cavitation and bubble dynamics feb 1995

Cavitation and Bubble Dynamics Christopher E. Brennen California Institute of Technology Pasadena, California New York Oxford Oxford University Press 1995 Oxford University Press Oxford New York Athens Auckland Bangkok Bombay Calcutta Cape Town Dar-es-Salaam Delhi Florence Hong Kong Istanbul Karachi Kuala Lumpur Madras Madrid Melbourne Mexico City Nairobi Paris Singapore Taipei Tokyo Toronto and associated companies in Berlin Ibadan Copyright c 1995 by Oxford University Press, Inc. Published by Oxford University Press, Inc., 200 Madison Avenue, New York, New York 10016 Oxford is a registered trademark of Oxford University Press, Inc. All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording, or otherwise, without the prior written permission of Oxford University Press. Library of Congress Cataloging-in-Publication Data Brennen, Christopher Earls, 1941- Cavitation and bubble dynamics / Christopher Earls Brennen. p.cm.—(Oxford engineering science series; 44) Includes bibliographical references and index. Cavitation and bubble dynamics / Christopher Earls Brennen. 1. Multiphase flow. 2. Cavitation 3. Bubbles I. Title II. Series TA357.5.M84B74 1995 620.1’064–dc20 94-18365 ISBN 0-19-509409-3 (alk. paper) Printing(lastdigit):987654321 Printed in the United States of America on acid-free paper Preface This book is intended as a combination of a reference book for those who work with cavitation or bubble dynamics and as a monograph for advanced students interested in some of the basic problems associated with this category of multi- phase flows. A book like this has many roots. It began many years ago when, as a young postdoctoral fellow at the California Institute of Technology, I was asked to prepare a series of lectures on cavitation for a graduate course cum sem- inar series. It was truly a baptism by fire, for the audience included three of the great names in cavitation research, Milton Plesset, Allan Acosta, and Theodore Wu, none of whom readily accepted superficial explanations. For that, I am immensely grateful. The course and I survived, and it evolved into one part of a graduate program in multiphase flows. There are many people to whom I owe a debt of gratitude for the roles they played in making this book possible. It was my great good fortune to have known and studied with six outstanding scholars, Les Woods, George Gadd, Milton Plesset, Allan Acosta, Ted Wu, and Rolf Sabersky. I benefited im- mensely from their scholarship and their friendship. I also owe much to my many colleagues in the American Society of Mechanical Engineers whose in- sights fill many of the pages of this monograph. The support of my research program by the Office of Naval Research is also greatly appreciated. And, of course, I feel honored to have worked with an outstanding group of graduate students at Caltech, including Sheung-Lip Ng, Kiam Oey, David Braisted, Luca d’Agostino, Steven Ceccio, Sanjay Kumar, Douglas Hart, Yan Kuhn de Chizelle, Beth McKenney, Zhenhuan Liu, Yi-Chun Wang, and Garrett Reisman, all of whom studied aspects of cavitating flows. The book is dedicated to Doreen, my companion and friend of over thirty years, who tolerated the obsession and the late nights that seemed necessary to bring it to completion. To her I owe more than I can tell. Pasadena, Calif. C.E.B. June 1994 3 Contents Nomenclature 9 1 PHASE CHANGE, NUCLEATION, AND CAV ITATION 15 1.1 INTRODUCTION 15 1.2 THELIQUIDSTATE 16 1.3 FLUIDITYANDELASTICITY 17 1.4 ILLUSTRATION OF TENSILESTRENGTH 19 1.5 CAVITATIONANDBOILING 21 1.6 TYPESOFNUCLEATION 22 1.7 HOMOGENEOUS NUCLEATION THEORY 23 1.8 COMPARISONWITHEXPERIMENTS 25 1.9 EXPERIMENTS ON TENSILE STRENGTH 28 1.10HETEROGENEOUSNUCLEATION 28 1.11NUCLEATIONSITEPOPULATIONS 30 1.12 EFFECT OF CONTAMINANT GAS . . . . 