peake and tobak 3d separation

8/12/2019 Peake and Tobak 3D Separation

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18 1:18

TOPOLOGY OF THREEDIMSIOAL SPARAEDOWS!Murra ak ad avd J Pak

erc ce e eerc ere e lr 90

INTDTINree-menl epre w repreen mn f u mecncf gre prccl nere ye beyn e rec f enveerecl nly r numercl cmpun A preen ur unernng f reemennl w eprn re prncplly n bervnrw frm epermenl ue ulzng w vulzn ecnque.rculrly ueful n regr bee e l-rek ece frmkng vble e pern f knfrcn lne n e urfce f wnunnel mel (Mlby 96) cmm bervn mng uenf ee pern ncery cnn fr e ccurrence f weprn e cnvergence f lrek lne n prculr lneWeer l ucen cn mer f curen ebee reuremen mke ene f ee per wn gvernng ype f ucen precn yel cnvcng ecrpn f reemennl w epr npre e er f number f nvegr. Of e numeru emp wever few f e cnenngrgumen len emelve prece memcl frmuln Herewe ll ngle u fr pecl enn e ype prpe by egenre (196) beng ne cpble f prvng mecl frewrk f cnerble ep.

egenre (956) prpe ern f remlne mmeelyjcen e urfce (n ermlgy wll remle") be c

ere rjecre vng prpere cnen w e f cnnuu vecr el e prncpl ne beng rug ny regulr(nnngul) pn ere mu p ne n nly ne rjecry On e

1 he US overnent h the ght t retn nonecve oytree cene nn to ny coyght coverng th er

61

Annu.R

ev.FluidMech.1982.14:61-85.

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62 TOB P

basis of his posulae i follows ha he eleenay snula pons of held ca b cagrid mamaically Tus he yps f sinula pinsir umbr ad ruls grig rlais bw m ca bsaid o haaeize he paen. Flow sepaaion in his view has beendened by he onveene o wall sealine ono a paiula wallsramli a rigias frm a sigular pi f paricular yp saddl pi. W suld wr a is iw f w sparaiis no univesally aeped, and, indeed, siuais xis wr i apparsa a mr uacd dscripi f w sparai ay b rquird

igill (96) addessn hiself speially o visous ows laied a nube o ipoan issues by yin he posulae o a onnuous

veo eld o he paen o skin-iion lines ahe han o sealineslyin us above he surfac. Paralll gdrs dii crgcf skifrici lis a paricular skifrici li rigiaig frma saddle pon was dd r as cssary cdii fr w spaaon. Moe eenly, Hun e al. (978) have shown ha he noions ofeleary sinula pins and e ules ha hey obey an be easily appy w a surfac pas f symmry prcis f cical ws (Smi 969) crss w plas c (sals rry airli 1974). urr applicais ad sis ca bfud i arius cribuis f gdr (1965 1972 977) Oswaisc (980) ad i riw aricls by bak Pak (1979) adPak bak (980).

As Leende (1977) hself has noed, his hypohesis was bu a eini wii a arrwr framwrk f rardinaril uiul linef rsarc iiiad by Pcar (92) udr il O CursDd by Dirial Equais." Y ar brac f sam lihas been he eseah beun by Andonov and his olleaues 97 1973)on he qualiave heoy of dieenial equaions, wihin whih he usful

is f plgical srucur" ad srucural sabiliy" wr irducd ialy frm sam li sms rapidly padig ld kwas bifurcai ry" (s cmprsi riw f Saigr 980)Applicais ydrdyamics ae eeplied by he woks of Joseph(1976) ad Bami (97). I as bcm clar a ur udrsadigo hee-diensional sepaaed w ay be deepened by plaing Leendes hypohesis wihin a aewok boad enouh o inude is f plgical suue suual sabiliy and bifuaon. Beaini mid a w sill awai a cicig dscripi f rdimsial

w sparai w may ask wr bradr framwrk will faciliahe eeene o suh a desipion. In he ollowin w sall y oanswe hs queson liiin ou aenion o hee-diensional visousows ha ar sady in ma

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THDMNSONAL SPAATD FLOWS 6

THEW cid ady icu w a m dimial dy.

ula a i-ici li uac dy m aci a ciuu c ld i aad mamaicalyas follows Le ( be enea uvnea oodnaes wh () benohoonal aes n he sufae and t deed ou of he sufae noal o(,). Le lg paaees be hl(t1) hit) xep a snupi i lw m adc cdii a y c uac cm lciy c aalll uac U>U2mu gw m z ialy wi . Hc a aicl a amlinea he sufae wl have veoy oponens of he fo

d Ui ( ) d = ( / , 0)= W2 L ) = P L /)

U2h2(L) d

= f-(L,O)

= fWI ( ) = ( )

()

whee WW ae he oponens of he sufae voy veo hespeaon of a ady w i cd y UU ig idpd ofim Wi f ad a a paam ad ad Q uci ly cdia Equai () a a pai aunmus diay diiaequaons The fo laes he n he sae aeoy as he equaonssudied y ona (2) n hs assa nvesaon he uvesdened y diial quai. ig

UT = L / 0)

(2)

U2T = ( / 0)be oponens of he skn fon paalle o and / espevely, wehave fo he equaon ovenn he aeoes of he sufae shea sessc m quai ()

hl =

h23)

Alaily ajci uac iciy c wic aa uac a c i uaon s

hd h2d1 (4)

Annu.R



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4 TOBK & PEKE

Snguar Pnt

iu pits i th ptt f si-fiti is u islt pits the surfae whee he ski fiti 'w w i quai 3) ate

aivel he sufae vrii 2) quai ) bemes ieiallyer iuar pits are lassiabe i tw mai tpes: es a salepits. es ma be furthe subivie w sublasses: al ptsa f f attahme r separai)

-

(bl:\

,

SKNFRCON LNE

Singular points: () node; b) focus () sdde Lighil 1963).


