Measurement of Lagrangian velocity in fully developed turbulence

We have developed a new experimental technique to measure the Lagrangian velocity of tracer particles in a turbulent flow, based on ultrasonic Doppler tracking. This method yields a direct access to the velocity of a single particule at a turbulent Reynold

MeasurementofLagrangianvelocityinfullydevelopedturbulence

N.Mordant(1),P.Metz(1),O.Michel(2),J.-F.Pinton(1)

CNRS&LaboratoiredePhysique,´EcoleNormaleSup´erieure,

46all´eed’Italie,F-69007Lyon,FranceLaboratoired’Astrophysique,Universit´edeNice

ParcValrose,F-06108,Nice,France

(1)

(2)

arXiv:physics/0103084v2 [physics.flu-dyn] 30 Jul 2001

WehavedevelopedanewexperimentaltechniquetomeasuretheLagrangianvelocityoftracerparticlesinaturbulent ow,basedonultrasonicDopplertracking.ThismethodyieldsadirectaccesstothevelocityofasingleparticleataturbulentReynoldsnumberRλ=740.Itsdynamicsisanalyzedwithtwodecadesoftimeresolution,belowtheLagrangiancorrelationtime.Weobserve

2

thattheLagrangianvelocityspectrumhasaLorentzianformEL(ω)=u2rmsTL/(1+(TLω)),inagreementwithaKolmogorov-likescalingintheinertialrange.Theprobabilitydensityfunction(PDF)ofthevelocitytimeincrementsdisplaysachangeofshapefromquasi-Gaussianaintegraltimescaletostretchedexponentialtailsatthesmallesttimeincrements.Thisintermittency,whenmeasuredfromrelativescalingexponentsofstructurefunctions,ismorepronouncedthanintheEulerianframework.

PACSnumbers:47.27.Gs,43.58.+z,02.50.Fz

Lagrangiancharacteristicsof uidmotionareoffun-damentalimportanceintheunderstandingoftransportandmixing.Itisanaturalapproachforreacting owsorpollutantcontaminationproblemstoanalyzethemotionofindividual uidparticles[1].Anothercharacteristicofmixing owsistheirhighdegreeofturbulence.Forprac-ticalreasons,mostoftheexperimentalworkconcerninghighReynoldsnumber grangianmeasurementsarechal-lengingbecausetheyinvolvethetrackingofparticletra-jectories:enoughtimeresolution,bothatsmallandlargescales,isrequiredtodescribetheturbulent uctuations.EarlyLagrangianinformationhavebeenextractedfromthedispersionofparticles,followingTaylor’sap-proach.Recentlynumericalandexperimentalstudieshavefocusedonresolvingthemotionofindividual uidortracerparticles.Theemergingpictureisasfollows.Theone-componentvelocityauto-correlationfunctionisquasi-exponentialwithacharacteristictimeoftheorderoftheenergyinjectionscale[2,3,4].ThevelocitypowerspectrumisexpectedtohaveascalingEL(ω)∝ω 2,asrecentlyreported[5,6]andexpectedfromaKolmogorovsimilarityarguments.Inthesamespirit,thesecondor-L

derstructurefunctionshouldscaleasD2(τ)=C0 τ,where isthethepowerdissipation.Measurementsofatmosphericballoons[7]havegivenC0=4±2andalimitC0→7hasbeensuggestedinstochasticmodels[8].Recentexperiments[9,usinghighspeedopticaltech-niqueshaveshownthatthestatisticsoftheLagrangianaccelerationarestronglynon-Gaussian.

