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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
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

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
Copyright
Copyright © Cambridge University Press 2007

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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