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THRDMOL PRTD LOW

A ol poi iu i h poi ommo o ii umbo iiio i A h poi l o h siiio lis p olbl i iu i o sil li BB A o

poi o hm l o h si-iio is i ouww om h o A o poi o pio l o h si-iiolis i iw ow h o. h xpiol s ih p o imm) h o io ioopi owh v siiio li i o lvin h iul poi h isi L 1977.

A ous iu Ib is om ol poi o iu i h ihs o ommo i. A ii umb o i-iio lispi ou h iu poi ih w om i ous o h

m) o io i ou o pio) oi o hm l oui h ps o oio ih o h ow o o h su wilo u s su. pol s os o ousom los ps ou h su po ous is l

A sl poi iu I, h o wo piul i CC h p houh h iul poi Th iios o ih io h siul poi iw o o piul i ouw o hoh piul i A o h oh i-iio lis miss h siupoi iios osis wih h iios o h piul is Th piu li bi i h o iiio is mi o s o si-iio is issibl o .

o h o h ps i ius h u vox om ssm o is ohool v poi o h ssm o si-iioli. Thus i is ws possib i piip o sib h ow i hviii o siu poi iv i ms o p o si

iio lis o p o su vox lis.v (61 Lihil (163 v bo o o l posib ps o siiio i o h su o bo o ho miibl whos siu poi ob opoloil u h umbo os o po o oi o boh) mu h umb o slpoi b wo sh mo hi ul is xio oh xl ow i umb o xmps

pgph Skn-tn Ln

h siu pois i i i ioio o i ombiio uli hisi uios mi h isibuio osi-iio is o h su d poi o hm is pil io poi ow-i h o

Annu.R



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TB P

w t xtnal w fm fa upstam attas itslf t t sufa nal pnt f attamnt tb ats as a su f sin-ftnlins tat m fm t pnt a spa ut v t sufa Cn

vsl t nal pint f spaatn s tpall a pnt a awafan sufa an ats as a snk w t skin-ftn lins tat avumsib t b sufa ma vans

T sal pnt ats tpiall t spaat t sinftn lns ssuinfm aant ns f xampl aant nal pints f attamntAn xampl f ts funtn is llustat n u 2 Ltll 196).knfitn lns mn m t al pints attamnt a pvnt fm ssin b t psn f a patula sknftn lnm fm t sal pnt Ltll 963) as labl t patiula

lin a line of separation an as nti t xistn f a sal pntfm w t ln ms as t ssa ntn f w spaatnAs u 2 nats sn-ftn lns fm it si tn t nvn t ln min fm t sal pnt Unftunatl t nf skntn lns n it s f a patula li us n tsais as wll pp, p sknin lnut f t innit st f lins manatn fm a nal pnt f attamntma ultimatl bm a li n wi ts f t st nv Allsas a tat t xstn f a patula sn-ftin li

wi t lns n is a nssa ntin f w spaatnT smn nunqunss f t ntn tfin t patula las nua t appaan f altnatv sptins f w spaatin tat n ntast t Litill t nsist n t psn f asal pint as t n f t in Wan 976) in patiula asau tat t a tw tps f w spaat pn" n w tsn-ftn ln n wi t ls nv s nt manat fm a

DD PRTION

N

u Adjacent noes n addle point (Lghthl 1963).

Annu.

Rev.FluidMech.1982.14:61-85

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THDMNSONL SPTD FLOWS 67

sae pnt an clse" n wc as n tll's entn t essee als an 1974 Han ael 1979 In wa llws we sal aresse quesn an apprprate escrpn w separatn b an appeal he her srucural sabl an burcan e an w shalln t ncessar t snush btwn ps separan but we shalapt a trmn that s sus b the theral ramwk sall sa ha a sknrctn ln emern rm a sale pn s a globallne separan an leas t gloal w separatn In te cnrar casewere e snrctn lne n wc her lnes cnere es nt rnate

rm a sal pnt we shall en the lne as ben a lcal n sparatn lan local w sparan hn n mr s us

wha s sa wll appl t ether ase. hus n eher case) an atnalncatr te n separatn s he behar he surace rexlnes. In e cn a lne separan e surace re lnes becmesrte rmn upstream-pntn lps wt e peas h lpsccurrn n e ne separan

he cnerse he lne sepaatn s e ne aacmen wlnes attacmen are llustrate n Fure 2, emanatn rm eac the nal pns attacmen. Snrcn lnes ten t ere rmlnes atachmen Ju as he ne separatn a rapc ncatr

te presence a lne aamen s e bear te surace rtxlnes Surace rex lnes rm wnstream-pntn lps n te cn a lne attachment wt e peas the lps ccurrn n te lne aachment.

