modelling the effect of suspended bodies on...

Modelling the Effect of Suspended

Bodies on Cavitation Bubbles near a

Rigid Boundary using a Boundary

Integral Approach

By Peter Stanley McGregor

Bachelor of Engineering (Mechanical)

Queensland University of Technology, Australia

For award of the degree of Doctor of Philosophy

2003

Tribology Research Group

School of Mechanical, Manufacturing and Medical Engineering

Queensland University of Technology

i

Keywords

Boundary Element Method, Boundary Integral Method, Cavitation, Suspensions, Emulsions,

Suspended Solids, Suspended Bodies, Contaminated Lubricants, Contaminated Hydraulic

Fluids, Multi-Phase Flow, Computational Fluid Modelling.

iii

Abstract

Cavitation is the spontaneous vaporisation of a liquid to its gaseous state due to the local

absolute pressure falling to the liquid’s vapour pressure (Douglas, Gasiorek et al. 1995).

Cavitation is present in a wide range of mechanical systems ranging from ship screws to

journal bearing. Generally, cavitation is unavoidable and may cause considerable damage

and efficiency losses to these systems.

This thesis considers hydraulic systems specifically, and uses a modified Greens equation to

develop a boundary integral method to simulate the effect that suspended solid bodies have

on a single cavitation bubble. Because of the limitations of accurately modelling cavitation

bubbles beyond touchdown, results are only presented for cases up to touchdown.

The aim of the model is to draw insight into the reasons there is a measurable change in

cavitation erosion rate with increasing oil-in-water emulsion percentage. This principle was

extended to include the effect that ingested particulates may have on cavitation in hydraulic

machinery.

Two particular situations are modelled; the first consists of stationary rigid particles in

varying proximity to a cavitation bubble near a rigid boundary. The second case is similar;

however the suspended particle is allowed to move under the influence of the pressure

differential caused by the expanding/contracting cavitation bubble.

Numerous characteristics of the domain are considered, including domain pressures and fluid

field motion, and individual boundary surface characteristics. The conclusion of the thesis is

that solid bodies, either stationary or moving, have little effect on the cavity from an energy

perspective. Regardless of size or density, all energy transferred from the cavity to the solid

body is returned indicating that there is no net change. As this energy is ultimately

responsible for the peak pressure experienced by the domain (and hence responsible for

eroding the rigid boundary) as the cavity rebounds, it then serves that a cavity with a solid

body will rebound at the same pressure as a cavity without a suspended body present.

If this is coupled with the observation that the cavity centroid at touchdown is largely

unaffected by the presence of a suspension, then it would appear that the bubble near a solid

would rebound at a very similar position as a cavity without a solid. Consequently, the

damage potential of a cavity is unaffected by a suspension.

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However, there is one point of contention as the profile of the re-entrant jet of the cavity is

altered by the presence of a suspension. As energy is radiated away from the cavity during

penetration, it is possible that the shape of the jet may alter the rate that energy is radiated

away during penetration. However, this requires further research to be definitive.

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Table of Contents

KEYWORDS...........................................................................................................................I

ABSTRACT..........................................................................................................................III

TABLE OF CONTENTS ......................................................................................................V

TABLE OF FIGURES.........................................................................................................XI

STATEMENT OF AUTHORSHIP ................................................................................XXV

ACKNOWLEDGMENTS ............................................................................................XXVII

NOMENCLATURE....................................................................................................... XXIX

CHAPTER 1: INTRODUCTION..................................................................................... 1-1

1.1 BACKGROUND ....................................................................................................... 1-1 1.2 CAVITATION.......................................................................................................... 1-3 1.3 EROSION................................................................................................................ 1-4 1.4 UNSOLVED PROBLEMS.......................................................................................... 1-5 1.5 ORIGINAL CONTRIBUTION .................................................................................... 1-6 1.6 THESIS SCOPE........................................................................................................ 1-7 1.7 ORGANIZATION OF THESIS.................................................................................... 1-7

CHAPTER 2: CAVITATION EROSION AND SUSPENSIONS ................................. 2-9

2.1 INTRODUCTION...................................................................................................... 2-9 2.2 LIFECYCLE OF A CAVITY....................................................................................... 2-9

2.2.1 Growth Phase.............................................................................................. 2-10 2.2.2 Initial Collapse............................................................................................ 2-11 2.2.3 The Re-entrant Jet ....................................................................................... 2-12 2.2.4 Post Touchdown.......................................................................................... 2-13 2.2.5 Rebound ...................................................................................................... 2-14

2.3 INFLUENCE OF THE SURROUNDING DOMAIN ...................................................... 2-14 2.3.1 Fluid Structure Interaction ......................................................................... 2-15 2.3.2 Multiple Cavities ......................................................................................... 2-15 2.3.3 Cavity Interaction with Suspended Solid Particles ..................................... 2-16 2.3.4 Cavity Interaction with Suspended Liquid Particles................................... 2-17

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2.4 PRESSURE AT THE RIGID BOUNDARY..................................................................2-18 2.4.1 Damage Mechanisms ..................................................................................2-18 2.4.2 Damage to the Rigid Material.....................................................................2-19

2.5 SIMPLIFICATIONS.................................................................................................2-20 2.5.1 Time and Size Scales ...................................................................................2-20 2.5.2 Damage Potential........................................................................................2-20

2.6 SUMMARY............................................................................................................2-21

CHAPTER 3: MATHEMATICAL FORMULATION.................................................3-23

3.1 INTRODUCTION ....................................................................................................3-23 3.2 THEORETICAL BASIS ...........................................................................................3-24

3.2.1 Conservation of Mass ..................................................................................3-24 3.2.2 Conservation of Momentum ........................................................................3-25 3.2.3 Pressure and Stress States...........................................................................3-26 3.2.4 Navier-Stokes Equation...............................................................................3-27 3.2.5 Governing Equations...................................................................................3-28

Reynolds Number....................................................................................................3-29 Inviscid Flow...........................................................................................................3-29

3.2.6 Non-condensable Gas Content and Surface Tension ..................................3-33 3.3 INTEGRAL EQUATIONS ........................................................................................3-33 3.4 SPECIAL CONSIDERATIONS..................................................................................3-38

3.4.1 Rigid Boundaries .........................................................................................3-38 3.4.2 Suspended Particles ....................................................................................3-39

Pressure Integration.................................................................................................3-39 Rigid and Deformable Particles ..............................................................................3-40

3.4.3 Vortex Sheets ...............................................................................................3-41 3.5 SUMMARY............................................................................................................3-43

CHAPTER 4: NUMERICAL IMPLEMENTATION...................................................4-45

4.1 INTRODUCTION ....................................................................................................4-45 4.2 DOMAIN DISCRETISATION ...................................................................................4-46

4.2.1 Linear Geometry – Constant Density Functions .........................................4-47 4.2.2 Linear Geometry – Linear Density Functions.............................................4-48 4.2.3 Higher Order Elements ...............................................................................4-49

4.3 ELEMENT INTEGRATION ......................................................................................4-51 4.3.1 Non-Singular Integration ............................................................................4-53 4.3.2 Singular Integration ....................................................................................4-54

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Singularity at the element beginning (ξ = 0)........................................................... 4-55

Singularity at the element End (ξ = 1) .................................................................... 4-56 4.4 MATRIX FORMULATION ...................................................................................... 4-58

4.4.1 Assembly and Rearrangement..................................................................... 4-58 4.4.2 Matrix Condition Improvement................................................................... 4-60 4.4.3 Combination Schemes for CBIE and HBIE ................................................ 4-62

4.5 TIME ADVANCEMENT.......................................................................................... 4-63 4.5.1 Euler Scheme .............................................................................................. 4-63 4.5.2 Predictor-corrector Scheme........................................................................ 4-65 4.5.3 Time Stepping Considerations .................................................................... 4-65

“Saw-tooth” Instability ........................................................................................... 4-66 The Penetration Process .......................................................................................... 4-67 Contacting Surfaces (Protection Zone) ................................................................... 4-67

4.6 SUMMARY ........................................................................................................... 4-68

CHAPTER 5: VERIFICATION OF NUMERICAL METHOD ................................. 5-71

5.1 INTRODUCTION.................................................................................................... 5-71 5.2 STATIC TEST CASES ............................................................................................ 5-72

5.2.1 A Spherical Source...................................................................................... 5-72 5.2.2 A Spherical Dipole ...................................................................................... 5-75 5.2.3 A Toroidal Cavity With Vortex Sheet .......................................................... 5-77

5.3 DYNAMIC TEST CASES........................................................................................ 5-79 5.3.1 A Rayleigh-Plessett Rebounding Bubble..................................................... 5-79 5.3.2 A Developing Toroid – CBIE/HBIE Comparison ....................................... 5-81 5.3.3 Penetration and Rebound............................................................................ 5-83

5.4 SUMMARY ........................................................................................................... 5-84

CHAPTER 6: CAVITIES NEAR STATIONARY SUSPENDED BODIES............... 6-87

6.1 INTRODUCTION.................................................................................................... 6-87

6.2 0.9 CAVITY STAND-OFF DISTANCE (γC) .............................................................. 6-90 6.2.1 Single Cavity ............................................................................................... 6-90 6.2.2 0.3 Diameter Solid Body – 0.9 Standoff from Cavity .................................. 6-93 6.2.3 0.3 Diameter Solid Body – 1.05 Standoff from Cavity ................................ 6-97 6.2.4 0.2 Diameter Solid Body – 1.05 Standoff from Cavity .............................. 6-100 6.2.5 0.1 Diameter Solid Body – 1.05 Standoff from Cavity .............................. 6-104 6.2.6 0.3 Diameter Solid Body – 1.2 Standoff from Cavity ................................ 6-106 6.2.7 0.3 Diameter Solid Body – 1.5 Standoff from Cavity ................................ 6-109

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6.2.8 Time Series Plots .......................................................................................6-111 6.3 1.2 CAVITY STAND-OFF DISTANCE (γC).............................................................6-117

6.3.1 Single Cavity..............................................................................................6-117 6.3.2 0.3 Diameter Solid Body – 0.9 Stand-off from Cavity ...............................6-120 6.3.3 0.3 Diameter Solid Body – 1.05 Stand-off from Cavity .............................6-122 6.3.4 0.3 Diameter Solid Body – 1.2 Stand-off from Cavity ...............................6-125 6.3.5 Time Series Plots .......................................................................................6-127

6.4 1.5 CAVITY STANDOFF DISTANCE (γC) ..............................................................6-132

6.5 1.8 CAVITY STAND-OFF DISTANCE (γC).............................................................6-133 6.6 SUMMARY..........................................................................................................6-133

CHAPTER 7: CAVITIES NEAR MOBILE SUSPENDED BODIES .......................7-135

7.1 INTRODUCTION ..................................................................................................7-135

7.2 0.9 CAVITY STAND-OFF DISTANCE (γC).............................................................7-137 7.2.1 0.3 Imaginary Surface – 1.05 Stand-off from Cavity.................................7-138 7.2.2 0.1 Diameter 1.0 Density Solid Body – 1.05 Stand-off from Cavity..........7-140 7.2.3 0.2 Diameter 1.0 Density Solid Body – 1.05 Stand-off from Cavity..........7-144 7.2.4 0.3 Diameter 1.0 Density Solid Body – 1.05 Stand-off from Cavity..........7-148 7.2.5 0.3 Diameter 0.7 Density Solid Body – 1.05 Stand-off from Cavity..........7-152 7.2.6 0.3 Diameter 2.8 Density Solid Body – 1.05 Stand-off from Cavity..........7-155 7.2.7 0.3 Diameter 7.5 Density Solid Body – 1.05 Stand-off from Cavity..........7-158 7.2.8 Time Series Plots .......................................................................................7-160

7.3 1.2 CAVITY STAND-OFF DISTANCE (γC).............................................................7-167 7.3.1 0.3 Imaginary Surface – 1.05 Stand-off from Cavity.................................7-167 7.3.2 0.3 Diameter 0.7 Density Solid Body – 1.05 Stand-off from Cavity..........7-170 7.3.3 0.3 Diameter 1.0 Density Solid Body – 1.05 Stand-off from Cavity..........7-174 7.3.4 0.3 Diameter 2.8 Density Solid Body – 1.05 Stand-off from Cavity..........7-178 7.3.5 0.3 Diameter 7.5 Density Solid Body – 1.05 Stand-off from Cavity..........7-180 7.3.6 Time Series Plots .......................................................................................7-181

7.4 1.5 CAVITY STAND-OFF DISTANCE (γC).............................................................7-187

7.5 1.8 CAVITY STAND-OFF DISTANCE (γC).............................................................7-188 7.6 LARGER DIAMETER PARTICLES – 0.9 INITIAL SURFACE SEPARATION..............7-188

7.6.1 Time Series Plots .......................................................................................7-188 7.7 SUMMARY..........................................................................................................7-195

CHAPTER 8: CONCLUSION ......................................................................................8-197

REFERENCES ................................................................................................................... 203

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APPENDIX A..................................................................................................................... 205

A.1 1.5 CAVITY STANDOFF DISTANCE (γC) TIME SERIES PLOTS ................................ 205

A.2 1.8 CAVITY STANDOFF DISTANCE (γC) TIME SERIES PLOTS ................................ 210

APPENDIX B ..................................................................................................................... 215

B.1 1.5 CAVITY STAND-OFF DISTANCE (γC) TIME SERIES PLOTS............................... 215

B.2 1.8 CAVITY STAND-OFF DISTANCE (γC) TIME SERIES PLOTS............................... 222

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Table of Figures

Figure 2.1. Bubble Profiles for a Simple Cavity Lifecycle................................................ 2-10 Figure 3.1. Coordinate System........................................................................................... 3-34 Figure 4.1. An axisymmetric coordinate system, representing a cavity surface and rigid

boundary.. ................................................................................................................... 4-46 Figure 4.2. Matrix assembly example consisting of a suspended solid and single toroidal

cavity........................................................................................................................... 4-59 Figure 4.3. Schematic showing approximate position of particular nodes ........................ 4-62 Figure 5.1. A spherical source of 32 elements. Left: Geometry of bubble surface with nodal

position. Right: Nodal potential. ................................................................................ 5-73 Figure 6.1. Schematic of plotted variables......................................................................... 6-88 Figure 6.2. Pressure and vector plot for a single cavity near a rigid boundary (at axial

coordinate 0) at time 0.1130. Note the reference vector represents one unit. ............ 6-90 Figure 6.3. Pressure and vector plot for a single cavity near a rigid boundary (at axial

coordinate 0) at time 0.9706. Note the reference vector represents one unit. ............ 6-91 Figure 6.4. Pressure and vector plot for a single cavity near a rigid boundary (at axial

coordinate 0) at time 1.9801. Note the reference vector represents ten units. ........... 6-91 Figure 6.5. Pressure and vector plot for a single cavity near a rigid boundary (at axial

coordinate 0) at time 2.0782. Note the reference vector represents ten units. ........... 6-92 Figure 6.6. A similar cavity is shown at time 0.1184, except that a 0.3 diameter solid body is

present at 0.9 from the initial cavity centre. Note the reference vector represents one

unit. ............................................................................................................................. 6-93 Figure 6.7. A similar cavity is shown at time 0.4223, except that a 0.3 diameter solid body is

present at 0.9 from the initial cavity centre. Note the reference vector represents one

unit. ............................................................................................................................. 6-94 Figure 6.8. A similar cavity is shown at time 1.9638, except that a 0.3 diameter solid body is

present at 0.9 from the initial cavity centre. Note the reference vector represents ten

units............................................................................................................................. 6-95 Figure 6.9. A similar cavity is shown at time 2.0559, except that a 0.3 diameter solid body is


units............................................................................................................................. 6-95 Figure 6.10. A similar cavity is shown at time 0.1176, except that a 0.3 diameter solid body

is present at 1.05 from the initial cavity centre. Note the reference vector represents one

unit. ............................................................................................................................. 6-97

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Figure 6.11. A similar cavity is shown at time 0.6068, except that a 0.3 diameter solid body


unit...............................................................................................................................6-98 Figure 6.12. A similar cavity is shown at time 1.9530, except that a 0.3 diameter solid body

is present at 1.05 from the initial cavity centre. Note the reference vector represents ten

units. ............................................................................................................................6-99 Figure 6.13. A similar cavity is shown at time 2.0711, except that a 0.3 diameter solid body


units. ..........................................................................................................................6-100 Figure 6.14. A similar cavity is shown at time 0.1170, except that a 0.2 diameter solid body


unit.............................................................................................................................6-101 Figure 6.15. A similar cavity is shown at time 1.0429, except that a 0.2 diameter solid body












units. ..........................................................................................................................6-105 Figure 6.21. A similar cavity is shown at time 2.072, except that a 0.1 diameter solid body is




unit.............................................................................................................................6-107

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unit. ........................................................................................................................... 6-107 Figure 6.24. A similar cavity is shown at time 1.9756, except that a 0.3 diameter solid body


units........................................................................................................................... 6-108 Figure 6.25. A similar cavity is shown at time 2.0771, except that a 0.3 diameter solid body










units........................................................................................................................... 6-111 Figure 6.30. Plots of cavity north pole coordinates, with respect to time, for various

stationary body cases. All cavities are initially centred at γC = 0.9.......................... 6-112 Figure 6.31. Plots of cavity south pole coordinates, with respect to time, for various

stationary body cases. All cavities are initially centred at γC = 0.9.......................... 6-112 Figure 6.32. Plots of cavity centroid coordinates, with respect to time, for various stationary

body cases. All cavities are initially centred at γC = 0.9. ......................................... 6-113 Figure 6.33. Plots of the distance between the cavity centroid and the solid body centroid,

with respect to time, for various stationary body cases. All cavities are initially centred

at γC = 0.9. ................................................................................................................. 6-113 Figure 6.34. Plots of the distance between the cavity surface and the solid body surface

along the vertical axis, with respect to time, for various stationary body cases. All

cavities are initially centred at γC = 0.9..................................................................... 6-114 Figure 6.35. Plots of the pressure at the rigid boundary directly below the cavity (coordinate

[0,0]), with respect to time, for various stationary body cases. All cavities are initially

centred at γC = 0.9. .................................................................................................... 6-115

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Figure 6.36. Plots of the maximum pressure in the domain, with respect to time, for various

stationary body cases. All cavities are initially centred at γC = 0.9. .........................6-115 Figure 6.37. Plots of the kinetic energy of the cavity, with respect to time, for various

stationary body cases. All cavities are initially centred at 0.9..................................6-116 Figure 6.38. Pressure and vector plot for a single cavity near a rigid boundary (at axial

coordinate 0) at time 0.1081. Note the reference vector represents one unit. ..........6-117 Figure 6.39. Pressure and vector plot for a single cavity near a rigid boundary (at axial

coordinate 0) at time 0.6678. Note the reference vector represents one unit. ..........6-118 Figure 6.40. Pressure and vector plot for a single cavity near a rigid boundary (at axial

coordinate 0) at time 1.9919. Note the reference vector represents ten units...........6-118 Figure 6.41. Pressure and vector plot for a single cavity near a rigid boundary (at axial

coordinate 0) at time 2.0719. Note the reference vector represents ten units...........6-119 Figure 6.42. A similar cavity is shown at time 0.1132, except that a 0.3 diameter solid body
















units. ..........................................................................................................................6-124

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units........................................................................................................................... 6-126 Figure 6.54. Plots of cavity north pole coordinates, with respect to time, for various

stationary body cases. All cavities are initially centred at 1.2. ................................ 6-127 Figure 6.55. Plots of cavity south pole coordinates, with respect to time, for various

stationary body cases. All cavities are initially centred at 1.2. ................................ 6-128 Figure 6.56. Plots of cavity centroid coordinates, with respect to time, for various stationary

body cases. All cavities are initially centred at 1.2. ................................................. 6-128 Figure 6.57. Plots of the distance between the cavity centroid and the solid body centroid,


at 1.2.......................................................................................................................... 6-129 Figure 6.58. Plots of the distance between the cavity surface and the solid body surface


cavities are initially centred at 1.2. ........................................................................... 6-129 Figure 6.59. Plots of the pressure at the rigid boundary directly below the cavity (coordinate


centred at 1.2............................................................................................................. 6-130 Figure 6.60. Plots of the maximum pressure in the domain, with respect to time, for various

stationary body cases. All cavities are initially centred at 1.2. ................................ 6-131 Figure 6.61. Plots of the kinetic energy of the cavity, with respect to time, for various

stationary body cases. All cavities are initially centred at 1.2. ................................ 6-132 Figure 7.1. Pressure and vector plot for a cavity near a rigid boundary (at axial coordinate 0)

and an imaginary surface at time 0.1165. Note the reference vector represents one unit.

.................................................................................................................................. 7-138 Figure 7.2. Pressure and vector plot for a cavity near a rigid boundary (at axial coordinate 0)

and an imaginary surface at time 0.9782. Note the reference vector represents one unit.

.................................................................................................................................. 7-138

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Figure 7.3. Pressure and vector plot for a cavity near a rigid boundary (at axial coordinate 0)

and an imaginary surface at time 1.9735. Note the reference vector represents ten units.

...................................................................................................................................7-139 Figure 7.4. Pressure and vector plot for a cavity near a rigid boundary (at axial coordinate 0)

and an imaginary surface at time 2.0745. Note the reference vector represents ten units.

...................................................................................................................................7-139 Figure 7.5. A similar cavity is shown at time 0.1166 with a 0.1 diameter 1.0 density solid

body present at 1.05 from the initial cavity centre. Note the reference vector represents

one unit. .....................................................................................................................7-140 Figure 7.6. A similar cavity is shown at time 0.5011with a 0.1 diameter 1.0 density solid


one unit. .....................................................................................................................7-141 Figure 7.7. A similar cavity is shown at time 1.9690 with a 0.1 diameter 1.0 density solid


ten units. ....................................................................................................................7-142 Figure 7.8. A similar cavity is shown at time 2.0719 with a 0.1 diameter 1.0 density solid














one unit. .....................................................................................................................7-149

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Figure 7.15. A similar cavity is shown at time 1.9697 with a 0.3 diameter 1.0 density solid


ten units..................................................................................................................... 7-150 Figure 7.16. A similar cavity is shown at time 2.0777 with a 0.3 diameter 1.0 density solid




one unit. .................................................................................................................... 7-152 Figure 7.18. A similar cavity is shown at time 0.5224 with a 0.3 diameter 0.7 density solid














ten units..................................................................................................................... 7-159 Figure 7.25. Plots of cavity north pole coordinates, with respect to time, for various moving

body cases. All cavities are initially centred at 0.9. ................................................. 7-160 Figure 7.26. Plots of cavity south pole coordinates, with respect to time, for various moving

body cases. All cavities are initially centred at 0.9. ................................................. 7-161 Figure 7.27. Plots of cavity centroid coordinates, with respect to time, for various moving

body cases. All cavities are initially centred at 0.9. ................................................. 7-161

xviii

Figure 7.28. Plots of solid body centroid coordinates, with respect to time, for various

moving body cases. All cavities are initially centred at 0.9. ....................................7-162 Figure 7.29. Plots of the distance between the cavity centroid and the solid body centroid,


at 0.9. .........................................................................................................................7-162 Figure 7.30. Plots of the distance between the cavity surface and the solid body surface

along the vertical axis, with respect to time, for various moving body cases. All cavities

are initially centred at 0.9. .........................................................................................7-163 Figure 7.31. Plots of the pressure at the rigid boundary directly below the cavity (coordinate

[0,0]), with respect to time, for various moving body cases. All cavities are initially

centred at 0.9. ............................................................................................................7-164 Figure 7.32. Plots of the maximum pressure in the domain, with respect to time, for various

moving body cases. All cavities are initially centred at 0.9. ....................................7-164 Figure 7.33. Plots of the kinetic energy of the cavity, with respect to time, for various

moving body cases. All cavities are initially centred at 0.9. ....................................7-165 Figure 7.34. Plots of the kinetic energy of the solid body, with respect to time, for various

moving body cases. All cavities are initially centred at 0.9. ....................................7-166 Figure 7.35. Pressure and vector plot for a cavity near a rigid boundary (at axial coordinate

0) and an imaginary surface at time 0.1115. Note the reference vector represents one

unit.............................................................................................................................7-167 Figure 7.36. Pressure and vector plot for a cavity near a rigid boundary (at axial coordinate

0) and an imaginary surface at time 0.6758. Note the reference vector represents one

unit.............................................................................................................................7-168 Figure 7.37. Pressure and vector plot for a cavity near a rigid boundary (at axial coordinate

0) and an imaginary surface at time 1.9859. Note the reference vector represents ten

units. ..........................................................................................................................7-168 Figure 7.38. Pressure and vector plot for a cavity near a rigid boundary (at axial coordinate

0) and an imaginary surface at time 2.0664. Note the reference vector represents ten

units. ..........................................................................................................................7-169 Figure 7.39. A similar cavity is shown at time 0.1130 with a 0.3 diameter 0.7 density solid




one unit. .....................................................................................................................7-171

xix





















one unit. .................................................................................................................... 7-181 Figure 7.51. Plots of cavity north pole coordinates, with respect to time, for various moving

body cases. All cavities are initially centred at 1.2. ................................................. 7-181 Figure 7.52. Plots of cavity south pole coordinates, with respect to time, for various moving

body cases. All cavities are initially centred at 1.2. ................................................. 7-182 Figure 7.53. Plots of cavity centroid coordinates, with respect to time, for various moving

body cases. All cavities are initially centred at 1.2. ................................................. 7-182

xx


moving body cases. All cavities are initially centred at 1.2. ....................................7-183 Figure 7.55. Plots of the distance between the cavity centroid and the solid body centroid,


at 1.2. .........................................................................................................................7-183 Figure 7.56. Plots of the distance between the cavity surface and the solid body surface





moving body cases. All cavities are initially centred at 1.2. ....................................7-185 Figure 7.59. Plots of the kinetic energy of the cavity, with respect to time, for various

moving body cases. All cavities are initially centred at 1.2. ....................................7-186 Figure 7.60. Plots of the kinetic energy of the solid body, with respect to time, for various

moving body cases. All cavities are initially centred at 1.2. ....................................7-187 Figure 7.61. Plots of cavity north pole coordinates, with respect to time, for various moving

body cases. All cavities are initially centred at 1.5. .................................................7-188 Figure 7.62. Plots of cavity south pole coordinates, with respect to time, for various moving

body cases. All cavities are initially centred at 1.5. .................................................7-189 Figure 7.63. Plots of cavity centroid coordinates, with respect to time, for various moving

body cases. All cavities are initially centred at 1.5. .................................................7-190 Figure 7.64. Plots of solid body centroid coordinates, with respect to time, for various

moving body cases. All cavities are initially centred at 1.5. See Figure 7.83 for legend.

...................................................................................................................................7-190 Figure 7.65. Plots of the distance between the cavity centroid and the solid body centroid,


at 1.5. See Figure 7.86 for legend..............................................................................7-191 Figure 7.66. Plots of the distance between the cavity surface and the solid body surface





moving body cases. All cavities are initially centred at 1.5. ....................................7-193

xxi

Figure 7.69. Plots of the kinetic energy of the cavity, with respect to time, for various

moving body cases. All cavities are initially centred at 1.5..................................... 7-194 Figure 7.70. Plots of the kinetic energy of the solid body, with respect to time, for various


.................................................................................................................................. 7-195 Figure A.1. Plots of cavity north pole coordinates, with respect to time, for various

stationary body cases. All cavities are initially centred at 1.5. ................................... 205 Figure A.2. Plots of cavity south pole coordinates, with respect to time, for various

stationary body cases. All cavities are initially centred at 1.5. ................................... 206 Figure A.3. Plots of cavity centroid coordinates, with respect to time, for various stationary

body cases. All cavities are initially centred at 1.5. .................................................... 206 Figure A.4. Plots of the distance between the cavity centroid and the solid body centroid,


at 1.5............................................................................................................................. 207 Figure A.5. Plots of the distance between the cavity surface and the solid body surface along

the vertical axis, with respect to time, for various stationary body cases. All cavities are

initially centred at 1.5. ................................................................................................. 207 Figure A.6. Plots of the pressure at the rigid boundary directly below the cavity (coordinate


centred at 1.5................................................................................................................ 208 Figure A.7. Plots of the maximum pressure in the domain, with respect to time, for various

stationary body cases. All cavities are initially centred at 1.5. ................................... 208 Figure A.8. Plots of the kinetic energy of the cavity, with respect to time, for various

stationary body cases. All cavities are initially centred at 1.5. ................................... 209 Figure A.9. Plots of cavity north pole coordinates, with respect to time, for various

stationary body cases. All cavities are initially centred at 1.8. ................................... 210 Figure A.10. Plots of cavity south pole coordinates, with respect to time, for various

stationary body cases. All cavities are initially centred at 1.8. ................................... 210 Figure A.11. Plots of cavity centroid coordinates, with respect to time, for various stationary

body cases. All cavities are initially centred at 1.8. .................................................... 211 Figure A.12. Plots of the distance between the cavity centroid and the solid body centroid,


at 1.8............................................................................................................................. 211 Figure A.13. Plots of the distance between the cavity surface and the solid body surface


cavities are initially centred at 1.8. .............................................................................. 212

xxii

Figure A.14. Plots of the pressure at the rigid boundary directly below the cavity (coordinate


centred at 1.8. ............................................................................................................... 212 Figure A.15. Plots of the maximum pressure in the domain, with respect to time, for various

stationary body cases. All cavities are initially centred at 1.8..................................... 213 Figure A.16. Plots of the kinetic energy of the cavity, with respect to time, for various

stationary body cases. All cavities are initially centred at 1.8..................................... 214 Figure B.1. Plots of cavity north pole coordinates, with respect to time, for various moving

body cases. All cavities are initially centred at 1.5. .................................................... 215 Figure B.2. Plots of cavity south pole coordinates, with respect to time, for various moving

body cases. All cavities are initially centred at 1.5. .................................................... 216 Figure B.3. Plots of cavity centroid coordinates, with respect to time, for various moving

body cases. All cavities are initially centred at 1.5. .................................................... 216 Figure B.4. Plots of solid body centroid coordinates, with respect to time, for various

moving body cases. All cavities are initially centred at 1.5. ....................................... 217 Figure B.5. Plots of the distance between the cavity centroid and the solid body centroid,


at 1.5. ............................................................................................................................ 217 Figure B.6. Plots of the distance between the cavity surface and the solid body surface along

the vertical axis, with respect to time, for various moving body cases. All cavities are

initially centred at 1.5................................................................................................... 218 Figure B.7. Plots of the pressure at the rigid boundary directly below the cavity (coordinate


centred at 1.5. ............................................................................................................... 218 Figure B.8. Plots of the maximum pressure in the domain, with respect to time, for various

moving body cases. All cavities are initially centred at 1.5. ....................................... 219 Figure B.9. Plots of the kinetic energy of the cavity, with respect to time, for various moving

body cases. All cavities are initially centred at 1.5. .................................................... 220 Figure B.10. Plots of the kinetic energy of the solid body, with respect to time, for various

moving body cases. All cavities are initially centred at 1.5. ....................................... 221 Figure B.11. Plots of cavity north pole coordinates, with respect to time, for various moving

body cases. All cavities are initially centred at 1.8. .................................................... 222 Figure B.12. Plots of cavity south pole coordinates, with respect to time, for various

moving body cases. All cavities are initially centred at 1.8. ....................................... 222 Figure B.13. Plots of cavity centroid coordinates, with respect to time, for various moving

body cases. All cavities are initially centred at 1.8. .................................................... 223

xxiii

Figure B.14. Plots of solid body centroid coordinates, with respect to time, for various

moving body cases. All cavities are initially centred at 1.8........................................ 223 Figure B.15. Plots of the distance between the cavity centroid and the solid body centroid,


at 1.8............................................................................................................................. 224 Figure B.16. Plots of the distance between the cavity surface and the solid body surface


are initially centred at 1.8............................................................................................. 224 Figure B.17. Plots of the pressure at the rigid boundary directly below the cavity (coordinate


centred at 1.8................................................................................................................ 225 Figure B.18. Plots of the maximum pressure in the domain, with respect to time, for various

moving body cases. All cavities are initially centred at 1.8........................................ 225 Figure B.19. Plots of the kinetic energy of the cavity, with respect to time, for various

moving body cases. All cavities are initially centred at 1.8........................................ 226 Figure B.20. Plots of the kinetic energy of the solid body, with respect to time, for various

moving body cases. All cavities are initially centred at 1.8........................................ 227

xxv

Statement of Authorship

The work contained in this thesis has not been previously submitted for a degree or diploma

at any other higher education institution. To the best of my knowledge and belief, the thesis

contains no material previously published or written by another person except where due

reference is made.

