On the front shape of an inertial granular flow down a rough incline

Granular material flowing on complex topographies are ubiquitous in industrial and geophysical situations. In this paper, we study the small-scale experiment of a granular layer flowing on a rough incline. The shape of the granular front is solved analytically by using depth-averaged mass and momentum equations with a fractional expression for the frictional rheology μ(I), which is a generalization of Gray and Ancey ["Segregation, recirculation and deposition of coarse particles near two-dimensional avalanche fronts," J. Fluid Mech. 629, 387 (2009)]. Unlike previous studies where a "plug flow dynamics" is assumed, a free shape factor α describing the vertical velocity profile is taken into account. The effect of inertia and shear rate on the front profile is evidenced through the introduction of the Froude number and the shape factor α. The analytical predictions are compared to experimental results published by Pouliquen ["On the shape of granular fronts down rough inclined planes," Phys. Fluids 11, 1956 (1999)] and with our new experimental data obtained at higher Froude numbers. A good agreement between theory and experiments is found for α = 5/4, corresponding to a Bagnold-like velocity profile. However, we observe a systematic deviation near the head of the front where the height vanishes: the theory predicts a continuous precursor layer, while a grain-free region is observed experimentally. This suggests that the vertical velocity profile is not uniform inside the front, but the shape factor α tends to 1 near the head of the front. This raises questions about the vertical velocity profile in granular flows and about the expression of the rheological function μ(I) and its calibration from experimental data.