Fundamentals of Cavitation FLUID MECHANICS and ITS APPLICATIONS Volume 76

Fundamentals of Cavitation FLUID MECHANICS AND ITS APPLICATIONS Volume 76

Series Editor: R. MOREAU MADYLAM Ecole Nationale Supérieure d'Hydraulique de Grenoble Boîte Postale 95 38402 Saint Martin d'Hères Cedex, France

Aims and Scope of the Series The purpose of this series is to focus on subjects in which fluid mechanics plays a fundamental role. As well as the more traditional applications of aeronautics, hydraulics, heat and mass transfer etc., books will be published dealing with topics which are currently in a state of rapid development, such as turbulence, suspensions and multiphase fluids, super and hypersonic flows and numerical modelling techniques. It is a widely held view that it is the interdisciplinary subjects that will receive intense scientific attention, bringing them to the forefront of technological advancement. Fluids have the ability to transport matter and its properties as well as transmit force, therefore fluid mechanics is a subject that is particulary open to cross fertilisation with other sciences and disciplines of engineering. The subject of fluid mechanics will be highly relevant in domains such as chemical, metallurgical, biological and ecological engineering. This series is particularly open to such new multidisciplinary domains. The median level of presentation is the first year graduate student. Some texts are mono- graphs defining the current state of a field; others are accessible to final year undergra- duates; but essentially the emphasis is on readability and clarity.

For a list of related mechanics titles, see final pages. Fundamentals of Cavitation by

JEAN-PIERRE FRANC Research Director (CNRS), Turbomachinery and Cavitation Research Group, Laboratory of Geophysical and Industrial Fluid Flows (LEGI) of the Grenoble University (Institut National Polytechnique de Grenoble (INPG) & Université Joseph Fourier (UJF), France and JEAN-MARIE MICHEL Presently retired, was Research Director (CNRS) and Head of the Cavitation Research Group in the Laboratory of Geophysical and Industrial Fluid Flows (LEGO) of the Grenoble University (Institut National Polytechnique de Grenoble (INPG) & Université Joseph Fourier (UJF), France

KLUWER ACADEMIC PUBLISHERS NEW YORK, BOSTON, DORDRECHT, LONDON, MOSCOW eBook ISBN: 1-4020-2233-6 Print ISBN: 1-4020-2232-8

No part of this eBook may be reproduced or transmitted in any form or by any means, electronic, mechanical, recording, or otherwise, without written consent from the Publisher

Created in the United States of America

Visit Springer's eBookstore at: http://ebooks.kluweronline.com and the Springer Global Website Online at: http://www.springeronline.com Grenoble Sciences

"Grenoble Sciences", directed by Prof. Jean Bornarel, was created ten years ago by the Joseph Fourier University of Grenoble, France (Science, Technology and Medecine) to select and publish original projects. Anonymous referees choose the best projects and then a Reading Committee interacts with the authors as long as necessary to improve the quality of the manuscripts.

(Contact: Tel: (33) 4 76 51 46 95 - E-mail: [email protected])

The "Fundamentals of Cavitation" Reading Committee included the following members:

® Professeur Hiroharu Kato, Faculty of Engineering, Tokyo University, Japan ® Professeur Kirill V. Rozhdestvensky, Saint-Petersburg State Marine Technical University, Russia ® Professeur Dr.-Ing. Bernd Stoffel, Darmstadt University of Technology, Germany

Front Cover Photo: Photograph Bassin d’Essais des Carènes, Délégation Générale à l’Armement (DGA), Val de Reuil, France. TABLE OF CONTENTS

Foreword by Dr. Hiroharu KATO...... XIII

Preface ...... XV

Symbols...... XIX

1. Introduction – The main features of cavitating flows...... 1 1.1. The physical phenomenon ...... 1 1.1.1. Definition...... 1 1.1.2. Vapor pressure ...... 2 1.1.3. The main forms of vapor cavities...... 4 1.2. Cavitation in real liquid flows...... 5 1.2.1. Cavitation regimes...... 5 1.2.2. Typical situations favorable to cavitation ...... 5 1.2.3. The main effects of cavitation in hydraulics ...... 6 1.3. Specific features of cavitating flow...... 7 1.3.1. Pressure and pressure gradient...... 7 1.3.2. Liquid-vapor interfaces ...... 8 1.3.3. Thermal effects ...... 10 1.3.4. Some typical orders of magnitude...... 10 1.4. Non-dimensional parameters...... 10

