acoustic cavitation - e.a. neppiras

ACOUSTIC CAVITATION

E.A. NEPPIRAS

Consultant,17, KinsbourneAvenue,Bournemouth,England

NORTHHOLLAND PUBLISHING COMPANY - AMSTERDAM

PHYSICSREPORTS(Review Sectionof PhysicsLetters)61, No. 3(1980)159—251.North-HollandPublishingCompany

ACOUSTIC CAVITATION

E.A. NEPPIRASConsultant, 17, KinsbourneAvenue,Bournemouth,England

ReceivedNovember 1979

Contents:

1. Introduction 163 7.4. Stability of an oscillating stablecavity 2222. Theequationsof cavitationbubbledynamics 165 7.5. Generaldynamical problem of the distortion of the3. Stablecavitation 172 surfaceseparatingtwo immiscible fluids 224

3.1. Undampedlinearoscillations 173 7.6. Experimentalresults 2243.2. Dampingof stableoscillations 174 8. High-speedphotographicstudiesof bubblemotion 2253.3. Couplingby mass-diffusionacrossthebubblewall 177 8.1. Photographicstudiesat California Institute of Tech-3.4. Oscillationsof stablecavitieswith thermalcoupling 181 nology 2263.5. Oscillations of stablecavities with thermalcoupling, 8.2. Photographicstudiesby Sovietworkers 226

including evaporation-condensationat the bubble 8.3. Photographicstudiesat Universityof Gottingen 227wall 184 8.4. Theory relatingto bubblescollapsingnearsolid sur-

3.6. Non-linearoscillationsof stablecavities 187 faces 2304. Thecollapseof transientcavities 193 8.5. Recent photographic studies using large bubbles

4.1. The collapsingempty cavity 195 excitedat low frequencies 2304.2. The collapsinggas-filled cavity 198 9. Cavitationfields 2324.3. The generationof shockwaves 204 9.1. Pioneer studies using visual and photographic4.4. Acoustically-generatedtransientcavities 205 methods 232

5. Cavitationthresholds 207 9.2. Cavitation studies using pulsed holographic tech-5.1. Structuralstabilityof cavitationbubbles 208 niques 2345.2. The transientthresholds 208 9.3. Acousticmeasurementsin thecavitatingfield 2365.3. The stablecavitationthreshold 213 9.4. Measurementstakenat theelectricalterminalsof the

6. Cycliccavitationprocesses 213 transducer 2376.1. The gaseouscavitationcycle 214 9.5. Concertedcollapseof cavity clouds 2396.2. The de-gassingcycle 214 9.6. Acoustic intensity-distribution in thepresence of6.3. The resonantbubble cycle with emissionof micro- cavitation 240

bubbles 214 9.7. Controlledbubblefields 2426.4. Bubble-growthabove theresonancesize 215 10. Acousticemissionfrom cavitationfields 2436.5. Cavitation relaxation 215 10.1. Emissionat low intensities:stablecavitationregime 2446.6. Cyclic behaviourinvolving exchangeof free gas,aided 10.2. Emission at high intensities: transient cavitation

by micro-streaming 216 regime 2467. Non-radialbubble motion 217 10.3. Recentcomputationsrelating to acousticemission

7.1. Taylor instability 220 from cavitationfields 2487.2. Instability at a sphericalgas—liquid interface 220 References 2487.3. Stability of a collapsingtransientcavity 221

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E.A.Neppiras,Acousticcavitation 161

Abstract:This article reviews the physicsof cavitation generatedby acousticfields, coveringthebasic theoreticaland experimentaldata neededfor a

properunderstandingof themanyeffectsof cavitationandtheirpractical applications.The dynamicsof bubblemotionaredeveloped,stressingtherelationbetweenstableand transienttypesof cavitationand theirthresholds.Non-radialtypesof bubblemotion arenow known to initiate severalimportanteffects, including erosiveaction,and thesearedealtwith in detail. Direct verificationof theoriesof bubbledynamicsis obtainedusinghigh-speedcinematography.The most recenttechniquesandresultsaredescribed,asarethehighly-sophisticatedexperimentalmethodsnow beingappliedto thebubble-fieldsandaggregates,including the acousticemissionfrom stableand transientcavitationfields.

List of symboLs

Only symbolsthat appearfrequently are listed. The first column gives the meaningswhich theyusuallyhave.

a Perturbationamplitudec Velocity of sound A real numberf Frequencyfr Resonancefrequencyh Specificenthalpyf dp/pk Wavenumber2ir/A An angularfrequency

Boltzmann’sconstantm Mass A numericn A numericp Pressure Dimensionlesspressure,PA/Po

Pmax Maximumcollapsepressureq Heatflux (energyperunit massper unit time)r Radialdistance,radial displacement DimensionlessradiusR/R0or R/R.rm Maximumvalueof rt Timex Spaceco-ordinatey Spaceco-ordinateB DampingcoefficientC Concentrationof dissolvedgasin liquid Velocity of soundat the bubblewallC0 Saturationconcentrationof dissolvedgas in

liquidC,. Specificheatat constantpressure(gas)C0 Specificheatat constantvolume (gas)D Diffusivity (mass-or thermal)of gas-phaseE EnergydistributionfunctionH Enthalpydifferencebetweenthe bubblewall An auxiliary function

and infinityK ThermalconductivityL Diffusion length= (2D/o)

1’12 Latent heatof vaporisation;LengthM AcousticalMach number= R/cN The gasconstant Functiondescribingthe number-distributionP Pressure

162 E.A.Neppiras,Acousticcavitation

PA Acoustic pressureamplitudePm Liquid pressureat transientcollapsePL Liquid pressurejust outsidethe bubblewallPT ThresholdacousticpressurePmax Maximumgaspressurein collapsingbubbleQ Gaspressurein bubbleatits maximumsize Quality factor=

Quantityof heatR Bubbleradius ReflectioncoefficientRm Maximumbubbleradius Re acousticReynoldsNo.Rmin Minimum bubbleradiusRr Resonanceradiusof bubbleRT Thresholdradiusof bubbleR’, R” Principalradii of curvatureS,,, Specific heatat constantpressurefor liquidS0 Specific heatat constantvolume for liquidT Temperature Acoustic period = 2rr/wTmax MaximumgastemperatureU Internalenergyper unit massV Particlevelocity of liquid GasvolumeV0 Volume of cavitationzoneW Total powerassociatedwith cavitationlossesY~ Sphericalharmonicof degreenZ Volume compressionratioa Henry’s coefficientfor a weaksolution Accommodationcoefficient

Ro/Lg; 2oIR0P0~R/WrRo

/3 w/o-10 (pv2R2/3yPo)”2

y Ratio of specific heats for gas6 A smallquantity ~ 1’ Dampingfactor; (Po/pw2R~)

RadiationdampingfactorThermaldampingfactorViscous dampingfactorA smallquantity~ 1OscillatoryamplitudeThresholdoscillatoryamplitude

A Acoustic wave-lengthA. Wave-lengthof surfacewaves

Shearviscosity of liquid11 Periodof non-linearvibrationsp Density of liquidcr SurfacetensionT Non-dimensionaltime = wt (3

7P0/p)1”2(t/2irRo)

Velocity potential in liquid Entropy2cr/R

0P0

xw Angular frequency


(Or Resonantradial angularfrequencyF Polytropicindex for gas

Suffix 0 refers to the Equilibrium valueg refers to the Permanent-gasphasev refers to the VapourphaseT refers to the Total gas-content~ refers to the conditionsata greatdistance

Prefix ~ refersto asmall increment

1. Introduction

It is difficult to give an adequatebrief definitionof cavitation.Somewould saythat cavitationoccurswhenevera new surfaceis createdin the body of a liquid. This broad definition would includesuchphenomenaas boiling andmereeffervescence.In the presenceof a soundfield not only expansion,butalsocontraction,of existingcavitieswill generallyoccur.The term “acousticcavitation” maythereforebe restrictedto caseswherebothexpansionandcontractionphasesarepresent.

In acousticcavitation,a time-varying,generallysinusoidal,pressureis superimposedon the steadyambientpressure.The effecton themotionof cavitiesmaybe violent,or rathergentle.In fact, weeasilyrecognisearathersharpdistinctionbetweentwo typesof cavitation,calledStableandTransient.Stablecavitiesarebubblesthat oscillate,often non-linearly,aroundsomeequilibrium size.They arerelativelypermanentandmay continueoscillating for manycycles of the acousticpressure.On the otherhand,transientcavitiesgenerallyexist for less thanonecycle.During this onecycle,they expandto at leastdouble,and often to manytimes, their original size.Then theycollapseviolently, often disintegratinginto a massof smallerbubbles.

Eithertypeof cavitymaycontaineitherpermanent-gas(generallyair) or vapour.We maythereforedefinefour simple modelsfor the free cavity: the gas-or vapour-filled stableor transientcavity. In atransient cavity it is usual to assumethat there is no time for any mass-flow, by diffusion, ofpermanent-gasinto or out of the bubble,whereascondensationand evaporationof vapourmayoccurmoreor lessfreely. So,for example,a collapsinggaseoustransientcavity is assumedto haveaconstantpermanent-gascontentover its lifetime, while a vaporoustransientcavity containsonly vapourwhichwill often remainat or nearits constantequilibriumpressure.The collapseof a vaporoustransientmaythereforebe veryviolent, as thereis no residualpermanent-gasto cushionthe implosion.With any typeof stablecavity, the time-scaleis typically so long that mass-diffusionof permanent-gasas well asthermal diffusion, with consequentevaporation-condensationof vapour, can occur, resulting insignificant long-term effects. If the liquid is entirely gas-free,so thatanycavities must be vaporous,itmight be thought that stablecavities could not persist;for it can be very easily shown that under asteady ambient pressure,spherical cavities containing only the liquid vapour must be unstable.However, as we shall see later, thereis a thermalmechanismwherebyvaporouscavities can achievestability in the presenceof asoundfield. In fact, undersuitableconditions,stablepure-vapourcavitiesarevery plentiful. Therefore,we concludethat all four of the proposedmodelscan exist in practice.

So far, we havedefinedmodelsfor the behaviourof isolatedcavities.In practice,cavitiesarerarelyproducedin isolation.We haveto contendwith bubble fields, consistingof many bubbles interacting

164 E.A. Neppiras,Acousticcavitation

with one anotherand undergoingtranslatory,as well as radial motions in the sound field. In whatfollows, we shall deal first with isolatedsphericalcavities,and later with their interactionsin the field.

In the earlydays of cavitationresearch,workersweremainlypreoccupiedwith what we havedefinedabove as transient cavitation. Interestwas centredon the spectaculardisruptive effects of collapsingtransients,effects such as: erosion, emulsification, moleculardegradation,sono-luminescence,sono-

chemical and biological effects. These are all related to the very high pressuresand temperaturesdevelopedin the transientimplosion.But the stabletypesof cavitationarenow knownto be at leastasimportant as transients.They cover a wider range of motions, and someof the side-effectsof stablecavitationarevery important. Theseinclude the initiation of surfaceoscillationsandmicrostreaming.Also, in anygeneralcavitationfield, the greatmajority of visible bubbleswill be oscillatingstably, andsince stablecavitiesby their very nature are long-lived, their integratedeffect can be substantial.Afurtherrelevantfact is that stablecavitiesoften evolve into transientsin the courseof time.This theydoby a processof mass-or heat-transferwhichresultsin bubble-growth.

In practice,the distinctionbetweenthe stableand transientforms of cavitationis not alwaysquiteclear-cut.The transition:stable—* transientmayoccur through the merepassageof time or throughchangesin the acousticalor environmentalconditions.The demarcationareasbetweenstable andtransientstates,sometimesrathervague, areknown as TransientCavitation Thresholds.Thereis alsoanotherthresholdregion, lying below the transientthresholds,which definesthe point wherestablecavitiescan grow to becometransients.This maybe termedthe StableCavitation Threshold.Studyofthesethresholdsandthe relationbetweenthem,is an active areaof research.

Our definition of acousticcavitation implies that cavities are in radial motion. Surfaceoscillationsandinstabilities,andall forms of translatorymotion,arestrictly excludedfrom thedefinition as theydonot involve radial motion.However,we now know that thesenon-radialmotions areoften initiated bystablecavitation,andthereforeclosely linkedwith it. In anycase,all of thesemotionsmaybeimportantin practicalapplications,sowewill dealwith themin somedetail.

As soonas cavitationstarts in a liquid, its acousticpropertieshavechanged,andit hasbecome,ineffect, a different medium, acoustically lossier and more compressible.The propertiesof the newmediumwill control the courseof subsequentevents.The cavitatingliquid will generallybeappreciablynon-linear.The conceptof “acousticalimpedance”thenloses its validity. Nevertheless,it is still usefulto refer to an “effective” acousticalimpedancefor the cavitatingmedium.We may thensaythat theonset of cavitation will generally increase the effective acoustic resistanceand compliance of themedium.

It is the high compressibilityof the gasbubblesrelativeto the liquid that permits the wide rangeofmotionsseenin cavitation fields. Oscillatingbubblesarevery effectivein redistributinganddissipatingthe acoustic energy.Stable and transientcavities behavedifferently in this respect.With transientcavities, greatconcentrationof energyoccurs towards the end of the implosion. In effect, energyisextractedfrom the soundbeam(that is, from the transmissibleform) and rapidly concentratedintosmall volumes:energyfrom progressivewavesis transformedto sphericalshocksemanatingfrom pointsources.But despitethe largepressuresinvolved, the effectiverangeof actionof the shocksis typicallyonly a few radial distancesof the collapsedcavity. We maycontrastthiswith stably-oscillatingresonantbubbles.The absorption-scatteringcross-sectionfor such a bubble is enormouslygreaterthan for aliquid or solid particleof comparablesize.The soundbeamis thereforeeffectively scatteredandlittlecoherentsignal is passedthrougha region of evenmild cavitation.

It is interestingto record that until about 15 years ago almost all theoreticalwork in acousticcavitationwas concernedwith the dynamicsof isolatedcavities.This problemcan nowbe regardedas


completelysolved for bubblesthat remainspherical.At the sametime,most of the experimentalworkwas concernedwith naturalbubble-aggregates,uncontrolledbubble-fields.Recently,this approachhasbeen largely reversed.Theoretical work is being concentratedon non-sphericalbubbles and onconditions for instabilities to form anddevelop. Experimentally,the dynamicsof single bubblesorcontrolledbubble-fieldsis nowbeing activelystudied,often usinghigh-speedphotographicmethods.

Publishedmaterialon acousticcavitationis now very extensive,andgood papersarestill appearing.A numberof useful reviews are availablecoveringwork publishedup to the late 1960’s [1—5]. Thesealso containexcellentreferencelists. The presentreview aims to bring the story up to date. The pastdecadehasseennew developmentsin bubbledynamics,especiallyrelating to pure-vapourcavities;detailedanalysesof bubbleinstabilities;muchvery high-speedphotographicwork; newwork relatingtothresholdsand cyclic cavitationprocesses;anddevelopmentof techniquesfor controlling andstudyingbubble fields. Partly through lack of space,we omit any discussionof such topics as: the effects,applicationsandassessmentof cavitation;andsomespecialistandperipheraltopicssuchas: cavitationin cryogenicliquids; nucleationmechanisms,etc. However,adequatereferencesarequotedin all cases.

2. The equationsof cavitation bubble dynamics

Interestin cavitationstretchesbackabout120years[6] but theimportantpioneertheoreticalpaperisthat of Rayleigh,1917 [7] which describesthe collapseof a sphericalcavity. Until the late 1940’s alltheoreticalwork was concernedwith hydrodynamically-generatedcavities,culminating in the importantpaperby Plesset[8]. The first systematictreatmentof acoustically-generatedcavitieswas by Blake in1949 [9], followed by Noltingk andNeppiras,1950—51 [10,11]. Sincethen,manygroupshavebecomeactive.

The basicdynamicalproblemof acousticcavitationis to determinethe pressureandvelocity fields inthe two-fluid medium, togetherwith the motion of the bubble wall, when under the influenceof atime-dependent(acoustic)pressure.A relatedproblemsometimesof interestconcernsthe temperaturefield within the bubble.

Evenin themostgeneralcase,wherethe cavity is not spherical,but mayassumeany arbitraryshape,it is not difficult to formulatethe setof equationsdefiningthe problem[see,for example,refs.5 and 12].But the exerciseis purely academic,as no progressis possiblewithout makingvery drasticassumptionsconcerningan initial shapeand the forcesresponsiblefor it. In the simplestcase,wherethe cavity issphericaland remainssphericalalways— which, fortunately, is the one of most practicalinterest— theproblemis greatlysimplified. When, later, we examineconditionsfor surfaceinstability, the approachwill be to imagineanoriginally sphericalsurfaceto undergoanarbitraryperturbation,thenexaminetheconditionsunderwhich it would grow.

Our model for the cavity is thereforea sphericalbubble,isolated in a liquid thatextendsto infinity.With sphericalsymmetry,everyphysicalquantity is a function of only onespaceco-ordinate,the radialdistance,r, from the bubblecentre,takenas the origin. The acousticpressure,superimposedon a steadyambientpressure,is appliedatagreatdistancefrom the bubble.The equationsfor this simple mo4elcan be formulatedto take accountof non-uniformconditions(that is, variable pressure,velocity andtemperaturefields) in both the gasandliquid phases.

The problemto be solvedis thereforea two-phasehydrodynamicalonein which the two phasesarecoupled through a moving boundary (the bubble wall). It is of the class of free boundary-valueproblems,as the positionof the boundarywill not be known until the final solution is obtained.Thedynamicalproblemis more or lesscomplicateddependingon the degreeof couplingassumedbetween

E.A.Neppiras,Acousticcavitation

the two phases.Apart from the purely mechanicalcoupling, mass-and heat-transfermay take placeacrossthe bubble wall. The mass-flowmay comprisetwo elements:diffusion of permanent-gasandevaporation-condensationof liquid, whichwill itself dependon the thermalconditions.

As with any hydrodynamicalproblem,we obtain generalrelationsby applying the laws of con-servationof Mass,Momentumand Energy.Conservationof mass,leading to equationsof continuity,can betakento hold separatelyin the liquid andgasphasesand— as a boundarycondition— acrossthebubblewall, sinceonly at the boundarydoesany mass-exchangeoccur. Conservationof momentumappliesat anypoint in eitherphaseand at the bubblewall. Energyconservationmustbe appliedto thecompletecoupledsystem.Apart from the moving boundary,a boundaryconditionexistsat t =0, r =

In addition to the conservationlaws, the set of relevantequationswill includethephysical lawsneededto defineelementsof the energyequation.Theseare the laws relatingto mass-andthermal-diffusion,evaporation-condensationanddissolution. Finally, the set is completedby the Equationsof State forthe liquid and gas phases,thesebeing relationsbetweenpressure(p) anddensity (p) of the formp =p(p).

The readeris referredto ref. [5], for example,for the completeset of equationsdefiningthis generalcase. Somenumericalsolutions have evenbeen obtained, after some simplifying assumptions[13].However,instanceswherethe flow fields within the bubbleareof interest,or needto be evaluated,arequite limited. On the other hand, greatsimplification is achievedby assumingspatial uniformity ofpressure,velocity and temperaturewithin the bubble during the motion. In what follows, we shallthereforeassumethatthe physicalquantitiesrelatingto the bubblecontentsarefunctionsof timeonly.Physically,this amountsto assumingthatthe bubblewall velocity,R, is at all timessmallcomparedwiththe ratesatwhich irregularitiessmooththemselvesout within thebubble.But evenwhenthis conditionceasesto hold, as nearthe endof a transientcollapse,the simplification is still broadlyjustified whenonly the flow fields in the liquid and the motion of the bubble wall are of interest. The relevantequationsfor this general caseare reproducedbelow. Thesecomprisethe equationsof Continuity,Motion andEnergy,deducedfrom the conservationlaws of Mass,Momentumand Energy,as well asthe physicallaws of mass-andthermal-diffusionin the liquid-phase.The boundaryconditionsareto beevaluatedat the bubblewall, at r = R,

+ -~- (pV)+ ~ = 0 Continuity (liquid) (1)ôt 3r r

+ V = — Motion (liquid) (2)

13U 3U\ iav 2V\ ~ ~ V\2 182T 2 ÔT\ .

~ ‘%,,-~~~-—~-) +KL~-~_~+_-~--)+pqEnergy(hquid) (3)

aC BC /d2C 23C\—+ V—= D~—~--+ —~ i Gas-diffusionin liquid (4)

Br \3r ron

with boundaryconditionsat r = R(t):

BC 1 d,4 3D~4R2~s1rR pg) (5)

E.A.Neppiras, Acousticcavitation 167

(p+~)=j~4(9~~Y) (6)

OT 1 d ~ I 4~IBV V\1 RdUTKLT4R2~ ~R p~)1L+3 ]+Pr~~+pTV (7)

andat r=oc: p =p~~(t)=Po—PAsinwt. (8)

As mightbe expected,theseequationsarestill verydifficult to manipulatewhenall forms of couplingare included. However,we can reasonablyexaminetransientand stablecavitation independentlybyseparatingout inertial coupling from that arisingfrom diffusion mechanisms.Transientcollapse,andanybubblemotion involving only a few cyclesof oscillation,can reasonablybestudiedby concentratingon the inertial terms, ignoring diffusion. For truly stable oscillations the diffusion terms must beincluded,but the problemis not necessarilymorecomplicated,as plausiblesimplifying assumptionscanbe madein this casealso.

The set of equations(1)—(8) do not includethe StateEquationsfor theliquid andgasphases.For thegas,theassumptionthat themediumremainsa perfectgas,obeyingthe equationpV= NT is realisticinmostcases.Thermaldiffusion maythenbe decoupledby taking oneor otherof the extremeconditions,isothermalor adiabatic.Then, in effect, the thermodynamicenergyequation(3) combineswith the StateEquationto give pV = constant(isothermal)or pl/” = constant(adiabatic).For the liquid, use of thesimplestequationof state(that is, p = constant,the incompressibleassumption)leadsto a relativelysimple formulationand a set of equationsthat are, in somecases,evenanalytic. However,we shall findthat this assumptioncannotbe justified, evenapproximately,for violent transients.Much theoreticalwork on cavitation in the decade1950—1960aimed at using more realistic equationsof statefor theliquid, introducingcompressibilityin variousways.

As well as the two majorsimplifications alreadymentioned(spatially uniform conditionswithin asphericalbubble)severalother“obvious” assumptionsareimplicit in the aboveformulation:

(i) The acoustic wave-length is large comparedwith the bubble dimensions; otherwise, radialsymmetrywould be lost and the problembecomesintractable.

(ii) No body-forcesarepresent:that is, weignore the effect of gravity andanysteadyforcesarisingfrom the soundfield.

(iii) The bulk viscosity coefficient, and also interaction betweenviscosity and compressibility,aresmallenoughto be ignored.

(iv) Thedensityof the liquid is large,andits compressibilitysmall, comparedwith that of the gas.In deriving theequationsof bubbledynamicsin formssuitablefor computation,we will follow the

historical development.We thereforestart with the simplestassumptionfor the liquid state, that ofincompressibility.Insertingtheconditionp = p~,aconstantin the equationsof continuityandmotion (1and2), integratingfrom r = R to and usingthe boundarycondition(8) gives:

V=R2R/r2 (9)

fR21~+2R1~2R4I~2\

p =p~(t)+po~ r — 2r4 )~ (10)

The motion of thebubblewall is now obtainedby usingtheboundarycondition(6) aftersubstitutingfor

168 E.A.Neppiras, Acousticcavitation

V from (9):

RR+1[pL(R) — p~(t)] (11)Po

wherepL(R) is the liquid pressurejust outsidethe bubblewall, givenby pL(R) = pT(R)— 4pi~/R — 2oJRand p-r(R) will in generalbe the sum of contributions from permanent-gasand vapour: p-~(R)=pg(R)+pv(R).

We notethat viscosity entersonly as a boundaryconditionandappearsas apressuretermanalogousto the surfacetensionpressure.Equation(11) can be solved as soonas the two functionsp~(t)andp-r(R) arespecified.For acousticexcitation,p~(t)= P

0— PAsinwt (boundarycondition8), while p.r(R),the total gas pressurewithin the cavity, can be evaluatedfrom the other defining equationsoncetheinitial gas-content(at R = R0) and the equationof statefor the gas havebeen specified. From thedefiningequations(1—8) we seethat considerablecomplicationwill arisefrom anyassumedcouplingviamass-diffusionand heat-flow with resulting evaporation-condensationat the bubble wall. It is con-venientto ignore thesein solvingfor the motion of thebubblewall. Theireffect can beincludedlater, ifnecessary,in an ad hocfashion.

In the specialcasewherep~,(t)= F0, a constant,p-r(R) = 0 (empty cavity) and surfacetensionandviscosity areneglected,eq. (11) reducesto a very simple form:

RR + = Po/po. (12)

This is the famousequationfirst obtainedby Rayleigh[7] by a simpleenergy-balanceargument.It is ofgreathistoricalimportanceandwe shall considerit in moredetail later.

First we assumethat the bubble is filled with a permanent-gasobeying the perfectgas equationp(4irR

3/3)= NT For the sakeof definingthe motion of the bubblewall over alimited numberof cycleswe ignoremass-andheat-flowacrossthe interface.We canthenusetheadiabaticrelationp(4irR3/3)1~=constantand pT(R) = pg(R)= (P

0 + 2o1R0)(R0/R)31~,since at t = 0 whereR = R0, the gas pressureinthe bubbleis just (P0+ 2o!R0). Equation(11) thenbecomes:

RR ~ I [(~0+~)(~)3Y _~_~_p~(t)]. (13)

This equation, with p~(t)= P0— PA sinwt but without the viscous term, was first derived and in-vestigatedby Noltingk—Neppiras[10,11]:

RR ~ = I [(~0+~)(~o)37 —~_ (Po—PA5111 wt)]. (14)

The viscous term was later addedby Poritsky [14].Equations(13) and (14) arestill valid in the otherextremecondition(isothermal,with y = 1). Also, no additionalcomplicationarisesif thecavitycontainsliquid vapour at its equilibrium pressurein addition to the permanent-gas.We find that eq. (14)accuratelydescribesthe motion of the cavity-wall overa limited numberof cyclesfor all typesof stablecavitation,andalsofor transientswherethe bubblewall velocity neverexceedsabout1/5 of the velocityof sound.


