Self-Similar Intermediate Structures in Turbulent Boundary Layers At Large Reynolds Numbers

Self-Similar Intermediate Structures in Turbulent Boundary Layers At Large Reynolds Numbers

Self-similarIntermediateStructures

inTurbulentBoundaryLayers

atLargeReynoldsNumbers

arXiv:math/9908044v1 [math.NA] 10 Aug 1999G.I.Barenblatt,A.J.ChorinDepartmentofMathematicsandLawrenceBerkeleyNationalLaboratoryUniversityofCaliforniaBerkeley,California94720,USAandV.M.ProstokishinP.P.ShirshovInstituteofOceanologyRussianAcademyofSciences36NakhimovProspectMoscow117218,RussiaAbstract.Processingthedatafromalargevarietyofzero-pressure-gradientboundarylayer owsshowsthattheReynolds-number-dependentscalinglaw,whichthepresentauthorsobtainedearlierforpipes,givesanaccuratedescriptionofthevelocitydistributioninaself-similarintermediateregionofdistancesfromthewalladjacenttotheviscoussublayer.The

appropriatelengthscalethatentersthede nitionoftheboundarylayerReynoldsnumberisfoundforallthe owsunderinvestigation.

Anotherintermediateself-similarregionbetweenthefreestreamandthe rstintermedi-ateregionisfoundunderconditionsofweakfreestreamturbulence.Thee ectsofturbulenceinthefreestreamandofwallroughnessareassessed,andconclusionsaredrawn.

Self-Similar Intermediate Structures in Turbulent Boundary Layers At Large Reynolds Numbers

1Introduction

Asymptoticlawsforwall-boundedturbulentshear owsatlargeReynoldsnumbersareconsidered.Classicalexamplesofsuch owsarethe owsinpipes,channels,andboundarylayers.Thisclassof owsisofmajorfundamentalandpracticalimportance.Allthese owsshareasdimensionalgoverningparameterstheshearstressatthewallτandthe uid’sproperties,itsdensityρanddynamicviscosityµ.Fromtheseparameterstwoimportant

1quantitiescanbeformed:thedynamicorfrictionvelocityu =(τ/ρ)

u =1

ν(1)

whereyisthedistancefromthewall;theconstantsκ(thevonK´arm´anconstant)andCshouldbeidenticalforallturbulentwall-boundedshear owsathighReynoldsnumbers,andthelaw(1)shouldbevalidinintermediateregionsbetween,ononehand,theviscoussublayerand,ontheother,theexternalpartsofthe ows,e.g.vicinityoftheaxisinpipe ow,orvicinityoftheexternal owintheboundarylayer.In1932,L.Prandtl,thegreatestmechanicianofthiscentury,cametothelaw(1)usingadi erentapproach,bute ectivelywiththesamebasicassumption.Thelaw(1)isknownasthevonK´arm´an-Prandtluniversallogarithmiclaw.Morerecentderivationswhich,however,followthesameideasandthesamebasicassumption,ofteninanimplicitform,canbefoundinmonographsbyLandauandLifshits(1987),MoninandYaglom(1971),Schlichting(1968)andinarecenttextbookbySpurk(1997).

AccordingtothevonK´arm´an-Prandtllaw(1),allexperimentalpointscorrespondingtotheintermediateregionshouldcollapseonasingleuniversalstraightlineinthetraditionalcoordinateslnη,φ.

