Granular flows on a dissipative base

We study inclined channel flows of sand over a sensor-enabled composite geotextile fabric base that dissipates granular fluctuation energy. We record strain of the fabric along the flow direction with imbedded fiber-optic Bragg gratings, flow velocity on the surface by correlating grain position in successive images, flow thickness with the streamwise shift of an oblique laser light sheet, velocity depth profile through a transparent side wall using a high-speed camera, and overall discharge rate. These independent measurements at inclinations between ${33}^{\circ }$ and ${37}^{\circ }$ above the angle of repose at $32.1\pm 0.8^{\circ }$ are consistent with a mass flow rate scaling as the $3/2$ power of the flow depth, which is markedly different than flows on a rigid bumpy boundary. However, this power changes to $5/2$ when flows are forced on the sand bed below its angle of repose. Strain measurements imply that the mean solid volume fraction in the flowing layer above the angle of repose is $0.268\pm 0.033$, independent of discharge rate or inclination.

Figure 1

Apparatus. (a) Position and orientation of the three FBGs. (b) Overview showing the bare geotextile laid at the base of the channel and the spring that tensions it at the exit. (c) Sketch showing reservoir, sluice gate, laser profiler, and overhead camera. (d) Detail near the exit showing the high-speed video camera trained to observe the flow through one of the transparent side walls.

Figure 2

Forces and lengths. Top: Force balance on a unit width of the geotextile in the flow direction $x$ at the inclination $\alpha$ along an infinitesimal length $dx$. The projection of geotextile weight along $x$ is $\omega gsin\alpha dx$. Sand of thickness $h$ flowing or resting overhead exerts a downhill shear force $\tau dx={\rho }_{s}gsin\alpha dx{\int }_{z=0}^h\nu dz$. The combined weight of sand and geotextile then presses on the base with normal force ${\sigma }_{b}dx=(\omega +{\rho }_s{\int }_{z=0}^h\nu dz)gcos\alpha dx$. This gives rise to an uphill shear force $\mu {\sigma }_{b}dx$. Bottom: Lengths of geotextile and spring at rest (top) and stretched after being joined (bottom).

Figure 3

Typical microstrain time histories from the top, middle, and bottom FBGs with $tan\alpha \simeq 0.70$ at a steady flow depth $(h_{\mathrm{stop}}-h)/\overline{d}\simeq 16.2$. The abscissa is time in seconds after opening the sluice gate. The ordinate is ${10}^6\times \epsilon$, translated to superimpose the three signals for $t<0$. (The three strains have widely different magnitude, but what matters is their relative change $\mathrm{\Delta }\epsilon$.) The loading, steady flow, and discharge phases are marked on the signal of the middle FBG, along with its residual strain after the flow stops. Gray dashed lines show the period of steady strain, lasting at least $4s$. The microstrain differences ${10}^6\mathrm{\Delta }\epsilon$ between these steady plateaus at a flow depth $h$ and the residual strain at $h_{\mathrm{stop}}$ (dotted lines) are plotted in the inset vs FBG distance $x$ to the upstream geotextile anchor in meters.

Figure 4

Residual depth of sand on the geotextile $h_{\mathrm{stop}}^{†}\equiv h_{\mathrm{stop}}/\overline{d}$, dimensionless with mean grain diameter, vs tangent of the angle of inclination $\alpha$. The solid line is the least-squares fit in Eq. (22). The vertical lines mark tangents of the asymptotic stop angle ${\alpha }_0$, the angle of repose ${\alpha }_r$, and the angle of spontaneous avalanching ${\alpha }_a$. Identical symbols are used in subsequent figures to identify inclination.

Figure 5

Dimensionless mass hold-up $\mathrm{\Delta }{H}^{†}=H^{†}-H_{\mathrm{stop}}^{†}$ in the flowing layer vs dimensionless flow depth $\mathrm{\Delta }{h}^{†}=h^{†}-h_{\mathrm{stop}}^{†}$. Symbols conform to inclination angles shown in Fig. 4. Error bars are $95\%$ confidence intervals. The slope is the mean solid volume fraction $\overline{\nu }$ in Eq. (27). The error bar at $\alpha ={31}^{\circ }$ ($\times$ symbol) is an estimate based on errors calculated at other inclinations.

Figure 6

Dimensionless discharge mass flow rate [Eq. (28)] relative to the $3/2$ power of dimensionless hold-up in the flowing layer vs excursion in $tan\alpha$ beyond its jamming value $tan{\alpha }_0$ for ${\alpha }_r<\alpha <{\alpha }_a$. The solid line is Eq. (29). The vertical dotted lines mark angles of repose ${\alpha }_r$ and spontaneous avalanching ${\alpha }_a$.

Figure 7

Dimensionless discharge mass flow rate [Eq. (28)] relative to $(tan\alpha -tan{\alpha }_0)$ vs dimensionless hold-up in the flowing layer. Symbols conform to inclination angles shown in Figs. 4 and 6. The main graph is for inclinations steeper than the angle of repose ${\alpha }_r$. The inset is for $\alpha ={31}^{\circ }<{\alpha }_r$. The solid line is inclined at a $3/2$ slope [Eq. (29)] and the dashed line at $5/2$ [Eq. (34)].

Figure 8

Dimensionless surface velocity [Eq. (30)] relative to $(tan\alpha -tan{\alpha }_0)$ vs dimensionless hold-up in the flowing layer. Symbols conform to inclination angles shown in Figs. 4, 5, 6, 7. The main graph is for inclinations steeper than the angle of repose ${\alpha }_r$, while the inset is for $\alpha =31.5^{\circ }$ ($\times$) and ${32}^{\circ }$ ($+$), both $<{\alpha }_r$. Solid lines are inclined at a $1/2$ slope [Eq. (31)] and the dashed line at $3/2$ [Eq. (35)].

Figure 9

Profile of local dimensionless velocity $\text{Fr}\equiv u^{†}/\sqrt{\mathrm{\Delta }{H}^{†}}$ over one half of the channel relative to $(tan\alpha -tan{\alpha }_0)$ vs dimensionless lateral distance $y/\overline{d}$ from the channel center plane and vs dimensionless distance $\zeta \equiv z^{\prime }/\mathrm{\Delta }h$ from the free surface, counted positive upward. The symbols $\times , \square , \circ$, and $\diamond$ correspond to measurements at $\mathrm{\Delta }{H}^{†}=2.4, 3.9, 6.5$, and $8.6$, respectively. The solid line is a best fit $\text{Fr}=(tan\alpha -tan{\alpha }_0)[1+tanh(2+4z^{\prime }/\mathrm{\Delta }h)]\{u_c^{†}-(u_c^{†}-u_w^{†})exp[-(W/2-y)/\ell ]\}/(1+tanh2)$, and the gray surface plot is its extrapolation throughout the flow. Lines parallel to the Fr axis represent velocity vectors. For these measurements at $\alpha ={34}^{\circ }>{\alpha }_r$, the surface profiles have $u_c^{†}=63\pm 7$ (represented by an open circle and its error bar on the midplane), $u_w^{†}=53\pm 6$, and $\ell /\overline{d}=21.3\pm 0.9$.