Lawinen. Physikalische Blatter 54 (1), 37–43.CrossRefGoogle Scholar
Gray, J. M. N. T., Irmer, A., Tai, Y. C. & Hutter, K. 1999 a Plane and oblique Shocks in shallow granular flows. In 22nd Intl. Symp. on Shock Waves, Imperial College, London, UK, July 18–23, 1999. Paper 4550, 1447–1452, Bound Proceedings ISBN 0854327118, CD-Rom ISBN 0845327061.Google Scholar
Gray, J. M. N. T. & Tai, Y. C. 1998 Particle size segregation, granular shocks and stratification patterns. In Physics of Dry Granular Media (ed. Herrmann, H. J., Hovi, J. P., Luding, S. P.), Nato ASI Series E: Applied Sciences, vol. 350, pp. 697–702.CrossRefGoogle Scholar
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, 161–181.CrossRefGoogle Scholar
Gray, J. M. N. T., Wieland, M. & Hutter, K. 1999 b Free surface flow of cohesionless granular avalanches over complex basal topography. Proc. R. Soc. Lond. A 455, 1841–1874.CrossRefGoogle Scholar
Grigorian, S. S., Eglit, M. E. & Iakimov, Iu. L. 1967 New state and solution of the problem of the motion of snow avalance. Snow, Avalanches & Glaciers. Tr. Vysokogornogo Geofizich Inst. 12, 104–113.Google Scholar
Hákonardóttir, K. M. & Hogg, A. J. 2005 Oblique shocks in rapid granular flows. Phys. Fluids 17, 0077101.CrossRefGoogle Scholar
Heil, P., Rericha, E. C., Goldman, D. I. & Swinney, H. L. 2004 Mach cone in a shallow granular fluid. Phys. Rev. E 70, 060301.CrossRefGoogle Scholar
Ippen, A.T. 1949 Mechanics of supercritical flow. ASCE. 116, 268–295.Google Scholar
Irmer, A., Schaefer, M., Gray, J. M. N. T. & Voinovich, P. 1999. An adaptive unstructured solver for granular flows. In 22nd Intl Symp. on Shock Waves, Imperial College, London, UK, July 18–23, 1999. Paper 4610, 689–693, Bound Proceedings ISBN 0854327118, CD-Rom ISBN 0845327061.Google Scholar
Iverson, R. M. 1997 The physics of debris-flows. Rev. Geophys. 35, 245–296.CrossRefGoogle Scholar
Jiang, G.-S., Levy, D., Lin, C.-T., Osher, S. & Tadmor, E. 1998 High-resolution nonoscillatory central schemes with nonstaggered grids for hyperbolic conservation laws. SIAM J. Numer. Anal. 35, 2147–2168.CrossRefGoogle Scholar
Jiang, G. & Tadmor, E. 1998 Non-oscillatory central schemes for multidimensional hyperbolic conservation laws. SIAM J. Sci. Comput. 19 (6), 1892–1917.CrossRefGoogle Scholar
Jóhannesson, T. 2001 Run-up of two avalanches on the deflecting dams at Flateyri, Northwest Iceland. Ann. Glaciol. 32, 2147–2168.CrossRefGoogle Scholar
Johnson, R. S. 1997 A modern introduction to the mathematical theory of water waves. Cambridge University Press.CrossRefGoogle Scholar
Kulikovskii, A. G. & Eglit, M. E. 1973 Two-dimensional problem of the motion of a snow avalanche along a slope with smoothly changing properties. J. Appl. Maths Mech. 37 (5), 792–803.CrossRefGoogle Scholar
Lie, K. A. & Noelle, S. 2003 An improved quadrature rule for the flux-computation in high-resolution nonoscillatory central difference schemes for systems of conservation laws in multidimensions. J. Sci. Comput. 18, 69–81.CrossRefGoogle Scholar
Nessyahu, H. & Tadmor, E. 1990 Non-oscillatory central differencing for hyperbolic conservation laws. J. Comput. Phys. 87, 408–463.CrossRefGoogle Scholar
Pouliquen, O. 1999 Scaling laws in granular flows down rough inclined planes. Phys. Fluids 11 (3), 542–548.CrossRefGoogle Scholar
Rericha, E. C., Bizon, C., Shattuck, M. & Swinney, H. 2002 Shocks in supersonic sand. Phys. Rev. Lett. 88, 014302.CrossRefGoogle ScholarPubMed
Rouse, H. 1938 Fluid Mechanics for Hydraulic Engineers. McGraw-Hill.Google Scholar
Rouse, H. 1949 Engineering Hydraulics. Wiley.Google Scholar
Saad, M. A. 1993 Compressible Fluid Flow. Prentice-Hall.Google Scholar
Savage, S. B. 1979 Gravity flow of cohesionless granular materials in chutes and channels. J. Fluid Mech. 92, 53–96.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.CrossRefGoogle Scholar
Sigurdsson, F. Tomasson, G. G. & Sandersen, F. 1998 Avalanche defences for Flateyri, Iceland. From hazard evaluation to construction of defences. Norwegian Geotech. Inst. Rep. 203.Google Scholar
Shinbrot, T. & Muzzio, F. J. 2000 Nonequilibrium patterns in granular mixing and segregation. Physics Today March, 25–30.Google Scholar
Sparks, R. S. J., Barclay, J., Calder, E. S. et al. 2002 Generation of a debris avalanche and violent pyroclastic density current on 26 December (Boxing Day) 1997 at Soufriere Hills Volcano, Monserrat. In The eruption of Soufriere Hills Volcano, Monserrat, from 1995 to 1999 (ed. Druitt & Kokelaar). Geological Society, London, Memoirs, vol. 21, pp. 409–434.Google Scholar
Spurling, R. J., Davidson, J. F. & Scott, D. M. 2001 The transient response of granular flows in an inclined rotating cylinder. Trans. Inst. Chem. Engrs 79 (A), 51–61.CrossRefGoogle Scholar
Stoker, J. J. 1957 Water Waves. Interscience.Google Scholar
Vallance, J. W. 2000 Lahars. In Encyclopedia of Volcanoes, pp. 601–616. Academic.Google Scholar
Weiyan, T. 1992 Shallow Water Hydrodynamics. Elsevier.Google Scholar
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.CrossRefGoogle Scholar