Self-Similar Intermediate Structures in Turbulent Boundary Layers At Large Reynolds Numbers

Subsequentinvestigationsshowed,however,thatthisisnotwhathappens.First,theexperimentsshowedsystematicdeviationsfromtheuniversallogarithmiclaw(1)evenifoneiswillingtotolerateavariationintheconstantsκandC(fromlessthan0.4to0.45forκ,andfromlessthan5.0to6.3forC).Furthermore,usinganalyticandexperimentalarguments,thepresentauthorsshowed[Barenblatt(1991,1993);BarenblattandProstokishin(1993);Barenblatt,ChorinandProstokishin(1997b);Chorin(1998)]thatthefundamentalvonK´arm´anhypothesisonwhichthederivationoftheuniversallaw(1)wasbased,i.e.theassumptionthatthein uenceofviscositydisappearstotallyoutsidetheviscoussublayer,isinadequate.Infact,thishypothesisshouldbereplacedbythemorecomplicatedoneofincompletesimilarity,sothatthein uenceofviscosityintheintermediateregionremains,buttheviscosityentersonlyinpowercombinationwithotherfactors.Thismeansthatthein uenceoftheReynoldsnumber,i.e.bothoftheviscosityandtheexternallengthscale,e.g.thepipediameter,remainsandshouldbetakenintoaccountintheintermediateregion.Forthereaders’conveniencewepresentherebrie ytheconceptofincompletesimilarity;amoredetailedexpositioncanbefoundinBarenblatt,(1996).Themeanvelocitygradient yuinturbulentshear owscanberepresentedinthegeneralformsuggestedbydimensionalanalysis

yu=u

u =(C0lnRe+C1)ηc/lnRe.(2)

wheretheconstantsC0,C1andαmustbeuniversal.Thescalinglaw(2)wascomparedwithwhatseemed(andseemstousuptonow)tobethebestavailabledataforturbulent

Self-Similar Intermediate Structures in Turbulent Boundary Layers At Large Reynolds Numbers

pipe ows,obtainedbyNikuradze(1932),undertheguidanceofPrandtlathisInstituteinG¨ottingen.Thecomparisonhasyieldedthefollowingvaluesforthecoe cients

c=3√2

whentheReynoldsnumberRewastakenintheform

Re=u¯d

√3/2lnRe2)η

or,equivalently

φ= √ α

2αη,α=3

αln 2αφ

3+5α =lnη,α=3(3)(5)

Self-Similar Intermediate Structures in Turbulent Boundary Layers At Large Reynolds Numbers

authors(Barenblatt,Chorin(1996,1997)),itwasdemonstratedthatthescalinglaw(2)iscompatiblewiththeproperly

modi ed

IMMprocedure.Themethodofvanishingviscosity(Chorin,(1988,1994))wasusedinthismodi cation.

Letusturnnowtoshear owsotherthan owsinpipes.Bythesamelogic,thescalinglaw(5)shouldbealsovalidforanintermediateregionadjacenttotheviscoussublayerforallgoodqualityexperimentsperformedinturbulentshear owsatlargeRe.

The rstquestionis,whatistheappropriatede nitionoftheReynoldsnumberforthese owswhichwillmaketheformula(5)applicable?Thisisaveryimportantpoint—iftheuniversalReynolds-number-independentlogarithmiclawwerevalid,thede nitionoftheReynoldsnumberwouldbeirrelevantprovideditweresu cientlylarge.Forthescalinglaw

(5)thisisnotthecase.Indeed,ifthescalinglaw(5)hasgeneralapplicabilityitshouldbepossibleto nd,foreveryturbulentshear owatlargeReynoldsnumber,anappropriatede nitionoftheReynoldsnumberwhichwillmakethescalinglaw(5)valid.

Thereexistsnowadaysalargeamountofdataforanimportantclassofwall-boundedturbulentshear ows:turbulentzero-pressure-gradientboundarylayers.Thesedatawereobtainedoverthelast25yearsbyvariousauthorsusingvariousset-ups.Forboundarylayersthetraditionalde nitionoftheReynoldsnumberis

UθReθ=

(9)ν

isproperlydetermined.Moreover,weshowthatforallthe owswheretheturbulenceintheexternal owissmall,thereexistsasharplydistinguishablesecondintermediateregion

Self-Similar Intermediate Structures in Turbulent Boundary Layers At Large Reynolds Numbers

betweenthe rstonewherethescalinglaw(5)isvalidandtheexternalhomogeneous ow.Theaveragevelocitydistributioninthissecondintermediateregionisalsoself-similarofscalingtype:

φ=Bηβ(10)

whereBandβareconstants.

