A conveyor belt experimental setup to study the internal dynamics of granular avalanches.

ABSTRACT: This paper shows how a conveyor belt setup can be used to study the dynamics of stationary granular flows. To visualise the flow within the granular bulk and, in particular, determine its composition and the velocity field, we used the refractive index matching (RIM) technique combined with particle tracking velocimetry and coarse-graining algorithms. Implementing RIM posed varied technical, design and construction difficulties. To test the experimental setup and go beyond a mere proof of concept, we carried out granular flow experiments involving monodisperse and bidisperse borosilicate glass beads. These flows resulted in stationary avalanches with distinct regions whose structures were classified as: (i) a convective-bulged front, (ii) a compact-layered tail and, between them, (iii) a breaking size-segregation wave structure. We found that the bulk strain rate, represented by its tensor invariants, varied significantly between the identified flow structures, and their values supported the observed avalanche characteristics. The flow velocity fields' interpolated profiles adjusted well to a Bagnold-like profile, although a considerable basal velocity slip was measured. We calculated a segregation flux using recent developments in particle-size segregation theory. Along with vertical velocity changes and high expansion rates, segregation fluxes were markedly higher at the avalanche's leading edge, suggesting a connection between flow rheology and grain segregation. The experimental conveyor belt's results showed the potential for further theoretical developments in rheology and segregation-coupled models.

Statistical description of sediment transport experiments.

A longstanding problem in the study of sediment transport in gravel-bed rivers is related to the physical mechanisms governing bed resistance and particle motion. To study this problem, we investigated the motion of coarse spherical glass beads entrained by a steady shallow turbulent water flow down a steep two-dimensional channel with a mobile bed. This experimental facility is the simplest representation of sediment transport on the laboratory scale, with the tremendous advantages that boundary conditions are perfectly controlled and a wealth of information can be obtained using imaging techniques. Flows were filmed from the side by a high-speed camera. Using image processing software made it possible to determine the flow characteristics such as particle trajectories, their state of motion (rest, rolling, or saltating motion), and flow depth. In accordance with earlier investigations, we observed that over short time periods, sediment transport appeared as a very intermittent process. To interpret these results, we revisited Einstein's theory on sediment and derived the statistical properties (probability distribution and autocorrelation function) of the key variables such as the solid discharge and the number of moving particles. Analyzing the autocorrelation functions and the probability distributions of our measurements revealed the existence of long-range correlations. For instance, whereas theory predicts a Binomial distribution for the number of moving particles, experiments demonstrated that a negative binomial distribution best fit our data, which emphasized the crucial role played by wide fluctuations. These frequent wide fluctuations stemmed particle entrainment and motion being collective phenomena rather than individual processes, contrary to what is assumed in most theoretical models.

Monte Carlo calibration of avalanches described as Coulomb fluid flows.

The idea that snow avalanches might behave as granular flows, and thus be described as Coulomb fluid flows, came up very early in the scientific study of avalanches, but it is not until recently that field evidence has been provided that demonstrates the reliability of this idea. This paper aims to specify the bulk frictional behaviour of snow avalanches by seeking a universal friction law. Since the bulk friction coefficient cannot be measured directly in the field, the friction coefficient must be calibrated by adjusting the model outputs to closely match the recorded data. Field data are readily available but are of poor quality and accuracy. We used Bayesian inference techniques to specify the model uncertainty relative to data uncertainty and to robustly and efficiently solve the inverse problem. A sample of 173 events taken from seven paths in the French Alps was used. The first analysis showed that the friction coefficient behaved as a random variable with a smooth and bell-shaped empirical distribution function. Evidence was provided that the friction coefficient varied with the avalanche volume, but any attempt to adjust a one-to-one relationship relating friction to volume produced residual errors that could be as large as three times the maximum uncertainty of field data. A tentative universal friction law is proposed: the friction coefficient is a random variable, the distribution of which can be approximated by a normal distribution with a volume-dependent mean.

Fluctuations of the solid discharge of gravity-driven particle flows in a turbulent stream.

Substantial variations in the particle flux are commonly observed in field measurements on gravel-bed rivers and in laboratory experiments mimicking river behavior on a smaller scale. These fluctuations can be explained by the natural variability of sediment supply and hydraulic conditions. We conducted laboratory experiments of particle transport down a two-dimensional inclined channel, for which the boundary conditions were properly controlled. Most flow variables and the features of particle trajectories were measured using a high-speed camera. The particles were 6-mm glass beads entrained by a rapid, turbulent, supercritical water flow. Even under these well-controlled experimental conditions and despite steady supply, solid discharge exhibited significant variations with time. The objective of this paper was to pinpoint the origins of these fluctuations by investigating different flow conditions. Two experiments were done with a fixed (smooth or corrugated) channel bottom and two others were run with a mobile bed (involving layers of closely packed particles lying along the channel base, which could be entrained by the stream); in the latter case, two particle arrangements were tested. It was found that, to a large extent, fluctuations reflected the finite size of the observation window. For fixed beds, the characteristic time scale of fluctuations and their probability distribution can be predetermined by evaluating the mean and fluctuating velocities of a single particle. Solid-discharge fluctuations were exacerbated when the bed was mobile because (i) the moving solid phase and the stationary bed exchanged particles and (ii) collective entrainment of particles occurred.

