The Pennsylvania State University

The Graduate School

College of Engineering

CHARACTERIZING LASER INDUCED CAVITATION: EFFECTS OF AIR CONTENT,

BEAM ANGLE, AND LASER POWER

A Thesis in

Mechanical Engineering

by

Minna L. Ranjeva

© 2012 Minna L. Ranjeva

Submitted in Partial Fulfillment

of the Requirements

for the Degree of

Master of Science

August 2012

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The thesis of Minna L. Ranjeva was reviewed and approved* by the following: Brian R. Elbing Associate Research Faculty Thesis Adviser Dan Haworth Professor of Mechanical Engineering PIC of MNE Graduate Programs Gary Settles Professor of Mechanical Engineering Karen A. Thole Professor of Mechanical Engineering Head of the Department of Mechanical Engineering *Signatures are on file in the Graduate School.

   

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Abstract

Laser-induced cavitation allows for cavitation bubbles to be systematically reproduced using

a high power laser and focusing the laser beam to a point. This provides the opportunity to study

the physics of the cavitation process under different circumstances in an experimental setting.

Laser-induced cavitation has many applications. It has been used successfully to study cavitation

near boundaries in an effort to understand the mechanisms of cavitation erosion. It has also found

new applications in the world of medicine, as well as other areas. Previous work has largely

focused on cavitation near a surface causing damage, but as new applications emerge,

characterization of bubbles in a bulk fluid will be useful, permitting a high level of control over

the bubble size, shape, and lifetime. With this in mind, laser-induced cavitation bubbles in a bulk

fluid are characterized. The focusing angle of the lens, the air content of water, and the laser

power are all varied to provide a comprehensive understanding of how these variables affect the

bubbles’ development. Scaling for the bubble behavior is developed.

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Table  of  Contents  List of Tables. .............................................................................................................................. vii

List of Figures ............................................................................................................................. viii

1 Introduction ............................................................................................................................. 1 1.1 Background .................................................................................................................................... 1 1.2 Physics ............................................................................................................................................. 2

1.2.1 General ...................................................................................................................................... 2 1.2.2 Laser-induced optical breakdown ............................................................................................. 3 1.2.3 Plasma expansion and production of cavitation bubble ........................................................... 4 1.2.4 Cavitation bubble growth and collapse ..................................................................................... 4

1.3 Earlier work ................................................................................................................................... 5 1.4 Recent attempts to characterize bubbles ..................................................................................... 8 1.5 Newer applications of cavitation .................................................................................................. 9

1.5.1 Laser cleaning ......................................................................................................................... 10 1.5.2 Micro-pumps .......................................................................................................................... 10 1.5.3 Lithotripsy .............................................................................................................................. 11 1.5.4 Drug delivery .......................................................................................................................... 12

1.6 Introduction summary ................................................................................................................ 13 1.6.1 Research Objectives ............................................................................................................... 13

2 Experimental Methods ......................................................................................................... 15 2.1 Test apparatus .............................................................................................................................. 15

2.1.1 Water Tank ............................................................................................................................. 15 2.1.2 Laser Optics ............................................................................................................................ 18

2.2 Instrumentation ........................................................................................................................... 22 2.2.1 Imaging setup ......................................................................................................................... 22 2.2.2 Laser power ............................................................................................................................ 23 2.2.3 Water dissolved gas content ................................................................................................... 24

2.3 Test Matrix ................................................................................................................................... 26 2.4 Measurement Uncertainty .......................................................................................................... 26

2.4.1 Camera .................................................................................................................................... 26 2.4.2 Laser ....................................................................................................................................... 27 2.4.3 Lenses ..................................................................................................................................... 27 2.4.4 Pressure ................................................................................................................................... 28 2.4.5 Air Content ............................................................................................................................. 28

3 Experimental Results ............................................................................................................ 30 3.1 General trends and overview ...................................................................................................... 30 3.2 Repeatability ................................................................................................................................ 35 3.3 Variation of bubble topology ...................................................................................................... 38 3.4 Bubble size .................................................................................................................................... 42

3.4.1 Bubble size: sensitivity to beam angle .................................................................................... 42

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3.4.2 Bubble size: sensitivity to air content ..................................................................................... 49 3.5 Bubble half-life ............................................................................................................................. 54

3.5.1 Bubble half-life: sensitivity to beam angle ............................................................................. 54 3.5.2 Bubble half-life: sensitivity to air content .............................................................................. 57

3.6 Bubble diameter time history ..................................................................................................... 60

4 Scaling of Laser Induced Cavitation ................................................................................... 70 4.1 Scaling ........................................................................................................................................... 70 4.2 Comparison with non-spherical bubbles ................................................................................... 80

4.2.1 Vertical diameter .................................................................................................................... 80 4.2.2 Horizontal diameter ................................................................................................................ 82 4.2.3 Bubble half-life ....................................................................................................................... 84 4.2.4 Behavior over time ................................................................................................................. 86

4.3 Error Propagation ....................................................................................................................... 89

5 Limitations and future work ................................................................................................ 91 5.1 Limitations ................................................................................................................................... 91

5.1.1 Pulsed laser ............................................................................................................................. 91 5.1.2 Equipment setup ..................................................................................................................... 91 5.1.3 Assumptions in deriving scaling ............................................................................................. 92

5.2 Future work .................................................................................................................................. 92 5.2.1 Pressure ................................................................................................................................... 92 5.2.2 Viscosity ................................................................................................................................. 93 5.2.3 Particulate matter .................................................................................................................... 93 5.2.4 Air content .............................................................................................................................. 94

6 Conclusions ............................................................................................................................ 95

Appendix: Uncertainty ............................................................................................................... 97

Bibliography .............................................................................................................................. 100

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List  of  Tables  Table 1 – Distances between components in experimental set up in Figure 1. ............................ 17 Table 2 – Lens specifications used to create bubbles ................................................................... 18 Table 3 – Average water-air content concentration (C) for each air content condition ............... 25 Table 4 – standard deviation for air content concentration measurements ................................... 28 Table 5 – Measured quantities and their corresponding measurement uncertainty ...................... 29 Table 6 – Lowest laser pulse energy for which bubbles could be seen for each of the test

conditions, E0 .................................................................................................................... 43 Table 7 – Coefficients for second order polynomial best fit curves based on nondimensionalized

lifetime profiles ................................................................................................................. 67 Table 8 – Beam waist for lenses ................................................................................................... 71 Table 9 – Variables of interest used to derive scaling rules for spherical cavitation bubbles. ..... 72 Table 10 – Results of error propagation through calculations for nondimensional variables ...... 90

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List  of  Figures  Figure 1 – Experimental setup used to generate laser-induced cavitation bubbles. The laser beam

is expanded and then focused through a lens. A camera was used to capture the cavitation bubbles. The dashed box in the side view represents the camera’s nominal field-of-view. ........................................................................................................................................... 16

Figure 2 – This figure illustrates the relationship between beam divergence and angular spread. In this work, the convergence or focusing angle refers to the angular spread. ................. 18

Figure 3 – Snell’s law applied to a ray path passing through air (n1 =1.00), glass (n2 =1.55) and water (n3 =1.33). Increasing index of refraction bends the beam towards the axis perpendicular to the interface between the two mediums. ................................................ 21

Figure 4 – Energy per pulse as a function of attenuator setting for the laser used in the current study. The flash lamp was fixed at the maximum level. ................................................... 24

Figure 5 – Example of a bubble lifetime produced at relatively low laser power (7.3 mJ per pulse) with the wide-angle lens configuration and intermediate air content level. Labelled data points correspond to labeled images at the top of the figure. In image A the bright white spot is produced from plasma generated by the focused laser beam, while the remaining images show the shadow produced from the backlighted bubble. .................. 32

Figure 6 – Example of a bubble lifetime produced at relatively high laser power (42 mJ per pulse) with the wide-angle lens configuration and intermediate air content level. Labelled data points correspond to labeled images at the top of the figure. The growth, collapse and rebound of the bubble is apparent from the plot. ....................................................... 33

Figure 7 – Comparison between low and high power bubble behavior. At lower pulse energies single, spherical bubbles are produced. At higher powers the bubble shape depends on the lens angle. The smaller angle lens produces one larger bubble formed from bubble coalesence. The wider angle lenses produce smaller, more elongated bubbles. In the figure SA, MA and WA refer to the small, medium and wide-angle lens configurations, respectively. ...................................................................................................................... 35

Figure 8 – Comparing bubble diameter, there is reasonable agreement between data collected in different setups from different points in time. This indicates that comparing results between the two different setups is appropriate. ............................................................... 37

Figure 9 – Narrow angle lens bubble patterns at high power (42 mJ) in high air content water. Smaller bubbles formed along the path of the laser beam coalesce to form a single, larger, elliptical bubble. ................................................................................................................ 39

Figure 10 – Medium angle lens bubble patterns at high power (42 mJ) in intermediate air content water. On the left is the bubble formation shortly after the laser pulse, on the right is the bubble formation at the time of maximum bubble diameter. ............................................ 41

Figure 11 – Wide-angle bubble patterns at high power. On the left hand side of (A), (B), and (C) is the bubble formation shortly after the laser pulse, and on the right hand side of (A), (B) or (C) are the bubble formations at the points of maximum bubble diameter. These images illustrate the variety of bubble patterns that can be formed. ................................ 42

Figure 12 – Vertical bubble diameter as a function of laser energy per pulse for (A) low, (B) intermediate and (C) high air content conditions. The solid lines represent the best fit curves for each condition. ................................................................................................. 47

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Figure 13 – Ratio of the horizontal to vertical bubble diameters plotted as a function of the laser pulse energy with (A) low, (B) intermediate and (C) high air content levels. .................. 49

Figure 14 – Vertical bubble diameter plotted versus the laser pulse energy for (a) narrow, (b) medium and (c) wide-angle lens configurations. Solid lines represent the best fit curves to the data. ............................................................................................................................. 52

Figure 15 – Ratio of horizontal to vertical diameter plotted versus laser pulse energy for (a) narrow (b) medium and (c) wide beam angles. ................................................................. 54

Figure 16 – Bubble half-life (time from laser pulse to achieve maximum bubble diameter) plotted versus the laser pulse energy with (a) low, (b) intermediate and (c) high or saturated air content levels. ................................................................................................................... 57

Figure 17 – Bubble half-life as a function of laser pulse energy with the (A) narrow, (B) medium and (C) wide-angle lens configuration. ............................................................................. 60

Figure 18 – Small angle, low air content lifetime curve ............................................................... 61 Figure 19 – Small angle, intermediate air content lifetime curve ................................................. 62 Figure 20 – Small angle, saturated condition lifetime curve ........................................................ 62 Figure 21 – Medium angle lens, low air content lifetime curve ................................................... 63 Figure 22 – Medium angle, intermediate air content lifetime curve ............................................. 63 Figure 23 – Medium angle, saturated air content lifetime curve .................................................. 64 Figure 24 – Wide-angle, low air content condition lifetime curve ............................................... 64 Figure 25 – Wide-angle, intermediate air content lifetime curve ................................................. 65 Figure 26 – Wide-angle, saturated lifetime curve ......................................................................... 65 Figure 27 – Comparison of lifetime polynomial fit for each condition. SA/L stands for small

angle, low air content. MA stands for medium angle, WA for wide-angle, I for intermediate air condition, H for high air content. ............................................................ 66

Figure 28 – Single lifetime curve for nondimensionalized bubble diameter vs. time. This curve describes the bubble diameter’s growth over time for th ≤ 2. The different colors represent different laser energy ranges, while the shapes indicate the air content condition (squares are low air content, circles are intermediate, and triangles are the high air content condition). ......................................................................................................................... 69

Figure 29 – An illustration of the beam profile of a Gaussian beam near the focal point. ........... 71 Figure 30 – The scaled horizontal diameter (Dh

*) is plotted versus the scaled vertical diameter (Dv

*), holding Π6 constant. This plot shows a linear relationship between Π1 and Π2 for the majority of bubbles. The outliers in this plot represent bubbles produced at high energies that do not maintain a spherical shape. These bubbles are indicated by open symbols and are not included when looking at the scaling relationships. ............................................. 73

Figure 31 – Relationship between th*/Dv* and ΔE* shows a logarithmic relationship that appears to be only minimally affected by air content for spherical bubbles. The average ratio for each condition (lens, air content, laser power) is plotted. ................................................. 76

Figure 32 – Average th* versus Dv* for various ranges of ΔE*. The relationship appears to be linear. ................................................................................................................................ 78

Figure 33 – The relationship between beam waist and Eo appears to be linear. ........................... 79 Figure 34 – Predicted Dv* versus actual Dv* when applying scaling to non-spherical bubbles. The

solid line shows where the predicted and actual Dv* values are equal. The scaling guidelines over predict the vertical diameter of the non-spherical bubbles. ..................... 81

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Figure 35 – Predicted vertical diameter plotted against the actual vertical diameter when scaling relationships are applied to non-spherical bubbles. The black line indicates where the predicted and actual values are equal. ............................................................................... 82

Figure 36 – Predicted Dh* versus actual Dh* when applying scaling to non-spherical bubbles. The solid line shows where the predicted and actual Dh* values are equal. The horizontal bubble diameter appears to be predicted relatively well with the scaling relationships used here. ................................................................................................................................... 83

Figure 37 - Predicted horizontal diameter plotted against the actual horizontal diameter when scaling relationships are applied to non-spherical bubbles. The black line indicates where the predicted and actual values are equal. ......................................................................... 84

Figure 38 – Predicted th* versus actual th* when applying scaling to non-spherical bubbles. The solid line shows where the predicted and actual th* values are equal. The scaling relationships used over predict the scaled half-life th* for non-spherical bubbles. .......... 85

Figure 39 – Predicted half-life versus actual half-life for scaling relationships applied to non-spherical bubbles. The black line indicates where the predicted and actual values are equal. ................................................................................................................................. 86

Figure 40 – Non-spherical bubble behavior over time. Each run represents the development of a single, non-spherical bubbles. The blue curve represents the curve in Figure 28, the generic lifetime curve obtained in chapter 3. .................................................................... 88

Figure 41 – The ratio of horizontal to vertical diameter for non-spherical bubbles. This graph represents the average ratio of vertical to horizontal diameter for six non-spherical bubbles. This illustrates that non-spherical bubbles begin as more elliptical shapes and then become more spherical over time. It also shows the oscillation in size that non-spherical bubbles exhibit. .................................................................................................. 89

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1 Introduction

1.1 Background

Cavitation is a phenomenon that occurs in nature when fluids develop areas of high speed

or local pressure drops. These forces cause the rapid formation and then collapse of a cavity

within a liquid, and this process is referred to as cavitation. Cavitation has traditionally been an

area of interest in the scientific community due to its erosive effects on mechanical parts, such as

propellers, and the fact that loud noise resulting from cavitation can cause an issue when vehicles

desire to go undetected.

