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Frictional hysteresis and particle deposition in granular free-surface flows

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

Abstract

Shallow granular avalanches on slopes close to repose exhibit hysteretic behaviour. For instance, when a steady-uniform granular flow is brought to rest it leaves a deposit of thickness $h_{stop}(\unicode[STIX]{x1D701})$ on a rough slope inclined at an angle $\unicode[STIX]{x1D701}$ to the horizontal. However, this layer will not spontaneously start to flow again until it is inclined to a higher angle $\unicode[STIX]{x1D701}_{start}$ , or the thickness is increased to $h_{start}(\unicode[STIX]{x1D701})>h_{stop}(\unicode[STIX]{x1D701})$ . This simple phenomenology leads to a rich variety of flows with co-existing regions of solid-like and fluid-like granular behaviour that evolve in space and time. In particular, frictional hysteresis is directly responsible for the spontaneous formation of self-channelized flows with static levees, retrogressive failures as well as erosion–deposition waves that travel through the material. This paper is motivated by the experimental observation that a travelling-wave develops, when a steady uniform flow of carborundum particles on a bed of larger glass beads, runs out to leave a deposit that is approximately equal to $h_{stop}$ . Numerical simulations using the friction law originally proposed by Edwards et al. (J. Fluid Mech., vol. 823, 2017, pp. 278–315) and modified here, demonstrate that there are in fact two travelling waves. One that marks the trailing edge of the steady-uniform flow and another that rapidly deposits the particles, directly connecting the point of minimum dynamic friction (at thickness $h_{\ast }$ ) with the deposited layer. The first wave moves slightly faster than the second wave, and so there is a slowly expanding region between them in which the flow thins and the particles slow down. An exact inviscid solution for the second travelling wave is derived and it is shown that for a steady-uniform flow of thickness $h_{\ast }$ it produces a deposit close to $h_{stop}$ for all inclination angles. Numerical simulations show that the two-wave structure deposits layers that are approximately equal to $h_{stop}$ for all initial thicknesses. This insensitivity to the initial conditions implies that $h_{stop}$ is a universal quantity, at least for carborundum particles on a bed of larger glass beads. Numerical simulations are therefore able to capture the complete experimental staircase procedure, which is commonly used to determine the $h_{stop}$ and $h_{start}$ curves by progressively increasing the inclination of the chute. In general, however, the deposit thickness may depend on the depth of the flowing layer that generated it, so the most robust way to determine $h_{stop}$ is to measure the deposit thickness from a flow that was moving at the minimum steady-uniform velocity. Finally, some of the pathologies in earlier non-monotonic friction laws are discussed and it is explicitly shown that with these models either steadily travelling deposition waves do not form or they do not leave the correct deposit depth $h_{stop}$ .

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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: andrew.edwards@manchester.ac.uk

References

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JFM classification

Type Description Title
VIDEO
Movie

Edwards Supplementary Movie 1
Movie corresponding to figure 5. The flow thickness $h$ and downslope velocity $u(x,z,t)$ for a slope inclined at $\zeta=36.3^{\circ}$ and with an initially uniform static layer of initial thickness $h(x,0) = 2.4$ mm, which is 0.5 mm greater than $h_{start}(36.3^{\circ}) $(solid green line). 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 (4.03). There is no further inflow at $x = 0$, but there is free outflow at the downstream boundary. A travelling wave at the rear of the steady uniform flow region passes through the material at a constant wavespeed that is greater than the surface velocity. This is followed by a slightly slower constant speed deposition wave that connects the static deposit and the transition thickness $h_*$ (solid orange line). These waves catch up with surface particles, which are shown with light blue circular markers, and they are deposited on the surface of the final deposit layer that has a thickness of approximately $h_{stop}$ (solid red line). The material properties are given in table 1.

 Video (4.3 MB)
4.3 MB
VIDEO
Movie

Edwards Supplementary Movie 2
Movie corresponding to figure 7(a). Numerical simulation showing the evolving flow thickness (solid black line) starting from $h(x,0) = h_*(\zeta)$ (dashed black line) at a slope angle of $\zeta=31.9^{\circ}$. The initial layer depths are equal to the thickness of the friction law transition (solid orange line), which is less than $h_{start}(\zeta)$ (solid green line). An initial momentum $m(x,0) = \bar{u}_*h_*$ is therefore imposed in order to start the flow, where $\bar{u}_*=\bar{u}_{\infty}$ is given by the steady uniform flow velocity relation (4.04). As the deposition wave travels through the system the flow eventually leaves a deposit that is close to $h_{stop}(\zeta)$ (solid red line). The material properties are given in table 1 and the slope angle-dependent properties in table 2.

