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UNIVERSITY OF MINNESOTA
This is to certify that I have examined this copy of a master�s thesis by
Travis Jon Schauer
and have found that it is complete and satisfactory in all respects, and that any and all revisions required by the final
examining committee have been made.
________________________ ________________________
Name of Faculty Adviser(s)
________________________ ________________________
Signature of Faculty Adviser(s)
________________________ ________________________
Date
GRADUATE SCHOOL
AN EXPERIMENTAL STUDY OF A VENTILATED SUPERCAVITATING VEHICLE
A THESIS SUBMITTED TO THE FACULTY OF THE GRADUATE SCHOOL
OF THE UNIVERSITY OF MINNESOTA BY
TRAVIS JON SCHAUER
IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE
MARCH 2003
i
ACKNOWLEDGMENTS
I would like to take the time to thank my advisors, Ivan Marusic and Roger Arndt,
for their assistance throughout this entire research project. In particular, I would like to
thank Roger for teaching me some of the vast knowledge he has in the field of cavitation.
Your thorough explanations helped me understand many of my experimental results. In
addition, I would like to thank Ivan for his assistance in conducting sound experiments,
especially when using the Particle Image Velocimetry system. Your ideas and
suggestions helped me gather valid experimental data that were crucial to the content of
this thesis.
I would also like to thank the gentlemen in the Aerospace Engineering shop, Dave
Hultman and Steve Nunnally. You both helped design and construct the test body used in
this experiment and did an extremely fine job.
I would like to thank Alex Cannan, an undergraduate research assistant, who
worked with me for one summer and helped me gather some of the data presented in this
thesis. Your quick and accurate work in helping with collecting and analyzing the test
data was greatly appreciated.
Finally, I would like to thank all of the other individuals, both at the Saint Anthony
Falls Laboratory and in the Aerospace Engineering Department, who helped me
throughout this project. These folks include shop personnel, graduate students, and
professors to whom I am sincerely thankful for their assistance throughout this project.
You were all very instrumental in assisting with the work presented in this thesis.
ii
ABSTRACT
A study was carried out to investigate some of the aspects of a supercavitating
vehicle. First, digital images of the cavity shape and wake are presented. These images
are used to qualitatively describe the cavity shape and details of the wake. Next, the
amount of air required to sustain an artificial cavity was investigated. This knowledge is
important because at low speed, which is the case when a torpedo is initially fired, a
natural cavity cannot be sustained. In this case, an artificial cavity can be created to
decrease the drag and allow the torpedo to accelerate to a point where it can sustain a
natural cavity. Finally, some of the wake details of artificially ventilated cavities are
characterized qualitatively. The results were obtained by using Particle Image
Velocimetry. With the aid of Particle Image Velocimetry, a new technique was
developed to measure the void fraction of gas to liquid in the wake of a cavitating body.
iii
TABLE OF CONTENTS
1. INTRODUCTION ...................................................................................................... 1 1.1 Fundamentals of Cavitation ............................................................................ 1 1.2 A Brief History of Supercavitation ................................................................. 3 1.3 PIV in Two-Phase Flows ................................................................................ 8
2. DESCRIPTION OF EXPERIMENT ........................................................................ 10 2.1 Objectives ..................................................................................................... 10 2.2 Experimental Facility.................................................................................... 11 2.3 Test Body Design.......................................................................................... 12 2.4 Experimental Setup....................................................................................... 14
3. EXPERIMENTAL RESULTS.................................................................................. 17 3.1 General Observations.................................................................................... 17 3.2 Air Entrainment Results................................................................................ 20 3.3 PIV Results ................................................................................................... 24
4. CONCLUSION......................................................................................................... 33 4.1 Conclusions................................................................................................... 33 4.2 Recommendations for Future Work.............................................................. 34
BIBLIOGRAPHY............................................................................................................. 37
APPENDIX A: Pressure Transducer Calibration Procedure ............................................ 87 APPENDIX B: Uncertainty Analysis ............................................................................... 88
iv
LIST OF FIGURES
Figure 1.1: Schematic of cavitator and cavity dimensions. .............................................. 40 Figure 1.2: Pictures of cavities in re-entrant jet and twin vortex regimes. ....................... 41 Figure 1.3: Cavity length versus cavitation number from various sources....................... 42 Figure 2.1: Schematic of water tunnel. ............................................................................. 43 Figure 2.2: Outline of test body. ....................................................................................... 43 Figure 2.3: Choking cavitation number vs. blockage ratio for a disk............................... 44 Figure 2.4: Problem description for choking phenomenon. ............................................. 45 Figure 2.5: Velocity ratio versus cavitation number in a bounded flow........................... 45 Figure 2.6: Cross-sections of elliptical and cylindrical test body struts. .......................... 46 Figure 2.7: Cross section of test body............................................................................... 46 Figure 2.8: Test setup for air entrainment measurements................................................. 47 Figure 2.9: PIV test setup.................................................................................................. 48 Figure 2.10: Side view of test body. ................................................................................. 49 Figure 2.11: Bottom view of test body with cylindrical strut. .......................................... 49 Figure 2.12: Bottom view of test body with elliptical strut. ............................................. 49 Figure 3.1: Pictures of oscillating cavity and wake. ......................................................... 50 Figure 3.2: Distortion of cavity shape due to cylindrical and elliptical struts. ................. 51 Figure 3.3: Re-entrant jet effects on cavity surface with the cylindrical strut. ................. 52 Figure 3.4: Re-entrant jet effects on cavity surface with the elliptical strut. .................... 53 Figure 3.5: Air entrainment results for 1 cm disk and cylindrical strut. ........................... 54 Figure 3.6: Cavity pictures for 1 cm disk. ........................................................................ 55 Figure 3.7: Air entrainment results for 1 cm disk and elliptical strut. .............................. 56 Figure 3.8: Comparison of air entrainment results of both struts with 1 cm disk. ........... 57 Figure 3.9: Air entrainment results for 1.5 cm disk and cylindrical strut. ........................ 58 Figure 3.10: Cavity pictures for 1.5 cm disk. ................................................................... 59 Figure 3.11: Air entrainment results for 1.5 cm disk and elliptical strut. ......................... 60 Figure 3.12: Comparison of air entrainment results of both struts and 1.5 cm disk......... 61 Figure 3.13: Brennen's data for blockage effects along with interpolated data. ............... 62 Figure 3.14: Air entrainment results for 1 cm disk using data from Brennen. ................. 63 Figure 3.15: Air entrainment results for 1.5 cm disk using data from Brennen. .............. 64 Figure 3.16: Air entrainment data for both disks and struts using data from Brennen..... 65 Figure 3.17: Schematic of wake profile. ........................................................................... 66 Figure 3.18: Wake profile in the non-cavitating regime for U∞ = 6.36 m/s. .................... 67 Figure 3.19: Power law relationships in non-cavitating wake for U∞ = 6.36 m/s. ........... 68 Figure 3.20: Non-dimensional velocity data in non-cavitating wake for U∞ = 6.36 m/s.. 69 Figure 3.21: Bubble velocity in cavitating wake for U∞ = 6.36 m/s and σ = 0.15. .......... 70 Figure 3.22: Measured velocities in cavitating and non-cavitating regimes. ................... 71 Figure 3.23: Normalized grayscale levels in the wake for U∞ = 6.36 m/s and σ = 0.15. . 72 Figure 3.24: Normalized grayscale levels after removing background noise................... 73 Figure 3.25: Calculated void fraction in wake for U∞ = 6.36 m/s and σ = 0.15............... 74 Figure 3.26: Wake profile in the non-cavitating regime for U∞ = 6.6 m/s. ...................... 75 Figure 3.27: Power law relationships in non-cavitating wake for U∞ = 6.6 m/s. ............. 76 Figure 3.28: Non-dimensional velocity data in non-cavitating wake for U∞ = 6.6 m/s.... 77
v
Figure 3.29: Comparison of measured velocities in non-cavitating wake........................ 78 Figure 3.30: Bubble velocity in wake for U∞ = 6.6 m/s and σ = 0.15. ............................. 79 Figure 3.31: Calculated void fraction in wake for U∞ = 6.6 m/s and σ = 0.15................. 80 Figure 3.32: Comparison of calculated void fractions...................................................... 81 Figure 3.33: Bubble velocity in wake for U∞ = 6.6 m/s and σ = 0.24. ............................. 82 Figure 3.34: Comparison of calculated void fractions for different cavitation numbers.. 83 Figure 4.1: Downstream distance versus error in static pressure measurement. .............. 84 Figure 4.2: Test setup for using rotameter to measure cavity pressure. ........................... 85 Figure 4.3: Preliminary data for rotameter cavity pressure measurement technique. ...... 86
vi
LIST OF SYMBOLS
A area
CD drag coefficient
0DC drag coefficient at a cavitation number of zero
mpC pressure coefficient
d diameter of cavitator
pd particle diameter
sd strut diameter
D maximum diameter of cavity
f frequency
Fr Froude number
g gravitational acceleration
h test section height/diameter
l characteristic length scale
L length of cavity
m& mass flow rate of gas
p pressure
p0 stagnation pressure
p1 pressure downstream of rotatmeter
patm atmospheric pressure
pc cavity pressure
pm minimum pressure in a fluid
pmeas static pressure at measurement location
pv vapor pressure of liquid
p∞ freestream static pressure
Q air entrainment coefficient
Q volumetric flow rate of gas at cavity pressure
r radial coordinate axis
S Strouhal number
vii
Tatm atmospheric temperature
u velocity in wake of test body
Uc velocity along cavity surface
Umax maximum velocity downstream of test body
U∞ freestream velocity
V velocity
x downstream coordinate axis
δu velocity defect in wake
η void (volume) fraction of gas to liquid
µ dynamic viscosity of fluid
ρ fluid density
gρ gas density
pρ particle density
σ cavitation number based on vapor pressure
cσ cavitation number based on cavity pressure
iσ incipient cavitation number
measσ cavitation number calculated based on measured flow quantities
trueσ actual cavitation number (no error)
fτ fluid time scale
1
1. INTRODUCTION
Cavitation is found to occur in many different hydrodynamic applications such as
pumps, hydrofoils, and spillways. In almost all cases, cavitation produces negative
effects such as loss in a hydrofoil�s lift, a drop in efficiency of a pump, and cavitation
damage to a variety of materials. For the cases just mentioned, cavitation may not be
avoidable. Therefore, the designer must try to minimize the effects of cavitation or take
advantage of the cavitation phenomenon. In the latter case, supercavitation may be the
answer. In fact, many devices have been designed that use supercavitation to their
advantage. Examples of these are supercavitating propellers, hydraulic turbines, and
hydrofoils. Many of these examples take advantage of the tendency towards steadiness
of supercavitation when cavitation cannot be avoided (Knapp et al, 1970).
A particularly interesting application of supercavitation is currently receiving a fair
amount of attention by the U.S. Navy. Supercavitation provides a means of reducing the
drag of an underwater body, leading to an increase in maximum speed. While this
concept may seem straightforward, implementing it is not an easy task. This is because
the dynamics of a supercavity are not completely understood.
1.1 Fundamentals of Cavitation
Cavitation is a dynamic phenomenon that occurs in a liquid when the local pressure
is reduced to a value near the vapor pressure of the liquid. Cavitation can be visually
observed by the formation of vaporous bubbles or cavities. The fundamental parameter
that describes cavitation is the cavitation number, which is defined as
2
v
ρU21
ppσ
∞
∞ −= , (1.1)
where p∞ is the freestream static pressure, pv is the vapor pressure of the liquid, ρ is the
density of the liquid, and U∞ is the freestream velocity. Under most circumstances,
cavitation is assumed to occur when the minimum pressure in the flow is equal to the
vapor pressure. For steady flow, the pressure coefficient is defined as
2
2
mp
ρU21
ppC
m
∞
∞−= , (1.2)
where pm is the minimum pressure. Assuming that pv equals pm, an incipient cavitation
number can be defined as
mpi Cσ −= . (1.3)
The incipient cavitation number can be thought of as a performance boundary such that
when σ > σi there are no cavitation effects and when σ < σi cavitation effects such as
noise and vibration can occur (Arndt, 2002).
At low cavitation numbers, a large cavity can attach to a solid boundary. In
extreme cases the cavity can entirely envelop a body. This is referred to as
supercavitation, following Tulin (1961) who first coined this phrase. When
supercavitation occurs, the drag of the body surrounded by the cavity can be greatly
reduced. This is because the skin friction drag, which depends on the viscosity of the
fluid, is lowered due to the vaporous pocket surrounding the body.
