Boundary-Layer Interactions in

AGARI'

ADVISORY GROUPFORAEROSPACE RESEARCH & DEVELOPMENT 7 BUEANCELLE 92200\EJILLY SURSE.NEFRANCE

PaperReprintedfrom AGARDReport792 SpecialCourseon

Shock-Wave/Boundary-Layer Interactions in SupersonicandHypersonicFlows (Interactions entreOndesdeChocet Couches LimitesdanslesEcoulernents Supersoniques etHypersoniques)

NOBTHATLANTIC TREATY ORGANIZATION

Interactions SweptShock/Boundary-Layer ScalingLaws,FlowfieldStructure,and ExperimentalMethods Gart S. Setfles GasDlaamicsLaboratory MechanicalEnginee ng Dopartmeff 303ReberBuilding PennStateUniversity UniversituPark.PA 16802USA

1. STIMMARY A geaoral rcview is given of severaldecadesof researchon the scalinglaws ard flou/fieldstructures of sw€pt shock wavo/turbulent bounda{y layer interactrons. Attentiotr is furth restrictedto the of the eryedme al studya.ndphysicalunderstanding flows. The hteraction steady-stateaspectsof these producedby a sharp,upright fin mountedon a flat plateis takenasan archetpg. Atr overallftamework of quasicoical sJrnmetrydescribingsuchitrteractions is first developed. Boundarylayerseparation,the interactiotr footprint, Mach nueber scalirg atrd Reynolds nurnber scaling are then considercd, foUowedby a discussionof the quasiconical similarity of iDteractionsproducedby geometrically-dissimilai shock generalors. The detailed $tructureof these interaction flo*delds is next roview€d, and is illustrated by both qualitative visualizationsand quantitativc flow images iD the quasiconical framework.Firaly, theexperimentaltechniques used to investigatesuchflows are reviowed,v,ith emphasis on modernnon-intrusive opticalflowdiagnoslics.

= 1t /6)Re6'4 DoD-dide$ional distance measurednormal to inviscidshocktlace oo t€stsurface = (!/6)Rer'i', nod-dimensionaldistance L, neasuredalonginviscidshocktraco on test surface Mn,l'I, = M@sMo,Machnumbernormalto inviscid shockwavetraceolr test surface = Machnumbornormal to soparationlhe M4" M6 incomingfreestreamMach numbor tr oil refractivoindex N' effectivefringonumber PAi\ primaryflow attacbmentlitre PSSr primaryflow separationline p MPa staticpressure, p@ incomingfreestreamstaticprgssure,MPa pr fregstrgam staticpressulebeforeshock,MPa p, fteest{eamstaticpressureafter shock,MPa q"-heatconvectedto flow, Wrn' q"b& heatefllux ftou heater,W/m' q'io" heat conductedthrough insulatorboard, 1.

distance measured from the virtualoriginio sphericalpolar coordinateframg mm r oil refractionange R resistancc, O Re fteestreamunit ReFolds number,m_' R.o Reytroldsnumber based on th€ hcoming, undisturbedboundarylayerthickness Rea Reytroldstrumber based on the incoming wdrsturbed bourdary-layer momeirtum dicknfss S separationline S,, opticalphasoshift SA secondary flow attachmentline flow separatiooLne SS,S1 socondary time t t' effectiveflow timc T temporature,'K To stagnationtempe.ature,'K U,tlI upsheaminlluenceline V velocity,m/q alsovolts R. r

2. LIST OF SYMBOLS 2.1Main Slmbols A amperes q incoming boundary-layer skin friction coefficient ircoming boutrdary-layer heat transfer cr coeflicrent D,d D

e H,h K k

Lt

specilic heat at constant pressure shock generator leading-edgediameter, mm themal difhrsivity heat generation rate, Wm3 step height, mm thernal coDductivity orfhonormal coordi.lates based on inviscid shock wave trace on tast sudacc inception length from VCO to codcal flow,

vitual conical odgin cartesial coordfumtesystem coordinate normal to plate in heat transfer aDalysjs o argle made by fin with respect to tle hcoming freestream directior, deg p alde made by surface'flow features vith freostream direction in spherical polar coordiDates,dsg. a.nglemade by inviscid shock-wavetrace on Bo test suface with freestream direction, deg. AFo = $" - p*), reducld itrviscid shock aagle parameter AB1 angular measure of )-foot defined in Fig. 28 AB, angular measure of )-foot deined in Fig. 28 AB,J = @" - t@), reduced upstream idluence angrc paramercr ratio of specific heats 1 incoming undisturbed boundary-layer 6 thickn€ss,mm As oil-film leadiry-edge distancg m incoming, undisturbed boundary-layer , momcntumthicknoss,mm ) shock generator loading-edgo swoopback angle, also laser wavolongth = sini(r/M@), Mach angle of iDcoming p6 freestrgam z oil viscosity,centistokcs = (p,/pr)Dincipiontseparationprossure ratio € p kg/m3 densitt r* wall shear stress, N/m' azimuthal angle in spherical polar d coordinates, deg, n Ohms

layerswhich are susceptrbleto disruptionby such gadients. Unfo unately,the nature of high-speed flow overpracticalmissiles,aircralt,re-ontryvehicles, and tubomachrnery components makes such with consequences ranging inreractioDs unavoidable. from tolerable to (occasionally) disastrous. Furthermorgthescaleatrdperfordanceconstrahtsof suchpracticalapplicationsalnost alwaysdictatethat

