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.