Gravity-driven, dry granular flows over a loose bed in stationary and homogeneous conditions

Flows involving solid particulates have been widely studied in recent years, but their dynamics are still a complex issue to model because they strongly depend on the interaction with the boundary conditions. We report on laboratory investigations regarding homogeneous and steady flows of identical particles over a loose bed in a rectangular channel. Accurate measurements were carried out through imaging techniques to estimate profiles of the mean velocity, solid concentration, and granular temperature for a large set of flow rates and widths. Vertical and transversal structures observed in the flow change as interparticle interactions become more collisional, and they depend on the bottom over which the flow develops. The lateral confinement has a remarkable effect on the flow, especially for narrow channels compared with the grain size, and a hydraulic analogy is able to show how the walls influence the mechanisms of friction and energy dissipation.

Figure 1

Sketch of the flow: the rigid bed corresponds to the channel base, while the loose bed corresponds to the interphase between moving grains and particles at rest. The $xyz$ system is based on the flow direction. $\beta$ is the tilting angle of the flume, while $\alpha$ is the slope of the free surface.

Figure 2

Behavior of the deposit ${\mathrm{\Delta }}_{\mathrm{dep}}$ and the flow depth $h_f$ measured from the fixed bed, for different flow rates and different apertures of the slit gate ($a_1<a_2<a_3$). The deposit keeps the same slope changing the aperture of the gate for the same flow rate.

Figure 3

Vertical profiles of velocity, concentration, and granular temperature for three different ranges of flow rate: (a) $\square :Q^{*}=2.8,\alpha =23.7^{\circ }$; $*:Q^{*}=5.8,\alpha =23.7^{\circ }, ◊:Q^{*}=6.8,\alpha =23.8^{\circ }$; (b) $\circ$: $Q^{*}=39.1,\alpha =24.4^{\circ }$; $◃$: $Q^{*}=46.2,\alpha =24.9^{\circ }$; +: $Q^{*}=82.4,\alpha =25.{87}^{\circ }$; (c) $\circ$: $Q^{*}=166.5,\alpha =26.7^{\circ }$; $*:Q^{*}=165.9,\alpha =26.5^{\circ }$; $▿:Q^{*}=182.9,\alpha =27.8^{\circ }$. For each group, different types of grain interactions are highlighted along the vertical profiles: (i) flow dominated by collisions (d.c.), (ii) intermediate region, (iii) flow dominated by friction (d.f.), and (iv) loose bed.

Figure 4

Detected particles in a single frame using the three-dimensional Voronoï diagrams. The different colors are relative to particles with a different distance from the sidewalls: blue points: $0<z_b<0.03\text{mm}$; red points: $0.03<z_r<0.2\text{mm}$; green points: $0.2<z_g<0.4\text{mm}$; and cyan points: $z_c>0.4\text{mm}$.

Figure 5

Surface profiles for different flow rates: (a) longitudinal velocity and (b) granular temperature measured from the top of the flow; (c) granular temperature normalized with the “surface” shear rate $\left[d\right(du/dz\left)\right]$.

Figure 6

Comparison of the transversal profiles between two base configurations: flow over an erodible bed and flow over a rigid and smooth bottom. (a) Longitudinal velocity from the top. (b) Granular temperature normalized with the mean longitudinal velocity. (c) Shear rate in the transverse direction. Both configurations refer to the same $Q^{*}=185,38$. Note that in the section where the flow develops over a rigid bottom, the bed slope is that of the channel (i.e., ${28}^{\circ }$) and the free surface has nearly the same slope, while in the reach over the loose bed, the free-surface slope is milder (i.e., $25.2^{\circ }$) and the bed line is parallel to the free surface.

Figure 7

Comparison of normal-to-bed profiles between two base configurations: flow over an erodible bed and flow over a rigid and smooth bottom from the lateral sidewalls. (a) Longitudinal velocity, (b) granular temperature, and (c) volume fraction profiles. Both configurations refer to the same $Q^{*}=185.38$.

Figure 8

Profiles of the longitudinal velocity and the granular temperature for two different widths ($W/d=42$ and $W/d=83$) and a single specific flow rate ($Q^{*}=13.7$), depicted together as they were performed simultaneously in the flume. The gray line divides the channel into two equal parts of 2.5 cm each and represents the imagery sidewalls for the narrower channels. The $z$ direction is normalized with respect to the channel width and spans from 0 to 1, while the granular temperature and the velocity are expressed in the international systems of units.

Figure 9

Experiments for two different widths ($W=83d$ and $W=250d$). The plots represent (a),(b) the longitudinal velocity normalized with respect to the mean value and (c) the evolution of the granular temperature along the width. The dotted line in (c) represents the limit over which the effect of the sidewalls on the granular temperature starts to vanish.

Figure 10

Tangent of the angle measured between the free-surface slope and the horizontal as a function of the dimensionless discharge, $Q^{*}=Q/\left(Wd\sqrt{dg}\right)$. The different symbols refer to different widths, as indicated in the legend. All the data are collected in the flume developed at the University of Trento.

Figure 11

Tangent of the free-surface slope against the ratio between the flow depth and width. Comparison of the experiments performed at the University of Trento (symbols) with the trend obtained by Jop et al. [6] (solid line) and Taberlet et al. [5] (dashed line). The friction angle of the plastic material used at the University of Trento is represented by the straight line $tan\varphi$.

Figure 12

(a) Profile of the flow depth across the width for $W=250d$ obtained through the erosion method and from the lateral recordings. (b) Evolution of the flow depth with the flow rates for two different widths.

Figure 13

Experimental points on the theoretical curves. All the parameters are reported on the plot.