Signed: ________________

Date: ________________

xxvii

Acknowledgments

When first starting this doctorate, I was convinced that a degree of this type was mainly a

single person’s effort. Now that I am at the end it is clear that, while in some aspects this

may be true, in others it could not be further from the truth. In this regard, the impact that

others have had is clear and direct, and to this end I would like to thank my associate

supervisors Dr Ian Turner and Dr Richard Clegg for their technical input and help

throughout this research. In addition, I wish to extend my appreciation to Dr Georges

Chahine and the other engineers at DYNAFLOW INC for all their efforts in helping me

understand better the nature of the problem.

Where the previously people were involved more in the technicalities, many others have

provided assistance in a less direct manner. To this end, special mention must be made to

the staff in the Mechanical, Medical and Manufacturing Engineering School at QUT for all

their support, especially Dave McIntosh, Neil Munro, Barbara Dabelstein and Joan

Whitham. In addition, I would like to thank all the postgraduate students for helping keep

me sane and listening to my prattle, in particular Dr Rosemary Thompson and Cameron Bell.

Special mention must also be made to my parents for helping provide the opportunity to be

self-indulgent in the pursuit of an advanced degree, as well as the inspiration to complete it.

Finally, I need to acknowledge the tremendous impact Dr Douglas Hargreaves has had on

my life to date. Without his faith in me and his assistance in the research, this degree would

simply not have been possible. Doug has been both mentor and patron to me, but more

importantly me has been terrific inspiration and friend. I will never be able to repay Doug

for all he has done for me. I just hope I have the opportunity to help someone in the same

way he has helped me. Thank you for everything Doug.

xxviii

xxix

Nomenclature

The following nomenclature is used unless otherwise noted.

eij = rate of strain tensor (where i,j = 1,2,3)

g = acceleration due to gravity

n = normal vector outward to a surface

n = unit normal

p = field point

q = source panel

r = radial coordinate

r = radial unit vector

s = unit vector tangent to a surface

u = velocity vector

z = axial coordinate

z = unit axial vector

Fi = force tensor (where i = 1,2,3)

S = arbitrary surface

α = angle from radial vector to vector normal to a surface

δij = Kronecker delta (where i,j = 1,2,3)

θ = azimuthal angle

κ = local curvature

µ = dynamic viscosity

xxx

ν = kinematic viscosity

ξ = parameterising variable

ρ = density

σ = surface tension coefficient

σij = stress tensor (where i,j = 1,2,3)

φ = velocity potential

∆ = u∇ = eij (where i,j =1,2,3) or ∆ = delta

Ω = arbitrary volume

∇ = Grad

DtD

= material derivative

1-1

Chapter 1

Introduction

1.1 Background

Cavitation is the spontaneous vaporisation of a liquid due to localised regions of absolute

pressure at or below a liquid’s vapour pressure (Douglas, Gasiorek et al. 1995). Cavitation

can occur almost anywhere within a hydraulic system including impellers, journal bearing,

valves, water jackets and other fittings (Peterson 1973). In many instances, cavitation cannot

be avoided as any feature that accelerates flow is subject to Bernoulli’s Equation, and hence

static pressure will fall. Generally speaking, cavitation adversely affects hydraulic systems

through loss of hydrodynamic efficiency and, in the case of cavitation erosion, damage to

components (Trevena 1987).

In the context of erosion, it is not so much the creation of cavitation bubbles that is of major

concern; rather it is the subsequent collapse of the bubble once the absolute pressure has

returned to that above the vapour pressure. These collapsing cavitation bubbles are capable

of producing highly directed dynamic pressure fronts that can produce stresses high enough

in a solid material’s surface to erode the surface (Gould 1973). This erosion, if of sufficient

intensity or present long enough, can not only change the macro scale flow fields, but also

lead to the catastrophic failure and disintegration of some plant equipment (Lecoffre 1999).

Consequently, cavitation has to be considered by design engineers.

Despite recent progress in cavitation modelling, design principles have not progressed

beyond general avoidance of cavitation (where possible) or extension of component lifetime

via material selection or over-engineering. Most of this “practical” engineering approach is

based on prediction of global flow patterns, and empirical data for material selection. While

the scattered nature of this data has not allowed all-encompassing theories to be developed, it

1-2

has highlighted the complex nature of cavitation and related erosion. Specifically, it has

been demonstrated that cavitation erosion is a combination of fluid rheology, solid

contaminants, solute gases, nucleation sites, turbulence, dislocation density, boundary

compliance, material microstructure and so on (Lecoffre 1999). It is this difficulty in

isolating the separate factors, which is perhaps the major motivation for fundamental

theoretical research into cavitation.

While Euler had postulated cavitation as a problem as early as 1754, it has only been since

the Second World War that the lifecycle of non-spherical collapse of bubbles has been

accurately documented (Philipp and Lauterborn 1998). These studies captured the growth,

collapse and rebound of bubbles near rigid boundaries using high-speed cinegraphs. Most

important of the observations made was the presence of a re-entrant jet directed toward the

boundary, piercing the bubble and leading to a toroidal collapse phase. Subsequent attempts

to model cavitation has brought about a good understanding of the collapse of single and

multiple cavities near rigid boundaries, compliant boundaries and free surfaces to the point

of touchdown of the re-entrant jet to the boundary surface and rebound of the cavity.

The effects of shear flow and large-scale vortices have recently been included in these

models (Hsiao 2001), demonstrating the progression toward capturing the characteristics of a

real system. This success of re-entrant jet modelling with real flow patterns has provided

sufficiently accurate data such that the surface pressure loading to solid material boundaries

can be used to include fluid structure interaction (Kalumuck, Duraiswami et al. 1995). As a

result, it is now possible to couple existing deformation theory to the cavitation models in

order to investigate the erosion mechanism present in material structures. However, these

cavity simulations tend to be conducted in single-phase liquid matrices without including the

effects of suspended solid particles or visco-plastic drops. It is the focus of this project to

include these effects to better model cavitation in multi-phase matrices such as contaminated

lubricants and emulsions.

These types of fluids are characteristic of real systems and are increasingly present in

modern hydraulic systems. Fire retardant oil in water (O/W) emulsions and moisture

contaminated gearbox lubricants are examples of the extremes of the oil and water lubricant

spectrum. It is common knowledge that changes in water and oil ratios (particularly at either

end of the spectrum) can have a remarkable effect in erosion rates of like materials. Theories

involving dampening of cavitation or increased cavitation inception have been postulated to

explain the observed change in erosion rates (Tomlinsen 1987). However, the sometimes-

contradictory natures of emulsion erosion rate tests make it difficult to subscribe to any

1-3

particular theory. It is the object of the model described in this thesis to provide fundamental

reasoning to support these hypotheses of changes of erosion rates in emulsions.

1.2 Cavitation

As mentioned previously, cavitation is initiated by the fall of absolute pressure to a level

where gas bubbles are produced in a liquid matrix. Ideally, the gas would be composed

entirely of vapour of the liquid matrix, however most real systems have dissolved gas present

in the liquid. These solute gases readily precipitate out of solution (usually at a pressure

higher than the vapour pressure of the liquid) to form micro-bubbles, which act as nucleation

sites for vaporisation, thus further enabling the cavitation growth phase. It is generally

assumed that this growth phase consists of an adiabatic expansion of the solute gas to the

saturated vapour pressure of the liquid (Lecoffre 1999). The cavity then continues to grow

with the liquid turning to vapour, for as long as the bubble is in a region of low pressure. It

should be noted that entire cavitation cycles usually take place within a few milliseconds.

The collapse of cavities creates “white” noise, which occupy a wide bandwidth of up to 1

MHz, with smaller bubbles resulting in higher frequencies (Pearsall 1972). In general, peak

noise amplitudes tend to correlate to peak erosion rates.

These regions of low pressure are commonly the product of accelerating liquid in hydraulic

systems. In fact, any machinery that moves liquid may experience cavitation. An alternative

to creating an area of low pressure is raising the saturated vapour pressure locally. It is this

principle exploited by some cavitation experimentalists where an electric spark or laser pulse

is used to create a cavity. Essentially, the laser or spark provides sufficient energy to a small

region to raise the vapour pressure above that of the ambient pressure, thus boiling an

extremely localised region of liquid.

The presence of nuclei can also have significant impact on the cavitating nature of a system.

In the absence of nuclei, it is possible for a liquid to withstand tensions similar to solids.

Theoretically, a pure liquid must have tensions applied to it sufficient to overcome

intermolecular forces and “crack” or “tear” the liquid (Trevena 1987). In most cases this

calculates to negative pressures at tens of atmospheres, rather than its vapour pressure. In

reality, it is virtually impossible to remove all of these nucleation sites; therefore it is more

important to measure the size and distribution of these sites. In most cases, this is also very

difficult to do, and as a result the total gas content of a liquid is commonly used as an

indicator. Unfortunately, the total gas content does nothing beyond indicating the relative

propensity for a liquid to cavitate, as size and distribution are not predicted.

1-4

Once a cavity is formed, there tends to be two alternatives. When time scales are long and

buoyancy is a consideration, a cavity may proceed to a free surface and burst. This is

common with simple boiling or large underwater explosions. Alternatively, when time

scales are short, the growing cavity may progress to a region of relatively high pressure

where the pressure differential causes the bubble to collapse. In the second case, the velocity

potential causes the gas-liquid interface (bubble surface) to accelerate toward the centre of

the bubble. It is generally assumed that any vaporised liquid condenses as the bubble

collapses, until only solute gases remain. The solute gas then undergoes compression, which

can be assumed adiabatic in most hydraulic situations due to short time scale involved. The

momentum of the liquid matrix is converted to potential, acoustic and thermal energy. It is

the compression of the solute gas that is responsible for the rebound and oscillatory nature of

some cavities, and localised high temperature observed with sonoluminescence (Margulis

1993).

This transfer of momentum can also result in high pressures being applied to rigid

boundaries, when the cavity collapses in their close proximity. Because the fluid flow is

restricted on the boundary side, there is an overall movement of the cavity toward the

boundary. This phenomenon is known as the Kelvin impulse. The translation of the bubble

centroid, and the restriction of fluid flow from the boundary side, results in an overall

deformation of an initially spherical bubble to a cresent-oid (half-moon shape) and later to a

toroid.

1.3 Erosion

Intuitively, only cavities collapsing at critical distances from the rigid boundary have erosive

ability. Some texts state that in a typical cavitating system, only 1 in 30,000 cavities are

capable of causing damage, although they are unclear exactly how this is ascertained.

Additionally, work conducted by Blake et al (1984-1997) and other authors seems to indicate

that a single collapsing bubble (of size typical of a hydraulic system) is incapable of eroding

materials found in these mechanisms. This is contradicted, however, by recent experiments

that indicate a single cavity is capable of damage (Philipp and Lauterborn 1998), provided

the rebound and subsequent oscillations of the bubble are taken into account. This was also

found to be the case for studies involving materials with very low Youngs modulus (Brujan,

Nahen et al. 2001; Brujan, Nahen et al. 2001), the limit of which are similar to rigid

boundary cases.

1-5

In any event, most of the practical information regarding erosion is qualitative. Cavitation

experiments usually involve creating cavities either through direct means (spark, laser),

indirect (venturi, inducers) or studying erosion using accelerated tests (rotating disks,

ultrasonic vibrators). Observations were then made regarding the profile of the collapsing

cavity (Blake, 1993) for the direct means, or the rate of erosion for the indirect (Pearsall,

1972).

While erosion is commonly a fatigue process (i.e. Incubation→Rapid increase in erosion

rate→stable linear rate→pitting and reduction in erosion rate), it is not well understood.

Incubation periods cannot be accurately predicted beyond hard materials usually having

longer incubation times and lower erosion rates than soft materials (Trevena 1987).

Furthermore, the impact that different fluids have on the erosion rate on similar systems,

while well documented, is also not well understood.

It is the very nature of cavitation that makes it so difficult to understand. The speed of the

growth and collapse of a bubble, as well as the areas where this occurs makes it a difficult

phenomenon to observe. Success has been enjoyed using high-speed photography which has

provided considerable understanding of the bubble, even at the very latest stages of the

collapse, which has verified the accuracy of computational modelling. However these

studies require extremely expensive equipment capable of imaging up to 100 million frames

a second. Thus, numerical models have been developed as these provide better observation

of the penetrating surfaces; allow simulation of cavitation process at much lower cost; and

allow a more controlled environment, thereby improving isolation of the respective

influencing factors.

Furthermore, steps have been made toward modelling complete systems, which have

included the effect of dynamic boundaries (Chahine and Kalumuck 1998), multiple bubbles

(Blake, Robinson et al. 1993), shear and large scale vortices (Hsiao 2001).

1.4 Unsolved Problems

When considering the modelling of cavitation and its related erosion, there are many areas

requiring further research. This includes the introduction of surface fatigue theory, the

extension of the duration of accurate cavity simulation and the inclusion of real fluid

rheology. It is the latter topic that in central to this thesis.

While the general mechanism is simple, the interaction of cavities, boundaries and other

conditions can be very complicated. As recent studies (Brogle, 1997; Kit, 1998; Pai, 2002)

1-6

at the Queensland University of Technology have revealed, the presence of suspended

droplets of oil in water can have measurable effect on the cavitation erosion rate of particular

test samples. The sometimes contradictory nature of these results has made it difficult to

determine whether the change in erosion rate is due to chemical interaction of the oil with the

test sample, or some effect of the droplet on the development on the cavity bubble. As both

tests were essentially measurement of cavitation erosion of metal test specimen in a

cavitating plume of varying oil percentages in water, it was not possible to witness the

cavitation bubble in detail. As a consequence, it was not possible to definitively answer

whether the presence of oil, or other contaminant, impacted on the cavity or not.

As a consequence, this thesis is concerned with the numerical analysis and simulation of a

cavity near a rigid boundary, specifically with respect to creation of cavitation bubbles with

erosive potential with scales typical of hydraulic system, and the influence that a suspended

solid body has on the cavity. To achieve this, a modified boundary element method is used.

1.5 Original Contribution

The results presented in this thesis are from original software based upon the mathematical

formulations developed by Taib (1985), Blake et al (1981-1993), Best et al (1992-1993) and

Zhang (1992-1993). All of these authors have extended the understanding of individual

cavity development; however they have not addressed the impact that small-suspended rigid

bodies of varying density have on the development on single cavities. It is precisely this area

that this thesis investigates.

As this area of research is novel, it was necessary to develop independent algorithms as the

presence of suspended bodies complicates the model and requires separate mathematical and

numerical treatment. It is in this sense, the numerical model used in this thesis extends the

work by Zhang (1992-1993) such that, once numerical stability has been improved, his

method can be used to model a cavity near a mobile solid body all the way to first rebound.

To achieve this, the mathematical formulation had to be modified to cope with the motion of

a solid surface. Furthermore, initiative had to be taken in many areas of the numerical

process to provide a mechanism to deal with the interaction of the cavity surface and solid

body.

These areas of progress have yielded a tool capable of analysing cavitation phenomena

specific to hydraulic systems, providing us with significant and original insight into the

behaviour and damage potential of cavities near suspended contaminants.

1-7

1.6 Thesis Scope

This thesis is limited to the modelling of small-suspended spherical bodies near a single

cavitation bubble in close proximity to a rigid boundary. The cavity and solid body is

sufficiently small such that it can be considered typical of those encountered in hydraulic

systems and the tests conducted by Pai & Hargreaves (2002). In this regard, the results of

this thesis can provide further evidence as to whether the presence of suspended bodies has

influence on damage potential of an individual cavity. As a result, only cavities that might

produce pressures sufficient to damage a material surface are considered, which limits the

distance that it is offset from the rigid boundary. Furthermore, the material surface is

considered rigid. Thus, interaction between the rigid boundary and the cavity is neglected;

consequently the problem considered in this thesis is reduced to the modelling of a single

cavity:

1. with initial centroid less than 1.8 bubble maximum radii from a rigid

boundary,

2. with stationary and mobile suspended rigid spheres of varying sizes,

3. where density is typical of low percentage oil and water emulsions with

solid contaminants, and

4. where the oil droplets or contaminant solid bodies are well separated.

Furthermore, cases where the cavity is between the solid and rigid boundary only are

considered. This was to avoid problems associated with the solid contacting the rigid

boundary, under the influence of the cavity.

1.7 Organization of Thesis

The next chapter of this thesis describes the cavitation process in greater detail. This is to

provide the reader with insight into the significant periods of a cavitation bubble as well as to

the significant impact that the cavity surroundings have on the bubble development. This

then leads to the probable damage mechanisms associated with cavity collapse, such that

some simplifications can be introduced.

The proceeding chapters then describe the mathematical basis for the numerical model used

in this thesis. As the case simulated in this thesis is unique, there are specific considerations

that have impact on the model, and subsequent algorithm development, not addressed

directly in previously referenced literature. After the description of the numerical scheme,

the algorithm used in this thesis is verified to demonstrate its accuracy.

1-8

Chapter 6 then presents the results for a single cavity below a stationary suspended body

near a rigid boundary. While it is acknowledged that this is not a natural phenomenon, this

case is used to investigate the effects that a body may have on a cavity where there is no

mechanism for energy transfer between the solid and cavity. The following chapter then

looks at similar cavities to Chapter 6, except that the near body is allowed to move, thus

providing data for a case where energy can be transferred to and from a cavitation bubble.

All chapters contain a summary of the relevant issues associated with each individual

chapter. The ultimate findings of this thesis are then summarised in the final chapter where

potential future directions of this work are discussed.

2-9

Chapter 2

Cavitation Erosion and Suspensions

2.1 Introduction

The surrounding environment can have a dramatic effect on the growth and collapse of

vapour cavities, which can take the form of free surfaces, gas bubbles, other cavities, solid

suspensions, rigid boundaries and other liquid drops. Much study has been focused on the

effect of rigid boundaries and multiple cavities, as well as bursting of bubbles and cavities at

free surfaces. For completeness, this past research shall be re-visited in this chapter with the

general findings used to provide further background to the hypothesis of this thesis.

This chapter will follow the erosion cycle from the perspective of a single damaging cavity.

Specifically, the growth and collapse of the cavity will be described, with digression made to

the impact that the environmental factors have on the destructive potential of the vapour

bubble. While a cavity’s damage potential should be ultimately based on the pressure that is

applied to the rigid boundary, its dynamic response, and how this relates to the fatigue or

immediate failure of the solid material at the boundary, it is beyond the scope of this thesis.

However, it is necessary to raise these issues to provide a framework within which to base

this thesis’ contribution.

Finally, the chapter shall be concluded with a discussion of the fundamental assumptions and

simplifications made in this thesis.

2.2 Lifecycle of a Cavity

In the simplest case, a single vapour cavity grows and collapses in an initially quiescent and

infinite domain. The lifecycle of the cavity consists of the perfectly spherical growth of the

2-10

cavity from a nucleus to its maximum radius, the subsequent collapse to a point, and the

emanation of a shock wave or acoustic front. If we consider the presence of non-

condensable gas in the bubble, the cavity can oscillate numerous times and dissipate energy

at each separate collapse. This phenomenon is well documented by Rayleigh and extended

by Plessett, leading to the famous Rayleigh-Plessett bubble equation (Benjamin and Ellis

1966).

In the event that the domain cannot be considered completely infinite, perfect sphericity is

not maintained and becomes more pronounced further into the collapse phase. This gives

rise to important events in the cycle including the development of a re-entrant jet, touchdown

or contact of opposite sides of the bubble and finally the product of an oscillating toroidal

bubble (Naude and Ellis 1961).

InitialSpherical

CavityEnd ofGrowth

Re-entrantJet Development Touchdown

ToroidalCavity &Rebound

Figure 2.1. Bubble Profiles for a Simple Cavity Lifecycle

2.2.1 Growth Phase

The growth phase of any cavity is normally assumed to begin with a nucleus. This nucleus,

sometimes referred to as a “seed”, is often considered to be a very small pocket of gas that is

not dissolved in the liquid matrix. Essentially, this provides an interface whereby the liquid

can spontaneously vaporise at the vapour pressure of the liquid. This fact is an important

initial assumption. If we were to consider an ideal liquid, that is; one with no microscopic

gas bubbles, then it is theoretically possible for the liquid to withstand tensile stresses that

correspond to a pressure lower than the vapour pressure (Pearsall 1972).

The size of this nucleus is generally considered to be microscopic. However, the gaseous

content can have considerable impact on the cavity, as it is this non-condensable gas that is

2-11

responsible for the bubble rebound. In the case of numerical modelling of bubbles, the

nucleus is usually grown spherically, using the Rayleigh-Plessett equation, to some small

fraction of the maximum radius of the cavity in question (Bevir and Fielding 1974). This is

justified, as any external factors have a negligible effect on the growth of the bubble at such

small radii, and to begin from a small sphere tends to improve numerical stability (Blake,

1981). From this small sphere, the cavity continues to grow to its maximum radius. Initially

the growth is extremely fast, but then slows considerably as the cavity approaches its

maximum volume.

If there are close proximity suspensions, rigid boundaries or any other domain elements that

constrain the growth, then the cavity will grow in a spherically asymmetric fashion and, if

sufficiently close, may even conform to the surface of these elements. In addition, there is a

slight relative movement of the cavity centroid away from the restricting surface due to

inhibited motion of the surrounding fluid (Taib 1985).

In the case of an infinite domain, the whole cavity interface will stop growing

simultaneously. However, this is not the case for a spherically asymmetric growth.

Generally, the areas of the bubble surface that are closest to a restricting boundary will stop

expanding first, with the converse true for cavity surfaces where motion is unopposed. The

result is an overall extension in time of the growth phase as compared to an infinite domain.

2.2.2 Initial Collapse

The initial collapse begins once the cavity has attained its maximum volume. This is due to

the energy available in the cavity having been converted to potential energy (in the case of a

spark or laser generated bubble), or the cavity has been moved to a region of higher pressure

(in the case of a typical hydraulic system). If the domain is considered infinite in all

directions, the collapse will continue until all vapour is converted to liquid and rebound,

and/or the generation of an acoustic pulse, occurs. This is due to the completely spherical

nature of the collapse.

When the collapse is not spherical, vapour is still converted to liquid, however the motion of

the surrounding liquid is essentially restricted in one area relative to another. Rather than

simply collapsing to a point, or being the complete reverse of the growth, the relative flow

field surrounding the cavity may cause the emergence of a re-entrant jet or side pinching.

This is defined as the moment when an area of concavity is formed at the fluid/cavity

interface with respect to the liquid domain. This phenomenon does not develop until some

2-12

time after the maximum volume of the cavity is attained and becomes more pronounced later

into the collapse phase.

In both spherical and non-spherical cases, the velocity of the interface increases throughout

the collapse. However, for spherically asymmetric collapse, interface velocity is naturally

not constant over the geometry with the re-entrant jet dominating the system and possessing

the highest velocity.

2.2.3 The Re-entrant Jet

Clearly, the re-entrant jet only pertains to a non-spherical collapse. At some stage during the

collapse, fluid from the surrounding liquid begins to penetrate the cavity on the side that has

the greatest pressure, and hence a greater propensity for motion. This phenomenon can be a

product of the geometry of the domain, or of buoyancy factors. However, buoyancy only

has time to be effective in systems where the cavitation bubble is large, such as those present

in underwater explosions (i.e. bubbles of the order or metres of tens of metres). In the case

of hydraulic systems, cavities are typically very small resulting in extremely fast motion.

Consequently, gravity may be ignored and will not be considered for the cases presented in

this thesis.

As the jet forms, its velocity and linear momentum increase dramatically. The jet continues

to penetrate the bubble until it contacts the opposite side, which is termed touchdown. Jet

width is affected by the surface tension at the interface, as well as the structure of the

domain. In addition, the domain has a profound effect on the duration of the collapse

compared to the collapse of a spherical cavity.

The unbalanced surface velocity gives rise to an overall movement of the centroid of the

cavity toward the domain surface responsible for inhibiting flow (Benjamin and Ellis 1966).

This motion, termed the Kelvin Impulse, is somewhat greater than the motion away from the

surface during the growth and is therefore not offset. Ultimately the cavity collapses at a

distance to the rigid boundary somewhat closer than that of its inception. Consequently there

is an increase in the damage potential of the cavity due to its ability to transmit higher

pressures at rebound to the rigid boundary (Philipp and Lauterborn 1998).

In addition, the momentum of the liquid domain is dependent on the forces acting on the

liquid. In this regard, it becomes possible for a cavity to apply damaging pressures to a rigid

surface after the first collapse as the bubble continues to move toward the surface. Clearly

this is of interest to the investigation of damage potential, and was demonstrated

experimentally by Philipp and Lauterborn (1998).

2-13

2.2.4 Post Touchdown

Touchdown is said to occur when an impinging jet from one side of the bubble touches the

other side. In an axisymmetric case, this is usually when the north and south poles come into

contact. As the jet continues through the opposite side, the contact of the opposing sides is

continuous and produces a shear layer that is propagated toward the boundary responsible for

the pressure differential (Zhang 1992). Linear momentum is maintained throughout the

penetration of the jet, and as the jet normally possesses a significantly higher velocity that

the opposing side, its velocity is reduced and approaches the average of the opposing liquid

surfaces due to the maintenance of the linear momentum.

The penetration of the re-entrant jet creates a multiply connected domain consisting of a

toroidal bubble and vortex sheet. The vortex sheet is the shear layer that separates the high-

speed jet’s penetration into the lower velocity liquid domain. At the shear layer, normal

velocity of the fluid is maintained. However, as the term suggests, tangential velocity is not

conserved across the sheet, which has a serious impact on the mathematical modelling of the

vortex sheet. This shall be addressed in a later chapter. In addition, it is generally accepted

that micro-bubbles are created and swept along at the shear layer (Best 1993). However, the

presence of these micro-bubbles is yet to be observed in experiment (Philipp and Lauterborn

1998).

Once the toroidal bubble is created, the Bjerkness forces, responsible for the Kelvin impulse,

continue to propel the cavity toward the rigid body surface. The shear layer also moves

toward the surface, normally at a higher rate, and the cavity continues to shrink until

rebound. Eventually the shear layer contacts the rigid surface and the momentum change is

responsible for applying water-hammer pressures, significantly higher than ambient, to the

solid surface. At one stage this water-hammer pressure was considered to be a potential

erosion mechanism. However, recent studies have conclusively demonstrated that this is not

the case (Philipp and Lauterborn 1998). Not only is the jet quickly dispersed by the liquid

layer separating the initial shear layer and the rigid surface, the velocities of typical cavity

jets are insufficient to produce water-hammer pressure capable of exceeding the flow stress

of even very soft metals.

The presence of the jet does affect the shrinkage rate of the cavity however, and this in turn

has impact on the rebound of the cavity.

2-14

2.2.5 Rebound

Rebound of a cavity represents the end of one cavitation cycle, and the beginning of another.

The first cycle is normally associated with a toroidal bubble’s collapse. This collapse is not

perfect due to instabilities in the cavitation process, and consequently the toroid will close

entirely as a range of points on an annular ring. This results in an annular collapse in the

circumferential direction of the toroid, which compresses the non-solute gases to points

where the momentum of the liquid is arrested, and secondary cavities begin to grow (Philipp

and Lauterborn 1998). This process results in a number of essentially spheroidal cavities

located on the annulus that was the original toroid, which may expand in star like shapes due

to chaotic domain instabilities (Benjamin and Ellis 1966). These cavities grow and collapse

again in what amounts to cloud cavitation. The dynamics of this phenomenon, while

interesting, are beyond the capability of an axisymmetric model and consequently will not be

addressed further in this thesis.

As the pressure that is responsible for damage to the rigid material is of greatest interest,

focus is placed on the pressure of the cavity at rebound (i.e., the initial shock front pressure).

As the Kelvin impulse draws the cavity toward the surface, the pressure inside the cavity is

easily transmitted to the material. It has been demonstrated that this peak pressure is of the

order of Giga-Pascals (Philipp and Lauterborn 1998), which is capable of damaging even the

hardest of metal alloys. As the pressure reduces proportionally to the square of the distance,

only those cavities, which rebound close to the surface, will damage the surface. In addition,

energy is radiated away both acoustically and thermally, thus only the first two or three

cycles of the cavities will be sufficiently energetic to significantly affect the rigid surface.

The nature of the rebound is highly dependent on the nature of the domain in question. This

includes the presence of other cavities, rigid surface and possibly other interfacial surfaces in

the case of emulsions and suspensions. These general effects form the basis of the

hypothesis presented in this thesis and shall be discussed in the next section.

2.3 Influence of the Surrounding Domain

The discussion presented in the previous section is largely based on observations made from

perfectly spherical cavities and those near a flat and near infinite rigid boundary. With a

near infinite rigid boundary, the collapse is dominated by the development of a re-entrant jet,

which penetrates the bubble and produces a toroid, which ultimately collapses and rebounds.

The re-entrant jet has much higher velocity than the opposing side, migrating the cavity

2-15

toward the boundary such that the damaging mechanism of the collapse (i.e., rebound)

occurs in relatively close proximity to the boundary surface. This is not always the case

when motion of the boundary wall or the presence of other surfaces is considered.

2.3.1 Fluid Structure Interaction

It has long been noted that pliable surfaces can reduce the rate of erosion (Benjamin and Ellis

1966; Pearsall 1972), which is in part due to the material’s ability to absorb impact-like

forces. However, computational modelling has also indicated that a boundary that can move

with respect to applied pressures actually changes the cavity dynamics (Chahine and

Kalumuck 1998; Brujan, Nahen et al. 2001). The effect of the boundary motion has far

reaching impact that includes reduction of the Kelvin impulse and the generation of a counter

jet. Clearly, these are related.

Essentially the mobile surface acts similarly to the surrounding fluid, and does not restrict

the growth of the cavity, as compared to a totally immobile surface. This also applies to the

collapse phase, and has the overall effect of reducing the propensity for the cavity centroid’s

movement toward the surface, indicating lower Bjerkness forces.

In addition to this, the pliability of the boundary allows the formation of a second jet on the

opposite side to the re-entrant jet described in the previous section. As these two jets

contact, the re-entrant jet is slowed considerably more than those of a rigid boundary case,

which can hasten the ultimate collapse of the toroid. While this can reduce the violence of

the collapse slightly, it couples with the reduced Kelvin impulse and reduces the damage

potential of the cavity by having it rebound further away from the boundary surface.

However, in the case of materials with very low elastic moduli, typical of biological systems,

the cavity is often still capable of disrupting the surface (Brujan, Nahen et al. 2001). Indeed,

the re-entrant jet often penetrates the solid surface and damages the material via tensile

loading, rather than shear (Brujan, Nahen et al. 2001). While this further highlights the

coupled and complicated nature of cavitation near boundaries, it is beyond the scope of this

thesis due to the large strains involved.

2.3.2 Multiple Cavities

Of equal importance to the fluid structure interaction is the presence of other cavities. When

two or more cavities are in close proximity they naturally affect one another’s growth and

collapse. If two cavities are identical, then the overall effect is the same as a rigid boundary

placed at the centre of the separation between them.

