Topology of 3D Separated Flow

7/27/2019 Topology of 3D Separated Flow

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- N qSA echnicai Memorandum 81294

Topology of Three DimensionalSeparated Flowsurray Tobak and David J. Peake

(NASA-TM-81294) TOPOLOGY iE N81-23037T H R E E D I E E N S I C N A L S E P A R A T E D YLOW S (NASA)b p HC A03/HF A01 C S C L 018 nclas

G3, 02 42 32 3

April 1981


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NASA Technical Memorandum 8 294

Topology of Three DimensionalSeparated FlowsMurray TobakDavid J. Peake, Ames Research Center, M o f f e t t Field. Cal i fornia

ICtP . '.h > I t


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CONTEPU TSPage

INTRODUCTION..~ E O R Y

ingular Pointsopography of Skin-Friction Lines 6

Forms of Dividing Surfaces 12Topography of Streamlines in Two-Dimensional Sections

f Three-Dimensional Flows 13

Topological Structure, Structural Stability, andifurcation 16

E X A M P L E S . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22Round-Nose Body of Revolution at Angle of Attack 22Asymmetric Vortex Breakdown on Slender Wing 25

SUNNARY 27Acknowledgments 28Literature Cited 29

iii


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INTRODUCTION

Three-dimensional 3D) separated flow represents a domain of fluidmechanics of great practical interest that is, as yet, beyond the reachof definitive theoretical analysis or numerical computation. At present,our understanding of 3D flow separation rests principally on observationsdrawn from experimental studies utilizing flow visualization techniques.Particularly useful in this regard has been the oil-streak technique formaking visible the patterns of skin-friction lines on the surfaces ofwind-tunnel models (Naltby 1962). It is a common observation amongstudents of these patterns that a necessary condition for the occurrenceof flow separaticn is the convergence of oil-streak lines onto a particu-lar line. Whether this is also a sufficient condition is a matter ofcurrent debate. he requirement to make sense of these patterns withina governing hypothesis of sufficient precision to yield a convincingdescription of 3D flow separation has inspired the efforts of a number ofInvestigators. Of the numerous attempts, however, few of the contendingarguments lend themselves to a precise mathematical fom~~lation. ere,we shall single out for special attention the hypothesis proposed byLegendre (1956) as being one capable of providing a mathematical frame-work of considerable depth.

Legendre 1956) ~roposed hat a pattern of streamlines immediately adja-cent to the surface (in his tersinology, wall streamlines ) be considered astrajectories having properties consistent with those of a continuous vectorfield, the principal one being that through any regular (nonsingular) pointthere must pass one and only one trajectory. n the basis of this postulate,it follows thai rne dlemcntary singular points of the field can be categorized


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mathematically. Thus, t he types of si ng ul ar po int s, t he ir numher, and theru le s governing the r e l a t io ns between them can be sa i d to ch arac ter i ze thepa t t e rn . Flow sep ar ati on i n t h i s view has been defined by th e convergence ofwa l l s t reaml ines on to a p a r t i c u l a r wall s t reaml ine tha t o r ig ina t e s f rom as i n g u l ~ r o i nt of p a r t i c u l a r t yp e, t h e s ad d le p o in t . W should note, however,t h a t th is view of f low sep ara t ion s not l in iver sa l ly accepted, and, indeed,s i t u a t i o n s e x i s t where t appears t ha t a more nuanced de sc r i pt i on of f lowsep arat ion may be req uired.

L ig h th i l l (1963), address ing h imsel f sp ec i f i c a l l y to v i scous f lows ,c l ar i f ie d a n ,, .sber of important i ss ues by ty ing t he pos tul a te of a con-

t i nu o u s v e c t o r f i e l d t o t h e p a t t e r n of s k i n - f r i c t i o n l i n e s r a t h e r th a nto s t reaml ines ly ing jus t above the sur face , * P a r a l l e l t o Legendre 'sde f in i t io n , convergence of sk i n - f r i c t ion l i ne s on to a p a r t i c u l a r s k in -f r i c t i on l i ne o r ig in a t in g from a sadd le po in t was de f ined he re as t h enecessary condi t ion f o r flow se pa ra tio n. More rec ent ly, Hunt e t a (1978)have shown that t h e not ion s of e lementary s ing ul ar point s and the rulesth at they obey can be ea s i l y extended t o apply t o th e f low above thesu rf ac e on planes of symmetry, on pr oj ect io ns of co ni ca l flows (Smith1969) , on crossf low planes, e t c (see a l so Perry F ai r l ie 1974) . Fur therapp l ica t ion s and extensions can be found i n the var ious cont r ib ut io nsof Lcgendre (1965, 1972, 1977) and i n th e review ar t i c l e s by TobakPeake (1979) and Peake Tobak (198 0).

s Legendre (1977) himself h as not ed, h i s hypo the sis was but are invent ion wi thin a narrower framework of the ext rao rdi nar i ly f r u i t f u ll i n e of research in i t ia te d by Poincart (1928) under the t i t l e , "On t h eCuntes Defined by Di f fe ren t i a l Equat ions ." Yet anoth er branch of th e


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s me line has been the research begun by Andronov and his colleagues(1971, 1973) on the qualitative theory of differential. quations, withinwhich the useful notions of t0p0108i~al structure and structuralstability1' ere introduced. Finally, from the same line stems therapidly expanding field known as bifurcation theory (cf the comprehen-sive review of Sattinger 1980). Applications to hydrodynamics are exem-plified by the works of Joseph (1976) and Benjamin (1978). It has becomeclear that our understanding of 3D separated f l w may be deepened byplacing Legendre's hypothesis within a framework broad enough to includethe notions of topological structure, structural stability, and bifurca-tion. Bearing in mind that we still await a convincing description of3D flow separation, we may ask whether the broader framework will facili-tate the emergence of such a description. In the following, we shalltry to answer this question, limiting our attention to 3D viscous flowsthat are steady in the mean.

TH ORY

We consider steady viscous flow over a smooth three-dimensional body.The postulate that the skin-friction lines on the surface of the bodyform a continuous vector field is translated mathematically as follows:Let [ , n , c ) be general curvilinear coordinates with C,n) being orthogo-nal axes in the surface and directed out of the surface n o m l to

Let the length parameters be h ( ) , h ) . Except at singul-irpoints, it follows from the adherence condition that, very close to the sur-face, the components of the velocity vector parallel to the surface (ul ui)must grow from zero linearly with c Hence, a particle on a streamline ncor


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the surface will have velocity components of the form

where wl,w2) are the components of the surface vorticity vector. Thespecification of a steady flow is reflected by ul,uZ) being independentof time. With treated as a parameter and P and Q functions onlyof the coordinates, equations 1) are a pair of autonomous ordinarydifferential equations. Their form places them in the same categorv sthe equations studied by Poinear4 1928) in his classical investigationof the curves defined by differential equations. Letting

be components of the skin friction parallel to and q respectively,we have for the equation governing the trajectories of the surface shearstress vector, from equations 1 ) .

Alternatively, for the trajectories of the surface vorticity vector, whichare orthogonal to those of the surface shear stress vector, the governingequation is


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Singular Points

Singular points in the pattern of skin-friction lines occur at isolatedpoints on the surface where the skin friction T ~ ~ , T ~in equation 3 ) ,or alternatively the surface vorticity w 1 , w 2 ) in equation 4 ) . becomesidentically zero. Singular points are classifiable into two main types:nodes and saddle points. Nodes may be further subdivided into two sub-classes: nodal points and foci of attachment or separation).

../A nodal point Figure la) is the point common O an infinite numberof skin-friction lines. At the point, all of the skin-friction linesexcept one labeled A-A in Figure la) are tangential to a single lineBB. At a nodal point of attachment, a11 of the skin-friction lines aredirected outward w y frorr; the node. .it a nodal point of separation, allof the skin-friction lines are directed inward toward the node.

A focus Figure lb) differs from a nodal point on Figure la in thatit has no conmon tangent line. An infinite number of skin-friction linesspiral around the singulai point, either away from it a focus o attach-ment) or into it a focus of separation). Foci of attachment generallyoccur in the presence of rotation, either of the flow r of the surface,and will not figure in this study.

