%0 Journal Article
%T THEEFFECTSOFSURFACECURVATUREONLAMINARBOUNDARY-LAYERFLOW
%J Á¦Ñ§Ñ§±¨
%P 375-378
%D 1957
%R 10.6052/0459-1879-1957-4-1957-030
%X Inatleastonecasewherethepotentialflowintheneighborhoodoftheouteredgeoftheboundary-layervariesrapidly,theboundary-layertheoryhasnotbeenwellunderstood.Theinterestinthisproblemledtheauthortothestudyofthelaminarflowofaviscousincompressiblefluidoveracurvedsurfacewhosecurvature,ashasbeenfoundpreviously,displaysratherlargeeffectsonthenatureofboundary-layerflow.Thespecificpointtobeinvestigatedhereisthequestionastohowtojointheboundary-layerwiththepotentialflow.Onthebasisofthefactthattheviscouseffectsdecayexponentiallyinthelateraldirection,ithasbeenintuitivelysuggestedthat,intheneighborhoodoftheouteredgeoftheboundary-layer,thevelocityinthemainstreamdirectionuishouldasymptoticallyapproachthatinthemainstream.Namely,wherexandyarethecoordinatesshowninFig.1.Rc(=1/¦Å2},KandRdenoteReynoldsnumber,thecurvatureofthesurfaceandalargeconstantrespectively.Byanorder-of-magnitudeanalysis,Murphyobtainedthefollowingdifferentialequationofthemotioninaboundary-layeroveraparticularcurvedsurfacewithcurvatureK=A/¦Å¦Á~1/2,alongwhichthesurfacepressuregradientiszero.whereAandfarerespectivelyasmallconstantandthestreamfunction;¦Çisdefinedasy/(2¦Åx~1/2),andfisafunctionofnnnnnnonly.Inthecaseofzeropressuregradient,Murphygaveanapproximaterepresentation,ofthepotentialflowasfollowswhich,byvirtueof(1),loadstoTheno-slipconditionadthewallyieldsHerotoforetheproblemmaybefurthersimplifiedtoasteaightforwardnumericalcom-pulation.SinceAisverysmall,assumeasolutionof(2)ofthisformByexpandingbothsidesof(8)intopowersericsofA,thereresultthefollowingjoiningconditionswhen¦Ç¡Ü1Fromtheno-slipconditionatwallitfollowsSituilarlythedifferentialequationsforvariousordersmaybodeducedfrom(2)and(5)Thefirstequationof(7)appearsinthecaseoftheflatplateproblemiswell-knownBlasiussolution.Itcanbeshownatthepresentstagethatthesolutionof(7)andconditionsgivenin(6)areconsistent.Substituting(6)in(7)andneglectingtermsofordere-¦Ç3itindicatesthatexpressionsin(6)arethesolutionsof(7)intheregionoflarge¦Ç.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-¦Ç3).2.thecurvaturehaslittleeffectonbound
%U http://lxxb.cstam.org.cn/CN/abstract/abstract138366.shtml