Granular collapse in a fluid: Different flow regimes for an initially dense-packing

Laboratory experiments of the granular collapse of an initially dense-packing column in a fluid are reported. By extracting the temporal evolution of the granular material height profile, both the dynamics of the granular flow and the final deposit are characterized depending on the Stokes number $\text{St}$, based on a dissipative process at the grain scale, the grain-fluid density ratio $r$, and the aspect ratio $a$ of the initial column. A full description of the granular collapse including the transient dynamics of the flow and the characterization of the final shape deposit is proposed. The main contribution of the present study is to provide (i) the $\text{St}$ dependence on the granular collapse beyond the effect of the aspect ratio $a$ already reported in previous studies, (ii) the characterization of granular flow regimes in the $({(d/H_i)}^{1/2}\text{St},{(d/H_i)}^{1/2}r)$ plane, where $d/H_i$ is the particle diameter to initial column height ratio, and (iii) simple correlations to describe the granular collapse which would be of interest for geophysical purposes.

Figure 1

Sketch of the experimental setup for the initial condition (granular column maintained by a vertical gate) and the final state (granular deposit).

Figure 2

Granular-fluid flow regimes in the $(\text{St},r)$ plane according to (5). Stars and circles correspond to the experimental series performed in the present study and in Ref. [7], respectively.

Figure 3

Snapshots of the granular collapse for regime FF, regime V, and regime I for three different aspect ratios and at three different times.

Figure 4

(a) Dimensionless runout length $(L_f-L_i)/L_i$ (inset $L_f/L_i$) and (b) final height $H_f/L_i$ (inset $H_f/H_i$) as a function of the aspect ratio $a$ for regime FF (black), regime I (blue), regime VI (red), and regime V (brown). Circles and squares correspond to experiments performed with reservoir widths of $L_i=10$ and $3\mathrm{cm}$, respectively.

Figure 5

Dimensionless runout $L_f/L_i$ as a function of $\text{St}$ for $r\sim 1.5$ and $a=[0.5,9]$. Circles and squares correspond to experiments performed with reservoir widths of $L_i=10$ and $3\mathrm{cm}$, respectively.

Figure 6

Temporal evolution of the front position $x_f-L_i$ for regime V; the inset shows a comparison for regime FF (black), regime I (blue), regime VI (red), and regime V (brown), with $a=1$. The dashed lines represent the runout length $L_f-L_i$. The trigger time $T_t$ and the time $t_{95}$ are indicated.

Figure 7

(a) Trigger time $T_t$ as a function of the initial height $H_i$ for regime FF (black), regime I (blue), regime VI (red), and regime V (brown). The dashed lines represent the mean value $\langle T_t\rangle$ for each regime. (b) Dimensionless trigger time $\langle T_t\rangle /{\tau }_{ff}$, with ${\tau }_{ff}={(2{\rho }_{p}d/\mathrm{\Delta }\rho g)}^{1/2}$, as a function of the Stokes number $\text{St}$. The white and gray areas correspond to experiments with $r\sim 1.5$ and 45, respectively.

Figure 8

Dimensionless collapse time $t_{95}-T_t$ as a function of the normalized initial height $H_i/d$ for regime FF (black), regime I (blue), regime VI (red), and regime V (brown). The characteristic times $T_{\text{FF}}$, $T_{\text{I}}$, and $T_{\text{V}}$ are defined according to (6). Circles and squares correspond to experiments performed with reservoir widths of $L_i=10$ and $3\mathrm{cm}$, respectively.

Figure 9

Maximum front velocity $U_m$ normalized by different characteristic velocities $U_{\text{FF}}$, $U_{\text{I}}^{\infty }$, and $U_{\text{V}}^{\infty }$ defined according to (7) as a function of the normalized initial height $H_i/d$ for regime FF (black), regime I (blue), regime VI (red), and regime V (brown).

