Unified Flow Rule of Undeveloped and Fully Developed Dense Granular Flows Down Rough Inclines

TL;DR

This paper proposes a unified flow rule for dense granular flows down rough inclines, reconciling inconsistencies in previous models by incorporating flow development length and universal behavior across materials.

Contribution

It introduces a new flow rule derived from dimensional analysis that unifies undeveloped and fully developed dense granular flows, addressing limitations of prior models.

Findings

The new flow rule fits experimental data across different granular materials.

It accounts for flow development length exceeding the chute length.

The model unifies flow behavior for various granular flows on inclined planes.

Abstract

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 $L\propto\mu_r^3\left[(\tan\theta-\mu_r)h\right]^{5/3}h$.