THEEFFECTSOFSURFACECURVATUREONLAMINARBOUNDARY-LAYERFLOW

Inatleastonecasewherethepotentialflowintheneighborhoodoftheouteredgeoftheboundary-layervariesrapidly,theboundary-layertheoryhasnotbeenwellunderstood.Theinterestinthisproblemledtheauthortothestudyofthelaminarflowofaviscousincompressiblefluidoveracurvedsurfacewhosecurvature,ashasbeenfoundpreviously,displaysratherlargeeffectsonthenatureofboundary-layerflow.Thespecificpointtobeinvestigatedhereisthequestionastohowtojointheboundary-layerwiththepotentialflow.Onthebasisofthefactthattheviscouseffectsdecayexponentiallyinthelateraldirection,ithasbeenintuitivelysuggestedthat,intheneighborhoodoftheouteredgeoftheboundary-layer,thevelocityinthemainstreamdirectionuishouldasymptoticallyapproachthatinthemainstream.Namely,wherexandyarethecoordinatesshowninFig.1.Rc(=1/ε2},KandRdenoteReynoldsnumber,thecurvatureofthesurfaceandalargeconstantrespectively.Byanorder-of-magnitudeanalysis,Murphyobtainedthefollowingdifferentialequationofthemotioninaboundary-layeroveraparticularcurvedsurfacewithcurvatureK=A/εα~1/2,alongwhichthesurfacepressuregradientiszero.whereAandfarerespectivelyasmallconstantandthestreamfunction;ηisdefinedasy/(2εx~1/2),andfisafunctionofnnnnnnonly.Inthecaseofzeropressuregradient,Murphygaveanapproximaterepresentation,ofthepotentialflowasfollowswhich,byvirtueof(1),loadstoTheno-slipconditionadthewallyieldsHerotoforetheproblemmaybefurthersimplifiedtoasteaightforwardnumericalcom-pulation.SinceAisverysmall,assumeasolutionof(2)ofthisformByexpandingbothsidesof(8)intopowersericsofA,thereresultthefollowingjoiningconditionswhenη≤1Fromtheno-slipconditionatwallitfollowsSituilarlythedifferentialequationsforvariousordersmaybodeducedfrom(2)and(5)Thefirstequationof(7)appearsinthecaseoftheflatplateproblemiswell-knownBlasiussolution.Itcanbeshownatthepresentstagethatthesolutionof(7)andconditionsgivenin(6)areconsistent.Substituting(6)in(7)andneglectingtermsofordere-η3itindicatesthatexpressionsin(6)arethesolutionsof(7)intheregionoflargeη.Consequentlythereisnodifficultyinsatisfyingthejoiningconditions.Withthejoiningconditionsgivenin(6)andno-slipconditions,theequationof(7)canbeeasilysolvednumerically.Forsmall17,thefollowingsolutionsarefoundorderA0orderA1whereorderA2wherewherea=1.32824,C1andC2arenumericallyfoundtobe-5.767and-2.3respectively.ItisnoticedthatMurphy'ssolutionmissesthepartC1h1,andthataisnotaconstant,butafunctionofA.Similarly,thesolutionsofhigherordercanbeobtained.InFig.3thetangentialvelocitiesarecalculatedtoorderofA2,Inconclusion,thefollowingpointsmaybenoted1.theboundarylayerflowjoinsthepotentialflowasymptotically,thatis,withanorderofo(e-η3).2.thecurvaturehaslittleeffectonbound