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.