33 1.13NUCLEATIONINFLOWINGLIQUIDS 34 1.14 VISCOUS EFFECTS IN CAVITATIONINCEPTION 36 1.15 CAVITATION INCEPTION MEASUREMENTS 37 1.16CAVITATIONINCEPTIONDATA 40 1.17SCALINGOFCAVITATIONINCEPTION 43 REFERENCES 43 2 SPHERICAL BUBBLE DYNAMICS 47 2.1 INTRODUCTION 47 2.2 RAYLEIGH-PLESSETEQUATION 47 2.3 BUBBLECONTENTS 50 2.4 IN THE ABSENCE OF THERMAL EFFECTS 53 2.5 STABILITYOFVAPOR/GASBUBBLES 57 5 2.6 GROWTHBYMASSDIFFUSION 61 2.7 THERMALEFFECTSONGROWTH 63 2.8 THERMALLYCONTROLLEDGROWTH 65 2.9 NONEQUILIBRIUMEFFECTS 67 2.10CONVECTIVEEFFECTS 68 2.11SURFACEROUGHENINGEFFECTS 70 2.12NONSPHERICALPERTURBATIONS 71 REFERENCES 75 3 CAVITATION BUBBLE COLLAPSE 79 3.1 INTRODUCTION 79 3.2 BUBBLECOLLAPSE 79 3.3 THERMALLYCONTROLLEDCOLLAPSE 83 3.4 THERMAL EFFECTS IN BUBBLECOLLAPSE 84 3.5 NONSPHERICAL SHAPE DURING COLLAPSE 84 3.6 CAVITATIONDAMAGE 91 3.7 DAMAGEDUETOCLOUDCOLLAPSE 94 3.8 CAVITATIONNOISE 96 3.9 CAVITATIONLUMINESCENCE 104 REFERENCES 107 4 DYNAMICS OF OSCILLATING BUBBLES 113 4.1 INTRODUCTION 113 4.2 BUBBLENATURALFREQUENCIES 114 4.3 EFFECTIVEPOLYTROPICCONSTANT 118 4.4 ADDITIONALDAMPINGTERMS 120 4.5 NONLINEAREFFECTS 122 4.6 WEAKLY NONLINEAR ANALYSIS . 123 4.7 CHAOTICOSCILLATIONS 126 4.8 THRESHOLD FOR TRANSIENT CAVITATION 127 4.9 RECTIFIEDMASSDIFFUSION 128 4.10BJERKNESFORCES 131 REFERENCES 133 5 TRANSLATION OF BUBBL ES 137 5.1 INTRODUCTION 137 5.2 HIGHReFLOWSAROUNDASPHERE 138 5.3 LOWReFLOWSAROUNDASPHERE 140 5.4 MARANGONIEFFECTS 145 5.5 MOLECULAREFFECTS 147 5.6 UNSTEADYPARTICLEMOTIONS 148 5.7 UNSTEADYPOTENTIALFLOW 151 5.8 UNSTEADYSTOKESFLOW 154 5.9 GROWINGORCOLLAPSINGBUBBLES 158 5.10EQUATIONOFMOTION 160 5.11MAGNITUDEOFRELATIVEMOTION 164 5.12 DEFORMATION DUE TO TRANSLATION 166 REFERENCES 171 6 HOMOGENEOUS BUBBLY FLOWS 175 6.1 INTRODUCTION 175 6.2 SONIC SPEED . . . 176 6.3 SONIC SPEED WITH CHANGE OFPHASE 179 6.4 BAROTROPICRELATIONS 183 6.5 NOZZLEFLOWS 185 6.6 VAPOR/LIQUIDNOZZLEFLOW 190 6.7 FLOWS WITH BUBBLE DYNAMICS . . . . 194 6.8 ACOUSTICSOFBUBBLYMIXTURES 196 6.9 SHOCKWAVESINBUBBLYFLOWS 199 6.10SPHERICALBUBBLECLOUD 205 REFERENCES 212 7 CAVITATING FLOW S 217 7.1 INTRODUCTION 217 7.2 TRAVELINGBUBBLECAVITATION 218 7.3 BUBBLE/FLOWINTERACTIONS 219 7.4 EXPERIMENTALOBSERVATIONS 220 7.5 LARGE-SCALE CAVITATION STRUCTURES 227 7.6 VORTEXCAVITATION 227 7.7 CLOUDCAVITATION 232 7.8 ATTACHEDORSHEETCAVITATION 233 7.9 CAVITATINGFOILS 237 7.10CAVITYCLOSURE 238 REFERENCES 240 8 FREE STREAMLINE FLOWS 245 8.1 INTRODUCTION 245 8.2 CAVITYCLOSUREMODELS 248 8.3 CAVITYDETACHMENTMODELS 251 8.4 WALL EFFECTS AND CHOKEDFLOWS 256 8.5 STEADY PLANAR FLOWS . . 259 8.6 SOMENONLINEARRESULTS 262 8.7 LINEARIZEDMETHODS 267 8.8 FLATPLATEHYDROFOIL 270 8.9 CAVITATINGCASCADES 272 8.10THREE-DIMENSIONALFLOWS 278 8.11NUMERICALMETHODS 278 8.12UNSTEADYFLOWS 280 REFERENCES 284 INDEX 291 Nomenclature RO MAN LETTERS a Amplitude of wave-like disturbance A Cross-sectional area or cloud radius b Body half-width B Tunnel half-width c Concentration of dissolved gas in liquid, speed of sound, chord c k Phase velocity for wavenumber k c P Specific heat at constant pressure C D Drag coefficient C L Lift coefficient ˜ C Lh , ˜ C Lp Unsteady lift coefficients C M Moment coefficient ˜ C Mh , ˜ C Mp Unsteady moment coefficients C ij Lift/drag coefficient matrix C p Coefficient of pressure C pmin Minimum coefficient of pressure d Cavity half-width, blade thickness to spacing ratio D Mass diffusivity f Frequency in Hz. f Complex velocity potential, φ + iψ f N A thermodynamic property of the phase or component, N Fr Froude number g Acceleration due to gravity g x Component of the gravitational acceleration in direction, x g N A thermodynamic property of the phase or component, N G(f) Spectral density function of sound h Specific enthalpy, wetted surface elevation, blade tip spacing H Henry’s law constant Hm Haberman-Morton number, normally gµ 4 /ρS 3 i, j, k Indices i Square root of −1 in free streamline analysis I Acoustic impulse 9 I ∗ Dimensionless acoustic impulse, 4πIR/ρ L U ∞ R 2 H I Ki Kelvin impulse vector j Square root of −1 k Boltzmann’s constant, polytropic constant or wavenumber k N Thermal conductivity or thermodynamic property of N K G Gas constant K ij Added mass coefficient matrix, M ij / 4 3 ρπR 3 Kc Keulegan-Carpenter number Kn Knudsen number, λ/2R  Typical dimension in the flow, cavity half-length L Latent heat of vaporization m Mass m G Mass of gas in bubble m p Mass of particle M ij Added mass matrix n Index used for harmonics or number of sites per unit area N(R) Number density distribution function of R ˙ N E Cavitation event rate Nu Nusselt number p Pressure p a Radiated acoustic pressure p s Root mean square sound pressure p S A sound pressure level p G Partial pressure of gas P Pseudo-pressure Pe Peclet number, usually WR/α L q Magnitude of velocity vector q c Free surface velocity Q Source strength r Radial coordinate R Bubble radius R B Equivalent volumetric radius, [3τ/4π] 1 3 R H Headform radius R M Maximum bubble radius R N Cavitation nucleus radius R P Nucleation site radius R Distance to measurement point Re Reynolds number, usually 2WR/ν L s Coordinate measured along a streamline or surface s Specific entropy S Surface tension St Strouhal number, 2fR/W t Time t R Relaxation time for relative motion t ∗ Dimensionless time, t/t R [...]... is due to Volmer and Weber (1926), Farkas (1927), Becker and Doring (1935), Zeldovich (1943), and others For reviews of the subject, the reader is referred to the books of Frenkel (1955) and Skripov (1974), to the recent text by Carey (1992) and to the reviews by Blake (1949), Bernath (1952), Cole (1970), Blander and Katz (1975), and Lienhard and Karimi (1981) We present here a brief and simplified version... practical difficulties involved in observing cavitation inception Further reduction in σ below σi causes an increase in the number and extent of vapor bubbles In the hypothetical flow of a liquid that cannot withstand any tension and in which vapor bubbles appear instantaneously when p reaches pV , it is clear that σi = −Cpmin (1.15) and hence the incipient cavitation number could be ascertained from... outward in order to create the bubble, and this implies work done on or by the system The pressure difference involved in this energy increment is the difference between the pressure inside and outside of the bubble (which, in this evaluation, is ∆pC , given by Equation (1.4)) The work done is the volume of the bubble multiplied by this pressure difference, or 4πR3 ∆pC /3, and this is C the work done by... Figure 1.8: Cavitation nuclei