^{1}and M. Dale Stokes

^{1}

### Abstract

An analysis of the acoustic emissions of air bubbles fragmenting in sheared fluid flow is presented. The fragmentation of bubbles into two products only is considered. While the measured pressure amplitude is highly variable, the partition of energy between fragmentation products is highly correlated. The partition of energy between products is, on average, approximately equal irrespective of the relative sizes of the bubble products. This observation suggests that the acoustic excitation mechanism is common to both bubbles immediately prior to fragmentation. A model for the excitation mechanism based on symmetric collapse of the neck of air joining fragmentation products is proposed and found to be sufficient to explain the range of observed bubble pulse amplitudes and the equal partition of energy.

The authors gratefully acknowledge helpful discussions with Professor David Farmer and the assistance of Cary Humphries who constructed the bubble smasher. We are grateful to Beswick Engineering who provided the pressure regulator for the bubble injector. This work has been supported by a grant from the Office of Naval Research, Ocean Acoustics Division, ONR Grant No. N00014-04-1-0728 and the National Science Foundation, Grant No. OCE 07-27140.

I. INTRODUCTION

A. Bubble fragmentation in breaking waves

B. Sound from a newly formed bubble

II. RESULTS FROM AN EXPERIMENT

III. SHAPE DISTORTIONS AND ENERGY CONSTRAINTS

IV. NECK COLLAPSE AND FORCING FUNCTION MODEL

V. MODEL CALCULATIONS

VI. SOURCES OF SYSTEMATIC ERROR

A. Bubble shape

B. Bubble-bubble interactions

VII. DISCUSSION

### Key Topics

- Bubble dynamics
- 64.0
- Surface tension
- 23.0
- Ocean waves
- 12.0
- Acoustic waves
- 10.0
- Acoustics
- 10.0

## Figures

Images of air entrainment in a laboratory breaker. (a) The main features of the wave seen through the sidewall of the SIO hydraulics laboratory glass-walled flume. The wave is moving toward the right. The plunging jet, splash-up region, and cavity of air trapped between the jet and wave face can be seen. Fragmenting filaments of air caught between the jet and wave face are shedding bubbles. (b) The upper left frame shows remnants of a filament of air trapped between the overturning jet and the wave face. The white box in the first of these images provides a context for the next three, which illustrate the progressive elongation and fragmentation of the bubble identified by the black arrow.

Images of air entrainment in a laboratory breaker. (a) The main features of the wave seen through the sidewall of the SIO hydraulics laboratory glass-walled flume. The wave is moving toward the right. The plunging jet, splash-up region, and cavity of air trapped between the jet and wave face can be seen. Fragmenting filaments of air caught between the jet and wave face are shedding bubbles. (b) The upper left frame shows remnants of a filament of air trapped between the overturning jet and the wave face. The white box in the first of these images provides a context for the next three, which illustrate the progressive elongation and fragmentation of the bubble identified by the black arrow.

Examples of the pressure pulses radiated by bubbles fragmenting at a depth of a few centimeters of water in the swash zone of La Jolla Shores beach.

Examples of the pressure pulses radiated by bubbles fragmenting at a depth of a few centimeters of water in the swash zone of La Jolla Shores beach.

Schematic of the geometry used for the bubble fragmentation experiment. Two opposing water jets create a region of turbulence and shear that fragment of all bubbles injected from below. A hydrophone, positioned from the fragmentation region and synchronized with an imaging camera, recorded the fragmentation noise.

Schematic of the geometry used for the bubble fragmentation experiment. Two opposing water jets create a region of turbulence and shear that fragment of all bubbles injected from below. A hydrophone, positioned from the fragmentation region and synchronized with an imaging camera, recorded the fragmentation noise.

The pressure pulse radiated by a bubble released from a nozzle below the water surface (broken black line), The least mean squares fit to an exponentially decaying sinusoidal pulse (solid gray line) and the difference between the observed and fitted pulses (solid black line).

The pressure pulse radiated by a bubble released from a nozzle below the water surface (broken black line), The least mean squares fit to an exponentially decaying sinusoidal pulse (solid gray line) and the difference between the observed and fitted pulses (solid black line).

The solid black line traces the acoustic emissions from the products of a binary fragmentation event. The vertical scale is the acoustic pressure at from the bubble products. The beating pattern is a result of interference between the two acoustic signatures. The gray line (see right-hand side of the trace) is a ten parameter fit of the emissions based on Eq. (1). The images below the acoustic trace show the fragmenting bubble and bubble products at time increments. For scale, the diameter of the bubble indicated by the arrow in the left-hand image is .

