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Bifurcations in Flow Patterns

Some applications of the qualitative theory ofdifferential equations in fluid dynamics

Proefschriftter verkrijging van de graad van doctor aande Technische Universiteit Delft, op gezagvan de Rector Magnificus, prof. dr. J.M . D irken,in het openbaar te verdedigen ten overstaanvan een comm issie door h et College vanDekanen daartoe aangewezen,op dinsdag 31 me i 1988 te 14.00 uu r.

door

Pieter Gerrit Bakker

vliegtuigbouwkundig ingenieurgeboren te Groningen

TR d iss1 6 3 2



Dit proefschrift is goedgekeurd door de promotorProf. dr. ir. J.W. R eyn



Aan Alida



Abstract

I n t he pr es ent t hes i s t he qual i t at i v e t heor y of di f f er ent i al equat i ons ' i s used

al ong wi t h t opol ogi c al c ons i der at i ons t o di s c us s pr obl ems i n f l ui d dynam c s and

gas dy nam c s .Spec i al at t ent i on i s gi ven t o t he qual i t at i ve as pec t s of f l ow f i el ds , i n par

t i c ul ar t o t he geomet r y , t he s hape and t he s t r uc t ur al s t abi l i t y of s t r eam i ne

pat t er ns .

The t heor y r el i es muc h on c r i t i c al poi nt anal y s es and on bi f ur c at i ons i n v ec t or

f i e l d s . Local so l ut i ons o f t he f l ow equat i ons a re der i ved t o di scuss changes i n

f l ow t opol ogy i n c onj unc t i on wi t h bi f ur c at i ons of c r i t i c al poi nt s .

T he t h eor y i s appl i ed t o t opi c s of i nvi s c i d, nonl i near c oni c al f l ows and of

s t eady v i s c ous f l ows ov er pl ane wa l l s .



C o n t e n t s

Page

Preface 1

Chapter I Some elements o f the qualitative theory of differential equations

1. Phase space r epr esent at i on of a dynam cal syst em 52. Phase por t r ai t s near si ngul ar poi nt s 93. Topol ogi cal s t r ucture of phase por t r ai t s , s t r uctural s tabi l i t y, 13

bi f urcat i on4. Hi gher - or der s i ngul ar i t i es i n R2 205. Bi f ur cat i on of vector f i el ds , unf ol di ngs 276. Cent er mani f ol ds 337. An appr oach t o physi cal unf ol di ngs i n f l ow pat t er ns 4l

8. Ref erences 43

Chapter II Topology of conical flow patterns

1. I nt r oduct i on 451. 1. Concept s and def i ni t i ons 451. 2. A sur vey of coni cal f l ow t heor y 461. 3. Coni cal st r eam i nes, coni cal stagnat i on poi nt s 481. 4. Tr ansi t i on phenomena i n coni cal f l ow pat t er ns 51

2. Local coni cal stagnat i on poi nt sol ut i ons i n i r r ot at i onal f l ow 552. 1. Coni cal pot ent i al equat i on 55

2. 2. Coni cal st agnat i on poi nt sol ut i ons 563. Cl ass i f i cat i on of coni cal s i ngul ar poi nt s i n coni cal f l ows 613. 1. Fi r st - or der coni cal st agnat i on poi nt s 613. 2. I r r ot at i onal at t achment s and separ at i ons 663. 3- Hi gher - or der coni cal st agnat i on poi nt s 70

4. Anal yt i cal unf ol di ngs i n coni cal f l ows 754. 1. Bi f ur cat i on par amet er s 754. 2. Appr oxi mat e sol ut i ons near r egul ar poi nt s 764. 3. Saddl e- node bi f ur cat i on 794. 4. Bi f ur cat i on of t opol ogi cal saddl e poi nt 834. 5- Bi f ur cat i on of t opol ogi cal node 90

5. Ext er nal cor ner f l ow; a nonanal yt i c unf ol di ng of a st ar l i ke node 92

5. 1. The f l ow ar ound an ext er nal cor ner 925. 2. Boundary condi t i ons and bi f ur cat i on modes 955. 3- Bi f ur cat i ons of t he st ar l i ke node 985. 4. Symmet r i cal ext er nal corner s 1045. 5- Tr ansi t i on of obl i que saddl e t o st ar l i ke node 109

6. Ref erences 112



- i i i -

Chapter III Topological aspects o f steady viscous flows near plane wa lls

1. Local sol ut i ons of t he Navi er - St okes equat i ons2. Steady vi scous f l ow near a pl ane wal l , el ement ar y si ngul ar poi nt s

2. 1. Appr oxi mat e sol ut i ons near a pl ane wal l2. 2. El ement ar y si ngul ar poi nt s l ocat ed at t he wal l

2. 3- El ement ar y si ngul ar poi nt s i n the f l ow3. Hi gher - or der s i ngul ar i t i es i n t he f l ow pat t er n3. 1. Hi gher - or der s i ngul ar poi nt s i n the f l ow f i el d3. 2. Hi gher - or der si ngul ar poi nt s on t he wal l

k. Unf ol di ng of a t opol ogi cal saddl e poi nt of t he thi r d or der4. 1. Local phase por t r ai t s of t he unf ol di ngk.2. I nci pi ent bubbl e separ at i on4. 3. Separ at i on al ong a movi ng wal lk.k. MRS- cr i t er i on f or separ at i on i n f l ows al ong a movi ng wal l4. 5- Unf ol di ng model f or movi ng wal l separat i ons

5. Unf ol di ng of a topol ogi cal saddl e poi nt of t he f i f t h or der5. 1. Descri pt i on of t he unf ol di ng5. 2. Bubbl e capt ur i ng by a secondar y separat i on

6. Unf ol di ng of a saddl e poi nt wi t h t hr ee hyper bol i c sectors i n ahal f pl ane6. 1. Uni ver sal physi cal unf ol di ng6. 2. Bi f urcat i on sets, f l ow pat t er ns

7. Unf ol di ng of a saddl e poi nt wi t h t wo or f our hyper bol i c sectors i na hal f pl ane7. 1. Uni ver sal physi cal unf ol di ng7. 2. Det er m nat i on of codi mensi on7- 3- Nei ghbour i ng si ngul ar poi nt s, l ocal bi f ur cat i on set s B and B7. 4. Fl ow pat t er ns and gl obal bi f ur cat i on set s B . and B _

8. Vi scous f l ow near a ci r cul ar cyl i nder at l ow Reynol ds number s8. 1. Descri pt i on of f l ow t opol ogy8. 2. Symmet r i cal bi f ur cat i ons8. 3. Asymmet r i cal bi f ur cat i ons, t r ansi t i on scenar i o' s

9• Ref erences

Samenvatting

About the author



- 1-

Preface

The mai n i dea of t he pr esent t hesi s i s t o demonst r at e t hat t he qual i t at i ve

t heor y of di f f er ent i al equat i ons, when appl i ed t o pr obl ems i n f l ui d- and gas-

dynam cs, wi l l cont r i but e t o t he under st andi ng of qual i t at i ve aspects of f l ui d

f l ows, i n par t i cul ar t hose concer ned wi t h geomet r i cal pr oper t i es of f l ow f i el ds

such as shape and stabi l i t y of i t s st r eam i ne pat t er ns.

I t i s obvi ous that i ns i ght i nt o the qual i t at i ve s t r ucture of f l ow f i el ds i s of

gr eat i mpor t ance and appear s as an ul t i mat e ai m of f l ow r esear ch. For , qual i t a

t i ve i ns i ght f ashi ons our knowl edge; i t ser ves as a good gui de f or f ur t her

quant i t at i ve i nvest i gat i ons. Mor eover , qual i t at i ve i nf or mat i on can become ver y

usef ul , especi al l y when i t i s appl i ed i n cl ose cor r espondence wi t h numeri calmet hods , i n or der t o i nt er pr et and val ue numer i cal r esul t s . A qual i t at i ve

anal ysi s may be cr uci al f or t he i nvest i gat i on of t he f l ow i n t he nei ghbour hood

of s i ngul ar i t i es where a numer i cal met hod i s not r e l i abl e anymor e due t o

di scret i sat i on er r or s bei ng unaccept abl e.

Up t i l l now, f am l i ar r esear ch met hods - f r equent l y based on r i gor ous anal yses,

car ef ul numer i cal pr ocedur es and sophi st i cat ed exper i ment al t echni ques - have

i ncreased cons i der abl y our qual i t at i ve knowl edge o f f l ows , al bei t t hat the

i nf or mat i on i s of t en obt ai ned i ndi r ectl y by a pr ocess of a car ef ul but cumber some exam nat i on of quant i t at i ve dat a.

I n t he past decade, new methods ar e under devel opment t hat yi el d t he qual i t at i ve

i nf or mat i on mor e di r ect l y.

These met hods, make use of t he knowl edge avai l abl e i n t he qual i t at i ve t heor y of

di f f er ent i al equat i ons and i n the t heor y of bi f ur cat i ons.

The qual i t at i ve t heor y of di f f er ent i al equat i ons as appl i ed t o dynam cal syst ems

of t wo- and t hr ee- di mensi onal vect or f i el ds appear s t o be ver y usef ul , i n or der

to determ ne t he t opol ogy and s t r uc tu ral s t abi l i t y of s t r eam i ne pat t er ns ,

occur r i ng i n f l ui d dynam cs. The t heor y or i gi nat es f rom t he wor k of Poi ncar é

( 1880) and i s f ur t her devel oped by Bi r khof f (~ 1927) . Lyapunov (- 19*»9) .

Andr onov ( 1937) and hi s co- wor ker s, Ar nol d ( 1963) and many ot her s.

For r ef erences see Chapt er I ( Guckenhei mer and Hol mes (1983))-

I n t he pr esent t hesi s t hi s t heor y i s appl i ed t o f l ows, whi ch can be descr i bed by

t wo var i abl es x and y and t he st r eam i nes are t he sol ut i on cur ves of a syst em of

t he second- or der , - ^ = P ( x , y ) , - ^ = Q( x, y) , wher e P and Q ar e r el at ed t o t he

component s of t he vel oci t y.



- 2 -

A survey of some elements of the q u a l i ta t i v e theo ry of d i f fe re n t i a l equa t ions i s

given in Chapter I . The review introd uc es and di sc us se s in some d e ta i l th e m ost

i m p o r t an t co n cep t s , t h eo r em s , an d m e t h o d s t h a t a r e u s ed i n t h e s u b s eq u en t

ana lyses ; fo r r igorous mathemat ica l p roof s the r eader wi l l be r e fe r r ed to the

l i t e r a t u r e . A ct ua l ly Ch apte r I may be consu l t ed in o rder to be in formed aboutn o t i o n s a s p h ase p o r t r a i t s o f a d yn am ical s ys tem , s i n g u l a r p o i n t s , t o p o l o g i ca l

s t r u c t u r e , s t r u c t u r a l s t a b i l i t y , d e g e n e r a te s y s te m s , b i f u r c a t io n of v ec to r

f i e l d s , un fold ing s , co-dimens ion and center -m anifold theory .

The f i r s t a p p l i c a t i o n o f t h e q u a l i t a t i v e t h eo ry i n t h i s t h e s i s , c on ce rn s t he

f low geometry of three-dimens ional inviscid conical gas f lows . I t i s t reated in

Ch ap te r I I . Co nic a l f low s have the spe c i f i c p roper ty tha t the ve loc i ty o f the

g as p a r t i c l e s and t h e q u an t i t i e s , d e f i n in g th e s t a t e o f t h e g a s , e . g . p r e s s u r edensity and temperature, are constant along rays originating from a common point

( co n i ca l c en t e r ) . A t h r ee - d im en s i o n a l c o n i c a l f lo w i s e s s e n t i a l l y t wo -d i m en

s i o n a l and i s de cr ib ed adequa te ly on a un i t sphere ( around the con ic a l c en te r )

on which the flow geometry i s d isplay ed by con ical s t re am lin es . Conical s t r ea m

l i n e s a re - de f in ed as fo l lows . For con ica l f lows , the f ami ly of sp a t i a l s t ream

l in es pas s in g the same ray form a co n i ca l s t r e am su r fa ce wi th ve r t e x in the

c o n i c a l c e n t e r . T he i n t e r s e c t i o n o f a c o n i c a l s t re a m s u r f a c e w it h t he u n i t

s p h e re i s a c o n i c a l s t r e a m l i n e . A lo ng a c o n i c a l s t r e a m l i n e t h e e n t r o p y i sconstant or i t jumps if a shock is passed.

The con ica l flow geom et ry on th e un i t s ph er e i s governed by a s e co nd -o rde r

dynamical system. The corresponding vector f ield is determined by the cross-f low

veloc i ty , be ing the ve loc i ty component t angen t i a l to the un i t sphere . In po in t s

on the un i t sphere where th i s component van i shes the f low i s pure ly r ad ia l ;

these po in t s a re ca l l ed con ica l s t agna t ion po in t s and appear as s ingu la r i t i es o f

the second-order dynamical sys tem governing the conical s t reamlines . In Chapter

I I , poss ib le f low pa t t e rns near con ica l s t agna t ion po in t s a re s tud ied and the i rs t r u c t u r a l s t a b i l i t y i s e x a m i n e d .

T r an s i t i o n s t o a d i f f e r en t co n i ca l flow p a t t e r n a re i n t e r p r e t ed a s b i f u r c a t i o n s

o f s t r u c t u r a l l y u n s t ab l e h i g h e r - o r d e r co n i ca l s t ag n a t i o n p o i n t s .

C ha pte r I I g i v e s a c om p le te c l a s s i f i c a t i o n o f f i r s t - o r d e r c o n i c a l s t a g n a t i o n

p o i n t s an d s t a r t s w i t h t h e c l a s s i f i c a t i o n of h i g h e r- o r d er c o n i ca l s t ag n a t io n

p o i n t s , by c o n s i d er i ng s ec on d- and t h i r d - o r d e r p o i n t s , t h e s e p o i n t s b e i n g o f

mos t p rac t i ca l in te res t in r ea l f low prob lems .

Chapte r I I ends up wi th a qu a l i t a t iv e examina tion o f th e i n v i s c id f low around

t h r e e d i f f e r e n t co n i ca l b o d i e s : c i r c u l a r co n es , d e l ta - w i n g s w i th a rr ow-s haped

c r o s s - s ec t i o n and ex t e r n a l co r n e r s .



- 3 -

In Chapter I I I the qual i t a t ive theory wi l l be appl i ed to s t eady two-dimensional

incompress ib le v i scous f lows a long the sur face of a p lane or a long a s l ight ly

c u rv e d w a l l . A s e c o n d - o r d e r d y n a m i c a l s y s t e m , w h ose t r a j e c t o r i e s r e p r e s en t

s t r eaml i nes , i s de r i ved .

A s i n g u l a r - p o i n t - a n a l y s i s i s per fo rmed i n o rde r t o ob t a i n de t a i l e d i nfo rma t ionabout the local flow topology.

I n p a r t i c u l a r , s i n g u l a r p o i n t s on t h e w a ll s u r f a c e a re o f i n t e r e s t , b e ca us e in

t h e s e p o i n t s t h e s h e a r s t r e s s v a n i s h e s i n d i c a t i n g flo w s e p a r a t i o n o r flow

at tachment .

As i n C ha pte r I I t h e s i n g u l a r p o i n t s a r e d i s t i n g u i s h e d i n t o f i r s t - o r d e r and

h i g h e r - o r d e r s i n g u l a r i t i e s , t h e form er a pp ea r o n l y a s c e n t e r p o i n t s and s ad dle

po i n t s . F i r s t -o rde r sadd l e po i n t s t ha t a re l oca t ed a t t he wa l l su r face r ep resen t

so lu t ions tha t a re recognized as the c l ass i ca l Oswat i t sch-Legendre so lu t ion forflow sepa ra t ion or a t t achment . The Oswat it sch-Legendre so lu t io n i s s t r u c tu r a l l y

s t a b l e i n t h e s e n s e t h a t a n a l y t i c a l p e r t u r b a t i o n s i n s tr e a m w i s e p r e s s u r e

g rad i en t s and / o r shea r s t r e s s g rad i en t s w i l l no t a f f ec t t he f l ow t opo l ogy nea r

the sep ara t ion (a t tachment ) p oi n t .

A part from t h e s t r u c t u r a l l y s t a b l e s o l u t i o n s , s t r u c t u r a l l y u n s t a b l e s o l u t i o n s

appea r a s w e l l , t hey occur a s h i ghe r -o rde r or degene ra t e s i n g u l a r i t i e s o f the

dynamical system that governs the s t reamline pat tern.

The s t udy o f t hese degene ra t e s i ngu l a r i t i e s ; t he i r un fo l d i ngs and b i fu rca t i ona lbehav i our , w i l l be t he main sub j ec t o f Ch ap t e r I I I . U nf o l d i ng s , r e s u l t i n g i n

d e g e n e r a t e s i n g u l a r i t i e s f a l l i n g a p a r t i n t o a num ber o f f i r s t - o r d e r s i n

g u l a r i t i e s , d e s c r i b e p o s s i b l e s t r u c t u r a l l y s t a b l e f l o w p a t t e r n s i n v i s c o u s

incompressible f low. These pat terns, being more complex for s ingulari t ies wi th a

high er degree of degeneracy, a re int er pr et ed p hy si ca l ly . Some of them app ea r to

be ve ry common in aero dyn am ics, o th er s ar e new and are con cerned w ith top ic s in

laminar flows as:

- genesis of laminar separat ion bubbles

- f low sepa rat io n on moving wa l ls (Moore-Roth-Sears c r i t e r io n )

- in t e r ference of separa t ions and a t t achments , and

- formation of asymmetric standing eddies in the near wake behind a body (vortex

shedding) .



- 5 -

Chapter I Som e elemen ts of the qual i tat ive theory of d i f ferent ia l equat ions

1. Phase space representation of a dynamical system

W t h t he ai m of easy r ef er ence we wi l l gi ve i n t hi s chapt er some el ement s of t he

qual i t at i ve t heor y of dynam cal syst ems whi ch we w l l use i n t he next chapt er s.

The t heor y of dynam cal syst ems has been extensi vel y st udi ed over a l ong per i od

of t i me. Al t hough many quest i ons r emai n as yet unsol ved, a l ar ge amount of

r esul t s has been obt ai ned and i s avai l abl e f or appl i cat i ons.

Numer ous t ext books, ar t i c l es and paper s t esti f y of t he pr ogr ess made i n t hi s

br anch of mat hemat i cs.

A compr ehensi ve sur vey of i mpor t ant achi evement s, i ncl udi ng r ecent devel opment sand advanced met hods, i s gi ven by Guckenhei mer and Hol mes (1983): ' Nonl i near

Osci l l at i ons, Dynam cal Syst ems and Bi f ur cat i ons of Vect or Fi el ds' .

Thi s book, wr i t t en wi t h a st r ong vi ew t o combi ne pur e mat hemat i cal r easoni ng and

appl i cabi l i t y t o pr act i ce, has i nspi red me i n wr i t i ng t hi s chapt er .

The t heor y of dynam cal syst ems ai ms t o st udy t he t i me behavi our of evol ut i onar y

syst ems whi ch are descr i bed mat hemat i cal l y by an equat i on of t he f orm

wher e X = X( t ) e R i s a n- vector and f ( X, t ) i s a suf f i c i ent l y smoot h f uncti on

def i ned on some subset U Ê R x R.

I n appl i cat i ons t i s usual l y i nt er pr et ed as t i me. The f unc t i on f def i nes a

vect or f i el d i n t he n- di mensi onal space t he so- cal l ed phase space.

The sol ut i ons of t he syst em r epr esent i ng sequent i al st at es of t he evol ut i onar yprocess, appear as i nt egr al cur ves of t he vect or f i el d i n t he phase space.

I n the l i t er at ur e on t hi s subj ect , one encount er s sever al names f or t hese

curves, we quot e: sol ut i on cur ves , t raj ect or i es , ( phase- ) pat hs and or bi t s . I n

t he same way, t he t er ms phase por t r a i t , phase pat t er n, f l ow or t r aj ect or y

pat t er n ar e used t o i ndi cat e t he whol e set of so l ut i on cur ves i n t he phase

space.

Dynam cal syst ems where f does not expl i ci t l y depend on t i me ar e cal l ed aut ono

mous and t he t r aj ect or i es of such syst ems do not change i f t i me goes on.

I n t he f ol l owi ng we onl y need t o st udy t hese aut onomous sys t ems, bei ng expr essed

by t he equat i on



x = f ( x ) , x e Rn(1.1)

In a classical t reatment of such a system, see for example Coddlngton and Levin-

son ( 19 5 5 ). t h e a t t e n t i o n i s m ain ly d i r e c t e d t o t h e p r o p e r t i e s o f i n d i v i d u a l

solut ion curves and culminates in to ques t ions about expl ic i t t ime behaviour anddependency on in i t i a l cond i t ions o f the so lu t ions .

A d i f f e r e n t ap p r o ach t o t h e s t u d y of d y n am i ca l s y s t em s i s o b t a i n ed i f o ne

cons ider s f ami l i es o f so lu t ion curves ; then qua l i t a t ive ques t ions a r i s e such as :

do t h e r e e x i s t s te a d y a n d /o r p e r i o d i c s o l u t i o n s , wh at a r e t h e i r s t a b i l i t y

p r o p e r t i e s .

Domains in the phase space where f (X) is a nonvan i sh ing vec to r func t ion a re

ca l l ed r egu la r ; the phase por t r a i t s in a r egu la r domain a re r e l a t ive ly s imple asth ey can be mapped by a homeomorphism onto a family of p a ra l l e l t r a je c t o r ie s ,

s ee F ig . 1 .1 .

Fi g. 1. 1. Mapping of the path s in a re gu la r domain int o a f ie ld

o f p a r a l l e l p a t h s .

An im po r ta n t c l a s s o f s o lu t i o n s o f Eq. (1 .1 ) a re the so-c a l l ed equ i l ib r ium or

s t e a d y s o l u t i o n s : X = X with X sa t is fy in g f(X ) = 0 . An equi l ibr ium or s t ea d y

s t a t e s o l u t i o n i s t h u s r e p r e s en t e d i n t h e p h as e sp ace by a s i n g l e p o i n t and

co rr e sp o n d s t o a c r i t i c a l p o i n t o f t h e v ec t o r f i e l d , i n t h e f o l l o w i n g s u ch a

po in t wi l l o f t en be r e fe r r ed to as a s ingu la r po in t o f the d i f f e ren t i a l equa t ion

( 1 . 1 ) . S t eady s o l u t i o n s : X= X can no t b e r each ed by a n e i g h b o u r i n g s o l u t i o n

X( t ) in f i n i t e t im e . A s t e ad y so lu t io n i s s t a b l e i f a so lu t ion X( t ) through a

po in t i n a neighbourhood of X , remains clo se to t h a t s tead y so lu tio n for t + ».

T he s t e a d y s o l u t i o n i s a s y m p t o t i c a l l y s t a b l e i f a l l n e i g h b o u r in g s o l u t io n s

- 7-

X( t ) ■ X as t ■ <». St eady sol ut i ons not sat i s f yi ng t he st abi l i t y condi t i on ar e

sai d t o be unst abl e.

Some exampl es of st abl e and unst abl e sol ut i ons i n t he phase space ar e depi ct ed

i n Fi g. 1. 2.

a: stabl e b: asympfof i catty sfabl e unst abl e

Fi g. 1. 2. St abl e and unst abl e steady sol ut i ons.

The phase por t r ai t of t r aj ect or i es near s i ngul ar poi nt s can be r at her compl i

cat ed and i s usual l y not homeomor phi c wi t h a f i el d of par al l el t r aj ect or i es.

I mpor t ant el ement s of phase por t r ai t s of syst ems ar e si ngul ar poi nt s and t he

l ocal t raj ector y pat t er n near t hese poi nt s.

Ther ef or e, a syst emat i c descr i pt i on of phase por t r ai t s near s i ngul ar poi nt s i sa usef ul t ool when anal ysi ng phase por t r ai t s of syst ems. A det ai l ed t r eat ment of

phase pat t er ns near si ngul ar poi nt s i s gi ven i n par agr aph 2 of t hi s chapt er .

Anot her cl ass of sol ut i ons of Eq. ( 1. 1) we want t o ment i on her e ar e t he so-

cal l ed per i odi c sol ut i ons sat i s f yi ng: X( t +T) = X( t ) wi t h per i od T. Per i odi c

sol ut i ons of a dynam cal syst em appear as cl osed pat hs i n t he phase space.

Cl osed pat hs i n R2, r epr esent i ng per i odi c sol ut i ons of t wo- di mensi onal systems,

must di vi de t he phase pl ane i nt o an i nner and an out er r egi on. I f t her e exi st s anei ghbour hood of a cl osed pat h whi ch does not cont ai n anot her cl osed pat h, t hi s

pat h i s cal l ed a l i m t cycl e. A l i m t cycl e i s s t abl e i f al l nei ghbour i ng pat hs

of t he l i m t cyc l e approach i t i f t ■> <■> ot her wi se t he l i m t cycl e i s unstabl e:

see Fi g. 1. 3.

An i mpor t ant quest i on, but of t en di f f i cul t t o answer , i s t o det er m ne whet her an

aut onomous syst em has per i odi c sol ut i ons and wher e t hey appear i n t he phase

space.



- 8 -

a: st ab le limit cycle b: un sf ab le limit cycle

F i g . 1 . 3 . L i m i t c y c l e i n R1

An e l e m e n t a r y c o n d i t i o n t h a t ca n b e u s e d f o r t h e n o n - e x i s t e n c e o f c l o s e d p a t h s

i s f o r m u l a te d by B e n d i x s o n ' s c r i t e r i o n .

B e n d i x s o n ' s c r i t e r i o n

L e t x = P ( x , y ) a n d y = Q ( x , y ) b e a n a n a l y t i c a l d y n a m i c a l

s y s t e m a n d l e t U b e a s i m p l y - c o n n e c t e d d om a in o f t h e p h a s e

p l a n e on w h ic h t h e d i v e r g e n c e o f t h e v e c t o r f i e l d : d i v ( P . Q )

d o e s n o t c h a n g e s i g n and i s n o t i d e n t i c a l l y z e r o . Then t h e r e

a r e n o c l o s e d p a t h s l y i n g e n t i r e l y i n U .

T h e p r o o f o f B e n d i x s o n ' s t he o r em g o e s a s f o l l o w s .

A p p l i c a t i o n o f t h e d i v e r g e n c e t h e or e m a l o n g a c l o s e d c u r v e T l y i n g e n t i r e l y i n U

g i v e s

; / i£ * ^ dx dy =

( P-

Q> •n ds

U Y 3 X 8 y T

wi t h U t he i nt er i or of T, n t he out war d nor mal and ds a l i ne el ement of T.

I f 7 i s a pat h of t he vect or f i el d (P,Q) , t hen ( P. Q) and n ar e perpendi cul ar so

t hat t he l i ne i nt egr al I vani shes i dent i cal l y. But s i nce the i nt egrand of t heT

i nt egr al II i s of one si gn, t he i nt egr al cannot be zer o. Then t he cur ve T cannotUT

be a pat h of t he vect or f i el d (P . Q ) .



- 9 -

2. Phase portraits near singular points

nAssume that a dynamical system X = f (X), X e R has a singular point X so that

f(X ) = 0 . In order to characterize the t r ajectory pat ter n near X we assume

f(X) to be suf f icient ly smooth and we expand f(X) near X . Retaining only the

linear par t in the expansion there follows with X - X = E

o

5 = Df(XQ) 6. 5 e Rn (1.2)

3 f .where Df(X ) denotes the Jacobian matrix [-—] of the f i r s t or de r p a r t i a l

o o x .J

derivatives of the vector function^(x j . . ..- xn)

f 2 ( x r . . . xn)

f n ( x l ' • s » /

eval uated i n the si ngul ar poi nt X

Equat i on ( 1. 2) i s a l i near system wi t h constant coef f i c i ent s and can be anal ysedwi t h cl assi cal met hods yi el di ng the cor r espondi ng ' l i near i zed' phase por t r ai t s.

However t hese, ' l i near i zed' phase por t r ai t s are not necessar i l y equi val ent wi t h

those near X of t he ori gi nal non- l i near system

The r el at i on between bot h phase por t r ai t s i s gi ven by the t heor em of Har t man-

Gr obman whi ch hol ds f or syst ems i n R .

Har t man- Gr obman

I f Df (X ) has no ei genval ues w t h zer o real par t , t hen the f am l yof t r aj ector i es near a si ngul ar poi nt X of a nonl i near syst em X =

o

f ( X ) * and those of the l oc al l y l i near i z ed s ys t em have the same

t opol ogi cal st r uct ur e; whi ch means t hat i n a nei ghbour hood of X

t her e exi s t s a homeomorphi c mappi ng whi ch maps t r aj ect or i es of the

non- l i near system i nt o t raj ect or i es of the l i near sys tem

When Df (X ) has no ei genval ues wi t h zero real par t , the si ngul ar poi nt X i s

cal l ed hyper bol i c or nondegenerat e.

* I n t hi s thes i s we r estr i ct our sel ves to the appl i cat i on of cases where f i sanal yt i c but the theorem can be ext ended to a wi der , cl ass of f unct i ons .

- 1 1 -

( i v) q > O, p = O: t he ei genval ues A and A? ar e pur el y i magi nar y. The t r a j ec

t or i es f or m cl osed cur ves sur r oundi ng t he or i gi n and t he s i ngul ar poi nt i s

cal l ed a cent er ,

( v) q = ( S - p ) ' (p * 0) : t he ei genval ues A. and A_ are r eal and equal (A. = A_ =

A ) . The nat ur e of t he phase port r ai t depends on t he J or dan f or m of A. I f A

has t he J or dan f orm (_ ) t he t r aj ec t or i es f orm a s t ar shaped node, whi ch

i s st abl e f or p < 0 and unst abl e i f p > 0.

I f A has t he J ordan f orm L . ) the phas e por t r ai t i s cal l ed an i nf l ect ed

node; al l t r aj ec t or i es tend t o t he or i gi n i n t he same d i r ec t i on and ar e

par al l el at i nf i ni t y.

( vi ) q = 0, p * 0: one of t he ei genval ues i s zer o ; t he phase por t r a i t cons i s t s

of a f am l y of par a l l e l pat hs a nd a l i ne of s i ngul ar poi nt s , whi ch i s

st abl e f or p < 0 and unst abl e i f p > 0.

Thi s cl ass i f i cat i on o f l i near sys tems i s conveni ent l y ar r anged i n t he p- q- pl ane,

shown i n F i g. 1. 4.

q = XvX

p = X , + \2

F i g . 1. 4. G e n e r a l c l a s s i f i c a t i o n of a l i n e a r s y s t e m i n R* .

-12-

The i sol at ed si ngul ar poi nt s for a l i near system are saddl es , nodes , f oc i and

cent er s .

Since for nodes, foci and saddle point s Re(A) * 0 ho lds , Hartman-Grobman's

theorem implies that adding sufficiently smooth non-linear it ies does not change

the phase portraits near these types of singular points.Centers in non-l inear systems sa ti sf y Re(A) = 0, and the ir existence cannot be

shown by linear izat ion. As an i l lustrat ion consider the nonlinear system

x = y

y = -x + G x2 y

wi t h ei genval ues A , A_ = ± i . Unl ess e = 0, the si ngul ar poi nt ( 0, 0) i s not acent er as i n the l i near i zed syst em but a nonl i near f ocus, stabl e i f E < 0.

Nonhyper bol i c si ngul ar i t i es as char act er i zed by a vani shi ng J acobi an (q = 0) can

appear i n the phase space i n di f f er ent f or ms: i sol at ed poi nt s , cur ves et c. The

i s ol at ed poi nt s are usual l y denot ed as mul t i pl e- equi l i br i um poi nt s, degener at e

s i ngul ar i t i es or hi gher - or der s i ngul ar i t i es . The t o pol ogi c al s t r uc t ur e of t he

l ocal phase por t r ai t near hi gher - or der s i ngul ar i t i es can be ver y compl i cat ed and

a gener al c l assi f i cat i on for si ngul ar i t i es i n R i s hard to gi ve.For R2 , Andr onov et . al ( 1973) of f er a cl assi f i cat i on for i sol at ed hi gher - or der

s i ngul ar i t i es havi ng a nonvani shi ng degener at e l i near par t . These poi nt s wi l l be

r evi ewed i n mor e det ai l i n paragraph k.

- 1 3 -

3.Topo logical structure of phase portraits structural stability bifurcation

ï2E2ï2Sï2?ï Ë truc£yE§_2?_E!}&Ë§_P2E£ï&iE§

The c o n ce p t of t o p o l o g i c a l s t r u c t u r e of p h a se p o r t r a i t s h a s a l r e a d y b e e n

in tr o d u c e d in paragra ph 1 when we mentioned th e fundamental theorem of H artman-Grobman.

Now a more thorough exam ination of t h i s co ncept w il l be given bec ause i t en ab les

us to in t roduce and exp la in the concept of s t ru c t u r a l s t a b i l i t y .

C h a r a c t e r i s t i c f e a t u r e s o f t h e p ha se p o r t r a i t w hich may b e c a l l e d q u a l i t a t i v e

properties are for example the number and type of singular points , the existence

of c lo se d p a ths and regions of a t t ra c t i o n . F ormal ly , one may def in e q u a l i ta t i v e

p r o p e r t i e s a s t h os e p r o p e r t i e s o f t h e p h a s e p o r t r a i t w h ic h r e m a i n i n v a r i a n t

un de r a to p o lo g ic a l mapping or homeomorphism. A to po lo gi ca l mapping between two

regions in de plane is a one-to-one and bicontinuous mapping, meaning that each

p o i n t M i s mapped ex ac tly onto one po int M' and th a t d i s t i n c t po in ts M. and M?

ar e mapped onto d i s t i n c t po int s M' and Mi, and the mapping i s c on tinu ou s e i t h e r

way.

An in tu i t iv e des c r ip t ion of a topo log ica l mapping o f the p la n e on t o i t s e l f i s

given by Andronov et. al (1973) as follows:

'imagine the plane is to be made from rubber which is deformed in some

way, s tr et ch in g and squeezing i t a t various po in ts, but without tearin g

or fol di ng . Any topo logical mapping of the plane int o it s e l f i s e ith er a

deformation of the above type (without tearing and folding) or a mirror-

reflection of the plane followed by such a deformation' .

I t may be c l e a r th a t a top o l og ic a l mapp ing o f a phase p o r t r a i t can re su l t in

d r a s t i c c h an g es i n t h e s ha pe o f t r a j e c t o r i e s , b u t c e r t a i n p r o p e r t i e s w i l l

n e v e r t h e l e s s b e p r e s e r v e d . As an e x a m p le : s i n g u l a r p o i n t s a r e mapped i n t o

s ingular poin ts and c losed curves remain c losed curves .

The concept of topologica l s t ruc ture of a phase por t ra i t in R2 is now ind i rec t ly

ind ica ted by the fo l lowing de f in i t ion .

Consider i n a regio n G d f f two dynamical systems D. and D_, give n by

X = f^X ) (D1)

x = f 2 ( x ) <D 2>



- 14-

The phase por t r ai t s of t he syst ems ( D. ) and ( D_) have t he same t opol ogi cal

st r uct ur e i f t her e exi st s a t opol ogi cal mappi ng ( homeomor phi sm : T whi ch maps G

ont o G and whi ch t akes pat hs of ( D. ) over i nt o pat hs of ( D_ ) * ) .

I f t wo poi nt s X. and X? l i e on t he same pat h of syst em D. , t hen t hei r i mages TX.

and TX_ l i e on the same pat h of syst em( D_) .Al s o, i f t wo poi nt s X1 and X_ l i e on t he same path of syst em(D_), t hen t hei r

- 1 - 1i mages T X. and T X_ l i e on t he same pat h of syst em( D. ) .

The gi ven def i ni t i on o f t opol ogi cal s t r ucture i s i n a cer t ai n sense i ndi rect

s i nce i t does not state exact l y what t opol ogi cal s t r ucture i s , but i t speci f i es

t he necessar y condi t i ons f or equal t opol ogi cal st r uct ur es.

St ruct ural _ st abi l i t y; bi f ur cat i on

Af t er t he def i ni t i on of equi val ent t opol ogi cal st r uct ur e, we now i nt r oduce t heconcept of s t r uct ural s tabi l i t y.

Consi der t he dynam cal syst em ( S) def i ned i n a regi on G C R .

X = f ( X) ( S)

Syst em ( S) i s sai d t o be st ructural l y s tabl e i f an i nf i ni t es i mal change of f ( X)

l eaves t he t opol ogi cal st r uct ur e unaf f ected i n G, ot her wi se t he syst em i s cal l ed

st ructur al l y unst abl e.

I t shoul d be not i ced, t hat st r uctura l s tabi l i t y i s not an i nt r i ns i c propert y of

a t opol ogi cal st r uctur e but i s rel at ed t o t he cl ass of i nf i ni t esi mal changes of

f ( X) t hat ar e al l owed. Bot h t he cl ass of pert urbat i ons of t he vector f i el d whi ch

ar e adm t t ed and f ( X) i t sel f det erm ne whet her t here i s s t r uct ura l s t abi l i t y or

not .

' ) I f t he t opol ogi cal mappi ng T ( wi t h i nver se T ) i s k t i mes di f f erent i abl e

(k > 0 ) , t he mappi ng i s cal l ed C di f f eomor ph and the t wo vect or f i el ds ar ek k

sai d t o be C - equi val ent ; C - equi val ence wi t h k > 0 i mpl i es t hat cer t ai n

s moot hnes s pr oper t i es ( k- t i mes di f f er ent i abl e) of t r aj ec t or i es r emai npr eser ved by t he mappi ng pr ocess .

- 1 5 -

This characteristic property of structural stability is illustrated in the next

example where two dif ferent types of per turbations are imposed on the linear

system (in R* ):

x = -x

(1.3)

y = -2y

which has a node at the origin.

Fi r s t we cons ider an an al yt ic p er tu rbat io n by e.^ing a linear term ux to the

right-hand side of y:

x = -x( 1 . 4 )

y = -2y + ux

wi t h u bei ng a smal l per t ur bat i on par amet er .

The or i gi n i s a st abl e node f or u = 0 as wel l for u * 0, t hus ( 1. 3) i s st r uc

t ur al l y st abl e wi t h respect to the per t ur bat i ons of the ( anal yt i c al ) c har ac t er

gi ven i n Eq. ( 1 . 4 ) .

Next we consi der a nonanal yt i c per t ur bat i on*) of ( 1. 3) sy addi ng a per t ur bat i on

of the f orm u / | x|

x = u / | x| - x

(1 .5)

y = -2y

I f l i > 0 s ys t em ( 1. 5) has two si ngul ar poi nt s on the x- axi s : ( 0, 0) and ( u2, 0 ) .

Near ( 0, 0) the t r aj ector i es behave par t i al l y l i ke t hose near a st abl e node and

par t i al l y l i ke t hose near a saddl e poi nt . At ( uJ , 0) t her e occur s a st abl e node.

For ii < 0 si m l ar r esul t s wi l l f ol l ow. The phase por t r ai t s obt ai ned i f u var i es

near zer o are shown i n Fi g. 1, 5.

* ) Al t hough such nonanal yt i cal per t ur bat i ons seem a bi t f ar - f et ched, t heyappear i n actual f l ow si t uat i ons, as can be observed i n Chapt er I I wher es t ructural s tabi l i t y i n coni cal f l ows ar ound an ext er nal cor ner i s t r eat ed.



- 1 6 -

M = 0 | j>0

F i g . 1 . 5 . N o n a n a l y t i c p e r t u r b a t i o n o f a n o d a l p o i n t .

T he e x a m p l e s i l l u s t r a t e t h a t t h e s t a b l e n od e a t ( 0 ,0 ) i s s t r u c t u r a l l y s t a b l e

a g a i n s t a n a l y t i c a l p e r t u r b a t i o n s (E q. ( 1 .4 ) ) b u t s t r u c t u r a l l y u n s t a b l e a g a i n s t

p e r t u r b a t i o n s ( n o n a n a l y t i c a l ) a s g i v e n by E q. ( 1 . 5 ) . E v i d e n t l y , s t r u c t u r a ls t a b i l i t y o f a t o p o l o g i c a l s t r u c t u r e c a nn o t b e e s t a b l i s h e d i n d e p e n d e n t of t h e

c l a s s o f p e r t u r b a t i o n s t h a t a r e i m po se d o n t h e d y n a m i c a l s y s t e m .

To g a t h e r m ore a s p e c t s a b o u t s t r u c t u r a l s t a b i l i t y a nd l e a v i n g t h e d i s c u s s i o n s a s

s i m p l e a s p o s s i b l e , l e t u s c o n t i n u e by c o n s i d e r i n g s y st em s i n R2 in some more

d e t a i l . Assum e t h a t t h e l i n e a r s y s t e m :

A X X 6 R2 ( 1 . 6 )

i s per t ur bed by addi ng a term u f ( X) yi el di ng t he nonl i near syst em

X = A X + u f ( X) u € R (1 .7 )

wher e f ( X) at l east Cl and f ( 0) not necessar i l y equal t o zer o.

I f t he ei genval ues A- , A_ of A have nonvani shi ng real par t s, Re( A. _) * 0 t hen

i t can be shown t hat near t he or i gi n t he phase por t r ai t of ( 1. 6) i s str uctural l y

s t abl e agai ns t t he C1- per t ur bat i ons gi ven i n Eq. (1 .7 ) . To t hi s end, t he phase

por t r ai t of Eq. ( 1. 7) near t he or i gi n has t o be det er m ned. A s i ngul ar poi nt : X

of Eq. (1. 7) sat i s f i es

A X + u f ( X ) = 0o K o'

Si nce A i s an i nver t i bl e mat r i x , and X =o

-u A - 1 f( X ) t h e i m p l i c i t f u n c t i o n

t h e o r e m c a n b e u s e d t o f i n d n e a r X

s u f f i c i e n t l y s m a l l p .

0 t h e u n i q u e s o l u t i o n X = 0 ( u ) f o r



- 17-

The phase port r ai t o f Eq. ( 1. 7) f ol l ows by consi der i ng t he l ocal l y l i near i zed

syst em near X , of whi ch t he coef f i c i ent mat r i x A + p Df ( X ) has ei genval ues

whi ch depend cont i nuousl y on t he per t ur bat i on par amet er p. For smal l p, compar ed

wi t h | Re( A. _ ) | , ( Re( A . ) * 0) t he ei genval ues of A + p Df ( X ) cannot cr oss t he

i magi nar y axi s so t hat X i s al so a hyper bol i c s i ngul ar i t y . Fr om Har t man-

Gr obman' s t heor em t hen f ol l ows t hat t he syst ems ( 1. 6) and ( 1. 7) have phase

por t r ai t s near ( 0, 0) whi ch ar e t opol ogi cal l y equi val ent .

Hence, phase por t r ai t s near hyper bol i c poi nt s a re s t r uc t ur al l y st abl e w th

r espect t o C1- pert ur bat i ons, phase por t r ai t s near nonhyper bol i c poi nt s may l oose

t hei r t opol ogi cal s t r uc t ur e i f C' - per t ur bat i ons ar e i mposed on t he dynam cal

syst em Thi s phenomenon wher e a smal l var i at i on of t he syst em causes a change of

t opol ogi cal s t r uc t ur e i s cal l ed a bi f ur cat i on. I f t he changes are caused byper t ur bat i ons cont ai ni ng par amet er s, t hose par amet er s whi ch act ual l y cause

bi f ur cat i on ar e cal l ed bi f ur cat i on par amet er s.

Two t ypes of bi f ur cat i ons i n phase por t r ai t s may be di s t i ngui shed: l ocal ^

bi f ur cat i ons and gl obal bi f ur cat i ons.

The f or mer appear i f t he t opol ogy of t he phase por t r ai t i s onl y . l ocal l y

af f ect ed. Such bi f ur cat i ons can be obser ved wi t h l ocal anal yses, t hey occur i n

par t i cul ar i f nonhyper bol i c poi nt s ar e pr esent ; bi f ur cat i on changes t he t opol ogyof t he phase por t r ai t onl y i n a smal l nei ghbour hood of t he nonhyper bol i c

s i ngul ar i ty . I n t hose cases wher e l ocal anal yses f ai l t o det ect bi f ur cat i on

ef fects , the bi f ur cat i on i s cal l ed a gl obal bi f ur cat i on.

Saddl e- connecti ons and mul t i pl e l i m t cycl es ar e wel l known exampl es of gl obal

bi f ur cat i ons. Let us di scuss them br i ef l y f or phase port rai t s i n R* .

Saddl e_connect i on

Consi der a dynam cal syst em i n R* havi ng a phase por t r ai t whi ch cont ai ns a

speci al t r aj ector y connect i ng two ( hyper bol i c) saddl e poi nt s. Suppose t hat t he

syst em i s pert ur bed such t hat t he saddl e connecti on br eaks up r esul t i ng i n i t s

di sappear ance. Al t hough t he t opol ogy i n t he phase pl ane i s s i gn i f i cant l y

al t er ed, see Fi g. 1. 6, t he di sappear ance of t he saddl e connecti on cannot be

det ect ed by onl y a l ocal exam nat i on of t he phase por t r a i t s ; t o observe t he

br oken saddl e connect i on a br oader vi ew, cont ai ni ng t he saddl e poi nt s t oget her

wi t h t he separ at r i ces, i s necessar y. The i mposed per t ur bat i ons have a so- cal l edgl obal ef f ect on t he t r aj ectory pat t er n and as a r esul t the t r ansi t i on pr ocess

i s cal l ed a gl obal bi f urcat i on.



- 1 8 -

saddl e connect i on br oken saddl e connect i on

F i g. 1. 6. Per t ur bat i on of s addl e c onnec t i on, gl obal bi f ur c at i on.

A n o t h e r e x a mp l e of a gl obal bi f ur c at i on may appear i f an i s ol at ed c l o s ed pa t h :

T i s embe dde d i n a t r aj ec t o r y pat t er n i n R' as shown i n F i g . 1. 7- A s s u me t h a tf or i n c r e a s i n g t t he t r a j e c t o r i e s i n t he o ut e r r e gi o n ar e s pi r a l l i ng i nwa r d s ,

al l t endi ng t o T i f t -» ». I n t he i nner r egi on of T t hey t end t o T f or t -> —•

and t o a f oc al p o i n t f or t ■ °> The i s o l at ed c l os e d pa t h T i s a ( s em s t a bl e)

l i m t c yc l e , whi c h c annot be r eached i n f i ni t e t i me al ong t r aj ec t o r i es s t a r t i n g

i n a ne i g hb ou r h o od of T .c

T he p ha s e po r t r a i t as s k e t c h ed i n F i g. 1. 7 c an be g en er a t e d by t he p a t h s of t he

s y s t e m

r =- r ( r - l ) 2 , 9 = 1

wh er e ( r , 8) ar e po l a r c oo r d i na t e s i n t he p hase p l ane . The c l o s ed p at h T i s t hen

f ound on t he c i r c l e r =1.

u = 0

7^1u>0

Fi g. 1. 7. Perturbati on of l i mt cycl e, gl obal bi f urcati on.

Consi der a perturbati on of the systemby addi ng the l i near term ur (u 6 R) to

the ri ght-hand si de of r; u appears to be a bi furcati on parameter.

- 19-

The phase por t r ai t of t he per t ur bed syst em

r = - r ( r - l )2 + ur , 9 = 1

has t wo cl osed pat hs: r. _ = 1 ±J~v i f u < 0 and no c l osed pat h i f p > 0. The

appear ance ( di sappear ance) of cl osed pat hs has a gl obal ef f ect on t he t opol ogi -cal str uctur e of t he phase por t r ai t s whi ch can not be est abl i shed by onl y a

l ocal exam nat i on of t he tr aj ector y pat t er n.