1.4.1. Cavitation number sv...... 10 1.4.2. Cavitation number at inception, svi...... 11 1.4.3. Relative underpressure of a cavity, sc ...... 12 1.5. Some historical aspects ...... 13 References ...... 14

2. Nuclei and cavitation...... 15 2.1. Introduction ...... 15 2.1.1. Liquid tension...... 15 2.1.2. Cavitation nuclei ...... 15 2.2. Equilibrium of a nucleus...... 17 2.2.1. Equilibrium condition [BLAKE 1949]...... 17 2.2.2. Stability and critical pressure of a nucleus...... 18 2.2.3. Nucleus evolution in a low pressure region...... 20 2.3. Heat and mass diffusion...... 21 2.3.1. The thermal behavior of the gas content ...... 21 2.3.2. Gas diffusion and nucleus stability ...... 23 2.4. Nucleus population ...... 27 2.4.1. Measurement methods...... 27 2.4.2. Conditions for inception of bubble cavitation...... 30 References ...... 32

3. The dynamics of spherical bubbles...... 35 3.1. Basic equations ...... 35 3.1.1. Introduction ...... 35 3.1.2. Assumptions ...... 35 3.1.3. Boundary and initial conditions...... 36 3.1.4. RAYLEIGH-PLESSET equation ...... 36 3.1.5. Interpretation of the RAYLEIGH-PLESSET equation in terms of energy balance...... 37 3.2. The collapse of a vapor bubble...... 38 3.2.1. Assumptions ...... 38 3.2.2. The interface velocity ...... 38 3.2.3. The pressure field...... 40 3.2.4. Remark on the effect of surface tension...... 41 3.3. The explosion of a nucleus ...... 42 3.3.1. The interface velocity ...... 42

3.3.2. The equilibrium case (p• = p•0) ...... 43 3.3.3. The case of nucleus growth (p•

4. Bubbles in a non-symmetrical environment...... 57 4.1. Introduction ...... 57 4.2. Motion of a spherical bubble in a liquid at rest...... 57 4.2.1. Translation of a solid sphere in a liquid at rest ...... 57 4.2.2. Translation with simultaneous volume variations...... 58 4.2.3. Application to bubbles...... 59

VIII 4.3. Non-spherical bubble evolution...... 60 4.3.1. Principle of PLESSET-CHAPMAN numerical modeling...... 60 4.3.2. Some general results...... 61 4.3.3. BLAKE's analytical approach...... 64 4.4. The path of a spherical bubble...... 67 References ...... 71 Appendix to section 4.3.3 ...... 72

5. Further insights into bubble physics...... 77 5.1. The effect of compressibility ...... 77 5.1.1. TAIT's equation of state...... 77 5.1.2. Basic equations ...... 78 5.1.3. The quasi acoustic solution [HERRING 1941 & TRILLING 1952]...... 79 5.1.4. The GILMORE approach (1952) ...... 80 5.2. Bubble noise ...... 83 5.2.1. Basic equations ...... 83 5.2.2. Weak bubble oscillations ...... 84 5.2.3. Noise of a collapsing bubble ...... 85 5.3. Some thermal aspects ...... 86 5.3.1. The idea of thermal delay...... 86 5.3.2. BRENNEN's analysis (1973) ...... 89 5.4. A typical numerical solution...... 92 References ...... 95 Appendix to section 5.1.3 ...... 96

6. Supercavitation...... 97 6.1. Physical aspects of supercavities...... 98 6.1.1. Cavity pressure...... 98 6.1.2. Cavity detachment...... 98 6.1.3. Cavity closure ...... 101 6.1.4. Cavity length...... 102 6.2. Supercavity flow modeling using steady potential flow theory...... 105 6.2.1. The main parameters...... 105 6.2.2. Equations and boundary conditions ...... 106 6.2.3. Cavity closure models...... 107 6.2.4. Overview of calculation techniques ...... 108 6.3. Typical results...... 110 6.3.1. Infinite cavity behind a flat plate in an infinite flow field...... 110 6.3.2. Finite cavity behind a symmetrical body in an infinite flow field ...... 111 6.3.3. Finite cavity behind a circular arc in an infinite flow field ...... 112 6.3.4. Variation of lift and drag coefficients with cavity underpressure...... 113

