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)