Dynamics of grain ejection by sphere impact on a granular bed

The dynamics of grain ejection consecutive to a sphere impacting a granular material is investigated experimentally and the variations of the characteristics of grain ejection with the control parameters are quantitatively studied. The time evolution of the corona formed by the ejected grains is reported, mainly in terms of its diameter and height, and favorably compared with a simple ballistic model. A key characteristic of the granular corona is that the angle formed by its edge with the horizontal granular surface remains constant during the ejection process, which again can be reproduced by the ballistic model. The number and the kinetic energy of the ejected grains are evaluated and allow for the calculation of an effective restitution coefficient characterizing the complex collision process between the impacting sphere and the fine granular target. The effective restitution coefficient is found to be constant when varying the control parameters.

Figure 1

Sequence of side-view images separated by 30 ms (with first image corresponding to $t=10\text{ms}$), showing the typical time evolution of ejected grains after the impact of a steel sphere of radius $R=6.75\text{mm}$ dropped from the height $h=30\text{cm}$ $\left(U_c=2.45\text{m}/\text{s},E_c=30\text{mJ}\right)$. To give the scale, a black disk of the same radius as the sphere is shown in the first frame.

Figure 2

(Color online) Time evolution of the total apparent surface of ejected grains $A_{\text{tot}}$ for an impacting steel sphere of radius $R=7.55\text{mm}$ dropped from different heights $h=13$, 23, 33, 43, 48, and 58 cm (from bottom to top).

Figure 3

(Color online) Geometrical parameters characterizing the corona of ejected grains: the bottom and the top diameters $D_b$ and $D_t$ of the corona, its height $H$, and its edge angle $\theta$. They are automatically extracted from the contour of the corona (see text).

Figure 4

(Color online) Time evolution of (a) the height $H$, (b) the top diameter $D_t$, (c) the bottom diameter $D_b$, and (d) the angle of the corona $\theta$ for the same experimental parameters as in Fig. 2. The horizontal line in (d) is the time and ensemble average of $\theta \left(t\right)$ for the 65 experiments.

Figure 5

(Color online) Time evolution of the apparent surface area of the corona $A_{\text{cor}}$ for the same experimental parameters as in Fig. 2.

Figure 6

(Color online) (a) Time $t_{A_{\text{cor}\text{max}}}$ for maximal value of the corona apparent area $A_{\text{cor}}$ as a function of time $t_{A_{\text{tot}\text{max}}}$ for maximal value of the total apparent area $A_{\text{tot}}$. The solid line is of slope 1. (b) Maximal values $A_{\text{cor}\text{max}}$ of $A_{\text{cor}}$ as a function of maximal values $A_{\text{tot}\text{max}}$ of $A_{\text{tot}}$. The solid line is a linear fit. Data symbols are for the 65 experiments with different dropping heights $h$ for impacting spheres of radius $R=5.15$ (○), 6.75 (◻), 7.55 (◇), and 9.50 (▽).

Figure 7

(Color online) (a) Ballistic model for grains ejected instantaneously from the same position $r_0=10\text{mm}$ in the same direction making the angle $\alpha =53°$ with the horizontal, with different velocity amplitudes. Trajectories are drawn in dashed lines. At any time grains align along a straight line forming the corona edge of angle $\theta =\alpha$ drawn in continuous lines. (b) Time evolution of the angle $\theta$ of the cone edge if grains are ejected during a finite time $\delta t=5\text{ms}$ in the direction $\alpha =53°$, with a linear, quadratic, or exponential decrease in ejection velocity from the initial value $u_0=1\text{m}/\text{s}$.

Figure 8

(Color online) Rescaled data of (a) heights $\left[H-H\left(0\right)\right]/\left[H_{\text{max}}-H\left(0\right)\right]$, (b) top diameter $D_t/D_t\left(0\right)$, and (c) bottom diameter $D_b/D_b\left(0\right)$ as a function of rescaled times appearing in Eq. (3) for all of the 65 experiments.

Figure 9

(Color online) Fitted ejection angle $\alpha$ as a function of the impact energy $E_c$ for the 65 experiments (same symbols as in Fig. 6).

Figure 10

(Color online) Variations of ejection parameters with the impact parameters: (a) Ejection velocity $u_0$ as a function of impact velocity $U_c$ and (b) kinetic energy of one grain $m{u}_0^2/2$ as a function of impact energy $E_c$ and the best power-law fit (4). Same symbols as in Fig. 6.

Figure 11

(Color online) (a) Number of ejected grains $N_{ej}$ as a function of impact energy $E_c$ for all experiments, and best power-law fit given in Eq. (8). (b) Kinetic energy of ejecta $E_{ej}$ as a function of impact energy $E_c$ and best power-law fit given in Eq. (9). Same symbols as in Fig. 6.