2-16

When two cavities are not identical, then some energy transfer takes place (Blake, Robinson

et al. 1993). The cavity that becomes dominant is largely case dependent. Despite this, the

presence of another cavity usually involves the delay of the collapse phase of one of the

cavities. This elongates the cavity along the axis of the re-entrant jet, which ultimately

increases the jet velocity. Energy is passed to this cavity with the increased jet velocity and

causes it to collapse more violently. Whether this causes the cavity to be more or less

destructive is completely case dependent.

If the cavity that feeds from the second is sufficiently close to the rigid material, then the

increased pressure at which it rebounds will be transferred to the solid material.

Alternatively, if the cavity that is further away from the surface feeds from the other, then it

will be too far to cause damage regardless of the pressure at which it rebounds. Furthermore,

the other bubble will collapse less violently, reducing its potential for erosion.

The nature of the cavitation process is sensitive to a bubble’s surroundings. By considering

the cases of cavities interacting with a structural interface, and cavities interacting with other

bubbles, it is then clear that the presence of solid and liquid suspensions must also have some

impact on the cavitation cycle and their subsequent erosion potential.

2.3.3 Cavity Interaction with Suspended Solid Particles

This section deals with the suspension of solid non-deformable particles that are of the same

order of magnitude as the considered cavity. In this thesis, this will focus on bodies of less

than a millimetre in diameter. In terms of real systems, these particles may be ingested

foreign particulates, recirculated previously eroded material or an analogy to second liquid

phase such as emulsified oils. In any case, neither the origin of these solids is considered

further nor is its chemical effect on the cavitating fluid.

We also consider these particles to move in the fluid stream with the cavity such that there is

no significant initial relative motion between the two. Any motion by the particle is caused

by the pressure differential across it due to cavity’s alteration of the domain flow field. This

is an important point as we would normally assume a no-slip condition to exist at the solid

body-liquid interface, as described in standard fluid dynamic analysis and boundary layer

theory. Notably, the no-slip condition cannot be enforced using potential theory, which

forms the basis for the mathematical model used in this thesis. Therefore, no viscous forces

are considered in the model at all. Normal velocities are maintained across interface

boundaries, however tangential velocities have no constraint, and shear forces within the

liquid domain are non-existent. We justify these conditions imposed by the use of the

2-17

potential model mainly due to the fast time scales involved, and the consequent dominance

of inertial forces over the viscous forces (i.e., the high Reynolds number).

Intuitively, if the solid particle is allowed to move and there is no dampening due to the

neglect of viscous forces, the solid would be expected to impact on the cavity in a similar

fashion to a movable boundary described in the fluid structure interaction section. The

actual effect of the solid particle’s presence shall be discussed in a separate chapter as it

constitutes this thesis’ contribution to the understanding of cavitation mechanics.

2.3.4 Cavity Interaction with Suspended Liquid Particles

The rationale of introducing liquid particles is to simulate the presence of an emulsified

phase on the cavity, exemplified in fire-retardant hydraulic fluids. The consideration of

liquid particles deals only with an encapsulated second liquid phase with some interfacial

tension. Electric double layers are not considered, thus the attractive/repulsive factors and

emulsifier chemistry does not impact on any coalescence or separation observed by a liquid

drop. However, one can consider the emulsifier to have some impact, namely on the value

of the interfacial tension (Friberg 1992).

It is beyond the scope of this thesis to deal with any case other than a highly dispersed liquid

in continuous phase. Thus, relatively few drops with fairly large separation distances are

considered. Additionally, three phase emulsions are not directly considered as in the case of

solid or crystalline material migrating to the liquid particle interface. However, one may

model the effect of third phase at the interface through inclusion of an inertial barrier.

As the surface is deformable, the interfacial tension acts primarily as a restoring agent,

attempting to minimise the drop’s surface energy and return the drop to a spherical shape

(Gopal 1967). As with solid bodies, only inertial terms are considered. Therefore, the only

parameters of importance of the liquid drop on the model are its interfacial tension, density,

and relative size of the drop to the cavity.

The size of the liquid drops is of the same order as the cavity. In real terms this applies to

liquid drops of less than 0.3 of a millimetre in diameter, which is a typical, albeit at the upper

limit, of emulsions (Gopal 1967).

Unfortunately, the limitations of the current model prevent the modelling of a deformable

body. As a consequence, a solid body must be substituted for an emulsion drop which

reduces this study to that of the effect of solid particles on cavitation bubbles. However, the

2-18

interest into oil-in-water emulsion droplets can be studied in the sense that the density of the

droplet can be varied from 0.7 – 1.0 relative density (i.e., approximate oil densities).

2.4 Pressure at the Rigid Boundary

The only damage mechanisms considered in this thesis are those that are a direct result of the

collapse of a vapour cavity. Any physio-chemical effects are ignored, therefore the ultimate

defining quotient for erosion damage is the pressure applied to the surface of the rigid

boundaries. As mentioned previously, while it was previously proposed that the re-entrant

jet’s water-hammer pressure may be responsible for damaging the solid material, Philipp and

Lauterborn (1998) have shown that this not the case, even considering the interaction of

multiple cavities.

2.4.1 Damage Mechanisms

In their study, Philipp and Lauterborn (1998) focused a laser in water to create a single

cavity. Their experimental set-up allowed the creation of identical vapour bubbles such that

the process could be observed with a temporal separation of microseconds through delay of a

flash-lamp/camera mechanism. In addition a collimated light source allowed the capture of

compression effects in the liquid, thus demonstrating the presence of pressure pulses or

acoustic waves in the water.

The objective of the experiment was to prove whether the jet or the pressure pulse was

responsible for the indentation and erosion of a metal specimen. It was proposed that if a jet

was responsible for damaging the solid boundary, then any indentation should be limited to a

circular region directly below the origin of the cavity. As this is the region where the re-

entrant jet penetrates the bubbles, it is clear that this is also the region of greatest water-

hammer pressure.

This phenomenon was not observed. Rather, the damage of a single cavity, in close

proximity to the surface, was limited to an annulus of the same approximate size of the

rebounding toroid. In cases where damage was observed in the central region, there was also

indentation at a second surrounding annular region. The damage to the central region

appeared to be due to the first near-singular (collapse to a point) collapse, while the outer

region was due to a second near-toroidal collapse. In all cases, the damage appeared to be

due to the emission of a high-pressure shock wave at rebound.

2-19

Furthermore, calculations carried out by Philipp and Lauterborn (1998) demonstrate that

even for the cases with fastest micro-jets of 200 m/s and maximum cavity diameter of 1.3

mm, the water-hammer pressure is still below the yield stress of the softest of metal

specimens used, namely 99.9% pure aluminium. This is in contrast to the calculated

pressures at rebound of the cavities, which ranged from 1-9 GPa. While the liquid separating

the cavity from the surface dissipates the peak pressure, tests conducted at distances of less

than two times the maximum cavity radius produced cavities that would eventually rebound

at the solid material’s surface. Hence, pressures very close to the rebound pressure would be

experienced by the material surface.

Now that the damage mechanism of the cavity has been identified for metal specimens, it is

clear that the rebound pressure should be the indicator of a simulated cavity’s erosion

potential.

2.4.2 Damage to the Rigid Material

Even though the high pressures at re-bound are considered to be predominantly responsible

for the erosion of the rigid material, it should be noted that cavitation erosion has properties

indicating a fatigue process (Benjamin and Ellis 1966; Pearsall 1972). The exception is a

system that undergoes catastrophic cavitation where the damage to components is extremely

fast. In the case of catastrophic erosion, the cavitation pressures clearly exceed the strength

of the material.

However, if a fatigue mechanism is assumed to be at work, it is possible for less severely

cavitating systems to produce rigid boundary pressures that do not exceed the flow stress of

the material, yet do produce stresses capable of propagating cracks either from surface or

sub-surface flaws.

This is significant in that we must now consider not only the flow stress of a material for the

compressive surface pressures, but also the shear stress produced below the surface and the

fatigue limit of the material. While it is beyond the scope of this thesis to completely

analyse the solid mechanics of this problem, these issues have been raised so that pressures

that fall below the yield strength of a material are not discarded prematurely. Even if a

cavity is unable to damage a completely homogeneous solid, it is quite probable that it can

have a significant effect once fracture analysis and material flaws are introduced to the

problem.

In this regard, it may even be possible for the re-entrant jet to play a direct role in the damage

to the solid boundary, albeit in only extreme situations. Regardless, the determination of the

2-20

pressure at which erosion takes place remains a solid mechanics and fracture analysis

problem, and already illuminates some of the possible directions for further work.

2.5 Simplifications

The purpose of this chapter was to give a basic understanding of the cavitation erosion

mechanism. Clearly this problem is complicated, and to successfully simulate this

phenomenon certain assumptions and simplifications of the problem must be made. These

are based upon the type of cavitating systems considered in this study, the nature of the

domain and the limitations of BEM cavitation modelling.

2.5.1 Time and Size Scales

Perhaps the most significant of the observations made, of cavitation in hydraulic systems, is

the short time scales involved in the process. The rapid growth and collapse of a cavity

makes the process one of a high Reynolds number, and as such, inertial terms dominate the

system. As a consequence, viscous forces are deemed negligible and are ignored, which is

significant, as the neglect of viscous terms is required for the use of potential theory.

A fast time scale has further impact as during growth and collapse, it is assumed that liquid

is instantaneously converted to vapour and vice versa. Due to the high speed involved,

diffusion of dissolved gas is considered to be insignificant, thus any non-condensable gases

are considered to originate from the nucleus of the cavity, thus the mass of these gases is

assumed to remain constant. As it is these gases that are responsible for the rebound, any

change in the non-condensable gas content would affect the point that the cavity rebounds.

As this thesis is primarily concerned with hydraulic systems, the size of the individual

cavities is considered to be relatively small (less than a millimetre). This couples with the

short time-scales to make gravity insignificant on the life of the cavity. However, unlike

large-scale problems, surface tension plays a potentially significant role due to the high

curvatures involved at rebound.

2.5.2 Damage Potential

Ideally, the damage potential would be measured by calculating the pressure at the rigid

boundary when the cavity rebounds. While it is acknowledged that the actual pressure where

damage occurs is dependent on the solid mechanics of the material in question, it is assumed

that damage will be proportional to pressure applied to the surface. Furthermore, if fatigue

2-21

mechanisms are considered, then the relatively low water-hammer pressures due to the re-

entrant jet are potentially damaging. Thus, pressure at the rigid boundary surface should be

monitored throughout the cavity lifecycle.

In addition, as this thesis looks specifically at cavitation in hydraulic systems, it is assumed

that the rigid boundary remains stationary throughout the lifecycle of the cavity. This then

makes it a simple matter to determine the pressure at the boundary at all times during the

simulation.

Finally, the unstable nature of modelling of the toroidal cavity collapse has prevented the

capture of toroidal rebound in this thesis. While some success has been enjoyed with regard

to continuing the cavity model beyond touchdown, the actual extent of this tends to be

problem dependent. The problematic nature of the post-touchdown phase provided the

impetus to establish a common point from which to compare separate cavities.

Consequently, data from the simulations contained in this thesis are presented only up to

touchdown. Furthermore total mechanical energy of the cavity is considered as the best

indicator of damage potential, as it is the mechanical energy of the cavity that must be

converted to potential energy during the compression of the non-condensable gas. This does

not, however, consider the point at which rebound occurs with respect to the cavity’s

proximity to the rigid boundary. The limitations of this indicator shall be discussed in a later

chapter.

2.6 Summary

In this chapter, the lifecycle of a cavity has been discussed. This can be broken down into a

number of stages including:

1. Growth from an initial cavity nucleus

2. Initial collapse

3. Development of the re-entrant jet

4. Touchdown

5. Development of a toroidal cavity

6. Rebound

It has also been demonstrated that the surroundings of the cavity can have a significant

impact on its lifecycle and ultimately the point at which the cavity rebounds. Furthermore, it

is the rebound of the cavity that is responsible for the maximum pressures experienced by the

domain. However, the inability to successfully model the toroidal rebound forces the use of

total kinetic energy at touchdown as a quantitative damage potential indicator.

2-22

In addition, the high-speed nature of the cavitation process investigated in this thesis allows

the neglect of:

• viscous forces

• effects of gravity

• mass diffusion

Furthermore, despite that fact that only relatively small bubbles are considered, it is

acknowledged that surface tension plays a negligible role until very late in the collapse (i.e.,

as the bubble approaches rebound).

Finally, as this thesis is limited to the effect that suspended bodies have on cavitation, two

main cases are of primary consideration. They are:

1. Stationary solid bodies or those of relatively high density

2. Suspended solid bodies of relatively low density that are permitted to move

thereby providing an analogy to liquid oil droplets

3-23

Chapter 3

Mathematical Formulation

3.1 Introduction

With the advent of digital computers, it has become possible to simulate a wide variety of

engineering problems. The most popular technique employed is a finite element/volume

analysis (FEA), which has proved successful in modelling a number of continuum mechanics

problems (Segerlind 1984). Generally, this method involves the discretisation of a relevant

domain, including the surface and interior, and the subsequent solution of linear equation

sets derived from the governing differential equations. This method is used with great

success in a wide number of engineering problems, especially with large-scale systems.

However, the discretisation of the entire domain can make this technique inappropriate for

analysing multi-phase flow, especially if the resolution of the phase-boundary is critical to

the solution (Beer and Watson 1992).

An alternative method to FEA is the boundary integral/element method (BEM) (Beer and

Watson 1992). This technique enjoys less popularity than FEA, mainly due to the constraint

that the fundamental solution must be known for the domain, which is not required for FEA

(Cartwright 2001). Furthermore, BEM schemes discretise only the boundary surfaces of the

relevant domain, and then solve the set of equations derived from the governing integral

equations. For the case of this thesis, potential theory is used such that the derivative of the

potential is determined from a known potential (or vice versa) using Green’s equation. As

the fundamental solution is known for all points in the domain, it is possible to then use the

resultant solution to determine the interior values, the details of which shall be discussed in a

later chapter.

3-24

BEM resolves the phase-boundary better than a FEA approach due to the collocation points

lying on the phase boundary. This is of great benefit for this project as it is the motion of the

cavity boundary, and the resultant pressure at the solid boundary surface, that is of primary

interest.

Much research has been conducted in the application of the BEM to cavitation simulation.

This section briefly describes the basis for the mathematical formulation to give background

to the assumptions made and the interrelated effect on the mathematical modelling. Once the

governing equations have been established, it will then be shown how Green’s Integral

Formula is put in terms of an axisymmetric half-space.

3.2 Theoretical Basis

With any simulation the physical principles must be expressed in an appropriate form. Here

the fundamental principles of fluid mechanics are listed, specifically with respect to the

BEM. This is given in order to define the mathematical impact of using potential flow

theory to model the growth and collapse of vapour cavities, beyond the ad hoc listing of the

assumptions given in Chapter 2.

3.2.1 Conservation of Mass

If we consider an arbitrary volume Ω, there must exist a surface S that encloses this volume.

We can then define the density ρ, at a point x and time t. Thus the mass enclosed by the

surface is expressed as:

∫Ω Ω= dρmass Equation 3.1

The net rate of mass flowing across the surface in the outward direction is then:

∫−= SdSnu ˆ.flow mass ρ Equation 3.2

where u is the velocity field and n is the unit outward vector normal to the surface S.

Conservation of mass yields (neglecting source terms):

( )∫∫∫ ΩΩΩ∇−=−=Ω ddSd

dtd

Sunu ρρρ .. Equation 3.3

3-25

Furthermore, by taking differentials under the integrands, the Equation of Continuity is

provided:

( ) 0. =Ω

∇+∂∂

∫Ω dt

uρρ Equation 3.4

which is true for all points within an arbitrary volume Ω (Brebbia and Walker 1980). In the

case of an incompressible fluid (where ρ is constant), ∇.u must equal 0, for the above to be

true (Power and Wrobel 1995).

3.2.2 Conservation of Momentum

Given that momentum can be expressed by:

∫Ω Ωdρu Equation 3.5

Through use of the material derivative, the rate of change of momentum can be expressed as:

( )∫ ∫∫ Ω ΩΩ

Ω∇+Ω=Ω ddDt

Dddtd ..uuuu ρρρ Equation 3.6

which reduces to the following:

∫∫ ΩΩΩ=Ω d

DtDd

dtd ρρ uu Equation 3.7

Through the expansion of the first right side integral and use of the mass continuity equation,

Equation (3.4).

By including a volume vector force per unit mass F, the total force due to body forces is:

∫Ω ΩdρF Equation 3.8

Surface forces imposed by the surrounding continuum can be written as:

∫∫ ΩΩ

∂

∂= d

xdSn

j

ij

S jij

σσ Equation 3.9

which is written in tensor notation where i,j = 1,2,3, and σij is a stress tensor. Using the

identities above, we can describe the momentum balance as (Power and Wrobel 1995):

3-26

∫ ∫∫ Ω ΩΩΩ

∂

∂+Ω=Ω d

xdFd

DtDu

j

iji

i σρρ Equation 3.10

which is termed the Equation of Motion.

Since Ω is arbitrary, we obtain:

j

iji

i

xF

DtDu

∂

∂+=

σρρ Equation 3.11

3.2.3 Pressure and Stress States

A fluid at rest has equal principal stresses with one third of the normal stresses equal to the

hydrostatic pressure. Naturally, the hydrostatic pressure is a function of position x. That is:

ijij pδσ −= Equation 3.12

3iip

σ−= Equation 3.13

where δij is the Kronecker delta, requiring δij = 1 when i = j, and δij = 0 when i ≠ j.

The above is not true for a moving fluid, however. For convenience, the stress tensor for a

fluid in motion is written as the sum of the isotropic and viscous components. Thus we

introduce a deviatoric stress tensor to represent changes due to a flowing continuum, thus:

ijijij dp +−= δσ Equation 3.14

where the deviatoric stress tensor dij (for a Newtonian fluid) can be expressed as:

( )ijijij ed δµ ∆−= 312 Equation 3.15

where µ = dynamic viscosity

eij = rate of strain

and 0. =∇=∆ u due to incompressibility.

Furthermore:

3-27

∂

∂+

∂∂

=i

j

j

iij x

uxu

e 21 Equation 3.16

and it should be noted that ∆=iie

Thus, the total stress tensor can be written as (Power and Wrobel 1995):

( )ijijijij ep δµδσ ∆−+−= 312 Equation 3.17

which, due to incompressibility, reduces to:

ijijij ep µδσ 2+−=

3.2.4 Navier-Stokes Equation

We now combine the result above into the momentum balance equation (Power and Wrobel

1995), while assuming µ is constant within our domain, such that:

∂∆∂

−∂

∂+

∂∂

−=ii

i

ii

i

xxu

xpF

DtDu

31

2

2

µρρ Equation 3.18

If the fluid is incompressible such that ∇.u = 0, and consequently ∆ = 0, the above reduces

to:

2

2

i

i

ii

i

xu

xpF

DtDu

∂

∂+

∂∂

−= µρρ Equation 3.19

or in vector notation:

uFu 2∇+∇−= µρρ pDtD

Equation 3.20

It can be seen that the assumption of incompressibility is far reaching, and is defined by the

density of the fluid remaining constant with respect to any change in pressure. In physical

terms this means that any flow velocity is significantly lower than the local speed of sound

(O'Neill and Chorlton 1986).

With the Navier-Stokes equation and the conservation equations, the set of equations is now

sufficient to determine velocity and pressure throughout any domain. We can use the

conservation equations and place the domain within a gravity field, as is typical to most fluid

systems. This leads to:

3-28

ugu 21∇+∇−= νp

DtD

ρ Equation 3.21

where ν = kinematic viscosity (µ/ρ)

g = acceleration due to gravity

Recalling incompressibility, gravity can then be balanced by pressure forces, ρg.x. Thus:

Ppp ++= xg.0 ρ Equation 3.22

where P is a modification due to the motion of the fluid

p0 is constant

The pressure modification, P, due to fluid motion is determined by the equation:

uu 2∇+∇−= µρ PDtD

Equation 3.23

The absence of gravity from Equation (3.23) indicates that gravity has no effect on the

velocity distribution, provided gravity is also absent from the boundary conditions (Power

and Wrobel 1995).

As was discussed in Chapter 2, gravity is neglected hence it is omitted from the boundary

conditions. For example, if a system involving a free surface was considered, one must use

the expression containing g. However, when considering systems where bubble radii are

small and the time scales are short (such as those in this thesis), gravity may be omitted as it

plays a negligible role.

3.2.5 Governing Equations

As is common practice, dimensionless parameters are used such that calculation results are

provided in terms of dimensionless units. Effects of scale are introduced by calculating

system properties, such as surface tension coefficients, in terms of these dimensionless

parameters. To achieve this we assume representative units for length and velocity, L and U

respectively, such that velocity, displacement, time and pressure can be respectively

expressed as:

3-29

20,,,

UPP

PandLUtt

LU ρ−

=′=′=′=′xxuu Equation 3.24

where P0 is a representative value of the modified pressure in the fluid (Power and Wrobel

1995).

REYNOLDS NUMBER

Using Reynolds Number, µ

ρLU=Re , we can write the continuity and motion equations in

dimensionless form (note that ‘ indicates a dimensionless parameter), such that:

,0. =′′∇ u Equation 3.25

and

′′∇+′′∇−=′

′∇′+

′∂

′∂ uuuu 2

Re1. P

t Equation 3.26

Generally, we can think of the Reynolds Number as the ratio of importance of viscous and

non-viscous forces. As pressure normally plays a passive role, it is customary to characterise

the flow as follows:

′′∇

′

′

=∇ u

u

u

u

22Re

tDD

DtD

µ

ρ Equation 3.27

Thus, the balance between the three terms in the equation of motion, Equation (3.10), can be

indicated by the ratio of the two terms in Equation (3.27). It is the relative ratio of the terms

above that designates which flow regimes are at work.

INVISCID FLOW

For the potential inviscid flow problems of cavitation, the Reynolds Number tends to be

large compared to unity, and therefore viscous forces can be neglected over the entire flow

field except for the boundary layers (Power and Wrobel 1995).

In the case where boundary layers do not separate from the rigid boundaries, viscous forces

are present in thin layers of the flow field and do not have significant impact on the overall

3-30

flow. Alternatively, where separation does occur, the limiting form of the flow is not the

same as completely inviscid flow. As a result, boundary layers must be assumed not to

separate when considering the collapse of a cavity; consequently we assume completely

inviscid flow.

As mentioned previously, we assume the fluid surrounding the cavity is incompressible,

which leads to a simplification of the equation of motion:

pDtD

∇−=ρ1gu

Equation 3.28

otherwise known as Euler’s equation.

However, the no-slip condition cannot be satisfied by Euler’s equation, as it is of one order

lower than the Navier-Stokes equation (due to absence of µ). As a result, the condition of

continuous tangential velocity component is relaxed, while the normal component of the

velocity field is still considered continuous. That means that the condition of no-slip can be

physically interpreted as the solid boundary becoming a streamline. Note that due to its

dynamic nature, the cavity interface is not a stream line.

A further implication of an inviscid flow, is that of irrotationality. Forces imposed by

adjacent elements of the domain depend on the deformation of the fluid flow. It is therefore

necessary to analyse the characteristics of the fluid flow. If we consider velocity as a

function of position and time, ( )t,xu , then the neighbouring point at rx + has velocity

uu ∂+ , where j

iji x

uru∂∂

=∂ . The geometrical character of the relative velocity can be

regarded as linear function of r, which leads to the decomposition of j

i

xu∂∂

into symmetrical

(superscript s) and anti-symmetrical (superscript a) parts (Power and Wrobel 1995). Thus:

ai

sii uuu δδδ += Equation 3.29

We can consider the symmetrical component as pure straining motion and a function of the

rate of strain tensor eij, such that:

ijjsi eru =δ Equation 3.30

and similarly, for the anti-symmetrical component:

3-31

ijjai ru ξδ = Equation 3.31

where

∂

∂−

∂∂

=i

j

j

iij x

uxu

21ξ Equation 3.32

ξij is the anti-symmetric stress tensor with only three independent components and can be

written as:

kijkij ωεξ 21−= Equation 3.33

Hence,

kjijkai ru ωεδ 2

1−= Equation 3.34

We can visualise δua as the velocity produced at a position, r, relative to a point about which

there is rigid body rotation of angular velocity ω/2. This can be written explicitly as:

2

1

1

23

1

3

3

12

3

2

2

31

xu

xu

xu

xu

xu

xu

∂∂

−∂∂

=

∂∂

−∂∂

=

∂∂

−∂∂

=

ω

ω

ω

Equation 3.35

or, in vector notation:

u×∇=ω Equation 3.36

However, for irrotational flow the curl (ω) is zero. Thus, the cavity flow can be considered

irrotational, inviscid and incompressible, which is often termed as solenoidal. A velocity

potential, φ, which satisfies Laplaces Equation is now introduced, such that:

0. =∇ u Equation 3.37

0u =×∇ Equation 3.38

φ∇=u Equation 3.39

thus 02 =∇ φ Equation 3.40

3-32

Although the equation of motion is non-linear in u, the restrictions of irrotationality and

mass conservation allows the velocity distribution to be completely determined by linear

equation sets. Therefore, the non-linear equations of motion are only used to calculate the

pressure after the velocity distribution has been determined (Power and Wrobel 1995). This

statement is justified through use of Kelvin’s Theorem, which states that:

For an inviscid fluid in which density is constant, or in which the pressure depends

on the density alone, and for which any forces that exist are conservative, the vorticity of

each particle is preserved. (Power and Wrobel 1995)

We can now write the equation of motion as:

uxguu 2221 . ∇+

−+∇−=×−

∂∂ νpq

t ρω Equation 3.41

through use of the vector identity:

( ) ( ) ( ) ( ) ( ) FGGFFGGFGF ×∇×+×∇×+∇+∇=∇ ... Equation 3.42

where F and G are any two vectors and q2=u.u

Thus,

( ) ( ) ω×+∇=∇ uuuuu ..21 Equation 3.43

which is substituted into the equation of motion that then simplifies to the previous identity.

Considering the solenoidal nature of the flow field (due to the viscous term in Equation

(3.41) being neglected), this identity further reduces to:

0.221 =

−++

∂∂

∇ xgρ

φ pqt

Equation 3.44

Integrating then gives:

constant.221 =−++

∂∂ xg

ρφ pqt

Equation 3.45

which is identified as Bernoulli’s equation. Recalling the quiescent nature of the domain at

infinity allows us to determine the constant, which tends to the pressure at infinity. Thus, the

equation can be written for the cavity surface as:

3-33

221 uρφρ +

∂∂

=−∞ tpp b Equation 3.46

In this case, buoyancy effects are neglected (due to speed of growth and collapse), p∞ is the

integral constant (the ambient fluid pressure), and pb is the pressure inside the bubble. This

ultimately assumes that the cavity originates from a small nucleus and grows to many times

its initial volume. This growth phase is sufficiently rapid such that mass-diffusion effects of

solute gases are negligible, thus the bubble is composed predominantly of liquid vapour and

any soluble gas content is taken into account only late in the collapse phase.

3.2.6 Non-condensable Gas Content and Surface Tension

The presence of non-condensable gas and surface tension forces affects the pressure inside

the cavity. To reflect this, Bernoulli’s equation written for the cavity surface is modified to:

221

0

uρφρσκγ

+∂∂

=−

−−∞ tV

Vppp gv Equation 3.47

where pv = saturation vapour pressure

pg = initial pressure due to non-condensable gas

V = bubble volume

V0 = initial bubble volume

σ = surface tension coefficient

κ = local curvature

γ = specific heat ratio of the non-condensable gas

noting that the compression of the non-condensable gas content is adiabatic.

3.3 Integral Equations

Most of the cavity simulations that employ a boundary integral approach are based on

Greens formula. For a sufficiently smooth function φ, which satisfies Laplace’s equation in

a domain Ω, having a smooth piecewise surface S, Green’s integral formula can be

reformulated into a Fredholm problem and can be written as (Taib 1985):

( ) ( ) ( ) ( )( )∫ ∫ −∂∂

=

−∂∂

+S S

dSdScqp

qnqpn

qpp 11 φφφ Equation 3.48

where p ∈ Ω and S

q ∈ S

3-34

∂/∂n = derivative with respect to the outward normal from S

and ( )S if ,2

if ,4∈Ω∈

=pp

pππ

c Equation 3.49

If p is on S, then Equation (3.48) yields a relation for either φ or ∂φ/∂n on S, if the other is

specified. Once both are known, φ can be generated at any point within the domain (Beer

and Watson 1992). In the axisymmetric case, φ and ∂φ/∂n are independent of the azimuthal

angle and analytical integration of this variable can be performed.

Using cylindrical coordinates, with p = [r0, 0, z0], and parameterising the surface by the

variable ξ, q = [r(ξ), θ, z(ξ)], the inverse of the distance between p and q can be written as:

( )( ) ( )( ) ( )( )[ ]

( )( ) ( )( ) ( )[ ]2122

02

02

0

21

20

220

cos4

1sincos

11

θξξξ

ξθξθξ

rrzzrr

zzrrr

−−++=

−++−=

− qp Equation 3.50

q

p

z

r

θ

Figure 3.1. Coordinate System

Furthermore, by integrating (Taib 1985),

3-35

( )( )( ) ( )( ) ( )[ ]

( )

( )( ) ( )( )[ ] ( )( )( ) ( )( )

∫∫

∫ ∫ ∫

−++−

−++

+

=

−−++

+

=

−

π

θ

π

θ

ξξξ

θ

ξξ

ξξξξ

ξξξ

θξξ

ξξ

2

0 21

20

20

22

0

1

0 21

20

20

21

22

1

0

2

0 21

22

02

02

0

21

22

cos41

cos4

1

zzrrrr

d

zzrr

ddr

ddzrd

rrzzrr

dddr

ddzrddS

S qp

Equation 3.51

by defining ( ) ( )( )( ) ( )( )2

02

0

02 4zzrr

rrk

−++=

ξξξ

ξ Equation 3.52

The integral with respect to θ becomes

[ ] ( ) ( )( )kK

k

d

k

d

k

d 4sin1

4cos1

4cos1

2

0 21

22

2

0

2

0 21

2221

222

∫∫ ∫ =−

=−

=−

ππ

π

θ α

α

α

αθ

Equation 3.53

where K(k) is the complete elliptic integral of the first kind. Equation (3.51) becomes (Taib

1985):

( ) ( )

( )( ) ( )( )[ ]∫∫−++

+

=−

1

0 21

20

20

21

22

41

zzrr

kKddr

ddzr

ddSS ξξ

ξξξ

ξqp

Equation 3.54

Noting that:

nn

.φφ∇=

∂∂

Equation 3.55

If we express the unit normal outward to the domain as:

3-36

21

22

,sin,cos

+

−

=

ξξ

ξθ

ξθ

ξ

ddr

ddz

ddr

ddz

ddz

n) Equation 3.56

and write that,

( ) ( ) ( )( )

( )( ) ( ) ( )( )[ ]23202

20

20

00

cos4

,sin,cos1

zzrrrr

zzrrr

−+−+

−−−=

−∇

ξξξ

ξθξθξ

θqp Equation 3.57

such that (Taib 1985),

( ) ( ) ( )( )

( )( ) ( ) ( )( )[ ]( )

( )( ) ( ) ( )( )[ ]∫ ∫

∫

∫ ∫

−−−+−

−−−+

−−−=

−∂∂

1

0

2

0 23

202

20

20

0

2

0 23

202

20

20

1

00

cos4

cos

cos4

1

π

θ

π

θ

ξξξ

θθ

ξξξ

ξξξ

θ

ξξ

ξξξξ

zzrrrr

rd

ddzrd

zzrrrr

d

ddrzz

ddzrrddS

S qpn

Equation 3.58

using Equation (3.52) and substituting into Equation (3.58) we find:

( ) ( ) ( )( )

( )( ) ( )( )[ ] ( )( )

( )( ) ( )( )[ ] ( )∫∫

∫ ∫∫

−−++−

−−++

−−

−=

−∂∂

π

θ

π

θ

θθ

ξξξ

ξξ

θ

ξξ

ξξ

ξξξξ

2

0 23

222

1

0 23

20

20

0

1

0

2

0 23

2222

32

02

0

0

cos1

cos

cos1

1

k

d

zzrr

rddzrd

k

d

zzrr

ddrzz

ddzrrd

dSS qpn

Equation 3.59

The first θ integral of Equation (3.59) can be written as (Taib 1985):

( ) ( ) ( )( )kk

k

d

k

d

k

d ,4sin1

4cos1

4cos1

2

0 23

22

2

0 23

22

2

0 23

222

Π=−

=−

=−

∫∫∫ππ

π

θ α

α

α

αθ

Equation 3.60

3-37

And the second θ integral of Equation (3.59) can be written as (Taib 1985):

( )( )( )

( )( ) ( )

( ) ( )( ) ( )kK

kkk

k

k

dkk

dk

k

dkk

dkk

k

d

k

d

22

2

0 21

222

2

0 23

222

2

0 23

222

2

0 23

22

22

2

2

0 23

22

22

0 23

222

8,48sin1

8

sin148

cos148

cos1

1cos8

cos1

1cos24cos1

cos

−Π

−=

−+

−

−=

−

−+

−

−=

−

−=

−

∫∫

∫∫

∫∫

ππ

ππ

ππ

θ

η

η

η

η

η

η

η

ηη

η

ηηθθ

Equation 3.61

where Π(k,k) is the complete elliptic integral of the third kind and can be written as:

( ) ( )21

,kkEkk

−=Π Equation 3.62

with E(k) the complete elliptic integral of the second kind. We can now write Equation

(3.59) as (Taib 1985):

( )

( )( ) ( )( )[ ]( )( ) ( )( ) ( )

( )( ) ( ) ( )

+−

−−−+

−++−=

−∂∂

∫ ∫

kKrddz

kkkEr

ddz

kzz

ddrrr

ddz

zzrr

rddSS

0220200

1

0 23

20

20

21

2

41

ξξξξξξ

ξξ

ξ

ξξ

ξξqpn

Equation 3.63

Ordinarily, elliptic K and elliptic E can be calculated as complete functions. However,

inspection of the elliptic terms shows that there exists a singularity as p approaches q. In

these cases it is necessary to decompose the elliptic functions into singular and non-singular

components in order for them to be accurately calculated. The method used to separate the

singular terms is discussed in the following chapter.