The pr evi ous r emar ks poi nt out t hat bi f ur cat i ons - l ocal or gl obal - may appear

i f t he phase por t r ai t i s s t ruct ural unstabl e.

Nonhyper bol i c si ngul ar poi nt s appear as el ement s i n phase por t r ai t s t hat may

cause l ocal bi f ur cat i ons; on the ot her hand saddl e- connect i ons and cl osed pat hs

can gi ve r i se t o bi f ur cat i ons wi t h a gl obal char acter.

Whet her a bi f ur cat i on act ual l y occur s i s not onl y det er m ned by t he pr esence of

st r uctur al unst abl e el ement s i n a phase por t r ai t but depends al so on t he cl ass

of per t ur bat i ons t hat i s adm t t ed.

The l ast s t at ement appl i es al so t o gl obal bi f ur cat i ons , as can be shown by3H 3H

per t ur bi ng a Ham l t oni an syst em x = —, y = - — such that t he per t ur bed syst em

r emai ns Ham l t oni an. Then cl osed t r aj ect or i es can r emai n c l osed pr event i ng a

gl obal bi f ur cat i on t o appear .



-20-

4. H igher-order singularities in R2

I n p a r a g r a p h 3 we h a ve s e en t h a t h y p e r b o l i c a nd n o n h y p e r b o l i c p o i n t s a r e

s t r u c t u r a l l y u n s ta b l e w ith r e sp e c t t o an a p p r o p r i a t e c l a s s o f p e r t u r b a t i o n s .

T r a j e c t o r y p a t t e r n s n e a r n o n h y p e r b o l i c p o i n t s c a n b e v e r y c o m p l e x a n d i nge ne ra l , a s w i l l be s een in the nex t paragraph , they w i l l change i n t o p a t t e r n s

near combinat ions of hyperbol ic s ingular points i f the sys tem is per turbed.

From t h i s po in t of view one might argue th at nonhy perbolic p o in ts a re fa r from

i n t e r e s t i n g b ecau s e t h ey a r e s o ' ex o t i c ' t h a t t h ey a r e h a r d l y m e t i n p r ac t i c a l

s i tua t ions where smal l per tu rba t ions , a lways p resen t , l ead them des in tegra te .

O bv iou s ly , a d i f f e re n t v iew re su l t s if■ non-hyperbo l ic po in t s can be he lp fu l to

an a l y s e t r a n s i t i o n p r o ces se s i n p h as e p o r t r a i t s , i . e . i f an ap p r o p r i a t e ch ang eof parameters in a dynamical sys tem changes the to polo gica l s t ru c tu re .

In tha t case , spec ia l combina t ions o f parameter va lues ex i s t a t which one o r

m o re s i n g u l a r p o i n t s i n t h e p h as e p a t t e r n b ecom e n o n h y p e r b o l i c . Fo r s uch a

parameter combinat ion the sys tem is cal led degenerate or nongener ic ; however , a

small change of these parameter values can convert the nongeneric system into a

non-degenerate or gener ic one.

Such a degenerate s ta te of the sys tem marks a point in the parameter space a t

which the topological proper t ies of the sys tem change qual i ta t ively .These o b se rv a t i o n s g ive us su f f i c i en t mot iva t ion fo r a more de ta i l e d t r ea tment

of these ' exo t i c ' nonhyperbo l i c po in t s .

The d i scuss ion wi l l be r es t r i c t ed to nonhyperbo l i c po in t s in R2 s ince only these

will be found to occur in the subsequent part of this work.

The theory of nonhyperbol ic s ingular points in R2 is well developed by Andronov

et. al (1973) and the following review of the most important results serves as a

b a s i s fo r ap p l i ca t io ns . For a p rofound t r ea tment inc lud ing mathematica l p roof s ,

we re fe r to Andronov e t . a l (1973)•

A n d r o n o v c o n s i d e r s a n i s o l a t e d n o n h y p e r b o l i c s i n g u l a r p o i n t o f a n a n a l y t i c

vec tor f i e ld in R2 such tha t the expans ion near the s ingu la r po in t invo lves a t

l e a s t one f i r s t - o rd er t e rm. Then a d i s t in c t io n can be made between s in g u la r i t i es

having one or both e igenvalues , zero .

-21-

Hi gher - or der si ngul ar poi nt s wi t h one zer o ei genval ue; p * 0

Suppose t hat 0(0, 0) i s an i sol at ed si ngul ar poi nt of a pl anar syst em wi t h one

nonzer o ei genval ue.

Then t he syst em can be wr i t t en as

x = P( x, y) = ax + by + P2(x, y)

y = Q( x, y) = ex + dy + Q2<x, y)

wher e P2( x, y) and Q_( x, y) ar e anal yt i c wi t h t er ms not l ower t han second degr ee

and f or t he ei genval ues i n 0( 0, 0) we have

p = A . + A ? = a + d * 0

q = A. A = ad - be = 0

Thi s syst em can be t r ansf or med by a nonsi ngul ar l i near coor di nat e t r ansf or mat i on

i n t he canoni cal f or m

x = P( x, y) = P2(x, y)

y = Q( x, y) = y + Q2( x , y )

For is ol a t ed po in ts i t i s assumed th at P_(x,y) ^ 0 .

F o r t h i s sy ste m Andronov h a s shown t h a t t r a j e c t o r i e s e x i s t t h a t t e n d to t h e

s i n g u l a r p o i n t i n a d e f i n i t e d i r e c t i o n ( s e m i p a t h s ) . T hese s em i pa th s te nd t o

0(0 ,0 ) on ly in the d i rec t ions 0 , p , n and ^ - ; o n l y on e s e m i p a t h e x i s t s i n t h e

d i r e c t i o n -z and on ly one in th e d i re c t io n ~ - ; these two sp ec ia l semipa ths a redenoted L1 and L_ respec t ive ly .

T o o b ta in th e p o s s ib l e t o p o lo g ic a l s t ru c tu re s n e a r t h e s in g u la r p o in t c o n s id e r

th e eq uat ion y + Q ?(x ,y) = 0 . By the im p li c i t funct ion theorem t h is e qu at ion has

e x a c t ly o n e so lu t i o n y = <p(x) i n a n e ig h b o u rh o o d o f 0 ( 0 , 0 ) , wh e re <p(x) i s

a n a ly ti c and obeys the co n d it io n <p(o) = <t> (o) = 0. The cu rv e y = <p(x) i s an

i so c l i n e o f h o r i z o n ta l d i r e c t i o n s (y = 0 ) o f t h e v e c to r f i e ld .

The next s tep is to def ine a funct ion:

*(x) = P 2 (x, <P(X))

- 2 2 -

w h ic h g i v e s t h e v a l u e o f x o n t h e c u r v e y = 0 . B e c a u s e 0 ( 0 , 0 ) i s i s o l a t e d

(?2 ^ 0 ) , i |i (x) assum es th e form

* ( x ) = A x m + o ( xm ) , m > 2

m —wher e A i s t he f i rst nonvani shi ng coef f i c i ent (A * 0) and m i s i nt eger .

Let us consi der t he nei ghbour hood U. of 0( 0, 0) bounded by a ci r cl e C wi t h r adi us

<5, s ee f i g. 1. 8.

The c u rv e y = <t>(x) d i v id e s t h e domain U,

i n t o tw o p a r t s . S i n c e y = y + Q 2 ( x , y ) ,

y > 0 i n t h e u p p e r r e g i o n , s o t h a t t h e

v e c t o r f i e l d p o i n t s u p w a r d s t h e r e .S i m i l a r l y w e h a v e do wnw ash i n t h e l o w e r

r e g i o n . T h i s i m p l i e s t h a t t h e se m i p a t h s

L . a n d L _ , l y i n g a b o v e a n d b e l o w t h e

c u r v e y = <p(x) r e s p e c t i v e l y , a r e u n

s t a b l e . S i n c e <t>(x) i s a n a l y t i c , e i t h e r

y = <p(x) c o i n c i d e s w i t h t h e x - a x i s

(<p(x) = 0 ) o r i t h a s , i n a s u f f i c i e n t l y

s m a l l n e i g h b o u r h o o d o f 0 ( 0 , 0 ) , n o o t h e rp o i n t s t h a n 0 ( 0 , 0 ) i n common w i t h t h e x -

a x i s .

* P ( x )

Fi g. 1. 8. Nei ghbour hood U- of 0( 0, 0)o

I n t he f i r st case the posi t i ve and negat i ve x- axes ar e obvi ousl y sem pat hs of

t he s yst em whi l e i n t he second case ( <p( x) 0) t he t r aj ect or i es must cross the

curve y = <P(x). These t r aj ect or i es ar e passed al ong i n a di r ect i on dependi ng on

t he si gn of t|>x).

W t h t he expr essi on i|i(x) A x

m

+ we obser ve t hat t hi s depends on t he

si gn A and on the par i t y of m whi ch i mpl i es t hat f our di f f er ent cases have t o

be consi der ed (A > 0 or < 0, m i s odd or even) . Accor di ng t o Andr onov, t hese

cons i der at i ons l ead to a c l ass i f i cat i on of t opol ogi cal s t r uc ture s whi ch i s

summar i zed i n Andr onov' s t heor em 65 ( p. 3 0, Andr onov et . al , (1973))-

Theor em 65. Hi gher - or der si ngul ar poi nt s wi t h onl y one zer o ei genval ue

Let 0( 0, 0) be an i sol at ed s i ngul ar poi nt of t he system x = P_(x, y) and y =

y + Qp( x, y) wher e P_ and Q? are anal yt i c and have ser i es expansi ons near

0( 0, 0) of whi ch t he l owest - or der t er ms ar e at l east quadr at i c.

-23-

Let y <p(x) be the sol ut i on of y + Q2( x , y ) 0 i n t he vi c i ni t y of 0( 0, 0) and

assume t hat t he ser i es expansi on of t he f unct i on * ( x) = P 2( x, <?(x)) has t he

form t|i(x) = A x + where m £ 2 and A * 0. Then:m m

1. I f m i s odd and A > 0, 0( 0, 0) i s a t opol ogi cal node wi t h an i nf i ni t e

number of sem pat hs i n t he di r ect i ons 0, n and exact l y one sem pat h t endi ngto 0 i n the di r ect i on -= and al so one i n t he di r ect i on ~- ,

2. I f m i s odd and A < 0, 0( 0, 0) i s a t opol ogi cal saddl e poi nt whose separ a-m _

t r i c e s ( s e m i p a th s ) a p pr oa c h 0 ( 0 , 0 ) i n t h e d i r e c t i o n 0 , -~, n and ~ - ,

r e s p e c t i v e l y .

3 . If m i s even , 0(0 ,0) i s a saddle-node , i . e . a s in gu lar poin t whose n e i g h

bourhood c o n s i s t s o f one p a ra b o l i c s e c t o r (no da l t ype ) and two hype r

bo l i c* ) se c t o r s ( s add l e - t ype ) . I f A < 0 t he hype rb o l i c sec t o r s co n t a i n asegment of th e p o s i t iv e x - a x i s , i f A > 0 they conta in a segment of the

nega t i ve x -ax i s .

By de f in i t i on the orde r of a s i ng ul ar i ty i s equal to m.

The fu l l p roof of t h is theorem may be found in Andronov e t . a l (1973) PP- 337

f f. T y p i c a l p h a s e p l a n e p i c t u r e s n e a r t h e s e m u l t i p l e - e q u i l i b r i u m p o i n t s a r e

shown in F ig . 1.9»

^ i < k -topological node topological saddle sadd le-nod e

Fi g . 1 .9 . H i ghe r -o rde r s i ngu l a r i t i e s i n R2 with one zero eigenvalue.

Higher^order s ingular goints_having both_eigenvalues_zero£_g_=_0

In th is se ct io n we con sider th e system

x = ax + by + P 2 (x ,y ) , y = ex + dy + Q 2(x,y)

wi t h t he assumpt i on

p = a + d = 0q = ad - be = 0

| a| + | b| + | c | + | d| * 0

* ) Ot her exampl es of hyper bol i c sect or s appear near a hyper bol i c saddl e poi ntwher e f our hyper bol i c sect or s can be di st i ngui shed.

-2k-

Suppose 0( 0, 0) i s an i sol at ed s i ngul ar poi nt at t he or i gi n and P, (x, y) and

Q2( x, y) are anal yt i c i n t he vi ci ni t y of 0. The ser i es expansi ons of P ( x , y ) and

Qp( x>y) i nvol ve t erms not l ower t han s econd order . Then a nonsi ngul ar l i near

t r ansf or mat i on exi st s ( c. f . Andr onov p. 3 7) br i ngi ng t he syst em i nt o t he nor mal

formx = y +P2(x, y) y =Q2( x , y )

The t opol ogi cal s t r uc t ur e of 0( 0, 0) i s f ound by consi der i ng t he i socl i ne of

ver t i cal di r ec t i ons : y =<p(n

On thi s cur ve t he f unct i ons

ver t i cal di r ec t i ons : y =<>x) wher e <p(x) sat i sf i es <p(x) +P-,(xt <p(x)) = 0.

3P- 3Q* ( x ) =Q2( x, <p(x) ) and p(x) =j ^ - (x, <(x)) +j ^ ( x, <p(x)J

may be eval uat ed and expanded as

i(i(x) = A x + , p(x) = b x +m r ' ' n

where A and b are the first nonvanishing coefficients in these expansions and

m and n are integer.

The topological structure of phase portraits in R2 near singular points with

zero as a double eigenvalue is then established by Andronov's theorems 66 and 67

(PP. 356, 362).

Theorem 66, 67. Higher-order singular points having both eigenvalues zero

Let 0(0,0) be an isolated singular point of the system x = y + P_(x,y), y =

Q2(x«y); l e t y = *(x) °e the solutio n of y + P?(x, y) = 0 near 0(0,0) and

assume that the series expansions of the function 4> (x) = Qp(x, <t>(x))

3P 2 3Q2and p( x) = - —( x, <p(x)) +- —( x, <p(x)) have t he r espect i ve f or msox oy

* ( x ) = A xm +. . . , A * 0m m

p( x) = b x + . . . , b * 0 .V ' n n

- 2 6 -

m = odd = 2l+ 1

'

bn*0<

<

'

A m > 0

bn=0

n=even

n=odc

,X<0

X = b £ + 4 A m ( H / )

' X>0

)

topological saddle

y

^ ¥ ^

foc us, center

topological node

ell ipt ic andhyperbolic sector

cusp

saddle-node

x = y + P2(x,y),y=Q 2(x,y)

y|x=0

= 4 J ( x , = A

m

x m

Ö P , da

èx óy x=0p ( x ) r b n X n -

Fig. 1.10. Higher-order si ng ul ar it ie s in R2 with two eigenval ues ze ro .



-27 "

5. Bifurcation of vector fields, unfoldings

In this paragraph we study local bi fu rc at io ns as they occur in vec to r f iel ds

near struc

defined as

near st r uc tu r al ly unstable singular po in ts . Consider a vector f ie ld in R

X = f (X), X 6 Rn, u e Rk

which depends on the k-dimensional parameter u = (p , u_, . . . . u, ) . The vector

field f (X) is assumed to be analytic.

The term bifurcation introduced in paragraph 3 was origi nally used by Poincare

to describe the s p l i t t i ng behaviour of st at io na r y s o lut io ns : X = X of thedynamical system X = f (X). These solutions, which are represented as singul ar

point s in the phase space, can be found by solving f (X ) = 0 for X as ap o o

function of u if the Jacobian Df (X ) has no zero eigenvalues. However, i f theu o

J acobi an has a zer o e i genval ue, occur r i ng at s ome u, say u , the syst em X =f (X) i s str uctur al l y unstabl e and sever al br anches of the s ol ut i on X = X (u)

n+kcan come t oget her at (X ,u ) i n R

The paramet er set u wher e the syst em l ooses i ts s t r uct ur al st abi l i t y i s cal l ed:

bi f ur c at i on s et , i t can be vi ewed as di vi di ng sur f aces bor der i ng domai ns i n the

paramet er space where the system i s nondegenerat e (gener i c ) .

Var i at i ons i n the par amet er space i nt er sect i ng the set (u| u = u } cause a change

of the t opol ogi cal st r uctur e of the phase por t r ai t .

Such changes are cal l ed bi f urcat i ons and the corr espondi ng paramet er val ues are

t he bi f urcat i on val ues.

A one- par amet er f am l y of syst ems wi t h k-1 r el at i ons bet ween the par amet er s u. ,u_ , .. . . u, i s represented at a curve A i n the k- di mensi onal par amet er space.

Ass ume t hat A i nt er s ec t s the bi f urcat i on set , u , and t hat the i nt er sect i on i sc

t r ansver sal . At the i nt ersect i on, s t r uct ural unstabi l i t y occur s , but due to the

t r ansver sal i t y condi t i on the degener acy can be r emoved by any smal l di spl acement

al ong A. The cor r espondi ng per t ur bat i on i s cal l ed a gener i c per t ur bat i on.

Var i at i ons i n the par amet er space t hat coi nci de wi t h u correspond to a non-

gener i c per t ur bat i on.



- 28-

I n t hi s par agr aph we f ocus t he at t ent i on on bi f ur cat i ons of i sol at ed si ngul ar

poi nt s. As we have seen t hese bi f ur cat i ons ar e r ef er r ed t o as l ocal bi f ur ca

t i ons , so t hat t he vect or f i el d i s st udi ed near degener at e si ngul ar poi nt s and

t he bi f ur cat i ng sol ut i ons ar e al so f ound i n t he nei ghbour hood of t hese poi nt s.

Let us s t ar t wi t h a s i mpl e exampl e of a dynam cal syst em i n R

1

. Consi der t heone- di mensi onal ' vector ' f i el d:

x = fu ^ = " * " x'

whi ch depends on t he scal ar paramet er u.

(1 .8 )

Her e Df ( x)P

3x 2, and t he onl y bi f ur cat i on poi nt i s (x , u (0 ,0 ) . I t i s

0) and t hr ee di f -asy t o check t hat f ( x) = 0 has one sol ut i on i f u i 0 (x

f er ent sol ut i ons i f u > 0 ( x =0 , ±/p~ . A qual i t at i ve pi ctur e of t hese sol u

t i ons i s gi ven i n Fi g. 1. 11a whi ch shows t he br anches of s i ngul ar poi nt s i n a( x, u) space. Thi s f i gur e i s cal l ed t he bi f urcat i on di agr am i t shows t he l oci of

si ngul ar poi nt s as a f uncti on of t he par amet er s.

The phase space i s one- di mensi onal and coi nci des wi t h t he x- axi s.

The poi nt x = 0 i s st abl e f or p S 0 and i t becomes unst abl e f or u > 0.

The phase por t r ai t s whi ch can occur af t er bi f ur cat i on are shown i n Fi g. 1. 11b.

bifurcationpoint

(b)

( j ï O:

U>0:- / J i *

Fi g. 1. 11. Bi f urcat i on of f ( x) = ux - x3.

( a) bi f urcat i on di agram (b) phase por t r ai t s .

The pr evi ous exampl e of a one- par amet er bi f ur cat i on gi ves t he r esul t of a one-

par amet er var i at i on on t he st r uctur al l y unst abl e ' vector' f i el d x = - x3.

The per t ur bed syst em x = +ux - x' shows some possi bl e bi f ur cat i ons and i s cal l ed

an unf o l di ng of t he unper t ur bed system x = - x3 . Now t he i mpor t ant quest i on

ar i ses : i s t he unf ol di ng gi ven by eq. ( 1. 8) a par t i cul ar one, show ng onl y somebi f ur cat i ons, or has t he unf ol di ng a mor e gener al char act er so t hat al l possi bl e

bi f urcat i ons of x = - x' , wi t hi n a cer t ai n c l ass of per t ur bat i ons ( f or exampl e

- 29-

anal yt i c ) , ar e descr i bed. I f t he unf ol di ng descr i bes al l poss i bl e bi f urcat i ons

of t he degener at e si ngul ar poi nt , t hen i t i s cal l ed a uni ver sal unf ol di ng.

He nc e a uni v er sa l unf ol di ng of a degener at e v ec t or f i el d i s a f am l y of

bi f ur cat i ng sol ut i ons whi ch cont ai ns t he bi f ur cat i on i n a per si st ent way. I t has

t he i mpor t ant pr oper t y t hat i t descr i bes t he bi f ur cat i on wi t h t he smal l estnumber of par amet er s. Thi s number i s cal l ed t he c odi mens i on of t he degener at e

vect or f i el d. The codi mensi on of a bi f ur cat i on i s t he smal l est di mensi on of t he

par amet er space whi ch cont ai ns t he bi f ur cat i on i n al l i t s as pect s. A uni ver sal

unf ol di ng i s not necessar i l y uni que si nce a coor di nat e t r ansf or mat i on can l ead

t o a di f f er ent l y f or mul at ed unf ol di ng havi ng t he same di mensi on. Ther ef or e t he

t er m ' uni ver sal unf ol di ng' i s not gener al l y ac cept ed i n t he l i t er at ur e; f or

exampl e Shi r er and Wel l s ( 1983) pr ef er t he term ver sal unf ol di ng t o poi nt out

t he nonuni queness of t he unf ol di ng.

Bear i ng th i s i n m nd we r et ur n t o t he syst em x = - x ' i n or der t o f i nd t he

uni ver sal unf ol di ng of t hi s sys tem Eq. ( 1. 8) gi ves an unf ol di ng wi t h one

par amet er , but does i t descr i be the bi f ur cat i on i n al l i t s aspect s or ar e mor e

par amet er s necessary f or a uni ver sal unf ol di ng?

I f the f unct i on f ( x) = - x1 i s subj ected t o a smal l per t ur bat i on f ( x) ■ f (x ) , i t

i s obvi ous t hat f ( x) possesses one, t wo or t hr ee zeroes near x = 0.

However , t he unf ol di ng ( 1. 8) gi ves ei t her one (u < 0) or t hr ee ( u > 0) zer oes,

whi ch i ndi cat es t hat ( 1. 8) i s probabl y not a uni ver sal unf ol di ng of x = - x1 .

Because i t i s not poss i bl e t o i nt r oduce mor e t han t hr ee zeroes l ocal l y , al l

t hese behavi our s can be capt ur ed by t he addi t i on of t he l ower - or der t er ms u- +

u_x , so t hat a uni ver sal unf ol di ng of x = - x' i s r epr esent ed by t he t wo-

par amet er f am l y

f u( x) = j i x + u2x - x' * ) ( 1. 9)

Equat i ng f ( x) t o zer o gi ves t he l oci of s i ngul ar poi nt s x as a f uncti on of t he

par amet er s u. and u_. Fi gur e 1. 12 shows t he cor r espondi ng bi f ur cat i on di agr am

wi t h these l oc i i n t he x , p1, u_ space.

h) Ot her uni ver sa l unf ol di ngs such as f ( x) = u. x + u2x2 - x* or f ( x) = u. +

u_x2 - x' have t he same bi f ur cat i on char act er i st i cs: t hey can be t r ansf or medi nt o eq. ( 1. 9) by a sui t abl e tr ansl at i on of t he x- coor di nat e.



-30-

A s i n g u l a r p o i n t be co m es d e g e n e r a t e i f

D f ( x ) h a s e i g e n v a l u e s w i t h z e r o r e a l

part , thus if u_ - 3xz = 0.

E l i m in at in g x from th is equat ion and the

equat ion f (x ) = 0 we f ind th e pa ra m et erco m b i n a t i o n s a t wh ich b i f u r c a t i o n t ak es

p l ace .

T h es e p a r am e t e r co m b i n a t i o n s a r e ca l l ed

t h e b i f u r ca t i o n s e t wh i ch s a t i s f i e s :

Fig. 1.12. Bifurcation diagram of

f (x) y l + P2 X " 0

Th is s e t c o n s i s t s of two branches i f u_ > 0 , the corresponding one-dimens ional

phase p o r t r a i t s a r e shown in F igure 1 .13 . We obs erve from th e phase p o r t r a i t s

th at on a branch (u_ > 0) the nonhype rbolic po int i s th e one-dimensional v a ri a n t

of a saddle-node. Cross ing a branch t ransversal ly impl ies the bi furcat ion of the

sa d d le -n o d e . Such a cros s ing is in fact d escr ibed by one parameter so tha t the

branches rep res en t a codimens ion-one bi fu rc at i on . At the p o in t (u . . ,u_) = (0 ,0 )

where both bran ch es te rm in at e w ith a common tan ge nt we have a nonh yperb olic one -

d imensiona l node ; i t s b i fu rca t ion i s descr ibed by two para m eter s im ply ing th a tthe po in t (u.. ,u_) = (0,0 ) re pr es en ts a codimension-two bi fu rc at io n.

This example of a codimension-two bifur

ca t i o n i l l u s t r a t e s t h a t th e u n f o ld i n g of

x = - x ' g i v e n by e q . ( 1 .9 ) c o n t a i n s ,

bes ides the gene r ic b i fu rc a t ion ( a l ready

g i v e n by e q . ( 1 . 8 ) ) , a l s o n o n g e n e r i c

b i f u r c a t i o n s . The p o s s i b i l i t y to d e

s c r ib e a l s o t h e s e n o n g e n e r ic b i f u r c a t ions i s real ized by int roducing the two

param eter un fold ing : f (x) = u 1 +y-x-x* ,

implying that the universal unfolding of

x = -x ' i s def ined wi th two p ar am et er s

involving a codimens ion- two bi furcat ion.

I f o n ly g e n e r i c b i f u r c a t i o n s a r e o f

i n t e r e s t i t i s en ou gh t o c o n s i d e r t h e

Fig . 1 .13. Bi furc at io n set and phase one-parameter unfolding as giv en by eq .p o r t r a i t s of x=u- +p_x-x' . (1 .8 ) .

{ # - ( # "

- 31-

The pr ec edi ng c o ns i d er a t i o n a bo ut b i f u r c a t i o ns o f a o ne - d i me ns i o na l s y s t em

i l l u s t r a t e s t h at c onc e pt s l i k e unf ol di ng and c odi mens i on c an be v al uabl e t ool s

i n bi f ur c at i on anal ys es . Es pec i al l y i f t he ai m i s t o de ve l o p a ge ne r a l t heor y

about degener ac i es and t he i r uni ver sa l unf o l d i ngs , t he concept codi mens i on pl ays

an i mpor t ant r ol e. I t o f f er s t he pos s i bi l i t y t o dev el op a s t r at egy f or t he s t u dy

of t hese degenerac i es whereby one s ta r t s by i nvest i ga t i ng codi mens i on- one bi f ur

c at i on, t hen c odi mens i on- t wo bi f ur c at i ons et c . S uc h a s t r a t e gy c a n u l t i ma t e l y

l ead t o a c l ass i f i cat i on scheme of bi f ur cat i ons based on gener a l cons i der at i ons

such as number of par amet er s, di mensi on of t he phase s pace and const r ai nt s whi ch

account f or t he cl ass o f per t ur bat i ons , see f or exampl e Shi r er and We l l s ( 1983) .

I n t he next t abl e we have summar i zed a f ew exampl es o f e l ement ar y uni ver sa l

un f o l d i n gs o f d eg en er a t e s i n gul a r i t i e s oc c ur r i ng i n one- and t wo- di mens i onal

syst ems.

One- di mens i onal uni ver sa l unf ol di ngs

name

f ol d

cusp

s wal l owt ai lx 5 but t er f l y

name

hyperbol i c -umbi l i c

e l l i p t i c - u m b i l i c

As we have ment ioned before, the presented form of the unfoldings i s not neces

s a r i l y u n iq u e ; s e v e r a l a l t e r a t i o n s c an b e f ou nd w h ic h a c c o u n t f o r t h e sam e

b i f u rc a t i o n behav i our . These a l t e ra t i o ns fo llow from su i t a b l e coord i na t e t r a ns

format ions and rearrangement of the parameters .

The lo c i of the nearby s in g u la r i t i e s : x (p , p p . ) , forming the b i furca

t i on d iagram can be po r t r ay ed a s su r f ac es (b ou nd ar i e s ) i n t he x - p - spac e

(p e R ) . The complex ge om et r i ca l s t ru c tu re of the se sur fa ces near the o r ig in

f o r m codi mensi on unf o l d i ng

1

U1 + u2x

u ? x + p x<P x + p2x + p3x J + p^x3

Two- di mens i ona l uni ver sa l unf ol di ngs

f or m codi mens i on unf ol di ng

x y

x 2 - y2

x '

P j + p , ,y + x y

U 2 + p . y + x 2 - j

| p x + p 3 x + p^y +

P 2 + y 2



/

- 32 -

can be a ss oc ia t ed wi th we l l known e l emen tary s i n g u la r i t i e s ap pe a r ing i n c a t a s

t rophe theo ry . Th i s co r respondence wi th e l emen ta ry ca t a s t rophes i s re f l ec t ed by

the names l i s t ed in the t ab l e .



-33 -

6. Center manifolds

Bef or e pr oceedi ng wi t h t he appl i c at i ons i n f l ui d f l ow pr obl ems , t her e i s a

gener al t echni que whi ch has to be di s cus sed f i r s t , as i t can s i mpl i f y the

anal ysi s of bi f ur cat i on pr obl ems consi der abl y. Thi s t echni que has the ef f ec t of

i nt r oduc i ng coor di nat e syst ems i n whi ch comput at i ons are mor e easi l y car r i ed

out. Af t er us i ng i t one i s l ef t wi t h a reduced system of di f f erent i al equat i ons

cont ai ni ng al l the qual i t at i ve f eatures of the bi f ur cat i on. I t must be empha

si zed t hat t hi s t echni que, cal l ed cent er mani f ol d t heor y, has a l ocal char ac t er

and i s onl y appl i cabl e to bi f ur cat i ons of si ngul ar poi nt s .

The cent er mani f ol d t heor y pr ovi des a means f or syst emat i cal l y r educ i ng the

di mensi on of the st at e spaces whi ch need to be consi der ed when anal yzi ng bi f ur

cat i ons of a gi ven t ype.

Suppose we have a system of nonl i near or di nar y di f f er ent i al equat i ons:

X =f (X ) X e Rn

f or whi ch f ( 0) = 0.

The l i near i zed syst em at X = 0 may be wr i t t en as

X =Df ( 0) . X

If Df(0) has no eigenvalues with zero real part, then Hartman-Grobman's theorem

s t a tes tha t the eigenvalues , with pos itive and negative real pa r ts , determine

the local phase po rt ra it near X = 0. If ther e are eigenvalues with zero rea l

par ts the to polo gi cal s t ructure can be quite complicated. Some examples in R*

have already been given in th is chapter (paragraph ^ ) .

A so lu t i on of the l i ne ar iz ed system, sat is fy in g the i ni t i al condition

X(t ) = X , X eR is the vector valued functiono o o

X(t;Xo) = exp(t Df(0)) XQ

A general solution of X = Df(0) X i s obtained by l inear super po si ti on of n

linea rl y independent solutions: ( Vf t ) , — v (t )}

n 1X ( t ) = X C. VJ (t)

j = l J



- 3 * » -

wher e C. ( j = 1, . . . n) i s a const ant .

A set of l i near l y i ndependent sol ut i ons can be obt ai ned f r om the r eal and i magi

n a r y p a r t s of t he v ect o r v al ue d f unc t i ons exp( A. t ) XJ wher e X"' are the

( gener al i zed) ei genvect or s associ at ed wi t h the ei genval ues ( r eal or c ompl ex)Aj ( j = 1, . . . n ) .

Var i ous subspaces spanned by ( gener al i zed) ei genvect or s and f i l l ed wi t h sol ut i on

curves may be di st i ngui shed.

nt he st abl e subspace, E : spanned by X' Xs

n

t he unst abl e subspace, E : spanned by X1

... X

nc ct he cent er subspace, E : spanned by X' ... X

ns •wher e X1 , . . . X are n ( gener al i zed) ei genvect or s of whi ch the ei genval uess s shave negat i ve r eal par t s,

nX1 , . . . X are n ( gener al i zed) ei genvect or s of whi ch the ei genval ues have

pos i t i ve real par t s andn

X1 , . . . X are n ( gener al i zed) ei genvect or s wi t h ei genval ues havi ng zer o r eal

par t s .

Si nce the di mensi on of the whol e phase space i s n we have n +n +n = n.

s u e

The adject ives stab le, unstable and center ref lect the behaviour of solutions in

their eigenspaces respectively.

Solutions lying in E a re char ac te riz ed by exponent ial decay (monotonie or

u co sc i l l a to r y) , those lying in E by exponential growth and those lying in E byneither. In Fig. 1.14 we show some examples of invar iant subspaces in R2 and R' .

s uFor nonlinear systems one can define the invariant subspaces W and W being the

s u s unonlinear analogues of E and E res pec t ivel y. The subspaces W and W are

g

f r equent l y denot ed as stabl e and unst abl e mani f ol ds, t hey are t angent to E and

E u at the si ngul ar poi nt wher e f ( X) = 0.

The ex i st ence of t hese mani f ol ds f or nonl i near syst ems i s est abl i shed by the

f ol l owi ng t heor em

- 3 5 -

D f ( 0 ) = ( 2 0l

E u = span{(1, O , <0, 1)}

( b) Df (0) = ( J . »)E u = (1, 0)

E s = (0, 1)

E c = 0

c o D f o , = - , o : . ? )EU=0, ES=( O. O, 1)

E c = span{(1, 0, 0), (0, 1, 0)}

F i g . 1. 14 . Subspaces of l i near f l ows .

St abl e Mani f o l d Theor em f or hyper bol i c s i ngul ar poi nt s

Suppos e t hat X = f ( X ) h a s a hy pe r b ol i c s i ngul ar poi nt X , t hen t her e exi s t

s t abl e and unst abl e mani f ol ds W ( X ) , W ( X ) , of t he s ame di mensi ons n , n aso' o s u

t hose of t he ei genspaces E , E of t he l i near i zed system at X , W (X ) i s

t angent t o E s and WU( X ) i s t angent t o Eu; WS( X ) and WU( X ) ar e as smoot h as

t he f unct i on f .

The pr oof of t hi s t heor em has been gi ven by Har t man ( 1964) and mor e r ecent l y by

Carr ( 1981) .

We not e t hat t hi s t heor em says not hi ng about t he exi st ence of cent er mani f ol dsc r

W f or nonl i near sys t ems . For t he case that t he nonl i near sys t em i s C the

exi stence of cent er mani f ol ds i s est abl i shed i n t he cent er mani f ol d t heor em

whi ch i nc l udes t he r esul t s of t he stabl e mani f ol d theor em The f i r st pr oof of

t he cent er mani f ol d t heor em has been gi ven by Kel l ey ( 1967) .

- 3 6 -

C en t e r M an i f o l d Th eo r em

L e t f b e a C v e c t o r f i e l d o n R v a n i s h i n g a t X = X s o t h a t f (X ) = 0 .o o

T h e s p e c t r u m o f e i g e n v a l u e s A- A o f Df(X ) i s d i v i d e d i n t o t h r e e p a r t s ,a , o , a w it hs u c

A e a i f Re A < 0s

A € a i f Re A = 0c

A 6 a i f Re A > 0u

T h e c o r r e s p o n d i n g g e n e r a l i z e d e i g e n s p a c e s a r e E , E a nd E u r e s p e c t i v e l y . Then

t h e r e e x i s t C s t a b l e a nd u n s t a b l e m a n if o ld s W a nd W t an g e n t t o E an d E a t

X ; r e s p e c t i v e l y an d a C c e n t e r m a n i f o l d W t a n g e n t t o E a t X . T hes u c

m an i f o l d s W , W an d W a r e e a c h f i l l e d w i t h s o l u t i o n c u r v e s .

T he c e n t e r m a n i f o l d t h eo re m i m p l i e s t h a t a n o n l i n e a r s y s t e m f (X ) c a n b e s p l i ts + u c "

u p , n e a r a s i n g u l a r p o i n t X , i n t o t w o s u b s y s t e m s f a n d f o f w h ic h t h e

l i n e a r p a r t s h a v e e i g e n v a l u e s w i t h n o n z e r o a n d w i t h z e r o r e a l p a r t s ,

r e s p e c t i v e l y .Th en o n e may w r i t e t h e s y s t em X = f (X ) n ea r t h e s i n g u l a r p o i n t X : (X ,Y ) f o r

m a l l y a s

X = f C (X,Y) = DfC(X ,Y ) .(X -X ) + F(X,Y)o o o

Y = f S + U ( X , Y ) = D f S + U (X ,Y ) .(Y -Y ) + G(X,Y)0 - 0 O

( 1 . 1 0 )

n n +nc s uwher e X € R , Y e R and F, G are nonl i near hi gher - order t er ms whi ch vani shat ( Xo. Yo) .

S i n c e t h e c e n t e r m a n i f o l d i s t a n g e n t t o E a t (X ,Y ) i t c an b e w r i t t e n ex -

p l i c i t e l y

Y-Y = h( X ) w it h h(X ) = Dh(X ) = 0o o o

T h e c e n t e r m a n i f o l d W may b e v ie w ed a s a p a r t i c u l a r s e t o f s o l u t i o n s o f t h e

o r i g i n a l s y s t e m X =f (X ) , s a t i s f y i n g t h e a d d i t i o n a l c o n s t r a i n t s f (X ) =Df(X ) = 0 .



- 37-

They may be cal cul at ed by subst i t ut i ng Y = Y + h( X) i nt o eq. ( 1. 10) , r es ul t i ng

i n

Dh( X) [ Df C( Xo. Yo) ( X - Xo) + F( X. YQ+h ( X ) ) ]

- Df S+U( XQ, Yo) h( X) - G( X, Y +h( X) ) = 0 ( 1. 11)

with the boundary co n di tio n s h(X ) = Dh(X ) = 0 .

E q u a t i o n ( 1 .1 1 ) i s i n g e n e r a l a p a r t i a l d i f f e r e n t i a l e q u a t i o n w hich c a nn o t b e

solved exact ly in most cases , but i t s solut ion can somet imes be approximated by

a s e r i e s e x p a n s i o n n e a r X . I f s uc h an a p p r o x i m a t e s o l u t i o n i s f o u n d , t h e

o r i g i n a l s ys te m may b e p r o j e c t e d on t h e c e n t e r m a n i f o l d r e s u l t i n g i n t o t h e

following reduced system

X = DfC(X ,Y ).(X-X ) + F(X,Y +h(X)) (1 .1 2)o o o o

Si nce h( X) i s t angent t o E at t he si ngul ar poi nt , t he sol ut i ons of t he r educed

syst em pr ovi de an appr oxi mat i on of ( 1. 10) near X . Thi s concl usi on was obt ai ned,

i ndependent l y, by Henr y ( 1981) and Car r ( 1981) when t hey pr oved t hat t he

pr oper t i es of ( 1. 10) near (X , Y ) and of ( 1. 12) near X ar e t he same.I t i mpl i es t hat t he bi f ur cat i on pr oper t i es of a degener at e s i ngul ar i t y can be

obt ai ned by anal ysi ng t he nonl i near syst em on i t s cent er mani f ol d. Al t hough t hi s

met hod l ooks st r a i ght f or war d, i t s succes depends l ar gel y on t he possi bi l i t y

whet her a cent er mani f ol d can be f ound or at l east appr oxi mat ed i n a sui t abl e

way.

Other d i f f i c u l t ie s th at can appear are the non-uniqueness and los s of smoothness

of s o l u t i o n s in th e ce nt er m anifold. The boundary co n di t io ns h(X ) = Dh(X ) = 0

a re no t su f f i c i en t fo r a un ique so l u t io n of eq . (1 .1 1) .

I n t h e f o l l o w i n g e x a m p l e s we p o i n t o u t som e of t h e s e a s p e c t s ( p o s s i b i l i t i e s ,

l imi ta t ions ) o f the use o f cen te r mani fo lds .

In th e f i r s t example c e n te r mani fo ld theory w i l l be used in o rder to f ind the

phase por t ra i ts of the two-dimensional sys tem

x = EX + xy - x'y = -y + y J - x2

i f t he r eal - val ued par amet er e var i es near zer o.



- 3 8 -

I f e = 0 t h e sy stem h as a d eg en er a te s i n g u l a r p o i n t a t t h e o r i g i n ( 0 , 0 ) ; t h e

c o r r e s p o n d i n g c o e f f i c i e n t m a t r i x o f t h e l i n e a r p a r t i s g iv en by L _1 . The

e igenspaces a re : E s = ( 0 , 1 ) , Ec = ( 1 , 0 ) , Eu = *.

Us ing theorem 65 (paragraph 4) the s ingu la r i ty a t the o r ig in i s a s t ab le topo-l o g i c a l n o d e ; an i n f i n i t e n um ber o f p a t h s a p p ro a c h t h e n od e a l o n g t h e x -

d i r e c t i o n ,

To obt ai n t he bi f ur cat i on sol ut i ons f or c * o we consi der t he cent er mani f ol d of

t he suspended system i n R1

x = ex + xy - x'

'c = 0 ( 1. 13)

y = - y + y* - x 1

where the 'pa ra m eter ' e i s seen as ' t r i v i a l ' dependen t va r i a b l e .

The co ef f ic ie nt ma tr ix of the l i n e ar p ar t becomes for c = 0

0 0 0^

0 0 0

0 0 - 1

wi t h ei genval ues A = A = 0, A

The cor r espondi ng ei genspaces ar e EC = ( 1, 0, 0) x (0 ,1 ,0 ) , Es = ( 0, 0, 1) and E = .

Now we seek a c ent er mani f ol d h = h( x, e) t angent t o E , whi ch i s a sol ut i on of

eq. ( 1. 11) , sat i sf yi ng t he boundar y condi t i ons h( 0, 0) = | ^ ( 0, 0) = | ^ ( 0, 0) = 0.

Equat i on ( 1. 11) becomes i n t hi s case

»3h 9h\ icx - x' + xhi . . , , _I —, —I + h - h2 + x' = 03x 3e' l o '

The cent er mani f ol d wi l l be appr oxi mat ed by

h( x, e) = eye2 + a2xc + a, e* + ^x ' + p2x2e + p xe2 + p c1 + 0( 4)

where 0( 4) denot es t er ms of 0[ ( | x| + | e | ) * ] .

- 3 9 -

Fig. 1.15. Center manifold W : h(x ,e) = -x2 + 2x2e + 0 (4) .

Equa t ing powers o f x ! , x t , . ,

1.15)

. E ' we f in d fo r th e cen te r manifold {see Fig .

W : h (x ,e ) = -x2 + 2x2 e + 0(4 )

The reduced system which governs the b i f u r c a t i o n cau sed by c , fo l lo w s as th e

projection of system (1.13) on W :

ex - 2x3 + 2ex3 + 0( 4) , E = 0

The v e c t o r f i e l d on W i s a o n e - p a r a m e t e r u n f o l d i n g o f t h e o n e - d im e n s io n a l

degeneracy: -2x ' .

The u n f o l d i n g h a s a l s o a s i n g u l a r i t y a t x = 0 a nd tw o s t a b l e n e i g h b o u r i n g

s ingu la r po in t s a t x = ±ff* + 0 (e ' ) but only for e > 0 . The s in gu la r p oin t x = 0

app ears to be s t a b le fo r E < 0 and u n s t ab le fo r e > 0 . The behav iour o f the

reduced system on W enab les us to d raw the t ra jec to r ies in the x ,y -phase p lane

i f e v a r i e s n e a r z e r o . S i n c e t h e u n s t a b l e e i ge n s pa c e E u appeared to be empty,

t h e t r a j e c t o r i e s t e n d t o W for increas ing t . see Fig . 1 .16.

- 40-

(a): C<0 (b): E>0

Fi g. 1. 16. Bi f ur cat i on of : x = ex - x' + xy, y = - y + y' - x2 .

The next exampl e i l l ust r at es t hat a cent er mani f ol d i s not al ways uni que.

Consi der t he t wo- di mensi onal syst em

x = - x' , y = - y

whi ch has a i sol at ed si ngul ar poi nt ( t opol ogi cal node) at t he or i gi n. The cent erc c

ei genspace E i s f or med by ( 1, 0) and t he cent er mani f ol d W : h( X) i s a sol ut i onof

- x' h' + h = 0 wi t h h( o) = h' ( o) = 0

The gener al sol ut i on of t hi s di f f erent i al equat i on i s

h( x) = C. exp( - — )

Si nce t he so l ut i on sat i s f i es h( o) = h' ( o) = 0 i r r espect i ve the val ue of C, i t

must be concl uded t hat an i nf i ni t e number of cent er mani f ol ds exi st s.

Si nc e al l der i vat i ves of exp( - ?— ) vani sh at x = 0 onl y t he cent er mani f ol d

h - 0 i s anal yt i c, and al l ot her cent er mani f ol ds are C .



- 4 1 -

An approach to physica l unfoldings in flow patterns

As announced in the in t r od uc t ion , th i s th es i s dea l s wi th b i fu rc a t i on s in v e c t o r

f ie lds a s they a r i se a s ve loc i ty f i e lds in gas - and f lu id mot ions ; e .g . con ica l

f lows (Chapter I I ) and v i scous f lows re ta rd ed ne a r a smoo th su r f a c e (C hap te rI I I ) . I n g e n e r a l a t s t a g n a t i o n p o i n t s t h e s e v e c to r f i e l d s w i l l h av e i s o l a t e d

s i n g u l a r i t i e s w hich c an d e g e n e r a t e u nd er c e r t a i n c o n d i t i o n s . To o b t a i n a

u n iv e rs a l un fo ld in g o f such a degen e racy one can fol low a ma themat ica l and a

p h y s i c a l a p p r o a c h . The f o r m e r , b e in g v e r y s t r a i g h t f o r w a r d , d e t e r m in e s t h e

u n i v e r s a l u n f o l d i n g o f a d e g e n e r a t e s t a t e on t h e b a s i s of i t s t o p o l o g i c a l

fea tur es such as codlmension , cen ter manifo ld and on the b a s i s of th e c l a s s of

pe r tu rba t ions admi t ted .

In th is way, the comple te b ifurca t ion can be descr ibed with the smal les t numberof p ar am et er s (say u.. , . . . u , , k e R) and gen era l unfo ld ing pr in c ip le s can be

used .

A l th o u g h t h i s a p p r o a c h i s v e r y a t t r a c t i v e i t s a p p l i c a t i o n i n v o lv e s p r o b l e m s o f

d i f f e r e n t k in d .

Fi r s t , degene rac ie s occu r r ing in f low f ie lds appea r sca rce ly in those conven ien t

form s b e in g n e c es s a ry f o r a s t r a ig h t f o r w a r d a p p l i c a t i o n o f u n f o l d i n g t h e o r y .

Th e s e d e g e n e r a c i e s - t o g e th e r w i th i t s u n f o ld in g s - a r e m ore o r l e s s o f p a r

t i c u la r type because the co r re spond ing degenera te so l u t io ns have to s a t i s f y th e

p a r t i a l d i f f e r e n t i a l e q u a t i o n s w h ic h g o ve rn t h e p h y s i c s o f t h e flo w ( e . g .

conservation of mass, momentum and energy).

C o n s e q u e n t l y , i t m eans t h a t we a r e l e s s i n t e r e s t e d i n f i n d in g t h e u n iv e r s a l

unfold ing of the degeneracy but more in those unfold ings which are a l lowed by

the flow eq ua t ion s . Let us c a l l such a perm it ted unfold ing a ph ys ic a l unfo ld ing .

A s ec on d d i f f i c u l t y s te m s from th e p ro ble m of i n t e r p r e t i n g t h e i n v o lv e d p a r a

meters of the universa l unfo ld ing .

The phy sica l in te rp re ta t i on of these parameters i s a ne ces sar y b u t by no means

ea sy ta sk , a s the phy s ica l ly r e lev an t pa ramete r s may be re la te d to the b i fu rca

t ion parameters u1 , . . . a in a very comp licated way. Fu rthe rm ore , th e number of

r e le v a n t ph ys ica l parameters may exceed the number of b i fu rc a t io n parameters so

t h a t a s i n g l e b i f u r c a t i o n p a r a m e t e r may b e c o n s t i t u t e d b y a c o m b i n a t i o n o f

s e v e r a l p h y s i c a l p a r a m e t e r s . To f i n d t h e s e c o m b in a ti on s , o ne n e e ds a d d i t i o n a lin f o rm a t i on ( e .g . from the f low p rob lem) wh ich can no t be d e r i v ed f rom th e

un ive rsa l un fo ld ing .