The solid black line traces the acoustic emissions from the products of a binary fragmentation event. The vertical scale is the acoustic pressure at from the bubble products. The beating pattern is a result of interference between the two acoustic signatures. The gray line (see right-hand side of the trace) is a ten parameter fit of the emissions based on Eq. (1). The images below the acoustic trace show the fragmenting bubble and bubble products at time increments. For scale, the diameter of the bubble indicated by the arrow in the left-hand image is .

(Color) The peak pressure for 505 binary fragmentation events (blue open circles are salt water; black open circles are fresh water) plotted as a function of bubble natural frequency. The gray dots are model predictions based on calculations described in Sec. V. The horizontal and vertical black lines indicate the estimated magnitude of systematic errors for one of the data points. The upper limit of the frequency band for the measurements was limited to by the recording equipment.

(Color) The peak pressure for 505 binary fragmentation events (blue open circles are salt water; black open circles are fresh water) plotted as a function of bubble natural frequency. The gray dots are model predictions based on calculations described in Sec. V. The horizontal and vertical black lines indicate the estimated magnitude of systematic errors for one of the data points. The upper limit of the frequency band for the measurements was limited to by the recording equipment.

Breathing mode radial oscillation amplitude normalized by bubble radius, plotted as a function of bubble radius.

Breathing mode radial oscillation amplitude normalized by bubble radius, plotted as a function of bubble radius.

A scatter plot of energy in the breathing mode of the smaller bubble vs the energy of the larger bubble for 505 binary fragmentation events. The scatter plot shows the strong correlation between breathing mode energies, suggesting an excitation mechanism common to both bubbles before fragmentation. The gray dots are model predictions based on calculations described in Sec. V.

A scatter plot of energy in the breathing mode of the smaller bubble vs the energy of the larger bubble for 505 binary fragmentation events. The scatter plot shows the strong correlation between breathing mode energies, suggesting an excitation mechanism common to both bubbles before fragmentation. The gray dots are model predictions based on calculations described in Sec. V.

Probability density function of energy distribution between fragmentation products. The horizontal and vertical black lines indicate the mean (0.43) and standard deviation of the distribution.

Probability density function of energy distribution between fragmentation products. The horizontal and vertical black lines indicate the mean (0.43) and standard deviation of the distribution.

Images of bubbles immediately before (top row) and after (bottom row) fragmentation. The time interval between the frames is . The leftmost and middle image pairs show binary fragmentation events. A small tertiary bubble can be seen in the bottom image of the right-hand pair. These pictures illustrate the high degree of distortion the occurs immediately before bubble fragmentation.

Images of bubbles immediately before (top row) and after (bottom row) fragmentation. The time interval between the frames is . The leftmost and middle image pairs show binary fragmentation events. A small tertiary bubble can be seen in the bottom image of the right-hand pair. These pictures illustrate the high degree of distortion the occurs immediately before bubble fragmentation.

A conceptual overview of the idealized fragmentation process showing the evolution of the bubble and fragmentation products over time. (a). The initially spherical bubble. (b) Flow forces distort the bubble into a prolate spheroid. (c) The distorted bubble assumes a form topologically similar to a dumbbell. (d) The neck pinches off resulting in two or three fragmentation products. The neck remnants collapse, initiating breathing mode oscillations and capillary waves on the bubble wall. The surface tension energies corresponding to the bubble state are shown below.

A conceptual overview of the idealized fragmentation process showing the evolution of the bubble and fragmentation products over time. (a). The initially spherical bubble. (b) Flow forces distort the bubble into a prolate spheroid. (c) The distorted bubble assumes a form topologically similar to a dumbbell. (d) The neck pinches off resulting in two or three fragmentation products. The neck remnants collapse, initiating breathing mode oscillations and capillary waves on the bubble wall. The surface tension energies corresponding to the bubble state are shown below.

(a) Surface tension energies normalized by energy in the initial spherical bubble as a function of major to minor radius ratio for the prolate spheroid. (b) Bubble product volume ratio for the fragmentation products. (c) The minimum ratio of radii required for an excited bubble undergoing a volume-conserving distortion to contain the same surface tension energy as the fragmentation products with a volume ratio .

(a) Surface tension energies normalized by energy in the initial spherical bubble as a function of major to minor radius ratio for the prolate spheroid. (b) Bubble product volume ratio for the fragmentation products. (c) The minimum ratio of radii required for an excited bubble undergoing a volume-conserving distortion to contain the same surface tension energy as the fragmentation products with a volume ratio .