Sreamlnes passn erclse e surace a s se ene bEquans () are calle limiing reamline n he cnt a lne separatn lmtn sramlns mus ae the surace rap as a smplarument ue Lhhll (93) xpans Rrrn quatn (3) leus aln wth exernal streamlne crnates s that ' are teresece sreamwse an crssw snrcn cmpnens. I n s tesance beween w aacen lmn sreamlnes see ure 3 an s he e a recanular sreamube ben assume small s ae lcal resultant elc ecrs are cplanar an rm a lnear prle)hn h mass ux hruh h stram ub s

m p

were

ps e ens an e mean elc he crss secn

e resulan skn rcn a he wall s he resulan WI an T " r

Annu.R



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8 TOB & P

s ha

_ TwhU Hene

h2nwm = - = nsanvedin

(2

h = c v ! n

hus as he line f sepaan s appahed h he heh f he minseamlne abe he suface nceases apdl hee ae w easns fhs inease n h: s whehe he ne f sepaain s lba lalhe disance n beween adjacen min seamnes fals apidl as eimn seamines cnee wads he lne f sepaan secnd heesulan snfin Tw dps wad a mnimum as he lne f sepaans appae an n e case e a ne sepaan auaappahes e as e saddle pi s appaced

Limin seamines sin n eihe sde f e line f sepaain ae

peened fm cssn b he pesene f a seam suface semminfm he ine f sepaan sef. he exisence f such a sam sufaceis chaaces f w sepaan hw i naes deemines whehehe sepaan is f lbal a fm. n he fme ase he pesenef a sadde pn as he n f he lbal ne f sepaan pides a

3D SEPARAONE

LIMTIG SREAMEDRCTOS PROJCDO SURFAC

r 3 g i h-o D) p .


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THRDMSIOAL SPARATD LOWS 69

mehanism o the eation of a new seam sufae that oiginates at thewll. mntn frm sadde point and teminating at nodal pnts fsprtn (t nds f) t ll in f sprtn trs

smt u on t wl w fms t s f t stm suf, thesteamlines of whih have l enteed the uid though the saddle point.We shall all this new steam sufae a dividin surface The dividingsufae extends the funtion of the global line of sepaation into the ow,tn s rrr sptn t st f lmitn strmlns tt rsn frm t surf n n sid f t ll ln f sptn frm tst sn fm t tr sd. On ts pss dwnstm, t ddnsufae olls up to fom the familia ld st aound a ntl tloe. Beause it has a welldened oe, we use the popula teminology,alling the ow in the viinity of the oiled-up dividing sufae a ortex.

Nw w nsdr t rin f t stm surf tst f lalw sprtn. W nt tt f snfritn ln mntn frm ndl pnt f ttmnt ultmtly ms ll ln f sprtnthen thee will be a point on the ln ynd w f t tnlsufae votex lines ossing the line foms an upsteam-ponting loop,signifying that the skin fition along the line has beome loally minimum.A sufae stating at this point and stemming fom the skin-ition line

dwnstrm f t pnt n nstrutd tt wll t us f stf lmtn strmlns rntn frm fr upstrm; ts surf myalso ol up on its development dwnsm.

his setion onludes with a disussion of the emaining type of sinua point, the fous aso ae spia noe. he fous invaaby appeas on the sufae n ompany with a sadde pont. Togethe they alow prtulr frm f ll w sprtn On l f t (ll) nf sprtn mntn fm t sdd pnt wnds nt t fus t fmt ntnuus u on t suf fom wi t ddn suf stms.

he fous on the wall extends into the uid as a onentated votexlament, whie the dividing sufae os up with the same sense of otationas the votex ament. When the dividing sufae extends downsteam itqulydrws t t lmnt nt ts r n t tn t tnsnnt t ud f t fus n t wl srs s t rtl r utw t ddn surf ls. s w r ws rst yptsidy Legende (1965, who also noted (egende 972 that an epeimentalonmation existed in the esuts of ealie expeiments aied out yWe (19 ur 4a shows Legende's oiginal sketh of the skin

fritn lns Fur 4 s ptrp lustrtin t pimntl nrmtn ddn suf tt ls rund t tnsn f t fus(Fur 4 wll trmd r ntyp ddn surf." On tthe hand t an happen that t ddn suf t wi t fus is


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70 TB

(b)

B C

(a) A - NODAL TTAHMEN POIN SAD NT

C FOU O SRTO

4 of seprtin: rgnal k f kn-fc n by Ld (1965); xpm ' 96 wter ; x u, nd (S.