-l\2-

On t he ot her hand, i f t he number of r el evant physi cal par amet er s i s smal l er t han

t he codi mensi on, i t i mpl i es t he exi st ence of a dependency bet ween t he bi f ur ca

t i on par amet er s. Thi s dependency can si gni f i cant l y r educe t he possi bl e bi f ur ca

t i on sol ut i ons wi t h respect t o t hose gi ven by t he uni ver sal unf ol di ng. Mor eover

i f t he number of physi cal par amet er s i s not known bef or ehand t her e i s a r i skt hat unal l owed bi f ur cat i ons wi l l creep i n, whi ch r emai n unnot i ced, and l ead t o a

di st or t ed v i ew about t r ansi t i on behavi our i n f l ow pat t er ns.

I n a physi cal appr oach, t he above ment i oned di f f i cul t i es ar e ci r cumvent ed si nce

a degener at e st at e i n the f l ow pat t er n and t he cor r espondi ng physi cal unf ol di ng

appear si mul t aneousl y i f t he gover ni ng f l ow equat i ons are eval uat ed near si n

gul ar poi nt s. Act ual l y, t he eval uat i on r esul t s i nt o appr oxi mat e sol ut i ons whi ch

sat i sf y t hese equat i ons t o a cer t ai n or der near t he si ngul ar i t y. These appr oxi mat i ons cont ai n sever al unknowns whi ch cannot be det er m ned i n a l ocal

anal ys i s .

For speci f i ed val ues of t he unknowns t he si ngul ar i t y at t ai ns a degener at e st at e

i ndi cat i ng t hat t he i nvol ved unknowns can be vi ewed as bi f ur cat i on par amet er s.

Then t he physi cal unf ol di ng of t he degener at e s i ngul ar i t y f ol l ows qui t e easi l y

by var yi ng t he i nvol ved par amet er s wi t h r espect t o bi f ur cat i on val ues at whi ch

t he degener at e si ngul ar i t y was f ound.

The aut hor i s i ncl i ned t o f avour t he physi cal appr oach as l ong as t he emphasi s

l i es on l ocal bi f ur cat i ons i n vec t or f i el ds t hat are gener at ed by phys i cal

syst ems.



- i » 3 -

8. References

Andr onov, A. A. , Leont ovi ch, E. A. , Gor don, I . I . and Mai er , A. G. ( 1973)

Qual i t at i ve t heor y of second- or der dynam cal syst ems. W l ey, NewYor k.

Coddi ngt on, E. A. and Levi nson, L. ( 1955)

Theory of or di nar y di f f er ent i al equat i ons. McGr aw- Hi l l , NewYor k.

Carr, J . ( 1981)

Appl i cat i ons of cent er mani f ol d t heor y. Spr i nger - Ver l ag, NewYor k , Ber l i n.

Guckenhei mer , J . and Hol mes, P. J . ( 1983)

Nonl i near osci l l at i ons, dynam cal syst ems and bi f ur c at i on of vect or f i el ds .Spr i nger - Ver l ag, NewYor k, Ber l i n, Hei del ber g, Tokyo.

Har t man, P. ( 1964)

Ordi nary di f f er ent i al equat i ons. W l ey, NewYor k.

Henr y, D ( 1981)

Geomet r i c t heor y of sem l i near par abol i c equat i ons . Lect ure not es i n mat he

mat i cs, Vol . 840. Spr i nger - Ver l ag, NewYor k, Hei del ber g, Ber l i n.

J or dan, D W and Sm t h, P. ( 1977)

Nonl i near or di nar y di f f er ent i al equat i ons. Cl ar endon Press, Oxf or d.

Kel l ey, A. ( 1976)

The stable , center st ab le , c ente r, cen ter unst able and uns table manifo lds.

J. Diff. Eqns, 3, 5**6-570.

Reyn, J.W. (1964)

Cl as si f ic at io n and de sc r ip ti on of the singu la r po in ts of a system of three

linear differential equations. J. Appl. Math. Phys., 15, 5^0-557•

Shirer, H.N. and Wells, R. (1983)

Mathematical structure of the singularities at the transitions between steady

states in hydrodynamic systems. Lecture notes in physics. Vol. I85. Springer-

Verlag, Berlin, Heidelberg, New York, Tokyo.



- 4 5 -

Chapter II Topology o f con ica l f low pat terns

1. Introduction

1 1. Concept s and def i ni t i ons

The qual i t at i ve t heor y of dynam cal syst ems wi l l be appl i ed t o t hr ee- di mensi onal

i nvi sci d f l ows wi t h coni cal symmet r y. Such f l ows, whi ch ar e cal l ed coni cal

f l ows , have t he speci f i c pr oper t y t hat t he vel oci t y and t he quant i t i es def i ni ng

t he st at e of t he gas, e. g. pr essur e and t emper at ur e, ar e const ant al ong r ays

emanat i ng f r om a common poi nt i n t he physi cal space. Thi s poi nt i s cal l ed t he

cent er of t he coni cal f i el d.

I nvi sci d coni cal gas f l ows embody an i mpor t ant cl ass of f l ows ar ound coni cal

conf i gurat i ons of pr ac t i cal i nt er es t , such as del t a wi ngs hav i ng a rbi t r ar y

c r os s - se c t i o n, i nl et c onf i gur at i ons , nose cones , wi ng- body j unc t i ons and

i nt er nal f l ows i n nozzl es and di f f usor s.

Due t o t he coni cal symmet r y, t hese f l ows can be descr i bed by f unct i ons of t wo

i ndependent var i abl es def i ni ng t he posi t i on of t he rays . The st r uc t ur e of aconi cal f l ow may be repr esent ed on a uni t spher e cent er ed at t he coni cal cent er .

The vel oci t y vect or V may be decomposed i nt o a r adi al component V nor mal t o t he

uni t spher e and a component V t angent t o i t ; t he l at t er def i nes a vect or f i el d

on t hi s spher e. I nt egr at i on of t hi s vector f i el d yi el ds l i nes on t he uni t spher e

whi ch wi l l be cal l ed coni cal st r eam i nes. The physi cal s i gni f i cance of a coni cal

s t r eam i ne may be expl ai ned qui t e eas i l y . Cons i der t he spat i al s t r eam i nes

pass i ng t hr ough a common r ay. Si nce t he f l ow i s assumed to be coni cal t hese

st r eam i nes f or m a coni cal st r eamsur f ace, whi ch i s bui l t up by sequent i al r aysemanat i ng f r om t he coni cal cent er . Taki ng the i nt er sect i on of a coni cal str eam

sur f ace wi t h t he uni t spher e, a coni cal st r eam i ne, i s obt ai ned.

Si m l ar l y, coni cal s t r eam i nes can be def i ned on a pl ane sur f ace Z at uni t

di stance f rom t he coni cal cent er . Such a pl ane i s usual l y cal l ed a cr oss f l ow

pl ane and t he coni cal st r eam i nes f ol l ow as t he i nt er sect i ons of coni cal st r eam

sur f aces wi t h Z. Anal ogous t o t he si t uat i on on t he uni t spher e, coni cal st r eam

l i nes i n t he cr oss f l ow pl ane Z can be t aken as i nt egr al cur ves of a vector

f i el d i n Z. Thi s vec t or f i el d may be obt ai ned i f t he vel oc i t y vec t or V i s

decomposed nonor t hogonal l y i nt o a r adi al component and i nt o a component i n t he

cr oss f l ow pl ane. The l at t er , whi ch i s cal l ed cr oss f l ow vel oc i t y , y i el ds t he

vector f i el d i n the cross f l ow pl ane.



- 46-

Poi nt s i n t he cr oss f l ow pl ane and on t he uni t spher e wher e t he cr oss f l ow

vel oc i t y vani shes, t hus t he vel oc i t y has onl y a radi al component , are cal l ed

coni cal s tagnat i on poi nt s . These poi nt s t he vector f i el d ar e a cr i t i cal poi nt s

wher e t he coni cal st r eam i ne di r ect i on i s undet er m ned.

l ; 2i _A_br i ef _sur vey_of _coni cal f l ow. t heor y

The ear l y t heor et i cal wor k on coni cal f l ows i s l ar gel y based on l i near i zed

t heor y and or i gi nat es f r om t he wor k of Busemann ( 1929) on ci r cul ar cones i n

super soni c f l ow. Fr om t hat t i me l i near t heor y of coni cal f l ows i s devel oped and

appl i ed t o conf i gur at i ons of aer odynam c i nt er est by sever al aut hor s. Much of

t he ear l y work cul m nat ed i n t he st udi es of Ger mai n ( 1949) and Gol dst ei n & War d

(1950). The devel opment of a nonl i near t heor y f or coni cal f l ows st ar t s wi t h t hework of Tayl or & Maccol l ( 1933) when t hey st udi ed t he axi symmet r i c f l ow ar ound a

c i r cul ar cone.

A gr eat var i et y of t heoret i cal aspects of nonl i near coni cal f l ows i s t reat ed by

Bul akh i n t he l at e f i f t i es . He concent rated hi s e f f or t s on t he f or mul at i on of

boundar y val ue pr obl ems and on f i ndi ng appr oxi mat e sol ut i ons f or t hem These

sol ut i ons ar e di scussed by Bul akh wi t h speci al emphasi s t o t hei r pr oper t i es and

si ngul ar behavi our , i n part i cul ar on t he Mach- cone and near shock waves. A

f ur t her s t udy on t he pr oper t i es of nonl i near i sent r opi c coni cal f l ows i sper f or med by Reyn ( i 960) , wher e he st udi ed t he sol ut i on of t he nonl i near par t i al

di f f er ent i al equat i on f rom t he poi nt of vi ew of di f f erent i al geomet ry . I n doi ng

so Reyn comment ed and cl ar i f i ed cer t ai n di scr epanci es, occur r i ng i n t he f l ow

ar ound a f l at del t a wi ng wi t h super soni c l eadi ng edges as gi ven by Masl en ( 1952)

and Fowel l ( 1956) .

Al l t hese i nvest i gat i ons cont r i but ed s i gni f i cant l y t o t he t heor y of nonl i near

coni cal f l ows whi ch, t o a cer t ai n ext ent , has reached a f i nal f orm by t he publ i

cat i on (i n Russi an) i n 1970 of Bul akh' s book: ' Nonl i near Coni cal Gas Fl ows' .

Ever si nce t hat t i me t he mai n at t ent i on di ver t s t o the appl i cat i on of numer i cal

met hods t o obt ai n sol ut i ons f or f l ows ar ound cones and del t a wi ngs at i nci dence.

Al t hough t hese met hods gi ve accur at e sol ut i ons at moder at e i nci dence, f or l ar ger

angl es of at t ack t wo f undament al di f f i cul t i es ar i se whi ch pr event a st r ai ght

f or war d numer i cal sol ut i on.

F i r s t , as t he angl e of at t ac k i s i nc r e as ed, t he c r o ss f l ow v el oc i t y wi l l

i ncr ease t o super soni c val ues i n some par t s of t he f l ow f i el d. Thi s changes the

nat ur e of t he governi ng par t i al di f f erent i al equat i ons f rom el l i pt i c t o a m xed

el l i pt i c- hyper bol i c t ype and r esul t s i n the appear ence of a coni cal supersoni c

r egi on t erm nat ed by an embedded shock wave.



-kl-

Anot her di f f i cul t y i s t hat some f l ow pr oper t i es such as ent r opy, densi t y and

radi al vel oci t y can become mul t i val ued at coni cal st agnat i on poi nt s . Si nce

ent r opy gr adi ent s can be i dent i f i ed wi t h rot at i onal i t y of t he f l ow ( Crocco' s

t heor em , t hese coni cal st agnat i on poi nt s can appear as vor t i cal s i ngul ar i t i es.

The pr esence of vor t i cal s i ngul ar i t i es , whi ch was a sever e obst ac l e f or ast r ai ght f or war d numer i cal cal cul at i on, see f or exampl e St ocker and Mauger

(1962) , di d r evi ve t he at t ent i on to nonl i near coni cal f l ow t heory wi t h speci al

emphasi s on t he coni cal f l ow st r uctur e near coni cal st agnat i on poi nt s. I t i s of

val ue t her ef or e, t o eval uat e as syst emat i cal l y as possi bl e by means of a l ocal

anal ysi s, t he possi bl e f l ow st r uctur es near such poi nt s, so t hat i n a par t i cul ar

f l ow pr obl em t he qual i t at i ve basi s f or a numer i cal pr ocedur e can be sel ected

wi t h mor e cer t ai nt y.

I n coni cal f l ows wi t h ent r opy gr adi ent s t he ent r opy r emai ns const ant on coni cal

str eam i nes. Then, i f i n a coni cal st agnat i on poi nt var i ous st r eam i nes mer ge, a

vor t i cal si ngul ar i t y i s f or med. Thi s conj ecture was put f or war d by Fer r i ( 1951)

when he di scussed t he super soni c f l ow past a ci r cul ar cone at i nci dence.

Si nce t he appear ance of Fer r i ' s paper i nvest i gat i ons of t he f l ow near coni cal

st agnat i on poi nt s show an emphasi s of i nt er est i n the possi bl e coni cal st r eam

l i ne pat t er ns t oget her wi t h t he rel at ed pr essur e di st r i but i ons near such poi nt s.

Mel ni k ( 1967) const r uct ed some appr oxi mat e sol ut i ons of t he nonl i near i nvi sci d

coni cal f l ow equat i ons near coni cal st agnat i on poi nt s at t ached t o a body sur

f ace. These sol ut i ons i nvol ve ent r opy gr adi ent s i n t he f l ow. When t he st r eam i ne

pat t er n i s r el at ed t o the cor r espondi ng pr essur e di s t r i but i on on the body

sur f ace no uni que cor r espondence was f ound. Bakker ( 1977) showed t hat f or t hese

sol ut i ons a uni que cor r espondence may be obt ai ned i f t he pr essur e di st r i but i on

nor mal t o t he body sur f ace i s al so t aken i nt o account . Bot h i nvest i gat i ons

i ndi cat e t hat t he pr esence of ent r opy gr adi ent s does not af f ect t he t opol ogi calst r ucture of t he coni cal st r eam i ne pat t er n t hat cor r esponds t o a gi ven pr essur e

di s t r i but i on. Thi s r esul t was f ur t her conf i r med i n t he spec i al case of the

coni cal st agnat i on poi nt s i n t he f l ow past sl ender c i r cul ar cones at hi gh i n

ci dence, as cal cul at ed usi ng sl ender body theor y ( Sm t h 1972) , or l i near i zed

t heory ( Bakker & Banni nk 1971* ) .

I t i s of i nt er est , t her ef or e, t o make a f ur t her study of t he topol ogi cal str uc

t ur e of coni cal f l ows near coni cal st agnat i on poi nt s usi ng t he ass umpt i on ofpot ent i al f l ow. An advant age of t hi s appr oach i s t hat t he nonl i near coni cal f l ow

equat i ons r educe to a si ngl e second- or der equat i on f or t he coni cal pot ent i al f or

whi ch sol ut i ons ar e si mpl er t o obt ai n. Mor eover i n a coni cal st agnat i on poi nt

t hi s equat i on becomes Lapl ace' s equat i on whi ch i s al so sat i sf i ed by t he vel oci t y



- 4 8 -

p o t e n t i a l i n i n c o m p re s s i b l e p l a n e f l o w . S t a g n a t i o n -p o i n t s o l u t i o n s fo r i n c o m

p r e s s i b l e p l a n e f l ow s a r e t h e n u s e d a s a g u i d e t o c o n i c a l s t a g n a t i o n p o i n t

so lu t ions . S ince con ica l s t agna t ion po in t s appea r a s c r i t i ca l po in t s i n a vec to r

f i e l d i n R2 t h e q u a l i t a t i v e t h e o r y of d y n a m i ca l s y s t e m s , a s o u t l i n e d i n t h e

p rev ious ch ap te r , appears t o be pa r t i c u l a r ly use fu l .

l12;_52Di5§i_5t reaml ines 1_conica l_stagfnat ion_Doints

We w i l l d i s c u s s some exam ples of con ica l f lows and consider the corresponding

topologica l s t ruc tures in the cross f low plane S . Let the f low be descr ibed by a

r ig h th an de d c a r t e s i a n coord ina t e sys tem x ,y ,z and l e t t he ve lo c i ty vec to r V be

decomposed into i ts components u,v,w along the x, 'y and z axes, respect ively.

The conica l center wi l l be taken a t the or ig in (0 ,0 ,0) and the cross f low plane1 is chosen normal to the x-axis . S ince the f low is conica l i t i s advantageous

t o u s e t h e c o n i c a l c o o r d i n a t e s n = y / x , t, = z /x to d es c r ib e the c ross flow

pa t t e rn in the p l ane I.

I f t h e v e l o c i t y v ec to r V = u i + v j + w k i s decomposed nono rthogonally i n to a

component g alon g the ray r = x i + y j + z k and a component g in the p la n e

X, then the l a t t e r de f ines the c ross f low ve loc i ty vec to r .

The d ir e c t i o n of q_ may be obtain ed as fol lo w s.

Because q_ l ies in I i t can be wri t ten as

g = o . i + v j + w k

si m l ar l y t he component al ong t he ray i s gi ven by

ls rlq = . (x i + y j + z k)" r | r |

>

The cr oss f l ow component s v and w f ol l ow f r om t he r equi r ement '

Y = 92 + 3r

whi ch yi el ds

l 9rl

k = u/ x

IsJr 'v = v - y = v - uu2 k |



- 49-

' 3 r 'w_ = w - z = w - UC2 k |

Thi s i mpl i es t hat t he coni cal st r eam i nes i n T ar e sol ut i ons of t he equat i on

dn _ v - uudC w - uC

or of t he equi val ent syst em

n = V j = v - un, C = wz = w - uQ ( 2 . 1 )

wher e t he dot r epr esent s di f f er ent i at i on wi t h respect t o a paramet er al ong t he

str eam i nes.Equat i ons ( 2. 1) wi l l be consi der ed as a dynam cal syst emi n R2 whose phase

t raj ectori es coi nci de wi t h t he coni cal s t r eam i nes i n t he cr oss f l ow pl ane.

Coni cal st agnat i on poi nt s occur on t hose r ays wher e t he vel oci t y vector i sal i gned w t h a l ocal r ay, yi el di ng

g = 0 or v - un = w - uC = 0

Coni cal st agnat i on poi nt s may be i dent i f i ed wi t h t he si ngul ar poi nt s of system(2 .1 ) ; they serve as t he i ndi spensabl e f l ow el ement s whi ch may be used t o com

pose mor e compl i cat ed coni cal f l ow pat t er ns. I n that way t he knowl edge about

di f f er ent t ypes of coni cal st agnat i on poi nt s cont r i but e t o t he underst andi ng oft he t opol ogy of coni cal f l ow pat t er ns of i ncr eas i ng compl exi t y. To obt ai n acer t ai n f am l i ar i ty wi t h t he char acter i st i c f eat ur es of coni cal f l ow t opol ogy weproceed by di s cus si ng f i r s t t he most si mpl e exampl e of a coni cal f l ow: t heuni f orm paral l el f l ow.

Consi der a uni f orm paral l el f l ow wi t h a const ant vel oci t y u i n t he pos i t i ve x-di rect i on. Thi s f l ow w l l be consi der ed i n t he cr oss f l ow pl ane I . Si nce u = u ,v = w = 0 t he coni cal st r eam i nes i n I may be obt ai ned f r om Eq. ( 2. 1) as

n = - u n; C = - u C

This sys tem has the genera l so l u t io n !; = con s t , n . The co n i ca l s t r e a m l in e s a r e

s t r a i g h t l i n e s p o i n t i n g t o t h e o r i g i n w here a c o n i c a l s t a g n a t i o n p o i n t i sformed, see Fig . 2 .1 .



- 51-

We have seen, i r r espect i ve of t he or i ent at i on of t he cross f l ow pl ane, t hat t he

uni f or m f l ow i n t he cr oss f l ow pl ane i s uni quel y r epr esent ed by a st ar l i ke node.

However t he opposi t e t hat any st ar l i ke node i n 1 r ef l ect s a uni f or m f l ow i s i n

gener al not cor r ect . Thi s may be ver i f i ed easi l y i f one consi der s t he exampl e of

an axi symmet r i c f l ow wher e ever y pl ane t hr ough t he x- axi s i s a st r eamsur f ace.Si nce the vel oci t y nor mal t o the st r eamsur f ace i s zero, t he coni cal st r eam i nes

must be st r ai ght l i nes f or m ng a st ar l i ke node at t he or i gi n ( 0 , 0 ) .

l ; 1_Tr ansi t i on_phenomenae i n coni cal _f l ow_gat t er ns

Consi der a ci r cul ar cone wi t h sem apex angl e 8 submer ged, at i nci dence a, i n a

super soni c f l ow wi t h f r ee st r eam Mach number M . The sem apex angl e, i nci dence

and f r ee st r eam Mach number ar e such that a coni cal bow shock or i gi nat es f r om

t he apex of t he cone, see Fi g. 2. 2. The f l ow f i el d wi l l be descr i bed i n a r i ght

handed car t esi an coor di nat e syst em x, y, z such t hat t he x- axi s coi nci des wi t h t he

cone axi s and t he f l ow f i el d i s symmet r i cal wi t h respect t o t he x- z- pl ane. The

cr oss f l ow pl ane 2 wi l l be chosen nor mal t o t he x- axi s at uni t di st ance f rom t he

cone apex {coni cal cent er ) .

Fi g. 2. 2. Ci r cul ar cone at i nci dence i n super soni c f l owwi t h at t ached bow shock.

The boundar y condi t i on on t he cone sur f ace r eads

(v - un) T + (w - uq) C = 0

For a = 0 t he f l ow i s axi symmet r i c havi ng a bow wave wi t h ci r cul ar cr oss sect i on

i n 2. The coni cal st r eam i nes out si de and i nsi de t he bow wave ar e st r ai ght l i nes

poi nt i ng t o t he or i gi n ( 0 , 0 ) .

- 52 -

Out s i de t he bow wav e t he f l o w i s u ni f o r m a nd i n s i d e of i t we h av e a T a y l o r -

Macc o l l f l ow. S i nce t he f l ow i s ax i symmet r i c wi t h no vel oc i t y component nor ma l

t o p l a ne s t h r o ug h t he x - a xi s t he c r o s s f l o w c o mpo ne nt s obey t h e r e l a t i o n

( v - u n ) <; - ( w - u C ) n = 0. T he bo und ar y c ondi t i on i mpl i es t h at on t he cone

s ur f ac e bot h e qu at i o ns c a n be s a t i s f i e d o nl y i f t he c r o s s f l ow v ec t o r q =(v - un , w - u O vani shes i dent i cal l y . As a consequence the cone sur f ace i s the

uni on o f coni c al s t agnat i on p oi n t s wh i c h ma y b e c a l l e d a c o ni c a l s t a gn at i o n

l i n e .

Al l c oni c al s t r e am i nes t er m nat e per pendi c ul ar l y a t t he c oni c al s t agnat i on

l i ne, s ee F i g. 2. 3a.

I f t he i nc i denc e i s sl i ght l y i nc r eas ed t he t opol ogy of t h e f l ow p at t e r n a l t e r s

dr as t i c al l y ; t he c oni c al s t r eam i nes enc ompas s t he c one s ur f ac e and t er m nat e att he l ees i de i n a common poi n t , s ee F i g. 2 . 3. At modera t e i nc i dence ( say a / 8 <

1) and i r r es pec t i v e o f t he i nf l uenc e of v i s c os i t y , t wo c oni c al s t a gnat i on poi nt s

may be obser ved i n t he symmet r y pl ane, one at t he wi ndwar d and anot her at t he

l eewar d genera t o r o f t he cone. The coni cal s t agnat i on poi nt a t t he wi ndwar d s i de

i s a s addl e poi n t , wher eas at t he l eewar d s i de a no da l poi nt ( vor t i c al s i ngul a r

i t y ) a ppear s .

i ncreasi ng i nci dence

(a ) (b) (c) (d)

v iscous ?

flow ,'

?►

( e) ( f )

F i g. 2 . 3- Co ni c al f l ow pat t e r n s f o r c i r c ul ar c one at i nc i denc e.



-55 -

2. Local conical stagnation point solutions in irrotational flow*)

2 1 l ^ _Con i ca l po t en t i a l _equa t i on

Consi der t he f l ow of an i nvi sci d, non heat conduct i ng, per f ect gas wi t h r at i o ofspeci f i c heat s Y = c / c .p vThe f l ow i s assumed t o be i r r ot at i onal so t hat a vel oci t y pot ent i al $ may be

i nt r oduced by V$ = V( u, v, w) .

The coni cal s i m l ar i t y al l ows t he i nt r oduc t i on of a coni cal pot ent i al F(n, t ; )

f r om whi ch t he vel oci t y component s may be der i ved by

V = V( x F( n. O) ( 2. 2)

From t he conservat i on l aws ( mass and moment um and t he pr essur e- densi t y r el at i onf or i sent r opi c f l ow, i t may be der i ved t hat F sat i s f i es t he nonl i near second-

or der par t i al di f f er ent i al equat i on, Bul akh ( 1970)

A F + 2B F _ + C Fr _ = 0 ( 2. 3)n u nC CC

wi t h

A = c2 ( 1+n2) - (v-un)2

B = c2 nC - ( V- UTI ) ( w- uO

C = c2 (1+; 2) - ( w- uq)2

wher e t he speed of sound c i s r el at ed t o t he vel oci t y by

c2 = ^ ( q2 - u2 - v2 - w2) ( 2 . 4 )2 max '

and q i s t he maxi mum speed, whi ch we ass ume t o be const ant t hr oughout t hemax ^ °

f l ow f i el d. Equat i ons ( 2. 3) and ( 2. 4) al l ow vel oci t i es t o be nondi mensi onal i zedby q ; as a r esul t we put q = 1 so t hat ( 2. 4) becomes .max max

= - ^ (1 - u2 - v2 - w2)" 2

The vel oci t y component s u. v. w f ol l ow f r om ( 2. 2) as

C F C= F -

v = F

w = F"

n FTl

* ) A conci se ver si on of t hi s par agr aph was publ i shed by Bakker , Banni nk & Reyni n t he J our nal of Fl ui d Mechani cs ( 1981) , Vol . 105, PP- 239- 260.

- 5 6 -

Then t he coni cal st r eam i nes obey t he equat i on

I> = V - U71 = -T ) F + ( 1+ T 1 2 ) F + TlC F

q = w - uq = - q F + nq Fn + ( 1+q* ) F

I n t he cour se of t hi s study i t appear s al so conveni ent t o use pol ar coor di nat es

n = p cos <p, C = P si n 0) i n t he c r oss f l ow pl ane X; then

t he vel oci t y component s become

u = F - p F ,P

v = F c o s <p - - F s i n <p,P p <P '

w = F s i n <p + — F c o s < fP P <P

and ( 2. 3) may be wr i t t en as

f cMl +P2) " [ pF- ( l +p*) F y) F + 2( F- ( p + -) F I f - F - V F 1 F +*• v r ' L r v r ' pJ ' pp l , r p ' p J l p p

2 . 2 . Coni cal _st agnat i on_goi nt sol ut i ons

Appr oxi mat e sol ut i ons of t he coni cal pot ent i a l equat i on ( Eqs. 2 . 3. 2. 5) near

coni cal st agnat i on poi nt s ar e obt ai ned by Bakker , Banni nk & Reyn (1981). These

sol ut i ons ar e l ocal l y val i d and ar e f ound i n t he f orm of a ser i es expansi on i n

t erms of t he di st ance to t he coni cal st agnat i on poi nt . The coor di nat e system i s

chosen wi t h i t s or i gi n i n a coni cal s t agnat i on poi nt . The f ol l owi ng ser i es

expansi on of t he coni cal pot ent i al i s used

F = Fo( l + pn F (<p) + pm F (<p) + o( pm) } , 1 < n < m

- 5 7 -

where F i s a const ant which equals the non-dimensionalized ra dial velocityo

component in the conical stagnation point.

The magnitude of F has a signif icant influence on the conical st reamline

pat ter n. For supersonic conical stagnation points we have c < |F | < 1, where

c i s the speed of sound at the conical stagnation po int. The solut ion F s F

yields a uniform parallel flow with a conical stagnation point at p = 0.

The condition n < m provides p F (q>) as the leading term in the expansion and

the condition 1 < n assures that p = 0 is a conical stagnation po int.

If Eq. (2.3) is evaluated in p = 0 (or n = q = 0) the relation

^„(0.0) FK(0 .0) = 0

r esul t s whi ch s hows t hat near the coni cal st agnat i on poi nt Eq. ( 2. 3) i s near l y

t he Lapl ace equat i on, whi ch i s al so s at i s f i ed by the vel oc i t y pot ent i al $ i n

i ncompr essi bl e pl ane f l ow. Ther ef or e, the pr oposed expansi on for F i s suggest ed

by i ncompr essi bl e pl ane st agnat i on poi nt sol ut i ons whi ch are gi ven by

$ = a p cos n 1n n

wher e a ,t|i are const ant s, t hey r epr esent the wel l known f l ows i n cor ner s wi t hopeni ng angl es w = n/ n.

I f the expansi on for F i s substi t ut ed i nt o the coni cal pot ent i al equat i on, Eq.

( 2. 5) and the r es ul t s are or dered wi t h r espect to power s i n p, the coef f i ci ent

of the l owest - or der t er m equat ed to zer o yi el ds

F" +n* F =0n n

w i t h the s o l u t i o n s

F (<P) = e cos(n ) , n > 1 (2.6)n n n

wher e e and t|> are ar bi t rar y const ant s .n Tn

Thus, t hi s t erm exactl y equal s the st agnat i on poi nt sol ut i on for pl ane f l ow.

Si nce a f r eedom i s l ef t i n the choi ce of the coor di nat e sys t em we may s t i l lro tate i t ar ound the x- axi s such t hat t(i =0.

n

- 59-

range of n

1 < n < 2

n = 2

n > 2

mc

3n - 2

4

n + 2

Tm

n' M2 e'o n4(2n-l)

c7° (M2 - 1 + te2 M2 16 l o n o '

n(n-l) (1P-1) En

2(n+2)

6m

0

G2 M

2

n o2

0

Tabl e: m , T , <5 .c m m

The sol uti ons, di scussed so far, are obtai ned for i nteger as wel l as f or non-

i nteger val ues for n and/ or m For non- i nteger val ues of one of these exponents,

the sol uti ons are not 2n-peri odi c; i mpl yi ng that i t i s i mpossi bl e to f i l l out a

ful l nei ghbourhood of the stagnati on poi nt w th sol uti ons such that the vel oci ty

i s conti nuous. Thi s i mpl i es that sol uti ons whi ch are not 2TI - per i odi c have onl y

physi cal rel evance f or coni cal stagnat i on poi nts whi ch are l ocated on thecontour of a coni cal body. Domai ns of mul t i pl i ci ty of the sol ut i ons are t hen

masked by the i nteri or of the body so that a si ngl e-val ued f l ow f i el d outsi de

the body remai ns. I n general thi s resul ts i n f l ow f i el ds over part i al l y smooth

bodi es w th di sconti nui ti es i n the sl ope where coni cal stagnati on poi nts occur.

Wth the ai d of the l i sted sol ut i ons for the coni cal vel oci ty potent i al the

coni cal streami ne pattern near coni cal stagnati on poi nts may nowbe determned.

I n addi ti on, the pressure di stri buti on wl l be determned and compared w th the

streami ne patterns.

I n i sentropi c f l ows the pressure p i s sol el y a functi on of the speed | v| and

fol l ows di rectl y fromBernoul l i ' s equati on

T-l p=const.

and Poi sson' s rel ati on

p =const, p



- 6 0 -

I f t he const ant s are eval uat ed i n t he coni cal st agnat i on poi nt t hese equat i ons

may be wr i t t en as

T- lYp_i 1 - ( u* ■■ v 8 w' )

lpJ = 1 - u2 (2.8)

wher e zero subscr i pt s r ef er t o condi t i ons i n the coni cal st agnat i on poi nt . Equa

t i on ( 2. 8) shows t hat t he i sobar s i n t he cross f l ow pl ane 2 coi nci de w t h con

st ant vel oci t y cur ves so t hat t hey can be det er m ned f r om

(—]v

dCp=const . 33n

l yl2 - uuq - wg - ww

|V|"uu + w + wwn r\ n

Usi ng u = F - nv - qw t o cal cul at e u and u , t he i sobar pat t er n i s obt ai ned

from

_ -(v-un) vc - ( w- uC) wql d£/ p=const . ( V- UTI ) v + (w ut; ) w (2.9)

Fr om t hi s equat i on i t may be obser ved t hat t he i sobar pat t er n i n t he cross f l ow

pl ane 2 has a si ngul ar i t y i n any poi nt ( rut ) i f

( v- un) v + ( w- uO w = 0

( v- un) v + ( w- uO w = 0' Tl Tl

t hus ei t her

q = 0 orv v^

w w,= 0

A coni cal st agnat i on poi nt al ways cor r esponds t o a si ngul ar i t y i n t he i sobar

pat t er n. The r ever se stat ement i s not t r ue, si nce i f t he second condi t i on i s

sat i sf i ed ( 2. 9) has a cri t i cal poi nt but q i s not equal t o zer o.

- 6 1 -

3.Classification of conical stagnation points in co nical flow

3 i l 1 _ F i r s t - o r d e r _ c o n i c a l _ s t a g n a t i o n _ p o i n t s

W i th t h e s o l u t i o n s , g i v en i n p a r a g ra p h 2 , we w i l l i n v e s t i g a t e t h e t h r e e c a s e sl < n < 2 , n = 2 a nd n > 2 s e p a r a t e l y . The d i s c u s s i o n w i l l be c o n c e n t r a t e d on

t h e d e s c r i p t i o n o f t h e r e s u l t i n g c o n i c a l s t r e a m l i n e p a t t e r n s a n d r e l a t e d

p r e s s u r e d i s t r i b u t i o n s ; f o r f u r t h e r d e t a i l s a b ou t t h e a n a l y s i s we r e f e r t o t h e

o r i g i n a l p u b l i c a t i o n o f B a k k e r , B an n in k & R ey n ( 1 9 8 1 ) .

C a s e 1 < n < 2 : o b l i q u e s a d d l e p o i n t s

S u b s t i t u t i o n o f t h e s o l u t i o n s f o r F (<p) a n d F (<p) i n t o t h e e q u a t i o n f o r t h e

c o n i c a l s t r e a m l i n e s y i e l d s

n+ 1 , „ , m+l>, n c p c o s n n . . , m . v '

-n c p s i n n<p + 0{p )

I n t r o d u c i n g a p a r a m e t e r T a l o n g t h e s t r e a m l i n e s , we i n v e s t i g a t e t h i s e q u a t i o n a s

a sys tem in the (p ,<p) p lane . Then

dp 3 _ n „ , m-n+1,-p- = n e p co s nip - p + 0 (p )

d < p . „ , m-n.T~ = -n e s i n n<p + 0 (p )dT n '

knT h e s i n g u l a r p o i n t s o f t h i s s y st em o n p s 0 a r e i n 9 = — (k = 0 , ± 1 , ± 2 , . . . )

a nd t h e c o e f f i c i e n t m a t r i x o f t h e l o c a l l y l i n e a r i z e d s y s t e m h a s a t r a c e

p = i n e ( 1 - n ) , a n d a J a c o b i a n q = - n ' e* . S i n c e q < 0 t h e s i n g u l a r p o i n t s a r eh y p e r b o l i c s a d d l e p o i n t s a nd t h e h i g h e r o r d e r t e r m s d o n o t c h a n g e t h e s a d d l e

p o i n t c h a r a c t e r o f t h e s e s i n g u l a r p o i n t s . T h e r e fo r e t h e l o w e s t o r d e r t er m s a r e

s u f f i c i e n t t o d e t e r m i n e t h e a p p r o xi m a t e s h a p e o f t h e c o n i c a l s t r e a m l i n e s :

p n s i n n<P = C, 1 < n < 2

wher e C i s const ant al ong a st r eam i ne.

Thi s equat i on r epr esent s t he st r eam i ne pat t er n f or a f l ow i n a cor ner w t h ani ncl udi ng angl e (o = n/ n; t hus n/ 2 < <> < n f or 1 < n < 2. The f l ow i n t he cor ner

has a saddl e poi nt behavi our and t he poi nt i s cal l ed an obl i que saddl e poi nt ;

i t s separ at r i ces f orm t he i nc l udi ng angl e <D = n/ n at t he coni ca l st agnat i on

poi nt . Si nce 2n i s not an even mul t i pl e of n/ n an obl i que saddl e poi nt can occur

- 6 2 -

o n l y on a b o d y . c o n t o u r w i t h o n e o r m ore s t r e a m l i n e s i n t h e s o l u t i o n t h a t

c o i n c i d e w i t h t h e b o d y .

T he p r e s s u r e d i s t r i b u t i o n n e a r t h e c o n i c a l s t a g n a t i o n p o i n t may b e o b t a i n e d from

( 2 . 8 ) a s

p o

T- lT 1 T -l „» , , 2n -2 n, n+m-2) .= 1 r - M2 e2 n2 p + 0( p )2 o n K v '

The pr ess ur e at t ai ns a maxi mum i n the coni cal st agnat i on poi nt s and t he i sobar s

ar e t o a f i r st appr oxi mat i on concent r i c ci r cl es ar ound the or i gi n. I n Fi g. 2. 6a

t he s t r eam i ne pat t er n i n a cor ner 0 i <p i n/ n i s shown. Fi g. 2. 6b gi ves the

r adi al pressur e di s t r i but i on f or var i ous val ues of n, near an obl i que saddl e

poi nt . I t appear s t hat t he pr essur e gr adi ent at t he coni cal st agnat i on poi nt i s

si ngul ar f or cor ner angl es 2 J I / 3 < <> < n, cor r espondi ng t o 1 < n < 3/ 2, f i ni t e

f or n = 3/ 2 and zero f or 3/ 2 < n < 2.

« P = 7 n

P / P c

t

-S^vniiu/niiiriDi, ^=0 0

/ ,<n<2

n = V ,

1 < n < 3 / ,

1 ///i

/ (a) streamline pa ttern (b) pressure distribution

F i g . 2 . 6 . O b l i q u e s a d d l e p o i n t ( 1 < n < 2 ) .

C as e n = 2 : s a d d l e , n o d esS u b s t i t u t i o n o f t h e s o l u t i

t h e c o n i c a l s t r e a m l i n e s y i e l d s :

S u b s t i t u t i o n o f t h e s o l u t i o n s F (<t>) an d F (<p) f o r n = 2 i n t o t h e eq u a t i o n f o r

d - ( 2 e2 + l ) C + (mFB s i n <> + F ^ c o s <p) pm ~ + e 2 q ( n2 - ( ; 2 ) + 0 ( pm + )

d n (2 e _ - l ) n + (mF co s <p - F ' s i n 9) p™"1 + e , n ( n2 - q 2 ) + 0 ( p m + 1 ) 'd. m m d.

( 2 . 1 1 )

w i th 2 < m £ 4 .

T he e i g e n v a l u e s o f t h e c o e f f i c i e n t m a t r i x o f t h e l i n e a r p a r t : A

y i e l d1,2 = - 1 ± 2 e 2

P = A l + A2 = ~2' q = A1 A 2 = x " '*e2

- 6 3 -

I f ( 2 . 11) i s wr i t t en i n pol ar coordi nat es and cons i der ed i n t he (p,<P) pl ane,

appl i cat i on of t he t heor em of Har t man- Gr obmann ( Chapt er I ) yi el ds t hat t he

l i near i zed system has t he same st r eam i ne pat t er n as t he nonl i near system i f | e|

* Y-I n the i\,Q pl ane t he l i near syst em r eads

ladt

(-1 + 2e2) T> dx (-1 2e2) Q

whi ch def i nes a cont i nuous st r eam i ne pat t er n al so i n t he n,Q pl ane. For | e_ | =

J- a degener at e si ngul ar poi nt occur s whi ch wi l l be i nvest i gat ed i n sect i on 3- 3

wher e hi gher - or der coni cal st agnat i on poi nt s are t r eat ed.

Equat i on ( 2 . 7 ) , shows t hat f or non i nt eger val ues of m t he f unct i on F (<p) i s not

per i odi c wi t h per i od 2n; t her ef or e, i n t hat case i t i s not poss i bl e to cover a

f ul l nei ghbour hood of t he st agnat i on poi nt such that t he vel oci t y i s cont i nuous.

I f m = 3 or 4 and i f al l hi gher - or der t er ms are 2n- per i odi c t he coni cal st agna

t i on poi nt can be real i zed i n t he f l ow f i el d away f rom t he body.

Fr om t he expr essi ons: p = - 2 and q = 1 - e* we obser ve t hat coni cal st agnat i on

poi nt s ar e repr esent ed i n t he p- q di agr am on t hat par t of t he l i ne p = - 2 wher e

pa S kq . I t i mpl i es t hat i n pot ent i al f l ow, coni cal s tagnat i on poi nt s wi l l be

nodal poi nt s i f |e | < J- (q > 0) and saddl e poi nt s i s i f | e 2 | > J- ( q < 0 ) . I nt he s pec i a l case e_ = 0 (q = 1) t he nodal poi nt i s a s t ar l i ke node wi t h

s t r eam i nes approachi ng the s i ngul ar i t y i n di f f er ent di r ect i ons . Sket ches of

t hese f i r st - or der coni cal f l ow si ngul ar i t i es ar e gi ven i n Fi g. 2. 7a; t hey may be

obser ved i n many pr act i cal f l ow si t uat i ons such as t he f l ow ar ound cones, del t a

wi ngs and axi al cor ner s.

^

(b)

L 2 <" / 2e2=0

: ^B0<£2<V2 t2>V2

Fi g. 2. 7. St ruct ural l y s tabl e coni cal f l ow pat t er ns, n = 2, | e_| = J -.

( a) coni cal s t r eam i nes , (b) i sobars .

- 6 4 -

Wi t h E q . ( 2 . 8 ) t h e p r e s s u r e d i s t r i b u t i o n b e c o me s :

T - lT

. ( E - ) = 1 - 2 ( T - 1 ) MQ c2 [ ( e 2 - J - ) n' ( e2 J - ) q « ] 0 ( p m ) ( 2 . 1 2 )

I f | e _ | >J - , and t h u s w h e n a s a d d l e p o i n t s i n g u l a r i t y o f t h e s t r e a ml i n e s o c c u r s ,

t h e p r e s s u r e a t t a i n s a ma x i mu m i n t he c o n i c a l s t a g n a t i o n p o i n t a n d t h e i s o b a r s

a r e t o a f i r s t a p p r o x i ma t i o n c o n c e n t r i c e l l i p s e s a r o u n d t h e o r i g i n .

I f 0 < | e2 l * i" ( n o d a l s i n g u l a r i t y ) t h e i s o b a r s a r e t o a f i r s t a p p r o x i ma t i o n

c o n c e n t r i c h y p e r b o l a e w i t h a s y mp t o t e s g i v e n by

L

i * E2

I n t he r egi on wi t hi n t he acut e angl e bet ween t he asympt ot es t he pr essur e i s

hi gher t han i n t he c oni cal st agnat i on poi nt , wher eas i n t he ot her r egi ons the

pr essure i s l ower ; see Fi g. 2. 7b. For e_ = 0 t he pr essur e does not change t o t he

or der mT/ ( T - l ) of t he di s t ance f r om t he coni cal s tagnat i on poi nt s ; t hi s case

act ual l y cor r esponds t o n > 2 whi ch wi l l be di scussed her eaf t er .

Case n > 2: st ar l i ke nodes

Subst i t ut i on of t he sol ut i ons F (<p) and F (q>), as obt ai ned f or n > 2 i nt o t he

equat i on f or t he coni cal st r eam i nes yi el ds

, n+1 . , m+1., - p' + n c p cos nq> + 0( p )S£ = n ( 2 . 1 3 )d

- n e p s i n n<? + 0( p )n

w h i c h , i n f a c t , i s e q u a l t o E q . ( 2 . 1 0 ) f or t he c a s e 1 < n < 2 .C o mp a r i s o n o f b o t h e q u a t i o n s h o w e v e r s h o w s t he d e c r e a s i n g i n f l u e n c e o f t h e c

t e r m w i t h r e s p e c t t o t he r a d i a l v e l o c i t y t e r m ( -p * ) w h e n n > 2 .

I n t r o d u c i n g t h e p a r a me t e r T a l o n g t h e s t r e a m l i n e s we i n v e s t i g a t e ( 2 . 1 3 ) as a

s y s t e m i n t he ( + 0 ( a )

- 65-

P=Po

P = Po

The l i ne o = O cont ai ns onl y r egul ar poi nt s , r esul t i ng i n a s i ngl e i nt egral

cur ve t hr ough any poi nt of t hi s l i ne. Cor r espondi ngl y, t he st r eam i nes appr oach

t he or i gi n i n t he cross f l ow pl ane S f rom

al l di r ec t i ons . To t he or der cons i der ed,

t he l i nes <p = k n/ n (k = 0, ±1, ±2, . . ) ar econi cal s tr eam i nes di vi di ng the f l ow f i el d

i nt o sect or s wi t h an openi ng angl e to = n/ n

(to <n / 2 ) . The coni cal st r eam i nes i n each

sect or ar e cur ved at t he si ngul ar poi nt and

t he c ur vat ur e i s o ppo s i t e i n adj a c ent

sectors. The f l ow i n t wo adj acent sect or s i s

shown i n Fi g. 2. 8. Si m l ar t o t he pr ev i ous

c a se s , t hi s t ype of c oni c al s t agnat i on

poi nt i n gener al cannot occur as a ' f ree' s i ngul ar i t y i n t he f l ow f i el d, cer

t ai nl y not i f n i s not an i nt eger . The pr essur e di st r i but i on, whi ch i s gi ven by

t he rel at i on

mum 0

Fi g. 2. 8. St ar l i ke node at n > 2.

T- l

( 2- ) = 1 + ( Y- l ) M ( n- l ) e pn cos nq> + 0( pm) + 0( p2n" 2)po o n

shows a saddl e behavi our near t he coni cal st agnat i on poi nt ( see Fi g. 2. 8) , wi t h

sectors cor r el at ed wi t h t he sector s i n t he st r eam i ne pat t er n.

A speci al exampl e of t hi s s i ngul ar i t y i s t he coni cal s t agnat i on poi nt i n a

uni f or m f l ow. Then E =0 and t he i sobar s shown i n Fi g. 2. 8 do not appl y s i nce

i n thi s par t i cul ar case t he pr essur e i s const ant t hr oughout t he f l ow f i el d.

The pr evi ous i nvest i gat i on of f i rs t - order coni cal s t agnat i on poi nt s i ngul ar i t i es

shows us t hat i n i r r ot at i onal coni cal f l ow di f f er ent t ypes of s i ngul ar i t i es may

be encount er ed such as obl i que saddl e poi nt s, saddl es whose separ at r i ces ar e

per pendi cul ar t o each ot her , nodal poi nt s and st ar l i ke nodes i n cor ner s wi t h an

openi ng angl e l ess t han n/ 2 r ad. However , ot her t ypes t hat pr obabl y coul d be

expect ed such as spi r al poi nt s and obl i que saddl es i n acut e cor ner s , as

ment i oned i n par agr aph 2. 1, ar e not f ound i n t hi s c l ass i f i cat i on. I n t hi s

r espect i t must be emphasi zed that t he r esul t s ar e obt ai ned f or i r r ot at i onalf l ows ; t he ext ens i on o f t he cl assi f i cat i on f or noni sent ropi c coni cal f l ows i s

subj ect of cur r ent r esear ch.

