Effect of finite container size on granular jet formation

When an object is dropped into a bed of fine, loosely packed sand, a surprisingly energetic jet shoots out of the bed. In this work we study the effect that boundaries have on the granular jet formation. We did this by (i) decreasing the depth of the sand bed and (ii) reducing the container diameter to only a few ball diameters. These confinements change the behavior of the ball inside the bed, the void collapse, and the resulting jet height and shape. We map the parameter space of impact with Froude number, ambient pressure, and container dimensions as parameters. From these results we propose an explanation for the thick-thin structure of the jet reported by several groups ([J. R. Royer et al., Nat. Phys. 1, 164 (2005)], [G. Caballero et al., Phys. Rev. Lett. 99, 018001 (2007)], and [J. O. Marston et al., Phys. Fluids 20, 023301 (2008)]).

Figure 1

Schematic representation of the impact of a ball into a sand bed, indicating the time and length scales that play an important role in the analysis of the experimental work in this paper, as described in the main text.

Figure 2

(Color online) Setup: (a) perspex container, $14\times 14\times 100{\text{cm}}^3$, (b) pneumatic release mechanism, (c) Photron Ultima APX-RS, (d) two light sources with diffusing plate, (e) pressurized, dry air source, (f) computer, and (g) vacuum pump with pressure gauges.

Figure 3

(I) Laser profilometer. A diode laser sheet (a) is directed onto the surface at an angle $\theta$. Using a mirror (b) and a high-speed camera (c), images of the surface are recorded. (II) Schematic view of the resulting surface. The dashed line represents the laser sheet when the surface is flat and the continuous line the laser sheet when the surface is perturbed. The local deviation $\delta x=x\left(y\right)-x_l$ of the laser sheet is related to the vertical coordinate $\delta z=z\left(y\right)$ of the surface. (d) is the center of the cavity, from which the cavity radius $R\left(z\right)$ can be deduced.

Figure 4

Influence of the height of the sand bed $h_{\text{bed}}$ on the shape and height of the jet for $\text{Fr}=70$ and $p=1\text{bar}$: Images of the jet, taken at 0.12 s after the ball impact for four different bed heights, decreasing from left to right. Below a threshold there is a clear change in height and width of the jet.

Figure 5

(Color online) Initial velocity of the jet, $v_{\text{jet}}$ as a function of the height of the sand bed, $h_{\text{bed}}$ for $\text{Fr}\sim 70$ and $p=1000\text{mbar}$ $(+)$ and $p=100\text{mbar}$ (○). There is a sharp threshold below which the initial jet velocity rapidly decreases. The dashed lines represent the undisturbed values of $v_{\text{jet}}$, measured in a deep bed $\left(h_{\text{bed}}=18.8d\right)$.

Figure 6

(Color online) The time $t_{\text{erup}}$ when the granular eruption at the surface starts is plotted as a function of the height of the sand bed $h_{\text{bed}}$, for $\text{Fr}=70$ and $p=1\text{bar}$ (black open circles). Measurement points with $t_{\text{erup}}=0$ correspond to those cases where no eruption was observed. The experimental regimes with and without eruption are separated by the vertical black lines. The grey region represents the region where no air bubble is entrapped. The thin blue and red lines represent the different time scales that are involved in the problem $t_1$ is the time the air bubble needs to reach the surface (black thin line) and $t_2$ is the time the air bubble needs to diffuse within the sand bed (red thin line). When $t_1$ is smaller than $t_2$, an eruption is expected; this is depicted by the continuous thick blue line. The different regions obtained from the timescale argument qualitatively correspond to the experimental results. More details about the way in which $t_1$ and $t_2$ are estimated are provided in the main text and in footnote [23].

Figure 7

(Color online) (top) Depth of the ball $z\left(t\right)$ (a) and its velocity $v\left(t\right)$ (b) as a function of time after impact for $\text{Fr}=25$, $p=1000\text{mbar}$, $D=10\text{cm}$ (○) and $D=6\text{cm}$ (◻). The lines correspond to a fit using Eq. (1) with $\kappa =4.525\text{N}/\text{m}$ and $\alpha =0.132\text{kg}/\text{m}$ for $D=10\text{cm}$ and $\kappa =1.695\text{N}/\text{m}$ and $\alpha =0.118\text{kg}/\text{m}$ for $D=6\text{cm}$. (bottom) Depth of the ball $z\left(t\right)$ (c) and its velocity $v\left(t\right)$ (d) as a function of time after impact $t$ for $\text{Fr}=75$, $p=50\text{mbar}$ $D=4.2\text{cm}$ (○) and $D=12.5\text{cm}$ (◻). Again, the lines correspond to a fit using Eq. (1) with $\kappa =14\text{N}/\text{m}$ and $\alpha =0.281\text{kg}/\text{m}$ for $D=4.2\text{cm}$ and with $\kappa =13.5\text{N}/\text{m}$ and $\alpha =0.111\text{kg}/\text{m}$ for $D=12.5\text{cm}$. Within the smallest container, and only at low pressure, we observe anomalous behavior: The ball reaches a plateau in which the velocity remains constant before going to zero again at larger times. Clearly, the model fails to describe the data in this case.