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THDMNSONL SPTD FLOWS 71

conncd do no xnd downa In ca vox lannang fo focu n dnc nd n p nyon coow pln downa of ogn on ufac. In an n

n ddonl npon of focu w n y condn pan of ln ooonal o a of n-fcon ln a pan of ufac vox ln. W a wa wa a focu fo pan of n-fcon ln bco ano focu of paaon fo pan of ufac vox ln ang appan naon of of ufc vox ln If w gn c of ufcvox ln ond p of oo vox n xnonno ud of focu on wall a a concnad vo lan n o pn cobaon no on lan of oo voxlg fo al of bound voc a av ndd a focu. On cannvg pobly of ncopoang dcpon of ow n vcny of focu no an appopa nvcd ow odl

Forms of vdg Surfaces

W av n ow conon of focu and a addl on n

pan of n-fcon ln allow a pacula fo of loal ow paaon caaczd by a on-yp dvdng ufac." nodal ponof cn and paon y lo con waddl pon o allowaddonal fo of gloal ow paaon aan caacd by pcul dvdng c ccc dvdng fc odfo conaon of a nodal pon of acn and a addl pon lluad n Fu Sa. fo of dvdn ufac ypcally occun ow bfo an obacl ( Fu 34 n Pa oba 19 In xapl lluad n Fgu Sa wll b nod a dvdng

ufc d of a pon n xnal o a wc d vlocy dnclly o dnonl ngul pon wc nFu S c ogn of ln unnn o voclco of olld up dvdng ufac

caacc dvdn ufac fod fo cobnaon of anodal pon of paaon and a addl pon lluad n Fgu 5b fo of dvdng ufac ofn occu n nonally wodnonalpd ow uc n pd ow nd cwd-fcngp ( gu 24 n o 17) nd pd ow cylnd- uncon (o wo nd dnonl co u47 nd n o ) no n o u 5 nd 5ba aln on dvdng ufac av all nd udou addl pon n pan of n-fcon ln.




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72 TOB & P

Tpgraphy tramn in Tw-Dmnnal tno Theemesoal los

Atr cctbl lg lp l rcl bcclr ht t thcl bi r t bvir ltr iglrpit d th tplgicl rl th th b i grl gh ppra uh wdr rg applia ha had rigiall b ralid Trul rprd b Si (969975) Prr Firl (197) d Hut al (1978) v d t vid t h rl gvrig k-ricl bhvr r l dptd d dd ild ilr rul gvrg bvir w tl. I prclr Hut t l. (978) vd th

=[u(yz) v(yz) w(yz)] t vlcit

D INGLAR POINT

N NODAL POINT OF ATANT

N NODAL POINT OF PARATON

(b)

Dvng surfces fome from combnons of no pon of cmen nse pon; no pon of sepron n se pon.


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EEES SEE WS 73

whos uv opots r surd i p z Zo ostt, bov surf sttd ty = x;o (s Fgur 6), th th strisi th p r soutos of th qutio

dx dyU

(5)

whih is drt outrprt of Equtio 3 for sk-frtio s o thsurf Hut t l (1978) utiod tht for gr thr-dsolow th stris dd by qutio (5) r o or th thtthyr ot ssrily th projtios of th thr-disio strsoto th pl z = z or r thy ssrily prtl pths v stdy ow Oly for spil psfor xpl, strwis pl of

sytry (whr w(xyo )r th strlis dd by quto(5) dtbl with prtl pth is i th p wh th ow s stdy,or wth sttous strs w t ow s ustd ssi [u(x y v (x y] is otuous vtor d V(xy with oly tubr of igur pots th itror of th ow t wh V 0 itfolows tht ods d sddls b dd i th p ust s thywr for sk-frtio lis o th surf Nods d sddls withi thow, xudi th boudry y = Yxz, r lbld d , rsptvly, d r show thr typil for i Fgur 6 oy w

ftur of t ysis tht is rqurd s th trtt of sigulr potso th boudry y (xzo Si for visous ow, V is ro vrywhr o th boudry, th boudry is itslf sgulr i i th p zo Sigur poits o th our whr th opot of thsurf vortity vtor orl to t pl z = is ro Thus, forxpl, t is surd tht sigulr poit wl our o t boudrywhrvr it psss through sgur poit th pttr of sk-frtiois s th surf vortity is dtly ro thr As trodud byHut t l (1978)sgulr poits o th boudry r dd s hf

ods d hlf-sddls (igur 6) With this sipl dt tot typs o sgulr poits lowbl, ll of th prvious otos ddsriptios rlvt to th ysis of sk-fritio lis rrovr to tysis of th ow wth th pl

prll v Hut t l (978) hv rogd tht ust s thsigulr pots th pttr of skifrito s o th surf oby topologl ul, so ust th sgur poits y of th stiol viwsof thr-disol ows oby topologil ruls Although vry grru ppyg to ultpy otd bodis b drivd (Hut t l

98) w st hr for ov oly thos spl ruls tht wll busfu i subsqut studs of th ow pst ws, bods, d obstlsI th v topologil rus std bow, w ssu tht th body is sipyotd d rsd i ow tht s uifor fr upstr

Annu.R



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74 TOBAK & PEAKE

1 Skfo e o ree-dmeo body Dvey 11 3

= 2 (6) Sfro e o reedmeo body B oeed mywou ) o e w P eer eed o y bo ustrm dow\tr a: o e ufe of oru P+B - 0 (7)