- 66-

3; 2i _I r r ot at i onal _at t achment s_and_segar at i ons

I t i s of i nt er est t o di scuss at t achment and separat i on phenomenae i n i nvi sci d

f l ows a l ong a smoot h coni cal body f r om a poi nt of v i ew of f i r s t - or der s i n

gul ar i t i es . Such si t uat i ons ar e l i kel y t o occur i n many f l ow pr obl ems, especi al l y i n f l ows around bodi es at hi gh i nci dences.

Let us consi der t he det ai l s of t he f l ow near a coni cal st agnat i on poi nt whi ch

l i es on t he cont our of a smoot h coni cal body i n such a f l ow.

The smoot hness of t he cont our i mpl i es t hat onl y c oni cal st agnat i on poi nt s can

occur whi ch bel ong t o t he cl ass n = 2.

Let t he body cont our i n t he cr oss f l ow pl ane I be def i ned by t; = t; (TI ) and

assume t hat t he coni cal st agnat i on poi nt i s l ocat ed on i; at n = C = 0. The body

cont our near t he or i gi n may be appr oxi mated by t; =0 ( T I J ) .

Ref er r i ng t o t he f i r s t - or der s i ngul ar i t i es , obtai ned for n = 2, the f ol l ow ng

di st i nct st r eam i ne pat t er ns may be consi der ed.

e_ > J- a saddl e poi nt of at t achment .

0 < e„ < J- . a nodal poi nt wi t h an i nf i ni t e number of st r eam i nes t angent t o

t he body cont our ( t angent node) .

G_ = 0 a st ar l i ke node.

-J - < e. < 0 a nodal poi nt havi ng an i nf i ni t e number of st r eam i nes per pendi cu

l ar t o t he body cont our ( nor mal node) .

c- < - J - saddl e poi nt of separ at i on.

Let us f i rst gi ve at t ent i on t o t he pr essur e di str i but i on over t he body sur f ace

i n r el at i on wi t h the cross f l ow str eam i ne pat t er n.

From Eq. (2. 12) , we obt ai n f or t he pr essur e over t he sur f ace:

T- l

T( j j - ) = 1 - 2( T- 1) M c2(e 2-i ) n2 0( n»)po

Thi s equat i on shows t hat t he pr essur e has a l ocal m ni mum i f a t angent node

occu r s , wher eas f or a normal node and a saddl e poi nt ( at t achment or separ at i on)

t he pr essur e at t ai ns a l ocal maxi mum I n t he speci al case of a st ar l i ke node the

pr essur e r emai ns const ant t o second or der .

Obvi ousl y onl y t angent nodes can be det er m ned uni quel y f r om a sur f ace pr ess ur e

di st r i but i on. To descri be al l possi bl e st r eam i ne pat t er ns i n a uni que sense t hepr essur e behavi our per pendi cul ar t o t he sur f ace has t o be known as wel l . A

conci se r esul t may be obt ai ned i f t he second or der pr essur e der i vat i ves ar e

eval uat ed i n t he coni cal st agnat i on poi nt . Fr om Eq. ( 2. 12) we obt ai n:

- 67-

*ni, = 0 (0'0) " S * 1 2c2>

316r « 3C

£ ( 0, 0) = -H z ( l 2c- )

wher e £ i s t he pr essur e nondi mensi onal i zed by t he dynam c pr essur e J- T p M2 i n

t he coni cal st agnat i on poi nt , p = . „ y.„, Tp Mo

I f e_ i s el i m nat ed t he f ol l owi ng rel at i on bet ween p and 6 r esul t s

(6 - 6 ) 2 + 4( 6 + p ) = 0

Fi g. 2. 9. Second- or der pr essur e gr adi ent s i nconi cal st agnat i on poi nt (n = 2 ) .

From F i g. 2. 9, whi ch shows t hi s r el at i onshi p, i t i s easy t o concl ude t hat a

l ocal maxi mum of t he sur f ace pr essur e {p < 0) cor r esponds t o:

a nor mal node i f * w > 0

- a saddle poin t of a t tachment i f T " > 1Tin

- a sadd le po i n t of sep a ra t io n i f r " < 1Jin

Fur t her mor e i t may be not i ced t hat p and 6 . never exceed t he val ue +\ i nnn CQ

i sent ropi c coni cal f l ows.

- 68-

The proper t i es of coni cal st agnat i on poi nt s on a smoot h body may al so be st udi ed

i n r el at i on t o the spat i al i nvi sci d str eam i ne pat t er n on the body surf ace near

t hose poi nt s. Fr om t he sol ut i on gi ven by Eq. ( 2. 6) t he vel oci t y component s may

be■der i ved as

u = uQ ( 1 - e2(n» - <')} 0( p»)

v = 2u c 2 n + 0( p2)

w = - 2u G2 q + 0( p* )

Sin ce the body su r fa ce i s ap p ro x im ate ly co inc iden t w i th the p l ane q = 0 , t he

spa t i a l su r face s t reaml ines fo l low f rom

ldx 'q=0 u 1 + 0(n 2 )

whi ch may be i nt egrated f or smal l n t o obt ai n t he spat i al st r eam i ne pat t er n

cl ose t o n = 0 wher e z er o c r oss f l ow appear s. The cor r espondence of sur f ace

st r eam i nes wi t h cr oss f l ow str eam i ne pat t er ns i s shown i n Fi g. 2. 10.

F i g. 2. 10. Sur f ace st r eam i nes and cor r espondi ng cr oss f l ow pat t er n.

I t appear s f or a tangent node that t he sur f ace st r eam i nes, di ver ge f r om t he

at t achment l i ne ( x- axi s) f or m ng a pat t er n wi t h a so cal l ed ' got hi c' behavi our .I n t he case of a starl i ke node t he sur f ace st r eam i nes ar e t o a f i r st appr oxi ma

t i on par al l el to each ot her .

- 69-

The nor mal node and the saddl e poi nt of separ at i on gener at e spat i al sur f ace

st r eam i nes whi ch conver ge to the ray wi t h zero c ross f l ow ( x- axi s ) . I n the

nodal case the rat e of conver gence i s l ess t han i n the saddl e case.

A saddl e poi nt of at t achment i s associ at ed by a sur f ace f l ow pat t er n di ver gi ng

f rom the at t achment l i ne.

F i n a l l y i t i s of pr ac t i c al i nt er es t to obt ai n the cor r espondence of the cross

f l ow str eam i ne pat t er n wi t h the spat i al sur f ace f l ow di r ect i on near the coni cal

st agnat i on poi nt .

The sur f ace f l ow di r ect i on, denot ed by * , i s def i ned as the angl e of the

vel oc i t y vec t o r wi t h the l ocal ray emanat i ng f r om the coni cal cent er , see Fi g.

2. 11, then there i s obt ai ned

t an *u + vn

The angle * passes through zero in a co nical s tagna tion po int . The ra te at

which t> changes along the surface i s determined by the par ticu lar cross flows d*

pattern since -z— = 2e_ - 1, so the cross flow pattern at a c oni cal stag na tionn d<>

p o i n t may be u n i qu e l y c h a r a c t e r i z e d by -z— (see Fig. 2.11) as f o l l o w s

conical center

local ray-2 "1 i *s

W/V/V.

\ \

Fi g. 2. 11. Di r ect i onal gr adi ent of sur f ace s t r eam i nes .

dV

di|i

> 0: saddle point of attachment

-1 < -;— < 0: tangent node

d*s

dnd t | >

- 1: st ar l i ke node

~2 <-z—<- 1: nor mal nodedn

d ^

dn< - 2: saddl e poi nt of separ at i on

- 7 1 -

n = vz = ( 2e2- l ) n + £ ( TI «- C* ) + u ^Q g 2 (?) v±+1 T i3 _ i e1 + 0(p*)

( 2. 15)

q = wz = - ( 2e2* l ) C " ux nC + / (" 2- i : 2) + g z (?) v£+i S3" * ^ + °(P*>

The coef f i c i ent s u. , v . and v* ar e new unknowns i nst ead of p. and 6. . Thei r

dependence on p. and 6. i s i r r el evant f or t he anal ysi s. Of mor e i mpor t ance ar et he mut ual r el at i ons bet ween v. and v* whi ch f ol l ow f r om Eq. ( 2 . 1 4 ) .

v» = - v» = - v = v2

V

l

+ V

3

= 4 e

2

M

o

( 1 ~ ' 2 e 2 ) '(2.16)

M* + v* = - 4 E2 M ( l + 2 E 2 ) 2

v* - v 1 = - 8e2 M ( l +4 E 2 ) 2 - 4 E2

The f l ow pat t er ns at Ep = +- may be der i ved f r om t hose at E_ = -J - by per f or m ng

t he t r ansf or mat i on

T> •* - <; v -> v*

S •» n vl + vl

1 * 2 v3 4 V3

p 2 * - n1 v» + v3

which shows t h a t th e n a tu re of the deg enera te s in g u la r i t i e s a t £., = +J- and a t

E - = -J- i s s i m i l a r and t h a t a r o t a t i o n o f i t/ 2 r a d t r a n s f o r m s th em i n t o e a ch

o t h e r . T h e r e f o r e , we r e s t r i c t t he d i sc u s si o n on h i g h er - o rd e r s i n g u l a r i t i e s t o

one va lue of e_ on ly , say e_ = -J-.

For e_ = -J- , sy s tem (2 .15) has a degen erate s i n g u la r i ty a t (0 ,0) w i th one zero

e i g e n v a l u e . Then A n d ro n ov 's T heo re m 65 a b o u t t h i s t y p e of m u l t i p l e s i n g u l a r

p o i n t s may be ap pli ed (c .f . Chapter I) from which follow s th at n = 0 on a curvewhich is approximated by

U ln = f ( c ) = - i p «:* + o(«;»)

- 7 2 -

and that on th is curve

i = Am qm 0 (qm + 1 ) = - / <:« (g 1 51) C' 0 ( O ) (2.17)

Accor di ng t o Andronov d i f f e r e n t d eg en e r a te s i n g u l a r i t i e s c an b e d i s t i n g u i s h e d

Case: u 2 * 0, saddle-no de

For u_ * 0 th e r e fo l low s m = 2 and 4 = -u _ / 2 so t h a t the m ul t ip le s in gu la r

po int i s a sad dle-n ode , i t has one pa rab ol ic sec tor (nodal type) and two hy pe r

bo l i c s ec to r s ( s add le type) , s ee F ig . 2 .12a .

An un l im i ted number o f s t r ea m l ine s , co ns t i tu t in g t he nod a l p a r t , approa ch th e

co n i c a l s t ag n a t i o n p o i n t a l o n g t h e d i r ec t i o n n = 0 wh ereas two s i n g l e s tr eam

l i n e s ap p r o ach t h e s ad d l e - n o d e a l o n g t h e d i r e c t i o n t; = 0 ; t h ey a r e co n vex

towards the nodal par t of the saddle-node. This impl ies that i f a saddle-node is

f i t t e d on a body su r fa ce , the nodal pa r t occurs for convex sur fa ces w hereas fo r

concave sur faces only the saddle par t appears .

The p a r t i c u la r s t r eam l ine which sepa ra tes the para bo l i c s e c to r f rom th e h yp er

bol ic sectors has the approximate form

q = " g u2 n* + 0 ( n , )

T h e o c c u r r e n c e o f a s a d d l e - n o d e i n c r o s s flo w s t r e a m l i n e p a t t e r n s may b e

expressed in terms of the cross f low veloci ty der ivat ives as fol lows .

3w 3v 32wF ro m E q . ( 2 . 1 5 ) w e o b t a i n e = - J - ( l + — - ) = J - (l + - — ) , u_ = - jpr~ which

impl ies tha t a s add le -node occur s i f the condi t ions :

1 . v 2 = wz = 0,

aJw

ar e sat i sf i ed i n a common poi nt i n t he f l ow f i el d.

- 7 f -

o

| = i - ^ Pj i! - \ u2 n t : + ïï u i c ' + 2 B i T , 3 ~ i q l

wher e T i s a par amet er al ong an i sobar , A and B ar e coef f i ci ent s whi ch depend

on u±, v and v* . They wi l l be spec i f i ed i n t he cour se o f t he anal ys i s i f t hei r

r es pec t i v e v al ues ar e needed. Bot h ei genval ues of t he l i near i zed syst em near

( 0, 0) ar e equal t o zero so t hat Andr ohov' s t heor em about t hi s t ype of mul t i pl e

s i ngul ar poi nt s may be appl i ed ( c . f . Chapt er I ) . The curve on whi ch - j * = 0 i s

u lit = - , j ± Q ' + 0( q»)

and on t hi s cur ve

g = A k qk ♦ 0(qk+1) .% ? ♦ ( J i - A3) q' ♦ 0<O)

wher e

A 3 = 12 + 4

Hence i t f ol l ows t hat f o r u_ * 0 , k = 2 whi ch i mpl i es t hat t he i sobar s f orma

cusp near t he si ngul ar i t y. As we have shown bef or e, t he case u? * 0 corresponds

t o a saddl e- node of coni cal s t r eam i nes : t he i sobar pat t er n i s sketched i n Fi g.

2. 12b.

For u_ = 0, there i s obt ai ned

u l 1k = 3 and Ak = g- - A3 = - ^ Am

i mp l y i ng t hat t he i sobar pat t er n exhi bi t s a cent er f or A < 0 and a t opol ogi cal

saddl e f or A. > 0.kSi nc e A, and A have opposi t e si gn we obt ai n as a f i nal r e sul t t hat a t opol ogi -

K mcal node of coni cal s t ream i nes ( A < 0) corresponds t o a t opol ogi cal saddl e of

i s obar s ( A. > 0) and a t opol ogi cal saddl e poi nt of coni cal s t r eam i nes (A > 0)

gi ves a cent er poi nt of i sobar s (A, <0 ) ; sket ches of t hese pr essur e f i gur es ar e

shown i n F i g. 2. 12b. I n t hese pi c t ur es , r egi ons wi t h a pr essur e exceedi ng t hat

i n t he st agnat i on poi nt ar e i ndi cat ed by a pl us s i gn, wher eas a l ower pr essur e

i s denot ed by a m nus s i gn.



- 75-

4. Analytical unfoldings in conical flows

The second- or der syst em Eq. ( 2. 15) . whi ch descri bes t he behavi our of coni calst r eam i nes near coni cal st agnat i on poi nt s, cont ai ns sever al unknown par amet er s

(e_, u- , v. , v* ) whi ch def i ne a mul t i pl e par amet er f am l y of f unct i ons i n the

r i ght - hand si de of t he syst em These f unct i ons are r epr esent ed by Tayl or ser i es

wher ei n t hese par amet er s may be i dent i f i ed as spat i al der i vat i ves of t he cross

f l ow vel oci t y component s i n t he coni cal st agnat i on poi nt .

The t opol ogi cal st r uct ur e near a coni cal st agnat i on poi nt wi l l depend on t he

act ual par amet er val ues of c_ , u. , v. , \>* whi ch can al so act as b i f ur cat i on

parameters.Si nce hi gher - or der st agnat i on poi nt s occur at | e_| = J- ( whi ch ar e saddl e- nodes

i f Up * 0 and t o po l o gi c a l no des or t opol ogi c al s addl es i f j i _ = 0) , the

bi f ur cat i on par amet er s ar e obvi ousl y e_ and y_ wi t h bi f ur cat i on val ues | e ? | = \

and Up = 0, r espect i vel y. The ri ght - hand si de of Eq. ( 2. 15) i s r epr esent ed by a

Tayl or ser i es , t her ef or e we cons i der onl y bi f ur cat i ons caused by anal yt i c

per t ur bat i ons.

The bi f ur cat i on of a saddl e- node i s descr i bed by at l east one par amet er ( e_)

wher eas t he bi f ur cat i on of a topol ogi cal node and a t opol ogi cal saddl e poi nt i s

gover ned by at l east t wo paramet ers ( c ? and u~) . So f ar t he par amet er s i n Eq.

( 2. 15) ar e r ecogni zed whi ch may cause bi f ur cat i on. However , t he bi f ur cat i on

pr ocesses t hat can be t r eat ed now ar e rest r i ct ed t o f l ow pat t er ns whi ch cont ai n

af t er bi f ur cat i on at l eas t one coni cal s tagnat i on poi nt . Thi s pr oper t y of the

bi f ur cat i on pr ocess i s due by t he f act t hat Tayl or expansi ons ar e per f or med wi t h

r espect t o a coni cal st agnat i on poi nt so that t he syst em has al ways a si ngul arpoi nt at t he or i gi n (0,0 ) . Thi s res t r i ct i on i s ra t her unf or t unat e, because i t

pr event s t he descri pt i on of spont aneous gener at i on of coni cal st agnat i on poi nt s;

f ur t her mor e i t pr event s al so t he obser vat i on of f l ow phenomenae wher e t wo or

mor e si ngul ar i t i es mer ge t oget her and di sappear af t er coal escence.

Thi s di f f i cul t y may be ci r cumvent ed i f t he bi f ur cat i on of degener at e s i ngul ar

i t i es a l l ows f or nonzer o cr oss f l ow at t he or i gi nal l ocat i on of t he hi gher - or der

si ngul ar i t y. Thi s means t hat t he magni t ude of t he cr oss f l ow vel oci t y | q | at

n = Q = 0 has t o be t aken as an addi t i onal i ndependent bi f ur cat i on par amet er .

- 76-

Toget her wi t h c_ and u ? we expect a codi mensi on- t wo bi f ur cat i on f or t he saddl e-

node and a codi mensi on- t hr ee bi f ur cat i on f or t he t opol ogi cal node and the

t opol ogi cal saddl e.

For a pr oper descr i pt i on of t hese bi f urcat i ons t he system of Eq. ( 2. 15) i s notappr opr i at e si nce t he t hi rd bi f ur cat i on par amet er | q | does not appear i n t he

r i ght hand s i de of ( 2. 15) . A modi f i ed f or mul at i on f ol l ows i f i n Eq. ( 2. 1)

coni cal f l ow sol ut i ons are used whi ch ar e obt ai ned by expandi ng F ( n, 0 near a

r egul ar poi nt i n the f l ow f i el d.

I n gener al t hi s expansi on may be per f or med near t he or i gi n n = C = 0 so t hat

F( n, 0 may be appr oxi mat ed by:

NF = F j l + 2 pn Fn(<p) 0( pN +±) ) ( 2. 18)n=

Among t he unknown f unct i ons F (<p), ( n = 1, . . N) ( whi ch wi l l be sol ved up t i l l

N = 4 i n t he next paragraph), onl y F. (q> cont r i but es to t he cross f l ow vel oci t y

q_ at the or i gi n, s i nce:

dF *l gzM( o. o) = P{F« ( ) } (2. i 9)

4; 21_Appr oxi mat e_sol ut i ons_near _r egul ar poi nt s

To const r uct appr oxi mat e sol ut i ons of t he coni cal pot ent i al equat i on, Eq. ( 2. 5)

near poi nt s wi t h nonvani shi ng cr oss f l ow and t o be i n accor dance wi t h stagnat i on

poi nt sol ut i on devel oped i n sect i on 2.2, we wi l l expand the coni cal pot ent i al i n

t er ms of t he di st ance t o t he or i gi n TI = q = 0 by t he power ser i es:

N ^F = F (1 I pn Fn(<p) + 0( pN+1) ) ( 2 .18 )

n=

I f t h i s e q u a t i o n i s s u b s t i t u t e d i n t o Eq . ( 2 . 5 ) a n d t h e r e s u l t i s o rde red i n

p o w er s o f p , o r d i n a r y d i f f e r e n t i a l e q u a t i o n s f o r t h e f u n c t i o n s F (<p) a r e

ob ta ined .

The equ at io n fo r F. (<p) g iv es

Y - l f , „, „ , , „ , T + lF ( 1 " P - F ^ ^ F i « ) } ( F ; ^ } = 0

- 77-

whi ch wi l l be sat i sf i ed by t wo di f f er ent t ypes of sol ut i ons

F^Ut) = a1 cos(<p - i ) ( 2. 20)

and

M - p* I •Fj 2)( <P) = - ^ s i n ( \ M « - <P0) (2 .21)

o v

(2)The second sol ut i on F. (<t) i s not 2n- per i odi c and shoul d t her ef ore be om t t ed

here. The sol ut i on descr i bes t he coni cal anal ogue of a Prandt l - Meyer f an andrepresents a possi bl e f l ow near a di scont i nui t y i n t he sl ope of a coni cal body.

The f i rs t sol ut i on F^ (<P) cont ai ns t wo ar bi t r ar y const ant s a- and i | ) . , whi ch mayi dent i f i ed w t h the magni t ude and di rect i on of t he cross f l ow vel oc i t y a t th eor i gi n, respect i vel y.

The r el at i on of | q | wi t h a1 i s gi ven by:

| gsl ( 0. 0) = FQ VF2! + F' > = Fo. f l l (2.22)

T he q u a n t i t y a . , w h ic h i s t h e b i f u r c a t i o n p a r a m e t e r we a r e l o o k i n g f o r , w i l l

a l s o a p p e a r i n t h e s o l u t i o n s f o r F ?(), cf . B ak ke r ( 1 9 8 4 ) .

S i n c e a . h a s t h e b i f u r c a t i o n v a l u e a . = 0 , o n l y s o l u t i o n s f o r s m a l l a , a p p e a r t o

b e s i g n i f i c a n t .

The l a b o r i o u s c a l c u l a t i o n s i n o b t a i n i n g t h e s e s o l u t i o n s , h a v e b e e n m ade by

B a kk e r ( 1 9 8 4 ) , h e r e we s u f f i c e t o l i s t t h e r e s u l t s

F_(<p) = CJK c o s * _ + cos(22)}

Isilwher e K i s a f uncti on of t he cr oss f l ow Mach number at t he o r ig i n M = lrt n, :c c( U, U)

a' F2 M

1 0 - c (2.23)( Y - l ) ( l - F2) - T a' F2 " 2 - M2v 'l o 1 o c

and e_ and t|> ar e ar bi t r ar y const ant s.

We may use t he f reedom st i l l exi st i ng i n t he choi ce of t he coor di nat e syst emt orotate i t ar ound t he x- axi s i n such a way t hat t|> - 2t|>. = 0 r esul t i ng i nt o

- 78-

F2( <p) = C2 ( K cos 2 ^ + cos 2<p) ( 2. 24)

Then t he sol ut i on f or F_(q> and F <p) are f ound t o be

a

l c

2 F

oFnf ' P) = c 3 1 si n 3<P +c cos 3<P +( T_ 1) ( 1_ F f . { ( 2 E2 - 1 ) COS <P COS +

+ ( 2e2+ ) si n <t si n * } + O( a ) ( 2. 25)

and

F1)W = Cj ^ s l n *** + c4 2 c o s 4) and Fj , (<p) are speci f i ed t o the order 0( a*) and

0( 8- ) r especti vel y; t he speci f i cat i on of hi gher - or der t erms has been om t t ed

here because t hei r i nf l uence on t he bi f ur cat i on pr ocess appears t o be absent .

The syst em def i ni ng the coni cal st r eam i ne pat t ern may now be wr i t t en as

n = P( n, 0 = FQ{ a1 cos ^ + ( 2e2- l ) n + j ^ j - + ü2 nq + p3 | - +

3

q = Q( n. q) = F Q[& 1 si n *1 - ( 2 c2+ ) q + ü2 j " + ü3 nq + ü, , § - +

3

* | I ( ?) v» +1 q3_ i n1 0(p»)

(2.27)

wher e

ê 2 = e2( l + J- a* M cos 2 ) + O( a )

e 2 = e2( l - J- a^ M cos 2* 1) + 0(a' 1)

1 = wl + 3 al E2 ( 2 c 2 " 1) Mo COS *1 + 0 ( ai )

P 2 = u2 + a1 e2^2e

2 +

^ M

o Si n

*1 +

° ^a

l ^j L = -Pj ^ + a± c2( 2e2 - 1) M cos ^ + Of a )

u/j = - u2 + 3ax e

2 ( 2 E2 + *' M

0 s i n *i + ° <ai )



- 80-

For t hat , s ys t em ( 2 . 27) i s ext ended wi t h the equat i on a = 0 i n order to obt ai n a

s y s t e m (i n R3) t hat c ont ai ns c as the onl y par amet er ; a i s nowcons i der ed as an

addi t i onal i ndependent v ar i abl e.

For e =0 t h i s extended sys tem becomes :

TIZ q2

q = a sin i\> 1 * | i2 2 — l^ nq - u2 f * °(3)

a = 0

• Tl2 C

2

n = a cos «j - 2n + ux 2~ + n2 nq - pj | - + 0(3)

which has a degenerate singularity at (0,0,0); 0(3) denotes third and higherorder terms composed of q, a and n.

The i n f l u e n c e of e i s an al yz ed in th e ce nt er manifold th at i s tangent to the

center eigenspace of:

/ 0 si n

0 0

* 1 o]

0 cos $ . - 2

wi t h ei genval ues A. = A_ =0, A- =- 2

The cor r espondi ng e i genspaces are

E c =( 0, 2, cos i l y) X( l . O. O) , Es =( 0 , 0 , 1 ) , Eu =*

c cThe cent e r mani f o l d W : n =h(q , c t ) , t angent t o E s at i s f i es t he equat i on

/ 3h 3h

3q 3a

0 0

/ . \n

i ° ;wi t h t he boundar y c ondi t i o ns : h( 0, 0) = h ( 0, 0) =0, h ( 0, 0) =J- cos i ^

The appr oxi mat i on of t he cent er mani f o l d near ( 0, 0, 0) becomes

W : h( q, a) =h^ +h 2 a 2 +h-aq +h q2 +0( 3)

wher e 0( 3) denot es t er ms of 0( q ' , q2

a, qa2

, a1

) .The c oef f i c i ent s h , h- , h and l u are det er m ned as

V cos *.



- 8 1 -

U l v2 Ulh p = -T 7 c o s2 i l ) , - K ~- s i n i j ) . c o s t l > - g - s i n2 1(1.

VU 1 s i n Tp + y2 c o s *1

S = " T

The es se n t i a l fe a tur es of the unfold ing fo llow i f (2 .27) i s pro je c ted on W :

p aC = a sin *. + 8 or cos 4 (*j) - (2e - 2^ " cos *1 ) C " P 2 f + 0(3 ) (2.29)

which shows that for a = 0 and e = 0 the C2- te rm is the lowest-order nonvanish-

ing te rm imply ing a cod imens ion-one b i fu rca t i on so th a t on ly one p e r tu r b a t i o npa r am ete r shou ld be ne ce ss a r y to dec r ibe the b i fu rca t ion in a pe r s i s te n t way.

H ow ever, E q. (2 . 2 9 ) a l s o show s t h a t t h e p h y s i c a l i n t e r p r e t a t i o n o f t h i s

b i f u rc a t i o n needs th e use of two pa ram et e r s ( a and e ) . The loc a t i on s of the

neighbour ing conica l s tagnat ion poin ts wi l l depend on these parameters and mayc c

be obtained from the expression for W and the projected system on W .

For sma l l va lues o f a and c the approx ima te lo c a t i o n i s g iven by th e l e ad in g

terms:

C i

-2c ± 2 Je 2 + J- u - a s i n tK

n. = i- a cos h - 4 ^ I V

i f s in iK * 0, i = 1,2 (2.30a)

and

C± = -CU: + Uja) ± J(4 e + uj a)2 + (u2 a)2

2 P

n i = L a

i f s in t|). = 0 (cos 4^ = +1 ), i = l , 2

(2.30b)

If s in I|I * 0 , the saddle-node f a l l s ap ar t in to two co nic a l s tag nat ion po in ts i f

t h e q u a n t i t y u _ a s i n ty-, which we wil l ca l l a , exceeds the va lue -2 e 2 . Thetype o f these po in t s fo llows by ev a l ua t in g the t r a c e p and the Jac ob ian q a t

t h e s e p o in t s ; i t a p p e a r s t h a t t h e p l u s s i g n i n Eq. ( 2 .3 0 a) c o rr e sp o n ds t o a

- 82-

st abl e node and the mnus si gn to a saddl e poi nt . For a = - 2 e2 a nongeneri c

bi f ur cat i on occurs; the saddl e- node remai ns present and i s shi f ted w th respect

to i t s or i gi nal posi t i on. I f a < - 2 c2 t he saddl e- node di sappear s w t hout

generat i ng any nei ghbour i ng si ngul ar i t y.

A qual i t at i ve i mpressi on of saddl e- node bi f urcat i on i n coni cal f l ow i s gi ven i nFi g. 2. 13 whi ch shows the bi f urcat i on set together w th coni cal f l ow pat terns.

I f si n *. = 0 saddl e- node bi f urcat i on w l l occur under the forced condi t i on that

t he cr oss f l ow vel oci ty component w remai ns zero. As a consequence, al ways two

coni cal stagnat i on poi nt s w l l appear af ter bi f ur cat i on: a saddl e- poi nt and a

st abl e node.

' a=- 2e2

Fi g. 2. 13. Bi f urcat i on of the saddl e- node.

Let us now di scuss f urther the case si n i|). * 0 and see how saddl e- node bi f urca

t i on may be used to expl ai n some t r ansi t i on phenomena i n coni cal f l ows. The

bi f ur cat i on set , Fi g. 2. 13, shows t hat a saddl e- node may bi f urcate i nto two

di f f erent ways. The f i rst possi bi l i ty i s i ts f al l i ng apart i nto two st ructural l y

stabl e si ngul ari t i es and the ot her i s i ts di sappearance l eavi ng no si ngul ari t i es

at al l . The exi st ence of t hese t wo bi f ur cat i on modes i s t he r eason f or two

di f f er ent t r ansi t i on processes i n coni cal f l ow pat terns. Each of themmay be

i dent i f i ed w t h a t raverse al ong a par t i cul ar curve i n t he a, E- di agr am Thef i rst t ransi t i on phenomenon occurs i f one moves f or exampl e al ong a = 0 f rome <

0 to c > 0. The transi t i on i s marked by t he f act that coni cal stagnat i on poi nt s

- 8 3 -

r e m a i n p r e s e n t d u r i n g t h e t r a n s i t i o n . I f e i n c r e a s e s from E < 0 t o c > 0 , a

sa dd le and a node move tog eth er and co ale sce i n to the sadd le-node (e = 0 ) ; a f t e r

coalescence the saddle and the node appear again and move away from each other .

Th i s type o f b i f u rc a t io n , which w i l l be ca l l e d : ' s add le -node b i fu rc a t io n o f th e

f i r s t t y p e ' , e x p l a i n s t h e ' l i f t -o f f ' phenomenon which was ob se rv ed a t theleeward s ide o f a cone a t su f f i c i e n t ly h igh inc idence { th i s c h a p te r , pa rag rap h

1 . 4 ) . At l i f t o f f, t h e c o n i c a l s t a g n a t i o n p o i n t a t t h e b ody s u r fa c e i n t he

leeward s i n gu la r i t y plane would be a sadd le-no de; th e no da l p a r t i s formed by

th e s t r e a m l i n e s o u t s id e t he body su r f ac e and the s add le pa r t i s found i f the

flow is extended ins id e th e body (Bakker & Bannink 197^) - A d e c re a se of i n

c idence l eaves a noda l po in t on the cone sur face and c rea tes a s add le po in t in

t h e s o l u t i o n i n s i d e t h e co n e . I f t h e i n c i d en ce i s i n c r ea s ed b ey o n d t h e v a l u e

wh e r e l i f t - o f f o c cu r s th e s ad d le -n o d e f a l l s ap a r t i n t o a s ad d le p o i n t a t t a ch edto the cone surface and a node moving away from the body.

A d i f f e r e n t t y p e o f t r a n s i t i o n e x i s t s i f we move i n t h e b i f u r c a t i o n s e t ( F i g .

2 .13) from a < 0 to a > 0 a lo ng e = 0 . We ob se rv e t h a t a re g u la r f low f i e l d

e x i s t s as long as a < 0 and tha t only a t a = 0 a saddle-node a pp ea rs . A fur the r

inc rea se to a > 0 re su ls i n to the format ion of a saddle po int and a s t a b le node.

I n c o n c l u s i o n , t h i s t r a n s i t i o n p r o c e s i s c h a r a c t e r i z e d by t h e sim u lta ne ou s

appearance o f two con ica l s t agn a t ion po in t s the ' b i r t h ' o f which i s cause d byth e sudden o cc u rr en ce of a degene rate saddle -node . This example of sad dle-node

b i f u r c a t i o n w i l l be c a l l e d : ' sa d d l e - n o d e b i f u r c a t i o n o f th e s e c o n d t y p e ' . I t

explains the observat ions of Bannink & Nebbeling (1978) that a saddle and a node

s u d d el y ap pea r i n t h e f lo w a r ou n d a c i r c u l a r co n e b ey o nd a p a r t i c u l a r h i g h

i n c i d en ce .

4 1 | l i_Bi furca t ion o f topo log ica l s add le_goin t

A t o p o l o g i c a l s a d d l e p o i n t o ccu r s in a co n i ca l flow p a t t e r n i f t h e c r o s s f low

components v and w obey th e follow ing co n di tio n s

1 .

2 .

3-4,

v 2 = wr = 0 (at = 0)3w

ÏT = ° ( e 2 ■ -* >3 l w

8 C , = 0 (u 2 = 0), 3 J W l 3 ' w



- 84-

The coni cal st r eam i ne pat t er n near t hi s degener at e si ngul ar i t y may be obt ai ned

f rom Eq. ( 2 . 27) as :

p lT = vs = - 2n +Y (" ' " S' ) + 0 ( p' )

Q = w_ = - p. nC + v« 5- + 0(p'( 2. 3D

1 6

wher e t he i mmat er i al ( posi t i ve) const ant F i s i ncl uded i n the par amet r i zat i on.

r» °On t he r i ght - hand si de, t he t erm v* g- i s wr i t t en expl i ci t l y wher eas t he r emai n

i ng t hi r d- and hi gher - or der t er ms ar e denot ed as 0( p' ) si nce t hei r cont r i but i on

t o t he qual i t at i ve f l ow pi ctur e appear s t o be a hi gher or der ef f ect .

The det ai l ed s t r uct ur e of t he t opol ogi cal saddl e poi nt depends on t he actualval ues of p. and v*. The t opol ogi cal saddl e poi nt has f our separ at r i ces, t wo of

t hem appr oach t o t he coni cal st agnat i on poi nt al ong t he n- axi s and the ot her s

al ong t he - di r ect i on. The l at t er ar e cur ved and can be appr oxi mat ed by t he

parabol a n = - r p1 t;* .

Bi f ur cat i on of t he topol ogi cal saddl e poi nt occur s i f t he bi f ur cat i on par amet er s

c- , a, and p_ ar e per t ur bed wi t h r espect t o thei r r espect i ve bi f ur cat i on val ues

E- = -J -, a. = 0, vip = 0. Anal ogous t o t he s addl e- node c ase we ass ume t hese

per t ur bat i ons t o be gi ven by c, a and u whi ch ar e smal l compar ed t o uni t y;

e 2 = -J- + E, a. = 0 + a, Up = 0 + u.

The unf ol di ng of Eq. ( 2. 31) sat i sf yi ng t he coni cal f l ow equat i ons may be der i ved

f r om Eq. ( 2. 27) by subst i t ut i ng t he per t ur bat i on par amet er s E, a and p.

To obt ai n t he l ocat i ons of nei ghbour i ng coni cal st agnat i on poi nt s P.(n. ,c;. ) as a

f unct i on of t he per t ur bat i on par amet er s: a, E and p we use cent er mani f ol d

t heor y. For t hat syst em ( 2. 27) i s ext ended w t h t he equat i on a = 0 i n or der t o

obt ai n a syst em ( i n R' ) t hat cont ai ns E and p as t he par amet er s , a i s nowconsi der ed as an addi t i onal i ndependent var i abl e.

For E = p = 0 t he ext ended syst em r eads:

C = a si n t ^ - p 1 nC + 0( 3)

• n2 C*n = a cos 4^ - 2n + p ^ ^ p 1 | - + 0( 3)

- 86-

wher e we have i nt r oduced t he shor t hand not at i on

- a s i n h -ze - j j _a = ~ 2 i — • c ■ 3 r« ^ p = u

m m mI f D < 0, t her e ar e t hr ee real and di f f er ent sol ut i ons; f or D = 0 agai n t hr ee

r eal sol ut i ons appear but at l east t wo of t hem ar e equal ; f i nal l y, i f D > 0 one

r eal sol ut i on appear s and two ar e conj ugat e compl ex. The bi f ur cat i on set of t he

t opol ogi cal saddl e poi nt consi st s of t he sur f ace D = 0 bor der i ng domai ns i n R

wher e t he f l ow t opol ogy i s di f f er ent . A vi ew of t hi s sur f ace i s gi ven i n F i g.

2. 14.

Fi g. 2. 14. Bi f ur cat i on set D = 0;

D = ( u3 + | êü - a)z - (6 +u ) ' •

The sur f ace D = 0 has t wo br anches whi ch t erm nate on t he cur ve ê =- p 2 , a=J - p'

where cr oss sect i ons of D = 0 w t h p = const ant f or m a cusp. I f u = 0 we observe

t hat a bi f ur cat i on sol ut i on w t h t hr ee coni cal st agnat i on poi nt s occur s i n a

' wedge' t ype r egi on near t he posi t i ve e- axi s. For u * 0 t hi s regi on def or mes

mor e or l ess, so t hat al so t hr ee coni cal st agnat i on poi nt s can occur f or an

appr opr i at e negat i ve val ue of e. At t he cusp poi nt t he t wo sol ut i ons on D = 0

coal esce i nt o t he tr i pl e sol ut i on: S. = p.

To obt ai n t he r esul t i ng coni cal f l ow pat t er ns i t suf f i ces t o anal yze t he bi f ur

cat i on on the cent er mani f ol d W and to i nvest i gat e t he st abi l i t y pr oper t i es of

t he occur r i ng si ngul ar i t i es.



- 88-

Fi g. 2. 15. Bi f ur cat i on of t opol ogi cal saddl e (j j > 0) bi f ur cat i on set (D = 0)

and f l ow pat t er ns near cent er mani f ol d W .

An exampl e of saddl e poi nt bi f ur cat i on may be encount er ed i n t he f l ow past an

el l i pt i c cone at i nci dence. Cr oss f l ow str eam i ne pat t er ns at var i ous i nci dences

wer e al r eady gi ven i n Fi g. 2. 4. Beyond a par t i cul ar i nci dence (say a = a ) the

f l ow st r uct ur e at t he wi ndwar d si de changes qual i t at i vel y when t he saddl e poi nt s

S. , ,.S_ and t he nodal poi nt N. ar e r epl aced by a s i ngl e saddl e poi nt S. The

t r ansi t i on t hat occur s may be expl ai ned as saddl e poi nt bi f ur cat i on occur r i ng at

a = a . The t opol ogi cal saddl e poi nt appear s at t he par t i cul ar i nci dence a = a .

A smal l decr ease of i nci dence causes bi f ur cat i on i nt o a st abl e node N. f l ankedby t wo saddl e poi nt s S. and S_.

I f t he i nci dence i s i ncreased beyond a t he t opol ogi cal saddl e poi nt changes

i nt o a saddl e poi nt S.

Our knowl edge about saddl e poi nt bi f ur cat i on wi l l be appl i ed t o show some

det ai l s of t hi s f l ow phenomenon. Fi r st we not e t hat t he f l ow i s symmet r i c wi t h

r espect t o t he m nor axi s and, mor eover , t her e i s al ways a coni cal st agnat i on

poi nt pr esent wher e t hi s axi s i nt er sect s t hé body cont our .Thi s i mpl i es t hat t he f l ow near t he w ndwar d si de may be descr i bed by t he syst em

(see Eq. ( 2. 27) )

- 89-

n = v z = (-2 + 2e) n + j= - ( n2 - q2) + vx £- + j 2 nq2 + 0( p* )

( 2. 34)v*

q = wz =- 2 e q - ux nq + ^ i ) 2 ; ; +u * |^ +o ( p » )

wher e t he bi f ur cat i on par amet er s a and u ar e om t t ed.

The n- axi s coi nci des wi t h t he m nor axi s of t he el l i pt i c body ( posi t i ve i nwar ds)

and t he coni cal st agnat i on poi nt at t he wi ndwar d si de i s t aken at t he or i gi n.

Near t he ori gi n t he body cont our wi l l be appr oxi mated by n = k. q2 + k_ q* + . . .

wher e k and k_ ar e posi t i ve quant i t i es whi ch depend on t he pr i nci pal ver t ex

angl es of the el l i pt i c cone.

The boundar y condi t i on t hat t he body cont our coi nci des wi t h a coni cal st r eam i ne

i s s a t i s f i e d i f

Ux = 4 ( 1- 36) + 0( e2)

Si nce e i s t he onl y bi f ur cat i on par amet er , saddl e poi nt bi f ur cat i on occur s at

e = 0.

Det ai l s of t he bi f ur cat i on pr ocess may be obser ved cl ear l y f r om t he behavi our of

t he cur ves v = 0 and w_ = 0 i f E var i es near zer o.

Apar t f r om t he symmet r y pl ane (q = 0) where w = 0 t hese cur ves wi l l be gi venby t he appr oxi mat i ons ( see Eq. ( 2. 3*0)

v r = 0: n = k1( l - 2 e ) q2 + ( k2 + iffl k ^ q' + . . .

wz = 0 : " = 2 ^ - ( A m - " k i > % + •••

The i nf l uence of e on t he cur ves v = 0 and w = 0 ar e sket ched i n Fi g. 2. 16 f or

A > 0. For e < 0 these cur ves do not i nt er sect at al l , meani ng t hat no coni calmst agnat i on poi nt s exi st out si de t he symmet r y pl ane. I n t he symmet r y pl ane wher e

w = 0 t her e i s a saddl e poi nt at t he body

si t uat i on beyond t he cri t i cal angl e a > a

w = 0 ther e i s a saddl e poi nt at t he body sur f ace, cor r espondi ng t o t he f l ow



- 90-

' _ - v r = 0

c ) : £>0

Fi g. 2. 16. F l ow near wi ndwar d s i de o f e l l i pt i c cone.

I f E i s i n c re a se d th e c urve w = 0 sh i f t s i n p o s i t i v e n -d i r e c t i o n a nd b ec ome s

ta ng en t to th e cu rv e v = 0 a t the o r ig i n i f c = 0 ; in th i s case we have to do

wi th the s t ru c tu ra l l y uns tab le f low a t a = a .

I f e i s in c re as e d beyond e = 0 , the w = 0 curve moves fu r the r and i n te rs ec ts

the v_ = 0 curve on the body co nt ou r a t C2 = —-. Thus fo r e > 0 two c o n ic a lZ A

mst agnat i on poi nt s appear at t he body sur f ace out s i de t he symmetr y p l ane.

4 5i _ § j f ur c at i on_ o f _ t opol ogi c al _ node

I n coni cal f l ow a t opol ogi ca l node occur s i f t he cr oss f l ow component s v and w

obey c ondi t i ons s i m l ar t o t hos e gi v en f or t he t opol ogi c al s addl e poi nt ( pr e vi

ous sec t i on) except t hat t he s i gn o f A has changed.

The coni cal s t r eam i ne pa t t e rn near t hi s degenera t e s i ngul ar i t y may be ob t a i n ed

f r om Eq. ( 2. 27) as

n = v = - 2n + ^p ( n2 - q2 ) + 0( p' )

q = wz = - u^q + v* g- + 0( p ' )

The de ta i l ed s tr u c tu r e of the flow depends on th e a c t u a l va lu e s of u . and v* .

The to po lo gi ca l node con tains two s tre am line s tendin g to the s tag na t ion poin t i nT i -d i r e c t io n a n d a n in f in i t e n u mb e r o f s t r e a ml in e s wh ic h a p p ro a c h in t h e C -

d i r e c t i o n .

- 91-

Anal ogous t o t he case of t he t opol ogi cal saddl e poi nt t he bi f ur cat i on phenomenon

i s governed by t he t hr ee paramet ers a, e and p. Appl i cat i on of t he cent er mani

f ol d t heor y l eads t o equat i ons exact l y t he same as t hose obt ai ned f or saddl e

poi nt bi f ur cat i on ( Eq. ( 2. 32) ) .

Consequent l y t he r esul t s obt ai ned f or saddl e poi nt bi f ur cat i on can be used

di r ect l y. To i nt er pr et t he f l ow on and near t he cent er mani f ol d i t suf f i ces t o

t ake i nt o account A < 0. Then t he f ol l owi ng concl us i ons about nodal poi ntmbi f ur cat i on can be est abl i shed.

Nodal poi nt bi f ur cat i on i n coni cal f l ow i s gover ned by t he l ocal quant i t y D ( Eq.

( 2. 33) ) as f ol l ows :

- For D < 0, a t opol ogi cal node bi f ur cat es i nt o a saddl e- poi nt f l anked by t wo

nodal poi nt s .

- For D > 0 t he t opol ogi cal node t r ansf or ms i nt o a hyper bol i c nodal poi nt .

- For D = 0 a nongener i c bi f ur cat i on occur s gener at i ng a nodal poi nt and a

st r uct ur al l y unst abl e saddl e- node. I n t he ver y speci al case D = 0 and e = - u2,

a = -£■u3 t he t opol ogi cal node remai ns as such and i s shi f t ed al ong t he pat hv l

" = - r «= •A qual i t at i ve i mpr essi on of t he f l ow at and near t he cent er mani f ol d may be

obt ai ned by consul t i ng Fi g. 2. 17.

Fi g. 2. 17. Bi f ur cat i on of t opol ogi cal node (j i > 0) bi f ur cat i on set (D = 0)

and f l ow pat t er ns near cent er mani f ol d W .

- 9 2 -

5. External corner flow; a nonana lytic unfolding of a starlike node*)

5 1l^_The_flow_around_an_external_corner

I n t h e p r e v i o u s p a r a g r a p h we h a ve d i s c u s s e d t h e e x i s t e n c e of h i g h e r - o r d e rc o n i c a l s t a g n a t i o n p o i n t s a n d t h e i r b i fu r c a t i o n d u e t o a n a l y t i c a l p e r t u rb a t i o n s .

In c o n t r a s t t o a n a l y t i c a l p e r t u rb a t i o n s a l s o n o n a n a l y t i c a l b i fu r c a t i o n s may be

encoun te red in con ica l f l ow pa t t e rns , e spec ia l ly in f low domains nea r d i scon

t i n u i t i e s i n t h e s l o p e of t h e c o n t o u r o f a c o n i c a l b o d y. The e x i s t e n c e of

no na na ly t ic a l unfold ings may be shown by d isc uss ing the su p er so n ic f low aroun d

an ex te rn a l co rn e r .

An ex te rn al cor ner co nfig ura t ion i s composed of two plane d e l t a wings 2. and !_,at tached to each other along a common edge such that the planes of the wings

inc lu de an ang le wi th each o th e r . The two rem ain ing f re e ( l e ad in g ) edg es a r e

supersonic (veloci ty component normal to the edge is supersonic) thus the flows

on e i t h e r s id e o f t he conf igu ra t ion can be co ns id e re d in de pe nd en t ly . The f low

will be assumed to be conical and the centre of the conical field coincides with

the apex of the co nfig ura t ion . We w i l l be in te re s t ed pa r t ic u la r l y in the flow in

t h e r e g i o n n e a r t h e e x t e rn a l a n g l e , i n c l u d i n g b o th t h e c a se o f t he e x t e rn a l

axia l corner , where the p lane wings are near ly perpendicular to each o ther andth e ca se where th e c o nf ig u ra t i o n i s s imi l a r t o one s id e (upper o r l ower) o f a

d el ta wing wi th an arrow shaped cro ss -se c t io n ( the p lane x = 1 in F ig . 2 . 18 ) .

