Air-cushioning below an impacting wave-structured disk: Free-surface deformation and slamming load

Prior to the impact of a horizontal disk onto a liquid surface, the air underneath flows radially outward across the liquid surface to escape from below the edge of the disk. Such airflow causes the surface to be elevated near the disk edge, creating a free-surface condition that influences the details of the subsequent impact dynamics. In this work, the nature of the surface elevation under an impacting disk is investigated by modulating the forcing of the free surface: The airflow below the disk is altered by imposing a radially symmetric wave structure of varying wavelength on the impacting disk surface. Subsequently, the liquid surface deformation before impact is measured experimentally using a total internal reflection technique. The experiments provide convincing evidence that supports the argument that the surface elevation is an instability of the Kelvin-Helmholtz type. In addition, the impact force exerted on the wave-structured disks is measured using a load cell. Due to the macroscopic wave structure on the disk, the maximum impact force is significantly reduced, and the results indicate that both the free-surface deformation before impact and the way in which the impacting surface is subsequently wetted influence the maximum impact force.

Figure 1

(a) Schematic of the experimental set-up and the geometry of a 3D printed disk with radially symmetric sinusoidal wave structure. All the wave-structured disks used in the experiment have the same radius $R$ and amplitude $a_0$ but different wavelength ${\lambda }_d$. The disk edge always coincides with a crest of the wave structures. (b) (i) Reflected reference grid pattern imaged by the high-speed camera using the total internal reflection (TIR) setup when the water surface is stationary. (ii) Distorted image of the reference pattern when the water surface is perturbed due to the approaching disk.

Figure 2

(a) Schematic to derive the theoretical estimate for the air velocity profile under a flat disk and a disk with radially symmetric sinusoidal wave structure. (b) (i) Temporal evolution of the theoretical velocity profile of air under a ${\lambda }_d=17$ mm disk (solid lines) and a $R=40$ mm horizontal flat disk (dotted lines). While the air velocity increases linearly across the radial coordinates for a flat disk, velocity maxima and minima are induced under a wave-structured disk as the time before impact $\tau$ decreases. The locations of the velocity maxima and minima are closely located under the crests of the wave structure (indicated by the vertical dashed lines) on the disk. (ii) Comparison of the theoretical velocity profile of air under a $R=40$ mm flat disk and ${\lambda }_d=2$, 17, 34 mm disks at $\tau =1.25$ ms. (iii) Pressure gradient of air for the selected disks at $\tau =1.25$ ms.

Figure 3

(a) Six snapshots from the temporal deformation of the water surface before impact taken at the $y$-center plane, for disks with imprinted wavelength ${\lambda }_d$ of 2, 8.5, 12, 17, 20, and 34 mm. The results for the other disks are shown in the Supplemental Material [20]. The vertical dashed lines indicate the location of crests on the impacting disk. In the figure for ${\lambda }_d=2$ mm disk, most of the dashed lines were omitted for clarity. The red vertical dashed lines denote the disk edge as well as the crest located at that point. Note that by virtue of the impact speed used in these experiments ($U_0=1.0$ m/s), times before impact $\tau$ denoted in ms can be easily converted to the distance of the crests of the disk to the undisturbed interface in mm. (b) Cross-sectional profile of the free surface taken at the $y$-center plane, right before the disk makes first contact with the water surface ($\tau =0.033$ ms) for four selected disks (${\lambda }_d=2$, 8.5, 17, and 34 mm).

Figure 4

(a) Time evolution of the growth of the primary instability $\mathrm{\Delta }h$ under the disk edge (P1) for selected wave-structured disks. The starting time, for which $\mathrm{\Delta }h=0$, was taken as $\tau =3.33$ ms as P1 started to increase monotonically around this time for all the wave-structured disks used in the experiment. (b) Plot of the time derivative $\dot{h}$ of $\mathrm{\Delta }h$ [red solid slope line in panel (i)] taken from $\tau =0.033$ ms to 0.5 ms, for all the disks. ${\lambda }_d=0$ m corresponds to a $R=40$ mm flat disk. Error bars represent the standard deviation of $\dot{h}$ over three repetitions of the experiment for each disk.

Figure 5

(a) (i) Decomposition of the measured free-surface profile at $\tau =0.033$ ms under a ${\lambda }_d=17$ mm disk into high (noise, not shown), intermediate (instabilities) and low (overall cavity) frequencies by discrete wavelet transform. (ii) Estimation of the wavelength of instability, ${\lambda }_f$ under the disk edge (P1) and under the subsequent crest on the disk (P2) by fitting the transformed profile to a sine-curve (red-dotted lines) locally. (b) (i) Estimated wavelength ${\lambda }_f$ of P1 and P2 at $\tau =0.033$ ms for all the disks. (ii) Difference between the distance $\mathrm{\Delta }r$ between the radial locations $r_{P1}$ and $r_{P2}$ of P1 and P2, respectively, on the one hand, and the wavelength of the disks ${\lambda }_d$, on the other hand. The pink solid line represents the difference ${\lambda }_d-{\lambda }_{\text{marg}}$ between the wavelength of each disk and the most unstable wavelength, ${\lambda }_{\text{marg}}=17$ mm. (iii) Ratio of P2 to P1 as taken from the transformed profile of all the disks with different ${\lambda }_d$. Error bars show the standard deviation of the peak ratio over three repetitions of the experiment for each disk.

Figure 6

(a) Snapshots of the impact process of a disk with (i) ${\lambda }_d=2$ mm, (ii) ${\lambda }_d=17$ mm and (iii) ${\lambda }_d=34$ mm, recorded at 50 000 f.p.s. from the bottom. $t_{\text{impact}}=0$ ms is the time at which the edge of the disks made contact with the water surface. With the disk edge being the first crest for all the disks, the dashed lines in the first images of (ii) and (iii) indicate the subsequent crests of the wave structure on the disk. A quarter of the disks are shown to get a better sight of the solid-liquid contact region and the water-air interface on the disk during impact. The surface of the disk that is wetted with water appears to be darker in the images while the brighter regions indicate a solid-air contact. Refer to Ref. [20] for false colored images to better visualize the wetted and nonwetted areas. Videos of the impact process featuring the whole disk as viewed from the bottom are also included in the Supplemental Material [20].

Figure 7

(a) Time evolution of the force experienced by the disk during impact, measured for (i) ${\lambda }_d=2$ mm, (ii) ${\lambda }_d=17$ mm, and (iii) ${\lambda }_d=34$ mm disks at $U_0=1.0$ m/s. The signals are centered around $t=0$ ms when the impact force reaches its first peak for each disk. (b) First peak impact force measured for all the disks with different ${\lambda }_d$. Error bars show the standard deviation of the first peak impact force over 10 repetitions of the experiment.

Figure 8

(a) Schematic of the half-body potential flow to estimate the inertial wetting velocity on a single wave structure $V_{\text{surf}}$. (b) Velocity on the surface of the wave structure $V_{\text{surf}}$ of several selected wavelengths at different heights of penetration into the voids $z$.