^{1,2}, Stephan Gekle

^{1,3}, Detlef Lohse

^{1}and Devaraj van der Meer

^{1}

### Abstract

We experimentally study the airflow in a collapsing cavity created by the impact of a circular disc on a water surface. We measure the air velocity in the collapsing neck in two ways: Directly, by means of employing particle image velocimetry of smoke injected into the cavity and indirectly, by determining the time rate of change of the volume of the cavity at pinch-off and deducing the air flow in the neck under the assumption that the air is incompressible. We compare our experiments to boundary integral simulations and show that close to the moment of pinch-off, compressibility of the air starts to play a crucial role in the behavior of the cavity. Finally, we measure how the air flow rate at pinch-off depends on the Froude number and explain the observed dependence using a theoretical model of the cavity collapse.

We acknowledge the Netherlands Organisation for Scientific Research (NWO) for financial support through the Spinoza program.

I. INTRODUCTION

II. EXPERIMENTAL SETUP

III. GEOMETRIC APPROACH

A. Cavity volume

B. Air flow rate

1. An analytical argument

IV. FLOW VISUALIZATION

A. Correlation technique

V. THE ROLE OF COMPRESSIBILITY

A. Stagnation point

B. Air flow rate

VI. CONCLUSIONS

### Key Topics

- Multiphase flows
- 15.0
- Cavitation
- 13.0
- Compressible flows
- 13.0
- Velocity measurement
- 12.0
- Boundary integral methods
- 11.0

##### G01P1/00

## Figures

Two snapshots of an experiment in which a disc with a radius of 2 cm hits the water surface and moves down at a constant speed of 1 m/s. A surface cavity is created that subsequently collapses under the influence of the hydrostatic pressure. Eventually, the cavity pinches off at the depth indicated by the dashed line, and a large air bubble is entrained. The arrows (red) indicate the direction of the air flow: On the left, volume is created, resulting in a downward air flow. On the right, the bubble volume below the pinch-off depth is decreasing, and therefore air is pushed upwards.

Two snapshots of an experiment in which a disc with a radius of 2 cm hits the water surface and moves down at a constant speed of 1 m/s. A surface cavity is created that subsequently collapses under the influence of the hydrostatic pressure. Eventually, the cavity pinches off at the depth indicated by the dashed line, and a large air bubble is entrained. The arrows (red) indicate the direction of the air flow: On the left, volume is created, resulting in a downward air flow. On the right, the bubble volume below the pinch-off depth is decreasing, and therefore air is pushed upwards.

(a) The volume of the cavity below the pinch-off depth (dashed line) is determined by tracing the boundary (red) and assuming symmetry around the central axis. (b) The volume decreases as the neck becomes thinner until the cavity closes. (c) After pinch-off a downward jet enters into the entrapped bubble, and the bubble shows volume-oscillations and cavity ripples.

(a) The volume of the cavity below the pinch-off depth (dashed line) is determined by tracing the boundary (red) and assuming symmetry around the central axis. (b) The volume decreases as the neck becomes thinner until the cavity closes. (c) After pinch-off a downward jet enters into the entrapped bubble, and the bubble shows volume-oscillations and cavity ripples.

Volume below the pinch-off depth as a function of time (dots (blue)), determined from an experiment with . The vertical dashed line indicates the moment of pinch-off. Close to pinch-off the volume decrease is well approximated by a linear fit (straight solid line (green)), after pinch-off the bubble oscillates with its resonance frequency (solid curve (red): fit with sine function). The steady growth in volume after the pinch-off is caused by the jet entering the bubble, which in our data analysis is not subtracted from the measured bubble volume, see main text.

Volume below the pinch-off depth as a function of time (dots (blue)), determined from an experiment with . The vertical dashed line indicates the moment of pinch-off. Close to pinch-off the volume decrease is well approximated by a linear fit (straight solid line (green)), after pinch-off the bubble oscillates with its resonance frequency (solid curve (red): fit with sine function). The steady growth in volume after the pinch-off is caused by the jet entering the bubble, which in our data analysis is not subtracted from the measured bubble volume, see main text.

Flow rate calculated from the volume changes as a function of the Froude number in a double logarithmic plot. Both the experimental data (black dots) and the numerical data (diamonds (red)) correspond to the maximum value of . The range of experimental data is limited to by the appearance of a surface seal. The solid curve (blue) represents the fit .

Flow rate calculated from the volume changes as a function of the Froude number in a double logarithmic plot. Both the experimental data (black dots) and the numerical data (diamonds (red)) correspond to the maximum value of . The range of experimental data is limited to by the appearance of a surface seal. The solid curve (blue) represents the fit .

A schematic view of the setup. A laser sheet shines from above on the disc, illuminating the interior of the cavity after the disc has impacted the water surface. We insert smoke in the top part of the container and when the linear motor pulls the disc through the water surface at a constant speed, the smoke is entrained into the cavity.

A schematic view of the setup. A laser sheet shines from above on the disc, illuminating the interior of the cavity after the disc has impacted the water surface. We insert smoke in the top part of the container and when the linear motor pulls the disc through the water surface at a constant speed, the smoke is entrained into the cavity.

A snapshot of the cavity with an overlay of a recording of the illuminated smoke. The smoke particles are artificially lightened (orange) in this figure. The size and position of the employed correlation window is indicated by the light-bordered square (yellow).

A snapshot of the cavity with an overlay of a recording of the illuminated smoke. The smoke particles are artificially lightened (orange) in this figure. The size and position of the employed correlation window is indicated by the light-bordered square (yellow).