3. Sree o wo-dmeo e u reedmeobody

(N + ) ( i s' -1 . (8)4 Sreme o ve e u urfe eed oy bo uem d dowrem

( + ) ( + ) = O (9) Sreme o e rojeo oo er urfe of o

ow reedmeo body Sm

(2N + i 2) - ( .) O (0)oooa Sruure, Sruura Saby adBurcan

e queo of deque dero of ree-dmeo eedo r prcur rp w o k ow redmo

L Z Z

HM

gur 6 Sg po co ecio o ow (Hu t 178




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HRMOL PR FOW 75

eprte w pttern rnte n hw the uee eh ther therelevnt prmeter f the prblem nle f ttk Renl numberMh number, et.) re vre. A tftr nwer t the quetn m

emere ut f the frmewrk tht we rete n th etn. We t urfrmultn n phl term thuh ur entn uht t be mptble wth mre pure mthemt tretment be, fr exmpe,n whtever tem f prt erentl equtn jue t vernthe u mtn In prtulr, we hne ur entn f tpltruture n truturl tblt retl t the prperte f pttern fkn-frtn ne ne th enble u t mke mxmum ue f reultfrm the prnpl ure f expermentl nfrmtn n three-menn eprte ww-vuztn experment utlzn the trek tehnque.

Aptn the termn Anrnv et (9) we tht pttern f kn-frtn lne n the ufe f b nttute the phsportrt f the urfe her-tre vetr w phe prtrt hve theme topoo trtr f mppn frm ne phe prtrt t thether preerve the pth f the phe prtrt It ueful t mnehvn mprnte phe prtrt n heet f rubber tht m be efrme n w wthut fln r tern. Ever uh ermtn pth-preervn mppn A topoo proprt n hrtertf the phe prtrt tht remn nvrnt uner ll pth-preervnmppn he number n tpe f nur pnt the extene f pthnnetn the nulr pnt n the extene f le pth re exmpe tp prperte The et tp prperte the phe prtrt erbe the tpll truture

We ene the trtr tbt f phe prtrt reltve t prmeter A flw (ee Anrnv et l. 7: A phe prtrt truturll tble t ven vlue f he prmeter A f the phe prtrt

reultn frm n nntem hne n the prmeter h the metpll truture the ntl ne. he prperte f truturl tbephe prtrt n be elute v mthemtl nl (Anrnvet l. 197) thuh the epen t me extent n whether pe ntn uh , fr exmple, emetr mmetre, re t be neretpl (e ener" ee Benjmn 1978) r untp (nnener")Here we wh t repet the ntn mpe b emetr mmetrewhenever the ext n th e tuturll tble phe prtrt fthe urfe hertre vetr hve tw prnpl prperte n mmn:

( the nulr pnt f the phe prtrt re l mtr nurpnt; n there re n le-pnt-t-le-pnt nnetn nthe phe prtrt. (e hul nte tht ntn (b prpert nlf the phe prtrt repreentn the trjetre f the urfe her-




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76 & E

strss vtor Sal-pont-to-sal-pont onntions otn our on twonsonal projtons o th rnal owy trnal ow w anth nir ow rior to th surau ths ar artiats o hpartiular projtions an o not prsnt onntions twn atual(thrinsional) singular points o th ui vloity vtor)

In spakng o th staty o th trnal ow, w n it nssary tostingush twn struturl stbilit an smptoti stbilit o thow. h nion o srutural staily ollows ro tha ntrou inrrn to th phas portrait o th sura shar-strss vtor. An trnal ow is all struturaly stal rlativ to a paratr A i a sallhang n th paratr os not altr th topologial strutur (.g. th

nur an typs o thr-insional singular points) o th trnalhr-nsonal vloity vtor l Asyptot staily s isussor ully latr. Hr w not siply that a an ow s all asyptoally sal i sall prturatios ro i at A) y to ro asti In spaking o strutual an asyptoti nabl w nit onvnit to stinguish twn lal an lal proprtis o thisailii all a iaili i mal als topologial strutur o ithr th trnal thr-insional vloity vor l or h phas portrait o th sura shar-srss vor W all

an nsaily lol i i os no rsul n an alraon o h topologialstrutur o thr vtor l Thus a strutural nstalty s nssarilygloal whil an asyptoti nsaity ay thr loal or gloal. Onth othr han, an asyptoti instaility nssarily iplis nonuniqunss(athatally spakng) in th solutons o th govrning ow quationswh a sruural nsailty n o iply nonuniunss

h ntrouion o isntons twn loal an gloal vnts hlpsto plan why w wr arlir to istnguish twn loal an gloallins o spration in th patrn o skin-rition lins. an (sypoti)

instality o th trnal ow os not altr th topologial strutur oth phas porrait rprsnting th sura sharstrss vtr thn thonvrgn o skn-rion lins ono on or svral pariular lins anonly a loal vnt so ar as th phas portrait s onrn aorngly,w lal h parular lins lol lins o sparaion On h ohr hani an (asyptoti or strutural) instaility o th trnal ow hangs thtopologial trutur o th phas portrait rsulting in th rgn oa sl pont in th pattrn o skin-rition lins, thn this is a gloalvnt so ar as th phas portra is onrn aorngly, w la

h sknrition ln anating ro h sal pont a lobl lno sparaon

Asyptoti instaility o th trna ow las to th notions obifurtion smmetr-rekin an dissipte strutures (Sattingr