Figure 10

Granular flow regimes in the $({(d/H_i)}^{1/2}\text{St},{(d/H_i)}^{1/2}r)$ plane with $\text{St}$ the Stokes number and $r$ the grain-fluid density ratio defined according to (5). Symbols (colors and sizes) correspond to the minimum dimensionless collapse time $T^{-}$ according to (8) and determined from Fig. 8 with, in particular, black symbols for $T^{-}=(t_{95}-T_t)/T_{\text{FF}}$, blue symbols for $T^{-}=(t_{95}-T_t)/T_{\text{I}}$, and brown symbols for $T^{-}=(t_{95}-T_t)/T_{\text{V}}$. The size indicates the deviation of $T^{-}$ from 1 (see the legend).

Figure 11

(a) Maximum front velocity $U_m$ as a function of the free-fall velocity $U_{\text{FF}}={(2\mathrm{\Delta }\rho g{H}_i/{\rho }_p)}^{1/2}$ for regime FF (black), regime I (blue), regime VI (red) and regime V (brown). (b) Evolution of $\zeta =U_m/U_{\text{FF}}$ as a function of the Stokes number $\text{St}$. The parameter $\zeta$ is obtained from the prefactor of the linear fit [dashed lines in Fig. 11]. White and gray areas correspond to $r\sim 1.5$ and 45, respectively.

Figure 12

Sketch of a final trapezoidal deposit with the runout length $L_f$ and the final height $H_f$, the angles ${\alpha }_s$ and ${\alpha }_f$ at the summit and at the foot, respectively, and $x_c$ corresponding to the longitudinal position of transition from a constant height profile to a decreasing height profile.

Figure 13

Height profile $h/H_f$ as a function of $(x-x_c)/(L_f-x_c)$ for (a) regime FF, (b) regime I, (c) regime VI, and (d) regime V. The dashed lines represent $h/H_f=1-(x-x_c)/(L_f-x_c)$.

Figure 14

Dimensionless vertical position of the mass center $(2y_f^G-H_i)/H_i$ as a function of its horizontal position $(2x_f^G-L_i)/L_i$ for regime FF (black), regime I (blue), regime VI (red), and regime V (brown). Symbols correspond to the final state $((2x_f^G-L_i)/L_i,(2y_f^G-H_i)/H_i)$ while the dashed lines correspond to the temporal trajectory $((2x^G-L_i)/L_i,(2y^G-H_i)/H_i)$ of the center of mass. The inset corresponds to $2y_f^G/H_i$ as a function of $2x_f^G/L_i$ and the dashed line is $2y_f^G/H_i=(8/9)(L_i/2x_f^G)$, which is the position of the mass center of any triangle having a surface $H_i{L}_i$.

Figure 15

Local angles ${\alpha }_s$ at the summit (closed symbols) and ${\alpha }_f$ at the foot (open symbols) as a function of the aspect ratio $a$ for (a) regime FF, (b) regime I, (c) regime VI, and (d) regime V.

Figure 16

Average values $\langle {\alpha }_f\rangle$ (open symbols) and $\langle {\alpha }_s\rangle$ (closed symbols) as a function of $\text{St}$.

Figure 17

(a) Dimensionless runout length $L_f/L_i$ and (b) final height $H_f/L_i$ as a function of the aspect ratio $a=H_i/L_i$ for regime FF (black), regime I (blue), regime VI (red), and regime V (brown). Symbols are experiments and lines are predictive models (12) (solid lines) and (13) (dashed lines). In the latter cases, ${\mu }_d=tan{\alpha }_r$, where ${\alpha }_r=22\circ$ is the angle of repose, and ${\mu }_{\text{St}}=tan\langle {\alpha }_f\rangle$, where $\langle {\alpha }_f\rangle$ is the mean angle at the foot of the deposit and determined from Fig. 16.

Figure 18

Height profile $h/h_b$ as a function of $x/L_f$ for regime I. The aspect ratio is in the range $a=[1,9]$ (from black to light gray, respectively). The inset shows $h_b/L_i$ as a function of $a$. No bump is observed in the gray area.

Figure 19

(a) Dimensionless runout length $L_f/L_i$ and (b) final height $H_f/L_i$ as a function of the aspect ratio $a=H_i/L_i$. Pluses correspond to experiments performed in the present study, with $\varphi \sim 0.64$, for regime FF (black) and regime V (brown), while circles are extracted from Ref. [7] with $\varphi =0.60$ (closed) and $\varphi =0.55$ (open) in the case of a granular collapse immersed in a viscous fluid.