Artificial cavitation is a process in which a cavity is sustained by ventilating it with
some form of gas. In the case of artificial cavitation, it is convenient to define a
cavitation number based on the cavity pressure,
2
cc
ρU21
ppσ
∞
∞ −= , (1.4)
where pc is the pressure inside the cavity. It can easily be seen that small cavitation
numbers can be achieved by increasing the pressure inside the cavity. The cavity details
in the case of a ventilated cavity are essentially the same as those of a natural cavity,
assuming the cavitation numbers are the same (Schiebe and Wetzel, 1961). However,
there is a fundamental difference between natural and artificial cavities. An artificial
cavity at a given cavitation number can be achieved at a much lower velocity than a
natural cavity with all other conditions being the same. Therefore, as might be expected,
gravity can play an important role since the gas phase has a tendency to rise due to
buoyancy. In this case the Froude number is an important parameter. It is defined as
3
lg
UFr ∞= , (1.5)
where g is the gravitation acceleration and l is a length scale. Usually, a characteristic
length scale of the cavitator (object which initiates the cavity), such as the diameter in the
case of a disk cavitator, is chosen for l.
The amount of gas required to sustain a ventilated cavity is an important parameter.
The air entrainment coefficient quantifies the gas required in non-dimensional form. It is
defined as
2UQQ
l∞
= , (1.6)
where Q is the volumetric flow rate of the injected gas at cavity pressure. Again, the
characteristic length scale of the cavitator is usually chosen for l. The characteristic
dimensions of a disk cavitator and associated supercavity are shown in Figure 1.1. Note
that the half-length of the cavity is measured from the cavity separation point to the
maximum diameter of the cavity. This is because the cavity closure region is difficult to
determine and measure with high accuracy.
1.2 A Brief History of Supercavitation
A history of the development of supercavitation theories and experimental studies
will now be discussed. Particular attention will be given to three-dimensional, axially
symmetric cavities since they are the main interest in the current research. The intent of
this section is not to list all of the theories developed or the experiments performed, but
rather give a brief overview of some of the more significant results.
The first calculations for the dimensions of an axisymmetric cavity were carried out
by Garabedian (1956). His theory was derived using asymptotic relations while
assuming a steady, axially symmetric, irrotational flow of an incompressible liquid. A
key assumption in the development of this theory is that of a Riabouchinsky model. This
model assumes that the cavity has a symmetrically shaped nose and tail. Garabedian�s
formulas for the dimensions of a cavity created by a disk are,
σ1ln
σC
dL D= (1.7)
4
σ
CdD D= (1.8)
σ)(1CC0DD += (1.9)
827.0C0D = , (1.10)
where d is the diameter of the disk, CD is the drag coefficient of the disk, and 0DC is the
drag coefficient of the disk at a cavitation number of zero.
Some of the first experiments performed to generate axisymmetric cavities were
those of Self and Ripken (1955). Their experiments consisted of testing various head
form shapes (zero caliber ogive, 45° cone, and sphere) and sizes under a variety of
conditions. Their results were compared to the semi-empirical formulas derived by
Reichardt (1946),
−=
78
D
0.132σσ
CdD (1.11)
( )1.7σ0.066σ0.008σDL
++= , (1.12)
which are valid for σ < 0.1, and found to be in fair agreement. An important observation
in the experiments of Self and Ripken was the presence of a re-entrant jet. The dynamics
of the re-entrant jet could be seen through the transparent cavity and its effects were
found to vary depending on the size of the cavity.
Some of the first results published for ventilated cavities were those of Cox and
Clayden (1956). First, they developed a theory for the case when the trailing end of a
cavity consists of two vortices, as opposed to the re-entrant jet case described above.
Their theory included an estimate for the amount of gas required to ventilate a cavity in
the twin-vortex regime as a function of the mean air velocity in the vortex tubes,
freestream velocity, Froude number based on cavity length, cavitation number, partial
pressure of the gas in the cavity, and the cavity pressure. A set of experiments was then
conducted at conditions corresponding to the twin-vortex regime (relatively high gas
injection rates) for a sharp-edged disk. Their experiments verified the large air
5
entrainment coefficient predicted by their theory. Pictures of cavities in the re-entrant jet
and twin vortex regimes are shown in Figure 1.2.
Further experimental investigations on the shape of a ventilated cavity around a
disk were performed by Waid (1957). His experiments were conducted with the disk
surface perpendicular to the oncoming flow as well as at various angles. His results for
cavity diameter and length agreed remarkably well with Garabedian�s theory discussed
previously. Waid also came up with empirical relations for the shape of the cavity based
on his experiments. The relations determined by Waid for disks at zero angle of attack,
dσ1.08L 1.118
c
= (1.13)
1σ0.534D 0.568
c
+= , (1.14)
are valid for cavitation numbers from 0.035 to 0.171. Unfortunately, Waid did not
measure the amount of gas required to sustain the cavities.
The equations found by Garabedian, Reichardt, and Waid for cavity length versus
cavitation number are compared in Figure 1.3. The agreement between the curves is very
good even though the equations were derived/developed over different ranges of
cavitation number.
The theory developed by Cox and Clayden was later modified by Campbell and
Hilborne (1958). Their modified theoretical model,
22
c4 d
LdD
σFr 32πQ
= , (1.15)
where Fr is the Froude number based on the cavitator diameter, does not contain any
empirical parameters, unlike the model proposed by Cox and Clayden. Their modified
model is also more applicable to lower air entrainment rates, whereas the model of Cox
and Clayden is for higher entrainment rates. Both models are for the twin vortex regime,
however. Campbell and Hilborne also carried out experiments to validate their theory.
Their experimental results showed some broad agreement with the theory. Campbell and
Hilborne noted that the air entrainment rate is not only a function of the Froude number
and cavitation number, but also the cavitator diameter. They noted that for a given
6
cavitation number and Froude number, as the cavitator diameter is increased the air
entrainment coefficient also increases. They also found that the re-entrant jet regime
occurred when the product of the cavitation number based on cavity pressure and Froude
number based on disk diameter was greater than one, whereas the twin vortex regime
occurred when σcFr < 1. A brief discussion of the cavity transparency was also given by
Campbell and Hilborne. They noted that clear cavities were associated with the twin
vortex regime. On the other hand, opaque cavities were observed for the re-entrant jet
regime. The opaque cavities were caused by the re-entrant jet splashing onto the cavity
wall.
The first observations and descriptions of unstable, pulsating cavities were made by
Silberman and Song (1959) and Song (1961). They noted that there are two distinct
classes of ventilated cavities: steady and vibrating. Steady cavities occur at relatively
small airflow rates and are similar to natural cavities in all respects. A vibrating cavity is
formed when excessive amounts of air are added to the cavity. Silberman and Song
developed the following empirical formula,
σ 0.19σ c ≤ , (1.16)
which describes when a vibrating cavity occurs. The presence of a wavy cavity surface
with wave fronts normal to the flow direction was observed in the vibrating regime.
Another important finding was that pulsation could only occur for a two-dimensional
cavity or a cavity in which a significant portion of the span was two-dimensional. The
presence of a free surface, other than the cavity surface, was also found to be essential for
pulsation to occur.
One of the first numerical simulations of cavitating flows was carried out by
Brennen (1969). His numerical solutions included the effects of a bounded flow, which
could be the case, for example, in a water tunnel. In this case the flow can become
choked. When the flow is choked, the velocity cannot exceed a certain maximum for a
given freestream velocity and cavity pressure. As Tulin (1961) notes, the cavity of a
body between solid walls will always be lengthened relative to the unbounded flow case.
Brennen compared the results of his simulation to experimental data and found the data to
agree very closely.
7
Unfortunately, there has been very little research conducted in the United States in
the field of axisymmetric supercavitation since 1970. However, countries such as Russia
and Ukraine have been investigating supercavitation actively throughout this time. One
main problem still exists when trying to learn from this research; most of the Russian
literature has not been translated to English. In the past few years, though, there have
been multiple international symposia in which the proceedings have been published in
English. Therefore, much can be learned from the research conducted in other countries.
Semenenko (2001) gives an excellent overview of research conducted in the field of
artificial supercavitation. Most of this research was conducted by Russians, and in fact
much of it is still only published in Russian. However, Semenenko does give a good
summary of different ventilation schemes, gas leakage types, and approximate calculation
techniques for cavity size, in addition to a wealth of other information, some of which
will be discussed later.
Finally, research conducted on high-speed, supercavitating, underwater weapons
(torpedoes and projectiles) has received much attention in the past few years (Vlasenko
1998, Braselmann et al 2002, Schaffar et al 2002, Spurk 2002). The results of these
experiments are very important since the velocities achieved during the research are very
high (up to 1300 m/s). Some of the more significant findings are the extremely long
cavities generated under these conditions, the stability issues that need to be overcome to
control the projectiles, and the amount of ventilation gas required to sustain cavities at
high speeds. The theory of Spurk is of particular interest to the current research. He
found that for long, slender cavities,
constant* σ1ln
σ1
σσ1
Qccc
c+= , (1.17)
where the constant in this equation can be found by performing a single experiment.
Braselmann et al performed experiments to validate this equation for cavitation numbers
based on cavity pressure in the range of 0.01 to 0.07 and found them to be in good
agreement.
8
1.3 PIV in Two-Phase Flows
Particle Image Velocimetry (PIV) is a tool used to determine velocity fields in a
two-dimensional plane. A detailed review of PIV for single-phase flows is given by
Adrian (1991). The basic idea behind PIV is to illuminate a plane of a flow field with a
laser sheet. The flow field is then seeded with particles that follow the flow accurately.
This can be assured by choosing particles whose Stokes number is significantly less than
one. The Stokes number is defined as
f
2pp
18µdρ
Stτ
= , (1.18)
where ρp is the density of the particles, dp is the diameter of the particles, µ is the
dynamic viscosity of the fluid, and τf is the fluid time scale. In the current research, the
fluid time scale is defined as
∞
=Udτ f . (1.19)
The laser is then pulsed two times and each pulse is recorded by a photograph. Based on
the time between pulses and particle displacements, a velocity field can be calculated.
Two-phase flows are significantly harder to measure with PIV since a method must
be developed to discriminate the two phases. Multiple methods for differentiating
between the phases have been developed over the years. A good summary of these
methods is given by Khalitov and Longmire (2002). In the case of bubbly, two-phase
flows, color discrimination is the main technique used. In this case, seed particles
embedded with fluorescing dye are used to track the liquid phase. The seed particles
therefore emit a different color light than the light reflected by the bubbles. Then, by
using two cameras with filters or a single, color camera, the two phases can be
distinguished. The fluorescing seed particles are then used to determine the velocity of
the liquid phase whereas the velocity of the gas phase can be determined by using the
bubbles as tracers. This technique has been used successfully by many researchers for
dilute, bubbly flows (e.g. Sridhar et al 1991, Sridhar and Katz 1995, Oakley et al 1997,
Chaine and Nikitopoulos 2002). Gopalan and Katz (2000) and Laberteaux and Ceccio
9
(2001) also used this technique in the near-field of cavitating bodies to determine the
liquid velocity field only.
Up to this point, there has been very little success in using PIV to measure the
velocity field of both phases (gas and liquid) simultaneously in flows where the gas to
liquid volume ratio is fairly high (>10%). Only recently have a couple of successful
techniques been developed for these situations. In some instances, Particle Tracking
Velocimetry, or PTV, is used for determining the bubble (gas) velocity. This is because
even though the void fraction is fairly high there may be only a single bubble in a PIV
interrogation region.
There has been success in using PTV and PIV for multiphase flows by Broder and
Sommerfeld (2001). They used this technique to simultaneously measure the liquid and
bubble velocities in a bubble column. Fluorescent tracers were used for the liquid and the
bubbles were used as tracers for the gas. Standard PIV techniques were then used to
determine the liquid velocity while a combined PIV/PTV technique was used for the gas
phase. 1000 to 2000 images were collected to determine the mean liquid and gas velocity
fields. In this case, only about 500 to 1000 vectors maps per phase were collected due to
interference between the two phases. In order to minimize the effects of bubbles
illuminated outside of the laser sheet, the camera was placed about 80° off the plane of
the laser sheet. This is because the scattering light intensity of air bubbles in water
decreases strongly for angles larger than 82.5°, which can be shown using optics theory.
Therefore, it is hard to distinguish between bubbles inside and outside of the light sheet
for a camera arrangement perpendicular to the light sheet.
Another technique that uses PTV for the gas phase has been developed by Lindken
and Merzkirch (2001). In this case, shadowgraphy is used to locate the bubbles in the
flow. Their velocities are then determined by using PTV. Again, fluorescent particles
are used to simultaneously measure the liquid velocity using standard PIV techniques.
Due to optical difficulties such as reflections of the laser off of the bubbles, only 14
bubbles were generated at a time, with a spacing between each bubble of one to two
bubble diameters. Therefore, the local void fractions were fairly high, but the number of
bubbles was fairly low compared to a cavitating flow.
