Journal
JOURNAL OF FLUID MECHANICS
Volume 869, Issue -, Pages 313-340Publisher
CAMBRIDGE UNIV PRESS
DOI: 10.1017/jfm.2019.215
Keywords
geophysical and geological flows; granular media; wave/free-surface flows
Categories
Funding
- NERC [NE/E003206/1, NE/K003011/1]
- EPSRC [EP/I019189/1, EP/K00428X/1, EP/M022447/1]
- Royal Society Wolfson Research Merit Award [WM150058]
- CNPq, Conselho Nacional de Desenvolvimento Cientifico e Tecnologico - Brazil
- EPSRC [EP/M022447/1] Funding Source: UKRI
- NERC [NE/K003011/1] Funding Source: UKRI
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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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