We can now approximate K(k) and E(k) as polynomials in the following:

( ) ( ) ( ) ( )xxQxPkK ln−= Equation 3.64

( ) ( ) ( ) ( )xxSxRkE ln−= Equation 3.65

3-38

where ( )ξ21 kx −= Equation 3.66

and the polynomials P,Q,R and S are tabulated in Hastings (1955).

3.4 Special Considerations

When collocating the surface of cavitation bubbles, it is generally assumed that the potential

is initially known; hence the normal velocity of the vapour-liquid interface is determined at

each time step. Once the presence of rigid boundaries, suspended particles and free surfaces

are introduced, it is clearly necessary to collocate these surfaces as well. The same

collocation method is used; however the assumed known and unknown variables are not

necessarily the same as for cavities.

In addition, any deviation from a perfectly infinite domain will result in non-spherical

collapse and the formation of a toroidal bubble toward the end of the collapse phase. This

complicates the system further by introducing a vortex sheet after touchdown, which requires

modification to the governing equations.

3.4.1 Rigid Boundaries

The presence of a semi-infinite rigid boundary can be accommodated in at least two different

ways. The first method is to consider the plane in which the rigid boundary lies to be a

mirror plane that reflects the domain elements. The presence of these reflected domain

constituents can be implemented directly by adding them at the initial problem formulation,

effectively doubling the domain members, or indirectly by including the reflection in the

calculation of the Green’s Equation kernels. In the indirect case, the modified Green’s

kernel becomes:

qpqpqp ′−+

−→

−111

Equation 3.67

where p = (r0, 0, z0)

q = (r, θ, z)

q’ = (r, θ, -z)

The second method is to create a surface where the rigid boundary lies and collocate in the

usual manner. This method is potentially less accurate than modifying the kernel, but does

have the benefit of allowing dynamic deformation of the surface and can reduce the number

of numerical operations in some cases. While it is possible to assume potential at the

surface, it is usually more meaningful to assume a normal velocity. For completely rigid

3-39

cases, this velocity is simply zero at all times. Otherwise, a normal velocity can be

calculated based on the pressure at the surface. In any case, velocity potential can be found

at each time step. It is then possible to use finite difference techniques to find the temporal

derivative of the velocity potential, and by rearranging Bernoulli’s equation, find the

pressure applied to the surface:

221 uρφ

−∂∂

−= ∞ tpp Equation 3.68

3.4.2 Suspended Particles

The modelling of suspended particles is similar to that used for treating a rigid boundary in

that the initial normal velocity is assumed. The potential at each time step is then determined

which leads to the calculation of the temporal derivative of potential so that the pressure may

be calculated. The motion of the particle is then resolved by finding the force, and hence

acceleration, applied to the body for a short time interval.

PRESSURE INTEGRATION

In order to find the resultant force applied to the body, the pressure applied to the surface of

the particle is integrated over the surface.

∫=S

dSpnF ) Equation 3.69

where F = force vector applied to the body

p = pressure at surface

n) = unit normal to surface

S = surface

Simple kinematics can then be applied to find the motion of the body, allowing it to be

stepped in time in a fashion similar to the cavity.

While the solid body is held stationary, the collocation points on the body surface are also

stationary (i.e., 0=∂∂

nφ

). Consequently, any numerical derivatives of potential with respect

to time determined at these points are in fact t∂

∂φ. Hence, the pressure in the fluid at the

solid body’s surface can be expressed as:

2

21

s∂∂

−∂∂

−= ∞φρφρ

tpp Equation 3.70

3-40

where s∂

∂φ = is the partial derivative of potential with respect to a

vector tangent to the surface.

When considering points that lie on the surface of the cavity, any numerical derivatives of

potential with respect to time determined at these points are in fact DtDφ

as these points are

considered Lagrangian particles (Blake, Taib et al. 1986). Through use of the definition of

the material derivative, and substituting into the Bernoulli equation, pressure can be

expressed as:

2

21 uρφρ +−= ∞ Dt

Dpp Equation 3.71

However, when considering points describing the surface of a solid body that is in motion,

any numerical derivatives of potential with respect to time determined at these points are

neither t∂

∂φ nor

DtDφ

, and consequently, Equations (3.70) and (3.71) are invalid. To remedy

this, the solid body is considered to act similarly to a compliant wall in that the motion of the

collocation points is purely vertical. This is significant in that the pressure can now be

written as (Duncan and Zhang 1991):

tpp

∆∆

−= ∞φρ Equation 3.72

However, testing demonstrated that the pressure determined at the surface of a solid body

using Equation (3.72) was incongruous with the surrounding domain. As the derivation of

Equation (3.72) was not provided, it is probable that while this approximation may be valid

for the very small deflections in Duncan et al (1991), it is not appropriate for the situation

described in this thesis. Consequently, sets of Lagrangian particles were tracked at the solid

body’s surface where the resultant pressure was then interpolated at the nodes describing the

solid surface. In this regard, the sets of particles tracked at the solid’s surface allowed the

use of Equation (3.71) and circumvented the need to directly calculate the pressure at the

solid surface using the solved potential.

RIGID AND DEFORMABLE PARTICLES

Unfortunately, pressure integration did not allow the modelling of deformable particles. The

modelling of rigid bodies reduces the unknowns by one degree allowing a unique solution to

be found. In the case of a deformable suspension, we must re-write the Greens Equation to

3-41

include the new interface, and write Bernoulli’s equation for the surface of the deformable

body.

This provides a very complicated problem. The interface can be modelled by assuming it to

be similar to a vortex sheet and allowing the existence of a potential jump across the

interface. However, we have no knowledge of the motion of the interface other than that

normal velocities must be conserved across it and the resultant motion must conserve

volume, and consequently mass, of the system. In addition, it is not possible to write

Bernoulli’s equation for the interior of the deformable particle as the constant is unknown

due to the inability to calculate the integral constant. Furthermore, despite knowing that the

pressure at the interface must be matched on either side, it has not been possible to model

anything other than a particle of the same liquid as the surrounding domain. In this case, the

problem reduces to situation where points lying on an imaginary surface are tracked through

time.

3.4.3 Vortex Sheets

As mentioned previously, the modelling of a cavity beyond touchdown is achieved through

the introduction of a vortex sheet. This sheet represents an interface in the domain where

normal velocities are maintained, but tangential velocities are not, which allows a jump in

potential to exist across this interface, and in the case of a toroidal cavity, has the added

benefit on ensuring the domain remains simply connected.

In any case, once a vortex sheet has been introduced, the Greens Equation must be modified

to take this into account. Zhang (1992) has listed this to be:

( ) ( ) ( )( ) ( ) [ ]

( )

∉Ω∈∈

=

−∂∂

−−

−∂∂

−−∂

∂=

+

+

+−+

∪∫∫+

Sor , and Sor ,

,4,2

111

wb

wb

S qqSS q

SSSS

c

dSdScwb

ppp

p

qpnqpnq

qpq

npp

ππ

φφφφφ

Equation 3.73

where Sb = cavity surface

Sw = wall surface

S+ = upper vortex sheet surface with +/- superscript indicating upper/lower vortex

sheet

Equation (3.73) is termed the conventional boundary integral equation (CBIE).

3-42

Inspection of Equation (3.73) shows that while the potential jump is accounted for, it is not

possible to analytically determine the normal velocity of the vortex sheet. To overcome this,

Zhang takes the derivative of Equation (3.73) with respect to the normal vector at p. This is

termed the hyper-singular boundary integral equation (HBIE) and reduces to:

( ) ( ) ( )( ) ( )

[ ]

( )

∉Ω∈∈

=

−∂∂∂

−−

−∂∂∂

−

−∂∂

∂∂

=∂∂

+

+

+−+

∪

∫

∫

+

Sor , and Sor ,

,4,2

1

11

2

2

wb

wb

S qp

qpSS pqp

SSSS

c

dS

dScwb

ppp

p

qpnn

qpnnq

qpnq

nnpp

ππ

φφ

φφφ

Equation 3.74

Equation (3.74) is determined using single and double layer potential theory, the details of

which are documented in Zhang (1992). Note that subscript p and q represent partial

derivatives taken with respect to that point.

The simplest method of determining the motion of the vortex sheet is to use the CBIE and

find the motion of the shear layer through finite difference techniques. By using the CBIE it

is possible to use linear geometric and density functions to attain sufficient accuracy in a

computationally efficient manner as demonstrated by Taib (1985). This leads to a relatively

robust matrix system, when compared to the HBIE. However, quite serious problems arise

when velocities are calculated for the vortex sheet, because the potential calculated close to

the sheet is dependent on the finite step used. In addition, the motion of the cavity close to

the point where the vortex sheet joins the bubble surface is very complicated and tends not to

be well captured using the CBIE. These are well known limitations of the CBIE approach,

and while it has been used to some success by Best et al (1993), it does not appear to match

well with results using the HBIE. Furthermore, it has not been demonstrated that the CBIE

approach for calculating the motion of the vortex sheet is accurate when it comes in close

proximity to a rigid boundary.

The HBIE tends to address most of the negative issues associated with the CBIE, and even

allows for simpler algorithm development in the matrix formulation, as the vortex sheet does

not have to be treated separately during calculation of the influence matrix. However, a

requirement of the HBIE is the use of cubic geometric and density functions as a minimum

due to the third order singularity introduced by the normal differentiation (Zhang 1992).

Furthermore, meticulous attention must be given to the touchdown stage, as perturbations

introduced by the penetration process can make the simulation totally unstable to the point

3-43

where it is no longer solvable. As a result, the time step used must ensure a smooth contact

between the surfaces to control these perturbations as much as possible, thus approximating

the true physical process (Zhang, Duncan et al. 1993).

3.5 Summary

For the purposes of modelling cavitation bubble collapse, the BEM is felt to be inherently

superior to FEA as it resolves the phase boundary. Correct determination of the phase

boundary is critical, as it is the collapse and rebound of the cavity that produces the peak

pressures responsible for erosion of the rigid boundary.

While some general assumptions were made in Chapter 2, the neglect of viscous forces due

to high Reynolds number flow has had significant impact on the model including:

• irrotationality of the fluid

• the notion that boundaries can be thought of as streamlines

• the relaxation of tangential velocity throughout the domain

• and the subsequent inability to transfer shear forces

Coupled with the assumption of incompressibility, it is then possible to use potential theory

to model the domain, where the governing equations are:

φ∇=u

and 02 =∇ φ

Green’s integral formula is then reformulated into a Fredholm problem allowing the solution

of velocity potential or its normal derivative, depending on which is initially known. Once

both have been determined, potential at any point throughout the domain can be found for a

given time period.

For simply connected domains, the CBIE is sufficient to describe the domain. However,

once the bubble become toroidal, leading to a multiply connected domain, it becomes

necessary to use a finite difference method to find the motion of the vortex sheet, or

differentiate the CBIE with respect to the normal at the field point (p), providing the HBIE.

In general, the potential on the cavity surface is normally given, and the normal derivative of

potential given for rigid boundaries (if they are explicitly included) or solid bodies. When

considering solid bodies, the linearised Bernoulli equation provided by Duncan (1991) does

not provide consistent calculations of the surface pressure with respect to the bulk of the

domain. As a consequence, Lagrangian particles are tracked in time to determine the

3-44

pressure at the surface such that a kinematic relation for the motion of the body can be

established.

Now that the fundamental equations and boundary conditions have been established, it is

necessary to develop a numerical scheme that can be implemented into a computer

algorithm. This will be discussed in the next chapter.

4-45

Chapter 4

Numerical Implementation

4.1 Introduction

This section deals with the development of a computationally convenient expression for the

mathematical foundation of cavitation modelling. To model any phenomenon, a set of

determinable linear equations must exist that approximates the governing equations of the

domain. Recalling Equations (3.39) and (3.40) from the previous chapter, the governing

equations for potential flow are:

0 , 2 =∇∇= φφu

It can be shown that the above is true for the reformulated Green’s Equation (again recalled

from Chapter 3), the elements of which were determined for an axisymmetric half-space with

integration about the azimuthal angle, θ (see Figure 3.1), carried out analytically, thus

reducing the numerical integration to only two dimensions.

( ) ( ) ( )( ) ( ) [ ]

( )

∉Ω∈∈

=

−∂∂

−−

−∂∂

−−∂

∂=

+

+

+−+

∪∫∫+

Sor , and ;4Sor ,;2

111

wb

wb

S qqSS q

SSSS

c

dSdScwb

ppp

p

qpnqpnq

qpq

npp

ππ

φφφφφ

Because integration about the axis of symmetry is analytical, it is only necessary to discretise

along a curve represented by the intersection of the relevant surfaces and the plane through

the origin and zero azimuthal angle, which also contains the radial and longitudinal axes.

4-46

n

s

r

z

Origin

Ω

SSouth pole

North pole

Figure 4.1. An axisymmetric coordinate system, representing a cavity surface and rigid

boundary. Note the outward normal (n) and tangential (s) vectors to the surface, S, of the

domain, Ω.

Finally, it is required that there exists some relation between the temporal derivative of

potential and surface velocity, such that the model may be progressed through time. For this,

the Bernoulli equation is used and written for the relevant surfaces. In most cases, this is for

the cavity surface and is recalled below.

221 uρφρ +

∂∂

=−∞ tpp v

4.2 Domain Discretisation

As the domain is represented by a collection of boundary surfaces in the BEM, it is usual to

represent these surfaces by a set of nodes where the relevant known variables are used to

solve for the unknown quantities.

It is usual to write these quantities in terms of dimensionless variables, mostly based upon

the size and lifetime of a spherical bubble. In this case, all coordinates and displacements are

scaled by the maximum radius of a cavity (Rmax) in an infinite domain, while the time is

scaled by the half-life of the same bubble ( ( )vPPR

−∞

ρmax ) in a fluid of density (ρ). The

pressure is then scaled by (P∞ – Pv), where Pv is the vapour pressure of the liquid. Thus the

dimensionless coordinates, time and pressure can be written as:

[ ] ( )

−−

=∞

∞

v

v

PPPPP

Rt

Rz

RrPtzr ,,,',',','

maxmaxmax ρ Equation 4.1

4-47

These quantities must be linked through expressions for the variation between consecutive

nodes or elements. The simplest case is to assign constant values for each element, however

this is usually a poor approximation to the actual domain, thus higher order elements tend to

be used (Brebbia and Walker 1980).

While higher order elements provide inherently better representations of a surface, it is

necessary to balance this with computational time as more numerous lower order boundary

elements may provide similar results and require less computational resources (Power and

Wrobel 1995). Unfortunately, the high problem dependency of cavitation modelling makes

the choice of the representative functions somewhat subjective. This is demonstrated by the

large variation between particular researchers. While linear functions were favoured for the

majority of the modelling work carried out in this thesis, some experimentation was

conducted into the use of quadratic and cubic functions, which shall be addressed later in this

section.

4.2.1 Linear Geometry – Constant Density Functions

In this case, N linear segments approximate the cavity surface and the potential and normal

derivative are held constant for each segment. Thus, the boundary integral equation is

collocated using the midpoint of each segment such that for the CBIE (Taib 1985):

( )∑ ∫ ∑ ∫= = −∂

∂=

−∂∂

+N

j S

N

j S jij

jiji

j j

dSdS1 1

112qpnqpn

φφπφ Equation 4.2

In order to reduce Equation (4.2) to matrix form, we denote n∂

∂ jφ by jψ so that:

∑∑==

=+N

jjij

N

jjiji CD

11

ˆ2 ψφπφ Equation 4.3

where ∫ −=

jS jiij dSC

qp1

Equation 4.4

dSDjS ji

ij ∫

−∂∂

=qpn

1 Equation 4.5

such that Ψ=Φ CD Equation 4.6

4-48

where Dij = ijD + 2πδij

δij = Kronecker delta

4.2.2 Linear Geometry – Linear Density Functions

Here, the segments are parameterised by a variable ξ, in the range (0,1), and φj and ψj are

assumed to have single values at the nodal points (end points of the segment). We can then

use the isoparametric linear shape function:

( ) ξξ −= 11M Equation 4.7

( ) ξξ =2M Equation 4.8

to define ( ) ( ) ( )ξξξ 211 MrMrr jjj ++= Equation 4.9

( ) ( ) ( )ξξξ 211 MzMzz jjj ++= Equation 4.10

( ) ( ) ( )ξφξφξφ 211 MM jjj ++= Equation 4.11

( ) ( ) ( )ξψξψξψ 211 MM jjj ++= Equation 4.12

As the collocation points have been moved to the end point of the segments, we are then left

with N+1 equations for N+1 unknowns. Segment integrals can be written as (Taib 1985):

∫ +=−∂

∂−

jSjijjij

ji

bbdS ψψφ211

1qpn

Equation 4.13

where ( ) ( ) 2,1 ,,

11

0

2

0

=−

= ∫ ∫ kddMSbi

kjkij ξθθξ

ξπ

qp Equation 4.14

Additionally ∫ +=

−∂∂

− jijjijji

aadS φφφ 2111

qpn Equation 4.15

where ( ) ( ) 2,1 ,,

11

0

2

0

=

−∂∂

= ∫ ∫ kddMSai

kjkij ξθθξ

ξπ

qpn Equation 4.16

4-49

4.2.3 Higher Order Elements

Higher order functions can be used to describe the surface geometry, as well as distribute the

potential and normal velocity between the collocation points. This includes various

combinations of parametric equations and spline interpolations.

Early experimentation demonstrated that the benefits of better description of the surface

geometry provided by higher order elements was offset as the model progressed through

time. This was due of problems related to undershoot and overshoot as a result of relative

extremes in curvature. In cases where the cavity surface was close to a boundary; the

element would occasionally overlap this boundary. This was clearly problematic, and often

caused the model to become singular and unsolvable.

Taib (1985) does provide a method using quadratic isoparametric interpolation functions,

however the improved accuracy attained by using these could be achieved by using a greater

number of linear elements, which ultimately required less computation (Taib 1985). As a

consequence, linear elements were preferred when using the CBIE. However, linear

functions were insufficient when using the HBIE to model the cavity.

In the HBIE case, cubic functions were required to ensure the correct solution of the matrix

elements, because of the third order singularity (Zhang, Duncan et al. 1993). The geometry

of each element is interpolated using cubic splines as per standard methods (Burden and

Faires 1997). Zhang (1992, 1993) then used cubic density functions for describing potential

over the bubbles surface, and then linear density functions for normal velocity, as this was

found to be the most stable combination.

While it is unclear exactly what method Zhang (1992, 1993) used to evaluate the matrix

elements, he states that the potential and its tangential derivative were written as functions of

arc length. One possibility is that he continues to use cubic splines to interpolate the cavity

potential. As his thesis looks only at cavities in close proximity to the solid boundary (using

an image bubble to emulate the rigid boundary), there is no need to solve directly for

potential at the rigid boundary or a solid body. Thus, potential and tangential velocity can be

evaluated with each matrix element and immediately assembled into a vector array.

This strategy is unsuitable for including the effects of suspended particles where velocity is

given. In this case, potential must be written in terms of it single value at the collocation

nodes, and the arc length or parameterising variable. To achieve this, isoparametric cubic

equations were used to describe potential, and consequently, the matrix elements were

4-50

rearranged to correspond to linear functions of nodal potential. The isoparametric cubic

equations, where ξ varies from 0 to 1, are given as follows:

( ) ( ) ( ) ( ) ( )ξφξφξφξφξφ 4231211 MMMM jjjjj ++− +++= Equation 4.17

where ( ) 321 6

121

31 ξξξξ −+−=M Equation 4.18

( ) 322 2

1211 ξξξξ +−−=M Equation 4.19

( ) 323 2

1 ξξξξ −+=M Equation 4.20

( ) 34 6

161 ξξξ +−=M Equation 4.21

Continuing with this form, we can also write the tangential derivative of potential in terms of

potential at the collocation points by taking the derivative of potential with respect to the arc

length. Thus:

( ) ( ) ( ) ( ) ( )j

jjjjj

dd

ddM

ddM

ddM

ddM

dd

+++= ++− ssξ

ξξ

φξξ

φξξ

φξξ

φξφ 4

23

121

1

Equation 4.22

where ( ) 21

21

31 ξξ

ξξ

−+−=d

dM Equation 4.23

( ) 22

232

21 ξξ

ξξ

+−−=d

dM Equation 4.24

( ) 23

231 ξξ

ξξ

−+=d

dM Equation 4.25

( ) 24

21

61 ξ

ξξ

+−=d

dM Equation 4.26

and ss 11

1

+− ++=

jjjj SSSddξ

Equation 4.27

4-51

This has a number of benefits. Firstly, because we are dealing with elements between two

nodes there are only two cases where the matrix elements become singular, and consequently

need to be separated. Singularity separation shall be discussed in the Element Integration

section.

Secondly, as both potential and its tangential derivative are in terms of potential at the nodes,

we can combine the CBIE and HBIE schemes by simply adding the respective matrices.

This then leads to a straightforward rearrangement of the matrix elements into the known and

unknown quantities, which shall be discussed in the Matrix Formulation section.

As with the linear elements, the higher order density functions must be incorporated into the

calculation of the individual elements of Cij, Dij, Eij and Fij. Thus, it is a simple matter to

assemble coefficient matrices with respect to the fixed nodal values for potential and its

normal derivative.

4.3 Element Integration

Integration of the elements of the reformulated Green’s Equation requires special attention

due to the presence of the singularity in the kernel. If we recall the CBIE and HBIE,

respectively:

( ) ( ) ( )( ) ( ) [ ]∫∫+

+−+

∪

−∂∂

−−

−∂∂

−−∂

∂=

S qqSS q

dSdScwb

qpnqpnq

qpq

npp 111 φφφφφ

( ) ( ) ( )( ) ( )

[ ]

( )

∉Ω∈∈

=

−∂∂∂

−−

−∂∂∂

−

−∂∂

∂∂

=∂∂

+

+

+−+

∪

∫

∫

+

Sor , and Sor ,

,4,2

1

11

2

2

wb

wb

S qp

qpSS pqp

SSSS

c

dS

dScwb

ppp

p

qpnn

qpnnq

qpnq

nnpp

ππ

φφ

φφφ

It is advantageous to write the above as (Zhang 1992):

( ) ( ) ( ) ( ) [ ]∫∫+

+−+

∪

−−−∂∂

=S

pqpqSS

pqq

dSDdSDCcwb

φφφφφ qn

qpp Equation 4.28

4-52

( ) ( ) ( ) ( ) ( ) ( )

( )

∉Ω∈∈

=

∂∂

+∂∂

−∂∂

−∂∂

=∂∂

+

+

+−+

∪∫∫+

Sor , and Sor ,

,4,2

wb

wb

Spq

qqpq

qSSpq

qp

SSSS

c

dSFdSFEcwb

ppp

p

sq

sq

sq

nq

npp

ππ

φφφφφ

Equation 4.29

where sq = unit vector tangent to surface

The coefficients Cpq, Dpq, Epq and Fpq can be defined as:

1IC qpq r= Equation 4.30

qqpq

IDn

r∂∂

−= 1 Equation 4.31

pqpq

IEn

r∂∂

−= 1 Equation 4.32

( ) ( )

∂∂

+∂∂

−=q

pq

pqpqII

Fzr

r 21 cossin αα Equation 4.33

The partial derivatives required for the above coefficients in Equations (4.30 – 4.33) are

calculated by Zhang (1992, 1993) and are repeated here for convenience.

( )211

4

A

mKI = Equation 4.34

( ) ( )

( )( ) ( )( )

q

qp

q rA

mEmK

mA

mErrI

21

223

1 2

1

4 −−

−

−=

∂∂r

Equation 4.35

( ) ( )

( )223

1

1

4

mA

mEzzI qp

q −

−=

∂∂z

Equation 4.36

( ) ( )

( )( ) ( )( )

p

qp

p rA

mEmK

mA

mErrI

21

223

1 2

1

4 −−

−

−−=

∂∂r

Equation 4.37

4-53

( ) ( )

( )223

1

1

4

mA

mEzzI qp

p −

−−=

∂∂z

Equation 4.38

( ) ( ) ( )( ) ( ) ( )

( )223

223

2

1

48

mA

mEzz

mA

mEmKzzI qpqp

q −

−+

−−−=

∂∂z

Equation 4.39

where K(m) and E(m) are elliptic K and elliptic E, defined in Chapter 3, and:

A

rrm qp42 = Equation 4.40

( ) ( )22qpqp zzrrA −++= Equation 4.41

r = radial coordinate

z = longitudinal coordinate

α = angle between outward normal and radial axis

p = field point

q = source panel

4.3.1 Non-Singular Integration

Inspection of the partial integrals indicates the singularity exists as p approaches q. When p

and q are not coincident, standard Gauss-Legendre quadrature is sufficient to evaluate the

integral with the elliptic functions evaluated using listed polynomial approximations

(Hastings 1955).

When p lies on the axis of symmetry, the elliptic quantities are known for all q, and

consequently do not need to be calculated because elliptic K and elliptic E tend toward π/2 as

rp approaches 0. Thus, Equations (4.34 – 4.39) reduce to:

211

2

AI π= Equation 4.42

4-54

( )( )22

31

1

2

mA

rrI qp

q −

−=

∂∂ πr

Equation 4.43

( )( )22

31

1

2

mA

zzI qp

q −

−=

∂∂ πz

Equation 4.44

( )( )22

31

1

2

mA

rrI qp

p −

−−=

∂∂ πr

Equation 4.45

( )( )22

31

1

2

mA

zzI qp

p −

−−=

∂∂ πz

Equation 4.46

( )( )22

32

1

2

mA

zzI qp

q −

−=

∂∂ πz

Equation 4.47

4.3.2 Singular Integration

Standard Gauss-Legendre quadrature is insufficient for the singular cases. To address this,

the elliptic functions are approximated through use of tabulated polynomials (recall Chapter

3) that contain a logarithmic quantity.

( ) ( ) ( ) ( )xxQxPmK ln2 −=

( ) ( ) ( ) ( )xxSxRmE ln2 −=

where x = (1 – m2)

These tabulated polynomials are substituted into the expressions for the partial differentials,

Equations (4.34 – 4.39), and then rearranged into components containing a non-logarithmic

and logarithmic quantity, to facilitate the separation of the regular and non-regular

integrands. Thus, Equations (4.34 – 4.39) can be expressed as:

( ) ( ) ( )[ ]xxQxPA

I ln4

211 −= Equation 4.48

4-55

( ) ( ) ( ) ( ) ( ) ( ) ( )

−

+

−−−

+

−=

∂∂ x

rxQ

rAxrr

xSr

xPrAx

rrxR

A

I

qq

qp

qq

qp

q

ln12122

21

1

r

Equation 4.49

( ) ( ) ( ) ( )[ ]xxSxRxA

zzI qp

q

ln4

23

1 −−

=∂∂z

Equation 4.50

( ) ( ) ( ) ( ) ( ) ( ) ( )

−

+

−−−−

+

−−=

∂∂ x

rxQ

rAxrr

xSr

xPrAx

rrxR

A

I

pp

qp

pp

qp

p

ln12122

21

1

r

Equation 4.51

( )( )

( ) ( ) ( )[ ]xxSxRmA

zzI qp

p

ln1

4

223

1 −−

−−=

∂∂z

Equation 4.52

( ) ( ) ( )[ ] ( ) ( ) ( )[ ] ( ) ( )

+

−−+

−−=

∂∂ x

xxS

mxQxS

xxR

mxPxR

A

zzI qp

q

ln22422

23

2

z

Equation 4.53

Recalling that each panel is parameterised by ξ, and that each panel consists of two nodes

and a cubic polynomial interpolating the coordinates between them, it is clear that there are

two cases where a singularity might exist. Namely, when p exists at the beginning of the

panel, and when p exists at the end. As ξ varies between 0 and 1 (for ease of the numerical

integration) this amounts to a singularity at ξ = 0 and ξ = 1.