In o r d e r t o d e s c r i b e t h e c o n f i g u r a t i o n i n more d e t a i l , we u se a r ig h t -h a n d e d

car t e s i an co -o rd ina t e sys t em x ,y ,z , w i th the o r ig in in the apex and the pos i t i ve

x - a x i s in th e d i r e c t i o n of the oncoming undis turbed f low. The y-a xis i s chosen

such that the leading edge of 2. i s i n t he x ,y -p l a ne and the z -a x i s i s pe rpe n

d i c u l a r t o i t . The l ead ing edge o f w ing Z ? and the x-axis determine a plane Q,which makes an ang le u wi th th e x , y - p l a n e measu red po s i t i v e a s i n d ic a t e d in

Figure 2 .18 . The leading edge of 2. has a sweep angle A. with the y-axis and 2.

i s inc l ined wi th respect to the x ,y-p lane a t an angle ó 1 measured in th e z ,x -

pla ne (see Fi g. 2. 18 ). The lea din g edge of Z_ has a sweep angle A_ with th e y .z -

pl an e and 2_ makes an an gle 6_ with Q (<5_ i s measured in a pla ne th ro ug h t h e x -

axis and perpendicular to the plane Q). The l ine where X.. and £_ meet wil l be

c a l l e d t h e c o rn e r l i n e .

' ) A su bs ta n t ia l p ar t of th is paragraph was publ i shed by:Bakker & Reyn in the AIAA-Journal, Vol. 23, no. 1, 1985.



- 94-

Kut l er and Shankar (1976), bot h f or symmet r i cal and asymmet r i cal conf i gur at i ons,

an obl i que saddl e poi nt was f ound at t he cor ner poi nt and a nodal poi nt on each

of t he wi ng sur f aces . Fur t her mor e i n t hese cal cul at i ons the pos i t i on of the

nodal poi nt cor r esponds t o the poi nt wher e t he t r ansver se pr essur e di st r i but i on

on t he wi ng sur f ace at t ai ns a maxi mum A qual i t at i ve i mpr essi on of t hi s f l owpat t er n may be obt ai ned f r om Fi g. 2. 19a.

conical streamlines

a) Conical stream line pa tt er n with an obl ique saddle po intin the corner point (C) and two nodal points IN, and N2)

bowshock

conical streamlines

b) Conical stream line pa tt er n with a single node atthe corner point.

Fi g. 2. 19. Possi bl e coni cal f l ow pat t er ns near t he cor ner poi ntof an ext er nal cor ner .

I n t he symmet r i cal case t hi s f l ow pat t er n was al so f ound i n t he exper i ment s of

Banni nk (1984). Despi t e t he numer i cal evi dence, Sal as and Dayw t t ( 1979) poi nt ed

- 95-

out t hat a f l ow pat t er n as shown i n Fi g. 2. 19b, havi ng a nodal poi nt at t he

cor ner poi nt as t he onl y coni cal st agnat i on poi nt i n the f l ow f i el d, shoul d al so

be consi der ed. I n f act a numer i cal st udy made by Sal as ( 1980) f or symmet r i cal

conf i gur at i ons (A. = K = 20° , 6 . = & ~ = 10° ) and a var i at i on of u f rom n/ 2

( ext ernal cor ner t ype) t o n ( del t a wi ng t ype) seems t o i ndi cat e t he occur ence of

t he f l ow pat t er n of Fi g. 2. 19b f or eo cl ose to it. Al so, t he cal cul at i ons suggest

a t r ansi t i on f rom t he obl i que saddl e pat t er n to t he st ar l i ke node pat t er n i f <o

i s var i ed f rom n/ 2 to n.

I n order t o c l ar i f y t hi s si t uat i on we want t o i nvest i gat e whet her l ocal coni cal

st agnat i on poi nt sol ut i ons appl i ed t o the r egi on near t he cor ner poi nt suppor t

t he i dea of a si ngl e node pat t er n and t he tr ansi t i on t o t he obl i que saddl e at

t he cor ner at some cri t i cal val ue of to.

We not e however , t hat t he real nat ur e of t he coni cal st agnat i on poi nt at t he

cor ner poi nt can onl y be f ound by sol vi ng t he f ul l nonl i near boundar y val ue

pr obl em wi t hout usi ng any appr oxi mat i ons such as numer i cal s ol ut i ons. Thi s , of

cour se i s a har d pr obl em unl i kel y t o be sol ved. However , t her e i s one not i ce

abl e except i on, namel y i f ó- = ó_ = 0, i n whi ch case 2. and Z? ar e al i gned wi t h

t he uni f or m f l ow. The resul t i ng f l ow i s t he undi stur bed uni f orm f l ow, whi ch i s

r epr esent ed by a si ngl e st ar l i ke node at t he cor ner poi nt . We wi l l consi der t hel ocal cor ner f l ow as a per t ur bat i on of t hi s f l ow and t he coni cal stagnat i on

poi nt s at and near t he cor ner poi nt as bi f ur cat i ons of t hi s st ar l i ke node. The

bi f ur cat i ng sol ut i ons have to sat i sf y t he boundar y condi t i ons on 1. and £_ and

moreover i n t he l i m t ó- , 6p ■ 0 t hey have t o rever t t o t he uni f or m f l ow.

5 2i _Boundary condi t i ons and_bi f ur cat i on modes

The anal ysi s i s per f or med i n t he cross- f l ow pl ane Z, t he r ef erence syst em n, t;

has i t s or i gi n i n t he cor ner poi nt C and t he n- axi s coi nci des wi t h 0. .

I f pol ar coor di nat es p,<p are used such t hat n = p cos <p, Q = P s i n <p t hen t he

cor ner boundar i es a. and cu i n 2 cor r espond t o <p = 0 and < = $ r espect i vel y and

t he cor ner poi nt C i s gi ven by p = 0.

Let a coni cal st agnat i on poi nt ex i st at the corner poi nt and l et the coni cal

pot ent i al F be expanded near t he cor ner poi nt by

F = FQ( 1 pn Fn( 9 ) Pm Fm( 9 ) + o( pm) ) .

-96-

T h e n t h e l o c a l s o l u t i o n s :

F (<p) = c c o s (n<t> + + X ) + Y co s(n ) + 6 , m = mm

v Km m ' m v n ' m c

o b t a i n e d i n p a r a g r a p h 2 may b e u s e d , w h ic h h a v e t o s a t i s f y t h e b o u n d a r y

c o n d i t io n s — = 0 a t <p = 0 and <p = $ .3<p e

W i th E q s . ( 2 . 6 ) a n d ( 2 . 7 ) t h e r e i s o b t a i n e d

t)i = X = 0n m

n = k ^ - , k = 2 , 3 , 4 . . . . ( 2 . 3 5 )e

m = t | - , E = 3,'t,5e

o r i f

m x £ — , f o r a ny 2 = 3 , 4 , 5 , . . . t h e n m = m , 6 = 0$ ^ - " c me

S i n c e e x t e r n a l c o r n e r f l o w s a r e c o n s i d e r e d f o r n S $ S 2n an d b e c a u s e m > n > 1e

i t f ol l ows t hat £ > k £ 2.Obvi ousl y, i mposi ng t he boundar y condi t i ons on a1 and a_ i s by i t s el f i ns uf f i ci ent t o ensure t he pr oper embeddi ng i n t he sur r oundi ng mai n f l ow of t he coni cals t agnat i on poi nt sol ut i on near t he cor ner . I n f ac t , t he f r eedom t o f ur t her

speci f y sol ut i ons i s al r eady expr essed by t he possi bi l i t y, of choosi ng t he val uesof k and S. i n t he l eadi ng t er ms of t he expansi on of t he coni cal pot ent i a l . Theexponent n, occur r i ng i n t he l eadi ng term i s i l l ustr at ed i n Fi g. 2. 20 as a f unct i on of $ f or var i ous val ues of k. The sol ut i ons f or t he var i ous choi ces of k

e

and 0. wi l l be consi der ed as per t ur bat i ons on t he uni f or m f l ow gi ven by F = F

whi ch has a s tar l i ke node at t he cor ner poi nt . Ther ef or e, t he coef f i c i ent s i n

t he expans i on f or F , such as e , 8 et c . wi l l b e i nt er pr et ed as smal lK

n ' mpar amet er s t endi ng to zero f or t he uni f or m f l ow. These smal l par amet er s var y i n

r el at i on t o each ot her such t hat p F (<p) r emai ns t he l eadi ng t er m i n t he

expansi on, even i f t he par amet er s t end t o zero.

-97-

kn

$e

4

3

2

11

0

k'/C

^|A'/,

/.'/

' l/

\/

i^

11

\ V k = k>4

\ i\

\ 1\ 1\

\ i \\ ' \

VN

VVi* \ 1

\ *j> N .

Ni* ! "--V//////A/////7?/

l e

■ k ' / 2

.__t__

t

nodal point at the

corner p oint (s ta r l ike )

saddle point at thecorner point

no conical stagnationpoints at the corner point

3n25 2n

Fi g. 2. 20. Dependence of n on the exte r nal angl e f or di f f er e ntbi f ur cat i on modes (k = 2, 3. 't. . . . ) .

As a r es ul t t he f l o w n ea r t he corner po i nt i s obt a i ned as an unf ol di ng of the

s t ar l i ke node, the st r e am i ne pat t er n of whi ch f ol l ows f r om the st r eam i ne equa

t i o n i n pol ar f or m p = (1 + p2) F -. pF, p*<V = F by t aki ng F = F :

P ="F0. P

P = FQ{- n cn p 1 si n (nq> +0( p m_ 1 ) )

wher e n and mare gi v en Eq. ( 2. 35) -Si nce n i s not necessary an i nt eger i t i s obvi ous t hat the unf ol di ng may be non-

anal yt i c . The c oef f i c i ent e i s the bi f ur c at i on par amet er wi t h bi f ur c at i on v al ue

- 9 8 -

e = 0 . S i n c e k c a n ta k e a l l i n t e g e r v a l u e s è 2 i t a p p e a r s t h a t v a r i o u s

b i fu rca t ion modes a r i se .

5 13 i l_Bifurca t ions_of th e_ s t a r l ik e node

Case k = 2: f i r s t b i fu rc at i on mode (n = —1$e

I n t he f i r st bi f ur cat i on mode, t he exponent n can t ake val ues 1 < n £ 2 correspondi ng t o ext ernal angl es n £ $ < 2n; t he exter nal angl e $ = 2n i s excl udedbecause t he exi st ence of a coni cal st agnat i on poi nt r equi r es n > 1.

hFrom Eq. ( 2. 35) i t f ol l ows that m = m = 3n- 2 f or 1 < n £ x and m = 3n/ 2 f orh c 3

S n $ 2.The l ocat i on of coni cal st agnat i on poi nt s may be f ound f r om t he condi t i ons t hatdp d<p-rr = 0 and -r- = 0 are sat i sf i ed si mul t aneousl y. For 1 < n < 2 t her e f ol l ows f r omEq. ( 2. 36) f or smal l val ues of e , B as l ocat i ons f or t hese poi nt s :

n m

C : p = 0 ( cor ner poi nt )

N r N2 : , x = 0, 92 = »e. P l = p2 = (n e^2 " " ; ^ > 0

N3 : , 3 = - »e, p3 = (-n E n ) 2 - n ; en < 0

wher e hi gher or der t er ms i n e and B ar e om t t ed.n ' m

I t appear s t hat apar t f r om t he coni cal st agnat i on poi nt at t he cor ner (C) , there

ar e ei t her t wo st agnat i on poi nt s N. and N_ l ocat ed on o. and o_ r espect i vel y, or

t her e i s one poi nt N_ , occurr i ng as a f ree s i ngul ar i ty i n the f l ow f i el d, F i g.

2. 21a.The poi nt s N. and N_ have an i nf i ni t e number of st r eam i nes t angent t o t he body

sur f ace. The nodal poi nt N_ i s s i t uat ed, t o a f i r s t appr oxi mat i on, on t he

bi sect or of t he ext er nal angl e such that an i nf i ni t e number of s t r eam i nes i s

t angent to t hi s bi sector .

Si nce 1 < n < 2, t he coni cal st agnat i on poi nt at t he cor ner i s an obl i que saddl e

poi nt ( t hi s chapt er , par agr aph 3 ) . The obl i que saddl e has t hr ee separat r i ces;

t wo of t hem coi nci de wi t h a. and a? , t he t hi r d wi t h t he bi sect or <p = J- $ .

E n < 0

-99-

Uniforrn flow

b) nr Z. grn

Fi g. 2. 21. Fi rst bi f ur cat i on mode of t he star l i ke node (k 2)

For n 2, t he ext er nal angl e $ equal s n and t he body cont our at C shows no

di scont i nui t y i n the s l ope. Theref ore t he cl ass i f i cat i on of f i r s t - or der s i n

gul ar i t i es ( par agr aph 3) may be consul t ed to est abl i sh t hat f or e * 0 (e < J -)

t he ' cor ner ' poi nt C becomes a t angent node i f e > 0 and a nor mal node i f

E < 0. Fur t her mor e we obser ve t hat i n t hi s case t her e ar e no nei ghbour i ng

coni cal st agnat i on poi nt s t endi ng to C i f E •» 0, Fi g. 2. 21b.

Thi s f undament al di f f er ence i n bi f ur cat i on behavi our bet ween t he t wo cases

1 < n < 2 and n = 2 yi el ds t he f ol l owi ng not i ceabl e r esul t .

Assume t hat a coni cal st agnat i on poi nt of nodal t ype exi st s on t he sur f ace of a

smoot h body. Fur t her mor e, l et t hi s node be hyper bol i c i n t he sense t hat anal yt i

cal per t ur bat i ons can not af f ect t he topol ogi cal st r ucture i n i t s nei ghbour hood.

Let t he cont our of t he sur f ace be per t ur bed sl i ght l y so t hat a di scont i nui t y i n

i t s s l ope i s obt ai ned ( $ > TI ) at t he l oc at i on of t he nodal poi nt . Thi s

di st ur bance causes a nonanal yt i c per t ur bat i on of t he f l ow f i el d and t he nodalpoi nt can not be mai nt ai ned. I n the case of a t angent node i t must f al l apar t

-100-

i n t o a n o b l i q u e s a d d l e p o i n t ( l o c a t e d a t a p o i n t w h er e t h e s l o p e i s

d i scon t inuous ) accompanied by two t angen t nodes on the body sur face one on

e i ther s ide o f the corner po in t .

I f a normal node i s a t t ached to the body th e node b i fu r c a te s in to an o b l iq ue

saddle point a t the corner and a nodal point away f rom the body. This normalnode b i f u r ca t io n looks very s im i la r to the l i f t - o f f phenomena which has been

expla ined ear l i e r as a s add le -node b i fu rca t ion (paragraph 4 , s ec t ion 3 ) .

Case k = 3: second b if u rc at io n mode [n = ~-)e

I n t h e s e c o n d b i f u r c a t i o n m ode (k = 3 ) . t h e e x p o n e n t n c an t a k e v a l u e s

3/2 £ n £ 3 correspon ding to ex ter na l angles n S $ £ 2n. The po ss ib le ran ge of

n , t o g e t h e r w i t h t h e b o u n d a r y co n d i t i o n s (E q. ( 2 . 3 5 ) ) . shows t h a t m s a t i s f i e s

t h e c o n d i t i o n m è 4 n / 3 . C o n s u l t i n g E q. ( 2 . 3 6 ) w e o b s e r v e a s t a r l i k e - n o d e

b i f u r ca t i o n f o r 3 /2 S n < 2 w i th t h e f o ll o win g s i n g u l a r i t i e s :

C : p = 0 (corn er po int )

N r N 3 : <f1 = 0, <P3 = 2/3 3>e. P 1 = 9^ = (n en ) 2 ~ n ; *n > 0

1N 2 . N / j : <P2 = 1/3 *e . ^ = *e . P 2 = P4 = (-n */'*; cfl < 0

where hi gher order t er ms i n e and B ar e om t t ed.n m

The second bi f ur cat i on mode has i n common wi t h t he f i r st bi f ur cat i on mode t hat

new coni cal st agnat i on poi nt s ar e gener at ed f r om t he or i gi nal st ar l i ke node. The

poi nt s N. and N. ar e l ocat ed at t he cor ner sur f aces a. and o? , r es pec t i vel y,

wher eas t he poi nt s N_ and N_ appear as f r ee si ngul ar i t i es, Fi g. 2. 22a.For c > 0 onl y t he coni cal st agnat i on poi nt s C, N. and N~ exi st and f or e < 0

t he poi nt s C, N_ and N ar e pr esent . I t appear s t hat al l t he poi nt s N. _ _ ^ ar e

st abl e nodes; N. and N^ have an i nf i ni t e number of st r eam i nes t angent t o t he

s ur f ac e, N_ and N_ have an i nf i ni t e number of st r eam i nes t angent t o t he

separ at r i ces <p = 1/ 3 $ and = 1/ 3 $ and <p = 2/ 3 $ . We concl ude t hat

i n t he second bi f ur cat i on mode f or 3/ 2 S n < 2 a st ar l i ke node bi f ur cat es i nt o

an obl i que saddl e poi nt at t he cor ner poi nt f l anked by t wo nodes. One of t hese

-101-

a ) 3 / 2 < n < 2 , ^ < $ e < 2 n

^ W7

77777y-,

conical streamlines

b)n =2 , $e=^

Fi g. 2. 22. Second bi f ur cat i on mode of t he st ar l i ke node (k = 3)«

nodes l i es on t he cor ner sur f ace (a . or a~), wher eas t he second node appear s as

a f r ee s i ngul ar i t y i n t he f l ow.The case n = 2 cor r esponds t o an ext er nal angl e $ = 3/ 2 n.

Si nce n i s an i nt eger t he cl ass i f i cat i on of f i rst - order s i ngul ar i t i es ( paragraph

3) may be consul t ed t o obt ai n the f l ow pat t er n f or var i ous val ues of e . For

smal l val ues of e a nodal poi nt i s f or med at C wi t h an i nf i ni t e number ofn *

s t reaml ines tangen t to a1 for E > 0 and tangent to o_ for e < 0 , F ig . 2 .2 2b .

In c o n t r a s t to th e ca se 3 /2 £ n < 2 we ob ser ve th a t fo r n = 2 the re a re no

conica l s tagna t ion po in ts b i fu rca t ing f rom the s ta r l ike node .In t h e ca se 2 < n $ 3 the ex te rn al ang le li e s i n th e range n £ $ < 3/2 ti . From

Eq. ( 2 .3 6 ) i t f o l l o w s t h a t t h e r e a r e no c o n i c a l s t a g n a t i o n p o i n t s t e n d i n g

tow ard s th e c or ne r p o in t C fo r e ■» 0 . For e * 0 the corne r po int remains an n

- 1 0 2 -

s t a r l i k e n o d e . T h e r a y s <p = 1 / 3 $ a n d 2 / 3 $ a r e t o a f i r s t a p p r o x i m a t io n

c o n i c a l s t r e a m l i n e s w hic h d i v i d e t h e flo w f i e l d i n t o t h r e e s e c t o r s . T h e c o n i c a l

s t r e a m l i n e s i n e a c h s e c t o r a r e c u r v e d a t t h e s i n g u l a r p o i n t , b u t t h e s i g n o f t h e

c u r v a t u r e i s o p p o s i t e i n a d j a c e n t s e c t o r s .

T he c o r r e s p o n d i n g f lo w p a t t e r n s a r e s k e t c h e d i n F i g . 2 . 2 2 c .T h is f i g u r e g i v e s a l s o an i m p r e s s i o n o f t h e p r e s s u r e d i s t r i b u t i o n n e a r t h e

co rn e r w hich may be ob ta in ed f rom Eq. (2 .8 ) and i s g iven by

~T~ F2 2 2 - 2(E- ) = 1 - Y ^ r ( - 2 ( n - l ) en pn cos (n«) + n2 e p2 n " 2 0 ( p3 )}

o o

1 3

T h e i s o b a r p a t t e r n s ho ws a s a d d l e b a h a v i o u r w i th s e p a r a t r i c e s a t <p = 7 $ , "k $and ? $ on wh ich p = p .b e o

I t may b e n o t e d t h a t i n c o n t r a s t t o t h e f i r s t b i f u r c a t i o n m o de , t he s ec on d

b i f u r c a t i o n m ode i s n o t s y m m e t r ic w i t h r e s p e c t t o <p = -z $ .

Ca se k = 4 : t h i r d b i f u r c a t i o n m ode fn = —)e

I n t h e t h i r d b i f u r c a t i o n m ode k = 4 , t h e e x p o n e n t n c a n t a k e t h e v a l u e s

2 S n £ 4 c o r r e s p o n d i n g t o e x t e r n a l a n g l e s n £ $ £ 2 n . T h i s p o s s i b l e r a n g e o f

n , t o g e t h e r w i t h t h e b o u nd a ry c o n d i t i o n s , E q. ( 2 . 3 5 ) . sh ow s t h a t m s a t i s f i e s t h ei n e q u a l i t y m > 5 n / 4 .

T hen E q . ( 2 .3 6 ) r e v e a l s t h a t , a p a r t f ro m t h e c o n i c a l s t a g n a t i o n p o i n t a t t h e

c o r n e r p o i n t C , t h e r e a r e n o n e i g h b o u r i n g c o n i c a l s t a g n a t i o n p o i n t s i n t h e f lo w

f i e l d which t en d to C fo r E -» 0 .n

F o r n = 2 , $ = 2n t h e c l a s s i f i c a t i o n o f f i r s t - o r d e r s i n g u l a r i t i e s may be

a p p l i e d a n d a s t r e a m l i n e p a t t e r n w i t h a s i n g l e n o d e a t t h e c o r n e r p o i n t r e s u l t s ,

F i g . 2 . 2 3 a .For n > 2 at t he cor ner poi nt t her e i s a star l i ke node s i m l ar to t hat f ound i n

t he sec ond bi f ur cat i on mode. However , t he number of sect or s i s now f our and the

rays <? = 1/ 4 $ , 2/ 4 $ and 3/ 4 $ ar e t he coni cal st r eam i nes bor der i ng t hese

sectors. The cor r espondi ng f l ow pat t er ns t hat occur i n t hi s bi f ur cat i on mode ar e

sket ched i n F i g. 2. 23b.

The pr ess ur e di st r i but i on near t he cor ner , whi ch may be obt ai ned f r om Eq. ( 2. 8)

i s gi ven by

r -1 _T F2 i p - 2

( J H =

F ^ V t2^ -

1)

£n pn 0S {m ) + n

' E

n p2n 2 + 0(p >)

- 1 0 3 -

tn<0

/ i * e

<ï=

U n i f o r m f l o w

E -O . f \ En> 0■=0

y. i,

isobars

a )n = 2 , * p = 2 n

Fig. 2 . 23 . Third bi f urc at io n mode of the s t a r l i k e node (k = k).

T h e i s o b a r p a t t e r n s h o w s a s a d d l e p o i n t b e h a v i o u r w i t h s e p a r a t r i c e s a t <p =

1/8 $ , 3/8 $ , 5 /8 $ and 7/8 $ on which p = p . We n o te in p a r t i c u l a r th at

th e p re s su re on th e co rn er su rf ac e (a- and a„) and on the bis ec to r 0 and d ec re a se s fo r

ne < 0.n

Case k > 5: hi gher bi f ur cat i on modes f n = k —)e

For k > 5 t he exponent n sat i sf i es n > 2 and n £ $ £ 2n. For t he coni cal

st r eam i ne pat t er n agai n t he l eadi ng t er ms i n Eq. ( 2. 36) may be used, l eadi ng t o

a st ar l i ke node at t he cor ner poi nt , havi ng k sect or s. I t may be not ed t hat as

k ■> => t he f l ow pat t er n r esembl es more and mor e t hat of t he uni f orm f l ow.

-loft

s' 4. _Svmmetrical external corners

In order to il lustrate the use of the classification of bifurcation modes of the

starlike node, the flow around a symmetrical external corner wil l now be dis

cussed. The symmetry implies that k = even, so that the generation of conicalstagnation points from the starl ike node only occurs in the f i r s t bifurcation

mode (k = 2) . In order to gain more insight into the question whether new

conical stagnation points actually arise or not we will invest igate the f i r s t

mode in more de ta i l ; in particular we direct our attention to the question how

the loca l corner flow f it s within the o ve ral l flow f ie ld . The pr essu re

distribution on the body surface and the location of the conical stagnation

points as function of freestream Mach number and body geometry will receive

special attention.

In this particular case the external angle $ is related to wedge angle 6, sweep

angle A, and <o by

^ -1 f2 tan 6 tan A sin u + (1 - tan2 Ó tan2 A) cos u - tan2 ö-i$ = 2n - cos I -. —T -I r—;—T-T—;—, 1e l 1 + tan2 6 + tan2 ö tan2 A '

where 0 £ cos £ it, since n £ $ £ 2n .e

For the fi rst bi furcati on mode the expansi on for the coni cal potenti al F may be

wri tten as (Eqs. ( 2. 6) , (2.7))

F = FQ( l +en pn cos (nq> +T p3" " 2 cos (n<p) +o(p3n"2) ) (2. 37)

(n e )J M _v v " o . . , _ . 2n

where Y,, =—777^—TV - . 1 < n s 2, $ = —.4 4(2n- l ) ' e n

The pressure di stri buti on may be obtai ned usi ng Eq. (2. 8)

T - lY

fE_] = ! . I l l M2 [ n * E2 p2 n "2 . 2(n-l) e pn cos (n<p) +

lp ' 2 oL n no

2n T^ e j n + 2(n- l) cos2 (n<p)] p '4"" '* + o(pkn~k)]

where p and Mrefer to val ues i n the corner poi nt.*o oSi nce we are part i cul arl y i nterested i n the f l ow around an external corner w th

compressi ve sur f aces (6 > 0) (and i t wl l be shown l ater that embeddi ng of the

l ocal corner f l ow i s onl y possi bl e for c >0) , we w l l restr i ct oursel ves here

- 105-

t o e > 0 i n whi ch case a saddl e poi nt occur s at the corner and t wo nodal poi nt sn _1_

on the body surface at p = p„ = (n e ) . I f th e p r e s s u r e d i s t r i b u t i o n i sN n

wri t t en i n t er ms of PN, we obt ai n

T- lf P \ , T- l „, r 2(2-n) 2n- 2 2( n- l ) 2- n n , , 1 . n. , _ -0,' D '

= ~T o^pN p " n —P N p c o s ( n < p )- l + o ( p * (2- 38)o

At t he cor ner poi nt t he i sobar s f or ma cent er poi nt and on t he body sur f ace

saddl e poi nt s occur . These saddl e poi nt s cor r espond t o a m ni mumi n t he wal l

pressure, whi ch, t o f i r s t order , i s gi ven by

P N , T * e " n m ,p- =l ~ 2 -J— Mo pN* o e

Matchi ng of l ocal cor ner f l ow wi t h t wo- di mensi onal wedge f l ows

To t he or der i ndi cat ed i n Eq. ( 2. 37) t he expr essi on f or t he coni cal pot ent i al Fcont ai ns t wo f r ee par amet er s e and M , whi ch may be used t o mat ch t he l ocal

cor ner f l ow wi t h t he t wo- di mensi onal f l ow f ound i n t he r egi on downst r eamo f t h e

super soni c l eadi ng edge. The f l ow i n t hi s r egi on i s si m l ar t o t he f l ow over awedge and wi l l t her ef or e f ur t her be ref er r ed t o as t wo- di mensi onal wedge f l ow.

The mat chi ng wi l l be per f or med by r equi r i ng cont i nui t y of t he vel oci t y ( i ndi rect i on and magni t ude) on t he r ay p = p, t he i nt er sect i on wi t h t he body sur

face of t he Mach cone of t he t wo- di mensi onal wedge f l ow emanat i ng f r omt he apex

of t he conf i gur at i on. We then obt ai n

F = ( F - pF ) . t an 6P P( 2. 39)

(1 I =i MS, D) {(1 p») F'p - 2pFFp + F») = "- f MD

at <p = 0, $ and p = pHer e 8 i s t he angl e bet ween t he di r ect i on of t he t wo- di mensi onal wedge f l ow andt he cor ner l i ne, and M_n i s t he Mach number of t he t wo- di mensi onal wedge f l ow.

Subst i t ut i on of Eq. ( 2. 37) i nt o Eq. ( 2. 39) yi el ds

- 1 0 6 -

n E n P n _ 1 + <3n-2) p3 n _ 3 = [ l - ( n - l ) CR pn - 3( n - l ) T ^ p3 n _ 2 o ( p2 n ) ] . t a n 8

( 2 . 4 0 )

1 =1 wi » f . . - , - 2 n - 2 _ , # _ l X „ ;n 0_ , 0 _ o ï v ■_ r4 n - 4 . _ , > » - ' » , ,

( 2 . 4 1 )

M 0 [ 1 * U ^ M ^ ) { n ' ^ p 2 n - 2 - 2 ( n - l ) c ^ n 2 n (3 n - 2 ) V ^ " 4 + ° <P » ] = M2D

w h i c h g i v e s c , M ; a s a r e s u l t a l s o t h e l o c a t i o n o f t h e c o n i c a l s t a g n a t i o nn o °

p o i n t s a nd t h e p r e s s u r e d i s t r i b u t i o n m ay b e d e d u c e d . I t m ay b e s e e n f r om E q .

( 2 . 4 0 ) t h a t , f o r s m a l l c , c a nd 8 h a v e t h e sam e s i g n w h ic h i m p l i e s t h a t f o r

c o m p r e s s i v e w e dg e a n g l e s ( 6 > 0 , 8 > 0 ) , e > 0 a nd f o r e x p a n d i n g w e d ge a n g l e s

(6 < 0, 8 < 0 ) , c < 0 .

F o r s m a l l v a l u e s o f 8 t h e e q u a t i o n s ( 2 . 4 0 ) a nd ( 2 . 4 l ) may b e u n c o u p l e d s o t h a t

t h e u n k n o w n p a r a m e t e r c c a n b e o b t a i n e d a s a f u n c t i o n o f 8 an d p from ( 2 . 4 0 ) ,

t h e n E i s a p p r o x i m a t e d b y

tan 8e = zn - n - l , - , ~n ;np + (n - l ) p t an 8

whi ch gi ves t he f ol l ow ng appr oxi mat i on f or t he l ocat i on of t he nodal poi nt s ont he wi ng sur f ace a. or cu i n t he f i r st bi f ur cat i on mode

PN =t an 8

i n- l ~ . x1 + . p t an 8n K /

.2- n j —

p n (2.42)

Si nce p can be expr essed i n t er ms of 8 and M? n by

VM» - 1 t an 8 + 1

P = —VM| D - 1 - t an 8

t h e s h i f t o f t h e n o d a l p o i n t aw ay fro m t h e c o r n e r p o i n t c a n b e d e t e r m i n e d i n

t e r m s o f t h e p h y s i c a l v a r i a b l e s $ , 8 , M _ n . F i g u r e 2 . 2 4 s h o w s f o r t h r e e

d i f f e r e n t M ach n u m b e rs o f t h e t w o - d i m e n s i o n a l w e d g e f l o w (M __ = 1 . 5 . 3 a n d 5 )

t h e l o c a t i o n o f t h e n o d a l p o i n t a s a f u n c t i o n o f t h e e x t e r n a l a n g l e , $ a nd t h e

f lo w d i r e c t i o n 8 . I t may b e o b s e r v e d t h a t f o r e x t e r n a l a n g l e s c l o s e t o n ( r e p r e -

s e n t a t i n g c o n f i g u r a t i o n s o f d e l t a w in g t y p e) t h e n o d a l p o i n t l i e s v er y c l o s e t ot h e c o r n e r p o i n t . T h i s s h i f t a wa y f ro m t h e c o r n e r b ec o m e s m o r e a p p a r e n t f o r

i n c r e a s i n g M ach n u m b e rs a nd a d i v e r g i n g of t h e f lo w t o w a r d s t h e l e a d i n g e d g e .

- 1 0 7 -

8 = 0.04 0.08 0.12 0.16

270

240

210

180

270

240

210

180

e

' 270

240

210

180

/

I// M2D=1.5

0.02 0.04 0.06 0.08 0.10 0.12- * r N

9=0.04 0.08 0.12 0.16

/

1 w/ v ^

»*

/

M2D=3

0.02 0.04 0.06 0.08 0.10 0.12

8=0.04 0.08 0.12 0.16

/

/

IA' ^

M2D=5

/

0 0.02 0.04 0.06 0.08 0.10 0.12'-- ^N

Pig. 2 .24. Locat ion of conical s tagnat ion pointon body surface as a function of flow

d i r e c t i o n (8) and Mach number (M2D)

of the two-dimensional wedge flow.

o r ig in a t equa l s lope . This e f fec t may be ver i f ie d

6 = 0 and tak ing in to account th at for <o = n /2 , A =

leading term for PN becomes

If the matching procedure is

a p p l i e d i n t h e c a s e o f a

s y m m e t r i c a l c o r n e r w i t h

c o m p r e s s i v e w e d g e a n g l e s ,

c h a r a c t e r i z e d by u = n / 2 ,A = 0 , t h e l o c a t i o n o f t h e

con ica l s tagna t ion po in ts on

t h e s u r f a c e s i s f o u n d t o

depend on the paramete rs 6

a n d M^ a s sh o wn i n F i g .

2 . 2 5 . No te th a t t h e r e su l t s

a re g iven in the coord ina tes

" (= f ) • Ü (= Z) wh ic hx x

d e s c r i b e t h e c r o s s f l o w

plane 2 per pen dic ular to the

x -a x i s . R e ma rk th a t i n t h e

l i m i t 6 ■» o the nodal p oin t

s h i f t s t o w a rd s t h e c o r n e r

po in t i f compared wi th the

p o s i t i o n of t h e n o de t h a twould appear fo r the f low

around a single wedge, since

then (ü) = 0.

F ig u re 2 . 2 5 s e e ms to i n d i

c a t e t h a t t h e b i fu rc a t io n o f

t h e s t a r l i k e n od e i n t o a n

oblique saddle point and two

n o da l p o i n t s i s a h i g h e r

order e f fec t in 6, since for

any f i n i t e Mach number M ,

the l i n e s for n = n_ and TI =

n N s ee m t o a p p r o a c h t h e

by expanding Eq. (2.42) near

0 , tan 8 = s i n 6; th e n th e

1-n 12-n „2-n

P N = P " 6 '

- 109-

T hi s c a l c u l a t i o n i n di c a t e s t ha t i n a us ual per t ur bat i on t heor y whe r e <5 i s t he

per t urbat i on par amet er , such as used by Vor ob' ev and Fedosov (1972) , bi f ur c at i on

of t he s t ar l i k e node i s unl i k el y t o appear .

I n or der t o compare t he pr essur e di s t r i but i on on t he wedge sur f aces wi t h numer i

cal and exper i ment a l r esul t s we cons i der t he case o f 6 = 10* , M =3 ; t hen ti- =

2.580 and p„ = 0. 0783- Appl y i ng Eq. ( 2 . 38) we obt a i n p/ PN whi ch i s shown i n F i g.

2. 26. The agr eement wi t h numer i ca l cal cul a t i ons o f Sa l as ( 1980) and exper i ment a l

r es ul t s of Bak ker e t . al ( 1981) i s qui t e s at i s f ac t o r y ev en at g r e at e r d i s t a nc e

f r om t he c or ne r po i nt . We not e t hat i n t he numer i c a l c a l c ul a t i ons ent r opy

gr adi ent s ar e t aken i nt o account , wher eas t he pr esent t heor y a ss umes po t en t i a l

f l ow nea r t he co r ne r , and onl y a l i m t ed number o f t er ms i n t he expans i on ar e

used.

I p r es e n t t h e o r y ( M - = 3 ,6 = 10°)

i G O O o exper imenrsfM =2.94,6 = 10.4°)

- 0 . 2 C - 0. 1 0. 0 0. 1 0. 2 0. 3 0. 4

t **I cornerpoi nr

Fi g. 2. 2 6. P r essur e di s t r i but i on al ong the wedge sur f ace o f

symmet r i cal ext er na l cor ner wi t h A = 0° , u = n / 2 .

Theor et i cal obser vat i ons

So f ar t he use of t he f i r s t bi f ur cat i on mode seems a cor r ect way t o descr i be t he

f l ow ar ound a symmetr i cal ext er na l cor ner wi t h compr ess i ve sur f aces . As a r esul t

- no-

t wo nodal poi nt s are f ound on the body sur f ace as wel l as a saddl e poi nt at the

corner . Thi s concl usi on i s unaf f ect ed by a change of I D f r om <o = n/2 ( the cor ner

t ype conf i gur at i on) to u « it (the del t a wi ng type). I n f act numer i cal cal cul a

t i ons wer e per f or med by Sal as for symmet r i cal conf i gur at i ons (6 =10°, A = 20°

and 40° ) , M = 3 and val ues of <o r angi ng f r om 90° to 180° . Accor di ng to Sal as ,t hese cal cul at i ons i ndi cat e t hat f or hi gher val ues of u> nodal poi nt s on the

sur f ace are not pr esent and that the cor ner poi nt i s then a nodal poi nt i nst ead

of a saddl e poi nt .

Thi s woul d i mpl y t hat f or a cer t ai n val ue of I D a t r ans i t i on f r om the f i r st

bi f ur cat i on mode (k = 2) to a hi gher bi f ur cat i on mode (k =4,6, . . . ) woul d t ake

pl ace. However such a tr ansi t i on al so i mpl i es the sudden di sappear ance of the

nodal poi nt s i t uat ed away f r om the cor ner . Thi s i s i mpossi bl e si nce the nodal

poi nt away f r om the corner i s str uctur al l y st abl e wi t h r espect to changes i n the

f l o w due to v ar i at i on of 9 . We f ur t her r emark t hat i f the nodal poi nt s are

c l ose to the cor ner poi nt , t hey are ver y di f f i cul t to det ect i n numer i cal

c al c ul at i ons , i n contrast to the si t uat i on when t hey can be cl ear l y obser ved

f ur t her away f r om the cor ner poi nt .

-0.2*.

*»n

Fig. 2.27. The conical stagnation point on o-

M 3 , <5 = 10corner node

A, O

Salas (1980);

Salas (priv. comm.)

In order to show how close

the nodal points are to the

corner point as u tends to

180° for the configurat ion

M = 3, 6 = 10° at A = 0°,

20° and 40°, we calculated

the position of the conical

stagnation points according

to the f i r s t bifur catio n

mode; as shown in Fig. 2.27.For these calculations it

220°

the nodal points and the

corner point are so close

that a numerical detection

would be rather unlikely.

For $ > -220° the re i s ae

substantial deviation of thenodal points away from the

corner point. If we compare

the results of the present

appears that for $ <

- I l l -

t heor y wi t h t he numer i cal r esul t s of Sal as ( 1980) t her e i s a good agr eement f or

A = 40° , wher eas f or A = 0 and 20° t he agr eement i s l ess. As ment i oned bef ore

i t shoul d be not ed t hat t he pr esent r esul t s ar e obt ai ned by us i ng a f i ni t e

number of t er ms i n a ser i es expansi on of pot ent i al f l ow sol ut i ons, wher eas t he

numer i cal cal cul at i ons use the Eul er equat i ons.

An experi ment al observat i on

As exper i ment al r esul t s conf i r m ng the pr edi cted f l ow pat t er n acc or di ng t o t he

f i r st bi f ur cat i on mode have al r eady been r epor t ed f or t he exter nal angl e con

f i gur at i on by Banni nk (1984), i t i s of i nt er es t t o i nves t i gat e whet her t he

occur r ence of nodal poi nt s, di st i nct f r om t he cor ner poi nt , can al so be obser ved

i n exper i ment s wi t h a del t a wi ng conf i gur at i on, even t hough t hese poi nt s may l i e

ver y cl ose t o t he cor ner . For t hi s pur pose a f l ow vi sual i zat i on st udy was madeon t he upper si de of a t r uncat ed del t a wi ng wi t h a f l at l ower sur f ace (<u = n

r ad, A = h0\ 6 = 10° , $ = 196. 5° ) at H = 3, Re = 2. 3 x 10s per cm Al t houghe w

one shoul d be caut i ous i n dr awi ng concl us i ons on i nvi sci d f l ow pat t er ns f r om

f l ow vi sual i zat i on t echni ques, exper i ment s on ext er nal cor ner s per f or med by

Sal as & Daywi t t ( 1979) and Banni nk ( 1984) have shown t hat t he coni cal nat ur e of

t he f l ow i s bor ne out by t he oi l f l ow st r eakl i nes . The oi l s t r eakl i nes on the

upper f ace of t he wi ng ar e phot ogr aphed when t he i nci dence a of t he pl ane l owersur f ace ( measur ed i n t he symmet r y pl ane) i s set at 0° .

From t hi s pi ctur e t he angl e <>of the l ocal s t r eam i ne wi t h t he l ocal r ay has

been car ef ul l y measur ed. The

r es ul t s ar e c ol l ec t ed i n

Fi g. 2. 28; t hey do suggest

t hat nodal po i nt s may be

di st i ngui shed i n t he coni cal

f l ow f i el d. The r el i abi l i t yo f t hese r esu l t s may be

j udged by c ompar i ng t hem

wi t h t hose obt ai ned f or a =

- 6 = - 10° , i n whi ch case the

upper su r f ace i s al i gned

wi t h t he f l ow. Compar i ng t he

r esul t s f or a = 0 and a = -

10°, a si gni f i cant s hi f t oft he nodal s i ngul ar i t y away

Fi g. 2. 28. Di r ect i on of oi l f l ow st r eakl i nes . f rom t he cor ner i s obser ved.

Y (deg

wing

disfance from apex

o 73,5 mm

a 88,0 mm

x 102,2 mm



- 1 1 2 -

6. References

Bakker , P. G. and Banni nk, W J . ( 197*0

Coni cal st agnat i on poi nt s i n the super soni c f l ow around sl ender ci r cul ar cones

at i nci dence. Del f t Uni ver si t y of Technol ogy, Repor t VTH- 184.

Bakker , P. G. ( 1977)

Coni cal st r eam i nes and pr essur e di st r i but i on i n the vi ci ni t y of coni cal stagna

t i on poi nt s i n i sent r opi c f l ow. Del f t Uni ver si t y of Technol ogy, Repor t LR- 244.

Bakker , P. G. , Banni nk, W J . and Reyn, J . W ( 198l )

Pot ent i al f l ow near coni cal s tagnat i on poi nt s , J . Fl ui d Mech. , Vol . 105•

Bakker , P. G. ( 1984)

St ructura l s tabi l i t y and bi f urcat i on i n coni cal f l ow f i el ds . Del f t Uni vers i t y of

Technol ogy, Repor t LR- 424.

Bakker , P. G. and Reyn, J . W ( 1985)

Coni cal f l ow near ext er nal axi al cor ner s as a bi f ur cat i on pr obl em AI AA- J our nal ,

Vol . 23, No. 1.

Banni nk, W J . and Nebbel i ng C. ( 1978)

Measur ement s of t he super soni c f l ow f i el d past a sl ender cone at hi gh angl es of

at t ack. AGARD Conf erence Pr oceedi ngs 247. Paper 22.

Banni nk, W J . ( 1984)

I nvest i gat i on of t he coni cal f l ow f i el d about ext er nal axi al cor ner s . AI AA-

J ournal , Vol . 22, No. 3-

Bul akh, B. M ( 1970)

Nonl i near coni cal f l ow. Tr ans l at ed f r om t he Russ i an by J . W Reyn and W J .

Banni nk, Del f t Uni ver si t y Pr ess ( 1985) .

Busemann, A. ( 1929)

D r ü ck e au f k eg e l f ö r m i g e S p i t z en b y B ew eg un g m i t U b e r s ch a l l g e s c h w i n d i g k e i t . ZAMM,

9 ( 6 ) .



- 113-

Fer r i , A. ( 1951)

Super soni c f l ow ar ound ci r cul ar cones at angl es of at t ack. NACA TR- 1045.

Fowel l , L.R. ( 1956)

Exact and appr oxi mat e sol ut i ons f or the super soni c del t a wi ng. J . Aer on. Sc i . 23

( 8 ) .

Ger mai n, P. ( 19 9)

La t heor i e géner al e des mouvement s coni ques et ses appl i cat i ons è 1' aérodynam -

que super soni que. ONERA, Publ . no. 3^

Gol dst ei n, S. &War d, G N. ( 1950)

The l i near i zed t heor y of coni cal f i el ds i n super soni c f l ow wi t h appl i cat i on topl ane ai r f oi l s . Aer on. Quar t . I I .

Gr ossman, B. ( 1979)

Numer i cal procedure for the comput at i on of i r r ot at i onal c oni c al f l ows . AI AA-

J our nal , Vol . 17, No. 8.

Kutler, P. and Shankar, V. (1976)

Computation of the inviscid supersonic flow over external axial corners.

Proceedings of the Heat Transfer and Fluid Mechanics Inst i tute, Davis. Calif.

Maslen, S. (1952)

Super soni c coni cal f l ow. NACA TN- 2651.

Mel ni k, R. E. ( 1967)

Vor t i cal s i ngul ar i t i es i n coni cal f l ow. AI AA- J our nal , Vol . 5i No. 4.

Reyn, J . W ( i 960)

Di f f er ent i al geomet r i c cons i der at i ons on the hodogr aph t r ansf or mat i on for

i r r ot at i onal coni cal f l ow. Ar chi ve Rat . Mech. Anal . 6( 4) .

Sal as, M. D. and Daywi t t , J . ( 1979)

S t r u c tu r e of the coni cal f l ow f i el d about ext er nal axi al cor ner s . AI AA- J our nal ,

Vol . 17, No. 1.



- 1 1 1 -

S a l a s , M.D. (1980)

C a r e f u l n u m e r i c a l s t u d y o f f lo w f i e l d s a b o u t s y m m e t r i c a l e x t e r n a l c o n i c a l

c o r n e r s . A I A A - J o u r n a l , V o l . 1 8 , N o . 6 .

S m i t h . J . H . B . ( 1 9 7 2 )R e m a r k s o n t h e s t r u c t u r e o f c o n i c a l f lo w . P r o g r e s s i n A e r o sp a c e S c i e n c e s , V o l.

1 2 , Pergamon.

St ocker , P. M and Mauger , F. E. ( 1962)

Super soni c f l ow past cones of gener al cr oss- sect i on.

J . Fl ui d Mechani cs, Vol . 13 ( 3) .

Tayl or , G. I . and Macol l , J . W (1933)The ai r pr essur e on a cone movi ng at hi gh speeds. Proc. Roy. So c , Ser . A 139-

Vor ob' ev, N. F. and Fedosov, V. P. ( 1972)

Super soni c f l ow ar ound a di hedr al angl e ( coni cal case). I zvest l ya Akadem i Nauk

SSR, Mekhani ka Zhi dkost i i i Gaza, No. 5.



-115-

Chapter II I Topo logic al aspects of steady visc ous flows near p lan e wal ls

1. Local solutions of the Navier Stokes equations

In th i s chap te r the qua l i ta t ive theory o f dynamica l sys tems wi l l be app l ied to

s t e a d y t w o - d i m e n s i o n a l v i s c o u s f lo w s o f c o n t i n u a n e a r f i x e d an d m ov in g

b o u n d a r i e s . The t 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 s e f l o w s may b e d e r i v e d from

s o l u t i o n s o f t h e g o v e r n i n g d i f f e r e n t i a l e q u a t i o n s . Q u a l i t a t i v e c o n s i d e r a t i o n s

a bo ut t h e se s o l u t i o n s i n t h e p h a s e p l a n e w i l l b e p r e s e n t e d w i t h t h e aim t o

p r o v i d e i n s i g h t i n t o t h e p h y s i c s of t h e flo w p a t t e r n p r i o r t o s o l v i n g

a p p ro p r i a t e , i n i t i a l / b o u n d a ry v a lu e p ro b le ms .