However,aReynoldsnumberdependenceofthepowerβwasnotobserved.Withintheaccuracyoftheexperimentaldataβiscloseto1/5.Whentheturbulenceintheexternalhomogeneous owbecomessigni cant,thesecondself-similarregiondeterioratesandthepowerβdecreaseswithgrowingexternalturbulenceuntilthesecondintermediateregiondisappearscompletely.

2The rstgroupofzero-pressure-gradientboundary

layerexperiments

Wewillexplainlaterwhywedividedtheexperimentaldataintothreegroups.Hereitissu cienttonotethatallavailablesetsofexperimentaldatawereeventuallytakenintoaccount.

Theoriginaldatawerealwayspresentedbytheirauthorsintheformofgraphsinthetra-ditional(lnη,φ)plane,suggestedbytheuniversallogarithmiclaw(1).TheshapeoforiginalgraphswasalwayssimilartotheonepresentedqualitativelyinFigure1a.Therefore,the rstrathertrivialstepwastoreplotthedatainthedoublylogarithmiccoordinates(lgη,φ)appropriateforrevealingthescalinglaws.Theresultwasinstructive:forallexperimentsofthe rstgroup(inchronologicalorder),speci cally:Collins,Coles,Hike,1(1978);ErmandJoubert(1991);Smith2(1994);Naguib3(1992),andNagibandHites4(1995);KrogstadandAntonia(1998),thedataoutsidetheviscoussublayer(lgη>1.5)havethecharacteristicshapeofabrokenline,shownqualitativelyinFigure1bandquantitativelyinFigures2–6.Thus,thetwostraightlinesformingthebrokenlinethatwererevealedinthelgη,lgφplanehaveasequations

(I)φ=Aηα;(II)φ=Bηβ.(11)

Thecoe cientsA,α,B,βwereobtainedbyusthroughstatisticalprocessing.

Self-Similar Intermediate Structures in Turbulent Boundary Layers At Large Reynolds Numbers

Weassumeasbeforethatthee ectiveReynoldsnumberRehastheform(9):Re=UΛ/ν,whereUisthefreestreamvelocityandΛisalengthscale.Thebasicquestionis,whetheronecan ndineachcasealengthscaleΛwhichplaysthesamerolefortheintermediateregion(I)oftheboundarylayerasthediameterdoesforpipe ow?Inotherwords,whetheritispossibleto ndalengthscaleΛ,perhapsin uencedbyindividualfeaturesofthe ow,sothatthescalinglaw(5)isvalidforthe rstintermediateregion(I)?ToanswerthisquestionwehavetakenthevaluesAandα,obtainedbystatisticalprocessingoftheexperimentaldatainthe rstintermediatescalingregion,andthencalculatedtwovalueslnRe1,lnRe2,bysolvingtheequationssuggestedbythescalinglaw(5):

1lnRe1+35

2lnRe2=α.(12)

IfthesevalueslnRe1,lnRe2obtainedbysolvingthetwodi erentequations(12)areindeedclose,i.e.,iftheycoincidewithinexperimentalaccuracy,thentheuniquelengthscaleΛcanbedeterminedandtheexperimentalscalinglawintheregion(I)coincideswiththebasicscalinglaw(5).

Table1showsthatthesevaluesareclose,thedi erenceslightlyexceeds3%inonlytwocases;inallothercasesitisless.Thus,wecanintroduceforallthese owsthemeanReynoldsnumber

Re= 2(lnRe1+lnRe2)(13)

andconsiderReasanestimateofthee ectiveReynoldsnumberoftheboundarylayer ow.Naturally,theratioReθ/Re=θ/Λisdi erentfordi erent ows.

