THELAMINARBOUNDARYLAYEREQUATIONS

N.Curle

DOVERPUBLICATIONS,INC.Mineola,NewYork

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This Dover edition, first published in 2017, is an unabridged republication of the work originallypublishedinthe“OxfordMathematicalMonographs”seriesbyTheClarendonPress,Oxford,in1962.

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PREFACE

THEconceptof theboundarylayer, introducedbyLudwigPrandtl in1904,hasbeenaparticularlyfruitfulone.Researchonthistopichasnowreachedthestagewhere there is a certain body of fundamental definitive information which isunlikelytobesupersededtoanygreatextent.Thisrelatesmainlyto thesteadyincompressible laminar boundary layer in two dimensions. Work proceeds,however,andaconsiderablenumberofpapersarebeingpublished,onunsteadyboundary layers,on three-dimensionalboundary layers in incompressible flow,and upon such topics as boundary-layer stability. In compressible flow, too,where additional important parameters arise, much is being done and moreremains.

This monograph is one of a series, each of which is being written by anauthor on the general field in which his own research interests lie. It isinevitable,therefore,thattherewillbeacertainamountofbiasinthechoiceofmaterial.Ihavetriedtomakethebookreasonablyself-sufficient,thoughlackofspace has led to the omission of a number of very interesting problems ofboundary layers. The topics so axed include unsteady boundary layers andboundary-layer stability, boundary layers on porous walls with suction orblowing, boundary layers in three dimensions (including axi-symmetric flow)andboundarylayerswithvorticityinthemainstream.Foradiscussionofthesetopics referencemay bemade tomore encyclopaedicworks, such asModernDevelopments in Fluid Dynamics (Oxford, edited by S. Goldstein), thecompanion volumes on High Speed Flow (edited by L. Howarth), LaminarBoundaryLayers(Oxford,editedbyL.Rosenhead),andVolumesIIItoVoftheseriesHighSpeedAerodynamicsandJetPropulsion.

The purpose of Chapter 1 of this book is to introduce the boundary-layerconcept,andtoshowhowtheequationsofviscousflowaresimplifiedthereby.The standard boundary-layer parameters, and the usual integral forms of theboundary-layer equations, are discussed, the incompressible flow forms beingintroducedasspecialcasesofthemoregeneralcompressibleforms.Chapters2to6dealwithvariousaspectsofsolutionsinincompressibleflow,commencingwithanalyticsolutionsforthevelocityfield,thesebeingsolutionswhichmaybeexpressed in terms of functions which satisfy ordinary differential equations(Chapter2).There followdiscussionsofhigh-accuracynumericalsolutions forthevelocityfield(Chapter3),practicalmethodsofcalculation(Chapter4),and

ananalysisof thefactorswhichmightgovernthechoiceofamethod(Chapter5).VarioustypesofsolutionofthetemperatureequationinincompressibleflowwithsmalltemperaturedifferencesareconsideredinChapter6.Chapters7to9dealwithcompressiblelaminarboundarylayers,considerationbeingfirstgivento flow with zero pressure gradient (Chapter 7), then to flow with zero heattransfer(Chapter8),andthentoflowinwhichbothpressuregradientandheattransferarepresent(Chapter9).Finallythereisabriefdiscussion(Chapter10)of some aspects of the problem of the interaction between shock waves andlaminarboundarylayers.

Itismyhopethatthisbookwillbeofvaluetoawidevarietyofworkers.Inthe firstplace Ihave tried topresent thematerial ina sufficientlyorderedandlogicalmannerastomakeitofvalueasanintroductiontoboundary-layertheoryfor young researchworkers who are new to the subject, or to undergraduateswho are familiar with the elements of classical inviscid fluid dynamics.Secondly, the book should be of somevalue to researchworkers in this field,sinceoneofthethingswhichhasgovernedmychoiceofmaterialhasbeenthequestion of whether a particular piece of work has been an end in itself orwhether it has assisted in opening up theway for further advances. Finally, Ihave borne in mind the needs of practising engineers, and have tried wherepossible to indicate the limitations, the likely accuracy, and the practicalcomplexity of the methods described for calculating the various properties oflaminarboundarylayers.

In conclusion, I have great pleasure in expressingmy thanks to themanypeoplewhohavehelpedme,directlyorindirectly,inthewritingofthisbook.ToDr.M.J.Lighthill,F.R.S.,DirectoroftheRoyalAircraftEstablishment,whosestudentIwasatManchesterUniversity,forthewisecounselhegavemetheninsomanybranchesoffluiddynamics,andwhoseinfluenceis,Ihope,evidentinthis book. To my colleagues at the National Physical Laboratory, for thestimulating discussions I have had with them at various times, and mostparticularlyDr.J.T.StuartandDr.G.E.Gadd.Thesetwocolleagueshavebeengoodenoughtoofferusefulcommentsonafirstdraftofthisbook,althoughtheresponsibility for its deficiencies remains entirelymyown.ToMrs.M.E.M.Sayer, for her patient and careful typing of the manuscript, and her cheerfulapproach to the difficult task of readingmywriting.ToProfessorG.Temple,F.R.S.,editorofthisseries,andthestaffoftheOxfordUniversityPressforthecourteouswaytheyhavedealtwiththevariousproblemswhichhavearisen.ToSirGordonSutherland,F.R.S.,DirectoroftheNationalPhysicalLaboratory,forpermissiontowritethisbook.Thewritinghasnotinfactbeenpartofmyofficialduties, and the views expressed are entirelymy own. Thanks are also due to

variousbodiesforpermissiontousecopyrightmaterialsuchaschartsandtables.Thesebodies include theAeronauticalResearchCouncil, theClarendonPress,Oxford,

theController,H.M.StationeryOffice,Prof.L.Crocco,theInstituteoftheAero/SpaceSciences,theEditor,JournalofFluidMechanics,theDirector,National Physical Laboratory, the Royal Aeronautical Society, the RoyalSociety,theUnitedStatesAirForce,theEditor,Zeitschriftfürangew.Math,undMech.

Finally,tomywifeand,thoughtheyknowitnot,tomychildren,forsoorderingtheir lives as tomake the task ofwriting this bookmuch less difficult.To allthesepeoplegomysincerethanksfortheirmuchappreciatedhelp.

Hanworth,MiddlesexJune1961

N.C.Hawker-SiddeleyReaderDepartmentofAeronauticsandAstronautics

UniversityofSouthampton(FormerlyPrincipalScientificOfficerAerodynamicsDivision,NationalPhysicalLaboratoryTeddington,Middlesex)

CONTENTS

1. INTRODUCTION

1. Theequationsofviscousflow2. Boundarylayers3. Thelaminarboundarylayeronaplanewall4. Thelaminarboundarylayeronacurvedwall5. Conditionsinthemainstream6. Somestandardboundary-layercharacteristics7. Themomentumintegralequation8. Thekinetic-energyintegralequation9. Thethermal-energyintegralequation10. Incompressibleflow11. Crocco’stransformation12. VonMises’stransformation

2. ANALYTICSOLUTIONSFORINCOMPRESSIBLEFLOW

1. Flowparalleltoasemi-infiniteflatplate2. Flownearthestagnation-pointofacylinder3. TheFalkner-Skansimilaritysolutions4. Seriessolutionsfromastagnation-point5. Seriessolutionsfromasharpleadingedge6. Görtler’smodifiedseriesexpansions7. Meksyn’stechnique

3. NUMERICALSOLUTIONSFORINCOMPRESSIBLEFLOW

1. ThesolutionofHowarth2. ThesolutionsofTani3. ThesolutionsofCurle4. ThesolutionsofHartree,Leigh,andTerrill

4. PRACTICALMETHODSOFCALCULATIONFORINCOMPRESSIBLEFLOW

1. Pohlhausen’smethod2. Timman’smethod3. ThemethodsofHowarthandWalz4. ThemethodofThwaites5. ThemethodofStratford6. ThemethodofCurle7. Useoftheenergyintegralequation. ThemethodsofTaniandTruckenbrodt8. ThemethodofHead

5. COMPARISONSOFAPPROXIMATEMETHODSOFSOLUTION

1. Informationprovidedbyvariousmethods2. Comparisonofaccuracy

3. Easeofcomputation4. Conclusions

6. SOLUTIONSOFTHETEMPERATUREEQUATIONATLOWSPEEDS

1. Forcedconvectionfromaflatplate2. Temperatureofplatethermometerinmovingfluid3. Heattransfernearastagnation-point4. ThesolutionsofFageandFalkner5. Lighthill’smethod6. TheworkofLiepmannandCurle7. Spalding’smethod8. ThemethodofDaviesandBourne9. TheMeksyn-Merkmethod10. Curle’sanalysisbyStratford’smethod11. Squire’smethod12. Freeconvectionfromaheatedverticalplate

7. THECOMPRESSIBLELAMINARBOUNDARYLAYERWITHZEROPRESSUREGRADIENT

1. ValuesforviscosityandPrandtlnumber2. ThesolutionsofBusemannandKármán3. ThesolutionsofKármánandTsien4. ThesolutionsofEmmonsandBrainerd5. ThecalculationsofCrocco6. Summaryofresultsforuniformwalltemperature7. ThesolutionsofChapmanandRubesin8. Lighthill’sanalysis

8. THECOMPRESSIBLELAMINARBOUNDARYLAYERWITHZEROHEATTRANSFER

1. Howarth’smethod2. Young’smethod3. TheStewartson–Illingworthtransformation4. Rott’smethod5. ThemethodofOswatitschandWeighardt6. TheworkofCopeandHartree7. TheworkofIllingworth,Frankl,andGruschwitz8. TheinvestigationsofGadd

9. THECOMPRESSIBLELAMINARBOUNDARYLAYERWITHPRESSUREGRADIENTANDHEATTRANSFER

1. Accuratenumericalsolutionsforspecialcases2. Kalikhman’smethod3. ThemethodofCohenandReshotko4. Monaghan’smethod5. Curle’smethod6. ThemethodofLuxtonandYoung7. ThemethodofPoots8. ThemethodsofLilleyandIllingworth9. Curle’smethodforcalculatingheattransfer

10. INTERACTIONSBETWEENSHOCKWAVESANDBOUNDARYLAYERS

1. Principalresultsofexperimentalinvestigations2. Summaryofearlytheoreticalinvestigations3. Gadd’sanalysesforinteractionscausingseparation4. TheanalysisofHäkkinen,Greber,Trilling,andAbarbanel5. TheworkofGaddandGreber6. ThemethodofCurle

REFERENCES

INDEX

1

INTRODUCTION

THIS book is about solutions of the laminar-boundary-layer equations. Theconcept of the boundary layer, one of the corner-stones of modern fluiddynamics, was introduced by Prandtl (1904) in an attempt to account for thesometimes considerable discrepancies between the predictions of classicalinviscid incompressible fluid dynamics and the results of experimentalobservations.Asanexample,wemayremark thataccording to inviscid theoryany body moving uniformly through an unbounded homogeneous fluid willexperiencezerodrag!

Nowtheclassicalinviscidtheoriesassumethattheviscousforcesinafluidmay be neglected in comparison with the inertia forces. This, indeed, wouldseemareasonableapproximation,sincetheviscosityofmanyfluids(andofairinparticular)isextremelysmall.However,incertainregionsofflow,fortunatelyoften limited, the viscous forces can still be locally important, as Prandtlobserved. The reason for this is that a typical viscous stress is of magnitudeμ(∂u/∂y), where μ is the viscosity, u is the velocity measured in a directionparallel to thatof thestress,andy isdistancemeasurednormal thereto,so thatwhen the velocity gradient (or shear) ∂u/∂y is large the viscous stress canbecome important even thoughμ itself is small. Itwas Prandtlwho remarkedthat in flow past a streamlined body, the region in which viscous forces areimportant is often confined to a thin layer adjacent to the body, and to a thinwakebehind it.This thin layer is referred toas theboundary layer.When thiscondition holds the equations governing the motion of the fluid within theboundary layer take a form considerably simpler than the full viscous-flowequations,thoughlesssimplethantheinviscidequations,anditisthesolutionoftheseequationswithwhichweshallbepresentlyconcerned.

Analternativemethodoflookingatthisconceptisasfollows.Theinviscid-flow equations are of lower order than the viscous-flow equations, so fewerboundary conditions can be satisfied in a mathematical solution of a givenproblem.Thusaninviscid-flowsolutionallowsafinitevelocityofslipatasolidboundary, whereas the solution of the viscous-flow equations does not allow

suchslip.Inotherwords,theinviscid-flowsolutionassumestheexistenceofanappropriatevortexsheetatthesolidboundaries.Nowinrealitythisvorticitywilldiffuse outwards from the boundary (in much the same way that heat woulddiffuse from a heated body) and will be convected with the stream. Thus,considering now the flow past a flat plate, the time t inwhich fluid travels adistancexparalleltotheplatewillbeoforderx/U,whereUisatypicalvelocity,andinthis timethevorticitywillhavediffusedoutwardsthroughadistanceoforder , where v = μ/ρ is the kinematic viscosity. This is anindicationoftheboundary-layerthickness.

Before turning toamorequantitativediscussion,mentionmustbemadeofthephenomenonofboundary-layerseparation.Whenthefluidisproceedingintoaregionofrisingpressure,itissloweddownbythisretardingforce.Intheouterpartoftheboundarylayer,wherethekineticenergyislarge,thisresultsonlyinarelativelysmallslowingdownofthefluid,buttheeffectontheslower-movingfluidnearertothewallcanbeconsiderable,andifthepressureriseissufficientitcanbebroughttorest,and,fartherdownstream,aslowback-flowbesetup.Insuch circumstances the forward flow must leave the surface to by-pass thisregion,andboundary-layerseparationissaidtohavetakenplace.Iftheregionofseparated flow is extensive, the separation can have a back-reaction on theexternalflow,whichisthenquitedifferentfromwhatitwouldhavebeenintheabsence of the boundary layer. If the separated region is limited, on the otherhand,theexternalflowmaynotbesignificantlyaffected,andtheflowfieldmaybe calculatedby calculating firstly the external flow (on the assumptionof noboundary layer) and then calculating the boundary layer appropriate to thisexternalflow.

Theabovequalitativeanalysisisrestrictedtolow-speedflows,butservestoindicatethenatureoftheboundarylayerinasimpleway.Atsupersonicspeeds,forexample,interactionsoftheboundarylayerwiththeexternalstreambecomemore important,and lead toconsiderable theoreticaldifficultieswhichwillnotbediscussedatthisstage.Inwhatfollowsaquantitativeanalysiswillbegivenofhowtheboundary-layerequationsmaybededucedfromtheexactequationsofviscous flow, and detailed discussions will be given of some of the pointstoucheduponbrieflyabove.

1. TheequationsofviscousflowWe take cartesian coordinates (x,y), with associated components of fluid

velocity(u,υ).Thefluidisassumedtohavepressurep,densityρ,andabsolutetemperature T, and these are functions of x and y only, in view of the

approximation(madethroughoutthisbook)thattheflowistwo-dimensionalandsteady.Theequationsofmotionthenexpressthebasicphysicalideasthatforagivenelementoffluidthereisconservationofmass,momentum(exceptinginsofarastheelementisacteduponbyvariousforces),andenergy(exceptinginsofar as work is done by these same forces). For a general derivation of theseequations,referencemaybemadetovolumeIofModernDevelopmentsinFluidDynamics,HighSpeedFlow (Howarth,1953). Insteady two-dimensional flowthe equation of conservation of mass takes the form

This equation is usually referred to simply as the equation of continuity. Theequations of conservation of momentum in the x and y directions (themomentum equations) become

and

where

andX,Y are theexternal forcesperunitmassof fluid.Finally theequationofconservation of energy (the thermal energy equation) takes the form

where

Therearethusfourequations,(1),(2),(3),and(5),forthefiveunknownsu,υ,p,ρ,T,andthesearesoluble,inprincipleatanyrate,whenanequationofstateisdefined, relating p, ρ, and T. For a perfect gas this takes the form

where isthegasconstant.In the above equations cp is the specific heat at constant pressure, usually

taken to be constant, and k is the thermal conductivity, related to thethermometric conductivity κ by the relationship

Foraphysical interpretationof thequantitiesκ,k,μ,andv=μ/ρ, thereader isreferredtothebookModernDevelopmentsinFluidDynamics(Goldstein,1938).Itwillsufficeforthepresenttoremarkthatvisaparameterdeterminingtherateat which vorticity is diffused, whilst κ determines the rate at which heat isdiffused. The ratio

accordinglydeterminestherelativeratesofthesetwotypesofdiffusion,andiscalled the Prandtl number. It is usual to assume that the Prandtl number isconstant, and this holds true for air over quite awide rangeof conditions, thevaluebeingabout0·72.ItwillbeseenlaterthatconsiderablesimplificationsareoftenpossibleifitisassumedthatthePrandtlnumberisunity,anapproximationthatisnotwithoutvalueforair.

2. BoundarylayersItwillbenotedthattheviscosityμappearsinequations(2)and(3)onlyasa

multiplicativefactorofvelocitygradients,orofpowersorproductsofvelocitygradients.Accordingly,iftheviscosityissmall,classicalfluiddynamicstheory,whichneglectsviscosity,willbevalidexceptinregionswherevelocitygradientsare large. Similarly, provided the Prandtl number σ is not too small, smallviscosityimpliessmallthermalconductivity,sothatthetermsinvolvingkin(5)areimportantonlywherethetemperaturegradientsarelarge.

Now it is often found in practice that the regions of high velocity andtemperaturegradientsareconfinedtoanarrowregionneartosolidwalls,knownastheboundarylayer,andtoathinwakebehindstreamlinedbodies.Insuchadomain, considerable simplifications of the equations of motion are possible,even though all the terms involvingμ and k cannot be neglected, aswas firstshown by Prandtl (1904) in a paper of fundamental importance. In low-speedflowitisusuallypossibletoconsiderthedevelopmentoftheboundarylayerasaseparate problem from that of the substantially inviscid external flow, the

exceptionbeingwhenthereisasubstantialregionofseparatedflow.Whentheexternal flow is supersonic, however, there is an interaction between theboundary layer and the external stream which must be taken into account.Crudelywemaysaythatthoughanartificiallyintroduceddisturbancecannotbepropagated upstream in awholly supersonic flow, the presence of a boundarylayer,inwhichtheflowsufficientlyneartothewallswillbesubsonic,providesamechanismforsuchupstreaminfluence.Accordinglytheexternalstreamdoesnot approximate to that obtained in the absence of the boundary layer, andcannotbeindependentlyprescribed.

Inspiteof thisdifficultyit isstilluseful tobeginbyconsideringseparatelytheboundary-layerapproximationandtheinviscidapproximation,asusecanbemade of these results even in certain problems in supersonic flow where theinteractionbetweenboundarylayerandexternalstreamisparticularlyimportant,as,forexample,whenashockwaveinteractswithaboundarylayer.

3. ThelaminarboundarylayeronaplanewallWe begin by deriving the boundary-layer equations for flow over a plane

wall.Thex-axis is takenalong thewall and they-axisperpendicular to it.Weassumethatthethicknessδυofthelayerinwhichthevelocitygradient∂u/∂yislarge,wherethevelocityurisesrapidlyfrom0atthewalltoavalueu1,ismuchsmaller than a typical length l in the flow field as a whole. Equally well weassumethatδt,thethicknessofthelayerinwhichthetemperaturegradient∂T/∂yislarge,andinwhichthetemperatureofthefluidchangesfromthetemperatureof thewallTw toavalueT1, isalsomuch less than l.Weshall assumeat thisstagethatδυandδtarethesameorderofmagnitude,eachbeingoforder .Thisrestrictioncanlaterberemoved,providedδvandδtareboth .Weletu0,ρ0,T0 be typical values of velocityu, densityρ, temperatureT, andmay thendeduce the order of υ from (1). This equation shows that

whichisoforderp0u0/l.Hence,uponintegratingacrossthelayerfromy=0toy= δ, we find that ρυ is of order (ρ0u0/l)δ, so that

Thusυissmallcomparedwithu,withinandattheedgeoftheboundarylayer.Wecannowexaminethemagnitudesofthevarioustermsin(2).Indoingso

werememberthatthederivative∂F/∂xofafunctionFwillbemuchsmallerthan

the derivative ∂F/∂y. In fact

Thus

and

Ideal fluid-dynamics theory rejects all the terms involving viscosity, namelythose in (9) and (10),butboundary-layer theory retains the former, sinceδ2 issmallaswellasμ0.Weseethatthesetermsarethesameorderofmagnitudeas(8) when

whereRistheReynoldsnumber

We have chosen δυ as the relevant value of δ since we are considering amomentum equation. Upon rejecting the terms in (10), then (2) simplifies to

Wenotethattherejectedtermsareoforder(δ/l)2timesthoseretained.Wenowdealinasimilarmannerwithequation(3).Wecanseethat

and

Accordingly, the terms in (15) areoforderδ/l times those rejected in (2), andthose in (13) and (14) are of order δ/l times those retained. It follows that

Now Y will usually be zero. For example body forces can be neglected inproblems of forced convection, and in problems of free convection the bodyforce(gravity)willactinthexdirection.Assuming,then,thatY=0,itfollowsfrom (16) that the pressure gradient ∂p/∂y is small, and the pressure changeacrosstheboundarylayerisverysmall,being ,whichisneglected.Thus (3) reduces simply to

We turnnow to the thermalenergyequation (5).By identical reasoning tothatgivenaboveitfollowsthatthetwotermsontheleft-handsideareofequalorderofmagnitude.Thetermυ(∂p/∂y)vanishesby(17),andtheterm∂(k∂T/∂x)/∂x is of order (δ/l)2 times ∂(k∂T/∂y)/∂y. Finally, in Φ, the term (∂u/∂y)2 is

, which is at least (l/δ)2 times the other terms; these are thereforeneglected. It follows that (5) becomes

When body forces are negligible, an alternative form of (18) is obtained byadding u times (12) to (18). This yields

Wenoteinpassingthatthetermsρu∂(cpT)/∂xand∂(k∂T/∂y)/∂yin(18)will

be the same order of magnitude if

or

where(Pe)isthePecletnumber

We deduce from (11) and (20) that

so that the assumption of similar velocity and thermal boundary-layerthicknesses is equivalent to the assumption that the Prandtl number is not toodifferentfromunity.

Theequations(1),(7),(12),(17),and(19)aretheboundary-layerequationsforthefiveunknownsu,v,p,ρ,T,whichmustbesolvedsubjecttothefollowingboundaryconditions.Atthewally=0wehaveu=v=0,sincethereisnoslipornormalvelocityatafixedsolidwall.FurtherthereisaboundaryconditiononT at y = 0, which is usually that either T or ∂T/∂y is prescribed there (ordeterminablefromasubsidiaryequation).Attheedgeoftheboundarylayerwemayassumethatu,T,parefunctionsonlyofxsincevariationsinthequantitieswith y outside the boundary layer are small compared with those within theboundary layer. Further, the boundary layer tends into the mainstreamasymptotically, so that the edge of the boundary layer is y → ∞, and theboundaryconditionsattheedgeareu→u1(x),T→T1(x),ρ→ρ1(x).Fromthepractical point of viewwemay regard the edge of the boundary layer as thepositionwhereu/u1=0.99say,thoughmathematicallythereisnoedgeassuch.

Thesolutionof(17)istrivial,forifwelety→∞inthemomentumequation(12), then provided there are no body forces we find that

sothatthepressureisgivenintermsofthemainstreamvelocityanddensity.

4. Thelaminarboundarylayeronacurvedwall

The equations as derived above are valid only for flow past a planewall.Physically one is often interested in flow past a curved surface, such as acylindrical obstacle say. To consider this flow it is necessary to take a moregeneral form of the Navier-Stokes equations than (1) to (6), by introducinggeneralorthogonalcoordinatesx,y,withxmeasuredparalleltoandynormaltothe wall. Since these coordinates are non-planar a considerable number ofadditionalterms(involvingthecurvatureκ1ofthewall)appearintheequations.Thedetailsarehereomitted,andthereaderreferredtoGoldstein’sbook(§45),ModernDevelopmentsinFluidDynamics,vol.I,wherethedetailedanalysisisgiven for incompressible flow; the analysis for compressible flow follows thesame lines. It is sufficient for us to remark that when the boundary-layerapproximationsaremadeexactlyasinsection3,subjecttotheassumptionthatκ1lisoforderunity,thentheequationsreducetotheforms(1),(12),(18),and

Accordingly the only difference between the equations for plane and curvedwalls lies here, since a normal pressure gradient is required to balance thecentrifugalforce.Nevertheless,thetotalchangeofpressureacrosstheboundarylayerisstillsmall,sincetheboundarylayeristhin,sothepressuremaystillbetaken to be a function of x only. Thus the equations for plane and curvedboundariesareidentical.

Thepracticalsignificanceofthisfactisthatincalculatinglaminarboundary-layerdevelopment,allcurvatureeffectsmaybeneglected,exceptinginsofarastheseaffecttheflowoutsideandattheedgeoftheboundarylayer.

5. ConditionsinthemainstreamCertain relationships between quantities in the external stream will be

constantly referred to in what follows. It is useful, therefore, to derive theserelationshipsherebriefly, so that theymaybeusedwhenrequired,and furtherdetailswillbefoundinanystandardtextbookonGasDynamics,suchasModernDevelopmentsinFluidDynamics,HighSpeedFlow,vol.I(Howarth,1953).

Itiscontinuallynecessarytointroducethelocalvelocityofsound,forwhichtheusual symbol isa, and in amoving stream this is the velocitywithwhichsound waves are propagated relative to the stream. It is given by

wherethederivativeistakenatconstantentropyS.Nowintheexternalstream,where viscosity and conductivity are negligible, the energy equation may be

usedtoshowthattheentropydoesnotvaryalongastreamline,exceptinpassingthrough a shock. Hence, outside the boundary layer, the derivative in (22) istaken along the streamlines. It also follows that if the entropy is uniform atinfinity itwillbe thesameeverywhere(upstreamofanyshockswhichmaybepresent). Such a flowwas formerly called isentropic, though Howarth (1953)uses the term homentropic and this term will be used here. The condition ofhomentropicflowmaybeused,alongwiththeequationofstateforaperfectgas,(7), to show that

inthemainstream,whereγistheratioofthespecificheatsofthegascp/cυ,andisequalto1·4foraperfectgas.Itisworthyofnotethatairbehaveslikeaperfectgas over a considerable range of conditions. With the notation used indeveloping the boundary-layer theory earlier, (23) is written as

and it then follows from (22) that a1, the local velocity of sound outside theboundary layer, is given by

thesecondequalitybeingdeducedfrom(7).Bymakinguseofthefactthat,foraperfect gas

wemaydeducefrom(25)that

Itmayalsobeshownfrom(7)and(24)that

and

FromtheNavier–Stokesequations,(1)to(6),itisnotdifficulttoshow,whenviscosityandconductivityareneglectedandtheexternalforcesareassumedtobe conservative and derived from a potential Ω, that

alongastreamline.Thisresultmaybeobtainedbyaddingtogether(5),utimes(2)andυtimes(3).Uponneglectingexternalforces,itfollowsthatattheedgeoftheboundarylayer(where

whichresultagreeswith thatobtainedby lettingy→∞in (19).Uponmakinguse of (26) it follows that (29) may be written as

whereTH1 is the total temperature, ormainstream stagnation temperature, andM1isthelocalMachnumberu1/a1attheedgeoftheboundarylayer.WhenM1isprescribedasa functionofx,T1 followsatonce from(30). If,however, thelocalvelocityu1 at theedgeof theboundary layer isprescribed, the followingrelationships will be useful, namely

and

6. Somestandardboundary-layercharacteristicsInpracticeweareoftennotinterestedinthefulldetailsofthevelocityand

temperature profiles, but only in some overall measure of boundary-layerthickness or in some quantity as calculated at the wall, such as the heattransferredfromthefluidtothewallorthefrictionaldragexertedbythefluidonthewall.Wewillnowdiscusssomeof themore interesting(andimportant)ofthesecharacteristics.