At saddle point Figure lc), there are only two particular lines,CC and DD, that pass through the singular point. The directions on eitherside of the singular point are inward on one particular line and outwardon the other particular line. All of the other skin-friction lines missthe singular point and take directions consistent with the directions ofthe adjacent particular lines. The particular lines act as barriers in


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the field of skin-friction lines, making one set of skin-friction linesinaccessible to an ad jaceat set.

For each of :he patterns in Figures la to lc, the surface vortexlines form a system of lines orthogonal at every point to the system ofskin-friction lines. Thus, it is always possible in principle to describethe flow in the vicinity of a singular point altcrnatively in terms of apattern of skin-friction lines or a pattern of surface vortex lines.

Davey 1961) and Lighthill 1963) have both noted that of all thepossible patterns of skin-friction lines on the surface of a body, onlythose are admissible whose singular points obey a topological rule: thenumber of nodes nodal points or foci or both) must exceed the number ofsaddle points by two. We shall demonstrate this rule and its recentextensions to the external flow field in a number of examples.

Topogriiphy of Skin-Friction Lines

The singular points, acting either in isolation or in combination, ful-fill certain characteristic functions that largely determine the distri-bution o skin-friction lines on the surface. The nods1 point of attach-ment is typically a stagnation point on a forward-facing surface, suchas the nose of a body, where the external flow from far upstrc.12 attachesitself to the surface. The nodal point of attachment thereby acts as asource of skin-friction lines that emerge from the point and spread outover the surface. Conversely, the nodal point of separation is typicallya point on a rearward-facing surface, and acts as a sink where the skin-friction lines that have circumscribed the body surface may vanish.


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7The saddle point acts typically to separate the skin-friction lines

issuing from adjacent nodes; for example, adjacent nodal points ofattachment. An example of this function is illustrated in Figure 2(Lighthill 1963). Skin-friction lines emerging from the nodal points ofattachment are prevented from crossing by the presence of a particularskin-friction line emerging from the saddle point. Lighthill (1963) haslabeled the particular line a line of separation and has ideniified theexistence of a saddle point from which the line emerges as the necessarycondition for flow separation. As Figure 2 indicates, skin-frictionlines from either side tend to converge on the line emerging from thesaddle point. Unfortunately, the convergence of skin-friction lines oneither side of a particular line occurs in other situations as well. Itcan happen, for exdmple, that one skin-friction line out of the infiniteset of lines emanating from a nodal point of attachment may ultimatelybecome a line on which others of the set converge. All researchers agreethat the existence of particular skin-friction line on which otherlines converge is a necessary condition for flow separation. The seemingnontiniquenrlss o the condition identifying the particular line hasencouraged the appearance of alternative descriptions of flow separationthat, in contrast to Lighthill's, do not insist on the presence of asaddle point s the origin of the li.~e. Wang (1976), in particular, hasargued that there arc two types of flow separation: open, in whichthe skin-friction line on which other lines converge does not emanatefrou saddle point, and closed, in which, s in Lighthill's definition,it docs (see also Wang 1974 an 6 Pare1 1979). In what follows, weshnll address the question of an appropriate description of flow separation


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by an appeal to th theory of structural stability and bifurcation. LikeWang, we shall find it necessary to distinguish between types of separa-tion, but we shall adopt a terminology that is suggested by the theoreti-cal framework. We shall say that a skin-friction line emerging from asaddle point is a ~l o b a l ine of separation and leads to global flowseparation. In the contrary case, where the skin-friction line on whichother lines converge does not originate from a saddle point, we shallidentify the line as being a local line of separation, leading to localflow separation. When no modifier is used, what is said will apply toeither case. Thus (in either case), an additional indicator of the lineof separation is the behavior of the surface vortex lines. In the vicin-ity of a line of separation the surface vortex lines become distorted,forming upstream-pointing loops with the peaks of the loops occurring onthe line of separation.

The converse of the line of separation is the line of attachment.Two lincs of attashrncn: are illustr-ted in Figure 2, emanating from eachof the nodal points of attachmcnt. Skin-friction lines tend to divergefrom lines of attaclmcnt. Just as with the line of separation, a grap5icIndicaLor of the presence of a line of attachment is the behavior of thesurface vortex lincs. Surface vortex lines fortn downstream-pointingloops in the vicinity of a line of attachmcnt, with the peaks of theloops occurring on the line of attachment.

Streamlines passing very close tc~ he surface, that is, those definedby equations (l', are called limiting streamlines. In the vicinity of alinc of scp~ration, imiting streamlines must leave ti.* surface rapidly,as a simple argument due to Lighthill (1963) explains. Referring to.


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equation 3 ) , let us align [,TI)ith external atreamline coordinates sothat T T are the respective streamwise and crossflow skin-friction1 2components. If n is the distance between two adjacent limiting stream-lines see Figure 3) and h is the height of a rectangular streamtubebeing assu~~edmall so that the local resultant velocity vectors are

coplanar and form a linear profile), then the mass flux through thestreamtube is

phnc-where p is the density and u the mean velocity of the cross section.But thc resultant skin friction a t the wall is the resultant of T

l

and T orw2

so that

Hence,

h nrw- - 2v constantyielding

Thus, as the line of separatinn is approached, h, the l~eight f thelimiting streamline above the surface, increases rapidly. There are tworeasons for this increase in h: first, whether the line of separationis global or local, the distance n between adjacent limiting streamlinesfalls rapidly as the limiting streamlines converge towards the line of


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1separation; second, the resultant skin-f iction w drops toward mini-mum as the line of separation is approached and, in the case of theglobal line of separation, actually approaches zero as the saddle pointis approached.

Limiting streamlines rising on either side of the line of separationare prevented from crossing by the presence of a stream-surface stemmingfrom the line of separation itself. The existence of such a stream-surfaceis characteristic of flow separation; how I originates determines whetherthe separation is of global or local form. In the former case, the pres-ence of a saddle point as the origin of the global line of separation

provides a mechanism for the creation of a new stream-surface that orig-inates at the wall. Emanating from a saddle point and terminating atnodal points of separation (either nodes or foci), the global line ofseparation traces a smooth curve on the wall which forms the base of thestream-surface, the streamlines of which have all entered the fluidthrough the saddle point. We shall call :his new stream-surface adividing surface. The dividing surface extends the function of the globalline of separation into the flow, acting as a barrier separating the setof liariring strea~ +nes thst have risen from the surface on one side ofthe global line of separation from the set arisen from the other side.n its passage downstream, the dividing surface rolls up to form thefamiliar coiled sheet around a central vortical core. Because it has awell-defined core, we shall invoke the popular tednclogy and call theflcw in the vicinity of the coiled-up dividing surface a vortex. Now weconsider the origin of the stream-surface characteristic of local flowseparation. We note that if a skin-friction line emanatirg from nodal


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poin t of a t tachment u l t imate l y becomes a l oc a l l i n e of sep ara t io n ,then there w i l l be a poi nt on th e l i n e beyond which each of th eor thogonal su r face vor tex l in es c ro ss in g th e l i n e forms an ups tream-p o i nt i n g lo op , s i g n i f y i n g t h a t t h e s k i n f r i c t i o n a lo n g t h e l i n e h asbecome l o c a l l y minimum. s u r f a c e s t a r t i n g a t t h i s p o i nt and s temmingfrom the sk i n- f r ic t ion l i ne downstream of th e poin t can be cons t ruc tedt h a t w i l l be the loc us of a s e t of l im i t ing s t r e a m l ine s o r i g in a t i ngfrom fa r upstream; t h is s ur fac e may al so r o l l up on t s developinentdownstream.

