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Retrogressive failure of a static granular layer on an inclined plane

  • A. S. Russell (a1), C. G. Johnson (a1), A. N. Edwards (a1), S. Viroulet (a2), F. M. Rocha (a1) and J. M. N. T. Gray (a1)...

Abstract

When a layer of static grains on a sufficiently steep slope is disturbed, an upslope-propagating erosion wave, or retrogressive failure, may form that separates the initially static material from a downslope region of flowing grains. This paper shows that a relatively simple depth-averaged avalanche model with frictional hysteresis is sufficient to capture a planar retrogressive failure that is independent of the cross-slope coordinate. The hysteresis is modelled with a non-monotonic effective basal friction law that has static, intermediate (velocity decreasing) and dynamic (velocity increasing) regimes. Both experiments and time-dependent numerical simulations show that steadily travelling retrogressive waves rapidly form in this system and a travelling wave ansatz is therefore used to derive a one-dimensional depth-averaged exact solution. The speed of the wave is determined by a critical point in the ordinary differential equation for the thickness. The critical point lies in the intermediate frictional regime, at the point where the friction exactly balances the downslope component of gravity. The retrogressive wave is therefore a sensitive test of the functional form of the friction law in this regime, where steady uniform flows are unstable and so cannot be used to determine the friction law directly. Upper and lower bounds for the existence of retrogressive waves in terms of the initial layer depth and the slope inclination are found and shown to be in good agreement with the experimentally determined phase diagram. For the friction law proposed by Edwards et al. (J. Fluid. Mech., vol. 823, 2017, pp. 278–315, J. Fluid. Mech., 2019, (submitted)) the magnitude of the wave speed is slightly under-predicted, but, for a given initial layer thickness, the exact solution accurately predicts an increase in the wave speed with higher inclinations. The model also captures the finite wave speed at the onset of retrogressive failure observed in experiments.

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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

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Type Description Title
VIDEO
Movie

Russell et al. supplementary movie 1
Full chute view of a planar retrogressive failure (approximately uniform in the cross-slope direction) of a thick layer in a small-scale laboratory experiment (similar to the flow shown in figure 1). Initally static glass beads in a layer of thickness $h_0 \approx h_{stop}(23^{\circ}) \approx $ 1.65 mm inclined at an angle $\zeta=24.0^{\circ}$ are perturbed causing a front to develop, which propagates upslope with constant speed $|u_w|$ progressively eroding the static layer. The transparent glass beads (125-160 $\mu$m diameter, appearing white) are seeded with red tracer particles (300-400 $\mu$m diameter) to aid visualisation.

 Video (34.1 MB)
34.1 MB
VIDEO
Movies

Russell et al. supplementary movie 2
Close up view of a planar retrogressive failure (approximately uniform in the cross-slope direction) of a thick layer in a small-scale laboratory experiment (the same flow as shown in figure 1). Initally static glass beads in a layer of thickness $h_0 \approx h_{stop} (23 ^{\circ}) \approx $ 1.65 mm inclined at an angle $ \zeta=24.0^{\circ}$ are perturbed causing a front to develop, which propagates upslope with constant speed $|u_w |$ progressively eroding the static layer. The transparent glass beads (125-160 $\mu$m diameter, appearing white) are seeded with red tracer particles (300-400 $\mu$m diameter) to aid visualisation.

 Video (1.6 MB)
1.6 MB
VIDEO
Movies

Russell et al. supplementary movie 3
A planar retrogressive failure on the chute shown in figure 2. Initially static glass beads in a layer of thickness $h_0 \approx h_{stop} (25 ^{\circ}) \approx $ 0.63 mm inclined at an angle $ \zeta=26.5^{\circ}$ are perturbed causing a front to develop, which propagates upslope with constant speed $|u_w |$ progressively eroding the static layer.

 Video (4.1 MB)
4.1 MB
VIDEO
Movies

Russell et al. supplementary movie 4
Overhead video showing the planar retrogressive failure front in figure 3. It separates the initially stationary layer of 125-160 $\mu$m glass beads of thickness $h_0 \approx h_{stop} (24 ^{\circ}) \approx $ 0.95 mm inclined at an angle $ \zeta=26.5^{\circ}$ from the flowing material below. The retrogressive front propagates upslope with speed $|u_w |$=11.7 cm s$^{-1}$, and continues to do so until it reaches the top of the static layer.

 Video (2.8 MB)
2.8 MB
VIDEO
Movies

Russell et al. supplementary movie 5
The retrogressive wave thickness $h(x,t)$ and downslope velocity $u(x,z,t)$ for a slope inclined at $ \zeta=27.5^{\circ}$ corresponding the simulations in figure 7. The initial stationary layer is of thickness $h(x,0)$=1.0 mm. The filled region shows the thickness and the contour scale within it denotes the velocity, which is reconstructed from the depth-averaged downslope velocity $\bar{u}(x,t)$ assuming an exponential profile (4.1) with $\lambda$=2.45. There is no inflow at $x$=0 and there is free outflow at the downstream boundary. The wave erodes the static surface particles, which are shown with light blue markers and they travel downslope on the surface of the steady uniform flow. The material properties are given in table 1.

 Video (4.5 MB)
4.5 MB
VIDEO
Movies

Russell et al. supplementary movie 6
The retrogressive wave thickness $h(x,t)$ for a slope inclined at $ \zeta=27.5^{\circ}$ for the simulations in figure 7. The initial stationary layer is of thickness $h(x,0)$=1.0 mm. The dots represent a uniform random distribution of particles which are tracked during the flow using the exponential velocity profile (4.1) with $\lambda$=2.45. There is no inflow at $x$=0 and there is free outflow at the downstream boundary. The material properties are given in table 1.

 Video (7.4 MB)
7.4 MB

Retrogressive failure of a static granular layer on an inclined plane

  • A. S. Russell (a1), C. G. Johnson (a1), A. N. Edwards (a1), S. Viroulet (a2), F. M. Rocha (a1) and J. M. N. T. Gray (a1)...

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