Erosion and deposition processes in surface granular flows

We report on experiments aiming at characterizing erosion and deposition processes on a tilted granular bed. We investigate the existence of the neutral angle, that is, the critical angle at which erosion exactly balances accretion after the passage of a granular avalanche of a finite mass. Experiments show in particular that the neutral angle depends on both avalanche mass and shape but is rather insensitive to the bed length. This result strongly suggests that the effective friction between the static and mobile granular phases cannot be taken as an intrinsic property that is only material dependent but should be considered a flow-dependent property. Interestingly, for a given avalanche mass, the net erosion rate increases linearly with the angular deviation from the neutral angle. We also compare our data with the predictions of the erosion-deposition model introduced by Bouchaud, Cates, Ravi Prakash, and Edwards (BCRE) [J. Phys. I 4, 1283 (1994)]. We show that the predictions drawn from the modified version of the BCRE model proposed by Boutreux and de Gennes, in which the local erosion rate between the static and mobile phases is independent of the flow thickness, are in remarkable agreement with the experimental results.

Figure 1

Sketch representing a granular pile topped by a flowing layer: $h(x,t)$ and $R(x,t)$ are the height of the static layer and the thickness of the mobile layer, respectively, at position $x$ and time $t$. The local angle $\theta$ is defined as the local slope of the static bed: $tan\theta =\partial h/\partial x$.

Figure 2

Schematic view of the experimental setup: (a) initial configuration before the release of the particles of the reservoir and (b) final configuration after the release. $L_b$ is the length of the static granular bed and $R_0$ and ${\mathrm{\Delta }}_0$ are respectively the height and the width of the reservoir, which can be varied.

Figure 3

Erosion efficiency $E=(M_f-M_0)/M_0$ as a function of the bed inclination $\mathrm{\Theta }$: (a) $M_0=110$ g and $R_0=L_0=205$ mm; (b) $M_0=55$, 110, 220, and 300 g and $R_0/{\mathrm{\Delta }}_0=1$; and (c) $M_0=110$ g and $R_0/{\mathrm{\Delta }}_0=0.57$, 1, 2.14, and 3.28. All above experiments have been conducted using the long granular bed: $L_b=1700$ mm.

Figure 4

Dependence of the neutral angle on the released mass $M_0\left(\sim R_0{\mathrm{\Delta }}_0\right)$ and the aspect ratio $\left(R_0/{\mathrm{\Delta }}_0\right)$ for the long bed configuration $(L_b=1700$ mm): (a) ${\mathrm{\Theta }}_n$ versus $R_0{\mathrm{\Delta }}_0$ for a given aspect ratio, (b) ${\mathrm{\Theta }}_n$ versus the aspect ratio for a given mass $M_0$, and (c) ${\mathrm{\Theta }}_n$ versus $R_0$ [inset: $1/(tan{\mathrm{\Theta }}_n-tan{\theta }_d)$ versus $R_0^2]$. The experimental data are fitted using Eq. (7).

Figure 5

Dependence of the erosion rate $e=dE/d\mathrm{\Theta }$ on the initial mass $M_0\left(\sim R_0{\mathrm{\Delta }}_0\right)$ and the aspect ratio $\left(R_0/{\mathrm{\Delta }}_0\right)$ for the long bed configuration $(L_b=1700$ mm): (a) $e$ versus $R_0{\mathrm{\Delta }}_0$ for a given aspect ratio and (b) $e$ versus the aspect ratio for a given mass $M_0$. In (a), the data are fitted using Eq. (8).

Figure 6

(a) Erosion efficiency $E$ versus $\mathrm{\Theta }$ obtained with the short and long bed lengths. (b) Dependence of the erosion rate $e=dE/d\mathrm{\Theta }$ on the bed length.

Figure 7

Mass percentage of particles from the reservoir that exit the chute versus the inclination angle (a) for a fixed aspect ratio $R_0/{\mathrm{\Delta }}_0=1$ and different initial masses $M_0$ (long chute, $L_b=1700$ mm) and (b) for a fixed initial mass $M_0=110$ g and different aspect ratios $R_0/{\mathrm{\Delta }}_0$ (long chute, $L_b=1700$ mm). (c) Average percentage versus $M_0$ for short and long chutes.

Figure 8

Avalanche profiles extracted from video images $(M_0=110$ g and $\theta ={30}^{\circ })$: successive profiles taken every 0.4 s. The solid line corresponds to the initial profile at the moment when the mass is released from the reservoir. After a short transition, the forefront reaches a constant speed.

Figure 9

Avalanche speed versus the initial height $R_0$ with released mass of unit aspect ratio. Bed length $L_b=1700$ mm.

Figure 10

Variation of $M_f/M_0$ as a function $\mathrm{\Delta }\theta$ derived from the BCRE and BDG models in the regime of large Péclet number [see Eqs. (A6) and (A9)]. Model parameters: $R_0={\mathrm{\Delta }}_0=10d,V/\gamma =2.5d,V_{\text{up}}/V=4$, and $L_b=100d$. The parameters have been chosen such that the slopes at the origin of both curves coincide.