《balance》观后感

时间：2024.5.4

《Balance》观后感

这是个微妙的平衡……

当一个人把音乐盒钓上来的时候，当一个人取得成功的时候，他身后有许多人来帮助他维持这个平衡。这就是团队，这个钓者他身后是一个团队，代表的是人与人之间的合作意识。团队的成员共同维系着木板的平衡，并在此基础上为自己谋福利。所以他们试图破坏平衡，让箱子滑向自己。影片的最后，“胜利者”成为最大的“失败者”，他将四个同伴驱逐，自己却只能与音乐盒分离在木板的两头。可望不可即。而五个人之间的平衡，却永远不能再恢复。死亡和遥不可及是他们最终付出了代价。影片中红色的音乐盒，它为这个枯燥的空间增添了一丝色彩。但它改变着人们的情绪，破坏着原有的平衡。利益是打破平衡的罪魁祸首，而自私和贪欲使得最后利益无法被人占有，还失去了平衡。

一个政治立场坚定，团结协作，以身作则的团队对于我们的进步起着重要的作用。一个团队如何在利益面前人人平等，如何实现“双赢”是值得我们深思的。

第四团队领袖队

自动化12-3班

孟元

第二篇：balance

Phil.Trans.R.Soc.A(2007)365,823–839

doi:10.1098/rsta.2006.1944

Publishedonline16January2007

Aphysicalmodeloftheturbulentboundary

layerconsonantwithmeanmomentum

balancestructure

BYJOEKLEWICKI1,*,PAULFIFE2,TIEWEI3

1ANDPATMCMURTRY4DepartmentofMechanicalEngineering,UniversityofNewHampshire,

Durham,NH03824,USA2DepartmentofMathematics,and4DepartmentofMechanicalEngineering,

UniversityofUtah,SaltLakeCity,UT84112,USA3DepartmentofMechanicalandNuclearEngineering,PennsylvaniaState

University,StateCollege,PA16802,USA

RecentstudiesbythepresentauthorshaveempiricallyandanalyticallyexploredthepropertiesandscalingbehavioursoftheReynoldsaveragedmomentumequationasappliedtowall-bounded?ows.Theresultsfromtheseeffortshaveyieldednewperspectivesregardingmean?owstructureanddynamics,andthusprovideacontextfordescribing?owphysics.Aphysicalmodeloftheturbulentboundarylayerisconstructedsuchthatitisconsonantwiththedynamicalstructureofthemeanmomentumbalance,whileembracingindependentexperimentalresultsrelating,forexample,tothestatisticalpropertiesofthevorticity?eldandthecoherentmotionsknowntoexist.Forcomparison,theprevalent,well-established,physicalmodeloftheboundarylayerisbrie?yreviewed.Thedifferencesandsimilaritiesbetweenthepresentandtheestablishedmodelsareclari?edandtheirimplicationsdiscussed.

Keywords:wall-turbulence;scaling;?owphysics;meanmomentumbalance

1.Introduction

Fluiddynamicboundarylayersformin?owstangentialtoano-slipwall.Afoundationalnotionpertainingtotheboundarylayeristhatthereisalwaysaregionnearano-slipsurfacewithinwhichthedirecteffectsofviscosityaredynamicallysigni?cant(Prandtl1904;Schlichting1979).WithincreasingReynoldsnumber,thisregionnecessarilybecomesadecreasingfractionoftheoverall?owdomain.Inthismanner,thesolutiontotheNavier–StokesequationapproachesthatoftheEulerequationastheReynoldsnumberbecomeslarge(e.g.alimitingmodelfortheboundarylayerisavortexsheetpositionedin?nitesimallyabovethewall).Giventhis,acentralobjectiveofboundarylayertheoryistodeterminethisrateatwhichtheeffectsofviscositybecomespatially

balance

localized.

*Authorforcorrespondence(joe.klewicki@unh.edu).

Onecontributionof14toaThemeIssue‘ScalingandstructureinhighReynoldsnumberwall-bounded?ows’.

823Thisjournalisq2007TheRoyalSociety

824J.Klewickietal.

Inconnectiontodynamics,attainingthisobjectivealsoservestorevealthescalingbehaviourofthemomentum?eld,sincedoingsoeffectivelydeterminesthenormalizationrequiredtowritethemomentumbalanceequationinaformthatremainsinvariantforvaryingReynoldsnumber.Perhaps,themostfamousexampleofthispertainstothe?atplatelaminarboundarylayer,asthewell-?p?????established1=Rxscalingbehaviourdirectlyre?ectstherateatwhichviscouseffectsdiminish(spatiallylocalize)withincreasingReynoldsnumber.Asmightbeexpected,identifyingthedominanttermsinthemomentumbalancealongwiththosenormalizationsthatretainthisbalanceindependentofReynoldsnumber(i.e.theReynoldsnumberp??????scalings)alsoprovidesabasisforeducing?owphysics.Forexample,the1=Rxbehaviourisalsodirectlyassociatedwiththedynamicsrealizedviathecompetitionbetweentheadvectionandthewall-normaldiffusionofaxialmomentum.

Ofcourse,determiningthescalingbehavioursassociatedwiththemeanmomentum?elddevelopmentinsmooth-wallturbulentboundarylayersisconsiderablymorechallenging.Primaryreasonsforthisarethat(i)themeanmomentumequationisindeterminateowingtotheappearanceoftheReynoldsstressesand(ii)insometurbulentwall-?ows,therearesubregionswithinwhichviscouseffectsarenegligibleinthemean.Thesechallengesnotwithstanding,signi?cantprogresshasrecentlybeenmadetowardsidentifyingtheappropriateapproximateformsofthemeanmomentumbalance(MMB)asthelayeristraversed,aswellasinidentifyingthosenormalizationsthatrenderthesesimpli?edformsoftheequationinvariantwithvariationsinReynoldsnumber(Fifeetal.2005a,b;Weietal.2005a).

TheprimarypurposesofthecurrenteffortaretopresentaphysicalmodeloftheturbulentboundarylayerthatisconsistentwiththestructureoftheMMBandtocompareitwiththeestablishedprevalentmodel.

2.Theprevalentphysicalmodel

Forthepurposeofcomparison,thissectionprovidesarelativelybriefdiscussionofthepredominant,well-accepted,modeloftheturbulentboundarylayer.

(a)Meanpro?le-basedlayerstructureandscalingbehaviours

Theprevalentphysicalmodelofthemeanstructureofthesmooth-wallturbulentboundarylayerhasratherdirectconnectiontothepropertiesofthemeanvelocitypro?le(e.g.Tennekes&Lumley1972;Pope2000;Davidson2004).(Herein,xistheaxialcoordinate,yisthewallnormalcoordinate,UandVarethevelocitycomponentsinthex-andy-directions,respectively,uppercaselettersrepresentmeanquantities,lowercaselettersdenote?uctuating

~ZUCu),anoverbarquantities,tildedenotesinstantaneousquantities(i.e.u

denotestimeaveragingandvorticitycomponentsareidenti?edbytheirsubscript.)Figure1ashowsrepresentativeturbulentboundary-layermeanvelocitypro?les.Thesepro?lesphave??????????beenmadenon-dimensionalusinginner

??w=risthefrictionvelocity,twisthewallshearvariables,utandn,whereutZt

stressandnisthekinematicviscosity,asdenotedbyasuperscript‘C’.Thus,aReynoldsnumberbasedonthefrictionvelocityandtheboundarylayerthickness,d,isgivenbydCZdut/n.Asisconventional,thedatain?gure1arePhil.Trans.R.Soc.A(2007)

Turbulentboundarylayermodel

(a)(b)1.0

AB0.8

0.6

0.4

110100

y+D10000.20.00100200300y+400500dU+/dy+<UV>+82530252015U+1050600

Figure1.Inner-normalizedmeanvelocityandstresspro?les.(a)Turbulentboundarylayermeanpro?lesandtheirassociatedlayerstructure:(A)viscoussublayer,(B)bufferlayer,(C)logarithmiclayerand(D)wakelayer(dataarefromKlewicki&Falco(1990)).(b)ViscousandReynoldsshearstresspro?lesinturbulentchannel?ow,dCZ590,whereinthiscasedisthehalfchannelheight.ThesedataarefromMoseretal.(1999).

plottedonsemi-logarithmicaxes.Themainfeaturesofthisgraphare(i)alayerimmediatelyadjacenttothesurfacewherethenormalizedpro?leislinear(A),(ii)alayerwherethedependenceofUConyCtransitionsfromlineartoapproximatelylogarithmic(B),1(iii)aregionofapproximatelylogarithmicvariation(C),followedby(iv)anextensiveouterregionwithinwhichUCvariesaccordingtoafunctionconsistingofalogarithmicpartplusawakepart,w(y/d).Dynamically,theviscoussublayer(A),0%yC%4,isidenti?edasaregionwhereviscosityhasamajoreffect.Inthebufferlayer(B),4%yC%30,theviscousandReynoldsstressesarebothdynamicallysigni?cant.Thelogarithmiclayer(C)extendsfromnearyCZ30toy/dx0.2andisseentobedominatedbytheeffectsofturbulentinertia.Withinthewakelayer(D),0.2%y/d%1,meanandturbulentinertiaarepredominant.

Attachingthesetime-averageddynamicalattributestolayersA–Distypicallyjusti?edbyexaminingtherelativemagnitudesofthemeanviscousstress,

(e.g.Tennekes&vUC/vyC,andtheReynoldsshearstress,TCeyCTZLumley1972;Pope2000;Davidson2004).Representativepro?lesofthesequantitiesareshownin?gure1b.ThesedatarevealthatthemagnitudeofTCiszeroatthewall,butrapidlyrisestoavaluethatisO(1)byyCx30.Conversely,vUC/vyCZ1.0atyCZ0,butdiminishestoaquantitythatismuchlessthanO(1)byyCx30.(Herein,theordersymbol,O($),willbeusedwithrespecttoe/0,whereeisasmallparameter,e.g.e2Z1/dC.Forexample,aZO(b)forpositivea(e)andb(e)istakentomeanthatbotha/bandb/aareboundedase/0.)Theseobservationsarecommonlyusedtosupportthenotionthat,inthemean,thedynamicaleffectsofinertiabecomegreaterthanthoseofviscositybeginninginlayerBandpredominantlysowiththeoutwarddistancefromlayerB.

ExaminationofthelayerthicknessesandthevelocityincrementsacrosstheselayersrevealstheReynoldsnumberdependenciesinherenttothisphysicalmodel.Thesedependenciesarelistedintable1.Asisapparentfromtheentries1Notethatapproximatelylogarithmicdoesnotexcludecertainpower-lawforms.Also,recentexperimentaldataprovideevidencethatthetraditionallyde?nedlogarithmiclayeriscomposedoftworegionshavinglogarithmic-likebehaviour,butnotnecessarilythesameslope(Osterlundetal.2000;McKeonetal.2004).

Phil.Trans.R.Soc.A(2007)

826J.Klewickietal.