6.3.5. Effect of submersion depth on the slope of the curve CL(a)...... 114

IX 6.4. Axisymmetric cavities...... 115 6.4.1. The GARABEDIAN asymptotic solution for steady supercavities...... 115 6.4.2. Momentum balance and drag...... 116 6.4.3. Approximate, analytic solution for steady supercavities...... 117 6.4.4. Unsteady axisymmetric supercavities ...... 121 6.5. Specific problems ...... 124 6.5.1. Unsteady 2D supercavities...... 124 6.5.2. Compressible effects in supercavitating flows...... 125 References ...... 126 Appendix: singular behavior at detachment...... 129

7. Partial cavities ...... 131 7.1. Partial cavities on two-dimensional foils...... 131 7.1.1. Main patterns...... 131 7.1.2. Cavity closure ...... 133 7.1.3. Cavity length...... 134 7.1.4. Three-dimensional effects due to an inclination of the closure line ...... 135 7.1.5. Multiple shedding on 2D hydrofoils...... 137 7.2. Partial cavities in internal flows...... 138 7.3. The cloud cavitation instability ...... 140 7.3.1. Conditions for the onset of the cloud cavitation instability...... 140 7.3.2. Global behavior ...... 141 7.3.3. Pulsation frequency...... 143 7.3.4. Jet thickness...... 144 7.4. Wakes of partial cavities...... 145 7.4.1. Mean pressure distribution...... 145 7.4.2. Production of vapor bubbles...... 146 7.4.3. Pressure fluctuations ...... 147 7.4.4. Wall pressure pulses at cavity closure ...... 148 7.4.5. Scaling of pulse spectra...... 150 7.4.6. Main features of the noise emitted by partial cavities ...... 152 7.5. Thermal effects in partial cavitation...... 153 7.5.1. The STEPANOFF B-factor ...... 153 7.5.2. The entrainment method ...... 154 7.6. System instability...... 159 7.7. Partial cavity flow modeling...... 161 References ...... 162 Appendix: sonic velocity in a liquid/vapor mixture with phase change...... 165

8. Bubbles and cavities on two-dimensional foils...... 169 8.1. Attached cavitation...... 169 8.1.1. Cavitation inception on a circular cylinder...... 169 8.1.2. Cavity patterns on a two-dimensional foil...... 172

X 8.1.3. Boundary layer features on a slender foil ...... 174 8.1.4. The connection between laminar separation and detachment...... 176 8.2. Traveling bubble cavitation ...... 179 8.2.1. The effect of water quality and nuclei seeding...... 179 8.2.2. Scaling law for developed bubble cavitation...... 182 8.2.3. Saturation...... 184 8.3. Interaction between bubbles and cavities...... 186 8.3.1. Effect of exploding bubbles on a cavity...... 186 8.3.2. Critical nuclei concentration for transition between attached cavitation and traveling bubble cavitation...... 187 8.3.3. The prediction of cavitation patterns ...... 188 8.4. Roughness and cavitation inception...... 189 References ...... 191

9. Ventilated supercavities ...... 193 9.1. Two-dimensional ventilated cavities ...... 193 9.1.1. Ventilated hydrofoils...... 193 9.1.2. The main parameters...... 194 9.1.3. Cavity length...... 196 9.1.4. Air flowrate and cavity pressure...... 199 9.1.5. Pulsation regimes...... 202 9.1.6. Pulsation frequency...... 205 9.1.7. Concerning the pulsation mechanism...... 206 9.2. Axisymmetric ventilated supercavities ...... 209 9.2.1. Different regimes of ventilated cavities...... 209 9.2.2. Gas evacuation by toroidal vortices ...... 210 9.2.3. Deformation of the cavity axis by gravity...... 210 9.2.4. Gas evacuation by two hollow tube vortices...... 211 9.3. Analysis of pulsating ventilated cavities...... 214 9.3.1. Basic equations ...... 214 9.3.2. Analysis of the pressure fluctuation equation...... 217 9.3.3. Comparison with experiments ...... 218 References ...... 220