22 Subscripts

3.1Pr€s€ntScopc Refs.1 and 2 are eachon the order of 25,000words long. Ref. 1 is a generalroviewof sweptinteractions, while Ref. 2 coDcedratesspecificallyuponturbulent interactionsproducedby sharp fins and inteiaction unsteadiress, Thopresentpap€rmaybethoughtof as an update and suppleBent rather than as a replacement for thesedocuments.Moroover,insofat as uNteadyphenomena,numericalsimulatioDs, and

VCO x,y,z x

is

adiabaticwall conditions incipiontseparation incidentshock

p pa pk ps

normal platoau primaryattachment peakvalue primaryseparation

s, 51 U,ui TP w oo 0 *

separation upstream influence triple-poiDt wall conditions ilcomirg freestreamconditions valueat beginniDgof intemction deDotosincipientseparationcondition

the interactions in questiod will bc both turbuled alld

This tbree-dimensionality tfuee-dimensronal. usually implie,stlat the interactiorsare (o! havecomponentj whicbare) moderatelyto highlyswept. Thepast20yearshaveseonintensiveresearchon tle subjectof sv/eptshockwave/turbulentboundarylayer htemctions. Ttis research was conductedatrd supportedbecauseit wasrecomizedthat suchflows are important, fundamental viscous/inviscid interactions,are key building'blocksin high*peed problems,aadare intemal andeKernalaerodynarnic primarytest casesfor numericalsimulations.From worldwide,a the effortsof a numberof investigators limiled undersranding of suchinteractioDs is now availablewhereas,only a few years ago, almost nothingwasknown. Several recert publications Lave attempted ar overviewof this newfoundundcrctanding,including tbosein 1986and 1990by the preseif authorand D. S.Dolling (Refs.1,2).The presentpaperreliesheavily uponRef. 2 for the interactionscalinglawsand part ol tho flowfield structuresections,while assembling materialnot previouslyroviewedfor thc soctionon expcrim€ntal m€thods.

laninar/transitional interactions are covcredby other authors h this report, they are omitted from consideration hcrc. Likewise, the present consideration of erperimontal methods forgoes any discussionof Laser Doppler Velocimetry in favor of the paper by J. Delery gived elsewherein this report.

This discussion is thus restricted to supersonic interactions with turbulent boutrdary layers, whence tho nair body of available data derives. Evetr so, a wide varioty of swept interaction tt?es exists. An 3. IIYTRODUCTION effort was made to classily aDd describo several of The iDteractionsof shockwavesand boundary-layers thesein Ref, 1, to which tLe reader is referred. Rel havelong been a fundamedtaland critical problem 2, on the otber hand, concentrated on the large body areaof fluid dynamics.Theyrepresentthe imposition of available information concernjng tle sharp-fuof thestroDgesL p.essurc adverse gadietrtsonviscous gen€ratedinteraction with a turbulent boundary layer.

As ia Ref. 2, the preseDtpaper wi[ also concentrate its attentron on the sieplest of all swopt icteractions, i? that getrerated by an equilibriun, adiabatic flat-plate turbulent bourdary layer inte.acting with thc swept, planar oblique shock wave generated by an upright, sha-rpleading-eGod fin at alrgle-of-attacko 'sharp 6a interaction' for G6o Fie. 1). Kro\vn as a brevity, this flow is a classic.alcase,atr archetlpo, of swept interacdons in general. (The reader should note that vadous auihors havo also referred to this case as a ?ancrng-shock" or "swept-normal-shock" intemction.) Emphasiswill bo plactd otr experimental resrlts which shed light on the physical behavior and phenomenolory of these intsractions rattrer than on prediction mothods or aerothermodynamic applicatrons per se.

Fig. 1 - Sharp Fin Interaction Test Geometry, This concedtration on sharp fin interactions neglects several other swept intoraction q?es-the blu fin, swept compressiod corner, etc.-which are certainly not ofnegligiblo inportance. However,relatively little new work has appeared on theso topici since they were reviewed in Rel 1 and other reviewsmentioned below. The only other rltcraction ty?e seeing signficafi recent activity is the ;rportant new area of crossing-shockinteractions. while this area is not covercd due to necessarylimitations oD tle present scope, it is expected that crossing-shockhteractroDs will bc the subject of a separatereview in due tirne. 32 Historical Sketch The hteractiotr of a shockwavewith a boundarylayer was appalcdtly tust observed by Fcrri (Ret 3) in 1939. Followitrg World War II, research on the subject concentratedotr 2-D interactions,which were thought to be more amenable to study than swept interactions. The work of Stalker (Ref. 4) in 1960was the fust detailed attempt to study hteractior swe€pback.There followed a 2s-yearporiod of steady increase in the number of paperspublished annually on svept hteractions, indicating tbat tlis subtopic of fluid dytraEics was st l a maturing research area during that time. The publication record reveals spurts of activity ia the late 1960s and nid-7os,