The topo logy of s teady v iscous f lows i s s tud ied on the bas is o f loca l so lu t ions

of the Navier-Stokes equations (NS) governing incompressible laminar flow. These

lo c a l s o lu t i o n s a r e o b ta in e d b y p er fo rming T a y lo r e x p an s io n s fo r t h e v e lo c i ty

vec to r f ie l d near a po in t in the flow f i e l d . The s teady s t ream l ine p a t t e r n ne ar

t h i s p o i n t i s r e p r e s e n t e d by t h e t r a j e c t o r i e s of t h e s e c o n d - o r d e r s y st em

x = u( x ,y ) , y = v (x ,y ) wi th u ,v deno ting the ve loc i ty components in a c a r t e s i a n

reference system x,y .

The T a ylor e x pa n sion of t h e v e lo c i ty f i e l d i s n o t r e s t r i c t e d to t h e n e ig h b o u r

hood of re g u l a r p o in ts (u * 0 , v * 0 ) on ly bu t may a ls o be perfo rmed near

sepa ra t ion o r a t t ac hm en t p o in ts (u = 0 , v = 0 ) be ca us e , a s Dean (1950) has

shown, th e NS-equat ions al low a lso fo r ana ly t ic a l so lu t ion s near po in ts where u

and v vanish s imultaneously .

C o n s id e r t h e f low p a t t e r n th a t r e su l t s i f an a n a ly t i c v e l o c i ty v e c to r f i e ld i s

exp an ded up to t h e N- th o rd e r n e a r an a r b i t r a r y p o in t 0 , t h e n th e t r a j e c to r ypattern near 0 is governed by the system (S):

N N-ix = u = 2 2 U x yJ + 0(N+1)

i=0 j=0 , J

(S)N N-i

y = v = 2 2 V x yJ + 0(N+1)i=0 j=0 , J

where 0( N+1) denot e hi gher - or der t er ms of at l east order N+l , composed by power s

of x and y. The coef f i ci ent s U. . and V. . ar e const ant s whi ch r emai n undet er -

m ned i n a l ocal anal ysi s. Si nce t he f l ow sat i sf i es t he cont i nui t y equat i on and



- 116-

t he NS- equat i ons, r el at i onshi ps bet ween U. . and V. . exi st , r educi ng t he numberi»J i, j

of coef f i c i ent s t hat can be chosen i ndependent l y . A f ur t her r educt i on of

unknowns i s obt ai ned by par t i al l y f ul f i l l i ng boundar y condi t i ons; f or exampl e i f

one st udi es f l ow pat t er ns near a wal l wher e no- sl i p condi t i ons hol d.Our mai n obj ect i ve now wi l l be to gi ve a uni f i ed descr i pt i on of al l t opol ogi -

cal l y di f f er ent f l ow pat t er ns that wi l l ar i se near 0 i f the remai ni ng coef f i

ci ent s ar e var i ed i ndependent l y. A syst emat i c t r eat ment of t hi s t ask, bei ng t he

subj ect i n t hi s chapt er , i s br i ef l y out l i ned bel ow and el abor at ed i n mor e det ai l

f or f i ni t e N (N S 4) i n t he sequel .

For f i ni t e N, we s t udy, i nstead of t he phase por t r ai t s of system ( S ) , t hose of

t he t r uncat ed syst em( SN) :

N N- ix = u = Z Z U. . x y J

i =0 j =0 , J

( sN)N N- i

y = v = Z I V x y J

i =0 j =0 , J

and s ince (SN) c o n t a i n s o n ly a f i n i t e n um b er o f t e r m s t h e q u e s t i o n a r i s e s

w h e th e r i t g iv e s a p r o p e r d e s c r i p t i o n o f t h e l o c a l fl ow to p o lo g y i n f lo w sgoverned by the full Navier-Stokes equations.

Wi th reg a r d to t h i s qu es t io n we assume th a t the im por tan t e lemen ts o f phase

p o r t r a i t s a r e t h e s i n g u l a r p o i n t s and t he l o c a l t r a j e c t o r y p a t t e r n n e a r t h e s e

p o i n t s .

I f s i n g u l a r p o i n t s (and t h e i r l o c a l t r a j e c t o r y p a t t e r n ) o f sy st em (S N) keep

t h e i r q u a l i t a t i v e l y c h a r a c t e r e ve n i f h ig h e r o r d e r t er m s a r e a dd ed , t he n t h e

truncated system (SN) s u f f i c e s t o o b t a in a q u a l i t a t i v e d e s c r i p t i o n of t h e l o c a l

flow topology.I f t h e s e s i n g u l a r i t i e s a r e h y p e r b o l i c, t h e t r u n c a t e d s y st em i s r a t h e r s i m p l e

b e c a u s e i n t h a t c a s e a l i n e a r e x p an s io n (N =l) i s a l r e a d y s u f f i c i e n t f o r

determining the loca l topology.

M o r e o v e r , n o n h y p e r b o l i c s i n g u l a r i t i e s i n t h e flow p a t t e r n can b e a n alys ed as

well using a truncated system (S N) p rov ided th a t N i s chosen su f f ic ie n t l y l a rg e

so t h a t 0(N+1) te rms do not d i s t ur b the top olo gie s! ch ara c te r of the degen era te

s ingular i ty as obta ined f rom (S, . ) .

T h i s i m p l i e s t h a t f o r a g i v e n N a c l a s s i f i c a t i o n o f d e ge ne ra te s i n g u l a r i t i e s ,can be established, and each of them represents a local f low topology which can

occur in f lows that are governed by the full NS-equations.



- 1 1 8 -

2. Steady viscous flow near a plane wall elementary singular points

2 Z 1 ^ A p p r o x i m a t e s o l u t i o n s _ n e a r _ a _ g l a n e _ w a l l

C o n s i d e r t h e i n c o m p r e s s i b l e t w o - d i m e n s i o n a l s t e a d y l a m i n a r f l o w i n t h e n e i g h

b ou rh oo d o f a p l a n e w a l l a t r e s t . T he f l o w w i l l b e d e s c r i b e d i n a c a r t e s i a n

c o o r d i n a t e s y s te m x , y s u c h t h a t t h e x - a x i s c o i n c i d e s w i t h t h e w a l l s u r f a c e ; x i s

i n c r e a s i n g i n d ow nstre am d i r e c t i o n . The y - a x i s i s c h o s e n p e r p e n d i c u l a r t o t h e

w a l l s u r f a c e s u c h t h a t t h e f l o w n e a r t h e w a l l o c c u r s i n t h e u pp er h a l f p l a n e

y i 0 .

T he v e l o c i t y c o m p o n e n t s i n t h e x an d y d i r e c t i o n w i l l be d e n ot ed by u and v

r e sp e c t i v e l y w h i l e t h e p r e s su r e i s i n d i c a t e d b y t h e sy mb o l p .The g o v e r n i n g e q u a t i o n s f o r s t e a d y f lo w a r e :

c o n t i n u i t y e q u a t i o n :

NS-equations conserving momentum:3u 3u 3p* „,u — + v — + r — = v V* u3x 3y 3x

3v 3v 3p* „,u — + v — + T * - = v V* v3x 3y 3y

(3 -2 )

p* d e n o t e s th e k i n em a t i c p r e s s u re % v i s t h e k i ne m a t ic v i s c o s i t y o f t h e f l u i dP

and V2 s tands for the we l l -known Laplace operator .The k i n e m a t i c p r e s su r e c a n b e e l i m i n a t e d f ro m Eq . ( 3 - 2 ) i n o r d e r t o o b t a i n a

dv 3us i ngl e equat i on f or the vor t i ci t y w = — - T—ox o y

3co 3to „, i0 0\u r - + V T - = v 7 l ID 3 o3x 3y

t h e s o - c a l l e d v o r t i c i t y tr a n s p o r t e q u a t i o n .

On t h e w a l l s u r f a c e w he re y = 0 t h e f lo w s a t i s f i e s t h e n o - s l i p b o u n d a r y

c o n d i t i o n s

u ( x , o ) = v ( x , o ) = 0



- 1 2 0 -

T he u nk no wn c o e f f i c i e n t s o c c u r r i n g i n E q. 3 ^ ca n b e u s e d f o r a p r o p e r f i t t i n g

o f t h e l o c a l f lo w i n t o a s u r r o u n d i n g ma in f l o w . T he n um b er o f t h e s e c o e f f i c i e n t s

t h a t c a n b e c h o s e n i n d e p e n d e n t l y i s r e d u c e d c o n s i d e r a b l y b y t h e r e l a t i o n s g i v e n

b y E q s . ( 3 - 5 ) a n d ( 3 - 6 ) . F o r a f o u r t h - o r d e r e x p a n s i o n , a s g i v e n i n E q . {3-^),

o n l y s e v e n c o e f f i c i e n t s {a., a - , a_ , a^., a , , a_ and ag ) a r e l e f t i n s t e a d o f t h eo r i g i n a l s i x t e e n .

Our m ain o b j e c t i v e h e r e w i l l b e a d e s c r i p t i o n o f a l l t o p o l o g i c a l l y d i f f e r e n t

f lo w p a t t e r n s t h a t w i l l a r i s e n e a r ( 0 , 0 ) i f t h e s e v e n u nk no w ns a r e v a r i e d

i n d e p e n d e n t l y . M o r eo ve r, t h e c o n d i t i o n s f o r w h i c h t h e s e f l o w p a t t e r n s a p p e a r

w i l l b e f o r m u l a t e d . T o i n t e r p r e t th e s e c o n d i t i o n s i n a p h y s i c a l c o n t e x t i t i s

w o r th w h i l e t o r e l a t e t h e c o e f f i c i e n t s a . t o w a l l s h e a r s t r e s s , p r e s s u r e an d

g r a d i e n t s o f w a l l s h e a r s t r e s s an d p r e s s u r e . The w a l l s h e a r s t r e s s i s u s u a l l y

d e f i n e d a s x = P ( T ~ ) _n w h e re p i s t h e d y na m ic v i s c o s i t y o f t h e f l u i d .

C o n s u l t i n g E q s {3.h) a nd ( 3 - 5 ) o n e c a n v e r i f y t h e r e l a t i o n s :

T

a. = -1 M

T p

a - — - - - ï2 p u

1 Pxa_ = = + = —

3 2 u

a 4 = 2 u = _ 2 u

1 p 1 p

1 xx 1 *yya 5 = " 2 u = " 2 p

i xxx 1 xxy _ jL yy y^ ~ 6 p 6 p ~ 6 p

1 xxx 1 xyya 8 = = 5 p = " 5 p

w h i ch h o l d i n t h e p o i n t ( 0 . 0 ) .

I n d i c e s a r e u s e d t o i n d i c a t e p a r t i a l d e r i v a t i v e s .

T h e s t r e a m l i n e p a t t e r n n e a r t h e w a l l i n t h e u p p e r h a l f p l a n e y i 0 may bed e r i v e d from t h e v e l o c i t y f i e l d u ( x , y ) , v ( x , y ) a s



- 121-

dx , .x = dï = u( x' y )

y = it = v ( x , y )(3-8)

t denot es a t i me- l i ke par amet er var yi ng al ong st r eam i nes.

I nt r oduci ng t he Tayl or expansi on of t he vel oci t y component s f rom Eq. (3-' t) i nt o

( 3. 8) we obt ai n a nonl i near aut onomous syst em f or t he descri pt i on of t he st r eam

l i ne pat t er n; t he phase pl ane r epr esent s t he r eal f l ow domai n, t he phase t r aj ec

t or i es may be i dent i f i ed wi t h t he st r eam i nes and t ser ves as a par amet er var y

i ng al ong a t r aj ector y.

Si nce y i s a common f act or i n u( x, y) and v ( x , y ) , t he l i ne y = 0 i s a l i ne of

s i ngul ar poi nt s i mpl yi ng a quasi nonhyper bol i c s i ngul ar char acter at t he wal lsur f ace. Si nce t he t r aj ector i es of system ( 3- 8) ar e i dent i cal wi t h t hose of t he

equi val ent syst em

• - 1 • - 1x = y u( x, y) , y = y v( x, y) ( 3- 9)

wi t h ( ) = -r-q = - -T-, we can r emove t he si ngul ar charact er of t he l i ne y = 0 by

i nvest i gat i ng Eq. ( 3. 9) i nstead of Eq. (3 .8 ) . I n Eq. ( 3 - 9 ) , t * i s a t i me- l i ke

par amet er var yi ng al ong t he t r aj ect or i es ( s t ream i nes ) ,The dynam cal system under consi derat i on becomes

x = a a +a y+aj j x' +a y- a +a x' +agx' y- a- xy r t - g^ ag) y' +0( 4)

y = - ^y - a x y - ^y ' - x ' y - ^a g x y' +^y ' +OCt )

( 3. 10)

The el ement ar y si ngul ar poi nt s of t hi s syst em wi l l be anal ysed bel ow ( sect i on2. 2 and 2 . 3 ) ; t he cl ass i f i cat i on of hi gher - order s i ngul ar i t i es wi l l f ol l ow i n

a separ at e par agr aph ( par agr aph 3 ) .

2i 2i _El ement ar y si ng l ar _goi nt s_l ocat ed_at _t he_wal l _ on- wal l _si ng^

Let us anal yse t he f l ow near a poi nt on the wal l wher e syst em ( 3- 10) di spl ays a

s i ngul ar i t y . We t ake t he or i gi n of our r ef er ence system i n the s i ngul ar poi nt ,

then a1 = 0 whi ch means t hat t he wal l shear st r ess vani shes i n t he si ngul ar i t y;

T ( 0 , 0 ) = 0. System ( 3- 10) may now be wr i t t en:

- 123-

Ham l t oni an syst em x = H , y = - H ; H( x, y) , t he Ham l t oni an, i s act ual l y equaly x

t o t he f am l i ar st r eam f unct i on * . St ream i nes i n t he f l ow pat t er n can be seen

as l evel cur ves: H = const ant on t he sur f ace H = H( x , y) i n t he H, x , y - space.

Si ngul ar poi nt s i n t he s t r eam i ne pat t er n appear i f u = v = 0, t hus i f H =

H = 0. The char acter of a s i ngul ar poi nt f ol l ows i n t he usual manner f r om t heei genval ues A ^ of t he coef f i ci ent mat r i x of t he l i near par t :

/ H H' yx yy

H - H\ xx xy

i n the s i ngul ar poi nt .

Then one f i nds A, + A. = H - H =0 and A. . . A- = H H - H» . El ement ary1 2 yx xy 1 2 xx yy xys i ngul ar poi nt s appear i f A. . . A- * 0, t hey wi l l be hyper bol i c saddl es i f t he

condi t i on H H - H* < 0 i s sat i sf i ed i n t he si ngul ar i t y.xx yy xyI n t he case H H - H* > 0 t he si ngul ar poi nt i s a cent er poi nt , at l east f or

wc yy xyt h e l o c a l l y l i n e a r i z e d s y s t e m . But t h e s i n g u l a r i t y i s a l s o a c e n t e r p o i n t f o rthe non l inear sys tem s inc e H H - H* > 0 im pl ies th at the H(x,y) su r f a c e has

xx yy xy

a l ocal ext r emum at t he si ngul ar i t y so t hat t he l evel cur ves H = const ant appear

as cl osed cur ves, at l east i n a nei ghbour hood of t he si ngul ar i t y.

So we may concl ude now t hat t he el ement ar y f r ee si ngul ari t i es ei t her appear as a

saddl e or as a cent er .

I n addi t i on, not e t hat i n t he spec i al case of an i nvi s ci d i r r otat i onal f l ow,

wher e t he st r eam unct i on i|i(x,y) and thus H( x, y) sat i s f i es Lapl ace' s equat i on,

t he ei genval ue pr oduct A-, . A_ i s negat i ve t hr oughout and consequent l y onl y saddl e

poi nt s wi l l appear i n such f l ows.



- 1 2 4 -

3. Higher-order singularities in the flow pattern

Hi g h e r - o rd er s i n g u l a r p o i n t s i n t h e fl ow p a t t er n o f a v i s co u s f l u i d may b e fo un d

i n t h e i n t e r i o r o f t h e fl ow a s w e l l o n a w a l l . To c l a s s i f y t h e d e g e n e r a t e f lo w

p a t t e r n s t h a t c a n o c c u r i n a v i s c o u s fl ow i t i s u s e f u l t o make a c l e a r d i s t i n c t i o n b e t w e e n p o i n t s i n t h e f l o w and p o i n t s on t h e w a l l . L e t u s s t a r t w i t h

s i n g u l a r i t i e s i n t h e f l o w a w a y f r o m t h e w a l l .

^ i l i . H i g h e r ^ o r d e r s i n g ^ l a r p ^ i n t s i n t h e ^ o w f i e l d

From t h e co n t i n u i t y eq u a t i o n ( 3 . 1 ) i t f o l l o w s t h a t t h e f lo w p a t t ern o f a s t ea d y

2D v i s co u s f l u i d ca n b e d es cr i b ed b y a Ha m i l to n ia n s y s t em

x = H , y = -Hy x

w h ere H( x , y ) i s a n a l y t i c a nd a c t u a l l y eq u a l t o t h e s t rea m f u n c t i o n $ .

A ssu me t h a t t h e s y s tem h a s a h i g h er - o r d er s i n g u l a r p o i n t a t ( 0 , 0 ) , s o t h a t t h e

c o n d i t i o n s

H = H

x y

and

A r A 2 H H - Hl = 0x x y y x y

a r e s a t i s f i e d s i m u l t a n e o u s l y a t t h i s p o i n t .

To s t u d y i t s t o p o l o g i c a l p r o p e r t i e s l e t th e H a m i l t o n i a n s y s t e m b e e x p a n d e d a s

f o l l o w s

/y x y y

-H -Hx x x y

f \X

y

1

+P 2 ( x , y )

Q 2 ( x . y )

w h e re t h e s e c o n d - o r d e r p a r t i a l d e r i v a t i v e s o f H ( x , y ) ar e tak en i n (0 , 0 ) and

where P _(x ,y ) and Q ? ( x , y ) c o n t a i n te rm s n o t l o w e r t h a n s e c o n d d e g r e e a nd o b e y

t h e r e l a t i o n



- 125-

Unl ess H and H ar e bot h zer o, a nonsi ngul ar l i near t r ansf or mat i on exi st sxx yy

whi ch t r ansf or ms t he syst em i nt o t he equi val ent f or m

x = y + P2( x , y )

y = Q2( x , y )

a n d we may a p p l y A n d r o n o v ' s c l a s s i f i c a t i o n s ch em e f o r h i g h e r - o r d e r s i n g u l a r

po in t s hav ing bo th e igenva lues ze ro ( see Chap te r I ) . As a re s u l t we o b ta in the

fol lowing degenerate f low pat terns :

3' Q2, * 0 : cusp poi nt s i ngul ar i t y

3 2 Q 2 3 2 Q 2

, , = 0 , - * 0 : th i rd -o rd er sadd le po in t■ ox oxoy

3 ! Q 2 3 » Q 2

, = . = 0 : flow p at te rn s wi th a h igh er degree of de ge ne rac y, t h a t

remain unspecif ied here .

Q u a l i t a t i v e s k e t ch e s of t h e c us p p o i n t an d t h e t h i r d - o r d e r s a d d l e p o i n t a r e

shown in Fig. 3-2.

cusp poi nt saddl e point (t hi rd- order)

Fi g. 3- 2. Hi gher - or der s i ngul ar i t i es away f r om t he wal l .

I f H and H ar e bot h zer o, t hen t he Ham l t oni an syst em has a vani shi ng l i nearxx yypar t i n t he hi gher or der s i ngul ar poi nt ; t he f l ow pat t er ns have a hi gher degr ee

of degeneracy and r emai n undi scussed.



-126-

3 ;2 ;_Higher -order s ingu la r_go in t s_on_the_wal l

I n p a r a g r a p h 2 i t w as show n t h a t t h e s t e a d y f lo w p a t t e r n n e a r a w a l l i s

d e s c r i b ed by an auto no mo us d y n am i ca l s y s t em , E q. ( 3 . 1 0 ) , co n t a i n i n g s e v e r a l

unknown co ef f ic ie nt s (a 1 , a_, a_, . . . e t c . ) . Moreover, i f a . = 0 and a_ * 0 th i ssy ste m h as a h y p e rb o l ic s i n g u l a r i t y a t t h e o r i g i n r e p r e s e n t i n g t h e o r d i n a r y

separation or at tachment phenomenon.

For a_ = 0 the c la s s i c a l Oswat i ts ch-Legendre so lu t i on near s epa ra t ion o r a t t a ch

m en t f a i l s a s a co n s eq u e n ce o f t h e n o n h y p e r b o l i c ch a r ac t e r o f t h e s i n g u l a r

p o i n t . M o re ov er ,th e s e p a r a t i o n a n g l e p r e d i c t e d b y t h i s s o l u t i o n E q . ( 3 . 1 1 )

becomes zero.

For a_ = 0 , a h igh er -ord er s ing u l a r i ty appear s on th e w a l l ; a de ge ne ra te f low

pat tern i s formed where shear s t ress and s t reamwise shear s t ress gradient vanishs i m u l t an eo u s l y . The s t r eam l i n e p a t t e r n n ea r s uch a s i n g u l a r p o i n t s a t i s f i e s t h e

nonl inear sys tem

x = a-y + a^x ' + a,.xy - ^ a^y1 + a_x" + ag x 'y - ^ a_xy2 - = agy' + 0(4)

(3-12)

y = - a^xy - x a -y1 - | a_xly - = agxy2 + ^ a,y ' + 0(4 )

The shear s t r es s d i s t r ibu t ion a long the wal l near the s ingu la r po in t (0 ,0 )

x = u(ajjX ! + a -x ' + 0 (x ' ) )

shows t hat at t he s i ngul ar poi nt t he di str i but i on of the shear str ess x at t ai ns

an extr eme val ue f or a^ * 0 and an i nf l exi on poi nt f or aj = 0, a_ * 0.

For t he case a * 0, t he appear ance of a shear st r ess ext r emum i ndi cat es t hat

f l ow r ever sal at t he wal l sur f ace i s not t o be expect ed. The f l ow r emai ns at t ached t o t he wal l r egar dl ess of t he occur r ence of a par t i cul ar poi nt wher e t he

shear st r ess and st r eamwi se shear st r ess gr adi ent vani sh.

I n t he case of aj , = 0, a_ * 0 t he shear st r ess changes si gn i ndi cat i ng t hat f l ow

r ever sal may be expect ed. The si mul t aneous occur r ence of an i nf l exi on poi nt i n

t he s hear st r ess di st r i but i on at t he separ at i on poi nt suggests separ at i ons of

hi gher - or der nat ur e.

I n or der t o obt ai n a cat al ogue of degener at e f l ow pat t er ns near a wal l , l et usanal yse t he nonl i near syst em ( 3- 12) near t he or i gi n.

- 127-

Thi s system has a nonhyper bol i c poi nt at ( 0, 0) wi t h a doubl e zer o ei genval ue. To

anal yse the var i ous possi bi l i t i es of degener at e f l ows Eq. ( 3. 12) wi l l be r ecas t

i nt o one of t he t wo f ol l owi ng f or ms.

A. I a_ * 0; nonzer o st r eamwi se pr essur e gr adi ent

x = y +a x2 +&5xy - | a y2 +a7x' +&gx*y - i éLxy2 - | i gy' +0(4)

( 3- 13)

y = - a xy - | a5y* - | a ^y - | agxy2 +| a?y' + 0( 4)

wher e a. =a. a,

B. | a_ =0; zer o st r eamwi se pr essur e gr adi ent

• 2 1 1

x = a^x2 + a^xy - Ö a^y2 + a_x* + a g x2 y - euxy* - ö a3y ' + 0(4)

(3.14)

y = - a^xy - ^ a -y2 - | a_x2y - x agxy2 + TJ a^y' + 0(4)

Note the dif ference between the two forms: for nonzero pressure gradient (a_ *

0) the system has a linear part whereas for zero pressure gradient (a~ = 0), a

system with at least quadratic terms results. Both forms will be investigated

below.

Case A: Singularities with a nonzero streamwise pressure gradient (a., * 0) .

For a nonzero pressure gradient Eq. (3.13) r esul ts which has a l inear part of

the form (_ _) . The character of the nonhyperbolic point may be analysed by

using Andronov's cl as si f icat io n for degenerate po ints with a double zero

eigenvalue (Chapter I) .• • •

On the isocline of vertical directions (x = 0) y will be expanded as y = tli(x) =k k k

A, x + o(x ) where x is the first nonvanishing term in the expansion

« ( x ) = a ^ x ' a 4 < | a y - | a ^ ) x- * | a ^ + a 4 ( . . . ) ) xs ♦ o ( x « )

The expansi on f or t|>x) poi nt s out t hat hi gher - or der s i ngul ar i t i es at nonzero

pr essur e gr adi ent may be di st i ngui shed, cor r espondi ng t o a * 0 and a = 0.

- 128-

Al : Thi r d- or der t opol ogi cal saddl e poi nt ( a^ « 0 ) , x ( 0, 0) * 0

For a^ * 0 we have k = 3, 4 = 41 so t hat t he degener at e si ngul ar i t y i s a

t opol ogi cal saddl e poi nt of t he thi r d or der .

The l ocal f l ow near t he t hi r d- or der saddl e poi nt i s descr i bed by t he system

x = y + &J.X1

V y

(b)Txx<0: r

w a l l

Fi g- 3- 3- Thi r d- or der saddl e poi nt s,

( px - ° ' Txx * 0 )

( a) one hyper bol i c sect or

( b) t hr ee hyper bol i c sect or s

A s k e t c h o f t h e f lo w p a t t e r n i s

show n in F ig u r e 3 - 3 - The s a d d l e

p o in t has fou r sep a ra t r i c es , two of

them coincide with the wall and two

s e p a r a t r i c e s a r e t a n g e n t t o t h e

w al l , forming a para bo l ic cu rv e . I fx > 0 th is para bol i c curve l i e s inxxthe lower ha l f p lane y < 0 i n d i c a t

ing that the f low in the upper half

p lane remains a t tached to the wal l ,se e F ig ur e 3- 3a . On the o ther hand

i f x < 0 , the pa rab ol ic a l ly curved

s e p a r a t r i c e s l i e i n t h e u pper h a lf

p lane so tha t a f low pat te rn resu l ts

as shown in Figure 3-3b.

A2: F i f t h - o r d e r t o p o lo g i c a l s a d d l e p o in t (&;. = 0 ) . ^ ( 0 . 0 ) = 0

Fo r &r = 0, we observe th at th e f i r s t n onvanishing term of *(x) i s of the f i f t h3degree (k = 5) having a po s i t i ve c oe ff i c ie n t A. = i a* if flu * 0.

A c c o r d i n g t o A n d r o n o v ' s c l a s s i f i c a t i o n s c h e m e , t h e d e g e n e r a t e s i n g u l a r i t y

appears to be a top olo gic a l saddle po in t of the f i f t h ord er .

The local f low near this point is adequately described by the system

x = y + É Lx '

2y = S y X ' y

- 129-

The condi t i on aj , = 0 i ndi cat es t hat t he second der i vat i ve of t he shear st r ess

T ___ vani shes at t he si ngul ar i t y. The separ at r i ces ar e f ound as y= - 4 a_x3+0( x*)xx

and y 0 .

Corresponding f low pat te rn s a re shown in F igu re 3 .4 . The cas e a_ < 0 re pr es en ts

a flow s e p a r a t i o n p h e n o m e n o n a s

(a)separat ion

(a 7<0)

(b)attachment

( a7>0)7777777777777777;

* x

Fi g. 3. 4. Fi f t h- or der saddl e poi nt s .

(p * 0, T = 0 )X XX

s k e t c h e d i n F i g u r e 3 - 4 a . Th e

se pa ra t in g s t re am line y = - r a_ x '

l i e s i n t h e u p p e r h a l f p l a n e ,

l e a v i n g t h e w a l l t a n g e n t i a l l y i n

s t reamwise d i rec t ion .

T he o t h e r c a s e a_ > 0 r e p r e s e n t s

f l o w a t t a c h e m e n t ; t h e s e p a r a t r i x i s

an a t t a c h m e n t s t r e a m l i n e w h i cht e r m i n a t e s a t t h e a t t a c h m e n t p o i n t

tangent to the wal l (see Fig . 3-4b) .

The stu dy of t h e cas e a_ = 0, i s not

t a k e n u p i n t h i s w o r k , a f u r t h e r

expansion of the velocity components would then be necessary.

Case B: S in gu la r i t i e s w i th ze ro p ressu re g rad ien t ( a - = 0 )

S in g u l a r f lo w p a t t e r n s , h a v in g z e r o p r e s s u r e g r a d i e n t ( a- = 0 ) i n st re am w ise

direc t ion are descr ibed by the sys tem (3 .14) :

x = aj.x* + a,-xy - = a^y2 + a^c ' + agx 'y - - a-xy* - -= agy' + 0(4)

y = a^xy - - a,-yl - | a_x*y - -z agxy* + -z a_y' + 0(4 ) (3.14)

Si nce t he l i near par t i s m ssi ng, t he cl assi f i cat i on of Andr onov cannot be used

i n obt ai ni ng t he degener at e f l ow pat t er ns . So we ar e l ef t wi t h t he bas i c

quest i on: whi ch t er ms i n t he r i ght - hand s i de of ( 3. l 4) are essent i al f or the

l ocal t opol ogy of t he degener at e st at e?

We wi l l pr ove t hat t he quadr at i c par t of ( 3. l 4) det er m nes t he qual i t at i ve

pat t ern pr ovi ded t hat aj , * 0.

The mai n t ool i s a t echni que cal l ed ' bl owi ng- up' , f or exampl e used by Andr onov

( 1973) and Takens ( 1974) t o study degener at e s i ngul ar i t i es wi t h at l east onel i near t erm I t wi l l be shown her e that t he bl owi ng- up met hod al so suf f i ces i f



- 1 3 0 -

t h e l i n e a r p a r t i s m i s s i n g . The i d e a c o n s i s t s o f i n t r o d u c i n g p o l a r c o o r d i n a t e

t r a n s f o r m a t i o n s t o e x p a nd d e g e n e r a t e s i n g u l a r p o i n t s i n t o c u r ve s c o n t a i n i n g a

f i n i t e n um ber o f s i n g u l a r p o i n t s . I f t h e s e a r e h y p e r b o l i c a f t e r b lo w - up t h e n t h e

l o c a l p h a s e p o r t r a i t n e a r t h e c u r v e , a n d h e n c e n e a r t h e o r i g i n a l d e g e n er a t e

p o i n t , i s u n a m e n a b l e t o t h e i n f l u e n c e o f h i g h e r - o r d e r t e r m s .

I n t h e c a s e o f E q . ( 3 -1 * 0 o n e b l o w - u p , p r o v i d e d aj, * 0 , i s s u f f i c i e n t t o o b t a i n

a s t a b l e t r a n s f o r m e d v e c t o r f i e l d .

H e re we d e s c r i b e t h i s b lo w - up i n d e t a i l t o i l l u s t r a t e t h e m e th o d .

L e t t i n g x = r cos 8 , y = r s in 9 ( r 2 0 , 0 £ 8 i n ) we g e t E q. ( 3 - l ' 0 r e c a s t i n

t h e p o l a r c o o r d i n a t e f o r m

• 5 1r = r ( a i , c ' + a_c2s - •£ aj.cs2 - ~ a _ s ' ) + 0 ( r 2 )

( 3 . 1 5 )

8 = - 2 a j . s e 1 - 5 a_cs 2 + 0 a ^ s ' + s.0(r)

w h e re c and s a r e s h o r t h a n d n o t a t i o n s f o r c o s 9 a nd s i n 9 , r e s p e c t i v e l y .

We now e xp an d t h e s i n g u l a r i t y a t r = 0 b y r e g a r d i n g ( r , 9 ) a s a c a r t e s i a n c o o r

d i n a t e s y s t e m .

On t h e i n t e r v a l 0 £ 8 £ 11 ( 3 . 1 5 ) h a s s i n g u l a r p o i n t s o n r = 0 a t 8 . = 0 , 9 _ = n

and 6- j . = t a n (ftj . ± V3 + a2 ) , where ft- = a ^ / a j . .

A s i m p l e e i g e n v a l u e c a l c u l a t i o n s ho ws t h a t a l l t h e s e p o i n t s a r e h y p e r b o l i c

p r o v i d e d t h a t a j, * 0 . M o r eo v e r,

i t a p p e a r s t h a t t h e e i g e n v a l u e

1 3 k 7 p r o d u c t i s n e g a t i v e t h r o u g h o u t,r

U ' ^i m p l y i n g t h a t t h e s e p o i n t s a r e

s a d d l e p o i n t s .

». Q F o r aj. * 0 t h e ' b l o w - u p ' p a t t e r n

i s s ho w n i n F i g u r e 3 - 5 ( a r r o w s

d e n o t e f l o w d i r e c t i o n s w h i c h

F i g - 3 - 5 - F i r s t b l o w - u p o f E q . 3 . 1 4 ( a , . > 0 ) . c o r r e sp o n d t o t h e c a se a ^ > 0 ) .

S i n c e f o r aj , * 0 a l l s i n g u l a r

p o i n t s a r e h y p e r b o l i c a f t e r t h e f i r s t b l o w - u p , t h e f lo w o f E q. ( 3 . 1 5 ) i s s t a b l e

w i t h r e s p e c t t o s m a l l ( h i g h e r - o r d e r ) p e r t u r b a t i o n s . C o n s e q u e n t l y , a s i m i l a r

c o n c l u s i o n f o l l o w s f o r E q . ( 3 . 1 4 ) ; t h e d e g e n e r a t e flo w p ro b le m i s d e te r m i n e d

t o p o l o g i c a l l y b y t h e q u a d r a t i c t er m s a nd i t s t o p o l o g i c a l s t r u c t u r e c a n n o t b e

d i s t u r b e d b y a d d i n g h i g h e r - o r d e r t e r m s .

http://aj.cs2/

http://aj.cs2/

http://aj.cs2/

http://-2aj.se1/

http://-2aj.se1/

http://-2aj.se1/

http://-2aj.se1/

http://aj.cs2/



- 1 3 1 -

B l : S a d d l e p o i n t w i t h t h r e e h y p e r b o l i c s e c t o r s (a ^ « 0 ) . T ( 0 . 0 ) * 0

T h e fl o w p a t t e r n s t h a t w i l l o c c u r i n t h e p h y s i c a l p l a n e fo r a j. * 0 may b e o b

t a i n e d i f F i g u r e 3 - 5 i s ' b l o w n d o w n ' fr om ( r , 8 ) -» ( x , y ) . T h e s e p a t t e r n s a r e

d ra w n i n F i g u r e 3 - 6 , t h e fl o w d i r e c t i o n c o r r e s p o n d s t o an > 0 .

T he b l o w i n g - u p m e th o d e v i d e n c e s t h a t t h e t 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 s e

p a t t e r n s a r e s u f f i c i e n t l y d e s c r i b e d b y t h e r e d u ce d sy st em

x = a^x* + a 5 xy - ^ a^y*

1 .y = - a^ x y - ^ a ^ 1

The degener at e f l ow pat t er n i n quest i on has a saddl e poi nt behavi our ; t her e ar e

f our separ at r i ces , di vi di ng t he upper hal f pl ane i nt o t hr ee hyper bol i c sect or s .

The separ at i on angl es a- and ou ( Fi g. 3. 6) sat i sf y t he r el at i on t an a-. tan cu=3-

tan avtan a2=3

wal l

Fi g. 3. 6. Saddl e poi nt wi t h t hr ee hyper bol i c

sect ors (p = 0, t * 0 ) .* X XX

B2: Saddl e poi nt wi t h t wo or f our hyper bol i c sect or s (a;, =0 ) . T ( 0, 0) = 0

For aj . = 0 t he f i r st bl ow- up gi ven by Eq. ( 3. 15) r educes t o

= r ( a c l s - ^ acs ' )

4= - -r a cs* + s. 0(r)

i 5

( 3- 16)

On t he i nt er val 0 £ 8 S n, ( 3. 16) has t hr ee si ngul ar poi nt s on r = 0, t hey occur

at 9 = 0, 6 = | and 8 = I T.

The si ngul ar poi nt at 8 = i s a hyper bol i c saddl e poi nt i f a , * 0. The poi nt s

at 8 = 0 and 8 = it appear t o be nonhyperbol i c and a second bl ow- up ( r , 8) ■> ( n, * )has t o be per f or med. Set t i ng 8 * o , r * 0 we expand ( 3. 1*0 i n Tayl or ser i es near

( r , 8) = ( 0, 0) t o obt ai n:

- 132-

r = a r 9 + a_ r2 + 0( 3)

ê = - a, n« - 5| a59« - | a? r 9 + 0( 3)

A s i m l ar f or m i s obt ai ned near ( r , 9) = ( 0 , u ) .Af te r t he second bl ow- up, def i ned by 9 = n cos t|> r = n s i n * (n > 0, 0 £ ty S )

t he f ol l owi ng vec to r f i el d i s obt ai ned

n = n f a ^ s i n ^ c o s * + a - s i n * * - ^ a ^ c o s ' * - | a ^ s i n ^ c o s 2 * } + 0 ( n * )

I3 "5*= £ a c s in i | i cos 2 r(i + £ a_cosi | i s inl * + s in i | i .0 (n )

( 3 . 1 7 )

W i t h i n t h e d om a in 0 £ t> S -r, t h e s i n g u l a r p o i n t s a r e f o u n d a t n = 0 ,i l> = 0 , ^- 1

a n d ^ = t a n ( - 2 / 3 . 8 5 - / 0 , ) . A l l t h e s e p o i n t s a r e h y p e r b o l i c p r o v i d e d t h a t a ,. * 0 ,

an d eu * 0 . H en ce t h e v e c t o r f i e l d o f ( 3 -1 7 ) i s s t a b l e w i t h r e s p e c t t o h i g h e r -

o r d e r p e r t u r b a t i o n s .

We now ' b l o w - d o w n ' o n ce t o g e t t h e

p h a s e p o r t r a i t s i n t h e r , 9 p l a n e

( F i g . 3 - 7 ) .

T h e f l o w d i r e c t i o n i n F i g u r e 3 . 7

co r r e s p o n d s t o a_ > 0 .

The n ex t ' b l o w - d o w n ' t o t h e p h y s i c a l

x , y - p l a n e r e s u l t s i n d e g e n e r a t e flo w

p a t t e r n s a s d e p i c t e d i n F i g u r e 3 . 8 .

The b low ing-u p method shows th a t t h e

f lo w t o p o l o g y o f t h e s e d e g e n e r a t e

s t a t e s i s f u l l y d e t e r m i n e d b y t h e

sy ste m (a_ * 0 , a_ * 0)

„ Oh<Q: UlLXln/2

r

a 5

UJ|1_J '0 i t / 2 1

F i g. 3. 7. Fi r st bl ow- up of Eq. 3. 14

(a,jj = 0 , a, . * 0 ) .

x = a-xy_xy + a_x*

5 Y ' "1 V Iy

T h e d e g e n e r a t e f l o w s h a v e a s a d d l e n a t u r e , i f a ^ a - > 0 (< 0 ) t h e r e a p p e a r tw o

( f o u r ) h y p e r b o l i c s e c t o r s i n t h e u p p e r h a l f p l a n e , r e s p e c t i v e l y .

- 133-

As we know, t he condi t i on a„ = a j = 0 cor r esponds t o poi nt s at t he wal l wher e

t he st r eamwi se pr essure gr adi ent ( p ) and t he second- or der shear st r ess gr adi ent

( T ) vani sh s i mul t aneousl y. Thi s i nvol ves that t he obser ved f l ow may be di s

cer ned accor di ng t o t he si gn of T and t he sor t of pr essur e ext r eme ( maxi mum

or m ni mum . At a l ocal maxi mum of t he wal l pr essur e (a_ < 0) t he f l ow separ at es

5

a 7 > 0 :7777777777777.

7 ^77777777777. -7777777777777.

minimumwall pressure

(a5>0)

maximumwall pressure

(a5<0)

Fi g. 3- 8. Saddl e poi nt s wi t h t wo or f our hyper bol i c sect or s

( Px = 0, T ^ - 0 ) .

per pendi cul ar l y f r om t he wal l sur f ace i f T < 0 (a_ < 0) ; but t her e i s a mor e

compl i cat ed separ at i on/ at t achment st r uct ur e i f T > 0 ( a_ > 0) . On t he ot her

hand i f a l ocal m ni mum of t he wal l pr essur e i s pr esent , t he f l ow at t aches

per pendi cul ar l y t o t he wal l i f T > 0 and a compl i cat ed at t achment / separ at i on

pat t er n occur s i f t < 0.X X X

I n c o n c l u s i o n t h e e l e m e n t a ry an d h i g h e r - o r d e r s i n g u l a r i t i e s on t h e w a l l ,encountered so far are summarized in the next table.

- 1 3 4 -

Hyperbo l ic

T „ ( 0 , 0 ) * 0

singular i ty vector f ie ld

sadd le po in t

O s w a t i t s c h - l e g e n d r e

so lu t i o n for e le me n ta r y

separa t ion and

a t t a ch me n t

f low pat terns

i ,«0 — - r ^ ^ C

a?>0 . „

No n -h yp e rb o l i c

n o n ze ro

p re ssu re g ra d ie n t

T X ( 0 , 0 ) = 0

p x(0.0M0

t h i r d -o rd e r sa d d le

f i f t h - o r d e r s a d d l e

y * a 4 x !

\-34Xy I

a\>0>//?/>////

4 >/?7yy7?;

t'<0

>f/7f//>/.

5 ' > 0 77777777;

No n -h yp e rb o l i c

ze ro p re ssu re

g ra d ie n t

■f* (0.01 = 0

p x(0,0) = 0

sa d d le p o in t w i t h t h re e

hyperbo l ic sec tors

sadd le po in t wi th two or

f o u r h yp e rb o l i c se c to r s

x 2 + a s x y - $ y 2

- x y - } a s y 2

/ a 5 x y a , x 3

a5<0 _- / f V_a,<o 777777577

a7>0 '//?}?///,

a5>0 - > 4 ^ -a7>o 77777777T.

a5>o Ja7<0 'TZ 777:

Tabl e: Hyper bol i c and nonhyper bol i c s i ngul ar i t i es on t he wal l

i n two- di mensi onal vi scous f l ow.

http://-34xy/

http://-34xy/

http://-34xy/

- 136-

P( x i , y i ; p1>u2) = O

( 3- 19)Q^. y ^Uj ^u ^ = o

whi ch sat i sf y t he condi t i on

11m ( x i . y±) = ( 0, 0)ux 0

U 2 + 0-

The sol ut i ons of ( 3- 19) « j = x ( p- . Pp) and y. = y. ( u1, u2) may be r epr esent ed by

sur f aces . Si nce p1 and p_ are assumed t o be smal l we ar e onl y i nt er est ed i n t he

l ocal geomet r y of t he sur f aces near t he or i gi n. Thi s enabl es us t o use apar amet r i sat i on t echni que by means of whi ch t he qual i t at i ve pr oper t i es of t hese

sur f aces can be f ul l y under st ood.

Usi ng t hi s t echni que one l ooks i n f act f or t he l eadi ng ter ms i n Eq. ( 3- 19) whi ch

det er m ne t he posi t i on of s i ngul ar poi nt s as f unct i ons of t he bi f ur cat i on par a

meters .

Let x ± = x±]i2, y± = y ^ ' Px = k p , t hen ( 3. 19) becomes

(k + x± + y± + &4xp p2 + O( p ) = 0

(- | y± - a4 x ^ ) p2 + O( p ) = 0

whi ch shows t hat t he l eadi ng t er ms f or x. and y. may be obt ai ned f r om

u l +p 2 x i +y i + Vi = °

2 p2 y i ■ V i y iÖ Voy< - &tlx^y ^ = °

These equat i ons may be sol ved f or x. and y. t o obt ai n t he f ol l owi ng si ngul ar

poi nt s

f - p 2 ± / p< • — 0S l , 2 - 2ft, , °

p 2 pc2f t 4' W ^

on t he wal l

away f r om t he wal l

-137-

where p i s a compound b if ur ca ti on par am eter : p = p* - ^a^p- .

L e t u s c o n s i d e r t h e two c a se s S j, > 0 a n d CU < 0 in t h e s e m i -p l a n e y £ 0

se p a ra t e ly , t a k in g a^ > 0 f i r s t .

Case &J, > 0: th ir d- or de r saddle p oin t w ith one hyp erbol ic se ct or

F or u < 0 n o s i n g u l a r p o i n t s o c c u r i n t h e s e m i - p l a n e y i 0 ; t h e o r i g i n a l

degenera te s ingu la r i ty d isappears and the f low remains a t tached to the wal l .

F or p > 0 th e re a p pe a r t h r e e d i s t i n c t s in g u la r p o in t s i n t h e u p pe r h a l f p l a n e .

The points S . and S_ are a t tached to the wall and both are saddle points ; above

th e wa l l and ly in g midway b e twe e n th e se s a d d le p o in t s a c e n te r p o in t C i s

found.

Away f r om t he wal l we al so observe a saddl e- t o- saddl e connect i on bet ween S. and

Sy encl osi ng t he cent er poi nt C. A shor t cal cul at i on shows t hat t hi s connect i on

i s gi ven by t he equat i on

= - K ( ( * - ^ > -Sr)

I n t he par t i cul ar case u =0 t he t hr ee si ngul ar poi nt s col l apse i nt o a si ngl e

poi nt on t he wal l (x = - u_/2aj , ) i n whi ch t he degener at e si ngul ar i t y, bei ng t he

or i gi nal t hi r d- or der t opol ogi cal saddl e poi nt , r emai ns. I t means t hat par amet er

changes sat i sf yi ng p = 0 do not r emove nor unf ol d t he or i gi nal degener acy;

consequent l y p = 0 r epr esent s a nongeneri c pert ur bat i on.

The bi f ur cat i on s et , t oget her wi t h the possi bl e f l ow pat t er ns of t he unf ol di ng

of Eq. ( 3. 18) ar e shown i n Fi gur e 3. 9a.

- 139-

Case &, < 0: t hi r d- or der saddl e poi nt wi t h t hr ee hyper bol i c sect or s

The unf ol di ng of t hi s degener at e si ngul ar i t y i s shown i n Fi gur e 3. 9b. For u > 0

t her e appear onl y si ngul ar poi nt s on t he wal l . Two di st i nct saddl e poi nt s S. and

S_ ar e obser ved t her e, r esul t i ng i n a f l ow pat t er n wher e f l ow at t achement (S.)

i s f ol l owed by f l ow separat i on ( S- ) .

For u < 0 t her e i s onl y one saddl e poi nt i n t he f l ow away f r om t he wal l and thel ocal f l ow st r ucture i n t he i nt er f ace bet ween opposi ng f l ow di r ect i ons r esul t s.

The case u =0 r epr esent s a nongener i c bi f ur cat i on, so t hat t he or i gi nal

hi gher - or der si ngul ar i t y r emai ns as a str uctural l y unst abl e el ement i n t he f l ow.

If • ? i _ï 52i Ei Sü£_ y ï ? _5SB5E5£ 25

Let us consi der t he unf ol di ng of t he t hi r d- or der saddl e poi nt havi ng one hyper

bol i c sect or (aj . > 0) i n mor e det ai l .

A gener i c per t ur bat i on of t he th i r d- or der saddl e poi nt can gi ve, at l eas t

l ocal l y, ei t her a f l ow f ul l y at t ached t o the wal l or a f l ow pat t er n wi t h separ a

t i on and r eat t achment . The l at t er i s r ecogni zed as t he wel l known t wo- di men

si onal l am nar separ at i on bubbl e.