Imagine inviscid flowpastabody, thevelocityof slipat the surfacebeingu1(x).Inarealfluidtheeffectofviscosityistoslowdownthefluidneartothesurface,bringingthefluidatthesurfacetorestandcausingthestreamlinestobedisplacedoutwardsbyanamountwhichmaybecalculatedasfollows.Considerastreamlinewhichisadistanceh(x)fromthewalland,butforviscosity,wouldhavebeenonlyh(x)—δ1(x)fromthewall.Thenthetotalmassoffluidflowinginunittimebetweeny=0andy=histhatwhichwouldbeflowingbetweeny=0and y = h–δ1 if the velocity were everywhere u = u1(x). Thus

or

Thusstreamlinesfarawayfromthewallhavebeendisplacedbyanamount

whichdistanceisknownasthedisplacementthickness.Inasimilarwaythedifferencebetweentheactualrateatwhichmomentum

isconvectedbetweeny=0andy=h,thatis andtherateatwhichthe

actualmomentumρuwouldbeconvectedbyaflowwithzeroviscosity,thatis

,maybecomparedwithinviscidflowmomentumbeingconvectedin

a region between y = 0 and y = δ2. Thus

or

Ash→∞thisyields

whichisknownasthemomentumthickness.Twoothermeasuresofboundary-layerthicknesswhichareofsomeinterest

are the kinetic-energy thickness, which is a measure of the defect of kinetic-energy flowwithin the boundary layer, and the enthalpy thicknesswhich is ameasureofthedefectofheatflowwithintheboundarylayer.Thekinetic-energy

thickness is defined as

andtheenthalpythicknessas

Theskinfrictionatthewall,τw,isdefinedastheviscousforceperunitareaacting at the surface, and is given by

A position of zero skin friction is one forwhich ∂u/∂y changes sign near thewall,andhence,sinceu=0aty=0,thelocalvelocitymustbeofoppositesignoneithersideofsuchapoint.Itfollowsthatwhentheboundarylayerisoffinitethicknesstheforwardflowmust‘separate’fromthewallinordertoby-passthebackwardflow.Forthisreasonapositionofzeroskinfrictionisreferredtoasaposition of boundary-layer separation. The serious effects of boundary-layerseparationhavealreadybeenreferredto.

Therateofheattransfertounitareaofwall,y=0,is

OfteneitherQworTwisgiven,anditisdesiredtocalculatetheother.However,at high speeds a relationshipbetweenQw andTwmaybeused, expressing thebalancebetweenheattransferandradiation.

7. ThemomentumintegralequationEquation(12)expresses localconservationofmomentumin thexdirection

atpointswithintheboundarylayer.Whenintegratedacrosstheboundarylayerfromy=0toy=∞theresultingequationexpressesconservationofmomentumintheboundarylayerasawhole.UponneglectingthebodyforceX,substitutingfor ∂p/∂x from (21), and integrating from y = 0 to y = δ we obtain

Now

andby(1)thisequals

Hence(39)yields

upon formally letting δ→∞. This equation is called themomentum integralequation,andwasfirstderivedbyKármán(1921)bymorephysicalarguments.Afterwards, K. Pohlhausen (1921) derived the incompressible form by theprocedureadoptedabove.

8. Thekinetic-energyintegralequationIfequation(12)isfirstmultipliedbythelocalvelocityu,andthenintegrated

fromy=0toy=δ,analternativeintegralequationisobtained.Upondoingthis

we find

The second terms on both sides of (41) are integrated by parts. Thus

and

Also

Upon substitution into (41) from (42) to (44) we find that

or

Since, for a perfect gas, the equation of state is (7), it follows that

sothat(45)becomes

This may be rewritten in terms of the energy thickness δ3 and the enthalpythickness δ4 as

TheincompressibleformofthisequationwasfirstderivedbyLiebenson(1935).

9. Thethermal-energyintegralequationThis equation, as its name implies, is obtained by integrating the thermal

energy equation (19) across the boundary layer. Now for any functionF, theintegral across the boundary layer of the expression {ρu(∂F/∂x)+ρv{∂F/∂y)}becomes

Uponmakinguseof(1)thisbecomes

Choosing

whereTHisthetotaltemperature,(19)becomes

or

This equation expresses the fact that the rate of change of flux of totaltemperaturemustbebalancedbytheheatbeingtransferredacrosstheboundary.

10. IncompressibleflowForflowofliquids,orofgasesatsufficientlylowspeeds,thedensityρwill

be sensibly constant unless the temperature differences are great. When thisapproximationismadetheboundary-layerequationstakeamuchsimplerform.The continuity equation (1) becomes

Similarlythemomentumequation(12)becomes

Here we have neglected the body forces, and have taken the viscosity asconstant, sinceμ is a function of the temperature, assumed not to varymuch.Finally, the thermal energy equation, (19), remains. If typical temperaturechangesaresmalllikethesquareoftypicalvelocities(innon-dimensionalform,of course), then (19) is trivial and vanishes.When, however,we consider thecase of ‘warm walls’, with

whereU/a is the relevant non-dimensional velocity, then (19) reduces to

Wenotice thatequations (50)and(51)are twoequations for thevelocity field

(u,υ),whichareuncoupledfromtheequation(52)forT.Therelevantboundaryconditions are

Inasimilarwaytheintegralequationssimplify.Wefindthat(42)becomes

or

Inthesameway(47)becomes

and(49)becomes

11. Crocco’stransformationVarious attempts have beenmade to transform the laminar-boundary-layer

equations into a formmore amenable to calculation. These have provedmostuseful in restricted fields, thoughnonehas been foundof universal value.Webeginbyconsideringa transformationdue toCrocco (1939), inwhichx anduaretakenasnewindependentvariables,withtheviscousstressτ=μ(∂u/∂y)andtheenthalpyI=cpTasdependentvariables.

Wemaywrite the continuity,momentum, and energy equations, (1), (12),and (18), as

and

These equations are then transformed to new independent variables, x, u, andaftersomealgebra(forwhichthereaderisreferredtoCrocco’soriginalpaper)they become

and

Equations(57)and(58)aresimultaneousequationsforIandτintermsofuandx.After these have been integrated, it is a straightforwardmatter to calculate

afterwhichυfollowsfrom(56).This form of the equations will be used in discussing the compressible

laminarboundarylayeronaflatplateinChapter7.

12. vonMises’stransformation

A related transformation, in which x and are taken as

independentvariables,wasgivenbyvonMises(1927)forincompressibleflow.Upon transforming, the equations become

and

Byaddingtheseequationstogetherweobtain,aftersomealgebra,

The equations in this form have proved useful in considering the heattransfer through a laminar boundary layer in incompressible flow (Chapter 6),alsothelaminarboundarylayeronaflatplateincompressibleflow(Chapter7).

3

NUMERICALSOLUTIONSFORINCOMPRESSIBLEFLOW

APART fromthosespecialcases,considered inChapter2,where theboundary-layer equations may be reduced to one or a number of ordinary differentialequations,oneisfacedingeneralwiththeproblemofsolvingnon-linearpartialdifferentialequations,andthisisanexceedinglytrickynumericalprocedure.Forthis reason very few precise numerical solutions have been obtained. Theimportance of these few may perhaps best be illustrated by reference to theapproximatemethodofPohlhausen,tobeconsideredinChapter4,section1.Formany years this method was regarded as being perfectly satisfactory forcalculating thedevelopmentofa laminarboundary layer in thepresenceofanarbitrarypressuregradient.Then, following theclassical experimentalworkofSchubauer(1935),itwasfoundthatwhenPohlhausen’smethodwasappliedtoSchubauer’s experimentally observed pressure distribution, it failed to predictseparationbyaconsiderablemargin,whereasseparationclearlytookplaceintheexperiments.Itwasthenapparentthatthemethodwasinadequateinregionsofrisingpressure. In consequence comparisonwith at least one accurate solutionhas since been regarded as essential before reliance could be placedupon anyapproximatemethod.

It was to answer this need that Howarth (1938) considered a precisenumericalsolutionfortheflowinwhichthevelocityattheedgeoftheboundarylayer is given as

Thissolutionwillbeconsideredinsection1ofthischapter.Later Tani (1949) considered the three cases for external velocities

with n = 2, 4, and 8. The method used was precisely that of Howarth, and,althoughtheaccuracyisprobablynotquitesogreat(particularlyforn=8),thereisnodoubtthatitisfarmoreaccuratethanonecouldhopeforbyapproximate

methods.InthemeantimeHartree(1939a,b)hadre-solvedHowarth’sproblem,using

adifferential analyser.Having thus confirmed the accuracybothofHowarth’scalculations and of his own procedure, he then applied it to a numericallyprescribedvelocitydistribution,derivedfromSchubauer’sexperimentalresults.Thisworkgivesaprecisenumericalsolutionforaflowcontainingastagnation-pointandseparation,whereasthesolutionsofHowarthandTanididnothaveastagnation-point. The only drawback to the solutionwas that arising from theuncertainties of twice numerically differentiating a tabulated function, namelytheexperimentalvelocitydistribution.

More recently, additional solutions have been obtained for flows withstagnation-pointandseparationforanalyticallyprescribedvelocitydistributions.Curle (1958 a) has considered the case

by a semi-analytic method similar to but a little simpler than that used byHowarthabove,andTerrill (1960)hasusedanelectroniccomputer toobtainasolution for the case

Weshallnowconsidersomeofthedetailsofthesevarioussolutions.

1. ThesolutionofHowarthAs remarked earlier, Howarth (1938) considered the case of an external

velocity

By reference to Chapter 2, section 5, we see that this is the special case forwhich

Accordinglythestreamfunction,

maybeexpandedinaseries

where

andHowarthderived theequationssatisfiedby thesefivefunctions(andbyfurtherfunctions in the expansion) and integrated several of them. The first threeequations take the form

and

withboundaryconditions

By integrating these equations in turn, together with the next four, and byroughlyintegratingafurthertwo,thedetailedvelocityprofilewasobtainedforaconsiderabledistancefromtheleadingedge.Inparticulartheskinfrictionatthewall, which is given by (81) as

wasfoundtoequal

Thevalues of the secondderivatives at thewall are perhaps best indicated byremarking that when 8ξ = 1 the series in brackets in (84) took the form

sothattheseriesisconvergingratherslowlyinthisneighbourhood.It is clear, however, that subsequent terms form only a relatively small

correction to this series. For this reason Howarth suggested representing theterms after say the seventh in an approximatemanner.Thus he approximated

by

Inotherwords,itwasassumedthattheshapesofthesubsequentterms (n> 6), could be adequately represented by a universal formB(η)), the variousmultiples of ξn being summed to give A(ξ). By examining the functions

,and,asfarashehadcalculatedthem, and Howarthfound that these were all multiples of the same functions, to a considerableaccuracy.Itremained,then,todetermineA(ξ),andthiswasdoneineachofthefollowingtwoways.

Firstly,Howarthsubstitutedfrom(85)intothemomentumintegralequation,anduponassuming theshapeB(η), this led toa firstordernon-linearordinarydifferential equation for A(ξ), which took the form

where P, Q, R, S, T are polynomials which are determined by the knowncoefficients in (85). This equation was integrated by standard numericaltechniques, and it turned out that S+TA became zero and dA/dξ infinite at apositionwheretheskinfrictionhaddecreasedalmost,butnotquite,tozero.Thishappeningisclearlyboundupwiththequestionofthesingularityatseparation,discussedearlier.Howarththencompletedthiscalculationoftheskinfrictionbyanalternativeprocedure,andconcludedthatseparationoccurredwhenξ=0·120.

ThefunctionA(ξ)was thencalculatedby the followingalternativemethod.By considering the various boundary conditions at the wall (implied by theboundary layer equations), it is easily seen that the simplest onewhich is not

automatically satisfied by (85) is that obtained by twice differentiating theboundary-layermomentumequationwith respect toy, thenputtingy=0.Thisyields

Bysubstitutingintothisrelationship,analternativeequationforA(ξ)isobtained.Byessentially thisprocess,Howarthobtainedalmost identical values forA(ξ),theseparationpositionagainbeingξ=0·120.

ItshouldbenotedthatthetwomethodsusedtodeterminethefunctionA(ξ)arequiteindependent,andthecloseagreementbetweenthetwosetsofresultsisvaluable confirmation of their accuracy. This has been further confirmed,beyondalldoubt,bythesubsequentworkofHartree(1939a)andLeigh(1955),whosolvedthesameproblemonadifferentialanalyserandamodernhigh-speedcomputingmachine,respectively.

2. ThesolutionsofTaniInanattempttoprovidefurtheraccuratesolutionsofthelaminarboundary-layerequations,forcasesinwhichthesecondstreamwisederivativeoftheexternalvelocity,d2u1/dx2,isnon-zero,Tani(1949)laterconsideredthecases

As inHowarth’s example, these are special cases of the family considered inChapter 2, section 5. When n = 2, so that

wehave

Thus,by(82),wemaywrite

with

and

Tani calculated the first seven non-zero functions F0, F2,..., F12, and thencontinued thesolutionbyassumingauniversal shape for thesubsequent termsand determining the amplitude function A(ξ) from the momentum integralequation.Similarly, for thecasesn=4,8,hecalculatedsixandfive functionsrespectively before the approximate continuation. In each case he obtained anequationoftheform(86),findingthatS+TAbecamezeroanddA/dξinfinitejustbefore separation was reached. It was easy, however, to extrapolate theextremelysmalldistance toseparation,and theseparation-points in these threecases were determined as

3. ThesolutionsofCurleAs has already been mentioned briefly, Curle (1958 a) has developed a

methodforcalculatingsolutionsof the laminarboundary-layerequationswhenthe external velocity is that appropriate to flow past a cylinder symmetricallyplacedinauniformstream.Theexternalvelocitythentakestheformofequation(73),withψ,u,and(∂u/∂y)wgivenby(74), (75),and(76)respectively. Inanychosencase the first six terms in these series expansionsareknown, since therelevantuniversalfunctionshavebeentabulatedbyTifford,andinallthecasesforwhich calculations have beenmade the series seem to convergewell untiljustupstreamofseparation.Forexample,incalculatingthesolutionforthecase

Curle estimated by reference to approximate solutions that separation wouldoccurinthevicinityofξ=0·66,andshowedthatatthispositiontheseries(76)for the skin friction takes the form

Wenotethatthefourthtosixthtermsareoforder1percent,ofthefirsttwo,so

thatevenifthesubsequent(unknown)termscontinuetodecreaseratherslowlytheywillpresumablybeonlyarelativelysmallcorrectiontothefirstsix.

FollowingtheideausedbyHowarth(1938),Curleassumedthattheshapeofthe subsequent functions could be approximated by F11 Theaccuracy of this approximation can only be tested by comparing say

, to see whether there is anyindication of a tendency towards universality in shape as n increases. In theexampleshecalculated,thistendencyappearedtobepresenttoacertaindegree.On the basis of this approximation we write the velocity as

andtheskinfrictionas

whereA(ξ)istobedetermined.Todothis,Curlesuggestedusingthesecondofthe two methods used by Howarth, that is by satisfying the first boundaryconditionatthewallwhichisnotidenticallysatisfiedbyaform(88).This,asinHowarth’s case, is

From(88),byfour-folddifferentiationwithrespectto ,wefind

Notingthat,in(89)and(91),A(ξ)ismultipliedonlybyconstants,wecaneasilyeliminate A(ξ) to give (∂4u/∂y4)w in terms of (∂u/∂y)w, and this yields

where

Upon substituting from (92) into (90), an equation for (∂u/∂y)w results. Thisequation may be put into non-dimensional form by writing

anddefining

whenceitbecomes

Uponformallyintegratingwithrespecttoξ,thisyields

where

Themethodof solution is thereforeas follows.Given theexternalvelocityu1(x), the coefficientsβ2n+1 are known, so that the first six terms in the series(89)forT,thefunctionQ(ξ),andtheconstant in(93)areeasilyobtained.TheseriesmaybeusedtoobtainthevaluesofTwhenξissmall,andthe solution is continued by numerical integration. A simple, and convenient,

procedurefordoingthisistoreplacetheintegral byitsSimpson’srule

equivalent, whereupon if T(ξ) and T(ξ+h) are known, T(ξ+2h) is given bysolutionofaquadraticequation.Thismethodappearstoworkparticularlywell,providedtheconstant isnotsogreatthattheerrorinSimpson’sruleismultipliedbytoolargeafactor.WhenT(ξ)isthusdetermined,thedetailsof thevelocityprofileeasily follow.ThefunctionA(ξ) isgivenfrom(89),andthenufrom(88).Anaposterioricheckontheaccuracyofthesolutionmaybemade,ifdesired,byusingthemomentumintegralequation.

Curle used thismethod to calculate an accurate solution for the case (87).

ThevaluesofT(ξ)atξ=0·250,0·275,0·300weregivenbytheknowntermsoftheseriestosixsignificantfigures.Withthesestartingvalues,theequation(93)was integrated with steps of Δξ = 0·025 and Δξ = 0·050, Richardson’s h2-extrapolationbeingused toobtain improvedvalues.Using smaller intervals asseparation was approached, the integration proceeded without difficulty, andseparationwaspredictedatξ=0·6551.

Curlealsogavesolutionsfortwocasesofthefamilyofvelocitydistributions

for which the value of the constant was zero. He showed that

iszerowhen

andthatineachcase(87)takestheform

Solutionswereobtained,therefore,withoutnumericalintegration,andseparationfound at

and

4. ThesolutionsofHartree,Leigh,andTerrillThesolutionsdiscussedabovewereallobtainedbysemi-analyticmeans.Wenowturntoanumberofprecisenumericalsolutionsobtainedbypredominantlynon-analyticmeans,eitherbynumericalanalysisorbymeansofanautomaticcomputingmachine.WebeginbyconsideringasolutionbyHartree(1939a)forthecaseofalinearlydecreasingvelocity,whichhadearlierbeenconsideredbyHowarth.ThemainideaofHartree’sapproachwasthatofapproximatingthepartialdifferentialequation(51)byanequivalentordinarydifferentialequation,derivativeswithrespecttoxbeingreplacedbyfinitedifferencesandtheintegrationbeingcarriedoutwithrespecttoy,thatisacross

theboundarylayer.Inotherwordsthedistributionofvelocityatonestationisusedtocalculatethedistributionatadownstreamstation.Theintegrationwasstartedat8ξ=0·40,Howarth’sserieshavingbeenused

togivethevelocityprofileatthatposition.Theprofileat8ξ=0·80wasderivedin one and two steps, after which Richardson’s h2-extrapolation formula wasusedandtheresultssmoothed.Similarcorrectionsforthefinitesizeofthestepsofintegrationwereappliedat8ξ=0·88,0·94,0·956,and0·958.NowaccordingtoHowarth(1938),separationoccurswhen8ξ=0·960,andHartreeencounteredconsiderable difficulties as this position was approached. These difficultiespointed very strongly to the existence of a singularity at the position ofseparation, and indeed it was because of them that Goldstein’s investigationsinto flow near to a position of boundary-layer separation were carried out.Hartreedeterminedthepositionofseparationbytwoindependentmethods,andconcludedthatitiscloseto8ξ=0·959,inexcellentagreementwithHowarth’sresult.

This same problem was later considered with even greater care by Leigh(1955),usingtheE.D.S.A.C.machineoftheUniversityofCambridge.Variousspecial tricksweresuccessfullyusedtogetoversomeof thedifficultieswhichHartree hadmet, and itwas established that separation tookplacewhere 8ξ =0·95854.ThemainobjectofLeigh’sworkwastoimproveontheaccuracyoftheearliersolutions,bymakinguseofthegreaterstoragecapacityofthemachine,soastoenablemoreinformationtobeobtainedregardingflowinthevicinityofseparation.FordetailsofthetechniquesseeLeigh(1955).

These precise numerical solutions are extremely valuable in that theyconfirm (by independent methods) the considerable accuracy of Howarth’sprocedure,aswellasthrowingsomelightonthesingularityatseparation.

WenowcommentonafurthersolutionobtainedbyHartree(1939b),onadifferentialanalyser,inwhichtheexternalvelocity(orpressure)distributionwasthat found experimentally by Schubauer (1935) in some experiments with anellipticcylinder.Theprocedureusedwasessentiallythatdevelopedforthecaseof a linearly retarded velocity save that in this case the integration was donemechanicallyratherthannumerically.Onepracticaldifficultyarose,however,inthat theassumedexternalvelocity,beinggiven fromexperimentaldata,had tobe smoothedand laterdifferentiatednumerically.There is a certain latitudeofinterpretationindoingthis,whichmaynotnormallybecriticalbutappearedtobeso in thisparticularcase.Forexample,Hartreesmoothedanddifferentiated

thevaluesofu1(x)givenbySchubauer,andthencalculatedthedevelopmentoftheboundarylayerbynumericalintegration.Withtheminoraxisoftheellipsecas the representative length, Schubauer’s experiments indicated that theboundary layer separated at roughly x/c = ξ = 2·0, whereas in Hartree’scalculationsno separationwaspredicted. Itwas found that anextremely smallchangeintheassumedvaluesofu1(x)inthevicinityofξ=2·0sufficedtocauseatheoreticalpredictionofseparationatthecorrectposition.Accordinglywemayregard Hartree’s work as providing an accurate numerical solution for a casewith a numerically prescribed external velocity, but it should be treated withsomecaution,inviewoftheuncertaintiesdiscussedabove.

Afinal,andimportant,solutiononanautomaticcomputingmachineisthatdue to Terrill (1960), who considered the case of an external velocity

Themethodused involved theapplicationofGörtler’s transformation(1957a)followed by the technique so successfully used by Hartree above, in whichderivativeswithrespecttoxarereplacedbyfinitedifferences.Thesizeofstepusedwasrapidlydecreasedasseparationwasapproached,andthiswaspredictedat

Terrillalsocarriedoutsomecalculationswhichthrowsomelightonthenatureofthesingularityattheseparationposition.FordetailsseeTerrill(1960).

4

PRACTICALMETHODSOFCALCULATIONFORINCOMPRESSIBLEFLOW

IN practice one may not be interested in a solution of the boundary-layerequationsas accurate aswouldbeobtainedbyusing themethodsdiscussed intheprecedingchapters,andcertainlyonewouldnotnormallywishtospendtheconsiderable time required to obtain such solutions. Accordingly it may besufficient to use methods which are quicker to apply, but yield less accurateresults.Someofthesewillnowbeconsidered.

1. Pohlhausen’smethodThemethoddevelopedbyK.Pohlhausen(1921)isnotnowusedmuchinits

originalform,havingbeenimproveduponandsupersededbyothermethods.Itisincludedhere,however,becauseitwashistoricallythefirstgeneralmethodtobe developed, and because the idea used by Pohlhausen has found so manyapplicationsinvariousbranchesoffluiddynamics.

Inthismethodtheboundary-layerequationsarenotsolvedeverywhere,butaresatisfiedatthewall,attheedgeoftheboundarylayer,andonanaveragebysatisfyingthemomentumintegralequation.Thevelocityuisassumedtotaketheform

whereδmayusefullybe regardedat this stageas theeffectiveboundary-layerthickness, but in the final formulae is no more than a convenient parameter.Then the skin-friction τw, the displacement thickness δ1, and the momentumthickness δ2 are equal to

Now the boundary conditions to be satisfied by the true local velocity u are

and

The idea used by Pohlhausen is to choose f(η) to be a polynomial satisfyingsomeof theboundaryconditions, (95)and(96), leavingδ tobedeterminedsothattheresultingapproximateusatisfiesthemomentumintegralequation.Ifwewrite

itcaneasilybeshownthattheaboveconditionsleadtotheequation

whereg(∧)andh(∧)areuniversalfunctionswhichdependonlyuponwhichoftheboundaryconditions(94)and(95)aresatisfied,andare independentof thedataoftheproblem.

TherearetwocriticismswhichcanimmediatelybelevelledatPohlhausen’smethodaspresentedabove.Thefirstisthearbitrarychoiceastowhichboundaryconditionsshallbesatisfiedandthefactthattheouterconditionsaresatisfiedaty=δ.Pohlhausenchoseaquarticforuwhichsatisfiedthefirsttwoconditionsof(95)atthewall,andthefirstthreeconditionsof(96)attheedgeoftheboundarylayer. This leads to a method which, when applied to the cases for whichaccurate solutions are known, yields accurate results in regions where thepressureisdecreasingbutbecomeslessaccurateasseparationisapproached,sothat the predicted distances to separation are typically 30 per cent, too high.

Curle(1957)suggestedthatbetterresultswouldbeobtainedintheregionneartoseparation if an additional boundary condition at the wall were satisfied bychoosing a quintic profile. Such a procedure leads to improved resultsthroughout the region downstream of the pressureminimum, so that a typicalpredicted separation distance is say 6 per cent, too high only. However, themethodthenbreaksdownentirelyneartoastagnation-point.EarlierSchlichtingandUlrich (1940), in connexion with somework on boundary-layer stability,hadchosentorepresentubyasextic,satisfyingstilloneconditionmoreat theedge of the boundary layer.This procedure also leads to better results near toseparation,say15percent,errorintheseparationdistance,butagainleadstoabreakdownofthemethodneartoastagnation-point.

AsecondcriticismofthePohlhausenapproachisthatthesecondderivativeof u1, that is , appears explicitly in the formulation. Now if u1 has beenobtainedfromexperimentaldata itwill inmanycasesbeextremelydifficult toobtain even roughestimatesof .Fortunately themethodcanbe recast, by aprocedureduetoHolsteinandBohlen(1940),sothat doesnotappear,anditisnecessaryonlytoderive bynumericaldifferentiation.HolsteinandBohlenintroduce a parameter λ, analogous to ∧, defined by

Themomentumintegralequationiswrittenintheform

whichwewriteas

where

Thus

Now from (94) and (99) we know τw, δ1, δ2, and λ as functions of ∧.Accordinglyby(101)and(102)wecandeducel,H,andLasfunctionsofλ,and(102)reducestoafirst-orderequationforλ,ascanbeseenbywritingitintheform

Thefunctionsg(∧),h(∧),forusewiththemethodinitsoriginalform,maybefound inPohlhausen’soriginalpaperor ineitherof thebooksbyGoldstein(1938)andSchlichting(1955),whilstthefunctionsH(λ),l(λ),L(λ),canbefoundinSchlichting’sbook.WemaymentionherethatTani(1941)indicatedthatthefunctionL(λ)isapproximatelylinear,sothat(103)integratesanalytically.Thisconsiderable simplification has been used by several other authors, notablyThwaites(1949).

2. Timman’smethodTimman (1949) attempted to obtain an improvedprocedure for calculating

laminarboundary-layerdevelopmentbyassumingavelocityprofileoftheform

where η = y/δ. This form of profile is an improvement over the polynomialsassumedinSection1inthatu/u1tendstounityasymptotically.Accordinglyonlythe boundary conditions at the wall need to be further considered, and theseservetodeterminea,b,c,d.

