bubble collapse near a solid boundary: a numerical study of the …zaleski/viscousjet.pdf ·...

Underconsideration for publicationin J. Fluid Mech. 1

Bubble collapsenear a solid boundary:A numerical study of the influenceof viscosity

By STEPHANE POPINET�

A ND STEPHANE ZALESKILaboratoiredeModelisationenMecanique,

CNRSandUniversite PierreetMarieCurie(ParisVI)8 rueduCapitaineScott,75015Paris,France,[email protected]@lmm.jussieu.fr

(Received28December2001)

The effect of viscosityon jet formationfor bubblescollapsingnearsolid boundariesis studiednumerically. A numericaltechniqueis presentedwhich allows theNavier-Stokesequationswithfree-surfaceboundaryconditionsto besolvedaccuratelyandefficiently. Goodagreementis ob-tainedbetweenexperimentaldataandnumericalsimulationsfor the collapseof large bubbles.However, thebubblereboundin our simulationis largerthanwhatis observedin laboratoryex-periments.This leadsusto concludethatcompressibleandthermaleffectsshouldbetakenintoaccountto obtaina correctmodelof the rebound.A parametricstudyof the effect of viscosityon jet impact velocity is undertaken. The jet impact velocity is found to decreaseas viscos-ity increasesandabove a certainthresholdjet impactis impossible.We studyhow this criticalReynoldsnumberdependson the initial radiusandthe initial distancefrom thewall. A simplescalinglaw is foundto link this critical Reynoldsnumberto theothernon-dimensionalparame-tersof theproblem.

1. Intr oductionViolent collapseof bubblesin asymmetricalgeometriesoccur in a numberof situationsof

practicalinterestincludingcavitation,shock-waveandlaserlithotripsy . Whencloseenoughto asolidboundarythesecollapsesareusuallyassociatedwith high-speedjet formationBlake& Gib-son(1987).While jet formationfor cavitationbubbleswasdemonstratedexperimentallyasearlyas1961(Naude & Ellis 1961)thereis still debateregardingthe importanceof jet impactasthemainmechanismfor cavitationdamage.Thefirst explanationwasgivenby Rayleigh(1917)whoconsideredthehighpressurecausedby thecollapseto bethemaindamagingmechanism.An al-ternativeexplanationwasgivenby Kornfeld& Suvorov (1944)who suggestedthejet formationeffect which wasexperimentallydemonstratedby Naude& Ellis. Benjamin& Ellis (1966)thenconcludedthat jet formationandimpactwasimportantandprobablythemainfactorfor cavita-tion damage.Recently, however, a numberof experimentshave castdoubton this explanation.In theseinvestigations,the damageswerefound to be distributedarounda circumferenceandnot on theaxisof symmetryaswouldbethecaseif jet impactwasthemainfactorfor cavitationdamage(Tomita & Shima1986).Philipp & Lauterbornconcludethat the main mechanismforcavitation damageare the high-pressuresandtemperaturesreachedinsidea bubblecollapsingverycloseto thesolidboundary(Philipp& Lauterborn1998).

A numberof pointsregardingbubblecollapsein boundeddomainsthusremainopenquestions.However, theanalyticalstudyof theproblemof bubblecollapsein thevicinity of asolidboundary�

Presentaddress:NIWA (NationalInstituteof Water& AtmosphericResearch),POBox 14-901Kilbirnie, Wellington,New Zealand

2 StephanePopinetandStephaneZaleski

is very difficult. Consequently, apartfrom theperturbationapproachof Rattray(1951),mostofthe resultswereobtainedusingnumericalsolutionsof boundaryintegral formulations(Plesset& Chapman1971;Blake & Gibson1987;Blake et al. 1993).Thesemethodsreston thesurfaceintegrationof apotentialsolutionfor thefluid flow andcanbeusedonly for vanishingviscositiesor vanishingadvectionterms(Stokesflow). In all the studiesso far the effect of viscosityhasthus beenneglected.While good agreementwas found betweennumericaland experimentalresults,due to obvious practicalconsiderationsexperimentsare usually performedwith largebubbles(millimeter sized).In thesecases,viscosityis unlikely to play a majorrole andinviscidcalculationsgive goodresults.However, onecanaskhow theseresultswould scalefor smallerbubbleswhereviscosityandthermaleffectsarelikely to comeinto play.

Thermalandviscouseffectsareimportantfor intermediatesizedbubbles(50to 2 � m) andvis-couseffectsdominatefor smallerbubbles(lessthan2 � m) (Plesset& Prosperetti1977;Brennen1995).Thepresentapproachincludesonly viscous,not thermaleffects.Thereareseveralreasonsfor this: themostimportantis theneedto build numericalmethodsincrementally, addingeffectsoneat a time.Thesecondis thatreducingthenumberof physicalparametersis importantto en-surea detailedanalysisof thephenomena.Thusour approach,at themoment,includesonly themostimportantdissipative effectsfor very smallbubbles.Thesesmallbubblesmaynot bepar-ticularly interestingin termsof damagein theusualsituations,but therearespecialcases,suchasdamageto living tissuesin the caseof high level ultrasoundexposure,wheresmall bubblesareinvolved.

It will probablybe difficult to comparethe resultsof this studywith experimentsgiven thesmallsizesinvolved.Howeverwe believe thatour numericaltechniqueasit is andtheresultsinthis papercangive usefulinsights.Furthermore,it maybea usefulstepon which to build morecompletemethods,involving thermalandcompressibleeffects.

An importantdifficulty whenimplementingthefreesurfaceconditionin viscousflow at finiteReynoldsnumberis to obtaina quantitative agreementwith theoreticalresults,for instancetheRayleigh-Plessetequation.We arenot awareof comparisonsof thatkind in theliterature.

Thereare indeedfew numericalstudiesof cavitation that take into accountthe free surfaceconditionandtheviscouseffects.Oneexceptionbeingthework by theTryggvasongroup(Po-Wenet al. 1995).Themethodemployedby theTryggvasongroup,which alsoinvolvesmarkersparticles,is rathersimilar to our methodhere,except for the fact that we implementthe freesurfaceboundaryconditionatahigherorderof accuracy. Ontheotherhandthework of Po-Wenet al. (1995)is fully three-dimensional,while our work assumesaxial symmetry.

In this article,we describeanoriginal numericaltechniquewhich allows theresolutionof theincompressibleNavier-Stokesequationsin axisymmetriccoordinates,without swirl, with free-surfaceboundaryconditions.We thenpresenttheresultsof comparisonsbetweenournumericalresultsand experimentsfor bubblecollapseand jet formation neara wall. Then,becausethecodeis fastenoughto allow aparametricstudyof theinfluenceof viscosityon jet formationandevolution,we presenta phasediagramfunctionof theindependentnon-dimensionalparametersillustratingtheeffectof viscosityon theimpactvelocity.

2. Numerical techniqueWe presenta numericalmethodfor solving the axisymmetricNavier-Stokesequationswith

free-surfaceboundaryconditions.While this hasbeendonein thepast,themethodsusedeithermadecrudeassumptionsaboutfree-surfaceboundaryconditions(Harlow & Welch1965;Chan& Street1970;Nichols& Hirt 1971;Hirt & Nichols1981)or usedboundaryfittedgrids(Blanco& Magnaudet1995;Legendre1996).Our methodis basedon a finite volumeformulationus-ing both a fixed grid anda front-trackingapproach.The free-surfaceis tracked usingsurfacepoints(markers) connectedwith cubicsplines.This allow usto dealwith surfaceintegral terms

Bubblecollapseneara solid boundary:influenceof viscosity 3

appearingin thefinite volumeformulationcorrectly. Moreover, thismethodis not limited to sim-ple geometriesandcandealefficiently with largedeformationsof the interface.This techniqueconstitutesanextensionof thetwo-fluid approachof Popinet& Zaleski(1999).

2.1. Theexplicit equations

TheincompressibleNavier-Stokesequationsfor anaxisymmetricflow withoutswirl in cylindri-cal coordinatescanbewrittenas�� (2.1)� �� !"#�� !�$�� &%"%� � (2.2)� �� #�� !�$'�� !�$�� (2.3)

where� �)(�*+ , + is constantandthecomponentsof thestresstensoraredefinedas " �-,/. � �� (2.4) &%"% �-,/. � � (2.5) �$� �-,/. � � �� (2.6) �" � "� �0. � � �� 1 � �� (2.7)

where . is thekinematicviscosity.