Since people began to study cavitation for its erosive consequences, new techniques to

both produce and record cavitation events have been developed. In the early stages of cavitation

study, scientists had difficulty controlling the occurrence of cavitation events due to the

statistical nature of this phenomenon in both space and time (Lauterborn & Bolle, 1975). Using a

laser allows cavitation bubbles to be produced at a known location. Early research focused on

cavitation events occurring near a boundary, since cavitation is well known for causing damage

to propellers. More recently laser-induced cavitation has been used in a variety of medical

applications such as lithroscopy (Kokaj et al., 2008). It is also being explored for use in other

areas, for example laser cleaning of surfaces or micropump technologies (Song et al., 2004;

Dijkink & Ohl, 2008).

As more and more applications for laser-induced cavitation bubbles emerge, a greater

understanding of how the environmental factors and controllable variables (such as laser power

or beam angle) affect the growth and development of cavitation bubbles is needed. As much

previous research has focused on the erosive consequences of cavitation bubble collapse, more

information on bubble formation away from a boundary in an infinite medium, or how changes

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in the properties of the fluids (such as air content or impurities) alter the size, shape, and lifetime

of cavitation bubbles produced via laser could prove useful. With this in mind, this research

focuses on the effects of laser power, water air content, and focusing lens angle. This information

will be useful in controlling bubble production in various environments, and will provide

particularly valuable knowledge as laser-induced cavitation bubbles are used in a variety of

fields, including medicine.

1.2 Physics

1.2.1 General

Optical energy of the high-intensity laser pulse is converted into mechanical energy,

allowing the formation and expansion of a plasma due to dielectric breakdown, propagation of a

shock wave and growth of a cavitation bubble. The laser energy causes the temperature and

pressure to increase at a point in space, producing plasma that expands rapidly. A cavitation

bubble results, and when it reaches its maximum diameter it is nearly empty. Thus the cavitation

bubble collapses due to the higher pressure outside its boundary. When the collapsing bubble

reaches a given size it may rebound and the process repeats until there is insufficient energy for

the bubble to rebound again. Bubble collapse in an infinite medium is symmetric, while bubble

collapse near a boundary is asymmetrical. Near a rigid boundary a liquid jet develops that is

directed towards the boundary, often causing damage. With a free surface as a boundary, the

bubble migrates away from the boundary as it collapses (Gregoric, Petkovsek, & Mozina, 2007).

Bubble dynamics near a surface have been of interest due to the erosive effects of cavitation, so

the majority of research has focused on these conditions.

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1.2.2 Laser-induced optical breakdown

Laser-induced breakdown can work in two ways: laser-induced thermal breakdown and

laser-induced optical breakdown. The first occurs due to continuous wave exposure or

repetitively pulsed lasers at high power in materials that are opaque at the laser wavelength. The

second mode, optical breakdown, occurs for short pulse durations between microseconds and

femtoseconds. Optical breakdown will be the focus of this discussion, as the breakdown and

resulting cavitation bubbles in this work are produced in water (transparent) with laser pulses in

the nanosecond range. Optical breakdown produces plasma, or a ‘gas’ of charged particles. Two

mechanisms of breakdown exist, multiphoton absorption and cascade ionization.

If a free electron already exists in the focal volume then photons from the laser pulse can

be absorbed by the free electron, and this energy can be used to ionize a bound electron via a

collision, resulting in two lower energy free electrons. This process repeats and leads to a

cascade that results in breakdown and the formation of a plasma. The initial free electron(s) can

be provided by impurities in the water sample. Multiphoton breakdown does not require a seed

electron or particle collisions. In this case, each electron is ionized simultaneously by absorbing

photons from the laser pulse. This type of breakdown does not require the presence of impurities

and can occur in a pure medium. It is much faster than cascade ionization and can occur during

shorter laser pulses. The irradiance threshold for multiphoton breakdown is much higher than

that for cascade ionization, and therefore cascade ionization is much more common (Kennedy,

Hammer, & Rockwell, 1997). Cascade ionization is assumed to be responsible for the breakdown

and subsequent appearance of cavitation bubbles studied in this work, due to the presence of

impurities in tap water and resulting lower irradiance threshold.

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1.2.3 Plasma expansion and production of cavitation bubble

Once breakdown occurs and plasma is formed the laser energy can cause the plasma to

expand rapidly at supersonic velocities resulting in an acoustic emission as well as a shock wave

and cavitation. Once the laser pulse is gone the plasma starts to cool. The plasma will cool by

losing energy to shock wave emission, spectral emission, and thermal conduction into the bulk

fluid. After significant cooling the plasma begins to decay through the process of electron-ion

recombination. This whole process creates a cavitation bubble of vapor water at the breakdown

site (Kennedy, Hammer, & Rockwell, 1997).

1.2.4 Cavitation bubble growth and collapse

The high temperature causes a bubble to form around the plasma volume and shock wave

velocity causes the bubble to grow rapidly. The bubble continues to grow quickly as the shock

wave detaches from the bubble due to the large pressure difference between the inside of the

bubble and the surrounding medium. As the volume of the bubble increases, the pressure inside

the bubble drops. Once the pressure reaches the saturated vapor pressure for the liquid the

cavitation bubble reaches its maximum size. This is because the rate of evaporation of liquid into

the bubble and the rate of condensation out of the bubble are equal for a brief moment in time.

The saturated vapor pressure inside the bubble is then lower than the pressure of the surrounding

liquid, resulting in the bubble starting to shrink. If there is enough energy in the bubble it can

rebound due to the increasing temperature and pressure of the gas inside the bubble as it shrinks.

The bubble can continue to oscillate in this manner until it finally collapses with no rebound

(Kennedy, Hammer, & Rockwell, 1997).

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1.3 Earlier work

The use of lasers to induce cavitation bubbles (a process referred to as laser-induced

cavitation or LIC) developed in the late 1960’s and through the 1970’s, with most experiments

focusing on generating cavitation bubbles near a boundary to study the physics and erosion

mechanisms of cavitation bubbles. At the helm of this research was Lauterborn, who coined the

phrase “optical cavitation” to describe the phenomenon of producing cavitation bubbles using a

laser. Lauterborn started in the early 1970’s, developing ways to study the growth and decay of

cavitation bubbles near boundaries.

In Lauterborn & Bolle (1975) some obstacles that previously had limited the

understanding of laser-induced cavitation were overcome. A spherical cavitation bubble collapse

near a solid surface could not be studied theoretically or experimentally until the early 1970’s.

Numerical methods could not predict bubble behavior in final stages of collapse, while

experimental investigations were encumbered by the fact that the appearance of cavitation

bubbles in most situations is statistical in both space and time. While a few experiments in the

1960’s provided evidence of jet formation resulting in cavitation erosion, laser-produced

cavitation bubbles were used starting in the early 1970’s to study cavitation under highly

controlled conditions. They were able to gain an understanding of cavitation damage

mechanisms for bubbles produced at varying distances from a brass plate and capture high speed

images at a frame rate of up to 75,000 frames per second (fps). They compared their results to

theoretical work on cavitation bubbles and damage (Plesset & Chapman, 1971), and saw good

agreement between the experimental and theoretical work for γ = 1.5, where γ = h/Rmax, Rmax is

the bubble’s maximum radius and h is the distance from the bubble center to the boundary.

Clearly, γ will influence the amount of damage that cavitation will cause to a surface. This was

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one of the first opportunities to use experimental work to validate the theory proposed in Plesset

& Chapman (1971) as methods to produce bubbles systematically were lacking (Lauterborn &

Bolle, 1975).

Interest in cavitation damage mechanisms continued, with many authors contributing to

the body of knowledge on cavitation events occurring near a boundary. Authors also sought to

understand how cavitation acted in the vicinity of different types of boundaries. Giovanneschi &

Dufresne (1985) studied laser-induced cavitation and discussed to some extent how to control the

bubble size and shape for reproducibility of results. Isolating a bubble was no longer an issue

thanks to the development of laser-induced cavitation, but controlling the specific parameters of

the bubbles generated via laser was still difficult. It is advised to use optics with a short focal

length and a small ratio of f/D (focal length/diameter). The setup Giovanneschi and Dufresne

used involved a Nd:YAG laser with wavelength of 1.06 µm. A beam of 7 mm diameter was

expanded by a telescope assembly (diverging and converging lens) and the resulting beam had a

diameter of 35 mm. This beam was then focused into water by a convex lens with focal length (f)

of 50 mm in air, resulting in a f/D ratio of 1.9. Bubbles in an infinite medium as well as near a

wall were photographed at a frame rate of 2×106 fps.

In an infinite medium it was found that a spherical bubble would remain spherical as it

expands. Near a solid wall, an initially spherical bubble will grow spherically but deforms during

the collapse phase, deforming more the closer the wall is to the bubble. The deformation can be

seen as an additional disturbance along an axis perpendicular to the wall; thus the bubble

becomes oval in a direction perpendicular to the wall, and this results in a jet directed at the wall

as the bubble rebounds (Giovanneschi & Dufresne, 1985).

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Lauterborn continued to delve into the area of laser-induced cavitation, and in Lauterborn

& Philipp (1998) the specific damage mechanisms were investigated. At that point in time there

had been some debate about specifically how cavitation bubbles cause damage to metal plates

and solid boundaries. In order to achieve this, the authors used a Q-switched Nd:YAG laser and

focusing optics to produce cavitation bubbles at known locations. They then varied the distance

between the cavitation bubbles and a solid metal plate (99.999% pure aluminum due to its

softness), and used a high-speed camera to track the bubble dynamics. One of the parameters

they looked at was γ, which was varied between ~0 and 3. The high-speed camera recorded at

two different frame rates: 56,500 frames/s for an overview, and one million frames/s for fast

events such as jet formation or shock wave emission. After being exposed to cavitation bubbles,

the metal sheets were analyzed for damage.

High-speed camera images of the cavitation bubbles were observed and analyzed. When

the bubbles reached their maximum bubble diameter the pressure inside the bubble was seen to

be much lower than the ambient pressure and the bubbles began to collapse. The fact that a

boundary exists on one side of the bubble (in the paper, the lower part of the bubble) caused

retardation of radial water flow and therefore lower pressure than for the part of the bubble

farther away from the wall (upper bubble wall). This caused the bubble to become elongated, and

the center of the bubble moves toward the boundary during collapse. A liquid jet developed,

directed toward the boundary, and the bubble became toroidal as the jet flows through the

bubble. As γ is decreased the jet formed earlier in time, and for γ < 1 the jet hit the metal plate

directly, with no deceleration from a water layer, which occurs at larger distances. The impact

velocity of the jet on the boundary determines the damage capability. The smaller the distance

between the bubble and the wall, the higher the jet velocity, and therefore the greater the

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potential for damage. It was determined that the diameter of the damage area scaled with the

maximum bubble radius. This work clarified that cavitation erosion is caused mostly by the

collapse of a bubble in contact with a material, as there was some doubt as to the specific

mechanism for cavitation damage (Philipp & Lauterborn, 1998).

1.4 Recent attempts to characterize bubbles

As cavitation becomes widely used in new areas, complete characterization and

understanding of the physics of cavitation is necessary. A lot of work has focused on

characterizing the physics of cavitation near a boundary, but less information is available on

laser-induced cavitation bubbles in a bulk liquid.

Peel, Fang & Ahmad (2011) used two methods, pump-probe beam deflection (PBD) and

high speed photography, to characterize cavitation bubbles induced in a bulk fluid (water) using

a Nd:YAG laser with a wavelength of 1064 nm, pulse energy of 140 mJ, pulse duration of 10 ns,

and repetition rate of 1 Hz. In both cases, the focus of the study was on the bubble interface

speed and bubble size, as well as the bubble lifetime. The laser pulse energy used in this paper is

higher than energies used previously. At lower energies the relationship between laser pulse

energy and bubble energy is linear. Using higher laser energies was desired to help determine if

this linear trend continues at much higher laser energies.

For the high-speed photography technique, a frame rate of 5.45×104 fps was used. At

this frame rate, the bubble’s growth was tracked over time, with still images of the bubble every

20 µs. Cavitation bubbles were formed along the trajectory of the laser beam, with smaller

bubbles initially forming as a result of multiple breakdown sites, and then merging to form single

bubbles after a certain amount of time (~300 µs from the laser pulse trigger). The maximum

bubble size was found to occur at about 320 µs, and the maximum bubble diameter was

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approximately 2.20 mm. The total bubble lifetime was estimated to be approximately 280 µs

(where the total image time is 480 µs, but the first bubble image is visible at 200 µs after the

laser pulse). The authors compared their experimental results for the laser-induced cavitation

bubbles with the Rayleigh equation, which is used to quantify cavitation in a bulk fluid based on

the bubble wall velocity (i.e. how fast the radius of the bubble grows). Comparison between the

experimental data and Rayleigh equation results was inconclusive, due to the fact that Rayleigh’s

equation assumes an incompressible liquid and the fact that in experiments the liquid density

near the cavitation bubble will change (Peel, Fang, & Ahmad, 2011).

This recent work illustrates that there is still much to learn about laser-induced cavitation

bubbles and their behavior under different conditions. Scientists have also been discovering new

ways of using cavitation in a positive way. While the majority of past research has focused on

the erosion caused by cavitation events, future work should examine bubble dynamics under

more varied and extreme conditions.

1.5 Newer applications of cavitation

More recently, cavitation has been a topic of interest for many researchers, finding

application in a wide variety of fields. It has been used in some biomedical applications, for

surface cleaning, and in lab-on-a-chip technologies as just a few examples. Cavitation is proving

to be a very versatile phenomenon with many uses, and while cavitation has traditionally been

viewed as an undesirable phenomenon due to the noise and damage it produces, these newer

applications aim to utilize cavitation in positive ways. For these newer applications,

characterization is extremely important, as accurate characterization will allow people to control

the size and effects of the cavitation process. This is particularly important in biomedical

applications where unexpected results could cause damage to the human body.

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1.5.1 Laser cleaning

Laser-induced cavitation bubbles can be utilized in a process referred to as “wet laser

cleaning” to remove particles from a substrate immersed in a liquid. When a high-power laser is

focused into the liquid and optical breakdown occurs, a cavitation bubble is produced. The shock

waves emitted as well as the liquid jet that develops during cavitation collapse produce high

pressures (i.e. several gigapascals). It is also known that a cavitation bubble collapsing near a

boundary (in this case the substrate) deforms and that a liquid jet is directed at the boundary.

This phenomenon allows surfaces to be cleaned via laser-induced cavitation bubbles, which can

prevent issues associated with other methods of laser cleaning. This wet laser cleaning method

circumvents issues associated with a substrate absorbing laser irradiation, which can induce high

temperatures that in turn can produce oxidization, melting, stress generation, and other changes

to the physical and chemical properties of the substrate. To further facilitate substrate cleaning

from the jets that result from cavitation bubble collapse, it is recommended that an organic

solution be used to lower the viscosity and consequently increase the impact pressure on

substrate contaminant particles. This will help increase the cleaning efficiency (Song et al.,

2004).