From (95), substituting from (104), these yield in turn

Initially Timman considers only the first three conditions of (105), and these

determinea,b,casfunctionsof∧whendissetequaltozero.ThemethodthenproceedsmuchasinPohlhausen’smethod.From(104)wecanobtainδ1/δ,δ2/δ,andτwδ/μu1asfunctionsof∧andalsoλasafunctionof∧.Thusweobtainl,H,andL,definedin(101)and(102),asfunctionsofλ,withthemomentumintegralequationreducingtotheform(103).

TimmanfoundthatthissolutionyieldedaccuraterepresentationsoftheflowinaBlasiusboundarylayerandoftheflowneartoastagnationpoint.Inregionsofretardedflow,however,themethodwasmuchlesssatisfactory,asapplicationtothecaseu1=u0{1–(x/c)}showed.Accordinglyhesuggestedthatinregionsof adverse pressure gradient the condition d = 0 should be replaced by

This condition was chosen to ensure that the complicated fourth condition in(105)shouldbesatisfiedattheseparationposition,andincidentallyledtovaluesofa,b,c,d,whichwerecontinuousat thejoin∧=0.Thisprocedureleads toresultswhichagreequitewellwith theaccuratecalculated resultsdiscussed inChapter3.Thevaluesof the relevant functions, required for integrating (103),aretobefoundinTimman’spaper.

3. ThemethodsofHowarthandWalzInthemethodsofPohlhausenandTimmanthevelocityprofileatanystationischosenasoneofasingly-infinitefamilyofvelocityprofileswhoseshapeischaracterizedineithercasebytheparameterAandwhosescaleischaracterizedby .Thechoiceastowhichprofileistoapplyateachstationismadebysatisfyingthemomentumintegralequation.Similarlytheaccuratesolutionoftheboundary-layerequationsforthecaseu1=u0{1–(x/c)},derivedbyHowarth(1938)anddiscussedinChapter3,section1,yieldsasingly-infinitefamilyofvelocityprofiles,whoseshapemaybecharacterizedbytheparameterλ,andwithscalerepresentedbysaythemomentumthickness

.AmethodontheselineswasgivenbyHowarth(1938),byregardingageneralvelocitydistributionasbeingreplacedbyaseriesoflinearportions.Hisgeneralresultcaneasilybereducedtotheform(103),thefunctionsH(λ),l(λ),L(λ)beingobtainedfromHowarth’stabulatedresults.Thesevalues,

ofcourse,willallbefornegativeλ,sincethevelocityiseverywhereretarded,butbyconsideringtheassociatedacceleratedflowu1=u0(1+(x/c)}itwouldbepossible,asWalz(1941)pointedout,toextendthefunctionstocoverarangeextendingfromthestagnationpointtoseparation.Walz (1941) also suggested a method in which the velocity profiles are

chosen to be those of the Falkner-Skan similarity solutions. As distinct fromHowarth’smethod,whereeachvalueofxcorrespondstouniquevaluesofH,l,L, and λ, each complete similarity solution for a given value of the velocityexponent yields single values ofH, l, L, λ for all x. However, the completefamily of similarity solutions yields values forH, l, L, which are continuousfunctionsofλ.

4. ThemethodofThwaitesItwaspointedoutbyThwaites(1949)thatifonewishestocalculateonlythe

boundary-layer thicknessesand the skin-frictiondistribution it isnotnecessaryto introduce explicit assumptions concerning detailed velocity profiles, as themethodspreviouslydiscussedalldo.Providedsuitablecorrelationsaredefinedbetween the overall boundary-layer characteristics H, l, L, and the shapeparameterλ,onecaneasilyobtainλasafunctionofxbynumericalintegrationof(103),afterwhichδ2followsfrom(99)andthenδ1andτwfrom(101).

Each accurate solution of the laminar boundary-layer equations willcorrespond to a different set of functions H(λ), l(λ), L(λ). Similarly anyapproximatemethodusingasingle-parameterfamilyofprofileswillcorrespondto a definite set of functions. It follows from this that an approximatemethodwillyieldanaccurateanswerforagivenproblemifandonlyifthevaluesofH,l,andL for theaccurateandapproximatesolutionsagreecloselyenough.NowtheideausedbyThwaiteswastoexamineandcomparethefunctionsH,l,andL,corresponding to all known exact and approximate solutions of the laminarboundary-layer equations.By considering the accuracy towhich these varioussolutions are known, and how typical each is of a practical boundary layer,Thwaites defined what are virtually the optimum values of the functions forgeneraluse.

Thwaites found that as regardsH(λ) and l(λ) the solutions layvery closelytogether forpositiveλ, that is for regionsof increasingvelocity—upstreamofthe pressureminimum in otherwords.Downstream of the pressureminimum,

for negative λ, the curves deviated considerably, so that values of λcorresponding to boundary-layer separation, that is values for which l(λ) = 0,wereobtainedrangingfromλ=–0·068toλ=–0·157.Fortunatelythetwomostimportant solutions of those available to him, namelyHowarth’s solution andHartree’s analysis of Schubauer’s experimental results, lay quite close to eachother, andThwaites’s choices ofH and l lay close to bothof these aswell asbeingaroughaverageoftheapproximatesolutions.

ThevaluesofL(λ),derivedfrom(102),forthevarioussolutionsweremuchcloserthanthoseofeitherH(λ)orl(λ),bothforpositiveandnegativeλ,andinaddition the functionL(λ) appeared to be linear to a high degree of accuracy.Thwaites accordingly suggested defining L(λ) to be

the coefficients being chosen to give the best overall agreement with theavailablesolutions.WithL(λ)givenby(106)itispossibletointegrate(103)interms of a single quadrature, to yield

Having;thusobtainedδ2,(101)thenyields

and

The ideaof expressingL(λ) as a linear functionhadbeen suggested earlier byboth Walz and Tani, but Thwaites’s coefficients are, if anything, moreacceptable, since they are not based upon a single solution, and yield thecomputationalsimplificationofrequiringonlyintegralpowersofu1.

Some improvements to Thwaites’s functionswere suggested byCurle andSkan(1957).Theirideamaybeillustratedthus.ThwaitesdeterminedhisH(λ)byconsideringtheratiosofexactδ1toexactδ2.Sincethepredictedδ1isobtainedbymultiplicationofH and the approximateδ2,Curle andSkan suggested that

moreacceptablevaluesforH(λ)wouldbederivedfromtheratiosofexactδ1toapproximateδ2.This argumentwouldbe complicated further if one accountedforthefactthatanapproximatevalueofλ,derivedfrom(107)and(108),isusedinpractice.Inthisway,andbysimilarargumentsappliedtol(λ),CurleandSkansuggested improvements to Thwaites’s functions in the region near separationwhere the functions for individual solutions differ most. These modifiedfunctionsaregiveninTable5.

TABLE5

UniversalfunctionsforThwaites’smethod

m l(m) H(m)–0·25 0·500 2·00–0·20 0·463 2·07–0·14 0·404 2·18–0·12 0·382 2·23–0·10 0·359 2·28–0·080 0·333 2·34–0·064 0·313 2·39–0·048 0·291 2·44–0·032 0·268 2·49–0·016 0·244 2·55

0 0·220 2·61

+0·016 0·195 2·670·032 0·168 2·750·040 0·153 2·810·048 0·138 2·870·056 0·122 2·940·060 0·113 2·990·064 0·104 3·040·068 0·095 3·090·072 0·085 3·150·076 0·072 3·220·080 0·056 3·300·084 0·038 3·390·086 0·027 3·440·088 0·015 3·49

0·090 0 3·55

ThemodificationsofCurleandSkanalsomadeuseoftheaccuratesolutionsgivenbyTani(1949),whichhadnotbeenavailable toThwaites.SubsequentlyCurle(1958a)hasgiventheadditionalaccuratesolutiondiscussedinChapter3,section3.Thegreaternumberofaccuratesolutionsnowavailablemakesitmucheasier to avoid the possibility of being unduly biased by the few accuratesolutions available to Thwaites. Curle and Skan (unpublished) have examinedthefunctionsHandlwhichwouldleadtoexactpredictionofδ1andτwforeachoftheaccuratesolutionsdiscussedinChapter3,andhaveindicatedthatwiththeexception of the modifications near separation, the original values given byThwaitesprobablyreflecttheseasaccuratelyasanysinglesetoffunctionscando.

5. ThemethodofStratfordThemethoddevelopedbyStratford(1954)isbasedupontheideaofdividing

theboundarylayerintoouterandinnerportions,foreachofwhichasolutionisobtained which joins smoothly onto the other. This idea was first used byKármánandMillikan(1934).

Stratfordbeginsbyconsidering theouterpartof theboundary layer,wherethe flow is nearly in viscid and so, by Bernoulli’s equation, the total head isalmostconstantalongstreamlines.Thusifthepressureisconstantbetweenx=0and x = x0, with a pressure rise downstream of x = x0, then we have

wherethestream-functionψisgivenby

andΔHisthesmallchangeintotalheadduetoviscosity.Nowtheshapeoftheouter part of the boundary-layer velocity profile is not greatly changed by thepressurerise,sotheviscousforcesaregivenapproximatelyasthoughtherewereno pressure rise. Hence

whereuBistheBlasiussolutionforaboundarylayerwithnopressuregradient.

From (111) and (109), remembering that u = uB when x = x0, we have

Stratford also indicates how this result may be deduced algebraically bywriting the boundary-layer equation (51) in the form

where∂/∂sdenotesdifferentiationalongastreamline,andconsideringtheTaylorexpansion

Thefirsttermontheright-handsideisindependentofthepressurerise,sincetheeffects have diffused outwards only downstream of x = x0. Similarly thecoefficient of (x–x0) is equal, by (113), to {μ(∂2u/∂y2)}x0,ψ, and so again isindependentofthepressurerise.Hence,iftermsoforder(x–x0)2areneglected,the left-hand side of (114) is independent of the pressure rise, so that

whichis(112)rewritten.By similar arguments,orbydifferentiationwith respect toψ, onecanalso

deduce from (112) or (115) that

and

intheouterpartoftheboundarylayer.Sufficiently near to the wall, in contrast, viscous forces become very

important,sothat theresultantviscousforceonanelementoffluidat thewallmust exactlybalance thepressure-gradient force.Thus,by settingy=0 in theboundary-layer equation (51) we find

By further considering the boundary conditions at thewall, Stratford indicatesthat in general

but

Bysatisfying(118)and(119),Stratfordsuggeststhatatseparationthevelocitydistribution near to the wall may be approximately obtained by writing

wherehistobedetermined.Withaninnerprofilegivenby(121),andanouterprofilegivenby(115)to

(117),itispostulatedthatthereshouldbecontinuityofψ,u,∂u/∂y,∂2u/∂y2atthepositionofthejoin,ψ=ψj,u=uj.Thesefourconditionsservetodeterminethevaluesofuj,ψj,hjandxatthepositionofseparation.Ingeneralthesolutionofthe fourequations isdifficult, sincesuchfunctionsasuB,∂uB/∂y,and∂2uB/∂y2

cannot be expressed in simple analytic terms.A considerable simplification iseffected,however, if theassumptionismadethat the inner layer issufficientlythin that the Blasius velocity profile uB(x, ψ) is sensibly linear throughout

. This condition, which restricts the application of the method in amanner which we shall discuss later, leads to the simple result

Byassumingthatthesevaluesholdatthejoin,Stratfordderivesthesimpleresult

whereCpisthepressurecoefficient,definedas

The condition on the inner layer is such that the method should strictly only

applywhenCp is less than about 0·04, with an absolute upper limit of about0·11. It has been found, however, that themethod yields useful results over alargerrangeofpressurecoefficients.

Stratford also gave an asymptotically exact solution for the casewhen theadverse pressure gradient downstreamofx =x0 is constant but asymptoticallylarge,andthedistancexs–x0 toseparationtendsasymptoticallytozero,asdoesthe thickness of the inner layer. For details of this solution referencemay bemade to Stratford’s paper. The principal result is that

Thenumerical coefficient isnotverydifferent from thatobtainedby themorephysical approach discussed earlier. Stratford suggested that in general (124)shouldleadtogoodpredictionsofseparation.

Stratfordalsoconsideredasecond,veryspecial, typeofflow, inwhich thepressuregradientiszeroupstreamofx=x0andissuchthattheskin-frictionisidenticallyzerodownstreamofx=x0.Hesuggeststhatamoreappropriateformthan (121) for the inner layer would be

An analysis along the same lines as that for the earlier case leads to a similarresult to (123) or (124). The numerical coefficient becomes 0·0049, and arigorous mathematical analysis for an asymptotically large adverse pressuregradient of the appropriate form leads to a value 0·0059. By carefullyinterpolatingbetween thesespecialcasesandempiricallyadding further terms,Stratfordalsoderivedaconsiderablylengthierformulathanthosequotedabove,which should yield exceedingly accurate predictions of separation. Later,however, Curle and Skan (1957) indicated that an equally accurate, butconsiderably simpler, empirical improvement of (124) could be obtained byreplacing the numerical coefficient by 0·0104; separation is then predicted tooccur where

Itshouldbepointedoutthatthemethodisbaseduponaconsiderationofthedownstreamdeformation of a knownBlasius velocity profile.Accordingly themethod is not, strictly speaking, applicable to flows containing a forwardstagnation-point. ItwaspointedoutbyStratford,however, that in sucha flowthe velocity profile at the pressure minimum will be reasonably similar to aBlasiusprofileofsuitablychosenscale.Accordinglythemethodmaybeapplied

downstreamof thepressureminimumifwereplacexbyx–xL say,wherexL ischosensothatthetruemomentumthicknessatthepressureminimumx=xmisequal to themomentumthicknessof theequivalentBlasius layeratx=xm–xL.Since the momentum thickness δ2 is usually given accuratelv bv Thwaites’sformula

thisconditioniseasilyseentoyield

whereumisthevalueofu1atthepressureminimum.Themethod as developed byStratfordwas designedmerely for predicting

separation. No estimate of the boundary-layer thicknesses or of the detaileddistribution of skin-friction between the leading edge and separation is given.This restriction has been partly removed by the method which will now bediscussed.

6. ThemethodofCurleThismethodisageneralizationofStratford’smethod,andleadstoasimple

methodforpredictingthedetaileddistributionofskin-friction.Theinnerlayerisassumed to be of the form

where τw is the skin-friction μ(∂u/∂y)w, and a(x) and the constant n are to bedetermined.ThisprofilegeneralizesStratford’s form in twoways,namely thattheskin-frictionisnotassumedtobezeroandthatthevalueofnisnotfixedapriori. The outer velocity profile is given, as in Stratford’smethod, by (112),(116), (117), and (122).The four joiningconditions, continuityofψ,u,∂u/∂y,∂2u/∂y2, are sufficient todetermine τw,a,yj,ψj as functionsofx,withn as anarbitraryparameter.Aftertherelevantalgebrahasbeenperformedthefollowingresults are obtained :

and

where τB(x) is the skin-friction at station x of a Blasius boundary layer. Theposition of separation, where τw = 0, is obtained from (128) as

so that the predicted separation position depends upon the choice of theparametern.Whenn=6theright-handsidebecomes0·0049,andwhenn=4itbecomes 0·0065, the values obtained by Stratford. In view of the fact that, asCurle and Skan have shown, the best overall predictions of separation areobtainedwhentheright-handsideof(129)issetequalto0·0104,itwouldseemmost natural to choose n accordingly, so that

Withthisvalueofn,thedistributionofskin-frictionisderivedfrom(128)whichnow becomes

Itfollowsthatwhenx2Cp(dCp/dx)2hasbeenobtainedatanystationx,thevalueofτw/τBfollowsbysolutionofasimplequarticalgebraicequation.

Forflowswithaninitialfavourablepressuregradient(130)maybeusedtodeterminethedistributionofskin-frictiondownstreamofthepressureminimumx=xm,providedτBisinterpretedastheskin-frictionofaBlasiusboundarylayeratastationx–xL,wherexLisgivenby(125).

ThemethodwasappliedbyCurle(1960) toconsidersomeof thecasesforwhich exact solutions are known, and good agreement was obtained, both asregards the distribution of skin-friction and, in particular, the position ofseparation.

7. Useoftheenergyintegralequation.ThemethodsofTaniandTruckenbrodtTheenergyintegralequation,duetoLeibenson(1935),is

where

and

Uponwriting

and

(131)becomes

Upon multiplication by , this becomes

whichintegratestogive

plusaconstantofintegrationwhichiszero.Thus

NowTruckenbrodt(1952)suggestedthatinmanycasesitshouldbesufficientlyaccuratetoregardαandβasconstants.Itisfairlyeasytoseethatαshouldnotvary verymuch, as the following argument indicates.Nowα, by (132), is theratio of the energy thickness to the momentum thickness, each of which areintegralswith integrandszeroat thewall.Accordingly littlecontributionarisesfromtheregionveryclosetothewall.Butitisjustinthisregionthatthegreatestchanges of profile shape take place, and the outer part of the boundary layerretains approximately the same general shape regardless of pressure gradient.Henceδ3/δ2=αdoesnotvaryverymuch.Thisargumentisborneoutbythefactthatforastagnation-pointα=1·63,foraBlasiusboundarylayerα=1·57,andforatypicalseparationprofileα=1·52,sothatαvariesbyamaximumofabout3percent,aboutitsmeanvalue.Similarlythevalueofβdoesnotvaryasmuchasmightbeexpected.Thereasonforthisispresumablythatthedecreaseinthecontribution to the dissipation integral as the skin-friction decreases to zero atseparation is partially balanced by an increase in momentum thickness and adecrease in the local mainstream velocity. The values of β are 0·209 for astagnation-point, 0·173 for a Blasius layer, and 0·157 for a typical separationprofile,sothatβvariesbynomorethan14percent,aboutitsmeanvalue.

Sinceαandβdonotvaryexcessivelyitseemsreasonabletoreplacethembytheir mean values, which will be close to the values appropriate to a Blasiuslayer. At positions downstream of the pressure minimum the most importantcontributions to the integral in (136) will arise from the region close to thepressure minimum, and the constant values chosen for α and β will beparticularly accurate in this region.Withα = 1·57,β = 0·173, (136) becomes

whichis identical toThwaites’sformula(107)savefora2percent.changeinthe numerical factor, corresponding to a changeof 1 per cent. in the resultingvalueofδ2.

Tani(1954)hasdevelopedamethodofcalculatinglaminarboundarylayerswhichusesboththemomentumandkinetic-energyintegralequations.Withthenotation already used we may write the momentum equation in the form

and the kinetic-energy integral equation in the form (135) may easily berewritten as

Hence, upon eliminating , we find that

Wemayregard(136)and(139)asnewformsofthetwointegralrelationships.Tani integrates (136) in the approximate form (137) to yield δ2 (and λ) as afunctionofx.He then assumes that l,H,α,β are related to eachother by thesame relationship as holds for Pohlhausen’s family of profiles, but determinesthedependenceuponλfrom(139).Thus,asafirstapproximationheignoresthelasttermandbyreplacingH,α,βwiththerelevantfunctionsofl,arelationshipbetweenlandλisobtained.†Havingthusrelatedl,H,α,β,andλ,thesevaluescan be used to estimate the neglected term, and a second approximationcalculated.When this procedure has been repeated until sufficient accuracy isobtained, the relationshipbetweenα,β,andλmaybesubstituted into (136) toobtainanimprovedrelationshipbetweenλandx,butTaniremarksthiswillnotin practice be necessary. The universal relationships between l, H, α, β areshowninTable6.

TABLE6

UniversalrelationshipsforTani’smethod

Truckenbrodt(1952)introducesanalternativeshapeparametertoH=δ1/δ2and α = δ3/δ2. He writes (139) in the form

andintroducesashapeparameterSdefinedby

andS=0whenλ=0.HealsodefinesT(S)as

so that from (139), putting λ = 0, T(0) = 0. Then (140) becomes

or

He then assumes that the relationship betweenT, S,H, l, α, β is exactly thatwhich holds for the Falkner–Skan similarity solutions, but determines thedependence upon λ and x from the integral relationships as expressed inequations(137)and(141).TheuniversalrelationshipsareshowninTable7.

TABLE7

UniversalrelationshipsforTruckenbrodt’smethod

By examination of the similar solutions, Truckenbrodt found that thefunction T(S) could be written as

sothat(141)becomes

whichmaybewritten

Now it may easily be shown that, with λ defined by (137),

where

and

Itthenfollows,aftersomealgebra,that

Theintegrationstartsfromtheleadingedge,whereξ=0,andifthisistakenasthe datum position, (143) yields

Ifu1(ξ)isanincreasingfunctionofξ,thenSwillbepositive,whereasSwillbenegative if u1(ξ) is a decreasing function of ξ. The value of d is determinedaccordingly from (142), and the integration proceeds easily until a point is

reached at which S changes sign. At this point ξ changes discontinuously(becauseofthechangedvalueofd),andtheintegrationthenproceedsbyputtingξ1inequation(143)asthedownstreamvalueofξatthispoint.

Itshouldberemarkedthatthetwomethodsdescribedabove,duetoTaniandTruckenbrodt, arose essentially as simplifications to a rather complex two-parametermethodduetoWeighardt(1946,1948).AfurthermethodbaseduponthatofWeighardt,designedinthiscasetoimproveratherthansimplifyit,isduetoHead(1957a),andweshallnowconsiderthismethod.

8. ThemethodofHeadThismethodwasdesignedtobeconsiderablymoregeneral thananyof the

preceding methods, in that velocity profiles are predicted with considerableaccuracy even when distributed suction is applied. Head prescribed a two-parameter family of velocity profiles dependent upon the parameters l and λ.Fromtheassumedprofiles,chosencarefullytogivethecorrectforminanumberofrepresentativecases,thevariousintegralproperties,α,β,H,canbededucedasfunctionsoflandλ,andHeadgiveschartsoftheserelationships.Healsomakesuseofboth themomentumandkinetic-energy integralequations,writing theseas

which are similar to (138) and (140) respectively. It is assumed that startingvaluesoflandλareknown.Thesewillbel=0·360,λ=0·085atastagnation-point,andl=0·221,λ=0atasharpleadingedge.Fromtheknownvaluesoflandλ,valuesofH,α,β,arereadfromthecharts,whence(144)and(145)yield

andd(α)/dx.Itfollowsthat,correspondingtoasmallincrementinx,the increments in δ2 and α are found. Once the new value of δ2 is known, λfollows.Then,byreferencetothechartforα(λ,l),thevaluesofαandλenablethenewvalueofltobereadoff.ThevaluesofHandβfollowandthesolutionat the newpoint is complete.By repeating these steps thedevelopment of theboundarylayermaybecalculatedbyastep-by-stepprocedure.

Headappliedhismethodtoseveralofthecases,consideredinChapter3,forwhichaccuratesolutionsareknown,andobtainedgoodagreement.Healsodidsome calculations (Head, 1957 b) for flows with distributed suction, and

estimatedtheeffectsofdistributedsuctionuponboundary-layerstability.Thesepoints,however,areoutsidethescopeof thisbook,andwillnotbeconsideredfurther.

†Itisinterestingtonotethataccordingtothisfirstapproximation,inwhichH,l,α,βarefunctionsofλalone,separationoccurswhenλ=–0·082,whichismuchclosertothevalueappropriatetotheknownexactsolutionsthanisPohlhausen’svalue,λ=–0·157.

5

COMPARISONSOFAPPROXIMATEMETHODSOFSOLUTION

INtheprecedingchapternolessthantendifferentmethodshavebeengivenforcalculating the development of a two-dimensional incompressible laminarboundary layer. In viewof the fact that littlemore remains to be done in thisparticular field, and there would appear to be little room for any improvedmethods, it seems desirable to make a critical comparison of the variousmethods,sothatreaderswhoareinterestedmainlyinobtainingnumericalresultswill be able to consider more readily which of the methods best suits theirrequirements.

Therewouldappeartobethreethingswhichmightdeterminethechoiceofamethod.Firstly,thequestionarisesastohowmuchinformationisrequired.Forexample, is it desired merely to calculate the position at which the boundarylayerseparates,orthemomentumordisplacementthicknessatvariousstations?Or,ontheotherhand,isitrequiredtoobtainanaccurateestimateofthedetailedmanner in which the boundary-layer velocity profile develops? Secondly, thequestion of accuracy arises. Is a rough estimate, say within 15 per cent.,adequate,or is itessential tocalculatesomepropertyorother towithinsay±5percent,atmost?Thirdly,whenthosemethodshavebeenrejectedwhichdonotmatch up to the requirements on these two counts, the choice between theremaindercanbedeterminedbyquestionsofspeedandsimplicity.Wewillnowconsider these three points in turn, and will first group the methods in thefollowingway.

A.Methodsbasedonthemomentumintegralequation:

Pohlhausen (section1)

Timman (section2)

Howarth (section3)

Walz (section3)

Thwaites (section4)

B.Methodsbasedonouterandinnersolutions:

StratfordandCurle (sections5and6)

C.Methodsbasedonthekinetic-energyintegralequation:

Tani (section7)

Truckenbrodt (section7)

Head (section8)

1. InformationprovidedbyvariousmethodsWebeginbyremarkingthatifonlythepositionofboundary-layerseparationisrequired,anyofthemethodsisabletoprovideanestimate.If,further,thedetaileddistributionofskin-frictionissought,againanyofthemethodsissatisfactory,withthequalificationthatforflowwhichstartsfromastagnation-pointratherthanasharpleadingedge,theStratford-Curlemethodisonlyuseddownstreamofthepressureminimum.If any further information is required the Stratford-Curle method must be

rejected, but any of the remainingmethods are able to give predictions of theboundary-layer momentum and displacement thicknesses. If, alternatively oradditionally,theenergythicknessisrequiredthemethodsingroupAfalldown,strictly speaking. However, it would be such a simple process to make therelevantextensions toeachof thesemethods thatwemayperhapsaccept themfromthepracticalpointofview.

The only remaining possibility (of any great likelihood) is the detaileddevelopmentofthevelocitydistribution,andthemethodsofThwaites,Tani,andTruckenbrodtfailtoanswerthisrequirement.

2. ComparisonofaccuracyBeforeassessingtheaccuracyofthevariousmethodsitisnecessarytoconsidercarefullyjustwhataccuracyisdesiredinanygivenproblem.Ontheonehanditshouldbepointedoutthatthepositionatwhichboundary-layerseparationtakesplaceissomewhatdifficulttodetermineexperimentally.Inthissenseitmaynotbenecessary,incertaincircumstances,tohaveverygreataccuracy.Ontheotherhand,ifoneisinterestedinatheoreticalpredictionofboundary-layerstability,extremelyaccuratedetailedvelocityprofilesmustbeobtained.Itfollowsthattheensuingremarksregardingaccuracymustbeinterpretedinthelightoftheparticularproblemunderconsideration.Anatural starting-point is tocompare themethodsof typeAabove,which

are all reducible to a form inwhich three functions, l,H, andL, are assumedgivenasfunctionsofλonly.Ofcourse,l,H,andLarenotuniversalfunctionsofλ. In fact, for anygivenpressuredistribution theexact solutionof the laminarboundary-layer equationswould yield a set of functions. In a sense, therefore,thequestionofaccuracyisoneofluck,whethertheassumedfunctionshappentobesufficientlyclosetothoserelevanttothecaseinquestion.

Animmediateconsequenceofthisisthatnoonemethodofthistypecanbedescribed absolutely as the most accurate or least accurate. For example,Pohlhausen’smethodwould probably be considered by almost everyone to bethe least accurate of the five methods of type A, yet there is no doubt thatpressure distributions could be devised forwhich Pohlhausen’smethodwouldyield more accurate solutions than any of the other methods. In practice,however,therewillbelimitstotherangeofpressuredistributionswhichmightarise,andaccordinglyintherangeofvariationofl,H,andLasfunctionsofλ.Itwas on this basis thatThwaites calculated these functions for all known exactandapproximatesolutions,andthensuggestedvaluesforgeneralusewhichare‘average’inthesenseofbeingfarremovedfromtheextremelimitsofvariation.