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HRDMNSONL SRD FLOWS 77

180, Nicos & Pigogi 77)_ uppos ha h uid moios vvaccodig im-dpd quaios o h ga om

U = G(u A) wh A gin is . Sins (UA) = 0 sn tadyn w h kind w hv bn cnidig w hv nd aa ow is a asypoicay sb ow i sa pubaions omi dcay o o as t - Wh paam A is vai, o maw ay sis [in h ahica sns h i ain vid souin f (A) 0] bu bcm usab sm pbis asA csss ciic v sch a nsiin in nw an w y

bifurcat h knwn w Th bhavi u dcibd is cnvninypoayd o a biucaio diaga, ypica aps o whih a iusd gu . ws h buc h kwn w nd by dina whch y b ny quniy h cacish bicin w n. Sb ws indicad by sid ns nsb ws b dshd ins. Ths v h ng A wh h knwnw is sb, is z, and h sb known ow is snd g hbsciss by sid in. Th knwn w bcs unsab vus

SABL

gur 7 Expe () upercrc bfurc ) ubcrc burcn

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18/25

78 K K

of X larger than e' a the ae ne along te abia nate ewmean ow bfurate from X e ether uperrtally or ubrtally

t a uerrtial bifuraton (iure 7a, a the arameter

h in

reae jut beyon the rtial ont e' the bfuration ow that relaethe untable known ow an ier oly nniteimally from t The bfuraton ow breaks the symmetry of the known ow aoptng a form ofleer ymmetry n whih dissipatve structures are to aborb jut theamount of ee aalable energy at the more ymmetrial known owno longer wa able to aborb. Beaue te bfuration ow intally eartonly nntemally from the untable known ow the trutural tablityofthe urfae hear tre intally i unaete However a X ontinue

to inreae beyon X the bfuration ow epart igniantly from theuntable known ow an begn to aet the trutural tablity of theurfae hear tre Ultimately a value of reahe at whih te urfaehear tre beome truturally untable eviene ether by one of theelementary ngular ont of t hae portrat beomn a nular pointof (o) multil orer or by the appearane of a ne ingular point ofeen multple orer In eter ae t ueful to oner te nular

oint of multple orer a being the oaleene of a number of elementarynular point wt the number ve among noal an ale ont

uh a to ontinue to atify the rt tooloal rule Equation (6) natonal nntemal inreae n te parameter X reult n the plittngof the ingular poin of multple orer into an equal number of elementaryngular pont. Thu there emere a ne truturally table phae portratof the urfae heartre vetor an a ne eernal o from hiaitonal ow ultimately will bifurate wt furter inreae of thearameter.

t a ub ritial bifuration igure 7b when te parameter i inreaejut beyon the rital point X tere are no ajaent bfuraton ow

that er only nnitemally from the untable known ow Here teremut be a nte jum to a new bran of ow that may rereent a raialhane in the tooloal truture of te eternal ow an era n tephae portrat of the urfae heartre vetor a well. urther wth 1on the new branh when ereae jut below te ow oe notreturn to the ornal table known ow Only when A ereae farenouh below to a o Fgure 7b) the table known ow reoereu ubrtal furation alway mle tat te bfuration ow wllehbit hysteress eet

Th omplete a framework of term an notion that houl ue toere ow the trutural form of three-menonal earate owornate an uee eah other e following eton eote tollutraton of te ue of t framework n two eamle inolin uerrtal an u rtal furaton.

Annu.



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19/25

EXPLE

TDlMNSN SRD FWS 79

RoudNosed Body o Revouo a Ae o Aack

Le us s cosde ow seed ow y oie o sledeoud-osed body o eoluo s oe o e ees o eoble le o ck is cesed o zeo cees Focuso e ow i e ose eo loe we do s exle o illuse sequece o ees wc supercrtcal bfurcatonis e e ledo e oo o le-scle dsse sucues

zeo le o ck (Fiue a) e ow s eeywee ced ll

ski-icio les oe e odl oi o ce e osed o scely soo slede body dse io odl oio seo e e ee oolol ule uo sssed e sles ossbe wy ( )

ey sll le o (Fiue b e oolocl suue oe e o sk-o es es ueed skco lesi oie odl o o ce d dse o odloi o seio Howee e oble cicueel essue die dies e ski-ico les leewd wee ey ed o coee oe skiicio lie ui lo e leewd y E o ode e sddle o d be e oo w oe sko es oee s cul le qules s lcal lne ofseparatonccod o ou deio e ow e ciy o e locle o seio odes e ocuous o o locl ow seoyicl o e ows le suces e e syey les o wkes

s e le o ck is icesed ue cil le is ecedus beyod wic e exel ow beoes loclly usble Coi oly ee s e well-kow suscebiliy o nexonal boudy-lyeelociy oles o sbly (Geoy e 955, Su 963, Tobk1973). Te exiol oles deelo o cossow les e slilyclied o e le ol o e exel isd ow deciolled w sby e ee s oe euso o boudylye sio yiclly occu Reyolds ubes us eei eso e McDe Mee199, ds 1971. eeo e biuo dis o Fue 7 d deyi e ee Awi le o ck we e e isbly occus e cicl

oi wee suercrtca bucio (Fue a) leds o ewsble e ow W e lol sce ueced by e isbliy eew e ow os y o dsse suues Te suuesllused secly o Fue ce illy o ey sl scle wsc o e ode o e boudy-lye ckess Resebl y