Table1.Scalingbehavioursofthelayerthicknessesandlayervelocityincrementsassociatedwiththepredominantlyacceptedphysicallayerstructureoftheturbulentboundarylayer(C,DandEareconstants).

physicallayer

D(viscoussublayer)(bufferlayer)(logarithmiclayer)(wakelayer)DyincrementO(n/ut)(x4)O(n/ut)(x26)O(d)(/0.2d)O(d)(x0.8d)DUincrementO(ut)(x4)O(ut)(x9)O(UN)(x(ut/k)log(d/C))

O(UN)(x(ut/k)log(d/D)CE)

intable1,thelayersassociatedwiththedirecteffectsofviscosity(AandB)haveathicknessthatremainsa?xednumberofviscouslengthsindependentofReynoldsnumber.Correspondingly,withincreasingdC,theselayerthicknessesbecomeadiminishinglysmallfractionofdataratedirectlyproportionalto1/dC.Similarly,thevelocityincrementsacrosstheselayersaremeasuredbya?xednumberofut,andthusasdC/N,theybecomediminishinglysmallrelativetoUNatarateproportionaltout/UN.Conversely,boththelogarithmiclayerandthewakelayergrowatarateproportionaltodasdC/N.Ofcourse,whenmeasuredinviscouslengths,theselayerthicknessesareunboundedasdC/N.Inasimilarmanner,thevelocityincrementacrosseitherlayerCorDapproachesa?xedfractionofUNasdC/N.

(b)Dynamicalconsiderations

Importantelementsassociatedwiththepredominantmodelrelatetothederivationofthelogarithmicmeanvelocitypro?le.Whileotherderivationsexist,theconstructioncommonlydeemedmostrigorouspostulatestheexistenceofatwo-layermathematicalstructure(innerandouter),andassumesthattheinnerandouterscalingshaveacommonregionofoverlapwithinwhichbotharesimultaneouslyvalid(e.g.Millikan1939;Tennekes&Lumley1972;Pope2000;Panton2005).Underfurtherassumptionthatthepro?lestrictlyincreaseswithy,themeanvelocitygradient(simultaneouslyexpressedinitsinnerandouterforms)ismatchedintheoverlaplayerasyC/Nandh/0.Theclassicalformsofthelogarithmiclawofthewallanddefectlawsubsequentlyfollow.Accordingtothisdescription,theinnerlayerextendsfromthewalltotheouteredgeofthelogarithmiclayer,andtheouterlayerextendsfromtheinneredgeofthelogarithmiclayertod.Animportantphysicalimplicationofthismathematicaldescriptionisthecorrespondencebetweentheoverlaplayerandaninertialsublayer.Thus,considerableresearchhasbeendevotedtowardsunderstandinglogarithmiclayerturbulenceanditssimilaritiestothespectralversionofinertialrangeturbulence(e.g.Townsend1976;Perry&Abell1977;Perry&Marusic1995;Morrisonetal.2004;Davidsonetal.2006;McKeon&Morrison2007;Metzger2006).

Otherconsiderationspertaintohowanygivenmodelsetstheconceptualframeworkforinterpretingmeasurementsandobservations.Notableamongtheserelatetodescribingthecharacteristicsanddynamicalbehavioursofboundary-layercoherentmotions.Acomprehensivereviewofboundary-layercoherentmotionsiswellbeyondthescopeofthepresenteffort.ItisusefulhowevertoidentifyPhil.Trans.R.Soc.A(2007)

Turbulentboundarylayermodel827

theimplicationsofthepredominantmodelrelativetotheinterpretationofdynamicalprocesses.Tothisend,issuesrelatingtotheself-sustainingmechanismsandtheso-calledinner/outerinteractionarenowbrie?ydiscussed.

Assupportedbythestresspro?lesof?gure1bandothernear-wallstatisticaldata(e.g.theturbulencekineticenergypro?le),theprevalentmodelplaceshighimportanceonthenear-wallregion(bufferlayerandbelow;e.g.Robinson1990).Numerouscoherentmotionshavebeenidenti?edasdynamicallysigni?cantinthisregion—includingstreaks,pockets,streamwisevortices,internalshearlayersandhairpinvortices.Indeed,thisregionisassertedbymanytobewheretheself-sustainingmechanismsofboundarylayerturbulenceprimarilyreside(e.g.seeanumberofcontributionstoPanton1997).Itisalsogenerallyaccuratetoattributethisfocusonthenear-wallregionwiththeidenti?cationofthebufferlayeraswherethedynamicstransitionfrombeingstronglyin?uencedbyviscositytowhereturbulentinertiadominates(Pope2000;Davidson2004).Thislastnotionalsohasbearingonhowtoconstructadescriptionoftheso-calledinner/outerinteraction.Properlycharacterizingthenatureoftheinteractionsbetweentheinnerandtheouterlayersisassertedbymanytobecentraltounderstandingturbulentboundarylayerdynamics(e.g.Kline1978;Falco1983;Thomas&Bull1983;Klewicki1989;Sreenivasan1989;Wark&Nagib1991).Thisassertion?ndssupportfromatleasttwocompellingarguments.Atperhapsthemostfundamentallevel,theturbulentboundarylayermaybeconsidereda?uiddynamicalmachinethat,onanaverage,convertsfree-streammomentumintotangentialforceactingatthe?uid/solidinterface.Giventhis,oneisthenfacedwithdescribinghowthenetmomentumtransferacrossthelayeroccurs,andthustheinner/outerinteraction.Similarly,aprimarycharacteristicoftheboundarylayeristhatwithincreasingdCtheratiooftheoutertotheinnerlength-scaleincreases.Theinner/outerinteractionisunavoidablyconfrontedifthedynamicalaccommodationtothisscaleseparationistobedescribed.Apparently,owingtoitsinherentcomplexities,theinner/outerinteractionremainsresistivetoathoroughcharacterization.Forexample,iftheviscous/inertialinteractionisseenascentraltotheinner/outerinteraction,thentheprevalentmodelplacesitinthebufferlayerforalldC.Ontheotherhand,byde?nition,theoverlapregioniswhereinnerandouterscalingsaresimultaneouslyvalid,andthusjusti?cationtoprimarilyassociateinner/outerinteractionswiththelogarithmiclayer.Doingso,however,requiresexplainingwhyinner/outerinteractionsshouldoccurinaninertialsublayer,2andnotintheregionwheretheviscousandReynoldsstressesareofthesameorderofmagnitude.