One of the important results obtained by Thwaites was that the functionsobtainedfromthevarioussolutionsagreewellforfavourablepressuregradients(λ>0),butdiffer as separation is approached in thepresenceofunfavourablepressure gradients (λ< 0).Thus itmatters littlewhich of themethods is usedfrom the forward stagnationpoint to thepressureminimum,butpredictionsofboundary-layerdevelopmentdownstreamofthispointwilldependvitallyuponthemethod used. Since the various curves do not differmuch in their overallgeneralshape,itisroughlytruethatthemethodwhichyieldsthebestpredictionof the position of separationwill also tend to yield the best overall picture ofskin-friction distribution. It is therefore natural to compare the predictedpositions of separation for cases in which the position has been accuratelydetermined.Thishasbeendone,notonlyforthemethodsoftypeAbutalsoforthe other methods discussed. The results, expressed non-dimensionally byquoting thepressurecoefficientsat separation,aregiven inTable8. ItappearsfromthisTablethatthemostaccuratemethods(atleastasregardspredictionsofseparation) are those of Stratford, Tani, and Head, and that there is little tochoosebetween these. It is also apparent that of themethodsof typeA, thoseduetoTimmanandThwaitesarethemostaccurate.

Withregardtopredictionsofmomentumthicknessitwouldprobablymatterrelatively little which of the methods is used. In methods of type A themomentumthicknessisdeterminedmainlybyassumptionsmadeconcerningthefunctionL(λ)inequation(102),and,asThwaitesshowed,thisfunctiondoesnot

varymuchbetweenvariousexactandapproximatesolutions.Thwaites’smethodpredictsthemomentumthicknesstowithin1or2percent,atallstationsinthecaseofthelinearlyretardedvelocity,butisinerrorbyabout6percent,neartoastagnation-point.

TABLE8

Pressurecoefficientsatseparation

Predictions of displacement thickness are given from the momentumthickness upon multiplication by the shape factor H. Accuracy in predictingdisplacement thickness is thus dependent upon accuracy in the value of H.Methods of typeA assume thatH depends only on λ, and are accordingly ofuncertainaccuracy.Thebestislikely,apriori,tobethatofThwaites,sinceitis‘geared’ to the known accurate solutions. Comparisons between Thwaites’smethod and the exact solutions indicate that displacement thickness is usuallypredictedfairlyaccuratelyexceptneartoseparationwhereerrorsofuptoabout10 per cent, can often arise. On the other hand, such comparisons as areavailable indicate that Head’s method predicts displacement thickness to anaccuracy of say 2 per cent, or thereabouts.Althoughdetailed comparisons arenotatpresentavailable,onewouldexpect thatmethods like thoseofTaniandTruckenbrodtwouldbeintermediateinaccuracy,sincesomeattemptismadetoimproveuponThwaites’sassumptionthatHdependsuponλalone.

Withregardtopredictionsofvelocityprofile,theonlymethodwhichmakesany serious attempt to consider this is that ofHead, andwe content ourselvesherewiththecommentthattheaccuracyachievedisconsiderable.

3. EaseofcomputationByfarthesimplestmethodofthoseconsideredisthatduetoStratfordandCurle.Todeterminetheskin-frictionatagivenpositionitisnecessaryonlytocalculate

thevalueofx2Cp(dCp/dx)2,afterwhichtheratioofskin-frictiontoBlasiusskin-frictionfollowsbysolutionof(130),asimplequarticequation.ThemethodsoftypeA,particularlythatofThwaites,arealsofairlysimple

toapply.Forthismethod,giventhevelocityu1(x)atanumberofstations,itisrequired that , and du1/dx be calculated, and a simple quadrature (107)then yields . The remaining characteristics then follow from prescribeduniversal functions.Other thanThwaites’smethod, themethodsof typeA arerather longer, in that either non-integral powers of u1 are required or thequadratureisreplacedbynumericalsolutionofadifferentialequation.

Themethods of Tani andTruckenbrodt are also rather longer than that ofThwaites. Thus, although δ2 is calculated as a function of x exactly as inThwaites’s method, the relationships betweenH and λ, l and λ, etc., must beestimatedbyapproximatesolutionofthekinetic-energyintegralequation.

Head’s method is the longest of all, as the simple method above forcalculatingδ2isrejected,andthemethodrequiresthesimultaneousstepbystepintegrationoftwofirst-orderordinarydifferentialequations.

4. ConclusionsInsummingupwhathasbeendiscussedabove,webeginbyremarkingthat

for calculations of skin-friction alone the obvious choice is the method ofStratford and Curle, since this method is both as accurate as any other andconsiderablysimplerthananyother.Bythismethodtheskin-frictionisgivenbyequation(130).

If, further, either themomentumordisplacement thickness is required it issuggestedthatthesebecalculatedbyThwaites’smethod,providedthaterrorsofup to about 5 per cent, in δ2 and 10 per cent. in δ1 are acceptable. If similarerrors in skin-friction are accepted the whole calculation can be done byThwaites’smethod.

If theaboveerrorsare rather toogreat, theymaybe somewhat reducedbyusingtheslightlylongermethodsofeitherTaniorTruckenbrodt.TheprobabilityisthatTani’smethodisslightlymoreaccurateandmarginallysimpler.

Finally, if greater accuracy than this is required, or if velocity profiles arecalledfor,itissuggestedthatthewholecalculationbedonebyHead’smethod.

One factor which has not been discussed above, which is fundamentallyimportantandveryrelevanttoanypossibleextensionsofamethod,istheextent

towhichtheapproximationsusedcorrespondtophysicalreality.Inthisrespectit should be remarked that the approximations of Stratford’s method arephysically realistic, and that Thwaites’s quadrature for momentum thicknessrests on a firm physical basis by virtue of the work of Leibenson andTruckenbrodt.

6

SOLUTIONSOFTHETEMPERATUREEQUATIONATLOWSPEEDS

IN low-speed flow, provided the difference between the temperature of thestream and that of the wall is not too great (so that the density is sensiblyconstant), thevelocity and temperature equationsmaybe solved separately, aswas previously shown in Chapter 1, section 10. The equations governing theflow then become

and

where the last term in (148), representing frictional heating, may usually beneglected.Then(146)and(147)maybesolvedtoobtainthevelocityfield(u,υ),the methods discussed earlier being available for this purpose, and the linearequation (148) for T remains. We now consider some solutions of theseequations.

1. ForcedconvectionfromaflatplateWebeginwithuniformflowparalleltoaheatedflatplatewheretheflowhas

zeropressuregradient,and thevelocityu1at theedgeof theboundary layer isconstant, u0. Then the solution of (146) and (147), discussed in Chapter 2,section 1, is

wheref(η)istoberegardedasaknownfunction.Wewrite

wheresuffixwreferstovaluesatthewallandsuffixzeroto(constant)valuesinthemainstream,andlookforasolutioninwhichθisafunctionofηalone.Then,uponsubstitutingforu,υ,Tfrom(149)and(150)into(148),wefind(whenthewall is at constant temperature)

after neglecting the frictional heating term. The boundary conditions, T = Twwhen η = 0, T → T0 as η → ∞, when substituted into (150), become

Thesolutionof(151),subjecttotheboundaryconditions(152),wasgivenbyE.Pohlhausen (1921), and is

whereSince f (η) satisfies (64) we may write

whence(153)becomes

Pohlhausencalculatedα0(σ)forarangeofvaluesofσ,whichareshowninTable

9, with the approximate formula

TABLE9

Thefunctionα0(σ)

σ α0(σ)

0·6 0·552 0·5600·7 0·585 0·5890·8 0·614 0·6160·9 0·640 0·6411·0 0·664 0·6641·1 0·687 0·6857·0 1·29 1·2610·0 1·46 1·4315·0 1·67 1·64

Thelocalheat-transferratetotheplate,

is

2. TemperatureofplatethermometerinmovingfluidTheproblemhereconsideredisthatofdeterminingthetemperaturewhichaflat-platethermometerwillreadwhennoheatisbeingtransferredtoitfromthefluid.Viscousheatingisnotneglected,asitistheonlyreasonwhythewalltemperaturerisesabovethatofthestream.SinceTwisnotprescribeditisconvenienttodefineanalternativenon-dimensionaltemperature,sowewrite

Thenuponsubstitutinginto(148),includingthefrictionalheatingterm,wefindthat

theboundaryconditionsbeing

and

Thesolution,firstgivenbyE.Pohlhausen(1921),is

sothatthetemperatureofthewallis

Thefunctionθ(0)hasbeencalculatedbyPohlhausen,andistabulatedinTable10, with the approximate interpolation formula

with which approximation (159) becomes

TABLE10

Thefunctionθ(0)

σ θ(0)0·6 3·08 3·090·7 3·34 3·34

0·7 3·34 3·340·8 3·58 3·580·9 3·80 3·791·0 4·00 4·001·1 4·20 4·197·0 10·06 10·610·0 11·86 12·615·0 14·14 15·5

Beforepassingon, twopoints areworthyofmention.Firstly,whenσ=1,(158) takes a particularly simple form, namely

so that the temperature distribution is given by (157) as

or

This result, in fact, holds when σ = 1 for flow at arbitrary Mach number.Secondly, since the thermal energy equation is linear, solutions can besuperimposed,sothatforflowpastaflatplatewithheat transferandfrictionalheating,thesolutionisasuitablelinearcombinationofthesolutionsgiveninthissectionandsection1ofthischapter.

3. Heattransfernearastagnation-pointNear to the forwardstagnation-pointofacylinder thevelocitydistribution,

as was shown in Chapter 2, section 2, may be written as

andasolutionsoughtbywriting

whereT1isthetemperatureinthemainstream,constantinlow-speedflow.Uponneglectingfrictionalheating,substitutionfrom(161)and(162)into(148)leadsto the equation

theboundaryconditionsbeing

and

Thesolutionis

where

This solution, due to Squire, is reported in the book by Goldstein (1938),togetherwithvaluesofα1(a)forarangeofvaluesofσ.Theseresultsareshownin Table 11, with the approximate form

TABLE11

Thefunctionα1(σ)

σ α1(σ) 0·570σ0·4

0·6 0·466 0·4650·7 0·495 0·4950·8 0·521 0·5210·9 0·546 0·5461·0 0·570 0·5701·1 0·592 0·5927·0 1·18 1·2410·0 1·34 1·43

15·0 1·54 1·68

Thelocalheat-transferratetothecylinderintheregionofthestagnation-pointis

4. ThesolutionsofFageandFalknerThesolutionsgiveninsections1and3ofthischapteraretwospecialcasesofamoregeneralclassofsolutions,consideredbyFageandFalkner(1931),Asbeforewewrite

where thewall temperatureTwmaybea functionofx.Thenuponsubstitutionintothetemperatureequation(148),andagainneglectingfrictionalheating,wefind

FageandFalknerconsider thecases inwhichthemainstreamvelocity isgivenby

sothat,asinChapter2,section3,thevelocitydistributionisgivenintermsofastream-function, and

where

and f(η) is the solution of (71). Fage and Falkner also assume that the walltemperature takes the form

Then upon substituting into (165), and looking for a solution inwhich θ is afunction of η alone, we find that

withθ(0)=0,θ(∞)=1.FageandFalknerintegrated(167)forarangeofvaluesofmandn,foraPrandtlnumberσ=0·77,andmakingtheapproximationthat

thatis,replacingthecurveofuagainstybyitstangentatthewall.Thevalueofαis,ofcourse,proportionaltotheskin-friction.

Forthecaseofawallatuniformtemperature,n=0,and(167)isintegrablefor arbitrarym without further approximation, as Squire has pointed out, theresultbeingshowninequation(175).

5. Lighthill’smethodLighthill(1950a)generalizedtheworkofFageandFalkner,subjecttotheirapproximationtothevelocityprofile,tothecaseofarbitrarymainstreamvelocityandarbitrarywalltemperature.LighthillbeginsbymakingatransformationofthevonMisesform,withnewindependentvariables(x,ψ).where

isthemassflowbetweenagivenpoint(x,y)andthewall(x,0).Thentheenergyequation (148), with frictional heating neglected, takes the form

FageandFalkner’sapproximationtothevelocity,

iswrittenbyLighthillintheform

sothat(169)becomes

This equation was integrated by Lighthill, using the Heaviside operationaltechnique, and his value for the local heat transfer to the wall is

whereT∊isdefinedasthedifferencebetweenthewalltemperatureTw(x)andthemainstream temperature T1 (constant at low Mach number), that is

The integral in (173) is a Stieltjes integral, which may be regarded as ashorthand notation for

whentheonlydiscontinuitying(t)occursatt=0.Lighthillpointsoutthattheapproximation(171)isessentiallyahighPrandtl

number approximation.As thePrandtl number increases the thermal boundarylayer becomes thinner, relative to the velocity boundary layer, so that for thepurpose of calculating the thermal boundary layer it becomes increasinglyaccurate to replace the velocity curve by its tangent at the wall. Accordingly(173) is expected to be asymptotically exact as the Prandtl number tends toinfinity.

Lighthill tests the accuracy of (173) at a Prandtl number σ = 0·7, bycomparison with the similarity solutions. With a mainstream velocity

and a uniformwall temperatureTw, it can easily be deduced from (167) that

afunctiononlyofmandσ.Exactresultsforσ=0·7andvariousvaluesofmarecomparedinTable12withvaluesdeducedbyLighthillfrom(173).

TABLE12

Thefunctionαm(0·7)

m αm(accurate) αm(approx.)

–0·0904 0·199 0

–0·0654 0·253 0·232

0 0·292 0·300

0·331 0·3600·384 0·435

1 0·495 0·587

4 0·813 0·995

It is immediately clear that agreement is excellent, about 3 per cent. error,when m = 0 (zero pressure gradient), and is reasonable in the vicinity of astagnationpoint,m=1,wheretheerrorisabout19percent.Ontheotherhandtheagreementispoornearapositionofseparation,wherem=–0·0904.Lighthillindicatesthereasonwhytheerrorvarieswithminthismanner.Whenm=0thevelocityboundary-layerprofile isessentially linearoveraconsiderableportionof theboundary layer.Forothervaluesofm,where there isacurvature in thevelocityprofileat thewall, there isagreaterdeviation from the tangentat thewall in the outer parts of the boundary layer. Thuswhenm > 0 the assumedvelocityisgreaterthanthetruevalue,sotheheattransferisoverestimated.Form<0 theassumedvelocity is less than the truevalue,and theheat transfer isunderestimated. As separation is approached this effect is excessivelyaccentuated, since Lighthill’s assumption is equivalent to putting the velocityequaltozeroeverywhere.

Theaerodynamicallydesirable range is , sincenegativevaluesof m lead to the possibility of boundary-layer separation, and further havevelocity profiles with a point of inflexion and a resultant possibility of

instability. Now in this range the error in the predicted heat transfer variesbetween+3percent,and+19percent.Lighthillpointsoutthatifthepredictedheat transfer be multiplied by a factor 0·904, so that the numerical factor

in(173)isreplacedby0·487,theerrorwillliebetween±7percent.Accordinglytheaccuracyisgoodwhenthisempiricalcorrectionismade.

6. TheworkofLiepmannandCurleLiepmann (1958) has derived an expression for the heat transfer which is

identicalinformtoLighthill’sresult(173),andisobtainedbyuseofthethermalenergy integral equation, (55). This equation may be written in the form

whereQw(x)istheheattransfertothewall

Liepmann, inplaceofLighthill’sapproximation(170),writes thevelocitynearto the wall as

andbywriting

(176)maybetransformedintotheform

whereθistheusualnon-dimensionaltemperatureasdefinedinequation(162).Liepmannthenconsidersthespecialcaseinwhichthewalltemperatureisequalto its zero heat-transfer value T1 when say, and increasesdiscontinuouslytoanewconstantvaluedownstreamofthisposition.Insucha

specialcaseQ/Qwisapproximatelyauniversalfunctionofθalone,sothat(177)becomes

where

is a constant. When τw is known as a function of x, (178) may easily beintegratedtoyieldQw(x).Liepmannexpressedthisintegralanalytically,andthenderived thevalueofQw(x) for anarbitrarywall temperatureby integrating thecontributionsfromadistributionofelementarychangessimilartotheabove,andsoobtaineda result identical to (173),with theconstantreplaced by . We note that the value a = 0·215 is obtained by setting

in(179),andthisleadstoavalue ,whichiswithin 3 per cent, of Lighthill’s value. Liepmann also shows how the sameapproach may be used to derive a formula which is valid in the vicinity ofseparation.

Curle (1961 a) has shown how, by an approach modelled on that ofLiepmann,Lighthill’smethodmaybeconsiderably improved inaccuracy.Thethermal-energyintegralequation(176)isused,andthevelocityuapproximatedby

Thisleadstotheresult

InthespecialcaseofaregionofzeroheattransferwithTw=T1followedbyadiscontinuity in wall temperature to a new constant value, the temperatureprofile is represented by a universal shape, characterized by a boundary-layerthickness δ which varies with position. It follows that

and

Thus

and

where a and b are positive constants. Substitution from (182) and (183) into(181) yields

whichistobecomparedwithLiepmann’sequation(178).Analternativewayofwriting this equation is to introduce a representative velocity U∞ and arepresentative length l, a local Nusselt number

andaReynoldsnumber

Thenthevalueof

isgivenbysolutionoftheequation

or

where

and

The constantsa,b are determined by reference to the similarity solutions.The values

are such that agreement iswithin about ±1 per cent, over thewhole range ofpressure gradients from stagnation-point to separation, andwith values of thePrandtl number of order unity (typical of gases) and order ten (typical ofliquids).Thenumericalaccuracyis,accordingly,animprovementonthatgivenbyLighthill’smethod,althoughthepresenceoftwotermsontheright-handsideof(184)or(185)precludesanalyticintegration.Asimplemethodofnumericalintegration,duetoThwaites,istoreplacetheintegralin(185)byitsSimpson’srule value, so that when K(x) and K(x+h) are known, K(x+2h) follows bysolutionofaquarticalgebraicequation.

Ithasbeen remarkedearlier that theaboveapproachesare essentiallyhighPrandtl number approximations. Curle (1961 a) has also examined thepossibilityofderivingacceptablesolutionsatPrandtlnumbersoforderunitybyempirical correction of a low Prandtl number solution. At low values of thePrandtl number the temperature boundary layer is thicker than the velocityboundarylayer,andCurlereplacesthelocalvelocityuin(176)byβu1,whereβwill tend to unity as the Prandtl number tends to zero. He also writes

with a0 constant, in the special case of a single discontinuous jump in walltemperature. Then substitution into (176) yields

whichmaybeintegratedandtransformedtoyield

Thegeneralizationtothecaseofarbitrarydistributionofwalltemperatureisexactly as described by Liepmann, and yields the Steiltjes integral

where T∊ is defined as Tw–T1 as in (174). The form of this equation is inagreement with that obtained by Morgan et al. (1958), who showed that theasymptotically exact value of as σ → 0 is

Curle remarks thatwhen this constant is changed to 0·313, the formula (186)yieldsvaluesoftheheattransferagreeingwiththesimilarsolutionsto±25percent.Providedthatthevelocityboundarylayerisnottoothinincomparisonwiththe thermalboundary layerCurle shows that a considerable improvementmaybe effected by writing

where λ is the pressure gradient parameter, defined by (108), which may becalculatedbymeansofequations(107)and(108).

Curle has applied these two methods to the case of flow past a heatedcylinder,andcomparedhisresultswiththeexperimentsofSchmidtandWenner(1941).Theagreementisgoodineachcase,thesolution(186)(derivedfromthelowσapproximation)givingparticularlygoodagreement.

7. Spalding’smethodAn alternative method of improving Lighthill’s method was given by

Spalding (1958). His method begins with the equation (180) for u,

andhe remarks that the correction toLighthill’smethodwill dependupon therelativeimportanceofthesecondtermontheright-handside.Nowatapositionwhich is representative of the thickness of the thermal boundary layer, say

theratioofthesecondtermtothefirstis

Accordingly he considers the case of a single step inwall temperature,writes(178) as

andsuggeststhattheright-handsideofthisequationbereplacedbyafunctionofthe parameter χ defined in (187), so that it becomes

whereχistheexpression(Δ4δ4/v)(du1/dx)ofSpalding’spaper.ThefunctionFisdeterminedbyreferencetotheavailableexactsimilarsolutions.ItisshownasagraphinSpalding’spaper,andistabulatedhereinTable13.Spaldingsuggeststhesolutionof(188)byiteration,TABLE13

ThefunctionF(χ)

χ F(χ)

–4 3·5

–3 3·8

–2 4·3

–1 5·1

0 6·4

+1 8·5

2 11·6

3 15·8

so that a first approximation toQw, obtained by setting F = 6·4, is used todetermineabetterapproximationtoF,andsoon.ForthecaseinwhichTw=T1when , and Tw = T1+ΔT∊(ξ) when , this yields the result

whereχ1isthevalueofχobtainedbyusingthepreviousapproximationtoQw.Thesolution forarbitrarywall temperature isobtainedbyadding the solutionsfor the relevant distribution of the above elementary increments in walltemperature.

It should be remarked at this stage that, although the method will workeffectively(andyieldanimprovementinaccuracyascomparedwithLighthill’smethod) provided thatF does not differ toomuch from6·4, itwill still breakdownasseparationapproachesandtheparameterχ→∞.

Spaldinggoesontoshowthatifonlythetotalheat-transferrateisrequired,it isnotnecessary to calculate (and then integrate) the localvalues.Aquickermethodistosetupanequivalentrelationshipto(188),whichwillallowthistobedonedirectly.Spalding takes the thermalenergy integralequation(55),andintegrates it with respect to x to obtain

whereΔ2 is ameasure of boundary-layer thickness.Considering, as usual, thesingle-stepdistributionofwalltemperature,Spaldingshowsthat(173),whichisLighthill’s approximation, may be written as

andthattheappropriategeneralizationmaybeexpressedas

Thefirststepinaniterativesolutionofthisequationis

whereagainabreakdownoccursasseparationisapproached.

8. ThemethodofDaviesandBourneWeconsidernowbrieflyone furthermethodwhichhasbeensuggested for

improvingtheaccuracyofLighthill’smethod,duetoDaviesandBourne(1956).Theirmethodistoassumethatthevelocityprofileu(x,ψ)isknown,andmaybeapproximated by the expression

sothat

The values of b1 and β are determined by the conditions of the particularproblemunderconsideration,sothattheexactandapproximatevelocityprofilesareingoodagreementoverthegreaterpartofthethermalboundarylayer.Asσ→ ∞ we have , as in Lighthill’s approximation. When consideringboundary-layerflowwithαoforderunity,DaviesandBournechoseb1andβtogivegoodagreementovermostof thevelocityboundary layer,whenapplying

theirmethodtothecaseofthesimilarsolutions.Thetemperatureequationinthevon Mises form (169), after substituting for u from (191), yields

which is a form amenable to analytic treatment. Davies and Bourne’s resultsindicate that the percentage error in the predicted heat-transfer rate is roughlyequaltoatypicalpercentageerrorintheassumedvelocityprofile.

Themethodisneat,butdoesnotappeartobeofgeneraluse,sinceitreliesupon a knowledge of the detailed velocity profile, which cannot easily beestimatedwithanyaccuracy.Furtherdifficultiesariseifinsufficientevidenceisavailableconcerningtherelativeextentsofviscousandthermalboundarylayers.

9. TheMeksyn-MerkmethodWe now turn our attention to a method which was devised by Meksyn,

applied originally to a number of viscous boundary-layer problems, andsubsequently adaptedbyMerk (1959) for calculating thermalboundary layers.Theapplicationstoviscousboundarylayerswillnotbediscussedhere,andthereader is referred to the book by Meksyn (1961) for details and for furtherreferencestothemethod.

Themethodasusedtocalculatethermalboundarylayersmaybeillustratedby reference to the problem of section 1 of this chapter, namely forcedconvection from a flat plate. Consider the function α0(σ), given by (153) as

WeintroducethefunctionF(η)definedas

sothat

Nowby(65)wehave

sothat

where α = 1·32824 and the series both converge for a finite range of η. Theseries (195) may be inverted to give

Then,uponsubstitutioninto(193)wehave

Thisseriesmaybeintegratedformallytermbyterm,andwhenthereciprocalofthe resulting series is calculated it is found that

Thelimitedrangeofconvergenceoftheseries(194)meansthat(196)isinthenatureofanasymptoticexpansion.However, even the three termsshownhereare sufficient to give excellent resultswhen .When , these threetermsyieldα0(0·5)=0·520,comparedwiththeexactvalue0·518,andwhenσ=0·8thethreetermsyieldthreecorrectdecimalsinα0(a).

Theaboveindicatesingeneral termstheprinciplesonwhichthemethodisbased.FormoregeneralcasesMeksynandMerkmakefirstatransformationofthevariableswhichisvirtuallyidenticaltothatusedbyGörtler(1957a),sothattheequationsforthevelocityandtemperaturefunctionstakeaforminwhichtheabovetypeofasymptoticexpansioncanbeapplied.

By developments along these lines, it has been possible to consider theboundary layer for Schubauer’s experimental pressure distribution (Meksyn,1956), and for retarded flow along a semi-infinite plane (Meksyn, 1950), theexactsolutionstotheseproblemsbeinggivenbyHartree(1939b)andHowarth(1938) respectively. For the thermal boundary-layer problemMerk (1959) hasderived solutions corresponding to the similar solutions of E. Pohlhausen and

Squire, considered in sections 1 and 3 of this chapter.He has also consideredmoregeneralcasesofflowpastcircularandellipticcylinders.Forthecaseofacircular cylinder, the predicted heat transfer over the front 70° of the cylinderagrees exceedingly well with experiment, but the method breaks down asseparation is approached at about 80°. Meksyn (1960) has also shown howaccurate solutions for the plate-thermometer problem (section 2) may beobtainedfor largevaluesofσ(>100)wherestraightforwardcalculationsof theintegral(159)areinaccurate.