10
2. DESCRIPTION OF EXPERIMENT
2.1 Objectives
A significant amount of research has been conducted to determine the air
entrainment coefficient in the twin vortex regime. However, very little data are available
in the case of the re-entrant jet regime, which could occur, for instance, when a
supercavitating torpedo is initially fired and accelerating to a final, steady-state velocity.
Therefore, one of the goals of the current research is to quantify the air injection
coefficient under a variety of different cavitation and Froude numbers corresponding to
the re-entrant jet regime (σcFr > 1).
Another goal is to examine some of the qualities of the cavity surface and wake
details. This will be done in two ways. First, some of the details will be examined from
digital images taken with short duration strobe lights (~3 µs). These images will be used
to qualitatively describe the cavity and wake details. Second, PIV will be used to
examine the wake details more closely.
The water tunnel used for the current research was originally designed to achieve
speeds of approximately 30 m/s. In spite of this, the velocities in the current research
were limited to approximately 10 m/s to extend the life of the aging electric motor and its
controller. However, a numerical simulation of a three-dimensional, axisymmetric,
cavitating body has been developed at the University of Minnesota. The numerical
simulation also has provisions for simulating gas injection for the case of a ventilated
cavity. This simulation could be used to simulate higher velocity flows than can be
achieved in the current research. Before this can be done, though, the numerical model
has to be verified. One way of verifying the model is to compare the numerical
simulation to the experimental results obtained in the current research. Namely, the air
entrainment coefficient, cavity shape details, measured velocities in the wake, and
general wake characteristics obtained from the experiments could be used for comparison
purposes.
11
2.2 Experimental Facility
The experiments were conducted in the high-speed water tunnel at Saint Anthony
Falls Laboratory, University of Minnesota. A schematic of the water tunnel is shown in
Figure 2.1. The test section is approximately 19 cm square and 125 cm long with circular
fillets installed in the lower corners. The entire test section is bounded by solid walls, i.e.
there is no free surface. The water tunnel also has three observation windows in the test
section, all of which nearly span the entire test section length and width/height. There is
one window on each side of the test section and one on the bottom of the test section.
The pressure throughout the tunnel can be adjusted by varying the pressure in a
chamber that is attached to and located above the tunnel. The static pressure in the test
section is measured with an absolute pressure transducer. The freestream velocity can be
calculated from a differential pressure measurement between the static pressure in the test
section and the stagnation pressure measured upstream of the test section. The
relationship between differential pressure and freestream velocity was found by Dugué
(1992) and thereafter verified by other researchers over the past ten years:
ρ
)p2(p0.972U 0 ∞
∞−
= , (2.1)
where p0 is the stagnation pressure.
Unfortunately, the static pressure port in the test section is located on a portion of
the roof that has a slight amount of curvature. This causes the pressure measured at the
port to be slightly lower at the measurement location than at the centerline of the test
section. Dugué measured this effect using a pitot-static probe located at the test section
centerline and found the difference to be
0.06ρU
21
pp2
meas =−
∞
∞ , (2.2)
where pmeas is the pressure measured at the static port location on the roof and p∞ is the
true static pressure at the test section centerline. To verify this result, a numerical
simulation was run using FLUENT. For the analysis, an Euler solution was obtained, i.e.
viscosity was neglected. This simulation gave the result
12
0.02ρU
21
pp2
meas =−
∞
∞ , (2.3)
which is fairly close to the result obtained by Dugué. However, the difference in the two
equations above can be significant when the cavitation number is low and is calculated
using the static pressure measurement. For the current research, the correction found by
Dugué, equation 2.2, was used to correct the static pressure measurement. Further details
on the possible errors resulting from the use of this equation versus the use of equation
2.3 in the present research will be discussed later.
2.3 Test Body Design
The general shape of the test body was designed so that its overall shape was
similar to other test bodies being tested at the time this research was conducted. This was
done so that future comparisons between the data collected in the current research and
data from other researchers could be easily compared. The overall shape of the body is
shown in Figure 2.2.
The size of the test body and the supporting strut were chosen based on the choking
phenomenon and structural considerations. As Tulin (1961) notes, when a body is
mounted between solid walls, the cavity will be lengthened relative to the unbounded
condition. Eventually, the cavity length will become infinite at a cavitation number
greater than zero, which is known as the choking condition. Once the tunnel is choked,
further reductions in the cavitation number are impossible to achieve. Choking cavitation
numbers for different tunnel blockage ratios are shown in Figure 2.3. Using this figure
and taking into account structural considerations, the test body was designed so that it
would be fully contained within a cavity generated by a 1 cm disk at cavitation numbers
below approximately 0.14.
The choking phenomenon in a water tunnel with solid walls can be described easily
by considering conservation of mass and the Bernoulli equation for steady,
incompressible flow:
constantAV = (2.4)
constantρV21p 2 =+ . (2.5)
13
The analysis that follows will be for the two dimensional case. However, the same
reasoning can be extended to a three dimensional problem with ease. The problem
description is shown schematically in Figure 2.4. A control volume (CV) is drawn so
that its upstream boundary passes through a region where the flow is undisturbed and has
a freestream velocity of U∞. The downstream boundary of the control volume passes
through the maximum diameter of the cavity while the upper and lower boundaries of the
control volume are drawn along the surface of the tunnel where the freestream velocity is
parallel to the surface of the control volume. Using equation 2.5, the velocity on the
surface of the cavity can be found to be
cc σ1
UU
+=∞
, (2.6)
where Uc is the magnitude of the velocity along the surface of the cavity. Next, assuming
the velocity along the downstream boundary of the control volume outside of the cavity is
constant with a value of Uc and that the gas velocity inside the cavity is zero,
conservation of mass (equation 2.4) can be used to write
hD1
1UUc
−=
∞
, (2.7)
where h is the height of the control volume. The cavity diameter as a function of
cavitation number for different shaped cavitators can be found in many references (e.g.
Tulin, 1961). For example, for a wedge with a 0DC of 0.5, one can generate the plot
shown in Figure 2.5 for various blockage ratios, d/h. The curve based on the Bernoulli
equation gives the maximum velocity ratio, Uc/U∞, that can be obtained at a given
cavitation number. As the cavitation number decreases for a given blockage ratio, the
velocity ratio increases up to a maximum value given by the Bernoulli equation. Once
the velocity ratio reaches this point, the flow is choked and further increases in the
velocity ratio or decreases in the cavitation number are impossible to achieve.
Multiple considerations were taken into account when designing the strut for the
test body. First, the cross-sectional area of the strut was minimized to reduce its effect on
blockage. The size of the strut was also kept small to minimize the effect it had on the
14
cavity boundaries. Another key factor in the design of the strut was symmetry. Initially,
it was thought that multiple angles of attack would be investigated. Therefore, a
symmetric strut was needed so that its characteristics would not vary with angle of attack.
Taking into account these considerations, a strut with a cylindrical cross-section was
chosen.
The cylindrical strut was found to have many negative influences on the test body,
as will be discussed later. Therefore, a second strut was designed that could be attached
to the cylindrical strut. The new strut was designed by overlaying the centers of two
ellipses with identical minor axes centered on the cylindrical strut. Cross-sections of
each strut are shown in Figure 2.6.
Provisions for artificial ventilation were made by boring out the inside of the test
body. Air could be supplied to the test body by means of the tubular strut that extended
outside of the water tunnel. Six cylindrical holes located symmetrically around the axis
of the test body were drilled so that they spanned from the interior of the test body to the
surface. A deflector redirects the air coming out of these ports so that the main
component of the gas velocity is in the freestream direction. This was done so that the
high velocity gas would not impinge on the cavity surface and cause disturbances in the
cavity shape. The test body was also designed so that different cavitator shapes and sizes
could be tested. This was accomplished by machining a threaded hole into the test body
that would accept a threaded rod attached to the cavitator. A schematic showing these
features is given in Figure 2.7.
2.4 Experimental Setup
The test setup used for measuring the air entrainment coefficient is shown in Figure
2.8. Each pressure reading is recorded by a simultaneous sample and hold (SS&H)
board, which is connected to a personal computer. The SS&H board allows for the
simultaneous reading of up to 16 channels at relatively high sampling rates (greater than
5000 Hz). Air from outside of the tunnel is used to ventilate the cavity. The air does not
need to be pressurized since the pressure inside the water tunnel is lower than
atmospheric pressure. The air flow rate is measured by a rotameter, Omega Model FL-
114. The air flow rate is adjusted be means of a valve.
15
The water tunnel is equipped with two pressure transducers for determining the
operating conditions. As previously described, an absolute pressure transducer measures
the static pressure, p∞, in the test section by means of a flush mounted static port in the
roof of the test section. A differential pressure transducer measures the pressure
difference, p0 - p∞, between the test section and the stilling chamber upstream of the test
section. From these two measurements, the cavitation number based on vapor pressure
can be determined. Calibration procedures for the pressure transducers are given in
Appendix A.
The cavitation number based on cavity pressure is a key quantity in the present
research. One way of determining this value is to calculate it directly by measuring all of
the quantities in equation 1.4. Two different techniques to directly measure the cavity
pressure were attempted in the current research. One method was using miniature
electrical transducers mounted slightly recessed to the surface of the body. The second
method was to use conventional static ports. The miniature transducers used were
manufactured by Entran, Model EPB-B0. These transducers were selected because of
their high frequency response (up to 11 kHz) so that unsteady cavity pressures could be
captured. The location of each transducer and static port is shown in Figure 2.7.
The Entran pressure transducers are basically cylindrical in shape, with a maximum
diameter of 3.18 mm. The sensing portion of the transducer is located at one end of the
cylinder, while the lead wires come out of the other end. The transducers were mounted
such that their flat sensing surface was slightly recessed in the test body. The transducers
could not be mounted flush since the surface of the test body has curvature while the
surfaces of the pressure transducers are flat. Therefore, a small gap existed between the
transducer and the test body surface. Measurements were made with the gap open and
also filled with a highly incompressible vacuum grease. These measurements will be
discussed later.
The test setup for the PIV measurements is shown in Figure 2.9. Note that the test
body was mounted to one of the side windows of the water tunnel to aid in the PIV
measurements. This was done by attaching the strut of the test body to a mounting plug
that fits into a hole in the side of the water tunnel. Some views of the test body in the
16
non-cavitating case are shown in Figure 2.10, Figure 2.11, and Figure 2.12. These
pictures point out some of the foreground and background items captured by the camera
so that these items do not confuse the reader in images presented later.
Measurements in the wake were made in both the cavitating and non-cavitating
regimes. In the non-cavitating case, 1-3 µm diameter titanium dioxide seed particles
were used. These particles gave a maximum Stokes number on the order of 0.002 over
the range of conditions tested, which implies that the particles accurately followed the
flow. In the cavitating case, only the gas velocity was measured. Due to the size of the
water tunnel used in the current research, fluorescing particles could not be used to
simultaneously measure the liquid phase because of cost considerations due to the large
volume of water that would need to be seeded. Other common techniques for
simultaneously measuring the liquid and gas phases could not be used due to the fact that
the bubbles scattered light which was much higher in intensity compared to the titanium
dioxide seed particles. The details of the measurements in the cavitating regime will be
discussed later.
The PIV images were processed using 2-frame cross-correlation. This method uses
two image frames with one pulse of laser light on each frame. The flow velocity is found
by measuring the distance the particles have traveled between two consecutive frames
(TSI Inc. Operations Manual, 2000). The software used to process the images was
Insight, Version 3.53, available from TSI Inc. Further details on the software and 2-
frame cross-correlation technique can be found in the TSI documentation materials (TSI
Inc., 2000). The digital cameras used were manufactured by TSI Inc., Model PowerView
4M. These cameras have 4 megapixels (2k x 2k pixels) with a 12 bit dynamic range.
17
3. EXPERIMENTAL RESULTS
3.1 General Observations
The first experiments were conducted with the cylindrical strut shown in Figure 2.6.
It was immediately noticed that the test body vibrated a significant amount, with the
largest amplitude vibrations occurring when the freestream velocity was approximately 7
m/s. These vibrations were transmitted to the cavity surface, which made the cavity
appear like it was oscillating. The test body vibrations also made the bubbly wake appear
wavy in nature. Examples of this are shown in Figure 3.1.
A reasonable question to ask at this point might be, �Is the test body oscillating due
to cavity oscillations or some other phenomenon?� As was previously mentioned,
supercavities have been found to oscillate. However, a criterion for these oscillations is
the presence of a free surface, which does not exist in the water tunnel used in the current
study. Another important observation is that the test body vibrated even without the
presence of cavitation. The frequency of oscillation, f, during cavitating and non-
cavitating conditions was determined by using a high speed strobe and found to be
constant for a given freestream velocity. Non-dimensionalizing this value by using the
Strouhal number,
∞
=Ufd
S s , (3.1)
where ds is the strut diameter, it was found that S ≈ 0.2. This value is very close to the
Strouhal number for a bluff obstruction, which is 0.21 (Fox and McDonald, 1998).