An analysis of the acoustic pressure pulse radiated by a bubble released from a nozzle. The bubble wall accelerations normalized by the bubble natural radian frequency squared, the bubble wall velocity normalized by the bubble natural radian frequency, and the bubble wall acceleration are, respectively, plotted vs time as the gray solid line, broken black line, and the dash-dot black line. The solid black line shows the forcing function normalized by the bubble natural radian frequency squared. In the figure legend, a single subscript “t” denotes differentiation with respect to time and a repeated subscript “tt” denotes double differentiation with respect to time.

An analysis of the acoustic pressure pulse radiated by a bubble released from a nozzle. The bubble wall accelerations normalized by the bubble natural radian frequency squared, the bubble wall velocity normalized by the bubble natural radian frequency, and the bubble wall acceleration are, respectively, plotted vs time as the gray solid line, broken black line, and the dash-dot black line. The solid black line shows the forcing function normalized by the bubble natural radian frequency squared. In the figure legend, a single subscript “t” denotes differentiation with respect to time and a repeated subscript “tt” denotes double differentiation with respect to time.

(a) Observed and modeled forcing functions plotted as a function of time. The solid black line is the average of 50 forcing functions obtained from the analysis of the acoustic pulses radiated by bubbles released from a nozzle. The four vertical gray lines show the standard deviation in the estimate of the forcing function. The broken black line shows the theoretical forcing function based on an analytical model for neck collapse. The dash-dot black line shows a theoretical forcing function that includes an *ad hoc* modification for the effects of dissipation. (b) Observed and modeled pressure pulses. The solid black line shows the pressure pulse for a selected average bubble. The solid gray, broken black, and dash-dot black lines, respectively, show calculated pulses for the observed, theoretical, and modified theoretical forcing functions.

(a) Observed and modeled forcing functions plotted as a function of time. The solid black line is the average of 50 forcing functions obtained from the analysis of the acoustic pulses radiated by bubbles released from a nozzle. The four vertical gray lines show the standard deviation in the estimate of the forcing function. The broken black line shows the theoretical forcing function based on an analytical model for neck collapse. The dash-dot black line shows a theoretical forcing function that includes an *ad hoc* modification for the effects of dissipation. (b) Observed and modeled pressure pulses. The solid black line shows the pressure pulse for a selected average bubble. The solid gray, broken black, and dash-dot black lines, respectively, show calculated pulses for the observed, theoretical, and modified theoretical forcing functions.

An analysis of the time-varying power law exponent of the observed forcing function. The black line shows the measured power law exponent vs time calculated from the mean forcing function plotted in Fig. 14(a), using Eq. (12). The broken black line shows a least mean squares linear fit to the observed exponent. The dash-dot black line shows the ideal theoretical exponent.

An analysis of the time-varying power law exponent of the observed forcing function. The black line shows the measured power law exponent vs time calculated from the mean forcing function plotted in Fig. 14(a), using Eq. (12). The broken black line shows a least mean squares linear fit to the observed exponent. The dash-dot black line shows the ideal theoretical exponent.

A schematic of the neck collapse geometry for a binary fragmentation event.

A schematic of the neck collapse geometry for a binary fragmentation event.

The fitted pressure amplitude vs bubble radius obtained from the analysis of synthetic pulses excited by the modified theoretical forcing function. Synthetic pulses were obtained from numerical integration of the Rayleigh–Plesset equation based on a Runge–Kutta integration scheme. The fitted pressure amplitude was determined using the same parameter fitting routine used to analyze the bubble fragmentation pulses.

The fitted pressure amplitude vs bubble radius obtained from the analysis of synthetic pulses excited by the modified theoretical forcing function. Synthetic pulses were obtained from numerical integration of the Rayleigh–Plesset equation based on a Runge–Kutta integration scheme. The fitted pressure amplitude was determined using the same parameter fitting routine used to analyze the bubble fragmentation pulses.

The predicted peak pressure vs observed peak pressure of the smaller bubble in a fragmentation product pair. The gray dots show a scatter plot of predicted vs observed pressure. The 14 black squares show the mean of the predicted peak pressure for all data points lying within an observed pressure bin centered on the square. The vertical black lines running through the center of the squares show the standard deviation of the data points. The solid gray line shows the a 1:1 correspondence between the predicted and observed pressures.

The predicted peak pressure vs observed peak pressure of the smaller bubble in a fragmentation product pair. The gray dots show a scatter plot of predicted vs observed pressure. The 14 black squares show the mean of the predicted peak pressure for all data points lying within an observed pressure bin centered on the square. The vertical black lines running through the center of the squares show the standard deviation of the data points. The solid gray line shows the a 1:1 correspondence between the predicted and observed pressures.

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