Figure 8

(Color online) Final depth $z_{\text{f}}$ as a function of the container diameter $D$, at different pressures, for (a) $\text{Fr}=25$ and (b) $\text{Fr}=75$. The final depth is divided by the final depth for the unconfined case in order to emphasize the deviations due to the proximity of the boundaries. The dashed lines are a guide to the eye to separate the different pressures.

Figure 9

(Color online) (a) $\kappa$ and (b) $\alpha$ as a function of cylinder diameter $D$ for different pressures $p$. For almost all values of $D$ and $p$ variations of both $\kappa$ and $\alpha$ are within the measurement error and each point is obtained from an average over a range of Froude numbers from 25 to 100. Only for the smallest container $\left(D/d=2.6\right)$ and the lowest pressure (50 mbar), there is a strong dependence of $\kappa$ and $\alpha$ on the Froude number; the model is not valid in this situation. Plot b) reveals that for large Fr the quadratic drag takes over for small cylinder diameters leading to less intrusion of the ball [Fig. 8b].

Figure 10

(Color online) Dynamics of the cavity collapse at closure depth for two container diameters $D=4.2\text{cm}$ (◻) and $D=10\text{cm}$ (○). Here, $\text{Fr}=70$ and $p=1\text{bar}$. The time has been rescaled by multiplying with a factor $2\sqrt{g{z}_c}/d$ in order to show the results in a single plot. The continuous line correspond to a fit using the 2D Rayleigh-Plesset equation [Eq. (2)].

Figure 11

(Color online) (a) Closure depth $z_c$ as a function of the container diameter $D$ for different pressures. (b) Closure time $t_c$ as a function of the container diameter $D$ for different pressures. For all measurements $\text{Fr}=70$.

Figure 12

(Color online) The jet height, $h_{\text{jet}}$ as a function of the container diameter $D$ for $\text{Fr}=25$ (a) and $\text{Fr}=50$ (b) at different ambient pressures. The jet height is divided by the jet height in the unconfined case in order to see the deviations due to the proximity of the boundaries. For all pressures and Froude numbers the jet height increases with increasing container diameter. The dashed lines are a guide to the eye to separate the measurement series at different pressures.

Figure 13

(Color online) The jet height $h_{\text{jet}}$ as a function of the container diameter $D$ for $\text{Fr}=100$ and different ambient pressures. Again, there is a clear change in jet height as function of container diameter. Measurements at the highest Froude numbers are not possible due to the surface seal (see text). The dashed lines are a guide to the eye to separate the measurement series at different pressures.

Figure 14

(Color online) (a) Phase diagram for the granular eruption at $\text{Fr}=100$ as a function of the pressure $p$ and the container diameter $D$. In both plots red open circles indicate parameter values where an eruption was absent, whereas blue plus signs stand for parameter values with an eruption. (b) The same phase diagram, now as a function of the volume of the entrapped air bubble $\left(\left[z\left(t_c\right)-z_{\text{c}}\right]/d\right)$ and the container diameter $D$. The latter plot clearly indicates that the presence of the eruption is a function of the entrapped air bubble size only [26].

Figure 15

Typical snapshots of the three distinct jet shapes observed in experiment: (1) Normal jet (for $D=10\text{cm}$, $\text{Fr}=100$, and $p=1000\text{mbar}$); (2) Thick-thin structure with sharp shoulder (for $D=8.5\text{cm}$, $\text{Fr}=100$, and $p=100\text{mbar}$); (3) Thick-thin structure with a transition (for $D=10\text{cm}$, $\text{Fr}=50$, and $p=50\text{mbar}$). All snapshots show the fully developed shape of the jet at its maximum height. The snapshots are not on the same scale.

Figure 16

Three snapshots of the shape of the jet at different values of the height $h_{\text{bed}}$ of the sand bed, taken 120 ms after the ball impact for $p=100\text{mbar}$ and $\text{Fr}=70$. For $h_{\text{bed}}=4.1d$ there is a clear thick-thin structure (with a transition region), which gradually disappears when the bed height is decreased to $3.4d$ and $2.6d$.

Figure 17

(Color online) Phase diagram of the observed jet shapes as a function of the ambient pressure $p$ and the container diameter $D$ for three different Froude numbers: (a) $\text{Fr}=25$; (b) $\text{Fr}=50$; and (c) $\text{Fr}=100$. The dashed lines are a guide to the eye to separate the different regions in the phase diagrams.

Figure 18

Schematic drawing of the proposed mechanism leading to the thick-thin structure. In case (a), the second collapse happens before a certain threshold time, such that the thickness of the layer of sand from the first collapse still is thin enough to be pushed up by the second jet and a thick-thin structure emerges. In case (b) we are above the threshold: The second jet collides with a thick layer of sand and is unable to disturb the formation of the first jet.

Figure 19

(Color online) Phase diagram with on the vertical axis the left hand side of Eq. (9) and on the horizontal axis the container diameter $D$. The plot contains all measurements from Fig. 17. Short, green dashes indicate normal jets, intermediate, blue dashes the thick-thin structure with a transition, and long, red dashes thick-thin structures with a shoulder.