Therefore, it was inferred that the periodic vortex shedding from the cylindrical strut
(Karman vortex street), which induces pressure oscillations to the strut, caused the test
body to oscillate. The elliptical strut was found to virtually eliminate the test body
oscillations, further supporting the hypothesis that the cylindrical strut induced the test
body oscillations.
The cylindrical strut also caused large distortions to the cavity boundary and
injected air to be entrained by the strut. Figure 3.2 shows a side by side comparison of
the cylindrical strut (left column) and elliptical strut (right column) for a constant
freestream velocity and pressure. Each row of pictures corresponds to a constant air
18
entrainment coefficient, with the air entrainment coefficient increasing from the bottom
row to the top row. The first thing to note in these pictures is the large distortion of the
cavity boundary near the cylindrical strut. This is due to the flow being perturbed far
upstream of the strut. The elliptical strut has less of an influence on the flow due to its
aerodynamic shape. Therefore, as the cavity reaches the elliptical strut, its boundaries are
perturbed less and the cavity extends over the length of the strut. This in turn leads to
results that are more characteristic and representative of a body without a strut, such as a
torpedo.
A second important difference to note in Figure 3.2 is the gas entrained by the strut.
As the cavity grows in length and reaches the cylindrical strut, injected gas is entrained
behind the strut due to the low pressure region there. This leads to an increased air
demand for a given cavity length when compared to the elliptical strut, which entrains a
very minimal amount of ventilation gas. In other words, this phenomenon leads to a
cavity length that is greater for the elliptical strut than for the cylindrical strut for a given
air entrainment coefficient. The elliptical strut entrains a much smaller amount of gas
than the cylindrical strut does because of the larger pressure recovery behind the elliptical
strut due to its more aerodynamic shape. This is also the reason why the cylindrical strut
has a natural cavitation number higher than the elliptical strut. In fact, in Figure 3.2 the
cylindrical strut is cavitating naturally whereas the elliptical strut is not cavitating.
Nonetheless, even when the cylindrical strut did not cavitate naturally, the same gas
entrainment phenomenon of the cylindrical strut was seen as the cavity length increased
to the point where it reached the strut.
Also note the Karman vortex street behind the cylindrical strut in Figure 3.2. For
large air injection rates, part of the injected air escapes to the core of the Karman vortices.
The pictures clearly show where the center of each main vortex core shed by the strut is
located. This further shows the significance of the Karman vortices, which cause both
the test body to oscillate and the air entrainment to increase as described previously.
An important observation to note from the cavity pictures is the effect of the re-
entrant jet on the cavity dynamics. As can be seen in Figure 3.1, some regions of the
cavity are transparent while others are opaque. A closer examination of the opaque
19
regions reveals that the cavity surface is not smooth in these areas. The re-entrant jet is
the cause of the surface disturbances. A typical picture of a supercavity is shown in
Figure 3.3. The body was supported by the cylindrical strut in this picture. The boxed
region in the upper picture has been magnified and shown in the bottom picture to show
some of the important features. Note that in the magnified picture, a drop of water on the
bottom of the test body directly behind the gas deflector is in contact with the cavity
surface. This drop perturbs the cavity surface and causes the cavity surface to become
opaque. The drop of water is caused by the re-entrant jet, which can be clearly seen
surging upstream in the magnified image. As the re-entrant jet surges forward, it loses
momentum due to friction. The re-entrant jet is also acted upon by gravity, which causes
drops of water to fall from the test body surface and impact the cavity boundaries. The
non-steady dynamics of the re-entrant jet also cause water to be splashed in all directions,
leading to even more opaqueness in the cavity walls.
The same type of phenomenon was observed when the elliptical strut was used, as
can be seen in Figure 3.4. Note that the re-entrant jet has surged farther upstream in this
picture when compared with Figure 3.3. This is due to the longer cavity (i.e. lower
cavitation number) in Figure 3.4. The lower cavitation number causes the velocity of the
re-entrant jet to increase. This leads to an increase in momentum and allows the jet to
surge farther forward. A secondary reason for the re-entrant jet surging farther upstream
might be due to the elliptical strut. As was previously mentioned, the elliptical strut has a
smaller influence on the cavity boundaries. Since the cavity boundaries are perturbed
less, the re-entrant jet can be assumed to be stronger when compared to the jet formed
with the cylindrical strut in place.
As was previously mentioned, two different techniques were used to try to measure
the cavity pressure. One technique was to use miniature Entran pressure transducers.
However, the use of these transducers led to many inaccuracies. First, the repeatability
and stability of the transducer�s output was poor when it was placed in water. This was
in part due to the fact that the transducer�s output is dependent on the temperature of the
sensor. Note that the transducers used were very small (3 mm in diameter), and therefore
have a small thermal mass. Thus, self-heating of the transducer occurs due to the power
20
required to operate the sensor. Eventually, the transducer reaches an equilibrium
temperature. However, as the flow of the surrounding fluid changes over the transducer,
the equilibrium temperature of the sensor varies, thus shifting its output and causing
inaccuracies. In an attempt to alleviate this problem, the gap between the transducer and
the test body was filled with an incompressible vacuum grease to insulate the sensor.
This dramatically decreased the change in output of the sensor due to varying flow
conditions. However, the uncertainty that still did exist due to temperature, along with
the nonlinearity and hysteresis of the transducer, was enough to make the cavity pressure
measurements obtained by using the Entran transducers too inaccurate for use.
A second attempt to measure cavity pressure was made by using conventional static
ports, whose locations are shown in Figure 2.7. Pressure tubing was attached to these
ports and sensed by an external pressure transducer. Cavity pressure measurements could
only be made for relatively long cavities because the pressure port locations were near the
downstream end of the test body. The pressure lines were filled with water and purged
between measurements to ensure no air bubbles were present in the pressure lines.
However, even with this careful practice, the pressure measurements were not repeatable.
The cause of this was most likely due to the re-entrant jet flowing over the pressure ports.
The re-entrant jet was also a likely contributor to the non-repeatability of the miniature
Entran transducers.
3.2 Air Entrainment Results
Even though the cavity pressure could not be measured accurately, the cavitation
number could still be determined. This was accomplished by measuring the cavity half-
length and then determining the cavitation number and cavity pressure. High quality,
digital images were used to measure the size of each cavity under different conditions.
First, a picture of the test body was taken in the non-cavitating case, which was used for
calibration purposes. Next, digital pictures were taken of the cavity for different
freestream conditions and air injection rates. The half-length of each cavity was then
measured manually for each picture using imaging software. Three to five pictures were
taken at each test condition and the results of each cavity length measurement were
averaged. The difference between the length determined from an individual picture and
21
the average value for each data set rarely exceeded 5%. Next, the cavitation number was
calculated. As was shown in Figure 1.3, relationships between cavity length and
cavitation number have been determined by a number of researchers. The experimental
and theoretical work does show good agreement over the range of cavitation numbers
tested in the current research. Therefore, accurate results for the cavitation number could
be obtained by simply measuring the cavity length.
Initially, the cavitation number was calculated from the formula derived by Waid
(equation 1.13). This equation was chosen in part due to the fact that it could be solved
explicitly for the cavitation number. Although this equation was derived for cavitation
numbers ranging from 0.035 to 0.171, the equation does agree well to the equations
developed by Garabedian and Reichardt for higher cavitation numbers. Note also that the
equations derived by Garabedian and Reichardt were derived for small cavitation
numbers, typically below 0.1. Considering the range of cavitation numbers tested in the
present study, the equation developed by Waid was used as a first estimate for calculating
the cavitation number.
Once the cavitation number was calculated, the cavity pressure could be obtained
using equation 1.4. The cavity pressure was required so that the volumetric flow rate at
cavity pressure of the injected gas could be obtained. After the volumetric flow rate was
determined, the air entrainment coefficient could be calculated. As was previously
mentioned, the formula found by Dugué, equation 2.2, was used for correcting the
measured static pressure. The uncertainty in the air entrainment coefficient due to the
uncertainty in the correction for the static pressure along with all of the other sources of
uncertainty is detailed in Appendix B.
The results for air entrainment coefficient versus cavitation number for a 1 cm disk
with the cylindrical strut are shown in Figure 3.5. The data are plotted for various Froude
numbers based on disk diameter. One thing to note is that the effect of the Froude
number is small except for the highest Froude numbers tested. For large Froude
numbers, the cylindrical strut began to cavitate. In these cases, more injected air was
entrained by the strut at low cavitation numbers than for the lower Froude number cases.
This led to a decrease in the cavity length behind the disk, thus leading to what appears to
22
be an increase in air demand for a given cavitation number compared to the lower Froude
number cases. Again, this was one of the reasons the elliptical strut was designed.
A second important thing to note from Figure 3.5 is the large increase in the air
entrainment coefficient for cavitation numbers below 0.14. Note that below a cavitation
number of about 0.14, the cavity length grows beyond the length of the test body. This
can be seen in Figure 3.6, which shows pictures of the cavity at different cavitation
numbers. It can be inferred from this data that the air entrainment coefficient for a given
cavitation number depends on whether or not the cavity closes on a solid object.
Results for the air entrainment coefficient for a 1 cm disk with the elliptical strut are
shown in Figure 3.7. The axes of this figure are the same as in Figure 3.5 for comparison
purposes. Note that the air entrainment coefficient at small cavitation numbers is
significantly lower for the elliptical strut when compared to the cylindrical strut data.
Also, the data collapse for the elliptical strut even for the high Froude number cases, in
contrast to the cylindrical strut data. Both of these observations can be attributed to the
aerodynamic shape of the elliptical strut. As was shown previously, the elliptical strut
did not entrain nearly the amount of air as the cylindrical strut. Therefore, the air
entrainment coefficient for a given cavitation number was smaller for the elliptical strut
than it was for the cylindrical strut. A comparison of the data for both struts is shown in
Figure 3.8.
Similar plots for a 1.5 cm disk are shown in Figure 3.9, Figure 3.10, Figure 3.11,
and Figure 3.12. Note that the general behavior of the data is the same for the 1.5 cm
disk as it was for the 1 cm disk. Another thing to note is that the large increase in the air
entrainment coefficient occurs at a larger cavitation number for the larger disk. This is
because the cavity reaches the back of the test body at the same cavity length, which
occurs at a larger cavitation number for larger and larger disks. For example, a cavitation
number of 0.20 for a 1.5 cm disk and a cavitation number of 0.14 for a 1 cm disk both
create approximately the same cavity length.
As was discussed previously, Brennen carried out numerical simulations of
cavitating flow behind a disk for bounded flows. Since blockage is important in the
current research, Brennen�s data were also used to calculate the cavitation number for
23
different length cavities so that the blockage ratio was taken into account. Brennen�s
numerical results are shown in Figure 3.13. Along with his data, results from Waid and
Garabedian are also plotted for reference.
Brennen�s calculations assumed that the flow was bounded by a cross-section that
was cylindrical in shape. This was not the case in the current study. However, the ratio
that is important is the blockage ratio based on area. Therefore, the cross-sectional area
of the water tunnel used in the current research was found and then converted to an
equivalent cylindrical diameter. This was done by finding the cross-sectional area of the
test section and finding what diameter, the equivalent diameter, would lead to the same
cross-sectional area. The equivalent diameter, h, was then used to determine the
blockage ratio.
Since the data of Brennen did not match up exactly to the blockage ratios of the
current study, Brennen�s data had to be interpolated. This was done using a cubic spline
interpolation. The interpolated values were found at h/d values of 21.5 and 14.3, which
correspond to disk diameters of 1 cm and 1.5 cm, respectively. The values given by
Tulin (Figure 2.3) for the cavitation number at choking were used as checks to ensure the
interpolated values were reasonable. All of these data are plotted in Figure 3.13. Note
that the interpolated data match up very close to the choking cavitation numbers given by
Tulin. Using the interpolated curves, new cavitation numbers were calculated for the
data in the current research. These results are shown in Figure 3.14 and Figure 3.15.
Note that in general the data collapse better at low cavitation numbers and have more
scatter at higher cavitation numbers when compared with the results when the cavitation
number was calculated by Waid�s formula. The large scatter at high cavitation numbers
can be explained by looking at the error in the measured cavity length. As can easily be
seen in Figure 1.3, as the cavitation number increases the curves of cavity length versus
cavitation number become very flat. Therefore, a small error in the measured cavity
length can lead to a large error in the calculated cavitation number. This leads to more
scatter in the data at large cavitation numbers.
