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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