number density distribution functions measured by holography in three different water tunnels (Peterson et al 1975, Gates and Bacon 1978, Katz 1978) at the cavitation numbers, σ, as shown) and in the ocean off Los Angeles, Calif (O’Hern et al 1985, 1988) Figure 1.8 are some typical distributions measured in the filtered and deaerated water of three different water tunnels and in... gas-filled microbubbles could exist for any length of time in a body of liquid that is not saturated with that gas It is not possible to separately assess the number of solid particles and the number of microbubbles with most of the existing experimental techniques Though both can act as cavitation nucleation sites, it is clear that microbubbles will more readily grow to observable macroscopic bubbles One... the flow where Cp and p i are a minimum, and that value of Cp(x∗ ) will be denoted for convenience by i Cpmin Note that this is a negative number Viscous effects within the flow are characterized by the Reynolds number, Re = ρL U∞ /µL = U∞ /νL where µL and νL are the dynamic and kinematic viscosities of the liquid and is the characterized length scale For a given geometry, Cp (xi ) and Cpmin are functions... liquid state and to remark on its comparison with the simpler crystalline solid or gaseous states The first and most obvious difference between the saturated liquid and saturated vapor states is that the density of the liquid remains relatively constant and similar to that of the solid except close to the critical point On the other hand the density of the vapor is different by at least 2 and up to 5 or... of 10−5 m, the subsequent tension required to expand the bubble beyond the envelope of the surface is only of the order of a tenth of an atmosphere and hence quite within the realm of experimental observation It is clear that some specific sites on a solid surface will have the optimum geometry to promote the growth and macroscopic appearance of vapor bubbles Such locations are called nucleation sites... Furthermore, it is clear that as the pressure is reduced more and more, sites will become capable of generating and releasing bubbles to the body of the liquid These events are readily observed when you boil a pot of water on the stove At the initiation of boiling, bubbles are produced at a few specific sites As the pot gets hotter more and more sites become activated Hence the density of nucleation... there is little difference between the two processes, and we shall attempt to review the two processes of nucleation simultaneously The differences in the two processes occur because of the different complicating factors that occur in a cavitating flow on the one hand and in the temperature gradients and wall effects that occur in boiling on the other hand The last sections of this first chapter will dwell . Earls, 1941- Cavitation and bubble dynamics / Christopher Earls Brennen. p.cm.—(Oxford engineering science series; 44) Includes bibliographical references and index. Cavitation and bubble dynamics. VISCOUS EFFECTS IN CAVITATIONINCEPTION 36 1.15 CAVITATION INCEPTION MEASUREMENTS 37 1.16CAVITATIONINCEPTIONDATA 40 1.17SCALINGOFCAVITATIONINCEPTION 43 REFERENCES 43 2 SPHERICAL BUBBLE DYNAMICS 47 2.1. 84 3.6 CAVITATIONDAMAGE 91 3.7 DAMAGEDUETOCLOUDCOLLAPSE 94 3.8 CAVITATIONNOISE 96 3.9 CAVITATIONLUMINESCENCE 104 REFERENCES 107 4 DYNAMICS OF OSCILLATING BUBBLES 113 4.1 INTRODUCTION 113 4.2 BUBBLENATURALFREQUENCIES