But we know that under violent transientconditions the bubble wall velocity may approach,orexceed,the velocity of sound.Our assumptionof incompressibilityfor the liquid thenceasesto hold. Afirst steptowardsa morerealistic treatmentis to postulateafinite but constantstiffnessfor the liquid(constantsound velocity). Its equationof state thenbecomesp/p = constant;Op/Op= constant= c2,defining the soundvelocity. Such a treatmentwas first carriedout by Herring [15]who wasconcernedwith the expansionof vapourcavitiesin underwaterexplosions.Later,Trilling [16]extendedthe theoryto gas-filledbubbles,includinganevaluationof the pressureandvelocity fields in the liquid.

The “Acoustic Approximation” implied by use of the State Equation Op/Op = c2 confines thetreatmentto caseswherethe velocity is alwayssmall comparedwith c, that is, the “Acoustical MachNumber”,M(=R/c)4 1. Disturbancespropagatein the liquid at the speedof sound,a finite constantunaffectedby the motion. The velocity potential,4 must thereforesatisfy the acoustic equationfordiverging sphericalwaves:

(~+cf)n~=o. (15)

Theequationsof continuityandmotion (1 and 2) musthold, andin addition to the boundarycondition(6) the following relationsmustapply atthe bubblewall:

dp~dt= Op/Ot +ROp/On (16)

dR/dt= 0V/Ot+ ROV/Or (17)

with equationsof state:

p/p = constant > OptOp = c2 (liquid) (18)

p(~irR~)= NT (gas). (19)

For many liquids, viscosity can be neglected,therebysimplifying eq. (6). Using (18), eq. (1) can be

written:

(20)pc2Bt pc2On Or r —

Equation(2) integratesto:

(21Ot 2 J p~

p~

Combining(21) with (15) gives:

~ J~=o. (22)Ot pOt 2 On pOr pp~


We can nowsolvethe four simultaneousequations(2), (16), (17) and(20) for the four partial derivativesOp/On, Op/Ot, 0V/On andOV/Ot in termsof quantitiesreferredto the bubblewall. Using thesein eq. (22)andretainingonly first-ordertermsin M gives a relationwhich reducesto:

RR(1 2M) + ~E2(1 — 4M) = I[po. — PL + (1 M) (23)

or, approximately:

RR (1 _2M)+~1~2(1— 4M) = ~-~-~f~+(Po~—PL) (24)

For a completelyempty cavity, neglectingsurfacetension,pL(R) = 0 andeq. (23) reducesto:

RR (1 — 2M) + ~J~2(1 — 4M) = poo/po. (25)

This is the analogueof Rayleigh’sequation(12), reducingto it as M —*0. (24) is the Herring—Trillingequation.The factors multiplying the inertial terms on the left-hand side are linear in the MachNumber.Theyexpresstheeffectof energy-storagein theliquid medium.The terminvolving dpL/dR onthe right-handsiderefersto the energyradiatedas sound,andincludesradiationdamping.Flynn [1] hasshownthat the simpler relationobtainedby omitting the correctionsto the inertial terms:

(26)

can be usedto give reliableinformationaboutnon-linearstablebubbleactivity andalsothe dissipativeeffectsof soundradiationon transientcavities.

The AcousticApproximationusedaboveis valid only wherethe liquid velocity remainsrathersmallcomparedwith c. To cope with conditions in rapidly-collapsingtransients,a more realistic StateEquationis neededfor the liquid. For manyliquids,includingwater, a realisticrelationis knownto be:

p=A(p/po)”—B (27)

with

Op/Op = c2 (28)

whereA andB are constantpressuresdiffering by the steadyambientpressureP0 and n is an integer.

For example,for waterA = (3000 + P0) Bar, B = 3000Bar andn 7. In this casealso,the equationsofcontinuityandmotion(1 and2) still hold andusingequations(27 and28) thesecan bere-writtenin theform:

(29)

EA. Neppiras,Acousticcavitation 171

(30)

It is convenientto introducenew co-ordinatesa and/3 suchthat:

On Ot Or Otand

so that equations(29) and(30) can be re-written:

(31)Oa\ n—i) n Ba

O 1V 2c \ = 2 Vc Ot (32)OJ3\~ n—i) n

Thesearethe basicequationsto be solvedfor the generalcompressibleliquid, subjectto the boundaryconditionsas before.

Numerical integrationof theseequationsmay be carried out by normal procedures.It is to beunderstoodthat this generaltreatmentis necessaryonly for studying the collapseof transients.Ingeneral,in the r, t plane,the characteristiccurveswill be fannedout from the bubblewall as the cavitycollapses.Numerical integrationsfor the caseof the empty cavity havebeen carriedout by severalinvestigators:Gilmore [17]; Hunter[18]; andHickling andPlessett[13].Hickling andPlessetthavealsodealtwith the caseof a collapsinggas-filledbubble.

Gilmore [17] hasshownthat in certaincasesgreatsimplification with, actual analyticalsolutionscanbeobtainedby makinguseof the Kirkwood—Bethehypothesis[19].This statesthat, for sphericalwavesof finite amplitudethe quantityn4i propagateswith avelocity equalto the sum of the fluid velocity andthe local velocity of sound.This is reasonableandis just a plausiblegeneralisationof a procedureusedin acousticalproblems.Thereforewe assumethat rçb propagateswith velocity (c+ V) wherec is nolonger a constant,but dependson the motion.Sincençb propagateswith velocity (c + V), thenso mustr Oçb/0t. Oçb/Ot maybe evaluatedfrom equations(1) and(2) andis foundto be just(h + V2/2) whereh isthe specificenthalpy,given by i;~dp/p. The quantity(h + V2/2) is anenergyper unit massandhasbeencalled the Kinetic Enthalpy.Since the quantity r(h + V2/2) propagateswith velocity (c+ V), we havethe following equationfor the diverging sphericalwave,analogousto eq. (15):

18 011 / V2\1(c + V)~j~r~h+-~--)j= 0. (33)

On evaluation,this becomes: -

(c+ V)(h+-~)+-~-[~+(c+v~~]+rV[~+(c+ v)~]=0.

As before,using(28) eq. (1) can be written:

172 EA. Neppiras,Acousticcavitation

(35)pc20t pc2or Or r —

Again, as before,the boundaryconditions (16) and(17) hold at the bubblewall and the four partialsOp/On, ap/Ot, 0 V/On and OV/Ot can be evaluatedfrom the four equations(1), (16), (17) and (35) andsubstitutedin eq. (34). Whenevaluatedat the bubblewall this gives the Gilmoreequation:

RR ~j(i _M)+~E2(1—~)= H(1 +M)+~~(1 —M). (36)

Here,H is thedifferencein theliquid enthalpybetweenthe bubblewall andinfinity. C alsorefersto thebubblewall andbothH and C arethereforefunctionsof R andt.

For a completelyemptycavity, dH/dR = 0 and eq. (36) reducesto:

R~(1_M)+~1~2(1_~)=H(1+M). (37)

These equationsaremore complexthan they look, and before they can be used,H and C must beevaluatedandexpressedas functionsof P. Using the equationof statefor the liquid (27) weobtain:

A’~’H = (n—l)p [(PL + B)(~~w~~— (p~c+ B)~’~”] (38)

and

C2=C~+(n—1)H (39)

which,for the emptycavityreduceto:

H = (n—i)p [(B)(~_l)/n — i] (40)

and

/ C\2 —

— (41)

if, as can usuallybe assumed,the collapseoccursat constantexternalpressure.In the generalcaseof the gas-filledbubblepL(R) is afunction of time andno integralscanbe found

for the Gilmoreequation.Numericalmethodsmust be used.On the otherhand,for the emptycavitythe Gilmore equationbecomesan ordinarynon-lineardifferentialequation.This is solvableanalyticallyin somespecialcases;but generally,as before,numericalmethodsarecalledfor.

3. Stablecavitation

In any generalcavitation field, most of the visible bubbleswill be oscillating stably. Far removedfrom the thresholds,oscillationsare linear, or nearly so. Below the radial resonancefrequencythe


radial motion is in phasewith the excitation pressure;above it, out of phase.Most stablecavitiescontainpermanentgas,generallyair. Only under certainspecialconditionscan a vaporouscavity bemaintainedin stableoscillation. Strictly speaking,permanent-gasbubblesare also not quite stable, asthereis a continualexchangeof gas, by diffusion, acrossthe bubblewall. But diffusion operateson along time-scale,soa relativelystableconditionholds.

3.1. Undampedlinear oscillations

We assumethat the bubble, equilibrium radiusR0, containspermanentgas in liquid at ambientpressureP0. Derivation of the equationof motionfor linearoscillationsunderan impressedsinusoidalpressureis a simple dynamicalproblem. It may be approachedin the usual way, calculating theequivalentmassandstiffnessof the systemandequatingthe sum of the inertial andrestoringforcestothe excitationforce. An alternativeapproachis to lineariseanyof the generalnon-linearequationswehavealreadyderived,neglectingdampingtermsif present.For example,usingeq. (14) andsubstitutingR = R0+ n, wheren/Ro> 1, expandingin powersof itR0 andretainingonly the first-orderterms, gives:

r+w~n=—~-sin(Ot (42)

wherethe resonancefrequency,(Or, is given by:

po~R~= 3y(Po+ 2u/Ro)— 2o/R0. (43)

For largebubbles,the surfacetensionterm becomeslessimportantandwe thenobtain:

po~R~= 3yP0 (adiabatic) or pw~R~= 3P0 (isothermal). (44)

On the otherhand,for bubblessosmall that surfacetensionpredominates:

pw~R~= 2(3y — 1)o (adiabatic) or pw~R~= 4o (isothermal). (45)

For large air bubblesin water undernormal atmosphericpressure,the relation (44) is approximately:frRo=300wherefr is the linear frequencyin Hz andR0 is in cm.

The generalsolutionof the linear equation(42) is:

PA 1. 0). 1n = 2 2 I sin ot — — sin0)rt I. (46)

PR0((OrW)L (Or J

Thisexpressionreducesto an indeterminateform atresonance.Furtherinvestigationgives,for (0 =

n=.., 0A 2(sinwt—otcoswt). (47)

The solutionshowsthatthe phaseof r with respectto PA changesthroughresonance.At resonancethesolutionshowsinstability, the amplitudeincreasingwith time. Neglectingdamping,aswe havedone,theamplitudecantheoreticallyincreasewithout limit.


It is of someinterestto comparetheresonanceradiuswith theacousticwave-lengthin theliquid (A).Since c5= 7P0/Pg andA = 2iw/w. using(44) we easilyseethat

7P~l(~p~)lI2(c) (48)

A irp c

For example,for an air bubblein water, takingp5 = 0.00i3; p = 1; Cs = 3 X i0~and c = 1.5 x 10~c.g.s.

units, 2R0/A= 0.004.The resonancediameteris thereforemuchsmallerthanthe acousticwave-length.As we shall seelater, it becomesdifficult to achievetransientcavitationfor bubblesaboveresonancesize,while stablebubblesof suchsizesarelikely to distort into surfacevibrationalmodes.This amountsto justification of our assumptionthat theacousticwave-lengthmustbelargecomparedwith the bubblesize under usefulcavitatingconditions.

3.2. Dampingofstableoscillations

We recognisethreesourcesof dampingof the bubblemotion:(i) Viscousforces are effective only at the bubble surface,where they exert an excesspressure

proportionalto the radial velocity. We haveseenin Poritsky’sequationthattheshearviscosity entersasa boundaryconditiononly. The viscousboundary-layerdistortsundervibration in oppositedirectionsdependingon whetherthe bubbleexpandsor contracts.On linearising the Poritsky equation(13) weobtain:

j+~t+(0~r=-~-sinwt. (49)\pRo pR0

Thus,the dampingcoefficientdueto viscosity is:

2 orP(0r’~0 3yP0

The viscous dampinghasonly a small effect on the resonancefrequency,which is now given by w~where

((01)2= (~r)2 — (~~)2 (51)

(ii) Acousticradiation damping. In a compressibleliquid, an oscillatingbubbleexpendsa portion ofits energyin radiating sphericalwaves. That energymust be lost is obviousfrom the fact that thepressurein thesphericalwaveat thebubblesurfacehasa componentin-phasewith the particlevelocity.We easilysee that if the velocity of the bubblesurfaceis R, the in-phasepressurecomponentwill begiven by pc(wRo/c)

2Rand the rate at which work is done by the bubble is thenpc(trR0R/c)

2.Thetime-averageof this expressionis not zero and the bubble thereforeloses energyto the liquid ingeneratingthe outgoingwave.The bubblewill lose a fraction (3pg/p)1”2(Cg/C) of its energyper cycle.Atresonance,wherepw~R~= 3P

0 thisloss factor is just6r = WrRo/C = krRo= (3P

0/poc2)1”2.This, of course,

appliesonly to largebubbleswheresurfacetensionis not a strongcontrolling influence.Under theseconditionsthereforeresonancedampingby soundradiationis independentof frequency.


(iii) Thermaldamping.As we haveseen,dampingdue to viscous forces and soundre-radiationiseasily evaluated.The third form of dampingon oscillating bubblesresults from thermalcoupling, afeaturewe haveso far ignored in formulating equations.It turns out that this is the most importantsourceof dampingin most cases,and alsothe most difficult to assess.We will discussthermalcouplingin detail later. Herewesimply reviewsomeearly history andquoterelevantresults.

Pioneertheoreticalstudiesof the dampingof free andforced oscillationsof bubbleswere carriedoutby Pfriem [20] andSaneyoshi[21].They predictedthat thermalcouplingwould contributemost to thedamping of gas bubbles in ordinary low-viscosity liquids at frequenciesbelow a few 100kHz. Thispredictionwas amply borne out in detailedexperimentalwork by E. Meyer and his group at theUniversity of Gottingen[22—26].Pfriem[20] showedthat the Q-vaiueof air bubblesre~-onantbetween200 and 300kHz in water would be as low as 6, thermalconductioncontributing about 80% of thedamping.The bestavailabletheoreticaltreatmentsof thermaldampingof stablecavities are thoseofPlessetand Hsieh[27] andHsieh [5]. A useful survey, comparingthe threedampingprocessesfor airbubbles in water hasbeenpublishedby Devin [28]. The review paperby Flynn [1] also containsadetaileddiscussionof the threeformsof dampingappliedto both stableandtransientconditions.

In the extremeisothermaland adiabaticstatesthe volume and pressurechangesare in-chaseandthere is no dissipation. Between theseextremes,the temperatureat the bubble wall may still beregardedas constant,as the liquid is an excellentheatsink. But therewill bea temperature-gradientinthe gasand aphase-differencewill exist betweenthe averagevolume andpressurechanges,resultinginloss. The averagethermalprocesscan then be regardedas polytropic, with the pressureand volumechangesrelatedthroughan “effective polytropicindex”, F, whichmayrangebetween1 and y. Then,forlargebubbles:

pg(R)= P0(R0/R)3~ (52)

Zwick [29],following PlessetandZwick [30],studiedthe effectsof heat-conductionon small-amplitudeoscillationsof gasbubblesin water,expressingresultsin termsof F definedby eq. (52). His systemofcurves,reproducedin fig. i relateF to frequencyand R0 for argon-filledbubblesin waterat normaltemperature.These results are instructive and revealing. They show, for example, that at lowfrequencies—20—40kHz, as usedin manypracticalapplications,largebubbleswith radii aboveabouti02 cm will behavepractically adiabatically,while thosewith radii below about iO~cm will behaveisothermally.At low mega-Hertzfrequencies,as usedin medicaldiagnostics,all bubbleswith radii lessthanabouti0 cm will behaveisothermally.By usingthe resonancerelationbetweento, R0 andF it isnot difficult to deducethe effective F to be usedin the formula (52) for any bubble size. This isindicatedby the brokenline in thefigure. Forexample,at 20kHz theeffectiveF at resonancewould heabout 1.45. The curvessuggestthat at sufficiently high frequenciesall bubblesmustbehaveadiabatic-ally. But of coursethe resultsarebasedon the usual assumptionthat bubblesaresmall comparedwiththe acousticwave-length.PlessetandHsieh[27]haveshownthat if this assumptionis denied,conditionsmaybecomeadiabaticagainat very high frequencies.

Devin, in his review, followed Pfriem in deriving an expressionfor the damping due to thermalconduction.Thedampingconstantis evaluatedin termsof R0,P0, ‘y anda wherea is the ratio of R0 toL5, the “thermal diffusion length” in the gas,given by L5 = (

2Dg/o)112, Dg being the thermaldiffusivityin the gas.Devin finally obtainedthe following resultfor the thermaldampingconstant:

~ (a—i) 3(y—1) 53ta+2a2/3(y_i)~ 2a


- APPROX, r ~i

LINEAR RESONANCE

• r~r~

.0 Ii .2 .3 .4 5 .6 .7

Effective gamma

Fig. 1. Effective gammafor stable cavities.

valid for largebubbleswherea ~ about 2.5. For small bubbleswith a � aboutunity, the result is:

& 2a2(y— i)/15y. (54)

Very large andvery small valuesof a correspondto adiabaticand isothermalbehaviourrespectively.Thermaldampingvanishesat both extremes,but peaksata critical valueof a betweentheseextremes.Figure2, plottedfrom Devin’s exactexpressionfor & shows6~as a function of a. Thermalconductionalso affects the real part of the bubblestiffness,changingthe resonancefrequency.But the effect israther small, the difference being no more than about20% as betweenthe extremeadiabatictoisothermalstates.Dampingis most importantat resonance,andthe resonancevalueof & is obtainablefrom (53) or (54) by substitutingthe appropriatevalue for R

0 containedin a.The total dampingconstant6 is simpiy the sum ôv + &~+ ö~.In fig. 3, reproducedfrom Devin’s

paper,the three dampingconstants,togetherwith the total damping,are displayedas functionsoffrequencyfor resonantair bubblesin water. Thermaldampingis dominantover mostof the frequencyrangefrom 10 to 1000kHz. Radiationdampingis almostindependentof frequencyandwould dominate

E•A• Neppiras, Acousticcavitation 177

___ 0’ 0~~V

Fro~u2ncy(kHz)Fig. 2. Thermaldamping constantfor gas bubblesasa function of Fig. 3. Frequency.dependenceof the theoreticalthermal, radiation,a(=Ro/L

5). viscous andtotal dampingconstantsfor resonantair bubblesin water.

at low frequenciesbelow about 1 kHz. The contribution from viscosity is almost proportional tofrequencyandbecomesvery importantat MHz frequencies.Fromfig. 3 we see that at low ultrasonicfrequencies,around20 kH,z, the0-valueof resonantair bubblesin water is about13, falling to about7at 1MHz.

3.3. Couplingby mass-diffusionacross the bubblewall

Over long periodsof time, mass-and thermal-flowsacrossthe bubblewall changethe size of thebubbleandmodify its motion. It is realisticto treatthesetwo effectsseparately.

Considerfirst gas-flowby diffusion in the absenceof the soundfield. If the bubbleis filled with apermanent-gasin an undersaturatedliquid, it will lose gas at a rate determinedby solving theconvectivediffusion eq. (4). With no soundfield, the bubblechangessizeslowly andit is reasonabletoomit the convectiveterm, giving:

(55)Ot \3r

2 rOnT

EpsteinandPlesset[3i] obtainedthe following solution:

= 4irR~D(CCO— C0) [~—+ (irDt)hhl2]

where C. is the mass-concentrationof gasdissolvedin the liquid at agreatdistancefrom the bubbleand C0 the saturationconcentrationunder the prevailingconditions.This result neglectsthe effect ofsurface tension but if this is included, it is easy to see that C0 must be multiplied by the factor(1+2u/RoPo):

dm 2 ~ 2cr\Ii 1 1--=4irR0C0D-— 1_~_~-)[~~~+(Dt)1/2j. (56)


Replacingdm/dtby 4~rpR2dR/dt, the rateof increasein bubblesize by diffusion becomes:

1 7dt p \CI ROPO) Lfl

0 (irDt)”2 (5 )

Evenwhenthe liquid is saturatedwith dissolvedgas(C~= C’0) the surfacetensionterm ensuresthat the

bubblewill eventuallydissolve away.The quantity (irDt~”2 in theseequationsis large at first, near

t = 0, but quickly decreasesand may generallybe ignored where diffusion occurs over long timeperiods.As numericalexamples:an air bubble,10’~cm radiuswill dissolveawaycompletelyin air-freewater in about 1.17sec: in about2 secin half-saturatedwater;andin 6.63secin saturatedwater.

It is not difficult to show that bubbles in stable oscillation in a sound field may grow by asecond-ordereffect of the field evenwhenthe relative concentrationC’~/C’

0is well below unity. Thisprocessis usuallyreferredto simply as “rectified diffusion” andmaybe envisagedphysically asfollows.On the positive pressurehalf-cycle the gas in a small bubblewill be compressed,while the relativeconcentrationof dissolvedgasin the liquid is reduced.Gasthendiffusesoutwardsfrom the bubbleintothe liquid. Conversely,during the negativehalf-cycle of pressure,while the bubbleis expanded,gasdiffuse from the Iiouid into the bubble.However,thesetwo ratesarenot equalas the surfaceareaofthe bubbleis greaterduring the negative(tension)half-cycle, andas diffusion ratesareproportionaltothe exposedarea,the bubblemustgainsomegasover a completecycle.

The first suggestionthat gasbubblesshouldgrow by a rectifiedeffect of this sortwas madeabout40yearsagoby Harvey [32]. This was followed by a mathematicaltreatmentby Blake [9] in his famousHarvardreport. Blake obtainedan approximatesolutionto the problemby assumingthat the bubblewall remainsstationarywhile its areaand the gas-concentrationvary as if the bubblewas oscillatingsinusoidally.Murray Rosenberg[33] continuedBlake’s work, using his results to estimatethe pulse-length dependenceof the cavitationthreshold.Pode [34] furtherextendedBlake’s theory by includingthe effectsof motion of the bubblewall as well asits changein area.His resultwasalmostidenticalwithBlake’s. The simplificationthat all theseworkerswere forced to makeimplied that the convectivetermin the diffusion equationcould be ignored.When, finally, measurementswere carriedout to assesstheeffect, the theoreticalexpressionswere found to be seriouslyin error. In fact, it is not very difficult toseethat the convectiveterm in the equationshouldbeimportantwhenthediffusion-lengthin the liquidis small comparedwith the bubbledimensions.

A more exact treatment,publishedby Hsieh and Plesset[35] showedthat the convectiveterm isindeedvery imnortant.They usedour simplemodel of a sphericalbubblewith uniform interiorsituatedin an infinite liquid. To makethe problemmoreamenableto analysis,theeffect of viscosityandthermalexchangewere neglected.Of the completesetof eauations(1—8), eq. (3) andboundarycondition(7)then becomeredundant,andthe analysisis basedon equations(1), (2), (4), (5), (6) and(8). The liquidmay he assumedsaturatedwith dissolvedgas at the pressureexisting within the bubble. For smalloscillations,we maywrite:

Pg PAl + � sin tot): R = R0(1— ~� sintot); C = C0(1 + e sin tot) (58)

where � 4 1. For the assumedsmall-amplitudeoscillations, the liquid may reasonablybe takenasincomnressihle.The bubbleis assumedfilled with a perfect isothermalgas. The full set of equationsthen becomes:

V=R2R/r2 (9)


j~j~~ =~(pL—p~) (11)

(4)Ot On \0n2 nOn!

with the boundarycondition:

4irR2D = (4irR3p5) (5)

at n = R, alongwith the linearising relations(58).The problem is solved by using a systemof successiveapproximationsin powersof e. Rectified

diffusion appearsas asecond-ordereffect (in termsof order �2).Assumingthat the diffusion length inthe liquid is small comparedwith R0, the authorsfind:

(4irR2D(OC/On)R)= ~irDCoRo�2 (59)

wherethe brackets( ) referto averagingover a completecycle.Making useof theboundarycondition

(5) gives the massflow-rate:dm/dt= ~iTDCoRo(t~P/Po)2. (60)

This resultshows,for example,that a smallgasbubblewould beexpectedto doubleits radiusin a time:

t = 9R~pg/4CoD(~) (61)

if the outwarddiffusion rateis smallenoughto be neglected.We must notethat the result, eq. (60), isvalid only for bubbles large comparedwith the diffusion length but still small comparedwith theresonancesize. Surfacetensionhasbeenignoredbut maybe accountedfor by multiplying C

0 by thefactor (i + 2oiR~P0),giving:

= ~irDC’~R0(i+ ~-) (v). (62)

As Kapustina[36] andSafar[37] haveshown,a correction-factorcan beapplied to includebubblesizesthroughresonance.The multiplying factor is:

[(1— 132)2 + $262]_1 (63)

wheref3 = (0/CUr and6’~is the 0-valueof the resonance.If the liquid is not over-saturatedwith dissolvedgas,the inflow by rectified diffusion, given by eq.

(62) with (63) competeswith the steadyoutflow, given by eq. (56). These two rates then define athresholdpressureor oscillatory amplitude,abovewhich the bubblewill grow. Below the threshold,the

i80 EA. Neppiras,Acousticcavitation

bubblewill decreasein sizeandmayeventuallydissolveaway.Adding to zero the mass-flowsgiven by

theseequationsdefinesthe pressure-threshold:f~P\23(

1~ 2o C~\(i+ 2u ~ Q2\2+Q262 64

\. P0) — 2\. R0P0 C0) ~ t~ i-’ .~ p

For example, for air bubbles in water well below resonancesize, with R0 = 6X 10~cm and P0=106 dyne/cm

2,we find ~P/P0 0.55. For very smallbubbles,such that2o-/R0P0~’1, ~P/P0—~1.22. For

large bubbleswhere2o/R0P04 1 but atfrequenciesstill well below resonance,we havet~P/P0-.* 0.32.We seethat the thresholdfor rectified diffusion falls as the bubblegrowstowardsits resonancesize,

but the growth rate increasesat any fixed value of excitationpressure.Above the resonancesize,thethresholdincreasesrapidly until, for bubblestwo or threetimesthe resonancesize,very high excitationpressuresare neededfor growth. A practical consequenceof rectified diffusion is that it reducestheeffective thresholdbubble size for transientcavitation to occur and causesthe thresholdto becometime-dependent.