This sec t ion conc ludes wi th a dis cus sio n of th e renainin g cype ofs ingu la r po in t, t he foc us ( a l so c a l l e d sp i r a l node) . The f c c us inva r i a b lyappears on the s urf ace i n company with a sad dle point . Together theyallow a pa rt ic ul ar form of glo bal flow sep ara tio n. One le g of the (g lobz l)l i ne of separa t ion enana t ing from the saddle poin t winds i n t o the focust o form the continuous curv e on th e su rf ac e from whFch th e di vi di ng s ur-face stems. The focus on the wa ll extends i n t o the f l ui d a s a concen-t r a t e d v o r t e x f i l am e n t, w h i l e t h e d i v i d in g s u r f a c e r o l l s u with the samesense of ro t a t i on as the vor tex f i lament . When the d iv id ing sur faceextends downstream t quickly ciraws th e vor tex f i lament i n t o t s core .I n e f f e c t , t he n, t he ex te ns ion in to th e f lu id of the f ocus on the wa l lse r ve s a s the vo r t i c a l c o r e a bout wh ich the d iv id ing su r f a c e c o i l s . Th isflow behavior was f i r s t hypothesized by Legendre (1965), who al s o noted(Legendre 1972) th at an experimental confirmation ex ist ed i n the re su lt sof ea r l ie r exper iments ca r r ie d out by Werl6 (1962). Figure 4a showsLegendre s o r i g i na l ske tch of the sk i n- f r ic t io n l i ne s ; Ffgure 4b s aphotograph i l l us t r a t in g the exper imenta l cocf i rmat ion . The dividing


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2s u r f a c e t h a t c o i l s a ro un d t h e e x t e n s i o n o f t h e f o c us ( F i gu r e 4 c) w i l l betermed h er e a horn- type d iv id i ng sur face . On th e ot he r hand, i t canh ap pen th a t t h e d iv i d in g su r f ac e t o w hich t h e f o cu s i s connected doesnot extend downstream. I n t h i s case t h e v o r t ex f i l amen t emana ting fr omt h e f o cus r emains d i s t i n c t , and i s seen as a s e p a r a t e e n t i t y o n c ro s s fl o wplanes downstream of i t s o r i g i n on t h e s u r f a c e . I n a n i n t e r e s t i n g a dd i-t i o n a l i n t e r p r e t a t i o n o f t h e fo c u s, we b eg in by co n s id e r in g t h e p a t t e r nof l i n e s o rt h og . wa l t o t h a t of t h e s k i n - f r i c t i o n l i n e s ; t h a t i s t h ep a t t e r n of s u r f a c e v o r t e x l i n e s . We s e e t h a t what was a f oc u s f o r t h e

p a t t e r n of sk in - f r i c t i o n l i n e s becomes an o th e r f o cu s of s e p a r a t i o n f o rth e p a t t e r n o f su r f a ce v o r t ex l i n es , mar king th e ap p a r ent t e r min a t io n ofa s e t of s u r f a c e v o rt e x l i n e s . I f we i ma gi ne t h a t e a c h of ~ h e s e u r f a c ev o r t e x l i n e s i s th e bound pa rt of a h o r sesh o e v o r t e x , t h en t h e ex t en s io ni n t o t h e f l u i d o f t h e f oc u s on t h e w a l l a s a c o n c e nt r a te d v o r t e x f il a m e nti s seen t o r ep r esen t t h e co mbin at ion i n to o n e f i lamen t o f t h e h o rsesho ev o r te x l e g s from a l l o th e bound v or t i ce s th a t have ended a t t he focus .ne c an e nv i sa g e t h e p o s s i b i l i t y o f i n c o r p o ra t i n g t h i s d e s c r i p t i o n of t h e

f lo w i n t h e v i c i n i t y o f 3 f o cus i n t o an ap p r o p r i a t e i n v i s c id f lo w modc l.

F o m of D i v i di n g S u r fa c e s

We hsve seen how the combinat ion of a f ocus and a sadd le point i n thep a t t e r n of s k i n - f r i c t i o n l i n e s a l l ow s a p a r t i c u l a r arm of g lo ba l f lowsep ara t io n cha rac te r iz ed by a horn-:ype di vi di ng surfa ce. The ngdalpo in ts of a t tachmer~tand se par a t i on may a l s o combine wi th sa dd l e po i n tst o a l l ow , ~ d d i t i o n a l orms of g l o b a l f l ow s e p a r a t i o n , a g a i n c h - r a c t e r i z e dy t h e i r p a r t i c u l a r d i v i d i ng s u r f a c e s . The c h a r a c t e r i s t i c d i v i d i n g


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su rf ac e formed from th e combination of a nodal poi nt of atta chment and asaddle point s i l l u s t r a t e d i n F ig ur e 5 a. This form of divi ding su rfa ce ./ A eF i g .typ i ca l ly occurs i n the flow before an obstac l e c f . F igure 34 i n PeakeC Tobak 1980). In the example i l l us t r a t ed i n Figure 5a t w l l be notedth at the divi ding surf ace admits of a poin t i n the ex t erna l f low a t whichth e f lu id v e lo c i ty s i d en t i c a l ly zero. This s a 3D s in g u la r p o in t ,which i n Figure 5a ac ts a s t he o r ig in of the s t reamline running throughthe vor t i ca l core of the ro l led up div id ing sur face .

The cha ra ct er is ti c divi ding s urf ac e formed from the combination ofa nodal point of sep ara tio n and a sadd le point s i l l u s t r a t e d i n F ig-ur e 5b. This form of di vid ing surfac e oft en occurs in nominally 2Dsepara ted flows such a s i n t he sep arated flow behind a backward-facingst ep cf . F igure 4 i n Tobak Peake 1979) 2nd the separ ate d flow a t acyl ind er-f lar e junction both 2D and 3D, c f . Fi gu re s 47 and 48 i n Peake6 Tobak 1980). We not e i n both Figures 5a and 5b t h a t th e str eam lin es onthe d iv id ing sur face have a l l en tered the f lu id through the sadd le poin tin the pa t te rn o f sk in - f r i c t io n l in es .

Topography of S tr eaml in es in Two-Dimensional Se ct io nsof Three-Dimensional Flows

Aft er an unaccountably long laps e of time, t has only recently becomecl ea r t hat the mathematical basis fo r the behaviar of elementary sing ularpoints and the topological rules that they obey s gene ral enough t osuppor t a much wider regime of ap pl ic at io n than had or ig in al ly beenrea l ized . he r es ul t s reported by Smith 1969, 1 9 7 5 , Per ry Fa i r l i e1974). and Hunt e t a1 1978) have made t ev iden t tha t the rules governing


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s k i n - f r i c t i o n l i n e b e h a v i or are e a s i l y a da p te d and e xt e nde d t o y l e l ds i m i l a r r u l e s go ve rn in g b eh av io r of t h e f l o ~t s e l f . I n p a r t i c u l a r , H u n te t a1 1978) h av e n o te d t h a t i f [ u x , y , z o ) , v x , y , z o ) , w x , y , z o ) ]s th e mean ve lo ci ty whose u,v components a r e measured in a plane

z = z o =t h e n t h e

c ons t a n t , a bove a s u r f a c e s i t u a t e d a t y Y x;zo) see F i gu r e 6 1mean s t r e a m l i ne s i n t he p l a ne are s o l u t i o n s o f t h e e q u a t i o n

which i s a d i r e c t c a u n t e r p a r t o f e q u at i o n 3) f o r s k i n - fr i c t i o n l i n e s o nt he s u r f a c e . Hunt e r a 1 1978) c a u ti o n ed t h a t f o r a g e n e ra l D f low thes t r e a m l i ne s de f i ne d by e qua t i on 5 ) a r c no more t ha n t ha t t he y a r e no tn e c e s sa r i l y t h e p r o j e c t i o n s of t h e 3n s t r e a m l i n e s o n t o t h e p l a n e z zn o r a r e t h e y n e c e s s a r i l y p a r t i c l e p a t h s ev en i n a s t e a dy f low . O nl y f o rs pe c i a l p l a ne s f o r e xa mple , a s trea~irwisepl an e of symmetry wherew x , y , z o ) 0) a r e t h e s t r e a m l i n e s d c f i n e d y e qua t i on 5) i d e n t i f i a b l ew i t h pa r t i c l e pa t h l i ne s i n t he p l a ne when t he f low s s t e a d y , o r w i t hins ta n tan eous s t r e aml ine s when t he f low i s uns te a dy. I n a ny c a s e , s i nc e[ u x , y ) , v x , y ) ] s c o n ti n u ou s v e c t o r f i e l d x x , y ) , w i t h on ly a f i n i t enumber of s i n g u l ar p o i n ts i n t h e i n t e r i o r of tl-rc f l ow a t w hi ch 0 ,t fo l lows th a t nodes and saddle s can be de f ined i n t h e p l a n e j u s t a s they

w er e f o r s k i n - f r i c t i on l i ne s on t he s u r f a c e . Nodes and s a dd l e s w i t h i nthe f low, exc lud ing th e boundary y Y x ; z o ) , a re 1abelc.d N and Sr e s p e c t i v e l y , and a r e shown i n t h e i r t y p i c a l f orm i n F i g u r e 6 . The onlynew f e a t u r e o f t h e a n a l y s i s t ha t s r e qu i r e d s t he t r ea t m en t o f s i n gu l a rpoi nts on t he boundary y Y x ; z o ). S i nc e f o r a v i s c ous f l ow , szer o everywhere on th e boundary, th e boundary s i t s e l f a s i r l gu l a r l i n