Self-Similar Intermediate Structures in Turbulent Boundary Layers At Large Reynolds Numbers

3Zero-pressure-gradientboundarylayerbeneathatur-

bulentfreestream:TheexperimentsofHancockandBradshaw

TheexperimentsofHancockandBradshaw(1989)revealedanewfeatureimportantforouranalysis.Examinationoftheseexperimentaldatasuggestedthatweseparatetheotherexperimentsintotwogroups.IntheHancockandBradshawexperimentsthefreestreamwasturbulizedbyagridinallseries,exceptone.Thus,processingthedatafromtheseexperimentswewereablenotonlytocomparethescalinglaw(5)withexperimentaldataonceagainbutalsotoinvestigatethein uenceoftheturbulenceoftheexternal owonthesecondself-similarintermediateregion.TheresultsoftheprocessingarepresentedinTable2andFigures7and8.InbothTable2andFigures7and8theintensityofturbulenceisshownbythevalueofu′/U,whereu′isthemeansquarevelocity uctuationinthefreestream.

Table2

Figure

Fig.8a

Fig.8b

Fig.8c

Fig.8d

Fig.8e

Fig.8f

Fig.8gReθ4,6802,9805,7604,3203,7103,1003,860αAlnRe1lnRe2lnReu′/UReθ/Re0.110.060.100.210.020.030.04β0.200.18––––Hancock,P.E.andBradshaw,P.(1989)0.1408.6610.6710.7110.690.00030.1388.7710.8610.9110.880.0240.1378.8010.9110.9510.930.0260.1508.229.9110.009.950.0410.1229.4912.1112.3012.200.0400.1289.1311.4811.7011.590.0580.1299.0711.3811.6311.500.058

Firstofall,ourprocessingshowedthatthe rstself-similarintermediatelayerisclearlyseeninalltheseexperiments,bothintheabsenceoftheexternalturbulence,andinitspresence.ThevaluesoflnRe1andlnRe2areclose.Thismeansthatthebasicscalinglaw

(5)isvalidintheintermediateregionadjacenttotheviscoussublayer.Atthesametime,thesecondself-similarregionisclearlyobservedandwell-de nedonlywhentheexternalturbulenceisweak(Figure8aandtoalesserextent,Figure8b)sothattheexternalturbu-lenceleadstoadrasticreductionofthepowerβ,andeventothedeteriorationofthesecondself-similarintermediateregionsothatβbecomesindeterminate.Weillustratethein uenceofthefreestreamturbulenceadditionallybyFigure7(b).

Self-Similar Intermediate Structures in Turbulent Boundary Layers At Large Reynolds Numbers

TheexperimentsofHancockandBradshawareinstructivebecausetheysuggestatleastonepossiblereasonforthedestructionoftheintermediateself-similarregionadjacenttotheexternal owthatisobservedintheexperimentsofthenextgroup.

Self-Similar Intermediate Structures in Turbulent Boundary Layers At Large Reynolds Numbers

4Theremaininggroupofzero-pressure-gradientbound-

arylayerexperiments

Inthissectiontheresultsoftheprocessingarepresentedforalltheremainingseriesofexperiments.ForallofthemweusedthedatapresentedintheformofgraphsinthereviewofFernholzandFinley(1996).TheresultsoftheprocessingarepresentedinTable3andinFigures9–15.

Allthedatarevealtheself-similarstructureinthe rstintermediateregionadjacenttotheviscoussublayer.ThescalinglawsobtainedforthisregiongivevaluesoflnRe1andlnRe2,closetoeachother,althoughthedi erencebetweenlnRe1andlnRe2issometimeslargerthanintheexperimentsofthe rstgroup.Thescalinglaw(5)iscon rmedbyalltheseexperiments.Atthesametime,forthisgroupofexperimentsthesecondself-similarstructureadjacenttothefreestreamturnsouttobelessclear-cut,ifitisthereatall.Therefore,forthisgroupofexperiments,wedidnotpresenttheestimatesforthevaluesofβ.Neverthelesswenotethatwhenitwaspossibletoobtainestimatesofβtheyalwaysgaveaβlessthan0.2.Notealsothatforalltheseexperimentsthenumberofexperimentalpointsbelongingtotheregionadjacenttothefreestreamwaslessthanfortheexperimentsofthe rstgroup:thiswasanadditionalargumentforourreluctancetoshowherethesecondself-similarlayer.AsexplainedinSection3,wesuggestthattheturbulenceoftheexternal owintheexperimentsofthisremaininggroupwasmoresigni cant.