 Video (704 KB)
704 KB
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Edwards Supplementary Movie 3
Movie corresponding to figure 7(b). Numerical simulation showing the evolving flow thickness (solid black line) starting from $h(x,0) = h_*(\zeta)$ (dashed black line) at a slope angle of $\zeta=34.1^{\circ}$. The initial layer depths are equal to the thickness of the friction law transition (solid orange line), which is less than $h_{start}(\zeta)$ (solid green line). An initial momentum $m(x,0) = \bar{u}_*h_*$ is therefore imposed in order to start the flow, where $\bar{u}_*=\bar{u}_{\infty}$ is given by the steady uniform flow velocity relation (4.04). As the deposition wave travels through the system the flow eventually leaves a deposit that is close to $h_{stop}(\zeta)$ (solid red line). The material properties are given in table 1 and the slope angle-dependent properties in table 2.

 Video (1.5 MB)
1.5 MB
VIDEO
Movie

Edwards Supplementary Movie 4
Movie corresponding to figure 7(c). Numerical simulation showing the evolving flow thickness (solid black line) starting from $h(x,0) = h_*(\zeta)$ (dashed black line) at a slope angle of $\zeta=36.3^{\circ}$. The initial layer depths are equal to the thickness of the friction law transition (solid orange line), which is less than $h_{start}(\zeta)$ (solid green line). An initial momentum $m(x,0) = \bar{u}_*h_*$ is therefore imposed in order to start the flow, where $\bar{u}_*=\bar{u}_{\infty}$ is given by the steady uniform flow velocity relation (4.04). As the deposition wave travels through the system the flow eventually leaves a deposit that is close to $h_{stop}(\zeta)$ (solid red line). The material properties are given in table 1 and the slope angle-dependent properties in table 2.

 Video (2.0 MB)
2.0 MB
VIDEO
Movie

Edwards Supplementary Movie 5
Movie corresponding to figure 7(d). Numerical simulation showing the evolving flow thickness (solid black line) starting from $h(x,0) = h_*(\zeta)$ (dashed black line) at a slope angle of $\zeta=38.5^{\circ}$. The initial layer depths are equal to the thickness of the friction law transition (solid orange line), which is less than $h_{start}(\zeta)$ (solid green line). An initial momentum $m(x,0) = \bar{u}_*h_*$ is therefore imposed in order to start the flow, where $\bar{u}_*=\bar{u}_{\infty}$ is given by the steady uniform flow velocity relation (4.04). As the deposition wave travels through the system the flow eventually leaves a deposit that is close to $h_{stop}(\zeta)$ (solid red line). The material properties are given in table 1 and the slope angle-dependent properties in table 2.

 Video (2.1 MB)
2.1 MB
VIDEO
Movie

Edwards Supplementary Movie 6
Movie corresponding to figure 7(e). Numerical simulation showing the evolving flow thickness (solid black line) starting from $h(x,0) = h_*(\zeta)$ (dashed black line) at a slope angle of $\zeta=40.7^{\circ}$. The initial layer depths are equal to the thickness of the friction law transition (solid orange line), which is less than $h_{start}(\zeta)$ (solid green line). An initial momentum $m(x,0) = \bar{u}_*h_*$ is therefore imposed in order to start the flow, where $\bar{u}_*=\bar{u}_{\infty}$ is given by the steady uniform flow velocity relation (4.04). As the deposition wave travels through the system the flow eventually leaves a deposit that is close to $h_{stop}(\zeta)$ (solid red line). The material properties are given in table 1 and the slope angle-dependent properties in table 2.

 Video (2.3 MB)
2.3 MB
VIDEO
Movie

Edwards Supplementary Movie 7
Movie corresponding to figure 8(a). Numerical simulation showing the evolving flow thickness (solid black line) starting from $h(x,0) = h_{start}(\zeta) + 0.1$ mm (dashed black line) at a slope angle of $\zeta=33.0^{\circ}$. The initially static layer has a thickness $h > h_{start}(\zeta)$ (solid green line) and so it gains momentum. As the deposition wave travels through the system the flow eventually leaves a deposit that is close to $h_{stop}(\zeta)$ (solid red line). The friction law has a transition thickness $h_*(\zeta)$ (solid orange line). The material properties are given in table 1 and the slope angle-dependent properties in table 2.

 Video (2.6 MB)
2.6 MB
VIDEO
Movie

Edwards Supplementary Movie 8
Movie corresponding to figure 8(b). Numerical simulation showing the evolving flow thickness (solid black line) starting from $h(x,0) = h_{start}(\zeta) + 0.1$ mm (dashed black line) at a slope angle of $\zeta=35.2^{\circ}$. The initially static layer has a thickness $h > h_{start}(\zeta)$ (solid green line) and so it gains momentum. As the deposition wave travels through the system the flow eventually leaves a deposit that is close to $h_{stop}(\zeta)$ (solid red line). The friction law has a transition thickness $h_*(\zeta)$ (solid orange line). The material properties are given in table 1 and the slope angle-dependent properties in table 2.