presMably due to the cold-war"spacerace"and the SSTandSpaceShuttledevelopmefiefforts. The Eost recentspurt of activitymaybe asqibedin part to the need for experimeffal berchmarks for CFD predictioN,and to the NASPprograrn. It may be tlat the freld of swept htemctions is nearingmatudtyat tte time of tijs writing. If so,it is principally the result of severalresearchgroups oD the problem,e,speciafly those havingconcentrated at the NASA-AmesResearchCenter, Penn State, Princeton,and RutgersUniversities,and the I$titute for ThooreticalandAppliedMechadcsin Novosibirsk. topicmay materialpertinenlto tbep(esetrt Additiotral papers, notably tho survey foud in several existing be (Ret work swept The oarly on by creen 5). one intoractionswas surveyed,along with other shock irterforence flows, by Korkegi (Ret 6) and mo{e recently by Peake and Tobal (Ref. 7). Othor portinentreviewsincludethoseof Delery and Marvitr (Rel 8) and Stolery (Ref. 9). 33 Initial Assumptions It shouldbe h€lpftl at the outse!to identiry some underlying characteristics,assumptioN, and h'?otheses conceming swept interactions. For example,it is traditioDally assumedthat these interactioDsdependprimarily upon thg freestream Mach number, the chamcteristicsof the incoming boundarylayer, and tho shockgenemtorgeometry (Rof. 5). However,thesedependerciescaitrot be determined by any closed-form solution of the governiDg equationsof motionon accoud of tle flow complexity.Experiments,similarityard dimensional reasoning and (within tho last 10 years) computatioDal simulationshavethusplayeda keyrole in the studyof sweptinteractions.The computatioDs, dealt with elsewherei! tbis report by D. D. Kaight, haverecendyreachedthe level at which they can of swept contributeto the fundamcntalundcrstanding interactions in ways not readily amenable to alsotrowservethe role experiments.The experinrents of providiry bencbmarksfor the validation of the computatiotrs.To this end a shock/boundary-layer interactiondatabase(Refs.10 & 11) is available. In the interactionstructuretlere is a hierarchy of infoimation that is also present ir the existiq exporimentaldata: Swepl interactionsare rich in patterns, and the most easily obtained and most prevalentdata are tlose of interaction "footpdnf' patternson solid surfacesobtainedby rtay of surface flow visualizatiotrmethod$ Moatr surfacepressure dishibutioDswere also talen in most of tle past expedments, anda fewexpeimeDtsalsoincludedwall shearstressandheattransferdata. Dolling el al have ill unsteadysurfacep]essuremeasurements specialized h theseflows. otr thc swface,flovideldvisualiation

tecbniquesand Fobe survels of mear pitot pressures ard flow angles have helped to map flo\\field slructures dd to provido a comection with the footprfut data. The highest level of the measuremeDt [ierarchy i&ludes d]namic data via hot-wiro, LDv, ard other quantitative methods. Very few of such measurcments have beea made h 3_D shock boundary-Iayerinteracttons. Much of the contert of this paper dedves ftom a parametric explontory approach to tle swept interaction probled. Since the problem is complex,it is a necessaryfust steP to ex?lore the rarye of possible flo\\6 by parametric studies. Only ttren catr detailed itrvestigationsof particular interactions be hterrelated h an overa.llftamework of urderstardirg. However, insofar as these ideractions are only partially determidstic due to their dopendonceupon turbulent traffport phetromela, the parametric explomtory approach is trot expected to lead to complet€ understandiog. Indeed, most of the ktrowledge gained thus far is dhectly related to the exent to which swept iDteractions depetd upon inviscid Dhenomena.

Fig. 2 " SphericalPolar coordinateSystem. An ossential first step ill understanding swept interactions is the choice of a proper coordinate systcm. Experiencedictates that it is easyto go astray with this problom if a poor choice is made. Many of the past sweptinteraction studies have revealedsomo degree of either quasiconicalor quasicylindricalflow stmmetry In principle, sphorical polar coordinatesare required for thc former (see Fig. 2) and normal-taagentialcoordinates for the latter, although experimeDtaldata are sometimes available only ir a streanwise-spalwise coord;nate frame. ln practiceit is often reasonableto male the approximation of au ortlogonal normaltangential frame attached to the interaction sweepline,which is usually the "footprint trace" or projection of the outer ("inviscid") shock waveupon tie interaction test surface. Fo owing the tradition of both swept aDd unswept jnteractiotr studies, lbls footpint trac€ of the hviscid shock is taler as rhe DroDer reference line ftom which to

measurethe interactiono'lent. Due to the limited scope of the presentpaper, discussedabove, the spherical polat cooid;aate frame of Fig. 2 tuns out to be the olly oue presentlyrequired. Its applicatiotr to sharp fi]1hteractions will be discusseditr detail bolow Next, a firm conrcctron is assumedbetween the swept hteractiotr footpdat topography and tle features of tle flowfield above tle surface. This hypothesis is supporled by recent experimeolal results to be describedin the section on flo$field sEucture, and by the applicatiotr of topolog/ to the sorface flow patterDs (Rel 12). Moreover, the topological rules (Ret 12) governingtheseflow pattcrns are recogdzed as impoltant aids to the utrdcrstanding of swept interactions. Finally, this paper coosiders only the mean-flow behavior of swept interactions. It appears that all interactions of shock waves with turbulent boundary-layersinvolvc somedegreeof unsteadiress, t ?ically manilestodas a "tremblhg" motion observed in flowfield visualizations. Howevor, there is no indicatiotr that such unsteadinessis the co rclling phetrometron in any overall sensg. A defmite, identfiable mean-flow structure is fouad in aI interaction casesconsidoredhere. It is thus assumed that the prsponderanceof mean-flow data on swgpt interactions is useful and pertident despito the prgsenceof unsteadrness. 3.4 lriteraction Classifi catrons A brief oveMow of the difforent tlpes of swept interactions is called for at the outset. Swgpt interactions \rrll be discussed in torms of the classificatioDsshown in Fig. 3 (from Rel 1), which result from elementary dimensional analysis. In all casesthe incoming flow provides a single length scale associatodwith the bourdarylayer (here taken as the incoming thrckness 60 for simplicity). when the overall dimeDsionsof the shock gon€mtor are large enough that further increasosifl these diiaensions do not change the interaction propertics, the resulting interactions arc classifiedas "serni-infinite." Further, cases witlin this classification in which the shock generator imposes no length dimension on the interaction are termed "dihellsionless." In such "dimensionloss"interactions it may be expectod fronl elementary dimensional analysis that the flow will respond to the imposed incoming boundarylayer length sc-alewith a single bala:rcinglength scale of its owh (later to be idontfied as the interactiod illception length). However, if a semi-h6nite shock generator does impose upon the interaction a length dimension comparableto 60(such as a fin leadi:rg-edgethickness D or a stop height H), the resulting hteraction is then temed "dimensioml." Herq D/60 or H/do js e4)ected to characterize the flow in the immediate

4

vicioity of the imposed lergth dimension. Such interactions are discussedin Rel 1 but are beyond the scope of the Fesent PaPer.

