Unified flow rule of undeveloped and fully-developed dense granular flows down rough inclines

We report on chute measurements of the free-surface velocity $v$ in dense flows of spheres and diverse sands and spheres-sand mixtures down rough inclines. These and previous measurements are inconsistent with standard flow rules, in which the Froude number $v/\sqrt{gh}$ scales linearly with $h/h_s$ or $(\tan\theta/\mu_r)^2h/h_s$, where $\mu_r$ is the dynamic friction coefficient, $h$ the flow thickness, and $h_s(\theta)$ its smallest value that permits a steady, uniform dense flow state at a given inclination angle $\theta$. This is because the characteristic length $L$ a flow needs to fully develop can exceed the chute or travel length $l$ and because neither rule is universal for fully-developed flows across granular materials. We use a dimensional analysis motivated by a recent unification of sediment transport to derive a flow rule that solves both problems in accordance with our and previous measurements: $v=v_\infty[1-\exp(-l/L)]^{1/2}$, with $v_\infty\propto\mu_r^{3/2}\left[(\tan\theta-\mu_r)h\right]^{4/3}$ and $L\propto\mu_r^3\left[(\tan\theta-\mu_r)h\right]^{5/3}h$.