 Video (1.7 MB)
1.7 MB
VIDEO
Movie

Edwards Supplementary Movie 9
Movie corresponding to figure 8(c). Numerical simulation showing the evolving flow thickness (solid black line) starting from $h(x,0) = h_{start}(\zeta) + 0.1$ mm (dashed black line) at a slope angle of $\zeta=37.4^{\circ}$. The initially static layer has a thickness $h > h_{start}(\zeta)$ (solid green line) and so it gains momentum. As the deposition wave travels through the system the flow eventually leaves a deposit that is close to $h_{stop}(\zeta)$ (solid red line). The friction law has a transition thickness $h_*(\zeta)$ (solid orange line). The material properties are given in table 1 and the slope angle-dependent properties in table 2.

 Video (2.3 MB)
2.3 MB
VIDEO
Movie

Edwards Supplementary Movie 10
Movie corresponding to figure 8(d). Numerical simulation showing the evolving flow thickness (solid black line) starting from $h(x,0) = h_{start}(\zeta) + 0.1$ mm (dashed black line) at a slope angle of $\zeta=39.6^{\circ}$. The initially static layer has a thickness $h > h_{start}(\zeta)$ (solid green line) and so it gains momentum. As the deposition wave travels through the system the flow eventually leaves a deposit that is close to $h_{stop}(\zeta)$ (solid red line). The friction law has a transition thickness $h_*(\zeta)$ (solid orange line). The material properties are given in table 1 and the slope angle-dependent properties in table 2.

 Video (2.9 MB)
2.9 MB
VIDEO
Movie

Edwards Supplementary Movie 11
Movie corresponding to figure 8(e). Numerical simulation showing the evolving flow thickness (solid black line) starting from $h(x,0) = h_{start}(\zeta) + 0.1$ mm (dashed black line) at a slope angle of $\zeta=41.8^{\circ}$. The initially static layer has a thickness $h > h_{start}(\zeta)$ (solid green line) and so it gains momentum. As the deposition wave travels through the system the flow eventually leaves a deposit that is close to $h_{stop}(\zeta)$ (solid red line). The friction law has a transition thickness $h_*(\zeta)$ (solid orange line). The material properties are given in table 1 and the slope angle-dependent properties in table 2.

 Video (2.8 MB)
2.8 MB
VIDEO
Movie

Edwards Supplementary Movie 12
Movie corresponding to figure 15(a). Flow thickness $h$ (solid black line) for a numerical simulation using the material properties for glass beads at a slope angle of $\zeta=23.0^{\circ}$. The initial conditions (dashed black line) are a stationary layer of thickness $h(x,0) = h_{start}(\zeta) + 0.1$ mm. The initial layer is of thickness $h > h_{start}(\zeta)$ (solid green line) should gain momentum before leaving a deposit of thickness $h_{stop}(\zeta)$ (solid red line). The material properties are given in table 3 and the slope angle-dependent properties in table 4.

 Video (3.8 MB)
3.8 MB
VIDEO
Movie

Edwards Supplementary Movie 13
Movie corresponding to figure 15(b). Flow thickness $h$ (solid black line) for a numerical simulation with the material properties for glass beads with $\kappa=1$ at a slope angle of $\zeta=23.0^{\circ}$. The initial conditions (dashed black line) are a stationary layer of thickness $h(x,0) = h_{start}(\zeta) + 0.1$ mm. The initial layer is of thickness $h > h_{start}(\zeta)$ (solid green line) should gain momentum before leaving a deposit of thickness $h_{stop}(\zeta)$ (solid red line). The material properties are given in table 3 and the slope angle-dependent properties in table 4.

 Video (2.2 MB)
2.2 MB
VIDEO
Movie

Edwards Supplementary Movie 14
Movie corresponding to figure 15(c). Flow thickness $h$ (solid black line) for a numerical simulation using the material properties for sand at a slope angle of $\zeta=35.0^{\circ}$. The initial conditions (dashed black line) are a stationary layer of thickness $h(x,0) = h_{start}(\zeta) + 0.1$ mm. The initial layer is of thickness $h > h_{start}(\zeta)$ (solid green line) should gain momentum before leaving a deposit of thickness $h_{stop}(\zeta)$ (solid red line). The material properties are given in table 3 and the slope angle-dependent properties in table 4.

 Video (2.5 MB)
2.5 MB
VIDEO
Movie

Edwards Supplementary Movie 15
Movie corresponding to figure 15(d). Flow thickness $h$ (solid black line) for a numerical simulation using the material properties for carborundum with $\beta_* = \beta - \Gamma$ at a slope angle of $\zeta=35.2^{\circ}$. The initial conditions (dashed black line) are a stationary layer of thickness $h(x,0) = h_{start}(\zeta) + 0.1$ mm. The initial layer is of thickness $h > h_{start}(\zeta)$ (solid green line) should gain momentum before leaving a deposit of thickness $h_{stop}(\zeta)$ (solid red line). The material properties are given in table 1 and the slope angle-dependent properties in table 2.