The vertical air velocity through the neck as a function of the neck radius R, measured in an experiment with in three different ways: (i) Directly, using smoke particles (diamonds), (ii) indirectly, using a smoothing polynomial fit to bubble volume of Fig. 3 (solid line (blue)), and (iii) indirectly, using a constant flow rate approximation, determined at pinch-off (cf. Fig. 3 , thick dashed line). The different gray scales (colors) of the diamonds correspond to different numbers of frames that are skipped in the cross-correlation (see main text). The inset shows the same vertical velocity data measured using method (i) for two different values for the peak-to-peak ratio λ: For λ > 1.5 (gray dots (orange)) we find strongly biased data, which are eliminated using a higher threshold (λ > 3.5, black dots).

The vertical air velocity through the neck as a function of the neck radius R, measured in an experiment with in three different ways: (i) Directly, using smoke particles (diamonds), (ii) indirectly, using a smoothing polynomial fit to bubble volume of Fig. 3 (solid line (blue)), and (iii) indirectly, using a constant flow rate approximation, determined at pinch-off (cf. Fig. 3 , thick dashed line). The different gray scales (colors) of the diamonds correspond to different numbers of frames that are skipped in the cross-correlation (see main text). The inset shows the same vertical velocity data measured using method (i) for two different values for the peak-to-peak ratio λ: For λ > 1.5 (gray dots (orange)) we find strongly biased data, which are eliminated using a higher threshold (λ > 3.5, black dots).

The location of the stagnation point z stag with respect to that of the pinch-off point z c as a function of the neck radius R. Note that when that the stagnation point lies below the pinch-off point, z c − z stag is negative. Time increases from right to left (decreasing R). The dots are experimental data, obtained by volume measurements of four different experiments, where each gray scale corresponds to a different experiment. All experiments were performed with disc radius R 0 = 2.0 cm and impact speed U 0 = 1.0 m/s, i.e., . The dashed line (green) is the result of a two-phase boundary integral simulation without taking compressibility into account [type (ii)]. The solid line (red) is obtained by a two-phase boundary integral simulation which includes a compressible gas phase [type (iii)]. Oscillations in the solid line are a numerical artifact due to wave reflections in the compressible domain, see Gekle et al. 8 for details.

The location of the stagnation point z stag with respect to that of the pinch-off point z c as a function of the neck radius R. Note that when that the stagnation point lies below the pinch-off point, z c − z stag is negative. Time increases from right to left (decreasing R). The dots are experimental data, obtained by volume measurements of four different experiments, where each gray scale corresponds to a different experiment. All experiments were performed with disc radius R 0 = 2.0 cm and impact speed U 0 = 1.0 m/s, i.e., . The dashed line (green) is the result of a two-phase boundary integral simulation without taking compressibility into account [type (ii)]. The solid line (red) is obtained by a two-phase boundary integral simulation which includes a compressible gas phase [type (iii)]. Oscillations in the solid line are a numerical artifact due to wave reflections in the compressible domain, see Gekle et al. 8 for details.

(a) The dimensionless derived air flow rate (from the time rate of change of the cavity volume) as a function of the dimensionless neck radius R/R 0 in an impact experiment with disc radius R 0 = 2 cm and impact speed U 0 = 1 m/s ( ). The black dots represent experimental data. The dashed-dotted line (red) is obtained using a one-phase simulation [type(i)], which excludes the air phase. The solid line (green) is a two-phase boundary integral simulation without compressibility [type (ii)]. Finally, the dashed line (blue) is the result of a two-phase boundary integral simulation which includes a compressible gas phase [type (iii)]. (b) Comparison of the dimensionless derived air flow rate [dashed line (blue); the same curve as in (a)] and the true air flow rate Φ*, both plotted versus R/R 0. The two curves diverge from each other below R/R 0 ≈ 0.2.

(a) The dimensionless derived air flow rate (from the time rate of change of the cavity volume) as a function of the dimensionless neck radius R/R 0 in an impact experiment with disc radius R 0 = 2 cm and impact speed U 0 = 1 m/s ( ). The black dots represent experimental data. The dashed-dotted line (red) is obtained using a one-phase simulation [type(i)], which excludes the air phase. The solid line (green) is a two-phase boundary integral simulation without compressibility [type (ii)]. Finally, the dashed line (blue) is the result of a two-phase boundary integral simulation which includes a compressible gas phase [type (iii)]. (b) Comparison of the dimensionless derived air flow rate [dashed line (blue); the same curve as in (a)] and the true air flow rate Φ*, both plotted versus R/R 0. The two curves diverge from each other below R/R 0 ≈ 0.2.

For the derivation in Appendix A , the cavity close to the pinch-off moment needs be divided into four regions: The expansion region (A), between the location of the disc z disc and the location of the maximum z M , where the cavity expands against hydrostatic pressure; the contraction region (B), between z M and the point z cross where the cavity reaches the disc radius again where the hydrostatic pressure approximation [Eq. (A2) ] is matched to the inertial approximation [Eq. (A4) ]; the collapse region (C) between z cross and z *, characterized by continuity; and the self-similarity region (D), between z * and the pinch off location z c , which is in addition characterized by a coupling between the vertical and horizontal coordinates.

For the derivation in Appendix A , the cavity close to the pinch-off moment needs be divided into four regions: The expansion region (A), between the location of the disc z disc and the location of the maximum z M , where the cavity expands against hydrostatic pressure; the contraction region (B), between z M and the point z cross where the cavity reaches the disc radius again where the hydrostatic pressure approximation [Eq. (A2) ] is matched to the inertial approximation [Eq. (A4) ]; the collapse region (C) between z cross and z *, characterized by continuity; and the self-similarity region (D), between z * and the pinch off location z c , which is in addition characterized by a coupling between the vertical and horizontal coordinates.

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