2.2. Finite volumeformulation

In orderto obtaina finite volumeformulationnecessaryfor numericalanalysis,we first needtointegratetheequationsover anarbitrarily moving domain.Let uscall thedomain 2 and

� 2 itsboundary. 3 is thevelocityof theboundary

� 2 . Integratingover 2 yieldstheintegralequations�� 465 �87/�97 � 46:'5 � � �� <; ��&7=� � 46:'5 � � �� <; ��>7 � �� (2.8)�� 465 � �87/�?7 � 46:'5 �� @; � �A7/� � 4�:#5 �� @; �&7 � � (2.9)4 :#5 � �97 � 4 5 � 7=�?7 � 4 :'5 �B ��"C7=� � 4 :'5 �� !"D7 � � 4 5 �%$%E7=�?7 � ��F� 4 5 � � �87=�?7 � 4 :'5 �� <; � �&7=� � 4 :'5 �� @; �&7 � � (2.10)� 4�:'5 � �?7/� 4�:'5 �� !�$�G7/� � 4�:#5 �B ��"H7 �JIThen,consideringa squarefinite domaincenteredat (

�K,�ML ) of side N , we seekthe integral

formulationabovefor the � componentof thevelocity. In thegeneralcaseof a free-surfaceflowproblemthis controldomaincanbecutby theinterface.In thiscasethedomainof integration 2is apieceof thesquarewith boundaryO?PRQTSVUWO (Figure1). Thevelocity 3 of theboundaryis0 on O?P-XYPZQ[XYSVU[X\UWO and ] on QTS . Equation(2.9)canthenbewritten�� 4 5 � �87=�?7 � 4�^>_A`�a&b �'� � � 7/� � 4�b!^�`�cEa �� 7 � � (2.11)4 b!^�`�cEa � �?7 � 4 _>c � �87 � 4�5 � 7=�?7 �


i,ja

i,jSθθ

ri,jv i,j

c

r

E D

A B

C

i,j+1/2 i,j+1/2

i,j+1/2i,j+1/2

r r rr

r

i+1/2,j

i+1/2,j

i+1/2,j

zr

z

rz

i,j−1/2

i,j−1/2

i,j−1/2

i,j−1/2r r rr

i−1/2,j

i−1/2,jz

i−1/2,j

zr

rz

vv S

s

s

vv

S

s

Svv

S

vv

s

r

z

FIGURE 1. Finitevolumediscretisation.

rv

θθS

r rvv

rs

rrSzv

zrvv rzS

rs

r

z

zs

zzSz zvv

zs

c

c

a

FIGURE 2. Discretisationof thevelocity componentsandpressureandcorrespondingcontroldomains.4 ^>_A`�a&b �B ��"C7=� 4 _>c �� !�$H7/� �4 b!^A`�cEa �� !"d7 � � 4 _&c �� !"d7 � � 4�5 �%$%d7/�?7 �FIWe thenassumethat thedifferentquantitiesneededaredefinedeitherat thecenter

�e� K � � L � ofthecell or at thecenterof thecell faces

�e� K � � LCf N *�, � and�g� K f N */,6� � L � , in a typical staggered

grid fashion(Peyret & Taylor 1983).We alsomake theassumptionthatquantitiesdefinedat thecenterof thecell areconstantover thewholecell whereasthequantitiesdefinedonthecell facesareconstanton thesefaces.

Thecontroldomainsanddiscretisationof thevelocitycomponentsandpressureareillustratedonfigure2. Also notethatin thefollowing theconventionis to alwaysdefinethe

�ih �kj � indexesatthecenterof thecontrolvolumesconsidered(i.e.eachof thetwo controlvolumesfor thevelocityin thissectionandthecontrolvolumefor thepressurein section2.3).

The products �l� , �� and � � � are computedby averagingthe velocity componentsin therequireddirections.Similarly, thecomponentsof thestresstensor !" , !�$� and �m� arecomputedusingfirst-orderfinite differences(whicharesecondorderon a regularCartesiangrid).


We alsointroducesomegeometricalquantities.Let n Kio L �-p b!^�`�cEa �?7 � wheretheintegral isoverthefluid regioninsideacontrol-domain(or cell) boundaryonly. It is proportionalto anarea:to beexplicit, it is thearea(cut by thefluid) of thevertical faceof anaxisymetric,3D-control-volume(or cell) generatedby the2D controldomain 2 . Similarly definethe3D-cell facesarean Kio L� � p ^A_>`�a>b �G7=� , 2D-cell areaq Kio L � p 5 7=�G7 � and3D-cellvolume r Kgo L � p 5 �G7/�G7 � . Eachcell is the control volumeof a componentof the velocity. ThesequantitiesarecomputedasinPopinet& Zaleski(1999),usingthemarkerdefinitionof theinterface.

Usingthesedefinitions,in thegeneralcaseweobtainthefollowing discretesolutionfor the � componentof thevelocity��F�Rs � r ��#� Kgo L�t � n �� KvuAwyx � o L � � n �#��z��B� Kg{!wmx � o L (2.12)� n z� � � Kio L u>wmx � � � n M� � � Kgo L {!wmx � �� n � � Kio L {�wyx � � � n � � Kgo L u>wyx � � q � � Kio L 4 K}|B~ � �87 � � n � �" � K}u>wmx � o L � � n � �" � Kg{!wmx � o L 4 Kv|�~ �� $ 7/� � n # !"�� Kio L u>wmx � � � n # �m'� Kgo L {!wmx � � 4 Kv|B~ �� !"d7 � � � q �%$%�� Kio L �where p Kv|�~ denotesthe integrationalongthepieceof interfacecontainedin thecontroldomain.If thecontroldomainis notcutby theinterface,thisequationreducesto asimpleclassicalMACscheme.Whenthecontroldomainis cut by theinterfacewe needto computetheintegral termsfor thepressureandstresses.Thetechniqueusedis detailedin section2.4.

Theequationfor the�

componentof thevelocity, obtainedin averysimilar manneris�� s � r ��B� Kgo L t � n �'� �� K}u>wmx � o L � � n �#� �� Ki{�wyx � o L (2.13)� n � � � � Kgo L u>wyx � � � n � � � � Kgo L {!wmx � �� n � � � Kg{!wmx � o L � � n � � � KvuAwyx � o L � 4 K}|B~ � �87/� � n �B !�$�'� KvuAwyx � o L � � n � !�$�B� Kg{!wmx � o L 4 Kv|B~ �� !�$�G7/� � n �" � Kgo L u>wmx � � � n �$ � Kio L {�wyx � � 4 K}|B~ �� $ 7 �FINoting that ��F� 4 5 �97/�?7 � � 4 :'5 � ; � 7=� � 4 :'5 � ; 7 � � (2.14)

(2.8)canberewrittenas 4�:#5 ��G7=� � 46:'5 ��C7 � �� (2.15)

which yieldsthediscretevolumeincompressibilitycondition� n � � � � Kvu>wmx � o L � � n � � � � Ki{�wyx � o L 4 K}|B~ �� 7/� (2.16)� n z��'� Kgo L u>wmx � � � n z��#� Kio L {�wyx � � 4 K}|B~ ��d7 � �0� I


2.3. TheDiscretePressure Equation

We usea projectionmethodto solve the incompressibilitycondition (Gueyffier et al. 1998;Brown etal. 2001).Givenavelocityfield at time � atemporaryvelocityfield ]�� is computedbyadvancing] in timeusingasimplefirst-orderin timediscretisationof equations(2.12)and(2.13)with thepressuregradienttermsomitted(notethattheintegralcontributionof thepressurealongtheinterfaceis includedat thispoint).Thepressureis thencomputedasthecorrectionnecessaryto ensurethe non-divergenceof the velocity field at time � � . The Poisson-like equationforthe pressurecanthenbe expressedasa function of the numericaldivergenceof the temporaryvelocityfield ]�� . Thevelocityfield at time � � is thenobtainedfrom therelations� Kio L � � � � � Kio L �� r Kgo L�� n � � Kgo L {!wmx � � � n � � Kio L u>wmx � � q � � Kgo Lz� I� Kio L� � � � �� Kio L � �r Kgo L�� n � � � Kg{!wmx � o L � � n � � � K}u>wmx � o L � IWe make theassumptionthatthetwo componentsof thevelocity vary linearly from onesideofthecell to theother, yielding theexpressionsfor thevelocity in a

�g�K � �zL'� cell� � �g� � N � �g� � � Ki{�wyx � �� Kvu>wmx � o L� � �g� � � KvuAwyx � �� Ki{�wyx � o L� � (2.17)�� N � � � � � L {!wmx � �� Kgo L u>wmx � � � � � � L u>wyx � �� Kgo L {!wmx � I(2.18)

Theintegralsalongthesectionof interfacein the�e� K � � L � cell canbecomputedas4 K}|B~i�i� � ��G7=� � (2.19)� Kvu>wmx � o L� N 4 Kv|B~ �i� � � �e� � � Ki{�wyx � �>7=� � � Kg{!wmx � o L� N 4 Kv|B~ �i� � � �e� � � K}u>wmx � �>7/� �4 K}|B~i�i� � ��C7 � � (2.20)� Kio L u>wmx � N 4 Kv|B~ �i� � � � � � � L {!wmx � �>7 � � � Kio L {�wyx � N 4 Kv|�~ �� L u>wmx � �>7 �FI