10. Vortex cavitation ...... 223 10.1. Theoretical results...... 223 10.1.1. Basic vorticity theorems...... 223 10.1.2. The main effects of cavitation on rotational flows...... 224 10.1.3. Axisymmetric cavitating vortex...... 226 10.1.4. Toroidal cavitating vortex ...... 227 10.2. The non-cavitating tip vortex ...... 231 10.2.1. Tip vortex formation ...... 231 10.2.2. Vortex models in viscous fluids ...... 232 10.2.3. Tip vortex structure ...... 234

XI 10.3. Cavitation in a tip vortex...... 239 10.3.1. Scaling laws for cavitation inception...... 239 10.3.2. Correlation of cavitation data with the lift coefficient ...... 240 10.3.3. Effect of nuclei content...... 242 10.3.4. Effect of confinement...... 244 References ...... 245

11. Shear cavitation...... 247 11.1. Jet cavitation...... 248 11.1.1. Some experimental results ...... 248 11.1.2. Some elements of analysis of jet cavitation...... 251 11.2. Wake cavitation...... 252 11.2.1. Cavitation inception in the wake of circular discs...... 252 11.2.2. Modeling of wake cavitation inception ...... 253 11.2.3. Cavitation in the wake of a two-dimensional wedge...... 256 References ...... 262

12. Cavitation erosion...... 265 12.1. Empirical methods...... 266 12.2. Some global results...... 267 12.2.1. Influence of flow velocity ...... 267 12.2.2. Time evolution of mass loss rate...... 267 12.2.3. Miscellaneous comments...... 268 12.3. Basic hydrodynamic mechanisms of energy concentration...... 269 12.3.1. Collapse and rebound of a spherical bubble...... 269 12.3.2. Microjet ...... 269 12.3.3. Collective collapse...... 270 12.3.4. Cavitating vortices...... 270 12.4. Aggressiveness of a cavitating flow ...... 271 12.4.1. Aggressiveness of a collapsing bubble ...... 271 12.4.2. Pitting tests...... 273 12.4.3. Force measurements...... 275 12.4.4. Scaling laws for flow aggressiveness ...... 278 12.4.5. Asymptotic behavior of pitting rate at high velocities...... 280 12.5. Insight into the material response...... 282 12.5.1. Interaction between the liquid flow and a solid wall...... 282 12.5.2. Cavitation erosion and strain rate...... 283 12.5.3. Correlation of volume loss with impact energy...... 284 12.5.4. Phenomenological model for mass loss prediction ...... 285 References ...... 289

Index...... 293

XII FOREWORD

This book treats cavitation, which is a unique phenomenon in the field of hydrodynamics, although it can occur in any hydraulic machinery such as pumps, propellers, artificial hearts, and so forth. Cavitation is generated not only in water, but also in any kind of fluid, such as liquid hydrogen. The generation of cavitation can cause severe damage in hydraulic machinery. Therefore, the prevention of cavitation is an important concern for designers of hydraulic machinery. On the contrary, there is great potential to utilize cavitation in various important applications, such as environmental protection. There have been several books published on cavitation, including one by the same authors. This book differs from those previous ones, in that it is both more physical and more theoretical. Any theoretical explanation of the cavitation phenomenon is rather difficult, but the authors have succeeded in explaining it very well, and a reader can follow the equations easily. It is an advantage in reading this book to have some understanding of the physics of cavitation. Therefore, this book is not an introductory text, but a book for more advanced study. However, this does not mean that this book is too difficult for a beginner, because it explains the cavitation phenomenon using many figures. Therefore, even a beginner on cavitation can read and can understand what cavitation is. If the student studies through this book (with patience), he or she can become an expert on the physics of cavitation. In conclusion, this book is very comprehensive and instructive for advanced students, scientists, and engineers, who want to understand the true nature of cavitation.

The authors, Dr. Jean-Marie MICHEL and Dr. Jean-Pierre FRANC, are professors at the University of Grenoble, although Dr. MICHEL retired recently. They have much experience in the teaching and study of cavitation. Dr. MICHEL and Dr. FRANC have presented many important papers in internationally recognized academic journals such as the Journal of Fluid Mechanics, which are referenced in this book.