Lastly,asigni?cantandgrowingbodyofresults(Wark&Nagib1991;Meinhart&Adrian1995;Adrianetal.2000;Ganapathisubramanietal.2003,2005;Tomkins&Adrian2003;Morrisetal.inpress;Priyadarshanaetal.2007)supporttheperspectivethatthelogarithmiclayerisinstantaneouslycomposedofahierarchyofmotions,andthatthesemotionsarenominallyarrangedasuniformmomentumzonessegregatedbyrelativelynarrowvortical?ssures.Atpresent,itisnotreadilyapparenthowsuchaninstantaneousstructuremightgiverisetoamathematicaldescriptionbaseduponanoverlaplayer.ThistypeofEmpiricalobservationsrelatingtothebehaviourofthelogarithmicmeanpro?lehaverecentlypromptedproposalsregardingtheexistenceofameso-layer(Wosniketal.2000)intheregion30%yC%300andanextended‘bufferregion’(Osterlundetal.2000)outtoaboutyCZ200.Phil.Trans.R.Soc.A(2007)2

828J.Klewickietal.

hierarchicalstructurehoweverwouldseemtohavearathernaturalconnectiontotheearlier,phenomenological,logpro?lederivationbaseduponpostulatingthedistancefromthewallastheappropriatelength-scale(Prandtl1925).

3.Analternativephysicalmodel

Recentempiricalobservationsandmultiscaleanalysisprovidethebasisforanalternativephysicalmodeloftheturbulentboundarylayer(Fifeetal.2005a,b;Weietal.2005a,b).Animportantpremiseunderlyingthelayerstructuretobedescribed,andinturnthealternativephysicalmodel,isthattheMMBinitsunintegratedform(andinthiscaseasappliedtoboundarylayer?owoveraplanarsurfacelocatedatyZ0),

CCv2UCvTCCvUCvUUCVZC;e3:1Tvxvyvyvy

providesthetime-averageddescriptionofthedynamics.Theleftsideofequation(3.1)representsadvectionbythemean?ow,whiletheright-sidetermsrepresenttheviscousandReynoldsstressgradients,respectively.Forthe?atplate?ow,thereareonlythesethreedistinctdynamicaleffects,andthustheratioofanytwodeterminesthenaturebywhichtheequationisbalanced.

(a)MMB-basedlayerstructureandscalingbehaviours

Weietal.(2005a)exploredthestructureofboundarylayer,pipeandchannel?owsbyexaminingtheratioofthelasttwotermsinequation(3.1).Thedynamicsre?ectedbyequation(3.1)mustarisefromabalanceofatleasttwonon-negligibleterms,andthusinterpretationofthisratioisasfollows.(i)Ifjev2UC=vyC2T=evTC=vyCTj[1,thentheReynoldsstressgradienttermisnegligibleandequation(3.1)sumstozeroessentiallythroughabalance

ofthemeanadvectionandviscousstressgradientterms.

(ii)Ifjev2UC=vyC2T=evTC=vyCTj/1,thenthemeanviscousstressgradienttermisnegligibleandequation(3.1)sumstozeroessentiallythrougha

balanceofthemeanadvectionandReynoldsstressgradientterms.

(iii)Ifjev2UC=vyC2T=evTC=vyCTjx1,thentheReynoldsstressandtheviscousstressgradientsbalanceandareeithergreaterthanorofthesameorderof

magnitudeasthemeanadvectionterm.

Availablehigh-qualityexperimentalandDNSdata(Zagarola&Smits1997;Moseretal.1999;DeGraaff&Eaton2000)weredifferentiatedandtheindicatedratiowasexaminedasafunctionofyfordifferingdC.Thesketchof?gure2depictsthebehaviourofthestressgradientratioatany?xeddC.Asindicated,thedynamicalbalanceisdescribedbyafour-layerstructure.LayerIessentiallyretainsthecharacteroftheviscoussublayer,andintheboundarylayerisaregionwheretheviscousstressgradientnominallybalancesmeanadvection.InlayerII,themagnitudeoftheratioisveryclosetounity,andthusthislayeriscalledthestressgradientbalancelayer.Acrossthemesolayer(layerIII),theReynoldsstressgradientchangessignandthetermsinequation(3.1)undergoabalancebreakingandexchange(Fifeetal.2005b;Weietal.2005a).WithinlayerIII,allthethreePhil.Trans.R.Soc.A(2007)

Turbulentboundarylayermodel

ratio of stress

gradients

peak Reynolds

stress location829

?1y+

IIIIIIIV

Figure2.SketchoftheratiooftheviscousstressgradienttotheReynoldsstressgradientinboundarylayer,pipeandchannel?owsatanygivenReynoldsnumber.ThedottedlineinlayerIisforaboundarylayer,andthesolidlineisforapipeorchannel.

Table2.Scalingbehavioursofthelayerthicknessesandvelocityincrementsoftheproposedalternativemodel.(NotethatthelayerIVpropertiesareasymptoticallyattainedasdC/N.)physicallayer

III

IVDyincrement3)O(n/ut)(x??p??????????????Ond=ut(x1.6)??p??????????????Ond=ut(x1.0)O(d)(/d)DUincrementO(ut)(x3)O(UN)(xUN/2)O(ut)(x1)O(UN)(/UN/2)termsinequation(3.1)arenominallyofthesameorderofmagnitude,3andfromtheouteredgeoflayerIIItoyZd(i.e.layerIV),equation(3.1)ischaracterizedbyabalancebetweenthemeanadvectionandtheReynoldsstressgradient.