@"R l/\) ,\-/r,2/

Fig. 3 - Interaction Classifications. Finally, casesin which the shock genorator fails to produce a semi-infirite interaction as defiled above are termod non-sgmliDfinite or "protuberance" 'fhese cascs typically irvolve shock ;ntcractions. genorators lvhose ovorall dimensionsare comparable in to 60. Protuberanceinteractionsare also discussed Rel 1 and a'Ie also beyotrd the scope of thc present paper. 3.5 Goals Taking the sharp fiII hteraction as an archetlT€ of swept interactrois in general, the present paper has three goals: 1) Survey the current knowledgeof the scaling laws which govcrn the shape, size, and behavior of the'footprint" of a swept interachons in t€rms of M- , Re, o, and 6, 2) Exanine in detail the intoraction flowfield structure which is rcsporsible for the obs€reedfootp nt behavior, and 3) Describe tle experimental methods, both modem and tiie-honored, which have been applied to learD much of what is trow ktrowr about swept interactrons.

4. SCALING I-AWS The unswept sharp fin interaction of Figs. 1-3 is, at least h the indscid sensq the simplest of all swept interactions. This simpliciry derivesfrom tho fact that the propsrdes of tle shock wave pdor to its interaction with the bolndarylaycr (tle "inviscid shock") are known inmediately from classical oblique-shock theory. As we will see,the sweptintenction necessarily leads to fundamental change,sof structure id both the shock wave and the boundarylayer, whercupon the flow may !o longer be regarded as "simple." As in all tlows where direct solutio$ of the goveming equationsare not possible,dimensional and similarity mothods are powerfirl tools ir uDderstanding ${,ept interactions. For dimensionlesssharp-filinteractioDs only a single paraneter, o, is rcquired to describe the shockgenerator. The other parameters of the problem aro entirely concemed with tle incoming floi,r', and indude M-, Re, arld 60. All are dimensionless parametersexcept 60,which is the only length dimension in the boundary conditions of the problem. Thus, as mentioned earlier, the intoraction is expected to respond to 60 by way of somo charactedstic length, brt otherwise to bo entirely dimensionless. Thus, despitc the noDlitrearity of the governing equations and a host of rolated complications, there is nonethelesshope from the outset of achieving some overall similarity ftamework for tho behavior of swept rntoractions. That ftamework is the slbject this sectionof the paper, whcrc the mean-flow s)'Inmetry and footpri.ot structue of the interaction are explored and the inlluence of M6, Re, a, and 60are examined, 4.1QuasiconicalSlrnmeary All previous invest8ators of sharp fin interactions havs found that the edent of the interaction grows with distance away from the fm leading edge. Many of these investigators (e8 Refs. 13-23) firrther obse ed that this growth appeared to be conical, or nearlyso, exceptfor an hitial region in the immediate vicinity of the juncture of thc fin leading-edgeand the flal plate. This observation is co.Irmcd by recent paramotdcstudies (Refs. 24-28) of sharp fin interactions over broad rangos of both Mach number and fin aryle. It is now ctear that tho salient characteristic of thii intcraction is its 4lasicohical a tr$e. The dimensiomt description given earlier describes this class of swept htemctions as having no leogth dimensionexcept for one characteristic length which arisesin responseto the initial condition imposed by 60. l-et this length be caledla, the inception length measuredoutward from the fin leading-edgealong the tle observed shockwavedirection, which encompasses hteraction (seeFig. 4). initid noncodcal region of the Thus !/66 is a natural nondimensional parameter describingthe class of interactions under study.

5

The supersonic or hypersonic potc'tial flow outside the interaction is well-known to exhibit co cal behavior (Ref. 29), wherein the velocity vector is constant along rays emanating from a common odgin, the collcal vertex. Further, aftor Lightlill (Rol 30) such a conical field will occur "ir atry flow where the boundary cotuhtions define naturaly no linear dinetrsion.'Tbe presentflow, outs;delbe incepdonzone.is very close to satis&iry this codition except for tie issue of the ooo-conical go\fi-h of tie incomi-ng boundary-layer \dth spanwise distance along the interaction sweer line.

dimensional.

