Air-cushioning effect and Kelvin-Helmholtz instability before the slamming of a disk on water

The macroscopic dynamics of a droplet impacting a solid is crucially determined by the intricate air dynamics occurring at the vanishingly small length scale between droplet and substrate prior to direct contact. Here we investigate the inverse problem, namely, the role of air for the impact of a horizontal flat disk onto a liquid surface, and find an equally significant effect. Using an in-house experimental technique, we measure the free surface deflections just before impact, with a precision of a few micrometers. Whereas stagnation pressure pushes down the surface in the center, we observe a liftup under the edge of the disk, which sets in at a later stage, and which we show to be consistent with a Kelvin-Helmholtz instability of the water-air interface.

Figure 1

(a) Experimental setup. A light source illuminates a reference pattern $O$ with printed lines, which reflects from the water-air interface into a high-speed camera. At high incident angle $\theta$, the water-air interface acts as a mirror, and the camera records the mirror image $O^{\prime }$, as indicated by the light rays (in gray). (b) Image of a stationary air-water interface reflecting the reference pattern. (c) Deformed air-water interface reflecting a distorted image of the reference pattern.

Figure 2

(a) Free surface profiles $h\left(r\right)$, azimuthally averaged about the disk center, shown at different times $\tau$ before impact (occurring at $\tau =0$) from an experiment with an 8-cm-wide disk (edge shown by vertical gray line) approaching the free surface at 1 m/s. (b) Free surface $\tau =0.033$ ms before impact for a range of disk diameters as indicated in the legend [shared with panel (c)]. (c) Nondimensionalized water-surface profiles at nondimensionalized time $\tau V/D=0.01$ before impact for the same disk sizes. See Supplemental Material [29] (Movies 2 and 3) for videos of the time evolution of the surface profile for $D=50$ and 80 mm.

Figure 3

Nondimensionalized cavity depth $h_{\text{min}}/D$ (measured at $r=0$) as a function of the nondimensionalized time before impact $\tau V/D$, with $D$ varying between 30 and 120 mm. The solid and dotted lines are the two-fluid boundary-integral simulation and analytical calculations results, respectively, from Peters et al. [25].

Figure 4

The instantaneous, dimensionless vertical deflection velocity $\mathrm{\Delta }h/D=\left(\partial h/\partial t\right)\left(\mathrm{\Delta }\tau /D\right)$ of the surface profile at dimensionless times (a) $\tau V/D=0.005$ and (b) $\tau V/D=0.003$ before impact. $\mathrm{\Delta }\tau =1/30000\text{s}$ is the time between two successive images in an experiment. The profiles are plotted with radial coordinates $r$ shifted by $D/2$ such that they are all centered under the disk edge.

Figure 5

(a) Time evolution before impact of the maxima of dimensionless free surface deflection velocities $\mathrm{\Delta }h/D=\left(\partial h/\partial t\right)\mathrm{\Delta }\tau /D$ for a disk of width 16 cm. The free surface starts to accelerate towards the disk close to ${\tau }_{\text{crit}}V/D\approx 0.11$ (gray dashed-dotted line), which is estimated by tracing the growth rate (using dotted lines). (b) ${\tau }_{\text{crit}}V/D$ measured over several $D$ are shown. The dashed line shows the mean of the measurements, while the gray box shows the range of standard deviations of the measurements.