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 (tanθ/μ_{r})^{2}h/h_{s}, where μ_{r} is the dynamic friction coefficient, h the flow thickness, and h_{s}(θ) its smallest value that permits a steady, uniform dense flow state at a given inclination angle θ. 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_{∞}[1-exp(-l/L)]^{1/2}, with v_{∞}∝μ_{r}^{3/2}[(tanθ-μ_{r})h]^{4/3} and L∝μ_{r}^{3}[(tanθ-μ_{r})h]^{5/3}h.