Much is already knol*'rl about the

behavior of conical 2-D flows, which are thoroughly revie$ed, for example,in Ref. 33. Briefly, the proper coordinate ftame for a conjcal flow is the spherical polar coordinate s)stem ir r, F, and C, sketchod itr Fig.2. Shce r is degoneratein true conical flow, the ilteractioD outside the inc€ption zone is described entirely h angular measureby p and d. Sheamlines and otler featur€s ol tle flow may be seen in tv,/o dimensiols by projectlon ftom tle vertex of the conical flow onto the surfaco of the sphere. Since experierce has sho\ln that such projections of swept interactions normally subtetrd small solid angles, the replacemeft of tbe appropriatespherical segmetrtwith a plane tatrgent to the sphere and normal to the inv;scid shock wave is a frequent simplfication. The pertinent Mach number describing the interaction is the compone ol M@ in this plane, namely M", rather than M& itself. This coordhate frame is adopted uniformly in the following discussion. Other thatr Lt and 60, all "dimetrsions"of swept interactions are discussedoDIy ir angular measure. mile it may be tempting to deline length dimonsions associatedwith, say, the upstream inlluenco of the interaction, er?erience shows this to be bighly misleading. For example,early investigators of this flow remarked pon tho streamwise lengtl of ths intcraction comparedto that bf a 2-D planar interactionof similarshockstrength.However,in the p(csentcontcxtof quasiconical flow this comparison is mcaningloss,since the stroamwise length of thc interaction can be as large or small as one wishes, dependingupotr the distanco,r, which ono takos ftom the conical vorter.

Fig. 4 - Sketchofsharp FiDInteractionFootprintwith NomenclatureDefinitions. An important contribulionby lrycr (Ret 31) largely resolvedthis problemthroughan order-of-magnitude analysisof the govcrniDgequations. IDger found that a swopt hteraction can approach a quasiconicalstatc at a large drstance from the fin leading-edge- This distance,itr fact, turns oul 1o be the inception length, Li, and how large it is dopeds upon 60. Inger also lound that L,/6. : rotBo.iD agreemrnlwittr experi mental dala for sharp fin interactions. Wlile some effect of the spanwise boundary-layer growth is probably presed, it is a second-ordereffect compared to the overall quasiconicalnature of the flow (Ret

The quasiconic2l symmetry of this class of swept interactions is now regarded as the most powertul simplification availablc itr a probleE of otheNise datnting complexity. As we will see, it allows a 3-D flow to be treated, in ma.tryrespects,as if it were two-

ppk*

or=45uu -r :95MM

ii I ?.r*-

i

I

pP nl9.a

60

40 o.

p

20

Fig. 5 - Surface ?Iessule Distributions (per Ret 16).

t

As a demonstratiotr of the codcal Mture of sharp-6n iateractions, surfa@-prGsurc data from a Mach 3 interactrod by Zubin and Ostapenko (Ref. 16) are shorux in Fig. 5. These data were measuredotr two concedric circular arcs on a flat plate at different radii. r. from the vertex of tle flow. Both moasulement arcs were outside tle inception zone. The data show excelle €reement when the prossue ratio is plotted in terms of the codical angle p. Such agreemenl cannot be bad. bowever. iI the lbear dimetrsiotr describing tho plessure-tap layout is used as ttre abscissaof the plot. Fiflally, note that quasiconicalinteraction symmetryis an appro)omate framework of utrdorstandingrather than a prccise law. It breaks down noar the fiD leading edge i:r the inception zone, atrd also along the intersectio! of the fin and the flat plate (which cannol lie on a generating ray of the conical flow). It is subject to the second-order3-D iDfluetrc€of the spatrwise go\ltl of 60, mentioned earlier. It is a mearllow ftamework for the description of intelactions which are kno\rb to have some time-dopendent characteristics, In eKromely weak interactions(Rgfs. 34 and 3t the conical spreadrngof the flow may be so small as to be undctectablo, In oxtremely strong interactions the approximation may once again break down, though experimental limitations prevent such a casefrom beiog observed. Between these two limits, and subject to the qualifications listed, a[ known dimensionless fin interactions exhibit quasiconical symmotry. 42 Boundary"Layer Separation A.ll shock/bourdarylayor interactions,whether2D or 3.D, involve the separation of the boundary-layer when the shock wave is of sufficient strengtb to briDg this about. Dcspite some controversyovor the semandcs of the word "separation,"the literature cited thus far leaves no room for doubt that it is a distinct and recognizablephenomenonin swept interactions. Duo to the nature of such flows, when separation does occurs it is ecessarilythree'dimensional and higbly swept with respect to the oDcomingflow. A proper treatment of the topological criteria for 3-D flow separation is well beyond the presentscope. The rcader is directed to Ret 12 for a thorough coverage of thjs topic. Bdefly, the present discussionaccepLs the I-€gendre-Lighthill view of 3-D separation as the cotrvergenceof limiting streamlines of the flow upon a particular (swept) streamline which comects singular poirts of separation located, in thjs case, somewLer€ oD the flat plate. The exact natrue of thoso singular poirts is a secondary issue for presert purposes. What js importallt is the observation(€g Rei 26) that the Doticoable offects of soparation develop gradually in sharp fin interactions, and that occunoDce of separation is clearly a furction of Mach number and shock wave strendh.

a

Fig 6 - Stagesiix the Developmentof tho Intoractiou el al.). Footprint(perzhehovodov Fig. 6, fton Ref. 26, shows8 stagesitr the development of the lisiting strcamlircs of the iDteraction footprinton the flat plate as a fuDctionof iDcreasing shockwavestrength, SuchpatternsaJeobtaircd by surfac!-flow visualizatioDtechniques,and will be further h later sections.For tle moment, discussgd ono seesin Fig. 6a the caseof a weak,urseparated interactior rll which the limiting streamlinesveer uderneath the inviscrdshockv,/ave but do not form a cotrvergence line. With increasingshockstrength,Fig. 6b,the separationline movesoutwardand eventually lies underneaththe inviscid shock, \rith limiting streamli.oes on eithersiderunnirg essontiallyparallel to it. (Statrbrook(Ref. 36) defined this condition ratherarbiharily as incipient sweptseparationof the laye..) A further iDcrcascof shockstrength, bouDdary Fig. 6c,leadsto aslmptoticconvorgence of the limiting streamlhes upon the line of 3-D separation. Finally,in Fig. 6d,a sufficiendy-sFong shockproduces an erdicit convergenceline, Sr, which lies wel outboardof lhe inviscidshockposition. Thrs gradual dov€lopment of the flow points out the disparity betwee! the stfict topologicrl defnition of 3-D separatioD,for which the present flow is always separated (Ref. 37), and more practic.al definitions which fldd no evideDceof lilt-ofi of the boundaryJayer Iron the fiat plate u$il the condition in Fig. 6d G reached. Should tlere be a.Dydoubt on this point, Fig. 7 showsthe kerosene-lampblackfootprint trace of a weal fitr interaction (Mach 2, a = 6"), in which the