If we assumethat� Kgo L u>wyx � � �� Kgo L � Kgo L u>w � */, , the incompressibilitycondition(2.16)canbe

writtenas�� n'�� K}u>wmx � o L � � n'�� Kg{!wmx � o L � n'� � � � Kgo L u>wmx � � � n'� � � � Kio L {�wyx � � (2.21)� Kgo L � n Kgo L�� n'�� * r � Kvu>wmx � o L � n'�� * r � Ki{�wyx � o L#� n Kio L�� n'� * r � Kgo L u>wmx � � n'� * r � Kgo L {!wmx � � � � Kgo L, � � q6n � * r � Kgo L u>wyx � � � q6n � * r � Kgo L {!wmx � � �� n � � � Kvu>wMo L � n#�� * r � K}u>wyx � o L � n � � � Ki{�w$o L � n#�� * r � Kg{!wmx � o L � Kgo L u>w � n'� * r � Kio L u>wmx �W� n Kgo L u>w � �, q Kgo L u>wyx �#� � Kgo L {!w � n � * r � Kio L {�wyx � � n Kgo L {!w �, q Kgo L {!wmx � � �where n �� and n � aredefinedas� n'�� K��>wmx � o L � n K��>wmx � o L� �N 4 K}|B~i�� g� � � K}�Awyx � �&7=� � (2.22)


2

3

4

5

6

7

8

9

10

11

8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27

FIGURE 3. Meshanddiscretisationof thevelocityandpressurefields.Themarkerpointsandtheconnectingcubic splinesarerepresented.The light arrows areextrapolatedvaluesof the velocity field. The squaresindicatethelocationof thepressurenodeswherethepressureequationis solved.� n � � Kio L �Awyx � � n Kio L �Awyx � � �N 4 Kv|B~i�i� � � � � � � L �>wmx � �>7 �FI (2.23)

Thecoefficientsappearingin equation(2.21)ensurethatthediscretisationschemetransitionscontinuouslyfrom a classicalsymmetricfinite differenceapproximationto anasymmetricalop-eratorin theneighborhoodof thefree-surface.

2.4. Numericalmethodof solution

The Poisson-like pressureequation(2.21) is solved using multigrid-acceleratedGauss-Seidelrelaxation(Brandt1982;Briggs1987;Wesseling1992).

Theinterfaceis discretisedusinga setof markerpointslinkedby cubicsplines.This descrip-tion allows thepreciseknowledgeof thepositionandcurvatureof theinterfacerequiredin orderto includethesurfacetensiontermsandall theothersurfaceintegral termsappearingin (2.12),(2.13)and(2.21).

As in all free-surfacecodes(Harlow & Welch 1965; Chan& Street1970; Nichols & Hirt1971;Hirt & Nichols 1981),the mostdelicatepoint is the treatmentof free-surfaceboundaryconditions.Thepressureon theinterfaceon thefluid sideis givenby(Y��( K ��Z �� I �<I � � (2.24)

where( K is the pressurein the bubble, � the surfacetensioncoefficient, � the curvature, � thedynamicviscosity, � thenormalto theinterfaceand

�thedeviatorypartof thestresstensor. This

boundaryconditionis useddirectly whencalculatingthe surfaceintegral pressurecontributionto (2.12)and(2.13)in orderto obtainthetemporaryvelocityfield ]&� .

Sincewe usea fixed grid to solve the Navier-Stokesequations,we needto extrapolatethevelocity field far enoughinsidethe bubbleto get the velocity valuesnecessaryfor the markerpointsadvectionandfor thesolutionof theNavier-Stokesequationsonthefluid boundary(Figure3). Moreover, this must be donewhile fulfilling the zero tangentialstressinterfaceboundarycondition � I �<I � �0�J� (2.25)

where

�is thetangentto theinterface.Givenapoint � neartheinterface,weassumethatlocally

around� thevelocity field canbedescribedas 3 � 3&� [�¡ ¢ where ¢ is thepositionvectorand � is a ,\£¤, matrix. For this particularvelocity field equation(2.25)canbe expressedas


P

L

In

FIGURE 4. Selectionof pointsfor theextrapolationtechniquearoundpoint ¥ .¦ K L � K � L �� with ¦ K L � � ; K��§ L � ; L�F§�K � O K L O L K I (2.26)

Givenasetof ¨ pointsin thevicinity of � andavector

�tangentto theinterface,3A© and � can

befoundby minimisationof thefunctionª �¬«|�®>w � 3 � ��¡ z¢ | � 3 | � � °¯ ¦ K L ��K � L � (2.27)

where is aLagrangemultiplier. Thesetof pointsis chosenasonFigure4. Line ± hasdirection� �e² � andgoesthroughpoints � and². A smallnumberof points(typically five)is chosenaround± by minimisationof thecostfunction 7 � � K � ± � � ´³ 7 � � K � � � � � (2.28)

where 7 is theEuclideandistanceand ³ is ageometricalparameterusuallysetto� *�, .

The marker points are advectedusing bilinear interpolationand a redistribution is doneatevery time stepto ensurea uniform distribution as the bubbledeforms.The averagedistancebetweenmarkersis of theorderof thegrid size.As underlinedby Popinet& Zaleski(1999)thecomputationalcostof themarker partof thealgorithmscalesas

� * � where � is thenumberofgrid pointsalongonedimensionandis thennegligible for reasonabledomainsizes.

2.5. Validation tests

The time evolution of the radiusof a sphericallysymmetricbubblesurroundedby an incom-pressiblefluid is describedby theRayleigh–Plessetequation(Plesset& Prosperetti1977).As weareinterestedin bubblesoscillatingradially, it is importantto obtaina goodagreementbetweendirectnumericalsimulationsandthenumericalsolutionof theRayleigh–Plessetequation.In thetestcaseillustratedon Figure5, a bubblewith anequilibriumradius µ9� �·¶ � m is setin a fluidinitially at rest.Theinitial radiusof thebubbleis 10 � m. A constantpressureis appliedon threesidesof the simulationdomain(the fourth side being the axis of symmetry)and the velocitygradientis setto zeroon thesethreesides.Thephysicalparametersareasfollows:dynamicvis-cosity � � 0.001kg/ms,surfacetensioncoefficient � �� I �l¸ kg/s� , +Z� � �=�/� kg/m¹ , (�º�� /»Pa. The pressurein the bubbleis given by a polytropic law of the form ( � µ � �¼( � � µ � * µ � ¹m½with ¾ �·¸/*/¶ . Thebubbleoscillatesradiallywhile keepingits sphericalshape.Theamplitudeoftheoscillationsdecreasesdueto viscousdamping.Theagreementbetweenthedirectnumericalsimulationandthenumericalsolutionof theRayleigh–Plessetequationis excellent.Therelativequadraticerrorbetweenthetwo solutionsillustratedis smallerthanonepercent.


0¿

2À

4Á

6Â

8Ã

10¿0

2

4

6

8

10

12

Rayleigh−PlessetNavier−Stokes

PSfragreplacements

Time ( Ä s)

Rad

ius(

Å m)

FIGURE 5. Freeradialoscillationsof a bubbleof 5 Æ m equilibriumradius.

Confinementratio Relativequadraticerror160 0.009580 0.03720 0.1925 0.564

TABLE 1. Evolutionof therelativequadraticerrorastheconfinementratio is variedfor theradialoscillationof a sphericalbubble.Theconfinementratio is theratio betweendomainsizeandbubble

diameter.

It is importantto notethat to obtainsuchanagreementit is necessaryto minimize the influ-enceof the boundaryconditionsin the simulation.This is doneby usingvery largesimulationdomains.Table 1 givesa summaryof the influenceof confinementon the obtainedsolution.By usingthe adaptive multidomaintechniquepresentedin the following sectionthe computa-tional costis still reasonable(approximatelyonehouron a Pentium350MHz for a basegrid of� ,/ÇR£ÉÈ�Ê ).

This simulationis alsoa goodvalidationtestfor theextrapolationtechniquepresentedin theprevious section.The pressurejump on the free-surfacedue to the normal componentof theviscousstresscontrolstheviscousdampingof thesolution.A 2% variationin viscosityleadstosolutionsof theRayleigh–Plessetequationvaryingby 1.3%.Giventhe1% error thatwe obtain,we canconcludethat local derivativesof thevelocity field nearthe interfacediffer by lessthan2%.