1.5.2 Micro-pumps

Dijkink & Ohl (2008) used laser-induced cavitation bubbles to pump water through a

small channel, applying LIC to lab-on-a-chip systems. Their cavitation-based technique is

capable of pumping 4000 µm3 in 75 µs. The cavitation bubble is induced above a boundary with

a small opening that creates a channel. Because cavitation bubbles close to a rigid boundary form

an asymmetrical jet, the jet forces water through the channel. If the jet did not occur as part of

the cavitation process, the bubble expansion and collapse would allow water to first move into

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the channel and then pull water out, resulting in an overall displacement of zero. The efficiency

of the pump varied with cavitation bubble size and the distance between the bubble and the

channel. One of the advantages of using a laser to create a pump is that no mechanical or

electrical connections are needed for the pump itself (Dijkink & Ohl, 2008). This type of

technology could utilize cavitation in a positive way for extremely small-scale systems.

1.5.3 Lithotripsy

Lithotripsy is a process used to break down stones that form in the body. These stones are

often referred to as calculi. Shock waves are used in lithotripsy to break up these stones. While

there are four main types of lithotripsy, laser lithotripsy is best suited and most commonly used

to break up bile duct stones and calculi in the ureter (tubes that transport urine from the kidney to

the bladder). Laser energy is directed to a specific point using an optical fiber. Pulsed Nd:YAG

lasers were considered as an option for breaking up these stones because they create shock waves

that are ideal for disturbing stones. The drawback to using Nd:YAG lasers is that the fiber

damage threshold depends on the irradiance, and transporting very short pulses places strict

limitations on size of the fibers that can be used. While some special probes have been developed

for use with Nd:YAG lasers, other types of lasers have also been investigated as alternatives

(Kennedy, Hammer, & Rockwell, 1997). Nd:YAG lasers have been looking promising as a

method of using laser lithotripsy to destroy stones. While research about this technology has

been progressing, traditional surgeries have still been needed to remove the majority of the

stones. This is largely due to a lack of understanding about the dynamics of the laser lithotripsy

process. With improved understanding of bubble dynamics laser lithotripsy could provide a

convenient, economical, and less painful technique than traditional surgeries (Kokaj, Marafi, &

Mathew, 2008).

  12  

Recently, a new frequency-doubled, double-pulse Nd:YAG laser (FRDDY) has been

developed and used to treat urinary stones. The frequency doubling means that wavelengths of

both 532 and 1064 nm are used for the laser pulses, which allows for very high pulse intensity at

a given point. The 532 nm wavelength light initiates the plasma formation and then the 1064 nm

wavelength light heats the plasma causing first expansion and then contraction, fragmenting the

stones in the body. The pulses are on the order of 1 µs, and therefore the surrounding tissues do

not experience thermal damage from the laser pulses (Kim et al., 2008).

1.5.4 Drug delivery

Using cavitation bubbles has been proposed as a method for improving drug delivery to

tumors and cancerous cells. The delivery of anti-cancer drugs from the bloodstream to cancer

cells is inhibited by blood vessel walls, interstitial space, and cell membranes. Laser-induced

cavitation could be used in specific locations to perforate tumor blood vessel walls and cancer

cell membranes, and produce microconvection within interstitial spaces (Esenaliev et al., 2001).

Dijkink et al. (2008) investigated the viability of cavitation-induced drug delivery.

Epidermal HeLa cells were grown in a medium. Cell permeabilization was tracked by using a

small fluorescent molecule (calcein) in the medium surrounding the cells. Apoptosis (cell death)

was also studied using cell-staining dyes. Cavitation bubbles were induced at various distances

from the cell boundary using a laser. The jet formed by cavitation bubble collapse near the

boundary caused some cells to become suddenly detached and die. Below these detached cells

another zone of cells was also impacted by the cavitation bubble jet resulting in large pores that

caused apoptosis, and these cells died within a few hours. The next zone of cells was porated in

such a way that drug delivery would be viable, and cells beyond this zone were unaffected by the

cavitation event. The number and size of pores that could be generated for drug delivery are

  13  

related to the distance between the cavitation bubble and the cell boundary. This distance is

characterized by the standoff distance γ, where γ = h/Rmax, where Rmax is the maximum bubble

radius and h is the distance between the bubble center and the boundary. Decreasing γ resulted in

more cells being detached, but an increase in the number of cells showing molecular uptake,

indicating an increase in pore size and/or number (Dijkink et al., 2008).

1.6 Introduction summary

Cavitation has been a topic of interest for over a hundred years. Initially considered an

undesirable phenomenon due to its erosive effect and loud acoustic signature, it has in more

recent years been utilized in a number of productive ways. The desire to study the physics of

cavitation and isolate bubbles for observation resulted in the use of lasers to produce bubbles, a

process referred to as laser-induced cavitation. While these bubbles initially allowed for

controlled study of cavitation events near boundaries, other possibilities for their use soon

emerged, and continue to be developed. Since many of the newer uses for laser-induced

cavitation have only recently been developed or are in the early stages of investigation, bubble

behavior under conditions not previously studied, such as much higher bubble energies, needs to

be studied.

1.6.1 Research  Objectives  

Based on the literature currently available, there is a wealth of knowledge about cavitation

events near surfaces. However, less information is available on laser-induced cavitation bubbles

in a bulk fluid and how to control the bubble dynamics. Therefore this research focuses on

controlling bubble development through the air content of the bulk fluid, pulse energy used to

generate bubbles, and the focusing angle of the lens used to produce bubbles. The combination

of these elements will influence bubble size, shape, and temporal development. Understanding

  14  

the effects of each variable will shed light on how to produce bubbles with a desired size, shape

or lifetime. This thesis presents the results of experiments done to reveal the effects of each

variable, as well as scaling analysis to guide the selection of lens, laser power, and air content to

produce bubbles with desired characteristics.

  15  

2 Experimental Methods

2.1 Test apparatus

2.1.1 Water Tank

Cavitation bubbles were produced in a tank using a focused laser beam. A laser beam

passed through a beam expander, which was used to expand and collimate the laser. The laser

beam diameter was expanded from 6 mm to the focusing lens diameter using a beam expander.

The expanded beam was then directed through the focusing lens, forcing the laser light rays to

converge at the focal point of the lens. The tank used was 0.30 m long, 0.15 m wide and 0.2 m

tall. The tank was filled to 0.1 m with water, and the bubbles were formed at a depth of 0.05 m

below the water surface. Figure 1 shows the distance between the components in the

experimental setup. Here A is the distance between the beam expander outlet and the focusing

lens, B is the distance between the focusing lens and the front wall of the tank, C is the distance

between the cavitation bubbles and the camera, D is the distance between the focusing lens and

the point where the cavitation bubble is produced, E is the distance between the cavitation bubble

and the bottom of the tank, and F is the depth of water in the tank. The distances between the

components for each test condition are given in Table  1. The lens angle referred to in this work

as the convergence angle is often referred to as the angular spread. This is equal to twice the

beam divergence, which is a term often used when discussing lasers (see Figure 2 for an

illustration of each angle).

  16  

Figure 1 – Experimental setup used to generate laser-induced cavitation bubbles. The laser

beam is expanded and then focused through a lens. A camera was used to capture the

cavitation bubbles. The dashed box in the side view represents the camera’s nominal field-

of-view.

  17  

Table 1 – Distances between components in experimental set up in Figure 1.

Lens angle

(nominal)

Air content

(ppm)

A

(m)

B

(m)

C

(m)

D

(m)

E

(m)

F

(m)

5° 10 0.08 0.14 0.24 0.25 0.05 0.10

5° 15 0.08 0.14 0.24 0.25 0.05 0.10

5° 20 0.11 0.14 0.16 0.25 0.05 0.10

10° 10 0.15 0.15 0.13 0.25 0.05 0.10

10° 15 0.15 0.15 0.13 0.25 0.05 0.10

10° 20 0.11 0.17 0.13 0.22 0.05 0.10

20° 10 0.25 0.02 0.24 0.14 0.05 0.10

20° 15 0.25 0.02 0.24 0.14 0.05 0.10

20° 20 0.25 0.02 0.24 0.14 0.05 0.10

  18  

Figure 2 – This figure illustrates the relationship between beam divergence and angular

spread. In this work, the convergence or focusing angle refers to the angular spread.

Table 2 – Lens specifications used to create bubbles

Lens angle

(nominal)

Lens angle

(actual)

Focal length,

f (mm)

Diameter,

D (mm)

Ratio

(f/D)

5° 4.3° 200 20 10

10° 10.7° 200 50 4

20° 21.0° 100 50 2

2.1.2 Laser Optics

Cavitation bubbles were successfully produced in the tank of stagnant fluid (water) by

focusing the beam of a Nd:YAG laser (Gemini PIV, New Wave Research) as shown in Figure 1.

  19  

The laser was operated at 532 nm wavelength with a nominal 6 mm diameter beam. The beam

diameter was expanded using a beam expander (NT64-419, Edmund Optics), which was adjusted

to produce a collimated beam at the diameter appropriate for the given focusing lens (see Table  

1). This beam was then focused into the water tank with varying levels of air content with the use

of three different focusing lenses (see Table 2). The laser was pulsed regularly at 15 Hz. The

fact that the laser was pulsed regularly at 15 Hz may have introduced some scatter into the

results of this experiment. Pulsing the laser at such a rate may cause the temperature around the

beam focal point to increase relative to the bulk fluid temperature due to residual heating from

earlier pulses. If single pulses were used instead of regular pulses, a larger separation between

test samples would have been produced allowing the local fluid temperature to return to the bulk

fluid temperature. This could have reduced the scatter in the results. The energy of each laser

pulse could be varied by changing the flash lamp power and/or the laser attenuation. In the

current study the laser flash lamp was fixed at the maximum power and the laser pulse power

was varied with the attenuator setting. The lower end of the power range for the laser used was

selected such that cavitation was barely audible to the human ear. The focusing angle of the lens

can have a great impact on the bubble shape. In particular, a narrow focusing angle can result in

the formation of multiple bubbles at one time along the laser path. A wider focusing angle, or

cone, promotes the formation of a single bubble. The beam-focusing angle is a function of the

laser beam initial diameter, the lens focal length and the optical path of the laser beam. To

achieve a wider focusing angle the initial beam diameter should be maximized, which results in

the limiting factor being the diameter of the focusing lens. The shorter the lens focal length the

wider the resulting focusing angle. However, this also limits the distance between the lens and

the cavitation bubble. The dependence on the optical path results in the beam angle being

  20  

sensitive to the fluid in which the bubble is being generated, which is due to Snell’s law based on

the refractive index of the medium.

The beam angle in water for each lens was calculated using Snell’s law (see Figure 3).

Snell’s law says that the angle of incidence of a ray on a boundary times the refractive index of

the medium through which the light propagates is constant or

𝑛! sin𝜃! = 𝑛! sin𝜃!  .

Since the laser light travels first through air, then glass, and then water, three different materials

must be accounted for,

𝑛! sin𝜃! = 𝑛! sin𝜃! =  𝑛! sin𝜃!.

All of the indices of refraction (n1, n2 and n3) are known. The initial angle of incidence θ1 is

calculated based on the known diameter and focal length of the lens,

𝜃! = tan!! !!!

,

where D is the diameter of the lens and f is the focal length. From this information the angle in

water, θ3, can be determined.

  21  

Figure 3 – Snell’s law applied to a ray path passing through air (n1 =1.00), glass (n2 =1.55)

and water (n3 =1.33). Increasing index of refraction bends the beam towards the axis

perpendicular to the interface between the two mediums.

For this work, three lenses were used with focusing angles of approximately 20°, 10° and

5° in water. These angles were chosen in order to explore the effects of using a relatively small,

medium, and large focusing angle on bubble size and shape. The 20° lens had a diameter of 50

mm and focal length of 100 mm, providing a ratio of focal length to diameter of 2. This is

comparable to previous work that has produced single, spherical bubbles. One such example is

from the work of Giovanneschi and Dufresne (1985) who used a lens arrangement with a focal

length to diameter ratio of 1.9. The small angle lens (22 mm diameter, 200 mm focal length) was

selected due to previous experiments that demonstrated the appearance of multiple bubbles

(Giovanneschi & Dufresne, 1985). The medium angle lens was selected so that the angle was

  22  

between the narrow and wide-angle, and a 50 mm diameter lens with 200 mm focal length was

selected.

2.2 Instrumentation

2.2.1 Imaging setup

A high-speed video (HSV) camera (Memrecam GX-3, NAC) was used to capture images

of the cavitation bubbles forming and collapsing in the water tank. The camera was mounted

perpendicular to the laser beam path, looking into the tank. A filter was used on the camera to

filter out the 532 nm wavelength light from the laser. A ring of halogen lights was mounted on

the opposite side of the tank to backlight the images. A sheet of white paper was hung between

the tank and the halogen light source to evenly diffuse the background light for the HSV images.

The camera was connected to the computer via an Ethernet cable. Images were captured and the

camera controlled via custom software (MEMRECAM GXLink). For lower power, images were

acquired with a frame size of 64 x 64 pixels at a frame rate of 80,000 frames per second. At

higher power (attenuator set at 200 and above) the image size was 96 x 64 pixels, and the frame

rate was 75,092 fps. The larger frame size was needed at higher power as the bubbles tend to be

larger, particularly in the horizontal direction as multiple bubbles merge together. The laser was

pulsed at 15 Hz while the HSV captured images of the cavitation event. The maximum frame

rate was dependent on the frame size (the smaller the frame size the higher the maximum frame

rate). The frame rate needed to be sufficiently high to observe the cavitation event through its

initial formation, growth and collapse. The final frame size was determined by slowly reducing

the frame size such that the cavitation bubble filled as much of the frame as possible to observe it

in the greatest detail while maintaining a sufficiently high frame rate.

  23  

In order to determine the bubble size, a target of known dimensions was placed in the

tank and calibration images were acquired. The calibration target was a black sheet with white

dots spaced 2.54 mm apart. The calibration image was used to determine the scale and therefore

the bubble dimensions. The physical distance between dots in the target image is known. Using

the measurement tool in the MEMRECAM GX Link software, the number of pixels between

each dot can be measured. Therefore the pixels per millimeter could be determined. The bubbles

were measured using the measuring tool to find the dimensions in terms of pixels, and the

measurement was then multiplied by the value determined from the calibration image to convert

the number of pixels to millimeters.

2.2.2 Laser power

The laser energy per pulse is controlled by the flash lamp and the attenuator on the

Gemini PIV Nd:YAG laser. Increasing either the flash lamp or the attenuator level can increase

the laser pulse energy. For this work, the flashlamp was set to the maximum value and held

constant and the attenuator was adjusted to vary the power. The laser energy per pulse in mJ was

extrapolated from existing information on the maximum energy per pulse of the laser and the

attenuation curve associated with the laser. The maximum laser power was determined to be 129

mJ, based on calibration information corresponding to the specific Gemini PIV laser used in this

experiment. The attenuation curve was acquired from the New Wave Research website. The

attenuation curve indicates the percent of the maximum energy that is reached with each

attenuation step of 100, where the attenuator can be set from 0 to 1000. Combining these two

pieces of information, the curve shown in Figure 4 was computed.

