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Granular jets and hydraulic jumps on an inclined plane

Published online by Cambridge University Press:  18 April 2011

C. G. JOHNSON  and
J. M. N. T. GRAY
C. G. JOHNSON*
Affiliation:
School of Mathematics and Manchester Centre for Nonlinear Dynamics, University of Manchester, Oxford Road, Manchester M13 9PL, UK
J. M. N. T. GRAY
Affiliation:
School of Mathematics and Manchester Centre for Nonlinear Dynamics, University of Manchester, Oxford Road, Manchester M13 9PL, UK
*
†Email address for correspondence: cjohnson@maths.manchester.ac.uk
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Abstract

A jet of granular material impinging on an inclined plane produces a diverse range of flows, from steady hydraulic jumps to periodic avalanches, self-channelised flows and pile collapse behaviour. We describe the various flow regimes and study in detail a steady-state flow, in which the jet generates a closed teardrop-shaped hydraulic jump on the plane, enclosing a region of fast-moving radial flow. On shallower slopes, a second steady regime exists in which the shock is not teardrop-shaped, but exhibits a more complex ‘blunted’ shape with a steadily breaking wave. We explain these regimes by consideration of the supercritical or subcritical nature of the flow surrounding the shock. A model is developed in which the impact of the jet on the inclined plane is treated as an inviscid flow, which is then coupled to a depth-integrated model for the resulting thin granular avalanche on the inclined plane. Numerical simulations produce a flow regime diagram strikingly similar to that obtained in experiments, with the model correctly reproducing the regimes and their dependence on the jet velocity and slope angle. The size and shape of the steady experimental shocks and the location of sub- and supercritical flow regions are also both accurately predicted. We find that the physics underlying the rapid flow inside the shock is dominated by depth-averaged mass and momentum transport, with granular friction, pressure gradients and three-dimensional aspects of the flow having comparatively little effect. Further downstream, the flow is governed by a friction–gravity balance, and some flow features, such as a persistent indentation in the free surface, are not reproduced in the numerical solutions. On planes inclined at a shallow angle, the effect of stationary granular material becomes important in the flow evolution, and oscillatory and more general time-dependent flows are observed. The hysteretic transition between static and dynamic friction leads to two phenomena observed in the flows: unsteady avalanching behaviour, and the feedback from static grains on the flowing region, leading to levéed, self-channelised flows.

JFM classification

Granular media: Granular media Geophysical and Geological Flows: Shallow water flows Compressible Flows: Shock waves
Type
Papers
Information
Journal of Fluid Mechanics , Volume 675 , 25 May 2011 , pp. 87 - 116
Copyright
Copyright © Cambridge University Press 2011

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References

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Johnson and Gray supplementary material

Time-dependent formation of the teardrop-shaped shock shown in figure 3. Hf=30cm, ζ=26.7°, D=15mm.

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Video 2.9 MB

Johnson and Gray supplementary material

Time-dependent formation of the teardrop-shaped shock shown in figure 3. Hf=30cm, ζ=26.7°, D=15mm.

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Video 2.6 MB

Johnson and Gray supplementary material

Time-dependent formation of a blunted shock.

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Video 2.1 MB

Johnson and Gray supplementary material

Time-dependent formation of a blunted shock.

Download Johnson and Gray supplementary material(Video)
Video 6.1 MB

Johnson and Gray supplementary material

A close-up of the steady breaking wave, or region of overturning, observed at the lowest part of a blunted shock. The overturning occurs as material in the inner region of fast radial flow collides with the streams of material that surround the shock. Towards the bottom of the shock, where the streams have curved towards each other, the shock is nearly normal to the flow in the inner region. Hf=38.4cm, ζ=25.5°, D=15mm

Download Johnson and Gray supplementary material(Video)
Video 5.7 MB

Johnson and Gray supplementary material

A close-up of the steady breaking wave, or region of overturning, observed at the lowest part of a blunted shock. The overturning occurs as material in the inner region of fast radial flow collides with the streams of material that surround the shock. Towards the bottom of the shock, where the streams have curved towards each other, the shock is nearly normal to the flow in the inner region. Hf=38.4cm, ζ=25.5°, D=15mm

Download Johnson and Gray supplementary material(Video)
Video 10.4 MB

Johnson and Gray supplementary material

Simulation of the time-dependent formation of a teardrop-shaped shock. The colour scale represents the flow depth h over a range of 0 (black) to 3cm (dark red). Parameters are identical to those of figure 9, i.e. ujet=0.99 ms-1, ζ=26.7°, D=15mm. The final steady state is that shown in figure 8 and figure 9(h).