Howeverthistestdoesnotinvolveany deformationof theinterfaceandtheinfluenceof surfacetensionis limited to a constantpressurejump. A secondsimpletestwherethe driving force issurfacetensionis illustratedon Figure6. A slightly ellipsoidalbubbleis setin a liquid initially


0.000Ë 0.003Ë 0.005Ë 0.007Ë 0.010ËTime (s)Ì−0.04

−0.02

0.00

0.02

0.04

0.06

Rel

ativ

e am

plitu

de o

f mod

e 2

Í

ProsperettiNormal modeNavier−Stokes

FIGURE 6. Temporalevolutionof thesecondmodeof deformationof a slightly ellipsoidalbubblesetin aliquid initially at rest.

at rest.Under the influenceof surfacetensionthe bubbleshapeoscillatesaroundits sphericalequilibriumposition.Theparametersareasfollows:equivalentradiusµ ��¶Î£ � � {FÏ m, surfacetensioncoefficient � �Ð� I �l¸ kg/s� , kinematicviscosity .<��¶V£ � � {�Ñ m� /s, +Ò� � �/�=� kg/m¹ .In order to limit the effect of confinementthe ratio betweenthe size of the domainand thebubblediameteris 240.Thediameterof thebubbleis about128grid pointsandthemultidomaintechniqueis used.

Thetemporalevolution of therelative amplitudeof thesecondmodeof deformationis illus-tratedtogetherwith two theoreticalsolutions.Thefirst oneis theclassicalnormalmodeanalysisof Lamb (1932).It supposesa stationaryregime of oscillationanddoesnot take into accounttransienteffectssuchasthediffusionof vorticity in theliquid. Thesecondsolutionis a numeri-cal inversionof aLaplacetransformobtainedby Prosperetti(1980)takinginto accounttransienteffects.Thissolutionis exactfor thisproblemin thelimit of avanishingamplitudeof oscillation.TheagreementbetweenthenumericalsimulationandProsperetti’s theoryis excellentwith arel-ativequadraticerrorof 0.4%for thefirst eightperiodsof oscillation.It is alsointerestingto notethat thedifferencebetweentheapproximatenormalmodesolutionandProsperetti’s solutionissignificant.This test further confirmsthat vorticity generationat the free surfaceis accuratelydescribedby our interpolationtechnique.

3. Comparisonwith experimentsIn orderto assesstheapplicabilityof our numericalmethodto realcases,we wantedto com-

pareour resultswith experimentalmeasurements.Lauterbornandcollaboratorshave developedaneleganttechniqueto generatehighly reproduciblebubblesnearsolid boundaries(Lauterborn& Bolle 1975; Lauterborn& Ohl 1997; Philipp & Lauterborn1998).The high speedphoto-graphicseriesof Figure7 illustrateoneof theseexperiments.A focusedshortlaserpulseis firedin waterneara solid wall, a gasbubbleis thenformedandexpands,eventuallyreachesa maxi-


mumradiusandthencollapsesviolently. A jet is formednearthepoint of minimumradiusandpenetratesthere-expandingbubble.

Themainproblemwe areconfrontedwith is thechoiceof initial conditionsfor thenumericalsimulation.In fact,neithertheinitial radius(or initial pressure),northeequilibriumradius(orgascontent)of thebubbleareknown. Wehavechosento useinitial conditionsgivenby thefit of theclassicalRayleigh–Plessetequation(Plesset& Prosperetti1977)to themeasuredtime-evolutionof theradius.Theexperimentaltimeevolutionhasbeenmeasureddirectlyusingadigital versionof thephotographicseriesandimageprocessingtechniques.

Thenumericalsimulationusesa 512£ 512grid. Themaximumbubbleradiusreaches50 gridpoints.Thebottomboundaryof thedomainis a free-slidingsolidwall (thenormalcomponentofthevelocity is zero,thetangentialstressis null) andtheright boundaryis theaxisof symmetry.On the top and left walls the pressureis constantand set to the ambientpressure(gravity isneglected).No constraintis imposedon the velocity. The dynamicviscosityfor water is

� � { ¹kg/ms,thesurfacetensioncoefficient for anair-waterinterface� �-� I �l¸/, kg/s� andthedensityof theliquid is 1000kg/m¹ . Theprocessis assumedto beadiabaticandthegaspressureis givenby (É�[( � ��Ó * Ó � � ½ where

Óis thevolumeof thebubble,

Ó � �-Ê I ¸�ÇÔ£ � � {�Õ m¹ , ( � � � �/�R�l¸�,Pa and ¾ �Ö¸/*/¶ . Thesurroundingfluid is initially at restandthe initial radiusof the bubbleis0.4mm.

SimulationresultsareshownonFigure7 with thesamespatiallayoutandinterframetime.Fig-ure8 illustratesthetemporalevolutionof theequivalentradius(definedas

�e×=Ó *Ê/Ø � wyx ¹ ) togetherwith theevolution measuredexperimentallyandgivenby theRayleigh–Plessetcalculation.Theagreementbetweentheexperimentandthenumericalresultis relatively goodwith initial jet for-mationsqualitatively andquantitatively similar. Both theexperimentandthesimulationclearlyshow the jet impactinganddeformingtheoppositesideof thebubble,while the jet is stretchedby the bubbleexpansion.The jet velocity is well reproducedby the numericalmodel, the tipof the deformedbubbletouchingthe wall at approximatelythe sametime. Anotherinterestingfeaturerevealedby thenumericalsimulationis the“splashing”effect of the jet impactwith theformationof anaxisymmetricrim expandingwith thebubble(Figure9).Thisobservationis verysimilarto resultsreportedby Blakeetal. (1997,1998,1999)usingaboundaryintegraltechnique.


FIGURE 7. Comparisonbetweena high-speedphotographicseriesof abubblecollapsingneara wall(Lauterborn& Ohl 1997)anda directnumericalsimulation.Samplingrateis 75 000framespersecond.


0.0 0.2 0.4 0.6 0.8Time (ms)

0.0

0.5

1.0

1.5

2.0

2.5

Equ

ival

ent r

adiu

s (m

m)

ÙNumerical simulationRayleigh−PlessetRayleigh−Plesset compressibleExperimental data

FIGURE 8. Temporalevolutionof theequivalentradiusof thebubbleasgivenby experimentalmeasurements,numericalsimulationandtheRayleigh–Plessetequation(with andwithout compressibility

terms).

FIGURE 9. Jetformationandimpact.


However, we canseebothon Figures7 and8 that the reboundof the bubbleis muchlargerin the numericalsimulationthanthat observed in the experiment.The Rayleigh–Plessetequa-tion givesa resultcomparableto thenumericalcalculation.Theadditionof a simplefirst-ordercompressibilitycorrection: µ+ rÛÚÚ � � ( � µ � � � �Y(�º � � (3.1)

to theRayleigh–Plessetequation(with r � � Ê=Ç � m/s, thevelocity of soundin water)doesnotchangetheresultsignificantly. It would beinterestingto seeif morecomplex modelsincludingthethermodynamicsof thephenomenon(Keller & Miksis 1980)canprovide betterpredictions.In any case,theseresultsconfirm the intuition that viscousdissipationis not the main sourceof dampingfor theselargebubblesandthat theeffectsof fluid compressibility, in particulartheemissionof acousticandshockwaves,needto betakeninto account.

4. Numerical study of the influenceof viscosityon jet formation and evolutionWhile interestingfor validationpurposes,thepreviousresult is of limited interestdueto the

small influenceof viscosityon the fastcollapseof relatively big bubbles.For this typeof sim-ulation,a boundaryintegral codewith a viscousboundarylayerapproximationwould probablygive goodresults(Plesset& Chapman1971;Boulton-Stone& Blake 1993).For this reason,wehave chosento focusour attentionon the influenceof viscosityon jet formationandevolutionfor moderateReynoldsnumbers.

In orderto studytheinfluenceof viscosity, wefirst needto findwhichcharacteristicparameterscontrol theproblem.We assumethat thepressurein thegasis describedby a polytropic law ofthetype ( � µ � �)( º � µ9�µ � ¹m½ � (4.1)

whereµ is theequivalentradiusof thebubble(radiusof asphericalbubbleof equalvolume), µ9�is theequilibriumradius,( º is theambientpressureand¾ isapolytropicexponent.Wecanchoseµ?� asa lengthscaleand Ü + µ �� *$( º asa time scale.TheRayleigh–Plessetequation(Plesset&Prosperetti1977)describesthe evolution of the radiusandcanbe written in non-dimensionalform as µ �?Ýµ � × ,ßÞµ �� Êl. � Þµ �µ � � µ { ¹m½� � � � , � �µ � � (4.2)

wherethe à denotesnon-dimensionalquantities.Thesetof characteristiccoefficientsis then. � � .µ �Aá +(�º � � � � �µ � (�º and ¾ � (4.3)

where� is thesurfacetensioncoefficient, . thekinematicviscosityand + thedensity. Weneedtoaddthecoefficientscharacteristicof theinitial conditions.Therelativeinitial radiusâ � µäã * µ �andthe relative initial distancefrom the boundaryå �¡æY* µTã , where æ is the distancefromthecenterof thebubbleto thesolidwall. Thuswehavefive independentparameters:â , å , ¾ , . � ,� � .