  24  

Figure 4 – Energy per pulse as a function of attenuator setting for the laser used in the

current study. The flash lamp was fixed at the maximum level.

2.2.3 Water dissolved gas content

Felix & Ellis (1971) used a laser to induce cavitation in tap water, deionized water, and

methyl alcohol. This work demonstrated that impurities in a liquid increase the number of “hot

spots” where plasma can form in the liquid. Similarly, the amount of air dissolved in the water

can also potentially affect the size and shape of the bubble. More air provides more nucleation

sites for cavitation bubbles and therefore the bubble size and shape may be affected at a given

laser power level. More nucleation sites may result in multiple bubbles being produced and

bubbles being formed at a lower laser power.

The air content of the bulk fluid was measured using a Van Slyke apparatus. A 10 ml

sample of the bulk fluid with unknown air content was tested in the Van Slyke apparatus to

measure the air content. First, a vacuum is applied to the sample and the sample is agitated for 4

0  

20  

40  

60  

80  

100  

120  

140  

0   200   400   600   800   1000   1200  

Energy  per  pulse  (m

J)  

A1enuator  se5ng  

  25  

minutes to separate the dissolved air in the original sample from the water. The vacuum was

produced by varying the height of a volume of mercury relative to the sample. The water and gas

are then forced into a 2 cm3 volume cavity. Now the air occupies a known volume at a given

pressure (measured from height of the mercury) and temperature (measured with a thermometer).

The ideal gas law is used to determine the volume at a standard temperature, which is then used

to determine the air content of the sample (Carl, 1977). The water air content is presented in

units of parts-per-million (ppm), which was defined in terms of mass parts.

In this work, the water air content concentrations are referred to herein by their nominal

values (10, 15 and 20 ppm). The average measured values for each condition are provided in

Table  3. The de-aerated (low air content) water was obtained by using a vacuum. A large tank

was filled with tap water. A vacuum was pulled on the tank and several hours elapsed. After

several hours water was drained from the bottom of the tank. Water taken directly from the tap

was used for the high air content condition. The intermediate air content condition was achieved

by leaving the deaerated water overnight.  

Table 3 – Average water-air content concentration (C) for each air content condition

Air content condition C (nominal)

(ppm)

Average C (actual)

(ppm)

Low 10 11.8

Intermediate 15 13.1

High 20 19.3

  26  

2.3 Test Matrix

The objective of the current study was to characterize laser-induced cavitation bubbles. It

was assumed that the cavitation bubble’s formation, growth and collapse is dependent upon the

optical arrangement (e.g. laser power, beam focusing angle) and bulk fluid properties (e.g. fluid

density, temperature, dissolved gas content, turbidity and pressure). In the current study the laser

beam focusing angle, laser pulse energy, and bulk liquid air content were varied. The pressure

was held constant at slightly above atmospheric pressure (cavitation bubble location was

approximately 50 mm below the tank free-surface), the bulk fluid temperature was held constant

at room temperature (approximately 25° C) and the bulk fluid used was tap water (density = 997

kg/m3). The test matrix varied the beam focusing angle from 5° to 20°, the laser pulse energy

varied between 0 and 42 mJ/pulse and the water air-content was varied between 10 and 20 ppm.

Future work will expand upon the current study to assess the influence of pressure, bulk fluid

density and distance from a solid surface.

2.4 Measurement Uncertainty

All measurements involve a degree of uncertainty that contributes to error and limits the

degree of accuracy of the results. In this section, the measurement uncertainty associated with

each component of the experimental set up is discussed, and the effect that this uncertainty has

on the final results of the analysis will be touched on in Chapter 4.

2.4.1 Camera

The camera setup involved uncertainty in both space and time. The spatial uncertainty

was due to the way in which the bubble diameters were measured from the images produced by

the camera. For each frame, the left, right, top and bottom of the bubble were identified visually

using the human eye, and then the measurement tool was used to determine the distance between

  27  

the left and right or top and bottom of the bubble. The measurement uncertainty for the bubble

length is 4 pixels (±2 pixels from the measured diameter). Based on the setup with the calibration

image that resulted in the greatest number of millimeters per pixel, the diameter measurements

for the cavitation bubbles have an uncertainty of ±0.158 mm. The temporal uncertainty has two

parts, the temporal resolution and the exposure time. In this case, the exposure was set equal to

the time between frames. Therefore the maximum possible uncertainty is associated with the

temporal resolution, which is the frame rate ±½ of the time between frames. With a frame rate of

75,092 fps, half a frame is equivalent to 6.6585 µs.

2.4.2 Laser

The laser provides another source of uncertainty. Each pulse can have slightly different

energies even at the same attenuator and flash lamp settings, resulting in bubbles that are

produced at slightly different energies. Residual heating in the area where cavitation is produced

could also cause bubbles to be produced at different energy levels even when the laser settings

are kept the same. Due to the way in which the energy pulse versus attenuator setting curve

(Figure 4) was obtained, this work assumes a measurement uncertainty for the laser pulse energy

of ±10%. For the highest laser pulse energy used, 42 mJ, the pulse energy would be 42 ± 4.2 mJ.

2.4.3 Lenses

The lens-focusing angle was calculated via Snell’s law, which assumes an ideal lens. For

these calculations the diameter and focal length of the lenses were used. Since the lenses are not

ideal lenses there will be some distortion towards the outer edges of the lenses, and therefore the

diameter is assumed to have an uncertainty of about 10%. The manufacturer (Edmund Optics)

specified focal length tolerance for the 10° and 20° lenses is ±1%. The focal length tolerance for

the 5° lens is assumed to also be ±1%.

  28  

2.4.4 Pressure

The experimental uncertainty in the pressure is a function of how accurately the distance

from the surface of the water to the point of cavitation was measured. This distance was

measured using a tape measure and evaluated using the human eye. It is estimated that this

distance has an uncertainty of ±5 mm, which corresponds to ±49 Pa.

2.4.5 Air Content

The uncertainty in the air content measurements was estimated by comparing multiple

water samples taken from the same source. The standard deviations for the three different air

content conditions were calculated and are presented in Table 4. The standard deviation for each

condition is converted into a percentage, and the average percentage (12%) is used as the

uncertainty for all three conditions. Table   5 summarizes the uncertainty associated with each

measured quantity.

Table 4 – standard deviation for air content concentration measurements

Nominal air content (ppm) 10 15 20

Standard deviation 0.76 0.65 1.32

  29  

Table 5 – Measured quantities and their corresponding measurement uncertainty

Quantity Uncertainty

Bubble diameter Dv, Dh ±0.158 mm

Frame rate ±6.7 µs

Laser pulse energy ±10% (or ±4.2 mJ max)

Lens diameter ±5% (or ±2.5 mm max)

Lens focal length ±1% (or ±2 mm max)

Pressure ±49 Pa

Air content ±12% (average)

  30  

3 Experimental Results

3.1 General trends and overview

A few general trends can be seen in the cavitation bubbles generated under the different

conditions from the test matrix. Below the trends will be listed in a qualitative sense, and the

specific trends will be discussed in more depth throughout the rest of the chapter.

1. Increasing the power increases the bubble size

2. Increasing the power increases the bubble lifetime

3. Lower power produces more spherical bubbles

4. Increasing the power produces more bubbles at a time

5. Increasing the focusing angle of the lens decreases the bubble size

6. Increasing the focusing angle decreases the bubble lifetime

7. Increasing the focusing angle produces more spherical bubbles

8. Smaller focusing angles allow bubbles to be produced at lower power

While the bubbles produced under the different conditions exhibited different maximum sizes

and a range of shapes from spherical to elliptical, they all follow a somewhat similar

development from the point of inception when the laser pulse appears at the focal point (denoted

as time t = 0 for each bubble), and then collapsing until the bubble is no longer visible. Figure 5

and Figure 6 show the bubble size versus time for a bubble produced at relatively low and high

laser energy per pulse, respectively. Please note that the images of the bubbles have been edited

to enhance visibility of the bubble formations. Both figures were produced using the wide-angle

lens configuration in water with the intermediate air content level. The first, run 211, was

produced at a power of 7.3 mJ, and the second, run 205, was produced with a laser power of 42

mJ. These runs were selected to demonstrate how the profiles of the bubble lifetime change with

  31  

increasing power and show the general trends in bubble behavior. The white spot in Figure 5a

shows the plasma that is formed by the laser light, and the rest of the frames illustrate the

bubble’s behavior over time.

At lower powers, the bubbles produced tend to be relatively smaller, singular, spherical

and have a shorter lifetime. They do not rebound or oscillate in size (i.e. the bubble forms,

grows, collapses, and then is no longer observed). The wide-angle lens produced the smallest

bubbles, so a bubble produced at 7.3 mJ with the small angle lens configuration would be

significantly larger than the bubble produced at the same power with the wide-angle lens

configuration, and may have multiple bubbles present. Figure 5 would be more indicative of

bubbles produced at very low powers for the small angle lens (around 2 to 5 mJ).

Figure 6 shows the temporal development of a bubble produced at the highest laser

energy used in this experiment (42 mJ). Compared to Figure 5 it illustrates a few key differences

in bubble behavior. Clearly, the bubble lifetime is longer (about 300 µs compared with around 65

µs) and the bubble size has increased, nearly doubling. The appearance of multiple bubbles and

the way they merge together to form larger, oblong bubbles is a trend at high powers. Another

interesting feature is that the bubble has two peaks in bubble diameter, with the first peak

occurring at 66 µs with a bubble diameter of 1.25 mm, and a second peak occurring at 187 µs

with a diameter of 1.17 mm. This is a significant difference between bubbles generated at higher

versus lower power. The higher power bubbles tend to rebound and oscillate while the lower

power bubbles simply collapse.

  32  

Figure 5 – Example of a bubble lifetime produced at relatively low laser power (7.3 mJ per

pulse) with the wide-angle lens configuration and intermediate air content level. Labelled

data points correspond to labeled images at the top of the figure. In image A the bright

white spot is produced from plasma generated by the focused laser beam, while the

remaining images show the shadow produced from the backlighted bubble.

  33  

Figure 6 – Example of a bubble lifetime produced at relatively high laser power (42 mJ per

pulse) with the wide-angle lens configuration and intermediate air content level. Labelled

data points correspond to labeled images at the top of the figure. The growth, collapse and

rebound of the bubble is apparent from the plot.

Figure 7 compares how bubble generation and growth differ for bubbles produced by all

three lens configurations at the upper and lower bounds of the laser pulse energy range tested.

Although the figure shows bubbles produced under different air content conditions, the

comparison of these bubbles is appropriate since it is shown subsequently that the results have

negligible sensitivity to the air content levels. In Figure 7 the run number is listed and the lens

condition is indicated by SA, MA or WA standing for small angle, medium angle, and wide-

  34  

angle, respectively. For low levels of power, all three lenses created single, spherical bubbles, as

can be seen in Figure 7 A, B, and C. The bubbles produced by the small angle lens at high

powers produce noticeably different shapes from bubbles produced by the medium and wide-

angle lens. At 42 mJ, all three lenses form a string of multiple bubbles along the path of the laser

beam, as illustrated in Figure 7 Da, Ea, and Fa. The bubbles produced by the small angle lens

coalesce and form a single larger bubble, which continues to grow outward in the vertical

direction. The medium and wide-angle lenses both produce a significant number of smaller

bubbles along the beam path (see Figure 7 D, E and F). Some of these combine to form larger

bubbles, while others grow as single bubbles. Figure 7 Db, Eb, and Fb show what the bubble

configurations look like at the time of maximum diameter. This results in a combination of

individual bubbles and longer bubbles that are two or three times the length of the single bubbles

in the horizontal direction, but are similar in diameter in the vertical direction. These trends are

discussed further in the following sections.

  35  

Figure 7 – Comparison between low and high power bubble behavior. At lower pulse

energies single, spherical bubbles are produced. At higher powers the bubble shape

depends on the lens angle. The smaller angle lens produces one larger bubble formed from

bubble coalesence. The wider angle lenses produce smaller, more elongated bubbles. In the

figure SA, MA and WA refer to the small, medium and wide-angle lens configurations,

respectively.

 

3.2 Repeatability

The bubble diameter was measured using the measurement tool available in the

MEMRECAM software. The measurements were made by clicking on the left and right-most

  36  

points of the bubble for the horizontal diameter and the top and bottom of the bubble for the

vertical diameter. The distance between the two points was then given in pixels, which was then

converted to millimeters using the calibration images. The time from frame to frame (in µs) was

determined based on the camera frame rate. When tracking the bubble diameter over time, it was

decided that the vertical diameter, and not the horizontal diameter, would be tracked. The

vertical diameter was selected due to the variability in the horizontal diameter and formation of

multiple bubbles that coalesce along the beam axis (horizontal direction). Since bubbles at higher

power sometimes coalesced into a larger bubble that was longer in the horizontal direction but

sometimes formed many individual bubbles, the vertical diameter was determined to be a more

reliable measure of bubble growth. However, comparison between the vertical and horizontal

bubble dimensions is provided subsequently. Whether many individual bubbles formed along the

path of the laser beam or one large bubble developed, the vertical bubble diameters were in the

same range.

Due to the availability of the laser, these data were collected in two phases, thus requiring

that the equipment be set up twice. Some differences in the distance between elements in the

experiment resulted, most notably the distance between the camera and the tank wall, though

other distances had some variation. Ideally all the data would have been collected at once, but

comparison of the two phases allows for an assessment of the repeatability of the experiment.

The narrow angle lens, intermediate air content condition at laser energy per pulse ranging from

20 to 50 mJ was repeated in both phases of testing. This was done to make sure that comparisons

could be made across the data sets collected at different points in time. The results for maximum

vertical diameter versus laser power are shown in Figure 8. The blue points represent values

obtained in the first phase, and the red squares indicate values obtained in the second phase.

  37  

Generally the two data sets overlap adequately. The average percent difference between data

collected in February and March was calculated to be about 7%, with values in March being on

average 7% lower than in February. There appears to be an outlier in the February data, where a

bubble had a diameter of ~1.8 mm when produced with a pulse energy of 42 mJ. The cause for

this outlier is unknown, but excluding it from the calculation the average percent difference

between data collected in February versus March drops to just 5%. The results support

comparison between bubbles that are produced in different experimental setups. The small

difference in bubble sizes between the two data sets allows conclusions about the effects of the

parameters being tested to be drawn.

Figure 8 – Comparing bubble diameter, there is reasonable agreement between data

collected in different setups from different points in time. This indicates that comparing

results between the two different setups is appropriate.

