Dynamics and Rheology of Finite Dry Granular Mass in Avalanche down an Inclined Smooth Reservoir

This work investigates the dynamics and rheology for finite number of nearly identical dry glass spheres in avalanche down a narrow inclined reservoir of smooth frictional bed via both experiments and parallelized discrete element simulation. The objective is first to study if the μ-I rheology law that describes bulk internal friction coefficient, μ, as a monotonically increasing function of dimensionless inertial number, I, for steady uniform flows can also be extracted from a non-uniform transient flow. Second, examine the effective sidewall friction coefficient in which most theoretical studies have assumed a constant value in the application of surface flow modelling. Finally, as an attempt to integrate experiment and simulation, the experimentally measured bulk discharge rate is used to calibrate the contact parameters in simulation while the detailed flow information gained from simulation is utilized to evaluate the assumptions made in the control volume analysis of experimental data. High-speed imaging technique is employed to measure bulk transient dynamics at the reservoir sidewall from which individual spheres can be traced to achieve particle tracking velocimetry. The individual sphere information is coarse-grained to estimate continuous bulk solid volume fraction, ϕ(t,x,y), and streamwise and transverse velocity components,U(t,x,y)andV(t,x,y)which are then applied in a quasi-two-dimensional control volume analysis to estimate internal friction, T(t,x,y). These flow properties are further analyzed to seek bulk instantaneous friction coefficient, μ^'', as a function of instantaneous flow inertial number and the obtained μ^''-I relation agrees qualitatively to that reported for steady flows, supporting its claim as a local rheology law and validating the applicability to unsteady flows. Complementary numerical simulation that reproduces the measured bulk dynamics at sidewall is conducted to extract flow information away from the sidewall and a μ^''-I in qualitative agreement to the measured data for I> 0.02 is obtained. A peculiar trend that μ^'' decays almost linearly with I at small I< 0.02 is revealed, suggesting stress thinning at the onset of bulk motion which should be related to different micro- mechanisms than that giving rise to the robust μ^''-I relation. To describe the decay trend, a simplified model considering the shear-induced dilation of initially compressed one-dimensional force chain upon yielding has been developed, predicting a linear decay at small I. On the other hand, the effective sidewall friction coefficient, μ_w, is not a constant but degrades with flow depth and exhibits a monotonically increasing function of quasi-two-dimensional inertial number at the sidewall, I_w. The fact that rotation at one sphere center can divert surface relative velocity across the contact area to render lower sliding friction is considered to develop a model describing how μ_w drops with the ratio between rotation-induced velocity and sliding velocity, Ω. The numerical data provides the degree of underestimation for flow properties under the quasi-two-dimensional approximation in control volume analysis and reveals anisotropic normal stress and their saturating, not fully hydrostatic, depth profile in addition to the varying μ_w. These findings from numerical simulation are discussed and used to construct a more feasible control volume analysis with in-depth understanding of bulk dynamics in narrow frictional confinement.