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