Wehavedevelopedanewexperimentalmethod,basedonsonartechniques[11],inordertostudyinalaboratoryexperimenttheLagrangianvelocityacrosstheinertialrangeoftimescales.Weobtainthe rstmeasurementofsingleparticlevelocityfortimesuptothe owlargescaleturnovertime,athighReynoldsnumber.InthisLetter,wereporttheresultsofthismeasurementsandcomparewithpreviousobservationsandnumericalpredictions.Ourtechniqueisbasedontheprincipleofacontinu-ousDopplersonar.Asmall(2mm×2mm)emittercon-tinuouslyinsoni esthe owwithapuresinewave,atfrequencyf0=2.5MHz(inwater).Themovingparticlebackscatterstheultrasoundtowardsanarrayofreceiv-ingtransducers,withaDopplerfrequencyshiftrelatedtothevelocityoftheparticle:2π f=q.v.Thescatter-ingwavevectorqisequaltothedi erencebetweentheincidentandscattereddirections.Anumericaldemod-ulationofthetimeevolutionoftheDopplershiftgivesthecomponentoftheparticlevelocityalongthescat-teringwavevectorq.Itisperformedusingahighresolu-tionparametricmethodwhichreliesonanApproximatedMaximumLikelihoodschemecoupledwithageneralizedKalman lterThestudyreportedhereismadewithasinglearrayoftransducerssothatonlyoneLagrangianvelocitycomponentismeasured.

Theturbulent owisproducedinthegapbetweentwocounter-rotatingdiscs[12].Thissetuphastheadvan-tagetogenerateastrongturbulenceinacompactregionofspace,withnomeanadvection.Inthisway,parti-clescanbetrackedduringtimescomparabletothelargeeddyturnovertime.DiscsofradiusR=9.5cmareusedtosetwaterintomotioninsideacylindricalvesselofheightH=18cm.Toensureinertialentrainment,thediscsare ttedwith8bladeswithheighthb=5mm.Inthemeasurementreportedhere,thepowerinputis =25W/kg.Itismeasuredontheexperimentcool-ingsystem,fromtheinjection-dissipationbalance.TheintegralReynoldsnumberisRe=R2 /ν=6.5104,where istherotationfrequencyofthediscs(7.2Hz),andν=10 6m2/sisthekinematicviscosityofwa-ter.AconventionalturbulentReynoldsnumbercanbecomputedfromthemeasuredrmsamplitudeofveloc-

We have developed a new experimental technique to measure the Lagrangian velocity of tracer particles in a turbulent flow, based on ultrasonic Doppler tracking. This method yields a direct access to the velocity of a single particule at a turbulent Reynold

ity uctuations(urms=0.98(λ=

m/s)andanestimateoftheTaylormicroscaleν/ =0.2ms)

issmaller,sothatwedonotexpecttoresolvethedis-sipativeregion.Thestatisticalquantitiesarecalculatedfrom3×106velocitydatapoints,takenatasamplingfrequencyequalto6500Hz.Theacousticmeasurementzoneisincentralregionofthe ow,10cmthickintheaxialdirectionandalmostspanningthecylindercross-section.Inthisregionthe owisagoodapproximationtoisotropicandhomogeneousconditions:atallpoints,themeanvelocityisnonzero,butequaltoaboutonetenthofitsrmsvalue.

We rstconsidertheLagrangianvelocityauto-correlationfunction:

RL(τ)=

v(t)v(t+τ) t

2

1+(T(2)

Lω)2

.WeobserveaclearrangeofpowerlawscalingEL(ω)∝ω 2.ThisisinagreementwithaKolmogorovK41pic-tureinwhichthespectraldensityatafrequencyωisadimensionalfunctionofωand :EL(ω)∝ ω 2.Toourknowledge,thisisthe rsttimethatitisdirectlyob-servedathighReynoldsnumberandinalaboratoryex-

periment,althoughithasbeenreportedinoceanicstud-ies[5]andinlowerReynoldsnumberdirectnumericalsimulations[6].DeparturefromtheKolmogorovbehav-iorisobservedatlowfrequenciesinagreementwiththeexponentialdecayoftheauto-correlation.Athighfre-quencies,thespectrumdeviatesfromtheLorentzianformduetotheparticleresponse.NoteinFig.1bthatthemeasurementismadeoveradynamicalrangeofabout60dB.