0.0  

0.2  

0.4  

0.6  

0.8  

1.0  

1.2  

1.4  

1.6  

1.8  

2.0  

0   5   10   15   20   25   30   35   40   45  

Ver7cal  diameter  (m

m)  

Pulse  energy  (mJ)  

first  set  (Feb)  

second  set  (Mar)  

  38  

3.3 Variation of bubble topology

When bubbles are generated near the minimum energy required to produce visible

cavitation, the cavitation bubbles that form are single, spherical bubbles for all the test

conditions. At higher power, the bubbles behave differently based on the lens used to generate

the bubbles. In fact, even for bubbles generated with the same lens configuration, there can be

pulse-to-pulse variation in the bubble generation, growth, and collapse. This is particularly

notable for the medium and wide-angle lens configurations.

The bubbles produced at the highest energy, 42 mJ, with the narrow angle lens

configuration showed minimal pulse-to-pulse variation. Figure 9 shows 5 different runs, and

what the bubble(s) looked like at the point of maximum vertical diameter. For all five runs, the

laser pulse produces around 5 small bubbles. These bubbles then coalesce into a single larger

bubble that grows in the vertical direction. While the bubble in Ah does not manage to form a

bubble as uniform in vertical diameter as the other runs, it still follows the general trend of the

bubbles coming together to form a single, elliptical bubble.

  39  

Figure 9 – Narrow angle lens bubble patterns at high power (42 mJ) in high air content

water. Smaller bubbles formed along the path of the laser beam coalesce to form a single,

larger, elliptical bubble.

As the laser-focusing angle became wider, the bubble behavior exhibited greater variety

and became less predictable. Figure 10 and Figure 11 show the bubble development at 42 mJ for

the medium and wide-angle lens configurations, respectively. Figure 11 also shows the bubbles

produced using the wide-angle lens under all three air content conditions. For each run, the

bubbles are shown shortly after the laser pulse and then at the point of maximum diameter. This

  40  

is done to illustrate how the bubbles change from individual bubbles into other configurations

over time. For all of the runs, many small bubbles are initially generated along the path of the

laser beam. Over time, the bubbles grow in different ways. A couple of different behaviors are

observed:

1. The individual bubbles that are produced initially may continue to expand as individual

bubbles, as demonstrated in Figure 10 f and h, as well as Figure 11 Af and Bh.

2. The bubbles may coalesce to form one very long bubble, which is illustrated in Figure 10

j as well as Figure 11 Ad, Af and Cj.

3. Some of the bubbles may coalesce while others grow as individual bubbles, as in Figure

10 b, and Figure 11 Ah, Aj, Bj and Cg.

The resulting bubbles may be smaller, as in Figure 10 h, or larger, as in Figure 10 j. The variety

of behaviors causes difficulty in quantifying the bubble time history, especially when multiple

bubbles coalesce during the lifetime.

  41  

Figure 10 – Medium angle lens bubble patterns at high power (42 mJ) in intermediate air

content water. On the left is the bubble formation shortly after the laser pulse, on the right

is the bubble formation at the time of maximum bubble diameter.

  42  

Figure 11 – Wide-angle bubble patterns at high power. On the left hand side of (A), (B),

and (C) is the bubble formation shortly after the laser pulse, and on the right hand side of

(A), (B) or (C) are the bubble formations at the points of maximum bubble diameter. These

images illustrate the variety of bubble patterns that can be formed.

3.4 Bubble size

3.4.1 Bubble size: sensitivity to beam angle

The sensitivity of the bubble diameter to beam angle was investigated by plotting the

bubble diameter versus laser energy for each lens angle configuration while holding the air

content constant. Bubbles produced by the three different lens angles produced bubbles of

different sizes. As mentioned earlier, the small angle lens produced larger bubbles at lower

  43  

power compared to the medium and wide-angle lenses. Bubbles were first visible at lower

energies using the smaller lens. In Figure 12 the maximum vertical bubble diameter for each

bubble is plotted on the ordinate while pulse energy minus the minimum energy needed to

produce a visible bubble for each lens condition is plotted on the abscissa (E – E0). Table 6

shows E0 for each of the conditions tested. This allows for comparison between the bubble size

at each laser power level for the three lens conditions, and allows a best-fit curve to be fit with a

power function.

Table 6 – Lowest laser pulse energy for which bubbles could be seen for each of the test

conditions, E0

Low air content

Intermediate air

content High air content

Small Angle 2.7 mJ 2.7 mJ 2.4 mJ

Medium Angle 5.8 mJ 3.5 mJ 5.8 mJ

Wide-angle 7.3 mJ 3.5 mJ 5.8 mJ

In general, as the laser power increases the bubble diameter also increases. The data are

fitted with a best-fit, power-law curve. For all of the low, intermediate and high air content

conditions, the small angle lens produced the largest bubbles (Figure 12, note the red squares

indicating the small angle lens condition). For the low and intermediate air content conditions,

the bubble size appears to asymptote at a bubble size close to 1.4 mm with the small angle lens.

For the high air content condition the asymptote for the small angle lens appears to be slightly

higher, possibly around 1.6 mm. The medium and wide-angle lens configuration asymptotes

appear to be at about 1.2 mm. In general, bubble sizes are extremely close for the intermediate

  44  

and high air content conditions with the medium or wide lens angle, with the best-fit curves

nearly overlapping at higher laser energy. Notably for the low air content condition the bubbles

created using the medium lens are larger than those produced with the wide-angle lens.

The ratio of the horizontal to vertical diameter at the point of maximum vertical diameter

is plotted against power in Figure 13 a, b, and c for the low, intermediate and high air content

conditions, respectively. This was done to examine how the bubble shape changed with power.

Based on the discussion earlier in 3.3, the bubbles may become less spherical and more

elongated as multiple bubbles coalesce at higher powers. This trend is apparent from Figure 13

as the ratio of horizontal to vertical diameter becomes larger and more scattered at higher power,

reflecting the appearance of the various bubble patterns and coalescence of multiple bubbles.

It is interesting to see that the wide-angle lens produces bubbles that are more spherical at

lower powers while producing bubbles that are more elliptical at higher powers. This is

illustrated in Figure 13 by the way that the green triangles representing the wide-angle lens start

out below the points for the small and medium angle lenses and then end up above these points at

high energies. Figure 13 shows that as the pulse energy increases the bubbles become less

spherical. This also suggests that the wide-angle lens can be used to produce spherical bubbles at

higher laser pulse energies compared to the smaller angle lens.

  45  

0.00  

0.20  

0.40  

0.60  

0.80  

1.00  

1.20  

1.40  

1.60  

0.00   5.00   10.00   15.00   20.00   25.00   30.00   35.00   40.00   45.00  

Ver7cal  diameter  (m

m)  

E  -­‐  E0  (mJ)  

small  lens  angle/low  AC   medium  lens/low  AC   wide  lens/low  AC   (A)  

  46  

0.00  

0.20  

0.40  

0.60  

0.80  

1.00  

1.20  

1.40  

1.60  

0.00   5.00   10.00   15.00   20.00   25.00   30.00   35.00   40.00   45.00  

Ver7cal  diameter  (m

m)  

E  -­‐  E0  (mJ)  

small  angle/med  AC   medium  angle/med.  Air  content   wide  angle/  med  air  content  (B)  

  47  

Figure 12 – Vertical bubble diameter as a function of laser energy per pulse for (A) low, (B)

intermediate and (C) high air content conditions. The solid lines represent the best-fit

curves for each condition.

0.00  

0.20  

0.40  

0.60  

0.80  

1.00  

1.20  

1.40  

1.60  

1.80  

0.00   5.00   10.00   15.00   20.00   25.00   30.00   35.00   40.00   45.00  

Ver7cal  diameter  (m

m)  

E  -­‐  E0  (mJ)  

small  angle/high  AC   medium  angle/high  AC   wide  angle/high  AC   (C)  

  48  

0.00  

0.20  

0.40  

0.60  

0.80  

1.00  

1.20  

1.40  

1.60  

1.80  

2.00  

0   5   10   15   20   25   30   35   40   45  

Horizon

tal  diameter/V

er7cal  diameter  

 

Pulse  energy  (mJ)  

small  angle  lens   Series1   wide  angle  lens  

0.00  

0.20  

0.40  

0.60  

0.80  

1.00  

1.20  

1.40  

1.60  

1.80  

2.00  

0   5   10   15   20   25   30   35   40   45  

Horizon

tal  diameter/V

er7cal  diameter  

 

Pulse  energy  (mJ)  

small  angle  lens   med  lens   wide  angle  lens  

(A)  

(B)  

  49  

Figure 13 – Ratio of the horizontal to vertical bubble diameters plotted as a function of the

laser pulse energy with (A) low, (B) intermediate and (C) high air content levels.

3.4.2 Bubble size: sensitivity to air content

The air content was varied from 10 ppm (low air content) to 20 ppm (high air content),

with a single intermediate air content level (15 ppm). To observe the effects of air content on

bubble size, Figure 14 presents the maximum vertical bubble diameter plotted versus the

difference between the laser energy and the minimum energy required to produce visible bubbles

for each of the three lens configurations (narrow, medium and wide-angle). Each graph has data

for all three levels of air content. As far as the air content, there does not appear to be a large

variation in the maximum ratio of horizontal to vertical diameter (see Figure 15). For the low air

content condition the maximum ratio is about 1.6, while for the medium and high air content

conditions the maximum ratio is about 2. Overall the air content did not have as significant of an

effect on bubble size as the lens configuration. However, a very interesting observation is that

0.00  

0.20  

0.40  

0.60  

0.80  

1.00  

1.20  

1.40  

1.60  

1.80  

2.00  

0   5   10   15   20   25   30   35   40   45  

Horizon

tal  diameter/V

er7cal  diameter  

 

Pulse  energy  (mJ)  

small  angle  lens   med  lens   wide  angle  lens   (C)  

  50  

the intermediate air content level resulted in the largest diameter bubbles. This is further

investigated in section 3.5.

The average ratio of the horizontal to vertical diameter at the time of maximum vertical

diameter is plotted in Figure 15 where A shows the ratios produced by the small angle lens at all

three air content conditions, B shows the ratios produced by the medium angle lens, and C shows

the ratios produced by the wide-angle lens. The general trend is that the ratio of horizontal to

vertical diameter is smaller, close to 1 indicating spherical bubbles, for low powers while the

ratios tend to be larger than one at higher powers. Within this general trend there appears to be a

lot of variability. For example in Figure 15 B and C around 23 mJ the bubble diameter ratio in

the intermediate air content condition jumps far above that of the low or high air content

conditions (to about 2 versus around 1.1).

0.00  

0.20  

0.40  

0.60  

0.80  

1.00  

1.20  

1.40  

1.60  

1.80  

2.00  

0.00   5.00   10.00   15.00   20.00   25.00   30.00   35.00   40.00   45.00  

Ver7cal  diameter  (m

m)  

E  -­‐  E0  (mJ)  

low  air  content   medium  air  content   high  air  content   (A)  

  51  

0.00  

0.20  

0.40  

0.60  

0.80  

1.00  

1.20  

1.40  

1.60  

0.00   5.00   10.00   15.00   20.00   25.00   30.00   35.00   40.00   45.00  

Ver7cal  diameter  (m

m)  

E  -­‐  E0  (mJ)  

low  air  content   medium  air  content   high  air  content   (B)  

  52  

Figure 14 – Vertical bubble diameter plotted versus the laser pulse energy for (a) narrow,

(b) medium and (c) wide-angle lens configurations. Solid lines represent the best fit curves

to the data.

0.00  

0.20  

0.40  

0.60  

0.80  

1.00  

1.20  

1.40  

0.00   5.00   10.00   15.00   20.00   25.00   30.00   35.00   40.00   45.00  

Ver7cal  diameter  (m

m)  

E  -­‐  E0  (mJ)  

low  air  content   medium  air  content   high  air  content   (C)  

  53  

0.00  

0.50  

1.00  

1.50  

2.00  

2.50  

0   5   10   15   20   25   30   35   40   45  

Horizon

tal  diameter/V

er7cal  diameter  

Pulse  energy  (mJ)  

low  air  content   med  air  content   high  air  content  

0  

0.5  

1  

1.5  

2  

2.5  

0   5   10   15   20   25   30   35   40   45  

Horizon

tal  diameter/V

er7cal  diameter  

Pulse  energy(mJ)  

low  air  content   med  air  content   high  air  content  

(A)  

(B)  

  54  

Figure 15 – Ratio of horizontal to vertical diameter plotted versus laser pulse energy for (a)

narrow (b) medium and (c) wide beam angles.

3.5 Bubble half-life

3.5.1 Bubble half-life: sensitivity to beam angle

The bubble half-life was investigated as a function of power. The bubble half-life is

defined as the time from the initial laser pulse to the point at which the bubble reaches its

maximum diameter. The laser pulse provides a clear initial point in time for each bubble. The

bubble half-life is used as a measure of time because it was a time that could be compared for

each bubble with minimal uncertainty. The half-life is useful because once the bubble reaches its

maximum diameter the bubble interface velocity is zero, as the bubble growth is changing from

positive to negative. This gives a well-defined reference time that can be compared across all the

bubbles produced in this work. Using the full lifetime of the bubble (from when it appeared to

when no sign of it was visible in the frame) was determined to be less comparable across data

0.00  

0.50  

1.00  

1.50  

2.00  

2.50  

0   5   10   15   20   25   30   35   40   45  

Horizon

tal  diameter/V

er7cal  diameter  

Pulse  energy(mJ)  

low  air  content   med  air  content   high  air  content   (C)  

  55  

sets due to the subjective nature of determining exactly when the bubble was gone from the

frame. Using the full bubble lifetime also introduces other complications, as some bubbles

rebound and oscillate. This would result in some bubble lifetimes (e.g. bubbles produced with

lower energy pulses) representing one bubble growth and decay, while others (e.g. bubbles

produced at the higher end of the energy spectrum) representing multiple cycles of collapse and

rebound. That being said, the majority of bubbles had a total lifetime between 200 and 300 µs.

The half-life sensitivity to beam angle was investigated by creating three plots, one for

each air content condition, where each plot showed the bubble half-life versus power for the

three different lens angles. For all of the air content conditions, the wide-angle lens configuration

produced the shortest bubble half-lives. This is related to the fact that the wider lens produced

smaller bubbles than the other two configurations. However, as the power increased the

difference between the half-life of bubbles produced with all three lenses appear to reach similar

lengths of time, and the difference in half-life becomes less pronounced. For all of the bubbles,

the maximum half-life is between 60 and 70 µs.

For the medium and high air content conditions the medium and wide-angle lens

produced bubbles with very similar half-lives at all powers. In the low air content scenario, the

medium angle lens appears to produce bubbles with significantly larger half-lives than the wide-

angle lens. For the low air content condition the medium angle lens produces bubbles with a

half-life slightly larger than the half-lives of bubbles produced with the small angle lens for

powers between 10 and 30 mJ. This is interesting because the small angle lens produced larger

bubbles for all levels of air content.