SINGULARITY AT THE ELEMENT BEGINNING (ξ = 0)

If we decompose:

( )

−

=

ξξ1ln2lnln 2

xx Equation 4.54

and substitute this into Equations (4.48 – 4.53), we can rearrange such that the non-regular

and regular components are separated as shown below:

4-56

( ) ( ) ( )

+

−=

ξξ1ln8ln4

212

211 xQ

A

xxQxPA

I Equation 4.55

( ) ( ) ( ) ( ) ( ) ( )

( ) ( ) ( )

−

+

−+

−

+

−−−

+

−=

∂∂

ξ

ξ

1ln124

ln12122

21

221

1

qq

qp

qq

qp

qq

qp

q

rxQ

rAxrr

xSA

xr

xQrAx

rrxS

rxP

rAxrr

xRA

Ir

Equation 4.56

( ) ( ) ( ) ( ) ( )

−+

−

−=

∂∂

ξξ1ln

8ln

4

232

23

1 xSxA

zzxxSxRxA

zzI qpqp

qz Equation 4.57

( ) ( ) ( ) ( ) ( ) ( )

( ) ( ) ( )

−

+

−−+

−

+

−−−−

+

−−=

∂∂

ξ

ξ

1ln124

ln12122

21

221

1

pp

qp

pp

qp

pp

qp

p

rxQ

rAxrr

xSA

xr

xQrAx

rrxS

rxP

rAxrr

xRA

Ir

Equation 4.58

( )( )

( ) ( ) ( )( )

( )

−

−−

−

−

−−=

∂∂

ξξ1ln

1

8ln

1

4

2232

223

1 xSmA

zzxxSxRmA

zzI qpqp

pz

Equation 4.59

( ) ( ) ( )[ ] ( ) ( ) ( )[ ] ( )

( ) ( ) ( )[ ] ( )

+

−−+

+

−−+

−−=

∂∂

ξ

ξ

1ln28

ln224

223

22223

2

xxS

mxQxS

A

zz

xxxS

mxQxS

xxR

mxPxR

A

zzI

qp

qp

qz

Equation 4.60

SINGULARITY AT THE ELEMENT END (ξ = 1)

We apply a similar technique for when p exists at the end of a panel. In this case we write:

4-57

( )( )

( )ξξ

−+

−= 1ln2

1lnln 2

xx Equation 4.61

By introducing η = (1 – ξ), Equation (4.61) can be written as:

( )( )

−

−=

ηξ1ln2

1lnln 2

xx Equation 4.62

and substituting this into Equations (4.48 – 4.53). Again we can rearrange such that the non-

singular and singular components are separated, as shown below:

( ) ( )( )

( )

+

−−=

ηξ1ln8

1ln4

212

211 xQ

A

xxQxPA

I Equation 4.63

( ) ( ) ( ) ( ) ( ) ( )( )

( ) ( ) ( )

−

+

−+

−

−

+

−−−

+

−=

∂∂

η

ξ

1ln124

1ln12122

21

221

1

qq

qp

qq

qp

qq

qp

q

rxQ

rAxrr

xSA

xr

xQrAx

rrxS

rxP

rAxrr

xRA

Ir

Equation 4.64

( ) ( ) ( )( )

( ) ( )

−+

−−

−=

∂∂

ηξ1ln

8

1ln

4

232

23

1 xSxA

zzxxSxRxA

zzI qpqp

qz Equation 4.65

( ) ( ) ( ) ( ) ( ) ( )( )

( ) ( ) ( )

−

+

−−+

−

−

+

−−−−

+

−−=

∂∂

η

ξ

1ln124

1ln12122

21

221

1

pp

qp

pp

qp

pp

qp

p

rxQ

rAxrr

xSA

xr

xQrAx

rrxS

rxP

rAxrr

xRA

Ir

Equation 4.66

( )( )

( ) ( )( )

( )( )

( )

−

−−

−−

−

−−=

∂∂

ηξ1ln

1

8

1ln

1

4

2232

223

1 xSmA

zzxxSxRmA

zzI qpqp

pz

Equation 4.67

4-58

( ) ( ) ( )[ ] ( ) ( ) ( )[ ] ( )( )

( ) ( ) ( )[ ] ( )

+

−−+

−

+

−−+

−−=

∂∂

ξ

ξ

1ln28

1ln224

223

22223

2

xxS

mxQxS

A

zz

xxxS

mxQxS

xxR

mxPxR

A

zzI

qp

qp

qz

Equation 4.68

4.4 Matrix Formulation

The method of matrix assembly is best illustrated through the example of a solid suspended

body near a toroidal cavity. Here we have a case where potential is initially known for one

surface (the cavity), and normal velocity is known for the other (solid body). Thus, the

matrices must be assembled and then rearranged such that we solve for the respective

unknowns.

4.4.1 Assembly and Rearrangement

Recalling the CBIE and HBIE, and writing in discretised form for each panel (Zhang 1992):

[ ]

( )( ) ( )

+=+=

=

−+

+

∂∂

−−++

+=

+−+

=∑ ∫∑ ∫

mniccnic

dSDdSDC

jj

i

m

nj Sjjij

n

j Sjjij

jij

jj

,...,1;,...,1;

11

φφφ

φφφφ

ppp

n Equation 4.69

( )

( )

( ) ( )

+=

∂∂

+

∂∂

=

∂∂

=

∂∂

+

∂∂

−

+

∂∂

=

∂∂

−

−−

+

++

+=

+−+

=∑ ∫∑ ∫

mnicc

nic

dSFdSFEc

jj

i

m

nj Sj

jjij

n

jj

Sjij

jij

j jj

,...,1;

,...,1;

11

np

np

np

ssnnp

φφ

φ

φφφφφ

Equation 4.70

where n = number of panels not on a vortex sheet

m = total number of panels

Sj = panel arc length

4-59

r

z

Solid Body

Toroidal Cavity

Ω

Figure 4.2. Matrix assembly example consisting of a suspended solid and single toroidal

cavity

If we consider Figure 4.2 and Equations (4.69) and (4.70), it is possible to construct a set of

arrays constituting the linear equation sets. As the potential and normal derivative density

functions can range up to cubic polynomials, it is clear that each matrix element is in fact an

aggregate dependent on the respective density function and adjacent elements, as described

in the Domain Discretisation section. Thus:

=Ψ

+

+

+

+++++++++

+++

+++++++++

+++

+++++++++

+++

1

1

1

1

1

1

111111111111

111111111111

111111111111

111111111111

1111111111111

111111111111

,,,,,,

,,,,,,

,,,,,,

,,,,,,

,,,,,,

,,,,,,

N

N

N

NNNNNNNNN

NNN

NNNNNNNNN

NNN

NNNNNTNNN

NNN

S

S

V

V

T

T

SSSSVSVSTSTS

SVSSVSVSTSTS

SVSVVVVVTVTV

SVSVVVVVTVTV

STSTVTVTTTTT

STSTVTVTTTTT

CCCCCC

CCCCCCCCCCCC

CCCCCCCCCCCC

CCCCCC

ψ

ψψ

ψψ

ψ

M

M

M

LLL

MOMMOMMOM

LLL

LLL

MOMMOMMOM

LLL

LLL

MOMMOMMOM

LLL

C

and similarly for D, E, and F, with subscripts T, V and S corresponding to toroid bubble,

vortex sheet and solid body of N nodes each respectively, and the sub-subscript referring to

the respective surface node number.

We now rearrange into known and unknown variables, such that:

4-60

−−

−−−−

−−−−

−−

=

−−

−−−−

−−−−

−−

+

+

+

+++++++++

+++

+++++++++

+++

+++++++++

+++

+

+

+

+++++++++

+++

+++++++++

+++

+++++++++

+++

1

1

1

1

1

1

111111111111

111111111111

111111111111

111111111111

1111111111111

111111111111

1

1

1

1

1

1

111111111111

111111111111

111111111111

111111111111

1111111111111

111111111111

,,,,,,

,,,,,,

,,,,,,

,,,,,,

,,,,,,

,,,,,,

,,,,,,

,,,,,,

,,,,,,

,,,,,,

,,,,,,

,,,,,,

N

N

N

NNNNNNNNN

NNN

NNNNNNNNN

NNN

NNNNNTNNN

NNN

N

N

N

NNNNNNNNN

NNN

NNNNNNNNN

NNN

NNNNNTNNN

NNN

S

S

V

V

T

T

SSSSVSVSTSTS

SVSSVSVSTSTS

SVSVVVVVTVTV

SVSVVVVVTVTV

STSTVTVTTTTT

STSTVTVTTTTT

S

S

V

V

T

T

SSSSVSVSTSTS

SVSSVSVSTSTS

SVSVVVVVTVTV

SVSVVVVVTVTV

STSTVTVTTTTT

STSTVTVTTTTT

CCDDDD

CCDDDDCCDDDD

CCDDDDCCDDDD

CCDDDD

DDCCCC

DDCCCCDDCCCC

DDCCCCDDCCCC

DDCCCC

ψ

ψφ

φφ

φ

φ

φψ

ψψ

ψ

M

M

M

LLL

MOMMOMMOM

LLL

LLL

MOMMOMMOM

LLL

LLL

MOMMOMMOM

LLL

M

M

M

LLL

MOMMOMMOM

LLL

LLL

MOMMOMMOM

LLL

LLL

MOMMOMMOM

LLL

and similarly for E and F.

Inspection of the matrices reveals that they actually consist of sub-matrices dependant on

which part of the domain the point (p or q) actually lay. Furthermore, distinction is made

between the vortex sheet (subscript V) and the bubble toroid (subscript T), as the aggregated

elements of the matrices are evaluated differently as demonstrated by Equations (4.69) and

(4.70).

4.4.2 Matrix Condition Improvement

For the constant, c(p), in Greens equation to equal 2π or 4π (for p existing on the surface or

within the domain respectively), the surface geometry must be smooth along the entire

surface. This constant, termed the solid angle, can be thought of as the limit of the region of

the domain witnessed by a point in the domain. If p does not lie on a boundary surface, then

there is no restriction to its view of the surroundings. Thus, the region of the domain it is

able to witness is unbounded in all directions, which corresponds to a whole sphere. In order

for this quantity to be general, it is expressed as the area of a spherical surface at unit radius

(i.e. 4π).

4-61

In the case of a point lying on a surface, the domain is limited by the surface. When the

surface is smooth, then the limit of the point’s view of the domain is restricted to a tangent at

that point. Thus, the witnessed domain is a hemi-sphere, corresponding to 2π in

dimensionless terms. However, if the geometry is not smooth, as is the case with linear

elements at a vertex, then the limit is in fact the angle between the two adjoining elements

and so the solid angle is not 2π and must be calculated.

This can be done in two ways. The first and most direct way is to simply calculate the angle.

This will correspond to either a conic (when p lies on the axis of symmetry) or a segment of

a sphere (when p does not lie on the axis) as the azimuthal angle is smooth due to the

analytical integration.

The second method is to introduce the Dirichlet and Neumann conditions for the interior and

exterior problems. Here, if we consider the interior problem and impose a constant potential

of unity along all boundaries (Dirichlet condition), the normal derivative of potential is zero

for all points of the domain. Additionally, if we consider the interior problem again, but

impose a specified normal derivative of potential (Neumann condition), the solution for

potential contains an additive constant of 4π. Thus, for the interior problem using the CBIE

(Taib 1985):

0=+∑≠ij

ijii DD Equation 4.71

However, for the exterior problem (Taib 1985):

∑

∑

≠

≠

−=∴

=+

ijijii

jiijii

DD

DD

π

π

4

4 Equation 4.72

In this regard, it is possible to find the solid angle indirectly through replacement of the

leading diagonal term of matrix D. Furthermore, even when the higher order geometric

elements are used, it is often advantageous to calculate the diagonal terms of D in this

manner as it improves the accuracy of the solution. This becomes especially important as the

re-entrant jet forms, as well as for special analytical cases such as the Rayleigh-Plessett

rebounding bubble.

In addition to this, the surface nodes were re-meshed to maintain relatively even individual

panel arc length across each surface. For the algorithm discussed in this thesis, re-griding

was conducted whenever the ratio between largest and smallest panel of any individual

4-62

surface exceeded 1.1. This was found to be a good compromise between maintaining good

separation between collocation nodes and computational time.

4.4.3 Combination Schemes for CBIE and HBIE

When using the HBIE, it is essential to write the CBIE for at least one point in the domain as

it recovers the additive constant lost during the differentiation for the HBIE (Zhang, Duncan

et al. 1993). The choice of point (or points) is dependent on the current state of the domain.

When dealing with simply connected domains (i.e. those not containing a toroidal cavity) it

is usually better to model using the CBIE only. This is primarily due to reduction in

computational resources required, as accuracy is not improved through use of a combined

scheme until the bubble approaches toroidal shape (we also reduce “saw-tooth” instability,

but that shall be discussed later). Once the bubble nears a multiply connected state, we must

use a combined scheme of some type.

Zhang (1992, 1993) uses the CBIE for all nodes until the collapse of the bubble brings the

north and south poles within 0.03 units separation. He then switches to a combined scheme

where the HBIE is used for all nodes except for the m/2 node (see Figure 4.3), where the

CBIE is used. This has the effect of reducing inaccuracies due to the increasingly singular

nature of the CBIE for the poles as they approach one another.

r

z

Toroidal Cavity

Ωp0

pm

pm/2

Figure 4.3. Schematic showing approximate position of particular nodes

4-63

Finally, once the cavity becomes multiply connected, Zhang (1992, 1993) applies the HBIE

to approximately half the cavity nodes beginning with the penetrating surface, while the

CBIE is used for the remaining collocation points.

While these schemes were experimented with, they enjoyed relatively little success. In

general, if the CBIE and HBIE were not applied to all nodes describing a particular surface,

instabilities tended to develop about the node where the boundary integral equation was

switched. It was for this reason that whenever the HBIE was used, it was applied to all

nodes of the domain in addition to the CBIE, in this thesis.

4.5 Time Advancement

Once the potential and its normal derivative are known for all the surfaces in the domain, it

can be marched in time.

4.5.1 Euler Scheme

Taib (1985) uses a simple Euler scheme for updating the surface geometry and potential.

( ) ( ) tdtdttt ∆+=∆+xxx Equation 4.73

where x = any vector

Points on the cavity surface are treated as Lagrangian particles, and are updated as:

( ) ( ) ttrttr ∆

∂∂

+∂∂

+=∆+ rss

nn

ˆ.ˆˆ φφ Equation 4.74

( ) ( ) ttzttz ∆

∂∂

+∂∂

+=∆+ zss

nn

ˆ.ˆˆ φφ Equation 4.75

In order to update potential on the cavity surface, it becomes necessary to introduce the

material derivative into the Bernoulli equation. By substituting the material derivative:

φφφ∇+

∂∂

= .utDt

D Equation 4.76

into 2

21 uρφρ +

∂∂

=−∞ tpp b

4-64

we can find the rate of change of potential noting that u is equal to ∇φ.

221 u+

−= ∞

ρφ bpp

DtD

Equation 4.77

For completeness, the tangential velocity must be calculated, but this is a simple operation

since the potential is known at the surfaces. Thus, the rate of change of potential for a

particle has been determined (i.e., we have an expression for change in potential including

the temporal and convection terms), leading to:

( ) ( ) tDtDttt ∆+=∆+φφφ Equation 4.78

Finally, the value for ∆t must be determined. Here we try to limit the maximum incremental

change in potential. Zhang (1992) conducted convergence studies to determine the most

accurate incremental change in potential, and found to this to range between 0.01 – 0.04.

These convergence studies amounted to modelling a classical problem and reducing the

potential increment until no appreciable difference in the model results was observed. Thus,

by finding the maximum velocity on any surface, we can limit this potential change

increment through:

max2

21

4.1

0

max u+−

−−

∆=

∆=∆

∞ σκ

φφφ

VVppp

t

gvDtD Equation 4.79

noting that Pb is replaced by the vapour pressure, Pv, and pressure due to non-condensable

gas, Pg, which can be written in dimensionless form as:

max2

21*

4.1

0

max 1 u+−

−

∆=

∆=∆

κσα

φφφ

VV

tDtD Equation 4.80

where v

g

ppp−

=∞

α Equation 4.81

v

m

ppR−

=∞

σσ * Equation 4.82

For the vortex sheet, nodes are moved only in the direction normal to the surface (Zhang

1992). In this case:

4-65

( ) ( ) tn

ttt rr ∆

∂∂

+=∆+ ++

++ rnxx ˆ.ˆφ Equation 4.83

( ) ( ) tn

ttt zz ∆

∂∂

+=∆+ ++

++ znxx ˆ.ˆφ Equation 4.84

( ) ( ) ( ) ( ) tss

ttttttt nnnn ∆

∂∂

−

∂∂∆

−−=∆+−∆+−+

−+−+ φφφφφφ2

Equation 4.85

where s = tangential direction

subscript n indicates a point following the a vector normal to the surface

superscript +/- indicating upper/lower vortex sheet respectively

subscript r and z indicate the r and z coordinate of point x

4.5.2 Predictor-corrector Scheme

An alternative to a simple Euler scheme is a predictor-corrector method (Zhang, Duncan et

al. 1993). The predictor scheme is given as:

( )iiiiii ytfttyy ,)( 1*

1 −+= ++ Equation 4.86

and the corrector:

( ) ( ) ( ) *1112

11 ,, ++++ +−+= iiiiiiii ytfytfttyy Equation 4.87

Clearly, the predictor step is the same as the Euler scheme presented previously. To use the

corrector step, it is necessary to solve the domain at the predictor interval and use these

results to advance the solution. This effectively increases the computational resources

required by over twofold, and it is as yet unclear whether significant gain in accuracy is

achieved to justify this other than that it is a higher order progression. As a consequence, the

Euler scheme is adopted as the update scheme in this thesis.

4.5.3 Time Stepping Considerations

Once a solution has been determined for a given time step, it is then possible to march the

solution using the Euler method. To do this, we find the velocity of each collocation point in

the domain by combining the solved normal velocity with the calculated tangential velocity.

Depending on the interpolation functions, this will be either a finite difference solution,

4-66

differentiated isoparametric cubic function, or differentiated cubic spline solution for linear,

isoparametric and cubic spline interpolation functions respectively.

With the maximum velocity found, it is a simple matter to find the maximum allowable time

increment such that the potential change is maintained at pre-determined maximum. When

using a completely CBIE scheme, the time stepping requires no special attention other than

keeping the maximum potential change less than 0.04 (provided the leading diagonal terms

in matrix D are recalculated according the Dirichlet-Neumann problems), in accordance with

the convergence studies conducted by Zhang (1992). This is not the case for schemes using

the HBIE.

“SAW-TOOTH” INSTABILITY

When using a combined scheme,”saw-tooth” instability is observed at the poles and extreme

maximum radial coordinate positions of the modelled cavity. The exact source of the

instability is unclear, however it was controlled using the smoothing algorithm provided

below (Zhang 1992).

Given a data set zi (i = 0, 1, …, m), a modified data set z’i (i = 0, 1, …, m), can be computed

by using the formulations:

( ) ( )[ ] 2,...,2for ,17123351' 1122 −=++++−= +−+− mizzzzzz iiiiii Equation 4.88

( )432100 353931351' zzzzzz +−−+= Equation 4.89

( )432101 5612139351' zzzzzz −+++= Equation 4.90

( )mmmmmm zzzzzz 9131265351' 12341 ++++−= −−−−− Equation 4.91

( )mmmmmm zzzzzz 319353351' 12342 ++−−= −−−−− Equation 4.92

While it is obvious to the human observer when “saw-tooth” oscillations begin, it is difficult

to write a general computer algorithm to identify this. Rather than try to identify the

oscillations themselves, and then commence smoothing, a method based upon local

curvature was adopted. At each time step, a search was conducted for extremes in local

4-67

curvature. In a similar fashion to the re-meshing; when the ratio between largest and

smallest curvature (for each individual surface) exceeded a certain limit (in this thesis, when

the tightest curvature was eight times the average), the nodes with sufficiently tight curvature

were smoothed using Equations (4.88 – 4.92).

This allowed the simulation to progress well, with little effect on the overall model when

compared to CBIE test cases. Without smoothing, however, the cavity model would

breakdown to the point where the numerical osculations would completely dominate the

solution, often causing the bubble surface to “fold” in upon itself in clearly erroneous ways,

and ultimately become singular.

THE PENETRATION PROCESS

The process of the re-entrant jet penetrating the opposing side deserves special attention. It

is necessary to approximate numerically a smooth penetration process, as closely as possible,

in order to reduce potential temporal derivative “spikes” throughout the domain.

Specifically, during the penetration process, the calculated potential in the domain can differ

slightly from a smooth penetration due to the incremental fitting of opposing nodes as they

approach one another.

To combat this, and also avoid any geometric abnormalities, a separation parameter is used.

When a node comes within a certain distance of an opposing surface, the time step is

changed such that this node should exactly contact the opposing surface in the given time

step. This differs from the previously discussed method, as we are no longer limited by a

maximum change in potential. However, we ensure the separation parameter is sufficiently

small that the maximum change in potential is less than that specified in the usual time step

algorithm.

CONTACTING SURFACES (PROTECTION ZONE)

The final area of the numerical scheme that requires discussion is the way in which

contacting surfaces were dealt with. When considering the case where a cavity is near a

stationary solid body, the technique employed is simple and involves incrementally reducing

φ∆ until a point where the contacting surface is no longer allowed to move. The rationale

is to insure that there always remains sufficient separation between opposing surfaces such

that there is no singularity developed.

When considering the case where the solid body is able to move, it was common for this

body to be drawn toward the cavity surface during the bubble’s growth phase, until the two

4-68

surfaces contacted. At this point, there are a number of options. The first is to allow the

solid to continue to move, and update the cavity surface appropriately. In this sense, the

solid body would continue into the cavity, and in some cases be completely enveloped. This

tended to create a plethora of other issues regarding how to handle the enveloping of the

solid and the multiple connection of the bubble surface.

An alternative to this would be to move the solid body with the cavity surface, thereby

allowing the cavity to reach maximum growth without enveloping the solid. However, in

this case the motion of the solid and pressures that propelled it would be significantly

reduced, if not ignored completely.

As the two possibilities presented each have their problems, it was decided to compromise

between the two. As the two surfaces approach each other, φ∆ is reduced incrementally as

in the stationary body case. Once again, once the two surfaces are sufficiently close, those

nodes in close proximity are halted. However, there is a very important numerical

abstraction. Even though the update of the solid body’s nodal coordinates is essentially

stopped, it is still allowed to retain its n∂∂φ

before contact. This allows the solid to still have

impact on the determination of the cavity surface velocity while avoiding problems

associated with complete enveloping of the solid.

4.6 Summary

Chapter 4 has demonstrated how the boundaries of the domain are discretised such that

matrix arrays can be assembled to allow the solution of the modified Greens equation. The

relative merits of various element combinations have been discussed, where linear geometric

and density functions are used for solution of the CBIE, and cubic geometric and linear

density functions are used for solution the HBIE.

An example, proposing the most complicated case considered in this thesis, was provided to

illustrate the assembly of the respective matrices, their combination and the improvement of

the matrix condition. Here it was noted that rearrangement must be made to reflect that the

potential is known for the cavity surface, while the normal derivative of potential (velocity)

is known for the solid body.

Temporal advancement schemes were then described, with the Euler scheme being selected

mainly due to its simplicity and lower computational cost. Furthermore, it was shown that

the time step is dependent on a maximum change in potential.

4-69

Finally, special considerations to maintain stability were discussed where a smoothing

algorithm was introduced, as well as the general penetration process, and a method of

separating contacting surfaces.

As mentioned in the introduction, the numerical algorithm used in this thesis was developed

independently, albeit based upon the mathematical formulation provided by previous

referenced authors. However, as this thesis considered the presence of suspended solid

bodies specifically, the numerical implementation had to be modified to include the presence

of these surfaces. Furthermore, as the use of the HBIE requires that cubic or higher order

functions be used, isoparametric density functions were provided such that the HBIE could

be used for suspended body cases. This is significant in that this particular combination of

equation and numerical implementation is original to this thesis, and provides a method of

progressing a cavity collapse beyond touchdown for cases involving a suspended solid body.

This is in addition to the unique treatment of the contacting bubble surface and solid body,

which together constitutes a significant portion of the originality of this thesis.

5-71

Chapter 5

Verification of Numerical Method

5.1 Introduction

As with all numerical analyses, once a mathematical model has been implemented in a

numerical form, the results must be tested for validity. Usually this involves comparison of

numerical solutions to problems where an analytical solution exists. Accurate solution of

these problems infers that the method is correct and will provide valid solutions to problems

where analytical solutions are impossible to find.

To verify the algorithm used in this thesis, five different element function and matrix

combination (i.e. method by which CBIE and HBIE matrix elements are combined) types are

tested. These consist of:

1. Linear geometric and density functions using the CBIE only.

2. Isoparametric cubic geometric functions, isoparametric cubic density functions

for potential, linear density functions for normal velocity and the CBIE only.

3. Cubic spline geometric functions, isoparametric cubic density functions for

potential, linear density functions for normal velocity and the CBIE only.

4. Isoparametric cubic geometric functions, isoparametric cubic density functions

for potential, linear density functions for normal velocity, and the CBIE plus the

HBIE for all collocation points.

5. Cubic spline geometric functions, isoparametric cubic density functions for

potential, linear density functions for normal velocity, and the CBIE plus the

HBIE for all collocation points.

5-72

Consequently, we are testing the accuracy of various element types, as well as the CBIE as

opposed to the CBIE plus HBIE combination. To achieve this, we must consider both static

and dynamic test cases.

For the static cases, we draw upon three different situations where analytical solutions exist.

These are:

1. A spherical source.

2. A spherical dipole.

3. A toroidal cavity with vortex sheet.

For the dynamic cases we are restricted to only one situation where an analytical solution

exists: the Rayleigh-Plessett rebounding cavity. For cases where a toroidal cavity develops

in time, there is no analytical solution. However, if we make an assumption that the CBIE

method is valid, we can use this solution to “bootstrap” verification for the HBIE.

Specifically, we compare the CBIE to the HBIE at selected similar time-steps to ensure

conformity between the two. Thus, there are two dynamic test cases:

1. A Rayleigh-Plessett rebounding bubble.

2. A developing toroid – CBIE/HBIE comparison.

5.2 Static Test Cases

5.2.1 A Spherical Source

A spherical source simply consists of a spherical surface where the potential is specified,

such that we solve for the normal derivative. For this test, the cavity is assumed to exist in

an infinite domain, thus the normal velocity is perfectly radial. As with all test cases, it is

assumed that the cavity starts from a Rayleigh bubble of given radius. The potential is

evaluated at an initial time, which is given as (Blake and Gibson 1981):

( )21

3

132

0

−

−= ∞

rRpp

rr mvt ρ

φ Equation 5.1

where Rm = maximum radius

r = radius at initial time (t0)

The initial time is evaluated using:

5-73

( )

−

=∞ 2

3,65

233

21

0 av

Bpp

t ρ Equation 5.2

where Ba = an incomplete Beta function

3

=

mRra Equation 5.3

If rebound of the cavity is not being considered, we assign an initial radius of 0.1

dimensionless units. This corresponds to a potential of 2.5807 at time 0.001553 from

inception, where the analytical solution for n∂∂φ

(velocity) is -25.807.

For this test case, we use a surface approximated using 32 elements. Results are provided

for each of the element and combination types described previously, which are identified as:

1. Linear CBIE

2. Isoparametric Cubic CBIE

3. Cubic Spline CBIE

4. Isoparametric Cubic HBIE Combined

5. Cubic Spline HBIE Combined

Radial Coordinate

Axi

alC

oord

inat

e

0 0.25 0.5 0.75 1 1.25 1.5 1.75-1

-0.5

0

0.5

1

Bubble SurfaceNodal Position

Arclength

Pot

entia

l

0 0.05 0.1 0.15 0.2 0.25 0.30

0.5

1

1.5

2

2.5

3

3.5

Analytical PotentialNodal Potential

Figure 5.1. A spherical source of 32 elements. Left: Geometry of bubble surface with nodal

position. Right: Nodal potential.

Figure 5.1 shows the initial bubble surface for an axisymmetric halfspace, and the applied

surface potential. All units are dimensionless in this figure, as is the case for all results and

figures presented in this thesis unless otherwise noted. Furthermore, arc length is measured

clockwise from the north pole.

5-74

Figure 5.2 shows the normal velocity with respect to arc length of the cavity surface,

measured from the north-pole node, for a 32 element spherical source. Regardless of which

element and combination scheme is used, we can see that the numerical solutions agree very

well with the analytical solution. Figure 5.3, however, gives us a better idea of the relative

accuracy of each individual scheme by changing the vertical axis scale. Here we can see that

the isoparametric cubic CBIE gives us the closest numerical approximation to the analytical

solution, and does not suffer problems associated with nodes lying on the axis of symmetry

as experienced by the linear CBIE scheme. The errors associated at the axis of symmetry are

more dramatic for the linear CBIE scheme, and even though the difference is less than 0.1

percent, these errors can accumulate over time. However, how these errors affect the model

over time is heavily influenced by the type of combination scheme used as well as the

magnitude of the relative error. This will be addressed in the temporal test cases.

Table 5.1 then shows the mean squared error term for the various combination schemes,

where it is confirmed that the isoparametric cubic CBIE provides a marginally better result

than the cubic spline CBIE.

Figure 5.2. Normal velocity for a 32

element spherical source, for all element

and combination schemes.

Figure 5.3. Normal velocity for a 32

element spherical source, with a

magnified axial coordinate scale, for all

element and combination schemes.

Arc Length

Nor

mal

Vel

ocity

0 0.1 0.2 0.3

-30

-25

-20

-15

-10

-5

0

AnalyticalLinear CBIEIsoparametric Cubic CBIECubic Spline CBIEIsoparametric Cubic HBIECubic Spline HBIE

Arc Length

Nor

mal

Vel

ocity

0 0.1 0.2 0.3

-25.85

-25.8

-25.75


5-75

Mean Squared Error Term (10-4)

Linear CBIE Isoparametric

Cubic CBIE

Cubic Spline

CBIE

Isoparametric

Cubic HBIE

Cubic Spline

HBIE

5.43968 1.06638 1.07808 1.9499 16.18127

Table 5.1. Mean squared error terms for a spherical source for various element and

combination schemes.

5.2.2 A Spherical Dipole

The spherical dipole for this test case consists of a spherical surface of unit radius with the

potential varying along the surface, such that the potential is opposite at the poles, thus:

( )2

sin αφ = Equation 5.4

Such that ( )αφ sin−=∂∂

n Equation 5.5

where α = angle between surface normal and positive radial axis

In this case s∂

∂φ is not zero, unlike the spherical source, and changes with respect to arc

length. Consequently, matrix F (see Chapter 4) is no longer zero and will have significant

impact on the numerical solution, once the CBIE and HBIE are combined. In this respect,

the spherical dipole provides a good test to determine the accuracy of the F matrix elements.

For completeness, the same element and combination schemes used for the spherical source

will be listed. It should be noted that the surface is now of unit radius, rather than 0.1 as in

the spherical source test.

5-76

Radial Coordinate

Axi

alC

oord

inat

e

0 0.5 1 1.5-1

-0.5

0

0.5

1

Bubble SurfaceNode Position

Arc Length

Pot

entia

l

0 1 2 3-0.5

0

0.5


Figure 5.4. A spherical dipole of 32 elements. Left: Geometry of bubble surface with nodal

position. Right: Nodal potential.

Arc Length

Nor

mal

Vel

ocity

0 1 2 3

-1

-0.5

0

0.5

1


Figure 5.5. Normal velocity for a 32 node spherical dipole with respect to arc length

measured from the north pole.

If we consider Figure 5.5, again we can see that the numerical approximations of normal

velocity agrees well with the analytical solution, although it is more difficult to ascertain any

problems associated with nodes lying on the axis of symmetry due to the scale of the vertical

axis.

In this case it is cumbersome to provide a graphical representation of the relative accuracy of

the various element and combination schemes provided for the spherical source. Instead, we

find the mean squared error term between each scheme and the analytical solution (see Table

5.2).

5-77

Mean Squared Error Term (10-6)

Linear CBIE Isoparametric

Cubic CBIE

Cubic Spline

CBIE

Isoparametric

Cubic HBIE

Cubic Spline

HBIE

3.07133 1.04158 0.102186 0.209099 0.116779

Table 5.2. Mean squared error terms for a spherical dipole for various element and


It is interesting to note that according to the mean squared error terms for the spherical

dipole, the cubic spline interpolation seems to produce the most accurate results. This

highlights the problem dependency of this kind of numerical modelling.

5.2.3 A Toroidal Cavity With Vortex Sheet

The toroidal cavity consists of a ring bubble of circular cross-section and a vortex sheet in

the plane of symmetry parallel to the radial axis (see Figure 5.6). If we transform the

geometry into toroidal coordinates, we can use Laplace’s equation in toroidal coordinates

such that an analytical solution is obtainable (Moon and Spencer 1961). Zhang (1992)

provides a particular solution for potential that satisfies the following conditions, which are

the same as those encountered in non-ideal toroidal cavities.