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1 5i n t h e p l a n e z . S i n g u l a r p o i n t s o n t h e l i n e o c c u r w h er e t h e com-p on en t o f t h e s u r f a c e v o r t i c i t y v e c t o r n o r m a l t o t h e p l a n e z i sz e r o . T h u s , f o r e x a m p le , i t i s e n s u r e d t h a t a s i n g u l a r p o i n t w i l 1 o c c u ron t h e b o u n da ry w h e r e v e r i t p a s s e s t h r o u g h a s i n g u l a r p o i n t i n t h e p a t t e r no f s k i n - f r i c t i o n l i n e s , s i n c e t h e s u r f a c e v o r t i c i t y s i d e n t i c a l l y z e r ot h e r e . A s i n t r o d u c e d by H unt e t a 1 ( 19 78 ), s i n g u l a r p o i n t s o n t h e b ou nd -a r y a r e d e f in e d as h a l f - n o d e s N a n d h a l f - s a d d l e s S ( F i g u r e 6 . W i t ht h i s s i m p l e am endm ent t o t h e t y p e s o f s i n g u l a r p o i n t s a l l o w a b l e , a o ft h e p r e v io u s n o t i o n s an d d e s c r i p t i o n s r e l e v a n t t o t h e a n a l y s i s of s k i n -f r i c t i o n l i n e s c a r r y o v e r t o t h e a n a l y s i s o f t h e f lo w w i t h i n t h e p l an e .

I n a p a r a l l e l v e i n , H un t e t a ( 19 78 ) h a v e r e c o g n iz e d t h a t , j u s t ast h e s i n g u l a r p o i n t s i n t h e p a t t e r n o f s k i n - f r i c t i o n l i n e s o n t h e s u r f a c eo be y a t o p o l o g i c a l r u l e , s o m us t t h e s i n g u l a r p o i n t s i n an y of t h e s e c -t i o n a l v ie w s o f 3 D f l ow s o be y t o p o l o g i c a l r u l e s . A l th o ug h a v e r y g e n e r a lr u l e a p p l y i n g t o m u l t i p l y c o n n e ct e d b o d i e s c a n b e d e r i v e d (H unt e t a 1 9 7 8 )w s h a l l l i s t h e r e f o r c on v en ie n ce o n l y t h o s e s p e c i a l r u l e s t h a t w i l l b e~ ~ s e f u ln d u b se q u en t s t u d i e s o f t h e f l o w p a s t w i n g s, b o d i e s , a nd o b s t a c l e s .I n t h e f i v e t o p o l o g i c a l r u l e s l i s t e d b el o w, w e a s su m e t h a t t h e b od y i ss i m p l y c o n n e c t e d an d i m me rse d i n a f l o w t h a t i s u n i f o r m f a r u ps t re a m .

1 . S k i n - f r i c t i o n l i n e s o n a t h r e e - d i m e n s i o n a l b od y (D avey 19 61 ;L i g l a t h i l l 1.; . \ :

2. S k i n - f r i c t i o n l i n e s on a t h r e e - d i m e n s i o n a l b o u y B c o n n e c t e ds i m p l y ( w it h o ut g a p s ) t o a p l a n e w a l l P t h a t e i t h e r e x t e n d s t o i n f i n i t yb o t h u p s t r e a m a n d do w n st re a m o r is t h e s u r f a c e of a t o r u s :


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3 S~reamlines n two-dimensional plane cutting a three-dimensionalbody:

4 Stieamline~ n a vertical plane cutting a 5urface that extendsto infinity both upstream and downstream:

5. Streamlines on the projection onto a spherical surface of aconical flow past a three-dimensional body (Smith 1969):

X opologici~l tructure, Structural Stability, and Bifurcation.

The question of an adequate description of 3D separated flow rises withparticular sharpness when one asks how 3D separated flow pattcrns origi-nate and how they succeed each other as the relevant parameters of theproblem (angle of attack, Reynolds number. Mach number, etc) are varied.

satisfactory answer tc the ques ti m say emerge out of the frameworkthat we s h l l try to create in this section. c shall cast our formula-tion in physical terns ~lthough ur definitions ought to be compatiblewith a more purely mathematical trcb,itment based, for example, on whatev~srsystem of partial differential equ3tions is judged to govern the fluidmotion. In particular, we shall hinge our deiinitions of topologicalstructure and structural stability dirpctly to the properties of pattcrnsof skin-friction lines, since this will enable us to make masinurn usc of


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17r e s u l t s f r om t h e p r i n c i p a l s o u r c e o f e x p e r i m e n t a l i n f o r m a t i o n o n 3D s e p a -r a t e d f lo w f l o w - v i s u a l i z a t io n e x pe r i m en t s u t i l i z i n g t h e o i l - s t r e a kt e c h n i q u e .

A d o p t i n g t h e t e r m i n o l o g y o f A nd ro no v e t a ( 1 9 7 3 ) , w e s h a l l s ay t h a ta p a t t e r n o f s k i n - f r i c t i o n l i n e s on t h e s u r f a c e o f a b o d y c o n s t i t u t e st h e p h a s e p o r t r a i t o f t h e s u r f a c e s h e a r - s t r s s s v e c t o r . Two phase por-t r a i t s h av e t h e sam e t o p o l o s i c a l s t r u c t u r e i f a map pin g f ro m o n e p h as ep o r t r a i t t o t h e o t h e r p r e s c r v e s t h e p a t h s of t h e p ha s e p o r t r a i t . t i su s e f u l t o im a g in e h a v i n g im p r i n te d a p h a s e p o r t r a i t o n a s h e e t o f r u b b e rt h a t flay b e d e fo rm e d i n an y w sy w i t h o u t f o l d i n g o r t e a r i n g . E v er y s u c hd e f o r m a t i o n i s a p a t h - p r e s e r v i n g m a pp in g. t o p o l o g i c a l p r o p e r t y i s a n yc h a r a c t e r i s t i c o f t h e p ha se p o r t r a i t t h a t re m ai ns i n v a r i a n t un d er a l lp a t h - p r e s e r v i n g m a pp in gs . T he n um ber a nd t y p e s of s i n g u l a r p o i n t s , t h ee x i s t e n c e of p a t h s c o n n ec t in g t h e s i n g u l a r p o i n t s , an d t h e e x i s t e n c e ofc l o s e d p a t h s a r e e xa mp le s o f t o p o l o g i c a l p r o p e r t i e s . T he set o f a l lt o p o l o g i c a l p r o p e r t i e s o f t h e ph as e p c r t r a i t d e s c r i b e s t h e t o p o l o g i c a ls t r u c t u r e .