  56  

0.00  

10.00  

20.00  

30.00  

40.00  

50.00  

60.00  

70.00  

0   5   10   15   20   25   30   35   40   45  

Time  of  m

ax  diameter  (μ

s)  

 

Laser  pulse  energy  (mJ)  

small  angle  lens   med  angle  lens   wide  angle  lens  

0.00  

10.00  

20.00  

30.00  

40.00  

50.00  

60.00  

70.00  

0   5   10   15   20   25   30   35   40   45  

Time  of  m

ax  diameter  (μ

s)  

 

Laser  pulse  energy  (mJ)  

small  angle  lens   med  lens   wide  angle  lens  

(A)  

(B)  

  57  

Figure 16 – Bubble half-life (time from laser pulse to achieve maximum bubble diameter)

plotted versus the laser pulse energy with (a) low, (b) intermediate and (c) high or

saturated air content levels.

3.5.2 Bubble half-life: sensitivity to air content

The effect of air content on bubble half-life is also of interest. To determine the bubble

half-life’s sensitivity to air content level, three graphs are provided in Figure 17 for each lens

angle configuration. Data for the low, intermediate, and high air content levels were plotted on

each graph. Overall, air content level appears to have minimal influence on the bubble half-life.

All the data sets display similar trends, with the low, intermediate, and high air content

conditions following the same curves. This is most significant in the small angle lens conditions,

where the data points appear to fall very closely on the same curve. The medium lens angle plot

shows increased scatter, especially at lower laser power. The wide-angle lens shows some

0.00  

10.00  

20.00  

30.00  

40.00  

50.00  

60.00  

70.00  

80.00  

0   5   10   15   20   25   30   35   40   45  

Time  of  m

ax  diameter  (μ

s)  

 

Laser  pulse  energy  (mJ)  

small  angle  lens   med  lens   wide  angle  lens   (C)  

  58  

sensitivity to air content. For the wide-angle lens plot, the low air content condition shows

noticeably shorter bubble half-lives relative to the medium and high air content conditions.

  59  

0.00  

10.00  

20.00  

30.00  

40.00  

50.00  

60.00  

70.00  

80.00  

0   5   10   15   20   25   30   35   40   45  

Time  of  m

ax  diameter  (μ

s)  

Laser  pulse  energy  (mJ)  

low  air  content   med  air  content   high  air  content  

0  

10  

20  

30  

40  

50  

60  

70  

0   5   10   15   20   25   30   35   40   45  

Time  of  m

ax  diameter  (μ

s)  

Laser  pulse  energy  (mJ)  

low  air  content   med  air  content   high  air  content  

(A)  

(B)  

  60  

Figure 17 – Bubble half-life as a function of laser pulse energy with the (A) narrow, (B)

medium and (C) wide-angle lens configuration.

3.6 Bubble diameter time history

The relationship between bubble diameter and time was examined in order to gain an

understanding of how bubbles produced under different conditions acted, and how to compare

their behaviors. Figure 5 and Figure 6 provide examples of individual lifetime curves. In order to

compare bubbles produced at different energies, the diameter and time were each

nondimensionalized. The diameter was normalized by the maximum diameter of the bubble, and

the time was scaled with the half-life (the time from the initial laser pulse to when the bubble

reaches its maximum diameter). Each of the plots in Figure 18 through Figure 26 provide the

scaled bubble time histories for a specific air content and lens angle condition. For each

condition the development of one bubble at each laser pulse energy level was tracked over time

0.00  

10.00  

20.00  

30.00  

40.00  

50.00  

60.00  

70.00  

0   5   10   15   20   25   30   35   40   45  

Time  of  m

ax  diameter  (μ

s)  

Laser  pulse  energy  (mJ)  

low  air  content   med  air  content   high  air  content   (C)  

  61  

and plotted. Therefore each different marker represents a single bubble’s normalized size versus

scaled time.

For all of the conditions tested, there appears to be convergence of the data along a curve

for the first part of the bubble lifetime, corresponding to twice the half-life (t/th = 2). After two

half-lives the behavior is much more inconsistent. This is probably due to the variety of bubble

patterns that form at high powers, especially when the bubble rebounds after the initial collapse.

Based on the convergence within the first two half-lives, curves were fit to each of the data sets

representing the bubble behavior from inception to two half-lives in time.

Figure 18 – Small angle, low air content lifetime curve

0  

0.2  

0.4  

0.6  

0.8  

1  

1.2  

0   0.5   1   1.5   2   2.5   3   3.5   4   4.5  

D/Dm

ax  

t/th  

2.68  mJ   3.51  mJ   4.53  mJ   5.78  mJ   7.30  mJ   11.26  mJ  

16.63  mJ   23.56  mJ   32.07  mJ   41.98  mJ   poly.  

  62  

Figure 19 – Small angle, intermediate air content lifetime curve

Figure 20 – Small angle, saturated condition lifetime curve

0  

0.2  

0.4  

0.6  

0.8  

1  

1.2  

0   0.5   1   1.5   2   2.5   3   3.5   4   4.5   5  

D/Dm

ax  

t/th  

11.26  mJ   16.63  mJ   23.56  mJ   32.07  mJ   41.98  mJ   2.68  mJ  

3.51  mJ   4.53  mJ   5.78  mJ   7.30  mJ   poly.  

0  

0.2  

0.4  

0.6  

0.8  

1  

1.2  

0   1   2   3   4   5   6  

D/Dm

ax  

t/th  

2.41  mJ   2.68  mJ   3.51  mJ   4.53  mJ   5.78  mJ   7.30  mJ  

11.26  mJ   16.63  mJ   23.56  mj   32.07  mJ   41.98  mJ   poly.  

  63  

Figure 21 – Medium angle lens, low air content lifetime curve

Figure 22 – Medium angle, intermediate air content lifetime curve

0  

0.2  

0.4  

0.6  

0.8  

1  

1.2  

0   0.5   1   1.5   2   2.5   3   3.5  

D/Dm

ax  

t/th  

5.78  mJ   7.30  mJ   11.26  mJ   16.63  mJ  

23.56  mJ   32.07  mJ   41.98  mJ   poly.  

0  

0.2  

0.4  

0.6  

0.8  

1  

1.2  

0   0.5   1   1.5   2   2.5   3   3.5   4   4.5  

D/Dm

ax  

t/th  

3.51  mJ   4.53  mJ   5.78  mJ   7.30  mJ   11.26  mJ  

16.63  mJ   23.56  mJ   32.07  mJ   41.98  mJ   poly.  

  64  

Figure 23 – Medium angle, saturated air content lifetime curve

Figure 24 – Wide-angle, low air content condition lifetime curve

0  

0.2  

0.4  

0.6  

0.8  

1  

1.2  

0   0.5   1   1.5   2   2.5   3   3.5   4  

D/Dm

ax  

t/th  

5.78  mJ   7.30  mJ   11.26  mJ   16.63  mJ  

23.56  mJ   32.07  mJ   41.98  mJ   poly.  

0  

0.2  

0.4  

0.6  

0.8  

1  

1.2  

0   0.5   1   1.5   2   2.5   3   3.5  

D/Dm

ax  

t/th  

7.30  mJ   11.26  mJ   16.63  mJ   23.56  mJ   32.07  mJ   41.98  mJ   poly.  

  65  

Figure 25 – Wide-angle, intermediate air content lifetime curve

Figure 26 – Wide-angle, saturated lifetime curve

0  

0.2  

0.4  

0.6  

0.8  

1  

1.2  

0   0.5   1   1.5   2   2.5   3   3.5   4   4.5   5  

D/Dm

ax  

t/th  

4.53  mJ   5.78  mJ   7.30  mJ   11.26  mJ   16.63  mJ  

23.56  mJ   32.07  mJ   41.98  mJ   poly.  

0  

0.2  

0.4  

0.6  

0.8  

1  

1.2  

0   1   2   3   4   5   6  

D/Dm

ax  

t/th  

4.53  mJ   5.78  mJ   7.30  mJ   11.26  mJ   16.63  mJ  

23.56  mJ   32.07  mJ   41.98  mJ   poly.    

  66  

Figure 27 – Comparison of lifetime polynomial fit for each condition. SA/L stands for small

angle, low air content. MA stands for medium angle, WA for wide-angle, I for intermediate

air condition, H for high air content.

The best fit curves using a second-order polynomial fit for all conditions tested are

provided in Figure 27, which indicates that the bubble lifetime behavior in Figure 18 through

Figure 26 follow a similar development over time during the initial growth. The equations for

these best-fit curves are of the form

𝐷𝐷!"#

= 𝐴𝑡𝑡!− 1

!+ 1

The value of A for each condition is provided in Table 7. The intercept is 1, since D/Dmax(1) must

be equal to 1 (since the bubble reaches its maximum diameter at one half-life). The derivative at

the point (1,1) is zero, indicating that the bubble reaches its maximum size at one half-life for the

bubble due to the definition of th and Dmax.

0  

0.2  

0.4  

0.6  

0.8  

1  

1.2  

0   0.5   1   1.5   2   2.5  

D/D m

ax  

t/th  

SA/L   SA/I   SA/H   MA/L   MA/I  

MA/H   WA/L   WA/I   WA/H  

  67  

Table 7 – Coefficients for second order polynomial best fit curves based on

nondimensionalized lifetime profiles

Lens angle Air content (ppm) A

5° 10 -0.6525

5° 15 -0.5505

5° 20 -0.6688

10° 10 -0.7364

10° 15 -0.3523

10° 20 -0.4244

20° 10 -0.6434

20° 15 -0.5922

20° 20 -0.4528

ALL ALL -0.577

These curves provide agreement on the behavior from t/th = 0.5 to 1.5 with little variation

between conditions. From t/th = 0 to 0.5 and 1.5 to 2 there is more variation between curve fits.

For the wide and medium lenses, the saturated air condition produced bubbles that remained

larger for greater lengths of time. The solid curves for the wide-angle lens under the different air

content conditions have the largest separation between them, indicating that the lower air content

may have caused bubbles to become smaller more quickly. The dashed lines representing the

medium angle lens conditions show a similar pattern to a lesser degree, and the dotted lines

representing the small angle lens conditions do not show much difference at all. With these

distinctions in mind, a single curve representing the bubble diameter behavior over time can be

  68  

generated. The variation between curves in Figure 27 is noticeable but not extreme, which may

be due in part to the fact that the lifetime curves were based on individual runs and not the

average of many runs at each level of laser energy. Therefore some variation is to be expected in

the curves in Figure 27, and creating a single curve, as shown in Figure 28, provides a nominal

representation of an average time history of a bubble growth and collapse without rebound.

Figure 28 also shows all the data points to illustrate the scatter around the single curve

representing the average bubble. In this figure blue points represent laser energies from 0 to 10

mJ, green represent 10 to 20 mJ, orange represent 20 to 30 mJ, and red points represent laser

energies from 30 mJ to 42 mJ. The squares represent low air content condition bubbles, circles

represent the intermediate air content condition, and the triangles represent the higher air content

condition. Now we can further investigate the sensitivity of Dmax and th on each experimental

parameter.

  69  

Figure 28 – Single lifetime curve for nondimensionalized bubble diameter vs. time. This

curve describes the bubble diameter’s growth over time for th ≤ 2. The different colors

represent different laser energy ranges, while the shapes indicate the air content condition

(squares are low air content, circles are intermediate, and triangles are the high air content

condition).

0  

0.2  

0.4  

0.6  

0.8  

1  

1.2  

0   0.5   1   1.5   2   2.5  

D/Dm

ax  

t/th  

0<laser  energy<10  mJ   10<laser  energy<20  mJ   20<laser  energy<30  mJ   30<laser  energy<  42  mJ  

  70  

4 Scaling of Laser Induced Cavitation

4.1 Scaling

The bubble size, shape and lifetime as a function of the laser power, beam angle, and

water air content are desired. In order to make these relationships clear, dimensional analysis is

used as a tool to help investigate these relationships. The value in dimensional analysis is well

summarized in Kundu, Cohen & Dowling (2012), which states that “the natural realm does not

need mankind’s units of measurement to function.” There are two essential conclusions drawn

from this statement: (1) if a physical relationship is valid it can be stated in dimensionless form

and (2) for a comparison to be valid the units of the items being compared must be the same. The

measured variables in these experiments were beam angle, laser energy, air content, vertical

bubble diameter, horizontal bubble diameter, and bubble half-life. The density of water and local

pressure are also considered, as they are a measure of the resistance from the surrounding

environment to bubble growth. With this in mind, Buckingham’s theorem was used to reduce the

number of parameters by finding the appropriate nondimensional Π-groups. These Π-groups

were then plotted against each other in order to reveal the functional relationships between the

reduced parameters in the problem.

The lens divergence angle was converted to an associated beam waist, which is the

minimum laser beam diameter and has units of length. Linear optics assumes that the laser light

is focused at a point, but in reality due to nonlinearities in laser beam physics near the focal point

the minimum beam diameter has finite size. It narrows with an intensity distribution that can be

represented by a Gaussian profile, which is why it is commonly referred to as a Gaussian beam.

Thus for a given optical configuration, there is an associated minimum beam diameter, as

  71  

illustrated in Figure 29. The beam waist (w) is determined based on the wavelength of the laser

light (λ = 532 nm) as well as the full-included focusing angle of the lens in radians (θ),

𝑤 =   !!!"

.

Table 8 provides the computed beam waists for each of the lenses used in the current study.

Figure 29 – An illustration of the beam profile of a Gaussian beam near the focal point.

Table 8 – Beam waist for lenses

Included beam angle (θ) Beam waist (w) (µm)

5° 4.5

10° 1.8

20° 0.9

  72  

Table 9 – Variables of interest used to derive scaling rules for spherical cavitation bubbles.

Dependent Variables Independent Variables Fluid Properties

Dv Dh th Eo El w C ρ P

M 0 0 0 1 1 0 0 1 1

L 1 1 0 2 2 1 0 -3 -1

T 0 0 1 -2 -2 0 0 0 -2

There were 9 variables in the problem, and 3 fundamental units (mass M, length L, and

time T). The variables are presented in Table  9. Since the horizontal diameter (Dh), the vertical

diameter (Dv), and the bubble half-life (th) were the variables of interest, they should not be

chosen as repeating variables for determining the Π-groups. The repeating variables selected to

derive the Π-groups were beam waist (w), the fluid density (ρ), and the pressure, (P). Density

and pressure were constant in the current experiment at 997 kg/m3 and 101 kPa, respectively.

The water air content concentration (C) is already nondimensional, as it is a ratio of the mass of

air to the mass of water. Here El is the laser pulse energy and E0 is the minimum pulse energy for

which cavitation bubbles were observed.

Using Buckingham’s theorem and equating powers of fundamental units the six resulting

fundamental Π-groups are

Π! =  !!!= 𝐷!∗,

Π! =  !!!= 𝐷!∗,

Π! =   𝑡!×𝑤!!×𝜌!! !×𝑃! ! = 𝑡!∗,

Π! =  𝐸!×𝑤!!×𝑃!! = 𝐸!∗,

  73  

Π! =  𝐸!×𝑤!!×𝑃!! = 𝐸!∗, and

Π! =  𝐶.