10. Curle’sanalysisbyStratford’smethodIthasbeenremarkedearlierthatoneofthesimplestandmostaccuratemethodsdevisedforcalculatingtheskin-frictioninanincompressiblelaminarboundarylayeristhatduetoStratford(1954),themethodbeingbasedontheideaofdividingtheboundarylayerintoinnerandouterportions.Curle(1961b)hasshownhowasimilartechniquemaybeusedforcalculatingthecorrespondingpropertyforthethermalboundarylayer,namelytheheat-transferrate,whenthewallisatuniformtemperature.Followingtheprocedureadoptedinthecaseoftheviscousboundarylayer,

the thermalboundary layer isdivided intoan innerandanouter region. In theouter region the thermal energy equation is written as

where (∂/∂s) denotes differentiation at constantψ, defined by (110).We alsoexpand T(x, ψ) in the form

Weconsiderthespecialcaseinwhichthereiszeropressuregradientupstreamofapositionx=x0.Thedownstreamvelocityprofile isdistortedby thepressuregradient,butonlyneartothewallinitially.Accordinglywededucefrom(197)thatthevalueof(∂T/∂s)x,ψ isunaffectedbythepressuregradient,exceptintheregionneartothewall,so(198)showsthatwhen ,say,T(x,ψ)isexactlyas it would have been in the absence of pressure gradient, provided terms oforder(x–x0)2,areneglected.NowitmaybereadilydeducedfromPohlhausen’s

solution (154) that in the absence of pressure gradient

and(anticipatingtheapproximationbelow)thismaybewritten

forsmallvaluesofψ,where arisesfromtheapproximation(155).Thereisnoa priori reason why ψj (determining the join in the inner and outer velocityprofiles) and (determining the join of the temperature profiles) should beequal. It is assumed, however, that both are sufficiently small that theBlasiusvelocity profile remains sensibly linear throughout and . Itfollows from (199) that the value of given from the outer profile is

andupondifferentiatingwithrespecttoψ

Turning now to the inner region, , we assume a form

which should be a good approximation, since both ∂2T/∂y2 and ∂3T/∂y3 areidenticallyzeroaty=0.ThevaluesofTand∂T/∂yatthejoin,asgivenby(202),may be equated to the values (200) and (201), yielding

and

These two equations connecting Qw, must be supplemented byrelationships giving and in terms of . To do this it is assumed forsimplicitythattheexpression(126)isadequatethroughout .Thismeans,in reality, that but since the inner and outer solutions for u havecontinuous values of u, ∂u/∂y, and ∂2u/∂y2, it is possible that (126) holds for

values of ψ rather greater than ψj. Then

and

withτwandaknownfromthesolutionforthevelocityprofile.ThevalueofQwmay be determined from (203) to (206), and is

Accordingly,oncetheskin-frictionτwhasbeendetermined,theheat-transferrateisgivenbythisexplicitformula.This,ofcourse,isbyfarthesimplestmethodofcalculatingheat transfer.The limitations of themethod are that it canonly beused when the wall temperature is uniform and the pressure gradients areadverse.

To estimate the accuracy,we remark that the formula (207) is exactwhenthereiszeropressuregradient.Forsmallpressuregradientstheaccuracyremainsgood. Thus, if the mainstream velocity is

the heat-transfer rate is predicted from (207) and (130) to be

whereas the exact solution replaces the coefficient of x/c by 1·86.We shouldexpecttheaccuracytodecreaseasweproceedfartherfromtheaccuratelyknownstartingconditions,thatisasseparationisapproached.Atseparation(207)yields

the numerical coefficient being equal to 0·72whenσ = 0·7. For the similaritysolution which corresponds to a separating boundary layer the numericalcoefficient is , which might typically equal about 0·57. Theagreementisstillfairlygood.

11. Squire’smethodWe shall now give a brief discussion of a method due to Squire (1942),

modelled on the integral methods first used by K. Pohlhausen (1921) for the

viscouslayer.Squire’smethodrestsinitiallyupontheideathatthe‘shape’ofthetemperatureprofileinalaminarboundarylayerdoesnotvarymuchoverawiderange of pressure gradients. Since the temperature profile is the same as theBlasiusvelocityprofile in the caseof zeropressuregradient,wemay supposetherefore that

where

is the Blasius velocity profile, being tabulated in Table 1. We thenassumethatthevelocityprofileisalwaysgivenby(210),withthedisplacementthickness δ1 suitably chosen. Squire chooses

which is the value given by an approximate method due to Young andWinterbottom(1942).Then,substitutingfrom(209)and(210) into the thermalenergy integral equation (55), we have

whichmaybewrittenas

or

where ϕ(Δ1/δ1), which was calculated by Squire, is defined as

Uponintegrating(212)wefindthat

andwhendividedby(211)thisyields

whichisanequationforΔ1/δ1.Squiresuggeststhatagoodfirstapproximationistotakeϕasaconstanton

theright-handsideof(213),whichmaytheneasilybecalculatedasafunctionofx. Then the values of Δ1/δ1 may be determined from a table of the function

,which is givenhere asTable14.A second approximation, ifrequired,isobtainedbyusingtheresultingvaluesofϕ torecalculatetheright-handsideof(213).

TABLE14

Thefunctionη2ϕ(η)

Squirecompares the resultsofhismethodwithsomeof theexact ‘similar’solutionsoverarangeofvaluesofσ.Hefindsagreementbetterthan1percent,asregardsheattransferintheabsenceofpressuregradient,butthepredictedheattransfer is about 5 per cent, too low near a stagnation-point. A similar

comparisonforthecaseoftheseparationprofileindicatesthatthepredictedheattransferis30percent,toohigh.Squirehasalsousedhismethodtocalculatetheheat transfer in flow past a circular cylinder, obtaining reasonable agreementwithexperiment.

12. FreeconvectionfromaheatedverticalplateToconcludethischapterweconsidertheproblemoffreeconvectionwhenaheatedplateisplacedinstillair.Itisassumedthatthetemperaturedifferencebetweentheplateandtheairissmall,sothatthefluidpropertiesmaybetakenasconstant,butthatthe(small)motionoftheairiscausedbyabuoyancyforceduetodensityvariations.Then,withxmeasuredupwardsfromtheloweredgeoftheplate,ynormaltoit,u,υ,theassociatedvelocitycomponents,Tw,T0,thetemperatureoftheplateandtheairrespectively,theequationsbecome

and

Uponintroducing

(215)and(216)become

and

These partial differential equations may be reduced to ordinary differentialequations, as was first shown by E. Pohlhausen (1921). We write

sothat

Then(217)and(218)become

withboundaryconditions

These equations were integrated numerically by Pohlhausen, with the Prandtlnumber σ = 0·733. From his results we deduce that

givingtheheat-transferrateattheplate.

7

THECOMPRESSIBLELAMINARBOUNDARYLAYERWITHZEROPRESSUREGRADIENT

THEessentialdifficultyindealingwithcompressiblelaminarboundarylayersisthat the momentum and thermal energy equations are coupled, since thetemperature field influences the density, which in turn influences the velocityfield. Mathematically we are faced with the immense task of solving twosimultaneous non-linear partial differential equations. For this reason the onlyflowwhichhasbeenconsideredindetailoverawiderangeofconditionsisthatwhen there is zero pressure gradient. For such a flow the velocity u1, thetemperatureT1,andthedensityρ1outsidetheboundarylayerareconstant.Eachof thesequantities is thusequal to itsvalueat the leadingedge,x=0,andweshall accordingly denote them by suffix 0. The boundary-layer equations, (1),(12), and (19), then become, upon neglecting body forces

and

Beforeproceedingfarther it isnecessarytodiscusswhatassumptionsaretobemadeconcerningthevaluesofμandσ,andtheirdependenceuponthevariousotherpropertiesofthefluid.

1. ValuesforviscosityandPrandtlnumberAn accurate representation of the variation of viscosity with absolute

temperature is given by Sutherland’s formula

where c is a constant, taken to equal 114° K for air. The complexity of thecompressiblelaminarboundary-layerequationsissuchthatitisoftennecessaryto use a simpler formula such as

By suitable choiceofω andof the constantofproportionality it ispossible tomake(227)agreewellwiththeaccuratevalue(226)overafairlywiderangeoftemperatures.Forexample,ω=0·76hasoftenbeenused,andCopeandHartree(1948) pointed out that good agreement in the rangecould be obtained by taking ω = 8/9. Considerable simplifications in theboundary-layerequationscanbeeffectedbytakingω=1,andthisfacthasbeenusedtogoodeffectinmanytheoreticalinvestigations.

ThevalueofthePrandtlnumber,σ,doesnotvarymuchwithtemperatureforair,andaconstantvalueofabout0·72or0·73hasusuallybeenassumed.Itmustbepointedout,however,thatconsiderablesimplificationsresultfromthechoiceσ=1,andtheorieshaveoftenbeendevelopedonthisbasis.

Wecaneasilyseefromequation(225) thatwhenσ=1 the thermalenergyequation always has a solution

This solution corresponds to the case of a thermally insulated wall, whosetemperatureisconstantwhenσ=1.Forthecaseofzeropressuregradientweseefurther,bycomparisonbetween(224)and(225),that(225)alwayshasasolution

providedthewallisatuniformtemperature.Thegeneralsolutionisthen

and sinceT =Tw, u = 0, when y = 0, andT =T0, u = u0, when y =∞, thisbecomes

or

introducingthetemperatureTzatwhichtherewouldbezeroheattransferatthe

wall.We note that when σ = 1, regardless of the viscosity-temperature

relationship,thesolutionof(225)isgivenby(229)forthecaseofzeropressuregradient, and it remains only to solve (224) for u. We now consider variousattempts to do this, with different assumptions regarding viscosity, as well assomesolutionswithvaluesofσotherthanunity.

2. ThesolutionsofBusemannandKármánBusemann (1935) assumed σ = 1, and so deduced as above that T is a

function only of u. He then reduced the equation (224) for u to an ordinarydifferential equationwith as independentvariable, the formbeingsimilartothatoftheBlasiusequation.ThisequationwasthenintegratednumericallyforthecaseofMachnumberM0=8·8andzeroheattransferatthewall.

As a result of these calculations he found that the velocity varies almostlinearlywithdistancefromthewallatthisMachnumber.SincealinearvelocityprofileisafairapproximationeveninincompressibleflowvonKármán(1935)considered the consequences of assuming a linear velocity profile at allMachnumbers, and calculated approximate solutions by the Kármán–Pohlhausenintegralmethod.Zeroheattransferwastaken,ofcourse,andthePrandtlnumberwassetequaltounity,sothatthetemperatureprofilewasgivenby(229),withTw = Tz, namely,

or

Nowwhen there isnopressuregradient, themomentum integralequation (40)may be written as

Uponsubstitutingforρ,

setting

and

wededucethat

Ifwelettheviscositylawbeμ∝Tω,thenμωisrelatedtothemainstreamvalueμ0 by

sothat(230)integratestogive

fromwhichwededucethattheskinfrictionis

This is effectively the form in which Kármán presented his results.We note,however, that it may alternatively be written as

Wenotethat increasesfrom0·290whenλ=0,i.e.M0=0,to0·393whenλ=1,i.e.M0→∞.Theaccuratevaluesare0·332and0·400respectively,sotheaccuracyisconsiderableathighMachnumbers,decreasingfrom13percent,at zero Mach number to less than 2 per cent, as the Mach number tends toinfinity.

3. ThesolutionsofKármánandTsienInviewof therelatively largeerroratsmallerMachnumbers,Kármánand

Tsien(1938)laterattemptedamoreaccuratesolutionoftheequationsofmotion.Theybeganwiththeboundary-layerequationsinthevonMisesform(59),(60),and in the absence of pressure gradient these become

and

Upon taking σ = 1, the solution of (234) is given by (229) as

so that, once the wall temperature Tw has been prescribed, the temperaturedistributionisgivenexplicitlyintermsofu,whichmaythenbedeterminedfrom(233). To solve this equation, Kármán and Tsien introduce non-dimensionalquantities,

and

where suffix zero refers to (uniform) values in the mainstream, and c is arepresentative length. Then (233) becomes

whichequationmaybesolvedbyiteration.Asafirstapproximationthefunctionu*(ψ*) appropriate to the incompressible Blasius solution is taken, with T*deduced from (235) and μ*, ρ*, from the relationships

and

this last result following immediately from theequationof state (7), sincep isconstant. Hence the product

isknown.Equation(236)thenyields

where

and

Thisimprovedvalueofu*(ψ*)maythenbeusedasastarting-pointforthenextapproximation. When u*(ψ*) has been determined to sufficient accuracy, thevalue of the true coordinate y normal to the wall is calculated as

KármánandTsiencalculatedsolutionsforflowwithzeroheattransferoverarangeofMachnumbersfrom0to10.Thevaluesoftheskin-frictioncoefficient

areshowninFig.1,togetherwiththeresultsofotherworkers.Theyalsocarriedoutsomecalculationsforcasesinwhichthewalltemperaturewaskeptconstantat one-quarter of the mainstream temperature. For details of these resultsreferencemaybemadetotheoriginalpaper.

4. ThesolutionsofEmmonsandBrainerdSome solutions for zero heat transfer, with various viscosity-temperature

lawsandat severalvaluesof thePrandtlnumber,weregivenbyEmmonsandBrainerd (1941,1942).Theseauthors introducedanumberofnon-dimensionalfunctions, defined as

andshowedthattheequationsofmotion(223)to(225)couldbereducedtothreeordinary differential equations for u*, T*, and

asfunctionsofη.Theseequationstaketheform

and

These equations were then integrated on a differential analyser under thefollowingassumptions.

Firstly, a range of solutions was obtained for μ* = 1, that is for constantviscosity and conductivity, equivalent to taking ω = 0 in the viscosity-temperature relationship (227). These solutionswere for values of the Prandtlnumberσ=0,0·25,0·733,1·00,1·20,respectively,andatMachnumbersM0=0,0·5,1,1·5,2,and√10.Someof the results for thevariationof skin-friction

withMachnumberareshowninFig.1.Aninterestingconclusionwhichmaybedrawn from the results of these calculations is that when there is zero heattransfer the relationship between wall temperature Tw and mainstreamtemperature T0, is given with considerable accuracy by

whichis inagreementwiththeresultobtainedbyE.Pohlhausenforlow-speedflow.

In their second paper, Emmons and Brainerd initially took σ = 0·733throughout, and examined various possible viscosity-temperature laws. Theyconcluded that thechoiceω=0·768,close to thevalue(237)usedbyKármánand Tsien, was the most satisfactory, and subsequently used only thisrelationship. They then calculated velocity and temperature distributions atvariousMachnumbers,andwiththesamePrandtlnumbersastheyusedintheirfirst paper. Some of these further results have also been included in the skin-frictioncoefficientsinFig.1.

FIG.1.Skinfrictiononaflatplatewithnoheattransfer.

—Calculatedbyvariousworkers;---accordingtoequation(272).C&H,CopeandHartree;K&T,vonKármánandTsien;B&E,BrainerdandEmmons.

5. ThecalculationsofCroccoAnexhaustiveinvestigationofthelaminarboundarylayerwithzeropressure

gradient has been carried out by Crocco (1946). Upon setting the pressure

gradient equal to zero and applying the Crocco transformation (Chapter 1,section 11), the equations (57), (58) become

and

thedependentvariablesbeingτandI,theindependentvariablesxandu.Croccolooks for a solution in which the velocity and temperature profiles are eachsimilar fordifferentvaluesofx (thoughnot similar toeachother).Thus I is afunction of u alone, and τ may be expressed as

After some straightforward analysis, Crocco deduces that , andaccordingly writes

whence substitution into (245) and (246) leads to

and

wheredashesdenotederivativeswithrespect tou.Non-dimensionalquantities,similar to those used in previously discussedmethods, are introduced, so that

and

Equations (248) and (249) then take the form

and

primesnowdenotingderivativeswithrespecttou*.Theboundaryconditionsare

and

Wenotethat

and

arefunctionsofI*alone.Croccoconsideredfirstsolutionswithσ=1.Whenthisholds(252)becomes

simply

so that the relationshipbetween I*andu* isquadratic,ashaspreviouslybeenindicated by other methods. It then follows that (251) takes the form

whereh is a known functionofu*.This equationwas solvedbyCroccoby asimilar iterativemethod to that used by Kármán and Tsien when they solved(236). Thus we have

and a first approximation G1, substituted into the right-hand side of thisequation, leads to a second approximation G2. Crocco remarked that thisprocedure would not, in general, converge but that it would do so rapidly ifeither or were taken as the starting value for the nextiteration.

Havingthusconsideredthecaseofσ=1,Croccoturnedtothecaseω=1,σ≠ 1. When this holds equation (251) takes the form

sothatGisindependentofI*andσ.Itisfairlystraightforwardtotransformthisequation so that it becomes identical with the Blasius equation for

incompressible flow. The required transformation is

whence(255)maybeshowntobecome

Theboundaryconditionsareeasilyshowntobesuchthattherelevantsolutionof(257)isthatgiveninChapter2fortheBlasiusprofile.Itfollowsfrom(255)thatGasafunctionofu*issimplytheknownrelationshipfor asafunctionof

.In particular, the skin-friction is given by (247) and (250) as

and by (256), substituting the value of f″(0) from (67), this yields

whichmayalternativelybewrittenas

sinceμ∝T.Oncethisrelationshipisknown,itremainstosolve(252)forI*.Sincethisis

alinearfirst-orderequationforI*′,itisastraightforwardproceduretoobtainI*by integration, and amongst the various forms in which the result can beexpressed is

whereΔ1,Δ2aretwofunctions,calculatedbyCrocco,andshowninTable15forvarious values of σ. They are defined as

and

In view of (256) thesemay alternatively be expressed in terms of theBlasiusfunctions. Crocco showed, in fact, that 8σΔ2 is the function θ(η) defined byequation (158)andcalculatedbyE.Pohlhausen (1921).Analternativemethodofexpressing(260),whichismoreimmediatelyusefulwhenI*′(0)isunknown,is

where

and

For themore general case,whenω ≠ 1,σ ≠ 1, it is necessary to integrate(251)and(252)simultaneously,andthiswasdonebyCroccoforalargenumberofcases,overarangeofMachnumbersfrom0to5,withω=0·5,0·75,1,and1·25,andalsoinanumberofcasesinwhichtheviscosity-temperaturelawwasgivenbyaparametricformofSutherland’slaw(226).Animportantconclusionfromthesecalculationsisthatthevariationofenthalpywithvelocity,thatisthefunction I*(u*), is virtually independent ofwhich viscosity-temperature law isassumed.For this reasonCroccosuggested that the relationship (260)or (261)couldbeassumed tohold ingeneral,evenwhenω≠1, so that thesolutionof(251) alone remains to be determined. This does depend, though, upon theviscosity-temperature law,and theaccuratevalueshouldbeused.Thesolutionoftheequationisformallyexactlyasinthecaseσ=1,sinceμ*ρ*isknownasafunction of I*, and hence of u* and σ, and (251) takes the form

TABLE15

ThefunctionsΔ1(σ,u*),Δ2(σ,u*)

Thisissimilar to(253),andmaybesolvedbyananalogousprocedure,sothatwe write

and substitute an approximationG1 in the right-hand side to obtain a secondapproximationG2.Theneither or thestarting-pointforthenextiteration.

This considerable simplification of the simultaneous equations, (251) and(252), was used by van Driest (1952), who carried out extensive calculationswithσ=0·75,Machnumbersfrom0to20andwall-to-free-streamtemperatureratios,I*(0),from0·25to6.

WemaydeducefromCrocco’swork,andinparticularfrom(260), that therelationship between heat transfer andwall temperature, that is between I*(0)and I*′(0), is

or

Now

wheresuffixzeroonΔ1orΔ2denotesvaluesatu*=0.Accordingly,whenthereis zero heat transfer the wall temperature Tw = Tz is given by

or

sinceitmaybededucedfromtheresultsofCrocco,vanDriest,andBrainerdandEmmons, and from the results of E. Pohlhausen for low speeds, that

AsimilarmethodtothatofCroccowasusedbyHantzscheandWendt(1940,1942)toconsidercaseswithandwithoutheattransfer.Theirresultsaremainlyfor eitherω = 1 or σ = 1, and, as they are less comprehensive than those ofCrocco,theywillnotbeconsideredfurtherhere.

We will, however, refer briefly to an approximate analysis byMonaghan(1949)fortherelationshipbetweenenthalpyandvelocityatarbitraryσclosetounity. Monaghan makes the major assumption that enthalpy and velocity arerelatedexactlyasifeachdependedonlyuponlocalconditions,anddeducesthat

Nowitisnotdifficulttoexpress(260)or(261)inthisform.Wewrite(261)as

andaddandsubtractthequantity

whereuponwehave

By making use of (262) and (263) this may be simplified to

Comparisonwith(267)showsthatMonaghan’ssolutionreplaces

Monaghancomparesthesevaluesfortwocases,σ=0·725andσ=1·25.Ineachcasehe findsagreementwithin1percent, forvaluesofu/u0 less than0·5, theerror increasingtoabout3or4percent,atu/u0=0·8,andtheorderof10percent,atu/u0=1·0.TheformulagivenbyMonaghanisexactwhenσ=1,andingeneral we can remark that for values of σ close to 1 the approximation isextremelygoodexceptperhapsattheveryedgeoftheboundarylayer.

6. SummaryofresultsforuniformwalltemperatureAllthesolutionsconsideredsofarinthischapterareforflowsinwhichthe

wall is maintained at a uniform temperature. Taken together they provide acomprehensivesetofresultsforvaluesofσbetween0·5and2andvaluesofωbetween 0 and 1·25, both with and without heat transfer. Young (1949) hascarriedoutadetailedanalysisoftheseresults,andhassucceededindevelopingsimple formulae for skin-friction and heat transfer which agree with theaccurately calculated values to within about 1 per cent, over the physicallyimportantrangeoftheparametersconsidered.

Young expresses his results as a formula for the product cf√Rx, where

and

aretheskin-frictioncoefficientandReynoldsnumber,respectively.Hededucesfrom (258) that

andshowsfromtheresultsofHantzscheandWendtthat

wherethecoefficient0·3andthefactor areapproximate.Thesetwoformulaeapply when there is zero heat transfer. For the case of heat transfer, Youngobtainsaroughsolutionof(251),whichindicatesthatwhenσ=1

where the numerical coefficients within the brackets are approximate. Byexaminationof (269) to (271)weconclude that forarbitraryω,σ, andTw, thevalue of cf√Rx may be expressed as

wherea,b,careroughlyequalto

Young adjusts the values of these constants to get the best overall agreementwith all of the available results, and thus finds that

The accuracyof this equation is considerable, since itmaybe shown to agreewiththeexactvaluestowithinabout1percent,overtherangeoftheparameterslikelytoariseinpracticewithairastheworkingfluid,asisillustratedinFig.1.

It is now easy to derive a similar formula for the heat transfer rate. From(265) and (266) we may deduce that

and, since Δ10 may be shown to be approximately this yields

Toconcludetheworkonthecompressiblelaminarboundarylayeronaflatplate with uniform wall temperature we should make mention of a paper byMack(1958)whohasprovidedaconsiderablecatalogueofresultsfor thecaseofzeroheattransferoverarangeofMachnumbersfrom0·4to5,withviscosityandthermalconductivityprescribedaccuratelyasfunctionsoftemperaturefromexperimentaldata,andthePrandtlnumberthereforenotabsolutelyconstant.

7. ThesolutionsofChapmanandRubesinAll the preceding solutions apply only when the wall is at a uniform

temperature. The problem becomes considerably more difficult when Tw isallowed to vary with x in a prescribed manner. The first serious attempt toconsider the high-speed boundary layer on a flat plate with non-uniformwalltemperaturewasmadebyChapmanandRubesin(1949),thoughsomesolutionsforlow-speedflowweregivenearlierbyFageandFalkner(1930).

ChapmanandRubesinbeginbywritingtheboundary-layerequationsinthe

von Mises form, (59) and (60). Upon approximating

which is equivalent to taking μ∝ T with the constant of proportionality leftarbitrary, and then setting the pressure gradient equal to zero, these equationsbecome

and

Theythenintroducenon-dimensionalquantities

and upon substituting into (274) and (275) this leads to

and

Wenotethatthemomentumequation(276)isnowuncoupledfromthethermal-energy equation, (277), so the solution of the former is simply the Blasiussolution,

where

Thissuggestsafurtherchangeofindependentvariablefrom(x*,ψ*)to(x*,η),whence (277) becomes

Nowthegeneralsolutionofthisequationisthesumofaparticularintegralandthecomplementaryfunction.Foraparticularintegralwemaychooseafunctionof η alone,

where

whichisverysimilartoPohlhausen’sequation.Asolutionis

whereθ(η)isgivenby(159)and

Thecomplementaryfunctionisasolutionof

andChapmanandRubesinwritethissolutionas

where

andYnisthesolutionoftheequation

withboundaryconditions

Itfollowsthat

Hence,whenthewalltemperatureisgivenbythesumof(279)and(281),thatis

thetemperatureprofileisgivenby

Inparticular,theheat-transferrateisgivenas

It should be remarked that the abovework holds for non-integral values ofn.Takingσ=0·72,ChapmanandRubesin integrated equations (280) for severalvalues of n, giving curves of the functions Yn(η). In particular they gave thevaluesof whichareshowninTable16.Thesevaluesmaybeinterpolatedfor intermediate values of n, to yield a solution for any wall temperaturedistributions which can be adequately approximated as a sum of arbitrarypowersofx*.

TABLE16†

Exactandapproximatevaluesof

n (exact) (approx.)

00·296 0·304

10·489 0·490

20·597 0·594

2

30·678 0·671

40·744 0·734

50·799 0·787

101·006 0·987

†ReprintedfromtheJournaloftheAeronauticalSciences,vol.16,p.547,bypermissionoftheInstituteoftheAero/SpaceSciences.

Amongst the conclusions which Chapman and Rubesin draw from theirresults is that when the wall temperature is not uniform, heat-transfer ratescannotbeestimatedfromuniform-temperatureresults.Inthiscontext,referenceshould be made to a range of solutions obtained by Baxter and Flügge-Lotz(1957,1958)includingflowswithzeropressuregradientandnon-uniformwalltemperature. These were obtained by means of a step-by-step solution on adigitalcomputer.Amongstthecasesconsideredareanumberinwhichthewalltemperature is uniform, or 3Tz upstream of a position x = l, withdTw/dxconstant(butnotzero)downstreamofthisposition.Theseauthorshavealso considered cases in which the wall temperature is uniform except in theregion ,whereitincreasescontinuouslybyamaximumof2percent,andthendecreasessymmetricallytoitsupstreamvalue.

8. Lighthill’sanalysisAmethod for calculating heat transfer in low-speed flow, due to Lighthill

(1950),wasdescribedinChapter6,section5.NowLighthillhasindicatedthatin equation (169), or equation (172), μ and ρ only appear in the product μρ.

Accordingly,ifthisproductisconstant,thelow-speedsolutionholdsatallMachnumbersprovidedtheappropriatevalueoftheconstantistaken.Thuswehave,rewriting(173)

Forthecaseofaflatplatewemaywrite

sothat(285)becomes

Thisis,strictlyspeaking,relevanttothecasewhenT=0inthemainstreamandT=T∊atthewall.Lighthillshowsthatathighspeedsitisalsothevalueoftheheat transfer for the case T = T1 in the mainstream and T = Tw at the wallprovided T∊ is defined as

which, of course, reduces to the form (174) when theMach number tends tozero. Accordingly, upon interpreting the Steiltjes integral in (286) we have

LighthillhascomparedhisresultswiththosegivenbyChapmanandRubesinwhen the wall temperature is (282). Upon setting

(287)yields

whichmayreadilybeshowntobeequalto(284)withtheexpressioninsquare

brackets replacing .Acomparisonof results isgiven inTable16forthevaluesofnconsideredbyChapmanandRubesin.Wenote that theerror isnowhere as great as 3 per cent., indicating the considerable accuracy ofLighthill’smethodforflowwithzeropressuregradient.

Having thus established the accuracy of his method, Lighthill uses it tocalculate the distribution of temperature which might arise in the high-speedflight of a projectile, when heat transfer to the body is balanced mainly byradiation from it.Now the rate atwhich heat is radiated at the surface of thebody is given by

whereα is theproductof theStefan–Boltzmannconstantand theemissivityofthe wall. Upon equating (287) and (289) we have

Wenoteat thispoint that theonlypossiblevalue forTw(0) isTz. If ithadanyotherfinitevaluethenitwouldbenecessaryfortheintegraltobeoforder asx→0,implyingthat ,whichcontradictstheassumptionthatTw(0)isfinite.Physicallytheboundarylayerisextremelythinnearx=0,sothatheattransferisextremelyeffectiveinforcingthewalltotakeupitsequilibriumtemperatureTzveryquickly.IfweneglectthetermTw(0)−Tz,theremainderofthe equation may be simplified by writing

and

whenceitbecomes

with

Lighthillshowedthatforsmallvaluesofzaseriessolution

could be calculated, and that for large z there is an asymptotic expansion

He also showed how the convergence of the series (292) could be extendedreasonablyaccurately,andsowasabletolinktheexpansionssuccessfully.