Wenowconsiderthesecondorderstructurefunction

We have developed a new experimental technique to measure the Lagrangian velocity of tracer particles in a turbulent flow, based on ultrasonic Doppler tracking. This method yields a direct access to the velocity of a single particule at a turbulent Reynold

ofthevelocityincrement

DL2(τ)= (v(t+τ) v(t))2 t= ( τv)2 .

(3)

Weemphasizethatthesearetimeincrements,andnot

spaceincrementsasintheEulerianstudies.Thepro leDLauto-correlation2(τ)isshownintheinsetofFig.2.byDtimesoneobservesthe2(τ)=2u2trivialscalingrms Itislinked1 RL

(τ) totheL:atsmall

DL

(τ)∝τ2andatlargetimesDL

2

2(τ)saturatesat2u2rms(asv(t)andv(t+τ)areuncorrelated).

)ετ(/L2

D10

10

t/τ10

η

FIG.2:SecondorderstructureDLfunction.Inset:pro le2(τ)asafunctionoftime,non-dimensionalizedTL.Inthemain gurethesecondorderstructurefunctionisnon-dimensionalizedbytheKolmogorovscaling τ.

Inbetweenthesetwolimits,oneexpectsaninertialrangeofscaleswithaKolmogorov-likescaling

DL2(τ)=C0 τ,

(4)

whereC0isa‘universal’constant.Suchabehavioriscon-sistentwithdimensionalanalysisandwithanω 2scal-ingrangeinthevelocitypowerspectrum.Fig.2shows

DL2(τ)/ τ;aplateauwithaconstantC0isnotobserved.NotethatthisalsothecaseinEulerianmeasurementswhenthethird

orderstructurefunctionisrepresentedinlinearcoordinates[13].Thefunctionreachesamaximumat20τη,forwhichC0～2.9.ThisvalueisinagreementwiththeestimationC0=4±2in[7]andintherangeofvalues(between3and7)usedinstochasticmodelsforparticledispersion[14].Inourcasetheremayalsobeabiasatsmalltimesduetoparticlee ects.Howeverifweassumetheexponential tforthevelocityautocorre-lationfunctiontobevaliddowntothesmallestscales,weobtainavalueC0=3.5asanupperboundforthe

maximumofDL

(τ)/ τ.Inoursetofmeasurementsbe-tweenR2

λ=100andRλ=1100,wehaveobservedanincreaseofC0(de nedinthesameway)from0.5to4.WepointoutthatintheabsenceofanequivalentoftheK´arm´an-HowarthrelationshipfortheLagrangiantimeincrements,alimitvalueofC0isnotapriori xed.

DimensionalanalysisyieldsDL

2(τ)=C0(Re) τandsimi-larityargumentsgiveC0(Re)→const.orC0(Re)→Reαinthelimitofin niteReynoldsnumbers.

3

TofurtherdescribethestatisticsoftheLagrangianve-locity uctuations,wehaveanalyzedthestatisticsofthevelocityincrements τv.TheirPDFΠτforτcoveringtheaccessiblerangeoftimescalesisshowninFig.3.

FIG.3:PDFστΠτofthenormalizedincrement vτ/στ.Thecurvesareshiftedforclarity.Fromtoptobottom:τ=[0.15,0.3,0.6,1.2,2.5,5,10,20,40]ms.

Toemphasizethefunctionalform,thevelocityincre-mentshavebeennormalizedbytheirstandarddeviationsothatallPDFshaveunitvariance.A rstobservationisthatthePDFsaresymmetric,inagreementwiththelocalsymmetriesthis ow.AnotheristhatthePDFsal-mostGaussianatintegraltimescalesandprogressivelydevelopstretchedexponentialtailsforsmalltimeincre-ments.Atthesmallestincrement,thestretchedexpo-nentialshapeisinagreementwithmeasurementsofthePDFofLagrangianaccelerationatidenticalReynoldsnumbers[10].Inourcase,thelimitformofthevelocityincrementsPDFisnotaswideasthatoftheaccelerationbecausetheKolmogorovscaleisnotresolved.NotethatinregardsoftheevolutionofthePDF,theintermittencyisatleastasdevelopedintheLagrangianframeasitisintheEulerianone[15].