In what follows we presenta systematicparametricstudy. We neglectsurfacetensionin thisstudy( � � �ç� ), aswe focuson themain topic of our investigation:how viscositycansuppressjet formation.Moreover, surfacetensionis probablystill negligible for rathersmallbubbles.Theanalysisof therelevanceof surfacetensioncanbemadesimply by evaluatingthedimensionless� � . For instancea 5 micronair bubblehas � �Îè � I � � Ê . A morerefinedanalysisinvolving smalltimeandspacescalesoccurringin thejet formationis deferreduntil theconclusionof thispaper.


FIGURE 10. Exampleof thehierarchyof gridsusedto dealaccuratelyandefficiently with largevariationsin bubbleradius.

We assumethatthegasis diatomicandtheprocessadiabatic( ¾ �1¸=*�¶ ). A Reynoldsnumberis definedas µ ¦ � � *�. � .

In orderto resolve thesmallscaleswhich canoccurfor high compressionratioscorrectlywehave usedanadaptivehierarchyof grids.Eachgrid hasa fixednumberof points(128£ 64 in allthe resultsillustratedhere)but is half thesizeof its parent.A typical setupis shown on Figure10.Gridsareaddedor removedasthebubbleshrinksandgrows.A singleglobaltimestepis usedto advanceeachgrid in time (seePopinet(2000)for furtherdetails).This techniqueallows theuseof relatively large simulationdomainsin order to minimise the influenceof the boundaryconditions.

Figures11, 12 and13 illustrate the influenceof the Reynoldsnumberon jet formationandimpactvelocity. For high Reynoldsnumbers(Figure11) the initial jet velocity is high andtheimpactoccursshortlyafter jet formation.For low Reynoldsnumbers(Figure13), the initial jetvelocity is smallandthevelocity of reboundof theoppositewall of thebubbleis largeenoughto prevent any impact.For intermediateReynoldsnumbers(Figure12), the initial velocity ofthe jet is comparableto thevelocity of reboundof the oppositewall, the jet is stretchedby theexpansionof the bubble, the impact is delayedand the impact velocity is small. The criticalReynoldsnumberµ ¦Bé is definedastheReynoldsnumberfor which theimpactvelocity is zero.If theReynoldsnumberis smallerthan µ ¦Bé , thejet never impactstheothersideof thebubble.

This transitionis illustrateddifferentlyon Figures14 and15. The lower curves(thick lines)representthetimeevolutionof thepositionof thesouthpole(closerto thewall) for thedifferentReynoldsnumbersshown in thelegend.Thethin linesarethetimeevolutionof thenorthpoleandtheuppercurvesdescribetheevolution of thepositionof thepoint farthestaway from thesolidwall. Theupperandlowercurvesdescribetheglobaldynamicsof thebubble:collapseuntil

�Eê ,followedby a re-expansionandmotionof the centerof gravity toward the wall. The thin linesdescribethejet dynamics.Initially thereis no jet andthesecurvesareindistinguishablefrom theuppercurves.At thetime of curvatureinversionoccurringat thenorthpole (andcorrespondingto jet formation)thecurvesseparate(Figure15).Thejet continuesto penetrateinsidethebubble


FIGURE 11. Time evolutionof a bubblecollapsingneara wall at highReynoldsnumber. Thewall is alongthebottomsideof thebox.Not all thesimulationdomainis shown. Interframetimeis 0.243.ëCìDí�î#ï�ð ï'ñ'ò ,ó í�î�ð ñî�ô , õÛíßîBð òî'ö .

FIGURE 12. Timeevolution of a bubblecollapsingneara wall at intermediateReynoldsnumber.Interframetime is 0.303. ëCìDí�î�÷=ð öBøMï , ó í�î�ð ñ'î#ô , õVí�î�ð òî#ö .

andeventuallyhits theotherside.If viscosityis toohigh thejet is sloweddown andnevermakesit to thesouthpole.

As canbeseenonFigure12,when µ ¦ is closeto µ ¦ é thejet becomesverythin andtheimpactoccurslate in the cycle of oscillation. If we want to find the valueof µ ¦ é with a reasonableaccuracy, weneedto useaveryfinegrid in orderto modelcorrectlytheflow insidethejet. Suchhigh resolutionswould beimpracticalfor a parametricstudyof µ ¦Bé � â � å � . We thereforesoughtanalternativeandmoreeasilycomputabledefinitionof µ ¦Bé .


FIGURE 13. Time evolution of a bubblecollapsingneara wall at low Reynoldsnumber. ëHìdí°ø$ù/ð ÷=ø$ô ,ó í�î�ð ñî�ô , õúíßîBð òî'ö .Figure16 illustratesthe time evolution of the relative velocity betweenthe top andbottom

polesfor variousvaluesof µ ¦ . The relative velocity is chosento be positive during collapseandnegative duringexpansion.Theblackdotsindicatethe instantof impactandthewhite dotsindicatethe instantof jet formation(curvaturechangessign at the top pole).As the Reynoldsnumberdecreasesthe impact is delayedand the impact velocity decreases.For valuesof µ ¦closeto µ ¦ é , thecurvehastwo extrema:a maximumrelativevelocity is reachedneartheendofthecollapseandaminimumrelativevelocityoccursnot longafterjet formation.Moreover, nearthecritical Reynoldsnumber, the relative velocity at impactis seento becloseto its minimumvalue.Thereforewe take thevalueof µ ¦ for which theminimumrelative velocity is zeroasanalternative definitionof µ ¦Bé (i.e. at somepoint closeto thebeginningof jet formationthetip ofthejet is moving exactlyasfastasthebubbleis re-expanding).

Figure17 confirmsthat both definitionsgive closevaluesfor µ ¦ é (17.15and17.77for theimpactandtheminimumrelativevelocityrespectively).Moreover, theimpactvelocityis stronglydependenton theReynoldsnumbernearthecritical value.This new criterion is mucheasiertotestnumericallysincewe only needthevalueof theminimumvelocity anddo not needto solvethejet impactdirectly.

The curveson Figure18 have beenobtainedfor variousvaluesof â (shown in the legend).Eachvalueof µ ¦ é is found usinga bisectiontechnique.On average,five simulationsarenec-essaryto find a valueof µ ¦é with a 3% accuracy (eachsimulationtakesapproximatelyfifteenminuteson a PC).As thevalueof å increases,it becomesmoredifficult to form a jet andvis-


0û

1ü 2ý

3þ

4ÿ

Time�1

2

3

4

5

Pos

ition

16.9317.6718.2419.3821.66ý34.21þ

FIGURE 14. Time evolution of thepositionof thepolesasa functionof theReynoldsnumberindicatedinthelegend.ó í°ø'ð �'öz÷ , õÛí°ø#ð ö .

couseffectsplay a moreimportantrole: thevalueof µ ¦Bé increases.On theotherhandwhen âincreases,thecollapsevelocity is largerandviscouseffectstendto besmaller:thevalueof µ ¦Bédecreases.

Thecurvesof Figure18 canberescaledasshown on Figure19. Theblackcurveson Figure18 aretherescaledversionsof theinterpolatingcurve of Figure19.Theagreementis excellent,andthediscrepanciesobservedin particularfor high valuesof å for â ��, I �=, × and â � � I � , �canbeattributedto theformationof thin jetswhich aredifficult to resolveaccurately.

We have obtainedthis scalingempirically by measuringnumericallythe correlation � �� ³ �betweenthetransformedpointsets,wherethetransformationis: å��å * â� andµ ¦ é ��µ ¦ é * â .The correlationis computedas follows. Given a setof points

� â K � å K � µ ¦ K � , for every pair ofcoefficients

�� ³ � thefollowing operationsareperformed:(a) thecoordinates

�g§FK � � K � areobtainedas§ K� å Kâ �K and � K�� µ ¦ Kâ K I(b) thesecoordinatesarenormalisedas§�K � §FK � §�� and � K � � K � ��

where§, � arethemeanvaluesand � � , � � thestandarddeviations.

(c) A polynomial ( �i§ � of ordereight is interpolatedthroughthesepointsthroughminimisa-tion of

ª �� ( �i§�K � �� K � � .(d) Thecorrelationis thendefinedas � �� ³ � �� ª�� .As illustratedon Figure20, themaximumcorrelationis obtainedfor valuesof

�and ³ close

to 3 and1/2 respectively.


1.8 1.9 2.0 2.1 2.2Time

2.64

2.84

3.04

3.24

3.44

Pos

ition

16.9317.6718.2419.3821.66!34.21"

FIGURE 15. Detail of Figure14.