Self-Similar Intermediate Structures in Turbulent Boundary Layers At Large Reynolds Numbers

5Checkinguniversality

1Theuniversalformofthescalinglawψ=√

Self-Similar Intermediate Structures in Turbulent Boundary Layers At Large Reynolds Numbers

intheseexperimentstheexperimentalpointsliemuchbelowthebisectrix.Furthermore,intheseexperimentslnRe1andlnRe2di eredsigni cantly,andwethereforepickedthevalueofαthatcorrespondstolnRe1.Theresultisapairoflinesparalleltothebisectrixbutfarbelowit(Figure16e,(b)).

Moregenerally,itisverylikelythatanyoutsidecausethatincreasesthelevelofturbu-lenceshouldalsoincreasethee ectiveviscosity,andthusshiftthepointsinthe(lnη,ψ)planedownwards.AcaseinpointisthesetofexperimentsofHancockandBradshaw(1989)discussedabove,whereturbulencewascreatedbyagridinthefreestream.Theparalleldownwardshiftisindeedobserved(Figure16f),anditisofthesameorderofmagnitudeastheshiftintheexperimentsofNagibetal.NotethatintheexperimentsofNagibetal.thesecondintermediateregionisintact,anditisthereforelikelythattheshiftintheuniversaldescriptionofthe rstintermediateregionisduetothedisturbanceclosetothewall,i.e.,toroughness,justasintheexperimentofZagarolaetal.(1996).

Self-Similar Intermediate Structures in Turbulent Boundary Layers At Large Reynolds Numbers

6Conclusion

uu yTheReynolds-number-dependentscalinglawφ=√)η3/2lnRe,2η=

Self-Similar Intermediate Structures in Turbulent Boundary Layers At Large Reynolds Numbers

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Barenblatt,G.I.andChorin,A.J.,1997.Scalinglawsandvanishingviscositylimitsforwall-boundedshear owsandforlocalstructureindevelopedturbulence,Comm.PureAppl.Math.50,381–398.

Barenblatt,G.I.,Chorin,A.J.,Hald,O.H.,andProstokishin,V.M.,1997.Structureofthezero-pressure-gradientturbulentboundarylayer,1997.Proc.Nat.Acad.SciencesUSA94,7817–7819.

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Barenblatt,G.I.,Chorin,A.J.andProstokishin,V.M.,1997,b.Scalinglawsinfullydevelopedturbulentpipe ow,AppliedMechanicsReviews50,no.7,413–429.

Barenblatt,G.I.andProstokishin,V.M.,1993.Scalinglawsforfullydevelopedshear ows.Part2.Processingofexperimentaldata,J.FluidMech.248,521–529.

Chorin,A.J.,1988,Scalinglawsinthevortexlatticemodelofturbulence,Commun.Math.Phys.114,167-176.

Chorin,A.J.,1994.VorticityandTurbulence,Springer,NewYork.

Self-Similar Intermediate Structures in Turbulent Boundary Layers At Large Reynolds Numbers

Chorin,A.J.,1998.Newperspectivesinturbulence.Quart.Appl.Math.56,767–785.Erm,L.P.andJoubert,P.N.,1991.LowReynolds-numberturbulentboundarylayers,J.FluidMech.230,1–44.

