Noncontinuous Froude Number Scaling for the Closure Depth of a Cylindrical Cavity

A long, smooth cylinder is dragged through a water surface to create a cavity with an initially cylindrical shape. This surface void then collapses due to the hydrostatic pressure, leading to a rapid and axisymmetric pinch-off in a single point. Surprisingly, the depth at which this pinch-off takes place does not follow the expected ${\mathrm{\text{Froude}}}^{1/3}$ power law. Instead, it displays two distinct scaling regimes separated by discrete jumps, both in experiment and in numerical simulations (employing a boundary integral code). We quantitatively explain the above behavior as a capillary wave effect. These waves are created when the top of the cylinder passes the water surface. Our work thus gives further evidence for the nonuniversality of the void collapse.

Figure 1

(a)–(c) Snapshots of the cavity for $\mathrm{Fr}=11.5$. The area designated by the yellow rectangle in (a) is magnified in (b) illustrating the rippled surface of the cavity walls. Red lines show the numerical simulation. In (a) and (b) video frames are connected to simulation time by matching the cylinder position; for (c) it was more convenient to match the cavity closure times instead. (d)–(g) The transition from one closing ripple to the other. At $\mathrm{Fr}=18.8$, (d) and (e), the upper of the two marked ripples closes. At $\mathrm{Fr}=20.6$, (f) and (g), it is the lower of the two ripples that pinches.

Figure 2

The experimental closure depth as a function of the Froude number. The asymptotic regime for low Froude numbers scales with ${\alpha }_1\approx 0.1$ (red solid line). Only in the limit of high Froude numbers does the data not contradict an exponent of $\frac{1}{3}$ (blue dashed line). For the intermediate regime the experiments show no systematic behavior.

Figure 3

(a) Capillary waves for $V=3\mathrm{m}/\mathrm{s}$ ($\mathrm{Fr}=45.9$) as obtained from the simulation. Wave crests are marked by red circles and allow an estimate of the dominant wavelength $\lambda =2\pi /k$ at a given position. The inset compares these wavelengths for the downward (blue circles) and outward (black crosses) waves to the theoretically expected behavior for capillary waves (red line). (b) The natural cavity dynamics (red solid) as a superposition of two hypothetical settings: without surface tension (black dashed) and without gravity (blue dotted) for $V=1\mathrm{m}/\mathrm{s}$ ($\mathrm{Fr}=5.1$).

Figure 4

Comparison of the experimental closure depth (black crosses) with the numerical data (blue diamonds). The insets illustrate the shape of the cavity at pinch-off for one representative of each regime (axes are the same for all insets). The regimes are determined by which local minimum first meets its counterpart on the central axis.

Figure 5

The numerical closure depth as a function of the Froude number and corresponding scaling exponents for different surface tension coefficients: ${\sigma }_1=728\mathrm{mN}/\mathrm{m}$ (black circles), ${\sigma }_2=72.8\mathrm{mN}/\mathrm{m}$ (blue diamonds), ${\sigma }_3=7.28\mathrm{mN}/\mathrm{m}$ (red crosses), and ${\sigma }_4=0$ (magenta squares). For $\sigma =0$ no data at low Froude numbers are shown, since axial velocity components during the collapse prohibit the application of the ${\mathrm{Fr}}^{1/3}$ theory. The inset shows the onset of the high Froude number regime as a function of the Bond number with the red line depicting the expected scaling law ${\mathrm{Fr}}_{\mathrm{trans}}\sim {\mathrm{Bo}}^{-3/4}$.