The principal conclusion to be drawn from Lighthill’s analysis is that thetemperature decreases extremely rapidlywith increasing x near to the leadingedge,withamoremoderaterateofdecreasefurtherback.Theactualtemperaturegradient predicted at x = 0 is infinite, because of the assumption of zeroboundary thickness. In practice this is countered by conduction of heatwithinthe solid and (though to a considerably lesser extent) by the fact that theboundary-layer approximation is inadequate there and the boundary-layerthicknessshouldberegardedassmallbutnon-zero.Evensooneexpectsthewalltemperature to fall fromamaximumat thenose, so thatmeltingmayoccurathighenoughspeedswithsolidificationfurtherback.Thereissomeevidencethatthishappensinthecaseofsomestonymeteorites.

Beforeconcludingthischapter,mentionshouldperhapsbemadeofamuchlessexactbutconsiderablysimplersolutionoftheaboveproblembytheauthor(Curle, 1958 b), based upon a method of calculating compressible laminarboundary layerswhichwill be considered later (Chapter 9, section5).By thismethod the heat-transfer rate through a laminar boundary layer is given as

the Prandtl number being taken as unity. This replaces the more complexexpression (290), involving an integral, andwhen equated to (289) leads aftertheappropriatealgebratothefollowingsimplealgebraicequationforF,namely

The solution of this equation is, of course, simple, and it follows fairlystraightforwardlythattheerrorinthepredictedwalltemperatureincreasesfromzeroattheleadingedgetoamaximumof18percent,fardownstream,wheretheconstant0·8409in(293)isapproximatedas0·9950.

8

THECOMPRESSIBLELAMINARBOUNDARYLAYERWITHZEROHEATTRANSFER

IThasalreadybeenseenthattheproblemofthecompressiblelaminarboundarylayer with zero pressure gradient becomes extremely difficult unless suitableapproximations aremade concerning the value of the Prandtl number and thevariationofviscositywith temperature. It is tobeexpected, therefore, that thesame will be true when there is zero heat transfer but an arbitrary pressuregradient.AccordinglyweconsiderfirstanapproachtothisprobleminwhichthePrandtlnumberistakentobeunity,andtheviscosityisassumedproportionaltotheabsolutetemperature.

1. Howarth’smethodTheequationsofmotionandcontinuityaregivenby(1)and(12),andwith

body forces neglected these become

and

Equation (297)maybe satisfied by introducing a stream-functionψ such that

wheresuffixsdenotessomestandarddatumposition.Howarth(1948a)assumesthat the viscosity and temperature are related by

andthenattemptspartiallytoreduce(296)toanincompressibleformbymakingthe transformations

and

Heshowsthat

andthatthemomentumequation(296)takestheform

withboundaryconditions

Thisequationdiffersfromtheincompressibleformonlyinthefactor

whichmultiplies thepressuregradient term,and in thealtered scalenormal tothewall.

ToproceedfartherrequiresknowledgeofthetemperaturefieldT(x,y),givenbythethermalenergyequation.WhenthePrandtlnumberistakentobeunityitfollows from (19) that

andHowarthwritesthisequationintheform

Itfollowsthat(301)becomes

sothat(300)becomesanequationforχ(x,Y)alone.Howarth then develops a momentum integral equation in the transformed

variables, by integrating (300) from Y = 0 to Y = ∞. This yields

where

and

are boundary-layer thicknesses measured in the transformed coordinate, Y,normaltothewall.Hethenseekstodevelopageneralmethodofsolutionalongthe lines followed by Pohlhausen for incompressible flow. Thus u/u1 isexpressed as a quartic polynomial in Y/δ′, where δ′ is the boundary-layerthickness in terms of theY-coordinate, and the coefficients of the quartic arechosen to satisfy the corresponding boundary conditions to those satisfied inincompressibleflow(Chapter4,section1).Howarththenfindsthattherelevantparameter is

whichmay be comparedwith the incompressible form (97).Corresponding to(98), the momentum integral equation (304) yields an ordinary differentialequation for Λ, which takes the form

where g(Λ),h(Λ) are the same universal functions as in incompressible flow.The inaccuracies inherent in the incompressible form of Pohlhausen’smethodare,ofcourse,perpetuatedinthecompressibleform.

Howarthusedhismethodtoconsiderflowwitha linearlyretardedexternalvelocity

forvariousvaluesoftheMachnumberM0=u0/a0.HefindsthatthepositionofseparationmovestowardstheleadingedgeastheMachnumberis increased,aresultlaterobtainedbythemoreacceptablemethodofStewartson(1949).

2. Young’smethodAnalternativemethodofgeneralizingPohlhausen‘smethodwassuggested

byYoung(1949).Thismethodwillnotbediscussedhere,as ithassincebeenimprovedandfurthergeneralizedtoflowwithheattransferatthewall(LuxtonandYoung, 1960). It is sufficient at this stage to point out that themethod isvalid for arbitrary σ andω, whereω is the temperature-viscosity relationshipindex,anditisdiscussedinChapter9,section6.

3. TheStewartson-IllingworthtransformationThe crux of Howarth’s method, which partially reduces the compressible

boundary-layerequationstoanincompressibleform,isthattheexternalvelocityremains unchanged. By relaxing this condition Stewartson (1949) andIllingworth (1949) have independently shown that it is possible to effect acomplete transformation between a compressible boundary layer and anincompressibleboundary layerwithadifferentmainstream.TheanalysisgivenherefollowsStewartson’sapproach.

Stewartsonassumesthatσ=1,sothatforthecaseofzeroheattransferthetemperature field is given by (302). He then transforms the y-coordinate bywriting

wheresuffix0referstoanystandardreferenceposition,sayattheleadingedgein the mainstream. He also introduces a stream function ψ, such that

which satisfies (297) automatically. Then upon substituting into (296),transformingfromcoordinates(x,y)to(X,Y),makinguseoftheassumptionthatω = 1, i.e. μ ∝ T, and the temperature profile (307), Stewartson finds that

where

and

SincetherelevantboundaryconditionswereshownbyStewartsontobe

it follows that an exact correlation has been established between thecompressibleboundarylayerinthe(x,y)planewithexternalvelocityu1(x),andan associated incompressible boundary layer in the (X,Y) planewith externalvelocity U1(X). It should be remarked that in deriving this same correlationIllingworth (1949) worked with the von Mises form of the boundary-layerequations.

Since many methods are available for calculating the development of theincompressible laminar boundary layer, the use of this transformation enablescompressiblelaminarboundarylayerstobecalculated,subjecttotherestrictionsmadeinthetransformation,namelyzeroheattransfer,σ=1,μ∝T.StewartsonhimselfusedthetransformationsimultaneouslywithHowarth’smethod(Chapter4, section3) to calculate the effect of compressibility on the separationof thelaminar boundary layer with linearly retarded mainstream. His results are inqualitativeagreementwiththelessaccurateresultsobtainedbyHowarth,usingthe method of Chapter 8, section 1, and with the results given by Young’smethodinChapter8,section2.Later,RottandCrabtree(1952)suggested thatfor general calculations of compressible laminar boundary layers thetransformationshouldbeusedwithThwaites’smethod,andthisisdonebymanypeople.Inviewofthesimplicityandoverallaccuracyofthelattermethod,thiscombinationprobablyleadstogoodresultsingeneral,thoughitispossiblethaterrorsintheincompressiblesolutionmaybeexaggeratedathighMachnumber,forexample.

This possibility is borne out by an accurate solution obtained by theMathematics Division, National Physical Laboratory, for a particular externalvelocity distribution,

the velocityu0 at the leading edge being such that theMach number isM0 =u0/a0=4.ItisknownthatwhenM0=0,boundary-layerseparationoccursatx/c= 0·120 (Howarth, 1938), and according to the precise numerical solution of

N.P.L., separation occurs at x/c = 0·04(5) whenM0 = 4. The correspondingsolutions by Thwaites’s method (as amended by Curle and Skan) and theStewartson-Illingworth transformation arex/c = 0·123whenM0 = 0 andx/c =0·06(7) whenM0 = 4. We see that the error increases somewhat with Machnumberinthisparticularcase.

Stewartson has also shown that, by applying the transformation, aconsiderable simplification can be effected evenwhen there is heat transfer atthe wall. In such a case, he writes the temperature as

whereS→0asY→∞,buttheboundaryconditiononSatY=0dependsuponthe temperature of the wall, and is S(X, 0) = 0 only when there is zero heattransfer. It may then be shown fairly easily that

and

andtheseequationsmustbesolvedsimultaneouslyforSandψ.SomesolutionsoftheseequationsweregivenbyCohenandReshotko(1956a),andweremadethe basis of a general method for calculating compressible laminar boundarylayers with pressure gradient and heat transfer, which will be discussed inChapter9,sections1and3.

4. Rott’smethodStewartson’s transformation holds rigorously onlywhenσ = 1 andμ∝T.

Rott (1953) showed how it may be generalized approximately to the case ofarbitrary σ and general temperature-viscosity law. The three approximationsupon which Rott’s method is based are as follows. Firstly the assumption ismadethatthewalltemperatureforwhichtherewillbezeroheattransferisgivenby

wheretherecoveryfactor,r(σ),isgiventoagoodaccuracyas

Thisformulaisexactwhenσ=1,andhasbeenfoundtobeofgoodnumericalaccuracy for other values of σ close to unity. Secondly, the expression

isacceptedasanadequateapproximatesolutionforthetemperatureprofile.Thissatisfies the boundary conditions at the wall, (315), and at the edge of theboundarylayer(whereT→T1),andisexactwhenσ=1.Thirdlyit isassumedthat the viscosity-temperature relationship is

sothatμisproportionaltoTatconstantx,butthevariationwithxischosensothat the viscosity at thewall is given accurately, by Sutherland’s law or by acarefully chosen power law.With these three approximations, the Stewartsontransformation can be generalized, and the equations (308), (311), and (312)become respectively

and

This transformationmay then be used, in conjunctionwith sayThwaites’smethod, to provide a convenient approximate method for calculating thedevelopment of compressible laminar boundary layerswith zero heat transfer,arbitraryσ,andgeneralviscosity-temperaturelaw.

5. ThemethodofOswatitschandWeighardtOneoftheearliestmethodsdevelopedforcalculatingcompressiblelaminar

boundarylayerswasthatduetoOswatitschandWeighardt(1943),theirmethodbeing a plausible generalization of the method of Walz (1941), using the‘similar’ solutionsof theboundary-layer equations.Theywrite themomentumintegral equation (40) in the alternative form

and further assume that thevelocityprofiles appropriate to specifiedvaluesofthe function

arethosegivenbythelow-speedsolutionsofFalknerandSkan(1930),andusedbyWalz. It is then fairly straightforward to obtain δ1/δ2 and δ2/u1(∂u/∂y)w asfunctionsofλ*andM1,thedependenceuponM1arisingbecauseofthevariationof density with Mach number. Then (316) becomes

andthisiseasilyintegratedbyastep-by-stepprocedure.AsnonumericalexampleswerecalculatedbyOswatitschandWeighardtitis

difficulttoestimatewhataccuracyistobeexpectedfromthemethod.However,

theassumptionsmadearesomewhatarbitrary,soitisunlikelythatthemethodisofparticularlywideapplication.

6. TheworkofCopeandHartreeAconsiderableattackontheproblemofthecompressiblelaminarboundary

layer with pressure gradient and zero heat transfer was made by Cope andHartree (1948), involving the use of a high-speed computing machine. Theseauthors make first the following transformation of variables

and

withψdefinedasin(168).Theythenderivetheequationswhichu*,r,Wmustsatisfy, and seek solutions of the form

withtwofurtherexpansions,

and

introducedforanalyticconvenience.It is found that the zero-order terms, h0, r0, f0, χ0, ɸ0 satisfy nonlinear

equations, and these have been integrated by Cope and Hartree for variousvaluesofM0,assumingσ=0·715andμ∝T8/9.Notsurprisinglythesolutionsarebarely dependent upon M0, since they would be identical under the similarassumption thatμ∝T. Some of the first-order functionswere also calculatedapproximately,but theseexhibitaconsiderabledependenceuponM0.Thenth-orderequations arealllinear,anddependuponthefunctionsoflowerorder. In principle, therefore, theymay be calculated successively in order toobtainacompletesolution.

Cope andHartree have also given some consideration to the possibility ofdevising a solution along the lines ofPohlhausen’smethod for incompressibleflow.TheyassumethatthePrandtlnumberisclosetounity,sothatanadequatesolutionofthethermalenergyequationisgivenby(302)whenthereiszeroheattransfer.Themomentumintegralequation(40)isfirstwritteninthealternativeform

andthevelocityprofileapproximatedas

where δ is a measure of the boundary-layer thickness, and the function f ischosentosatisfyanumberoftheboundaryconditionsonthevelocityuanditsderivatives. In the absence of a pressure gradient Cope and Hartree take

andobtainresultsfortheskin-frictionwhichagreewellwiththecalculationsofBrainerdandEmmons.

When there is a pressure gradient the algebra becomes considerablymorecomplicated, andCopeandHartreeconcluded that thePohlhausenmethoddidnot offer much promise of progress, a conclusion which was perhaps a littlepessimistic.

7. TheworkofIllingworth,Frankl,andGruschwitzWemightperhapsmentionat thispointsomeoftheotherinvestigationsof

the Pohlhausen-type which have been made by various authors. Illingworth(1946) has considered the case of zero heat transfer and arbitrary pressuregradient, with the assumption that viscosity μ and thermal conductivity k areconstant. The velocity was defined by

andthetemperature(ordensity)by

where χ = δ′/δ is the ratio of thermal boundary-layer thickness to velocityboundary-layer thickness.Thefunctions fandgarechosen tosatisfy theusualboundaryconditions,derived from thedifferential formsof the equations.Themomentumintegralequationandthethermalenergyintegralequationthenyieldtwo relationships for the three unknownsρw/ρ1, χ, andδ2/u1(∂2u/∂y2)w, a thirdrelationshipbeinggivenbytheboundaryconditionwhichfollowsfromsettingy=0inthethermalenergyequation(18).

Illingworthhascarriedoutdetailedcalculationsforflowwithzeropressuregradient,usinganumberofpossibleformsofthefunctionsfandg,Inparticularheconsideredcaseswhenfandgwereofthesameorderinη,namelyquadratic,cubic,andquartic,andonecaseinwhichfwasquinticandgsextic.Further,healso considered trigonometric forms, in which f, q are

Upon comparing his results with the accurate calculations of Emmons andBrainerd (1942), Illingworth concludes that the trigonometric and cubic formsappear togive themostaccuratevalues.Henotes,however, that theparticularform of the function g is critical and that the predictions of wall temperatureleavemuchtobedesiredinallthecasesconsidered.

Illingworth’sworkmayberegardedasageneralizationoftheearlierworkofFrankl(1934),whomadethefurtherapproximationthattheviscousandthermalboundarylayershaveequalthickness(χ=1),althoughthePrandtlnumberwasnotassumedtobeunity.Variouspossibleformsofthefunctionsf(η),g(η)wereconsidered,andinspiteoftheapparentadditionalassumption(ascomparedwiththelaterworkofIllingworth)Frankl’sresultsforzeropressuregradient,with f

andgbothquartics,werebetterthananyofthoseobtainedbyIllingworth.Thisis perhaps indicative of the uncertainty which besets attempts to make directextensions of Pohlhausen’s method without first making a suitabletransformationofcoordinatesand/orvariables.

A slightly different approach was adopted by Gruschwitz (1950) whointroduced a new boundary-layer thickness

andexpressedu/u1asaquarticin

Further,insteadofexpressingρ/ρ1intermsofη,hechosetowritetheproduct{l− (u/u1)}(ρ1/ρ) as a quintic in η, satisfying appropriate boundary conditions.Substitutionintothemomentumintegralequationandthekinetic-energyintegralequation leads, after some considerable calculation, to a first-order nonlinearordinary differential equation for the momentum thickness δ2 in terms of theknownu1andM1andtheunknownTw/T1,andanexplicitexpressionforTw/T1,in terms of M1 and δ2. These two equations involve universal functions,analogous to those arising in Pohlhausen’smethod,which are dependent onlyupon the Prandtl number σ and the shape parameter

It should be remarked that themethod does notmake use of the thermal-energy integral equation, and accordingly is suspect unless σ = 1, when thekinetic-energyintegralequationisidenticalwithit.

8. TheinvestigationsofGaddWe conclude this chapter with a brief discussion of some work of Gadd

(1953a),whichinvestigatesapproximatelytheeffectsontheseparationpositionofvariations inPrandtlnumberσ and the indexω in theviscosity-temperaturelaw(227).

He begins by considering an artificial case in which σ = 0. In suchcircumstances it is clear, by inspection of (19), that temperature gradients can

remain finite only if

sothat

whentheheattransferatthewalliszero,andthus

With this solution of the thermal-energy equation, Gadd shows how themomentumequationmaybetransformedintoanincompressibleformbymeansof a transformation analogous to that of Stewartson and Illingworth. Thisequation may then be solved approximately by any convenient method, andGaddgivessomesolutions(atvariousMachnumbers)byThwaites’smethodforthecasewhenω=1andtheexternalvelocityisoftheform(307).Comparisonswith solutions for the caseω = σ = 1 suggest that the distance to separationincreaseswithdecreasingσ,andthesolutionsforσ=0provideanupperboundtothisvariation.

Gaddthenconsidersapproximatesolutionsforcasesinwhicheitherωorσisequal to unity and the other does not differmuch therefrom. In each case, hewrites the equations in theCrocco form,with independent variables x,u, andthenfurthertransformstovariablesxandf=u/u1.

Whenσ=1,ω≠1,thethermalenergyequationhasthesimplesolution(228)for temperature in terms of velocity. The solution of themomentum equationwasthenobtainedbyanapproximatemethod.Whenω=1,σ≠1,theprocessisslightlymoredifficult,asit isnecessarytoobtainfirstanapproximatesolutionfor the temperature distribution, and Gadd did this by means of a smallperturbationaboutthesolution(228)validforsmallvaluesof|1−σ|.

TheconclusionstobedrawnfromGadd’scalculationsmaybeillustratedasfollows.Whenω = 1, the proportionate increase in the distance to separationwhenσ = 0·72 (as comparedwithσ = 1) is about 8 per cent,when theMachnumber at the leading edge isM0 = 4, and about 25 per cent,whenM0 = 10,theseresultsbeingforthecaseofalinearlyretardedexternalvelocity.Further,whenσ=1,theproportionateincreaseinthedistancetoseparationwhen(ascomparedwithω=1)isabout2percent,whenM0=4andabout6percent,when M0 = 10. Gadd has also derived results by direct application of theapproximatemethodofYoung.AtthelowerMachnumber,M0=4,theresultsarealmost identicalwith thosequotedabove,butatM0=10Young’smethod

predicts proportionate changes in separation distance which are in each caseapproximatelyhalfthosederivedbyGadd’smethod.

Wededuce from these results that,when there is zero heat transfer at anyrate,theerrorsinducedbyassumingω=σ=1willbenomorethanabout10percent,whenM0=4and30percent,whenM0=10,15percent,ifYoung’svaluesaretobepreferred,inpracticalapplicationswithairastheconstituentgas.Theseconclusionsarebroadlysimilartothosewhichwouldbereachedif themethodofRottwereusedinthecalculations.

9

THECOMPRESSIBLELAMINARBOUNDARYLAYERWITHPRESSUREGRADIENTANDHEATTRANSFER

AS far as the formal boundary-layer problem is concerned, the casewith bothpressure gradient and heat transfer is by far the most difficult to deal withtheoretically.Themomentumandenergyequationsaresimultaneousnon-linearpartial differential equations for which no simple approach is known. Theproblembecomesevenmoredifficultifthewalltemperatureisnotuniform(orverynearlyso),asituationwhichcouldariseinpracticeatveryhighspeeds.Themathematicalandnumericaldifficultiesencountered insolving theseequationsare such that, even with the assistance of modern high-speed computingmachines, very few precise numerical solutions for special cases have beenattempted,andsomeoftheseareofdoubtfulaccuracy.

Weshallbeginthischapterbydiscussingbrieflythesenumericalsolutions,continuing with some of the approximate methods developed for this verygeneral problem. In incompressible flow,where the pressure gradient alone issufficient to prescribe the problem, there are now about eight really accuratesolutionsforparticularcases,halfofwhichareforflowsinwhichthepressuregradientiseverywhereadverse.Theseeightsolutionsareonlyjustsufficienttoenable us tomake a reliable assessment of the accuracy and value of variousapproximatemethods. In compressible flow things are further complicated bythe presence of additional parameters such as Mach number and walltemperature. Clearly at least six or eight accurate solutions are required forseveral values of these parameters beforewe can assess the usefulness of theapproximatemethodswithanycertainty,andthispointwillaccordinglynotbeconsideredasthoroughlyasintheincompressiblecase.

1. AccuratenumericalsolutionsforspecialcasesA rangeof accurate solutions are, of course, available for the caseof zero

pressure gradient, covering various values of the Prandtl number σ, severalpossible viscosity-temperature laws, and a fewvalues of thewall-temperature.

These were discussed and summarized in Chapter 7, and will not be furtherconsideredhere.

Forflowswithpressuregradient,theearliestaccuratesolutions(asfarastheauthorisaware)wereobtainedbytheMathematicsDivision,NationalPhysicalLaboratory,andarefirstreferredtoinapaperbyGadd(1952).Threeparticularcaseswereconsidered,theassumptionsthatσ=1andμ∝Tbeingmadeineachcase. All were such that the wall temperature was uniform and the pressuregradienteverywhereadverse,sothatflowneartotheleadingedgewaslikethatonaflatplatewithzeropressuregradient.Furtherdetailsofthethreecasesareasfollows:(i)Inthefirstcaseauniformadversevelocitygradientwasassumed,sothattheexternalvelocitywasgivenby(307),theMachnumberattheleadingedgewasM0=4andzeroheattransferwasconsidered,(ii)Thedataforthiscasedifferedfromcase(i)onlyinthatthewallwastakenatatemperatureT0,whichis theexternalstreamtemperatureat the leadingedge.SinceTz/T0 isequal,by(244) to , upon putting γ = 1·4 we see that the ratio of walltemperatureTw(=T0)toitszero-heat-transfervalueTzis .Inotherwordsthewall has been considerably cooled, (iii) In the third case, a uniform adversepressure gradient was assumed, so that

TheMachnumberattheleadingedgewasM0=2,andthewalltemperaturewasT0 (defined as above), equal to five-ninths of its zero-heat-transfer value; thecoolingatthislowerMachnumberislesssevere.

Foreachofthesecasesdetailedcalculationsweremadeofboththeviscousand thermal boundary layers in the region between the leading edge and thepositionofzeroskin-friction,andsomeoftheseresultshavebeenreportedinapaper by Curle (1959). For further details, reference may be made to theMathematicsDivision,NationalPhysicalLaboratory.

Some time later a series of ‘similar’ solutionswere derived byCohen andReshotko(1956a),analogoustothosefoundforincompressibleflowbyFalknerand Skan and by Squire. Following Li and Nagamatsu (1953), Cohen andReshotko look for solutions of (314) in which the external velocity in thetransformed plane is equal to

andintroducenewvariablesf,η,where

and

Thenequations(314)reducetotheform

Hereprimesdenotederivativeswith respect toη, and theboundaryconditionsare

Equations(317)and(318)wereintegratednumericallyforaseriesofvaluesofthe parameters m and Sw, corresponding respectively to different pressuregradientsandwalltemperatures,andtheresultsaretabulatedandplottedintheirpaper. Although the same reservations must be observed in applying theseresultstonon-similarflowsaswererelevanttotheFalkner–Skansolutions,thereis no doubt that the solutions ofCohen andReshotko are extremely valuable,nonetheless.

SubsequenttotheabovesolutionsBaxterandFlügge-Lotz(1957,1958)havereported a series of some sixty examples which were calculated on a digitalcomputer bymeans of a step-by-step solution, starting fromCrocco’s formofthe equations, and have given detailed results for the skin-friction and heattransfer.Asan illustrationof the typeofexampleconsideredby theseauthors,wemention four cases inwhich the external velocity is constant, equal tou0,when , and varies when according to the law

Thefourcasesdifferonlyinthat intwoofthem,withTw=3Tzintheother two,and in twocases,withM0=3 in theother two. Inall theexamplesconsideredthePrandtlnumberwasσ=0·72,andSutherland’slawwas

used for the viscosity-temperature relationship.For details of themethodusedandforthenumericalresults,referencemaybemadetothetwopapersbytheseauthors.

2. Kalikhman’smethodTurning now to approximate methods for calculating the compressible

laminarboundarylayerwithheattransferandpressuregradient,webeginwithageneralizationofthePohlhausenmethod,duetoKalikhman(1946).Themethodis valid for arbitrary constant Prandtl number, and assumes that the viscosity-temperature relationship is

Kalikhmanintroducesanewindependentvariableηinplaceofy,where

and suffix s denotes a standard reference condition, which he takes to be thefree-streamstagnationvalue.Healsointroducesnewdependentvariables ,E*,defined by

whereTH is the total temperature, asdefined in equation (48).The continuity,momentum,andthermalenergyequations(1),(12),and(19)thentaketheform

and

Itwillbeobservedthatthevarioustransformationshavegonesomewaytowardsreducing theseequations toan incompressible form, inmuch the samewayastherelatedtransformationofHowarth(1948a).Kalikhmanthenintegratestheseequations with respect to η, and obtains the results

and

where

and , , and are analogous to the displacement thickness, momentumthickness, and enthalpy thickness of the boundary layer as previously defined.Following the idea of Pohlhausen, he then makes the approximations

where the ten coefficients An, Bn are chosen to satisfy the usual boundaryconditions, namely

and

togetherwiththetworelationshipsobtainedbysettingη=0inthemomentumand energy equations, (320) and (321). Then (322) and (323) yield twosimultaneous ordinary differential equations relating the boundary-layerthicknesses, δ and Δ, which appear in the parameters

and

where

NonumericalresultsweregivenbyKalikhman,sotheaccuracyofthemethodisuncertain. Itmay be remarked, however, that the partial transformation of theequationstoincompressibleformisapointinfavourofthemethod,althoughthearbitrary choice of the order of the polynomials in (324) and (325) isunfortunatelysomewhatrestrictive;anexaminationofwhichorderwouldyieldbestresults(bycomparisonwithaccuratesolutions)wouldbeextremelylengthy.

3. ThemethodofCohenandReshotkoMentionhasalreadybeenmadeofsome‘similar’solutionsforcompressible

laminar boundary layers with pressure gradient and heat transfer. Thesesolutions were calculated by Cohen and Reshotko (1956 a) and weresubsequentlyused (1956b) as thebasisofanapproximatemethod forgeneraluse.

