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.