Interfacial Instability during Granular Erosion

The complex interplay between the topography and the erosion and deposition phenomena is a key feature to model granular flows such as landslides. Here, we investigated the instability that develops during the erosion of a wet granular pile by a dry dense granular flow. The morphology and the propagation of the generated steps are analyzed in relation to the specific erosion mechanism. The selected flowing angle of the confined flow on a dry heap appears to play an important role both in the final state of the experiment, and for the shape of the structures. We show that the development of the instability is governed by the inertia of the flow through the Froude number. We model this instability and predict growth rates that are in agreement with the experiment results.

Figure 1

(a) Successive lateral views of a typical experiment. The shallow flowing granular layer appears brighter above the wet heap. The first frame is the initial state, and the others are snapshots taken during the erosion. The black arrow follows one step. The flow rate is $3.3\mathrm{g}/\mathrm{s}$ and the initial inclination 35.8°. The width of one picture corresponds to 27 cm. (b) Successive profiles of the interface, from dark blue to light red, every 15 min in the 6 mm channel, showing the steady propagation of steps over an initial slope angle of 39.4°.

Figure 2

(a) Superimposition of steps from a given experiment (black and blue dashed lines, solid green), and of a bigger step (dotted black), built from a higher initial slope angle: 43.7°. The shifted red lines are a fitting curve composed of a straight line of slope 31° followed by a parabolic curve corresponding to a free fall. (b) Angle of the heap at the end of the experiment ${\theta }_f$, versus the steady angle of flow of the same granular material down a dry heap ${\theta }_{\mathrm{eq}}$. Different flow rates are used to modify ${\theta }_{\mathrm{eq}}$, and the cohesive heap is made of glass beads.

Figure 3

(a) Evolution of the interface with the initial Fr. The symbols depict the evolution of the interface. The borders of the gray strip are 0.7 and 0.8. (b) (${\theta }_i$-flow rate) phase diagram in the 12 mm-wide channel. The solid line is the heap flow angle ${\theta }_{\mathrm{eq}}$, under which erosion is not possible. The dashed line is the curve $\mathrm{Fr}=0.87$. (c) Dimensionless growth rates of the most unstable mode as a function of the Froude number in experiments for glass beads in the 6 mm channels (filled circle) and 12 mm channel (open circle) and steel beads in 12 mm channel (diamond). The curves correspond to a prediction of the theoretical growth rate [Eq. (2)] for the most unstable mode $\tilde{k}=\infty$ (dashed line) and for a wave number in agreement with the experiment $h_{0}k=0.1$ (solid line) at an inclination of ${\theta }_i=30°$. Inset: Direct comparison of growth rates using the measured wave numbers $\tilde{k}$ as a parameter.