^{1}, Tadd T. Truscott

^{2}, Alexandra H. Techet

^{3}and John W. M. Bush

^{4}

### Abstract

We present the results of a combined experimental and theoretical investigation of the vertical impact of low-density spheres on a watersurface. Particular attention is given to characterizing the sphere dynamics and the influence of its deceleration on the shape of the resulting air cavity. A theoretical model is developed which yields simple expressions for the pinch-off time and depth, as well as the volume of air entrained by the sphere. Theoretical predictions compare favorably with our experimental observations, and allow us to rationalize the form of water-entry cavities resulting from the impact of buoyant and nearly buoyant spheres.

J.W.M.B. gratefully acknowledges the financial support of the National Science Foundation through Grant No. CTS-0624830; J.M.A. acknowledges the National Science Foundation Graduate Research Fellowship Program; A.H.T. and T.T.T. acknowledge the Office of Naval Research University Laboratory Initiative under Grant No. N00014-06-1-0445.

I. INTRODUCTION

II. EXPERIMENTAL STUDY

III. THEORETICAL MODEL

IV. DISCUSSION

### Key Topics

- Cavitation
- 9.0
- Bubble dynamics
- 6.0
- Z pinch
- 6.0
- Surface dynamics
- 5.0
- Water vapor
- 4.0

## Figures

Schematic of the impact parameters. The advancing contact angle is , and the cavity cone angle is .

Schematic of the impact parameters. The advancing contact angle is , and the cavity cone angle is .

Schematic of the experimental apparatus.

Schematic of the experimental apparatus.

Image sequences showing the water-entry cavity formed by four spheres with different densities. The radius (1.27 cm) and impact speed were held constant , while the density ratio was increased through (I) 0.86, (II) 1.14, (III) 2.30, and (IV) 7.86. Times since the sphere center passed the free surface are shown.

Image sequences showing the water-entry cavity formed by four spheres with different densities. The radius (1.27 cm) and impact speed were held constant , while the density ratio was increased through (I) 0.86, (II) 1.14, (III) 2.30, and (IV) 7.86. Times since the sphere center passed the free surface are shown.

Measured mean sphere depth vs time for the four impact sequences shown in Fig. 3. Every fifth data point is shown. The solid curves denote the theoretically predicted trajectories and are given by Eq. (8) for and . The pinch-off event is denoted by .

Measured mean sphere depth vs time for the four impact sequences shown in Fig. 3. Every fifth data point is shown. The solid curves denote the theoretically predicted trajectories and are given by Eq. (8) for and . The pinch-off event is denoted by .

Characteristics of the water-entry cavity formed by decelerating spheres. The hollow symbols denote the dependence on of the (a) pinch-off depth , (b) pinch-off time , (c) sphere depth at pinch-off , and (d) ratio between the pinch-off depth and the sphere depth at pinch-off . The black symbols denote these same quantities when corrected for the average sphere deceleration , specifically at (a) , (b) , (c) , and (d) . The solid lines denote the theoretical predictions and are given respectively by Eqs. (18)–(21) for . Symbols indicate different density ratios : ◻, ; △, ; ▽, ; ○, ; and ◇, . The error in measurement is of the order of the symbol size.

Characteristics of the water-entry cavity formed by decelerating spheres. The hollow symbols denote the dependence on of the (a) pinch-off depth , (b) pinch-off time , (c) sphere depth at pinch-off , and (d) ratio between the pinch-off depth and the sphere depth at pinch-off . The black symbols denote these same quantities when corrected for the average sphere deceleration , specifically at (a) , (b) , (c) , and (d) . The solid lines denote the theoretical predictions and are given respectively by Eqs. (18)–(21) for . Symbols indicate different density ratios : ◻, ; △, ; ▽, ; ○, ; and ◇, . The error in measurement is of the order of the symbol size.

Dependence on of the volume of air entrained by the sphere, . The hollow symbols denote the measured bubble volume. The black symbols denote this same quantity when corrected for the average sphere deceleration , specifically, . The solid curve denotes the theoretically predicted bubble volume and is given by Eq. (23) for . Symbols indicate different density ratios : ◻, ; △, ; ▽, ; ○, ; and ◇, . The error in the measurement of volume is , that is, roughly 1% of the sphere volume.

Dependence on of the volume of air entrained by the sphere, . The hollow symbols denote the measured bubble volume. The black symbols denote this same quantity when corrected for the average sphere deceleration , specifically, . The solid curve denotes the theoretically predicted bubble volume and is given by Eq. (23) for . Symbols indicate different density ratios : ◻, ; △, ; ▽, ; ○, ; and ◇, . The error in the measurement of volume is , that is, roughly 1% of the sphere volume.

Image sequence of the water-entry cavity formed by a hollow polypropylene sphere with density , radius , and impact speed . and . This is an example of obstructed collapse in which the sphere reverses direction prior to pinch-off.

Image sequence of the water-entry cavity formed by a hollow polypropylene sphere with density , radius , and impact speed . and . This is an example of obstructed collapse in which the sphere reverses direction prior to pinch-off.

The dimensionless average deceleration, , of a sphere upon water entry. The theoretically predicted deceleration is given by Eq. (10) for and . Symbol types correspond to the density ratios : ◻, ; △, ; ▽, ; ○, ; and ◇, . The error in the measurement of average deceleration is ±0.1%.

The dimensionless average deceleration, , of a sphere upon water entry. The theoretically predicted deceleration is given by Eq. (10) for and . Symbol types correspond to the density ratios : ◻, ; △, ; ▽, ; ○, ; and ◇, . The error in the measurement of average deceleration is ±0.1%.

## Tables

Densities of the spheres used in our study. Each sphere has a diameter of , an advancing contact angle of , and a characteristic roughness of 1 .

Densities of the spheres used in our study. Each sphere has a diameter of , an advancing contact angle of , and a characteristic roughness of 1 .

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