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UNIVERSITY OF MINNESOTA This is to certify that I have examined this copy of a masters 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

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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

© Travis Jon Schauer 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.

Physics of Fluids, 7, No. 2, pp. 389-399, 1995.

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.

TSI Inc. Insight Particle Image Velocimetry Software Version 3.3 Instruction Manual.

TSI Inc., St. Paul, 2000. TSI Inc. Particle Image Velocimetry (PIV): Operations Manual. TSI Inc., St. Paul,

2000. Tulin, M.P. Supercavitating Flows. In Streeter, V. (ed.), Handbook of Fluid Dynamics,

McGraw-Hill, New York, pp. 12-24 to 12-46, 1961. Vlasenko, Y.D. Experimental Investigations of High-Speed Unsteady Supercavitating

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,

New Jersey, 1996. White, F.M. Viscous Fluid Flow. McGraw-Hill, New York, 1991.

40

Figure 1.1: Schematic of cavitator and cavity dimensions.

d D

2L

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.

44

Figure 2.3: Choking cavitation number vs. blockage ratio for a disk. Taken from Tulin (1961).

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.

48

Figure 2.9: PIV test setup.

Flow

Dual Nd:YAG lasers and optics

Laser sheet

Image field

Camera

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

50

Figure 3.1: Pictures of oscillating cavity and wake.

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.

66

Figure 3.17: Schematic of wake profile.

U∞

x

r Umax

u(r)

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.

85

Figure 4.2: Test setup for using rotameter to measure cavity pressure.

patm

pc

p1

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.