Thefeaturesof?gure2,depictedfor?xeddC,persistforthedCrangecurrentlyaccessibletoinquiry(e.g.spatialresolutioninthePrincetonSuperpipelimitedtheReynoldsnumbertoRC%41235).Inthisregard,itisalsorelevanttonotethatboundarylayerandpipe/channel?owsexhibitthesamebehaviourstowithinthedifferencesbetweenthemeanadvectionandpressuregradientpro?les.Thus,forexample,inlayerIItheirstructureisexpectedtobehighlysimilarsinceinthislayerthesetermsaremuchsmallerthanthedominantstressgradientterms.Quantitatively,thelayerthicknessesandthevelocityincrementsacrosstheselayershavebeenshownbothempiricallyandanalyticallytoexhibitdistinctReynoldsnumberdependencies.Table2presentsthesescaling3Ofcourse,rightatthepeakintheReynoldsstresstheReynoldsstressgradientpassesthroughzero,andthusinanarrowzonearoundthispointwithinlayerIIItheReynoldsstressgradientissmallerthantheothertwoterms.

Phil.Trans.R.Soc.A(2007)

830

(a)

×10?3200

?20

?40

?60

?80IIIIIIJ.Klewickietal.(b)III～(d +)1/2III～(d +)1/2IVlineU +?U = 1utIV?U = U∞/2

110100log(y+)

Figure3.AttributesoflayersIIandIII:(a)innernormalizedReynoldsstressgradient(solidline)andviscousstressgradient(dashedline)pro?les(dataarefromtheRqZ1410DNSofSpalart(1988))and(b)schematicdepictionoftheconnectionbetweenthegrowthratesoflayersIIandIIIandtheirassociatedvelocityincrements.

behaviours.Asisevident,layersIandIVadheretoinnerandouterscaling,respectively.Ontheotherhand,layersIIandIIIexhibitmixedscalingproperties.TheinnernormalizedthicknessoflayerIIgrowslikethegeometricp?????CmeanoftheReynoldsnumber(i.e.wd),whileitsvelocityincrementp?????remainsaCC?xedfractionofUN,independentofd.Similarly,DIIIywdC,whileitsvelocityincrementisonlyabout1.0ut,independentofdC.Thesescalingbehavioursdifferconsiderablyfromtheprevalentmodelandareassociatedwith????????????ptheexistenceofathirdfundamentallength-scale,nd=ut.Thislengthisintermediateton/utandd,requiredtoscaletheMMBinlayerIIIandbecomesincreasinglydistinctasdC/N.

ThemeandynamicsandscalingbehavioursassociatedwithlayersIIandIIIarecentral(andapparentlyunique)totheproposedphysicalmodel,andthuswarrantfurtherdiscussion.LayerIIiscalledthestressgradientbalancelayersincethedominantdynamicalmechanismsarethetermsontherightofequation(3.1);theirratiobeingK1in?gure2.Contrarytotheprevalentnotionthat,onanaverage,boundarylayerdynamicsaredominatedbyturbulentinertiaoutsidethebufferlayer(independentofdC),momentumbalancedatarevealthatanequalcompetitionbetweentheviscousforceandtheturbulentforcepersiststoay-locationnearthepeakintheReynoldsstress,Tmax.ConsistentwiththemathematicalhierarchyofscalinglayersrevealedbyFifeetal.(2005a),inthemodelposedbelowthiscompetitionisassociated(inatime-meansense)withthevorticalmotionsformingandevolvingfromthenear-wallvorticity?eld.Itissigni?canttonote,however,thatthebalanceinlayerIIcomesaboutviatwooppositesign,nearlyequal,butdecreasingmagnitudefunctions.ThesefunctionslosedominanceovermeanadvectionaslayerIItransitionsintolayerIII(?gure3a).

Thescalingsoftable2revealthatthelayerIIandIIIthicknessesarecoupled,suchthattheirvelocityincrementsfollowouterandinnerscaling,respectively.Thesepropertiesunderlienewinterpretationsrelatingto,forexample,thenatureoftheinner/outerinteractioninboundarylayers.ItisrelevanttonotethatthemajorportionsoflayersIIandIVandalloflayerIIIresidewithintheboundsofthetraditionallyde?nedlogarithmiclayer.TheloweredgeoflayerIIis?xedneartheedgeoftheviscoussublayer(independentofdC),whilethepositionofitsPhil.Trans.R.Soc.A(2007)

Turbulentboundarylayermodel

detached eddies

(?T/?y < 0)

831fully three-dimensionalvorticity field

balance

IVIIIII

IFigure4.Schematicofsomeofthedynamicalattributesoftheproposedmodelfortheturbulentboundarylayer.Layernumbersp?????arethesameasthoseidenti?edin?gure2.Notethattheposition

CofTmax,ymax,varieslikedC.

p?????outeredgeextendstoincreasingyvalueslikedC,suchthatDIIUZUN/2.

(NotethatunderouterpnormalizationthepositionoftheouteredgeoflayerII?????Cmoves‘inward’like1=d.)Owingtothispositioningbehaviourp?????forlayerII,CboththeendpointsoflayerIIIvarywithd,particularly,likedC.Thus,whilethelayerIIIthicknessexhibitsthesameReynoldsnumberscalingbehaviouraslayerII,itsvelocityincrementisonlyx1ut.ThisarisesowingtothefactthatwithincreasingdC,layerIIIispositionedatincreasingyC-locationsinaregionwhereUCxlog(yC)(?gure3b).C

(b)Physicalinterpretation

Elementsofanewphysicalmodelarerepresentedintheschematicof?gure4.Unlikethephysicalpicturepromotedbytheprevalentview,thismodelisspeci?callyconstructedtobeconsistentwiththeMMBproperties.Thisleadstoanumberofnewinterpretationsrelatingtothedescriptionof?owphysics.Ofcourse,itisrecognizedthatanydefensiblephysicalmodelshouldalsoembracethenumerousindependentempiricalobservationsrelating,forexample,totheboundary-layercoherentmotionsdiscussedpreviously.Existingexperimentalobservationsarenowinterpretedinthecontextofthisnewmodel.