W s h a l l d c f i n c t h e s t r u c t u r a l s t a b i l i t y o f a p ha s e p o r t r a i t r e l a -t i v e t o a p a r a m e te r a s f o l l o w s ( c f . A nd ro no v e t a 1 1 97 1) : p h a s ep o r t r a i t is s t r u c t u r a l l y s t a b l e a t a g iv e n v a l u e of t h e p a ra m et er i ft h e p ha se p o r t r a i t r e s u l t i n g f ro m a n i n f i n i t e s i m a l c ha n ge i n t h e param -e t e r ha s t h e same t o po l o g ic a l s t r u c t u r e a s t h e i n i t i a l o ne . T he p r o p e r t i e sof s t r u c t u r a l l y s t a b l e p ha se p o r t r a i t s c a n b e e l u c i d a t e d v i a m a th e m a t ic a la n a l y s i s (A nd ro no v e t a 1971 a l t h o u g h t h e y d e p en d t o som e e x t e n t o nw h e th e r s p e c i a l c o n d i t i o n s s u ch a s , f o r e xa m ple , g e o m e t ri c sy m m e tr ie s , a r et o b e c o n s i d e r e d t y p i c a l ( i . e . g e n e ri c i ' ; c f . B e n j a s i n 19 78 ) o r u n ty p ic cz l


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P( nongeneric ) . Here we s h a l l w i sh t o r e s pe c t t he c ond i t i ons imposed byge ose t r i c symmet r i es whenever they ex i s t . I n t h i s c as e s t r u c t u r a l l ys t a b l e p ha se p o r t r a i t s o f t h e s u r f a c e s h e a r - s t r e s s v e c t o r ha ve two pr in-c ip a l p r op er t i e s i n common: a) t h e s i n g u l a r p o i n t s of the phase por-t r a i t a re a l l e le me n ta r y s i n g u l a r p o i n t s; and ( b) t h e r e are no saddle-p o i n t - to - s a d d le - ~ o i n t c o n ne c ti o n s i n t h e p h as e p o r t r a i t . We should notet h a t c o n d i ti o n ( b ) i s a pr ope r t y on l y o f t he pha se po r t r a i t r e p r e s e n t i ngt h e t r a j e c t o r i e s of t h e s u r f a c e s h e a r - s t r e s s v e c t o r . Saddle-point-to-saddle -poin t connec t ior l s o f t e n occur on D p r o j e c t i o n s of t h e e x t e r n a lf l ow $ b ut t h e s e a r e a r t i f a c t s of t h e p a r t i c u l a r p r o j e c t i o n s and d o n o tr e p r e se n t c o n n c ~ t i o n s etw een a c t u a l 3D) s i n g u l a r p o i n t s of t h e f l u i dv e l o c i t y v c c t o r ) .

S t a b i l i t v of t h e e s t e r n a l t lo w a l s o c an be d e f i n e d i n t e r ms o f i t st o p o l o g i c a l s t t u c t u r e . T he re i s however, a u s e f u l d i s t i n c t i o n t h a ts l ~ou l d e made bet w ee r~ oc a l a nd ~ 1 o b a 1i n s t a b i l i t y of t h e e x t e rn a l t iow.We s h a l l sa y th a t i f an i n s t a b i l i t y o i t he ex te rn a l f low o ccur s t h ~ t oesno t r e s u l t i n t he a ppe ar a nc e of a new 3D) s i n g u l a r p o i n t f t h e f l u i d*; t- loci ty vc ct or , then the t o p o l o g ic a l s t r u c t u r e of t h e e x t e r n a l f lo w h a sbccn unnl t c rcd clnd the in s ta b i l i t y i s lo ca l onc. On th e ot hc r hand,t h c dppe.lrnncc of a nc\J 3D) s i n g u l a r p o i n t mt :lns t t l a t the topologica ls t r uc t u r e of t h e e x t e r na l f low has b cc n a l t e r e d and t h e i n s t a b i l i t y i s s~ 1 1 l b ~ i Ine. I n c o n t r a s t , we s h a l l n o t make t h i s d i s t i n c t i o n f o r t h es u r f a c e s h e a r - s t r e s s v e c t o r . We s h a l l s a y t h a t t h e s u r f a c e s h e a r - s t r e s sv e c t o r e xp er ie rl cc s slo b;1 1 i n s t a b i l i i c s o n ly ; t h o se i n s t a b i l i t i e s o c c urwhen t he t c ~ po l og i c a l t r uc t u r t o f i t s p h a s e p o r t r a i t is a l t e r e d ..

By e x t e r na l flow wc mean t b e e n t i r e f l o w e x t e r i o r t o t hc s u r f a c e .


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19The introduction of distinctions betwren local and global cvcnts

helps to explain why we were led earlier to distinguish between local andglobal lines of separation in the pattern of skin-friction lines. f dninstability of the external flcw either local or global) does not alterthe topological structure of the phase portrait representing the surfaceshear-stress vector, then the convergence of skin-friction lines onto oneor several particular lines can only be a local event so far as the phaseportrait is concerned; accordingly, we label the particular lines locallines of separation. On the other hand, if an instability of the externalflow changes the topological structure of the phase portrait, resultingin the emergence of a saddle point in the pattern of skin-friction lines,then this is a global event so far as the phase portrait i5 concerned;accordingly, we label the skin-friction line emanating from the saddlepoint global line of separation.

Instability of the cxtrrr,al flow leads to the notions of bifurcation,syrmnctry-breaking and dissipative structures Sattinger 1980; Nicolis 6Prigogine 1977 . Suppose that the fluid motions evolve according totime-dependent equations of the general form

u G u,X)where again is parameter. Soluticlns of G u,X) 0 representsteady mean flows of the kind have been considering. A mean flow uOis an asymptotically stable flow if small perturbations from it decay tozero s t - a When thr? parameter is varied, one mean flow maypersist [in the mathematical sense that it remains a valid solution ofG u.1) ] but hecome unstable to small perturbations as crosses acritical value. At such a transition point, new mean flow may bifurcate


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2from t h e known flow. The behavior jus t descr ibed i s convenient ly por-t r ayed on a b i fu r ca t i on d iag ram, t yp i ca l examples of which a re i l l u s t r a t e di n F i gu re 7. Flows t h a t b i fu rc a t e f rom th e known f low a r e repre sent ed by /t h e o r d i n a t e 9 which may be a ny q u a n t i t y t h a t c h a r a c t e r i z e s t h e b i f u r -ca t i on f l ow a lone . S t a b l e f lo ws a r e i n d i c a t e d by s o l i d l i n e s , u n s t a b leflows by dashed l i ne s . Thus, ove r th e range of X where t he known fl owi s s t a b l e , is zero , and t he s ta b l e known f low i s r e p r e s e n t e d a l o n g t h eabsc i s sa by s so l i d l i ne . The known f low becomes unstable fo r a l l value sof X l a rgcr t han X c a s t h e da sh ed l i n e a l o ng t h e a b s c i s s a i n d i c a t e s .New mean flows b i f u r c a t e from = c e i t h e r s u p e r c r i t i cn l l y o rs u b c r i t i c s l l y .

t a s u p r r c r i t i c a l b i fu r ca t Lo n F i g br e 7 3 , a s the parameter Ii s increased jus t beyond t h e c r i t i c a l p oi nt X c t h e b i f u r c a t i o n f lo wtha t r ep l aces t he uns t ah lp known f low can d i f f e r on ly i n f i n i t e s im al l yfrom i t . The bi f ur ca ti on flow brc.iks t he sqmn:t.try of :he known flow ,adopt ing a form of l c s s c r s p m t r ) ; i n which d i s s i p a t i\ * c s t r u c t u r e s a r i s et o abso rb j u s t t he Jmount o f excess ava i l a b l e encrgy tha t t he no rc SFmptrical known flow no lnnger w s ab l e t o abso rb . Because t he b i fu rc a-t i o n f low in i t i a l l y dep ar t s on ly i r l f i n i t c s i rna l l y from the uns t ab l e knownflow, t h e g lo ba l s t a b i l i t y of t he s urf :l cc s h e a r s t r e s s i n i t i a l l y i sunclffectcd. Howcver, s con t i~ lue s o i ncre ase beyond I C , t h ebi furcnt i -on f low de pa r t s s i gn i f ic a r~ t l y rom t he unsta ble known f low andeg i n s t o a f f e c t t h c gl ob a l s t a b i l i t y of t h e s u r f a ce s h ea r s t r e s s .

U l t i m , ~ t c l v v a l u e of \ i s reack.c.d a t which t h e s u r in c e s h e ar s t r e s s--- .becomes global ly unstable tlvidr-:need e i t h e r by one of t11. elementarys i n g u l a r p o i n t s of i t s phase p o r t r a i t becoming a s i n g u l a r p o i n t of odd)


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multiple order or by the appearance of a new singular point of evcn)multiple order. In either case, it is useful to consider the si::gulnrpoint of multiple order as being the coa~escence f a nunber of ele~:lt~lt~il-ysingular points, with the number divided among nodal and saddlc pointssuch as to continue to satisfy the first topological rule, equation 6).An additional infinitesimal increase in the parameter results in thesplitting of the singular point of multiple order into an equal number ofelementary singular points. Thus there emerges a new structurally stablephase portrait of the surface shear-stress vector and a new external flowfrom which additional flows ultimately will bifurcate with furtherincreases of the parameter.