Fernholz,H.H.andFinley,P.J.,1996.Theincompressiblezero-pressure-gradientturbulentboundarylayer:anassessmentofthedata.Progr.AerospaceSci.32,245–311.

Hancock,P.E.andBradshaw,P.,1989.Turbulencestructureofaboundarylayerbeneathaturbulentfreestream,J.FluidMech.205,45–76.

Krogstad,P.- A.andAntonia,R.A.,1998.Surfaceroughnesse ectsinturbulentboundarylayers,ExperimentsinFluids,inpress.

Landau,L.D.andLifshits,E.M.,1987.FluidMechanics,PergamonPress,NewYork.Monin,A.S.andYaglom,A.M.,1971.StatisticalFluidMechanics,vol.1,MITPress,Boston.

Nagib,H.andHites,M.,1995.HighReynoldsnumberboundarylayermeasurementsintheNDF.AIAApaper95–0786,Reno,Nevada.

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Self-Similar Intermediate Structures in Turbulent Boundary Layers At Large Reynolds Numbers

Zagarola,M.V.,Smits,A.J.,Orszag,S.A.,andYakhot,V.,1996.ExperimentsinhighReynoldsnumberturbulentpipe ow,AIAApaper96-0654,Reno,NV.

Zagarola,M.V.andSmith,A.J.,1998.Mean- owscalingofturbulentpipe ow,J.FluidMech.373,33–79.

Self-Similar Intermediate Structures in Turbulent Boundary Layers At Large Reynolds Numbers

Table1

FigureReθαAlnRe1lnRe2lnReReθ/Re

Collins,D.J.,Coles,D.E.,andHiks,J.W.(1978)

Fig.2(a)5,9380.1299.1011.4311.6311.530.06

Fig.2(b)6,8000.1259.2311.6612.0011.830.05

Fig.2(c)7,8800.1239.4111.9712.2112.090.04

Erm,L.P.andJoubert,P.N.(1991)

Fig.3(a)6970.1637.839.239.209.220.07

Fig.3(b)1,0030.1597.969.469.439.450.08

Fig.3(c)1,5680.1567.979.479.629.540.11

Fig.3(d)2,2260.1488.269.9810.1410.060.10

Fig.3(e)2,7880.1408.6610.6710.7110.690.06

Naguib,A.M.(1992)andHites,M.andNagib,H.(1995)

Fig.4(a)4,5500.1567.879.309.629.460.36

Fig.4(b)6,2400.1488.249.9410.1410.040.27

Fig.4(c)9,5900.1438.3710.1710.4910.330.31

Fig.4(d)13,8000.1318.9411.1511.4511.300.17

Fig.4(e)21,3000.1388.6110.5810.8710.730.47

Fig.4(f)29,9000.1308.9911.2411.5411.390.34

Fig.4(g)41,8000.1249.3011.7812.1011.940.27

Fig.4(h)48,9000.1249.2811.7412.1011.920.33

Smith,R.W.(1994)

Fig.5(a)4,9960.1468.3610.1510.2710.210.18

Fig.5(b)12,9900.1299.1911.5911.6311.610.12

Krogstad,P.-A.andAntonia,R.A.(1998)

Fig.612,5700.1468.3810.1810.2710.230.45

β0.2030.1950.2020.2020.1920.2020.2140.2060.220.200.2060.1930.220.2040.2010.1920.200.1670.201

Self-Similar Intermediate Structures in Turbulent Boundary Layers At Large Reynolds Numbers

Table3

Figure

Fig.9(a)

Fig.9(b)

Fig.9(c)

Fig.9(d)

Fig.9(e)

Fig.9(f)

Fig.9(g)ReθαAlnRe1lnRe2lnReReθ/Re0.450.240.130.080.070.050.06Winter,K.G.andGaudet,L.(1973)32,1500.1338.8611.0211.3211.1742,2300.1229.3711.9012.3012.1077,0100.11510.3013.5113.0413.2796,2800.10710.5613.9614.0213.99136,6000.10310.8314.4314.5614.50167,6000.10111.2015.0714.8514.96210,6000.10011.1514.9815.0014.99