Separ at i on bubbl es ar e f eat ur es i n l am nar f l ows t hat have been st udi ed f or many

year s and i t i s wel l - known t hat near bot h t he separ at i on poi nt and t he reat t ach

ment poi nt t he f l ow sat i sf i es t he Oswat i sch- Legendr e condi t i on as gi ven by Eq.( 3. 11) . However , i n t hi s st udy t he separ at i on bubbl e al so appear s i n a l ocal

sol ut i on of t he Navi er - St okes equat i ons showi ng a coher ent f l ow st r uct ur e i n

c l udi ng t he essent i a l f eat ures as separ at i on poi nt , r eat t achment poi nt , r eci r

cul at i ng regi on, vor t ex cent er and di v i di ng s t r eam i ne. Moreover , t hi s f l ow

st r uc t ur e ar i ses as a bi f ur cat i on so l ut i on of a nonhyperbol i c s i ngul ar i t y so

t hat t he genesi s of separ at i on bubbl es can be descri bed i n t er ms of an unf ol di ng

of thi s s i ngul ar i t y .

A necessar y condi t i on f or i nci pi ent bubbl e- t ype separ at i on can now be der i ved.

I t i s u > 0, or expr essed i n ter ms of wal l shear st r ess:

x' - 2t t > 0X XX

Thi s condi t i on poi nt s out t hat separ at i on bubbl es ar i se i n t he f l ow near a poi nt

wher e T = T = 0 i f t he f l ow i s di st ur bed so, t hat xz - 2TT > 0 i n t hi s poi nt .X ' X X

The bi f ur cat i on sol ut i on f or bubbl e- t ype separ at i on yi el ds i nf or mat i on on t he

geomet r y of smal l separ at i on bubbl es. Act ual l y we can cal cul at e t he shape of t he



- 140-

bubbl e for t he l i m t i ng condi t i on u -» 0. The l eadi ng t er ms f or t he hei ght of

t he bubbl e h. and f or t he hei ght of t he vor t ex cent er h above t he wal l w t hD c

r espect t o t he bubbl e l engt h £ become:

bubbl e hei ght h. / 2

vor t ex cent er h / £c'

Expr essed i n t er ms of separ at i on angl e 8 we obt ai n

bubbl e hei ght h. / g = JT tan 9

vor t ex cent er h / £ = -? t an 0c' 6 s

The f i r st expr essi on shows t hat bubbl es i n embr yoni c st at e ar e ver y sl ender i n

t he sense t hat t hey are mor e ext ended i n f l ow di r ect i on than i n t he di r ect i on

normal t o t he wal l .

Fur t her mor e, t he cent er poi nt of t he vor t ex al ways l i es i n t he upper par t of t he

rec i r cul at i ng r egi on.

I n t hi s sect i on we addr ess t he pr obl em t o descri be f l ow separat i on near a wal l

movi ng i n i t s own pl ane as an unf ol di ng of t he t hi r d- or der t opol ogi cal saddl e

poi nt havi ng one hyper bol i c sect or , f t ^ > 0. The set - up of t hi s p robl em i s

basi cal l y t he same as gi ven f or t he f i xed- wal l case wi t h t he except i on t hat t he

wal l moves wi t h a const ant wal l vel oci t y u i n i t s own pl ane. Posi t i ve val ues of

u mean t hat t he wal l moves i n downst r eam di r ect i on wher eas negat i ve val ues ofwu r epr esent an upst r eam movi ng wal l .

Separat ed f l ow pat t er ns near t he movi ng wal l wi l l be i nvest i gat ed by appr opr i at e

unf ol di ng t he t hi r d- or der t opol ogi cal saddl e poi nt ( Eq. 3- 13) so t hat t he

unf ol di ng sat i sf i es t he boundar y condi t i ons

u = u , v = 0 on t he wal l .w

In o rde r t o ob ta in these un fo ld ings sys t em (3 -10) , wh ich i s u sed in the f ixed

wal l case , has to be rep laced by

t ' " c

= A IVn



- 1 4 1 -

x = u + yfa j+apX+a-y+aoc '+a j -xy+a^-y1) + 0( 4)

( 3 - 20)

y = y f - ^ y - a ^ x y - ^ 'W *) + 0W

whi ch sat i sf i es t he cont i nui t y equat i on ( 3 -1 ) .The vor t i c i t y t r anspor t equat i on, Eq. ( 3- 3) r equi r es :

u a_T 1 W 22 a

4+ 3 a e = -*r

Not e that ( 3. 20) r educes t o ( 3. 10) i f u =0 .

Let us exam ne t he phase por t r ai t s ( f l ow pat t er ns) of Eq. ( 3- 20) near t he or i gi n

( 0 ,0 ) , whi ch i s a nonhyper bol i c t hi r d- or der saddl e poi nt i f a. = a. = u =0 . To

obt ai n f l ow pat t er ns val i d near a movi ng wal l we t ake, i n addi t i on t o a- and a_,

al so u as a bi f ur cat i on par amet er havi ng a bi f ur cat i on val ue u =0 . Smal l

var i at i ons wi t h r espect t o t hi s bi f ur cat i on val ue wi l l be denot ed by u = u / a_.

Consul t i ng Eq. ( 3- 20) we f i nd t hat we have t o study t he t hr ee- par amet er unf ol d

i ng

2 u u

?x = uQ + yt uj x+y+a x' + xy - ( j ^ - a^ -^~) yz + 0( 3) ] = P( x, y)( 3- 21)

y = yM- | u2 - V 3 S y + 0 ( 3 )1 =Q(x,y)

wi t h a. = a. (a_) and wher e p , u. and p_ ar e bi f ur cat i on paramet er s.

Si ngul ar poi nt s ( x. , y. ) of ( 3. 21) whi ch appear near ( 0, 0) af t er bi f ur cat i on w l l

be f ound by usi ng t he par amet r i sat i on t echni que adopt ed i n par agr aph ( 4. 1) f or

t he f i xed wal l case.Ref er r i ng t o t hi s case we pr opose

xi = V 2 ' yi = h»2' Vo = koU2' "l = kl 4

then

P( x±, y±) = 0,

Q( x i . y i ) = 0

-1 4 3 -

The unusual b i furcat ion is expla ined i f sys tem (3 .20) is unders tood to unfold a

l i n e o f s i n g u l a r i t i e s , l y i n g a t y = 0 a nd a p p e a r i n g i n t h e v e c t o r f i e l d x =

y1 + a ^ y , y = - a ^ y ' x .

The loc a t io n of the s in gu la r po in t s i s e s se n t i a l l y d e te rm ine d by th e compound

b i f u r c a t i o n paramete r p and the wa l l ve lo c i t y p which then appear to desc r ib e

the unfolding complete ly .

F o r f i x e d u a nd p a v a r i a t i o n of p _ r e s u l t s o n l y i n a s h i f t p2/2&j. in x-

d i r e c t i o n o f t h e s i n g u l a r p o i n t s and t h u s a s h i f t o f t h e c o m p l e t e f lo w

s t r u c t u r e .

B i f u r c a ti o n s e t s and p h as e p o r t r a i t s i n t h e u p p e r h a l f p l a n e a r e d i s p l a y e d i n

Figure 3 .10.

'& & //////777;

Fi g. 3. 10. Fl ow pat t er ns near movi ng wal l s.

For p =0 t he f i xed- wal l case emer ges as a par t i cul ar sol ut i on, r epr esent i ng a

bubbl e- t ype separ at i on i f p > 0 and at t ached f l ow condi t i ons i f p < 0.

Movi ng t he wal l wi l l af f ect t hese f l ows as f ol l ows ( see Fi gur e 3- 10) .

Let us star t wi t h t he case that t he f l ow i s i ni t i al l y (p =0 ) separ at ed (p >0 ) . Movi ng t he wal l downst r eam ( p > 0) wi l l at once r emove the saddl e poi nt s S-

and S_ and wi l l gener at e a f r ee saddl e poi nt S, and a cent er poi nt C i n t he

-In

f l ow. The c l osed l oop f or med by t he separ at r i ces above S, i s a st r eam i ne

enci r c l i ng t he c l osed st r eam i nes around t he cent er poi nt C, f orm ng a regi on

wi t h reci r cul at i ng f l ow above t he wal l . The l ower separ at r i ces of S~ ext end f ar

upst r eam and downst r eam di v i di ng t he f l ow i n an upper and a l ower par t . The

l ower par t accomodat es wi t h t he movi ng wal l and f l ows under neat h t he r eci r

cul at i ng bubbl e; t he upper par t of t he f l ow passes over t he bubbl e. A f ur t her

i ncr ease of t he wal l vel oci t y at const ant p (u > 0) wi l l move t he cent er poi nt

C and t he saddl e poi nt S-, cl oser t oget her , t he bubbl e s i ze di m ni shes unt i l i t

shr i nks i nt o a s i ngl e nonhyperbol i c s i ngul ar i t y when p r eaches t he cr i t i cal

val ue

P *uo = lg|") ( 3* 22 )

° c r i t . öf t4

or expr essed i n physi cal t er ms

i k _u = Q - 2 T

J 8u p Tc r i t . x | xx

wher e p , T, T , T ar e t o be eval uat ed i n ( 0 , 0 ) .X X XX

Beyond th i s c r i t i ca l wal l ve loc i ty the bubble has d i sappeared ; the f low pa t t e rnad ap t s s u f f i c i e n t l y t o t h e m o vin g wa l l an d d o es n o t co n t a in s i n g u l a r p o i n t s

anymore.

I f t he f l ow i s i ni t i al l y (u = 0) at t ached, t hus p < 0, and t he wal l i s moved

downst r eam p > 0, t he unf ol di ng shows t hat t he at t ached f l ow condi t i ons ar e

mai nt ai ned.

Consequent l y, unf ol di ng t heory r ef l ect s t he wel l - known pr i nci pl e t hat separ at i on

near a wal l wi l l be pr event ed i f t he wal l i s movi ng downst r eam at a suf f i ci ent l y

hi gh speed.

Next , i f t he wal l i s moved upstr eam (p < 0) , i r r especti ve of the val ue of u a

f l ow pat t er n i s obser ved havi ng a s i ngl e cent er poi nt above t he wal l . Thi s

cent er poi nt i s par t of a rec i r cul at i ng bubbl e ext endi ng f rom f ar upstr eam t o

f ar downst r eam at l east accor di ng t o t he l ocal appr oxi mat i on. To el uci dat e t hi s

l ar ge st r eamwi se ext ensi on of t he bubbl e l et us obser ve t he i socl i nes x = 0 and

y = 0 i n t he phase por t r ai t s of Eq. ( 3. 21) . These cur ves ar e sket ched i n Fi gur e

3. 11 f or t he cases p = 0 ( f i xed wal l ) , p < 0 ( upst r eam movi ng wal l ) and p > 0

( downst r eam movi ng wal l ) .

- 145-

Fi g. 3- 11- I nf l uence of wal l vel oci t y on i socl i nes x = 0 and y = 0.

The x = 0 cur ve has t wo br anches; i n t he f i xed- wal l case one br anch coi nci deswi t h t he wal l sur f ace, t he ot her f or ms a par abol a i nt er sect i ng t he wal l at Sj

and S_. I mposi ng a smal l wal l vel oci t y (u < 0) wi l l cause t hese br anches t o be

per t ur bed as f ol l ows. Apar t f r om a br anch i n the l ower hal f pl ane t her e appear s

an x = 0 cur ve i n t he upper hal f pl ane, l yi ng above t he unper t ur bed posi t i on and

ext endi ng f r om f ar upstr eam t o f ar downstr eam ( Fi gure 3- H) -

To t he or der consi der ed, t he y = 0 cur ves (y = 0 and x = - p_/ 2aj , ) r emai n

unaf f ect ed by t he movi ng wal l .The behavi our of t he x = 0 and y = 0 cur ves i n t he upper hal f pl ane i ndi cat e t he

f or mat i on o f a r ec i r cul at i ng bubbl e c l ose to the wal l sur f ace i f t he wal l i s

moved upst r eam The bubbl e i s smal l i n hei ght but ext ends f r om f ar upst r eam t o

f ar downst r eam

The pr evi ous obser vat i ons concer ni ng t he unf ol di ng of Eq. ( 3- 21) seem t o suppor t

t he i dea t hat :

- Movi ng a wal l downst r eam can pr event or at l east del ay separ at i on and bubbl ef or mat i on.

- Movi ng a wal l upstr eam wi l l al ways l ead to a ' separ at i on' of t he mai n f l ow; a

smal l r eci r cul at i ng l ayer appear s under neat h t he mai n f l ow.

The f l ow pat t erns al ong movi ng wal l s , obt ai ned by unf ol di ng a t hi r d- or der saddl epoi nt w t h one hyper bol i c sect or , wi l l now be used t o comment on t he so- cal l ed

MRS- cr i t er i on, devel oped t o pr edi ct separ at i on i n unst eady f l ow si t uat i ons.



-146-

I t i s g e n e r a l l y a g re ed t h a t P r a n d t l ' s s e p a r a t i o n c r i t e r i o n : 3 u/3 y = 0 a t y = 0 ,

i s adequa te to p red ic t sepa ra t ion in s teady f lows .

However , i n un s te ad y f lows wi th moving sepa ra t ion reg io ns , Pr an d t l ' s c r i t e r io n

f a i l s and a sugges t ion was made to re p la ce i t by th e so ca l le d M R S- c r i t e r io n

de v el op ed ind ep en de ntl y by Moore (1958 ), Ro tt (1956) and Se ars (19 56 ). The MRS-c r i t e r io n p re d i c t s sep a ra t io n in unsteady f lows i f the co nd i t io n 3u /3y = u = 0

i s s a t i s f i e d in some po in t o f th e flow f i e l d . The MRS-c r it er ion (3u/3y = 0 ,

u = 0) reduces to Pr a n d t l ' s c r i te r i o n (3u/3y = 0 , y = 0) i f the wa l l (y = 0) i s

at rest in the reference frame in which u is measured.

Po in ts in th e flow f ie ld where the cond i t ion u = 0 , 3u/3y = 0 is f u l f i l l e d ar e

ca l led MRS-poin ts .

I t i s no t c l ea r , however , what i s to be unders tood p r e c i s e ly by se p a r a t i o n in

t h i s c o n te x t . This w i l l be ev iden t in the s teady flow a long a moving w al l , th isflow i s un s te a dy i n a r e f e r e n c e s y s te m f i x e d t o t h e w a l l , b u t n o s t r e a m l in e

separa tes f rom the wal l .

Using the MR S-criterion (3u/3y = 0, u = 0) Se ars and T el io ni s (1975) proposed a

model fo r se pa ra ti on in th e steady flow nea r a moving w al l . This so- ca ll ed MRS-

m o d e l i s p r e s e n t e d i n F i g u r e 3 - 1 2 , w h e r e s t r e a m l i n e p a t t e r n s a n d v e l o c i t y

d is t r ibu t ions u (y ) a re shown .

Ir re sp e ct iv e of whether the wa ll moves up stre am o r downstream t h e MRS-model

r e v e a l s th e form ation of a si ng le sadd le po in t in the f low away from the moving

w al l . The ve lo c i t y d i s t r i bu t i on s u (y ) ind ic a te th a t th i s sadd le po in t i s a MRS-

po in t , sa t is fy in g the MR S-cr ite rion 3u/3y = 0 , u = 0 .

Se ve ral num erical and expe rime ntal attemp ts have been made to v e r if y th e MRS-

m o de l t o g e t h e r w i t h t h e c o r r e s p o n d i n g v e l o c i t y p r o f i l e s b y c o n s i d e r in g t he

st ea dy flow alo ng a moving w a ll . A con cise and profound review of most of t h e s e

a t t e m p t s i s giv en by W illiams (197 7). We do not fe el the need to summarize them

here in de ta i l , apar t f rom some remarks suppor t ing Wil l iam's conclus ion tha t the

MRS-model is applicable for the downstream moving case; for upstream moving

w a l l s t h e q u e s t i o n i s s t i l l o p en .

C o n c e r n i ng t h e a p p l i c a b i l i t y i n t h e d ow n str ea m c a s e t h e c o n t r i b u t i o n s of

Telionis and Werlé (1973), Tsahalis and Telionis (1973) and Danberg and Fansler

( 1 9 7 5 ) h a v e t o b e m e n t io n e d i n p a r t i c u l a r . Th e i r r e s u l t s a r e b a s e d o n t h enumerical solution of the boundary layer equations and show a close agreement of

the calculated velocity profiles with those expected from the MRS-model.



- 147-

Fi g. 3- 12. St eady st r eam i ne pat t er ns and vel oci t y pr of i l es near

a movi ng wal l accor di ng t o Moor e Rot t and Sear s.

Fur t her mor e t he br eakdown of t hese cal cul at i ons near a st at i on i n t he f l ow wher et he vel oci t y and t he shear st r ess vani sh si mul t aneousl y, seems t o i ndi cat e t he

exi st ence of an MRS- poi nt i n t he f l ow.

More r ecent l y, I noue ( 1981) present ed a numer i cal st udy of t he st eady f l ow al ong

a movi ng wal l . I ns t ead of us i ng t he cl ass i cal boundar y l ayer equat i ons , he

so l ves t he Nav i er St okes equat i ons appr ox i mat el y by negl ec t i ng onl y t he

di f f usi on t er ms i n t he mai n f l ow di r ect i on. Thi s appr oxi mat e ver si on of t he NS- -

equat i ons has some advant ages i n compar i son wi t h t he boundar y l ayer equat i ons.F i r s t , t he equat i ons can be sol ved t hr ough r egi ons of r ever sed f l ow wi t h a

pr escri bed ext er nal vel oci t y di st r i but i on wi t hout meet i ng any si ngul ar behavi our

near t he separ at i on poi nt ( Gol dstei n s i ngul ar i t y) .

Next , whi l e t he st eady boundar y l ayer equat i ons do not per m t sol ut i ons wi t h u =

3u/ 3y = 0, see Danber g and Fansl er (1975). the appr oxi mat i on of Navi er - St okes,

as used by I noue, may per m t such sol ut i ons because pr essur e var i at i ons i n y-

di r ect i on, bei ng rel at ed t o t r ansver se di f f us i on, ar e al l owed.



-148-

i!i5i_yo?2ï§ïDS_52^el f ° r movinS_ï?ïï_§§E§£a£i225

In th i s section we continue the discussion on flow patterns along moving walls.

A comparison is made between the MRS-model and the unfolding model, as obtained

by unfolding the third-order saddle point (section 4.3). Let us start with thedownstream moving wall.

Downstream moving wall

Concerning a downstream moving wall Inoue 's numerical solut ion yie lds flow

patterns very similar to those obtained by unfolding the third order saddle

point (unfolding model). In the unfolding model, a closed recirculating bubble

i s formed above the wall when separation i s already present a t zero wall

velocity, see Figure 3-10.The bubble is bounded by a single streamline which forms a saddle loop with the

saddle point (S_) underneath the bubble. Two stagnation points appear in the

flow, the saddle point S_ just mentioned and a center point C in the interio r of

the bubble region, see Figure 3-13-

Looking for MRS points in the unfolding

model we find two of them, one upstream

and one downstream of the bubble; neither

of them coincides with S_, as suggested bythe MRS-model. These observations confirm

a flow pattern as suggested by the numeri

cal solution given by Inoue. Let us calcu

late the MRS-points (u = 3u/3y = 0) from

the unfolding model.

The leading terms describing the stream

line pattern are (Eq. (3.21))

x = u = uQ + j^y + u2xy + y' + fl^x'y

(3-23)• iy = v = - g P2

y ' " ft4xy' • % > °

and t he unf ol di ng model pr edi ct s MRS- poi nt s i n t he upper hal f pl ane at

m 3i I" 2 ^ * 4 \/Pc 8ft4 *V W (3 -24)

downstream moving wall

F i g. 3- 13- Fl ow near downst r eam

movi ng wal l accor di ng

t o unf ol di ng model .



-11,9-

Based on the un fo ld in g m odel th e f o l l ow in g remarks on the ex is t en ce and s ig

nificance of MRS-points are made:

- Real so lu t io ns for MRS-points occur only i f p > 0 , in di ca t in g t h a t MRS-points

may appear in flows a long downstream moving w a l l s ; th ey w i l l n o t a pp ea r inflows along an upstream moving wall.

- MRS-points wil l appear in the downstream case i f the wall veloci ty parameter ,

p , d o e s n o t e x c e e d t h e c r i t i c a l v a l u e p = ( p /Ö êL )1 . Th is c r i t i c a lc r i t .

v a lu e e q u a l s t h e v a lu e o f p where a nd a b o v e wh ich n o r e c i r c u l a t i n g r e g i o n

could be observed in the unfolding model . I f the bubble disappears due to an

increase in wall veloci ty the MRS-points wil l vanish a t the same t ime.

- In agreement with Inoue 's observat ion, the unfolding model shows the exis tenceof two MRS-points, one downstream and one up st re am of th e bu b b l e . How ever,

I - P 2 p c - V P J . - u 0 ( 8 a 4 ) '

nei t her of them wi l l coi nc i de wi t h S_ : \-zr-, K Z

3 \ 24 8^

suggested by the MRS-model.

O n l y i n t h e tw o l i m i t i n g c a s e s , n am e ly z e r o w a l l v e l o c i t y ( p = 0 ) an dc r i t i c a l wa l l ve lo c i t y (p = p ) , MRS-poin ts co inc id e wi th sad d le po in ts

c r i t .in the f low pa t te rn .For zero wall veloci ty the two MRS-points are located on the wall surface , one

of them co inc ides wi th the separa t ion po in t and the o ther wi th the a t tachment

po in t (po in ts S j and S_ respec t ive ly in F igure 3 -10) .At t h e c r i t i c a l v a lu e o f t h e wa l l v e l o c i ty p a rame te r p = p b o th MRS-

c r i t .p o in t s , t h e s a d d le p o in t (S _) and th e c e n te r p o in t (C) c o a l e s c e in a s in g lepoint above the wall forming a degenerate point in the f low.

E x c ep t i n som e p a r t i c u l a r s i t u a t i o n s wë o b s e r v e t h a t M R S - p o in ts c a n n o t b e

i d e n t i f i e d w i t h s t a g n a t i o n p o i n t s i n t h e flo w f i e l d s i g n a l l i n g t h e o n s et of

s e p a ra t io n . However, MR S -po in ts may b e c o n s id e re d a s s ig n p o s t s (p r e c u r s o r s )

ind ica t ing the ex is tence o f domains wi th rec i rcu la t ing f low fur ther downs t ream.

Upstream moving wall

T he u n fo ld in g mo de l p r e d i c t s f lo w p a t t e rn s n e ar u p st re am mov ing wa l l s r a th e r

different f rom those ar is ing in the MRS-context . The MRS-model shows a saddlepoint (MRS-points) above the wall (Figure 3-12) and a reversed flow region close

t o t h e w a l l . T he u n f o l d i n g m o de l g i v e s a flo w p a t t e r n w i t h a c e n t e r p o i n t

http://-zr/

http://-zr/

http://-zr/

- 150-

above t he wal l . Thi s cent er poi nt i s par t of a reci r cul at i ng f l ow whi ch ext ends

f ar i n st r eamwi se di r ect i on f or m ng a l ayer under neat h t he mai n f l ow. The cent er

of r ot at i on i ns i de the l ayer i s mar ked

by a st agnat i on poi nt ( cent er poi nt ) ,

as shown i n Fi gur e 3•14. Mor eover , noMRS- poi nt s ar e f ound i n the unf ol di ng

model .

The posi t i on of t he vor t ex cent er (y )

upst ream movi ng wal l

Fi g. 3- 11*- Fl ow near upst r eam movi ng

wal l accor di ng t o unf ol di ng

model

r el at i ve t o the maxi mum hei ght of t he

l ayer ( y. ) wi l l be det erm ned f or the

f l ow pat t er n def i ned by Eq. (3. 23) .

The hei ght of t he l ayer i s found f rom

t he di vi di ng st r eam i ne bet ween mai n

f l ow and t he r eci r cul at i ng r egi on. The

maxi mum hei ght yff appear s at t he vor t ex

pos i t i on.

The l i m t i ng val ue of the hei ght rat i o h = y / y . f or zero wal l vel oc i t y (u * 0)

i s f ound t o depend on t he si gn of u :

h = 2/ 3 i f u c > 0

h = 1/ 3 JÏ i f uc = 0

h = 1/ 2 i f u < 0

The di f f er ences ar e r el at ed to the l i m t i ng f i xed wal l sol ut i ons, namel y ei t her

bubbl e t ype separ at i on ( u > 0) degener at e f l ow ( u = 0) or f ul l y at t ached f l ow

(p < 0 ) . Ev i dent l y an ups t r eam movi ng wal l i nt er f er es mor e s t r ongl y wi t hsepar at ed f l ows t han wi t h at t ached f l ows.

Fi nal l y, we consi der t he vel oci t y pr of i l es u = u( y) al ong the wal l .

Typi cal prof i l es , as obt ai ned f r om numer i cal cal cul at i ons ( I noue, 198l ) , ar e

shown i n Fi gur e 3- 15 f or var yi ng wal l shear st r ess. I n t he unf ol di ng model t he

u- vel oci t y near t he wal l may be appr oxi mated by t he l eadi ng t er ms

u ( x , y ) I V - 5i H (& , > o)

- 1 5 1 -

-0.2 0.0

where x has to be measured wi th

r es pe ct to the vor tex ce nt er .

According to th i s equation th ree

typical velocity distributions can

be o bs erved i n r e c i r c u l a t i n g

layers , each one corresponding to

an x-posi tion where the wall shear

stress

u l3yJy=0 V 4;c

Fi g. 3' 15. Typi cal vel oci t y prof i l es for

the case of an upst r eam movi ngwal l accor di ng to I noue ( 1981) .

i s ei t her negat i ve, zer o or pos i

t i ve. The unf ol di ng model suggest st he pos si bi l i t y to have two di f f e

r ent t ypes of reci rcul at i ng l ayers :

one char act er i zed by pos i t i ve wal l shear s t r ess over the whol e l engt h of the

l ayer (u < 0) and a second t ype i n whi ch the wal l shear st r ess changes s i gn.

Compar i ng the r es ul t s we f i n d a qual i t at i ve agr eement of the unf ol di ng model

wi t h I noue' s cal cul at i ons.

- 1 5 3 -

The wall shear st re ss dis tr ibutio n near the sep ar at io n (at tachment) po int i s

given by T ( X , O ) = Êux' + o( x' ).

Let us invest igate the physical unfo lding of Eq. (3-26) s at i sfying the flow

equations together with the no-slip boundary conditions on a fixed wall.

The discussion wil l be res tric ted to the case of separat ion, where a_ < 0 (say

V-D.The physical unfolding of Eq. (3-26) i s obtained i f this equat ion i s supp le

mented with the lower-order terms occurring in Eq. (3.10); the coefficients of

these terms serve as the bifurcation parameters. The physical unfolding of Eq.

(3.26) becomes

x = ]i^ + u2x + y + u_x2

- x» = P(x,y)

• 1 3y = - 2 u2 y " u3 x y + 2 x* y = Q( x , y )

(3.27)

The unf ol di ng, Eq. ( 3. 27) , cont ai ns t hr ee bi f ur cat i on paramet ers , p1, u_ and u_

whose ef f ect on the f l ow t opol ogy have to be s t udi ed. Char ac t er i s t i c f or the

unf ol ded s t r eam i ne pat t er ns are number , t ype and l ocat i on of the si ngul ar

poi nt s near (0,0 ) . There are two set s of such nei ghbour i ng poi nt s. Fi r st we have

t he si ngul ar poi nt s on the wal l sur f aces ( on- wal l poi nt s) l ocat ed at

u.. +u-,x +( i - x1 - x' = 0, y = 0

The second set descr i bes the of f - wal l s i ngul ar i t i es gi ven by

1 3

2 u2 +p3x - 2 x2 =0, y =- j - x( u2 +]i?x - xJ )

The on- wal l s i ngul ar poi nt s f ol l ow f rom a cubi c equat i on, i ndi cat i ng t hat one,

t wo or three s i ngul ar i t i es can occur on the wal l . I f two s i ngul ar i t i es appear ,

t he bi f ur c at i on i s nongener i c si nce one of the s i ngul ar i t i es i s nonhyper bol i c.

Nonhyper bol i ci t y of a si ngul ar i t y on the wal l can al so occur i n the case t hat

t hree s i ngul ar i t i es coi nci de. Thi s occur s i f the bi f ur cat i on par ameter s obey the

r el at i ons: p_ =- 3( / 3) * , ]i. =( p - , / 3 ) ' . Then i t turns out that no of f - wal l

s i ngul ar i t i es wi l l ar i se and the onl y ef f ect of the bi f ur cat i on i s a di spl ace

ment of the f i f t h- or der saddl e poi nt al ong the wal l to t he l ocat i on x = x u_ .Thi s nongener i c bi f ur cat i on i s of no i nt er est and can be i gnor ed wi t hout l oosi ng

essent i al f eat ur es of the unf ol di ng by t aki ng u- = 0 t hr oughout .

- 154-

Thi s may be expl ai ned as f ol l ows. The nongener i c per t ur bat i on u-, = - ö (u. , )* •1 * i 5

p. = 27 ( M Q ) ' occurs i n the t hr ee- di mensi onal space (p.. , u_, u_) on a spat i al

cur ve whi ch appr oaches the or i gi n ( 0, 0, 0) i n the di r ect i on of the p_ - axi s . The

nongener i c subspace has di mensi on one so that the uni ver sal unf ol di ng Eq. 3-27

i s f ul l y det erm ned by the two par amet er s u. and \i havi ng a span whi ch i nt er

sects the nongener i c subspace t r ansver sal l y at the or i gi n.

Then the on- wal l s i ngul ar i t i es are gi ven by

A±: \ix * p2x ± - x^ = 0, y± = 0, i = 1,2,3

wher eas the of f - wal l s i ngul ar i t i es sat i sf y

B B. , : (±[f] , -Ul 2 [f) ) , u2 > 0

The on- wal l s i ngul ar i t i es A. are el ement ary or t hi r d- or der saddl e poi nt s whi l e

t he o f f - wal l si ngul ar poi nt s B . and B_ appear to be a saddl e poi nt and a cent er

poi nt , r espect i vel y, or a cusp i f t hey coi nci de.

Nongener i c bi f ur cat i ons of t hi s unf ol di ng occur on the cor r espondi ng bi f ur cat i on

s et s :

± ul ,v2s*

V F " * (3 ) Bc : v2 = ° <ul < °)

These sets , t oget her wi t h the var i ous r es ul t i ng f l ow pat t er ns , are shown i n

Fi gur e 3. 17-

The generic bifurcation of a fi fth-order saddle point wi ll give one of the

following three topologically different structurally stable flow patterns.

The first and simplest case consists of a single hyperbolic saddle point at the

wall, corresponding to the ordinary separation case already described as the

classical Oswatitsch-Legendre solution. It occurs i f u_ < 0 and i f (i. > 0,

^ > A *^ l

3 J

"

The second f l ow pat t er n l ooks l ess f am l i ar ; i t cont ai ns a wel l - known separ at i on

poi nt on the wal l but i n addi t i on a cent er poi nt and a saddl e poi nt i n the f l ow

downst r eam of the separ at i on po i nt . The ups t r eam separ at r i ces of the saddl e

poi nt f or m a st r eam i ne whi ch enc i r c l es the cent er poi nt by a cl osed l oop. The

f l ow r egi on behi nd the separat i on s t r eam i ne cont ai ns a c l os ed r ec i r c ul at i ngy l ,P2, *r egi on of f the wal l sur f ace. I t occurs for p_ < 0, p- <-['T -] •

- 155-

Fi g. 3. 17. Unf ol di ng of f i f t h or der saddl e poi nt , bi f ur cat i on s et s :

± P l ,v2,'Bs: F = ± ^ ' B

c: ^ 2 = ° < » > i < ° > -

Final ly, the third structural ly stable f low pattern shows the famil iar laminar

separation bubble fol lowed by a secondary separation further downstream. It

,P2,J" u

l ,u

2 , *occurs for u 2 > 0 , - ( ^ j < + < (y) .

Transi t ional f low patterns between these three types of f low are described by

the nongeneric b i f u rc a t i o n s . They correspond to s t r u c t u r a l l y un s ta b l e f low

p a t t e r n s sh ow n i n F i g u r e 3 - 1 7 - A g a i n , we d i s t i n g u i s h t h r e e d i f f e r e n t

p o s s i b i l i t i e s :

1 . B : In th e flow re gio n downstream of the separ ation stream line a nonhyper-bo l i c cusp s i ngu l ar i ty occ ur s . The cusp s i n g u l a r i t y may b i f u rc at e i n two

ways: i t w i l l f a l l a p a rt i n t o a c e n t e r p o i n t and a s a d d l e p o i n t , o r i t

disappears leaving no singularity in its neighbourhood.

- 1 5 6 -

2 . B : A t h i r d - o r d e r s ad d l e p o i n t w i th t h ree h y p e r b o l i c s e c t o r s o cc u r s o n t h e

wal l sur face. Bifurcat ion of the th i rd-order saddle point ( see paragraph 4 .1)

r e s u l t s e i t h e r i n two h y p e r b o l ic s ad d l e s on t h e w a l l o r i n a s i n g l e s a d d l e

point above the wal l .

3 . B : A th i rd -o rd er sa ddle point wi th one hype rbol ic se cto r occurs on th e w al lsu r face . B i fu rca t ion g ives e i ther a l aminar s epara t ion bubble on the wal l , o r

i t d i s ap p ea r s l e av i n g no s i n g u l a r i t y i n i t s n e i g h b o u r h o o d . No t e t h a t t h i s

b i f u r c a t i o n p r o c e s s h a s a l r e a d y b ee n d e s c r i b e d i n p a r a g r a p h 4 o f t h i s

Chapter .

§ 12 i_Bubble_cagtur ing_by_a_secondary_segarat ion

I t w i l l be a t tempted now to in te rp re t the unfo ld ing of the f i f th -or de r topo log i -c a l s a d d l e p o i n t i n a p r a c t i c a l flo w s i t u a t i o n w h e re a s e p a ra t io n b ub ble i s

fol lowed by a secondary separat ion. For th is purpose i t i s convenient to observe

th e v a r i o u s f low pa t t e rn s i f p_ va r i e s a t cons tan t p . . . I t su f f i ce s to take p . <

0 s ince a l l poss ible f low pat terns wi l l then be encountered and the var ia t ion of

p_ may be r e s t r i c t e d to go from an appro pr ia te pos i t iv e va lue p_ > 3 [- 2~l to

u 2 < 0.

These f low pat terns may be thought to be embedded in the f low f ie ld over ana i r f o i l , more in p a r t ic u la r they would appear in the process of the capture of a

laminar separat ion bubble by a secondary separat ion. Figure 3-18.Ul f

For p_ > 3( - p - ) we h a v e t h e s i t u a t i o n a s g i v e n i n F i g u r e 3 - l 8 a . I f p _d ec re as e s be low t h i s v a lu e , th e s econdary sepa ra t ion po in t s moves in upstr eam

r " Vd i r e c t i o n , a p p r o a c h i n g t h e l a m i n a r s e p a r a t i o n b u b b l e . A t p - = 3 1 - 2~J th e

s e c o n d a r y s e p a r a t i o n s t a r t s t o i n t e r f e r e w it h t h e b u b bl e: t he q u e s ti o n a r i s e s

how the bubble wil l be 'captured' by the upstream moving separation.

T he u n f o l d i n g o f t h e f i f t h - o r d e r t o p o l o g i c a l s a d d l e p o i n t s u g g e s t s ( Fi gu re

3 . l8b) the fo llowing answer: before a f ina l s i tu a t i o n i s r each ed i n which th e

bubble i s completely los t i n the sepa rated flow region (p_ < 0 ) , t he re appears a

t r a n s i t i o n a l s t a g e w i t h a n o f f - w a l l r e c i r c u l a t i n g r e g i o n do w ns tre am o f t h e

s e p a r a t i o n s t r e a m l i n e . T h e a p p e a r a n c e o f s u c h a t r a n s i t i o n s t a g e , w h i c h i s

s t r u c t u r a l l y s t a b l e , i s r e f l e c t e d v e r y w e l l by t h e w a l l s h e a r s t r e s s

d i s t r i b u t i o n , F i g u r e 3 . l 8 b .

When the s econdary sepa ra t ion po in t S_ meet s th e r e a r p a r t o f th e s e p a ra t i o n

bubble at the at tachment point A the wall shear s tress reaches a local maximum

value T = 0.

-160-

nongenencsubspace

Fig. 3-20. Nongeneric subspace of Eq. 3-29.

b y two p a r am e t e r s i f t h e s e t wo

p a r am e t e r s a r e ch o s en s u ch t h a t

the i r span in the parameter space

in te r sec t s the nongener ic subspace

t r an s v e r s a l l y a t t h e o r i g i n .Since f t_ is f ini te the plane u, .=0

s a t i s f i e s t h e t r a n s v e r s a l i t y

co n d i t i o n , s o t h a t t h e f o l l o wi n g

t wo - p a r am e t e r f am i l y p r o v i d es a

universal unfolding of Eq. (3 .28) :

x = ji1 + u3y + x* + ft xy - ^ y 2 + 0(3)

y = -x y - = è y z 0(3)

W t h sui t abl e rescal i ng of x , y and t , al l poss i bl e cases f or a, . < 0 can be5

reduced to ft. > 0.

6 i 2 i _ B i f u r c a t i o n _ s e t s i _ f l o w _ g a t t e r n s

To f ind the b i f u rc a t io n se t s fo r which the flow pa t t e r n i s s t ru c t u ra l ly uns tab lewe seek the s in gu la r poi nt s near (0 ,0) on the wal l as wel l as in the flow f i e l d .

The o n- w al l s i n g u l a r i t i e s S . ,S_ a re g iven by y = 0 , x = ± J -u . and w i l l occur

on ly i f vi < 0 .

The o f f -w a l l s i n g u l a r i t i e s s a t i s f y

' 1 , 2 ' x =c

y p 3 - 9 / S '

where K = V(3+ai) ' .

For t he gener i c bi f ur cat i ons the on- wal l s i ngul ar i t i es S. . S- are saddl e poi nt s ,

whi l e t he of f -wal l si ngul ar i t i es C. , C_ appear t o be a cent er poi nt (C.) and a

saddl e poi nt ( C_) r especti vel y.8

The nongener i c bi f ur cat i ons ar e speci f i ed by p. = 0, and p* 7 P p . =0 and

wi l l be denot ed as B and B r especti vel y. Thi s bi f ur cat i on set , t oget her wi t hs ct he var i ous f l ow pat t er ns i n di f f er ent r egi ons ( I - V) are present ed i n Fi gure

3. 21. At p- = 0 and p^ > 0 we have a t hi r d- or der saddl e poi nt at t he wal l ( wheret he poi nt s S. . S- and C. coal esce) and a si ngl e saddl e poi nt ( C_) above t he wal l .



- 161-

h

Fi g. 3- 21. Unf ol di ng of saddl e poi nt wi t h t hr ee hyper bol i c sect or s,4 . .

bi f urcat i on sets: B : p = 0, B : u. = V- 2u. , B : p~ = 2J - U- .

Smal l per t ur bat i ons due t o p1 var i at i ons wi t h r espec t to p . =0 ( u, * 0 ) gi ve

t hi r d- or der saddl e poi nt bi f ur cat i on as descr i bed ear l i er i n par agr aph 4. I f y.

i s negat i ve and u_ > 0 a f l ow pat t er n resul t s wi t h a separ at i on bubbl e at t he

wal l underneat h t he saddl e poi nt C_ ( r egi on I V ) ; i f u, i s t aken as pos i t i ve,

onl y t he saddl e poi nt C_ r emai ns i n t he f l ow above t he wal l ( r egi on V) .

At B cusp poi nt bi f ur cat i on occur s i n t he f l ow, causi ng ei t her t he f or mat i onof a reci r cul at i ng bubbl e ( r egi on I I I ) i n the f l ow domai n i n bet ween a separ a

t i on and an at t achment st r eam i ne i f u_ i s i ncreased or a di sappear ance of t he

cusp poi nt i f j i , decreases ( regi on I I ) .



-162-

>/;///;;;//;/;;;.

Fig . 3 .22 . Globa l b i fu rca t ion o f sadd le -

p o i n t - t r i a n g l e .

We now note th at the phase p o r t r a it s

in re g i on I I I near B and in reg ion

IV ne ar u.. 0 ( u , > 0) a r e no t

homeomorph ic , fo r the f low in IV

p o s s e s s e s a h e t e r o c l i n i c c y c l ew he rea s t h e flow in I I I has a homo-

c l i n i c c y c l e .

Hence the re mus t be an add i t iona l

b i f u r c a t i o n w h ic h t r a n s f o r m s t h e

p h a se p o r t r a i t s i n I I I and IV i n t o

each o the r . The sadd le C- and the

c e n t e r C . s t i l l e x i s t i n t h e r e g io n

I I I and IV and thus a g l ob al b i f u r

c a t i o n m us t t a k e p l a c e , p e r h a p s a

s a d d l e - p o i n t - t r i a n g l e fo rm ed by

s a d d l e - s a d d l e c o n n e c t i o n s ( s s )

between the sad dle po in ts S. , S_ and

C_ and surrounding the center poin t

To s t u d y t h i s g l o b a l b i f u r c a t i o n ,

shown in Figure 3.22, we determine

the Hamiltonian of the vector f ield:

P ^ u 3 y + x» + a^ y I y» + 0(3)

y = - xy - I ^y ' + 0(3)

whi ch gi ves t he l eadi ng t er ms of t he st r eam f uncti on 4i (x, y) near ( 0, 0) :

1>( x, y) = \-[lVx + 2p3y + 3x' + 2&5xy - y1 + 0( 3) ) (3-3D

Because the sadd le po in t s S 1 and S_ are on-wal l s ingular i t ies , the presumed ss-

con nec tions S.-C- and S--C- have to sa ti s fy the cond ition t|>(x,y) = 0.

A s im ple c a l c u l a t io n shows th a t these cond i t ions a re fu l f i l l e d i f u_ = K J -p . , .

then a sadd le -po in t - t r i ang le i s fo rmed by sadd le

J-P-!and —— ("*(;• 3 ) . an d a c e n t e r p o i n t i n 2 K

a p p ea r t o b e s t r a i g h t s t r e a m l i n e s i f t h e h i g h e r - o r d e r t e rm s i n E q. ( 3 - 31 )

denoted by 0(3) , a re not taken in to account .

Id le poin ts a t ( - V-U j .0) , ( J - u . , 0 ) ,

(-ft,., 3) . The s s -connec t ions5

- 1 6 4 -

7. Unfolding of a saddle point with two or four hyperbolic sectors in a half plane

2; l ; _yni ver sal phys i cal _unf ol di ng

The l as t unf ol di ng we want t o s tudy i n chapt er I I I i s t he unf ol di ng of the

hi gher - or der s i ngul ar i t y occur r i ng i n a vi scous f l ow i f t he str eamwi se pr essur e

gr adi ent and t he shear st r ess gr adi ent x vani sh si mul t aneousl y i n t he si ngul ar

poi nt .

Thi s s i ngul ar i t y, and par t s of i t s unf ol di ng, seems t o be of par t i cul ar i nt erest

because i t descr i bes f l ow pat t er ns t hat we meet i n sever al pr act i cal si t uat i ons,

such as t he f l ow i n t he near wake of smoot h bodi es. The appl i cat i on t o t hese

f l ows wi l l be deal t wi t h i n the next par agr aphs, but we shal l f i r s t pay at t ent i on t o the unf ol di ng i t sel f .

The degener at e s i ngul ar i t y we want t o unf ol d her e, and whi ch was act ual l y

i nvest i gat ed i n par agr aph 3- 2, sat i sf i es t he nonl i near syst em

x = a, . xy + a_x*

(3-32)

• 1 3y = - ö a y* - £ «y' y. a. * 0, a_ * 0

T h e co rres p o n d i n g p h a s e p o r t ra i t s i n t h e s em i p l a n e y i 0 , see Figure 3 -8 , showed

t h a t f o u r d i f f eren t f l o w p a t t ern s w i l l a r i s e f o r v a r i o u s ch o i ce s o f a _ a n d a _ .

A s e p a r a t e t r e a t m e n t o f a l l t h e s e f ou r flo w p a t t e r n s i s n o t n e c e s s a r y b e c au s e

Eq- ( 3 - 3 2 ) i s i n v a r i a n t f o r t h e t ra n s f o rm a t i o n s ( a ^ . y ) ■ » ( - a , - y ) an d ( a ^ . y . t ) •»

( - a ^ - y . - t ) .

Thus i f Eq. (3 -32 ) i s s t ud ied both in the upper (y 2 0 ) a s w el l as in th e l ow er

( y £ 0 ) h a l f p l a n e , i t i s s u f f i c i e n t t o d i s c u s s th e p o s s i b l e u n f o l d i n g s o f E q.

( 3 . 3 2 ) f o r a t m o st o n e c a s e , s a y

a_ < 0 and a_ < 0 . The co rr es po nd ing

h i g h e r - o r d e r s i n g u l a r i t y i s s h o w n i n

F i g u r e 3 • 2 4 , i t r e p r e s e n t s fl o w

/ ^ V \ x s e p a r a t i o n w h e r e t h e p r e s s u r e

a t t a i n s a m ax im um a t t h e w a l l ;

s u i t a b l e r e s c a l i n g a l l o w s t o t ak e

F i g . 3 . 2 4 . D eg en era t e f lo w a t p = 0 , a c = a_ = - 1 . S i m i l a r l y t o f or m e rx 0 1

T = 0 . c a s e s , t h e p h y s i c a l u n f o l d i n g may b e

y y \



-165-

obtained from Eq. (3.10) by talcing the 'lost' parameters (a^.a-.a-.a^) as the

bifurcation parameters. All of them have zero as the bifurcat ion value, and as

usual the parameters Uj, p2> u_ and pj, will be introduced to denote small

find 1

x = ul * P2 X + u 3 y + p* »

x ~ x y " x' + ° ^

perturbations of a. (i = 1 to 4). Then we find the following physical unfolding:

(3-33)• 1 y* 3y = - 2

u 2 y "

p4

x y + f"

+ 2

x'

y * ° ^

2i21_Determination_of_codimension

The number of relevant parameters determining the unfolding completely can be

reduced if the four-dimensional parameter space M: (u. ,u_,u_ .u^.) conta ins anongeneric subspace where the original degeneracy (Eq. (3.32)) is preserved

after perturbation.

Such a situation will occur if the effect of the perturbations is limited to a

shif t of the degenerate point along the wall surface. Other possibili t ies which

preserve the degenerate point, such as a rotation in the x,y-plane or a shift

off the wall, must be ruled out for their v iola tion of the no-slip boundary

conditions.

Let us elaborate the shift along the wall and write Eq. (3-32) with a_ = a_ = -1

in the translated coordinate system x = x - x , y = y

x y\j \J \j \J

(3.34)1 3 — y* 3y = 2 x

Qy + 3xQxy * f " + 2 y x '

Compar i ng t hi s r esul t wi t h t he unf ol di ng ( Eq. ( 3- 33) ) we observe t hat t he par

t i cul ar per t ur bat i on

" l

V2

w3

=

=

=

-kx'0

-3kx^

-kx0

" „ = 3 * 0

wher e k i s a constant, i s t he nongener i c pert ur bat i on whi ch l eaves t he degener

at e phase por t r ai t unaf f ect ed.