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20/25

0 TOB & P

= 0

b)

(e)

(d)

(e)

SON

TN -

SCON -

8 Suence of ows leading to global three-dimension ow sepo on onnose y of evolon s nge of ck is ncsed.

Annu.R

ev.FluidMech.1982.14:61-85.Downloadedfromwww.annualreviews.org

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21/25

HR-DIMNSONL SPRTD FLOWS 1

am v hav a hy kd m h d h a h b ad tial hh snin f scs n cssw n n i c is

inn b y scic, nss sc siss cl l f slins n cssw n, in () s lsad h d v 8 h aay va d h pa k by h appaa a snin y f aa ln f aahm ad (a) paaBcs bfcin s scic, w, ic scsinilly f innis sn n cnn c liclsc f pa k- . Th aa ha lcal paa ah hh ad a ay paad ha ay vy ma a.

l nl l l nf sn, bin s n nnnif cssw Viw in cssw ln, sn f scs ncss f sinfm h dad ay ah a mamm a haay ad addmh ad h ad ay. Ra ha h paam a ppd haa h ba d v da h mamm vy dd

b s f icl scs Ts, w f incs inn f c, incss ccnly, s Fi inics yscly, ncss bcs inn icl sc cs pa h m d h hby hh aby dmh ad a da h b h dma Th a h a aak a h mb ic scs n inn sc iniss wi inn sc ws iy. Mnwil, wi incs in n fc, w n n cs ns bcs sbc

aby ad dvp a aay ma va ma h ha had dvpd h dam a a a aak. Th a ad 8d W bv ha hsciin is a snn f y f w Wn (94 s cc s n "n in W n l

inn icl sc nw s sn fll-a f w aan, nss sfc s-sss c smad ay ab ha m h a a alal paa.

W f ncs in n f c, cssw nsbiy in in s f inn icl sc s wyf fw n f s n s ssc c in fsin ny n nl f c s c wc innc




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22/25

82 E

he vrica rucure i rea enuh aer he rucura abiiy h urace harr cr in h iia iciniy he n. ew (uabe) inuar pi ec rer appear a he rii each he ca ie eparai Wih a igh urher icrae i age aackhe uabe iuar pi pi i a pair eemeary iguarpina cu eparain an a ae pin hi cmbinai pruce he hrn-ype iiing urace ecribe earier (iure 4 an iura aain in igure 8 ( a igure 1 an 1 in Wr 1)We w have a glbal rm w eparain. new ab mean wha emrge rm which aiina w uimay wi biurcae wihfurher increae f h ae aac

symmer Vorex Breakdo o Sleder Wg

n cra upercriica bifurcai which are rmay beig evebegiin a hey mu wih he apearance ny iieima iipaiverucurubcriica bifurin y b ric ev ivvin uean aaic chan in w uc huh w ar ny binnin appreciae he re biurcain in he uy epara w wecan anicipa ha un arg-ca en uch a h in inaircra bue an a wi be cribabe i er ubcriica biur

cai. Here we ci ne exape where i i areay vien ha a uiyamica pheme invvig a ubcriica biurai ca ignicany inuce he aircraf ynaica bhavir. hi i h ca aymeric vrex breawn which ccur wih ener we wing ahih an aac.

We eave aie he exing quein he echanim unerying rxbrakwn i (e Ha 12) a w a i pgica rucure fcu v ubequn he breaw f h wig' primary vrice Lwn (6) ne ha whe a ener ea wi wa wy

ich a ucieny arge ang f aac wih iei ane he xea r h brakwn h ai ain- ic which a wrange ha ccurre yericay (ie i by i)beca aymrcwihhe iin f n r brakw ving cr h win aexha he her Which he w pibe aymmeric paern wa berve aer ay ige ichup wa prbabiiic bu ce eabihe heraiv iin h w rx brwn wu ri r h winen a h an f aack wa uc vau a which h brawnha ccurr iniiay wnrea h win raiin r ien

iyin r w hw ha h brain a prcy caibewih ur previu ecripi f a ubcriica bifurcai (iure b)

Le u ene by he irenc bwee he chrwi piin he eha a rihhan vrx breawn n A b iiv

Annu.



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HRDMNONA ARAD OW 3

wh th thad akdow poto t co o th two to th wgap. g w to th uctca ucato dagam gu 7w dt th ucato aamt wth ! ad th aamt A wthagl attac W s tat accdac wt sat t s aag a < c, whch th t akdw ostos ca costd sd a ta tat std ! = O. t th ctca ago attack th akdow ca o g uta thm d d,so tat o > o og a sta tat Th g oadact ucato ows ut od !mut ump to a dtatach o ta ws, whch psts th sudd sht owad o oo th ot akdow pto th wth ! o th w ach,as th ag attack ducd

!d t tu to zo at

aut o

at ha pasd a ma au o o th accodac wthat (ws 1964). t ay ag attack wh ! ca zo u smmtc ua cto, t aat wtsdslp o ll agl must cssal sttc. alo as dmtatd pmta (E 196) uth, c ! must dct potoa to th og momt, th coqut httc hao o th og momt wth dp o o ag mak th acatssct t t damca m wg-ck (c t a 9)

