Role of Air in Granular Jet Formation

A steel ball impacting on a bed of very loose, fine sand results in a surprisingly vigorous jet which shoots up from the surface of the sand [D. Lohse et al., Phys. Rev. Lett. 93, 198003 (2004)]. When the ambient pressure $p$ is reduced, the jet is found to be less vigorous [R. Royer et al., Nature Phys. 1, 164 (2005)]. In this Letter we show that $p$ also affects the rate of penetration of the ball: Higher pressure increases the rate of penetration, which makes the cavity created by the ball close deeper into the sand bed, where the hydrostatic pressure is stronger, thereby producing a more energetic collapse and jetting. The origin of the deeper penetration under normal ambient pressure is found to lie in the extra sand fluidization caused by the air flow induced by the falling ball.

Figure 1

(a) Maximum height of the jet $h$, and (b) final depth of the ball $z_{\mathrm{final}}$, as a function of the ambient pressure $p$ for different Froude numbers: $\mathrm{Fr}=7$ (◂), 18 (★), 35 (▪), and 132 (●). $D=1.6\mathrm{cm}$ is the diameter of the impacting ball. (c) Maximum height $h$ of the jet versus the penetration depth $z_{\mathrm{final}}$ of the ball. The lines in this plot are guides to the eye.

Figure 2

(a) Experimental trajectories of the ball in the sand (thick lines) for $\mathrm{Fr}=35$ compared to the drag force model, Eq. (1) (dashed lines), with $\kappa$ the only fitting parameter. The vertical dotted line indicates the measured closure time $t_c$ of the cavity. The black dots mark the location $z_i$ of the ball at closure. The star indicates the calculated closure time $t_c$ and closure depth $z_c$ using the model [Eqs. (1, 2)]. (b) Location of the ball at closure $z_i$ (squares) obtained from the top figure, and the closure depth $z_c$ (diamonds) predicted by the model, both as a function of pressure. In order to show that the two quantities follow the same trend, we plotted them shifted and normalized such that they go from 0 at 25 mbar to 1 at 1 bar, with $z_c(25\mathrm{mbar})=1.43D$, $z_c(1\mathrm{bar})=1.57D$, $z_i(25\mathrm{mbar})=2.68D$, and $z_i(1\mathrm{bar})=4.33D$.

Figure 3

(a) Final depth as a function of the Froude number at different pressures $p=1$ (●), 0.4 (★), and 0.025 bar (□). The lines are fits using the prediction of $z_{\mathrm{final}}$ by the force model, where the drag force coefficient $\kappa$ is the only free parameter. The values of $\kappa$ that result from the fits are plotted as circles in (b), suggesting a relation of the form $\kappa \propto p^{-1/2}$. The stars in (b) result from the fitting of Eq. (1) to the experimental trajectories in Fig. 2a (dashed lines).

Figure 4

$\Delta V=V(t)-V(t=0)$ for $\mathrm{Fr}=132$, where $V(t)$ is the total volume occupied by the sand bed and $t=0$ is the moment of impact. $V_b$ is the volume of the impacting ball with $D=1.6\mathrm{cm}$. Each curve is the average of three independent experiments. The vertical lines show, in chronological order and for 0.4 bar, the closure time obtained with the model, and the first time at which the jet can be seen ($\approx 5\mathrm{cm}$ above the surface). The jet tip reaches its maximum height at $\approx 330\mathrm{ms}$.