1.5 2#

2.5 3$

Time%−5

0

5

10

15

Rel

ativ

e ve

loci

ty

18.41324.519#29.906#52.892&ImpactJet formation'

FIGURE 16. Relativevelocitybetweenthetopandbottompoleof acollapsingbubbleasa functionof timefor variousvaluesof theReynoldsnumber(indicatedin thelegend).Thewhitesquaresandblackdotsmarkthetime of jet formation(curvaturechangessign)andjet impactrespectively. ó í�îBð ñ'î#ô , õVí�î�ð òî#ö .


15( 25( 35( 45( 55(Reynolds

−1

0

1

2

3

4

5

6

Rel

ativ

e ve

loci

ty

ImpactMinimum

FIGURE 17. Relative impactvelocity andminimumof the relative velocity asa functionof theReynoldsnumber. Taking ascritical Reynoldsnumberthe valueof the Reynolds for which the minimum relativevelocity is zerogivesa goodapproximationof the Reynoldsnumberfor which the jet impactvelocity iszero. ó í°ø'ð �'öz÷ , õVí°ø#ð ö .

1.0 1.5 2.0 2.5 3.0 3.50

25

50

75

100

125

1.5371.6031.6701.7541.8371.9212.023)2.505)2.672)3.340*

PSfragreplacements

+ , -

.FIGURE 18. Valueof thecritical ReynoldsnumberëHì0/ asa functionof therelative distanceto thesolid

wall õ . Eachcurve correspondsto a differentvalueof therelative initial radius ó (indicatedin thelegend).


0.0 0.1 0.2 0.3 0.4 0.50

20

40

60

801.5371.6031.6701.7541.8371.9212.02312.50512.67213.3402interpolation3 Impact

No impact

PSfragreplacements

4 5 678 9

õ;: ó;<FIGURE 19. Rescaledversionof Figure18.Theblackline is aninterpolatingpolynomial.


2.7

0.8

1.2

1.6

2.0

0.0

0.4

3.3

−0.2

5

−0.5

−0.5

−1.0

−1.5

−5

5

4

3

2

1

0

−1

−2

−3

−4

−5−4 −3 −2 −1 0 1 2 3 4 5

−0.2

5

−0.

25

−0.25

−0.25

−2.5

−2.0

−1.5

−2.5

−2.0

−2.9

5

3.02.8 2.9 3.1 3.2

PSfragreplacements

=

=

>

>

FIGURE 20. Numericalevaluationof thecorrelation?A@ =CBEDGF betweentherescaledpoint setsof Figure18.Thelower plot is anenlargedview of a portionof theupperplot.


0.0 0.1 0.2 0.3 0.4 0.5 0.60

20

40

60

801.5371.6031.6701.7541.8371.9212.023H2.505H2.672H3.340I

PSfragreplacements

4 5 678 9

õJ: óAKMLONEPFIGURE 21. Rescaledversionof Figure18 for thetheoreticalexponent= í�î�øQ:#ù .


5. TheoryIn whatfollowswedescribeatheorybasedontheapproximationof arelatively largeReynolds

number, a largeinitial bubbleradius â anda largedistanceto thewall å . Thusin our estimateswe first neglecttheviscouseffects.Thenwe discussstability andviscouscorrections.

5.1. Velocitycondition

Theasymmetricalbubblecollapsethatheraldsjet formationis influencedby thepresenceof thewall. While theapproximatelysphericalbubbleshrinksin size,its centerof massis attractedto-wardsthewall by theimagebubble.As thebubbleapproachesthewall, amomentumR is gainedmostlyby theliquid phasesurroundingit, theso-calledaddedmassmomentum.In equationsÞR �1� Ê×!Ø µ ¹ � (��D� (5.1)

wherewe have usedthe dimensionlessvariablesas in (4.2) but whereas in what follows wedroppedthe à subscriptsfor simplicity. Theadded-massmomentummaybeexpressedin termsof theverticalpositionof thebubble

�GS��g� � as

R � Q ã Ê × Ø µ ¹ Þ�GS � (5.2)

where Q ã is theaddedmasscoefficient,approximatelyQ ã � � *�, at high Reynoldsnumbers.To estimateconditionsfor jet formation and impact,we distinguishtwo effects: the stabilityor instability of thenear-sphericalbubblemotion,andthe possiblemismatchbetweenthe timescaleof the reboundandthe time scaleof the jet traversingthe bubble.Thefirst conditionis ajet formationcondition,while thesecondis asufficient velocitycondition.Webegin with thejetimpact.

In what follows, we first investigatethe sphericalcollapse,thenthe effect of the imagebub-ble. We neglectviscosity. As in thenumerics,surfacetensionis left aside.ThenEquation(4.2)becomes µ Ýµ × , Þµ � � µ { ¹m½ � � I (5.3)

This equationintegratesto UG� � µ ¹ Þµ � ,× ¾ � × µ { ¹m½ u ¹ ,× µ ¹ I (5.4)

Wehave µ � µUT atminimumradiusand µ � â atmaximumradius.Assumingâ�V �wehaveUG� � �¹ âA¹ and µUT èXW � ¾ � � �ZY[ â�\ YY^]`_ I (5.5)

This formula is a goodapproximationfor high µ ¦ asconfirmedby Figure22 wherewe haveplottedtheratioof themaximumto minimumradiusâ * µ T for differentvaluesof â and µ ¦ . Allthesimulationswehaveperformedarerepresentedwhichalsoshowsthesmallinfluenceof å onthecompressionratio.

In whatfollowsweomit ¾ -dependentprefactors.A full solutionmaybefoundfor theinviscidmotion in theform of an integral, but we shallonly needsomebasicasymptoticfeaturesof thesolutionat large â . We have a time scaleduringwhich theradiusremainscloseto µ T andthetypical accelerationis ÝµUT . From(5.3) Ýµ T è µ { ¹m½ {�wT (5.6)

andthus � T è � µaT * ÝµaT � wyx � è âcbed [ _b ] b _ � ÝµaT è â [ _ d Y_f]ZY I (5.7)


0g

20g

40g

60g

80g

100g

120h

10

5

15

20

30

40

2

3

4

6789

1.5371.6031.6701.7541.8371.9212.023h2.505h2.672h3.340i

PSfragreplacements ëCì

974fj

FIGURE 22. Ratio betweenthe maximum(initial) radius ó andthe equivalentminimum radius ëlk asafunctionof theReynoldsnumberandfor thedifferentvaluesof ó indicatedin thelegend.Theoreticalvaluesin theinviscidcasegivenby equation(5.5)arerepresentedby thedashedlines.

We setthereferencetime� �1� at thepoint of minimumradius.Thereis thewell-known outer

solutionfor m � m è � of theform µ è m � m � x » â ¹ x » � (5.8)

andaninnersolutionfor m � mAnè � T . We arenow in positionto integratetheadded-massequation(5.1).Thepressuregradientmaybeestimatedfrom thepressurefield associatedwith theimagebubble.FromBernouilli’sequationthedominanttermis, in dimensionlessunits(Y� � �, Þµ � µ� � �, Þµ � µ Ï� Ï I

(5.9)

At large å andat distanceæ (still in dimensionlessunits)from thewall theleadingorderis

m o ( m è Þµ � µæ � I (5.10)

Thus

R è 4 �{;p Þµ � µ Ïæ � 7 �MI (5.11)

Using the innerandouterscalings,we find that the leadingordercontribution comesfrom theoutersolution(5.8),thus,sinceå ��æY* â weget

R è â ¹ å { � I (5.12)

The jet forms at time � nearminimum radius.(This is obviously an approximation,ascanbeseenfrom numericalresults,Fig. 15. Varying viscosity may advanceor delay the time of jetformation)A kind of equipartitionprinciple leadsus to assumethat it getshalf theaddedmass


momentum,on aspatialscaleof theorderof µUT . Thusthejet velocityÓrq

is givenbyÓ�q è R * µ ¹T (5.13)

andthus Ó qts Q q â [ __f]ZY å { � I (5.14)

We have measuredthe jet velocity directly from the data:the results,althoughrelatively noisyareconsistentwith the above scalingfor µ ¦vu � �/� with a prefactor Q qws � I �l¶ . We cannowstateour first conditionfor theimpactof thejet: thetime thatthejet takesto traversethebubblemustbeshorterthanthesmall time scale

� T . For if it wereotherwise,thesouthandnorthpoleof thebubblewould separateever faster, while the jet, dueto viscouseffect,would slow down.In equations,thecritical velocity is of theorderÓ�q è µUT * � T � (5.15)

which leadsto å � â ] [b __f]ZY è � I(5.16)

A ¾ -dependentnumericalconstantis boundto appearon the right hand-side.For ¾ �¡¸/*/¶ wethushave å&â { � wmxMx � Qd� I (5.17)

In a graphhaving å&â { � wmxMx on thehorizontalaxisand µ ¦ * â wmx � on theverticalaxis,theinvisciddynamicsyield a verticalline, to theright of which thereis no jet impact.

