Length of standing jumps along granular flows down smooth inclines

Granular jumps—the change in height and depth-averaged velocity during granular flows—occur during transitions from thin and fast flows (supercritical) to thick and slow flows (subcritical). The present paper describes discrete element method simulations inspired by recent laboratory experiments, which produce standing jumps in two-dimensional free-surface dry granular flows down a slope. Special attention is paid to characterizing and measuring the finite length of those standing granular jumps, as well as to deciphering their internal structure. By varying macroscopic quantities, such as slope angle and mass discharge, and microscopic properties, such as interparticle friction and grain diameter, a rich variety of granular jump patterns is observed. Hydraulic-like granular jumps with an internal waterlike roller are identified, in addition to diffuse laminar granular jumps and steep colliding granular jumps. Moreover, a recently established depth-averaged relation for the prediction of jump heights is fed with the measured jump length and fitted against the numerical simulations to examine the effective friction in the granular medium for each type of jump. The dominant component of the general friction law is found to be different when transitioning from one jump pattern to the others. This study demonstrates that the granular jump pattern, and particularly its geometry and internal structure, can offer a stringent test for addressing the dissipation mechanisms that govern the flowability of granular media.

Figure 1

Discrete element method numerical setup (top) and example of a volume fraction field obtained by coarse-graining (bottom).

Figure 2

Three different jump lengths identified for a single standing (laminar) granular jump, based on the depth-averaged variable considered: $h$ (blue line), $\overline{u}$ (green line), or $\varphi$ (pink line). This jump was obtained with $\zeta ={28}^{\circ }, H=0.5$ m, $\mu =0.5, d=4.0$ cm.

Figure 3

Numerical and experimental phase diagrams of jump patterns for various slopes $\zeta$ and discharge height ratio $H/d$. (a) Phase diagram obtained numerically in the present study. The streamlines inside the jumps are drawn with a color indicating the local velocity. All the simulation use an interparticle friction $\mu =0.5$ and a bottom friction ${\mu }_b=0.25$. (b) Phase diagram obtained experimentally in Ref. [16] using glass beads.

Figure 4

Transition from (a) a laminar granular jump ($\mu =0.8>{\mu }_{\mathrm{cr}}$) to (b) a hydraulic-like granular jump with the presence of a waterlike roller ($\mu =0.1<{\mu }_{\mathrm{cr}}$). The streamlines are drawn in each jump as in Fig. 3 with colors representing the corresponding velocity.

Figure 5

Velocity profiles of a laminar granular jump ($\zeta ={24}^{\circ }, d=4\mathrm{cm}, H=0.5\mathrm{m}, \mu =0.5$) before (red), inside (green), and after (blue) the jump. The velocity profiles shown were measured every 0.12 m along the inclined plane.

Figure 6

Transition from a laminar granular jump [cases (a) and (b)] to a steep colliding granular jump [cases (c), (d), and (e)] by varying the slope: streamlines (first row) and longitudinal variation of the depth-averaged volume fraction (second row).

Figure 7

Velocity profiles of a steep colliding granular jump ($\zeta ={38}^{\circ }, d=4\mathrm{cm}, H=0.5\mathrm{m}, \mu =0.5$) before (blue), inside (green), and after (red) the jump. The velocity profiles shown were measured every 0.12 m along the inclined plane.

Figure 8

Transition from a laminar granular jump (a) to a hydraulic-like granular jump with a waterlike roller [(b), (c), (d), and (e)] by varying the interparticle friction coefficient: Streamlines (first row) and longitudinal variation of the depth-averaged volume fraction (second row).

Figure 9

Velocity profiles of a hydraulic-like jump ($\zeta ={25}^{\circ }, d=4\mathrm{cm}, H=0.5\mathrm{m}, \mu =0.04$) before (red), inside (green), and after (blue) the jump. The velocity profiles shown were measured every 0.12 m along the inclined plane.

Figure 10

Jump length $L$ (a) and jump length relative to incoming flow thickness $L/h$ (b) versus the slope angle $\zeta$. Inset of (a): Incoming thickness of the flow $h$ as a function of $\zeta$. Inset of (b): Jump length relative to the incoming flow as a function of the Froude number. The circles correspond to the granular laminar jumps and the squares to the steep colliding granular jumps with recirculation.

Figure 11

Jump length $L$ (a) and jump length relative to incoming flow thickness $L/h$ (b) versus the opening height $H$ at the exit of the tank (which controls the mass discharge). Inset in (a): Thickness $h$ of the incoming flow as a function of the opening height $H$ of the reservoir. Since $L$ versus $H$ and $h$ versus $H$ have the same slope in (a), $L/h$ in (b) is independent of the mass discharge.

Figure 12

(a) Jump length $L$, (b) jump length relative to incoming flow thickness $L/h$ versus the grain diameter $d$. Inset in (a): Thickness $h$ of the incoming flow as a function of $d$. $L$ versus $d$ and $h$ versus $d$ in (a) have the same slope, thus $L/h$ in (b) is independent of $d$.

Figure 13

(a) Jump length $L$ and (b) jump length relative to incoming flow thickness $L/h$, both against the interparticle friction coefficient $\mu$. The circles represent laminar granular jumps and the stars denote hydraulic-like jumps with a waterlike roller.

Figure 14

The transition from laminar granular jumps to hydraulic-like jumps, as visible in Fig. 4, seen through the change in different variables as a function of $\mu$: (a) the Froude number Fr, (b) the incoming volume fraction $\varphi$, (c) the relative height of the jump $h_{*}/h$, and (d) the angle of the free surface after the jump ${\zeta }_0$. Stars represent hydraulic-like jumps and dots denote laminar jumps.

Figure 15

Jump height ratio, $h_{*}/h$, as a function of the Froude number, Fr, for all jumps investigated in the present study. The continuous line shows the prediction of the Bélanger equation used in hydraulics (see text).

Figure 16

Dimensionless length of the jump $L/h$ versus the Froude number Fr for all jumps. The orange-colored line shows that $L/h$ is rather constant (within the uncertainty for measuring the jump length) for laminar granular jumps.

Figure 17

Effective friction inside the jump ${\mu }_e$ extracted from Eq. (10) as a function of the Froude number for all types of jumps.

Figure 18

Volume fractions measured before ($\varphi$) and after (${\varphi }_{*}$) the jump as a function of the range $c$ of the Lucy function scaled by the grain diameter ($c/d$). Simulations done for $H=0.5\mathrm{m}$ and $\zeta ={25}^{\circ }$, considering the default values given in Table 1 for $d, \mu , {\mu }_b$, and $e$.

Figure 19

Evolution of the depth-averaged volume fraction along the chute: comparison between time-averaged results sampling data every 0.3 s (continuous line) or 1 s (dashed line). Simulations done for $H=0.5\mathrm{m}$ and $\zeta ={25}^{\circ }$, considering the default values given in Table 1 for $d, \mu , {\mu }_b$, and $e$, and taking $c=2.5d$ for the range of the Lucy function.