All of the Π-groups are nondimensional, and when all other parameters in the problem are held

constant

Π! = 𝑓!(Π!,Π!,Π!,Π!,Π!)

or

𝐷!∗ = 𝑓!(𝐷!∗ , 𝑡!∗ ,  𝐸!∗,  𝐸!∗,𝐶).

Figure 30 – The scaled horizontal diameter (Dh*) is plotted versus the scaled vertical

diameter (Dv*), holding 𝚷6 constant. This plot shows a linear relationship between 𝚷1 and 𝚷2

for the majority of bubbles. The outliers in this plot represent bubbles produced at high

energies that do not maintain a spherical shape. These bubbles are indicated by open

symbols and are not included when looking at the scaling relationships.

0  

500  

1000  

1500  

2000  

2500  

3000  

3500  

4000  

4500  

0   200   400   600   800   1000   1200   1400   1600  

D h*  

Dv*    

Π6  =  10   Π6  =  15   Π6  =  20   Linear  (equal)  

  74  

Figure   30 shows Dh* plotted versus Dv

*, which shows a clear grouping near that of

spherical bubbles. This is illustrated by how most of the data points appear to lay along the black

line indicating Dh* = Dv

*. The majority of bubbles conform to this spherical geometry, or

geometry very close to spherical. At some points there appears to be branching away from this

spherical trend toward more elongated bubbles with larger horizontal diameters. These points

have been identified with open symbols and this branching occurs at higher laser powers

(typically between 23 and 42 mJ). These bubbles are not included in calculating the scaling

relationships. Above the spherical bubble line, the bubbles produced are elongated in the

horizontal direction, while below the line bubbles are elongated in the vertical direction. This is

expected given the observations of typical bubble shapes discussed in Chapter 3.

Since the majority of points follow a linear trend, for this work the generalization is made

that Dh* and Dv

* are directly proportional. This means that the scaling rules developed here are

for spherical bubbles and the analyzed data are limited to these conditions. The scaling rules

follow the assumption that Dv* ≈ Dh

*, therefore the scaling law of interest reduces to

Π! = 𝑓!(Π!,Π!,Π!,Π!)

or

𝐷!∗ = 𝑓!(  𝑡!∗ ,  𝐸!∗,  𝐸!∗,𝐶).

From Chapter 3, it was observed that the vertical diameter can be related to the laser power by a

power function of the form

𝐷! ≈ A(E− 𝐸!)!.

In this relationship A and B are constants for a given setup. Based on this observation, two of the

Π-groups can be combined to form a new group (ΔE*), which is equal to

  75  

П! = ∆𝐸∗ = 𝐸!∗ − 𝐸!∗

=(𝐸! − 𝐸!)𝑤!𝑃

.

The combination of El* and E0* into ΔE* results in Dv* being dependent on only three variables

and allows the scaling law to be reduced to

𝐷!∗ = 𝑓!(  𝑡!∗ ,∆𝐸∗,𝐶).

The Rayleigh equation provides a relationship between bubble wall velocity and the

bubble radius for cavitation bubbles. Integrating the Rayleigh equation provides a relationship

between the bubble collapse time τc, which is the time it takes for a bubble to change from its

maximum to minimum diameter, and the bubble’s maximum radius R0 (Peel, Fang, & Ahmad,

2011),  

𝜏! = 0.915𝑅!×!!

! !.

This equation can be rearranged such that all of the variables are on one side of the equation and

a single nondimensional constant is on the other side,

𝜏!𝑅!×

𝑃𝜌

! !

= 0.915 = 𝑐𝑜𝑛𝑠𝑡.

For the scaling in the current study, Dv is the bubble’s maximum diameter, and therefore should

be twice R0 in the Rayleigh integral equation. Furthermore, it is assumed that τc and th are

directly proportional (note that the bubble growth and collapse in Figure 18 through Figure 26 is

represented by a parabola). This suggests that th* and Dv* can be combined in a similar manner

to form a new nondimensional group,

П! =

!!∗

!!∗= !!

!!× !

!

! !

or

  76  

!!∗

!!∗= 𝑓(𝛥𝐸∗,𝐶).

The number of П-groups has now been reduced to 3 for this problem (П6 = C, П7 = ΔE*, and

П8 = th*/ Dv*). Reducing to three П-groups allows the relationships between these three П

groups to be readily compared on a single plot.

Figure 31 – Relationship between th*/Dv* and ΔE* shows a logarithmic relationship that

appears to be only minimally affected by air content for spherical bubbles. The average

ratio for each condition (lens, air content, laser power) is plotted.

Figure 31 illustrates that th*/ Dv* has a weak logarithmic dependence on ΔE*, which is

nearly constant as predicted from the Rayleigh integral analysis. Air content concentration, C,

appears to have only a minimal effect on this logarithmic relationship, as can be seen by the

linear best fit and scatter for each level of air content in the figure. Therefore in this scaling

y  =  0.0103ln(x)  +  0.2886  

0  

0.1  

0.2  

0.3  

0.4  

0.5  

0.6  

0.7  

0.8  

1.0E+06  

1.0E+07  

1.0E+08  

1.0E+09  

1.0E+10  

1.0E+11  

1.0E+12  th*/Dv

*  

ΔE*    

C=10   C=15   C=20   Log.  (ALL)  

  77  

approach air content will not be included from here on, however more air content concentration

conditions that represent a wider range should be examined. The concentrations considered in

this work represent a very limited range, and further work would validate whether scaling rules

should be affected by air content for more extreme conditions. Based on the assumption that the

effects of air content are minimal, a single logarithmic best fit line is used to relate ΔE* and th*/

Dv*,

П! =

𝑡!∗

𝐷!∗= 0.0103 ln 𝛥𝐸∗  ∗ + 0.2886

While the scaled ratio of the reference time to the radius in the integrated Rayleigh equation is a

constant value (0.951), the ratio between half-life and diameter is related to ΔE* by a logarithmic

function in the scaling presented here, as shown in Figure 31. The two approaches yield similar

results – dividing the Rayleigh equation constant by two shows that the ratio of

nondimensionalized half-life to bubble diameter is a constant (0.458), while the intercept in

Figure 31 is around 0.5. While this information is useful, another piece of information is needed

in order to determine the bubble diameter and half-life from the ratios determined in Figure 31.

Different approaches can be taken to address this problem. In this work, a graph of th* vs Dv* is

generated showing different ranges of ΔE* (see Figure 32). This can be used to find Dv* and th*

based on the ratio of the two.

  78  

Figure 32 – Average th* versus Dv* for various ranges of ΔE*. The relationship appears to

be linear.

Thus the desired th* and Dv* can be determined from ΔE*. ΔE* depends on the pressure, the beam

waist (w), the laser pulse energy (El), and the minimum energy at which cavitation occurs (E0).

However, based on the data assessed in this work, E0 depends on the lens used, and is a function

of the beam waist.

y  =  0.5922x  -­‐  29.646  

-­‐100  

0  

100  

200  

300  

400  

500  

600  

700  

800  

0   200   400   600   800   1000   1200   1400  

th*  

Dv*  

0≤ΔE*≤9.71E08   1.53E09≤ΔE*≤9.40E09   1.33E10≤ΔE*≤5.28E11  

  79  

Figure 33 – The relationship between beam waist and Eo appears to be linear.

Figure 33 illustrates the linear relationship between E0 and beam waist (w),

𝐸! = −834188𝑤 + 6.3854.

The minimum energy for visible cavitation bubbles (E0) decreases with increasing beam waist. It

is important to note that this is not a fundamental relationship between E0 and w, but rather an

empirical relationship established based on the range of conditions tested in this experiment.

Figure 33 has the average values of E0 for each lens plotted on the y-axis, since in this work the

assumption is made that the effects of air content are minimal. Again, this assumption will need

to be evaluated in future work to ensure its validity for wider ranges of air content. Given all of

these pieces of information, a single bubble with particular characteristics can be generated.

y  =  -­‐834188x  +  6.3854  R²  =  0.99357  

0  

1  

2  

3  

4  

5  

6  

0.E+00  

5.E-­‐07  

1.E-­‐06  

2.E-­‐06  

2.E-­‐06  

3.E-­‐06  

3.E-­‐06  

4.E-­‐06  

4.E-­‐06  

5.E-­‐06  

5.E-­‐06  E 0

 

Beam  waist  (m)  

  80  

Combining these scaling rules with Figure 28, the behavior of the bubble over time is also

known.

4.2 Comparison with non-spherical bubbles

In order to understand how the behavior of non-spherical bubbles compares with the

spherical bubbles used to develop the scaling relationships, the scaling was applied to each of the

non-spherical bubbles represented by open symbols in Figure  30. Since these bubbles were not

used to calculate the scaling relationships due to their non-spherical geometries, their actual

behaviors can be compared to the behaviors predicted by the scaling to shed light on how non-

spherical bubbles act and when the scaling presented here may not be applicable.

For each bubble, E0 is first determined based on the lens used. El is known as are the

pressure and density so ΔE* can be calculated. From ΔE*, the ratio th*/Dv* can be found. This,

combined with Figure 32, allows for th* and Dv* to be determined. The numbers are then

converted from nondimensional quantities into their dimensional forms.

4.2.1 Vertical diameter

The vertical bubble diameter predicted by applying the scaling relationships to the non-

spherical bubbles results in predicted diameters that are larger than the actual measured vertical

bubble diameters. Figure 34 illustrates how the predicted nondimensionalized bubble diameter,

Dv*, is larger than the actual measured vertical diameter of the bubbles. The points in the figure

lie far above the line illustrating where the predicted and actual bubble diameters are the same.

When converted from the nondimensional form to the dimensionalized bubble diameter in

millimeters, as in Figure 35, the predicted bubble diameters range from 2 to 5 mm, while the

actual measured bubble diameters range between 0.75 and 1.3. In general the scaling appears to

be predicting diameters that are about twice as large as the actual bubble diameter, but

  81  

sometimes predicts even larger bubble diameters, notably for the point where the actual bubble

diameter is about 1.3 mm but the scaling predicts a vertical bubble diameter of 5 mm.

Figure 34 – Predicted Dv* versus actual Dv* when applying scaling to non-spherical

bubbles. The solid line shows where the predicted and actual Dv* values are equal. The

scaling guidelines over predict the vertical diameter of the non-spherical bubbles.

0  

500  

1,000  

1,500  

2,000  

2,500  

3,000  

0   200   400   600   800   1,000   1,200   1,400   1,600  

Dv*  pred

icted  

Dv*  actual  

  82  

Figure 35 – Predicted vertical diameter plotted against the actual vertical diameter when

scaling relationships are applied to non-spherical bubbles. The black line indicates where

the predicted and actual values are equal.

4.2.2 Horizontal diameter

Interestingly, while the predicted vertical diameter tended to be larger than the actual

vertical diameter, the predicted horizontal diameter was not far off from the actual horizontal

diameter of the non-spherical bubbles. Keep in mind that the horizontal and vertical diameter

predicted from the scaling results will be the same, since the bubbles considered in the scaling

derivation are spherical. Figure 36 illustrates how the actual and predicted nondimensionalized

horizontal diameters compare. The data points show some scatter, but tend to be close to the line

representing where the predicted and actual nondimensionalized horizontal diameters are equal.

When these nondimensional values are converted into their dimensional quantities the predicted

and actual measured horizontal bubble diameters can be compared. While the data points do not

0.00  

1.00  

2.00  

3.00  

4.00  

5.00  

6.00  

0.00  

0.20  

0.40  

0.60  

0.80  

1.00  

1.20  

1.40  

1.60  Dv

 predicted

 (mm)  

 

Dv  actual  (mm)    

  83  

adhere as closely to the black line as they do for the nondimensionalized quantities, they still

cluster around it. While there is scatter among the points and a linear trend is not as strong as for

the nondimensional quantities Dh*, there is no shift as there was for the predicted vertical

diameters. The scaling appears to give a general idea of what the horizontal diameter is for non-

spherical bubbles generated in the energy range considered here (up to 42 mJ).

Figure 36 – Predicted Dh* versus actual Dh* when applying scaling to non-spherical

bubbles. The solid line shows where the predicted and actual Dh* values are equal. The

horizontal bubble diameter appears to be predicted relatively well with the scaling

relationships used here.

0.00E+00  

1.00E+03  

2.00E+03  

3.00E+03  

4.00E+03  

5.00E+03  

6.00E+03  

0.00E+00  

1.00E+03  

2.00E+03  

3.00E+03  

4.00E+03  

5.00E+03  

6.00E+03  

Dh*  pred

icted  

Dh*  actual  

  84  

Figure 37 - Predicted horizontal diameter plotted against the actual horizontal diameter

when scaling relationships are applied to non-spherical bubbles. The black line indicates

where the predicted and actual values are equal.

4.2.3 Bubble half-life

The nondimensional half-life, th*, and the dimensional half-life th were also investigated

when the scaling was applied to non-spherical bubbles. Figure 38 shows that the predicted

nondimensional half-life is significantly higher than the actual nondimensional half-life. When

converted into the dimensional form th as in Figure 39, the predicted half-lives appear to be two

to three times higher than the actual measured half-lives. This is not surprising considering that

the size of the bubble was largely overestimated by the scaling due to the predicted vertical

diameter being much larger than the actual measured vertical diameter. A larger bubble would

0.00E+00  

1.00E+00  

2.00E+00  

3.00E+00  

4.00E+00  

5.00E+00  

6.00E+00  

0.00  

0.50  

1.00  

1.50  

2.00  

2.50  

3.00  

3.50  

4.00  

4.50  

5.00  

Dh  predicted

 (mm)  

Dh  actual  (mm)  

  85  

take longer to reach its maximum size, and therefore the scaling over predicts the length of time

it takes for the bubble to reach its maximum diameter.

Figure 38 – Predicted th* versus actual th* when applying scaling to non-spherical bubbles.

The solid line shows where the predicted and actual th* values are equal. The scaling

relationships used over predict the scaled half-life th* for non-spherical bubbles.

0.00E+00  

2.00E+02  

4.00E+02  

6.00E+02  

8.00E+02  

1.00E+03  

1.20E+03  

1.40E+03  

1.60E+03  

1.80E+03  

0.00E+00  

1.00E+02  

2.00E+02  

3.00E+02  

4.00E+02  

5.00E+02  

6.00E+02  

7.00E+02  

8.00E+02  

9.00E+02  th*  pred

icted  

th*  actual  

  86  

Figure 39 – Predicted half-life versus actual half-life for scaling relationships applied to

non-spherical bubbles. The black line indicates where the predicted and actual values are

equal.