7

pdmary separation line lies well i$ide the zone bounded by tLe fi.o and tle irviscid shock. (This pattem was obtained by placiry sDrface-lracermaterial on the flat plate only upstream of the Et prior to the e)perimeut.) Sirce a line of 3-D separationb cledly presenti! Fig. 7. Slanbrooks crileriotr (Ret 36) must be strictly inconect. lt serves.oonettreless, to iadicate the approximate conditionwherc sizableor signficant 3-D separation occuls,insofar as effects on the outer florvfield are concerned. More will be said about thjs b9low.

15

ro ci' oeg _ 5

\. tr\ \^

Korkeqi .

Mccabe O

Mccobe -

pcokc O Oskom o

zuui" * o"r.p""i.i zh"tro'od; --".'*i;

*.--".,*-fua_=\-

\

\

^

i

J

Fig. 8 - IncipientSeparationCriterionfor Sharp-Fitr Interactions.

Fig. 7 - Footprht Trace of a Weak SeparatedFin Interaction. put forwardby Mccabe, (Rel Basedon aJg:uments 38) Korkegi (Refs. 39 and 40) proposeda practical criterion for iacipientsweptseparationin the form: M.a, - 0.3

o)

for M@ > 1.6 and 1 = 1.4 whore ar is moasuredin radians. Lu (Rel 41) rocendy re-derived this result on somewhat firmer ground. As the most v,/ell-knowll and popular criterion for incipient swept separation, Eqn. (1) is illustratedhere in Fig. 8. It corresponds ro thc flow cordition sho\vnin Fig.6b and delermines the condition in which the limiting streamlines run parallel to the inviscid shock wavc, not true incipie separation, a^swas already mentioned in connection with Fig. 7 (se€ also Rers.21,37,and 42). It may be seetr from Fig. 8 that lhe practical cotrdition of iDcipient swept separation not only occurs for woaker shocks thao does incipient 2-D separation at the same M@, but also happensevenmore reailily as Mo incroases. Early on, the lack of a physical er?lanation for this was tho source of somo consternation and disb€liei Howev€r, the physical explanation is not alif{icult to urderstand if one acceptsthat it makes no sonseto compare 2-D al1d3-D sepaiation at the same value of M-. Accoding to the above quasiconical syrnmetry discussioA it is the rolrr1ol Mach number, lvl, which is significart in determining tte properties of a swept interaction, rather tlan M6.

Whetrcomparedon thisbasis,2.D andsweptilcipiedt sopa.rationcliteria are in close agregment. For example, in broadterms,incipiontturbulentboundary layerseparationis found at normal Mach Dtmbers beMeen1.2 and 1.3 in a wide varietyof swept and hteractioDs(Refs.26, unsweptshock/boundary-layer pres41,and 43). The conespondinginviscid-shock sureratios, (pr/pr)r, in air are bet*een 1.5 and 1.8. Thus the seemingaromaly of incipient separation morelikelywith increasilgMachnuqber is becomi:rg in termsof the highors\eoepback allgloof exTlainable betweon2-D and the shockwavg. No discrepancy sweptsoparationcritoria occurs,so long as one pays attentionto the proper coordinatoframo. The recentwork of Zheltovodovel 41. (Refs.26 and 43) has devolopedthis approachconsiderably. In re-workingthe classicalfree-irteractio[ analysisof Chapmanet al. (Ref.44),theyobtainedthe following for the "plateau'pressurein a separated expression interaction: pr- p--

KMn:@rn: - D-u4crlt'+ |

(2)

where1 = 1.4 is assumed,K = 5.94for turbulent atrd 1.216for lamhar flows, and M4 is the Mach numbernormal to the separationline. This expression was verfied by comparisonwith experimental datafor 2-D turbulentinteractions.It embodiesthe pressure-rise requiredto produceincipientseparation, if ore acceptsthis to bc cqualto the plateaupres$ue riseof a well-separated ifteraction. WLencombined with th€ classicalexpressionfor oblique-shockpressure ratio ir terms of M", Eqn. 2 can be used to predict incipient swept separation as well. Zheltovodovet al. fov)d the result to be itr good agreement with the data for whatthoy call incipie separation,ie tie conditionrepresented "sma.l1-scale" by Fig. 6b wheresurfacestreamlinesrun parallel to

