Indirect measurement of $\mu (I)$ relation from finite granular avalanche down an inclined narrow reservoir of smooth bed

Abstract

This work examines the algebraic $\mu -I$ relation proposed for steady uniform dry granular flows via unsteady granular avalanche experiments of finite nearly identical dry glass spheres down an inclined narrow reservoir of smooth bed. Lateral high-speed digital imaging permits particle tracking velocimetry with which we can evaluate bulk local instantaneous volume fraction and velocity components to conduct a quasi-two-dimensional control volume analysis of streamwise momentum assuming an internal shear stress based on the $\mu -I$ rheology, a hydrostatic normal stress and a Coulomb yielding condition at lateral walls. Hence, the desired $\mu$ is a function of flow dynamics and a wall friction coefficient $\mu _w$. Complementary sliding table experiments were conducted to estimate an upper bound of $\mu _w=0.17$ which was used with a chosen nonzero lower bound $\mu _w=0.05$ to extract possible range of $\mu$ at a local instantaneous inertial number I. The so-obtained local instantaneous $\mu -I$ data conform to the non-linear monotonically increasing trend proposed for steady inertial flows above a crossover value $I_c\approx 0.03$. Below $I_c$, a peculiar segment of decaying $\mu$ with I was revealed agreeing to the rheology tests in quasi-static regime.

Appendix

This appendix describes how we estimated the variation of U across the current reservoir width, w / D = 9.4, based on the experimental results by Jop et al. [12]. These authors report the surface velocity for a steady heap flow across a channel of various widths, w / D=19-570, and the maximum and minimum velocity were measured at the flow center line, $U_c$, and at the sidewalls, $U_w$, respectively. Based on Fig. 3a therein, we measured both $U_c$ and $U_w$ to compute a surface velocity decay percentage by $\varepsilon \%=(U_c-U_w)/U_c \times 100\%$ for each examined width. The obtained $\varepsilon \%$ is examined with respect to w / D in Fig. 12 showing monotonic decrease with w / D but at a much pronounced degree at small w / D. This trend suggests a milder velocity variation across a narrower channel width since a greater portion of the flowing mass is under the influence of lateral wall friction. We used the data from the narrowest two channel widths, w / D = 19 and 57, to extrapolate the decay percentage for the current reservoir width to obtain $\varepsilon \%\approx 7\%$ which value led to the assumption of nearly two-dimensional flow in the control volume analysis.