1. Potential and its tangential derivative are discontinuous across the vortex

sheet.

2. The normal derivative is continuous across the vortex sheet.

3. The fluid must be quiescent at infinity.

4. Potential must be sufficiently smooth throughout the domain.

Here, potential is calculated using:

( ) ( ) ( )[ ]

−=

2sincoscosh, 2

1 θθηθηφ Equation 5.6

thus ( ) ( )[ ]surface bubble on the points allfor ,

22

sin)sinh(coscosh

sheet vortex on the points allfor ,0

21

a

−

=∂∂

=∂∂

θηθηφ

φ

n

n

Equation 5.7

where a = the distance from [0, π, 0] to [∞, π, 0] in the toroidal coordinate system.

5-78

For the case where the vortex sheet and bubble radii equal 1.0 and 0.124556 respectively, a

equals 1.117637 (Zhang, Duncan et al. 1993).

Figure 5.6 shows the toroidal geometry and the bubble surface potential, while Figure 5.7 is

a plot of the resultant velocity.

Radial Coordinate

Axi

alC

oord

inat

e

0 0.25 0.5 0.75 1

-0.1

-0.05

0

0.05

0.1 Bubble SurfaceNodal Position

Radial Coordinate

Pot

entia

l

0 0.25 0.5 0.75 1-3

-2

-1

0

1

2

3


Figure 5.6. A toroidal cavity of 32 elements consisting of a ring bubble of circular cross-

section and flat vortex sheet. Left: Geometry of bubble surface with nodal position (note that

the horizontal and vertical axes are of differing scale). Right: Nodal Potential.

Radial Coordinate

Nor

mal

Vel

ocity

0 0.2 0.4 0.6 0.8 1 1.2

-20

-10

0

10

20 AnalyticalLinear CBIEIsoparametric Cubic CBIECubic Spline CBIEIsoparametric Cubic HBIECubic Spline HBIE

Figure 5.7. Normal velocity for various element and combination schemes for the 32-

element toroidal cavity presented in Figure 5.6.

From Figure 5.7 we can deduce that the methods using the CBIE appear to be more accurate

than those employing the HBIE. This is somewhat misleading, for a number of reasons.

Firstly, any scheme using the CBIE can only solve for the normal velocity on the ring

5-79

bubble. The velocity for the vortex sheet must be determined using an alternative method (a

finite difference scheme in this case), which may not be accurate for all cases.

Secondly, as was shown in Chapter 4, the vortex sheet is treated quite separately and

effectively provides an additive constant during the multiplication of the known matrix

elements and known quantities vector. As it is treated separately, we are able to manipulate

the geometry of the sheet to better reflect the actual shape. In this case, the numerical

approximation of the vortex sheet geometry, for all methods using the CBIE, is exact.

This is not the case for the HBIE. Chapter 4 demonstrated that there is no separation

between the vortex sheet or bubble surface beyond that of the calculation of the solid angle.

Hence, there is some inaccuracy at the point where the vortex sheet attaches to the ring

bubble (Attachment Point), due to the inability of higher order element to capture the

geometric step change. Therefore, the numerical approximation of the geometry is

significantly different to the actual shape and it is this that is responsible for the deviation of

the normal velocity around the Attachment Point. This is one limitation of numerical

approximation to systems where a step change is present. Fortunately, the formation of

toroidal cavities in this project is assumed to be smooth, consequently we do not encounter

any step changes except for this toroidal test case.

5.3 Dynamic Test Cases

5.3.1 A Rayleigh-Plessett Rebounding Bubble

The numerical approximation of the Rayleigh-Plessett rebounding bubble is a good test of

the temporal accuracy and stability of the simulation. It consists of a completely spherical

bubble in an infinite medium with a portion of non-condensable gas. For the case of

cavitation, we assume the expansion and collapse to be adiabatic, consequently the bubble

theoretically undergoes an infinite series of rebounds as the non-condensable gas is

compressed to the point where it arrests the collapse and begins a new expansion cycle. We

shall only look at one cycle, however.

For the dimensionless cases presented in this thesis, we use a modified Rayleigh equation

listed as Equation (5.9) (Best and Kucera 1992):

( ) ( ) ( )( ) 01

23

32

2

2

=

−+

+

kg

tRr

dttdR

dttRdtR Equation 5.8

5-80

where R(t) = bubble radius as a function of time

rg = non-condensable gas content as an equivalent radius

k = adiabatic expansion coefficient (1.4)

For consistency, results for one cycle using the same element and combination methods as in

the static cases are presented in Figure 5.8. All cavities begin as a spherical Rayleigh bubble

of 0.35 dimensionless units in radius, completely filled with non-condensable gas. Temporal

iterations are halted once rebound has been achieved.

Figure 5.8 demonstrates the accuracy of the present model. Indeed, all the element

combination methods, except for the cubic spline HBIE, agree with the solution of the

Rayleigh-Plessett equation to within 3%, at the time of rebound.

The cubic spline HBIE does not capture the rebound well at all, as the instability at the poles

requires heavy smoothing to keep the model progressing in time. Clearly, this is far from

ideal and represents one limitation associated with using cubic spline elements.

Furthermore, the dramatic effect of excessive smoothing of the cavity surface shows us that

this technique of keeping the model under control through time must be used as infrequently

as possible as it causes the simulation to deviate significantly away from the actual problem

being solved.

Dimensionless Time

Ave

rage

Rad

ius

0 0.5 1 1.5

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Rayleigh-Plesset EquationLinear CBIEIsoparametric Cubic CBIECubic Spline CBIEIsoparametric Cubic HBIECubic Spline HBIE

Figure 5.8. Average radius for an initially spherical 32-element cavity, over dimensionless

time up to rebound.

5-81

5.3.2 A Developing Toroid – CBIE/HBIE Comparison

In this case, an initially spherical bubble is allowed to grow in close proximity to a rigid

boundary. This causes a re-entrant jet to form that will eventually touch the opposing bubble

surface, thus creating a toroidal bubble. Considering the successful modelling of the

previous test cases, we are confident that the CBIE method is functioning correctly. As a

final test, it is helpful to compare the CBIE to a HBIE combination to demonstrate the

correct evaluation of the E and F matrices (see Chapter 4) through time where the tangential

derivative of potential is not zero (as with the Rayleigh-Plessett bubble) throughout the

simulation.

Again, the same five element and combination schemes are used, the results of which are

presented in Figures 5.9 and 5.10. The cavity is initially a spherical source with potential of

2.5807, and is offset from the boundary by 1.25 dimensionless units. The cavity then grows,

and snapshots (for similar time-steps) at selected times are shown in Figure 5.9. The

collapse is then shown for various times in Figure 5.10.

The model is halted upon touch down, as comparing a finite difference estimate of the vortex

sheet movement to a HBIE solution is the subject of a following chapter. The purpose of

this test is merely to demonstrate parity between the two boundary integral equation

methodologies, for the relatively simple case of simply connected domain with re-entrant jet

formation.

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Radial Coordinate

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alC

oord

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-1 -0.5 0 0.5 10

0.5

1

1.5

2

Linear CBIEIsoparametric Cubic CBIECubic Spline CBIEIsoparametric Cubic HBIECubic Spline HBIE

Figure 5.9. Growth of a 32 element cavity near a rigid boundary, using various element and

combination types, at selected times.

From Figure 5.9 and Figure 5.10, as the bubble surfaces overlap for given times, it is clear

that all the element and combination types are equivalent, except for the isoparametric cubic

HBIE. Due to the increased size of the plot, the instability at the poles of the isoparametric

cubic HBIE is more evident. Interestingly enough, once smoothing has been applied the

model using the isoparametric cubic HBIE recovers and provided results very similar to the

CBIE schemes as shown in Figure 5.10.

This is not the case for the cubic spline HBIE. Despite the growth being constant, the early

collapse diverges at the poles, which becomes further compounded during the collapse phase

presented in Figure 5.10. Here we notice the much more developed re-entrant jet, which

ultimately causes the cavity to touchdown at a much more reduced volume than the other

element and combination schemes.

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Radial Coordinate

Axi

alC

oord

inat

e

-1 -0.5 0 0.5 10

0.5

1

1.5

2

Linear CBIEIsoparametric Cubic CBIECubic Spline CBIEIsoparametric Cubic HBIECubic Spline HBIE

Figure 5.10. Collapse of a 32 element cavity near a rigid boundary, using various element

and combination types, at selected times.

5.3.3 Penetration and Rebound

Reliable modelling of the penetration is not possible. Greatest success was achieved using a

linear CBIE scheme with the vortex sheet motion determined using a finite difference

technique. However, instability was encountered at the poles and Attachment Point, and as

such aggressive smoothing was required throughout the penetration process to maintain the

model progression. In any case, the model would always become unsolvable before rebound

with the actual point of divergence depending on element type and number, and the shape of

the re-entrant jet at touchdown. In general, the closer the cavity was to the rigid boundary,

the earlier the model would become un-continuable. Furthermore, the dependence of the

5-84

vortex sheet motion on finite difference step raised serious issues regarding the accuracy of

the simulation.

Despite the HBIE schemes theoretically providing better solutions for the motion of the

vortex sheet, relatively little success was enjoyed by these schemes in modelling the

penetration. Problems regarding stability have already been raised in Section 5.3.2. After

touchdown, the model diverged very quickly where in many cases only 10-20 iterations

could be performed before the model became singular. Once again, the source of the

instability centred on the poles and the Attachment Point, although in this case the

Attachment Point seemed to be primarily responsible.

The precise source of this instability is difficult to determine. Similar problems were

reported by Zhang (1992 and 1993) and are believed to be a product of high-order elements

in Hyper-Singular Boundary Equations (Zhang 1992). Despite numerous efforts to

numerically improve stability, these problems could not be overcome and the complexity of

this particular problem puts it beyond the scope of this thesis. In any event, significant

information regarding the behaviour of cavities near suspended bodies could be gained from

modelling the cavity only to touchdown, to which point the model has been demonstrated to

perform quite well.

5.4 Summary

In this chapter, a number of test cases have been presented, the purpose of which was to

determine the accuracy of the numerical approximation for various element type and


In the static cases, it was shown that all the combinations performed very well. Even those

using the CBIE plus HBIE scheme approximated the toroid fairly well, especially when we

considering that the Attachment Point was not captured well due to step change present in

the analytical geometry. Furthermore, this was not considered to be critical, as cavities

growing to toroidal bubbles tend to do so in a smooth manner.

When considering the dynamic tests, the limitations of the CBIE plus HBIE schemes became

more evident. Considerable instability was observed at the poles (“Sawtooth” instability)

that required heavy use of a smoothing algorithm to keep the solution from diverging.

Ultimately this resulted in these particular schemes not finding the desired solution in either

dynamic test case.

5-85

Furthermore, the inconsistency in modelling the penetration process using either a CBIE, or

HBIE method scheme, raised serious issues regarding the accuracy of the results beyond

touchdown. This was primarily due to instability in the multiply connected bubble at the

poles and around the attachment point.

Numerous efforts were made to improve this instability, as it was clear that any problems

associated with the simply connected domain would only be exacerbated once the bubble

became a toroid. This was confirmed by testing, as once the bubble because multiply

connected the solution would diverge very quickly unless very aggressive smoothing was

performed. This was unacceptable, as it raises considerable questions regarding the accuracy

of the model beyond touchdown.

As a consequence, only linear geometric interpolating functions are used for the remainder of

this thesis. This means that the CBIE only could be used. Furthermore, as the only

consistent point of accurate modelling was touchdown, all subsequent simulations were

halted at this point. However, this still provided valuable insight into the effects that

suspensions have on individual cavitation bubbles, as is demonstrated in the following

chapters.

6-87

Chapter 6

Cavities near Stationary Suspended Bodies

6.1 Introduction

This chapter is concerned with solid bodies that are held in space with respect to the initial

cavity centre. In this regard, the solid body is a small rigid boundary where the velocity is

known at all times and is constant at zero. To investigate these cases, two types of plots are

presented. The first consist of Vector and Pressure plots that are shown for selected bubble

profiles that approximate to:

• halfway through growth

• end of growth

• halfway through collapse

• immediately prior to touchdown

The aim of these plots is to give the reader an understanding of the bubble shape during its

lifetime, as well as show the relative domain properties. The four selected profiles provide

instances for comparison at the most significant periods of the bubble evolution, while

limiting the number of graphs so as not to be overwhelming and confusing. Furthermore, the

actual range of cavitation cases investigated in this chapter extends to:

• cavity offset from the rigid boundary (γC) 0.9 – 1.8,

• solid body offset above the cavity (γSb) 0.9 – 1.5, and

• diameter of solid body (DSb) 0.1 – 0.3.

6-88

Figure 6.1. Schematic of plotted variables

This band constitutes the typical formation and damage range for cavitation bubbles (Philipp

and Lauterborn 1998), where the solid body has significant influence on the cavity and is

still sufficiently far from the cavity nucleus to not act as a seed itself. However, for the

Vector and Pressure plots the cavity offset is limited to 0.9 – 1.5 and the solid body offset is

limited to 0.9 – 1.2, as there is no significant difference between separate plots above these

limits. For these results to have relevance, it is necessary to develop a frame of reference.

To achieve this, a similar cavity, without a solid body present, is provided such that

comparison can be made to the case where a solid body does exist.

Colour pressure contours are provided with a legend for the relative pressures, while a vector

plot is overlayed with a reference vector provided at the top of the pressure contour legend.

Note that the black outline represents the actual respective surface.

To aid the reader’s interpretation, consistency in the contour legend and reference legend has

usually been maintained for each respective bubble profile. However, there are exceptions

for cases where the legend extents do not fully capture the pressure in the domain.

The second type of plot is a time history (time series data) of the cavity and domain

characteristics such that comparison can be made between the listed various cavitation cases.

The variables plotted are shown schematically in Figure 6.1 and include (subscripts Sb and C

represent solid body and cavity respectively):

• Cavity north pole (Np)

Cavity

Solid Body

Rigid Boundary

DSb

γC

γSb NpC

SpSb SpSb - NpC

6-89

• Cavity south pole (Sp)

• Cavity centroid (CC)

• Cavity and solid body centroid separation (CSb – CC)

• Separation between cavity and solid body surfaces along axis of symmetry

(SpSb – NpC)

• Pressure at the rigid boundary and axis of symmetry intersection(P0,0)

• Maximum domain pressure (PMax)

• Cavity kinetic energy (KE)

It should be noted that all variables presented are in dimensionless terms. Furthermore, the

cavities were discretised using 50 linear elements, while the solid bodies were approximated

using 24 linear elements. The rigid boundary was included via an image bubble as described

in Chapter 4.

Table 6.1 is a reference grid to plotted result figures with respect to the variables discussed

previously.

γC = 0.9 γC = 1.2

Pressure

Contour and

Vector Plots

Time Series

Data

Pressure

Contour and

Vector Plots

Time Series

Data

No Solid Body 6.2 – 6.5 6.38 – 6.41

γSb = 0.9 DSb = 0.3 6.6 – 6.9 6.42 – 6.45

DSb = 0.3 6.10 – 6.13 6.46 – 6.49

6.54 – 6.61

DSb = 0.2 6.14 – 6.17 – – γSb = 1.05

DSb = 0.1 6.18 – 6.21 – –

γSb = 1.2 DSb = 0.3 6.22 – 6.25 6.50 – 6.53 6.54 – 6.61

γSb = 1.5 DSb = 0.3 6.26 – 6.29

6.30 – 6.37

– –

Table 6.1. Reference grid for figures contained in this Chapter 6.

6-90

6.2 0.9 Cavity Stand-off Distance (γC)

Cavities presented in this section have an initial standoff from the rigid boundary of 0.9.

6.2.1 Single Cavity

Radial Coordinate

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-1 0 10

0.5

1

1.5

2

2.5

1.31.21.110.90.80.70.60.50.40.30.20.10

Figure 6.2. Pressure and vector plot for a single cavity near a rigid boundary (at axial

coordinate 0) at time 0.1130. Note the reference vector represents one unit.

6-91

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1.5

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2.5

0.60.550.50.450.40.350.30.250.20.150.10.050



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2.5

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coordinate 0) at time 1.9801. Note the reference vector represents ten units.

6-92

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This cavity cycle typifies that which was explained in chapter two. The presence of the rigid

boundary causes the generation of a re-entrant jet that ultimately touches the opposite side of

the bubble. Figure 6.4 illustrates a regular early jet formulation and the subsequent

generation of a high-pressure region in the domain directly behind the jet. In Figure 6.5, a

regular jet immediately before touchdown is shown, demonstrating the intensification of the

high pressure region in the domain, and a slight increase in jet velocity.

6-93

6.2.2 0.3 Diameter Solid Body – 0.9 Standoff from Cavity

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1.31.21.110.90.80.70.60.50.40.30.20.10


is present at 0.9 from the initial cavity centre. Note the reference vector represents one unit.

6-94

Radial Coordinate

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alC

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-1 0 10

0.5

1

1.5

2

2.5

0.60.550.50.450.40.350.30.250.20.150.10.050

Figure 6.7. 1 A similar cavity is shown at time 0.4223, except that a 0.3 diameter solid body


1 Note that the deviation away from a spherical shape for the solid particle in Figure 6.7 is an aberration from the domain grid, developed for pressure contour presentation. The true solid surface is denoted by the black outline.

6-95

Radial Coordinate

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-1 0 10

0.5

1

1.5

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2.5

2522.52017.51512.5107.552.50


is present at 0.9 from the initial cavity centre. Note the reference vector represents ten units.

Radial Coordinate

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1.5

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6-96

By comparing Figure 6.6 to Figure 6.2, it is clear that the presence of a solid body causes

disruption the fluid flow. A pressure build up on the inner surface demonstrates the

propensity for movement during the growth phase of the cavity. However, this reverses once

the cavity surface comes in close contact with the solid body and envelops it, thereby

attempting to draw the solid into the cavity, as shown in Figure 6.7. Figure 6.8 shows the

early re-entrant jet development. In this case it is somewhat narrower and apparently more

focused than that shown in Figure 6.4. Furthermore, there is a very localised and extremely

high-pressure region developed immediately behind the jet, which then dissipates toward

touchdown as shown in Figure 6.9.

While not shown here, the time series plots demonstrate that the collapse motion of the

cavity’s north pole is delayed significantly by the presence of a solid body. This results in

very rapid motion of the north pole away from the solid body that also undergoes a high

level of smoothing due to extreme local curvature. Unfortunately it is unclear whether the

high pressure spike observed in Figure 6.8 is a product of the converging fluid flow causing

the high velocity of the north pole, or an indirect result of the smoothing from previous

iterations (smoothing is switched off prior to any domain pressure plots). In any case, the

spike is no longer present at touchdown. However, the presence of the body has a dramatic

effect on the shape of the jet tip, which is now much narrower and has a more intense and

broader high-pressure region behind it.

6-97


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-1 0 10

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1

1.5

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1.31.21.110.90.80.70.60.50.40.30.20.10



6-98

Radial Coordinate

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-1 0 10

0.5

1

1.5

2

2.5

0.60.550.50.450.40.350.30.250.20.150.10.050

Figure 6.11. 2 A similar cavity is shown at time 0.6068, except that a 0.3 diameter solid

body is present at 1.05 from the initial cavity centre. Note the reference vector represents

one unit.

2 Note that the deviation away from a spherical shape for the solid particle in Figure 6.11 is an aberration caused by the domain grid, developed for pressure contour presentation. The true solid surface is denoted by the black outline.

6-99

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-1 0 10

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units.

6-100

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units.

The growth phase presented in Figures 6.10 and 6.11 is very similar to those shown for the

0.9 standoff case, although it is slightly less severe. Figure 6.12 shows a more narrow

development of the re-entrant jet as compared to the single cavity case, again very similar to

the 0.9 standoff case, except the pressure spike is not present in this particular plot. Despite

the difference in standoff distance being only 0.15, there is quite dramatic change in jet tip

profile at touchdown, as shown in Figure 6.13. Rather than a narrower tip profile, as in the

0.9 standoff case, the jet tip is much broader and squarer in shape than the single cavity case.

Furthermore, the pressure zone behind the jet is slightly broader and more intense than the

0.9 standoff case.


A slight digression is made now into the effects of solid body size. The following are plots

for the same standoff distance, however the solid body width is reduced incrementally.

6-101

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0.5

1

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2

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1.31.21.110.90.80.70.60.50.40.30.20.10



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6-102

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units.

6-103

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units.

As would be expected, the results are very similar to that of the 0.3 diameter case shown

previously. However, Figure 6.16 indicates a slightly earlier flattening of the re-entrant jet,

which is ultimately less broad than for the case of a 0.3 diameter solid body. The pressure

zone is also very similar.

6-104


Radial Coordinate

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-1 0 10

0.5

1

1.5

2

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1.31.21.110.90.80.70.60.50.40.30.20.10



Radial Coordinate

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-1 0 10

0.5

1

1.5

2

2.5

0.60.550.50.450.40.350.30.250.20.150.10.050



6-105

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units.

6-106

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-1 0 10

0.5

1

1.5

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units.

Further reduction in solid body size produces results that are very similar to the single cavity

case. All the profiles are very regular and the solid body seems to have little effect on the

growth and collapse of the cavity, despite the fact that the bubble does in fact contact the

solid body (as evident in the time series plots at the end of this section). There is, however, a

slight intensification of the pressure region behind the jet as compared to the single cavity

case.

It is clear that as size of the solid body decreases, so does its effect. From the results

presented previously, it would appear that a 0.1 diameter solid body (or 0.05 of maximum

bubble radius) approaches the lower limit for a solid body to have impact on the cavity, at

least for cases where the bubble only just contacts the solid. While this is by no means

conclusive, this does provide evidence that larger particles will affect the cavity from further

away and vice versa.


The remainder of the plots is limited to 0.3 diameter bodies, as any smaller bodies tend to

have limited effect on the bubble profile.

6-107

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6-108

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6-109

In this case, the cavity only just makes contact with the solid body. While there is some

disruption of flow by the solid (and consequent impact on domain pressure), this does not

seem to affect the cavity profile until collapse.

In Figure 6.24 it is evident that there is some flattening of the early re-entrant jet similar to

previous cases. However, the final tip profile is quite different again. Rather than a square

profile, the jet in Figure 6.25 is quite irregular. Finally, despite the jet’s dissimilarity to a

single cavity, the pressure region behind the jet is very near that of a case where no solid

body is present.


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6-110

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All the profiles shown in Figures 6.26 – 6.29 are nearly indistinguishable to that of a single

cavity, despite a slight pressure build-up at the inner surface of the solid body. It can be

concluded that 1.5 standoff from the cavity is the approximate limit for a 0.3 diameter

stationary body.

6.2.8 Time Series Plots

The following are plots for the previously listed variables. However, each separate case is

overlayed so that comparison can be made.

6-112

CAVITY NORTH POLE

Time

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0 0.5 1 1.5 20

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

Single Cavity0.3 Wide Stationary Body - 0.9 Standoff from Cavity0.1 Wide Stationary Body - 1.05 Standoff from Cavity0.2 Wide Stationary Body - 1.05 Standoff from Cavity0.3 Wide Stationary Body - 1.05 Standoff from Cavity0.3 Wide Stationary Body - 1.2 Standoff from Cavity0.3 Wide Stationary Body - 1.5 Standoff from Cavity

Figure 6.30. Plots of cavity north pole coordinates, with respect to time, for various

stationary body cases. All cavities are initially centred at γC = 0.9.

Figure 6.30 shows the motion of the north pole. Generally, the closer or larger the solid

body, the more delayed the collapse motion of the north pole. Furthermore, with large or

close bodies, the collapse motion of the north pole occurs at a greatly increased initial

acceleration, resulting in a marginally earlier touchdown.

CAVITY SOUTH POLE

Time

Axi

alC

oord

inat

e

0 0.5 1 1.5 20

0.2

0.4

0.6

0.8

1


Figure 6.31. Plots of cavity south pole coordinates, with respect to time, for various


The south pole is unaffected by the presence of solid bodies.

6-113

CAVITY CENTROID

Time

Axi

alC

oord

inat

e

0 0.5 1 1.5 20

0.2

0.4

0.6

0.8

1


Figure 6.32. Plots of cavity centroid coordinates, with respect to time, for various stationary

body cases. All cavities are initially centred at γC = 0.9.

Motion of the cavity centroid is very similar for all cases, however close inspection of Figure

6.32 seems to indicate that the cavity migrates toward the rigid boundary sooner with large

or close solid bodies, which is constant with the earlier touchdown.

CENTROID SEPARATION

Time

Dis

tanc

e

0 0.5 1 1.5 20

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

2.2

0.3 Wide Stationary Body - 0.9 Standoff from Cavity0.1 Wide Stationary Body - 1.05 Standoff from Cavity0.2 Wide Stationary Body - 1.05 Standoff from Cavity0.3 Wide Stationary Body - 1.05 Standoff from Cavity0.3 Wide Stationary Body - 1.2 Standoff from Cavity0.3 Wide Stationary Body - 1.5 Standoff from Cavity

Figure 6.33. Plots of the distance between the cavity centroid and the solid body centroid,

with respect to time, for various stationary body cases. All cavities are initially centred at γC

= 0.9.

6-114

The centroid separation profiles are also very similar, as these plots are really just shifted

centroid motion graphs due the stationary nature of the solid body. However, the increased

velocity of the centroid is perhaps more evident.

SURFACE SEPARATION ALONG VERTICAL AXIS

Time

Dis

tanc

e

0 0.5 1 1.5 20

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

2.2

0.3 Wide Stationary Body - 0.9 Standoff from Cavity0.1 Wide Stationary Body - 1.05 Standoff from Cavity0.2 Wide Stationary Body - 1.05 Standoff from Cavity0.3 Wide Stationary Body - 1.05 Standoff from Cavity0.3 Wide Stationary Body - 1.2 Standoff from Cavity0.3 Wide Stationary Body - 1.5 Standoff from Cavity

Figure 6.34. Plots of the distance between the cavity surface and the solid body surface

along the vertical axis, with respect to time, for various stationary body cases. All cavities

are initially centred at γC = 0.9.

The surface separation plots are similar to the north pole plots for the same reasons

mentioned for the centroid separation. However, it is more evident that the 1.2 standoff case

is the limit where the cavity surface is not artificially halted by the protection zone

subroutine of the model.

6-115

PRESSURE AT RIGID BOUNDARY

Time

Pre

ssur

e

0 0.5 1 1.5 20

1

2

3

4

5

6


Figure 6.35. Plots of the pressure at the rigid boundary directly below the cavity (coordinate


centred at γC = 0.9.

Pressure at the rigid boundary directly below the cavity centroid is unaffected by the

presence of a solid body.

MAXIMUM DOMAIN PRESSURE

Time

Pre

ssur

e

0 0.5 1 1.5 20

5

10

15

20

25




Here we can see the pressure spike captured in Figure 6.8. More interestingly, there is a

smaller spike for the 1.05 case with the 0.3 diameter body. These spikes occur during a

period where the motion of the north pole is most rapid, which is accentuated by the

6-116

smoothing sub-routine of the model due to the high curvature at these times. Therefore, it is

likely that these pressure spikes are also accentuated by the smoothing, but are not actually

caused by it. In any event, these spikes dissipate before touchdown, where there is a slight

trend for slightly higher maximum domain pressure with larger or closer solid bodies.

CAVITY KINETIC ENERGY

Time

Ene

rgy

0 0.5 1 1.5 20

1

2

3

4



stationary body cases. All cavities are initially centred at 0.9.

Except for the 0.9 standoff case, the kinetic energy plots are indistinguishable. As there is no

mechanism for energy to be dissipated at this period of the bubble development, it is likely

that this discrepancy is a product of smoothing the re-entrant jet tip. This highlights the

difficulties of modelling cavities with solid bodies nearer than those shown here, as well as

the necessity to be cautious in manner by which the surface is smoothed.

6-117


Cavities presented in this section have an initial offset from the rigid boundary of 1.2.

6.3.1 Single Cavity

Radial Coordinate

Axi

alC

oord

inat

e

-1 0 10

0.5

1

1.5

2

2.5

1.41.31.21.110.90.80.70.60.50.40.30.20.10



6-118

Radial Coordinate

Axi

alC

oord

inat

e

-1 0 10

0.5

1

1.5

2

2.5

0.60.550.50.450.40.350.30.250.20.150.10.050



Radial Coordinate

Axi

alC

oord

inat

e

-1 0 10

0.5

1

1.5

2

2.5

1816.51513.51210.597.564.531.50



6-119

Radial Coordinate

Axi

alC

oord

inat

e

-1 0 10

0.5

1

1.5

2

2.5

1816.51513.51210.597.564.531.50



Figures 6.38 – 6.41 show the growth and collapse of a single cavity near a rigid boundary.

In this case, the southern surface of the cavity does not make contact with the rigid

boundary. A high speed re-entrant jet develops, as in the 0.9 cavity offset case, however the

high pressure region behind the jet is of significantly higher pressure and touchdown occurs

at a slightly earlier time and smaller volume.

Furthermore, the re-entrant jet is different in profile when compared to the 0.9 offset case

and possesses a more parabolic shape at touchdown.

6-120

6.3.2 0.3 Diameter Solid Body – 0.9 Stand-off from Cavity

Radial Coordinate

Axi

alC

oord

inat

e

-1 0 10

0.5

1

1.5

2

2.5

1.41.31.21.110.90.80.70.60.50.40.30.20.10



Radial Coordinate

Axi

alC

oord

inat

e

-1 0 10

0.5

1

1.5

2

2.5

0.60.550.50.450.40.350.30.250.20.150.10.050



6-121

Radial Coordinate

Axi

alC

oord

inat

e

-1 0 10

0.5

1

1.5

2

2.5

1816.51513.51210.597.564.531.50



Radial Coordinate

Axi

alC

oord

inat

e

-1 0 10

0.5

1

1.5

2

2.5

1816.51513.51210.597.564.531.50



6-122

The results in Figures 6.42 – 6.45 are generally very similar to those for a 0.9 offset cavity

and a similar solid body. Once again there is a high-pressure region at the inner surface,

indicating a propensity for early motion away from the cavity, which then reverses once the

cavity partially envelops the cavity. However, in this case the pressure at the inner surface is

slightly higher for a similar-diameter growing cavity.

During the collapse, there is a very localised extremely high-pressure region directly behind

the early re-entrant jet, which is consistent with plots discussed previously. In addition, the

jet profile is very similar to the 0.9 cavity offset case for a similar solid body, both in the

early jet development as well as at touch down.


Radial Coordinate

Axi

alC

oord

inat

e

-1 0 10

0.5

1

1.5

2

2.5

1.41.31.21.110.90.80.70.60.50.40.30.20.10



6-123

Radial Coordinate

Axi

alC

oord

inat

e

-1 0 10

0.5

1

1.5

2

2.5

0.60.550.50.450.40.350.30.250.20.150.10.050



Radial Coordinate

Axi

alC

oord

inat

e

-1 0 10

0.5

1

1.5

2

2.5

1816.51513.51210.597.564.531.50



units.

6-124

Radial Coordinate

Axi

alC

oord

inat

e

-1 0 10

0.5

1

1.5

2

2.5

1816.51513.51210.597.564.531.50



units.

The plots in Figures 6.46 – 6.49 are consistent with those discussed for a 0.9 offset cavity.

The growth is similar to the immediately preceding case (0.9 solid body offset), although the

solid is not enveloped to the same degree. Once again, the narrowing of the early jet

development is observed and progresses to a squaring of the jet tip at touchdown. In

addition, there is again a broadening and intensification of the high pressure region behind

the jet.