- 1 6 6 -

T h i s n o n g e n e r i c p e r t u r b a t i o n i s a o n e - d i m e n s i o n a l s u b s pa c e o f M and app roach es

t he b i fu r c a ti o n p o in t ( 0 , 0 , 0 , 0 ) i n t h e n o n g e n e r ic d i r e c t i o n ( 0 , 0 , - 1 , - 3 ) . T n e

u n f o l d i n g o f Eq . ( 3 - 3 2 ) i s f u l l y d et e r mi n e d b y t h r e e p a r a me t e r s , wh ic h h a v e t o

be chosen in such a way that the ir span in M a c t u a l l y i n t e r s e c t s t h e n o n g e n e r i c

s u b s p a c e t r a n s v e r s a l l y i n t h e b i f u r c a t i o n p o i n t . W ith r e ga r d t o t h e n o n g e n er i cd i r e c t i o n , th e t r a n s v e r s a l i t y c o n d i t i o n w i l l b e s a t i s f i e d t h r o u g h o u t b y t a k i n g

Uj. = 0 so t h a t t h e f o l l o w i n g t h r e e - p a r a m e t e r f a m i l y p r o v i d e s a c o d i me n s i o n - t h r e e

u n f o l d i n g o f E q . ( 3 . 3 2 ) .

x = ]i± + u 2 x + u .y - xy - x ' + 0 (3 )

• 1 1 3

y = - 2

u

2

y +

3

y

'

+

2

x

'

y +

° *

3

'

(3-35)

2 1 3 i _ N e i g h b o u r i n g s i n g u l a r g o l n t s ± _ l o c a l _ b i f u c a t i o n _ s e t s _ B a n d _ B

T o o b t a i n i n s i g h t i n t o t h e v a r i o u s p r o p e r t i e s o f t h e p ha se p o r t r a i t s o f Eq.

( 3 - 3 5 ) we s e e k t h e s i n g u l a r p o i n t s , x . = ( x . , y . ) , n e a r ( 0 , 0 ) o n t h e w a l l a s w e l l

a s i n t h e f l o w f i e l d . V a l u e s o f t h e p a r a me te r s f o r w h ic h t h e s e s i n g u l a r p o i n t s

a r e n o n h y p e r b o l i c a r e t h e b i f u r c a t i o n s e t s a t w h ic h t h e fl ow p a t t e r n s a r e

s t r u c t u r a l l y u n s t a b l e . S i n c e we o n l y d e a l w i t h p h a s e p o r t r a i t s c l o s e t o t he

o r i g i n ( x , y ) = ( 0 , 0 ) we a r e o n l y i n t e r e s t e d i n t h e l o c a l g e o m e t r y o f t h e

s u r f a c e s x = x . ( p , p _, p _) and y . = y ^ p ^ . p ^ p , ) n e a r t n e b i f u r c a t i o n p o i n t p 1 =

p_ = p . = 0 where x = 0 .

T he p a r a m e t r i s a t i o n x . = x . p ' , y . = VJIU-I )' . P? = ^ P ^ l ' ' u 3 = ^ j ^ l ' ' a l l o w s t o

f i n d t h e se su r f a c e s and p h a se p o r t r a i t s f rom t h e t r u n c a t e d sy s te m

Ux + u 2 x + u_y - x y

1 1 1 2 1■ 5 p , y 5 y + o x y

(3.36)2 H 2 J 3

T h e o n - w a l l s i n g u l a r i t i e s h a v e x - l o c a t i o n s o n t h e w a l l f o u n d a s s o l u t i o n s o f t h e

cu bic equ at io n p . u_x - x ' = 0 .

Th i s e q u a t i o n p o i n t s o u t t h a t t h e number o f o n - w a l l s i n g u l a r i t i e s d e p e n d s o n u .

and p _ ; a c t u a l l y t h r e e s i n g u l a r i t i e s a pp ea r i f ( p 2 / 3 ) ' > ( u j / 2 ) 1 whereas one

s i n g u la r po in t may be found i f ( u 2 / 3 ) ' < {]x±/2)'. I f ( p 2 / 3 ) ' * ( P j / 2 ) 1 i t may be

v e r i f i e d t h a t t h e s i n g u l a r p o i n t s a r e s a d d l e p o i n t s and t h a t a l l o f them a reh y p e r b o l i c . T h e c o n d i t i o n

-167-

B g : (u- , /3) ' = (U l / 2 ) '

forms a b i f urc a t i on s e t , ca l led B .s

A t t h e b i f u r c a t i o n s e t B , t h e r e w i l l be a t m ost tw o o n -w a l l s i n g u l a r i t i e s ofw hich a t l e a s t o ne i s n o n h y p e r b o l i c . T he t y p e of t h e n o n h y p e r b o l i c p o i n t i s

e s t ab l i shed a s fo l lows .

Let us denote the x- po s i t io n of the nonh yperbol ic p o in t by x- and th a t of th e

rem ain ing s ad d le p o in t by x_ . S ince x . i s a t l e a s t a double roo t of the cubic1/3eq ua tion P j + u_x - x ' = 0 we find e a s il y th a t x_ = -2 x. and x. = -(u . , /2 ) ' .

I f (x . ,0 ) i s taken as a new o ri gi n in the phase pl an e, system (3-36) becomes

1/3 1/3

* = (H3 + ij1) ) y + 3 [ j1] ( x -X l ) ' - (x-X l ) y - (x -X l)>

(3-37)p 1/3

y = -3 (2-) (x -x x ) y + g y' * 2 (x " x i ) ' y

Unless (u _,u ,) = (3(u. , /2) , -(u . , /2) ' ) , we can use And ronov's theorems 66 and

67 to e s t ab l i sh t ha t (x ,0 ) i s a t h i r d -o rde r s add le po in t i f p . * 0 o r a f i f t h -

order saddle poin t i f \i . = 0.

The pa r t i cu l a r ca se : (u 2 > p , ) = ( 3 ( ^ / 2 ) ' 3 , - ( U j / 2 ) 1 ' 3 ) , b e i n g e l e m e n t o f B s

w i l l be l a b e l l e d B - . I t d e s c r i b e s a n o n h y p e r b o l i c s a d d l e p o i n t w i t h t h r e e9 1/3

se pa ra t r i ce s having the angular co ef f ic ien ts 0 , - 5 (u ., /2 ) and » .

F i n a l l y we c o n c l u d e t h a t t h e b i f u r c a t i o n s e t B \B .. i n d i c a t e s s a d d l e p o i n t

b i f u r c a t i o n s on t h e w a l l : of t h i r d o r d e r f o r p . * 0 a n d o f f i f t h o r d e r f o r

u- = 0. Both t y p es of bif ur ca t io n have alre ady been dis cus sed (paragrap h 4 and

5) so that the resul ts can be appl ied here without fur ther comments .The pos i t i ons o f o f f -wa l l s i ngu la r po in t s o f Eq . (3 . 36 ) fo l low f rom the r e l a

t i ons ;

7x' - 9u 3x l - u2x + 2ux + 3 u2 u 3 - 0, y - I (u2 - 3 x ') (3-38)

The x-component s a t i s f ie s a cubic equ at ion , w i th the re su l t th a t i f ( 7l / 3 ' U-i +

7/3 P2U3 - u ' )The condi t ion

V <V vl + 3 3 ' W = (

3 + ¥ U

2 > '

7/3 P2Po " v\)* * (uó + 7/3' \i 2)' o n e o r t n r e e

hyperbolic points appear.



- 168-

form a bi furcati on set i ndi cati ng the exi stence of nonhyperbol i c poi nts i n the

fl ow fi el d.

Let us examne the topol ogi cal type of thi s nonhyperbol i c poi nt i n order to

determne the character of B .

cI f ( xj . y^ i s a doubl e root of Eq. (3 - 3 8 ) , x^sat i sf i es the rel at i on 21x» -

l S p ^ - p =0 and the l i near part of the vector f i el d (Eq. (3. 36)) :

' - " 3(u- -x)Df = |

y

9xy

' 3y

has zero as a doubl e ei genval ue i n ( x . , y . ) ,

The l i near transformati on

3 0

3^y l 3( U3- Xl )

which i s nonsingular on B \B- allows Eq. (3-36) to be brought i n to t he normal

form

x = y - (iy l

9 Pj-x^x* xy .

U3"xi

y = Z7—"—C Oui-lZy^ x.+u-) x* + 5 — xy + ^ T — - — r y2 + | x' y + \ y-x'6 (11-^) WK3 K3 1 K2' 9 Uo-Xj 3 3( u3-x1) J 2 J 6 Jl

where x = x-xx and y = - x y1(x-x1)+(u_-x1)(y-y1) .

Using Andr onov 's theorems 66 and 67 i t f ol lo ws t ha t the nonhyperbolic point(x-. y.) i s a cusp point s i n gu l a r i t y pro vided t ha t t he y-e quat io n has a non-

vanishing finite x2 term, thus if 9PÓ- 12u_x.. + u_ * 0.

This condi ti on i s ful f i l l ed except for th e ca se u- = u_ = 0 and fo r t he case

labelled B_ represented by the parameter combinations:

v [ü 2.u3] ■ [-3 x/g £ i 2 / 3 . j y? (^)V3i. B 2C ,c

For u. = p? = 0 there occurs a fi f th-order saddl e poi nt that coi nci des wth a

si ngul ari ty on the wal l whereas B- represents a thi rd-order saddl e poi nt i n the

hal f pl ane bel ow the wal l .

-169-

In conclusion: the bifurcation set B \(B. u B_) corresponds to cusp point

bi furcation in the flow off the wall. In paragraph 5-1 cusp point bifurcations

have already been discussed and we have seen there that they cause the spon

taneous generation of singular points in regular domains of flow fields. Since

the vector field (Eq. (3-36)) i s Hamiltonian, these new singular it ies will

always appear as a center-saddle pair.

71 ti_Flow P.atterns_and_global bifu££ation_sets_B 1 and_B ?

The bifurcation sets B and B divide the parameter space (p..,Pp,p_) in several

subdomains where flow patter ns of di fferen t topology may occur. Standard

methods, such as parametrization and singular -p oint analysis (c.f . previous

paragraphs) may be used to es tabl ish these patterns in e i ther subdomain

separately. The task to obtain a clear three-dimensional view of the parameter

space and of i t s par tition by B and B in subdomains is not an easy one;

especially the geometry of B is rather complicated.

Therefore, we investigate the case p.. = 0 fi rst and then the influence of u- * 0

will be considered.

For p. = 0 Eq. (3-36) reduces to

x = p x + u_y - xy - x1

3-39)

2 M

2y

1 p_y + £- + \ x'j

The bi f urcat i on set B : ( p - / 3 ) ' =(p j / 2 ) 2 whi ch concerns the on- wal l si ngul ar

i t i e s , r educes f or u1 = 0 to the si mpl e form u_ = 0 and coi nci des w th t he pr

axi s . However, i f p. = 0 the p. - axi s al so

For p.. = 0 the bi f urcat i on set B becomes1 c

axi s . However, i f p. =0 the p. - axi s al so f orms part of the B bi f ur cat i on set .l 3 c

( | P 2 P 3 - p'3)' = (P3 j r n j l '

which consists of three branches p2 = 0, pi = 0,00334 p2 and pi = 0.95963 P2-

A sketch of B and B in the p_,p_-plane is shown in Figure 3-25. This figures c 2 i

shows also the flow patterns caused by unfolding the degenerate singularity near

the bifurcation point (p2>p,) = (0,0). I t suff ices to take p, > 0 only. Flow

pat terns corresponding to p_ < 0 can be obtained from those at p_ > 0 by a

mirror reflection with respect to the y-axis.

- 170-

Fi g. 3. 25. Par t i al unf ol di ng of Eq. 3. 36 ( ^ = 0, u. . > 0) bi f ur cat i on sets :

Bg: u2=0, Bc : u3=0. 9796 / p2 . u3=0. 0578 Ju 2. Bg]_: Vy0, \iyh2-

Let us descr i be t he var i ous f l ow pat t er n i f u ? i ncr eases at const ant u„ > 0.

I n regi on I we f i nd t wo hyper bol i c saddl e poi nt s, one on t he wal l and t he ot her

i n t he l ower sem - pl ane (y < 0 ) . Goi ng to r egi on I l a we appr oach t he l i neu_ = 0 wher e t he saddl e poi nt on t he wal l becomes a degener at e f i f t h- or der

saddl e poi nt . Cr ossi ng t he l i ne p_ = 0 wi l l unf ol d t hi s degener at e saddl e poi nt

as descr i bed i n par agr aph 5. Fi gur e 3- 17. wher eas t he hyper bol i c saddl e i n t he

l ower hal f pl ane r emai ns qual i t at i vel y unaf f ect ed.

Consequent l y, t he f l ow r egi on i n I l a cont ai ns t wo separ at i on bubbl es, one on

ei t her s i de of t he wal l , and a ' f r ee' saddl e poi nt bel ow t he wal l . I n r egi on

l i b , we f i nd t hat t he ón- wal l separ at i on bubbl e i n t he l ower hal f pl ane ( pr esent

i n I l a) does not occur but i s r epl aced by a cent er - saddl e pai r bel ow t he wal l .Obvi ous l y t here i s a t r ans i t i on f rom I l a to l i b (and v i ce ver sa) whi ch i s not

pr edi ct ed by B or B . Det ai l ed cal cul at i ons show us t hat t hi s t r ansi t i on may be

- 171-

char act er i zed as t he f or mat i on and br eaki ng up of a separ at r i x t r i angl e f or med

by two on- wal l saddl e poi nt s , t he ' f ree' saddl e poi nt i n t he l ower hal f pl ane

and t hei r connect i ng st r eam i nes. Si nce number and t ype of s i ngul ar poi nt s

r emai n unchanged dur i ng t hi s t r ans i t i on, t he p r ocess wi l l be a g l oba l

bi f ur cat i on. The cor r espondi ng gl obal bi f ur cat i on set i s cal l ed B ., and f ol l ows

glby assum ng t hat saddl e connect i ons exi st bet ween t he saddl e i n the f l ow and t he

t wo saddl es on t he wal l . These connect i ons wi l l appear i f t he condi t i on p. +

UpU- - p i = 0 h o l d s .

T he t e r m i n a t i n g s a d d l e p o i n t s ( 0 , 0 ) , ( u ^ , 0 ) a nd ( p . , , 4 (p - 3 p? ,) ) a r e

hyperbo l ic . The cond i t ion :

V >»i + W i - u3 = °

f or ms t he gl obal bi f ur cat i on set B . .

For p =0 , B . r educes t o t he cur ves p. , = 0 and p_ = ±/ pT as depi ct ed i n Fi gur e

3- 25-

Goi ng f rom r egi on I I ( i n ( F i gur e 3- 25) t o r egi on I I I a B - cur ve i s crossed

i ndi cat i ng cusp poi nt bi f ur cat i on i n t he l ower hal f pl ane; bel ow t hi s cur ve t he

cent er - saddl e pai r i n the l ower hal f pl ane has di sappear ed. Ent er i ng r egi on I V

f rom I I I , agai n a br anch of B i s passed on whi ch a cusp i n t he upper hal f pl anebi f ur cat es i nt o a cent er - saddl e pai r .

As a r e s u l t t h e flo w in r e g io n IV i s c h a r a c t e r i z e d b y a n o n - w a l l s e p a r a t i o n

bubble together wi th a center-saddle pa ir above the wal l .

On B . where p_ = 0 and p_ > 0 the f low i s s t ru c tu ra l l y un s ta ble and co ns is ts of

tw o c o u n t e r - r o t a t i n g v o r t i c e s , s u rr ou n de d e ac h by t h r e e s a d d le c o n n e c ti o n s i n

the upper ha lf p lan e . Pass ing B . (p_ = 0 ) to p .. < 0 w il l rev ers e th e m utual x-Sl 3 J

p o s i t i o n s o f t h e o n - w a l l s e p a r a t i o n b u b b l e a n d t h e c e n t e r - s a d d l e p a i r a s t h e yappear in region VI.

Pr oc ee di ng now to th e more gene ra l case y- * 0 we con sider f i r s t the inf lu enc e

of \i . o n t h e b i f u r c a t i o n s e t s a n d t h e n o n t h e p h a s e p o r t r a i t s . Fo r n o n v a n i s h in g2/3 1/3

p . we may use t he p a ram et r i s a t io n p_ = k ( p . / 2 ) , p - = £ (p . /2 ) J to obta in a

un i f ied p i c t u r e of the b i fu rca t ion se t s in a p lane p . = con s tan t . Due to p . * 0

th e b i f u r c a t i o n s e t s w i l l b eco me p e r tu r b e d , a s i s shown in F ig u r e s 3 -2 6 and

3-27.

- 1 7 2 -

The par amet r i zed f or m of t hese s et s becomes f or u- * 0:

B s : k = 3

B„ : {98+63 k« - 2 7 V }2 = {9£! + \ k}>

B . : e1 - k£ - 2 = 0gl

Two speci al par amet er combi nat i ons to be denot ed by B. and B_ , appear at ( k, £)

( 3 . - D and (-3 ^/ I s ' 3 V F ' r e sP e c t i v e l y -

The bi f ur cat i on set s B and B . are t angent t o B i n B. . Fur t her mor e B_ i s a

c gl ° s 1 2cusp poi nt of B .cThe bi f ur cat i on set s di v i de t he par a

met er pl ane ( u- , u, ) i n sever al domai ns

( l abel l ed I , I ' , I I , I I ' , I I I , I I I ' , I V,

I V' , V and VI ) wher e t opol ogi c al l y

di f f erent phase por t r ai t s occur . These

phase por t r ai t s ar e obt ai ned by usi ng

st andar d met hods such as l ocal l i near i s at i on and s i ngul ar - poi nt a nal ys i s .

Si nce t hese met hods have been f r equent l y

di scussed i n pr evi ous par agr aphs we om t

det ai l ed cal cul at i ons her e and we con

f i ne oursel ves to s tat i ng the r esul t s .

The i nf l uence of a nonvani shi ng \i. on

t he qual i t at i ve pr oper t i es of B , B and

B . i s mai nl y concent r at ed ar ound t he u - axi s . Mor e pr eci sel y, beyond thei nt er v al p? ( B_ ) < p_ < u? ( B. ) t hey appr oach, f or i ncreasi ng | up| , mor e cl osel y

t o t he f orm t hey have i n t he case u1 =0 . Thi s means t hat corr espondi ng domai ns

H, = 0 : . , h>0: .

Fi g. 3- 26. Bi f ur cat i on set Bgl '

( equal l y l abel l ed i n F i gur e 3- 27) at p. 0 and u.. * 0 wi l l represent s i m l ar

( t opol ogi cal l y equi val ent ) phase por t r ai t s of Eq. (3 -36 ) . However , i f up i s

suf f i c i ent l y sma l l , t he i nf l uence of u. i s more severe and gi ves r i se to the

appearance of t wo new domai ns: V ( near B_) and VI ( near B. ) f or u1 * 0.

The phase por t r ai t s i n V and VI wi l l be deduced f r om t hose f ound i n adj acent

domai ns and by passi ng t he separ at i ng bi f ur cat i on set . I n t hi s way i t f ol l owst hat t he f l ow i n r egi on VI i s model l ed by a hyper bol i c saddl e poi nt on t he wal l ,

a cent er - saddl e pai r above the wal l and a f r ee saddl e poi nt i n t he l ower hal f

pl ane. For r egi on V mor e di scussi on i s needed.



- 173-

Fi g, 3. 27. Bi f ur cat i on set s B and B .

Region V i s subdivided by B in to Va and Vb (for d e ta i l s se e Figure 3-28) . The

f low in Va i s ob ta ined by app ly ing a sadd le po in t b i fu rca t ion in the f low in

H a . Then i t i s found th a t a hyperb o l ic sadd le p o in t S 1 , o c c u r s o n th e w a l l ,

t o g e t h e r w i t h a c e n t e r - s a d d l e p a i r ( C - S - ) and a h y p e r b o l ic s a d d l e p o i n t S _,

b o t h i n t h e l o w e r h a l f p l a n e ; s t r e a m l i n e s e n t e r i n g t h e r e g i o n b e t w e e n t h e

c e n te r - s a d d le p a i r a nd th e s a d d le p o in t S_ a r e p r e v e n te d from c ro s s in g by th eseparatr ix emanating from S 1 .

The f low p a t t e r n in reg ion Vb m er i t s fu r th er co ns ide ra t io n . We can ob ta in th i s

p a t t e r n from th e p a t t e r n in Va by c ro s s in g th e g lo b a l b i f u rc a t io n s e t B . , t h e

ef fe c t of which w il l be th a t (O S -) and S- w il l now be found on th e same si de of

the se p ar at r i x through S. . On B . the re w il l now be a sa dd le- co nn ec t io n betw een

S . a nd S _ , i n d i c a t i n g s t r u c t u r a l i n s t a b i l i t y i n t h e g l o b a l s e n s e . T h is flow i s

ob ta i ne d as w el l when go ing th rou gh th e upp er b ran ch o f B f rom I ' to Vb .

Ho we ve r, d e r i v in g th e f lo w p a t t e rn from th e p a t t e rn in l i b by c ro s s in g B (ora l t e r n a t i v e ly from I ' , by c ros s ing the lower b ranch o f B ) , we ob ta i n a p a t t e r n

in which (C-S-) forms a center-saddle pair , ins tead of (C-S 2 ) . Consequently, two

- 1 7 5 -

B 2 :

< V

- 1 . 9 6 3

- I . 9 0 2

- I . 7 4 7

-1.2*»5- .657

0 . 0

.O85

.223

1.217

2 . 3 6 9

3 . 0 0 0

. 7 1 3

.729

. 7 8 1

.9141.078

1.259

1 .283

1.317

1 .558

1.864

2 . 0 0 0

Tabl e: Numer i cal appr oxi mat i on of B _.

Al l t he pr evi ous r esul t s concer ni ng u. * 0 may be br ought t oget her t o obt ai n a

t ot al vi ew of t he phys i cal unf ol di ng of a hi gher - or der s i ngul ar i t y wher e t he

st r eamwi se pr essur e gr adi ent and the shear st r ess gr adi ent , T , vani sh si mul

t aneousl y. The resul t i ng uni f i ed pi ct ur e i s gi ven i n Fi gur e 3- 29 di spl ayi ng t he

var i ous pr oper t i es ( bi f ur cat i on set s and phase por t r ai t s ) of t he unf ol di ng.Si nce par amet r i sat i on i n t er ms of k and ft appear ed t o be possi bl e, t he unf ol di ng

i s act ual l y gover ned by t wo parameter s and has codi mensi on t wo.

- 177-

Fi g. 3. 29b. Codi mensi on- t wo degener at e phase por t r ai t s of Eq. 3- 36.

Cusp point bifurcations G lobal bifurcations Saddle point bifurcations

wrsI -5 a

w

I'-Ïb1

~>r^ ^

I ' - H

W-J \^ ~^2-

nr-iib

fV.s :

i-i'

v^ \cV£-Aö>

^

V

ffa-Db

j ,

Ïa-?b2

E'-U'

m'-ib'

Ïb2-2b1

m-m'

A o

2 <^->v

Ïïa'-Hb'

Db-?b1

I'-m

XI-H

Ha'-H

Fi g. 3. 29c. Codi mensi on- one degener at e phase por t r ai t s of Eq. 3- 36.

-178-

8. Viscous flow near a circular cy linder at low Reynolds numbers

8. 1. Descr i pt i on of f l ow t opol ogy

Al t hough t he r esul t s present ed above have been der i ved f or f l at wal l s, i t may beat t empt ed t o di scuss t he st eady i ncompr essi bl e f l ow behi nd a ci r cul ar cyl i nder

as an exampl e of bi f ur cat i on at zer o pr essure gr adi ent . We know t hat t he

nondi mensi onal quant i t i es descr i bi ng t he f l ow f i el d wi l l depend onl y on t he

Reynol ds number R = p. D. U / u ( wher e D i s t he di ameter of t he cyl i nder and U t he

undi st ur bed f r ee st r eam vel oci t y) and not on D, U p and u separ at el y.

The val ue of R char act er i zes t he f l ow f i el d around the cyl i nder and i t i s of

i nt er est t o recal l how t he f l ow t opol ogy changes w t h Reynol ds number . At smal l

Reynol ds number (R < 1) i ner t i a f orces are negl i gi bl e compar ed t o vi scous f or cesover most of t he f l ow f i el d. The dom nant pr ocess i n t he f l ow i s t he di f f usi on

of vor t i c i t y away f r om t he body. At ver y l ow Reynol ds number (R << 1) the

vor t i c i t y spr eads out f ai r l y evenl y i n al l di rect i ons so that the f l ow has fore-

and- af t symmet r y near t he cyl i nder . Thi s symmet r y i s ref l ected ver y wel l by

Oseen' s appr oxi mat i on of t hi s f l ow.

0 20 40 60 80

R e = U „ D / v

(a) length of the wake

5 10 50 100 500Re=U„D/ v

(b) positi on of separati on poi nt

Fi g. 3- 30. Wake geomet r y of ci r cul ar cyl i nder• Taneda ( 1971) ■ Korom l as & Tel i oni s ( 1980) 1 Homann ( 1937)

t heor y: Pr uppacher , Le Cl ai r & Ham el ec ( 1970)

- 179-

At moder at e Reynol ds number , speci f i ed ver y r oughl y as 1 < R < 100, i ner t i a

f orces and vi scous f or ces become comparabl e i n magni t ude and nei t her of t hem can

be negl ect ed. Vor t i c i t y di f f used away f rom t he body w l l be car r i ed downst r eam

and di st ur bs t he af or e- ment i oned f or e- and- af t s ymmet r y, s ee Bat chel or ( 1970) .

Fur t her mor e t her e appear s t o be a def i ni t e Reynol ds number , R = R , above whi ch

a r egi on of sl ow y r eci r cul at i ng f l ui d occur s, f or m ng a wake i mmedi at el y behi nd

t he cyl i nder ; i f R i s f ur t her i ncreased t hi s wake r egi on becomes l onger .

Taneda ( 1971) and Kor om l as and Tel i oni s ( 1980) have measur ed t he l engt h of t he

ci r cul at i ng regi on f or di f f er ent Reynol ds number s and t hei r r esul t s suggest t hat

t hi s r egi on f i r st appear s f r om R = 5- 2 onwards, see Fi gur e 3- 30a- The exper i men

t al obser vat i ons compar e wel l wi t h t heor et i cal r esul t s as obt ai ned f r om a

numer i cal sol ut i on of t he st eady st at e NS- equat i ons by Pr uppacher , Le Cl ai r and

Ham el ec ( 1970) .A s i m l ar conc l us i on may be dr awn f r om F i gur e 3. 30b whi ch di sp l ays some

numeri cal and experi ment al r esul t s concer ni ng t he l ocat i on of t he separ at i on

poi nt on t he cyl i nder sur f ace f or var i ous Reynol ds number s.

8. 2i _SYmmet r i cal bi f ur cat i ons

The changi ng f l ow t opol ogy at a cri t i cal val ue of t he Reynol ds number wi l l be

r econs i der ed and seen as a l ocal bi f ur cat i on occur r i ng at t he rear of t hecyl i nder .

At t he r ear a second st agnat i on poi nt appear s and t he f l ows over t he upper and

l ower si de meet and l eave t he sur f ace of t he cyl i nder . Due t o symmet r y t he

pr essur e gr adi ent al ong t he sur f ace wi l l be zer o i n t he r ear st agnat i on poi nt .

The f l ow near t hi s poi nt can be descri bed, at l east appr oxi mat el y, by Eq. ( 3- 10)

whi ch, f or symmet r i cal f l ow condi t i ons:

u( x, y) = - u( - x. y) , v(x ,y ) = v( - x, y) ,

reduces to

x = a_x + a- xy + a_x* - r a_xy* + 0( 4)

( 3. 40)

y = - § a2y - i a5y* - \ a?xsy + | «y» + 0( 4)

The st r eam i ne y = 0 may be r egar ded as an appr oxi mat i on of t he cyl i nder sur

f ace. Separ at i on f r om t hi s sur f ace occur s at (x. ' y) = ( 0, 0) whi ch i s a hyper bol i c



-180-

s a d d l e p o i n t i f a ? * 0 . Due to symmetry the she a r s t r e s s and the p re ss u r e

gr ad ie nt a lon g the wall wi l l van ish th e re . For a . = 0 Eq. (3•'JO) shows a hig he r-

o r d e r s i n g u l a r i t y a t t h e s e p a ra t io n p o i n t, i n d ic a t in g a s t r u c t u r a l i n s t a b i l i t y

o f t he flow pa t t e r n p robab ly co r respond ing to the c r i t i c a l s t a t e R = R . F low

p a t t e r n s o c c u r r in g a t ne a r -c r i t i c a l Reynolds numbers may be found by un fo ldingthe h igh er -o rde r s in gu la r i t y (a_ = a_ = -1 ) :

- x y -

£ ! * •

under t he r est r i ct i on of symmet r i cal f l ow condi t i ons.

Thi s means t hat u1 = u- = 0 so t hat t he number of bdi m ni shed and t hat t he f ol l owi ng one- par amet er unf ol di ng resul t sThi s means t hat u- = u, = 0 so t hat t he number of bi f ur cat i on par amet er s i s

U2x - xy

1 v* 32 p2'

where u ? denotes sm al l pe r tu rb at io ns of a_ wi th resp ect to a . = 0 ,

Symmetrical b if u rc at io n s as caused by p 2 a r e shown in Figure 3-31a-

rear of cylinder

" r

streamlines

vor t i c i ty

Fi g- 3- 31- Symmet r i cal bi f ur cat i ons at t he rear of t he ci r cul ar

cyl i nder i n st eady f l ow.

-181-

Obviously the case u_ = 0 reflec ts a c r i t ica l st at e very similar to that

occurring at R = R .

Beyond the cr it ical state (u2 > 0) a cluster of s ingu lari ties (saddles and

centers) arises forming a closed recirculating region behind the cilinder; three

stagnation points S.: (0,0), S- _: (±Ju_, 0) appear on the surface and threestagnation points appear in the flow field: S^: (0, 4 uo ' a n d C1 2 ?

6u2/7), see Fig. 3.31a-

The relative positions of the saddle points S2, S, and Su determine the general

shape of the recirculating region. In the limit p_ •» 0 this region shrinks in an

'oval' fashion such that i t s dimension normal to the cylinder (S.) is smaller

than along the cylinder surface (s). The ratio of the pr incipal dimensions,

É/s = § Jpp, may be expressed in local variables

0 -J-6r .Ti _ 5 x x x io iii \

s 2p U - H I )X X

and wi l l depend on t he Reynol ds number .

Thus unf ol di ng t heor y l eads t o the concl usi on t hat £/ s ■ 0 i f R ■ R f r om pos t -

cri t i cal val ues and t hat t he bi f ur cat i on par amet er u_ i s r el at ed t o t he Reynol ds

number such t hat u- ■ 0 i f R +R .2 c

Anot her i nt er est i ng aspect of t he unf ol di ng concer ns t he vor t i c i t y di st r i but i on

o)( x, y) near t he r ear par t of t he cyl i nder . For symmet r i c bi f ur cat i ons ( u. =u, =

0) , equat i on ( 3. 0) can be used t o der i ve t he appr oxi mat e vor t i ci t y di st r i bu

t i on.

<o(x, y) =- VUx +2xy +x3 +| xy2 +0(4)

Li nes of const ant vor t i ci t y are di spl ayed i n Fi g. 3- 31b and yi el d t he concl usi on

t hat t he vor t i c i t y pat t er n near t he r ear stagnat i on poi nt i s mai nl y det er m ned

by t he posi t i on of a saddl e poi nt on t he axi s of symmet r y at y = 5 P2 * 0( ul ) .

Thi s saddl e poi nt i s f ound i n t he act ual f l ow domai n ( wake r egi on) f or u_ > 0

onl y. The l evel cur ve of zer o vor t i c i t y: <o =0, has a par abol i c shape, and

i nt ers ect s t he sur f ace of t he cyl i nder i n t he separ at i on poi nt s S. . and S_. The

f or mat i on of t he wake beyond t he cr i t i cal s t at e R = R c o nc ur s wi t h t he

appear ance of a saddl e poi nt i n the vor t i ci t y pat t er n.



-182-

§ i 2 ^ _ A s Y 5m e t r ica l_b i fu rca t ions i t r a n s i t i o n s c e n a r i o ' s

I n t h i s p a r a g r a p h we c o n s i d e r a s ym m e t r ic b i f u r c a t i o n s of t h e h i g h e r - o r d e r

s i n g u l a r i t y :

x = - x y - x '

y 2 3 ,y = | - + | x 2 y

i n or der t o i nvest i gat e some char act er i s t i c pr oper t i es of smal l asymmet r i c

ef f ects as they can appear i f t he f l ow near a ci r cul ar c i l i nder i s obser ved

exper i ment al l y . Ac t ual l y , dev i at i ons i n the uni f orm ty of the oncom ng f l ow,

i mper f ect i ons of c i r cul ar shape and i nt r udi ng measur ement t echni ques mayi nt r o du c e i r r egul ar i t i es di s t ur bi ng a per f ec t l y symmet r i c f l ow pat t er n.

Mor eover , at hi gher Reynol ds number s unst eadi ness and f l ow i nst abi l i t y may

appear so t hat symmet r i cal f l ow condi t i ons, shoul d t hey appear , cannot be

mai nt ai ned i n t i me ( Von Karman vor t ex s t r e e t ) .

The i nf l uence of asymmet r i c di st ur bances on the steady f l ow near t he cri t i cal

st at e may be st udi ed f r om t he i nf l uence of t he bi f ur cat i on par amet er s u1 and p. .

on t he physi cal unf ol di ng:

x = p l + P2 X + p 3 y " x y " x'

1 y2 3y = - 2 u2 y + 3 + 2 x' y

Not e t hat t he bi f ur cat i on par amet er s u- and u_ may be i dent i f i ed w t h t he shear

st r ess T and t he ci r cum er ent i al pr essur e gr adi ent , r espect i vel y, at t he rear oft he cyl i nder .

The phase por t r ai t s and bi f ur cat i on set s of t hi s unf ol di ng ar e ext ens i vel y

st udi ed i n par agr aph 3- 7; a conci se assembl y of t he r esul t s i s pr ovi ded i n

F i gur e 3- 29- I n or der t o appl y t hese r esul t s t o t he f l ow at the rear of the

cyl i nder we onl y need t he phase por t r ai t s i n t he sem pl ane y £ 0. The por t r ai t s

at y i 0 t oget her wi t h t he r el evant bi f ur cat i on set s ar e r epr oduced i n Fi gur e

3. 32 usi ng t he scal ed var i abl es k = u_(^) and i = p, ( - ) whi ch ar e r el at edt o t he ci r cum er ent i al gr adi ent s of shear st r ess t and pr essur e p r especti ve

l y. I t shows a gr eat var i et y of f l ow pat t er ns whi ch can occur at t he r ear of t he



- 183-

cyl i nder and a l so var i ous poss i bi l i t i es of sequences of f l ow pat t er ns , when

i ncr easi ng t he Reynol ds number . I n or der t o di scuss t he asymmet r i c bi f ur cat i ons

i n a s yst emat i c way, we pr ef er t o gi ve t her ef or e more at t ent i on t o sequences of

f l ow pat t er ns t han t o t he i ndi vi dual f l ow pat t er ns. However , sequences can be

chosen i n var i ous, ways. Her e we want t o di scuss t hose sequences whi ch appear i f

t he bi f ur cat i on par amet er u_ ( r el at ed to the c i r cum er ent i al shear s t r ess

gr adi ent ) i ncreases whi l st y. and u_ ar e hel d const ant . Thi s i dea i s t aken f r om

par agr aph 8. 2 wher e a si m l ar appr oach i s f ol l owed t o di sc uss t he symmet r i cal

bi f ur cat i ons (p1 = 0, u_ = 0 ) . Mor eover , i n t hat case an i ncr ease of u? coul d be

associ at ed wi t h an i ncr ease of t he Reynol ds number .

Fi g. 3. 32. Asymmet r i cal bi f ur cat i ons at t he rear of t he ci r cul ar

cyl i nder i n steady f l ow.

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Consu l t ing F igure 3 -32 one observes tha t a con t inuous increase o f u p , and alsoof k , causes the in te r se c t io n of a s eque l o f b i fu rca t ion se t s of d i f f e re n t type

(B , B , B ) . I t i n v o l v e s t h e e x i s t e n c e o f a s c e n a r i o of s e q u e n t i a l flows c gp a t t e r n s , l i n k e d t o g e t h e r by b i f u r c a t i o n s of s t r u c t u r a l l y u n s t a b l e f lo w

s i t u a t i o n s .The o r d e r i n which co n s ecu t i v e b i f u r c a t i o n s a r e p a s s ed t h r o u g h d e t e r m i n e s a

p a r t i c u l a r s c e n a r i o , m o r eo v e r s ev e r a l s ce n a r i o ' s seem t o o ccu r. Ac t u a l l y f iv e

d i f f e r e n t s c en a r i o ' s : S . , S 2 . . . . S , - are dis t inguished in Figure 3 .32.

A l l th e se sc en ar io ' s revea l the common fea tur e tha t asymmetr ic dis tu rba nc es as

int rodu ced by p . and p .. are qu a l i ta t i v e ly uno btrus ive as long as p . < 0 , th at i s

t o s ay i f t h e c i r c u m f e r e n t i a l s h ea r s t r e s s g r ad i en t T rem ain s n eg a t i v e . W ith

r e s p e c t t o t h e s ym m etr ical s i t u a t i o n t h e flo w p a t t e r n i s s l i g h t l y d e fo r m ed i nthe sense tha t the s epara t ing s t r eaml ine l eaves the body ob l ique ly d iv id ing the

wake f low into an 'obtuse ' region and an ' acute ' region.

If u~ is in cr ea se d beyond u_ = 0 th e flow behind the cyli nd er i s subje cte d to a

sequence of b i furcat ions , making the inf luence of p . and p , more severe .

Dep ending on the o rd er in which b i f u rc a t i o n s o f d i f f e r en t kind succeed each

other , var ious scenar ios can be observed in Figure 3-32.

Le t u s e l a b o ra te some d e ta i l s o f these s c e na r i o ' s , fo r example those occur r ing

in S. if Up increases beyond Up = 0.

At pp = 313—J (k = 3) a sa d d le p o in t b i f u r c a t io n i s passed, announcing the

format ion of a separat ion bubble in the 'obtuse ' par t of the f low. Next a global

b i f u r c a t i o n ap p ea r s which t u r n s t h e s ep a r a t i n g s tr eam l i n e co u n te r -c l o ck wi s e so

th a t the bubble i s now in the ' ac u t e ' r eg ion . F in a l ly a cusp po in t b i f u rc a t i o n

terminates the sequence and generates a c losed f low domain wi th c i rculat ing f low

between the s epara t ion bubble and the s epara t ing s t r eaml ine .

The r e su l t in g flow pa t t e rn i s to po log ica l ly d i f f e r en t from th a t in a symmetr ica ls i t u a t i o n wh ere tw o c o u n t e r r o t a t i n g v o r t i c e s fo rm a c l o s e d d o m a in w i t h

c i r cu la t ing f low beh ind the cy l inder .

The o c c ur re n c e of a f r e e c i r c u l a t i n g r e g i o n u n d e r s t e a d y flo w c o n d i t i o n s i s

e s s e n t i a l l y an a sy m m e tri c e f f e c t , s i g n a l l i n g v o r t e x s h ed d in g t y p i c a l f o r

uns teady quas i -per iodic f lows at h igher Reynolds numbers .

C o n s i de r i n g th e s c e n a r i o ' s S_, S . ., Sj. and S- s im i la r conclus ions may be drawn

from the possible sequences of f low patterns which can occur if p ? i n c r e a s e s t op o s t c r i t i c a l v a l u e s . A u n i f i e d p i c t u r e of t h es e s e qu en ce s ( t r a n s i t i o n

scenar io ' s ) i s shown in Figure 3-33-



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transitional" flow patterns

Cusp point bifurcationSaddle point bifurcationGlobal bifurcation

Fig- 3 •33- Trans i t ion pa t te rns in the s teady f low of a c i rcu la r cy l inder .

On th e ba s is o f th es e r e s u l t s we conclude th a t asymmetrica l e f fe c t s in t rodu ce

t h e p o s s i b i l i t y of m ore t h an one t r a n s i t i o n s c e n a r i o . M o r e o v e r , s u c h a n

a sy mme t r i c a l t r a n s i t i o n in c lu d e s s e v e ra l b i fu rc a t io n s su g g e s t in g th a t a c e r t a in

range of Reynolds numbers has to be passed through before a fully developed wake

reg ion i s es tab l i shed beh ind the cy l inder .

L a s t l y , t he a s y m m e t r i c a l e f f e c t s i n c l u d e t h e f o r m a t i o n o f a f r e e r e g i o n of

c i r c u l a t i n g f low, remind ing o f an es se n t ia l fea tu re o f vor tex shedd ing , however

without including the actual effect and unsteady nature of vortex shedding.



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9. References

Bat chel or , G. K. ( 1970)

An I nt r oduct i on t o f l ui d dynam cs, Cambr i dge Uni ver si t y Pr ess, 255- 263.

Danber g, J . E. and Fansi er , K. S. ( 1975)

Separ at i on- l i ke s i m l ar i t y sol ut i ons on t wo- di mens i onal movi ng wal l s , AI AA-

J our nal , Vol . 13, no. 1; 110- 112.

Dean, W R. ( 1950)

Not e on t he mot i on of a l i qui d near a posi t i on of separ at i on, Pr oc. Cambr i dge

Phi l . Soc. 46, 293- 306.

Van I ngen, J . L. ( 1975)

On t he cal cul at i on of l am nar separ at r i x bubbl es i n t wo- di mensi onal i ncompr es

s i bl e f l ow, AGARD Conf . Pr oc , no. 168, Göt t i ngen.

I noue, D. ( 198l )

MRS cr i t er i on f or f l ow separ at i on over movi ng wal l s , AI AA- J our nal , Vol . 19. no.

9, 1108- 1111.

Kor om l as, C. A. and Tel i oni s , D. P. ( 1980)

Unst eady l am nar separ at i on: an exper i ment al st udy, J . Fl ui d Mech. , Vol . 97-

Legendr e, R. ( 1955)

Decol l ement l am nai r e r égul i er , Compt es Rendus, Acad. Sci . Par i s 241, 732- 734.

Moor e, F. K. ( 1958)

On t he separ at i on of t he unst eady l am nar boundar y l ayer s, i n Boundar y Layer

Research ( ed. H. Gör t l er ) , Spr i nger Ver l ag Ber l i n, 296- 3I O.

Oswat i t sch, K. ( 1957)

Di e Abl ösungsbedi ngung von Gr enzschi cht en, Symposi um on boundary l ayer r esearch,

I UTAM Fr ei bur g.

Pr uppacher , H. R. , Le Cl ai r , B. P. and Ham el ec, A. E. ( 1970)Some r el at i ons bet ween dr ag and f l ow pat t ern of vi scous f l ow past a spher e and a

cyl i nder at l ow and i nt er medi at e Reynol ds number , J . Fl ui d Mech. Vol . 44.



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Rott, N. ( 1956)

Uns t eady vi s cous f l ow i n t he vi c i ni t y of a s tagnat i on poi nt , Quar t . J . Appl .

Mech. , Vol . 13, 444- 451.

Sears, W R. ( 1956)Some r ecent devel opment s i n ai r f oi l t heor y, J . Aer onaut . Sci . , Vol . 23, 490- 499.

Sears, W R. and Tel i oni s , D. P. ( 1957)

Boundar y l ayer separ at i on i n unst eady f l ow, SI AM, J . Appl . Mat h. , Vol . 28, 215-

235.

Takens, F. ( 1974)

Si ngul ar i t i es of vect or f i el ds , Publ . Mat h. I HES, 43, 47- 100.

Taneda, S. ( 1971)

Vi sual i zat i on exper i ment s on unst eady vi cous f l ows ar ound cyl i nder s and pl at es.

I n: Recent Resear ch on Unst eady Boundar y Layer s ( ed. E. A. Ei chel br enner ) , Vol .

2, Les Pr esses de 1' Uni ver si t é Laval , Quebec.

Tel i oni s , D. P. and Wer l e, M. J . ( 1973)

Boundar y l ayer separ at i on f r om downst r eam movi ng boundar i es, J . Appl . Mech. ,

Vol . 40, 369- 374.

Tsahal i s , D. T. and Tel i oni s , D. P. ( 1973)

The ef f ect of bl owi ng on l am nar separ at i on, J . Appl . Mech. , Vol . 40, 1133- H34.

W l l i ams, J . C. I l l ( 1977)

I ncompr essi bl e boundar y l ayer separ at i on, Annual Revi ew of Fl ui d Mechani cs ( ed.M van Dyke) , Vol . 9, 113- 144.



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en bi f ur cat i e wor den hi er aanger ei kt en t oegel i cht . Ook de t heor i e over cent r um

var i ët ei t en wor dt vanwege haar t oepassi ngsmogel i j kheden, i n di t hoof dst uk kor t

beschr even.

I n hoof dst uk I I wor dt een kl ass i f i kat i e van st r om ngspat r onen i n de buur t vanconi sche st uwpunt en i n ni et - l i neai r e coni sche st r om ngen gegeven. Naast de

bekende t ypen coni sche st uwpunt en zoal s knooppunt en zadel punt , zi j n ver der

gevonden: zadel - knoop, t opol ogi sch zadel punt , t opol ogi sch knooppunt en scheef

zadel punt . Met behul p van deze coni sche st uwpunt en wor den ver s chi l l ende

st r om ngsf acet t en i n coni sche st r om ngsvel den ver kl aar d en t oegel i cht zoal s: het

' l i f t - of f ' verschi j nsel bi j ci rkel kegel s onder i nval shoek , het gener er en van

coni sche st uwpunt en, ' ni et - vi sceuze' l osl at i ng van de st r om ng l angs kegel opper -

vl akken en het coni sch st r oom i j nenpat r oon nabi j het hoekpunt van een ui t wendi gehoek of del t avl eugel met gekni kt opper vl ak.

I n hoof dst uk I I I wor dt een beschouwi ng gegeven over de t opol ogi e van een st at i o

nai r e st r om ng van een vi sceus medi um i n de nabi j hei d van een wand. Lokal e

opl ossi ngen van de Navi er - St okes ver gel i j ki ngen worden geconst r ueer d met behul p

van de kwal i t at i eve t heor i e. De bi j behor ende st r om ngspat r onen zi j n ver kregen

al s bi f ur cat i es van gedegener eer de hoger e- or de s i ngul ar i t ei t en. Naar mat e de

degener at i egr aad van de s i ngul ar i t ei t hoger i s bl i j ken s t r om ngspat ronen te

ont st aan waar van de t opol ogi sche s t r ukt uur kompl exer wor dt .

Ver vol gens wor dt i n hoof dst uk I I I een kl assi f i kat i e gegeven van st r om ngspat r o

nen, di e i n de buur t van een st i l st aande of bewegende wand kunnen opt r eden; deze

kl assi f i kat i e bi edt een ver schei denhei d aan mogel i j ke st r om ngsvor men.

Naast bekende pat r onen, zoal s de kl assi eke Oswat i t sch- Legendr e opl ossi ng voor

l os l at i ng, doen z i ch ni euwe st r ukt ur en voor ; o. a. di e wel ke geschi kt bl i j ken

voor de beschr i j vi ng van:

- het ont st aan van l okal e l osl aat bel l en i n uni f or m gesl aagde st r om ngen;

- de i nt er f er ent i e van een l osl aat bel met een st r oomaf waar t s gel egen, t egen de

st r oom i n bewegende, secundai r e l osl at i ng;

- l osl at i ngsver schi j nsel en i n de st at i onai r e st r om ng l angs een bewegende wand;

- de str ukt ur el e stabi l i t ei t van de str om ng i n een ' zadel punt - dr i ehoek' ;

- de vor m ng van asymmet r i sche st at i onai r e ' eddi es' i n het zog van een st omp

l i chaam

De r esul t at en van het onder zoek al s beschr even i n di t pr oef schr i f t onder st eunende s t e l l i ng dat i n s t r om ngsvel den opt r edende, hoger e- or de s i ngul ar i t ei t en,

hoewel nauwel i j ks waar neembaar i n de pr akt i j k, bel angr i j ke bouwst enen zi j n voor

het konst r uer en van l okal e opl ossi ngen van de gel dende st r om ngsver gel i j ki ngen.

Al s z odani g kunnen ze bi j dr agen t ot het ver schaf f en van i nzi cht i n de f ysi ca van

st r om ngsver schi j nsel en.

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