At a subcritical bifurcation Figure 7b). when the parameter isincreased just beyond the critical point Xc, there are no adjacentbifurcation flows that differ unly infinitesimally from the unstable knownflow. H e w , there must be a finite jump to a new branch of flows that nayrepresent a radical change in the topologica: structure of the externalflow and perhaps in the phase portrait of the surface shear-stress vectoras ~ 1 1 urther, with on the new branch, when is decreased justbelow l the flow does not return to the original stable known flow.Only wll n is decreased far enough below c to pass o Figure 7b)is the stable known flow recovered. Thus, subcritical bifurcation alwnvsimplies that t h e bifurcation flows will. exhibit hysteresis effects.

This complttcs a framework of terns and notions that should sufficeto describe haw the structr;r,ll forms of 3D separated flovs originate andsucceed each other. The following section will be devoted to illustra-tions of the use of this fraaework in two examples involving supcrcritic.11and subcritical bifurcations.


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Round-Nose Body of Rev olu tio n a t Angle of At ta ck

L e t u s f i r s t c ons i de r how a sep arat ed f low may or i gi na te on a s l e n d e rround-nose body of r evo lut ion a s one of th e main parameters of th e prob-lem, angle of a t t ac k , i s inc rea sed f rom ze r o in inc rements . Focus ing onthe f low in th e nose reg ion a lone , w a do pt t h i s e xample t o i l l u s t r a t e as eq ue nc e of e v e n t s i n wh ic h s u p e r c r i t i c a l b i f u r c a t i o n i s t he a ge n t lead-i n g t o t h e f o r ma ti on of l a r g e- s c a l e d i s s i p a t i v e s t r u c t u , e s .

A t zero angle of a t t ac k F igr: rc 8a ) th e flow i s everywhere a t ra .hed. .----

A l l s k i n - f r i c t i o n l i n e s o r i g i n a t e a t t h e n o da l p o i n t of a t ta ch me n t a tt he nos e and , f o r a s u f f i c i e n t l y smooth s l e nde r body, d i s a ppe a r i n t o anodal poin: of s e pn r a t i on a t the t a i l . The r e l e va n t t opo l og i c a l r u l e ,equa t ion 6 ) , i s s a t i s f i e d i n t h e s i m p l e s t p o s s i b l e way N 2 7).

A t a very s m s l i a n g l e of a t t a c k F i gu r e 8 b) t h e t o p o l o g i c a l s t r u c t u r eof the p t cl-n o f s k i n - f r i c t i on l i n t s re ma ins una l t e r e d . A l l s k i n - f r i c t i o i il i n c s ag a in o r i g i n a t e t 3 ncda l po in t o f a t t achment and d i sappear in tonoda l po in t of s epa ra t i on . However, t he favor able c i rc umf ere n t i a l p res -s u r e g r a d ic n t d r i v c s t h e s k i n - f r j c t i o n l i n c s l e ew ar d whcrc t hey t end t oc onve rge on t he s k i n - f r i c t i on l i n c running a long the l eeward ray .Fmanating from a node r, l ther than a sadd le po in t and be ing a l i n e o n t owhich o th e r s k i n - f r i c t i o n l i n e s co nv er ge , t h i s p a r t i c u l a r l i n e q u a l i f i e sa s a lo ca l l i n e of s t ~ : ~ r a t i o n- a c c o r di ng t o ou r de f i n i t i on . The f l owi n t h e v i c i n i t y o f t h e l o c a l l i n e of s e p a r a t i v n p r ov i d es a r a t h e r i n no cu oa sonn of l oc a l f l ow separation t y p i c a l of t hc f l ow s l e a v i ng s u r f a c e s ne a r

t h e syc netry planes oi wakes .


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appearance of a c o rr e sp o nd ing a r r a y of a l t e r n a t i n g l i n e s o f a t t a c hm e ntand ( l o c a l ) s e p a r a t i o n . The b i f u r c a t i o n b e in g s u p e r c r i t i c a l , ho we vc r,

A s t h e a n g l e of a t t a c k i s i n c r e a s e d f u r t h e r , a c r i t i c a l a n gl e sci s r eached j u s t beyond which the ex t e rn a l f low becomes lo ca l l y uns ta b le .Coming i n t o p lay he re i s the wel l-known su sc ep t i b i l i ty of --nfleuior.:i1b ou nd ar y- la ye r v e l o c i t y p r o f i l e s t o i n s t a b i l i t y ( Gregor y e t a i 7 9 5 5S t u a r t 1963 Tobak 1973). The in f l ex io na l p r o f i l es deve lop on c ross f l owp la n e s t h a t a r e s l i g h t l y i n c l i n e d f ro m th e p l a n e normal. t o t h e e : .t er na li n v i s c i d f lo w d i r e c t i o n . C a l l e d a c r o s sf l o w i n s t a b i l i t y , t h e cqk7ent s o f t e n

p r e c u r s o r of b ol ~n da ry -l a ye r t r a n s i t i o n , t y p i c a l l y o c c u rr i n g a t k e y n ~ l d snumbe rs j u s t e i ~ t c r i n g h e t r a n s i t i o n a l r a ng e (Mc De vi tt 6 M e l l e ~ 1969dams 1971) . Refer r i ng t o the b i fu r ca t ion d iagrams o f F igu re 7 .

ide n t i fy i ng the p:irarl;eter w i t h ang l e of a t ta ck , we have th a t thei n s t a b i l i t y o c cu rs a t t he c r i t i c a l p oi nt a= where a s u p e r c r i t i c a lb i fu r ca t i on (F igu re 7a) leads t o a new s t a b l e mesn f low. Wi thin thelo ca l space inf lue nced by the ins 'a b i l i ty , the new mean f low con ta ins a na r r a y of d i s s i p a t i v e s t r u c t u r e s . he s t r u c t u r e s , i l u s t r a t e d s ch em at i-c a l l y on Fi g ur e 8 c , a r e i n i t i a l l y of v e ry sm ll s c a l e w i th s p a ci ng o lthe orde r of th p boundary- layer th i ckne ss . Zesembling an ar r ay of s t rearc-wise vo r t i ce s having axes s l i gh t l y skewed f rom the d i r ec t i on of thee x t e r n a l f lo w, tt;e s t r u c t u r e s w i l l be c ~ l l e d o r t i c a l s t r uc t u re s . Althoughthe r e p res en t a t i on of the s t ru c t u r es on a c ross f l ow p lane i n F igu re Bci s i nt e nd ed t o b e m er el y s c h c m t i c , n e v e r t h e l e s s , t h e s ke t c h s a t i s f i e st h e t o p o lo g i c a l r u l e f o r s t r e a ml i n e s i n a c r o s s f l o w p l a n e , e q u a t io n 8 ) .

s i l l u s t r a t e d i n t h c s i d e v ie w of F i g u re 8 c , t h e a r r a y of v o r t i c a ls t r u c t u r e s i s reflected i n t h e p a t t e r n of s k i n - f r i c t i o n l i n e s by t h e


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t h e v o r t i c a l s t r u c t u r e s i n i t i a l l y a r e of i n f i n i t e s i m a l s t r e n g t h andc an no t a f f e c t t h e t o p o l o g i c a l s t r u c t u r e of t h e p a t t e r n of s k i n - f r i c t i o nl i n e s . T h e r ef o r e , on ce a g a i n , t h es e a r e o c a l l i n e s of s e p a r a t i o n , e a c hof which leads to a l o c a l l y s e p a r a t e d f lo w t h a t i s i n i t i a l l y of v e rys m a l l s c a l e .