Finally, all of the experimental data for both disk sizes and strut shapes are plotted
in Figure 3.16. Again, the cavitation number was calculated using Brennen�s data. This
24
plot clearly shows how the location of the steep increase in the air entrainment coefficient
depends on the cavitator size. However, as previously discussed, the cavity length where
the air entrainment increases dramatically is approximately the same for both cavitator
diameters.
It should be noted that the data where the cavitation number is calculated from data
interpolated from Brennen�s numerical results should be considered more accurate. This
is because it takes into the account the effect blockage whereas the equation of Waid does
not. The best way to determine the cavitation number would be to measure it directly.
However, this was not possible in the current study due to aforementioned reasons.
Therefore, using the data of Brennen was considered the next best option.
3.3 PIV Results
As was previously mentioned, wake measurements were made in both the
cavitating and non-cavitating regimes. Measurements were made in the non-cavitating
regime for two reasons. First, these measurements were made to ensure that accurate
results could be obtained with the PIV system since this was the first time a PIV system
was used in the facility. Second, the non-cavitating wake was measured so that it could
be compared both qualitatively and quantitatively to the cavitating wake characteristics.
All of the PIV data presented in this section correspond to the cylindrical strut and the 1
cm disk.
The growth of a wake�s width and the velocity profile is well known for a non-
cavitating, circular wake, so comparisons to these known results were made to ensure the
PIV system gave accurate results. For a circular wake, which is the case in the current
experimental study because the body is symmetrical and at zero angle of attack, the
growth of the width of the wake is proportional to x1/3, whereas the velocity defect, δu, is
proportional to x-2/3 (White, 1991). These results are known as the power laws for wakes.
Note that for the bounded flow case under consideration,
δu = Umax � u(r), (3.2)
where Umax, the maximum velocity at a given downstream location, is greater than U∞
due to blockage effects. A schematic showing the applicable coordinate system and the
definitions of the terms just described is shown in Figure 3.17.
25
A 60 mm lens was used to capture the images for the non-cavitating data discussed
below. This led to images with a resolution of 142 pixels/cm. The interrogation area
used for analyzing the PIV images was 32 x 32 pixels with 50% overlap in both the
vertical and horizontal directions. During data processing, conventional rules for
correlating the data were used based on suggestions from the software developers, TSI
Inc. This included using an interrogation area in which the seed particles did not move
more than one-fourth of the way through the interrogation region between two
consecutive images (TSI Inc. Operations Manual, 2000).
Post processing tools were used to eliminate spurious vectors. The order of the
filters used was based on recommendations from TSI (TSI Inc. Instruction Manual,
2000). First, a global range filter was used to eliminate vectors that were obviously
incorrect. In other words, vectors were removed if their absolute magnitude was much
larger than the freestream velocity. Next, a global standard deviation filter was used.
This filter was setup to remove vectors if either of their velocity components were more
than three standard deviations away from the mean. Next, a median filter was used. This
filter looked at a 5 x 5 region around the vector of interest. If either of the velocity
components of the vector of interest were farther than two positions away from the
neighborhood median, the vector was removed. Finally, a mean filter was used to
eliminate the remainder of the spurious vectors. This filter also looked at a 5 x 5 region
around the vector of interest, only this time an average value for each velocity component
was calculated. If either of the velocity components of the vector of interest varied by
more than 2 m/s from the neighborhood mean, it was removed.
Liquid velocity data in the wake for the non-cavitating regime are shown in Figure
3.18. For this figure and the following non-cavitating results, 500 image pairs were used
to calculate an average velocity field. The freestream velocity in this case was 6.36 m/s.
Note that Umax is greater than U∞ because the flow is bounded. The plot clearly shows
the velocity in the wake approaching its freestream value as the downstream distance
increases.
Figure 3.19 shows the same data plotted in Figure 3.18 using the power law
relationships. Note the collapse of the data is quite good for the wake�s width and
26
velocity decay. However, there is one region where the data do not collapse very well.
This region is in the upper portion of the wake and even outside of the wake, namely for
values of r/x1/3 greater than approximately 0.6. It is believed this is due to the non-
uniform velocity field in the water tunnel where the test body was located. The test body
was mounted in a region of the test section where the roof had a slight amount of
curvature (near the static port location). Therefore, the velocity field was not exactly
uniform at the test body location. However, the velocity field was more uniform than
what it may first appear to be from Figure 3.19. As can be seen in Figure 3.18, the
difference in velocity from 0.5 cm to 9.0 cm downstream of the test body outside of the
wake is less 1.5% of the freestream velocity. Ultimately, it is important to note the good
collapse of the data in Figure 3.19, which indicates that the velocity data from the PIV
instrumentation are accurate.
Finally, the velocity data are plotted in non-dimensional form in Figure 3.20. The
power law relationships were used as a starting point for converting the data to non-
dimensional form. Then, the cavitator diameter and maximum velocity in the wake were
used to non-dimensionalize the power law terms. Again, the data at various downstream
locations correlate well.
Measurements in the cavitating wake were made next. As was previously
discussed, fluorescing seed particles could not be used because of cost considerations.
However, a new technique was developed that uses PIV to measure the void fraction of
gas to liquid in the wake without the need to use seed particles. The details of this
technique are discussed below. Experimental data are also presented to show the results
of this new technique.
The test setup for the void fraction measurements is shown in Figure 2.9. For this
technique, traditional seed particles were not used to generate velocity fields. Instead, the
bubbles were used as the particles from which the velocity of the gas phase was
determined. When using this technique, the aperture of the camera was set such that high
intensity reflections of laser light from the bubbles did not saturate the digital camera�s
pixels. In addition, since the bubbles reflected light that was much higher intensity than
the light reflected by microscopic particles in the water, such as seed particles from
27
previous experiments, the microscopic particles were lost in the background noise in the
final image.
Standard PIV images were collected and processed in a manner analogous to the
non-cavitating case to determine the velocity of the gas in the wake. This was possible
since the void fraction of the gas phase was relatively high. Therefore, enough bubbles
were present in the interrogation regions in the bubbly wake to provide a usable signal for
cross-correlation. However, the number of validated vectors was not as high for the
cavitating case as it was for the seeded, non-cavitating case. In the non-cavitating
regime, the number of validated vectors was approximately 90% over the entire flow
field. However, in the cavitating regime, the number of validated vectors was lower due
to bubble reflection and refraction and bubble density. At the center of the bubbly wake,
the number of validated vectors was in the range of 25% to 70%, with the percentage
increasing with downstream distance. The number of validated vectors dropped off fairly
rapidly moving radially outward from the center of the wake.
The first step in calculating the void fraction in the wake is to analyze the grayscale
levels of the PIV images. This is done by first calculating the average grayscale value for
each pixel from a series of PIV images. Once the average grayscale value for each pixel
is determined, the background noise must be subtracted off so that a grayscale value of
zero corresponds to a void fraction of zero. Next, it is assumed that the intensity of the
reflected light is linearly proportional to the void fraction. Therefore, once the
background noise is subtracted off, the general shape of the radial position versus void
fraction curve is obtained by plotting the grayscale value for each pixel. At this point, the
curve is only qualitatively correct as the magnitude of the void fraction must still be
determined. However, calculating the void fraction is a simple process since the velocity
of the gas phase can be measured with PIV and the air injection rate can be measured
with a rotameter.
The void fraction can be determined by solving the equation
dA u(r)ηρmA g∫=& (3.3)
for η, the void fraction of gas to liquid. Note that the values for all of the other terms are
known except for the density of the gas, ρg. Its value can be estimated in the following
28
manner. First, the pressure of the gas is assumed to equal that of the surrounding fluid,
with the pressure being a constant for a given downstream location. The pressure is
calculated from the Bernoulli equation, equation 2.5, by setting V equal to Umax and
considering the freestream conditions U∞ and p∞. Once the pressure is known, the
density of the gas can be calculated from the ideal gas law. The temperature of the gas is
assumed to equal the temperature of the injected air, which is approximately the same as
the water temperature inside the tunnel. Finally, since the shape of the void fraction
curve is known from the PIV images, the magnitude can be determined by numerically
integrating equation 3.3.
Experimental results will now be shown to illustrate the procedure for determining
the void fraction in the wake as outlined above. The same procedure for cross-correlating
the PIV images in the non-cavitating flow was used in the cavitating flow. In addition,
the same post processing tools were used to eliminate spurious vectors.
Bubble velocity data in the cavitating wake are shown in Figure 3.21. The
cavitation number for this case was approximately 0.15. These data were taken at the
same freestream conditions as the data shown previously in the non-cavitating regime
(U∞ = 6.36 m/s). In addition, these data were collected and generated using the same
camera lens and interrogation area as the data discussed previously in the non-cavitating
regime. Again, 500 image pairs were used to calculate an average velocity field. Points
are plotted only for interrogation areas that had validated vectors in at least 15% of the
500 images. Therefore, each data point represents, at a minimum, an average of 75
vectors. As can be seen, the data are not nearly as smooth as the data in the non-
cavitating regime. This can be attributed to the fact that not as many vectors were
averaged in the cavitating regime. Therefore, not enough data points were taken to
represent a true, averaged vector field. However, the data at least shows good qualitative
results, if not quantitative, as discussed later. As can be seen, the data show the velocity
in the wake tending towards the freestream velocity as the downstream distance
increases.
Since the data presented in Figure 3.21 were taken at the same freestream
conditions as the data shown previously in the non-cavitating regime, a comparison
29
between the two data sets can be easily made. Velocity data in the wake for both the
cavitating and non-cavitating regimes are shown in Figure 3.22. Note that liquid
velocities for the non-cavitating case and bubble velocities for the cavitating regime are
shown for the same downstream locations. It can be seen that the velocity data correlate
quite well between the two regimes. This is true even though a fairly small number of
vectors were validated for the cavitating regime compared to the non-cavitating regime.
The good agreement between the data at the same freestream conditions lends support to
the qualitative validity of the PIV results in the cavitating regime.
Next, the calculated void fraction for the conditions described above will be
presented. First, however, the step by step process of determining the void fraction from
the grayscale levels of the images will be shown graphically to illustrate the method.
Again, 500 images were used to obtain averages in the data that follow.
The average grayscale values for various downstream locations are shown in Figure
3.23. The data are normalized such that a value of zero corresponds to black and a value
of one corresponds to pixel saturation (white). Note that the grayscale levels are highest
for downstream locations greater than six centimeters. At first this may seem odd since
previously it was mentioned that the void fraction is proportional to the grayscale level.
Therefore, since the void fraction is highest closest to the body, (where the wake has not
had a chance to spread and the velocity is lowest) it would seem like the grayscale levels
should be highest there as well. However, for this case the laser was centered at a
downstream location of approximately seven centimeters. Since the laser is brightest at
the center and decays in intensity from the centerline of the sheet, the bubble reflections
are the most intense at the center of the sheet. Remember, though, that at this point we
are only interested in obtaining the relative shape of the void fraction curve. The
magnitude of the curve will be determined later. Note also that the curves do not
approach zero away from the center of the body. This is because there is some
background noise in the images, which in this case is equal to an average grayscale value
of approximately 400.
Figure 3.24 shows the grayscale levels at various downstream locations after the
average background noise was removed. Again, the grayscale data have been
30
normalized. Note how the curves now approach a grayscale level of zero, indicating a
void fraction of zero percent. Here again, the curves only show the general shape of the
void fraction curves, not the absolute magnitude.
The void fraction results are shown in Figure 3.25. The void fraction was
calculated by using equation 3.3 and the midpoint rule for numerically approximating the
integral. The velocity data used to determine the void fraction are shown in Figure 3.21.
Note that the velocity data do not extend to the edge of the wake since the number of
validated vectors there was low. Therefore, as an approximation the velocity values at
the outer edges of the profiles at each downstream location were extended radially
outward (in the r direction) and kept constant so that the void fraction could be
determined.
Note that the magnitudes of the void fraction curves shown in Figure 3.25 are now
correct and show, at a minimum, the qualitatively correct result of the maximum void
fraction decreasing with downstream distance. However, continuity for the gas phase
still holds. In other words, the mass flow rate of the gas is a constant at each downstream
location. Note from Figure 3.21 that since the velocity in the wake increases with
downstream distance, the void fraction must decrease. Also, since the velocity data for
the gas phase correlated well with the non-cavitating case, the magnitudes of the void
fraction curves should also be correct.
Additional PIV data were also taken on a different day from the data described
above. For this case, the freestream velocity was 6.6 m/s, which is slightly higher than
the previous case. These data were actually taken prior to the data discussed previously
so it was not known how many images were needed to obtain true averages of various
quantities. Therefore, for the data presented below, only 100 image pairs for each test
condition were taken and analyzed. Also, the PIV images were taken using a 105 mm
lens. This led to images with a resolution of 265 pixels/cm. Therefore, the total area of
the wake captured in each image was about half of what it was when the 60 mm lens was
used. Because of this, an interrogation area of 64 x 64 pixels was used to ensure the
bubbles did not move one than one-fourth of the way through the interrogation region
between two consecutive images.
