Experimental configuration. The axisymmetric granular column is released by rapidly removing the confining cylindrical tube. Granular material flows down an unconfined wooden slope. Definitions of variables are shown.
Final deposit shapes for various slope angles (θ) using initial granular mass of 2.0 kg. (a) θ = 0°, (b) 5°, (c) 10°, and (d) 15°. Down-slope direction is to the left.
Cross sections of final deposits in laboratory experiments for particles with an average diameter of 100 μm. Left: x-direction profile taken along the line of y = 0; right: y-direction profile taken across the line of x = 0. The mass in the release is given for each plot in the captions.
Perimeter of the deposits using grains with diameters of (a) 100 μm and (b) 400 μm in laboratory experiments. Radial distances are in cm.
The relationship between slope angle (θ) and axial ratio (y ∞/x ∞) of an elliptically shaped deposit for initial granular mass (M) with grain diameters of (a) 100 μm and (b) 400 μm.
The maximum down-slope runout distance (x d) as a function of the mass of granular material released (M), for slope inclinations θ = 0° to 15°. Grain diameter is 100 μm.
A time-distance plot of the flow front position in a down-current direction for an initial mass of granular material (0.5 to 2.5 kg). Slope angle θ is 15°. Grain diameter is 100 μm.
Cross section of flow deposits in the x-direction from numerical simulation and observation for M = 0.5 kg, 1.5 kg, and 2.5 kg with θ = 0° (upper), and M = 0.5 kg with θ = 5°, 10°, and 15° (bottom). For a constant friction coefficient model, δ = 28° (thin solid line, black), 30° (medium solid line, red), and 32° (thick solid line, orange) were examined. For the I-dependent (varying friction coefficient) model (s), δ s was set to be 22° (long dashed line, green), 24° (medium dashed line, blue), or 26° (short dashed line, purple) with a constant value of δ2 = 30°.
The depth of the final deposit as a function of the radial distance from release for a friction angle of 30° and for various shape factors Γ = 1.0 (thin solid line, black), 1.1 (medium solid line, red), and 1.25 (thick solid line, orange). Also plotted are the deposit for friction angles 28° (short dashed line, blue) and 32° (long dashed line, purple) with Γ = 1.0. The initial conditions consist of a 1.0 kg mass of particles held within a cylinder of radius 5 cm. The flow was computed over a horizontal boundary.
Cross sections of flow deposits in the x-direction from observation (open squares) for M = 0.5 kg with θ = 15° and numerical simulation using the I-dependent friction model where δ s was set to be 22° (solid line), 24° (long dashed line), or 26° (short dashed line) and δ2 was set to be 26°, 28°, 30°, or 32°. A model with δ s = δ2 = 26° is equal to a constant friction coefficient model with δ = 26°.
Cross sections of flow deposits in the x-direction from numerical simulation and observation for M = 1.0 to 2.5 kg and θ = 15°. Legend is the same as in Fig. 8 .
Comparison of the maximum runout distance in the down-current direction for laboratory observation (open squares) and numerical simulation for a constant friction coefficient model (solid circles, solid squares, and diamonds) and the I-dependent model (upward/downward triangles and open circles). Colors in legend corresponding to the friction models are the same as in Fig. 8 .
Relationships between slope angle (θ) and axial ratio (y ∞/x ∞) for elliptically shaped deposit consisting of different initial granular mass (M). Open squares are laboratory data; closed circles are numerical results for two different friction models using different friction angles.
Comparison of time-distance plots of the flow front position in the down-current direction. Initial mass of granular material is 0.5 to 2.5 kg). Slope angle θ is 15°. A non-dimensional distance and non-dimensional time scale are used. They are normalized by the maximum runout distance and the duration of flow observed in laboratory experiments, respectively. Legend is the same as in Fig. 8 .
Magnification of cross section (x-direction) in the central (a) and the frontal (b) parts of the deposit from numerical simulation and observation (M = 1.5 kg, θ = 15°). For the maximum deposit height, h ∞, the I-dependent friction model (especially with δ s = 22° and 24°) shows a better agreement with observation than the constant friction coefficient model does. The frontal shape is also reproduced well by the I-dependent model. Legend is the same as in Fig. 8 .
Results of numerical simulation using M = 0.5 kg (initial column radius is 5 cm) on unconfined slopes. (a) θ = 0°, constant friction model (δ = 30°), (b) θ = 5°, I-dependent friction model (δ2 = 30°, δ s = 26°), (c) θ = 10°, I-dependent friction model (δ2 = 30°, δ s = 24°), (d) θ = 15°, I-dependent friction model (δ2 = 30°, δ s = 22°). Contour interval is 1 mm.
The shape of the final deposit, characterized by the ratio of its cross-slope to down-slope extent, y ∞/x ∞, as a function of ɛ/λ. Circles show numerical data. Squares and triangles show experimental data with 100 and 400 μm-diameter grains, respectively, assuming δ = 30°. Typical deposit profiles at some values of ɛ/λ.
(a) h ∞/h 0 as a function of λ. (b) h ∞/h 0 as a function of λ 2/9 ɛ 2/3. Circles (red) for ɛ/λ ≥ 0.5 and diamonds (blue) for ɛ/λ < 0.5 for numerical results. Squares and triangles (gray) show laboratory data with 100 and 400 μm-diameter grains, respectively, assuming δ = 30°. Some of the data are indicated with values of ɛ/λ. Solid lines indicate the scaling relationships for numerical results, and broken lines indicate the scaling relationship for laboratory data.
y ∞/x ∞ as a function of λ−1/9(ɛ/λ)2/3. Circles (red) for ɛ/λ ≥ 0.5 and diamonds (blue) for ɛ/λ < 0.5 for numerical results. Squares and triangles (gray) show experimental data with 100 and 400 μm-diameter grains, respectively, assuming δ = 30°. Solid line indicates the scaling relationship for numerical results with ɛ/λ < 0.5. Broken line indicates the scaling relationship for laboratory data.
Material properties of the granular materials: mean particle diameter d, grain density ρ b, angle of repose φ, dynamic bed friction angle δ.
Friction angles of granular material used in numerical simulation: dynamic bed friction angle δ, minimum critical friction angle δ s , and maximum friction angle δ2.
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