Purtell,L.P.,Klebanov,P.S.,

Fig.10(a)1,0020.1707.39

Fig.10(b)1,8370.1647.62

Fig.10(c)5,1220.1498.11

Fig.11(a)

Fig.11(b)andBuckley,F.T.(1981)8.478.828.640.189.148.879.000.239.7210.079.890.269.809.749.700.149.750.16Erm,L.P.(1988)2,2440.1538.049.602,7770.1548.139.75

Petrie,H.L.,Fontaine,A.A.,Sommer,S.T.andBrungart,T.A.(1990)

Fig.1235,5300.1199.7612.5712.6112.590.12

Bruns,J.,Dengel,P.,Fernholz,H.H.(1992)and

Fernholz,H.H.,Krause,E.,Nockemann,M.andSchober,M.(1995)

Fig.13(a)2,5730.1518.4610.329.9310.130.10

Fig.13(b)5,0230.1448.8511.0010.4210.700.11

Fig.13(c)7,1390.1488.4910.3710.1410.250.25

Fig.13(d)16,0800.1428.4510.3110.5610.430.47

Fig.13(e)20,9200.378.5110.4110.9510.680.48

Fig.13(f)41,2600.1328.6310.6211.3610.980.70

Fig.13(g)57,7200.1308.7110.7611.5411.140.84

Fig.14(a)

Fig.14(b)

Fig.15(a)

Fig.15(b)Djenidi,L.andAntonia,R.A.(1993)1,0330.1548.209.879.749.810.061,3200.1508.3710.1710.0010.080.06Warnack,D.(1994)2,5520.1528.2910.039.874,7360.1498.209.8710.079.950.129.970.22

Self-Similar Intermediate Structures in Turbulent Boundary Layers At Large Reynolds Numbers

Figure1.(a)Schematicrepresentationoftheexperimentaldataintraditionalcoordinateslnη,φ.(b)Schematicrepresentationoftheexperimentaldatain(lnη,lnφ)coordinatesforexperimentsofthe rstgroup.

Figure2.(a)TheexperimentsbyCollins,Coles,Hike,(1978).Reθ=5,938.Bothself-similarintermediateregions(I)and(II)areclearlyseen.

Figure2.(b)TheexperimentsbyCollins,Coles,Hike,(1978).Reθ=6,800.Bothself-similarintermediateregions(I)and(II)areclearlyseen.

Figure2.(c)TheexperimentsbyCollins,Coles,Hike,(1978).Reθ=7,880.Bothself-similarintermediateregions(I)and(II)areclearlyseen.

Figure3.(a)TheexperimentsbyErmandJoubert,(1991).Reθ=697.Bothself-similarintermediateregions(I)and(II)areclearlyseen.

Figure3.(b)TheexperimentsbyErmandJoubert,(1991).Reθ=1,003.Bothself-similarintermediateregions(I)and(II)areclearlyseen.

Figure3.(c)TheexperimentsbyErmandJoubert,(1991).Reθ=1,568.Bothself-similarintermediateregions(I)and(II)areclearlyseen.

Figure3.(d)TheexperimentsbyErmandJoubert,(1991).Reθ=2,226.Bothself-similarintermediateregions(I)and(II)areclearlyseen.

Figure3.(e)TheexperimentsbyErmandJoubert,(1991).Reθ=2,788.Bothself-similarintermediateregions(I)and(II)areclearlyseen.

Figure4.(a)TheexperimentsbyNaguib,(1992).Reθ=4,550.Bothself-similarinterme-diateregions(I)and(II)areclearlyseen.

Self-Similar Intermediate Structures in Turbulent Boundary Layers At Large Reynolds Numbers

Figure4.(b)TheexperimentsbyNaguib,(1992).Reθ=6,240.Bothself-similarinterme-diateregions(I)and(II)areclearlyseen.