Download Johnson and Gray supplementary material(Video)
Video 252.2 KB

Johnson and Gray supplementary material

Simulation of the time-dependent formation of a teardrop-shaped shock. The colour scale represents the flow depth h over a range of 0 (black) to 3cm (dark red). Parameters are identical to those of figure 9, i.e. ujet=0.99 ms-1, ζ=26.7°, D=15mm. The final steady state is that shown in figure 8 and figure 9(h).

Download Johnson and Gray supplementary material(Video)
Video 1 MB

Johnson and Gray supplementary material

Simulation of the time-dependent formation of a blunted shock. The colour scale represents the flow depth h over a range of 0 (black) to 3cm (dark red). Parameters are identical to those of figure 10, i.e. ujet=0.99 ms-1, ζ=24.5°, D=15mm. The final steady state is that shown in figure 10.

Download Johnson and Gray supplementary material(Video)
Video 424.5 KB

Johnson and Gray supplementary material

Simulation of the time-dependent formation of a blunted shock. The colour scale represents the flow depth h over a range of 0 (black) to 3cm (dark red). Parameters are identical to those of figure 10, i.e. ujet=0.99 ms-1, ζ=24.5°, D=15mm. The final steady state is that shown in figure 10.

Download Johnson and Gray supplementary material(Video)
Video 1.8 MB

Johnson and Gray supplementary material

Unsteady flow, as shown in figure 13. Hf=25cm, ζ=26.5°, D=10mm. Initially, two regions of material have stopped downstream of the impingement point. Before 00:16 (16 seconds into the movie), the flowing material on the plane collides with the stationary material downstream and forms two shocks which propagate upslope towards the point of impingement. As the shocks approach the impingement point, they touch, forming a closed granular jump. At 00:16, the shock coincides with the impinging jet, and the flow regime at the impingement point switches to a regime of conical pile formation. The pile grows, through periodic avalanching down its flanks, until 00:37, when a spontaneous collapse occurs. This collapse enables the creation of a new region of fast radial flow and closed shock surrounding the impingement point. The material from the collapse spreads across the plane, thins, and eventually stops flowing due to friction. At 00:47, the material downstream is completely stationary, and the shock propagates inwards. At 00:52, the shock reaches the impingement point, and a new conical pile is formed. The cycle of pile formation and collapse continues until around 1:50 (one minute, 50 seconds into the movie), when enough stationary material has been deposited on the ramp to channelise the flow. This restriction of the flow width allows a flow deeper than hstop to exist, and the flow is consequently steady. The steady-state configuration reached is asymmetric.

Download Johnson and Gray supplementary material(Video)
Video 9.2 MB

Johnson and Gray supplementary material

Unsteady flow, as shown in figure 13. Hf=25cm, ζ=26.5°, D=10mm. Initially, two regions of material have stopped downstream of the impingement point. Before 00:16 (16 seconds into the movie), the flowing material on the plane collides with the stationary material downstream and forms two shocks which propagate upslope towards the point of impingement. As the shocks approach the impingement point, they touch, forming a closed granular jump. At 00:16, the shock coincides with the impinging jet, and the flow regime at the impingement point switches to a regime of conical pile formation. The pile grows, through periodic avalanching down its flanks, until 00:37, when a spontaneous collapse occurs. This collapse enables the creation of a new region of fast radial flow and closed shock surrounding the impingement point. The material from the collapse spreads across the plane, thins, and eventually stops flowing due to friction. At 00:47, the material downstream is completely stationary, and the shock propagates inwards. At 00:52, the shock reaches the impingement point, and a new conical pile is formed. The cycle of pile formation and collapse continues until around 1:50 (one minute, 50 seconds into the movie), when enough stationary material has been deposited on the ramp to channelise the flow. This restriction of the flow width allows a flow deeper than hstop to exist, and the flow is consequently steady. The steady-state configuration reached is asymmetric.

Download Johnson and Gray supplementary material(Video)
Video 21.7 MB