Dr. MICHEL and Dr. FRANC are the most suitable persons to write a book on cavitation such as this. I take great pleasure in being the first to congratulate them on their most recent contribution to this very unique and fascinating field.

March, 2003, Tokyo, Japan Dr. Hiroharu KATO, Professor Emeritus, University of Tokyo PREFACE

The present book is aimed at providing a comprehensive presentation of the phenomena involved in cavitation. It is focused on hydrodynamic cavitation, i.e. the kind of cavitation which occurs in flowing liquids, contrary to acoustic cavitation which is induced by an oscillating pressure field in a liquid almost at rest. Nevertheless, the principles which govern the hydrodynamic bubble and the acoustic bubble are basically the same. Briefly, cavitation is the occurrence of vapor cavities inside a liquid. It is well known that in static conditions a liquid changes to vapor if its pressure is lowered below the so-called vapor pressure. In liquid flows, this phase change is generally due to local high velocities which induce low pressures. The liquid medium is then "broken" at one or several points and "voids" appear, whose shape depends strongly on the structure of the flow. This book deals with all types of cavitation which develop in real liquid flows. This includes bubble cavitation (spherical bubbles in the simplest case), sheet cavitation, supercavitation and superventilation, cavitation in shear and vortex flows and some other patterns. It covers the field of cavitation inception as well as developed cavitation, which is encountered in advanced hydraulics at high speed. It is intended for graduate students, research workers and engineers facing cavitation problems, particularly in the industrial fields of hydraulic machinery and marine propulsion. A special effort has been made to explain the physics of cavitation in connection with various phenomena such as surface tension, heat and mass transfer, viscosity and boundary layers, compressibility, nuclei content, turbulence, etc... In addition to the physical foundations of the phenomenon, various methods of investigation, either experimental or computational, are presented and discussed so that the reader can deal with original problems. The book results from about 40 years of research carried out at Grenoble University in various fields of cavitation science, with the financial support of several firms and institutions, particularly the French Navy. Initially, two main influences converged to stimulate the creation by Pr. J. DODU of the cavitation research group: the strong hydraulic experience of private Companies in Grenoble and the advice of renowned foreign scientists (M.S. PLESSET, M.P. TULIN, B.R. PARKIN, A.J. ACOSTA... and so many others) who delivered a detailed account of the state of the art to Pr. J. DODU. Many of those initial scientific and industrial relationships have remained active over the years. Here we particularly wish to acknowledge the very fine contribution we received from Mr Y. LECOFFRE, either in the design of experimental rigs or the initiation of new research programs. We must also remember the name of Pr. A. ROWE, whose acute insight into hydrodynamics and pioneering work on numerical modeling of cavitating flows are still present in our minds. The book is made of rather short chapters, each designed to correspond to one or two lectures. It is the result of our teaching program given over many years including a number of seminars. The lists of references (at the end of each chapter) are limited to major contributions, while the very abundant literature devoted to cavitation can be found from the quoted review papers. After an introductory chapter, classic results relative to liquid breakdown are recalled in chapter 2. It is shown that a liquid can actually sustain absolute pressures smaller than the vapor pressure (and even tensions) without cavitating. This leads to the idea of nuclei (i.e. points of weakness in the liquid continuum) which is a fundamental concept in cavitation. The physics of the microbubble as a nucleus is presented in detail with a special emphasis on stability. Chapter 2 ends with the definition of the quality of a liquid sample in terms of nuclei content, a key concept for the prediction of cavitation patterns in real liquid flows. Chapters 3 to 5 are concerned with the isolated bubble. In chapter 3, basic results on the dynamics of the spherical bubble are presented and the famous RAYLEIGH- PLESSET equation is derived. Throughout the book, this fundamental equation is used to throw light on essential questions, such as scale effects. The evolution of a bubble in a non-symmetrical environment will result in deviations from sphericity which are discussed in chapter 4, together with the problem of the path of a bubble within a liquid flow. The effects of liquid compressibility and thermal diffusion are presented in chapter 5. Chapters 6 to 9 address sheet cavitation, which appears on blades of propellers, foils of boats, or behind axisymmetric bodies such as torpedoes. Chapters 6 and 9 are devoted to the neighbouring problems of supercavitation and superventilation respectively. A special effort has been made to present the analytical approach derived by our colleagues from Russia and Ukraine on the basis of the so-called “Logvinovich independence principle of cavity expansion“ in the case of axisymmetric cavities. We are especially grateful to Pr. V.V. SEREBRYAKOV, Pr. Y.N. SAVCHENKO and their colleagues from the Kiev University. Through them, we became aware of the very significant research which was carried out in those countries over the years. One example (chapter 9) is the theoretical modeling of ventilated cavity pulsations by Pr. E.V. PARISHEV of Moscow University. Partial sheet cavitation is addressed in chapter 7 with special attention given to cloud cavitation, re-entrant jets and more generally to cavitation instabilities. Because of their practical importance, those subjects have been studied in a number of laboratories in the recent past, in Europe, Japan and the USA. We would like to thank especially Pr. Y. TSUJIMOTO (Osaka University) for numerous and fruitful discussions on cavitation instabilities. The interaction between traveling bubbles and attached sheet cavities is addressed in chapter 8. It is shown that the boundary layer on the wall together with the water nuclei content strongly influence the type of cavitation that can occur. Basic principles for the prediction of cavitation patterns on hydrofoils or pump blades are proposed.