31
Velocity data in the non-cavitating wake are shown in Figure 3.26, Figure 3.27, and
Figure 3.28. It can easily be seen that the curves are not as smooth as the data shown
previously for which 500 vector fields were averaged. However, the data are starting to
collapse fairly good even though only 100 vector fields were averaged. A comparison of
these data to the non-cavitating wake data shown previously is shown in Figure 3.29.
Note that these data are plotted using the power law relationships. As can be seen, the
two data sets do agree well.
Next, bubble velocity and void fraction data are shown. These data corresponds to
a cavitation number of 0.15. The velocity data in the wake for various downstream
locations are shown in Figure 3.30. Again, the velocity data are not very smooth.
However, in general it shows the correct trend of the velocity increasing with
downstream distance. The calculated void fractions are shown in Figure 3.31. Here, the
data are significantly less smooth than the void fraction data presented previously. This
is due in large part to the average grayscale values. Since only 100 images were
averaged, the grayscale values did not reach a true average value. This in turn caused the
void fraction curves to look jagged. A comparison between the current void fraction data
to the data presented previously is shown in Figure 3.32. Note that although the
cavitation numbers are the same, the air injections rates were slightly different between
the two cases. This is because the freestream velocities were slightly different between
the two cases. The mass flow rate of the injected gas, m& , is about 8% higher for the
lower freestream velocity case. Note that the data for the two cases do show some broad
agreement even though only 100 images and vector fields were averaged for the case
where U∞ = 6.6 m/s compared to 500 averaged images and vector fields for the case when
U∞ = 6.36 m/s.
Next, data are presented for a cavitation number of 0.24 and U∞ = 6.6 m/s. These
data were also obtained from 100 averaged vector fields and images. Bubble velocity
data in the wake are shown in Figure 3.33. As before, clearly not enough vector fields
were averaged to obtain an accurate representation of the average velocity in the wake.
A comparison between the calculated void fraction for this case and the data shown
previously at the same freestream conditions but a cavitation of 0.15 is shown in Figure
32
3.34. The data show the general trend of a higher void fraction for the data
corresponding to the lower cavitation number. This should be expected since the air
injection rate increases as the cavitation number decreases (for a given freestream
velocity).
33
4. CONCLUSION
4.1 Conclusions
The lack of recent research in the field of supercavitation led to a fair amount of
work required to obtain valid experimental results. Some of knowledge obtained during
the initial phases of the current research was probably well known thirty years ago, but
due to the lack of recent studies these basic ideas had to be revisited. This included
designing an appropriate mounting strut for the test body. At first, a cylindrical strut
seemed like a logical choice. However, it was quickly found that the cylindrical strut led
to quite a few problems. These included test body vibrations and air entrainment behind
the strut. In short, it was found that the strut shape was very important and that an
aerodynamically shaped strut is necessary to obtain accurate air entrainment results.
The shape of the afterbody was found to have a dramatic effect on the air
entrainment coefficient. Once the cavity extended beyond the end of the test body, the
air entrainment coefficient increased at a much higher rate than when the cavity collapsed
on the body. It was found that while the sharp increase in the air entrainment coefficient
occurs at a given cavity length, the sharp increase may occur at different cavitation
numbers depending on the size of the cavitator.
Blockage effects were also found to be very important. Since the cavity length was
used to determine the cavitation number in the current research, ignoring blockage effects
led to an underestimation of the cavitation number for a given cavity length. This is
because blockage effects cause the cavity length to increase versus an unbounded flow
case. By using the numerical results generated by Brennen, the air entrainment data in
the current research were found to collapse quite well.
Experiments conducted using PIV helped to quantify some of the details in the
wake of the test body. Initially, a non-cavitating case was investigated to ensure the PIV
results were accurate. The data collected showed excellent agreement with the power
laws for axisymmetric wakes.
Experiments using PIV were then extended to the cavitating regime. In this case,
only the bubble velocities were obtained. Agreement between the liquid velocity in the
non-cavitating regime and the bubble velocity in the cavitating regime for a given
34
freestream condition was found to be very good and lends support to the use of PIV in
accurately measuring bubble velocities in cavitating flows. However, it was found that a
large number of vector fields, at least 500, were needed to obtain accurate average
velocities in the wake in the cavitating regime.
Finally, a new technique was developed that uses PIV to measure the void fraction
of gas to liquid in the wake of a cavitating body. This technique uses the velocity
information obtained using PIV along with analyzing the grayscale levels of the images
collected. This simple technique does show promise in accurately determining the void
fraction in the wake of cavitating bodies even when the void fractions are relatively high.
4.2 Recommendations for Future Work
The uncertainty in the freestream static pressure was a major contributor to the
overall uncertainty in the data and is therefore something that should be investigated so
that its value can be known more precisely. There are two ways that this problem could
be approached. First, a detailed study of the flow characteristics in the water tunnel could
be performed to fully characterize the static pressure in the test section at various
locations. Obviously, this is not a trivial task and would take a fair amount of time. A
quicker and most likely better solution would be to relocation the static pressure tap
farther downstream in the test section and also mount the test body further downstream
where the velocity field and pressure distribution is more uniform. With a carefully
drilled static port in the proper location, the static pressure could be measured with a high
degree of accuracy. Based on the FLUENT analysis performed, the static port should be
moved approximately 25 cm downstream of its current location to eliminate the error due
to roof curvature. This is assuming the port would still be located on the roof of the test
section. A graph showing the error based on the distance from the current static port is
shown in Figure 4.1.
As was discussed, there were also problems measuring the cavity pressure in the
current research. However, this measurement would be useful so the cavitation number
could be obtained directly. One simple idea for measuring the cavity pressure would be
to use the rotameter that was used for measuring the air entrainment coefficient. The
proposed test setup is shown in Figure 4.2.
35
Initially, the test body would be mounted into an empty water tunnel. The pressure
in the tunnel, pc, would then be set to a given value. Next, the air airflow rate would be
adjusted and the pressures p1 and pc would be recorded. Here, p1 is the pressure directly
downstream of the rotameter. Since the volume of the tunnel is large, the value of pc
would remain fairly constant. Multiple values of pc could then be tested to generate
calibration curves of patm � pc versus patm � p1, where patm is the atmospheric pressure. A
preliminary set of calibration data are shown in Figure 4.3 for various rotameter readings
(air injection rates).
Next, the water tunnel would be filled and multiple cavities generated for various
freestream conditions and air injection rates. During this testing, the value of p1 would be
recorded. Digital images of each cavity would also be taken so that the cavity length
could be measured for each test case. Since the value of p1 and patm would be known, the
cavity pressure could be obtained using the calibration curves generated previously.
Finally, a curve of cavity length versus cavitation number could be obtained. This curve
would take into account the blockage effects of the tunnel. Namely, the effect of the strut
on blockage would be taken into account. Note that this effect was assumed negligible in
the current research.
One of the downfalls of this technique is that the cavity length would still need to
be measured for each test condition. In addition, different curves of cavity length versus
cavitation number would need to be generated for different test setups since the blockage
effects would vary depending on the geometry of the test body.
One interesting observation noted from the air entrainment results was the effect the
afterbody had on the air entrainment coefficient. Namely, the air entrainment coefficient
increased significantly faster once the cavity extended beyond the body. Useful data
could be obtained by measuring the air entrainment coefficient for different afterbody
shapes and sizes.
More void fraction experiments should also be performed at various freestream
velocities and cavitation numbers. In addition, data should be collected using the
elliptical strut since it leads to a wake that is more representative of a real life situation.
In the current research, the number of test conditions was limited due to the aging electric
36
motor that powered the water tunnel. However, the experiments that were run did show
promise in the new procedure for measuring the void fraction.
As was mentioned earlier, other researchers have had success at using a non-
perpendicular camera arrangement to measure bubbly flows with PIV. This is something
that maybe worthwhile pursuing in the future.
Finally, the results of this research should be compared to the numerical simulation
currently being developed at the University of Minnesota to simulate ventilated,
supercavitating flow. At the time this thesis was written, the numerical simulations were
still converging so comparisons could not be made.
37
BIBLIOGRAPHY
Adrian, R.J. Particle Imaging Techniques for Experimental Fluid Mechanics. Annual
Review of Fluid Mechanics, 23, pp. 261-304, 1991. Arndt, R.E.A. Cavitation in Vortical Flows. Annual Review of Fluid Mechanics, 34, pp.
143-175, 2002. Braselmann, H., Buerger, K.H., Koeberle, J. On the Gas Loss from Ventilated
Supercavities � Experimental Investigation. International Summer Scientific School on High Speed Hydrodynamics, Cheboksary, Russia, 2002.
Brennen, C. A numerical solution of axisymmetric cavity flows. Journal of Fluid
Mechanics, 37, pp. 671-688, 1969. Broder, D., Sommerfeld, M. Experimental studies of the hydrodynamics in a bubble
column by an imaging PIV/PTV-system. 4th International Symposium on Particle Image Velocimetry, Gottingen, Germany, 2001.
Campbell, I.J., Hilborne, D.V. Air Entrainment Behind Artificially Inflated Cavities.
Second Symposium on Cavitation on Naval Hydrodynamics, Washington, 1958. Chaine, G., Nikitopoulos, D.E. Multiphase Digital Particle Image Velocimetry in a
Dispersed, Bubbly, Axisymmetric Jet. Proceedings of ASME Fluids Engineering Summer Meeting, Montreal, Quebec, 2002.
Cox, R.N., Clayden, W.A. Air Entrainment at the Rear of a Steady Cavity. Proceedings
of the N.P.L. Symposium on Cavitation in Hydrodynamics, London, 1956. Dugué, C. Preliminary Investigation of the Tip Vortex Cavitation and the Lift and Drag
of Four Elliptic Foils. St. Anthony Falls Hydraulic Laboratory, 1992 (unpublished). Fox, R.W., McDonald, A.T. Introduction to Fluid Mechanics, 5th ed. John Wiley &
Sons, Inc., New York, 1998. Garabedian, P.R. Cavities and Jets. Pacific Journal of Mathematics, 6, No. 4, pp. 611-
684, 1956. Gopalan, S., Katz, J. Flow structure and modeling issues in the closure region of
attached cavitation. Physics of Fluids, 12, No. 4, pp. 895-911, 2000. Khalitov, D.A., Longmire, E.K. Simultaneous two-phase PIV by two-parameter phase
discrimination. Experiments in Fluids, 32, pp. 252-268, 2002. Knapp, R.T., Daily, J.W., Hammitt, F.G. Cavitation. McGraw-Hill, New York, 1970.
38
Laberteaux, K.R., Ceccio, S.L. Partial cavity flows. Part 1. Cavities forming on models
without spanwise variation. Journal of Fluid Mechanics, 431, pp. 1-41, 2001. Laberteaux, K.R., Ceccio, S.L. Partial cavity flows. Part 2. Cavities forming on test
objects with spanwise variation. Journal of Fluid Mechanics, 431, pp. 43-63, 2001. Lindken, R. Merzkirch, W. A novel PIV technique for measurements in multiphase flows
and its application to two-phase bubbly flows. 4th International Symposium on Particle Image Velocimetry, Gottingen, Germany, 2001.
Oakley, T.R., Loth, E., Adrian, R.J. A Two-Phase Cinematic PIV Method for Bubbly
Flows. Journal of Fluids Engineering, 119, pp. 707-712, 1997. Reichardt, H. The Laws of Cavitation Bubbles at Axially Symmetrical Bodies in a Flow.
Ministry of Aircraft Production (Great Britain), Reports and Translations No. 766, 1946.
Schaffar, M.J., Rey, C.J. Boeglen, G.S. Experiments on Supercavitating Projectiles
Fired Horizontally into Water. Proceedings of ASME Fluids Engineering Summer Meeting, Montreal, Quebec, 2002.
Schiebe, F.R., Wetzel, J.M. Ventilated Cavities on Submerged Three-Dimensional
Hydrofoils. St. Anthony Falls Hydraulic Laboratory, University of Minnesota. Technical Paper No. 36, Series B, 1961.
Self, M.W., Ripken, J.F. Steady-State Cavity Studies in a Free-Jet Water Tunnel. St.
Anthony Falls Hydraulic Laboratory, University of Minnesota. Project Report No. 47, 1955.