Figure4.(c)TheexperimentsbyNagibandHites,(1995).Reθ=9,590.Bothself-similarintermediateregions(I)and(II)areclearlyseen.

Figure4.(d)TheexperimentsbyNagibandHites,(1995).Reθ=13,800.Bothself-similarintermediateregions(I)and(II)areclearlyseen.

Figure4.(e)TheexperimentsbyNagibandHites,(1995).Reθ=21,300.Bothself-similarintermediateregions(I)and(II)areclearlyseen.

Figure4.(f)TheexperimentsbyNagibandHites,(1995).Reθ=29,900.Bothself-similarintermediateregions(I)and(II)areclearlyseen.

Figure4.(g)TheexperimentsbyNagibandHites,(1995).Reθ=41,800.Bothself-similarintermediateregions(I)and(II)areclearlyseen.

Figure4.(h)TheexperimentsbyNagibandHites,(1995).Reθ=48,900.Bothself-similarintermediateregions(I)and(II)areclearlyseen.

Figure5.(a)TheexperimentsofSmith,(1994).Reθ=4,996.The rstself-similarinter-mediateregion(I)isclearlyseen,thesecondregion(II)canberevealed.

Figure5.(b)TheexperimentsofSmith,(1994).Reθ=12,990.The rstself-similarintermediateregion(I)isclearlyseen,thesecondregion(II)canberevealed.

Figure6.TheexperimentsofKrogstadandAntonia,(1998).Reθ=12,570.Bothself-similarintermediateregions(I)and(II)areclearlyseen.

Figure7.(a)TheexperimentsbyHancockandBradshaw,(1989)–ageneralview. ,seeFigure8(a);+,seeFigure8(b);×,seeFigure8(c);2,seeFigure8(d); ,seeFigure8(e); ,seeFigure8(f); ,seeFigure8(g).

Self-Similar Intermediate Structures in Turbulent Boundary Layers At Large Reynolds Numbers

Figure7.(b)ThesamedataasinFigure 7(a)withthecoordinates ,φ.Thedeviationsfromtheaxisx=0re ectthex=lnη 253

in uenceoftheturbulenceofthefreestream.

Figure8.(a)TheexperimentsbyHancockandBradshaw,(1989).Reθ=4,680,u′/U=0.0003.Bothself-similarintermediateregionsareclearlyseen.

Figure8.(b)TheexperimentsbyHancockandBradshaw,(1989).Reθ=2,980,u′/U=0.024.Bothself-similarintermediatestructures(I)and(II)areclearlyseen.

Figure8.(c)TheexperimentsbyHancockandBradshaw,(1989).Reθ=5,760,u′/U=0.026.The rstself-similarintermediateregion(I)isclearlyseen,thesecondisnotrevealed.Figure8.(d)TheexperimentsbyHancockandBradshaw,(1989).Reθ=4,320,u′/U=0.041.The rstself-similarintermediateregion(I)isclearlyseen,thesecondisnotrevealed.Figure8.(e)TheexperimentsbyHancockandBradshaw,(1989).Reθ=3,710,u′/U=0.040.The rstself-similarintermediateregion(I)isclearlyseen,thesecondisnotrevealed.Figure8.(f)TheexperimentsbyHancockandBradshaw,(1989).Reθ=3,100,u′/U=0.058.The rstself-similarintermediateregion(I)isseen,althoughwithalargerscatter.Thesecondisnotrevealed.

Figure8.(g)TheexperimentsofHancockandBradshaw,(1989).Reθ=3,860,u′/U=0.058.The rstself-similarintermediateregion(I)isseen,althoughwithalargerscatter.Thesecondisnotrevealed.

Figure9.(a)TheexperimentsofWinterandGaudet,(1973).Reθ=32,150.The rstself-similarintermediateregion(I)isseen,althoughwithalargerscatter.Thesecondisnotrevealed.

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