XVI Chapters 10 and 11 are devoted to vortex cavitation, tip vortex cavitation and shear cavitation respectively. Several results presented in chapter 10 were obtained in the framework of a joined program supported by the French Navy. We are particularly grateful to Pr. D.H. FRUMAN who directed that research program, and to the colleagues of the associated laboratories : Bassin d’Essais des Carènes (Val de Reuil, France), École Navale (Brest, France), Institut de Machines Hydrauliques (Lausanne, Switzerland). The difficult subject of shear cavitation is approached in chapter 11, with a special emphasis on the physical analysis derived by Pr. R.E.A. ARNDT (University of Minnesota). The main effects of cavitation on hydraulic equipments are also examined in this book. An overview on cavitation erosion is given in chapter 12, whereas the reader will find information on cavitation noise in several chapters.

We are very indebted to H. KATO (formerly Professor at the Tokyo University), Pr. K.V. ROZHDESTVENSKY (Saint Petersburg State Marine Technical University) and Pr. B. STOFFEL (Darmstadt University of Technology) who kindly accepted to review our manuscripts and made precious comments and suggestions. We are also grateful to our colleague Pr. F. MCCLUSKEY who accepted the difficult charge of correcting the English and spent a considerable time improving our manuscript, far beyond matters of pure form.

Thanks also to our colleagues in the Grenoble Cavitation research group (J.C. JAY, M. MARCHADIER, M. RIONDET and J.F. VERDYS) whose technical competence allowed us to obtain a significant number of the experimental results presented in this book. Our editors, Kluwer and Grenoble Sciences, deserve special acknowledgements. We are particularly grateful to Pr René MOREAU, Scientific Editor of the series Fluid mechanics and its applications and to the local team of Grenoble Sciences : Pr Jean BORNAREL (Grenoble University) who supported this project since its very beginning, Nicole SAUVAL who ensured the administrative work, Sylvie BORDAGE, Julie RIDARD and Thierry MORTURIER for their constant and exceptional care in the production of the manuscript.

Finally, we would like to thank once more our colleague and friend Hiro KATO whose scientific road crossed our own one so many times and who accepted to write the foreword for this book.

J.P. FRANC & J.M. MICHEL November, 2003

XVII LIST OF SYMBOLS

> Numbers in square brackets refer to the corresponding chapters. a Constant in the VAN DER WAALS state law [1] ML5 T–2 Thermal diffusivity [2, 7] m2/s L2 T–1 Viscous core radius [11] m L [3] an Amplitude of spherical harmonics mL b Constant in the VAN DER WAALS state law [1] m3 L3

B STEPANOV factor [5, 7] c Chord length [6, 7] mL Speed of sound [5, 7] m/s LT–1

[2, 5] 2 –2 –1 cv, cp Heat capacities at constant volume (resp. constant pressure) J/kg/°K L T q

[2] 3 –3 Cs Concentration of a gas dissoved in a liquid kg/m ML [1, 10, 11] Cp Pressure coefficient [4, 6] CD Drag coefficient [4, 6] CL Lift coefficient [5, 7, 9] CQ Flowrate coefficient D Coefficient of mass diffusion [2] m2/s L2 T–1 Diameter [6, 9] mL