Semenenko, V.N. Artificial Supercavitation. Physics and Calculation. RTO AVT
Lecture Series on �Supercavitating Flows,� Brussels, Belgium, 2001. Silberman, E., Song, C.S. Instability of Ventilated Cavities. St. Anthony Falls Hydraulic
Laboratory, University of Minnesota. Technical Paper No. 29, Series B, 1959. Song, C.S. Pulsation of Ventilated Cavities. St. Anthony Falls Hydraulic Laboratory,
University of Minnesota. Technical Paper No. 32, Series B, 1961. Spurk, J.H. A Theory for the Gas Loss From Ventilated Cavities. International Summer
Scientific School on High Speed Hydrodynamics, Cheboksary, Russia, 2002. Sridhar, G., Katz, J. Drag and lift forces on microscopic bubbles entrained by a vortex.
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39
Sridhar, G., Ran, B., Katz, J. Implementation of Particle Image Velocimetry to Multi-Phase Flow. ASME Cavitation and Multiphase Flow Forum, pp. 205-210, 1991.
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Flows. Proceedings of the Third International Symposium on Cavitation, Grenoble, France. J.M. Michel and H. Kato, ed. Vol. 2, pp. 39-44, 1998.
Waid, R.L. Cavity Shapes for Circular Disks at Angles of Attack. California Institute of
Technology. Report No. E-73.4, 1957. Wheeler, A.J., Ganji, A.R. Introduction to Engineering Experimentation. Prentice Hall,
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41
Figure 1.2: Pictures of cavities in re-entrant jet and twin vortex regimes. Left two pictures correspond to re-entrant jet regime. Leftmost picture shows re-entrant jet surging forward (arrow) while center picture shows that the re-entrant jet has lost its
momentum and fallen back. Right picture corresponds to twin vortex regime. The curvature over the length of the cavity is due to the test body being mounting on a
rotating arm. Re-entrant jet pictures taken from Self and Ripken (1955). Twin vortex picture taken from Campbell and Hilborne (1958).
42
0
5
10
15
20
25
30
35
40
0.05 0.15 0.25 0.35 0.45 0.55 0.65
Cavitation Number, σσσσ
Cav
ity L
engt
h/D
isk
Dia
met
er, L
/d
GarabedianReichardtWaid
Figure 1.3: Cavity length versus cavitation number from various sources.
43
Figure 2.1: Schematic of water tunnel.
Figure 2.2: Outline of test body. Dimensions are given in centimeters. Cavitator size is variable.
45
Figure 2.4: Problem description for choking phenomenon.
0
1
2
3
4
5
6
0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6
Cavitation Number, σσσσ
Uc/U
∞
Bernoullid/h = 0.01d/h = 0.05d/h = 0.1
Figure 2.5: Velocity ratio versus cavitation number in a bounded flow.
U∞, p∞ Uc, pc
U = 0 p = pc
CV
h d D
46
Figure 2.6: Cross-sections of elliptical and cylindrical test body struts. Flow is from left to right, dimensions given in centimeters.
Figure 2.7: Cross section of test body. Entran transducers at 3.5 and 5.5 centimeters, static ports at 7.95 centimeters.
47
patm, Tatm
Rotameter
Valve
SS&H board
pc
U∞, p∞
p∞ p0 - p∞
Figure 2.8: Test setup for air entrainment measurements.
49
Figure 2.10: Side view of test body.
Figure 2.11: Bottom view of test body with cylindrical strut.
Figure 2.12: Bottom view of test body with elliptical strut.
o-ring in mounting plug
holes in mounting
plug
metal plate in roof of test
section
clay that covers screw
holes
51
Figure 3.2: Distortion of cavity shape due to cylindrical and elliptical struts. Q increasing from bottom to top.
52
Figure 3.3: Re-entrant jet effects on cavity surface with the cylindrical strut. Arrow points to the re-entrant jet.
53
Figure 3.4: Re-entrant jet effects on cavity surface with the elliptical strut. Arrow points to the re-entrant jet.
54
0.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
0.05 0.10 0.15 0.20 0.25 0.30 0.35
Cavitation Number
Air
Entr
ainm
ent C
oeffi
cien
t
Fr = 19.8Fr = 20.4Fr = 27.2Fr = 28.7Fr = 28.7Fr = 30.3Fr = 36.2Fr = 32.6Fr = 36.7Fr = 37.7
Figure 3.5: Air entrainment results for 1 cm disk and cylindrical strut.
Cavitation number calculated using formula from Waid.
55
0.00
0.10
0.20
0.30
0.40
0.50
0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45Cavitation Number
Air
Entr
ainm
ent C
oeffi
cent
Figure 3.6: Cavity pictures for 1 cm disk.
Pictures were taken with elliptical strut in place.
56
0.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
0.05 0.10 0.15 0.20 0.25 0.30 0.35
Cavitation Number
Air
Entr
ainm
ent C
oeffi
cien
t
Fr = 20.4
Fr = 21.1
Fr = 23.9
Fr = 26.7
Fr = 31.6
Figure 3.7: Air entrainment results for 1 cm disk and elliptical strut.
Cavitation number calculated using formula from Waid.
57
0.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
0.05 0.10 0.15 0.20 0.25 0.30 0.35
Cavitation Number
Air
Entr
ainm
ent C
oeffi
cien
t
Cylindrical Strut
Elliptical Strut
Figure 3.8: Comparison of air entrainment results of both struts with 1 cm disk.
Cavitation number calculated using formula from Waid.
58
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50 0.55 0.60
Cavitation Number
Air
Entr
ainm
ent C
oeffi
cien
t Fr = 15.2
Fr = 17.9
Fr = 21.0
Fr = 25.1
Fr = 27.6
Fr = 30.7
Figure 3.9: Air entrainment results for 1.5 cm disk and cylindrical strut.
Cavitation number calculated using formula from Waid.
59
0.00
0.05
0.10
0.15
0.20
0.25
0.30
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7Cavitation Number
Air
Entr
ainm
ent C
oeffi
cien
t
Figure 3.10: Cavity pictures for 1.5 cm disk.
Pictures were taken with elliptical strut in place.
60
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50 0.55 0.60
Cavitation Number
Air
Entr
ainm
ent C
oeffi
cien
t Fr = 17.2
Fr = 19.3
Fr = 21.9
Fr = 25.7
Figure 3.11: Air entrainment results for 1.5 cm disk and elliptical strut.
Cavitation number calculated using formula from Waid.
61
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50 0.55 0.60
Cavitation Number
Air
Entr
ainm
ent C
oeffi
cien
t Cylindrical Strut
Elliptical Strut
Figure 3.12: Comparison of air entrainment results of both struts and 1.5 cm disk.
Cavitation number calculated using formula from Waid.
62
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7
Cavitation Number, σσσσ
Dis
k D
iam
eter
/Cav
ity L
engt
h, d
/L
h/d = infinityh/d = 10.5h/d = 7.5h/d = 6.0h/d = 5.0h/d = 4.4h/d = 21.5h/d = 14.3h/d = 21.5 (Tulin)h/d = 14.3 (Tulin)GarabedianWaid
Figure 3.13: Brennen's data for blockage effects along with interpolated data.
Disk diameters of 1 cm and 1.5 cm in the current research correspond to h/d values of 21.5 and 14.3, respectively.
63
0.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
0.05 0.10 0.15 0.20 0.25 0.30 0.35
Cavitation Number
Air
Entr
ainm
ent C
oeffi
cien
t
Cylindrical Strut
Elliptical Strut
Figure 3.14: Air entrainment results for 1 cm disk using data from Brennen.
64
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50 0.55 0.60
Cavitation Number
Air
Entr
ainm
ent C
oeffi
cien
t
Cylindrical Strut
Elliptical Strut
Figure 3.15: Air entrainment results for 1.5 cm disk using data from Brennen.
65
0.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50 0.55
Cavitation Number
Air
Entr
ainm
ent C
oeffi
cien
t
Cylindrical Strut, 1 cm Disk
Elliptical Strut, 1 cm Disk
Cylindrical Strut, 1.5 cm Disk
Elliptical Strut, 1.5 cm Disk
Figure 3.16: Air entrainment data for both disks and struts using data from Brennen.
67
-3
-2
-1
0
1
2
3
3 3.5 4 4.5 5 5.5 6 6.5 7
Water Velocity, u (m/s)
Dis
tanc
e Fr
om T
est B
ody
Cen
terli
ne, r
(cm
)
x = 0.5 cmx = 3.0 cmx = 4.5 cmx = 6.0 cmx = 7.5 cmx = 9.0 cm
Figure 3.18: Wake profile in the non-cavitating regime for U∞ = 6.36 m/s.
Average of 500 vector fields.
68
-3
-2
-1
0
1
2
3
-0.02 0 0.02 0.04 0.06 0.08 0.1 0.12
δδδδu/x-2/3
r/x1/
3
x = 0.5 cmx = 3.0 cmx = 4.5 cmx = 6.0 cmx = 7.5 cmx = 9.0 cm
Figure 3.19: Power law relationships in non-cavitating wake for U∞ = 6.36 m/s.
Average of 500 vector fields.
69
-3
-2
-1
0
1
2
3
-0.1 -0.05 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4
δδδδu x2/3/(Umax d2/3)
r/(x1/
3 d2/3 )
x = 0.5 cmx = 3.0 cmx = 4.5 cmx = 6.0 cmx = 7.5 cmx = 9.0 cm
Figure 3.20: Non-dimensional velocity data in non-cavitating wake for U∞ = 6.36 m/s.
Average of 500 vector fields.
70
-3
-2
-1
0
1
2
3
3 3.5 4 4.5 5 5.5 6 6.5 7
Bubble Velocity in Wake (m/s)
r (cm
)
x = 0.5 cmx = 3.0 cmx = 4.5 cmx = 6.0 cmx = 7.5 cmx = 9.0 cm
Figure 3.21: Bubble velocity in cavitating wake for U∞ = 6.36 m/s and σ = 0.15.
Average of 500 vector fields.
71
-3
-2
-1
0
1
2
3
3 3.5 4 4.5 5 5.5 6 6.5 7
Velocity in Wake (m/s)
r (cm
)
x = 0.5 cm, cavitatingx = 3.0 cm, cavitatingx = 6.0 cm, cavitatingx = 9.0 cm, cavitatingx = 0.5 cm, non-cavitatingx = 3.0 cm, non-cavitatingx = 6.0 cm, non-cavitatingx = 9.0 cm, non-cavitating
Figure 3.22: Measured velocities in cavitating and non-cavitating regimes.
Open symbols correspond to bubble velocity in cavitating regime, filled symbols correspond to liquid velocity in non-cavitating regime. U∞ = 6.36 m/s, σ = 0.15.
Average of 500 vector fields.
72
-3
-2
-1
0
1
2
3
0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16
Normalized Grayscale Level
r (cm
)
x = 0.5 cm
x = 3.0 cm
x = 4.5 cm
x = 6.0 cm
x = 7.5 cm
x = 9.0 cm
Figure 3.23: Normalized grayscale levels in the wake for U∞ = 6.36 m/s and σ = 0.15.
Average of 500 images.
73
-3
-2
-1
0
1
2
3
-0.02 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16
Normalized Grayscale Level
r (cm
)
x = 0.5 cm
x = 3.0 cm
x = 4.5 cm
x = 6.0 cm
x = 7.5 cm
x = 9.0 cm
Figure 3.24: Normalized grayscale levels after removing background noise.
U∞ = 6.36 m/s, σ = 0.15. Average of 500 images.
74
-3
-2
-1
0
1
2
3
0 0.02 0.04 0.06 0.08 0.1 0.12 0.14
Void Fraction, ηηηη
r (cm
)
x = 0.5 cm
x = 3.0 cm
x = 4.5 cm
x = 6.0 cm
x = 7.5 cm
x = 9.0 cm
Figure 3.25: Calculated void fraction in wake for U∞ = 6.36 m/s and σ = 0.15.
Average of 500 vector fields and images.
75
-3
-2
-1
0
1
2
3
6 6.1 6.2 6.3 6.4 6.5 6.6 6.7 6.8 6.9 7
Water Velocity, u (m/s)
Dis
tanc
e Fr
om T
est B
ody
Cen
terli
ne, r
(cm
)
x = 5.0 cm
x = 6.0 cm
x = 7.0 cm
x = 8.0 cm
x = 9.0 cm
Figure 3.26: Wake profile in the non-cavitating regime for U∞ = 6.6 m/s.
Average of 100 vector fields.
76
-3
-2
-1
0
1
2
3
-0.02 0 0.02 0.04 0.06 0.08 0.1 0.12
δδδδu/x-2/3
r/x1/
3
x = 5.0 cm
x = 6.0 cm
x = 7.0 cm
x = 8.0 cm
x = 9.0 cm
Figure 3.27: Power law relationships in non-cavitating wake for U∞ = 6.6 m/s.
Average of 100 vector fields.