a

the shock but troticeablelift-off of the boundarylaver from the flat plate has yet to develop. These authors then proposed an additional new criterion for incipient "large-scale"separation. Plots of the angle between the upstream-influeDceand primary separation lines of the interactior show &at it decreases with i&reasing interaction strength, eveatually tenditrg toward a constantvalue. Otrcetlis constancyoccurs, it 1sargued that the iDteractionhas reachod a condition (frg. 6d) where the sweptseparatiotr is frrlly developedalrd whete its gross effects aro clearly manifested in the interaction flo*'fielo. Finally, Zheltovodov e/ 4/. used the above quasi-2-D freo-hteraction analysisto demonstrate that (p,/pr)i tends to decrease ftom 1.7 to 1.5 as Re, idcreases ftom 1d to 5x10," as illustratcd ir Fig.9, Vaiations in the shape parametor of the incomiry boulrdarylayers itr the experimental data points shown (numbered srmbols 4-9) further required that K in Eqtr. 2 be varied over tho range 5.9 to 9.4 producing the band shown in the Figure. This view of hcipiert separation is clearly an improvcment over Eqn. 1, Korkegi's critcrion, which embodies the simple implicit assumptionthat (pr/pJt = 1.5, and thus ignores the Reynoldsnumber and shapefactor depenof the incoming dcnceinherentio thc characteristics boundarylaycr. Korkogi'scriterion is also shownin Fig. 9 for comparison.

2.6

ffiZ-D

.-4 ^-6 x-8 r-5 +-7 t-9

t.5

1

1.'

4 log Reu

4.'

Fig. 9 - Incipient SeparationCriteria vs. Rep (per zhelto\odo\ et al.). 43 Th€ Intemction Footprint 43.1 Li mi ting Sueamline Pattet n Returning to Fig. 4, the 'footprht" of a sharp-fingetrerated swept interaction, ie its pattem of limiting streamliacs on the flat plate, is ilustratod with defini tions of nomonclature. Four importa features are identfied by their conical angles: the upstream influetrce line U, tbe p! |mary separalionline S,. tbe primary attachmefi lile A! aad the secotrdarysepaiation line Sr. The upstream influence lhe, being the forwardmost o'1ent of the interaclon, has be€n the

subject of various attempts to relate the respoDseof the interaction footprint to charyes in Mach number, Reynolds number, ard fir angle (Refs. 1, t5, n-22, '%, n, 34, 45, and a6). Sinilar but Eore limited attempts have been made to correlate t[e positioDsof the pdmary separationand attachmentlines. Pdmary attachment, which tust appearsin tle shock-sbenet! proglession in Fig. 6d, marks the position where tho flow, haviag left the surface of the flat plate at the pdmary separation line, reattach€s to the plate (though the stream-surfacewhich separatesis Dot tlie orc which reattaches). Betwe€tr those two lines lies a region of reverse flow in the conical projection, which also has a strong sparwise (r-direction) component. This leads to a helical separation vortex. One may view this vo ex as a reorganization of tho spanwiso vorticity of the incoming boundaryla]€r. Zheltovodo! et al. (Rel 26) not€d an inllection in the primary separationline which they ascribedto tia$itiod from lamhar to turbulent reverse-flowwithitr the separatedregion. The aryular differ€ncepu - Bsr betweenthe upstream influence and primary separation linos is sizable for weak interactions, but shrinks quickly to a small value as the interaction gows stronger. Tho achiovementof this asymptoticvalue (Refs. 26 and 43) has been usod as an indicator of incipiotrt "large-scale"flow separation, as noted earlier. In fact, the reglon betveen these two defining lines of the interaction has been found to be highly unsteady. Since thc current discussiotrcolcorns only mean-flow properties, this unsteaditressis corered elsewherein this rcport by D. S. Dolling. If tho definition of primary separation has caused somc past confusioD,as noted oarlier, this is doubly the case for socoDdaryseparation. AccordiDg to zheltovodov e, a/. (Refs. 18,2-6),the footprint feature known as secondary separationfirst appearsoncethe interaction has achieveda certain strength (Fig. 6d), showing up in the conical region of the flow but not in tho inception zone. Its spanwse o.tent gows with increasingshockstrength (Fig. 6e) but then dininishes again (Fig. 60, eventually appearing only in the inception zone and then disappeadngaltogether (Fig. 69). Secondary separation then rcappears in the strongest interactions obsorvedto date (Fig. 6h), but in a differcnt position, noticeablycloser to the fir thaa previously. Zheltovodov et al. werc the first to observethis behavior, ard have investedconsidenble €ffort in trying to undorstand it. Their experiments with sand-grain rougbnessapplied to thc interaction region appear to demonstratethat the initial behavior of secondaryseparation is rclated to lamhar, tmnsitional, and tlen turbulent reverse-flow in the swept separation bubble. They also ascribe the reappearanceof secondaryseparation (Fig. 6h) to the developmod of supelsonicreverseflow in the separat-

ed region with ao inbedded lordal

shock wave. This

wastust reportodby Zubh andOstapeqlo(Ref.16), tiough Do clear imageof it wasshowD. Very clear imagesconfumingthis hypothesis haverecentlybeeD obtainedby Alvi and Settles(Refs.16and 47). Arotler, evenmore serious,problemwiih secotrdary separatiodjs that it usuallyappearswitLoutthevisible accompariEert of secondaryattacbmentl This topologic.ally'impossible situationis probablydueto a very small argle betweetrsecotrdaryseparatronand attachmeDt, suchthat theyappearasa singlefeatule. Zheltovodov(Rol 48) hasobtalredevidenceof both secolrdary sgparatiotrandattachmentin ar e]ffemelyshongfin-inducedturbuleDtboudary-layerhteraction at Mach 4 anda = 30.6".Thosudaceflow pattemof this ilteractioD,shownrn Fig. 10,wastraceddiectly ftom ao enlargemontof an oil-flow photograph prolided by Zheltovodov,siDcethe photoitselfwould rct be likely to reproducewell enoughto showthe featuresdescribed. Nonetheless, distirct secoddary separatio!and secondary attachmentlinesare clearly visiblewith al] angleof 1 or 2 degreesbetweenthom. This evidenceconlirms,itr the ophion of the present author,lhat secondary separationactuallycaDoccurin jnteractions, give! the proper circumthis classof stances.A local maximumappearsat this secondary separation/attaclhent location itr measuredskinfriction distributions Refs. 49 and 50). A local flowfield disturbancealso appearsin the vicinity of this location in cooicalshadowgrams aDdintgrferogramsof the flo$field (Refs,24. 25,aad 47).