Drag force [6] N LMT–2 e Cavity thickness [6, 7] mL

[11] –1 –1 eij Deformation rate s T f Frequency [3, 7, 9] s–1 T–1

Fr FROUDE number [6, 9] h Enthalpy [5] J/kg L2 T–2 Heat convective transfer coefficient [7] W/m2/°K MT–3 q–1

H HENRY constant [2] s–1/m2 k Polytropic exponent [5] l, dl Curvilinear distance, length element m L l Cavity length [6, 7, 9] mL L Latent heat of vaporization J/kg L2 T–2

Lift force [6] N MLT–2

m˙ Mass flowrate through a unit surface area [1] kg/m2/s ML–2 T–1

Mass loss rate [12] kg/s MT–1

M Virtual mass of an immersed body [4] kg M

n(R) Density distribution of nuclei size [2] /cm3/DRL–4

N Density concentration of nuclei [2] /cm3 L–3

p Absolute pressure [1] Pa ML–1 T–2

[1, 6, 7, 9] –1 –2 pc Cavity pressure Pa ML T

[1, 9] –1 –2 pg Partial gas pressure inside a cavity Pa ML T

[1] –1 –2 pr Pressure at the reference point Pa ML T

[1] –1 –2 pv(T) Vapor pressure at temperature T Pa ML T q Mass flowrate through a unit surface area [7] kg/m2/s ML–2 C

Q Heat transfer [1] JML2 T–2

[9] –1 Qm Mass flowrate of air kg/s MT

r Radial coordinate [3, 6, 9] mL

R Spherical bubble radius [3] mL

Radius of an axisymmetric cavity [6, 9] mL

R˙ Bubble interface velocity [3] m/s LT–1

Re REYNOLDS number r [4] –2 R p Force exerted by the liquid on an immersed body N MLT s Curvilinear distance[3, 8] mL

S Cavity cross-sectional area [6, 9] m2 L2

STROUHAL number [7, 9, 11]

Surface tension of the liquid[2, 3]

t Time s T

[3] –1 –2 trr Radial stress Pa ML T T Absolute temperature [1, 7] °K q

Period [3, 9] sT

u Radial component of the velocity [3, 5] m/s LT–1

V Velocity [4] m/s LT–1

ᐂ Bubble or cavity volume[5, 9] m3 L3

XX W Relative velocity [4] m/s LT–1

We WEBER number [3] x, y, z Cartesian coordinates [3, 6, 10] mL

> Greek characters

␣ Angle of attack [6]

[5] 2 2 –1 ␣l Liquid thermal diffusivity m /s L T

␦ Boundary layer thickness [8] mL

⌬ Increment operator

␧ Small parameter

Strain [12]

␸ Non-dimensional frequency [9]

Velocity potential [4, 6] m2/s L2 T–1 ␥ [2, 3, 9] Ratio of heat capacities (g = cp /cv) ⌫ Circulation [10, 11] m2/s L2 T–1

␭ Thermal conductivity [2, 5] W/m/°K MLT–3 q–1

Wavelength [9, 11] mL

␮ Dynamic viscosity [3, 11] kg/m/s ML–1 T–1

␯ Kinematic viscosity [3, 11] m2/s L2 T–1

␳ Density [1, 2, 3...] kg/m3 ML–3

␴ Normal stress [12] Pa ML–1 T–2 ␴ [1] v Cavitation number ␴ ␴ [1, 2, 8, 10, 11] vi, vd Incipient (resp. desinent) cavitation number ␴ [9] a Relative pressure of air inside a ventilated cavity ␴ [1, 6, 7, 9] c Relative underpressure of a developed cavity

⌺ BRENNEN thermodynamic parameter [5] m/s3/2 LT–3/2

␶ Characteristic time [2, 3, 6, 9] sT

RAYLEIGH time for the bubble collapse [3] sT

␻ Rotation rate [11] /s T–1

⍀ Vorticity [10, 11] /s T–1

XXI > Subscripts c Cavity min Minimum value g Gas v Vapor l Liquid

0 Initial value. Mean value r Reference point

XXII