5.2. Viscouseffects

Whathappenswhenasmallviscosityis added?Ononehandmostof our estimatesinvolveonlythe dynamicsof a sphericalbubble,without the imagebubbleforcing. All theseestimatesareaffectedby correctionsof theform .zy � â � were y � â � is somefunctionthatdependson â but noton å . Theexactexpressionsarecomplex sincethey involve theintegralsof theinviscidmotion.On theotherhandtheadded-massmomentumcomputedin equation(5.1) remainsproportionalto å { � . In thatequationthepressuregradientcouldnow involveadditionalviscousterms.Theseinvolve the pressurefield of the imagebubble and could thus dependon the distanceto thewall å , introducingcorrectionsof the form .zy w�� å � . We claim thereare no suchcorrectionsfor the following reason:thefluid velocity aroundthebubble,imposedby incompressibility, is;\� ÞµTµ � * �B� onwhichtheviscousterm . o � ; vanishes(viscositycomesaboutin theRayleigh–Plessetequationonly becauseof surfaceterms).Thus the pressurefield createdby the imagebubbleremainsthe onecomputedexceptfor boundarylayersnearthe wall. Keepingnow ¾ �¸/*/¶ , all the .zy � â � correctionsamounttoå&â { � wyx{x � Q � y � â �µ ¦ }| � �µ ¦ � � I (5.18)

Thusthe impactcondition,dependenton threevariablesâ � å and µ ¦ may be collapsedontoasinglegraphin thevariables

§ � å&â { � wmxMx and �Ô� µ ¦ *~y � â � . In thatgraphtheimpactconditionasymptotesto thevertical inviscid condition,in a mannerconsistentwith our numericalfinding(Figure23). In thenumericaldata, y � â � s â wyx � providesa goodfit (althoughthereis someun-certaintyontheexponent,seeFigure20).At thistimewehavenotfoundaconvincingtheoreticalargumentyielding y � â � . It is possiblethat y � â � combinesseveraleffectsthatyield aneffectivescalinglaw in therangeof â considered.We discusstheseissuesbelow.

5.3. Jet formationcondition

Thejet formationarisesthroughaninstabilityof theRayleigh–Taylor type:asthebubblewall isacceleratedtowardstheheavier, liquid phase,it becomesinstableto deviationsfrom sphericity.


velocity

β/α 21/8

Re/ α)f(

ΙΙΙ

ΙΙΙ IIIa b

IIIc

insufficient velocity

unstable (jet forms)

stable (no jet)

sufficient

PSfragreplacements

FIGURE 23. A graphicalsummaryof our scalingtheory. Curve I is theasymptoticlimit at smallviscosity,separatinga region wherethe velocity of the jet is too small for impact from a region of large enoughvelocity. With viscosity, theseparationappearsascurve II. Jetformationis possibleif thereis instability,it occursabove curvesIIIa,b,c.However thesecurvesarenot scalingwith õ , thuseachõ yieldsa differentcurve.Fromournumericalresults,thereis goodreasonto assumethatall curvesof typeIII arebelow curveII, sothestability conditionis not relevant.

Usually the RayleighTaylor instability will be presentfor a broadbandof sphericalharmonicmodesof which thehigh orderonesarethemostunstable.In viscousflow howevermostof themodeswill bedamped.

Viscositymaypreventthat instability providedthat the time scalefor viscousdiffusion� � �µ �T *�. is smallerthanthetime scaleassociatedwith theRayleigh–Taylor instability for a mode

of lengthscaleµaT . For planewavesof wave number � thegrowth rate n is n � �� wmx � . Herethe mostdangerousmodehasapproximatelya wavenumber� è � * µaT andthe equivalentofgravity is the acceleration ÝµUT so the time scale

� * n of the instability is the short time scaledefinedabove:

� * n � � T . Theinstability is thusmarginalwhen� T è � � .

Equatingthe viscousandinstability time scales� T and

� �yields a critical conditionfor jet

formation: µ ¦ è â [ _f] bb ] b _ I (5.19)

In the diatomiccase¾ � ¸/*/¶ this yields µ ¦ � â {�wmwmxmÏ . Notice that this scalingdoesnot in-volve å (theinfluenceof å on theexponentialgrowth phaseof theinstability yieldslogarithmiccorrections.).Thusin thevariablesof our scalingdiagram,retainingthetheoreticalvalueof theprevioussectionfor the exponentof

§,§ � å&â { � wmxMx and �<� �A�p YO� b we have �<� �g§ * å � �"� x � w .

This is almostastraightline andmayhavesomeconnectionwith thelowerpartof ournumerical-experimentalcurve.

5.4. Discussion

We find two conditionsto observe jet impact,oneof sufficientvelocityandoneof stability.


Thecritical linesfor thetwo conditionsareshown on Figure23.As å decreases,thestabilityconditionmovesup.However, ournumericsshow thatit doesnot intersectthesufficientvelocitycondition:nearthecritical µ ¦ , the jet alwaysformsandwhat is relevant is the time it needstoreachtheothersideof thebubble.Thusthesufficient velocityconditionis theonly relevantone.However, with surfacetensionadded,it is possiblethatthejet formationconditioncouldbecomerelevant.

An interestingconsequenceof ourtheoryis thatit runscounterto aconventionalpointof view.It is generallyconsideredthatthemoresphericalthebubble,thelaterthejetdevelopsandthusthestrongerjet impactis. Nearawall å controlsbubblesphericity:thefurtherthebubbleis from thewall, themoresphericalit remainsat leastinitially. Thusthejet velocity shouldincreasewith å(Brennen1995;Blake& Gibson1987).Thisis notconsistentwith ourasymptoticscaling,whereat large å thejet velocity is small(Equation(5.14)).It seemsthattheconventionalpointof viewis basedon relatively smallvaluesof å for which jetsform very early. Thenimpactmayhappenwaybeforeµ reachesµaT Kv| whichwould reducethejet velocityat thetimeof impactcomparedto the theoreticaljet velocity at µ � µaT Kv| . Indeedrecentnumericalresultsby Blake & Keen(2000)show that jet velocity first increases,thendecreasesasthebubbleis locatedfurtherfromthewall.

Our theoryis only partially in agreementwith thenumericalexperiments.Theexponent, � *BÇthatwe find in our theorydiffers from the exponentthatbestcollapsesthe datain figure18. Ifweusethevariable

§ � å&â { � wyx{x insteadof å&â { ¹ weobtainfigure21.Thecollapseis markedlyworse.Choosingy � â � to be somethingotherthana power law couldperhapsimprove thecol-lapsein view of theabovetheory. We leavehowever this andsimilarattemptsto furtherstudy.

We alsopoint out that the theoryis asymptoticallyvalid for large â , å and µ ¦ only. In ourparametricstudy, â and å reachat most

×. Thus it is unlikely that the theory will be very

accurate,and the relatively poor agreementof Figure21 is not surprising.What is in a sensesurprisingis the very goodagreementobtainedwith the empiricalexponentin Figure19. Aninviscid numericalstudymayshedfurther light on this issue,asthe valueof the exponent

�is

determinedby inviscideffectsalone.

6. ConclusionsWehavepresentedanoriginalnumericaltechniqueto solveaccuratelytheNavier-Stokesequa-

tionswith free-surfaces.Thismethodis notlimited to simplegeometriesor smallinterfacedefor-mations.Particularemphasishasbeenput on theaccuratedescriptionof free-surfaceboundaryconditions.Validationtestshave shown anexcellentagreementwith theRayleigh-Plessetequa-tion anda theoreticalsolutionobtainedby Prosperettifor thesmallamplitudeshapeoscillationof anellipsoidalbubble.

Direct comparisonsbetweenhigh-speedphotographicseriesand numericalsimulationsofbubblecollapseneara solidboundaryhaveshown a goodqualitativeandquantitativeagreementwhile giving accessto thedetailsof theprocess.However, thesimulationsalsoshow thatviscousdissipationalonecannotexplain thestrongdampingof radialoscillationsobservedin theexper-iments.Acousticandthermaldissipation—nottaken into accountin the presentcode—shouldbe includedin orderto capturecorrectlythedynamicsof the processafter the first rebound.Asimple solution would be to usea more sophisticatedmodel equationfor the pressurein thebubble.

A detailedparametricstudyof the influenceof viscosityhasdemonstratedthe existenceofa critical valueof the Reynoldsnumberbelow which jet impact is no longerpossible.A sim-ple scalinglaw is shown to relatethe valueof this critical Reynoldsnumberto two othernon-dimensionalparameterscontrolling theproblem:the relative stand-off distanceandtherelativeinitial radiusof thebubble.


We havepresentedasimpletheorywhich describescorrectlytheoverall characteristicsof thephasediagramwe obtain.In particular, we demonstratetheexistenceof a verticalasymptoteinthe parameterspaceof the rescalednon-dimensionalcontrol parameters.This providesa sim-ple upper-boundfor thedomainin which jet impactis possible,independentlyof theReynoldsnumber.