Eller andFlynn [38] avoidedthe small-amplituderestrictioninherent in the Hsieh—Plessettreatmentby getting computersolutionsof the set of non-linearequations(5), (9) and (11). They also obtainedanalyticalsolutions for the asymptotic small-amplitudecondition through resonance.Dampingwasneglected,but theyconsideredboth isothermalandadiabaticconditions,usingseveralarbitraryvaluesof y. Their approximateanalyticalsolution is complex,but resembleseq. (64). When the liquid issaturatedwith dissolvedgas,the approximateresultfor adiabaticconditionsreducesto:

(~P\2~~~(1_132)2 (65)

\P0) R0P0(1—fl

2/8)

where,as before 13 is written for to/to.= (pto2R2/37P0)”

2.This result was publishedby Eller [39] andmaybecomparedwith themodifiedHsieh—Plessettthreshold,equation(64)aftersettingC

0.= C0andô = 0.

Thresholdmeasurementsby Strasberg[40and41] at afrequencyof 24.5kHz; by Eller [39and42] at11 and26.6kHz; andby Gould [43]at 20 kHz agreereasonablywell with eithereq. (64) or (65). In all ofthesemeasurements,the bubbleswere well below resonancesize. Dampingwas thereforesmall and13 4 1. Eller measureda growth-ratefor small air bubblesin watermuchgreaterthanpredictedby theHsieh—Plessettheory,despiteagreementon thresholds.Gould [43]found that growth ratesagreedwiththeory only when precautionswere taken to preventacoustic microstreamingaround the bubble.Microstreamingoccursin the acousticboundary-layerand is associatedwith the formationof a shearwaveduethe retardingeffect of viscosity. Microstreamingis greatly influencedby anyasymmetry,likethe presenceof solid boundariesor the onsetof surfacewaves.Kapustina[44] also studiedthe growthrate of a small, 0.02cm diameter,air bubblein water in soundfields of increasingintensity (~P= 0—1.4Bar peak). In this case,the bubblewas suspendedon the measuringmicrophoneandwas thereforeprobablynot vibratingsymmetrically.As might be expected,the measuredgrowth rateswereappreci-ably greaterthan predictedby theory.

Microstreamingaffects the convectiveterm in the diffusion equationby adding a steadystreamingvelocity to the oscillatoryvelocity. Kapustinaand Statnikov[45] evolveda theory applicablewhenthebubbleis largecomparedwith the diffusion length in the liquid andwheremostof the flow velocity isdueto streaming.Fig. 4 comparestheoreticalresultsfor the growth of anair bubblein water,oscillatingradially. Curve (1) wasplottedfrom the modified Hsieh—Plessetformula, eqs. (62) and (63), andcurve

EA. Neppiras,Acousticcavitation 181

dm/dt

‘0-a

/\°

0—u ~, ,,2-= . 0 — 2=

‘0-02~~

lo_1~— —0 0.8 1.6 2.4 3.2

R0\R,

Fig. 4. Rate of inflow of air by diffusion into stably oscillating bubbles in water in termsof the dimensionless ratio Ro/Rr: (1) plottedfrom theresult,eqs.(62, 63); (2) from ref. [45],taking accountof acoustic streaming.PA = 0.144Bar; f = 26.5ki-Iz.

(2) from the authors’ theory, assumingthat the influx of gas is entirely due to the effect of thestream-flow. The calculations refer to an acoustic pressureamplitude of 0.144Bar at frequency26.5kHz. At sucha low pressuremicrostreaminghasaprofoundeffect, the growthratebeingabouttwoorders of magnitudegreaterthanpredictedfrom the Hsieh—Plessettheory. The experimentalpointsshown on the graph were obtainedwith the bubble supportedon the measuringmicrophone.Theauthorswerealsoable to showthatwhile the steadymicrostreamingflow is themost importantfactoratlow excitation levels, the oscillatory velocity will assumethe major role at sufficiently high levels.Despitereasonableagreementwith observations,this theory hasbeencriticised. It seemsthat unlesstheliquid wasover-saturatedwith dissolvedgasto an extentsufficient to permit growth in the absenceofthe soundfield, microstreamingalonewouldsimply havethe effectof increasingthe outflow of gasfromthe bubble.However, as pointed out by Gould [43], the authors interpret their resultsfor C0. = C0,

underwhich conditionthe bubblemustbe losinggas.Observationshaveshownthat the bubblesdo not continueto grow indefinitely by rectified diffusion.

Hsieh and Plesset[35] suggestthat this is due to instabilities developingon the bubble surface.Kapustinapreferstheexplanationthatgrowth will continueonly until theliquid hasbecomedepletedofdissolvedgas.But in anycase,as we haveseen,the growth ratemustslow down after the bubblehasreachedits radial resonancesize.

3.4. Oscillationsofstablecavities with thermal coupling

The pioneerpapersdealingwith forced quasi-linearoscillationsof bubbles in a soundfield, whichincludethe effectsof heatconductionandconvectionarethoseof: PlessetandZwick [30]; PlessetandHsieh[27]; Hickling andPlesset[13]; andHsieh[5]. PlessetandHsieh[27]includedthermalconductionin both the liquid and gas-phases.But, as usual,we will first assumespatialuniformity within thebubble.This assumptionis equivalentto restrictingthe analysisto bubblessmall comparedwith boththe acousticwave-lengthand the thermaldiffusion-lengthin the gas.The set of equations(1—8) then

I 52 E.A.Neppiras, Acousticcavitation

apply. Plesset and Hsieh also neglectedthe effects of viscosity, mass-transferand evaporation-condelisaLion at the bubble wall. Restriction to small-amplitude oscillations justify assuminganincompressiolemedium.

Time goveining equations are (1), (2) and (3) with boundaryconditions(7) and (8) modified by thea~Jovesimplifying assumptions.As before,(1) and (2) with the liquid stateequation(p = constant),evaluated at the bubblewall, give eq. (11), repeatedherefor convenience:

RR ~ = (~ Pu/P. (11)

Equauon (3) reduces to:

82T 201’ 1 lOT OT\

-o-~i----~—=-—t---mV---I (66)or r or DL\dt Or!

aiiu the uoundaiycondition(7) to:

3KL(~) = 3PgR+ pC’vi’g. (67)r=i~

Here, DL is the thermaldiffusivity in the liquid, given by KL/PSV whereKL and S,, are the thermalconductivity and specific heatat constantvolume for the liquid. C’~ is the specific heatat constantvolume for tile gas.The equationof statefor the assumedperfectgasis: Pg(41TR3/3)= NTggiving:

PL = (~0 ±~)(~)— ~. (67a)

Following PiessetandHsieh, a linearisationprocesscan be applied to solve the setof four equations(11), (67a), (66) and (67), writing P = P

0(1+ �) where � is a small quantity 41, with similarapproximationsfor R/R0, T/T0, T~/T0 andPg/Po. Solutionsof theselinear equationscanbe expressedintermsof error-functions.The formal solutionsarenot of muchpracticalvalue, however,and we shalltherefore not quote the results here. But the steady-statesolution, expressingthe average ther-modynainicbehaviourof the bubble,can readily be found. The main conclusionis that the averagebehaviouris isothermalor adiabaticdependingwhether

~PC)L LL)

This implies that the oehaviouris isothermalif the bubbleis smallor thefrequencylow, andvice versa.For large bubblesand/or high frequencies,the assumptionof spatialuniformity within the bubble

niay not be realistic. Plessetand Hsieh [27] also examinedthis more complexcaseand found theasymptoticsteady-statesolution. The significant practicalresult is thatwhile conditionsareisothermalat low frequencies,they also become isothermal again at very high frequenciesif the condition(Kg~’oiKLSv)1~~2~ 1 holds,as it usuallydoes.As the frequencyis increasedfrom low values,the averagethermodynamicbehaviourchangesfrom isothermalto adiabaticand backagainto isothermalat suchhigh frequenciesthat the acousticwave-lengthis small comparedwith the diffusion lengthin the gas.

E.A. Neppiras,Acousticcavitation 183

Essentiallythe samegeneralconclusioncan be reachedby a simpler argument,which is instructiveandworth repeatinghere (see refs. [27] and [46]). The increasein internal energyin the bubblecorrespondingto any incrementin temperatureE~Tis:

AQ2~1TR~pgCp~T.

On the otherhand,the lossof heatfrom thebubbleduringthecompressionhalf-cycleis approximately:

= KL~41TR~2~.

Writing DL = KL/pSP this becomes:

= 4~R~SP(DL/w)112i~T.

Then:

i~Qi3 ~~jDL/w)

112

~Q2 ~pgCp R0

If i~Q24i~Q1sothat

3 ~ (D1Jto)1”~~

PgCp R0

only a smallportionof the energytransferredis availablefor increasingthe internal energyof the gas.In this case,thermal diffusion is so efficient that temperaturechangesremainsmall, and the ther-modynamicbehaviouris essentiallyisothermal.Conversely,if 1~Q2~‘ L~Q~the averagebehaviourisadiabatic.The detailedanalysisagreeswith thesefindings whenconditionsarespatiallyuniform withinthe bubble.But whenthe temperaturefield in the bubblecan becomenon-uniform,the high-frequencylimit doesnot leadto adiabaticbehaviour.An intuitive pictureof the physicalsituationis still possibleby comparingthe magnitudesof the threequantities:Ag (the acousticwave-lengthin the gas);L5 (thediffusion lengthin the gas);andthe bubbleradiusR0. Of thesethreelengths,the first two involve theacoustic frequency. Table 1 summarisesthe approximatethermodynamicbehaviourin the variousregimes.The high-frequencytransition from adiabaticto isothermalconditionsrequiresfrequenciesin

Table 1Thermodynamicbehaviourof anoscillating gas bubble in liquid

Frequency Comparisonof Relevant Thermodynamicrange lengths criterion behaviour

Veryhigh Ag4 Lg 4 R0 LgSO Ag IsothermalHigh Lg<Ag<Ro Lg.cZAg AdiabaticModeratelyhigh Lg < Ro<Ag Lg< R0 AdiabaticLOW R0 < Lg < A~ SPLL i’ C~R0 Isothermal


the Giga-Hertzrange, and rather large bubbles.Over a very wide rangeof conditionscommonly

encounteredin practicalultrasonics,the averagebehaviouris approximatelyadiabatic.

3.5. Oscillationsofstablecavitieswith thermal coupling, including evaporation-condensationat the bubble

wall

The dynamicalproblembecomesmorecomplicatedwhenmass-transferby evaporation-condensationcan occur alongsideheat-flow. Theseprocessesarecoupledandcannotbe dealtwith in isolationfromone another.In someimportantapplicationsof ultrasonicsthe vapourpressureis high, with the liquidin a near-boilingcondition.Vapour-exchangeis thenclearlyimportant.The set of equations(1—8) applywhenspatially-uniformconditionscan beassumedwithin the bubble.

The first thing to decide is whetherthe stable oscillations will permit free exchangeof vapourorwhetherthe vapourpressuremustbe regardedas a function of the motion.The criterion is whetherthequantity aP~(2rrkT)”2is > or <pR. Here, a is the “accommodationcoefficient”. Applying thiscondition to the bubblewall velocitiesthat we can expectfor low-amplitude stableoscillations,it seemsreasonableto assumethat P~remainsnearto its equilibrium value,therebysimplifying the problem.

As long ago as 1952, Plessetand Zwick [30] carriedout a theoreticalstudyof cavity formation inboiling liquids. But the first detailedstudyof the dynamicsof vapourcavities underacousticexcitationis dueto Trammell [471.Trammell was interestedin acousticdetectionof bubblesin nuclearreactors.As a first step towards quantifying the problem he derived an expressionfor the dynamic com-pressibilityof pure-vapourbubbles.He madeseveraldrastic assumptions,over-simplifyingthe problem.But he appearsto havebeenthe first to appreciatethe possibility thatrectified heattransfercould occurunder acoustic excitation. Soviet workers have also been interestedin the acoustical problem, inconnectionwith the operationof the hydrogenbubblechamber(see, e.g. ref. [48]).Shadley [49] andalso Wang and Hsieh [50] haveindependentlyre-examinedthe behaviourof a bubblecontainingamixture of permanent-gasandvapour in a soundfield, includingthe effectsof rectifiedheat transfer.

In view of the partial natureanddiscrepanciesin previouspublishedwork, we decidedto developalinearisedtheory of the acoustically-excitedpure-vapourcavity, taking into accountthermalconductionand evaporation-condensationat the bubblewall [51]. The relevantequationsare similar to thoseofPlessetandHsieh [27] but with an addedterm to take accountof vapour-exchangeat the bubblewall.Both the bubbleandliquid were assumedfree of permanent-gas,viscosity was neglectedandspatially-uniform conditionsassumedwithin the bubble.The vapourpressurewasrelatedto temperaturevia datatakenfrom measuredvapour-pressurecurves.The resultingsetof five equationswere linearisedin themanneralreadydescribedandan expressionobtainedfor the motion of the bubblewall. By relatingthisto the excitationpressure,an “effective mechanicalimpedance”can be cited for the vibrating bubble.The effectivestiffnessis a complexexpression,involving thefrequencyandbubble-sizeimplicitly.

Detailed study of this expressionshowsthat abovea certaincritical size the bubblebehavesas iffilled with permanent-gas.Smaller bubbles will collapseunder the influenceof surfacetension.Thecritical size increaseswith decreasingfrequency.A resonancecondition is inferred by equatingthestiffnessreactanceof the bubble to the mass reactanceof the surroundingliquid. The theory wasoriginally developed in an attempt to explain cavitation phenomenaobservedin low-temperatureliquids. In fig. 5 the “effective acousticadmittance”for liquid helium I at 3 K is shownas a function ofthebubblesize over arangeof excitationfrequencies.Fromsuchcomputedresults,the bubbleresponsecan bere-expressedas a resonancefrequency— bubblesize plot. This is shownfor helium at 3 K in fig. 6and for nitrogen at 77.4K in fig. 7. The remarkableconclusionto be drawn from thesetheoretical


100

10~2

::‘~ \ \\\ \ * ‘ a

—so ‘~\\1012

10~ 10~ 10~ 100 102 io4 Ro

Fig. 5. Effective acousticadmittancefor pure-vapourcavities in helium I at 3K in terms of the equilibrium cavity radius (Ro): (a)f= 102Hz; (b)102Hz;(c) lO4Hz; (d) i0~Hz.

resultsis that over awide rangeof frequencies,resonancecan apparentlyoccur for two widelydifferentbubble sizes. For e)ample, for helium I at 20 kH.z the resonanceradii are about 2 x 102 and6 x 10~cm. The larger size is very nearthat given by the formula (43), while the otheris very small.Although this double resonancephenomenonis difficult to visualise physically, it is in qualitativeagreementwith experiment[52,53]. The prediction that a high densityof very small andvery activebubblesmay be presentin acoustically-excitedvolatile liquids is important in a numberof practical

fr(Hz) f1-(Hz)

-

• \ \\

LOC . 10’ ~ \

I \io

4 \\ 1

\

i03 - 102

02 - . 0

~ 0—2 ~—‘ I io~ io~ o’ ioa io~ IO2tO~

R0(crfl) R0(cm)

Fig. 6. Resonancefrequency(fr) for pure-vapourcavitiesin helium I Fig. 7. Resonancefrequency(fr) for pure-vapourcavities in liquidat 3 K as a function of the equilibrium cavityradius (Ro). nitrogenat 77.4 K asa function of the equilibrium cavity radius (Ro).


applications,notably ultrasoniccleaning, wherelow boiling-point fluorcarbonsolvents arefrequentlyused.

A more completestudy of the dynamicsof the acoustically-drivenpure vapourcavity hasbeenpublishedby Akulichev andcolleagues[54].Theseauthorsshowthat themotion of the cavitywall canbeexpressedby the compactequation:

RU+2~U—~--+1 [P0._PL+~ (i _P)] = 0. (68)

Here, the velocity of the bubble wall, R, is not identified with the particle velocity of the liquid incontactwith it (U) becauseof the couplingby mass-flowdueto evaporation-condensation.This eq. (68)combinesthe equationsof motion,continuity and statefor the liquid, evaluatedat the bubblewall. Inthe extreme case where U = R and p~.,4p it reducesto the Noltingk—Neppiras equation (11).Evaporation-condensationis expressedby the equation

4irR2p(R— U) = dm~/dt (69)

wherem~is the massof vapourin the bubble.Thetreatmenttakesaccountof the variabletemperatureandpressurefields within the bubble, so the completeset of governing equationsinclude those ofmomentum-, mass- and energy-conservationfor the vapour-phaseas well as the heat-conductionequationand state-equationfor the vapour. The completeset of equationsis naturally difficult tohandle,but numericalsolutionswereobtainedandapplied to experimentalresults.The growth ratesofvapourbubblesin nitrogenandhydrogenwere foundto agreecloselywith the theory for smallbubbles,but beyonda certain size, wide discrepanciesappeared.The authorsattributedthis to the effects ofbubbleinteractions.

It is easyto seethat understaticconditionsa purevapourcavitycannotremainin stableequilibrium.At any temperaturebelow the boiling-point, bubblesmust collapseunderthe excessinward pressuredue to surfacetension.The growthandstabilisationof acoustically-excitedvapourbubblespredictedbythe abovetheoriescan be understoodphysically as follows. On the expansionphaseof the oscillation,vapour in the bubbleis cooled slightly and heat diffuses into it. On compression,the reverseprocessoccurs. But as the diffusion rateis proportionalto the exposedarea,andas the areais slightly greateron the expansionphase,the vapourwill gainsomeheatover completecycles, maintainingthe interiorof the bubble at a temperaturea little abovethe surroundingliquid, which may be regardedas aconstant-temperatureheat-sink.The increasedtemperatureimplies an increasein the vapourpressure,so that evaporationmustoccur to restoreequilibrium. Butevaporationitself is subjectto a rectificationdue to the area effect, acceleratinggrowth. The bubblewill thereforegrow, in a mannersomewhatanalogousto the rectified mass-diffusionpreviously discussed.Just as for mass-diffusion,a thresholdacousticpressurewill exist, the growthmechanismcompetingwith condensationimpelledby the surfacetensionexcesspressure.Also, as for mass-diffusion,andfor thesamereasons,radial resonancewill be amore or less stablecondition. Rectified thermaldiffusion is important only for mainly-vapour-filledbubbles.The effect is small for gas bubbles in ordinary liquids at normal temperatures,for whichrectified mass-transportpredominates.

The thresholdand rate of growth by rectified heat flow can readily be obtainedfrom the abovetheories.For example,fig. 8, reproducedfrom ref. [55] shows the thresholdacousticpressure(PA)requiredto maintainbubbles of radiusR

0 in thermal and mechanicalequilibrium in helium I at a


10’

106,

1~

1o~

102

10

1 ~ i~-~- 1O_2 10’~ R0

Fig. 8. Theoreticalacousticpressureamplitude (dyne/cm2)required to maintain a vapourcavity at radiusR

0 cm in helium I at 3K anddrivingfrequency10 kHz.

temperatureof 3 K at 10 kHz. At the main resonance,PA reachesa critically low value of about7 dyne/cm

2peak.Interest in stable oscillationsof pure vapour cavities arosefrom studieson the low-temperature

liquids, nitrogen,hydrogenandhelium. Theseliquids arenormallyusedneartheir boiling-points,whereanybubblesmustbe mainly, or entirely, vapour-filled.Cavitationstudiesin liquid hydrogenare relevantto the designof bubblechambers.In the caseof helium, acousticstudieshavethrownmuchlight on thewhole questionof nucleation.This is a specialisedareawhich will not be reviewedin detail here.However,the importantrefs.[56—70]areincludedin the bibliographyto enablethe interestedreadertopursuethe subject.

3.6. Non-linearoscillationsof stablecavities

In the abovetheories, useful results havebeenobtainedby linearising the governingequations.Direct numericalsolutionsof the non-linearequationsoften showcomplexstableoscillationsthatmaynever evolve into transients.In all these cases,eq. (14), which is based on the incompressibleassumption,can safely beused to predictthe motion.

To discussthesecases,it is convenientto re-writeeq. (14) in anon-dimensionalform:

rF+ ~t2= 8[r3~’ — l+~Sifl T + a(r3’~— r1)]. (70)

Here r = R/Ro; p = PA/PO; r = ~t; ~ = Po/pw2R~a = 2cr/R0P0.For largebubbles,the resonance

frequencyis given by pw~R~= 3yPoso that & = w~/3ya2.The coefficient S is thereforesmall for large

bubblesat frequencieswell aboveresonance.Undertheseconditions,wecan expecta stablenon-linearsolutionwhenp is alsonot too large. Largebubbles,well abovethe resonancesize will thereforetendto oscillate in aquasi-linearfashion.For suchlargebubbles,surfacetensionis not likely to be a strong


controlling force.Then,with a —~0, eq. (70) reducesto:

,~+~j2=8(r3~_i+psinr). (71)

Onthe otherhand,if thebubblesaresmallenoughfor surfacetensionto dominateoverthegas-pressureterm,weobtain:

n.+~t2= S[psinr+a(r3”—r’)] (72)

wherea and S are now bothlarge. This conditionmayrefer to bubblesthat are not only well belowresonancesizebut alsobelow the transientthresholdsize (seesection5).

Noneof theseequations(70—72) is analyticwhile the time-dependentpressureterm is present.If weomit the forcing term in eq. (70) the equationfor the free oscillationsis:

3~2 —3 —3 —1rr+3r =~—~-d—(rr )=5[r “—l+a(r “—r )]. (73)

Integrationnow gives r in termsof r, essentiallythe energy-balanceequation:

~2~3 = J 2r S[r3v —1+a(r3v— r1)] dr. (74)

The r—t characteristiccan be obtainedby a numericalintegration.The stationarypoints occurat r = 0andso the period of the non-linearfree oscillation is obtained.In the energyequation(74) the termrepresentingthe internalenergyof the gasbecomessignificant only in thefinal stagesof compression.Ifwealso neglectsurfacetension,eq. (74) becomes,on revertingto the original variables:

pR3R2=~_~PoR3where W=4~” [Po+(’~l)]. (75)

Herring [711hassolvedthisequationin termsof beta-functions.ApproximatingRmin to zero,the periodH is given by:

H6 2.14(p/Po)3(W/P0)

2. (76)

Under strictly linear acousticexcitation, the free oscillation associatedwith the linear resonancefrequency, although present, remains small and quickly dies out. But when conditions becomenon-linear,coupling exists and can be maintainedbetweenthe free and forced vibrations. For thisnon-linearcouplingto generateperiodic motion, the ratio of forcing to resonancefrequencymustbe arational fraction, that i5, U/CUr = n/rn. The coupling is particularly strongwhen n and m are smallintegers.In thegeneralcase,the vibration will containboth the free andforcingcomponents,with theirharmonicsand sum and difference frequencies.Many frequency-componentstherefore arise, andwheneverharmonicsof the forcing andfreevibrationscomewithin eachother’sbandwidths,energycantransferfrom the forcing to the free vibration. As the bandwidthfor air bubbles in water is typicallylarge,strongcoupling is possible.


According to Stoker [72] the free oscillation appearsas a sub-harmoniccomponentif m = 1 andn> 1 (that is, 0) = nwr); as a harmoniccomponentif n = 1 andm > 1 (that is to = wr/rn); and as anultra-harmoniccomponentif neithern or m = 1. Flynn [11presentssolutionsfor the two cases:whereCU/CUr = ~, reproducedin fig. 9; and whereCU/CUr = ~ (fig. 10). In eachcase,PA/PO was takento be 0.33,well within the stableregime, and isothermalconditionsassumed.In the first case,fig. 9, wherethebubbleis driven below resonance,the responsecontainsa strongcomponentat frequencyw/4. Whenthe samebubble is driven above resonance,fig. 10, the amplitude-responseis noticeablyweaker,asexpected,with a strongcomponentat w/9. For thesecaseswhere n and m are small integers, thecorrespondinglineshavebeenobservedin the spectrumof noiseemittedby the cavitatingfield.

Many computersolutionsof the generaleq. (14) havebeenobtainedfor stableoscillationsover awide rangeof non-linearconditions(see refs. [1], [10], [73—79]).Typical exampleswere publishedbyBorotnikov and Solukin [79].They computedradius—timecurvesoverseveralcyclesof oscillationfor awide rangeof valuesof PA and R0. For small PA/Po andR0 no greaterthanRr the oscillationoccursapproximately at the excitation frequency. But with R0> Rr the bubble oscillation has a strongcomponentat its own natural resonancefrequency.For small bubbles,with R04 Rr, typical transientconditions set in as PA is increasedabove P0. But if PA is further increased,a point is reachedeventuallywherethe bubblehasgrown solargeduring the tensionphasethat it hasno time to collapsecompletelybefore the end of the pressurecycle. It will thenusually collapse as a transientnearthesecondpositive pressurepeak. Furtherincreasein PA will thendelaythe transientcollapseuntil thethird pressurepeak;andso on. Eventually,with PA/PO ~ 1 the bubblewill neverbe able to collapseas atransient.This meansthat an upperthresholdin PA hasbeenreached.This behaviouris well illustratedin the R—t curvesof fig. 11, reproducedfrom ref. [79]. Theserefer to an air bubble in water, radius10~cm driven at a frequencyof 500 kHz. The bubble was thereforemuch smaller than its radialresonancesize.The numbersattachedto the curvesrefer to the ratio PA/Po and the time-variationofthe acousticpressureis shown below the curves.The changein the form of the curvestakes place

0.0

-J 0420~*is.c 5O~iucI I

0 lOGO’ 2000 13000 4000TIME, lit0

Fig. 9. Radius-time curvefor a cavity driven below resonance:R0 = 2.6 x iO~cm; PA = 0.333Bar;f=83.4 kHz.


0.8 -

~ 0.7 -

ZOMUC 30~,.c 40~..c ~ii I I $ I

0 000’ I 2000 ‘ ‘3000TINE, t/t~

Fig. 10. Radius-timecurvefor a cavity driven aboveresonance:Ro = 2.6 X iO~cm; PA = 0.333Bar;f = 191.5kHz.

suddenly,andthe hatchedregionsof the diagramindicatethe regionsof instability wherea verysmallincreasein PA will trigger off an additionalstableoscillationbefore theeventualtransientcollapse.Theswitch occurswhenthetheoretical“collapse-time”approacheshalf theacousticperiod.Dataof thissortareimportant in offering an explanationfor the strongsubharmonicemissionsfrom the cavitationfield.