Rolling motion of a bead in a rapid water stream.

This paper investigates the two-dimensional rolling motion of a single large particle in a shallow water stream down a steep rough bed from both an experimental and a theoretical point of view. The experiment is prototypal of sediment transport on sloping beds. Two theoretical models are presented. The first model uses the mean kinetic energy balance to deduce the average particle velocity and the bounds of the flow-rate range within which a rolling regime occurs. This range is found to be narrow, which means that the fully rolling regime is a marginal mode of transport between repose and saltation. In the second model, the particle state (resting, rolling, saltating) is considered as a random variable, whose evolution constitutes a jump Markov chain. This makes it possible to deduce the mean particle velocity as a function of the flow conditions without explicit mention of its state. The theoretical results are finally compared to the experimental data. The second model provides correct estimates of the particle velocity and the probability of finding the particle in a given state for various flow conditions (bead material, slope, and roughness).

Saltating motion of a bead in a rapid water stream.

This paper experimentally and numerically investigates the two-dimensional saltating motion of a single large particle in a shallow water stream down a steep rough bed. The experiment is prototypical of sediment transport on sloping beds. Similar to the earlier experimental results on fine particles entrained by a turbulent stream, we found that most features of the particle motion were controlled by a dimensionless shear stress (also called the Shields number) N(Sh) defined as the ratio of the bottom shear stress exerted by the water flow to the buoyant weight of the particle (scaled by its cross-sectional area to obtain a stress). We did not observe a clear transition from rest to motion, but on the contrary there was a fairly wide range of N(Sh) (typically 0.001-0.005 for gentle slopes) for which the particle could be set in motion or come to rest. When the particle was set in motion, it systematically began to roll. The rolling regime was marginal in that it occurred for a narrow range of N(Sh) (typically 0.005-0.01 for gentle slopes). For sufficiently high Shields numbers (N(Sh)>0.3), the particle was in saltation. The mean particle velocity was found to vary linearly with the square root of the bottom shear stress and here, surprisingly enough, was a decreasing function of the channel slope. We also performed numerical simulations based on Lagrangian equations of motion. A qualitative agreement was found between the experimental data and numerical simulations but, from a quantitative point of view, the relative deviation was sometimes substantial (as high as 50%). An explanation for the partial agreement is the significant modification in the water flow near the particle.

Dry granular flows down an inclined channel: experimental investigations on the frictional-collisional regime.

This paper presents experimental results on dry granular flows down an inclined rough channel. Different flow regimes were identified depending on the Froude number. For Froude numbers exceeding a critical value (function of the channel slope), flow was characterized by a fairly linear velocity profile and a discharge equation in the form q varies with h(n) with q the flow rate per unit width, h the flow depth, and n an exponent in the range 2-3 (regime A). When the Froude number was lower than the critical value, the flow was characterized by a convex velocity profile and a discharge equation of the form q varies with h(n), with n ranging from 0.97 to 1.16, producing the striking result that the mean velocity was constant for a given inclination of the channel (regime B). Experimental data were used to test three theoretical models developed to describe dry granular flows in a frictional-collisional regime. Savage's model provides results that capture experimental trends well and yield the correct magnitude for velocity and discharge for regime A, but it reproduces the dependence of the discharge on the channel slope for only a narrow range of slopes [S. B. Savage, in U.S./Japan Seminar on New Models and Constitutive Relations in the Mechanics of Granular Materials, Ithaca, 1982, edited by J. T. Jenkins and M. Satake (Elsevier Science Publishers, Amsterdam, 1982), p. 261]. In contrast, Mills et al.'s model is less refined and requires fitting an input parameter to give the correct magnitude of velocity but it successfully accounts for the variation in the discharge with slope for regime A for a wide range of slopes [Mills, Loggia, and Tixier, Europhys. Lett. 45, 733 (1999); Eur. Phys. J. E 1, 5 (2000)]. Ancey and Evesque's model is also crude in determining the density profile but manages to provide velocity profiles and discharge equations in good agreement with experimental data for regime B [C. Ancey and P. Evesque, Phys. Rev. E 62, 8349 (2000)].