Dynamics of dry granular avalanches

A fine analysis of the statistics of dry granular avalanches in a rotating drum setup reveals that, beyond the fluctuations, the angle at which an avalanche ends is correlated experimentally to the angle at which the avalanche starts. This correlation resulting from inertia defines an intermediate “neutral” angle that characterizes the corresponding granular pile. In addition, an intensive study of the dynamics of the avalanche shows that the time duration of the avalanche is correlated to its amplitude, being smaller for higher amplitude. The time relaxation of the pile slope during any avalanche, governed by the deviation of the starting angle from the neutral angle, follows a master curve. A simple model recovers most of the results and contributes to a better understanding of the physics of the avalanche flow.

Figure 1

Time evolution of the pile slope $\theta$ for glass beads of diameter $d=1\mathrm{mm}$ in a drum of diameter $2R=50\mathrm{cm}$ and width $b=5\mathrm{cm}$ rotating at the angular velocity $\Omega \simeq 0.018°∕\mathrm{s}$.

Figure 2

(Color online) Histograms of ${\theta }_m$ and ${\theta }_r$, and of $\Delta \theta$ in inset, for 340 avalanches of glass beads $(d=1\mathrm{mm})$ in a rotating drum ($2R=50\mathrm{cm}$, $b=5\mathrm{cm}$). (—) Best Gaussian fits.

Figure 3

Stopping angle ${\theta }_r$ as a function of the starting angle ${\theta }_m$ for each of the 340 avalanches of Fig. 2. (◇) Experimental results and (---) best linear fit of slope $-0.22$ and intercept $\overline{{\theta }_n}\simeq 23.7°$ with the bisector ${\theta }_r={\theta }_m$.

Figure 4

(Color online) Experimental time evolution of $\theta$ for two avalanches (●, ○) of different amplitudes $\Delta \theta$, together with the corresponding Gaussian fits (—): (●) ${\theta }_m=25.64°$, ${\theta }_r=23.25°$, and $\tau =0.64\mathrm{s}$ $({\theta }_n=23.68°)$; (○) ${\theta }_m=24.71°$, ${\theta }_r=23.59°$, and $\tau =0.83\mathrm{s}$ $({\theta }_n=23.79°)$. The dashed line corresponds to the neutral angle ${\theta }_n\simeq 23.74°$. Inset: same data in the normalized plot [$\tilde{\theta }=(\theta -{\theta }_n)∕({\theta }_m-{\theta }_n)$, $\tilde{t}=t∕\tau$].

Figure 5

(Color online) Normalized time evolution of the reduced pile angle for the 340 avalanches: Experimental average time evolution (—) and standard deviation (⋯). Inset: Same data (—) with the “free fall” parabolic approximation (— —), the cosine “collisionless” approximation (–⋅–) and the adjusted solution from complete model equation (4) (---).

Figure 6

(Color online) Avalanche time duration $\tau$ as a function of $g\Delta \theta ∕R$ for 1500 avalanches. Each data point corresponds to one avalanche and the different colors correspond to different drum radius $R$ from $4\text{to}25\mathrm{cm}$ as indicated in the gray range. Inset: Same data as a function of ${(R∕g\Delta \theta )}^{1∕2}$. (—): Best fit by equation $\tau =0.83{(R∕g\Delta \theta )}^{1∕2}$.