TheseauthorsbeginbyapplyingtheStewartson–Illingworthtransformation,in a formwhich neglects the factor appearing in the stream functionψ asdefined by Stewartson. Thus they write

andintroducethenewindependentvariables,

which differ from (308) and (311) only as regards the factor . Then with

thecontinuity,momentum,andthermal-energyequationsmaybewrittenas

and

whereSisdefinedbyequation(313).Theboundaryconditionsare

Upon defining boundary-layer thicknesses, measured in the Y-coordinate, asfollows,

and

equations(326)to(328)maybeintegratedwithrespecttoYfromY=0toY=∞to yield

and

CohenandReshotkonowintroduceanumberofnon-dimensionalquantities,analogous to those introduced by Thwaites (1949) for incompressible flow,namely

and

Upon substitution into (330) it is found that the following result may beobtained, namely

whichisanalogoustothelow-speedform(103).CohenandReshotkomaketheassumptionatthisstagethatL,H*,andlare

allfunctionsofnandSwonly, thesefunctionsbeingderivedfromthe‘similar’solutionswhichtheyhadcalculated.ThisassumptionisanalogoustothatmadebyWalz(1941).Oneconsequenceoftheassumptionisthatequations(330)and(331) cannot be satisfied simultaneously, and Cohen and Reshotko choose tosatisfy (330) whilst rejecting (331). One expects intuitively, therefore, that incertaincircumstancesthemethodmaygivethetemperaturefieldwithonlylowaccuracy,whilstpredictingthevelocityfieldmoresuccessfully.WhenL(n,Sw)isthus prescribed, equation (336) may be integrated numerically as in theincompressiblecase.WhenthewalltemperatureisuniformitisoftenpossibletoapproximateL asa linear functionofn, andananalytic solutionof (336) thenbecomespossible.Havingcalculated , followsfrom(335),(∂U/∂Y)w from(332)and (∂S/∂Y)w from (334). It is thena simplematter toderive from theseresults the true boundary-layer thicknesses, the skin-friction, and the heat-transfer as measured in the physical plane. The relevant formulae are

and

The functions l,H*, L, and r are plotted in Cohen and Reshotko’s paper asfunctionsofnforvariousconstantvaluesofSw.

Comparisonshavebeenmadewithsomeoftheaccuratesolutionsdiscussedin section 1, and the agreement is quite good. For example, for the case of auniformadverseexternalvelocitygradientataMachnumberof4,withthewallcooled to a temperature , separation is predictedwhen the externalvelocity has decreased by about per cent., as comparedwith the accuratevalueof percent.Thiscomparisonindicatesfairlywelltheoverallaccuracyofthemethodforthisparticularexample.

4. Monaghan’smethodImportant simplifications and improvements to the method of Cohen and

Reshotko have been suggested byMonaghan (1960). In the course of amoregeneralstudyofthecompressiblelaminarboundarylayer,Monaghanfoundthatbycareful choiceofparameters someofCohenandReshotko’s curves for thefunction l, at different values ofSw,may be collapsed onto a single curve. Inparticularitappearsthatinregionsoffavourablepressuregradientthefunctionl(m, Sw), where

isalmostindependentofSw,whilstinregionsofadversepressurethefunctionsl{(m/msep),Sw}fallonasinglecurve,msepbeingthevalueofmatwhichl=0.Asimilarcollapseontoasinglecurveisfoundforafunctionwhichisrelatedtother of Cohen and Reshotko’s method. Monaghan further generalizes thetransformation of coordinates used by Cohen and Reshotko, and writes

Hisfinalequationstaketheform

and

where

andthefunctionsl(m),h(m)aregiventoagoodapproximationas

and,whenm>0,

andlisgivenasafunctionofm/msepbymeansofagraphinMonaghan’spaper.Itremainstodefineg1,g2,andmsep.Toafirstapproximationg1andg2are

given by

Thevalueofmsepisdeterminedbymeansofagraphrelatingmsep/m0toTw/Tz,where m0 is theoretically equal to 0·0681. However, Monaghan suggests anempiricalimprovementinwhichm0istakentoequalthevalueofmatwhichthesame external velocity distributionwould lead to separation in incompressibleflow.Ifwemayjudgebyexperienceinincompressiblelaminarboundary-layertheory,itshouldbetakentoequalabout0·090.

Comparisons with the exact solution for the case considered earlier (,uniformadversevelocitygradient)indicatethat,withall

the empirical improvements noted above, Monaghan’s method is remarkablyaccurate.

5. Curle’smethodThe two methods above, namely those of Cohen and Reshotko and of

Monaghan, may be regarded as possible generalizations of the method ofThwaites(1949),ormorepreciselythatofWalz(1941).AnalternativemethodofdoingthiswaspresentedbyCurle(1959)inapapergivingageneralizationofThwaites’s method starting from the Howarth transformation, as defined byequations(298)and(299).Uponformallyapplyingthis transformationwefind(following Howarth’s analysis) that

andthismaybeformallyintegratedwithrespect toY, fromY=0toY=∞, toyield

where and aredefinedbyequation(305).Furtherprogressdependsuponaknowledgeofthetemperatureprofile.

Now it iswell known thatwhenσ = 1 the total temperature is constant ifthere is zeroheat transfer, and is a linear functionof thevelocityu if there iszeropressuregradient.Forthepurposeofobtainingastraightforwardsolutionof(337) for thevelocityboundary layer (butnot forcalculating theheat transfer,for example) Curle suggests writing

wherethelasttermshouldbezerowheneitherthepressuregradientortheheattransfer is zero. The value

where

and

issuggestedforuseinregionsofretardedflow.ThetemperatureT0,velocityu0,andMachnumberM0arevaluesattheleadingedgeifthevelocitythereisnon-zero,otherwisetheyarevaluesatthepressureminimum.Theanalysiscannotbedirectlyappliedtoregionsofacceleratedflow,whereCurleindicatesthatCohenandReshotko’smethodshouldgivegoodaccuracy.

Withtheapproximation(338)itiseasytoshowthat

whence(337)becomes

Curlenowintroducesnon-dimensionalparameters

whereupon(339)becomes

where

ByconsideringtheconsequencesofdeterminingrelationshipsbetweenL′,H′,l′,andλ′bymeansofvelocityprofilesprescribedintheY-plane,CurleconcludesthatthefunctionsL′(λ′),H′(λ′),l′(λ′)arelargelyindependentofwalltemperatureandMachnumber,andaregivenapproximatelybytheirincompressiblevalues;the values appropriate to Thwaites’s method are accordingly recommended.Since we can now use the approximation to L′(λ′) analogous to the Thwaitesvalue (106) we have

and (340) is integrable in the formof a simplequadrature.Upon transformingfromtheY-planeintothephysicalplane,thefinalequationsofthemethodare

where

Themethodisperfectlystraightforwardtoapply,andcomparisonswiththethreeaccurate solutions calculated by the Mathematics Division, National PhysicalLaboratory,indicatethattheaccuracybearsreasonablerelationshiptotheworkinvolved.

Themethodasdiscussedaboveisvalidforσ=1,μ∝T,anduniformTw.IntwolaterpapersCurle(1958b,1961b)hasremovedtheserestrictions.Forfulldetails reference may be made to these papers. Briefly, the removal of theserestrictions leaves the general procedure unaltered, but amends some of the

details,sothat(forexample)thecoefficientsofTw/T1and in(342)and(345)areprescribedfunctionsofσ,andanadditionalfactortoaccountapproximatelyforthegeneralizedviscosity-temperaturerelationshipappearsin(344)andinthedefinitionofthefunctionG1.

6. ThemethodofLuxtonandYoungWe now consider a method which is essentially a generalization of the

Pohlhausen procedure, due to Luxton and Young (1960). It was originallydeveloped by Young (1949) for the case of zero heat transfer, but was latergeneralized and improved by these two authors for use when there is heattransfer,andisapplicableforarbitraryvaluesofthePrandtlnumberσandwiththeviscosity-temperaturerelationshipgivenby(227)forarbitraryω.

Themethodassumesthatthevelocityprofileisoftheform

where

and a, b, c, d are chosen to satisfy the usual boundary conditions. Thus

andΛisgivenby

Itfollowsthat

Theformparameter

isnotchosenfromtheassumedvelocityprofile,butistakentobe

whichisapproximatelycorrectinthecaseofzeropressuregradient(whenσ isclose to unity), and agrees fairly well with values derived from Cohen andReshotko’ssolutionsprovidedk2isasuitablychosenfunctionofTw/Tz.Furthertheratiooftotalboundary-layerthicknesstomomentumthicknessisnotderivedfrom the velocity profile, but is given by

which again is correct in the case of zero pressure gradient, and givesapproximatelythecorrecteffectofpressuregradientwhencomparedwithCohenandReshotko’svaluesprovidedk1iscorrectlychosenasafunctionofTw/Tz.

Uponsubstitutingfrom(346)to(349)intothemomentumintegralequation,written first in the form

itisfoundthat

which may be formally integrated to give δ2 as a function of x. Luxton andYoung suggest a numerical method of integration, but upon introducing apreliminary quadrature to give

itmaybeintegratedas

Havingobtainedδ2,δ1followsfrom(348)andthenτwfrom(347),makinguseof(349).

ItwillberecognizedthatthismethodcontainselementsofbothPohlhausen‘smethod and of the similar solutions.At first sight the relative contributionsmight appear to be rather arbitrary. On the other hand, Pohlhausen’smethod,applied formally in incompressible flow, usually yields values of the skin-frictionwhicharetoohigh,withtoolateaseparation,thevalueofthepressuregradientparameteratseparationbeingλ=−0·157.Methodsbasedonthesimilarsolutions,suchasthatduetoWalz,usuallyerrintheoppositedirection,withλ=−0·068 at separation. It is interesting to note that the method of Luxton andYoungyieldsλ=−0·090atseparationinincompressibleflow,whichwasshownbyCurleandSkan(1957)tobetheoptimumvalue.Comparisonswithsomeofthe accurate solutions of section 1 suggest that at a Mach number of 4 themethodstillyieldsresultsofgoodaccuracy.

7. ThemethodofPootsAmethod has recently been given by Poots (1960) which is theoretically

more acceptable than any of the methods given in this chapter, but becomesmuchmore complicated and lengthy in practice. Themethod begins with theStewartson transformation, so that upon further introducing transformedvelocities

equations(314)become

and

whereX, Y are defined by (308) and (311), and the boundary conditions aregivenby(329).Byformallyintegrating(351)and(352)withrespecttoYfromY=0toY=∞,andbymultiplying(351)byUandthenintegrating,threeequationsare obtained, analogous to themomentum, kinetic-energy, and thermal-energyintegral equations. These may be written as

and

where

and

thedefinitionof beingdifferentfromthatusedbyCohenandReshotko.PootsnowexpressesU/U1andSasquarticsinY/δ*,whereδ*isaboundary-

layerthicknessinthetransformedplane.Thesequarticsarechosentosatisfytheconditions

and

but, followingthemethoddevisedbyTani(1954)for incompressibleflow, theconditionsobtainedbysettingY=0in(351)and(352)arenotsatisfied,afreecoefficientbeing left in eachof thequartics.Thus, for the casewhenSw=1,

and

Equations (353) to (355) must be solved simultaneously for δ*, a, b. Theproblem has thus been reduced to that of solving three simultaneous ordinarydifferentialequations.

Pootshasexpressed thevarious factors in (353) to (355) in termsofδ*,a,andb(forthecasewhenSw=1),andhasgoneontoshowhowthesimultaneousintegration might be done, taking as an example the case when the externalvelocity in the transformed plane is

thewalltemperature,ofcourse,beingTw=2Tz.Hefindsthatseparationoftheboundary layer occurs at

andthisvalueagreeswiththatobtainedbyexpandingapowerseriessolutionofthepartialdifferentialequations,validatsmallvaluesofX,andcontinuingthesolutionbyanumericalstep-by-stepmethod.

This method provides simultaneously solutions for skin-friction, heattransfer, and indeed any desired boundary-layer property. It is, therefore, ofwider application than the methods previously considered, where the thermalenergy equation is sacrificed in an attempt to obtain a quicker approximatesolutionfortheviscousboundarylayer.Ifthecomputingfacilitiesareavailablethe method should yield good accuracy, even in cases of non-uniform wall-temperature.

8. ThemethodsofLilleyandIllingworthTwo rather similar methods of calculating the skin-friction and the heat-

transfer rate in a compressible laminar boundary layer have been given byIllingworth (1954) and Lilley (1959). In what follows a brief summary ofLilley’sapproachwillbegiven,togetherwithsomediscussionofthenumericalexamplesconsideredbyboththeseauthors.

ThemethodadoptedbythesetwoauthorsisdesignedtoextendtheanalysisofLighthill(1950a)tothecaseofcompressibleflowwithbothheattransferandpressure gradient. Unlike Illingworth, Lilley applies first the Stewartson-Illingworth transformation and then, following Lighthill, transforms theequationsfurthertothevonMisesform,inwhichtheindependentvariablesare

X and ψ, and the dependent variables are S and Z, where

andS,X,ψ,aredefinedrespectivelyby(313),(311),and(309).Themomentumand energy equations then take the form

and

andtheassumptionsσ=1,μ∝T(notmadeinLilley’sanalysis)aremadeherefor simplicity. Lilley rewrites (356) in the form

andnotesthatatbothψ=0andψ=∞thismaybewrittenas

where

Lilley assumes that (358) is an adequate approximation right across theboundary layer, and solves it subject to the boundary conditions

andforsmallψ

andsince,by(313),

thisfurthersimplifiesto

Lilley indicates thatequation (358)maybe integrated, subject to theboundaryconditions(360)and(362),byamethodanalogoustothatusedbyLighthill,theterm U2(X, ψ) in (362) being first replaced by

whereμ0 is theviscositycalculatedat thereferenceconditions.Hissolutionis

Lilleyfurthertransformsbackintothephysicalplane(forwhichseehispaper),and also shows how (363) may be approximately inverted to yield τw as anexplicit functionofU1(X) andTw(X); againnodetailswill begivenhere. It iseasily seen that (363) reduces identically to the corresponding expression inLighthill’spaperwhenincompressibleflowisconsidered.

Lilleydealswithequation(357)inasimilarmanner,theboundaryconditionsbeing

and

where

ThesolutionfollowsLighthill’ssolutionfortheheattransfer,andis

Lilleyindicateshowthesolutionsgivenabovemaybeimprovedbytakingσ≠ 1, by introducing a factor C(X), following Rott (1953), to accountapproximatelyfortheviscosity-temperaturelaw,andbyadaptingthetechniqueofSpalding(1958)soastoextendtherangeofconditionsoverwhichthetheoryisaccurate.

Both Lilley and Illingworth (by his basically similar method) haveconsidered in detail the numerical results for a number of special cases, andobtain fair agreement with the accurate solutions. In particular, Lilley hasconsidered the three cases considered by the Mathematics Division, NationalPhysicalLaboratory,andobtainsanaccuracywhich,forthevelocityfieldatanyrate, is comparable with that given by the other methods of this chapter. Nocomparisons were made for heat-transfer rates. Illingworth has not made anycalculationsforflowswithbothheattransferandpressuregradientathighMachnumbers. He has, however, made some comparisons with the low-speed‘similar’ solutions,and finds that theskin-friction is satisfactory inacceleratedflow but not in retarded flow, especially in the vicinity of the true separationposition. On the other hand, the heat-transfer rate as calculated using theerroneousskin-frictionisexceedinglyaccurateowingtoafortuitouscancellationof errors. It is unlikely that this will happen in general. Illingworth has alsocalculated the separation positions for a uniform adverse external velocitygradient with zero heat transfer for several values of the Mach number. Hissolution at zeroMach number differs from the exact value by less than 8 percent, and at M0 = 4 appears to agree with the solution calculated by theMathematicsDivision,NationalPhysicalLaboratory,towithinabout1percent.Itisdifficulttoestimatewhataccuracymightbeexpectedinageneralcase,in

viewoftherespectivefailureandsuccessofthemethodinpredictingseparationinthetwoexamplescited.

9. Curle’smethodforcalculatingheattransferIn view of the predominance of methods which aim at calculating skin-

friction as accurately as possible, whilst sacrificing accuracy of prediction ofheattransfer,itisvaluabletoconcludethischapterwithamethodwhichaimsatcalculatingtheheat-transferratewhentheskin-frictionisassumedknown.Themethod,duetoCurle(1962a)isanextensiontohigh-speedflowofthemethoddescribedinsection6ofChapter6.

ThemethodbeginswiththeStewartsontransformation,andforconveniencethe factor is neglected in the definition of the stream function, so that thetransformedvelocityandtemperaturefieldsaregivenby(326)to(329).Ifit isassumed that the velocity field is adequately knownby some othermethod, itremains to calculate the temperature field, which is given by the equation

Followingthetechniqueusedatlowspeeds(Curle,1961a),basedontheideasof Lighthill and Liepmann, this equation is integrated to yield

NowLighthill indicated that forPrandtlnumbersoforderunityandabove thevelocitycouldbetakeninanapproximateformvalidneartothewall,sincethethermalboundarylayerisusuallythinnerthantheviscouslayer.Thuswewrite

where the first term only was retained by Lighthill, and the third term isnegligible in the low-speedcaseprovidedtemperaturedifferencesaresmall. Inthecaseunderconsiderationweassumethat (∂U/∂Y)w isknown,(∂2U/∂Y2)w isobtained by setting Y = 0 in (327), and is

whilst (∂3U/∂Y3)w, obtained by differentiating (327) with respect to Y beforesetting Y = 0, is equal to

Thus, writing (∂S/∂Y)w as S′w, (366) becomes

andhence(365)maybewritten

FollowingLiepmann,wewrite

for the case of a wall-temperature distribution which is either uniform orpossiblyuniformwithzeroheattransferupstreamofsayX=X1,withasteptoanew uniform value downstream of this position. Then (367), after formalintegration with respect to X, and substitution from (368), takes the form

wherea,b,c arepositiveconstants,whichwillbedeterminedat a later stage.Finally, if we write

whichisaNusseltnumberinthetransformedplane,(369)becomes

Thethreetermswhosecoefficientsincludea,b,careofdecreasingordersofimportance.ThusLighthill found thatuponretainingonly the first termon the

right-handsideof(370)(thussettingbandcequaltozero)theequationcouldbeintegrated analytically, and a formula obtained forF which agreed well withaccurate solutionsexceptnear to separation,where (∂U/∂Y)w iszero.SimilarlyCurle(1961a)foundthatfor thecaseofsmall temperaturedifferences(i.e.Swverynearly zero) the retentionof the second term, involving the coefficientb,enabled the accuracy to be maintained right up to separation. The valuessuggested by Curle for the two coefficients are

Itmightbeexpectedthatthesetwotermsalonewouldbeequallysatisfactoryin more general cases, with Sw not almost zero, unless both almost vanishsimultaneously. This could happen near to separationwhenSw is close to −1,that is for very low wall temperatures. Now examination of the Cohen andReshotkosimilarsolutionsbearsout thispoint,andCurlesuggestsretentionofthe third term tocover this relativelyunlikelyoccurrence. Itwillbenoted thatthesimplicityofequation(369)isnotatallreducedbytheretentionofthisterm.Thevalueofc is chosen toobtaingoodagreementwith the accuratelyknownsimilar solutions, and the term in square brackets in (369) becomes

wherebisgivenby(371)and

Theapplicationofthemethodislimitedstrictlytothecaseσ=1(althoughafactor mightwelladequatelyconvertthevaluesofFintothoseappropriatetoalternative values of σ close to unity), and to the case of uniform walltemperature, unless the assumption be made that the skin-friction is littledependent upon wall temperature, when the heat-transfer rates appropriate toeachofanumberofelementarystepsinwalltemperaturemaybeadded.Withinthese limits the accuracy is considerable, and for the similar solutions heat-transferratesaregiventowithinabout±2percent,fortheconsiderablerangeofwalltemperaturesandpressuregradientsconsidered.

10

INTERACTIONSBETWEENSHOCKWAVESANDBOUNDARYLAYERS

IThasalreadybeenremarked,inthefirstchapterofthisbook,thatinteractionsbetweentheboundarylayerandtheexternalstreamaremuchmoreimportantinsupersonicflowthaninsubsonicflow.Thus,althoughitispossibletotreattheboundary layer and the external flow separately inmanypractical problemsatlowspeeds,thisissoonlyrarelyinthesupersoniccase,sincethepresenceoftheboundary layer can make a tremendous difference to the external flow,particularly if the layer isa laminarone.Thus,although the inviscid theoryofsupersonic flow does not allow a disturbance at a point to have an upstreaminfluence,theboundarylayerprovidesthemeanswherebyanupstreaminfluencecanexist.

OneofthemechanismswherebythiscanoccurwassuggestedbyOswatitschand Weighardt (1943), who remark that if a disturbance causes a positivepressuregradienttheboundarylayerwillthicken,sothattheexternalstreamlinesmustbegintocurveslightlyupstream.Nowthiscurvaturewill itselfproduceapressuregradientinthesamesense,andtheboundarylayergrowsinequilibriumwith the pressure gradient caused by its owngrowth.When the disturbance isstrongenoughtocausetheboundarylayertoseparatethepresenceofaregionofreversed flow (possibly extensive) makes a considerable difference to themechanism, and, indeed, upstream spreading of a position of separation canoccur in subsonic flow equally well. An important practical example of theinteraction of a boundary layer with the external flow is the problem of theinteractionbetweenashockwaveandaboundarylayer.Inpracticethenatureofthe interactionwill dependvitally upon the state of the boundary layer.Threemain possibilities exist, namely (i) when the boundary layer is laminarthroughouttheregionofinteraction;(ii)whentheboundarylayergoesturbulentdownstreamof the separation position, but is laminar upstream; (iii)when theboundary layer is turbulent throughout the interaction region. In this chapter itwillonlybepossibletoconsiderthecaseofboundarylayerswhicharelaminar

throughouttheregionofinteraction(andtheseonlyverybriefly).Fordetailsoftheothercases,referencemaybemadetoreviewsbyHolderandGadd(1955)andbyGadd(1957a).

1. PrincipalresultsofexperimentalinvestigationsIn practical cases of shock-wave and boundary-layer interaction, such as

those occurring on an aerofoil surface at transonic speeds, it is sometimesdifficulttoseparatethefundamentaleffectsoftheinteractionfromthosewhichareduetothelimitedextentofthesupersonicregionortothepressuregradientsassociated with surface curvatures. For this reason the majority of the basicexperimentalinvestigationshavebeencarriedoutwithaboundarylayeronaflatsurface,sothatthepressuregradientswouldbeentirelyabsentiftherewerenoshockwaves.Theshockisproducedeitherbyplacingawedgeinthesupersonicstreamabovetheplate(Fig.2)orbyattachingawedge(Fig.3)orasteptothesurface of the plate. Ifθ is the angle of deflexion through the external shock,then the streamlines at the edge of the boundary layer are turned through anangle2θinpassingthroughtheshockandthereflectedexpansion.Awedgeofangle2θontheplatewillgiveroughlythesamepressurerise,andinthiscasethe external streamlines change direction continuously but are deflecteddiscontinuouslythroughanangle2θrelativetothewall.

FIG.2.Boundarylayerandexternallygeneratedshock.

FIG.3.Shockgeneratedfromwithinboundarylayerbywedgeonsurface.

It is found experimentally that the pressure distribution is roughlysymmetrical about either the apexof thewedgeon the surface or the positionwheretheshockstrikestheboundarylayer.Whenseparationoccursthepressuredistributionhasthreepointsofinflexion,whichariseforthefollowingreasons.The pressure gradient rises from zero well upstream of the interaction to amaximumvalue at separation.Downstreamof separation thepressuregradientdecreaseswith distance due to the thickening of the dead-air region, and thenincreasesagainwhenthedead-airregionbecomesthinnerdownstreamofwherethe shock strikes the layer. It increases to amaximumvalueat thepositionofreattachment and then gradually decreases to zero as the pressure tends to itslimitingdownstreamvalue.

In the case of zero heat transfer a considerable body of experimentalevidence has been accumulated concerning the dependence of this pressuredistribution upon such parameters as Mach number, Reynolds number, andshock strength. Chapman,Kuehn, and Larson (1958) have correlated some oftheseresults.InparticulartheyfindthatforarangeofupstreamMachnumbers1·1<M0<3·6,andwithaReynoldsnumber(basedonexternalconditionsintheinteractionregion)104<R<106,thepressurecoefficientCpatseparationtakesthe value

thepressurecoefficientbeingdefinedas

The scatter in the experimental pressure coefficients is of order 10 per cent.†Various theoretical investigations give formulae like (373)with the numerical

coefficienttakingvaluesrangingfrom0·83to1·15.Morerecentlyattentionhasbeenturnedtowardstheeffectsofheat transfer

uponthepressuredistribution.Earlyexperiments(Gadd1957b)suggestedthatthepressuredistributionwasbarelyaffectedbyheatingorcoolingthewall.Thisconclusion did not agree with the theoretical treatments then available, andfurtherexperimentscarriedoutbyGaddandAttridge(1961)withconsiderablygreater rates of heat transfer have indicated that, provided the boundary layerremains laminar throughout, the lengthof the interaction region increaseswithwall temperature, with a corresponding decrease in the pressure gradients.Agreement between these experimental results and the latest theoreticalinvestigationsispromising.

2. SummaryofearlytheoreticalinvestigationsOneofthefirsttheoreticalattemptstothrowsomelightontheinteractionof

a shockwavewith a laminar boundary layerwasmade byHowarth (1948b).Howarth’smodelconsistedoftwosemi-infinitestreamsmovingparalleltooneanother, one at a supersonic speed and the other at a subsonic speed.A smallsteadydisturbance originates in the supersonic region and is propagated alongtheappropriateMachline.Attheboundarywiththesubsonicregionitispartlyreflectedandpartlytransmitted.Theportiontransmittedintothesubsonicregionaffects the whole of that part of the flow, including the interface, and causesfurther disturbances in the supersonic region which are propagated along theother family of Mach lines. For certain particular forms of incident wave,Howarthwasabletocalculatethedetailsofthecompleteflow.

Howarth’sresultsweresomewhatinconclusive,becauseoftheassumptionofaninfinitedepthofsubsonicflow,ascomparedwiththethinsubsonicregioninapracticalboundarylayer.AccordinglyTsienandFinston(1949) improvedonHowarth’s model by making the subsonic region of only finite thickness andboundedontheothersidebyaplanewall,butagainnoconclusiveresultscouldbeobtainedasnoindicationwasavailableastowhatMachnumberorthicknesswouldbeappropriateforthesubsonicregion.Lighthill(1950b)thereforefurtherrefined the analysis by replacing the subsonic layer by a region in which theMachnumberfallscontinuouslyfromM1inthemainstreamtozeroatthewall.Unfortunatelytheseassumptionsledtopredictionsoftheupstreaminfluenceofthe disturbance which were an order of magnitude smaller than experimentalvalues.

NowthesourceoftheerrorinLighthill’swork,ashehimselfpointedoutinlater papers (Lighthill 1953 a, b), was one of inconsistent approximation. All

three contributions discussed above assume that viscosity may be neglectedexcept in the actual setting up of the basic steady flow, and that squares ofdisturbancesmaybeneglected.Nowinaboundarylayerthepressuregradientatthewall isexactlyequal to the lateralderivativeof theviscousshearstress,sothe neglect of viscosity will force the pressure at thewall to remain constantunless a non-zero Mach number at the wall,M2, is postulated, allowing thepressuregradienttobebalancedbyinertialforces.ThusthecontinuousvariationinMachnumber fromM1>1 toM2<1 is an improvementon theTsien andFinstonmodel,butthevalueofM2stillremainstobefound.