77
-3
-2
-1
0
1
2
3
-0.1 -0.05 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4
δδδδu x2/3////(Umax d2/3)
r/(x1/
3 d2/3 )
x = 5.0 cm
x = 6.0 cm
x = 7.0 cm
x = 8.0 cm
x = 9.0 cm
Figure 3.28: Non-dimensional velocity data in non-cavitating wake for U∞ = 6.6 m/s.
Average of 100 vector fields.
78
-3
-2
-1
0
1
2
3
-0.04 -0.02 0 0.02 0.04 0.06 0.08 0.1 0.12
δδδδu/x-2/3
r/x1/
3
x = 0.5 cm
x = 4.5 cm
x = 6.0 cm
x = 9.0 cm
x = 5.0 cm
x = 6.0 cm
x = 9.0 cm
Figure 3.29: Comparison of measured velocities in non-cavitating wake.
Open symbols correspond to U∞ = 6.36 m/s and 500 averaged vector fields, filled symbols correspond to U∞ = 6.6 m/s and 100 averaged vector fields.
79
-3
-2
-1
0
1
2
3
5.5 5.6 5.7 5.8 5.9 6 6.1 6.2 6.3 6.4 6.5
Velocity in Wake (m/s)
r (cm
)
x = 4.0 cmx = 5.0 cmx = 6.0 cmx = 7.0 cmx = 8.0 cmx = 9.0 cm
Figure 3.30: Bubble velocity in wake for U∞ = 6.6 m/s and σ = 0.15.
Average of 100 vector fields.
80
-3
-2
-1
0
1
2
3
0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1
Void Fraction, ηηηη
r (cm
)
x = 4.0 cm
x = 5.0 cm
x = 6.0 cm
x = 7.0 cm
x = 8.0 cm
x = 9.0 cm
Figure 3.31: Calculated void fraction in wake for U∞ = 6.6 m/s and σ = 0.15.
Average of 100 vector fields and images.
81
-3
-2
-1
0
1
2
3
0 0.02 0.04 0.06 0.08 0.1 0.12 0.14
Void Fraction, ηηηη
r (cm
)
x = 0.5 cm
x = 3.0 cm
x = 6.0 cm
x = 7.5 cm
x = 4.0 cm
x = 6.0 cm
x = 8.0 cm
Figure 3.32: Comparison of calculated void fractions.
Filled symbols correspond to U∞ = 6.36 m/s, σ = 0.15 and 500 averaged vector fields and images, open symbols correspond to U∞ = 6.6 m/s, σ = 0.15 and 100 averaged vector
fields and images.
82
-3
-2
-1
0
1
2
3
5.7 5.8 5.9 6 6.1 6.2 6.3 6.4 6.5
Velocity in Wake (m/s)
r (cm
)
x = 5.0 cm
x = 6.0 cm
x = 7.0 cm
x = 8.0 cm
x = 9.0 cm
Figure 3.33: Bubble velocity in wake for U∞ = 6.6 m/s and σ = 0.24.
Average of 100 vector fields.
83
-3
-2
-1
0
1
2
3
0 0.01 0.02 0.03 0.04 0.05 0.06 0.07
Void Fraction, ηηηη
r (cm
)
x = 5.0 cm
x = 7.0 cm
x = 9.0 cm
x = 5.0 cm
x = 7.0 cm
x = 9.0 cm
Figure 3.34: Comparison of calculated void fractions for different cavitation numbers. Filled symbols correspond to U∞ = 6.6 m/s and σ = 0.15, open symbols correspond to U∞ = 6.6 m/s and σ = 0.24. 100 images and vector fields were averaged in both cases.
84
-0.05
0
0.05
0.1
0.15
0.2
0.25
-0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8
Distance from Static Port (m)
(p∞∞ ∞∞ -
p mea
s)/(0
.5ρρ ρρU
∞∞ ∞∞2 )
Figure 4.1: Downstream distance versus error in static pressure measurement.
Data is based on FLUENT analysis and assumes static port is located on the roof of the test section.
86
0
5
10
15
20
25
30
35
0 5 10 15 20 25 30 35
patm - p1 (cm Hg)
p atm
- p c
(cm
Hg)
0251015202530
Rotameter Reading
Figure 4.3: Preliminary data for rotameter cavity pressure measurement technique.
87
APPENDIX A: Pressure Transducer Calibration Procedure
The pressure transducers that measured the static pressure in the water tunnel test
section, p∞, and the differential pressure, p0 - p∞, were calibrated every day before an
experiment. The calibrations were performed using two mercury manometers, one for
each transducer. Before calibration, the pressure lines were purged to ensure no air
bubbles were trapped in them.
When calibrating the absolute pressure transducer that measured p∞, one leg of the
manometer was connected to the static pressure port while the other leg was connected to
a tank filled with water whose level was at the same height as the static port in the water
tunnel test section. The pressure in the tunnel was then varied by pulling a vacuum in the
water tunnel. At each calibration point, the height of the mercury column and the
pressure transducer output were recorded. Approximately six calibration points were
taken each time the transducer was calibrated. After calibration, the pressure in the
tunnel was relieved to atmospheric pressure.
After calibrating the absolute pressure transducer, the differential pressure
transducer was calibrated. When calibrating the differential pressure transducer, one leg
of the manometer was connected to the stagnation pressure in the tunnel while the other
leg was connected to the static port in the test section. The differential pressure in the
tunnel was then varied by changing the freestream velocity. Again, at each calibration
point the height of the mercury column and the pressure transducer output were recorded.
After calibrating the two pressure transducers, the calibration curves were created
by plotting the differential pressure given by the manometers versus transducer output. A
straight line was then fit through the data for each transducer using a least squares fit.
The pressure transducer calibrations produced curves that were consistently linear,
with R-squared values typically 0.9999 or higher for both the absolute and differential
transducers. Maximum errors due to the least squares fit line were approximately 0.6
kPa, with typical errors being closer to 0.12 kPa. These errors lead to a maximum error
in the measured velocity of 0.11 m/s, with typical errors being closer to 0.02 m/s.
88
APPENDIX B: Uncertainty Analysis
The uncertainty analysis outlined below will be conducted using a root of the sum
of the squares (RSS) estimate. This estimate is of the form (Wheeler and Ganji, 1996)
1/22n
1i ixR x
Rwwi
∂∂= ∑
=, (B.1)
where wR is the uncertainty in the quantity R, which is of the form
)x,...,x,(xR n21f= . (B.2)
Sources of Error and Uncertainty
As was previously mentioned, the static pressure measurement in the water tunnel
was one major source of uncertainty due to the location of the static pressure port. The
static port was located on a curved portion of the roof of the test section so the pressure at
the port location was different than the freestream value. As was previously mentioned,
Dugué found the difference between the static pressure at the center of the tunnel
compared to the value at the roof to be
0.06ρV
21
pp2
meas =−∞ . (2.2)
Neglecting this correction factor can lead to large errors in the calculated cavitation
number, especially for low cavitation numbers. This can be seen by looking at the
difference between the true cavitation number and the cavitation number calculated by
assuming the pressure at the static port is the true static pressure:
0.06σσ meastrue += . (B.3)
For example, by ignoring the correction factor for the static pressure, the error in the
calculated cavitation number would be 60% at a cavitation number of 0.1.
As was previously mentioned, a numerical simulation was run to estimate the
pressure difference between the static port and the static pressure at the centerline of the
tunnel. The difference between these locations was found to be
89
0.02ρV
21
pp2
meas =−∞ . (2.3)
If this correction is assumed correct, the error in the calculated cavitation number by
ignoring the correction would be 20% at a cavitation number of 0.1.
In the current research, the static pressure was not used to calculate the cavitation
number because an accurate measurement of the cavity pressure could not be obtained.
However, the static pressure was still used for two purposes. First, once the cavitation
number was found based on the cavity length, the static pressure was used along with the
cavitation number to calculate the cavity pressure and ultimately obtain the volumetric
flow rate of the injected gas at cavity pressure and the air entrainment coefficient.
Second, the static pressure was used to determine the downstream pressure in the wake of
the body so that the density of the gas in the wake and the void fraction could be
calculated. Therefore, uncertanties in the measured static pressure did lead to
uncertainties in the air entrainment coefficient and void fraction. For a worst case
analysis, the maximum uncertainty in the static pressure measurement will be assumed to
equal the difference between the formula obtained by Dugué and the result from the
numerical simulation, namely
=−∞
2meas ρV
210.04pp . (B.4)
Other sources of error and uncertainty also existed. These include the error in curve
fitting the calibration data to determine the velocity (0.11 m/s, see Appendix A), the
uncertainty in the measured volumetric flow rate (estimated at 2% of reading), and the
uncertainty in the calculated cavitation number (estimated at 0.01). All of these sources
of error and uncertainty were included to determine the overall uncertainty in the air
entrainment coefficient and void fraction as discussed below.
Uncertainty in the Air Entrainment Coefficient
The density of the gas inside of the cavity was needed to calculate the volumetric
flow rate of the gas at cavity pressure. For this case, it was assumed that the air behaved
as an ideal gas. Therefore, once the cavity pressure and temperature were known, the
90
density could be calculated. The temperature was assumed to equal atmospheric
temperature since the air was injected from atmosphere and the water temperature was
roughly equal to the ambient air temperature. To calculate the cavity pressure, the
cavitation number was used. As was previously noted, the cavitation number was
determined based on the cavity length.
Velocities up to approximately 11.8 m/s were tested in the current research.
Therefore, the difference between the true static pressure and measured static pressure
was no greater than 2.8 kPa, or about 7%. The uncertainty in the calculated cavity
pressure was largest for large cavitation numbers and high velocities, with a maximum
uncertainty of about 8%. It follows that the maximum uncertainty in the calculated air
density is also 8% for the high velocity and cavitation number conditions. This leads to a
maximum uncertainty in the calculated volumetric flow rate of the gas at cavity pressure
to be about 10%. Finally, taking into account the uncertainties in the measured velocity
and calculated volumetric flow, the maximum uncertainty in the air entrainment
coefficient is about 10% also. This maximum uncertainty again occurs when the velocity
and cavitation number are large. For lower velocities and/or cavitation numbers, the
uncertainties in the cavity pressure and density, volumetric flow rate, and air entrainment
coefficient are lower, with a minimum uncertainty in the air entrainment coefficient of
about 2%. The overall uncertainty is almost entirely due to the uncertainty in the static
pressure measurement, with the uncertainties in the other variables such as the freestream
velocity and cavity number contributing only a minor part to overall uncertainty.
However, overall the fairly large uncertainty in the static pressure measurement does not
have a large impact on the calculated air entrainment coefficient for the majority of the
test conditions. Only the very high velocity and cavitation number conditions are
affected most.
Uncertainty in the Void Fraction
Unlike the air entrainment coefficient, the uncertainty in the static pressure
measurement was not the only major contributor to the uncertainty in the void fraction
calculation. This is because the maximum velocities tested during the void fraction
91
experiments were only 6.6 m/s. This made the error in the freestream velocity due to the
curve fitting just as important as the uncertainty in the freestream pressure.
In order to determine the void fraction, the static pressure and freestream velocity
were needed to determine the pressure in the wake and eventually the density of the gas
in the wake. In order to calculate the density of the gas, it was assumed the pressure
throughout the wake at a given downstream distance was a constant. This was an
accurate assumption since the bubbles were fairly large (average diameter of 0.4 mm as
measured from digital pictures). Therefore, the pressure inside the bubbles was
approximately the same as outside of the bubbles. For the purposes of this analysis, a
maximum error of 1% will be used for the pressure inside the bubble due to the
assumption that it is the same as the pressure in the wake.
The uncertainty in the calculated pressure in the wake was found to be
approximately 1%. This uncertainty was small since the maximum velocities tested were
only 6.6 m/s, which were much lower than the velocities tested during the air entrainment
coefficient experiments. The error in the freestream velocity and the uncertainty in the
static pressure measurement both contributed about the same amount to the uncertainty of
the pressure in the wake. The uncertainty in the pressure in the wake led to an
uncertainty of about 2% in the density of the gas in the wake. The uncertainty in the
density is higher than the uncertainty in the pressure due to the assumption that the
pressure inside the bubbles was the same as outside of the bubbles, which leads to an
additional amount of uncertainty in the pressure calculation.
In determining the uncertainty in the void fraction calculation, it will be assumed
that the uncertainties in both the shape of the void fraction curves and the density of the
bubbles are both 2% when 500 images and vector fields were averaged. These
uncertainties, along with the uncertainty in the density of the gas in the wake, lead to a
maximum uncertainty of about 3% in the void fraction calculation. For the data where
only 100 vector fields and images were averaged, the uncertainty is much higher since
not enough vector fields and images were averaged to obtain a true representation of the
data.
Recommended