An interestingstudyof the free oscillationsof a gas-filled bubblewas carriedout by RobinsonandBuchanan [78]. They used eq. (14) with PA zero. If surface tension is also neglected,the non-

2x

Fig. 11. Radius—timecurvesfor anair bubblein water. R0 = i0~cm; f= 500kHz. The numberson thecurvesrefer to theratio PA/Pa.


dimensionalform of eq. (71) becomessimply:

r ~ + 3(dr)2= ~(r3~ — 1). (77)

The authorsusedr = (3’yPo/p)”2t/2irR0 asthe dimensionlesstime, so that x is just 4~.2/3y,independent

of frequency.In fact, the equationrelatesonly two parameters,r andT. Equation(77) is analytic,with afirst integral:

~ (~J~)2= J (1 — r37)r2 dr. (78)

The oscillationsaresetoff from a bubblein equilibrium initially, with Pg= P0. In the solutionwe must

now retain the term representingthe energyof the gas.The R—t curveswere obtainedby numericalintegration.On examiningthe trajectoriesovera rangeof valuesof y andbubble-wallvelocities,dr/dr,the authors found that the period of the non-linear oscillations increasedsteadily with increasingexcitation amplitude. Also, as might be expected,increasing y had the effect of reducing thecompressionratio.

Anotherform of non-linearoscillation, which also illustratescouplingbetweenthe free and forcingvibrations, may occur when conditions are very non-linear with bubbles well below their radialresonancesize.When PA is not far below the transientthreshold,the first collapsewill occurnearthepeakof the excitationpressure,at phase3ir/2. On collapse,the bubblemaynot disintegrate,asit wouldif it were transient,but may oscillate for the remainderof the cycle at its own natural resonancefrequency. The sequencemay be repeatedat the next, and subsequentcycles. This is a case ofshock-excitationandof coursethe resonancefrequencyof the bubbleneednot be integrally relatedtothe excitationfrequency.The acousticemissionspectrumwill thencontainastrongcomponentat thebubbleresonancefrequency.The exampleshownin fig. 12 was reproducedfrom ref. [2]. The R—t curverefersto an air bubblein waterof equilibrium radius10~cm driven at 28 kHz with an acousticpressurevery close to the transientthreshold.In this case,the initial phaseof expansionandcollapse wasphotographedusingavery high-speedcamera,and,as can beseenfrom the figure, measurementsof thebubblesize takenfrom the film agreewell with the theoreticalresult.

Sovietworkers [80]haveshownthat eq. (14) can beexpressedin a frequency-independentform for awide rangeof drive conditions.They illustratethisby writing the equationin the following dimension-lessform:

fd2a\ nfda\2 Po I 2a~/ to \ 1 ~ ~ 3a—r)+~~—) ~ (1 Cu) ‘Iwr”a )~) (79)

R/R

0

Acouatlc periods

Fig. 12. Radius—timecurve for anair bubblein watershowingoscillationsat thebubbleresonancefrequency.R0 = i0~cm; f =28 kHz; PA is nearthetransientthreshold.(1) theoreticalplot; (2) experimentalpoints.


where: a = wR/wrRo; r = wt; p = PA/PO; and (.Ll~= (P0+ 2oiRo)/pR~.If w/tora 4 1 or R/R0>> 1 thisequationreducesto a particularly simple form which does not dependexplicitly on the acousticfrequency:

fd2a\ 3/da\2 P

0 / 2o .a~-T) + = pw~R~ + P sinT) — 1. (80)

It can easilybe seenthat the condition to/wra 4 1 implies the two conditionsCUr/CU ~‘ 1 and (PA — P0)~

(P0+ 2ff/Ro). If theseinequalitieshold simultaneously,thenanysolutionobtainedat onefrequencywillmodel that for any otherfrequencyfor the sameequilibrium bubblesize. The authorssupply manyexamples.

So far, our discussionshavebeenbasedon eq. (14). This equationis suitable for dealing with allforms of stable cavitationover a limited numberof cycles wherethe time-scaleis short enoughtoneglectthermalcoupling andmass-diffusion.Theoreticaltreatmentsby Plessetandcolleaguescan beappliedto the generalproblem,but the exactsetof equationsis not readily handled.Flynn [1,81] hasalso researchedthis problemand succeededin producingsetsof equationsthat can be programmed,and results obtained relatively quickly, using a modern high-speedcomputer. To this end, someempirical data, and “plausible” equationsof state are used. Thermal coupling and evaporation-condensationcanbe included,along with acoustic-radiationandviscous damping.In oneformulation,useis made of the acousticapproximation,eq. (26), which includesthe acousticradiationterm. Theeffectsof thermalconductionareintroducedin asimple way: the entropyof the cavity-contents(~) iscalculatedas a function of time by the useof first-orderdifferentialequations:

= .!a~i. ~!~i= — ~- ~ ~ + ~,~ pp2, ,~ 2 g

g

wherePg is givenby eq. (82) below.H(t) is an auxiliary functionandk1 andk2 areconstantsof thegasinvolving C~,N and y. Theseequations(81) were derivedfrom the differential equationsfor heatconductionwithin the cavity. Uniform pressurewithin the cavity was assumedand somesimplifyingassumptionsmadeaboutthe velocity andtemperaturedistributions.The gas pressurein the cavity canthenbe written:

Pg= exp(4IC~)(P0 + 2u/Ro)(R0IR)3~. (82)

This can be insertedin the boundaryconditionat the bubblewall:

FL = Pg+ Pv — 2cT/R— 4ILR/R. (83)

Using eq. (26) thengives:

~ (84)

which with (81) describesthe motion of the bubblewall.Solutionsof theseequationsprovide informationon the relative importanceof thermal,radiation,


0 ‘000 2000 3000 4000

Fig. 13. Radius—time curves for an air bubble in water, comparingadiabaticwith heat-conductingconditions. Ro =2.6 x iO~cm; f =83.4kHz;PA/PO~1.

and viscousdampingon the motion of bubblesoscillating non-linearlyfor long periods.At moderateamplitudes,soundradiationandviscosity havelittle effect,but thermalconductionis importantenoughto dampout muchof the harmoniccontent,giving R—t curveswith the sameperiod as the excitationfield. But underhighly non-linearconditions,the effect of the dampingbecomesmore marked.Heatconductiontendsto increasethe kineticenergystoredin the liquid surroundinga collapsingcavity, thecollapse-speedof the heat-conductingcavity beingalwaysgreaterthanthat of the adiabaticcavity. Onthe otherhand, soundradiationandviscosity must always tendto reducethe violence of the collapse.Clearly also,thermalconductionwill alwaysdecreaseboth the maximum temperatureandgaspressurewithin the cavity comparedwith the adiabaticcase.These resultsare important for explaining thethermaldependenceof cavitation effects. Figure 13 reproducedfrom Flynn’s review [1], comparestheR—t curvesfor the heat-conductingand adiabaticcasesfor an air bubble,radius2.6 x 10~cm, drivenjustbelowthe transientthresholdat a frequency83.4kHz. Anotherinstructiveresult, reproducedin fig.14 comparesthe extremecasesof the isothermaland adiabaticmotion of the samestably oscillatingbubbledriven at the samefrequencybut at much lower excitationpressure(PA/Po = 0.2). The R—tcurvesaresubstantiallydifferentevenat thislow drive level.

4. The collapseof transient cavities

It is in the final stagesof the collapseof transientcavities that the well-known disruptive effectsoccur. Thesearethe resultof the high concentrationsof energyandconsequenthigh liquid pressuresand velocities. The most important of theseeffects— metal-erosion— was well known to hydraulicengineerslong before acoustically-generatedcavitationarousedanyinterest.Early investigatorsthere-fore tendedto concentrateon the dynamicsof the collapseof cavities,without referenceto their modeof generation.


I I I I I I I I I I 1

I • ,/ISOTHERMAL -

/ ‘

I I I I I I I I I I0 1Q00 2000 3000 4000

TIME, t/I0

Fig. 14. Radius—time curvesfor an air bubble in water, comparing adiabatic and isothermal conditions. R0 = 2.6 x i0~cm; 1 = 83.4kHz;P.8.JP0=0.2.

Thetime-scaleof an imploding cavity is soshortthatwe can usuallyignore anyeffect of thermalormass transfer by diffusion. The motion is essentiallyinertia-controlled.Empty cavities will collapsecompletely.In a gaseoustransientthe motion is cushioned,in the final stages,by compressionof theresidualgas. Fora vaporoustransient,the implosion will proceedto completiononly if vapourhastimeto condensefreely soas to hold thevapourpressurenearto its equilibriumvalue.This possibilityneedstesting.We hereusethe formuladerivedfrom kinetictheory giving the maximumrateatwhich vapourcan condenseat a specifiedtemperatureandpressure.Unlessthe quantity aP~(2irNT)

112is greaterthanp~R,condensationwill not occur freely and the vapour pressuremust increase.Here, a is the“accommodationcoefficient”, a numberless thanunity. We can easily see that for typical transients,someincreasein F,., mustoccur towardsthe end of the collapse.Nevertheless,it is usual to specify aconstant,or zero,vapourpressureover the collapsephase,therebysimplifying the analysis.

In studyingcollapsingtransients,interestis centredmainly on: (i) the velocity field in the liquid, butespeciallythe velocity of the bubblewall; (ii) the pressurefield in the liquid: and (iii) the temperaturereachedby the permanent-gas,if any, in thebubble.Thesecan all be evaluatedonceweknow the threequantities;(i) the initial cavity size from which the collapsestarted(Rm); (ii) the gas pressurein thecavity at this initial size (0); and (iii) the ambientliquid pressureresponsiblefor collapsingthe cavity(Pm). In an acousticfield, the cavity will havegrownfrom someinitial radiusR

0, and Rm and Q willdependon R0, PA, P0 and to. Many solutionsof the generalequationshaveshown that the collapsegenerallyoccupiesless than 20% of the acousticcycle, while the pressureis nearits peak.We maythereforeregardPm as nearlyconstantover the collapse.For the ideal caseof asingle smallbubbleinan infinite liquid, Pm mustbe closeto (PA+ Fo); here,we assumethat the isolatedbubbledoesnot itselfinterfere with the ambientpressurefield, relieving the pressureto any extent. However, in a realcavitation field, with a high densityof bubbles,the presenceof the bubble cloud must relieve the


ambientpressureto someextent.The acousticpressurebecomespeak-rectified,reducingPm below thevalue(PA + F0). In the extremecaseof a very intensecavitationfield, measurementshaveshown [82]thatPm hardly risesaboveF0 on the compressionhalf-cycle.On the negativehalf-cyclealsothe tensionis relieved to the point wherethe externalpressuremaynot fall much below vapourpressure.Theexternalpressurefield can thenbe representedas a square-wave,with pressure-excursion(P0— P~).

4.1. The collapsingemptycavity

In his famous1917 paper“On the Pressuredevelopedin a Liquid during the Collapseof a SphericalCavity”, Rayleigh [7] quotedBesant’sformulation of the problem [61: “an infinite massof homo-geneousincompressibleliquid actedon by no forces is at rest and a sphericalportionof the fluid issuddenlyannihilated.It is requiredto find the instantaneousalterationof pressureat anypoint in themass,and the time in which the cavity will be filled up, the pressureat an infinite distancebeingassumedto remainconstant”.

In this case,the equationof motion of the cavity wall is:

R1~+~I~2=Po/p. (12)

A first integralgives the bubblewall velocity:

2P R3 2P R3

R2r~_Q[(_~) —i] or ~ —1]. (85)

This is justthe energyequation,whichRayleighset down immediatelyby equatingthekinetic energyofthe liquid mass to the work done by the externalpressure,F

0, in collapsingthe cavity. A secondintegrationof the equationgives the collapsetime, ‘r, as the following incomplete/3-function:

— Rm R312dR (86)

~~2P0) J (R~,,_Rs)u

2.

The time for completecollapse,as R -~0, is readily obtained:

0.915Rm(p/Po)1”2. (87)

This is the simplestpossiblemodel for the collapsing’sphericalcavity. It is importantto noticethat alltransientcavities,whateveradditional complicationsare introduced,start their collapselike this. Thisfeaturecan be included in the definition of a transient;we say that thesecavities start their collapse“Rayleigh-like”.

Rayleighexaminedthe distributionof pressurein the liquid using the equationof motion (2). Thepartial derivatives8V/3t and u9V/3r can be obtainedas functionsof r and R by using eq. (85) andthecontinuity relation r2V = R2R. This gives the pressuregradient,which on integrationyields:

(88)


Here, Z = (Rm/R)3,the volume compresssionratio. The distribution of pressurein the liquid, expres-

sion (88), is displayedin fig. 15 using Z as parameter.As the collapseproceeds,the pressurepeakincreasesin heightand movescloserto the bubblewall. The thicknessof the high pressureregion alsodecreases.Eventually, in the final stagesof collapse,the cavity is surroundedby a thin shell of highlycompressedliquid. At high compressionratios, the maximumliquid pressure,Pmax, obtainedby settingdp/dr = 0, is given by:

Pmax/Fo = Z/44”3 (89)

andthispeakoccursat distance4”3R from the cavity centre.Rayleigh’sanalysisloses noneof its simplicity if weassumethat the cavity is filled with vapourat its

constantequilibrium pressureF,.,. (P0— P~)then simply replacesP0. Also, surface tensionmay be

includedwith little additionalcomplication,the problemremaininganalytic.Rayleigh’s treatmentcan be applied to the expansionphase of an empty cavity, as well as the

collapse.The ambientpressurethen hasthe form of a rectangularwaveandwe mustassumethat thecavity is drawnout from a nucleusin the form of an unwettedsphere.It is easyto see that the radialvelocity is approximatelyconstantovermostof the expansion-phase,at the value R = (2P~/3p)”

2whereF~is the constanttension.It is alsoeasyto showthat whenF

0 = F~thecollapsetime is approximately3/4 of the expansiontime.

For a completelyempty cavity the liquid pressuresand velocity near the cavity wall reachinfinitevaluesas R —*0. In water,radial velocitiesreachthe velocity of soundat a compressionratio of about32. Long before this the assumptionof incompressibilityfor the liquid would havebecomeunrealistic.To examineconditionsnearthe end of the collapse,therefore,we must revert to treatmentsusing amorerealisticequationof state.

The Herring—Trilling equation (24) assumesa constantvelocity of sound, and with a constant

I’ ‘•••‘-‘•~•“•-.

/ •• \-\s’

2 (

Fig. 15. Pressuredevelopedin the liquid surroundinga collapsingRayleighcavity; Z =volume compressionratio.


ambientpressure,reducesto:

RR(1 — 2R/c)+ ~J~2(1+ 4EIc)= Fo/p. (90)

A first integralis:

)D r/D ~3 1 / AD~—1

~ 1111 ~“ 913pRR) 1k 3c

These revert to the correspondingRayleighequations(12 and 85) as lVc-+0. From (91) E canbeobtainedas a function of R and a numerical integrationthen gives the R—t characteristic.This willdiffer from the Rayleighsolution only nearthe endof the collapse,as we see from plots of R as afunctionof the radial compressionratio (fig. 16).

The pressurefield surroundingthe collapsingcavity can beobtainedin termsof r andR by a processsimilar to that usedfor the Rayleigh case.The result is bestexpressedas the eliminantof R betweentheexpression(91) and:

p = -~+~ (i _~~)(~pJ~2)+:~ (i _~)(pJ~2_2P) (92)

When the collapsespeedsapproachthe steady-statevelocity of soundwe areforced to usethe Gilmoreequation,which,for the emptycavity, reducesto:

R(1_~)+~-f(1_~)=H(1+~). (37)

Gilmore obtainedexactsolutionsof this equationfor a constantexternalpressure.A first integral is

M

10 I

0.__0.001 0.01 0.I

R/ RmFig. 16. Comparisonof the cavity-wall velocities developedon implosion for threemodelsof the empty cavity: (1) the Rayleigh model; (2)Herring—Trilling model; (3) Gilmore model.


obtainedby separationof variables:

ln ~-° =2 1 RU?—C) dl (93)R J j~2(j~- 3C)+ 2H(R + C)

andthis can be evaluatednumerically or graphically. In mostcases,H 4 C2 andeq. (93) then simplifiesto:

~2~Q ~ 11 (943pRR)k 3C) I

reducingto the Rayleigh formula (85) as M —*0. In fig. 16 the radial velocitiesgiven by eq. (93) arecomparedwith the RayleighandHerring—Trilling predictions.The effect of including compressibilityisclearlyseenin the slowing up of the radial velocity at high Mach numbers.Also shown in fig. 16 areplotsobtainedby Schneider[83]from a numericalintegrationusingthe exactexpressions,eqs.(31) and(32). Gilmore’s resultsagreesurprisinglywell with the exactsolutions,evenup to M 2. The pressurefield surroundingthe collapsingcavity can be evaluatedby a processsimilar to that used in previousformulations[17].

4.2. Thecollapsinggas-filledcavity

The presenceof permanent-gasin the collapsing cavity preventsthe singularity present in theRayleighmodel.Wemaythereforeconsiderasimple extensionof the Rayleighcavitywherethebubbleis filled initially with gas at somepressure0 at its maximum radiusRm. As before, we will assumeconstantambientpressurePmover the collapseperiod.At present,wewill not considerhow this statewas reached.Neglectingsurfacetensionandviscosity and assumingadiabaticcompression,eq. (14) forthe motion of the bubblewall becomes:

RR+~2[O(~)”Pm]. (95)

Thefirst integralgives the energyequationfor the collapse:

= Pm(Z 1) Q(Z — ZT)/(1 — y). (96)

A numericalintegrationwill nowgive the R—t curve.This follows the Rayleighsolutionfor mostof itstrajectory.Differentiation of eq. (96) gives the accelerationof the bubblewall:

— — ~ Ii — QY Z~~1)l (97)— pR 1 Pm(

71) j~

SettingR = 0 in (96) the minimumradius,Rmin, reachedby the collapsingcavity is given by:

1? - I () ll/

3(v—l)

98Rm ~LFm(y1)


approximately,assuming04 Fm for a typical acoustically-generatedcavity. The maximum collapse

speed,given by R = 0 occursat:(R\3~~’ Fm(y—1) (R\3(~1)

or %D\L ml ‘..,~‘7 ‘. mini

approximately,andits valueis:

J~2 ~2Pm(Yl)[Pm(Y1)1~7~ 100max ~ L Qv ~ ( )

Under adiabaticcompression,we easily find the maximum pressure(Pmax) and temperature(Tmax)reachedby the gas:

Pmax Q[Pm(y — 1)/Q])4~1) QZ~ (101)

Tmax ToPm(y— 1)10= T0Z~

1~ (102)

to good approximations.The Mach numberfor the bubblewall motion can be written downfrom eqs.

(96) or (100). Expressedin termsof theradial compressionratio, its maximumvalue is:Mmax (0.Ol5Rm/Rmin)3”2(Pm/Fo)1”2 (103)

if Rm/Rminis greaterthanabout4. In fig. 17 Mmax is displayedas afunction of theratio Pm/Q with Pmas parameter.We know that the incompressibleassumption,on which the theory is based,is strictlyvalid only up to M 0.2. From fig. 17 it is clear that the validity of the assumptioncan also beexpressedin termsof Pm or 0. For example,the assumptionis justified for 0/Fm as low as 1.5% when

Mma x

__ /[0.6 ~ioBcr//

02

0 __-0 100

~r ~“0

Fig. 17. Maximum Mach numberfor thebubble-wallvelocityfor a collapsinggas-filled bubble.


the collapsepressureis 1 Bar, but only for Q/Pm4 about4% for a collapsepressureof 10 Bar. Table 2indicatesthe rangeof validity of the simple expression(95) for the collapseby comparingits predictionof the minimum bubblesize with theoriesthat take accountof the liquid compressibility.The figuresconfirm that theincompressibletheory is reasonablyreliablefor valuesof 0/Pmup to about0.01. As wehaveseen,thiscorrespondsto maximumMachnumbersup to about0.2—0.3.

Khoroshev[84] hascalculatedthe collapsetime for a gas-filledtransientfrom the formula

R,r.jn

I dRT0= I —

JRRm

andfinds:

Tm 0.915Rm(p/Pm)1”2(1+ 0/F) (104)

with “ taken to be 4/3. This differs from the correspondingRayleigh collapsetime by the factor(1+QIPm).

The pressurefield surroundingthe collapsinggas-filledbubble can be evaluatedusing a proceduresimilar to that for the emptycavity. In termsof r and R theliquid pressureis given by [10]:

p — Fm = — 4~[QZ (~j 4~ ~ (z 4)Fm] — ~4 [Fm(Z 1) (105)

This functionis shownplottedout in fig. 18 for severalvaluesof the volumecompressionratio, andwithPm/Q = 10. As the collapseproceeds,the highly-compressedshell of liquid follows the bubblewall, thepeakgetting closer to the boundary,until, at maximum compression,the maximumliquid pressureoccursatthe bubblewall, equalto thegas-pressureinside,givenby: QZ7 or Q[Fm(y — 1)IQ]~~’’~.Thevery high liquid pressuresdevelopedaroundthe imploding bubbleareimportantin certainwell-knowneffectsof ultrasonics,suchas erosion,dispersionandmoleculardegradation.The photographof fig. 19shows the erosion pitmarks that develop on the radiating surface of a metallic magnetostrictiontransducerdriven for long periodsat high intensity.The erosionpatternfollows the cavitationstreameractivity, the significanceof which we will discusslater. The high temperaturesdevelopedwithin the

Table 2Comparisonof compressibleand incompressibletheories

for collapseof gas-filledcavities (y = 1.4)

0/Pm RffJRmm

Incompressibletheory Compressibletheory

PmlBar Pm’lOBar

10’ 0.262 0.2810-2 0.047 0.060 0.074iO~ 0.0069 0.018 0.025io~ 0.0010 0.006 0.009


z-to

0 Rm2Rm 3~m

Distancefrom Bubble Centre

Fig. 18. Pressuredevelopedin the liquid surroundingacollapsinggasfilled bubble;Z = volume compressionratio; Pm/Q = 10.

Fig. 19. Erosionpitmarkson theradiatingsurfaceof a nickelmagnetostrictiontransducerdrivenat high intensity in water;f= 25kHz.


collapsinggas-filled transientare responsiblefor sono-luminescenceand sono-chemicaleffects. Themaximumtemperaturereachedis just T0Z°’

11whereT0 is the ambienttemperature.Re-expressedin

terms of Pm and Q the maximum pressureand temperatureare respectively 0(Pm/30)4 and

To(Fm/30), wherewe havetakenY = t With Pm 1 Bar and 0 = 0.01Bar, the maximum pressureisabout 1.2 x 10~Baror 120kg/mm2.An intermittentor alternatingpressureof this magnitudewould besufficient to fatigue most metals.With T

0 = 300K, the maximumtemperaturewould be about10~K,sufficient to accountfor sono-luminescentandsono-chemicaleffects.

The equation(95) gives reasonablyaccurateresultsfor the collapseof gas-filledbubbleswhile fluidvelocities remainbelow about 0.2M. An even closer approachto reality is obtained by using theHerring—Trilling equation(24). PuttingFc.o= Fm andPL Q(Rm/R)

3~in (24) gives:

RR (1—2M)+ ~2(1 — 4M) = — .q (~)~‘(1+ 37M) (106)

with M i~/c.R is obtainedin termsof R by anumericalintegrationanda secondintegrationgivestheR—t characteristic.Exampleshave been given by Trilling [16]. Gilmore [171obtainedseveral verycomplicatedexpressionsfor thepressurefield in this case.Whenthe acousticapproximationis used(eq.(26)), the pressurefield can be obtainedexplicitly in termsof R and R:

p = (FL — Pm) + ~. (i — ~ + ~- (i — ~-)[pi~2 — 2(FL — Fm) — R (107)

whereFL Q(Rm/R)3~.This equationmustbe solvedsimultaneouslywith (26) to obtainthe pressureas

a function of r andR.

WhenM is knownto approachor exceedunity, it is necessaryto go one stepfurtherandmakeuseof the Gilmore approximation (36) or the exact equations(31 and 32). Hickling and Plesset [13]

obtainedthe most completeset of computedresults availablefor the bubble-wall motion and thepressurefield in the liquid. They useda combinationof the Gilmore theoryand solutionsof the exactequations.Fig. 20, reproducedfrom their paper,shows the dependenceof the cavity-wall velocity onthe radial compressionratio for several values of -~and ambient pressure.The exact solution iscomparedwith that obtainedfrom Gilmore’s theory (which usesthe Kirkwood—Bethe hypothesis)andwith the incompressiblecase.Gilmore’s theorygives resultsin closeagreementwith the exact theory,exceptin the final stagesof collapse.

The distribution of pressurein the liquid during the implosion is shown in fig. 21afrom the samesource.The initial pressureof gas in the bubblewas takenas 0.003Bar. The results are given forsuccessivetimes, measuredfrom the time of minimumbubblesize,expressedas a fraction of the timefor completecollapse.Theseresultsmaybe comparedwith thoseshownin fig. 18, derivedon the basisof the incompressibleassumptionandwherethe compressionratio ratherthanthe collapsetime is usedas the parameter.Also shown,in fig. 21b, is the pressurewaveformed on the reboundandits progressas it propagatesinto the liquid andgeneratesa shockfront. The pressurepeakforms very closeto thebubble wall and attenuatesapproximatelyas 1/r in travelling outwardsfrom the source.The peakpressuresarethereforeconfinedto the immediatevicinity of the collapsedbubble.Extrapolatingfromthe limit to which the computationsapply, a peakpressureof about200Baris obtainedat r/Rm= 2 witho = 10~Bar. This is hardly sufficient to causeanyseveredamageto tough materials.

Ivany andHammitt [85]carriedthroughcomputationssimilar to thoseof Hickling andPlesset.They


~ io~

Empty c.aVIty”-. p~ 1 atmC

K~rkwood-Exact Bethe —

.c 1 0 ~a

~ -1

~ 10

I I I I I I

10 io~~ 102 lOl

AIR0. Bubble radius

(a) Gasconstanty = 1.4; ambientpressurep~= I atm.

~ io~

E : -~ Pm=l0atm~rnpty cavity~’-. y = 1 4C -

IncompressIble10 ~. Kirkwood‘-. Exact

.~ 1~

~ 10_li

ii IIIi..

111111

1o_2 10—i

R/R0. Bubble radius

(b) Gasconstanty = 1.4; ambientpressurePx = 10 atm.

~ io~ —... f~=1atm

E -~Emptycavuty~. V = 1.0IncompressibleC Kirkwood- Beth.C 10C., Exact

~ 1

.C -1~ 10

—4 310 10 1o_2 io~

R/R0, Bubble radius(c) Gasconstanty = 1.0; ambientpressurep,,~= I atm.

Fig. 20. Comparisonof cavity-wallvelocitiesdevelopedon implosion for threemodelsof emptyandgas-filled cavities,andfor variousvaluesof y,ambientpressure(P, = Fo) andgas-content(Q). The numberson thecurvesrefer to 0 in Bar.


