Crater formation during raindrop impact on sand

After a raindrop impacts on a granular bed, a crater is formed as both drop and target deform. After an initial, transient, phase in which the maximum crater depth is reached, the crater broadens outwards until a final steady shape is attained. By varying the impact velocity of the drop and the packing density of the bed, we find that avalanches of grains are important in the second phase and hence affect the final crater shape. In a previous paper, we introduced an estimate of the impact energy going solely into sand deformation and here we show that both the transient and final crater diameter collapse with this quantity for various packing densities. The aspect ratio of the transient crater is however altered by changes in the packing fraction.

Figure 1

(a) Experimental setup of droplet impact on a granular bed. The height profile, $z_c(x,y,t)$, is measured with laser sheets and high speed camera. (b) Relevant length scales of the crater at the moment it reaches its maximum depth and at its final shape.

Figure 2

Height profiles, $z_c(x,y,t^{\infty })$, obtained by scanning the crater after impact. Typical liquid-grain residues are denoted as truffle (a), doughnut (b), and pancake (c), and of each an example picture is added, where the liquid was colored red, to increase contrast. The bar at the top left of each figure represents 2 mm. White areas in the contour plot lack height information as either laser or camera is blocked due to a sharp height change (as the scan is executed with a translation of the substrate only).

Figure 3

Various radial height profiles $z_c(r,t^{\infty })$ of the final crater are shown for increasing impact velocities and, from left to right, for increasing packing fraction. Note that the center region is occupied by the granular residue, which for all packing fractions changes from a doughnut, through a truffle to a pancake shape for increasing velocity.

Figure 4

Dimensionless final crater diameter is plotted against (a) the kinetic energy $E_k$ of the impacting droplet; (b) the estimated amount of kinetic energy transferred to the sand $E_s=[Z_c^{*}/\left(Z_c^{*}+D_0/2\right)]E_k$. The inset shows the same data on a doubly logarithmic scale. The dashed line indicates $D_c^{\infty }/D_0\propto E_s^{0.16}$.

Figure 5

Transient and final radial height profiles $z_c(r,t^{*})$ (dashed lines) and $z_c(r,t^{\infty })$ (solid lines) for two impact velocities and three different packing fractions. As the droplet impact location is usually at a distance from the laser lines, data for the transient height profile near the center ($r=0$) is missing.

Figure 6

(a)–(c) Relevant length scales nondimensionalized with the initial drop diameter $D_0$ are plotted against packing fraction in (a) the maximum depth $Z_c^{*}$, in (b) the crater diameter $D_c^{*}$ at the moment the maximum crater depth is reached, and in (c) the maximum diameter $D_c^{\infty }$ of the final crater. The legend in (a) corresponds to all three subfigures. (d)–(e) The transient diameter $D_c^{*}/D_0$ is plotted against the energy $E_s$ transferred to the sand (d) and against the energy $E_d$ transferred to the droplet (e). Clearly $E_s$ does collapse the data, whereas $E_d$ does not. For both figures the color bar at the top indicates the packing fraction.

Figure 7

Final crater angle ${\theta }_c^{\infty }$ ($◯$) and the transient crater angle ${\theta }_c^{*}$ ($\blacksquare$) at the moment the maximum crater depth is reached, measured around the maximum crater slope, and plotted against the packing fraction ${\varphi }_0$. The first varies around a slightly smaller value than the angle of repose of ${28}^{\circ }\pm 2^{\circ }$ indicated by the horizontal solid line.

Figure 8

Differential volume change $\mathrm{\Delta }V$ between the final $z_c(r,t^{\infty })$ and transient $z_c(r,t^{*})$ crater profiles are shown per radial distance for increasing impact velocity. It is denoted as $\mathrm{\Delta }V=2\pi r\mathrm{\Delta }r[z_c(r,t^{\infty })-z_c(r,t^{*})]$ with $\mathrm{\Delta }r=0.18\mathrm{mm}$. The packing fraction is increasing from (a) to (c).

Figure 9

Ratio between the maximum crater depth $Z_c^{*}$ and the crater diameter at the moment at which the maximum crater depth is reached, $D_c^{*}$, is plotted against packing fraction. The light (green) area shows the location of the data cloud for the ratio of $Z_c^{*}$ to the final crater diameter $D_c^{\infty }$, i.e., $Z_c^{*}/D_c^{\infty }$.