(c)Characteristicvorticalmotions

~es0providesperhapsthemostusefulcriterionfordistinguishingtheSinceu

boundarylayerfromthefree-stream,theprocessofdescribingtheproposedmodelbeginsbyattributingthecharacteristicvorticalmotionstolayersI–IV.SincethepropertiesoflayerIIIarelargelyattributedtotheinteractionbetweenlayersIIandIV,thediscussionproceedsaccordingtoaI,II,IV,IIIorder.(i)LayerI

Existingphysicalandnumericalexperimentsrevealthatthevorticity?eldintheregion0%yC%3hasaperturbedsheet-likecharacterthatpredominantlymeandersinthex-andz-plane,buthaslittlewall-normalcomponent(e.g.Kim

~z(i.e.oppositesigntoetal.1987;Balintetal.1991;Klewicki1997).Positiveu

Phil.Trans.R.Soc.A(2007)

832J.Klewickietal.

themeanvorticity)isvirtuallynon-existentinthisregion(Klewickietal.1990;Rajagopalan&Antonia1993;Metzger&Klewicki2001).Perturbationsofthissheet-likedistributionofvorticityarebelievedbymanytoconstitutetheinitialconditionsthattriggertheevolutionofmotionsarisingoutoflayerI(Offen&Kline1974;Perry&Chong1982;Wallace1982;Smithetal.1991).

(ii)LayerII

AttheloweredgeoflayerII,thevorticity?eldishighlyalignedinthespanwisedirection.Acrossthislayer,thevorticity?eldthreedimensionalizes.TheassociatedcharacteristicsarearapidreductioninjUzj,theappearanceof000~zandatrendtowardsuxpositiveuZuyZuz,wheresuperscript‘0’denotesr.m.s.(e.g.Klewickietal.1990;Balintetal.1991).Asindicatedin?gure4,thislayerisprimarilyassociatedwiththeattachededdies;attachedinthesensethattheiretiologyisattributabletothethreedimensionalizationofthenear-wallvorticity?eld.Notethatforallofthedescriptionsgiven,theeddiesandthedynamicsattributedtoanygivenlayershouldbeviewedasprobabilisticwithachangingmixtureofcharacteristicpropertiesintraversingfromonelayertothenext.Thus,attachededdiesareviewedaspredominantlypopulatinglayerII,andtoadecreasingdegreewithincreasingyCacrosslayerIIIandintolayerIV.Asigni?cantbodyofresearchsupportsthehypothesisthathairpin-likevorticesconstituteabasicbuildingblockofwallturbulence(e.g.Theodorsen1952;Head&Bandyopadhyay1981;Wallace1982;Perryetal.1986;Smithetal.1991;Adrianetal.2000;Ganapathisubramanietal.2005).Broadlyspeaking,hairpinvorticesareseentoformviatheredistributionofnear-wallvorticity(mainly

~z),andsubsequentlyevolveoutward.Whileearlier?owcomposedofu

visualization-basedevidenceindicatedthatthesemotionsmightextendfromthesublayertotheedgeoftheboundarylayer(Head&Bandyopadhyay1981),anincreasingandpredominantlymorerecentbodyofresultsindicatethatatsomey-positionthevorticalmotionsloseconnectionwiththeirnear-wallorigin.Consistentwiththesestudies,theprimarycandidateeddiesforlayerIIarehairpin-likevorticesthatcollectivelyorganizeintohierarchicalpackets.Inthisregard,theevolutionofthevorticityinthesemotionsisgenerallyattributedtoacompetitionbetweenvortexstretchingandviscousdiffusion.

(iii)LayerIV

Propertiesofthevorticity?eldinlayerIVincludeessentiallyequalvorticitycomponentintensitiesandanegligiblemeanvorticity.Thevorticityprobabilitydistributionshoweverarerelativelylongtailed—re?ectingtheincreasinglyintermittentnatureofthevorticity?uctuations(e.g.Balintetal.1991).Giventhesecharacteristics,thevorticalmotionsassociatedwithlayerIVareidenti?edasdetachededdies;detachedinthesensethatthesespatiallycompactmotionsareuncorrelatedwiththenear-wallvorticity?eld,andnominallyadvectwiththelayerIVmean?ow.Thoughaspeci?cgeometricformforthedetachededdiesis

~zstructure,thenotcompletelyestablished,directmeasurementsofnear-wallu

increasinglythree-dimensionalnatureofthevorticity?eldwithincreasing~e$ueZ0(KlewickidistancefromthesurfaceanddataconsiderationsrelativetoV

etal.1990;Rajagopalan&Antonia1993)supporttheexpectationthatatsomey-positionthecharacteristicvorticalmotionsbecomespatiallylocalizedandPhil.Trans.R.Soc.A(2007)

Turbulentboundarylayermodel833

topologicallyformclosedloops.Detachededdiesarehypothesizedtopredomi-nantlypopulatelayerIV,andtoadecreasingdegreewithdecreasingyCacrosslayerIIIandintolayerII.Thesimplestformofsuchaneddyisavortexring-likemotion.Falco’searlier‘typicaleddy’observations(Falco1977,1983,1991)supporttheexistenceofintermediatescalering-likeeddiesinboththeinnerandouterregions.Regardlessoftheexactgeometricformofthedetachededdies,a

~z,knowntobehypothesizedcharacteristicfeatureisthattheycontainpositiveu

prevalentinlayerIV.

(iv)LayerIII

GiventheattributesoflayersIIandIV,layerIIIisviewedasazonewithinwhichthecharacteristiceddytransitionsfromattachedtodetachedwithincreasingy.Thus,withinlayerIIItheexpectationisto?ndanearlyequalmixtureofattachedanddetachededdies.Similarly,theprocessbywhichattachededdiesmightevolveintodetachededdiesisexpectedtobecharacteristicoflayerIII.Inthisregard,therecentsimulationsofBakeetal.(2002)providecompellingevidencefortheformationofvortexringsfromthepinch-offofthelegsofhairpin-likevorticesduringthelateststagesoftransition.ThisprocesswaspreviouslyproposedbyFalcoasamechanismforring-likemotionformationintheturbulentboundarylayer,andwasexplorednumericallybyMoinetal.(1986).Theattached–detachededdydecompositionofthevorticity?eld?ndssupportfromvisualstudies(Falco1983,1991;Klewicki1997),two-pointvorticitycorrelations(Klewicki&Falco1996;Metzger&Klewicki2001)andDNSandPIVstudies(Jimenez&delAlamo2004;Christensen&Wu2005;Ganapathisubramanietal.2005).Furthermore,theinclusionofthedetachededdyconcepthasbeenfoundtoimprovecoherentmotion-basedmodelperformance(Perry&Marusic1995).Overall,layerIIIisnominallyviewedastheregionwheretheattachedanddetachededdiesinteractwithhighestprobability.