In this case, the additional offset of the cavity from the rigid boundary makes the southern

pole surface more convex. This indicates that the touchdown will not occur at the poles, as

observed in single cavities, but rather it will occur along an annulus thereby separating the

bubble into a small inner bubble connected to a toroidal bubble via a vortex sheet. It is likely

that this inner bubble would rebound first, providing an interesting question as to how this

would interact with the collapse of the toroidal cavity and its ultimate rebound.

6-125


Radial Coordinate

Axi

alC

oord

inat

e

-1 0 10

0.5

1

1.5

2

2.5

1.41.31.21.110.90.80.70.60.50.40.30.20.10



Radial Coordinate

Axi

alC

oord

inat

e

-1 0 10

0.5

1

1.5

2

2.5

0.60.550.50.450.40.350.30.250.20.150.10.050



6-126

Radial Coordinate

Axi

alC

oord

inat

e

-1 0 10

0.5

1

1.5

2

2.5

1816.51513.51210.597.564.531.50



Radial Coordinate

Axi

alC

oord

inat

e

-1 0 10

0.5

1

1.5

2

2.5

1816.51513.51210.597.564.531.50



6-127

Even though the cavity is further away from the rigid boundary, it appears that a 1.2 offset is

near the limit of effect for this size of solid body, again consistent with a cavity nearer to the

rigid boundary. The bubble profiles are very similar to those for a single cavity, although

there is some narrowing of the jet tip at touchdown. Most significant is the broader and

higher pressure region behind the jet, as compared to a single cavity.


CAVITY NORTH POLE

Time

Axi

alC

oord

inat

e

0 0.5 1 1.5 20

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

2.2

Single Cavity0.3 Wide Stationary Body - 0.9 Standoff from Cavity0.3 Wide Stationary Body - 1.05 Standoff from Cavity0.3 Wide Stationary Body - 1.2 Standoff from Cavity0.3 Wide Stationary Body - 1.5 Standoff from Cavity

Figure 6.54. Plots of cavity north pole coordinates, with respect to time, for various


As expected, the north pole motions have the same characteristics as those for a 0.9 offset

cavity. Generally, the closer the solid body, the more delayed the collapse motion of the

north pole. With close bodies, the collapse of the north pole occurs at an increased initial

acceleration, resulting in a slightly increased velocity that results in marginally earlier

touchdown.

6-128

CAVITY SOUTH POLE

Time

Axi

alC

oord

inat

e

0 0.5 1 1.5 20

0.2

0.4

0.6

0.8

1

1.2




For the most part, the cavity south pole is indistinguishable for each case. However, there is

a marginal tendency for the south pole to move away from the rigid boundary slightly more

as the solid body offset distance is reduced.

CAVITY CENTROID

Time

Axi

alC

oord

inat

e

0 0.5 1 1.5 20

0.2

0.4

0.6

0.8

1

1.2


Figure 6.56. Plots of cavity centroid coordinates, with respect to time, for various stationary

body cases. All cavities are initially centred at 1.2.

Once again, there is only marginal difference between the centroid motions for the different

cases, with closer solid bodies generally causing an earlier migration toward the rigid

boundary.

6-129

CENTROID SEPARATION

Time

Dis

tanc

e

0 0.5 1 1.5 20

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

2.2

0.3 Wide Stationary Body - 0.9 Standoff from Cavity0.3 Wide Stationary Body - 1.05 Standoff from Cavity0.3 Wide Stationary Body - 1.2 Standoff from Cavity0.3 Wide Stationary Body - 1.5 Standoff from Cavity


with respect to time, for various stationary body cases. All cavities are initially centred at

1.2.

The centroid separation plot essentially contains the same information as the centroid motion

when the solid body is stationary. However the increased velocity of the 0.9 solid body

offset case is more apparent.


Time

Dis

tanc

e

0 0.5 1 1.5 20

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

2.2

0.3 Wide Stationary Body - 0.9 Standoff from Cavity0.3 Wide Stationary Body - 1.05 Standoff from Cavity0.3 Wide Stationary Body - 1.2 Standoff from Cavity0.3 Wide Stationary Body - 1.5 Standoff from Cavity



are initially centred at 1.2.

6-130

As for the 0.9 cavity offset case, it is more evident that the 1.2 solid body offset is the limit

where the cavity surface is not halted by the protection zone subroutine of the model.


Time

Pre

ssur

e

0 0.5 1 1.5 20

2

4

6




centred at 1.2.

In this case, there is already significant build up in pressure at the rigid boundary prior to

touchdown, that was not present in the 0.9 cavity offset cases. Furthermore, while the

pressure build up begins earlier with reduced solid body standoff, the actual pressure at

touchdown is greatest for the 1.05 offset case. This tends to suggest that there may be a

standoff distance that maximises the pressure at the rigid boundary, which is not simply the

nearest to the initial cavity centroid.

6-131


Time

Pre

ssur

e

0 0.5 1 1.5 20

5

10

15

20

25

30

35

40




As in the 0.9 cavity standoff case, there is a maximum domain pressure spike for the cavity

with the nearest solid body. As was observed in the pressure contour plots, the presence of a

solid body has impact on pressure intensity. However, from Figure 6.60 it is evident that the

pressure builds earlier with reduced solid body standoff, with maximum domain pressure

behaving similarly to the rigid boundary pressure. That is; there may be a solid body

standoff distance that maximises the domain pressure, which is not simply the nearest to the

initial cavity centroid.

6-132


Time

Ene

rgy

0 0.5 1 1.5 20

0.5

1

1.5

2

2.5

3

3.5

4




As expected, the kinetic energy for each case in Figure 6.61 is almost indistinguishable until

touchdown. As mentioned for the 0.9 cavity offset, any discrepancy is likely due to

smoothing occurring more often as the local geometry becomes more extreme toward

touchdown.

6.4 1.5 Cavity Standoff Distance (γC)

For consistency, time series plots are provided in Appendix A for 1.5 cavity offset distances.

As inspection of the plots demonstrates, the general properties of the following case are very

similar to those already discussed. Consequently, any commentary will be limited to the

summary of this chapter.

6-133


For consistency, time series plots are provided in Appendix A for 1.8 cavity offset distances.




6.6 Summary

From the plots presented in this chapter a number of observations can be made. If the

pressure contour plots are considered initially, it is clear that an initial pressure differential

builds up across a stationary solid body. This pressure differential then tends to reverse as

the cavity surface approaches the body and then attempts to draw the solid body into the

cavity. Depending on the proximity of the cavity to the solid body, the motion of the north

pole during collapse is delayed longer with nearer or larger cavities. Further delay in the

collapse motion of the north pole results in a very rapid acceleration of this portion of the

cavity that causes a narrowing of the early re-entrant jet development. The profile of the re-

entrant jet at touchdown is then dependent on the actual offset distance of the solid body, and

is less affected by the offset of the cavity from the rigid boundary. The jet tip profile then

ranges from a general narrowing, to a squaring or flattened profile, to quite irregular shapes.

For a 0.3 diameter body, the standoff limit of the body from the cavity for significant impact

on the jet development is approximately 1.2. Any “wrinkling” of the jet profile is a result of

minor instability at the north pole and is adequately controlled by the smoothing algorithm.

For cavities where the lower surface contacts the rigid boundary, the solid body has little

effect on the motion of the southern pole. When this is not the case, there is a trend for

closer solid bodies to cause earlier collapse motion of the south pole. Furthermore, as the

lower surface of cavities, that do not contact the rigid boundary, are more convex it is

possible for the jet tip profile to produce an annular touchdown. The creation of a small

central bubble may further complicate the touchdown of the toroidal bubble, as it is likely

that the central bubble will rebound first and may interact with the toroid.

Of note is an early high-pressure spike in the domain witnessed by all cases involving a 0.9

standoff of the solid body and some cases for 1.05 solid body offsets. This is interesting as

the duration of the spike is very brief and is always dissipated at touchdown. Because the

spike appears across the time where the cavity surface is heavily smoothed due to high

curvature, it is felt that this spike is real, and not just a numerical artefact. However, it is

likely that the smoothing affects the magnitude of the spike in some manner. In any case, as

6-134

touchdown is the latest point where information regarding the cavity is deemed relatively

accurate, some inferences can be made regarding the pressure in the domain and at the rigid

boundary. Although the pressure at the boundary appears unaffected by the presence of solid

bodies for cases where the bubble’s lower surface contacts the boundary, there is still a trend

for pressure rise in the domain to occur earlier with nearer solid bodies. This trend is more

prominent where the cavity does not contact the rigid boundary, and is present for both the

maximum domain pressure and the rigid boundary. Moreover, it would seem that there

exists an optimum offset distance for a given size of stationary body that maximises the

pressure (both at the rigid boundary and in the domain) at touchdown.

Given that the development of the toroidal cavity must be affected by the pressure in the

domain and the shape of the re-entrant jet, there may be some optimal size and position of

the solid body that will result in a maximum rebound pressure. This is despite the

observation that there is only marginal change in the cavity centroid motion with respect to

solid body size and standoff, and is primarily to do with the radiation of energy away from

the cavity during the re-entrant jet penetration phase as postulated by (Zhang, Duncan et al.

1993). This issue shall discussed further in a later chapter

7-135

Chapter 7

Cavities near Mobile Suspended Bodies

7.1 Introduction

In this chapter the effects of mobile solid bodies on single cavitation bubbles are

investigated. The mobile bodies are no longer held stationary, as in Chapter 6, and are

allowed to move from an initially stationary position as a result of the growth and decay of

the cavity. A range of cases are studied, similar to the preceding chapter, however density of

the mobile solid body is now included. As in Chapter 6, both Pressure Contour and Vector

plots are presented, as well as time series plots.

Pressure Contour and Vector plots are presented for only two cavity standoff distances (γC =

0.9 and 1.2), as this provides sufficient information for the reader to gain insight into the

different pressure profiles while reducing the number of graphs to a digestible number. Only

one solid body standoff distance (γSb = 1.05) is presented as the effects of this parameter

were investigated with sufficient depth in the previous chapter, to make inferences as to how

this impacts on the cavity lifecycle. However, the effect of solid body size is investigated as

well as solid body density. The particle sizes considered range from 0.1 to 2.0 diameter

(DSb), while the densities (ρSb) investigated include:

• 0.7

• 1.0

• 2.8

• 7.5

These choices are to reflect a very light oil, neutrally dense solid, a light alloy and a ferrous

alloy respectively, while the size range is designed to give insight into relatively small

7-136

contaminant sizes, all the way to cloud cavitation where particle size is approximately the

same at the cavity bubble.

Time series data plots are presented for the similar variable ranges as in Chapter 6 (see

Figure 6.1 for a schematic of the variables), except that density is also investigated. Thus,

the variable list now includes:

• Cavity north pole (Np)

• Cavity south pole (Sp)

• Cavity centroid (CC)

• Solid body centroid (CSb)

• Cavity and solid body centroid separation (CSb – CC)

• Axial separation between cavity and solid body (SpSb – NpC)

• Axial pressure at the rigid boundary (P0,0)

• Maximum domain pressure (PMax)

• Cavity kinetic energy (KE)

• Solid body kinetic energy (SE)

It should be noted that all variables presented are in dimensionless terms. Furthermore, the

cavities were discretised using 50 linear elements, while the solid bodies were approximated

using 24 linear elements. The rigid boundary was included via an image bubble described in

Chapter 4.

γC = 0.9 γC = 1.2

Pressure

Contour and

Vector Plots

Time Series

Data

Pressure

Contour and

Vector Plots

Time Series

Data

Imaginary Body (DSb = 0.3) 7.1 – 7.4 7.34 – 7.38

DSb = 0.1 ρSb = 1.0 7.5 – 7.8 –

DSb = 0.2 ρSb = 1.0 7.9 – 7.12 –

ρSb = 0.7 7.17 – 7.20 7.39 – 7.42

ρSb = 1.0 7.13 – 7.16 7.43 – 7.46

DSb = 0.3

ρSb = 2.8 7.21 – 7.22

7.25 – 7.34

7.47 – 7.48

7.51 – 7.60

7-137

ρSb = 7.5 7.23 – 7.24 7.49 – 7.50

ρSb = 0.7

ρSb = 1.0 DSb = 0.6

ρSb = 2.8

ρSb = 0.7

ρSb = 1.0 DSb = 1.0

ρSb = 2.8

ρSb = 0.7

ρSb = 1.0 DSb = 1.4

ρSb = 2.8

ρSb = 0.7

ρSb = 1.0 DSb = 2.0

ρSb = 2.8

– 7.61 – 7.70 –

Table 7.1. Reference grid for figures contained in this Chapter 7.


Figures 6.1-6.4 depict a single cavity with an originally spherical imaginary surface 0.3

diameter offset at 1.05 from the cavity centroid. This is the configuration considered for

much of this thesis. The cavity presented here is exactly the same as the single cavity (0.9

from the rigid boundary) shown in Chapter 6, however the inclusion of the imaginary surface

shows how a deformable particle of same liquid density, no interfacial tension and constant

volume reacts to a close proximity cavity bubble. As in the previous chapter, this is to

provide some basis for comparison for the following mobile body cases.

7-138

7.2.1 0.3 Imaginary Surface – 1.05 Stand-off from Cavity

Radial Coordinate

Axi

alC

oord

inat

e

-1 0 10

0.5

1

1.5

2

2.5

1.31.21.110.90.80.70.60.50.40.30.20.10

Figure 7.1. Pressure and vector plot for a cavity near a rigid boundary (at axial coordinate

0) and an imaginary surface at time 0.1165. Note the reference vector represents one unit.

Radial Coordinate

Axi

alC

oord

inat

e

-1 0 10

0.5

1

1.5

2

2.5

0.60.550.50.450.40.350.30.250.20.150.10.050



7-139

Radial Coordinate

Axi

alC

oord

inat

e

-1 0 10

0.5

1

1.5

2

2.5

7.576.565.554.543.532.521.510.50


0) and an imaginary surface at time 1.9735. Note the reference vector represents ten units.

Radial Coordinate

Axi

alC

oord

inat

e

-1 0 10

0.5

1

1.5

2

2.5

7.576.565.554.543.532.521.510.50



7-140

Inspection of the above plots demonstrates how the imaginary surface deforms to a disk-like

shape as the bubble expands in a divergent manner, and then elongates to an egg-like shape,

as the bubble collapses and the re-entrant jet forms under axial convergent flow. There is

relatively little change in imaginary surface profile from the early jet development to

touchdown as more fluid is drawn laterally immediately above the cavity as indicated by the

vectors.

In addition, the centroid of the imaginary surface moves up during the bubble expansion, and

then down as the bubble collapses. However, this is more noticeable in the time series plots

contained in Section 7.2.8.

7.2.2 0.1 Diameter 1.0 Density Solid Body – 1.05 Stand-off from

Cavity

Radial Coordinate

Axi

alC

oord

inat

e

-1 0 10

0.5

1

1.5

2

2.5

1.31.21.110.90.80.70.60.50.40.30.20.10


body present at 1.05 from the initial cavity centre. Note the reference vector represents one

unit.

7-141

Radial Coordinate

Axi

alC

oord

inat

e

-1 0 10

0.5

1

1.5

2

2.5

0.60.550.50.450.40.350.30.250.20.150.10.050

Figure 7.6. A similar cavity is shown at time 0.5011with a 0.1 diameter 1.0 density solid


unit.

7-142

Radial Coordinate

Axi

alC

oord

inat

e

-1 0 10

0.5

1

1.5

2

2.5

7.576.565.554.543.532.521.510.50


body present at 1.05 from the initial cavity centre. Note the reference vector represents ten

units.

7-143

Radial Coordinate

Axi

alC

oord

inat

e

-1 0 10

0.5

1

1.5

2

2.5

7.576.565.554.543.532.521.510.50



units.

The presence of a 0.1 diameter 1.0 density mobile body has negligible effect on the profile of

the cavity, except for some squaring of the cavity jet tip at touch down and broadening of the

pressure region behind the re-entrant jet (see Figure 7.8). There is, however, significant

motion of the particle, which clearly is in response to any pressure differential across it. This

is a product of the small size, and hence low inertia, of the body that allows it to be more

responsive to the pressure differential as compared to larger diameter or denser bodies.

Thus, making it react more similarly to the surrounding fluid, thereby having less effect on

the cavity.

7-144


Cavity

Radial Coordinate

Axi

alC

oord

inat

e

-1 0 10

0.5

1

1.5

2

2.5

1.31.21.110.90.80.70.60.50.40.30.20.10



unit.

7-145

Radial Coordinate

Axi

alC

oord

inat

e

-1 0 10

0.5

1

1.5

2

2.5

0.60.550.50.450.40.350.30.250.20.150.10.050



unit.

7-146

Radial Coordinate

Axi

alC

oord

inat

e

-1 0 10

0.5

1

1.5

2

2.5

7.576.565.554.543.532.521.510.50



units.

7-147

Radial Coordinate

Axi

alC

oord

inat

e

-1 0 10

0.5

1

1.5

2

2.5

7.576.565.554.543.532.521.510.50



units.

In a similar manner to the 0.1 diameter body, the mobile particle in Figures 7.9-7.12 moves

in reaction to the pressure differential placed across it, thereby moderating pressuring on

either side when compared to the stationary body cases. Close inspection reveals that the 0.2

diameter body does not move as much as the 0.1 diameter body, thus it tends to have greater

impact on the jet development (as indicated by the flattening of the re-entrant jet in Figure

7.11) as well as increasing the breadth of the pressure region behind the re-entrant jet. This

is a product of the more significant altering of the flow field near the cavity, caused by the

larger body, which then interrelates with the pressure field.

7-148


Cavity

Radial Coordinate

Axi

alC

oord

inat

e

-1 0 10

0.5

1

1.5

2

2.5

1.31.21.110.90.80.70.60.50.40.30.20.10



unit.

7-149

Radial Coordinate

Axi

alC

oord

inat

e

-1 0 10

0.5

1

1.5

2

2.5

0.60.550.50.450.40.350.30.250.20.150.10.050



unit.

7-150

Radial Coordinate

Axi

alC

oord

inat

e

-1 0 10

0.5

1

1.5

2

2.5

7.576.565.554.543.532.521.510.50



units.

7-151

Radial Coordinate

Axi

alC

oord

inat

e

-1 0 10

0.5

1

1.5

2

2.5

7.576.565.554.543.532.521.510.50



units.

As the size of the mobile body is further increased, it is evident that the body has further

impact on the pressure and flow fields (as compared to cases with smaller bodies) as shown

by Figure 7.13. The motion of the body is further reduced, thereby disrupting the fluid flow

and creating pressure differentials more similar to a stationary solid body. Furthermore,

there is greater squaring of the re-entrant jet and intensification of the high-pressure region

above the re-entrant jet.

7-152


Cavity

Radial Coordinate

Axi

alC

oord

inat

e

-1 0 10

0.5

1

1.5

2

2.5

1.31.21.110.90.80.70.60.50.40.30.20.10



unit.

7-153

Radial Coordinate

Axi

alC

oord

inat

e

-1 0 10

0.5

1

1.5

2

2.5

0.60.550.50.450.40.350.30.250.20.150.10.050



unit.

7-154

Radial Coordinate

Axi

alC

oord

inat

e

-1 0 10

0.5

1

1.5

2

2.5

7.576.565.554.543.532.521.510.50



unit.

7-155

Radial Coordinate

Axi

alC

oord

inat

e

-1 0 10

0.5

1

1.5

2

2.5

7.576.565.554.543.532.521.510.50



unit.

As density is reduced, it has greater ability to react to the pressure differential across it

thereby moderating the pressure better than the 1.0 density case. However, the collapse

bubble profiles are very similar, as are the pressure contours.


Cavity

As can be seen, the growth phase, of the cavities with different density mobile bodies, is very

similar. There is a tendency for pressure differential to build up with increasing density.

However, this does not result in equalising the motion of the solid body, and consequently

the profile of the cavity looks increasingly similar to the stationary body cases from Chapter

6. To demonstrate this, only the collapse phases are shown for higher density mobile body

cases.

7-156

Radial Coordinate

Axi

alC

oord

inat

e

-1 0 10

0.5

1

1.5

2

2.5

7.576.565.554.543.532.521.510.50



units.

7-157

Radial Coordinate

Axi

alC

oord

inat

e

-1 0 10

0.5

1

1.5

2

2.5

7.576.565.554.543.532.521.510.50



units.

Figures 7.21 and 7.22 show that as the density is increased there is slightly greater narrowing

of the early jet development that results in further squaring of the re-entrant jet at touchdown.

7-158


Cavity

Radial Coordinate

Axi

alC

oord

inat

e

-1 0 10

0.5

1

1.5

2

2.5

7.576.565.554.543.532.521.510.50



units.

7-159

Radial Coordinate

Axi

alC

oord

inat

e

-1 0 10

0.5

1

1.5

2

2.5

7.576.565.554.543.532.521.510.50



units.

Even with density raised significantly to that equivalent to a metal particle, there is only

marginal change in the jet development and pressure in the domain. The general narrowing

of the jet development continues, as well as the squaring of the jet tip at touchdown.

7-160


CAVITY NORTH POLE

Time

Axi

alC

oord

inat

e

0 0.5 1 1.5 20

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

0.3 Wide Imaginary Surface - 1.05 Standoff from Cavity0.3 Wide 0.7 Density Moving Body - 1.05 Standoff from Cavity0.1 Wide 1.0 Density Moving Body - 1.05 Standoff from Cavity0.2 Wide 1.0 Density Moving Body - 1.05 Standoff from Cavity0.3 Wide 1.0 Density Moving Body - 1.05 Standoff from Cavity0.3 Wide 2.8 Density Moving Body - 1.05 Standoff from Cavity0.3 Wide 7.5 Density Moving Body - 1.05 Standoff from Cavity

Figure 7.25. Plots of cavity north pole coordinates, with respect to time, for various moving


Figure 7.25 demonstrates that as density or particle size increases, the motion of the north

pole is further restricted resulting in a more deformed bubble surface. This then delays the

collapse motion of the north pole that results in increased initial acceleration of the collapse

motion. However, the speed and position of the north pole at touchdown is generally the

same regardless of whether there is a mobile body present or not.

7-161

CAVITY SOUTH POLE

Time

Axi

alC

oord

inat

e

0 0.5 1 1.5 20

0.2

0.4

0.6

0.8

1



moving body cases. All cavities are initially centred at 0.9.

The motion of the cavity south pole is unaffected by the presence of a mobile body.

CAVITY CENTROID

Time

Axi

alC

oord

inat

e

0 0.5 1 1.5 20

0.2

0.4

0.6

0.8

1


Figure 7.27. Plots of cavity centroid coordinates, with respect to time, for various moving


The motion of the cavity centroid is unaffected by the presence of a mobile body.

7-162

SOLID BODY CENTROID

Time

Axi

alC

oord

inat

e

0 0.5 1 1.5 2

1.4

1.6

1.8

2

2.2




The motion of the mobile body is clearly reduced as density or width is increased, where

width appears to be a more significant factor during the collapse of the bubble. Furthermore,

as density is raised to 7.5, motion of the mobile body is insignificant when compared to the

other cases, as would be expected.

CENTROID SEPARATION

Time

Dis

tanc

e

0 0.5 1 1.5 2

0.8

1

1.2

1.4

1.60.3 Wide Imaginary Surface - 1.05 Standoff from Cavity0.3 Wide 0.7 Density Moving Body - 1.05 Standoff from Cavity0.1 Wide 1.0 Density Moving Body - 1.05 Standoff from Cavity0.2 Wide 1.0 Density Moving Body - 1.05 Standoff from Cavity0.3 Wide 1.0 Density Moving Body - 1.05 Standoff from Cavity0.3 Wide 2.8 Density Moving Body - 1.05 Standoff from Cavity0.3 Wide 7.5 Density Moving Body - 1.05 Standoff from Cavity



0.9.

7-163

Figure 7.29 shows that the centroid separation generally increases throughout the cavity life

time, albeit at differing rates. 0.7 and 1.0 density particles, all have initially increasing

centroid separations until which time as the mobile body is drawn toward the cavity surface.

Once the surfaces are very near, the protection zone subroutine halts the update of the

contacting surfaces, thereby allowing the cavity to expand as the cavity centroid moves away

from the mobile body. As the cavity collapses, the centroid separation for 0.3 diameter body

cases increases indicating that the cavity is more attracted to the rigid boundary than the

mobile body is attracted to the cavity. While the 0.1 and 0.2 body cases are initially attracted

more to the cavity during the collapse of the cavity (indicated by the decreasing centroid

separation), the cavity centroid ultimately moves toward the rigid boundary at a greater rate.


Time

Dis

tanc

e

0 0.5 1 1.5 20

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8



along the vertical axis, with respect to time, for various moving body cases. All cavities are

initially centred at 0.9.

For the cases considered in Figure 7.30, the cavity surface always approaches the mobile

surface such that the protection zone subroutine halts the update of displacement. Otherwise,

there is a general trend for the axial separation increase with increased density or particle

width, at touchdown.

7-164


Time

Pre

ssur

e

0 0.5 1 1.5 20

1

2

3

4

5

6



[0,0]), with respect to time, for various moving body cases. All cavities are initially centred

at 0.9.

The presence of a mobile particle does not affect pressure at the boundary, similar to the

stationary body case.


Time

Pre

ssur

e

0 0.5 1 1.5 20

1

2

3

4

5

6

7

8




Figure 7.32 shows that maximum domain pressure at touchdown generally increases with the

presence of a mobile body, as compared to the imaginary surface/single cavity case. Once

7-165

again, a pressure spike is witnessed for cases where the cavity surface contacts the solid

body.


Time

Ene

rgy

0 0.5 1 1.5 20

1

2

3

4




There is a marginal reduction in kinetic energy for cases where a mobile body is present.

This is believed to a product of the smoothing of the cavity surface due to the small kinetic

energy possessed by the mobile particle.

7-166

SOLID BODY KINETIC ENERGY

Time

Ene

rgy

0 0.5 1 1.5 20

0.001

0.002

0.003

0.004


Figure 7.34. Plots of the kinetic energy of the solid body, with respect to time, for various


The most significant observation to be made from Figure 7.34 is a rigid body tends to absorb

more energy from the cavity than does a deformable particle. This is evident by the higher

peak kinetic energy for all rigid particles regardless of particle density. Furthermore, during

the cavity growth phase, a denser particle absorbs more energy than a lighter one.

Interestingly enough, this phenomenon is reversed during the collapse phase.

In any case, however, the kinetic energy of the mobile body is insignificant when compared

to the total energy of the system, at least for 0.3 diameter bodies. Moreover, this energy

must then be returned to the cavity at touchdown, as indicated by the reduced particle kinetic

energy at touchdown, as the bubble is the only other entity to which energy can be

exchanged.

7-167


7.3.1 0.3 Imaginary Surface – 1.05 Stand-off from Cavity

Radial Coordinate

Axi

alC

oord

inat

e

-1 0 10

0.5

1

1.5

2

2.5

1.41.31.21.110.90.80.70.60.50.40.30.20.10



7-168

Radial Coordinate

Axi

alC

oord

inat

e

-1 0 10

0.5

1

1.5

2

2.5

0.60.550.50.450.40.350.30.250.20.150.10.050



Radial Coordinate

Axi

alC

oord

inat

e

-1 0 10

0.5

1

1.5

2

2.5

1816.51513.51210.597.564.531.50



7-169

Radial Coordinate

Axi

alC

oord

inat

e

-1 0 10

0.5

1

1.5

2

2.5

1816.51513.51210.597.564.531.50



Figures 7.35-7.38 of a single cavity 1.2 away from a rigid boundary with a 0.3 diameter 1.05

offset imaginary body. The phenomena observed are the same as for the 0.9 standoff cavity.

7-170


Cavity

Radial Coordinate

Axi

alC

oord

inat

e

-1 0 10

0.5

1

1.5

2

2.5

1.41.31.21.110.90.80.70.60.50.40.30.20.10



unit.

7-171

Radial Coordinate

Axi

alC

oord

inat

e

-1 0 10

0.5

1

1.5

2

2.5

0.60.550.50.450.40.350.30.250.20.150.10.050



unit.

7-172

Radial Coordinate

Axi

alC

oord

inat

e

-1 0 10

0.5

1

1.5

2

2.5

1816.51513.51210.597.564.531.50



units.

7-173

Radial Coordinate

Axi

alC

oord

inat

e

-1 0 10

0.5

1

1.5

2

2.5

1816.51513.51210.597.564.531.50



units.

Figures 7.39-7.42 demonstrate that a 0.7 dense rigid mobile particle in unable to move

sufficiently away from the cavity to stop contact of the surface. This then results in a

broader pressure zone behind the re-entrant jet and a squaring of the tip profile at touchdown.

7-174


Cavity

Radial Coordinate

Axi

alC

oord

inat

e

-1 0 10

0.5

1

1.5

2

2.5

1.41.31.21.110.90.80.70.60.50.40.30.20.10



unit.

7-175

Radial Coordinate

Axi

alC

oord

inat

e

-1 0 10

0.5

1

1.5

2

2.5

0.60.550.50.450.40.350.30.250.20.150.10.050



unit.

7-176

Radial Coordinate

Axi

alC

oord

inat

e

-1 0 10

0.5

1

1.5

2

2.5

1816.51513.51210.597.564.531.50



units.

7-177

Radial Coordinate

Axi

alC

oord

inat

e

-1 0 10

0.5

1

1.5

2

2.5

1816.51513.51210.597.564.531.50



units.

As density of the mobile particle increases, the jet profile becomes more similar to that of a

stationary body case of the same standoff distance and particle diameter (see Figures 6.46 –

6.49). This is further exemplified in the following plots contained in Figures 7.47 – 7.50.

7-178


Cavity

Radial Coordinate

Axi

alC

oord

inat

e

-1 0 10

0.5

1

1.5

2

2.5

1816.51513.51210.597.564.531.50



unit.

7-179

Radial Coordinate

Axi

alC

oord

inat

e

-1 0 10

0.5

1

1.5

2

2.5

1816.51513.51210.597.564.531.50



unit.

7-180


Cavity

Radial Coordinate

Axi

alC

oord

inat

e

-1 0 10

0.5

1

1.5

2

2.5

1816.51513.51210.597.564.531.50



unit.

7-181

Radial Coordinate

Axi

alC

oord

inat

e

-1 0 10

0.5

1

1.5

2

2.5

1816.51513.51210.597.564.531.50



unit.


CAVITY NORTH POLE

Time

Axi

alC

oord

inat

e

0 0.5 1 1.5 20

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

2.2

0.3 Wide Imaginary Surface - 1.05 Standoff from Cavity0.3 Wide 0.7 Density Moving Body - 1.05 Standoff from Cavity0.3 Wide 1.0 Density Moving Body - 1.05 Stand from Cavity0.3 Wide 2.8 Density Moving Body - 1.05 Standoff from Cavity0.3 Wide 7.5 Density Moving Body - 1.05 Standoff from Cavity



7-182

The same phenomena of the cavity north pole is observed in Figure 7.51, as for the 0.9

cavity standoff case. Furthermore, the motion of the north pole for each cavity offset case is

indistinguishable except for the offset.

CAVITY SOUTH POLE

Time

Axi

alC

oord

inat

e

0 0.5 1 1.5 20

0.2

0.4

0.6

0.8

1

1.2




The general motion of the cavity south pole is negligibly affected by the presence of a

mobile body.

CAVITY CENTROID

Time

Axi

alC

oord

inat

e

0 0.5 1 1.5 2

0.6

0.8

1

1.2




7-183

The general motion of the cavity centroid is negligibly affected by the presence of a mobile

body.

SOLID BODY CENTROID

Time

Axi

alC

oord

inat

e

0 0.5 1 1.5 2

1.6

1.8

2

2.2

2.4




Interestingly, the motion of the solid body in Figure 7.54 is practically the same as the 0.9

cavity offset case, despite the difference in cavity offset. It therefore seems that any

deformation of the cavity due to a rigid boundary has negligible effect on the mobile body.