 Video (2.2 MB)
2.2 MB
VIDEO
Movie

Edwards Supplementary Movie 16
Movie corresponding to figure 15(e). Flow thickness $h$ (solid black line) for a numerical simulation using the material properties for carborundum with $\beta_*(\zeta) = 2.05 > \beta$, i.e. Edwards \textit{et al.}'s (2017) friction law that has a transition thickness $h_*(\zeta) = (h_{stop} + h_{start})/2$ (solid orange line), at a slope angle of $\zeta=33.0^{\circ}$. The initial conditions (dashed black line) are a stationary layer of thickness $h(x,0) = h_{start}(\zeta) + 0.1$ mm. The initial layer is of thickness $h > h_{start}(\zeta)$ (solid green line) should gain momentum before leaving a deposit of thickness $h_{stop}(\zeta)$ (solid red line). The material properties are given in table 1 and the slope angle-dependent properties in table 2.

 Video (931 KB)
931 KB
VIDEO
Movie

Edwards Supplementary Movie 17
Movie corresponding to figure 17(a). Flow thickness $h$ (solid black line) for a numerical simulation using the material properties for glass beads at a slope angle of $\zeta=23.0^{\circ}$. The initial conditions (dashed black line) are a stationary layer of thickness $h(x,0) = h_{start}(\zeta) - 0.1$ mm. The initial layer is of thickness $h < h_{start}(\zeta)$ (solid green line) and should remain static. The solid red line denotes $h_{stop}(\zeta)$. The material properties are given in table 3 and the slope angle-dependent properties in table 4.

 Video (3.6 MB)
3.6 MB
VIDEO
Movie

Edwards Supplementary Movie 18
Movie corresponding to figure 17(b). Flow thickness $h$ (solid black line) for a numerical simulation with the material properties for glass beads with $\kappa=1$ at a slope angle of $\zeta=23.0^{\circ}$. The initial conditions (dashed black line) are a stationary layer of thickness $h(x,0) = h_{start}(\zeta) - 0.1$ mm. The initial layer is of thickness $h < h_{start}(\zeta)$ (solid green line) and should remain static. The solid red line denotes $h_{stop}(\zeta)$. The material properties are given in table 3 and the slope angle-dependent properties in table 4.

 Video (908 KB)
908 KB
VIDEO
Movie

Edwards Supplementary Movie 19
Movie corresponding to figure 17(c). Flow thickness $h$ (solid black line) for a numerical simulation using the material properties for sand at a slope angle of $\zeta=35.0^{\circ}$. The initial conditions (dashed black line) are a stationary layer of thickness $h(x,0) = h_{start}(\zeta) - 0.1$ mm. The initial layer is of thickness $h < h_{start}(\zeta)$ (solid green line) and should remain static. The solid red line denotes $h_{stop}(\zeta)$. The material properties are given in table 3 and the slope angle-dependent properties in table 4.

 Video (2.3 MB)
2.3 MB
VIDEO
Movie

Edwards Supplementary Movie 20
Movie corresponding to figure 17(d). Flow thickness $h$ (solid black line) for a numerical simulation using the material properties for carborundum with $\beta_* = \beta - \Gamma$ at a slope angle of $\zeta=35.2^{\circ}$. The initial conditions (dashed black line) are a stationary layer of thickness $h(x,0) = h_{start}(\zeta) - 0.1$ mm. The initial layer is of thickness $h < h_{start}(\zeta)$ (solid green line) and should remain static. The solid red line denotes $h_{stop}(\zeta)$. The material properties are given in table 1 and the slope angle-dependent properties in table 2.

 Video (1.1 MB)
1.1 MB
VIDEO
Movie

Edwards Supplementary Movie 21
Movie corresponding to figure 17(e). Flow thickness $h$ (solid black line) for a numerical simulation using the material properties for carborundum with $\beta_*(\zeta) = 2.05 > \beta$, i.e. Edwards \textit{et al.}'s (2017) friction law that has a transition thickness $h_*(\zeta) = (h_{stop} + h_{start})/2$ (solid orange line), at a slope angle of $\zeta=33.0^{\circ}$. The initial conditions (dashed black line) are a stationary layer of thickness $h(x,0) = h_{start}(\zeta) - 0.1$ mm. The initial layer is of thickness $h < h_{start}(\zeta)$ (solid green line) and should remain static. The solid red line denotes $h_{stop}(\zeta)$. The material properties are given in table 1 and the slope angle-dependent properties in table 2.

 Video (608 KB)
608 KB

Frictional hysteresis and particle deposition in granular free-surface flows

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

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