FIG.4:EvolutionoftheexcesskurtosisfactorK(τ)= ( τv)4 / ( τv)2 2 3forthePDFsofthetimevelocityincrements.

Thecontinuousevolutionwithscalecanbequanti ed

We have developed a new experimental technique to measure the Lagrangian velocity of tracer particles in a turbulent flow, based on ultrasonic Doppler tracking. This method yields a direct access to the velocity of a single particule at a turbulent Reynold

usingthe atnessfactor.WeshowinFig.4thevari-ationexcesskurtosisK(τ)= ( τv)4 / ( τv)2 2 3.ItisnullatintegralscaleasexpectedfromtheGaus-sianshapeofthePDFandincreasessteeplyatsmallscales.Belowabout5τη,theincreaseislimitedbythecut-o oftheparticle;anextrapolationofthetrendtoτηyieldsK(τη)

～40inagreementwithaccelerationmea-surementsin[10].

10

L

q

D10

10

DL

10

FIG.5:ESSplotsofthestructurefunctionvariation(indou-blelogcoordinates).Thesolidcurvesarebestlinear tswith

slopesequaltoξL

q=0.56±0.01,1.34±0.02,1.56±0.06,1.8±0.2forp=1,3,4,5fromtoptobottom.Coordinatesinarbitraryunits.

Moregenerally,onecanchoosetodescribetheevolu-tionofthePDFsbythebehavioroftheirmoments(or

‘structurefunctions’)DL

changeofq(τ)= |δτv|q .Indeed,acon-sequenceoftheshapeofthePDFswithscaleisthattheirmoments,asthe atnessfactorabove,varywithscale.ClassicallyintheEulerianpicture,oneex-pectsscalingintheinertialrange,DE

q(r)∝rζq,atleastinthelimitofverylargeReynoldsnumbers.Atthe -niteReynoldsnumberwheremostexperimentsaremade,thelackofatrueinertialrangeisusuallycompensatedbystudyingtherelativescalingofthestructurefunc-tions–theESSansatz[16].Weusethesecondorderstructurefunctionasareference.Indeedthedimensional

estimationofDL2(asthatofDE

3)dependslinearlyontheincrementandonthedissipation.Fig.5showsthat,asintheEulerianframe,arelativescalingisobservedfortheLagrangianstructurefunctionsoforders1to5,DLq(τ)∝DL2(τ)ξq.Weobservethattherelativeexpo-nentsfollowasequencecloseto,butmoreintermittentthanthecorrespondingEulerianquantity.Indeed,we

obtain:ξLL

L/ξLξ1/ξ3=0.42,ξ3=0.75,ξL/ξLL3=1.17,5

/ξL

2

43=1.28tobecomparedtothecommonlyac-4

ceptedEulerianvalues[17]ξEξ1/ξE3=0.36,ξE2/ξE

3=0.70,E4/ξE3=1.28,ξE5/ξE

3=1.53.

Inconclusion,usinganewexperimentaltechnique,wehaveobtainedaLagrangianvelocitymeasurementthatcoverstheinertialrangeofscales.OurresultsareconsistentwithKolmogorov-likedimensionalpredictionsforsecondorderstatisticalquantities.Athigherorders,theobservedintermittencyisverystrong.HowtheLagrangianintermittencyisrelatedtothestatisticalpropertiesoftheenergytransfersisanopenquestion.Fromadynamicalpointofview,theNavier-StokesequationinLagrangiancoordinatescouldbemodeledusingstochasticequations.WorkiscurrentlyunderwaytocomparethedynamicsoftheLagrangianvelocitytopredictionsofLangevin-likemodels.

acknowledgements:WethankBernardCastaingforinterestingdiscussionsandVermonCorporationforthedesignoftheultrasonictransducers.Thisworkissup-portedbygrantACINo.2226fromtheFrenchMinist`eredelaRecherche.

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