(d)Characteristicdynamics

ThepropertiesoftheMMBsuggestthatspeci?cdynamicalattributesmaybeassociatedwiththeattached/detachededdystructureproposed.Forexample,undertheproposedmodelattachededdiesformandevolveacrosslayerII,andthustheirdynamicalsignatureisthattheyproduceinstantaneouscontributionstopositiveKv=vy.Similarly,thecharacteristiceddiesoflayerIVaredetached.Therefore,theirdynamicalsignatureisthattheyproducenegativeKv=vy.Thisidenti?cationofbothasourceandasinkcharacterwiththeReynoldsstresstermintheMMBisapparentlydistincttothepresentmodel.

Inthecontextofthesedynamicalsignatures,itisusefultoexaminetheequation??vv??KvCwKu:ZzKyCe3:2Tvyvx

Forturbulentchannel?ow,thelasttermisidenticallyzero,whileforboundarylayersthistermisgenerallysmall.4Toagoodapproximation,however,thegradientoftheReynoldsstressislargelyestablishedbythedifferenceofthe4Hinze(1975)associatedthevorticaltermswithTownsend’sactivemotionsandthestreamwisederivativetermswiththeinactivemotions.

Phil.Trans.R.Soc.A(2007)

834J.Klewickietal.

indicatedvelocityvorticitycorrelations.Giventhis,theinterpretationisthatinlayerIItheattachededdiesinteractwiththevelocity?eldtogenerateanetpositivesum,andinlayerIV,thedetachededdiesgenerateanetnegativesum.Thedominanttermsinequation(3.1)indicatethatthedynamicsunderlyingtheevolutionofattachededdiesischaracterizedbyacompetitionbetweenviscousshearforcesandturbulentinertia.Duringthisevolution,thesemotionsactasasourcetothemeanmomentum,andateachReynoldsnumberthissource-likecharacteris,onanaverage,depletedatymax,thepositionofthepeakintheReynoldsstress,Tmax.Similarly,detachededdydynamicsinlayerIVischaracterizedbyacompetitionbetweenmeanadvectionandturbulentinertia.Onaverage,thissink-likecharacterextendsfromymaxtod.The?ow?eldinteractionsunderlyingequation(3.2)ineitherlayerIIorIVhavebeenshowntohavesigni?cantcontributionsfromintermediatescalemotions(Priyadarshana&Klewicki2003).Physically,itisrationaltoattributethistothefactthatasdC/Nvelocityspectrapeakatdecreasinglylowwavenumber,whilevorticityspectrapeakatincreasinglyhighwavenumber.Thus,accordingtoequation(3.2),thevelocityandvorticity?eldsmustcorrelateoversomeintermediatewavenumberrangeinorderfortheretobeanetmomentumtransportviaturbulentinertia.Thisargument,however,apparentlyonlypartiallyholds.MeasurementsoveraverybroadrangeofdCindicatethatwithincreasingdCaspectrallylocalscaleselectionoccurs.Thisresultsinsigni?cantcontributionstothelong-timecorrelationtoarisefromportionsofthecospectranearthepeaksintheparticipatingvelocityandvorticityspectra,respectively(Priyadarshanaetal.2007).

(e)Inner/outerinteractions

TheMMB-basedmodelsupportsaclearsetofperspectivesregardinginner/outerinteractions,someofwhicharenowbrie?ydiscussed.OneinterpretationofthelayerI–IVvelocityincrementsisthatthenetcirculationassociatedwiththeoutwardtransportofvorticityfromlayerII(supportingboundarylayergrowth)isasymptoticallybalancedbyanetinwardtransportofmomentumfromlayerIV(requiredforthegenerationofasurfacedragforce;?gure4andtable2).Theseattributesrathernaturallyalignwiththemomentumsource/sinkcharacterascribedtotheattachedanddetachededdies,respectively.Speci?cally,theattachedanddetachededdiesareviewedastheactivevorticalmechanismsbywhichlayerIIturbulenceissustained(andnear-wallvorticityistransportedoutwardtolayerIV),andbywhichfree-streammomentumisextractedandtransportedintolayerII,respectively.Thesephysicalattributes?ndconsistencywithperturbedboundarylayerstudies,indicatingthattheinnerregionrapidlyadjusts,andfollowingperturbation,rapidlyrecovers(e.g.Smits&Wood1985).Conversely,recoveryoftheouterlayerrequiresverylongredevelopmentlengthsdownstreamofaperturbation(e.g.Eaton&Johnston1981).

Anotherimportantfeatureisthatthestatisticalcentreoftheinner/outerinteractionisinlayerIII.ThiscentrestheinteractiononthezonewheretheMMBundergoesabalancebreakingandexchange;thenetresultbeingthatthedynamicschangefromabalancebetweentheviscousforceandtheturbulentinertiatoabalancebetweentheturbulentinertiaandthemeanadvection.Similarly,thepresentmodelalsoattachesdynamicalimportancetotheinteractionbetweenthemotionsthatproducepositiveKv=vyinlayerIIandthosethatproducenegativeKv=vyPhil.Trans.R.Soc.A(2007)

Turbulentboundarylayermodel835

inlayerIV.5Inconnectionwiththis,increasingscaleseparationrequirestheaforementioneddynamicalaccommodationwithincreasingdC,andthusthepresentmodelidenti?esanintermediaterangeofscalesthatcontinuallyadjustswithdCasplayinganessentialroleintheinner/outerinteraction.