Although t h e v o r t i c a l s t r u c t u r e s a r e i n i t i a l l y a ve ry sma l l , t heya r e n o t of e q u al s t r e n g t h , b e i n g i m m e r s e i n a nonuniform crossflow.V i c ~ e d n a c r o ss f l ow p l a ne , t h e s t r e n g t h o f t h e s t r u c t u r e s i n cr e: sf rom zer o s t a r t i n g fro in t he windward ray, reaches a maximum near halfwayaround, and diminishes toward zer o on the leeward ray . ke ca l l in g th a tthe pa ramc t r r :I i n F i ba r e 7 was s up po se d t o c h a r a c t e r i z e t h e b i f ~ s - c a -t io n f lows , we sh a l l f ind i t c o nv e ni e nt t o l e t J des ignate the maxinunc r o ss f l o w v c l o c i t y i nd uc ed by t h e l a r g e s t o f t h e v o r t i c a l s t r u c t u r e s .Thus, w i t h f u r t h e r i n c r e a s e i n a n g l e o f a t t a c k , i n c r e a s e s a c c ~ r d i ~ i g l y ,a s F igure 7 3 ind ic a te s . Phys i ca l ly , inc r eases because the dominan tvor t i c i l l s t r uc tu re cap tu re s the g re a t e r pa r t o f the oncoming f low feed ingt he structures the reby g rowing whi le the nea rby s t ru c t u r e s d imin i sh anda r e drawn in to the o r b i t of the dominant s t ruc tu r e . Thus, a s the ang l eof at t a c k inc:-cases, th e ntunbcr of vertic l s t r u c t u r e s n e a r t h e dominants t ru c t u r e d imin i shes whi le the don inan t s t ru c t u r e grows rap i d ly . ?lean-w h i le , w i t h t h e i n c r e a s e i n a n g l e of a t t a c k , t h e f l ow i n a r g i i n c l o s e rt o the nose bocomes s ub je ct t o t h e c ro s s fl o w i n s t a b i l i t y a nd d ev e lo p s a na r r a y o f s m al l v o r t i c a l s t r u c t u r e s s i m i l a r t o t h o se t h a t h ad d ev el op cdfu r the r dohns tream a t lower ang le of a t t ac k . T h e s i t u a t i o n s i l l u s t r a t e don Figure 8 d We b e l i e v e t h a t t h i s d e s c r i p t i o n i s a c r u c r e p r e s e n t a t i o r ~of thc type of f low th a t \ J~rrg 1974 , 1976) h a s c h a r a c t e r i z e d a s an open


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5s e p a r a t i o ~ We note t ha t a l though th e dominant vo r t ic a l s t ru c t ure nowappears to r epresen t a fu l l- f ledged case of f low separa t io n , never the lessth e su r f ace sh ea r - s t r e s s v ec to r has remained g lo b a l ly s t ab l e so t h a t , i n

our te rms , t h i s i s s t l l a case of a l oc a l f low separa t io n .With f u r th e r i n c r ease i n t h e an g le o f a t t ack , t h e c r os s f lo w in s t a -

b i l i t y in the r eg ion upst ream of t he dominant vor t i ca l s t ru c t ur e p reparesth e way fo r the forward movement of th e s tr uc tu re and t s as so c i a t edlo ca l l i n e of s ep a ra t i o n. Eventual ly an ang le o f a t t ac k s r each ed a twhich r he i n f F a c e of t h e v o r t i c a l s t r u c t u r e s s g r e a t e nough t o a l t e rt h e g lo b a l s t ab i l i t y of t h e su v lace sh ea r - s t r e s s v ec to r i n t h e immed ia tev i c i n i ty of the nose . A new (uns tab le) s in gu la r poin t of second ord erapp ea rs a t t h e o r ig in of each o f t h e l cc a l l i n es o f s ep a r a t i o n . With as l i g h t f u r t h e r i n c re a s e i n a n gl e of a t t a c k , t h e u n st a b l e s i n g u l a r po i nts p l i t s i n t o a pai r of elementary s in gu l ar po in ts a focus of separa t ionand a sad dle poin t. This combination produces th e hcrn-type d iv id in gsu r f ace descr ibed e a r l i e r ( F ig u re 4 and i l l u s t r a t ed ag a in i n F ig u r e e( cf . a l so Figures 11 and 12 i n WerlC 1979 ). e now have a gl o b a l formof f low sepa rati on. A new s t a b l e mean flow h as emerged from which addi -t ional f lows u l t imate ly w i l l b i f u r ca t e w i th f u r th e r i n c r ease of t h e an gleof a t tack .

Asymmetric Vortex Bre akdom on Sl en de r Win1

I n ccl :+ras t to su pe rc r i t i ca l b i fu rca t ions , which ar e normal ly benigneve? ; : eg inning a s they must wi th the appearance of on ly i~ t in i c e s i m a l

- .-. bat '.ve s t ~ u cr rres su bc r i t i ca l b i fu r ca t i ons may be d r as t i c even ts ,i n : : . i n g sudden and dramztic changes . f low st ru ct ur e. Although w a r e


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6on ly be ginn ing t o a ppr e c ia t e the r o l e o f b i f u r c a t ions i n th e s tudy ofsepara ted f lows, we can an ti c i pa te th at sudden large-s cale event s, such

s t ho se i nvolved i n a i r ~ r a f t u f fe t and s t a l l , w i l l be de sc r iba b le interms of s u b c r i t i c a l b i f u rc a t i on s . Here we s h a l l c i t e one o,xample wheret s a l ready evident t k t a fl u i d dynamicdl phenomenon inv ol vi ng a sub-

c r i t i c a l b i fu rc at -Lon c an s ig n i f i c a n t ly in f lue nc e the a i r c r a f t ' s dynam ic albehavior. This s the case of asymmetric vortex breakdown which occurswith slen der swept wings a t high angle s of a t t ack .

We leave as ide the vexing ques t ion of the mechanisms underlyingvor tex breakdown i t s e l f (c f . Ha ll 1972), a s we l l a s t s topologica l

st ru ct ur e , t o focus on events subsequent t o th e breakdown of th e wing'sprimary vo rt ic es . Lovson (1964) noted t h at when a sle nde r d el t a wing wass lowly p i t che d t o a su f f i c i e n t ly l a r ge ang le of a t t a c k wi th s id e s l i pangle held f ixe d a t zer o, th e 5reakdown of t he pa ir of leading-edge vor-t i ce s, which a t lower angles had occurred symmetr ical ly ( i .e ; s i de by sid e) ,became asymmetric, w it h t he p os it io n of one vo rte x breakdown moving cl o s e rto the wing aplx than th e o ther . W lich of th e two p oss ib le asymmetricpa t t e r ns w s observed a f t e r any si ng le pi tchu p was p r o b a b i l i s t i c , b u tonce es ta bl is i~ ed , he re la t i ve po sit ion s of the two vortex breakdownswould persist over the win even as th e ang le of at ta ck was reduced t ovalues a t which th e breakdowns had occurred i n i t i a l l y downstream of thewing t r a l l i ng edge. Af te r ide rc t i fy ing te rn s , we s ha l l se e th a t theseobservat ions a r e per fec t ly compat ib le wi th our previous de scr ip t i on of as u b c r i t i c a l b i f u r c a t i o n ( Fi gu re 7b .

Let us den ote by Ac th e di ffe ren ce between t he chordwise posit i onsof t ho lef t-h and and right-hand vo rte x breakdowns and l e t Ac be p o s i t i v e


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when th e left- hand breakdown po si ti on s the c lo ser of the two to th ewing apex. Referr ing now to the su bc ri t i ca l bi fu rcat ion diagram inFigure 7b, we id en t i fy the bi f ur cat ion parameter wi th Ac and theparameter with ang le of at t ack . We see tha t , i n accordance wi thobservat ions , there s a range of a , a ac, i n which the vortex break-down pos it io ns can coexis t s id e by side , a s t ab le s t a t e represen ted byI C = 0. A t t h e c r i t i c a l a n gl e of a t t a c k ac, the breakdowns can nolonger sus ta in themselves s ide by s id e , so th a t fo r a > ac, I C I =s n o longer a s t ab le s t a t e . There be ing no ad jacent b i furca t ion f lows

just beyond a = ac, ldc] must jump t o a di st an t branch of s t a bi e flows,which repre sen ts th c sudden s h i f t forward of one of th e vorte x breakdownpos i t ions . Fur ther , w i th I A C on th e new branch, a s th e ang le ofa t t a c k s reduced [ C ~ doe s not r e t u r n t o z e r o a t a c bu t on ly a f t e ra has pzssed a smal ler value ao A l l of t h i s s i n accordance withobservations Lowson 1 9 6 4 . A t any angle of a t ack where AC can benonzero under symmetric boundary co nd iti on s, th e va ri at io n of Ac wi ths i d e s l i p o r r o l l a ng le must ne c e s s a r i l y be hys t e r e t i c . This a l s o hasbeen demonstrated experimentally El le 1961). Fu rt he r, si nc e Ac mustbe di re ct ly proport ional t o the r ol l i ng moment, the consequent hy s te re t icbehavior of t he ro l l in g moment with s id es l i p or r c l l ang le m kes t hea i r c r a f t s u s c e p t i b l e t o t h e dynamical phenomenon of wing-rock Sc hi ffe t a1 1980) .