4.2.4 Behavior over time

The non-spherical bubbles behave differently over time than spherical bubbles do. Figure 40

shows the behavior of 12 non-spherical bubbles over time. The vertical diameter divided by the

maximum vertical bubbles diameter is plotted on the y-axis, and the time divided by the half-life

is plotted on the x-axis. The blue line is the polynomial fit derived in chapter 3 for the bubble

size versus time (Figure 28). This curve appears to slightly over-predict the size of the bubble

diameter over time, but still gives a reasonable idea of the bubble behavior for t/th ≤1.5. After t/th

= 1.5 the bubbles behave with greater variation between each run. Run 165 appears to rebound

quickly and then collapse. Run 67 grows and expands three times before finally collapsing. The

bubbles can take anywhere from 2.5 to 4.5 half-lives to finally collapse. Interestingly some

0  

50  

100  

150  

200  

250  

300  

350  

30.00  

35.00  

40.00  

45.00  

50.00  

55.00  

60.00  

65.00  

70.00  

75.00  th  predicted

 (μs)  

th  actual  (μs)  

  87  

bubbles (run 157, for example) reach a maximum diameter within two half-lives, collapse to a

smaller diameter, and then rebound to an even larger diameter before collapsing. This illustrates

how varied the behavior of non-spherical bubbles is, especially in the latter part of the bubble’s

lifetime.

While Figure 40 illustrates the behavior of the vertical diameter, an important variable to

look at is the behavior of the horizontal diameter. Figure 41 shows the average ratio of horizontal

to vertical diameter for six non-spherical bubbles over time. The bubbles initially start off very

long in the horizontal direction and thin in the vertical direction. Over time the bubbles become

shorter in the horizontal direction and larger in the vertical direction. There appear to be some

points in time where the ratio of horizontal to vertical diameter decreases quickly (e.g. 0 ≤ t/th ≤

.3) followed by plateaus where the ratio remains constant until it begins to decrease again.

  88  

Figure 40 – Non-spherical bubble behavior over time. Each run represents the

development of a single, non-spherical bubble. The blue curve represents the curve in

Figure 28, the generic lifetime curve obtained in chapter 3.

0  

0.2  

0.4  

0.6  

0.8  

1  

1.2  

1.4  

0   0.5   1   1.5   2   2.5   3   3.5   4   4.5  

D/D m

ax  

t/th  

run  67   run  418   run  276   run  271   run  282   run  285   run  157  

run  191   run  167   run  199   run  165   run  201   poly.  

  89  

Figure 41 – The ratio of horizontal to vertical diameter for non-spherical bubbles. This

graph represents the average ratio of vertical to horizontal diameter for six non-spherical

bubbles. This illustrates that non-spherical bubbles begin as more elliptical shapes and

then become more spherical over time. It also shows the oscillation in size that non-

spherical bubbles exhibit.

4.3 Error Propagation

The uncertainties associated with the measured quantities in chapter 2 will affect the

uncertainty of the nondimensional Π groups derived in this thesis. Using error propagation

analysis the uncertainty associated with each nondimensional Π group is calculated, and the

results are presented in Table  10. The calculations for the error propagation are available in the

Appendix. The quantities in Table   10 that say “max” next to the uncertainty represent the

condition that resulted in the maximum uncertainty. For example for the lenses, since the

uncertainty depends on both the focal length and the diameter, each lens will have a different

0  

1  

2  

3  

4  

5  

6  

7  

8  

9  

10  

0   0.5   1   1.5   2   2.5   3  

horizon

tal/ver7cal  diameter  (a

vg.)  

t/th  

  90  

uncertainty value for the focusing angle and beam waist. The value presented as the uncertainty

is the largest of these values.

 

Table 10 – Results of error propagation through calculations for nondimensional variables

Quantity Uncertainty

Bubble diameter Dv, Dh ±0.158 mm

Frame rate ±6.6586 µs

Laser pulse energy ±10% (or ±4.2 mJ max)

Lens diameter ±5% (or ±2.5 mm max)

Lens focal length ±1% (or ±2 mm max)

Lens focusing angle ±.047 rad (2.7°) max

Beam waist ±.60 µm max

Pressure ±49 Pa

Air content ±12% (average)

Dv* ±25% (average)

Dh* ±24% (average)

th* ±23% (average)

E0* ±48% (average)

El* ±48% (average)

C ±12% (average)

th*/Dv* ±34% (average)

  91  

5 Limitations and future work

5.1 Limitations

As with all experiments, there are some limitations associated with these results. This

section focuses on things that could have resulted in error or account for the variability in the

data.

5.1.1 Pulsed laser

The laser being pulsed at 15 hertz is one of the factors that may have introduced scatter

into the results. Each pulse of the laser causes the water to heat up around the focal point of the

lens. This localized heating is what causes the cavitation bubble to occur. This also causes

heating of the adjacent region. When the laser is pulsed rapidly residual heating from the

previous pulse is still present, leaving the water in the area of the focal point warmer than the rest

of the water in the tank. With the water around the focal point already at a higher temperature,

the bubbles produced by the next pulse may be larger or less spherical than they would have

been if a single pulse were used at a consistent temperature. The laser pulse energy also varies

slightly with each pulse. Therefore each bubble in a series may be produced with a slightly

different laser pulse energy, resulting in a slightly different size.

5.1.2 Equipment setup

For consistency across the various conditions, it would be ideal to set up once and run all

the conditions in the same setup. However, that was not possible for this experiment. The data

were collected in two different batches spaced approximately a month apart in time. Since the

components for the experiment needed to be set up twice, comparisons between the data taken in

the first round of testing and data taken during the second round of testing may have some small

differences. Most significantly, the camera was placed at a slightly different distance from the

  92  

plane of the laser beam. Therefore the bubbles in the first and second round of experiments

appear to be different sizes. Using the calibration images allows the measurements made

between the different sets of data to be compared, but it would have been better to allow direct

comparison by having the camera at the exact same distance from the bubbles for each round of

images.

5.1.3 Assumptions in deriving scaling

The scaling derived in this work involved a few simplifications. These include constant

density and no visible effect of air content. The scaling also focused on spherical bubbles that

occur at lower power, although the effect of higher power is discussed in a more qualitative

sense.

5.2 Future work

The results from the work presented here provide valuable insight about how to control the

size, shape, and lifetime of laser-induced cavitation bubbles. While this work is certainly useful,

it is by no means all encompassing. Many other factors can affect the development of laser-

induced cavitation bubbles. Future work should focus on exploring the effects of some other

factors that could be used to manipulate cavitation bubbles for various applications. A few

factors for exploration are suggested, such as effects of pressure and viscosity.

5.2.1 Pressure

Focusing a laser beam induces cavitation by increasing the local temperature of the water

where the laser beam rays converge. When the temperature is high enough at a given pressure,

the water at the specific location turns into a vapor. As the pressure is changed the temperature

needed to produce a vapor also changes. The higher the temperature the lower the pressure needs

to be for the water to turn into steam and cavitate. Some work in this area has been done.

  93  

Tanibayashi et al. (2003) looked at the effect of both pressure and air content by depressurizing

water. They concluded that as pressure is increased, the number and size of cavitation bubbles

decreases. It would be interesting to see how changes in pressure would affect the scaling

derived in this work.

5.2.2 Viscosity

The viscosity of the fluid in which cavitation is being induced can affect both the size and

lifetime of a cavitation bubble. It has been shown that a higher viscosity causes the bubbles to

expand and collapse less violently than in a liquid with a lower viscosity. In pure glycerin, which

has a higher viscosity than water, the maximum bubble radius was smaller than in water, the

minimum bubble radius was larger than in water, and the bubble lifetime was longer than in

water. A higher viscosity fluid will work to oppose bubble expansion by exerting a radial force

in the opposite direction of the bubble’s expansion. Therefore bubbles will reach a smaller

maximum radius. However the higher viscosity liquid will dissipate more energy during the

collapse phase of the cavitation bubble, resulting in a larger minimum bubble diameter.

Increasing the viscosity also increases the oscillation time for cavitation bubbles (Xiu-Mei et al.,

2008). As the majority of work has been done on laser-induced cavitation bubbles in water, more

information on how cavitation bubble dynamics change in liquids of higher or lower viscosity

could prove useful in newer applications.

5.2.3 Particulate matter

Particles in water (or whatever fluid is being used as the bulk liquid) could affect the

development of cavitation bubbles in a number of ways. Small particles in the water may change

the bulk viscosity of the fluid, and particles can provide nucleation sites for cavitation. Impurities

in a fluid can greatly affect the amount of energy needed to cause breakdown in a medium.

  94  

Impurities can provide seed electrons that can then be used to ionize other molecules, reducing

the energy threshold for generating a plasma (Kennedy, Hammer, & Rockwell, 1997). Varying

the size and concentration of particulate matter, as well as varying the materials used as the

particles themselves, could provide useful information about bubble inception and dynamics.

5.2.4 Air content

The results of this work showed that laser induced cavitation bubbles were not

significantly influenced by the water air content, but only a narrow range of air concentrations

were tested. Future work should investigate a greater difference in air content concentration.

Looking at fully saturated, supersaturated, and under saturated conditions would be informative.

Additionally more than three conditions need to be tested to gain a clear understanding of the

relationship between air content and bubble dynamics.

  95  

6 Conclusions

This work provides valuable information about laser-induced cavitation bubble behavior.

Observations in chapter 3 discuss the behavior of both spherical and non-spherical bubbles in a

qualitative sense, while chapter 4 focuses on scaling guidelines for spherical bubbles that could

be of use for future studies and applications involving laser-induced cavitation bubbles. The

scaling provided allows for bubble size and behavior to be controlled by selecting the appropriate

optics and laser energies.

It has been shown that increasing the laser pulse energy will result in larger bubbles, and

at higher laser pulse energies the bubbles become more elliptical, as opposed to the spherical

bubbles produced at lower energies. There is also a minimum laser pulse energy with which

cavitation bubbles can be produced. This minimum energy is a function of the lens used and

depends on the beam waist. A wider angle lens will result in a smaller beam waist and a higher

minimum energy for cavitation. The wider angle lens also produced spherical bubbles at higher

energy compared to the smaller angle lens.

The results of changing the air content of the fluid show minimal influence on the bubble

behavior in this work. A wider range of air content concentrations should be considered to assess

the universality of the current results. Based on this, the scaling guidelines derived in this work

should only be used for air contents within a reasonable variation from the air content

concentrations used in this work. Future work deriving scaling rules over a wider range of air

content concentrations could be compared to this work and shed insight into the exact range for

which the scaling derived in this work is valid.

The scaling relationships were applied to non-spherical bubbles as a comparison and to

understand the differences in behavior between spherical and non-spherical bubbles. The scaling

  96  

relationships overestimated the size of the bubbles, predicting a larger vertical diameter than the

measured diameter for these bubbles. The scaling relationships also predicted a longer half-life

than the actual half-life of the bubbles. Interestingly, the vertical diameter predicted based on the

scaling was similar to the actual horizontal diameter of the bubbles investigated.

The scaling relationships presented here can be used to produce cavitation bubbles of

known size and half-life using a focused laser beam. This information is valuable for the many

current and emerging laser-induced cavitation applications. This information can be used to

produce spherical bubbles, and also provides some initial (though brief) investigation on non-

spherical bubble production and behavior.

  97  

Appendix: Uncertainty

Uncertainty calculations/propagation of error

Beam angle

11 tan

2Df

θ − ⎛ ⎞= ⎜ ⎟

⎝ ⎠

0.1D Dσ =

0.01f fσ =

2 22 22 2

/0.1 0.01 0.1 0.01 0.1fD

D fD D D f D Df D f f D f f f

σσσ

⎛ ⎞⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞⎛ ⎞ ⎛ ⎞= + = + = + ≈⎜ ⎟⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠⎝ ⎠

( )1

/21 2

D f

D fθ

σσ =

+

Beam waist

( )1 113 1

3

sin sin sinnn

θ θ η− −⎛ ⎞= =⎜ ⎟

⎝ ⎠

( )1 1 1sin 1 1sin cosθ θ θσ σ θ σ θ

θ∂

= =∂

1 1 ( )n n assumedσ <<

3 3 ( )n n assumedσ <<

31 1 1 1

22 2sin 1 11 1

1 1 13 1 3 1 3 1 3

cossin sin cos

sin sinnn nn n

n n n n nθ θ θ

η

σσ σ σ θ σσ θ θ θ

θ θ

⎛ ⎞ ⎛ ⎞ ⎛ ⎞⎛ ⎞ ⎛ ⎞ ⎛ ⎞= + + =⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟ ⎜ ⎟

⎝ ⎠ ⎝ ⎠ ⎝ ⎠⎝ ⎠ ⎝ ⎠ ⎝ ⎠;

( )( )3

1

2sin

1η

θ η

σσ σ η

η η−∂

= =∂ −

  98  

3

2w λπθ

=

( )assumedλσ λ<<

3 3

22

3 3 3 3

2 2w

θ θλσ σλ σ λ

σπθ λ θ πθ θ

⎛ ⎞ ⎛ ⎞ ⎛ ⎞⎛ ⎞⎛ ⎞= +⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎜ ⎟⎜ ⎟⎝ ⎠⎝ ⎠ ⎝ ⎠ ⎝ ⎠⎝ ⎠

;

Dv*, Dh*

𝐷!∗ =𝐷!𝑤

𝜎!∗ =𝐷𝑤

𝜎!𝐷

!+

𝜎!𝑤

!

th*

𝑡!∗ = 𝑡!𝑃! !𝑤!!𝑃!! !

=𝑡!𝑤

𝑃𝜌

! !

Let 𝐵 = !!

! !.

𝑡!∗ =𝑡!𝑤 𝐵

𝜎!!∗ =𝑡!𝑤 𝐵

𝜎!𝑡

!+

𝜎!𝑤

!+

𝜎!𝐵

!

𝜎! =12𝑃𝜌

! ! 𝜎!𝑃 =

12

1𝜌𝑃

! !

𝜎!

𝜎!!∗ =𝑡!𝑤

𝑃𝜌

! ! 𝜎!𝑡

!+

𝜎!𝑤

!+

𝜎!𝐵

!

  99  

𝜎!!∗ =𝑡!𝑤

𝑃𝜌

! ! 𝜎!𝑡

!+

𝜎!𝑤

!+

𝜎!2𝑃

!

E0*, El

*

𝐸!∗ =𝐸!𝑤!𝑃

Let 𝐴 =  𝑤!.

𝐸!∗ =𝐸!𝐴𝑃

𝜎!∗ =𝐸!𝐴𝑃

𝜎!𝐴

!+

𝜎!!𝐸!

!+

𝜎!𝑃

!

𝜎! = 3𝑤! 𝜎!𝑤 = 3𝑤!𝜎!

𝜎!!∗ =𝐸!𝑤!𝑃

3𝜎!𝑤

!

+𝜎!!𝐸!

!+

𝜎!𝑃

!

th*/Dv* = R

𝜎! =𝑡!∗

𝐷!∗𝜎!!∗𝑡!∗

!

+𝜎!!∗𝐷!∗

!

  100  

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