Gray and Cui supplementary movie

Movie 1. A video sequence showing the development of a weak shock in a laboratory experiment. The camera is positioned normal to a chute inclined at 38 degrees to the horizontal. Non-pareille sugar grains are stored in a hopper that is located upstream of the left-hand boundary. The avalanche is released by raising the hopper gate and flows downslope from left to right, quickly establishing a steady non-uniform flow. A wedge inclined at an angle of 20 degrees deflects the flow and a weak shock is formed at an angle of approximately 29 degrees. The Froude number just before the wedge is equal to 5. Note that the flow after the oblique shock lies parallel to the wedge.

Download Gray and Cui supplementary movie(Video)
Video 6.9 MB

Gray and Cui supplementary movie

Movie 1. A video sequence showing the development of a weak shock in a laboratory experiment. The camera is positioned normal to a chute inclined at 38 degrees to the horizontal. Non-pareille sugar grains are stored in a hopper that is located upstream of the left-hand boundary. The avalanche is released by raising the hopper gate and flows downslope from left to right, quickly establishing a steady non-uniform flow. A wedge inclined at an angle of 20 degrees deflects the flow and a weak shock is formed at an angle of approximately 29 degrees. The Froude number just before the wedge is equal to 5. Note that the flow after the oblique shock lies parallel to the wedge.

Download Gray and Cui supplementary movie(Video)
Video 2.1 MB

Gray and Cui supplementary movie

Movie 2. A video sequence showing the development of a strong shock in a laboratory experiment. The parameters are exactly the same as before, but now a second gate is placed downstream of the right-hand boundary. Initially this gate is closed and the chute is empty. Once the avalanche is released a weak shock forms along the wedge and an upslope propagating time-dependent strong shock is formed by the second blocking gate. Just before the shock reaches the wedge tip, the second gate is partially opened, and a steady strong shock forms provided the inflow and outflow mass fluxes balance.

Download Gray and Cui supplementary movie(Video)
Video 9.6 MB

Gray and Cui supplementary movie

Movie 2. A video sequence showing the development of a strong shock in a laboratory experiment. The parameters are exactly the same as before, but now a second gate is placed downstream of the right-hand boundary. Initially this gate is closed and the chute is empty. Once the avalanche is released a weak shock forms along the wedge and an upslope propagating time-dependent strong shock is formed by the second blocking gate. Just before the shock reaches the wedge tip, the second gate is partially opened, and a steady strong shock forms provided the inflow and outflow mass fluxes balance.

Download Gray and Cui supplementary movie(Video)
Video 3.8 MB

Gray and Cui supplementary movie

Movie 3. The formation of a detached oblique shock in a laboratory experiment. To acheive this, the downstream gate is discarded, the inflow Froude number is reduced to 4 and the wedge angle is increased to 44 degrees. A steady state is rapidly established with the flow detaching upstream of the wedge.

Download Gray and Cui supplementary movie(Video)
Video 9.5 MB

Gray and Cui supplementary movie

Movie 3. The formation of a detached oblique shock in a laboratory experiment. To acheive this, the downstream gate is discarded, the inflow Froude number is reduced to 4 and the wedge angle is increased to 44 degrees. A steady state is rapidly established with the flow detaching upstream of the wedge.

Download Gray and Cui supplementary movie(Video)
Video 3 MB