10~

10~~ __

Bubble walI~-~ 10 ~:~v=1.4

1 11111111 I I~IIII 1 I 11111111 I

10 10 1 to_2 1o~2 1

nA0, Radial coordinate dR0.Radialcoordinate

(a) Bubble collapse. (bi Bubble rebound.

Fig. 21. Pressuredevelopedin the liquid surroundinga gas-filled bubblecollapsingand reboundingin water. P = Pm= 1 Bar; 0 = iO~Bar;= 1.4. Numberson thecurvesareproportionalto thetime elapsedfrom thetime at minimumbubblesize.

includedthe effectsof surfacetensionandviscosity.Theirresultsconfirmtheassumptionwehaveusuallymade,that neithersurfacetensionnor viscosity affect the generalbehaviourof the collapsing bubble.

4.3. Thegenerationofshockwaves

Following Flynn [1] it is instructive to considerthe conditionsunderwhichthe cavitatingbubbleactsas a sourceof soundwavespropagatinginto the liquid as the drive frequencyand amplitude areincreased: (i) For very slow changesin bubble size, disturbancescan be regardedas propagatinginstantaneously.This is a stablecondition and the appropriatemodel for the liquid would be anincompressibleone. The cavity must thenbe regardedas non-radiating,no work being done on theliquid. All of the energypassedto the liquid during the expansionphaseof the bubbleoscillation isreturnedto the bubbleon contraction.This is just equivalentto sayingthat the radiationresistanceisnegligible whenw is verysmall; (ii) For morerapid changesin the bubblesize,it will eventuallybecomemeaningful to say that the pressuredevelopedin the liquid at anypoint will dependon the motion ofthe bubbleat someearlier time; that is, thereis atime-lagbetweenpressuremeasuredin theliquid andthe motion of the bubblewall. This time-lagimplies energydissipation,this energybeing propagatedinto the surroundingliquid, which thencan beconsideredto presenta finite radiationresistanceto thebubble.The appropriatemodel in this casewould be the acousticapproximation;(iii) However, weknow that the velocity of soundin the liquid is an increasingfunction of the pressure.In a transientevent, the liquid pressurearoundthe bubble as it implodesand reboundsmay increaseto the pointwherethe velocity of sound is seriouslyaffected.Disturbanceswould thentendto overtakethosethatstartedearlier.Thewaveswould thenstartto crowdone anotherandthisis just the conditionrequiredfor a shock-waveto develop.The appropriatemodel to use would thenbe one, like the Gilmoreapproximation,wherethe velocity of sound is afunction of the motion.


The strengthof the cavitationshocksandtheir rangeof actionhavebeenstudiedby: Schneider[83],Brand[86],Brook Benjamin[87],Akulichev andcolleagues[88],andin detail, by Hickling andPlesset[13]. As we have seen,Hickling andPlessetfound radiatedpressurewavesobeying the geometricattenuationlaw (amplitudex 1/r). However,Akulichev and colleagues,basingtheir computationsonthe Gilmore approximation,found the attenuationmuch more rapid than this as the pressurepeakmovesaway from the cavity wall. Theseauthorsalsoshowthat if a shockwaveis to form at all it mustdevelopwithin about r = 20R0from the cavity. In anycase,as Hickling and Plessetalso showed,theshock will remainstrongonly within a few radial distancesfrom the bubble. Therefore,cavitationeffectsthat dependon shockpressurescan takeplace only in the immediatevicinity of the collapsingbubble.Thishasbeenconfirmedexperimentally.

Brooke Benjamin[87] useda very simple model to studywhere,on the rebound,the shockwouldform. He found a criterion in terms of the maximumgaspressure,Fmax, developedin the collapsingbubble.With Fmax in kilo-Bar he found the following simple formula for the distancer at which theshockwould form:

/ rFmax 13.6(~logh—) (108)

mm

On this basis,when Fmax is muchlessthan2 kilo-Bar the shockcan form only in remoteparts of thefield andwill be very weak.Butwith Fmax between2 and3 kilo-Bar it is likely thatthe shockwill format a reasonabledistance.The more recentand more exact calculationsof Hickling and PlessetgiveT/Rmin about5—6 whenPmax> 1 kilo-Bar.

In all the above theory we have assumedthat the imploding bubble will remain spherical.Photographicevidence,to be discussedlatershowthat the sphericalsurfacecanbe expectedto distortandthe implosion is usuallyviolent enoughto fragmentthe bubble.

4.4. Acoustically-generatedtransientcavities

We haveseenthat the collapseof gas-filled transientscan be analysedin detail oncethe threeparametersPm, Rm and 0 are known. In an acousticfield, the cavity will havegrown from someabove-thresholdsize,R0, andtheseparameterswill be functionsof the four “given” quantities:R0, PA,

P0 and w. Severalresearchgroupshaveobtainedmany numericalsolutionsof eq. (14). Examplesofstable,andnearly-stable,solutionsaregiven in fig. 11. The resultsreproducedin fig. 22 refer to truetransients,that is, cavitiesthat collapse completelywithin a single acousticcycle. These,andsimilar,data provide useful information on the dependenceof Pm, Rm and 0 on the primary variables.Forexample,we havetakenthe co-ordinates(Rm, tm) at the peakof bubble-growthandgraphedtheseasfunctionsof R0, F0 andw~.Resultsaredisplayedin figures23—25.Of specialinterestis fig. 25 showingRm and tm roughly proportionalto the acoustic period over the rangeconsidered.Collapse alwaysoccurswhile the acousticpressureis nearits peak.This justifiesour approximation:

Fm ~(FA + F0).

Although we have ampleevidencethat typical gas-filled transientscollapseadiabatically,furtherinvestigationis neededto establishthe growth conditions.But we easilysee that nowhereduring theexpansionphasecan R exceed[

2(Pg+ PA — Fo)/3p]”2 or approximately(2PA/3p)112for large bubbles.


TIME (MICROSECS)

Fig.22. Radius—timecurvesshowingexpansionandcollapseof air bubblesin waterundertransientconditions.PA= 4 Bar; P0 = 1 Bar;f = 15 kHz.

Forexample,for the conditiondepictedin fig. 22,which correspondsto intensetransientcavitation,theMach numberfor the expansion-phasecannotexceedabout 0.01. Clearly, we are justified in assumingisothermalconditions for the growth-phase,and R0, Rm, F0 and 0 are then simply relatedby theformula 0 = (F~+ 2CiRo)(RoIRm)

3.An approximaterelation is therefore establishedbetween theparametersRm, Fm, 0 andtheprimaryvariablesR

0,PA, F0 andthe collapseconditionscan1e analysedin termsof thesevariables.

Flynn [811hasproduceda precisemathematicalformulation coveringthe problemof the acoustic-ally-excitedcavity, whichembracesstableandtransientconditions.Thetreatmentincludesthe effectsofthermalconduction,viscosity,surfacetensionandcompressibilityof the liquid to a limited extent,using

600 -

~~___ ~

lea 40Co 10 20 30 40 50 60

_____________________________________________ ~ (microns)o I I I

0 I 2 3 4 5 6 7&jatmospheres)

Fig. 23. Dependenceof Rm andCm on R0 for air bubblescollapsingas Fig. 24. Dependenceof Rm and tm on PA for air bubblescollapsingastransientsin water.PA = 4 Bar;! = 15 kHz. transientsin water. R0 = 3.2 x 10~

4cm;f= 15 kHz.


107/co

Fig. 25. Dependenceof Rm and Cm on or’ for air bubblescollapsingastransientsin water.R0 =3.2 X 10~cm; PA = 4 Bar.

Gilmore’s equation. A comprehensivetreatmentof this sort necessarilyinvolves assumptionsandapproximations,someplausible,otherslessso. Evaporation-condensationat the bubble-wallis neglec-ted and it is assumedthat the soundvelocity is not affectedby the motion. The result is a set of 11simultaneousequationsthat can be programmedto give answersin a few minutesusing a modernhigh-speeddigital computer.Resultsarein reasonableagreementwith the exacttheory of Hickling andPlesset,which relates only to the collapse and rebound of transients.For full details of Flynn’sformulation,the readeris referredto ref. [81].

5. Cavitationthresholds

For studyingthe thresholds,it is sufficientto usethe simplestnon-linearequationfor the motionofthe bubblewall (eq. (14)). The thresholdscan be locatedfrom solutionsobtainedover a wide rangeofthe parametersF0, PA, R0 and o. To illustratethis we mayreferbackto fig. 11 which showshowthebubblemotioncanswitch from stableto transientandeventuallybackagainto stable,as the excitationpressureis increased.Upper and lower thresholdsthereforeexist for PA betweenwhich transientconditionshold, while outsidethe motionis stable.If FA hadbeenheld constantand R0 usedas thevariable quantity, the thresholdcould have beendefinedin terms of R0 instead of PA. Very smallbubblesoscillatestably,atafrequencycloseto the drive frequency,their motion controlledby surfacetension.Bubbles too large to collapse as transientsin the time availablealso oscillate stably, at afrequencycloseto their own naturalresonancefrequency.Similarly, the thresholdscould equallywellbe expressedin termsof the staticambientpressure,F0, if both PA andR0 are regardedas constant.Summarising,therefore,the thresholdscan be statedin termsof anyof the threeparametersPA, F0 orR0, althoughnaturally PA is generallyused,this beingthe easily-variablequantity.

The transition betweenstable and transientconditions is often sharply defined.When the term“cavitation threshold”is usedwithout qualification,it is the lower of the two thresholdsthat is meant.An ability to predict this thresholdmaybe quite important.When transientcavitationwould be toodamagingor annoying,as in underwatersignallingor medicaldiagnosticsor therapy,it will generallybeessentialto operatewithin or below the stableregime.On the otherhand, aswe haveseen,transientconditionsaredesirablein severalindustrialapplicationsof macrosound.Much experimentalwork hasthereforebeen directed to locating these thresholds in terms of the acoustic and environmentalparameters.Much of this work was carriedout in the years1946—1960andis well summarisedin earlierreviews.We shall thereforenot reviewit here.


5.1. Structuralstability ofcavitation bubbles

For bubblesbelow resonancesize,yet so largethat surfacetensionis unimportantin controlling themotion,the thresholdPA is very closeto P0. To illustrate the sharptransitionin thisimportantcase,fig.26, from ref. [11,showssolutionsof eq. (14) takenover a single acousticcycle,wherePA/PO is increasedover the range0.75—1.0 in small steps.IncreasingPA acceleratesthe growth of the bubble,while themaximumcollapsespeedincreasesover the range0.02—1.6M. The non-linearstableoscillationrevertsto transientwhenPA/POreachesaboutunity. Theliquid is thenaboutto go into tensionon the negativehalf-cycle. At the threshold,the bubblehasgrown to a maximumsize Rm suchthatRm/R0 2, aswemight expect intuitively. This is an important threshold condition, deducible from the collapseconditions.At the threshold,the collapsecurve is practically identicalwith the Rayleighsolutionoverall but the last stages.

Soviet workers[2] and[76]havedescribeda picturesqueway of showingthe transitionfrom stabletotransientconditions.Solutionsof eq. (14) areplottedout on the phase-plane.This involves recordingdR/dt as afunction of R. Stablesolutionsgive closedcurves,transientsshow discontinuities,the curverunningaway to infinity in somedirection.An exampleis reproducedin fig. 27.The plotsrefer to an airbubblein waterradiusR0 = 5 x 10~cm driven at a frequencyof 500kHz. The numberson the curvesrefer to the ratio PA/Fo andx is written for the dimensionlesstime, wt. Conditionsare stableforPA/Po� aboutunity; transientfor PA/FO between1 and about 4; and transient,but with a stablecomponent,for PA/Fo greaterthanabout10. Althoughnot shownin thisexample,for PA/Fo very large,theplots mustdegradeinto a seriesof loopsthat closeup as the oscillationsrevert to the stableform.

5.2. Thetransientthresholds

Bubblesbelow resonancesize are stiffness-controlled.In this case,the lower transientthresholdisvery sharplydefined.As it can be expressedin termsof either PA, P0or R0 it is not surprisingthatasimplerelation shouldexist betweenthesethreequantities,defining the threshold.

~/dx (i/ia)

2 __

~2Q~*)J~~= .0

R/~~

0.56 ~i

0-if

0

wt

Fig. 26. Radius-timecurvesfor an air bubble in water, plotted for Fig. 27. Solutions of eq. (14) representedin the phase-plane.R0 =

increasingP....JP0 in therange0.75—1.0, showing the developmentof 5 X iO~cm; / = 500kHz; x = wt. The numberson thecurvesrefer totransientconditions.R0 = 2.6 x iO~cm; / = 24.5kHz. theratioP~JPo.

E.A.Neppfras,Acousticcavitation 209

The liquid pressurejust outsidethe bubble: FL = (P0+ 2oiR0)(Ro/R)

3— 2oiR is a minimum whendPJdR= 0, thatis, where

— R — [3R~ / 2o.\1112R— TL2o ~Po+R)j

An alternativeexpressionfor FL is (P0— PAsinwt) with a maximumnegative value (PA— F0). The

condition for instability is therefore:PT> F0 + (P0+ 2o/Ro)(Ro/RT)3— 2o-/RT,or, substitutingfor RT:

PT> F0 + ~[3R~(p

2~2/R)]112~ (109)

This equationgives the pressure-thresholdin termsof the othertwo relevantparameters,F0 andR0. It

can be written: FT — P0=4a/3RT. If the bubble is so small that the surfacetensionpressure,2o-/Ro

overwhelms F0, the following simpler result is obtained: PT— F0 4o-/3\./~R0.Similarly, if Po is

significantly greaterthan2oiRo but with the bubblestill belowresonancesize,we obtain: (PT — F0)2

32o-3/27FoR~.For P0~‘2ajRo weobtain FT F0, as wehavealreadyseen.

These thresholdconditions apply only to gas-filled bubbles.Similar conditions do not exist forvaporouscavities. However,if the nucleuswas anunwettedsolid sphere,the thresholdtensionneededto separatethe liquid from the solid would be just: (PT — F0) = 2o/R0. This shows that the tensionrequiredto expanda gas-filledbubbleis lessby afactor of about 2.5 thanfor a sphericalnon-wettedsolid nucleusof the samesize.

It is interesting to comparethe thresholdgiven by eq. (109) with that obtainedby solving thecompleteequationof motion(14). This is bestdisplayedby plotting out R0 againstRm takenfrom theR — t curves. Figure 28 shows the result for a typical caseof air bubblesin water with PA = 4 Bar;F0 = 1 Bar, over awide rangeof bubblesizes.A suddenincreasein Rm is seenas the thresholdR0 ispassed.This value agreescloselywith thatgivenby eq. (109)on insertingtheabovevaluesof PA andF0.

400 - I: -

L 0300- 000 2 U

R~(MIcRoNS)

Fig.28. Dependenceof Rm (x) and Cm (s)on Rofor air bubblesinwater,withR0 variableoverthethresholdrange.PA = 4 Bar;P11= 1 Bar;/ = 15kHz.


It is interestingto note that all bubbles small comparedwith resonantdimensions grow to ap-proximatelythe samemaximumsize.

In derivingthethresholdcondition (109)it hasbeenpossibleto neglecttheinertial terms, asit relatesonly to smallbubbles,well belowresonancesize.Therefore,it doesnot involve frequencyas a variable.However,we know that for large bubblesthe thresholdmaybecomevery frequency-dependent,andbubblesaboveresonancesize tendto revertto stableoscillation.For example,the full-line curveof fig.29 is a solution of eq. (14) for an air bubble in water, radius 8x 10~cm, driven at a frequencyof15MHz, with PA = 4 BarandP0= 1 Bar.The plot is takenover four completecyclesof pressure.In thiscase,astableform of oscillationis obtained.Thebroken-linecurverefersto identicalconditions,exceptthat the frequencyhasbeenreducedto 5 MHz. At this frequencythe bubbleis below resonancesize,and,as expected,the cavitationis transient.Clearly, for anygiven bubblesize, it is alwayspossibletotakethe frequencyhigh enoughto inhibit transientcavitation.In fact, this can be extendedto refertoany bubblefield with anydistributionof bubblesizes. It is a well-known observationthat it becomesvery difficult to cavitatewaterat frequenciesaboveabout3MHz.

Having extractedsolutionsof the non-linearequationof motion,eitheroneof two simple criteriacanbe used to identify the transientthreshold: Rm/Ro 2 andRmax/c= M 1. Lauterborn[89] usedthelatter criterion andproducedthresholdcurvesrelatingFT to R0 or o for the conditionM = 1. Resultsagreewell with plotsof eq. (109) at frequenciesbelowresonance.Throughresonance,the Lauterbornthresholdswill dependon whateverdampingis assumed.More recently,lernetti andcolleagues[90—92]measuredtransient thresholds through resonanceusing the onset of the half-order subharmonicemissionandsono-luminescenceas the thresholdcriteria. In place of the smoothPT—RO characteristicpredictedby Lauterborn, they found many small discontinuities,correspondingto resonancesatharmonicsandsubharmonicsof theexcitationfrequency.This suggesteda theoreticalstudyin whichthe

Time (rflicrasec

Fig. 29. Radius—timecurvesfor anair bubblein water:R0 = 8 x iO~cm; PA = 4 Bar;P0 = 1 Bar;full-line curve15 MHz; broken-linecurve5 MHz.


equations(14) and (26) were both used.This confirmed the presenceof the discontinuitiesand thereasonsfor them.Figure 30 displays a set of resultsobtainedby Creschiaet al. [92] (full-line curve)along with the correspondingLauterbornthresholdcurve (broken-line) for air bubbles in water at700kHz.

The simple thresholdcondition (109) appliesonly in the region of stiffness-control.Apfel [93] hasshown that a simple approximationto the thresholdconditioncan also be obtainedfor largebubblesabovethe resonancesize. We first expressRm in termsof R0 and to this endwe examinethe bubblegrowth in two stages:(i) Considergrowth while the liquid is still in tension,representedby time t1.

Then, writing z~mP= (PA— F0), the averagebubblewall velocity is approximately(2~F/3p)112and the

radiuswill increaseby an amountR1 = t1(2~FI3p)

112.Approximatevaluesfor t1 andL~Pcan beobtained

by consideringthe second-orderapproximationto the pressurevariationabout the phaseof maximumtension(~t= irI2). In this way, writing p = PA/Fo we find:

(wti)2 = 8(p— 1)/p and L~F= 2F

0(p— 1)13.

Thus, R1 = (4/3w) (p — 1) (2Fo/pp)112.(ii) Considerthe period from stage(i) until the momentumof the

bubble-wallbecomeszero,whengrowthwill cease.Thefurther increasein bubblesize can befound byequatingtheinitial kinetic energyof the liquid surroundingthe bubbleto the changein potentialenergyof its contentswhile it expandsto itsmaximumsize Rm. As thebubbleis assumedto belarge,weneglectsurfacetension.We also assumethat the ambientpressureremainsat .l’~during this period.Then

(4irR~p)V2/2= 4ir(R~.— R~)P0/3

where V2 = 2~P/3p= 4Po(p— l)19p. This gives (Rm/Ri)3= 1 + 2(p — 1)/3 andon substitutingfor R

1 we

P1 (Bor)

IC

5 7/

i~/

0.510 5 2 I 0.5 0.2 0.1

R0(Microns)

Fig. 30. Transientcavitationthresholdfor air bubblesin water(ref. [92]).


obtain:

Rm= (4/3w) (p — 1) (2Fo/pp)”2[1 + 2(p — 1)/3]”~.

Making useof the resonancecondition pw2R~= 31’o and using the criterion Rm = 2R0 to define the

transientthreshold,we get:

= ~(p — 1) (2/3p)112[l + 2(p — 1)/3]h/3. (110)

Thisconditionis independentof all gasandliquid constants.It canonly be evenapproximatelyvalid forp appreciably>1.

In fig. 31 the expressions(109) and(110) areplotted out overthe appropriaterangesof bubblesizesfor air bubbles in water at the frequency1 MHz. Over their limited range, theseeasily-calculatedthresholdsagreequitewell with computedresultsusingeither of the two criteria: Rm/Ro= 2 or M 1.

The abovetheory refers to the lower transientthreshold.The upperthresholdis of lesspracticalimportance.Also, it cannotbe expressedin anysimple formula,but mustbe inferred from numericalsolutions of the non-linear equations.For example, fig. 32 shows the calculatedmaximum liquidpressure(Pm) developedin a transientcollapseas a functionof the ambientstaticpressure(F

0). p~was

calculatedon the assumptionthat the bubble, with equilibrium radiusR0 = 1.6x i0~cm, containedonly vapourat a constantpressureof 10~dyne/cm

2.The acousticpressureamplitude was takento beconstantat 4Baratfrequency15 kHz. The correspondingmaximumradiusreachedby the bubble(Rm)is also indicated.The curve for Pm showsupper and lower thresholdsin Fo outsidewhich transientpressureswill not bedeveloped.In this case,the lower thresholdfor PA/PO is about 1.1 andthe upperthresholdabout6.0. The peakof Pm, indicatingmaximumcavitationintensityrequiresFo to beincreasedwell abovethe normalambientlevel. This theoreticalpredictionof anoptimumoperatingpointbetween

100

!~0,1 I lO 00 P~ (ATMOSPHERES)R~/RrFig. 31. Transientcavitation thresholdfor air bubblesin water,plot- Fig.32. Maximumliquid pressure(pm) andmaximumradius(Rm) fortedfrom the approximateformulae(109) and(110). / = 1 MHz. a collapsingvapour-filledbubble asfunctionsof theambientpressure.


102

I~ — _____

I 0’

Rr

10’ __________ 3R~

1o’ Io~’ io~ IP

0(C rn)

Fig. 33. Stableandtransientcavitationthresholdsfor air-saturatedwaterat/ = 20kHz.

thethresholdshasbeenamplydemonstratedin practice,andis importantin severalindustrialapplications

of macrosound.

5.3. Thestablecavitation threshold

Thethresholdfor bubble-growthby rectifieddiffusion, eq. (64), appliesoverthe full rangeof bubblesizes,throughresonance.Abovethis threshold,bubblescangrow in the soundfield to becomeactive.Below it, theywill dissolveawayor becomestabilisedagainstdissolutionin someway. At all events,theywill ceaseto be active. For thesereasons,the diffusion thresholdis frequentlyreferredto as theStableCavitationThreshold.Thestableandtransientthresholdsrelatethesameparameters:PA, P0andR0. It is instructiveto see themdisplayedtogether,as in fig. 33 wherethe plotsrefer to air bubblesinair-saturatedwater ata frequencyof 20 kHz.

6. Cyclic cavitationprocesses

In fig. 33 the TransientThreshold curve has been plotted out from eq. (109). For the StableThresholdcharacteristic,eq. (64) is re-formedto displaythe thresholdpressure,PT, in termsof thebubbleradius,R0:

= ~[APO~/(A+ R0)] [(1—R~B)2+ R~BS] (111)

where A = 2a/P0 and B = w

2p/3yPo.The curvesrefer to air-saturatedwater at normal atmosphericpressure(p= 1; y = 1.4; P

0= 106c.g.s units) at 20 kHz, where the damping factor, ö, for a resonantbubble is 0.0645.


6.1. Thegaseouscavitation cycle

Any bubble entering the region below the Stable Thresholdcannotgrow by diffusion, and willshrink, eventuallypassinginto solution. Bubbleslocatedin the region abovethe TransientThresholdmust immediatelyexpandandcollapseastransients.But anybubblefinding itself in theregion betweenthe thresholds,say at A, in fig. 33, can grow by rectified diffusion. To the right of A the slopeof thetransientthresholdcurve is negative,so the growing bubble will eventuallyreach it and will thenimmediatelyexpand,implode anddisintegrate.Thedebriswill consistof smallbubbles,manyof whichwill reachthe “dead”region below the stablethresholdand disappear.But somemaybelargeenoughto re-nucleatethe region betweenthe thresholds.Thesewill thengrow againandthe cycle will repeat.A repetitivecyclic processis thereforeset up, with the transientactivity locatedat the threshold.This iscalled the GaseousCavitation Cycle. It is easily picked out by examining successiveframes ofhigh-speedphotographsof cavitation in gassyliquids. However, it is not very regular, as the smallbubblesarevery variable in size. Neitherdo theynecessarilyreturn to the samepressureregion of thefield.

6.2. Thede-gassingcycle

lithe bubbleis locatedin a region of the field wherePA < P0, sayat B, it can alsogrow in the soundfield. But it is unlikely to reachthe transientthreshold.It will continuegrowing and may eventuallybecomelarge enough to separateout under gravity. This illustratesthe De-gassingProcess,a well-knowneffect of stablecavitation. It is usedindustrially in applicationssuchas: degassingopticalglass,metalmelts, photographicemulsions,etc. De-gassingmay also becomeintermittent or cyclic. This isseenin the focal regionsof ultrasonicconcentratorswhere activity will ceaseafter a time until theregioncan bereplenishedby gasdiffusing in from surroundingareas.It is clearthat de-gassingcan onlyoccurwithin a restrictedrangeof R0 and PA andits rate can be maximisedby choosingappropriatetreatmentconditionsrelatedto the bubble-sizedistribution.