SUMMARYHdg stct to th ot that patts o sk-ct s ad ta stams ct th popt o cotuou cto d aus to characterize the patterns on the surface and on particular projections t w (t cow pa, am) a stctd m sgula pot (od, sadd pot ad oc). It s uul to cosd thtctd m o gula pot ad th tpologcal ul tat tho as cmt a gazg pcpl: a ow gamma ws

t um o mts ca comd mad was t dc,udtad, ad coct togth th popts commo t all tadth-dmoal cou ow Itducg a dtcto tw ocaad glal ot o th ow ol a amgut th ppdt a th-dmal paatd ow. doptg th ts otgca stuctu tuctua stat, ad ucat gs us a amwk wc t dsc w th-dmsal saatd ws gatad ow t uccd ac ot a t lat paamt o t plma ad

CNOWDGMN

W a gatu to . V oko uggtg a uul thought pmt ad to G. T. apma, who ctuto w tumta




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24/25

84 K

rg dertdg o erody hyere er o buro heoy

trtur td

Adams, J C r 197 1 Thredimensionalamina boundary-aye analysis of upwash patens and entrained vorex foration on shap cones at angle of atack.AEDTR71215

Andonov, A. A, eontovich, E. A, Gordon,I I., Maier, A G 97. Thor ofBur

to of Dm tm o PlNAA TT F56Andronov A A eonovich E A, Gordon,

I I., Maier, A G 1973 Qulttv Thor of odOrdr Dm tmNew Yok Wiley

Benjamin, T. B 1978 Bifurcation phenomena in steady ows of a viscous uid ,Theoy. II Expeiments. Pro R ooo 359:-3

Davey A 1961 Boundayaye ow at asaddle poin of attachmen. J ludMh 05960

Ele, B J 1961. An investgation at lospeed of the ow near the ape of thindeta wings with shap leading edges. Brth AR 19780 R M 3176

Gregory, N, Stuar, J T, Walker, W S1955 On te sabiity of heedimensiona bounday layers wih appicaion tothe ow du to a otating disc Ph/oTr R o odo r A 2485599

al, M. G 972 Votex breakdown. ARv lud Mh 4:9528

an, T, atel V C 979 ow separaionon a p a ncdnc u 92:64357

Hun J. C. R Abell . J, Peerka J A,Woo, 198. Kinematica sudies of heows aroun fe o sufaceouned obstaces; applying opoogy to ow visualization J ud Mh 86: 179200

Joseph, 96 tblt oflud Moto 1 Berin: Spinge

Legende R 956 Sparation de lcolemen laminaie tidimensionnel. RhAro 5438

egende, R. 965 Lignes de couran d'unulement ntinu Rh Arop 05

egendre R 1972 a condition de Jou

kowski en coulement tidimensionneRh Arop 52448Legendre R. 977. Lignes de courant dun

coulemen peanen dcollemen etsparation Rh Arop 1977-6:32755

Lighthill, M. J 963 Attachment and separation in hree-dimensiona ow In mr Boudr r ed Rosenhead,, 2.67282 Oxfod Univ Press

owson M V 964 Some experimentswih vorex beakdown J R Aro o6834346

Malby R. 1962 Fow isuaiaon inwind tunnels using indicaors AGARDorph No 70

McDevitt J. B., Melenthin, J A. 1969pwash paerns on ablating and nonabang cones at hypesonic speedsNAA TN D5346

Nicois, G., Prigogine 1977. lfOrzto Noqulbrum tmew o Wey

Oswatitsch, K. 1980. The conditions for thesepaaion of boundary ayers In otrbuto to th Dvlopmt of Gdm ed. W Schneider, M atze, pp.618 Baunschweig/Wiesbaden Vieweg

Peake , Tobak, M. 980 Thee-dimensional interactions and vorical ows withemphasis on high speeds AGARDorphNo 252

Perry, A E., Fairie B. 1974 Citicapoins in ow patens. n Adv Goph 18B:29935. New York:Academic

Poincar, H. 928. Ouvr d Hr Pocr Tom aris Gautier-Vilars

Satng, D H. 980 Burcain and symetry breaking in appied mathematicsBull (Nw r) Am Mth o377989

Schi, L B, Tobak, M , Malcom, G N 980 Mahematica modeling of the aeodynamics of high-angle-ofatack maneuvers. AAA Ppr 80-18P

Sith, J. . B. 1969 Rmas on the stucure of cnca ow RAE TR 69119

Smih, . H B 1975 A eview ofseparationn steady, theedmensona ow AGARDP168

Stuat, J T 196 Hydodynaic stablityn mr Boudr r ed . Ro

sen head IX492579. Oxfod niv PressTobak M. 973 On loca inexiona insabiiy in boundary-ayer ows Z AwMth Ph 24054

Tobak, M, Peake, D 1979 Toogy of


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THRDMSA SARATD FWS 5

twdml d th-dmptd w AIAA Paper 79140

Wg K C. Bo e oe bl bd t hgh id with p

typ pt oc. R Soc. Londoner. A 05Wg K C. 76 Spt thd

e w I Rw Vou

ow, Po Lokhee-Geogia Co SympLG 77ER0044 pp

We H epo e bdi t lw pd R Ar.

03W, H 7 Tbill e p l d v. L'Aero. L'Astro 6

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