Lighthill then makes the crucial point that viscous forces have not beenentirelyneglected,butonlydisturbances to theviscous forces.Now, followingtheideasofStratford(1954),weknowthattheviscousforcesintheouterpartofthe boundary layer are almost exactly what they would be in the absence ofpressuregradient;inotherwords,thedisturbancestotheviscousforcesarenon-negligibleonlyinaninnerportionoftheboundarylayer.Lighthillassumes(andverifiesafterwards)thatthisinnerlayerissufficientlythinfortheapproximationof incompressibility to be made (because velocities very near to the wall arelow).Heassumesalsothatthevelocitymaybeassumedtobelinearwithinthisregion, which is probably adequate provided that the disturbance is not greatenoughtocause theboundary layer tobenear toseparation,andfinds that thevalueoftheeffectiveMachnumberatthewallforuseintheinviscidtheoryis

whereM′(0) is the lateralMachnumbergradientat thewall in theundisturbedflow,whichisproportionaltotheskin-friction,andLisalengthdefinedintermsofthekinematicviscosityandskin-frictionatthewall,andthewavenumberofthedisturbance.

Comparisonsofthemagnitudeoftheupstreaminfluencewiththoseobtainedexperimentally show reasonable agreement. As an example, the upstreamdisturbancefallsbyafactore−1inabout5or6boundary-layerthicknesseswhentheReynoldsnumberisoforder105to106andthefree-streamMachnumberisabout2to3.Accordinglyitwillhavefallento5percent,inabout3timesthisdistance,say15to20boundary-layerthicknesses.

3. Gadd’sanalysesforinteractionscausingseparationProvided that the incident shock is strong enough the boundary layerwill

separate and the assumptions of the preceding analysis will break down.Alternativemethodshavethereforebeendevelopedforconsideringshock-wave

andboundary-layer interaction incaseswhere separationoccurswellupstreamof the agencywhichcauses it.This situation isof someconsiderablepracticalimportanceandcanarise inavarietyofways, suchaswhenaboundary layerencountersasteporawedgeonthesurface,orwhenseparationoccursupstreamof a sufficiently strong externally generated shock. In such circumstances thepressureatthewallandthedevelopmentoftheboundarylayeraregovernedbythe equilibrium between the pressure gradient, the resulting boundary-layerthickening and the deflexion of the external streamlines. The relationshipbetweenpressureand thedeflexionof theexternal streamlines isderived fromthetheoryofsupersonicflowand,providedthedeflexionangleisnottoogreat,that given by linear theory will be adequate, namely

wheresuffixzeroreferstoconditionsintheundisturbedstream,aboutwhichthelinear perturbation is taken, and α is the angle through which the externalstreamlines are deflected. In the shock-wave and boundary-layer interactionproblemitisusualtoequatethedeflexionoftheexternalstreamlines(causedbyboundary-layerthickening)totherateofgrowthofthedisplacementthickness,that is

whered{δ10(x)}/dxrepresentstherateofgrowthofdisplacementthicknessintheabsence of pressure gradient, which is often neglected if the interaction takesplacefarenoughfromtheleadingedgeoftheboundarylayer.Combining(374)and (375) we have

A second relationship connecting p and δ1 is given from the basic theory oflaminarboundary-layerdevelopmentinthepresenceofapressuregradient,andthreepapersbyGadddiffermainly in themethodused to treat thispartof theproblem.

Intheearliestpaper(Gadd,1953b)itisassumedthatintheregionupstreamof the interaction,where there is no pressure gradient, the velocitywithin theboundary layer is given by

where the boundary-layer thickness δ is chosen so that the displacementthicknessisinagreementwiththatgivenbyCrocco’scalculations.Ifweassume

that there isnoheat transferat thewall,and thatσ=1andμ∝T, thisyields

Intheregionwheretheboundarylayerthickensundertheinfluenceoftheshockit is assumed that

when

and

where

The form of r1 is determined by the condition that in the outer part of theboundary layer changes in the profile caused by the pressure gradientmay bedetermined by essentially inviscid considerations, following the ideas used byStratford(1954).Theformofr2ensurescontinuityofu,∂u/∂y,and∂2u/∂y2,atthe join,y =nλ,with the inner profile, andoneof the twoparametersn,λ, isdeterminedbythebalancebetweenpressuregradientandviscousstressesatthewall.Theconditiononmassflow(correspondingtoStratford’sconditionthatψiscontinuousatthejoin)isalsoapproximatelysatisfiedprovidedthat(p/p0)−1and λ both remain small. The other relationship between n ond A, which isrequiredtocompletethesolution,isgivenby(376).

Gaddmadeanumberofcomparisonsbetweenhistheoryandexperiment.Inparticularthepredictedvalueofthepressurecoefficientatseparationisoforder10percent,greaterthanthemeanexperimentalvalue(373),whichisextremelysatisfactory.

InasecondpaperGadd(1956)considersagain thecasewhenthere iszeroheat transfer and the Prandtl number is equal to unity, and assumes that thevelocity profiles are given either by

orbyasimilarformwhichwillnotbeconsideredhere.Thevirtuesofthisfamilyofprofiles are firstly that it is simple, and secondly that for largevaluesofmthereisanextensiveregionoflow-velocityflowneartothewall,ratherlikethatoccurringinawell-separatedlaminarboundarylayer.Themaindrawbackisthatnoregionofnegativevelocityoccurs,sothatthepositionofseparationcannotbepredicted.

Thevalueof l isdetermined from thedisplacement thickness in the regionupstream of the interaction, and is assumed to be constant. Thus

sothatjustupstreamoftheinteraction,wherem=0,wehave

Theparametermisdeterminedfromtheconditionthatthepressuregradientat the surface is given by

whichfollowsfromthemomentumequation(12)uponsetting(∂μ/∂y)wequaltozeroforthecaseofzeroheattransfer,andasecondrelationshipbetweenmandpis given by (376) with δ1 given by (377). Simultaneous solution of theseequations yields p as a function of x. Gadd remarks that in the case of anexternally generated shock (strong enough to deflect the external streamlinesthrough an angle θ) the streamlines are deflected instantaneously through anangle 2θ at the position where the shock strikes the boundary layer, and thiscorresponds to the discontinuous change in the slope of the wall when theinteractionisduetoawedgeonthesurface.

Gadd’sresultsmaybeexpressedas

inwhichdQ/dsisanumericallydefineduniversalfunctionof ,sisanon-dimensionalvalueofx,and isthevalueofscorrespondingtotheupstreamendof the interaction region. A comparison of the predictions of the upstreaminfluencewiththeexperimentallyobservedvaluesshowsfairagreement.

In a third paperGadd (1957 c) presents amore complete treatment of the

problem (including the effects of heat transfer) by a similar approach to thosediscussedabove,but involving fewerapproximations.For theouterpartof theboundarylayer,followingmorecloselytheanalysisusedbyStratford(1954)inlow-speedflow,it isassumedthatthedistortionoftheupstreamzero-pressure-gradientprofilemayagainbedeterminedbylargelyinviscidconsiderations,andintheinnerpartoftheboundarylayer(closetothewall)theprofileisassumedgiven by a form which is essentially equivalent to

Thethreecoefficientsa,b,c,andthepositionofthejoinyj,arethendetermined

by the conditions that , and ∂2(ρu)/∂y2 are all

continuous at the join.Thus the complete profile is determined in principle atanyrate.Inpracticetwofurtherapproximationsaremade,namelythattheinnerprofile is thin (so that the usual simplifications may be made to the joiningconditions) and that the changes indisplacement thicknessunder the actionofthepressuregradientaremainlyduetochangesintheshapeoftheinnerprofile.Subjecttotheseapproximationsit isindicatedthatthepressureatseparationisgiven by an expression similar to the empirical form (373), but with thecoefficient0·93 replacedby1·13. It isalsopredicted that thewall temperatureTw has no influence on the pressure coefficient at separation, but that thestreamwise extent of the region of interaction varies with Tw as . As hasalready been remarked, the earliest experiments on the effects of walltemperature suggested that the pressure distribution is barely dependent uponwall temperature.Morerecentexperiments,however,withgreaterratesofheattransfer, have shown that the streamwise extent of the interaction region doesindeedincreasewithTw,thoughprobablynotasquicklyas .

4. TheanalysisofHäkkinen,Greber,Trilling,andAbarbanelAninterestinganalysis,basedonsimilarideastotheabove,hasbeengivenbyHäkkinenetal.(1959).Theapproximationsmade,thoughnumerous,areallphysicallysound,andtheanalysisdeservesmentionbecauseofitsessentialsimplicity.It is assumed first that the boundary layer consists of an outer portion,

differinglittlein‘shape’fromtheupstreamundisturbedlayer,andaninnersub-layerofsmallmomentum.Thejoinbetweenthelayersoccursaty=Δjsay,and

the apparent displacement of the undisturbed layer is Δ, with . Twoconsequences of this assumption are that

and

whereτisthelocalviscousstressandτ0theskin-frictionimmediatelyupstreamoftheinteraction.Sincethemomentumthereinissmall,itmaybeassumedthatpressure and viscous forces balance throughout the inner layer. Thus

andhence

whereτwisthelocalskin-frictionand,moregenerally,

Theconditionsofcontinuityinuandτatthejoin,y=Δj,maythenbeshowntoyield

Ifitisassumedthatthepressureriseduetothethickeningoftheboundarylayeris mainly due to the growth in Δ, then

and elimination of Δ between (378) and (379) yields

or

SinceitisfoundexperimentallythatdCp/dxisverynearlyconstant,exceptintheregion where Cp is very small, we may write approximately

sothat(380)becomes

or

where

areskin-frictioncoefficients.Thus,atseparation,(382)yields

and, it μ ∝ T, it follows from (270) that this becomes

whereR is the localReynolds number based on conditions at the edge of theboundary layer.Thisvalueof thepressurecoefficientatseparation isabout20percent,higherthanthemeanexperimentalvalue,sotheagreementisfair.Thisis probably partly fortuitous, since the approximation (381) alone couldintroduceanerroroforder20percent,inCp,sosomecancellationofthevarioussourcesoferrorhaspresumablybeenpresent.

These sameauthorshave also carriedout a similar analysis,makingdirectuseof themomentum integral equation, andGreber (1960)has shownhow totake account of convex wall curvature. This analysis has been somewhatimproved upon byGadd (Bray, Gadd, andWoodger, 1960), andwill now bediscussed.

5. TheworkofGaddandGreberThedescriptiongivenherefollowsthatofGadd,andwillberestrictedtothe

caseofsolidwalls,althoughGaddinfactdevelopedthemethodinaformwhich

makes it applicable when suction or injection takes place. Gadd writes thedisplacement thickness δ1 as

where

andTH is the total temperature. If, for simplicity, it is assumed that to agoodapproximation we may write

which holdswhen there is zero heat transfer andwhen there is zero pressuregradient (provided σ = 1), then (384) becomes

where

Thisformwillbeacceptedinwhatfollows.Now in an interaction which takes place over a distance which is small

compared with the distance from the leading edge, changes in displacementthicknessarelikelytobelargecomparedwithchangesinmomentumthickness.This is borne out by the experimental results and was surmised by Gadd byinspectionofthemomentumintegralequation.Accordinglyitmaybeassumedthat δ2 is approximately constant, as isM1. Then it follows from (386) that

sothat

where suffix zero refers to (constant) values in the region upstream of theinteraction.

Gadd further deduces from the momentum equation (12) that

where

and

He then assumes that g is a function only of r2, being independent of suchparameters as the ratio of wall temperature to mainstream temperature, anddeduces from (376) and (387) that

By equating this value of dp/dx to that given by (388) it follows that

Multiplication by 2(dr2/dx) and formal integration with respect to x yield

sothatdr2/dxisknownasafunctionofr2.Thisrelationshipisuniversalexceptforthescalingfactor,whichissuchthattheextentoftheregionofinteractionisproportionaltoTw.Thisis inbetteragreementwithexperimentthantheearlierpredictions,inwhichtheextentoftheinteractionwasfoundtobeproportionalto

.Gaddassumes that theuniversal relationshipbetweeng and r2 is precisely

that appropriate to the similar solutions with zero heat transfer, and deduces

from(390)therelationshipbetweenr2anddr2/dx,afterwhichthevalueofr2(orp)asafunctionofxeasilyfollows.Hefinds,inparticular,thatthevalueofthepressure coefficient at separation is equal to

whichdiffersfromthemeanexperimentalvaluebyonly1percent.,inthecaseofzeroheattransferatleast,overarangeofMachnumbersuptoabout4.

Gaddhasalsogivenasecondmethodwhichdoesnotinvolvetheuseoftheapproximatetemperaturerelationship(385),butratherassumesthatthefunctiong is dependent upon r1 and Tw, the dependence being given by the similarsolutionswithheattransfer.TheresultsarebroadlythesameasthosegivenbythemethoddiscussedabovebutindicateslightlylessdependenceuponTwinthecase of the extent of the interaction region, and a slight dependence of thepressurecoefficientatseparationuponTw.

In the same paperBray,Gadd, andWoodger have also given a simplifiedoutline of a method originally due to Crocco and Lees (1952), and havediscussed solutions of the shock-wave and boundary-layer interaction problemobtained by that method, making use of the National Physical LaboratoryDEUCEcomputer.Thisworkwill notbediscussedhere,partly for reasonsofspace,andpartlybecause it is theauthor’sview that, for investigationsof thistypeof interaction, themethod is less convenient thanothersdiscussed in thisbook.

6. ThemethodofCurleWe conclude this bookwith a brief description of amethod due to Curle

(1961b),whichbears considerable similarity to that discussed in theprevioussectionand, indeed,wasdevelopedalmost simultaneously.Themethodbeginswith the approximate equation (341) for the momentum thickness of acompressible laminar boundary layer, from which it is deduced that if thepressure rise is a short sharpone (ashasalreadybeen seen tobeoften true ininteractions between shock waves and laminar boundary layers) then itautomatically follows that approximately

Underthesameconditionsitfollowsfrom(345)that

where λ′ is defined by (343) and H′(X′) is a universal function. From this

definitionofλ′itfollowsthatwhenthepressure(andvelocity)changesaresmallwe have

Theneliminationofδ1andλ′betweenequations(376),(392),and(393)leadstoasingleequationforporu1.Thisequationmaybeexpressednon-dimensionallyby writing

and

wherexsrepresentsthepositionofseparation(oranyotherconvenientpositionwithin the region of interaction), the Reynolds number R is

and2θistheangleofdeflexionofthestreamlinesrelativetothewallcausedbyeither the shock (Fig. 2) or the wedge (Fig. 3). We then find that

where ∊ = 0 upstream of the position X = X1 where the shock strikes theboundarylayer,and∊=1downstreamofthisposition.Solutionof(397)yieldsFandF′ as functions ofX. This is done bymultiplying the equation byF″ andformallyintegratingwithrespect toX from−∞toX,usingintegrationbypartswhere necessary. This yields

whereP(F′)mayeasilybedeterminedonceH′isknown.ThusFisknownasafunction of F′, from which we easily deduce X as

The condition thatF is continuous atX =X1 implies, byvirtueof (398), that

It is theneasy to see that in taking the square rootof equation (398)wemustwrite

and

Itwillbenotedthat thepressuredistributionissymmetricalaboutX=X1.Theresults are shown inTable 17,where the relationship betweenF,F′, andX isgivenforX<X1.

TABLE17

Universalfunctionsforshock-waveandboundary-layerinteractions

F′ X

0·0103 0·02 −7·030·0237 0·05 −5·120·0351 0·08 −4·090·0479 0·12 −3·140·0612 0·17 −2·210·0736 0·23 −1·320·0832 0·29 −0·550·0900 0·338

0

0·0885 0·40 +0·700·0828 0·50 1·860·0645 0·60 3·210·0465 0·70 5·03

003230·80 7·61

0·0174 0·90 11·75

001010·95 15·52

0·0042 1·00 23·33

0 103 ∞

At separation F′ = λ′ = 0·090, and from Table 17 we note that

andwemaythendeducefrom(394)that

Thisvalueisingoodagreementwiththemeanexperimentalvalue(373)forzeroheattransfer.ThatCpsisindependentofwalltemperatureisalsopredictedbythemethoddescribedintheprecedingsection,duetoGadd.

It follows from(395) that the lengthof the regionof interaction isdirectlyproportionaltoTw,whichresultisalsopredictedbyGadd’smethod,andisinatleastreasonableaccordwiththemostrecentexperimentalresults.

A similar approach has also been used (Curle, 1962 b) to consider theinteractionwhena laminarboundarylayermeetsanexpansivecornerfollowedbyacompressiveagency,suchasastep.

†ArecentpaperbySterrettandEmery(1966)indicatesthatthenumericalcoefficientappropriatetotheirexperimentalobservationsdecreasesasM0increasesfrom4to6.

REFERENCES

BAXTER, D. C., and FLÜGGE-LOTZ, I. 1957. Stanford University, Div. of Eng.Mechs.,Tech.Rep.110.

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FRÖSSLING,N.1940.LundsUniv.Årsskr.N.F.Avd2,36,No.4.GADD,G.E.1952.ARCC.P.312.______1953a.Aero.Quart.4,123.______1953b.J.Aero.Sci.20,729.______1956.Ibid.23,225.______1957a.Proc.IUTAMSymposium,Freiburg.______1957b.J.FluidMech.2,105.______1957c.J.Aero.Sci.24,759.______andATTRIDGE,J.L.1961.ARCC.P.569.GOLDSTEIN,S.1930.Proc.Camb.Phil.Soc.26,19.______1938.ModernDevelopmentsinFluidDynamics.O.U.P.(2vols.).______1948.Quart.J.Mech.Appl.Math.1,43.GÖRTLER,H.1957a.J.Math.Mech.6,1.GÖRTLER,H.1957b.DeutscheVers.fürLuft.E.V.34.GREBER,I.1960.NASATech.NoteD–512.GRUSCHWITZ,E.1950.ONERAPublication47.HAKKINEN, J., GREBER, I., TRILLING, L., and ABARBANEL, S. S. 1959. NASA

Memo.2–18–59W.HANTZSCHE,W.,andWENDT,H.1940.Jahrbuchdeut.Luftfahr.1,517.____________1942.Ibid.p.40.HARTREE,D.R.1937.Proc.Camb.Phil.Soc.33,223.______1939a.ARCR.&M.2426.______1939b.ARCR.&M.2427.HEAD,M.R.1957a.ARCR.&M.3123.______1957b.ARCR.&M.3124.HIEMENZ,K.1911.DinglersPolytech.Journ.326,321.HOLDER,D.W.,andGADD,G.E.1955.Proc.N.P.L.Symposiumon‘Boundary

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(2vols.).ILLINGWORTH,C.R.1946.ARCR.&M.2590.

______1949.Proc.Roy.Soc.A,199,533.______1954.Quart.J.Mech.Appl.Math.7,8.KALIKHMAN,L.E. 1946.Prikl.Matem, iMech.10, 449.TranslationN.A.C.A.

T.M.1229.KÁRMÁN,T.VON.1921.Z.angew.Math.Mech.1,244.______1935.AttiAccad.d’Italia(5thVoltaCongress),p.255.______andMILLIKAN,C.B.1934.NACAReport504.______andTSIEN,H.S.1938.J.Aero.Sci.5,227.LEIGH,D.C.F.1955.Proc.Camb.Phil.Soc.51,320.LI,T.Y.,andNAGAMATSU,H.1953.J.Aero.Sci.20,653.LIEBENSON,L.S.1935.C.R.Acad.Sci.U.R.S.S.(Nouv.Série),2,22.LIEPMANN,H.W.1958.J.FluidMech.3,357.LIGHTHILL,M.J.1950a.Proc.Roy.Soc.A,202,359.______1950b.Quart.J.Mech.Appl.Math.3,303.______1953a.Proc.Roy.Soc.A,217,344.______1953b.Ibid.p.478.LILLEY,G.M.1959.CollegeofAeronauticsNoteNo.93.LUXTON,R.E.,andYOUNG,A.D.1960.ARCR.&M.3233.MACK,L.M.1958.JPLProgressReport20–352.MEKSYN,D.1950.Proc.Roy.Soc.A,201,279.______1956.Ibid.237,543.______1960.Z.angew.Math.Phys.11,65.______1961.NewMethodsinLaminarBoundaryLayerTheory.PergamonPress.MERK,H.J.1959.J.FluidMech.5,460.MISES,R.VON.1927.Z.angew.Math.Mech.7,425.MONAGHAN,R.J.1949.ARCR.&M.2760.MONAGHAN,R.J.1960.ARCR.&M.3218.MORGAN,G.W.,PIPKIN,A.C.,andWARNER,W.H.1958.J.Aero/SpaceSci.25,

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______andULRICH,A.1940.Lil.-Ges.fürLuft.S10,75.SCHMIDT,E.,andWENNER,K.1941.Forschungs-GebieteIng.12,65.SCHUBAUER,G.B.1935.NACAReport527.SPALDING,D.B.1958.J.FluidMech.4,22.SQUIRE,H.B.1942.ARCR.&M.1986.STERRETT,J.R.,andEMERY,J.C.1960.NASATND–618.STEWARTSON,K.1949.Proc.Roy.Soc.A,200,84.______1954.Proc.Camb.Phil.Soc.50,454.______1958.Quart.J.Mech.Appl.Math.1,43.STRATFORD,B.S.1954.ARCR.&M.3002.TANI,I.1941.J.Aero.Res.Inst.TokyoImp.Univ.Rep.199.______1949.J.Phys.Soc.Japan,4,149.______1954.J.Aero.Sci.21,487.TERRILL,R.M.1960.Phil.Trans.Roy.Soc.A,253,55.THWAITES,B.1949.Aero.Quart.1,245.TIFFORD,A.N.1954.WADCTech.Rep.53–288.TIMMAN,R.1949.NLLReportF.35.TÖPFER,C.1912.Z.fürMath.undPhys.60,397.TRUCKENBRODT,E.1952.Ing.-Arch.20,211.TSIEN,H.S.,andFINSTON,M.1949.J.Aero.Sci.16,515.ULRICH,A.1949.Arch.Math.2,33.WALZ,A.1941.Lil.-Ges.fürLuft.141,8.WEIGHARDT,K.1946.MAPVölk.Rep.andTrans.89.______1948.Ing.-Arch.16,231.YOUNG,A.D.1949.Aero.Quart.1,137.______andWINTERBOTTOM,N.E.1942.ARCR.&M.2068.

INDEX

Thisindexcombinessubjectsandauthors,thereferencesbeingtopagenumbers.Whereagivenauthororsubjectisreferredtoonconsecutivepagesonlythefirstpagenumberisgiven,unlessthereferencesare

independent.

Abarbanel,S.S.,150.Attridge,J.L.,145.

Baxter,D.C.,104,122.Bernoulli,D.,48.Blasius,H.,20,22,25,48,50,82,92.Böhlen,T.,43.Bourne,D.E.,79.Brainerd,J.G.,93,116.Bray,K.N.O.,152.Busemann,A.,90.

Chapman,D.R.,102,105,144.Cohen,C.B.,121,125.Conservationofenergy,3.—mass,3.—momentum,3.Cope,W.F.,89,115.Crabtree,L.F.,112.Crocco,L.,17,94,154.Croccotransformation,17,94.Curle,N.,32,35,38,42,47,52,60,63,74,81,107,112,121,129,138,154.

Davies,D.R.,79.Displacementthickness,11,21,27,41,60,84,124,127,131.Driest,E.R.van,99.

Emery,J.C.,144.Emmons,H.W.,92,116.Enthalpythickness,12,124.Entropy,9.

Fage,A.,70,102.Falkner,V.M.,23,56,70,102.Finston,M.,145.Flugge-Lotz,I.,104,122.Frankl,F.I.,117.Frictionalheating,65,67.Frössling,N.,25.

Gadd,G.E.,118,121,143,145,147,152.Goldstein,S.,4,20,24,43,69.Görtler,H.,29,30,40,81.Greber,I.,150,152.Gruschwitz,E.,117.

Hakkinen,J.,150.Hantzsche,W.,99,101.Hartree,D.R.,23,32,34,38,39,81,89,115.Head,M.R.,58,62.Heattransferrate,12,67,69,71,78,83,87,99,101,104,105,127,135,141.Heimenz,K.,22,25.Holder,D.W.,143.Holstein,H.,43.Howarth,L.,3,20,22,25,28,29,32,36,39,45,81,108,145.Howarthtransformation,108,123,129.

Illingworth,C.R.,110,116,135.

Kalikhman,L.E.,123.Kármán,T.von,48,90,91.Kineticenergyintegralequation,13,54,63,117,134.Kineticenergythickness,12,53,58.Kuehn,D.M.,144.

Larson,H.K.,144.Lees,L.,154.Leigh,D.C.H.,34,39.Liebenson,L.S.,15,53,64.Liepmann,H.W.,73,76,138.Lighthill,M.J.,71,76,105,135,138,145.Lilley,G.M.,135.Li,T.Y.,121.Luxton,R.E.,110,131.

Machnumber,10.Mack,L.M.,102.Meksyn,D.,30,80.Merk,H.J.,30,80.Millikan,C.B.,48.Mises,R.von,17,71.Misestransformation,17,71,102,136.Momentumintegralequation,13,43,54,58,90,109,114,117,132,134.Momentumthickness,12,21,41,51,54,124,127,130.Monaghan,R.J.,100,127.Morgan,G.W.,77.

Nagamatsu,H.T.,121.

Oswatitsch,K.,114,142.

Platethermometer,67,81.Pohlhausen,E.,66,67,82,86,97,103.Pohlhausen,K.,31,41,55,61,110.Poots,G.,133.Prandtl,L.,1,4.Prandtlnumber,4,89.Pressurecoefficient,50.

Radiation,106.Reshotko,E.,121,125.Reynoldsnumber,6.Rott,N.,113,137.Rubesin,M.W.,102,105.

Schlichting,H.,42,43.Schmidt,E.,77.SchubauerG.B.,31,39,81.Separation,2,34,35,49,60,73,84.Seriessolutions,24,28,35.Shockwave/Boundarylayerinteractions:Deflexionofstreamlines,143.Heattransfereffects,144,149,154,156.Pressurecoefficientatseparation,144,148,151,154,156.Pressuredistribution,144,156.Thickeningofboundarylayer,147,151.Upstreaminfluence,146.

Skan,MissS.W.,23,47,56,112.Skinfriction,12,27,34,36,41,52,61,63,83,91,94,101,127,131,135.Spalding,D.B.,77,137.Specificheats,4,10.Squire,H.B.,69,81,84.Stagnation-point,22,24,68.Stagnationtemperature,10.State—equationof,3,14.Sterrett,J.R.,144.Stewartson,K.,24,110,112.Stewartson–Illingworthtransformation,110,113,125,133,138.Stratford,B.S.,63,81,146.

Tani,I.,29,31,35,47,54,62,134.Terrill,R.M.,32,38,40.Thermalconductivity,4.Thermalenergyintegralequation,73,78,134.Thermometricconductivity,4.Thwaites,B.,45,62,112,126.Tifford,A.N.,22,25,28,36.Timman,R.,44.Töpfer,C.,20.Totaltemperature,10,15.Trilling,L.,150.

Truckenbrodt,E.,53,55,60,62.Tsien,H.S.,91,145.

Ulrich,A.,25,42.Universalfunctions,25,35.

Velocityprofile,36,42,44,48,63,72,74,82,90.Viscosity,88,97,113.Viscousstress,1.

Walz,A.,45,47,114,126.Weighardt,K.,57,114,142.Wendt,K.,99,101.Wenner,K.,77.Winterbottom,N.E.,84.Woodger,M.,152.

Young,A.D.,84,100,110,119,131.

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