Hold ing s t r i c t l y to the no t ion tha t pa t t e rns of sk in - f r i c t io n l i ne s andexte rna l s t reaml ines re f l ec t the pr oper t i e s o f cont inuous vec tor f i e l d s


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2enables us to characterize the patterns on the surface and on particularprojections of the flow the crossflow plane, for example) by a restrictednumber of singuiar points nodes, saddle points, and foci). It is usefulto consider the restricted nmber of singular points and the topologicalrules that they obey as components of an organizing principle: a flowgrammar whose finite number of elements can be combined in myriad ways todescribe, understand, and connect together the properties common to allsteady three-dimensional viscous flows. Introducing a distinction betweenlocal and global properties of the flow resolves an ambiguity in theproper dtfinition of a 3D separated flow. Adopting the notions of topo-logical structure, structural stability, and bifurcation gives us aframework in which to describe how 3D separated flows xiginate and howthey succeed each other as the relevant parameters of the problem arevaried.

Acknowledgments

e are grateful to M. V. Morkovin for suggesting a useful thought experi-ment and to G. T. Chapman, whose contributions were instrumental in ourframing an understanding of aerodynamic hysteresis in tenns of bifurca-tion theory.


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Literature C i .

Adams, J. C. Jr. 1971. Three-dimensional laminar boundary-layeranalysis of upwash patterns and entrained vortex lormation onsharp cones at angle of attack. AEDC-TR-71-215

Andronov, A. A., Leontovich, E A. Gordon, I. I., Maier, A. G.1971. heory f Rifurcations of Dynamic Systems on a Plane,NASA TT F-556

ndronov, A. A . Leontovich, E. A., Gordon, 1 I . Msier, A. d1973. kalitative Theory of Second-Order Dynamic Systems,New York: Wiley

Benjamin, T. B. 1978. Bifurcation phenomena in steady flows of aviscous fluid. I, Theory. 11, Experiments, Proc. R. Soc.London Ser. A. 359:l-43

Davey, A. 1961. Boundary-layer flow at a saddle point of attach-ment. J Fluid Mech. 10:593-610

Elle, B. J 1961 An investigation at low speed of the flow nearthe apex of thin delta wings with sharp leading edges. BritishARC 19780 R 3176

Grezory, N., Stuart, 1. T., Walker, W. S. 1955. On the stabilityof three-dimensional boundary layers with application to theflow due to a rctating disc. Philos. Trans. R. Soc. ondcnSer. A. 248:155 199

Hall, M. G. 1972. Vortex breakdown. Ann. Rev. Fluid Mech.4:195-218

Han, T., Patel, V. C. 1979. Flow separation on a spheroid atincidence. J Fluid Mech. 9: :643-657


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30Hunt, J. C. R. Abell, C J. Plterka, J. A., Woo, H. 1978 . Kine-

matical studies o the flows around free or surface-mountedobstacles; applying topology to flow visualization. J. FluidMech. 86 :179 -200

Joseph, D. D. 1976. Stability of Fluid Motions 1 Berlin: SpringerLegendre, R 1956 . Sdparat on de 1'4coulement laminaire tridimen-

sionnel. k c h . A6ro. 5 4 : 3 - 8Legendre, R. 1965. Lignes de courant d'un t coulemcnt continu.

Rech. AtSrosp. 105 : 3 - 9

Legendre, R. 1972 . La condition de Joukowski en ecoulement tridi-mensionnnl. Rech. Aerosp. 5 :241 -248

Legendre, R. 1977 . Lignes de courant d'un 6coulement permanent:d6collement et s6paration. Rech. A6rosp. 1977-6:327-355

Lighthill, P1 J. 1963. Attachment and separation in three-dimensional flow. In Laminar Boundary Lavers, ed. L. Rosenhead,11 2 . 6 : 7 2 - 8 2 Oxford Univ. Press

Lowson, M. V. 1964. Some experiments with vortex breakdown.J. R. Aero. Soc. 68 :343 -346

Maltby, R L. 1962 Flow visualization in wind tunnels usingindicators. AGARDograph KO. 7 0

McDevitt, J B., Mellenthin, J. A. 1969. Upwash patterns on ablat-ing and non-ablating cones at hypersonic spccds. NASA T i D-5346

Nicolis, G., Prigogine, I. 1577 . Self-organization in onequ i l i b r i umSystems. New York: Wiley

Peake, D. J., Tobak, M. 1980. Three-dimensional inter~~ctionsndvort ical flows with emphasis on high speeds. ACt1rd)ogr:iphNo. 252


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3Poincare, H. 1928. Oeuvres de Henri Poincar6, Tome 1. Paris:

Gauthier-Vi larsPerry, A. E., Fairlie, B. D 1974. Critical points in flow yat:erns.

In Advances in Geophysics, 18B:299-315. New York: AcademicSattirger, D. H. 1980. Bifurcation and symmetry byeaking in applied

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of the aerodynamics of high-angle-of-attack maneuvers. IAAPaper 80-1583-CP-

Smith, J. H. 3 1969 Remarks on the structure of conical flow.RAE TR 69119

Smith, J. H. B. 1975. A review of separation in steady, three-dimensional flaw. AGARC CP-168

Stuart, J. T. 1963. Hydrodynamic stability. In Laminar BoundaryLayers, ed. L. Rosenhead, IX:492-579

Tobak, M. 1973. On local inflexional instability in boundary-layerflows. 2 Angew. Math. Phys. 24:330-35h

Tobak, M., P e s k c D. J. 1979. Topology of two-dimensional andthree-dimensicnal sepalsated flows. AIAA Paper 79-1480

Wang, K. C. 1974. Boundary layer over a blunt body at high inci-dence with an open type of separation. Proc. R. Soc LondonSer 11 3 4 0 : 3 3 3 5

Wane, K. C. 1976. Separation of three-dimensional flow. In Reviews inViscous Flcw Proc. Lockheed-Ceoreia Co. S v m ~ . LG 7 7 E R O U :


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32WerlB, H, 1962. Separation on axisymmetrical bodies at low speed.

Rech. A6ro. 90:3-14Werlb, H. 1979. Tourbillons e corps fuseles aux incidences

ilevees. L Aero. et ~ Astro., no. 79, 1979-6


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FIGURE C PTIONS

Figure 1 Singu lar points : a) node; b) focus; c) saddle .

Figure Adjacent nodes and saddl e point Li gh th i l l 196. .Figure 3 Limi t ing s t reaml ines near 3D separa t i on l i ne .

Figure 4 Focus of separation: a) or i g in a l ske t ch of sk in - f r i c t i on l i n esby Legendre 1965); b) experiment of Werle 1962) i n wate r tunn el;c) extension of focus, Legendre 1965).

Figure 5 Dividing surfaces formed from combinations of: a) nodal pointof attachment and saddle point; bj nodal point of sepa rat io n and saddlepoin t .

Figure 6 S i r~gu l a rpoint s in cross s ec t i on of f low Hunt e t a1 1978).

Figure 7 Examples of a) s u p e r c r i t i c a l b i f u r c a ti o n ; b ) s u b c r i t i c a lb i fu rca t i on .

Figure 8 Scqucnce of lows l ead ing to g lobal D flow separation onround-nose body of revo lu t ion a t ang l e of a t t ack .


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LINE

c)

Fig


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SADDLE SEPARATIONLINE

Fig 2


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ON SURF CF

Fig


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N O D L T T C H M E N T P O I N TB S D D L E P O I N TC FOCUS OF S E P R T I O N


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N NO DAL POINT OF SEPARATION

PLANE

Fig


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PLANE z z

SEPARATION ATTACHMENT

Fig


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Fig


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SECTION A-Ac) a

SECTION 6 6dl a a

Topology of 3D Separated Flow

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