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The kinematics of bidisperse granular roll waves

  • S. Viroulet (a1) (a2), J. L. Baker (a1) (a3), F. M. Rocha (a1), C. G. Johnson (a1), B. P. Kokelaar (a4) and J. M. N. T. Gray (a1)...

Abstract

Small perturbations to a steady uniform granular chute flow can grow as the material moves downslope and develop into a series of surface waves that travel faster than the bulk flow. This roll wave instability has important implications for the mitigation of hazards due to geophysical mass flows, such as snow avalanches, debris flows and landslides, because the resulting waves tend to merge and become much deeper and more destructive than the uniform flow from which they form. Natural flows are usually highly polydisperse and their dynamics is significantly complicated by the particle size segregation that occurs within them. This study investigates the kinematics of such flows theoretically and through small-scale experiments that use a mixture of large and small glass spheres. It is shown that large particles, which segregate to the surface of the flow, are always concentrated near the crests of roll waves. There are different mechanisms for this depending on the relative speed of the waves, compared to the speed of particles at the free surface, as well as on the particle concentration. If all particles at the surface travel more slowly than the waves, the large particles become concentrated as the shock-like wavefronts pass them. This is due to a concertina-like effect in the frame of the moving wave, in which large particles move slowly backwards through the crest, but travel quickly in the troughs between the crests. If, instead, some particles on the surface travel more quickly than the wave and some move slower, then, at low concentrations, large particles can move towards the wave crest from both the forward and rearward sides. This results in isolated regions of large particles that are trapped at the crest of each wave, separated by regions where the flow is thinner and free of large particles. There is also a third regime arising when all surface particles travel faster than the waves, which has large particles present everywhere but with a sharp increase in their concentration towards the wave fronts. In all cases, the significantly enhanced large particle concentration at wave crests means that such flows in nature can be especially destructive and thus particularly hazardous.

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Copyright

This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited.

Corresponding author

Email address for correspondence: nico.gray@manchester.ac.uk

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VIDEO
Movies

Viroulet et al. supplementary movie 1
Experiment showing the formation and evolution of bidisperse granular roll waves in a chute inclined at $29^{\circ}$. An initially homogeneous mixture consisting on a 80\% small glass ballotini (white, 75-150 $\mu\textrm{m}$ diameter), 20\% large ballotini (green, 200-250 $\mu\textrm{m}$ diameter) flows steadily from a hoper through a gate raised 3 mm from the bed.

 Video (12.2 MB)
12.2 MB
VIDEO
Movies

Viroulet et al. supplementary movie 2
Experiment showing the formation and evolution of bidisperse granular roll waves in a chute inclined at $29^{\circ}$. An initially homogeneous mixture consisting on a 40\% small glass ballotini (white, 75-150 $\mu\textrm{m}$ diameter), 60\% large ballotini (green, 200-250 $\mu\textrm{m}$ diameter) flows steadily from a hoper through a gate raised 3 mm from the bed.

 Video (20.2 MB)
20.2 MB
VIDEO
Movies

Viroulet et al. supplementary movie 3
Numerical simulation showing the temporal evolution of a regular wavetrain that forms by sinusoidally perturbing a steady uniform inflow of thickness $h_0=2$ mm and depth-averaged small-particle concentration $\bar{\phi}_0=0.8$ at frequency $f=2$ Hz. Bold black line denotes the flow thickness $h$ and the thin line represents the interface $\eta = h\bar{\phi}$ separating regions of pure small and large particles, with the latter being shaded in green. The value of shear parameter $A=0.1$ used corresponds to the low shear regime with `continuous' solutions for the concentration $\bar{\phi}$.

 Video (7.0 MB)
7.0 MB
VIDEO
Movies

Viroulet et al. supplementary movie 4
Numerical simulation showing the temporal evolution of a regular wavetrain that forms by sinusoidally perturbing a steady uniform inflow of thickness $h_0=2$ mm and depth-averaged small-particle concentration $\bar{\phi}_0=0.8$ at frequency $f=2$ Hz. Bold black line denotes the flow thickness $h$ and the thin line represents the interface $\eta = h\bar{\phi}$ separating regions of pure small and large particles, with the latter being shaded in green. The value of shear parameter $A=0.5$ used corresponds to the high shear regime with `internal shock' solutions for the concentration $\bar{\phi}$.

 Video (6.8 MB)
6.8 MB
VIDEO
Movies

Viroulet et al. supplementary movie 5
Numerical simulation showing the flow kinematics of the low shear `continuous' regime, which is calculated by sinusoidally perturbing a steady uniform inflow of thickness $h_0=2$ mm and concentration $\bar{\phi}_0=0.8$ at frequency $f=2$ Hz, with shear parameter $A=0.1$. Bold black lines represent flow thickness $h$ and the green shading shows the region of large particles between $h$ and interface $\eta$. Red-shaded region tracks a small amount of large `tracer' particles, which lie between $h$ and interface $\eta_R$, and red markers show the path of two large particles at the surface of the flow.

 Video (6.3 MB)
6.3 MB
VIDEO
Movies

Viroulet et al. supplementary movie 6
Numerical simulation showing the flow kinematics of the high shear `full internal shock' regime, which is calculated by sinusoidally perturbing a steady uniform inflow of thickness $h_0=2$ mm and concentration $\bar{\phi}_0=0.8$ at frequency $f=2$ Hz, with shear parameter $A=0.5$. Bold black lines represent flow thickness $h$ and the green shading shows the region of large particles between $h$ and interface $\eta$. Red-shaded region tracks a small amount of large `tracer' particles, which lie between $h$ and interface $\eta_R$, and red markers show the path of two large particles at the surface of the flow.

 Video (4.4 MB)
4.4 MB
VIDEO
Movies

Viroulet et al. supplementary movie 7
Numerical simulation showing the flow kinematics for a chute flow where a steady uniform inflow of thickness $h_0=2$ mm and concentration $\bar{\phi}_0=0.8$ is randomly perturbed, with shear parameter $A=0.5$. Bold black lines represent flow thickness $h$ and the green shading shows the region of large particles between $h$ and interface $\eta$. Red-shaded region tracks a small amount of large `tracer' particles, which lie between $h$ and interface $\eta_R$, and red markers show the path of two large particles at the surface of the flow.

 Video (3.9 MB)
3.9 MB
VIDEO
Movies

Viroulet et al. supplementary movie 8
Numerical simulation showing the flow kinematics for a chute flow where a steady uniform inflow of thickness $h_0=2$ mm and concentration $\bar{\phi}_0=0.4$ is randomly perturbed, with shear parameter $A=0.5$. Bold black lines represent flow thickness $h$ and the green shading shows the region of large particles between $h$ and interface $\eta$. Red-shaded region tracks a small amount of large `tracer' particles, which lie between $h$ and interface $\eta_R$, and red markers show the path of two large particles at the surface of the flow.

 Video (8.1 MB)
8.1 MB

The kinematics of bidisperse granular roll waves

  • S. Viroulet (a1) (a2), J. L. Baker (a1) (a3), F. M. Rocha (a1), C. G. Johnson (a1), B. P. Kokelaar (a4) and J. M. N. T. Gray (a1)...

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