A numberof further studieswould be possibleand useful. In order to reducethe numberof free parametersin the problem,we have neglectedsurfacetensionand chosena constantpolytropicexponentfor thegaslaw. It is interestingto discussthevalidity of thisapproximation.Thesimplestanalysisof theeffect of surfacetensioninvolvesthedimensionlessratio � � (goingback to our first notations).The number � � comparesthe capillary time scaleto the naturaloscillationperiodof the bubble.A still simple but moresubtleanalysisis to computea timescalerelatedto surfacetensionnear µaT . If this timescaleis longerthan

� T thensurfacetensionis negligible with respectto the only time scaleappearingin the analysisof stability and jetvelocity. Therelevantdimensionlessnumberis,¨ � � � � �Tµ ¹T I

(6.1)

Using the above estimates,for ¾ � ¸/*/¶ , ¨ � � � â {�wyxmÏ � W � * � µ9� ( º � \ � µ ã * µ9� � {!wmxyÏ . It isinterestingto applythis to thestrongcompressionratiosin sonoluminescentair bubbles,whichareverysmall.For a 5 micronair bubble,with â � � � , ¨ è � I �=�/Ç and � � è � I � � Ê . For typicalexperimentson bubblecollapsewith larger(1 mm) bubbles,thesenumbersareevensmaller. Sofor several practicalapplicationsneglectingsurfacetensionin the analysisof jet formation isjustified.

On the otherhand,a parametricstudyof the influenceof the polytropic exponentwould al-low us to confirm the generalityof the scalinglaws we have found numericallyandpredictedtheoretically. Moreover, while adequatefor describingthegeneralpicture,thesimpletheoryweproposeis not ableto explain the fact that the scalingwe find numericallyis clearly valid notonly in theinviscid limit but acrossthewholerangeof Reynoldsnumberswe investigated.Thisremainsanopenquestion.

From a more practicalpoint of view, it would be interestingto investigatehow our phasediagramfor jet impactinfluencesour understandingof cavitation damagefor real distributionsof bubblesizesin experimentsof hydrodynamiccavitation. If a majority of cavitation bubblesfall in the zoneof the phasediagramwhereno jet impact is possiblethencavitation damagewould mostprobablybedueonly to theoverpressurecausedby thebubblecollapse.

REFERENCES

BENJAMIN, T. B. & ELL IS, A. T. 1966Thecollapseof cavitation bubblesandthepressuretherebypro-ducedagainstsolidboundaries.Phil. Trans.RoyalSoc.LondonA 260, 221–240.

BLAKE, J. R. & GIBSON, D. C. 1987Cavitation bubblesnearboundaries.Annu.Rev. Fluid Mech. 19,99–123.

BLAKE, J. R., HOOTON, M. C., ROBINSON, P. B. & TONG, R. P. 1997Collapsingcavities, toroidalbubblesandjet impact.Phil. Trans.R.Soc.Lond.A 355(1724),537–550.

BLAKE, J. R. & KEEN, G. S. 2000Fluid mechanicsof singlecavitation bubbleluminescence.In IUTAMSymposiumon freesurfaceflows, pp.55–62.London:Kluwer AcademicPublishers.

BLAKE, J. R., KEEN, G. S., TONG, R. P. & WILSON, M. 1999Acousticcavitation: thefluid dynamicsof non-sphericalbubbles.Phil. Trans.Roy. Soc.A357, 251–267.

BLAKE, J. R., ROBINSON, P. B., SHIMA , A. & TOMITA , Y. 1993Interactionof two cavitation bubbleswith a rigid boundary. J. Fluid Mech. 255, 707–721.

BLAKE, J. R., TOMITA , Y. & TONG, R. P. 1998The art, craft andscienceof modellingjet impactin acollapsingcavitationbubble.Appl.Sci.Res.58, 77–90.


BLANCO, A. & MAGNAUDET, J. 1995 The structureof the axisymmetrichigh-reynolds numberflowaroundanellipsoidalbubbleof fixedshape.Phys.Fluids 7 (6), 1265–1274.

BOULTON-STONE, J. & BLAKE, J. R. 1993Gasbubblesburstingat a free surface.J. Fluid Mech. 254,437–466.

BRANDT, A. 1982Guideto multigrid development. Berlin: Multigrid Methods,SpringerVerlag.BRENNEN, C. E. 1995CavitationandBubbleDynamics. Oxford UniversityPress.BRIGGS, W. L. 1987A multigrid tutorial. SIAM Philadelphia.BROWN, D. L ., CORTEZ, R. & M INION, M. L. 2001Accurateprojectionmethodsfor theincompressible

Navier-Stokesequations.J. Comput.Phys.168, 464–499.CHAN, R. K. C. & STREET, R. L . 1970A computerstudyof finite-amplitudewaterwaves.J. Comput.

Phys.6, 68–94.GUEYFFIER, D., NADIM , A., L I , J., SCARDOVELLI , R. & ZALESKI , S. 1998Volumeof fluid interface

trackingwith smoothedsurfacestressmethodsfor three-dimensionalflows. J. Comput.Phys.152,423–456.

HARLOW, F. H. & WELCH, J. E. 1965Numericalcalculationof time-dependentviscousincompressibleflow of fluid with freesurface.Phys.Fluids 8, 2182–2189.

HIRT, C. W. & NICHOLS, B. D. 1981Volumeof fluid VOF methodfor thedynamicsof freeboundaries.J. Comput.Phys.39, 201–225.

KELLER, J. B. & M IKSIS, M. 1980Bubbleoscillationsof largeamplitude.J. Acoust.Soc.Am.68 (2).KORNFELD, M. & SUVOROV, L . 1944On thedestructive actionof cavitation.J. Appl.Phys.15, 495–506.LAMB, H. 1932Hydrodynamics, 6thedn.CambridgeUniversityPress.LAUTERBORN, W. & BOLLE, H. 1975Experimentalinvestigationsof cavitation-bubblecollapsein the

neighbourhoodof a solidboundary. J. Fluid Mech. 72, 391–399.LAUTERBORN, W. & OHL , C. D. 1997Cavitationbubbledynamics.UltrasonicsSonochemistry4, 65–75.LEGENDRE, D. 1996Quelquesaspectsdesforceshydrodynamiqueset destransfertsde chaleursur une

bulle spherique.PhDthesis,InstitutNationalPolytechniquedeToulouse,France.NAUDE, C. F. & ELL IS, A. T. 1961Onthemechanismof cavitationdamageby non-hemisphericalcavities

collapsingin contactwith a solidboundary. Trans.ASMED: BasicEng. 83, 648–656.NICHOLS, B. D. & HIRT, C. W. 1971 Improved free surface boundary conditions for numerical

incompressible-flow calculations.J. Comput.Phys.8, 434–448.PEYRET, R. & TAYLOR, T. D. 1983ComputationalMethodsfor Fluid Flow. New York/Berlin: Springer

Verlag.PHIL IPP, A. & LAUTERBORN, W. 1998Cavitationerosionby singlelaser-producedbubbles.J. Fluid Mech.

361, 75–116.PLESSET, M. S. & CHAPMAN, R. B. 1971Collapseof an initially sphericalvapourcavity in theneigh-

bourhoodof a solidboundary. J. Fluid. Mech. 47, 283–290.PLESSET, M. S. & PROSPERETTI , A. 1977Bubbledynamicsandcavitation. Annu.Rev. Fluid Mech. 9,

145–185.PO-WEN, Y., CECCIO, S. L . & TRYGGVASON, G. 1995Thecollapseof acavitationbubblein shearflows

– a numericalstudy. Phys.Fluids7, 2608–2616.POPINET, S. 2000Stabiliteet formationdejetsdanslesbullescavitantes: Developpementd’unemethode

dechaınedemarqueursadapteeautraitementnumeriquedesequationsdeNavier-Stokesavecsurfaceslibres.PhDthesis,Universite PierreetMarie Curie,Paris,France.

POPINET, S. & ZALESKI , S. 1999A front trackingalgorithmfor the accuraterepresentationof surfacetension.Int. J. Numer. Meth.Fluids 30, 775–793.

PROSPERETTI , A. 1980Freeoscillationsof dropsandbubbles:the initial-valueproblem.J. Fluid Mech.100, 333–347.

RATTRAY, M. 1951Perturbationeffectsin bubbledynamics.Technicalreport.CaliforniaInstituteof Tech-nology.

RAYLEIGH, L . 1917On thepressuredevelopedin a liquid during thecollapseof a sphericalcavity. Phil.Mag. 34, 94–98.

TOMITA , Y. & SHIMA , A. 1986Mechanismsof impulsive pressuregenerationanddamagepit formationby bubblecollapse.J. Fluid. Mech. 169, 535–564.

WESSELING, J. 1992An introductionto Multigrid Methods. Chichester:Wiley.

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