Transverse instability of avalanches in granular flows down an incline

Avalanche experiments on an erodible substrate are analyzed using the “partial fluidization” model of dense granular flows. The model identifies a family of propagating solitonlike avalanches with shape and velocity controlled by the inclination angle and the depth of the substrate. At high inclination angles, the solitons display a transverse instability, followed by coarsening and fingering similar to recent experimental observation. A primary cause for the transverse instability is directly related to the dependence of the soliton velocity on the granular mass trapped in the avalanche.

Figure 1

(Color online) $h$ (a) and $A$ (b) for various values of $m$ and $\alpha$. The solid line is for $m=147.7$, $V=2.72$; the dashed line is for $m=211$, $V=3.12$, for $\delta =1$, $\alpha =0.08$, $\beta =2$; the dotted-dashed line is for $\alpha =0.025$, $\delta =1.15$, $m=62$, $V=0.86$. Inset to (a): Representative height profiles for avalanches in $300\mu \mathrm{m}$ sand, $\overline{\phi }=32.3°$ (solid line), and $500\mu \mathrm{m}$ glass beads, $\overline{\phi }=22°$ (dashed line), [4]. Inset to (b): $V$ vs $m$ (solid line), diamonds depict data for sand avalanches, $\overline{\phi }=32.3°$.

Figure 2

(Color online) $\lambda \left(q\right)$ vs $q$ for $\delta =1.15$, $\alpha =0.08$, and $m=102$, $q$ is scaled by $q^{*}$, and the growth rate by ${\lambda }^{*}=\lambda \left(q^{*}\right)$. Solid line: $\lambda \left(q\right)$ obtained by numerical stability analysis of one-dimensional solution Eq. (6). The dashed line is the solution of Eq. (13). Symbols depict experimental data for sand avalanches. Inset: optimal wave number of $q^{*}$ vs $\alpha$ for $\delta =1.15$.

Figure 3

(a) Typical fingering patterns for underwater avalanches; [4] (b) and (c) Gray-coded images of $h(x,y)$ (white corresponds to larger $h$), with (b) $t=300$ and (c) $t=500\text{units}$ of time. Domain size is $600\text{units}$ in $x$ and $450\text{units}$ in the $y$ direction, the only part of domain in the $x$ direction is shown. Parameters: $\delta =1.16$, $\alpha =0.14$, $\beta =2$, and initial height $h_0=2.285$.