CHAPTER 4 LAMINAR BOUNDARY LAYER INVISCID FLOW PAST WEDGES AND CORNERS FALKNER-SKAN SOLUTION In fluid dynamics, the Falkner–Skan boundary layer (named after V. M. Falkner and Sylvia W. Skan) describes the steady two-dimensional laminar boundary layer that forms on a wedge, i.e. flows in which the plate is not parallel to the flow. It is a generalization of the Blasius boundary layer. Falkner and Skan (1931) found that similarity was achieved by the variable = ^ , which is consistent with a power-law free stream velocity distribution. = a = 2 + 1 The exponent may be termed the Falkner-Skan power-law parameter. The constant must make dimensionless but is otherwise arbitrary. The best choice is g = h ija gk , which is consistent with its limiting case for = 0 , the Blasius variable.
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CHAPTER 4 LAMINAR BOUNDARY LAYER INVISCID ...syahruls/resources/MKMM-1313/Chapter-04/...CHAPTER 4 LAMINAR BOUNDARY LAYER INVISCID FLOW PAST WEDGES AND CORNERS FALKNER-SKAN SOLUTION
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EFFECTOFPRESSUREGRADIENTSEPARATIONANDFLOWOVERCURVEDSURFACESAllsolidobjectstravelingthroughafluid(oralternativelyastationaryobjectexposedtoamovingfluid)acquireaboundarylayeroffluidaroundthemwhereviscousforcesoccurinthelayeroffluidclose to the solid surface. Boundary layers can be either laminar or turbulent. A reasonableassessmentofwhethertheboundarylayerwillbelaminarorturbulentcanbemadebycalculatingtheReynoldsnumberofthelocalflowconditions.Flowseparationoccurswhentheboundarylayertravelsfarenoughagainstanadversepressuregradientthatthespeedoftheboundarylayerrelativetotheobjectfallsalmosttozero.Thefluidflowbecomesdetachedfromthesurfaceoftheobject,andinsteadtakestheformsofeddiesandvortices.Boundarylayerseparationisthedetachmentofaboundarylayerfromthesurfaceintoabroaderwake.Boundarylayerseparationoccurswhentheportionoftheboundarylayerclosesttothewallorleadingedgereversesinflowdirection.Theseparationpointisdefinedasthepointbetweentheforwardandbackward flow,where theshear stress is zero.Theoverallboundary layer initiallythickenssuddenlyattheseparationpointandisthenforcedoffthesurfacebythereversedflowatitsbottom.
Wehavesofarconsideredflowinwhichthepressureoutsidetheboundarylayerisconstant. If,however, thepressure varies in thedirectionof flow, thebehaviourof the fluidmaybe greatlyaffected.Let us consider flow over a curved surface as illustrated below. The radius of curvature iseverywherelargecomparedwiththeboundarylayerthickness.
A turbulent boundary layer can survive an adverse pressure gradient for some distance beforeseparating.Foranyboundarylayer,however,thegreatertheadversepressuregradient,thesoonerseparationoccurs.Theboundarylayerthickensrapidlyinanadversepressuregradient,andtheassumptionthat𝛿issmallmaynolongerbevalid.
Inalmostallcasesinwhichflowtakesplaceroundasolidbody,theboundarylayerseparatesfromthesurfaceat somepoint.Oneexception isan infinitesimally thin flatplateparallel to themainstream.Downstreamoftheseparationpositiontheflowisgreatlydisturbedbylarge-scaleeddies,andthisregionofeddyingmotionisusuallyknownasthewake.Asaresultoftheenergydissipatedbythehighlyturbulentmotioninthewake,thepressuretherereducedandthepressuredragonthebodyisthusincreased.Themagnitudeofthepressuredragdependsverymuchonthesizeofthewakeandthis,inturn,dependsonthepositionofseparation.Iftheshapeofthebodyissuchthatseparationoccursonlywelltowardstherear,andthewakeissmall,thepressuredragisalsosmall.Suchabodyistermedastreamlinedbody.Forbluffbody,ontheotherhand,theflowisseparatedovermuchofthesurface,thewakeislargeandthepressuredragismuchgreaterthantheskinfriction.
In a certain range of Reynolds number above the limiting value, eddies are continuously shedalternatelyfromthetwosidesofthecylinderand,asaresult,theyformtworowsofvorticesinitswake,thecentreofavortexinonerowbeingoppositethepointmidwaybetweenthecentresofconsecutivevorticesintheotherrow.Thisarrangementofvorticesisknownasavortexstreetorvortextrail.Von Karman considered the vortex street as a series of separate vortices in an ideal fluid anddeducedthattheonlypatternstabletosmalldisturbance,andthenonlyif:
Whereas earlier workers would have proposed a family of profiles to evaluate the parametricfunctionsinEq.4-135,Thwaites(1949)abandonedthefavorite-familyideaandlookedattheentirecollectionofknownanalyticandexperimentalresultstoseeiftheycouldbefitbyasetofaverageone-parameterfunctions.AsshowninFigure4-22,hefoundexcellentcorrelationforthefunction𝐹 𝜆 andproposedasimplelinearfit.