(f)Thelogarithmiclayer

Asnotedin§2b,theprevalentmodelhasratherdirectconnectionstothemeanpro?leanditsoverlaplayer-basedderivationofthelogarithmicpro?le.Conversely,multiscaleanalysesoftheMMB(forchannelandCouette?ow)revealthatalogarithmicpro?leispossible,butnotbyassumingtheexistenceofanoverlaplayer.Speci?cally,throughtheuseofanadjustedReynoldsstressfunction,Fifeetal.(2005a,b)revealthattheMMBadmitsascalinglayerhierarchythatnaturallygivesrisetoalength-scaledistribution,andthatthesecharacteristiclengthsasymptoticallyscalewithy.Thus,theseanalysesprovideajusti?cation,rigorouslyfoundedintheMMB,forthedistancefromthewallscaling-basedderivationofthelogarithmicpro?le.TheyalsoidentifytheconditionsrequiredfortheMMBtoadmitanexactlogarithmicmeanpro?le.Brie?y,foreachmemberofthefamilyofadjustedReynoldsstresses(associatedwiththevalueofaparameterb),therecorrespondsalayerIII-likestructureacrosswhichtheMMBundergoesabalancebreakingandexchange.Thelength-scalesassociatedwitheachoftheselayerscanberigorouslyderivedaccordingtotheconditionthatwhentheyareusedtonormalizetheMMB,theMMBattainsaninvariantform.Itfollowsthatanexactlylogarithmicmeanpro?lewilloccurwhenthelocallynormalizedsecondderivativeoftheReynoldsstress(divergenceoftheLambvector)remainsinvariantoverarangeofy(i.e.forarangeofb).ThelayerhierarchyinitiatesnearyCZ30andterminatesaty/dx0.5(Fifeetal.2005a).Fromthisanalysis,itissurmisedthatpurelylogarithmicbehaviourisexpectedonlyoverarangeofyinteriortothesebounds,i.e.where‘endeffects’donotdisrupttheself-similarityofadjacentblayers.Ingeneral,however,thescalehierarchycoverspartoflayerII,alloflayerIIIandpartoflayerIV.Interestingly,asnotedinWeietal.(2005a),thiscoverageofdifferingmomentumbalancelayersprovidesanaturalexplanationfortheexistenceandtheobservedbreakpointsofthevaryinglogarithmic-likepro?lebehavioursnotedin§2.Itisalsoworthmentioningthesimilaritiesbetweenthismathematicalstructureandthe?ndingsregardingthehierarchicalmotionpropertiesofthelogarithmiclayer.Amongtheseisthepotentialcorrespondencebetweenagivenblayerandthestatisticalensembleofhairpinvortexpacketsthatrisetoagiveny-location.

4.Summary

AphysicalmodeloftheturbulentboundarylayerthatisbaseduponthepropertiesoftheMMBhasbeendescribedandcomparedwiththeprevalent,well-established,model.SomedistinctionsbetweentheperspectivesprovidedbythetwomodelsareInterestingly,supportforparticularlyintensevorticalmotioninteractionsatintermediatescaleisgivenbytheremarkableobservationthatasdCvariesthepeakinthedissipationspectraexhibitsthesameReynoldsnumberdependenceasdoesthewidthoflayerIII(Tsuji1999).

Phil.Trans.R.Soc.A(2007)5

836J.Klewickietal.

nowbrie?ynoted.Intheprevalentmodel,theregionwheretheeffectsofviscosityaredeemedsigni?cantshrinkslike1/dCwithincreasingdC.Inthepresentp?????model,theregionwheretheeffectsofviscosityaresigni?cantshrinkslike1=dC.Forthereasonsdiscussedin§1,theimplicationsofthisdifferencearefarreaching.TheprevalentmodelderivesperspectivesrelativetotheReynoldsstressasasourceofturbulentmomentumtransport.Thepresentmodelderivesperspectivesfromthesource/sinkpropertiesoftheReynoldsstressgradientrelativetoaffectingatimerateofchangeofmomentum.Distinctfromtheprevalentmodel,thepresentmodelcentrestheinner/outerinteractioninthetraditionallogarithmiclayerandintheregionwherethemeanviscousforcetransitionstohavingahigher-ordercontributiontothedynamics.Theprevalentmodelismostoftendiscussedinconnectionwiththeassumptionofanoverlaplayerandthecorrespondingderivationofthelogarithmicmeanpro?le.Thepresentmodelincorporatesthescalehierarchy-basedderivationadmittedbytheMMB.

Lastly,somesimilaritiesanddifferencesbetweenthescalehierarchyandtheoverlaplayerareworthnoting.Onesimilarityisthattheyconstituteaninternalintermediatelayer,insulatedfromboundaryconditioneffects,thatexhibitsself-similarbehavioursaccordingtolocaldynamics.Thenotionthatalogarithmiclayerisaninertialsublayer,however,requiresclari?cation.ThescalehierarchyexistsinlayersII,IIIandIV.AccordingtotheMMB,IIandIIIarelayerswherethemeanviscousandReynoldsstressgradientshavethesameorderofmagnitude.Thus,oftheselayers,onlyIVcanrationallybeconsideredaninertiallayer.Interestingly,recentempiricalresultsrelatingtowherethe‘true’logarithmiclayerstartsareconsistentwiththisnotion(seeWeietal.2005a).Lastly,althoughitisnotreadilyapparenthowtheoverlaplayerandthescalehierarchydescriptionsmightbeequivalent,thepossibilityremainsthatbotharevaliddescriptionsofthesamephysics.

ThisworkwassupportedbytheNationalScienceFoundationundergrantCTS-0555223andtheDepartmentofEnergyundergrantW-7405-ENG-48.

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