CENTROID SEPARATION

Time

Dis

tanc

e

0 0.5 1 1.5 21

1.051.1

1.151.2

1.251.3

1.351.4

1.451.5

1.551.6

1.651.7

1.751.8




1.2.

7-184

As expected, the centroid separation plots have the same general profile as for the 0.9 cavity

offset case, except that magnitudes are slightly increased due to the reduced deformation of

the cavity by the rigid boundary.


Time

Dis

tanc

e

0 0.5 1 1.5 20

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8





Considering that the north pole and the solid body centroid plots are largely indistinguishable

for both the 0.9 and 1.2 cavity standoff cases, it is expected that Figure 7.56 is also the same

as for the 0.9 cavity standoff case.

7-185


Time

Pre

ssur

e

0 0.5 1 1.5 20

1

2

3

4

5




at 1.2.

As the cavity surface is now away from the rigid boundary, it is now possible to observe a

pressure rise at touchdown, similar to the stationary body case. However, as the plot profiles

are the same for different density, we may conclude that density of the solid body has a

negligible effect on pressure at the boundary, although the presence of a body does increase

the pressure slightly.


Time

Pre

ssur

e

0 0.5 1 1.5 20

2

4

6

8

10

12

14

16

18




7-186

The same phenomenon at the rigid boundary is observed for the maximum domain pressure,

except that the magnitude is increased. Once again, the presence of a mobile body increases

the pressure slightly at touchdown.


Time

Ene

rgy

0 0.5 1 1.5 20

0.5

1

1.5

2

2.5

3

3.5

4




Kinetic energy is unaffected by the presence of a mobile body.

7-187


Time

Ene

rgy

0 0.5 1 1.5 20

0.002

0.004

0.006




As the motion of the mobile particle was the same as for the 0.9 offset case, so is the solid

body kinetic energy, as demonstrated by Figure 7.60.


For consistency, time series plots are provided in Appendix B for 1.5 cavity offset distances.




7-188


For consistency, time series plots are provided in Appendix B for 1.8 cavity offset distances.




7.6 Larger Diameter Particles – 0.9 Initial Surface Separation

The following cases investigate a system more applicable to cloud cavitation. This is similar

to a system using ultrasound to create cavitation bubbles where the bubbles are similar in

size to the contaminant bodies. This is different to the previous cases where the moving

bodies are significantly smaller than the cavity, typical of an inducer type cavitating system.

As the particles vary considerably for these cases, the systems are initiated with a constant

distance between the cavity seed surface and the solid particle. Specifically, the initial

distance from the initial cavity north pole and the initial solid body south pole is 0.9. As in

the previous cases, the bodies are spherical and of diameter ranging from 0.3 to 2.0

dimensionless units. In addition, the effects of density are considered with ranges from 0.7

to 2.8 relative densities.


CAVITY NORTH POLE

Time

Axi

alC

oord

inat

e

0 0.5 1 1.5 20

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

0.3 Wide 0.7 Density Moving Body - 0.9 Initial Separation0.3 Wide 1.0 Density Moving Body - 0.9 Initial Separation0.3 Wide 2.8 Density Moving Body - 0.9 Initial Separation0.6 Wide 0.7 Density Moving Body - 0.9 Initial Separation0.6 Wide 1.0 Density Moving Body - 0.9 Inital Separation0.6 Wide 2.8 Density Moving Body - 0.9 Initial Separation1.0 Wide 0.7 Density Moving Body - 0.9 Initial Separation1.0 Wide 1.0 Density Moving Body - 0.9 Inital Separation1.0 Wide 2.8 Density Moving Body - 0.9 Inital Separation1.4 Wide 0.7 Density Moving Body - 0.9 Initial Separation1.4 Wide 1.0 Density Moving Body - 0.9 Initial Separation1.4 Wide 2.8 Density Moving Body - 0.9 Initial Separation2.0 Wide 0.7 Density Moving Body - 0.9 Initial Separation2.0 Wide 1.0 Density Moving Body - 0.9 Initial Separation2.0 Wide 2.8 Density Moving Body - 0.9 Initial Separation



7-189

As was observed in sections 7.2 – 7.5, the larger the mobile body, the more the collapse

motion of the north pole is delayed and there is, therefore, a general increase in initial

acceleration of the north pole. The resultant motion of the north pole does not adhere to such

a simple proportionality, however. As the cavity continues to collapse, we can see that the

velocities of the cavity jet normalise between the various mobile body cases. Furthermore,

complexities in the jet profile ultimately result in the 1.0 diameter body case having an

earlier touchdown than all the others, thereby demonstrating a complex interaction between

the mobile body and the cavity surface such there is body size which will result in a

minimum time to touchdown for this particular separation distance.

Finally, density has a less significant, but complex effect on the motion of the north pole

such that no clear inferences can be made.

CAVITY SOUTH POLE

Time

Axi

alC

oord

inat

e

0 0.5 1 1.5 20

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2




As expected, the south pole is unaffected by body size as the cavities lower surface makes

contact with the rigid boundary.

7-190

CAVITY CENTROID

Time

Axi

alC

oord

inat

e

0 0.5 1 1.5 20

0.2

0.4

0.6

0.8

1




Figure 7.63 shows that the larger suspended bodies tend to delay the motion of the cavity

centroid toward the rigid boundary. However, the velocities of the centroid normalise for

each case, toward touchdown. It is difficult to draw further inferences, as the individual jet

shape affects the point at which touchdown occurs. Therefore, the fact that touchdown

occurs when the cavity centroids for smaller suspended solid cases is closer to the rigid

boundary is likely due to the squaring of the re-entrant jet tip as observed previously.

SOLID BODY CENTROID

Time

Axi

alC

oord

inat

e

0 0.5 1 1.5 21.8

2

2.2

2.4

2.6

2.8

3



7-191

In Figure 7.64 the trends are quite clear, with larger and denser particles moving less.

Furthermore, the interaction between the solid and the cavity becomes more complicated

with body diameter, as a phase extension of the body motion is observed as the inertia of the

suspended body is increased. This is in addition to the increased ability of the solid to slow

the expansion of the upper cavity surface.

CENTROID SEPARATION

Time

Dis

tanc

e

0 0.5 1 1.5 21

1.2

1.4

1.6

1.8

2

2.2

2.4

2.6



1.5. See Figure 7.86 for legend.

Centroid separation in Figure 7.65 demonstrates complex interaction between the two

surfaces observed in Figure 7.64. Furthermore, it becomes clear that density plays an

increasing small role in the centroid separation as the suspended body size increases.

7-192


Time

Dis

tanc

e

0 0.5 1 1.5 20

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8 0.3 Wide 0.7 Density Moving Body - 0.9 Initial Separation0.3 Wide 1.0 Density Moving Body - 0.9 Initial Separation0.3 Wide 2.8 Density Moving Body - 0.9 Initial Separation0.6 Wide 0.7 Density Moving Body - 0.9 Initial Separation0.6 Wide 1.0 Density Moving Body - 0.9 Inital Separation0.6 Wide 2.8 Density Moving Body - 0.9 Initial Separation1.0 Wide 0.7 Density Moving Body - 0.9 Initial Separation1.0 Wide 1.0 Density Moving Body - 0.9 Inital Separation1.0 Wide 2.8 Density Moving Body - 0.9 Inital Separation1.4 Wide 0.7 Density Moving Body - 0.9 Initial Separation1.4 Wide 1.0 Density Moving Body - 0.9 Initial Separation1.4 Wide 2.8 Density Moving Body - 0.9 Initial Separation2.0 Wide 0.7 Density Moving Body - 0.9 Initial Separation2.0 Wide 1.0 Density Moving Body - 0.9 Initial Separation2.0 Wide 2.8 Density Moving Body - 0.9 Initial Separation




The axial separation in Figure 7.66 shows the propensity for larger bodies to slow the motion

of the cavity’s upper surface during the expansion process, as well as the further delay of the

surface separation during collapse. Furthermore, the two surfaces are artificially separated

by the protection zone subroutine for all cases.


Time

Pre

ssur

e

0 0.5 1 1.5 20

1

2

3

4

5




at 1.5.

7-193

As observed previously, the contact of the cavity’s lower surface to the rigid boundary

means that the presence of the solid body has no effect on pressure witnessed at the

boundary.


Time

Pre

ssur

e

0 0.5 1 1.5 20

10

20

30

40

50




The maximum domain pressure presented in Figure 7.88 is affected by the presence of solid

bodies. Once again, we observe the pressure spike associated with the greater initial

acceleration of the north pole. What is more significant, is the trend for the maximum

domain pressure to increase as particle size is increased. This is very significant when

combined with Figures 7.69 and 7.70, as it provides a mechanism whereby energy of the

suspended body is transferred back to the cavity.

7-194


Time

Ene

rgy

0 0.5 1 1.5 20

0.5

1

1.5

2

2.5

3

3.5 0.3 Wide 0.7 Density Moving Body - 0.9 Initial Separation0.3 Wide 1.0 Density Moving Body - 0.9 Initial Separation0.3 Wide 2.8 Density Moving Body - 0.9 Initial Separation0.6 Wide 0.7 Density Moving Body - 0.9 Initial Separation0.6 Wide 1.0 Density Moving Body - 0.9 Inital Separation0.6 Wide 2.8 Density Moving Body - 0.9 Initial Separation1.0 Wide 0.7 Density Moving Body - 0.9 Initial Separation1.0 Wide 1.0 Density Moving Body - 0.9 Inital Separation1.0 Wide 2.8 Density Moving Body - 0.9 Inital Separation1.4 Wide 0.7 Density Moving Body - 0.9 Initial Separation1.4 Wide 1.0 Density Moving Body - 0.9 Initial Separation1.4 Wide 2.8 Density Moving Body - 0.9 Initial Separation2.0 Wide 0.7 Density Moving Body - 0.9 Initial Separation2.0 Wide 1.0 Density Moving Body - 0.9 Initial Separation2.0 Wide 2.8 Density Moving Body - 0.9 Initial Separation



From Figure 7.69, it is clear that in general the cavities’ kinetic energy converge for each

case at touchdown. This is consistent with observations already made, thus it can be

concluded that even though there is initial energy exchange between the cavity and the solid

body, any energy gained by the body is then mostly transferred back to the cavity prior to

touchdown.

7-195


Time

Ene

rgy

0 0.5 1 1.5 20

0.1

0.2

0.3

0.4

0.5

0.6



Figure 7.70 confirms the return of energy to the cavitation bubble, where it is evident that

regardless of solid size or density, there is a definite trend for the body’s kinetic energy to

fall during the collapse of the cavity.

7.7 Summary

Allowing the solid body to move has demonstrated some interesting phenomena. While it

has been observed that an imaginary surface moves in phase with the cavity cycle, this is not

the case for a solid. All solid bodies move less than an imaginary deformable volume, and

consequently they tend to come in contact with the cavity surface. In fact, suspended solids

tend to be drawn toward the cavity by a pressure differential that develops late into the

7-196

growth phase. In general, the larger or denser the solid the less the body moves. This has a

number of effects, including the delay of the collapse motion of the north pole of the cavity

(as discussed in Chapter 6) and an increase in the magnitude of kinetic energy gained by the

moving body.

The delay of the collapse motion of the north pole tends to create pressure spikes in the

domain. However, these dissipate toward touchdown resulting in an overall tendency in

pressures to build up earlier in the domain, which are ultimately higher at touchdown for

larger or denser solid bodies.

The kinetic energy possessed by the solid body is somewhat more complicated. For small

bodies, the energy gained is insignificant at all stages when compared to the kinetic energy

of the cavity. It is only when the solid approaches the same maximum size of the cavity that

any significant energy transfer takes place. However, in all cases the energy of the solid

tends toward zero as the cavity approaches touchdown.

Assuming that this trend continues during the penetration of the cavity, it is logical to

conclude that the motion of the solid body has no significant effect on the kinetic energy of

the cavity bubble at touchdown. Furthermore, despite that there exists a marginal tendency

for the cavity centroid to migrate toward the rigid boundary earlier with smaller moving

bodies, the motion seems to converge again toward touchdown. In any case it is unclear how

this will impact on the cavity centroid at rebound.

This is significant as it is the proximity and kinetic energy of the cavity that is ultimately

responsible for the peak pressure experienced by the rigid boundary at rebound. Therefore, it

would appear the peak pressure, and hence damage potential, of a cavity is unaffected by the

presence of a solid body contaminant, at least when only considering kinetic energy.

Lastly, the behaviour of the energy transfer between the cavity and the solid seems to be

independent on the initial offset of the cavity from the rigid boundary.

8-197

Chapter 8

Conclusion

In this thesis, the general lifecycle of a cavitation bubble was described, with digression

made toward the damage mechanisms of cavitation erosion. In addition, the significant

periods in a cavitation bubble cycle were discussed and listed as:

1. Growth from an initial cavity nucleus

2. Initial collapse

3. Development of the re-entrant jet

4. Touchdown

5. Development of a toroidal cavity

6. Rebound

It was also demonstrated that the surroundings of the cavity has significant impact on its

lifecycle and ultimately the point at which the cavity rebounds. Furthermore, it is believed

the rebound of the cavity is responsible for the maximum pressures experienced by the

domain, and consequently any erosion of the rigid boundary.

The scope of this study was to model cavitation typical of that present in hydraulic systems,

specifically those with particulate contaminants or those using oil-in-water emulsion fluids.

This made the problem one of a very high-speed nature with relatively small cavities. This

had a number of effects, including the neglect of:

• viscous forces,

• effects of gravity,

• mass diffusion,

• and surface tension until rebound.

These general assumptions were then formalised in the mathematical formulation, leading to:

8-198

• irrotationality of the fluid

• the notion that boundaries can be thought of as streamlines

• the relaxation of tangential velocity throughout the domain

• and the subsequent inability to transfer shear forces

Coupled with the assumption of incompressibility, it is then possible to use potential theory

to model the domain, where it is governed by:

φ∇=u

and 02 =∇ φ

For the purposes of modelling cavitation bubble collapse, the BEM was shown to be

inherently superior to FEA as it resolves the phase boundary due to the collocation points

lying directly on the boundary. Correct determination of the phase boundary is critical, as it

is the collapse and rebound of the cavity that produces the peak pressures responsible for

erosion of the rigid boundary. This resulted in Green’s integral formula being reformulated

into a Fredholm problem allowing the solution of velocity potential or its normal derivative,

depending on which is initially known. Once both variables have been determined, potential

at any point throughout the domain could be found for a given time period.

The inability to accurately model the toroidal rebound forces the use of total kinetic energy

at touchdown as a quantitative damage potential indicator. Thus, only the Convectional

Boundary Integral Equation (CBIE) was used as this was sufficient to model to touchdown,

which was the only accurate common period in the bubble cycle for all cases.

For cases where a solid body was present, the potential on the cavity surface was given, and

the normal derivative of potential given for solid bodies. The domain was then marched in

time using a modified Bernoulli’s equation such that the material derivative was included.

As a consequence, Lagrangian particles were tracked in time to determine the pressure at the

surface such that a kinematic relation for the motion of the body could be established. All

variables were then advanced using a Euler scheme where the time step was based upon

minimising the maximum incremental change in potential in the domain.

This ultimately allowed the development of a model to investigate the effect that suspended

bodies have on cavitation. This was in order to derive some insight into whether a

suspended contaminant would affect the characteristics of a single cavity, thereby

influencing its erosion potential. To achieve this, two distinct cases were considered:

1. Stationary solid bodies

8-199

2. Suspended solid bodies of differing density which were allowed to move

according to the mechanics of the cavity growth

Although deformable contaminants (i.e., those in an oil-in-water emulsion) were not

modelled, the use of rigid bodies still provided an analogy to the effects that liquid oil

droplets, as well as ingested particles, have on cavitation.

From the cases involving a stationary body, a number of observations were made. From the

pressure contour plots, it was clear that an initial pressure differential built up across the

stationary solid body. This pressure differential then tended to reverse as the cavity surface

approached the body, thus attempting to draw the solid body toward the cavity. Depending

on the proximity of the cavity to the solid body, the motion of the north pole during collapse

was delayed longer with nearer or larger cavities. Further delay in the collapse motion of the

north pole resulted in a very rapid acceleration of this portion of the cavity that caused a

narrowing of the early re-entrant jet development. The profile of the re-entrant jet at

touchdown was dependent on the actual offset distance of the solid body, while the cavity

offset from the rigid boundary tended to have greater influence on the opposing side. In

some cases, a squaring of the jet tip was observed, which has also been witnessed in cavities

near a free surface.

In some cases, a more convex lower bubble profile caused the jet tip profile to produce an

annular touchdown. It was then postulated that the resultant creation of a small central

bubble would rebound first, thus interacting with the toroidal collapse.

Of special note was an early high-pressure spike in the domain witnessed by all cases

involving a 0.9 standoff of the solid body, and some cases for 1.05 solid body offsets. This

is of interest as the duration of the spike was very brief and was always dissipated at

touchdown. Because the spike appears during and after the time where the cavity surface is

heavily smoothed, it was felt that this spike is real, and not just a numerical artefact.

However, it is likely that the smoothing affects the magnitude of the spike in some manner.

In any case, inferences were made regarding the pressure in the domain and at the rigid

boundary as the spike dissipated at touchdown. In general, there was a trend for pressure rise

in the domain to occur earlier with nearer solid bodies. This trend was more prominent

where the cavity did not contact the rigid boundary, and was present for both the maximum

domain pressure and the rigid boundary. Moreover, it would seem that an optimum offset

distance existed for a given size of stationary body that maximised the pressure (both at the

rigid boundary and in the domain) at touchdown.

8-200

Allowing the solid body to move demonstrated some interesting phenomena, but was in

general not dissimilar to the stationary cases. While it was observed that an imaginary

surface moves in phase with the cavity cycle, this is not the case for a solid. All solid bodies

moved less than an imaginary deformable volume, and consequently they tended to come in

contact with the cavity surface. In general, the larger or denser the solid the less the body

moved. This has a number of effects, including the delay of the collapse motion of the north

pole of the cavity and an increase in the magnitude of kinetic energy gained by the moving

body. This was consistent with the pressure differentials observed in the stationary cases.

As already observed, the delay of the collapse motion of the north pole tended to create

pressure spikes in the domain, which dissipated toward touchdown resulting in an overall

tendency in earlier pressures to build up in the domain, that were ultimately higher at

touchdown for larger or denser solid bodies.

The kinetic energy possessed by the solid body was somewhat more complicated. For small

bodies, the energy gained was insignificant at all stages when compared to the kinetic energy

of the cavity. It is only when the solid approaches the same maximum size of the cavity that

any significant energy transfer took place. However, in all cases the energy of the solid

tended towards zero as the cavity approached touchdown.

Assuming that this trend continued during the penetration of the cavity, it can be concluded

that the motion of the solid body has no significant effect on the kinetic energy of the cavity

bubble at touchdown. Furthermore, despite that there was a marginal tendency for the cavity

centroid to migrate toward the rigid boundary earlier with smaller mobile bodies, the motion

seemed to converge again toward touchdown. In any case it is unclear how this would

impact on the cavity centroid at rebound. Furthermore, the behaviour of the energy transfer

between the cavity and the solid appeared to be independent of the initial offset of the cavity

from the rigid boundary.

Lastly, discussion should be made regarding the energy in the re-entrant jet. It has been

suggested that the energy in the jet will be unavailable for compression of the toroid. If this

were the case, then analysis of the toroidal jet at touchdown may provide valuable

information regarding the damage potential of the cavity at rebound. It is, however, the

opinion of this thesis that no sweeping statements can be made, regarding what portion of the

jet energy will be available for compression of the solute gas content. This will be case

dependant. If we consider the case of a cavity very near a rigid boundary, then the jet will be

redirected in a radial fashion; directly compressing the bubble. If the cavity was further

away from the boundary, then the jet would tend to continue in a more linear path, thereby

8-201

producing a more circulatory flow. How this then affects the late stages of collapse is

unclear, which highlights the questionable value of this analysis. Consequently, analysis of

the jet was not carried out in this thesis.

Thus, it is conclusion of this thesis that the only significant impact that suspended body, of

any size or density, has on the development of a cavitation bubble is on the profile of the re-

entrant jet. As the kinetic energy, and ultimate motion and position of the cavity centroid is

unaffected by the solid’s presence, it is logical that the only way the solid can affect the

amount of kinetic energy that is converted to potential energy at rebound (i.e., pressure at

rebound) is via radiation of energy during penetration. The issue then becomes; does the

shape of the re-entrant jet have significant impact of the energy dissipation of the cavity

during penetration? To answer this would require the accurate modelling of the toroidal

cavity to rebound, which is one area of further research.

Thus, it can be concluded that it is unlikely that the presence of any suspension near a

cavitation bubble directly alters the damage potential. The observed erosion rate change by

Pai and Hargreaves (2002) must therefore be the product of some other mechanism.

However, modelling of the penetration process would be required to make this statement

definitive. This would involve determining the source of the instability or improving the

“cut method” utilised by (Best 1993).

Further investigation could also be conducted into bubble growth very near solid

contaminants; however this work should be coupled with laboratory experiments of this

bubble development. Finally, cloud cavitation phenomenon would benefit from studies

considering larger particles, the limit of which would be two parallel rigid boundaries.

203

References

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Blake, J. R., B. B. Taib, et al. (1986). "Transient cavities near boundaries. Part 1. Rigid boundary." J. Fluid Mech. 170: 479-497.

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Brujan, E.-A., K. Nahen, et al. (2001). "Dynamics of laser-induced cavitation bubbles near an elastic boundary." J. Fluid Mech. 433: 251-281.

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Cartwright, D. J. (2001). Underlying Principles of the Boundary Element Method. Southampton, WIT Press.

Chahine, G. L. and K. M. Kalumuck (1998). "The influence of structural deformation on water jet impact loading." Journal of Fluids and Structures 12: 103-121.

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Duncan, J. H. and S. Zhang (1991). "On the intercation of a collapsing cavity and a compliant wall." J. Fluid Mech. 226: 401-423.

Friberg, S. E. (1992). Emulsion stability. Emulsions - A Fundimental Approach. J. Sjoblom. Dordrecht, Kluwer Academic Publishers: 1-24.

Gopal, E. S. R. (1967). Principals of Emulsion Formation. Emulsion Science. P. Sherman. London, Academic Press: 1-69.

Gould, G. C. (1973). The Role of Mechanical Properties in Cavitation Erosion Resistance. Mechanical Failures Prevention Group: The Role of Cavitation in Mechanical Failures, Boulder, Colorado, USA, National Bureau of Standards.

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Hastings, C. (1955). Approximations for Digital Computers. Princeton, Princeton University Press.

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205

Appendix A

A.1 1.5 Cavity Standoff Distance (γC) Time Series Plots

CAVITY NORTH POLE

Time

Axi

alC

oord

inat

e

0 0.5 1 1.5 20

0.5

1

1.5

2

2.5

Single Cavity0.3 Wide Stationary Body - 0.9 Standoff from Cavity0.3 Wide Stationary Body - 1.2 Standoff from Cavity0.3 Wide Stationary Body - 1.5 Standoff from Cavity

Figure A. 1. Plots of cavity north pole coordinates, with respect to time, for various


206

CAVITY SOUTH POLE

Time

Axi

alC

oord

inat

e

0 0.5 1 1.5 20

0.5

1

1.5


Figure A. 2. Plots of cavity south pole coordinates, with respect to time, for various


CAVITY CENTROID

Time

Axi

alC

oord

inat

e

0 0.5 1 1.5 20

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6


Figure A. 3. Plots of cavity centroid coordinates, with respect to time, for various stationary


207

CENTROID SEPARATION

Time

Dis

tanc

e

0 0.5 1 1.5 20

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

0.3 Wide Stationary Body - 0.9 Standoff from Cavity0.3 Wide Stationary Body - 1.2 Standoff from Cavity0.3 Wide Stationary Body - 1.5 Standoff from Cavity

Figure A. 4. Plots of the distance between the cavity centroid and the solid body centroid,


1.5.


Time

Dis

tanc

e

0 0.5 1 1.5 20

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8


Figure A. 5. Plots of the distance between the cavity surface and the solid body surface



208


Time

Pre

ssur

e

0 0.5 1 1.5 20

1

2

3

4

5

6

7


Figure A. 6. Plots of the pressure at the rigid boundary directly below the cavity (coordinate


centred at 1.5.


Time

Pre

ssur

e

0 0.5 1 1.5 20

5

10

15

20

25

30

35

40

45


Figure A. 7. Plots of the maximum pressure in the domain, with respect to time, for various


209


Time

Ene

rgy

0 0.5 1 1.5 20

0.5

1

1.5

2

2.5

3

3.5

4


Figure A. 8. Plots of the kinetic energy of the cavity, with respect to time, for various


210

A.2 1.8 Cavity Standoff Distance (γC) Time Series Plots

CAVITY NORTH POLE

Time

Axi

alC

oord

inat

e

0 0.5 1 1.50

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

2.2

2.4

2.6

2.8


Figure A. 9. Plots of cavity north pole coordinates, with respect to time, for various


CAVITY SOUTH POLE

Time

Axi

alC

oord

inat

e

0 0.5 1 1.50

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8


Figure A. 10. Plots of cavity south pole coordinates, with respect to time, for various


211

CAVITY CENTROID

Time

Axi

alC

oord

inat

e

0 0.5 1 1.50

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2


Figure A. 11. Plots of cavity centroid coordinates, with respect to time, for various


CENTROID SEPARATION

Time

Dis

tanc

e

0 0.5 1 1.50

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8


Figure A. 12. Plots of the distance between the cavity centroid and the solid body centroid,


1.8.

212


Time

Dis

tanc

e

0 0.5 1 1.50

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8


Figure A. 13. Plots of the distance between the cavity surface and the solid body surface




Time

Pre

ssur

e

0 0.5 1 1.50

1

2

3

4

5

6

7

8

9

10

11

12


Figure A. 14. Plots of the pressure at the rigid boundary directly below the cavity

(coordinate [0,0]), with respect to time, for various stationary body cases. All cavities are


213


Time

Pre

ssur

e

0 0.5 1 1.5

10

20

30

40

50

60

70


Figure A. 15. Plots of the maximum pressure in the domain, with respect to time, for

various stationary body cases. All cavities are initially centred at 1.8.

214


Time

Ene

rgy

0 0.5 1 1.50

0.5

1

1.5

2

2.5

3

3.5

4


Figure A. 16. Plots of the kinetic energy of the cavity, with respect to time, for various


215

Appendix B

B.1 1.5 Cavity Stand-off Distance (γC) Time Series Plots

CAVITY NORTH POLE

Time

Axi

alC

oord

inat

e

0 0.5 1 1.5 20

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

2.2

2.4

0.3 Wide Imaginary Surface - 1.05 Standoff from Cavtiy0.3 Wide 0.7 Density Moving Body - 1.05 Standoff from Cavity0.3 Wide 1.0 Density Moving Body - 1.05 Standoff from Cavtiy0.3 Wide 2.8 Density Moving Body - 1.05 Standoff from Cavity0.3 Wide 7.5 Density Moving Body - 1.05 Standoff from Cavity

Figure B. 1. Plots of cavity north pole coordinates, with respect to time, for various moving


216

CAVITY SOUTH POLE

Time

Axi

alC

oord

inat

e

0 0.5 1 1.5 20

0.2

0.4

0.6

0.8

1

1.2

1.4


Figure B. 2. Plots of cavity south pole coordinates, with respect to time, for various moving


CAVITY CENTROID

Time

Axi

alC

oord

inat

e

0 0.5 1 1.5 20

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6


Figure B. 3. Plots of cavity centroid coordinates, with respect to time, for various moving


217

SOLID BODY CENTROID

Time

Axi

alC

oord

inat

e

0 0.5 1 1.5 22

2.1

2.2

2.3

2.4

2.5

2.6

2.7

2.8

2.9

3


Figure B. 4. Plots of solid body centroid coordinates, with respect to time, for various


CENTROID SEPARATION

Time

Dis

tanc

e

0 0.5 1 1.5 2

0.8

1

1.2

1.4


Figure B. 5. Plots of the distance between the cavity centroid and the solid body centroid,


1.5.

218


Time

Dis

tanc

e

0 0.5 1 1.5 20

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6


Figure B. 6. Plots of the distance between the cavity surface and the solid body surface




Time

Pre

ssur

e

0 0.5 1 1.5 20

1

2

3

4

5

6

7

8


Figure B. 7. Plots of the pressure at the rigid boundary directly below the cavity (coordinate


at 1.5.

219


Time

Pre

ssur

e

0 0.5 1 1.5 20

5

10

15

20

25

30

35


Figure B. 8. Plots of the maximum pressure in the domain, with respect to time, for various


220


Time

Ene

rgy

0 0.5 1 1.5 20

0.5

1

1.5

2

2.5

3

3.5

4


Figure B. 9. Plots of the kinetic energy of the cavity, with respect to time, for various


221


Time

Ene

rgy

0 0.5 1 1.5 20

0.002

0.004

0.006


Figure B. 10. Plots of the kinetic energy of the solid body, with respect to time, for various


222

B.2 1.8 Cavity Stand-off Distance (γC) Time Series Plots

CAVITY NORTH POLE

Time

Axi

alC

oord

inat

e

0 0.5 1 1.5 20

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

2.2

2.4

2.6

2.8

0.3 Wide Imaginary Surface - 1.05 Standoff from Cavity0.3 Wide 0.7 Density Moving Body - 1.05 Standoff from Cavity0.3 Wide 1.0 Density Moving Body - 1.05 Standoff from Cavity0.3 Wide 2.8 Density Moving Body - 1.05 Standoff from Cavity0.3 Wide 7.5 Density Moving Body - 1.05 Standoff from Cavity

Figure B. 11. Plots of cavity north pole coordinates, with respect to time, for various


CAVITY SOUTH POLE

Time

Axi

alC

oord

inat

e

0 0.5 1 1.5 20

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8


Figure B. 12. Plots of cavity south pole coordinates, with respect to time, for various


223

CAVITY CENTROID

Time

Axi

alC

oord

inat

e

0 0.5 1 1.50

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2


Figure B. 13. Plots of cavity centroid coordinates, with respect to time, for various moving


SOLID BODY CENTROID

Time

Axi

alC

oord

inat

e

0 0.5 1 1.5 22.2

2.3

2.4

2.5

2.6

2.7

2.8

2.9

3

3.1

3.2


Figure B. 14. Plots of solid body centroid coordinates, with respect to time, for various


224

CENTROID SEPARATION

Time

Dis

tanc

e

0 0.5 1 1.5 21

1.05

1.1

1.15

1.2

1.25

1.3

1.35

1.4

1.45

1.5


Figure B. 15. Plots of the distance between the cavity centroid and the solid body centroid,


1.8.


Time

Dis

tanc

e

0 0.5 1 1.5 20

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1.1

1.2

1.3

1.4

1.5


Figure B. 16. Plots of the distance between the cavity surface and the solid body surface



225


Time

Pre

ssur

e

0 0.5 1 1.5 20

1

2

3

4

5

6

7

8

9

10

11

12


Figure B. 17. Plots of the pressure at the rigid boundary directly below the cavity

(coordinate [0,0]), with respect to time, for various moving body cases. All cavities are



Time

Pre

ssur

e

0 0.5 1 1.5 20

10

20

30

40

50

60

70


Figure B. 18. Plots of the maximum pressure in the domain, with respect to time, for

various moving body cases. All cavities are initially centred at 1.8.

226


Time

Ene

rgy

0 0.5 1 1.50

0.5

1

1.5

2

2.5

3

3.5

4


Figure B. 19. Plots of the kinetic energy of the cavity, with respect to time, for various


227


Time

Ene

rgy

0 0.5 1 1.5 20

0.002

0.004

0.006


Figure B. 20. Plots of the kinetic energy of the solid body, with respect to time, for various


modelling the effect of suspended bodies on...

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