6.3. The resonantbubblecyclewith emissionof micro-bubbles

At sufficiently high frequencies,bubbleswill reachtheir resonancesize beforeseparatingout undergravity. At radial resonancethereis a high probability that instability will set in, encouragingsurfacemodesof vibration. The strong surfacevibrations are parametricallyexcitedat half the excitationfrequencyandarestrongly coupledto the radialmotion.The equationfor the motion of the bubblewallthenbecomes[94]:

R1i~0+3R1R0= [(n — 1)1~~— (n — 1)(n + 1)(n + 2) ojpRflR() (112)

whereR1 is the radiusof curvatureof the surfacedisturbanceand n the mode-orderof the surfaceoscillation. The free resonancefrequencyfor modeswith symmetry abouta diameter(n � 2) is givenby:

w~=’(n—1)(n+1)(n+2)o~/pR~. (113)

If damping is included, a thresholdmust exist. This has been evaluatedby Sorokin [95] and byEisenmenger[96] for the caseof a planesurface.The thresholdoscillatory amplitude,~T, is given by


= 2B/kw where B is the damping coefficient. For surfacevibrations, only viscous damping isimportantandB = 2p~k2/pandk is given by the ripple-waveformula: k3 = ~ Then

& = (16j~/wo-p2)113. (114)

Although this appliesto a planesurface,the result is also approximatelyvalid for a sphereif themode-orderis high. For largeresonantbubbles,theapproximaterelationbetweenacousticpressureandoscillatory amplitude is: FT/eT 3yôFo/Roand using (114) the thresholdpressurefor exciting surfacewaveson a resonantair bubble in water at 20 kHz is found to be about0.0025Bar. Clearly, in ourexample,fig. 33, thereis no difficulty in exceedingthis threshold.In an intensefield, the surfacewavesmaygrow to largeamplitudesandthrowoff micro-bubblesfrom the crests.This can occurvery rapidly,the parentbubbleapparentlyexploding,thusaccountingfor the “disappearingbubbles”thathavebeenreported[97]. Thesemicro-bubblesare all nearly constantin size, radii near A

5/4 where A~is thewave-lengthof the surfacevibration. This is about 10~cm in our case.Micro-bubblescan thereforereturn to the region betweenthe thresholds,re-nucleatingit. A cyclic processmay then becomeestablished,wherethe bubblesizesat the beginningand endof the cycleareknown. In the presenceofa strongstanding-wavefield, the small bubbleswill collect at the pressureanti-nodeswhile they aregrowing, but will tendto moveawayon reachingresonancesize.

6.4. Bubble-growthabovethe resonancesize

Fromeqs.(62) and(63) weseethat the growth-rateof bubblesincreasesas resonanceis approached.If the acousticpressureis too weak to disintegratethem,the bubblescan growthroughresonance.Butaboveresonancesize,the thresholdincreases,while the growth-ratedecreasesrapidly. Referringto fig.33,II Rr is the resonanceradius,the thresholdpressuresfor growthby diffusionto radii Rr, 2Rr and3Rrareseento be 0.009,0.1 and0.4Bar respectively.Bubblesatthesesizes would resonateat frequenciesnearw, w/2 and w/3. The probability of bubblesreachingtheselargesizesis clearlynot high, but if theydo they can oscillate as stable cavities generatingthe correspondingsub-harmonicsignals. Thesesub-harmonicemissionshaverecently been studiedenergetically,and severalauthorshavepredictedthresholdsfor the w/2 emission (seesection 10). For example,Eller and Flynn [98] give:

PT = 6Po[(w/w. — 2)2+ ~2]1/2 (115)

This showsthat the thresholdis just65F0whenthe bubbleresonancefrequency,w. is half the excitationfrequency.In our case,with S 0.0645at20kHz, thethresholdforgeneratingthehalf-ordersubharmonicis about0.4 Bar.As wehaveseen,thethresholdfor theformationof suchbubblesis0.1Bar.Furthergrowthto thelargersizerequiredfor emittingthe 1/3 rd ordersubharmonicis lesslikely andthisline is thereforerarelyseenin the emissionspectrum.

We shouldemphasisethat this mechanismfor generatingthe sub-harmonicsappliesonly to bubblesin stableoscillation. A very different sequenceof eventsis responsiblefor the strongsignal at thehalf-ordersubharmonicthat accompaniestransientevents(seesection10).

6.5. Cavitationrelaxation

De Grois and Badilian [99—101]detecteda regularcyclic changein a cavitatingliquid, due to themediumswitchingbetweentwo stateswhich theyassociatedwith “gaseous”and“vaporous”cavitation


conditions.They useda frequencyof 1 MHz and acousticintensitiesno greaterthanabout4 watt/cm2.They foundthatin agassyliquid, with manyactivebubblespresent,on increasingthe acousticintensitya critical level was reachedwherethe large gas bubbleswould suddenlydisappear.The liquid wouldthenrevertto the previousgaseousstateandbackto the bubble-freestatein aregularcyclic fashionontime-constantsthat couldbevariedbetweenafew tenthsof a secondandseveralminutesdependingonthe drive level. The gaseousstatewas characterisedby adrop in temperatureandreductionin acousticintensityas measuredby thermocoupleprobes.In this stateit was possibleto demonstratewell-knowneffects of transientcavitation, such as sono-luminescence,sono-chemicaleffects and increasederosiveaction. However, therewas little noise.In the vaporousstate,theseeffectswere absentand erosiveactionreduced.But the noiseincreased,alongwith the “fountain” effect andatomisationof the liquid.The authorscalled this cyclic effect “cavitation relaxation”. It is clearly seenonly at high frequenciesand cannotbe distinguishedat the lower frequenciesused in industrial applications.The authors’explanationof this phenomenonis as follows. When gaseouscavitationstarts, the medium becomeslossy andlesstransmissive,the bubblesbeinggood scatteringcentres,causingmostof the energyto beabsorbed.The resultingtemperatureriseevaporatesvapourinto the bubbles,causingrapidtransitiontoa vaporousstate.The propertiesof the medium havenow changed,as vaporouscavities havea low0-valueand do not oscillateandscatterin the sameway as gas-filled bubbles.The mediumis moretransmissiveandlessenergyis absorbedin it. Vapourcondensesin the lower-temperatureenvironmentand thereis a return to gaseousconditions. The cycle repeatson time-constantsdeterminedby theheat-exchangemechanismsinvolved. Severalother effects of transientcavitation, other than thosenoted by the authors,are now thoughtto correlatewith this type of cycling; notably certainelectricaleffects accompanyingbubble activity, studiedby Chincholle [102] and the time-dependenceof theacousticemissionfrom cavitatingfields observedby Lauterborn[103].

Accordingto thetheory presentedby the authors[99]the time-constantscontrolling this cyclic effectdependon the rates of evaporationand condensationand all bubblesinvolved are operatingas truetransients.A possiblealternativeexplanationis thatthe cyclic behaviouris simply an effect of bubblesmoving into andout of resonance,the “gaseous”statecorrespondingto resonance.The scatteringcross-sectionfor resonantair bubblesin water is large.Absorptionof thescatteredenergywill increasetemperature,evaporatingvapourinto the bubblesso that they grow beyondresonancesize. It is notdifficult to calculate the energyneededto evaporatethis small quantity of vapour. This fixes onetime-constantof the cycle; the otherrefersto the condensationstageandis just as readilydeduced.Theeffect can be visualisedby tracing the changesin bubblesize by referenceto the diagram,fig. 33.

Figures 34 and 35, reproducedfrom ref. [101], relate to this relaxationprocess.Figure 34 showsmeasuredchangesin acoustic intensity and temperatureas functions of time. In fig. 35 the time-durations,t

5 andt2, of the gaseousandvaporousstatesareshownasfunctionsof the inputpowerto thetransducer.The acoustic intensity correspondingto the “cross-over” point was not greater than3 watt/cm

2, well below the transient thresholdat the 1 MHz frequency used.The cavitation wasthereforestable.

6.6. Cyclic behaviourinvolvingexchangeoffreegas, aidedby micro-streaming

A useful type of cyclic behaviour,involving only stable cavities, can be observedby trapping abubbleon the surfaceof a radiator,the dimensionsof which arecomparablewith the resonantbubblediameter[104].Whenthe surfaceis set into vibration, smallbubbles,moving underradiation forces,collectnearthe centreof the surface.Growth occursmainly by accumulatingsmallerbubblesuntil theresonancesize is reached.Thensurfaceactivity setsin, becomesviolent anddisintegratesthebubbleas

E.A.Neppira~s,Acousticcavitation 217

ISO-

• I I

1J~J~L° ?2o~\

6O~

L J PeIo,ct~ns12 I

I. I I ‘

0 1 2 3 ~~•

0 ‘2 -~ -.-

Time [mir~J p—.~. ~

Fig. 34. Temperatureand acoustic intensity as functions of time Fig. 35. Durationsof thegaseouscavitationregime (t1), andvaporousduring cavitation relaxation. t1 duration of gaseous-cavitation; C2 cavitation (t2) asfunctionsof the input electricalpowerduringcavita-vaporous. tion relaxation.

alreadyexplained.A smallbubbleremainsandthis growsagainby accumulatingmicro-bubblesformedfrom the debriswhich arerecirculatedby strongmicrostreamingflows. Radiationforcesare attractive,as all bubblesinvolvedarebelow the resonancesize.

This cyclic processhasbeenstudiedin detail [104]. It is not only spectacularto watch,but alsouseful.It suggestsa methodof generatinglargenumbersof smallbubblesof a controlledsize from freegasinanyform. As wehaveseen,the emittedmicro-bubblesareall nearauniform size relatedto the surfacewave-length. The cycling is very regular, its frequency varying typically between 1—10 per sec,dependingon the drive level. It can be generatedin most gassyliquids, but is most regularandeasilyproducedin silicon oils, which havealow surfacetensionandlow solubility for air. This permitsverysmallbubblesto remainin suspensionfor long periods.The photographsof fig. 36, takenat 1/200secexposure,show three stagesin the catatostrophicexplosionphase.The micro-bubblesare projectedfrom the parentbubbleat a speedof about 1 cm/sec.The surfacedistorts violently as it approachesresonancesizeandthis accountsfor the emissionof strongsignalsat harmonicsof the drive frequency.Figure37 is a c.r.o. traceof the emissionat the third harmonic(63 kHz) for two valuesof the excitationpressure.Here, the total time-baseduration is 2sec. Some white noise has also been recorded,associatedwith the actual emissionof the micro-bubbles.The cycling is very regular,repeatableandpermanent.For example,fig. 38 showsthe changesobservedin the cycling frequencyin atypical caseover a periodof 26 hoursof continuousoperation.

7. Non-radialbubble motion

For largebubblesin stableoscillation,surfacetensionis a relativelyweakcontrolling force and thesphericalshapeis easily lost. In the caseof transients,the sphericalshapeis usuallyretainedover the


=

j ~-- ~ ~

Fig. 36. Bubble distortion andmicro-bubble evolution for a bubbleof near-resonantsize: liquid-silicon oil, viscosity 1 Stoke; exposuretime0.005sec. (a) nearthestartof theexplosion;(b) and(c) at laterstages.


lti~•~— 4

-~ - ~ ~“J~

Fig. 37. Acousticsignal at thethird harmonicof theexcitation frequencyI63kHn. Liquid — siliconeoil. vIscosity 0.5 Stoke;total time-baseduration2 sec; (a) PA = 3.6milli-Bar; (b) PA = 3.4milli-Bar.

growth period, but irregularitiesoften developtowardsthe endof the collapse.These becomemorepronouncedon the rebound,whenthe cavity usuallybreaksup into a massof micro-bubbles.Surfacedistortion of an originally sphericalbubble must arisefrom some asymmetryin the environment.Typical asymmetriesare:gravity; proximity to boundarywalls or otheroscillatingbubbles;or perhapsjustthe pressure-gradientacrossthe bubblewhen its diameteris no longer very small comparedwiththe acousticwave-length.

Surfacedistortionshaveseveralimportant consequences.For transientsit must set a limit to themaximum pressuresand temperaturesdevelopedon collapse.With stable cavities, the surfaceoscil-lationsenhancemicro-streamingwhich hasimportantside-effects.In both cases,micro-bubblesresultingfrom the final disintegrationof the parentbubblere-seedthe liquid with freshnuclei.

4

CS 10 IS 20 25 30

fl~f (NnJRS}

Fig. 38. Time.dependenceof the cyclingfrequency.

220 E.A.Neppiras.Acousticcavitation

7.1. Taylor instability

The theoreticalproblemof stability of the surfaceseparatingtwo fluids acceleratedrelativeto oneanotherwas first treatedby Taylor [105].Here, we follow Plesset’soutline of Taylor’s solution [106].The surfaceis assumedto beplane,acceleratednormal to itself.Supposethe interfaceis representedbyy = H(t) andthatthe fluid in the region y > H is a gaswith densitynegligible comparedwith the liquidin the region y <H. Supposethe interfaceis perturbedinto the form: y~= H + a coskx wherea (t) is

takento be a smallquantity.The velocity potential4’ is perturbedto 4’ = 4~+ 4~where4~= —Hy and4’i = b expk(y + H) coskx. At the boundary y = H, we have: V. = dy5/dt= —a4’/9y=—(34o/ay+ acb1/ay)giving ~a/3t= —kb, defining b. Using the Bernoulli integralto satisfythe boundaryconditionatthe interfacegives: a d

2H/dt2— db/dt = 0. The resultis therefore:d2a/dt2+ ka d2H/dt2= 0.The interfaceis thereforestablewhenthe accelerationis directedfrom the liquid to the gas-phaseandunstablewhen oppositely directed. Lewis [107] successfullytested this conclusion experimentally.However, observationshave shown that for a sphericalsurfacethe opposite is true: the interfacebecomesunstableon collapse,whenthe accelerationis directedinwards,from liquid to gas.

7.2. Instability at a sphericalgas—liquidinterface

In attemptingto extendTaylor’s treatmentto the caseof a sphericalbubbleboundary,we note thatthe essentialdifferenceis that whendistortion developsin the form of aprotuberanceextendingintothe liquid, this extendsinto a region wherenot only the density,but alsothe flow velocity, is changing.The tip andbaseof the protuberancewill be subjectto differentflow velocities.This complicatestheproblemto the extentthat it becomesimpossibleto predict, using Taylor’s simple argument,whenconditionswill becomeunstablefor a bubblein radial oscillation.

Plesset[106]hasstudiedthis caseof the stability of a sphericalbubble.We simplify the problembyignoring viscosity and assumingthat both fluids are incompressible.It is then natural to expresstheperturbationin termsof sphericalharmonics,andwemaytakethe equationfor the bubblewall to be:

r5=R+aY~ (116)

whereR (t) is the unperturbedbubbleradius,given by eq. (11) with PL = (Pg— 2oiR) andp,. = P~:

R~+~2(FgFo~~). (117)

Y~is a sphericalharmonicof degreen anda is the amplitudeof the perturbation,with a 4 R. To makethe problem tractableit is necessaryto introduce a further simplification by assumingthat the flowconfigurationis unaffectedby the surfaceinstability. The flow velocity is then given by the usualcontinuity relation: V = R

2R/r2.Plessetthendevelopedhis theory by an argumentsimilar to Taylor’s. Assumingthat the flows in the

gas-and liquid-phasescan be describedby velocity potentials4’~and4’2, the kinematicconditionat theundisturbedsurfaceis:

= —(t94’2/3r)= R + âY~.


Thedynamicboundaryconditionis satisfiedby usingthe Bernoulli equationto evaluatethe pressureoneither side of the interface,thesebeingconnectedby the relation (F1 — P2)= o-(1/R’+ 1/R”), whereR’andR” arethe principal radii of curvatureat the interface.Using eq. (117) and applyinga linearisationprocedure,the stabilityconditionfor n >0 is found to be:

a + 3I~á/R+ Aa = 0 (118)

where:

A — [n(n — l)PL— (n + lXn + 2)pg]1~—(n — 1)n(n+ lXn + 2)oJR2

— R[npL+(n+1)pg]

This implies thatwhereasstability can be correctly inferred,the instability inferred from an increaseofa with time is just a reasonableconjectureandnot a necessaryconsequence.For a gasbubble in aliquid, Pg4PL andafter writing p for PL, A reducesto:

A=(n—1)R/R—(n—1)(n+1)(n+2)u/pR3.

If we use the substitutionb/a = R213eq. (118) transformsto:

b+Gb=0 (119)

where:

G— —1 1 2 .r3(RE)2(2n+1)1~—(n )(n )(n+ ~ R3 4 2R

This stability conditioncan nowbeinvestigatedfor thetwo casesof interest,that of transientandstablebubblemotion.

7.3. Stability ofa collapsingtransientcavity

Towardsthe endof atransientcollapsewe knowthat R x R312 as R —~0. Substitutingin eq. (119)weseethat only thelast term is relevantwhenR is very smallandG -= c2/R5as R —~0, wherec is a realnumber.Asymptoticallywe find:

a =~m~~R_h/4exp(±icJR_5/2dt) (120)

as R —~0. a thereforeincreasesas R114 in amplitudeandoscillateswith increasingfrequencyas R -÷0.Thedetailedanalysishasbeengiven by PlessetandMitchell [108].In fig. 39 theperturbationamplitudeis shownas a functionof the relativecavityradiusfor two valuesof the mode-order.Instabilitybecomesviolent for R/RO less thanabout0.1. Collapsingtransientsmaythereforebecomeunstablebefore theeffects of compressibility,viscosity or surfacetension becomevery great. As the theoretical radial


I COLLAPSING BuBSLS

4~2O4O~~8HOn/n.

Fig. 39. Perturbationamplitudefor a collapsingcavity in termsof therelativeradius(R/Ro) for modeordersn = 3 and n = 6.

compressionratio is often greaterthan 10, a transientthat starts its collapse containinggasat lowpressuremay be expectedto becomeunstablebefore reachingits theoreticalminimum size. On theotherhand,small, relatively-stablebubbleswith highergaspressuremay remainstable,the compressionnot proceedingfar enoughfor instabilitiesto develop.

We also see from the stability condition (119) that in the expansionphaseof the transientcavity,sinceR2—~2~P/3p,a—~ constantas R becomeslarge,so that a/R-÷0as R -÷oc. The expansionphaseisthereforestable.

In an independentstudy, Brook Benjamin [87] looked for a criterion for instability in termsof theratio of the bubblesize neededfor instabilitiesto form, to its minimumsize. The collapseconditionswereassumedadiabatic.Hefound the rathersimple result that a largedisturbanceof thesurfacewouldnot be expecteduntil

(R/Rmin)3°’1~becomes<y + (y — 1)/2n (121)

where n is the order of the supposedsurface spherical harmonic in which the perturbationisrepresented.With y = 1.4 andn = 2 this predictsthat R/Rmin would reachabout1.3 beforeanysurfacedistortionwould grow. This is nearthe point of maximumdecelerationof thebubblewall.

7.4. Stabilityof an oscillating stablecavity

Historically, Kornfeld and Suvarov [109] were the first to observesurface oscillations on stablecavities. Many workershavesubsequentlystudiedthesevibrations. Applied to stableconditions, eq.(11) for the motion of the bubble wall can be linearisedby writing P = P

0(1+ esin wt) where e4 1.

Using PL = (P0 + 2oiR0)(R/R0)3— 2aiR the linear solution for R canbe written: R = R

0(1+ S sino.t)

where S is of the sameorder as �, and a constantphase-differencebetweenP and R hasbeenneglected.The function G in the stability relation (119) can now be evaluated.Following Plesset,retainingonly the leadingterms, G can be written:

G=a+f3sinwt (122)


with

a =(n—1)(n+1)(n+2)o-/pR~

and

= [(n+~)w2_(n_ 1)(n+ 1)(n+2)3o-/pR~]S.

The stability condition (119) is now justthe Mattieu equation.Consideringthe solution in very simpleterms,wecan seethat it will be essentiallyunstableif G <0. As stability will increasewith decreasingmode-order,we maytake n = 2, the lowestapplicableto distortionmodes.The eq. (119) thengives:

l2o 15w2 36o1G=_SL—~--_~ik-5jslnwt. (123)

An order-of-magnitudecriterion for stability is therefore:

24o-= 5Sw2pR~ giving R0 — (24oi5pSw

2)1~’3. (124)

Forexample,for an air bubblein water,taking S= 10_2 andfrequency20 kHz (w = 1.26x 10~persec),the critical radiusis found to be about2 x 10_2 cm.

Benjaminand Ursell [110]werethe first to studythe stability of solutionsof the Mattieuequationinrelation to oscillating bubbles. They found that the surface waves most likely to grow will beparametricallyexcitedat frequencyw/2 wherew is the excitationfrequency.Thetreatmentin ref. [110]refers to aplaneliquid-gasinterface.A theoryfor the stability of waveson asphericalsurface,similarto Plesset’swas alsoindependentlypublishedby BenjaminandStrasberg[111].Theyobtainedthe samegeneral condition for stability (118), reducing to a Mattieu equation on applying a linearisationprocedure.

With linear, non-parametric,excitation, the amplitude of the surfacedisturbanceincreaseslinearlywith the drive pressure.However,with parametricexcitation,a thresholdamplituderelatedto dampingmust be reachedbefore the parametricmechanismcan be triggered.This is given by eq. (114).Following Nyborg [112] it is instructiveto comparethis thresholdamplitudewith a representativevalueof the bubbledimensions,in particular,the resonantradiusof a largebubble,given by (44). We find:

/ 1~ ‘~1/3/ ~1/2T~ I IU 1 1 A

2 I IRr \P OK!! \37P0

For air bubblesin water, thisreducesto eT/Rr 105f213with f in Hz. This ratio increasesratherrapidly

with frequency.At 20 kHz it is 0.0074,but at 500 kHz it has increasedto 0.06. We mayexpressthisthresholdin termsof the drivepressurevia the relation PA/Po = 3

75&/Rr. Using appropriatevaluesofy and S for air bubblesin water, the thresholdpressurefor exciting surfacewavesparametricallyonresonantbubblesis foundto be only 0.0025Bar at 20 kHz and 0.037Barat 500kHz.

Surfacewaves are always easy to excite on bubbles much larger than resonancesize. Then themode-orderis largeandthe equationfor free oscillations(113) reducesapproximatelyto:

w~R3~o-n3. (126)


In the limit this must revert to the ripple-waveformula pw~ k3o which refers to a plane surface,wherek is the wavenumberfor the surfacewaves.Then kR= n and2irR = nA.

Equation(113) showsthat, for example,an air bubblein waterthat would be in radial resonanceat1 MHz, of radius3.5x 10~cm wouldneedto be drivenat0.78MHz to excitethe order-2surfacemode;at 1.43 for order-3;andat 2.15 for order-4. For a bubble,radius2 x 10~cm the frequencieswouldbe 1.67, 3.04 and4.56MHz respectivelyfor the samemodes.Strongcouplingresultswheneverasurfacemode comeswithin the bandwidthof the radial resonance.Such coupled oscillations are said to be“linearly driven at resonance”.The abovefigures suggestthat there must exist a certain minimumbubblesize belowwhichsurfacemodescannotcoupleto the radial resonance.The implication is thatthesurfacewavelengthis thenof the order2ITRr, that is kRr — 1 where k refers to the surfacevibration.

BrookeBenjamin[111]hasstudiedin detail the dampingto be expectedwhenthe bubblesurfaceisdistortedin generatingsurfacewaves. He finds that in manyphysical situationsthe main effect of thefilm that constitutesthe interfaceis to makethe surfaceincompressible,andincreaseddampingoccurswithin a boundary-layerin the liquid ratherthanthroughquasi-viscousdissipationwithin the film itself.He finds the simple result that, if a defines the damping,so that a/a

0= p~°’tsinwt thenaRe/w

(n + 2)2/2\ñnwhereRe is the acousticReynold’snumber.

7.5. Generaldynamicalproblemof the distortionof the surfaceseparatingtwo immisciblefluids

Using avariational approach,Hsieh [113,114] hastackled the difficult problem of the distortionmodespossiblein the surfaceseparatinganytwo immisciblefluids, taking non-linearitiesinto account.Among severalimportantresultsobtainedso far is an extensionof the Plesset—MitchellandBenjamin—Strasbergstability relation(118) to the nexthigher orderof approximation:

R0d+3~0á+~ [(n —1)(n +2)(n

2+ n +9)~~~_(n2+ n — 1)1~o]a=0. (127)

Thisresult is actuallyidenticalwith (118)for n = 2, but of coursedivergesfrom it for n >2. Hsiehhasalso exploredthe feed-backmechanismwherebythe surfacemodesaffect the radial oscillation, withinterestingresults[113].

7.6. Experimentalresults

There have beenseveral reportedmeasurementsof thresholdsfor generatingsurfacewaves onbubblesexcitedat low ultrasonicfrequenciesin the range20—30kHz: Strasbergand Benjamin[115];Eller andCrum [116];andGould [117].In all cases,bubbleswerebelowresonancesize,with radii in therange(15—60)x 10~cm. The experimentalresultsdo agreereasonablywell with theory for the largerbubbles,but for thosewith radii belowabout25 x 10~cm, the measuredthresholdsarenotably higherthan the theoreticalpredictions.A discrepancyis expectedin view of the ratherdrastic assumptionsmadein deriving the expressions.

Surfacewave activity generatesintense micro-streamingwhich in turn aids thermal- and mass-transferacrossthe bubblewall. Anotherphysicaleffect of intensesurface-waveactivity is thegenerationof uniform-sizedmicro-bubbleswhen the parent bubble disintegrates.This probably haspracticalapplications[104]. It is analogousto the reverseprocess,i.e., the break-up of liquid films in the“atomisation”of liquids, which hasbeendevelopedcommercially.Here, the atomisedliquid droplets


are very uniform in size, with diametersclose to 0.34A5. We may assumethat in our casethemicro-bubblesareof comparablesizeanddistribution.

The form takenby thesurfacedistortion that developswhen a transientcollapsesmust dependtosome extent on the type of asymmetryresponsiblefor it. In the presenceof a solid surface theimploding bubblescauseerosion. In this case,we find that the type of distortion is predictableandinvolvesa jettingprocess.We shall dealwith this in detail in the nextsection.

8. High-speedphotographicstudiesof bubblemotion

Confirmationof theoriesof cavitationbubbledynamicscanonly be obtainedby direct observationofthe expansionand collapse.At ultrasonic frequencies,this demandsthe useof high-speedcameras.Startingin theearly 1930’s manystudieshavebeenmadeof isolatedbubblesoscillating in normal,andgravity-free conditions, and of bubblescollapsing near to solid surfacesand interacting with otherbubbles.Threegroupsof workers havebeenprimarily involved, at: California Instituteof Technology(Cal-Tech),U.S.A.; Universityof Gottingen,WestGermany;and at theAcoustics Institute,Academyof Sciences,Moscow,U.S.S.R.