Flg. 4 illustrates how the co"ical lines of the interaction footprint all emanate&om a conmoD vertex, the virtual cooical origin or VCO. This origin, which is the center of the sphedcalcoordinate frame showtr in Fig. 2, lies somewhatahead of the actual fir leadingedge due to the presence of t-he iiceptio! regioD, whid is also itrdi(ated in Fig. 4 by irs length, L. zheltovodovel al. (Ref. 26) proposedcorelations for the conical angles of upstream inlluence f" and pdnary separation p.r iD tems of the invjscid shock arglep0. Itrcipient "sma[-scde' separatioD,as defined in the previous section (Fig. 9), is used as a basis for thosecorrplations,and is dmoted by an asterisk. The correlations are: 0,_0J*_lj3(0o_pot

ln -,n

- 0o+F (4) =zr5(1o- 00, - 0or44(po

one confirsingfactor is that two forms of the B,l conelatiotr wore given, one for laminar and the secondfor tubulent reverse-flow,Siice it maydot bo €asyto detelmingwhicbform to use,it is suggested herothat supersodcinteractioDs with freestrea$udt Relnoldsmrmberson the ordcr of 10'/m or higher useoDlythe turbulent forn, which is the one given aboveasEqn.4. The corrolations ofEqns.3 and 4 pointsobtainedat arebasoduponmanyexperimental Novosibirsk irl thoMach2-4nnge. Howover,Sdnissour (Rel 51) also comparedthem with his data, obtainedat Mach5, aDdfoundgoodagreoment. 432 SurlacePrcssureDistribulion havemeasurodthe sudacepresMany investigators sure beneaththe footprirt of the interaction. Its classicalshape,measureditr the proper crossflow coordinate framcfor awell-separated interact;on, was atreadyshownin Fig. 5. The peak pressure,just aheadof the 6n, is of interestbocauseit overshoots the maximum pressrue,p, expectedbehind the inviscidshockwave.Severalattemptshavebeenmade to derive expressionsfor the peak pressue ratio, p,/p6. For exanple,accordingto Hayes(Ret 52):

tab- = M,lt

Fig. 10- Tracingof SurfacePatternof Zheltovodofs Mach 4 d = 30.6' Interaction.

(t

Scuded(Ref. 53), Zheltovodov(Ref. 18), and Lu (Ret 27)proposedsimilarexpressiobs. Mostrecently, Lu (Rel 54) hasreexamircdthis issuein the light of curIentkllowledge'to be djscussed below,about the detailedstluctuloof the 6n interactionflowfield. He arrivedat a physicallymore meaniogful,ihough less cxplicil approachto predictingpp/po. His summary graphis reproducedhorc as Fig. 11,which showsa comparison of aBilable dataatrdgediction methods for this ouartiw.

l(.

nature of the fin illeracliotr was generally rccognized, so thcy cannotbe taultedtor overlooki-og this.

' z u b i n& o s r a p e n k o( 1 9 7 s ) . Lo{ (1s7s)

In light of the present knowledge of quasiconical interaction s,'mmetry I€e er al. (Ret 5? assuaod tiat q aslmptotcs to a constart value along a conicnl ray outside the intoraction irception zoae, as was sho$1 experiDent.tly by Rodi and Dolling (Rer 58). This embled them to propose a Duch-simpler data correlation in terms of q..o/co- vs. M" o y. This corrolatiotr i6 demoDstratedin the gaph of Fig. 12. The data of Lee (Refs. 57 atrd 59) are shonn by scilid srnbols, while other available data (cited by l,eo e, al. in Ref.5?) are 6hown by open slmbols. Talen togethor, these data approximatcly describe a linear relationshipwitlin their overall scattcr. The equation of tho line sbown i! Fig. 12 is:

o Neuno.n & Burkc (1969)

-

zheltovodov (1s32)

-- L'r0s33)

2.O

2.5

Fig. 11 - InteractionPeal PressureRatio vs Normal Mach Number(per Lu). zheltovodovet 41. (Rel 5t also proposeda new for the peakpressureratio basedon their expressiotr quasi-2-Dfree interactionanalysis: palp. = 1.3@2lpt)- o.3(pzlPt\

(6)

This relationshipwasproposedby Lge el al. as a sinple enpirical guidefor poa-kheatingin sharp-fingeneratod interactions$,ith turbulentboundarylaye$ oulboad ol the inceptionzonenearthe li4 leading edge. No suchsimple relationshipis possible insidethe hceptionzone. 10

They devolopeda mothodto predict not oDlyppbbut also the cntire shape of the hteraction-footprint pressuro distribution. 433 Heat Trantlet Distribution Tho peakheattransferratein the interactionfootprint occursat tle samglocationas Pek,andis oftenlinked to it by simple calculationschemes. For example, Holden (Ref. 13) correlated the resultsof soveral heattransfer experimentswith: swept-interaction crnlcr. = @6lp)oss

(8)

CUICh_- 3.1M,-2J

6