Early photographicstudies concernedhydrodynamically-producedcavities. Much of this pioneerwork wascarriedout at Cal-Tech.ProfessorKnappandcolleagues[1181studiedcavitationgeneratedinwaterflowing pastmodel projectilesandsuitably-shapedhydrofoils in a watertunnel. In studiesof thissort, experimentalconditionsdiffer from thosein typical acoustically-generatedcavitation.Neverthelessthework is historically importantand worth discussingin outline here.The camerasusedstroboscopiclight sourceswith flash-durationabout 1 micro-second.A batteryof 7 synchronisedcameraswasused,overlappingpicturestakenat up to 3000 per second.In this way, effective repetitionratesof about20 000 per secondcould be achieved.The film was exposedat constantspeed.Figure 40, reproducedfrom ref. [118]is a plot of a typical radius—timecurvefor the motion of a single bubble.The bubblesizeswere takendirectly from successiveframesof the film and thevolume wascalculatedassuminga

0.160 - 0.016 _L - ____ - _L ~adius I

0.120 0.012 — — — ~ — — — —.‘ \ Volume.E 0.080 - ~ 0.008 — — -i---- — — — —

::20.040 ~0.004- ——

0.080- ~0.00e — — — —h-1— ~1 J± —

0.120 0.012 — —I— — _______ — —io.16o. ooie——————— L.._...._...L I.000 0.001 0.002 0.003 0.004 0.005 0.006

I I I Time,sec I i iFig. 40. Radius— andvolume—timecurvesfor an oscillatingair bubblein water.


sphericalshape.Theframe-speedwasthe maximumobtainable,20 000 per second.The first collapseisrapidby comparisonwith theinitial growth.Onthe first reboundthebubblerecoversto about65% of itsoriginal volume.Butafterthe third reboundit wasreducedto nearits equilibriumsize.Theradius—timecurvefor the first collapseagreesquite closelywith Rayleigh’stheory.In the figure, the odd-numberedcyclesof oscillationaredrawnabovethe axis andeven-numberedonesbelow for clarity.

& 1. Photographicstudiesat California Institute of Technology

Ellis [119,120] developeda camerasystemusing a Kerr Cell as the shutter.The Kerr Cell usesaliquid that becomespolarisedfor light-transmissionwhenplacedin a suitableelectric field. Combinedwith polaroidplatesthe cell can thereforebeusedas alight switch, triggeredby voltagesappliedto theelectrodes.In Ellis’s work, the illumination was continuousat very high intensityandthe strip of filmremainedstationary,woundon the insideof a drum. Light from the bubblefield was projectedalongthe axisof the drum,anda mirror mountedat the centreandinclined at an angleof 45°to the axiswasrotatedat high speed.The images,shutteredby the Kerr Cell wereprojectedon to the stationaryfilm.By 1955 Ellis was achievingexposuretimes of iO~sec at repetition rates of 106 per sec. He laterimprovedthe resolutionby usinglaserillumination. Thelimitation with this typeof camerais the smallnumberof framesit can record in a single run. Recently,Ellis [1201and NaudeandEllis [121]havestudiedcavitiescollapsingon to solid surfaces.In somecases,the bubblesareseensimplyto flatten,butin othersthe flatteningcontinuesto dimpling andpenetrationof the liquid as a jet into, and through,the bubble.In effect, the bubbleturnsitself insideout. Neara solid boundarythe jet forms on collapse.But if the initial asymmetryis simply a hydrostaticfield, for example, gravity, it will form on thereboundbefore the bubble disintegrates.Initially, the bubblealwaysdistorts first by flatteningon thehigh-pressureside,keepingits flattenedsurfacealwaysparallelto a surfaceof constantpressurein thefield. The steeperthe pressuregradient the more pronouncedthe effect. To photographthe shockproducedwhenbubblesimplodedon a solid surface,they weremadeto collapseonto a photoelasticsolid illuminated through crossedpolaroids.The maximum shockrecordedin this way occurredjustbeforethe completecollapseof the bubble.

8.2. Photographicstudiesby Sovietworkers

Reportsof high-speedphotographicstudiesfrom U.S.S.R.refer to the motion of compactgroupsofcavitation bubbles.Using a radial-modefocusing transducerresonantat 15 kHz, Akulichev [1221obtainedphotographsof the growth and collapseof bubbleclouds in water using a camerasystemsimilar to Ellis’s, that is, usingfilm fixed in anon-rotatingdrum with rotatingmirror sitedat the centreof the drum. The cavitatingfield was illuminated by a focusedlight beamstrobedat the appropriatespeed.This systemwas capableof recording800 consecutiveframesat frame-speedsup to 300000 persecond.Thewholecloudof bubblesexpandedandcollapsedtogetherin-phase.Froman estimateof theaveragesize of the bubblesat eachphaseof the motion, the radius-timecurvescouldbe plottedout.For the examplesshownin fig. 41 the framespeedwas 200 000 per second,giving about12 framespercycle. In the figure the measuredplots are comparedwith theoreticalcomputationsbasedon (i) theHerring—Flynn and Gilmore theories (solid lines); and (ii) the simpler Noltingk—Neppirassolution(broken lines). The measurementsagreequitewell with either theory.


R xid2 (cm)

2.75

Fig. 41. Measuredradius—timecurvesfor air bubblesin watercomparedwith theories;R

0 = i0~cm;I = 15 kHz. The full-line curve wasplottedfrom theHerring—FlynnandGilmore theories,andthebroken-linecurvefrom theNoltingk—Neppirastheory.

8.3. Photographicstudiesat the University of Gottingen

The mostrecenthigh-speedphotographs,with exceptionallyclear pictures,havebeenpublishedbythe teamworking at theUniversity of Gottingen[123—128].Photographshavebeenobtainedof isolatedstable and transientbubbles,and also of bubble fields. Figure 42 is a schematicdiagram of theexperimentalset-up.The liquid containerwas a small rectangulartankwith transparentsides andthecavities wereproducedby focusingan intenselight beamfrom a ruby laser,Q-switchedby meansof aKerr Cell. The energy involved was in the region of 1 joule with pulse-durationabout 30—50nanoseconds.In this way, existingsmallair bubbleswereexpandedto a diameterof a few millimeterand allowed to collapse and rebound under the steady ambient pressure.The cavities thereforecontaineda mixture of air andvapour.The liquids used includedwater andvarious silicone oils withviscosities around5—10 Poise. Both rotating-mirrorand rotating-drum cameraswere used to takepicturesat frame-speedsup to 900000per second.

In water,evenwhenthe bubbleswereformed in the body of the liquid awayfrom boundaries,theytendedto distort after oneor two rebounds.But in silicone oils of high enoughviscosity, theyretaintheir sphericalshapethroughseveralrebounds.For the photographsof fig. 43 the bubblewasgeneratedin silicone oil of viscosity 4.85 Poiseand the picturesweretakenat frame speed75000per second.Threereboundsareobtained,afterwhich the bubblehasstill not lost its sphericalshape.Themaximumbubblediameterwas about4 mm.

~FLASH LAMP

___________ PLATE

ROTATINGSTART DRUM OR MIRROR

CAMERA

Fig. 42. Experimentalarrangementfor studyingthedynamicsof laser-inducedcavitationat Universityof Gottingen.


.000••0000000•00000 00000000.0000000000000000000... 000

00 00000....

•.• 0000 00Fig. 43. Oscillationof laser-inducedcavitationbubblein silicon oil of viscosity 4.85 Poise.The framespeedwas 75000/sec.

Measurementstakenfrom the film can be comparedwith theory.The full-line curve in fig. 44 hasbeenplottedfrom eq. (13)with P = F

0. The fit is very goodconsideringthat thedampingeffectsof heatconductionandsoundradiationare neglectedin this equation.In the computation,an arbitraryfigurefor y was used.

The studyof cavitiescollapsingnearboundariespresentsno problemswhentheyare generatedbythe laser technique.For example,fig. 45, takenat frame-speed305 000 per second,showsan initiallysphericalbubblein watercollapsingneara rigid boundary.As the bubbleapproachesto within a fewradial distancesof the surfacea jet develops,projectedtowardsthe surface.Onrebound,acounter-jetis produced.After that,the bubbleinvariablydisintegrates.This jetting is now acceptedas an initiatingcauseof erosionof surfaces.We showlater that theory canpredictthe initial jet-formation,but thereisat presentno theory to explain the counter-jet.

When collapsingbubblesof similar sizesarewithin a few radial distancesof one anothertheywillattractandusuallycoalesce.But bubblesof widely differing sizeswill developjets. Theywill thennotcoalesce,but the smaller of the two will collapse first and disintegrateon the reboundas if it wereapproachinga solid surface.

By noting the effect on smallstraybubblesin the vicinity, Lauterborn[126]hasbeenable to estimatethe magnitudeof the shock pressuresdevelopedaroundthe collapsing sphericalbubbles.Figure 46gives anevaluationof the radius—timecurvefor a smallbubblecompressedby theshockwave as alarge

.S•S_~••~~

~ ~O.4 0 ~4O8L2~e. 1.6

TIME

Fig. 44. Decayingnon-linearbubbleoscillationin silicon oil comparedwith theory.

E.A. Neppiras.Acousticcavitation 229

•0Fig. 45. Dynamicsof a sphericalcavitation bubble in waterneara plane solid boundary,after the first collapse. The bubble wasgeneratedat adistanceof 2.3mmfrom theboundary.The framespeedwas305 000/sec.

300

p =1200

0 2 4 6 (ll7usec) 10• time

Fig. 46. Radius—timecurvefor asmall air bubblecompressedby theshockpulsefrom an implodingtransient.0 and x refer to thesplitting of thevaluefor theradius dueto deformationof thebubble by theshockwave.


cavity collapsesin water closeto a rigid boundary.The curvesshownin fig. 46 havebeencomputedfrom eq. (14), using for ~ ashortpulse of sinusoidalshapeand30 nanosecduration.The resultsforpulseswith amplitudes1200, 1500 and 1800 Bar areshownand thesespanthe measuredpoints. Thevalueof 30 nanosecwas chosenfor the pulse-durationbecauseof the lengthof the laserpulse.The truedurationof the pressurepulse is unknown, but there is someevidencethat it may be as low as 10nanosec.For pulsesas short as this, the peakpressurerequiredto generateanygiven R—t trajectorywill be approximatelyinverselyproportionalto thepulse-duration.This meansthatthe truevalueof thepeakpressurein the shockmaybe as high as 4500Bar.

In all thework reportedabove,the steadyambientpressurewas usedto collapsethe bubbles.Goodphotographshavealso beenobtainedof bubblesforced into oscillationin anacousticfield. Air bubblesformed at the focus of a tubular piezoceramictransducerwere driven at a frequencyof 13 kH.z andphotographedusing a frame-speedof 100000 per second. Bubbleswere never evenapproximatelysphericalbut an equivalentradiuscouldbe inferredandthe measuredradius—timecurvecomparedwitha theoretical prediction based on eq. (14). After applying an appropriate fitting procedure,theagreementwas found to be quitegood.

&4. Theoryrelating to bubblescollapsingnear solid surfaces

PlessetandChapman[129]haveshownhow the problemof an initially sphericalbubblecollapsingnear a rigid boundarycan be solved using numericalmethods.Two examplesfrom their results areshown in fig. 47. The liquid is water and the collapse occurs under a steady pressureof 1 Bar.Dimensionsof the collapseare scaledin termsof the initial bubbleradiusR0. In case(1) the bubblecollapsesin contactwith aplanesurface,while in (2) thebubblesurfacewasinitially at adistanceR0/2from the boundary.The computationgives the velocity of the liquid jet as it reachesthe boundaryas128 and 170rn/sec in the two cases.Assumingthat the characteristicimpedanceof the liquid (pc)L issmall comparedwith that of the surfacematerial,and taking V as the collapsevelocity, the pressuredevelopedon impactis approximatelyF (pc)LV. If nowwe adoptthe acousticapproximation,that is,using a constantvelocity of soundat the usual value (1500m/secfor water),with V = 130rn/sec,weobtain F = 1930Bar. Plessetshowedthat evenwhenamorerealisticfigure was usedfor the velocity ofsoundunderthe prevailingconditions,the theoreticalshockpressureis not increasedvery much. Thisstressis effective for a very short time, of the order of the time for the impactsignal to traversetheradiusof thejet, that is —10~sec.The timefor the jet to propagatecompletelyis justits lengthdividedby its velocity,andfor mostof thistime thepressurewill be given by ~pV

2or about800Barin thiscase.Thesepressuresarehardly greatenoughto causeerosionin resistantmetals.

8.5. Recentphotographicstudiesusinglarge bubblesexcitedat low frequencies

Crum [130]hasbeen able to observeand photographthe jetting phenomenawithout resorting tohigh-speedcameras.Largebubbleswereexcitedattheir radial resonancefrequencieswith the ambientpressurereducedto nearvapourpressure.This ensuredthat the bubbleswereessentiallyvapour-filled.The motion of the cavitieswasthenmainlycontrolledby surfacetension.The bubbleswere maintainedin resonancevibrationby forming them on a vibratingplatform.Jettingwas observed,similar to thatobtainedin the work alreadyreported.By strobingat the drive frequency,still pictureswere takenatvariousphasesof the bubblemotion.

Storm [131] seededlarge air bubbles,with volumes 0.05—1.0cm3, in a gel so as to hold them


~~ SPIiE6E

/ .7.—~ ~

/ / a/

(i) / /‘~:

\ \ ~I / I~I\ \ \~‘( / I / / /

.~ALL

(2)

WALL

Fig. 47. Surfaceprofiles for bubblescollapsing(1) initially in contact with a plane boundary;(2) initially at distanceR0/2 from the boundary.

Pm=1B~.

stationary. The bubbles were driven at low frequenciesin the range 500—1050Hz. Using a Fastexcamerawith frame-speedsup to 5000 per second,he was ableto examinein closedetail the formationof distortion modes and the emission of micro-bubbles. He observeda tendencyfor originallynon-sphericalbubbles to becomemore sphericalat low andmedium levels of excitation. A notableresultwas the detectionof anacousticemissionfrom the bubbleat the half-ordersub-harmonicwhichcould be relatedto the surfaceoscillation.


9. Cavitationfields

The study of isolated cavitation bubbles must now be extendedto the complex associationsofbubblesthat makeup the bubblefieldsencounteredin practice.Theliteraturecontainsmanyqualitativedescriptionsof what can be seenand inferredwhen liquids are cavitatedunder typical conditions,inpapersdatingbackto the 1930’s.We haveno spaceto repeattheseaccounts,but merelypoint out someof the more useful references[9,132—136]. The detailed descriptionof the forces controlling theinteractionsand translatory motions of bubblesin sound fields is also a large subject on its own,relevantto, but ratheroutsidethe scopeof this review.Here,wemerelylist thetypesof forcesthat maybe involved, with some useful references.The chief forces responsiblefor translatory motion are:gravity; acousticradiation [2,112, 137]; acousticstreamingandmicro-streaming[2, 112, 138, 139]; andforces due to electric chargescarriedby small bubbles [102].The chief interactionforces betweenbubbles are: Bernoulli attraction; radiation pressure,including the so-called Bjerknes-typeforces[112,140]; andvarious steadyforcesdue to wave-form distortion.All of theseforces areopposedbyStokes-typeresistanceto motion. Clearly, with so many different forces involved, the interactionsofbubbleswith the acousticfield, and with themselves,is a theoreticalproblemof greatcomplexity.

Descriptionsof thetransducersandinstrumentationusedin generatinganddetectingcavitationfieldscovers a large technologicalarea lying outside the scope of our review. However, the successofexperimentalstudiesof cavitationfields often doesdependvery muchon the designof transducersandtheir associatedinstrumentation.The emittersusuallyemployedare focuseddevices,chosennot merelyto achievethe requiredhigh intensities,but also to ensurethat the cavitatingregion is isolatedfrominterfering boundaries. Again, we have to be content with quoting the useful references:[133,135, 136, 141]. Detectorsand receiversare many and varied; publicationscovering design andperformancewill be referencedlater.

9.1. Pioneerstudiesusingvisualandphotographicmethods

Beforediscussingrecentresearch,wemust mentiontwo earlystudiesof cavitationfields which areofhistoricalimportanceandhaveencouragedlaterwork. Blake, in hisfamous1949Harvardreport [9] wasthe first to distinguishclearly betweenstablebubbleactivity, gaseoustransientsand vaporoustransients.He useda focusingtransducerresonantat 60 kHz, operatedinto water. Stablecavitationactivity andgaseoustransientsareobservedin gassywaterthe former as isolatedbubbles,or smallgroups,formedaroundthe focal region,and the latter in the “streamers”or branchingstructuresobtainedat higherintensities.On the other hand,vaporoustransientsare mosteasilygeneratedin degassedwater. Theyappearas small, short-lived,almostexplosive,ruptures,accompaniedby a snappingnoise.Blake did notobservestablevaporousbubbles,as his experimentalconditionswereneversuitablefor generatingthese.To illustrate typical cavitationactivity at high intensityin gassywater, fig. 48 showsstreamerformationswith the generationof micro-bubble clouds in the focal region of a radial-modemagnetostrictiontransducerresonantat about 18 kHz.

In another historically-important paper, Willard [135] described in detail a particular type ofcavitation, obtainableat high frequenciesin focusedfields. A sphericalbowl transducer,resonantat2.5MHz was used to generateintensefields in water. Focal intensitiesup to about 1800watt/cm2,correspondingto pressuresof about70 Bar, were achieved.Willard photographedthe cavitationeventsusing a cameraoperating at a frame-rate that we would now classify as slow-to-medium (24—1000persec).He alsouseda continuousfast-movingfilm with steadyillumination. He concludedthata


Fig. 4.S. Streamerformation at thefocusof an IN kHz radial-modemagnetostrietiontransducer.

definitesequenceof eventswas involved in the typeof cavitationhe observed,building up to a violentexplosiveeventat the focus. First, the liquid mustcontainsuitablenuclei, which are draggedinto thefocal region by radiationpressureandstreamingmotion.This is called the “pre-initiation” stage.Thesecondstage,“initiation”, occurswhen a suitably weak nucleusstreamsinto the focal region. Theseinitiating nuclei arevisible gas-filledbubblesandthereforewell abovethe radial resonancesize at thefrequencyused.The third, or “catastrophic”stageoccurswhenthe initiating nucleus,on enteringthefocal region,breaksup into a massof micro-bubbles,accompaniedby a loud snappingnoise.We nowknow that in this third stage,the large initiating bubbledevelopsviolent surfaceactivity, throwing offmicro-bubblesat an acceleratingrate,as explainedin anearlier sectionof thisreview.Thesecondandthird stagesmayoccurwhateverthe air contentof the water. The cloud of micro-bubblesfrom theexplosionform a plume-shapedobject.The micro-bubblecloud is projectedthrough the focal regionvery rapidly underradiation forces.This suggeststhat a large proportion of the micro-bubblesareofnear-resonantsize. Willard measuredtranslatoryspeedsup to 10 rn/sec through the focus. In aeratedwater, two further stagestermedthe “bubble-phase”and the “post-cavitation” condition are recog-nised.A few relatively large, non-collapsing,bubbles are generatedby, andconcurrentlywith, thecatastrophicphase.They remainafter the Willard eventhaspassed,to be carriedoff downstream.Thecatastrophiceventis usuallyoverin a few millisec. Fig. 49 showsthreeframesselectedfrom Willard’sresults,illustrating the violent catastrophicphaseof the cavitation.For thesephotographs,degassedwaterwas used.The plume-likeshapeof the micro-bubblecloud can be seenas it streaksthroughthefocal region.The exposuretime was about22 millisec, but Willard was able to showthat the eventwascompletedin lessthan4 millisec. As the lengthof the plumeis about 1 cm, this correspondsto a speedthroughthe focus of atleast 2.5rn/sec.


Fig.49. Typical Willard eventsin degassedwater;f = 2.5 MHz.

9.2. Cavitationstudiesusingpulsedholographic techniques

In a detailedstudyof cavitationfields, we needto follow the pathsof individual bubblesin motionunder the steadyforces of the field. For this purpose,ordinaryhigh-speedphotographyis not quitesuitable.The reasonis simply thatbubblestendto moveout of focusduring successiveexposures.Thisdifficulty has beenovercome by using holographic recording methods, developedrecently at theUniversity of Gottingen [142,143]. Short pulsesof intenseillumination are neededand in the workreportedin [142]pulsesof about20 nanosecdurationweregeneratedby meansof a Q-switchedlasertoform the holograms.The light pulse could be triggeredat a pre-selectedphaseof the bubblemotion.Hologramscould thenbe reconstructedfor differentdepthsin the field and at any given phaseof themotionfrom asingle hologram.

Extendingholographictechniquesto studyingthetemporaldevelopmentof thefield requiresdouble-or multiple-pulse equipment. Details of the instrumentsdevelopedby the Gottingen group arecontainedin refs. [142,143] to which the readeris referred.Figure 50 from ref. [142] shows theappearanceof a radially-focusedcavitationfield in gassywaterat two instantsseparatedby an intervalof 320 microsec.The excitation frequencywas 19 kHz, so the interval representsjust 6 acousticcycles.The photographsshow that althoughthe basicconfigurationof the bubblefield hasremainedthe samein this shortinterval, with a commoncentralcore of bubbles,the detailedarrangementof bubbleshasalteredconsiderably.Hologramreconstructionsof this sort arevery revealing.They showvery clearlythatthe unaidedeye is unableto follow eventhe translatorymotionsof bubble-aggregatesin cavitating


~1~ .. ___

I . . ~ -~ .‘~t~

~ ~ . .. .T •.

Fig. 50. Temporaldevelopmentof cavitation bubbleconfiguration.Plate (b) wastaken320 microsecafter (a).

fields. The optical equipmentdevelopedat Gottingen permits time-separationsdown to about 10microsec.But since the holographicplate cannotbe movedin sucha short time, the imagesmust besuperimposedon oneanotheron a single plate.As the imagesmustbe separablewhenreconstructed,an acousto-opticaldeflection systemis used to shift the direction of the laser beam betweentheexposures.This involvesusingtwo switchablebeam-splitterswith the pulsedlaser.

This techniqueis capableof extensionto holo-cinematography,wheremultiple imagesareproducedat highspeed.The authors[144]describea systemfor producingup to 8 hologramsat20 kHz speed,allon a single plate,but separatedby spatialmultiplexing, that is, usingadifferentportionof the platefor


ruby laserfor breokdown

1000mm /

hoI~.:beaimfwr4~m~f~07

cuvetterotating disk groundwith apertures glass plate

Fig. 51. Experimentalset-upfor recording hologramsusing spatialmultiplexing.

eachexposure.The separationis achievedby using a rotatingdisc with aperturesas ashutter,placeddirectly in front of the holographicplate.Thus,at each laserpulse a differentportion of the plate isexposed.The completeexperimentalset-upis showndiagrammaticallyin fig. 51.

9.3. Acousticmeasurementsin the cavitatingfield

Precautionsare necessarywhen using conventional acoustic instruments to obtain informationdirectly from a cavitating field. Two types of receiving devicehave been found useful: very smallprobe-typehydrophonesandradiometers.

Thevalueof probehydrophonesis thattheymeasureinstantaneousvaluesof theacousticpressureintheir vicinity. They pick up the acousticemissionfrom the field. The complexwave-formscan beanalysedinto their harmonic, sub-harmonicand white-noise components.These signals are charac-teristicof the type andintensityof the cavitationandare so importantandrevealingthat section10 ofthis review will be devotedto interpretingthem. However, it mustbe emphasisedthat theseprobeinstrumentswill not usually respondto the direct shock-wavesemitted from collapsingcavities. Thereasonis that the rangeof the strongshocksis limited to a few radial distancesfrom the collapsingcavity. Mostof the receivedsignal will refer to the lower-frequencycomponentsof noisefrom themoredistantenvironment.

For usein cavitatingfields, probehydrophonesmust be physically very small, so as to interfereaslittle as possible with the field. The surfaceof the probe should be such that bubblescannoteasilycollect on theprobeitself to interferewith measurements.Precautionsmustalsobe takento ensurethatthe probe cannotbe damagedby cavitationerosive action.Thesehydrophonesareessentially“point-probes”but of coursethey arealso capableof being mechanicallyscannedto recordthe field pattern.Detailed information on the construction and use of suitable small hydrophonesis given in refs.[133,145—149].

Wheninterestis centredon the totalenergyexpendedin generatingcavitation,a differentapproachis used.In any non-uniform hydrodynamic field, streamingflows are generatedby a force whichdependson the energy-gradientset up in the medium.In a cavitatingliquid, providedthe mediumitselfis not too lossy, the steadystreamingcan be attributedto the cavitationfield. It can be shownthat in a


closedregion,the sumof the energy-densityof the acousticfield and the kineticenergyof the streamflow is constant.This sum is readily measuredusinga radiometerbalance[150].It is expectedthat theenergyused in generatingcavitationgives rise to an equivalentstreamflow. Thus,a simpleexperimentusing a radiometerplaceddownstreamfrom the cavitationzonemeasuresthe energyassociatedwiththecollapseof cavities interceptedby the beam[151].It is necessaryto separateout the effect of theacousticfield, namely, thatportionof the energydueto radiationforce. This is doneby interposingathin, acoustically-transparent,film in the pathof the beam.This cuts out the streamflow, passingtheacousticcomponent.The differencein the two radiometerreadingsthenrefersto the energycarriedbythe streaming.In atypical case,quotedin ref. [1511,about20% of the total radiatedpowerwas usedingeneratingcavitation.

9.4. Measurementstakenat the electricalterminalsof the transducer

Whencavitationstartsin a liquid, its physicalpropertieschange.It becomesmorecompressibleandmorelossy. Its “effective impedance”haschangedandonemethodof studyingcavitationis to monitorthesechanges.In this method it is possibleto take all the information neededfrom the electricalterminalsof the drive-transducer.Unlike probemethods,the acousticfield is not interferedwith in anyway. It is surprisinghow muchusefulinformationcan beobtainedby measurementsof this sort.

The techniqueis describedin detail in references[136]and[152].The drive transduceris wired up asonearmof anadmittancebridge.Thebridge is balancedinitially atthe clampedpoint of thetransducer,that is, with the transducernon-motional,just off-resonance.It can be shown [136] that when thetransduceris tunedto resonancethe voltage appearingacrossthe detectorterminalsof the bridge isproportionalto, andin phasewith, the oscillatoryamplitudeof the transducersurface.If the transduceris driven from ahigh-impedancesource,thenthe detectorsignal will be proportionalto the motionalimpedance,and inverselyproportionalto the acousticimpedanceof theliquid.

Measurementsobtainedby severalgroupsof workers [153—155]haveshown that when cavitationstarts, the effective radiation resistanceof the medium begins to fall off rapidly. With increasingintensityin degassedwaterthe resistanceeventuallyreachesa low level at approximatelyone-thirdofthe nominal pc. The examplegiven in fig. 52 hasbeen reproducedfrom ref. [1541.In this case,themeasuredquantitieswere the acousticpowertransmittedto the liquid (Wa) andthe particlevelocity of

Rradxlo5 I (W/cn~)

(a) LO (b) /~ ~ 1500 0 12

V~(cm/s)~ 0 (W/crn~’)

Fig. 52. (a) Effective radiationresistanceversusthe squareof the velocityof theradiatorsurface;(b) actualradiatedintensityversustheoreticalintensityundernon-cavitatingconditions.


the vibrator surface (vm). The effective radiation resistanceper unit area of the vibrator is thenRrad = 2WJv~,SwhereS is the area. In fig. 52a this quantity is shownplotted out as a function ofv~,,which would be proportional to the acoustic intensity in the absenceof cavitation. Below thecavitationthresholdRrad 1.5 x 10~c.g.s.units, asexpected,but aftercavitationhasset in, this dropsoffrapidly, reachingaconstantvalueof about5 x 10~at high intensities.Anotherway of presentingtheseresultsis to showthe intensityactuallyradiatedinto the liquid (WJS) as afunction of the intensitythatwould be obtainedin the absenceof cavitation. This is shown in fig. 52b. In this case,the measuredradiatedintensityis almost constantat about 1.5W/cm2 over a wide rangeof valuesof the transducerface velocity.

Undercavitatingconditions,the voltagesignal receivedat the detectorterminalsof the immittancebridge is ripple-modulated,reflecting bubbleeventsoccurring at frequenciesdiffering from the drivefrequency.The signals due to thesecavitation events can be extractedby simply filtering out thefundamental.We have found that this bridge-filter arrangementis a very sensitive indicator ofimpedancechangesdue to a wide rangeof bubblemotions. An interestingexample,fully exploredinref. [1361,relatesto the Willard-eventsreferredto above.With suitablefiltering of the receivedsignal,the Willard event is recordedas a low-frequency,short-durationvoltagepulse.An exampleof atypicalc.r.o. traceis givenin fig. 53.The explanationis that manyof the micro-bubblesthat makeup the cloudarein radial resonance.As thesebubblespassthroughthe nodesandanti-nodesof the quasi-standing-wave field theyareexcitedto emitstrongly at the pressureanti-nodes,but becomequiescentagainonpassingthe nodes.The receivedpulse thereforeconsistsof a modulatedcarrier, the wave-lengthofwhichcorrespondsto the half-wave-lengthseparationbetweenthe anti-nodesof the standing-wavefield.As the acousticwave-lengthandc.r.o. time-basespeedareknown, the c.r.o. signalgives the distanceover which the bubblesareactive andalso the time-durationof the event.In this way, the velocity ofthe bubblecloudthroughthe focal region was found to be severalmeter/sec,in substantialagreementwith Willard’s direct measurements.

TheWillard eventsarespectacularandreflectalargeimpedancechange.But eveneventssoweakasto be undetectableby the nakedeye can be recordedin this way. For example,it is easyto detecttranslatorymotionof single resonantbubblesmoving throughthe field. At the 1 MHz frequencyusedinthe work reportedin ref. [136] the resonantdiameter is only about 6 micron and isolated bubbles

Fig. 53. Acoustic impedancechangedue to a typical Willard e%ent.Total time-base length = 10 millisec.


Fig. 54. Acoustic impedancechangedueto a singleresonantbubble moving in thefield. Total time-baselength = 10 millisec.

streakingthroughthe field would not normally be detected.Figure54 showsa typical c.r.o. traceof thesignal receivedfrom a single bubblemoving underradiationforces.The eventcould not beresolvedonphotographstakenin synchronismwith the c.r.o. time-base.

9.5. Concertedcollapseof cavity clouds

The acousticfield conditionsoften encouragecavitationbubblesto collect into groups.When thisoccurs in the body of the liquid, away from boundaries,the bubbles typically expandand collapsein-phaseor approximatelyso.However,an importantpracticalcaseis whenbubble-cloudscollapseonor neara vibratingsurface,whereerosionmayoccur. In this case,it hasbeenshown that whereasthebubble may expandapproximatelyin-phaseon the tensionhalf-cycle, the collapse will start in theoutermostlayersof the cloud andpropagateinwards, increasingthe violenceof the implosion.

Morch and colleagues[1561have studiedthe dynamics of a systemof bubbles collapsing as ahemisphericalcloud on the surfaceof an ultrasonicvibrator.The dimensionsof the cloud areassumedsmall compared with the acoustic wave-length. Under these conditions the bubbles will expanduniformly asthe vibrating surfaceis acceleratedaway from the liquid. But on thepossitivehalf-cyclethepressureis relieved by the bubbles to the extentthat the acousticwave is full-wave rectified. Thebubbleswill thencollapseunderthe ambientstaticpressureonly. The collapsestartsin anouterregionof the cloud, propagatinginwardsand triggering the collapseof consecutivelayers of cavities. Theauthorsassumethat the velocity of soundin the bubbly mixture is smallenoughfor the outershelltocollapse completelybefore the next is affected, andso on. A convergentsphericalpressure-waveisthereforegenerated,so the final collapseof the bubblesin contactwith the surfaceis greatlyamplified.Only in thisway, the authorsbelieve,can collapsepressuressufficient to causeerosionbe developed.

The authorsproducedtwo simplified theories, representingextremecases,where (i) a definitefractionof the totalenergyradiatedfrom anysymmetricalshell of bubblesis propagatedinwards;and(ii) the energyof collapseof eachindividual bubblein a shell is transferredto the centreof the cloudwithoutattenuating.Case(i) seemsthe morerealistic,andhereit can be shownthat the energycarriedby the implosion on the solid surfacemay be enhancedby a factor of 10 or more. In case(ii) theboundariesof the cloud areanalogousto a sphericalacousticlensfocusingtowardsthe centre,andhere


also, an appreciablegain is achieved.The authorsconsider that their results show why cavitationerosion can manifestitself on a vibratingsurfaceright from the initial onsetof transientcavitation,whereasotherwisewe might expectan “incubation” period,as observedin damageassociatedwithhydrodynamiccavitation.

9.6. Acousticintensity-distributionin thepresenceofcavitation

In anytheoreticalapproachto the studyof bubblefields, we must be contentwith examiningtheaveragepropertiesof the cavitatingregion.A phenomenologicaltreatmentis necessary.Rozenberg[157and1581 showedhow to estimatethe space-dependenceof the acousticintensityin a beamgeneratingcavitationin a definedregion.Considerfirst a planewave.The numberof cavitatingbubblesin a thinslice,thicknessdx normalto the beam,maybe written:

dx Jd~{~L)dJ

whereN(I) is a functiondescribingthe number-distributionof stationarybubblesat soundintensityI.

Assumingthat the growth andcollapseof bubblesoccursin just onecycle of thefield, andwriting E(I)for the energyof formationof onebubble,thenthe total energylost by the wavein forming all bubbleswithin the elementthicknessdx, is:

E~= dx JE(I) dN(I)dl.

But wecanequatethis with — T dl whereT is the acousticperiod (2ir/w). We maythenwrite:

- = -~- I E(I)dN~I)dl (128)dx T~ dlIt

andx can be expressedexplicitly in termsof I by further integration.We notethat in consideringonlyenergy-flowin the x-directionwe needtakeno accountof wave-scatteringor anydetailedbehaviourofthe wave.The functionN(I) can bestbe obtainedby direct countingfrom high-speedmotion-pictures.The function E(I) is not readily obtained,however.

The expression(128) is just the volume-densityof the energylost per unit time in sustainingthecavitationzonein the steady-state.It can be written W/Vo where W is the total powerassociatedwiththecavitation lossesand V0 is the totalvolume of the cavitatingzone.We havepreviouslyshownhowto measureW usinga radiometerprobe.Figure55 gives typical measurementsobtainedby Rozenbergshowingthedependenceof W/Vo on the field intensityI, both expressedin W/cm

2. The quadraticcurveis a good fit to the experimentalpoints, so we can write: W/V

0= A(I — I~)2 —dI/dx for I> thethresholdintensityI~.Then:

I

I dlx=—j A(I—I5)2~ (129)


W/v0 (W/cmt)

I (W/cm~

bOO

400 / 1100

20: i::: /2

900 000 1100 0 2

I (W/crn’) X (cm)

Fig. 55. Powerdensityassociatedwith cavitationlossesversusacous- Fig. 56. Intensity-distributionon the axisof a one-dimensionalcavi-tic intensity;I = 500kHz. tation zone:(1) Jo = 1000W/cm2 (2) 1100WIcm2 (3) 1200W/cm2.

This equationis readily solvedfor I in termsof x. Figure56 displaysthe resultsgraphicallyfor the threecases:I~= 1000, 1000 and 1200W/cm2.

The above discussionapplies to plane waves. Rozenberg[158] has extendedthe treatment tosphericalwaves and applied it to experimentalresults obtained using his high-intensity focusingtransducer.With a transducerof this sort, the acousticenergyis utilised muchmore efficiently thaninplane-wavesystems.

Further relevant experimentalresults obtainedby Rozenberg[158] are displayed in fig. 57. Afocusingconcentratorwasused,resonantat 15 kHz andthe cavitationzonewasgeneratedat thetip of awire, to locatethe region for photographicpurposes.The soundpressurewasmeasuredby a miniaturehydrophoneplacedatasuitabledistance(about4—5 mm)from the investigatedzone.Figure57a showsthe hydrophonereceived signal as a function of the transducerdrive voltage. The departurefrom

P(Bor) N/cm’xio’

_____ I_____

0 ~ ~ ~50 ~00 0 10 (00 sb ~oo

VD VDFig. 57. (a)Acousticpressure;and(b) Bubble density;atthefocusof aradial-modefocusingtransducer,as functionsof thetransducerdrivevoltage(VD).


linearity occursat the cavitationthresholdandreflectsthe loss,by scattering,of the transmittedpowerwhenthe bubblesfirst appear.The bubbledensity(N) as a function of the drive voltageis displayedinfig. 57b. N wasobtainedby direct countingfrom photographs,usinga high-speedcamera.N increasesrapidly from the thresholdand passesthrough a maximum which coincideswith an upturn in thepressure-voltagecharacteristic.

9.7. Controlledbubblefields

In the work reportedsofar in this section,cavitationfields havebeenallowedto developnaturally,thatis, from pre-existingnuclei,underthe influenceof the appliedacousticfield. Preciseobservationsofthe motionof bubblesis really only possiblewhenbubblesof knownsize andcompositionareseeded,or placed,in the liquid at specifiedpoints. Someobvioustechniquesfor studyingsingle bubbleshavealreadybeenmentioned.A successfulmethodof producingan arrayof bubblesof knownsmall sizes,with diametersdown to a few tenthsof a micron,hasbeendescribedby Nyborg andMiller [159,160].This method makes use of commercially-availableNuclepore filters [161]. These are sheetsof apolycarbonatematerial about 10 micron thick, containing randomly-spacedholes. The holes arecylindrical and of a very uniform size down to about 0.2 micron diameter.The sheetscan be madehydrophobicand in this case,whenthe filters aresimply immersedin an aqueousliquid, the air carriedin the holesis trapped,forming anarrayof smalluniformly-sizedcylindrical bubblesof knowndensity.Miller, Nyborg andothersatthe Universityof Vermont,U.S.A. havereportedphysicalandbiologicaleffectsoccurringnearstably-oscillatingbubblestrappedin this way. Theseinclude: acousticemission;micro-streamingof liquid around the bubbles;migration of small particles towardsthe bubbles as aresultof radiationforces;andvarious effectson biological cells.Thereis evidencethat bubblescan beexcited into violent non-linearmotion before they leave their sites. But, as expected,a limiting drivelevel existsbeyondwhich the bubblesdisintegrateor aredisplacedfrom their sites.

It is not difficult to evolvean approximatetheory for the dynamicsof thesecylindrical bubbles.Forexample,the resonancefrequencyof a cylindrical bubble,radiusR0, lengthL, is found to be:

= (2yPo/pRoL)1”2. (130)

This derivationneglectsthe effect of surfacetensionand is analogousto the resonanceconditionforlarge sphericalbubbles(eq. (44)). As an example,a cylindrical bubble,diameter5 micron, length 10micron will resonateat about 0.5MHz. The resonantamplitudeof the oscillationis given by:

= LP~j2yöP0(1+ R) (131)

wherePA is the acousticpressure,& the loss angleandR the reflectioncoefficient from the surfaceofthe bubble.

Reasonableagreementwith theoryhasbeenfound in experimentaltestsusing methodsbasedonmeasuringthe radiationscatteredfrom the bubbles.Modificationsto the theory areneededwhenthebubblesin the arrayarecloselyspaced.Whenthe holesaretoo close,interactionreducesthe pressurein the vicinity of eachhole.A theorydevelopedby Weston[162]can be used to obtainthe necessarycorrection.


10. Acousticemissionfrom cavitationfields

Cavitating bubbles act as secondarysourcesof sound, emitting sphericalwaves. This acousticemissioncanbe analysedandusedasa meansof distinguishingbetweenstableandtransientcavitation.The noisemayrepresentahazardfor usersof industrialandmedicalultrasonicequipment,andis veryimportantin military sonar.For all thesereasons,the analysisandinterpretationof thesoundemittedfrom cavitationfields hasbeen,andstill is, an activeareaof research.

Historically, thegroupworking at theUniversity of Gottingenwasthe first to attempta systematicstudy of cavitation noise. Esche[163]recordedmany noise spectraat various drive frequenciesandintensity-levelsin gassywater. He wasthe first to reportthe sub-harmonicemissions.Strongsignalsatharmonicsof the drive frequencywereseenat intensitieswell below the transientthreshold.Lines atsub-multiplesof thedrive frequency(the sub-harmonics)anda continuumof “white” noise appearedtogether,apparentlyat the transientthreshold.The noisespectrumshownin fig. 58 is reproducedfromEsche’s results. It refers to intense transient cavitation in gassy water, generatedby a 15 kHzmagnetostrictivetransducer.The half-ordersubharmonicis prominent,alongwith harmonicsup to highorder,superimposedon awhite-noisecontinuum.Esche’spaperwasfollowed by othersfrom thesamegroup[73,164]. Later,Negeshi[165]also recordedharmonicandsub-harmonicemissions.But neitherEschenor Negeshicould offer explanationsof thestrongsub-harmoniclines.

Much of the recentresearchhasbeenconcernedwith identifying thesourceof thesesub-harmonicemissions. In U.K. these studies have been mainly experimental [166—169],and in U.S.A. boththeoreticaland experimental[98,170—173]. Workers in U.S.S.R. have helpedto explain the strongsub-harmonicemissionandthedistributionof white-noisein the transientcavitationspectrum.Recentwork by Lauterborn at Gottingen[175,176] hasresolvedsome remainingmysteriesrelating to thesub-harmonicandultra-harmonicemissions.

Very small probehydrophonesare generallyusedto record the acoustic emission.The receivedsignal is takenthrough a wide-bandamplifier andspectrum-analyseranddisplayedon a c.r.o.or chartrecorderwhile the frequencyis scannedthrough the appropriaterange.The contentof the emissiondependsmainly on: the stateof gassificationof the liquid; ambientpressureand temperature;theacousticintensity;and,to a lesserextent,on thedrive frequency.It dependsvery little on the positionof thehydrophonein the liquid.

- ~od~

Fig. 58. Intensity—frequencyplotof theacousticemissionfrom stronglycavitatinggassywater;resonancefrequency,fo = 15 kHz.


10.1. Emissionat low intensities:the stablecavitation regime

At very low excitationlevels,the receivedsignal consistsof the fundamentalonly. The acousticwaveis partially scatteredby any bubblespresent,especiallythoseof resonancesize.At higher intensities,but still belowthe transientthreshold,the largerbubblesbeginto oscillatenon-linearly.The spectrumthen contains lines at harmonicsof the drive frequency,up to high order. The second harmoniccomponentis the most prominent.Its amplitude is proportionalto the squareof the fundamental,arelation that is predictabletheoretically.Someof the harmoniccontentcan usuallybe attributedto theinherentnon-linearityof the medium, but the gassierthe liquid, the greater the contribution fromoscillatingbubbles.A low level of white noiseis usuallypresent,increasingto an impressivelevel as thetransientthresholdis approached.The half-ordersubharmonic(at f/2) may also appearin short burstsseparatedby long intervals and therefore at low averagelevel. This subharmonic,and also theultra-harmonics(at frequencies(2n + 1)f/2) aremorelikely athigh drive frequencieswherethe bubblesresponsiblefor them aresmallenoughto remainin suspensionin the liquid. Sometimes,linesareseenat frequenciesunconnectedwith the drivefrequency.It is thought that theseareemissionsfrom largebubblesof arbitrarysize,shock-excitedinto radial vibration [172,173].

There are threepossibleexplanationsfor the weak sub-harmonicsignalsseenin stablecavitationfields:

(i) We havealreadyseenhow strongsurfacevibrationscan beexcitedparametricallyat half the drivefrequency. These are pure distortion modes, not involving any area-changeand thereforeweaklycoupledto the liquid. Sub-harmonicemissionsfrom this causehavethereforeonly beendetectedbyhydrophonesheldvery closeto the bubblesurface[131].

(ii) Above acertainthreshold,bubblesof suchasize that theywould resonateatafrequencyclosetof/2 arecapableof emitting a strongsignal at their ownnaturalresonancefrequency.Threetheorieshavebeenpublisheddefining the thresholdfor this typeof emission.Eller andFlynn [98]obtainedthe result:

PT= 6P0[(.~-—2)+ 62]’ (132)

giving PT = 6P06 at W =2Wr, where6 is the total dampingcoefficient for the radial motion.Safar[169]

publishedthe result for Cs) = 2Wr

PT=6yPo6,~. (133)

NayfehandSaric [171]obtained:

PT= 6Po[(~I~r— 2)2+ (kRr)2]112 (134)

giving PT = 6Po(kRr) at to = 2Wr, where (kR1) is just the radiationdampingof the bubbleat frequency

tor. NayfehandSaricobtainedtheir resultdirectly from the equationsdescribingthe naturalamplitudeand phaseof the radial pulsationsof the bubble.Eller and Flynn and also Safar derived theirs byexaminingstability after applying a perturbationto a steady-statesolution. Strong emissionsat thesub-harmonicfrequencyhavebeenmeasuredwhenbubblesof the appropriatesize arepresent.In thework reportedin [172,173] bubblesof the correctsize were seededinto the liquid usinga mechanicalbubble-generator.The bubbles were allowed to rise under gravity in the acoustic field pasta small


hydrophone,andthe filtered responserecordedon eitherachart-recorderor c.r.o. time-base.Examplesof the emissionat the half-ordersubharmonicfor air bubblesin waterare reproducedin fig. 59. Here,the drive frequencywas 89 kHz and the total time-basedurationis about20000cycles(0.22sec). Theexcitationlevel wasnot knownaccurately,but could not havebeenfar abovethe thresholdsindicatedby the abovetheories,that is, well below the transientthreshold.The photographsshow that a singlebubblecan sometimesgive aseriesof emissions,eachlastingmanycycles,separatedby inactiveperiods.Thesetracesarequite different in appearancefrom thoseobtainedwhena bubbly liquid is allowedtocavitatenaturally.Thenthe subharmonicresponseappearsas a massof short overlappingtransienta.Ifa bubble of the appropriatesize attacheditself to the surfaceof the measuringhydrophone,theemissionwouldcontinuelong enoughto recordsignalsover arangeof increasingdrive levels.Figure 60recordsthe resultsof suchmeasurements.In this casethe liquid was a 90% glycerol—watermixture andthe indicatedthresholdfor the emission is appreciablygreaterthansuggestedby the formula (133).Eventually, the bubbledetacheditself. Flynn [1] haspublishedsolutionsto the non-linearequationofmotion (14) for bubblessatisfyingthe sub-harmonicconditionto =

2Wr. In the exampleshownin fig. 61the bubblewas driven below,but closeto, the transientthreshold.The oscillationat the sub-harmonicfrequencycontinuesfor manycycles, building to large amplitude.In this case,the bubble eventuallycollapsesas a transientwhenRm approaches2R

0.(iii) When the liquid medium is sufficiently compressibleand non-linear,a signal at fl2 may be

generatedby a parametricamplificationprocess,evenin the absenceof bubbles[174].Thissometimesoccurs in water left standing long enough to removebubbles of the size required to excite theoscillationsdescribedin (ii).

Therearetwo possibleexplanationsfor the white-noisesignalsobtainedfrom stablecavitation:(i) Under highly non-linear conditionsmost of the emissionconsistsof discrete pulsescontaining

harmonics of the fundamental,repeatedat the excitationfrequency.Thesepulsesare emittedduringthe compressionphaseof the bubbleoscillation. Figure12 illustrateshow, undertheseconditions,thebubblemaybe shockedinto oscillatingfor a time at its own naturalfrequencyduring eachcycle of theacousticpressure.However, the sequenceis neverrepeatedexactlyon successivecycles. The receivedsignalwill thenappearasnoise.However,this noisewill haveastructure.Its frequency-dependencewill

:~:~

Fig. 59. Signalsreceivedat 1/2 from waterseededwith air bubblesof the appropriatesize;I = 89 kHz; total time-baselength= 0.22 sec.


v)0

~40 ~I .8

~20 2 _____

0I 0~ ~ 10 20

Fig.60. Signalat fo/2emittedfrom asingle bubbleof theappropriate Fig. 61. Unstablemotion of a cavity driven at twice its resonancesize caught on the hydrophone,as a function of the transducer frequency; R

0 = 2.6 x 102cm; PA = 0.75Bar; P0 = I Bar; f=

excitation current (Jo); to = 38kHz and the liquid is 90% glycerol- 24.5kHz.water; 1 — Bubble trappedon the hydrophone;2— Bubble removed.

berelatedto the sizesof the smallbubblesresponsiblefor it. The recordednoise-distributionthengivessomeindication of the size-distributionof bubble-nucleiin the liquid.

(ii) The emissionof micro-bubblesduring strong surface-waveactivity mayresult in a low level ofrandomnoise[104].

10.2. Emissionat high intensities: transientcavitation regime

Theemissionspectrumchangesdramaticallyassoonas transientcavitationstarts.Thereis a suddenincreasein intensity of the subharmonicsand their harmonics.Many workers have recordedanddiscussedtheseemissions[167,168, 172and 175].The intensityof the white noisealsoincreasesrapidlywith increasingdrive level. The harmonicoutput,as recordedon the hydrophone,is usuallyweaker,aneffect that can be explainedin termsof scatteringby the developingbubblescreenwhich shieldsthehydrophonefrom radiationemanatingfrom stablesources.Measurementsreproducedin fig. 62 refer toordinary tap-waterleft standinglong enoughto removethe largerbubbles.Underthe conditionsof theexperiment,the recordedtransducercurrentwas almostproportionalto the acousticfield pressure.Thesubhannonicsignal comesin very abruptlyat the transientthreshold.Up to thispoint the white noiseisatalow level, but beyondthe thresholdincreaseslinearly with increasingexcitationpressure.In fig. 63the second and third harmonic componentsof the received signal are displayed along with thesubharmonic.Again, tap-waterwas used,freedof large bubbles.The secondharmonic is much thestrongestsignal, increasing approximately as the squareof the drive pressureuntil the transientthresholdis reached,whenit levelsoff. For similarconditions,that is, air-saturatedwater left standingfor severalhours, the frequencyspectrumhasthe appearanceshown in fig. 64. This coversonly therangebelow the excitation frequency,40 kHz. The excitationpressurewas 1.5 times the transientthresholdpressure.At this level, the half-ordersub-harmonicis always prominent,showing clearlyabovethe noise.This tracealsoshowsa responseat the one-third-ordersubharmonic(at about13 kHz).This is rarelyseenat suchalow drive frequency,andmusthavebeendueto the chancepresenceof alargebubbleof the appropriatesize.


3

0.1 0.IS 0.2 0.25

‘D (A) 04 08 2lo (A)

Fig. 62. Emissionat: 1— the half-ordersubharmonic;and 2—white Fig. 63. Emissionat fl2, 2f and3fin air-saturatedwater;!= 181 kHz.noiseas functionsof thetransducerexcitationcurrent(SD).

Therearetwo possibleexplanationsof the strongsub-harmonicsignal emittedby transientcavities:(i) We havealreadyseenthat situationsoften occurwherebubbleshaveinsufficient time to collapse

completelybefore the end of the acousticcycle. In such cases,they maycomplete their collapseastransientsat the second,or subsequent,pressuremaximum. If the bubbleremainsintact, this motionwill becomeperiodic. Thisbehaviouris seenclearlyin the theoreticalR—t curvesof fig. 11. Evenverysmallbubbles,well belowresonancesize,will behavein this way. The bubbleseventuallycollapseastransientsand the intensity of the sub-harmonicemission may be comparablewith that at thefundamental.Clearly, the probability of exciting any sub-harmonicdecreasesas the order of thesub-harmonicincreases,andthisis confirmedfrom observation.

- ~ ..:...,

.4~.. :~ .

J.O~.3 20 40

Ftg. 64. Frequencyspectrumfor cavitatingair-saturatedwater.PA is approximately1.5 timesthetransientcavitationthreshold;I = 40 kHz.


(ii) The type of motion illustrated in fig. 61 can be regardedas a near-thresholdcondition. Thebubble-motion is part stable, part transient. In this case, though, the bubble must be near theappropriatesize for resonanceat ff2. As the suddenincreasein responseat the half-ordersubharmonicoccurs at, or very near, the transientthreshold,it is now generallyregardedas the best availablethresholdindicator[166—168].

The shock wave generatedby a collapsingtransientandpicked up by the hydrophonewould intheory be recordedby the wave analyseras an infinite seriesof harmoniccomponents,decreasinginamplitudeas f2 at high frequencies.However, in practice,the higher frequencycomponentswill notreachthe hydrophone,and with many bubblesemitting, the remainderwill contributeto the generalnoise.

10.3. Recentcomputationsrelating to acousticemissionfrom cavitationfields

By examining many solutions of eq. (14), Lauterborn[175] hasconcludedthat when bubbles aredriven below their natural resonancefrequencies,the thresholdsfor the 3fl2 and 5f/2 emissionsareactuallylower thanfor ff2. This suggeststhatthe 3ff2 and5fl2 emissionsaremorefundamentalthanf/2.In anycase,their thresholdsbeingsomewhatlower, Lauterbornsuggeststhat it would be logical to usethem,ratherthanthe ff2 threshold,to indicatetransientcavitation.

In anotherrecentpaper,LauterbornandHaussmann[176]havestudiedtheway the cavitationnoisespectrumbuilds up as afunction of time. Specialattenuationwas paidto emissionsat the frequencies(2n + 1)fl2. Resultsfor aeratedwateratf = 20 kHz show that theseemissionsareall generatedat nearlythe sameamplitude,with similar time-dependence.

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