vortex singularities, numberical solution, and a xi symmetric vortex breakdown

8/8/2019 Vortex Singularities, Numberical Solution, And A Xi Symmetric Vortex Breakdown

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I

.". .e.,-"

C O N T R A C T O RR E P O R T

.

S O L U T I O N , A N DB R E A K D O W N

by

Barbara, Calif. 93106r Langley Research Center

LOAN COPY: RETURN 7'0AFWL (DOUL)

KIRTLAND AFB. N. M.

E R O N A U T I C SN DP A C ED M I N I S T R A T I O N W A S H I N G T O N ,. C. JULY 1972


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TECH LIBRARY KAFB,NM~ -.

" .. . . -~1. R e p a t No. 3. Recipient's C a t a l o g No.. Govkment Accession No.NASA CR-2090

". ~7. itle and subtitle6. PerformingOrgmnization CodeAXISYMMETRIC VORTEX BREAICDOWNVORTEX EQUATIONS: SINGULARITIES, NUKZXICAL SOLUTION,ND5. Report Date

July 1972

7. Author(s) 8. Performing Organization Report No.Hartmut H. Bossel". . ~ 10. Work Unit No.9. Perfwming Organization Name and AddressUniversity of CaliforniaSanta Barbara,alifornia

I 126-13-10-0111. Contract or Grant No.

NGR 05-010-02513 . Type ofReportandPeriodCovered-I 12.SponsoringAgencyNameandAddress I ContractoreportNational Aeronautics and Space AdministrationWashington, D. C 20546 14.SponsoringAgency Code

16: AbstractA method of weighted residuals for the computationf rotationally symmetric quasi-cylindricalviscous incompressible vortex flow is presented and usedo compute a wide variety of vortex flows.The method approximates the axial velocity and circulation profiles by series of exponentials having(N + 1) and N free parameters, respectively. Formal integration results inset of (2N + 1)ordinary differential equations for the free parameters. The governing equations are showno have aninfinite number f discrete singularities correspondingo critical values of the swirl parameter.The computations point o the ontrolling influence of the inner core flow on vortex behavior. Theyalso confirm the existence of two particular critical swirl parameter values: one separates vortexflow which decays smoothly from vortex flow which eventually breaks down, and the second is the singularity of the quasi-cylindrical system, at which point physical vortex breakdown is thought to

ortex Equationsisymmetric Vortex Breakdownip and Leading-Edge Vortices

18.DistributionStatement

Unclassified - Unlimited19. Security aar rif. (of t h r report)

$3.004nclassifiednclassified22. Rice'1. No. of Pages0. Security Clacsif. (of this page)

For a l e by the Natio nal echnical Inform ation Service, Springfield, Virginim 22 15 1


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V OR T E XQU A T I ON S :I N GU LA R I T I E S ,U M E R I C A LOLU T I ON ,AND AXISYMMETRICVORTEX BREAKDOWN'

Hartmut H . Bosse lU n i v e r s i t y o f C a l i f o r n i aS a n t a B a r b a r a , C a l i f o r n i a

Mechanica lEngineeringDepar tment '

SUMMARY

A v i s c o u sp a ra b o l i c s u b s e to f h e n c o m p r e s s i b le N a v i e r -S t o k e s e q u a t i o n s , h e q u a s i - c y l i n d r i c a lv o r t e xe q u a t i o n s ,c a n b eused t o compute vor t ex f l o w sa s o n g a s t h e i r stream s u r f a c eang le emains smal l . A t low valueso f a swir l p a ra m e te r ,c o m p u t a t i o np re se nt s no d i f f i c u l t i e s . A t h i g h s w i r l v a l u e s ,s i n g u l a r i t i e s a r e e n c o u n t e r e dw h i c h n d i ca t e a f a i l u r e of t h eq u a s i - c y l i n d r i c a la p p r o x i m a t i o n a n d ' a r e h o u g h t o b e a s s o c i a t e dw i thphys ica la x i s y m m e t r i cvor texb re a k d o w n .V o r t e xb re a k d o w n 'i s normal ly o l lowed by a ' v o r te xbu bb le ' , and a ' v o r te x jump 'may occurfa r the rd o w n s t r e a m .

I t is found th a t he q u a s i - c y l i n d r i c a lv o r t e x e q u a t i o n s h a v ean i n f i n i t e number of d i s c r e t e s w i r l - d e p e n d e n t s i n g u l a r i t i e sw h o s e c o r r e s p o n d in g c r i t i c a l swi r l va luesd e p e nd on v e l o c i t y andc i r c u l a t i o np r o f i l e h a p e s . The b e h a v i o ro f l o w d e c e l e r a t io nvs'. a c c e l e r a t i o n on t h ev o r t e xa x i s ) i s o p p o s i t e on b o t h s i d e sof a s i n g u l a r i t y (i e . a t h i g h e r o r lower s w i r l ) . A t h igh s w i r l s ,t h e a p p r o a c h o s i n g u l a r i t i e s c a n o n l yb e a v o i d e d by a p p l i c a t i o no f s p e c i f i c e x t e r n a l a x i a l v e l o c i t y a n d / o r c i r c u l a t i o n g r a d i e n t s .T h e s i n g u l a r i t i e s a p p e a r o c o r r e s p o n d o h e c r i t i c a l s w i r lv a l u e so f h e e q u a t i o no f n v i s c i d r o t a t i n g f l o w .

The f i r s t s i n g u l a r i t y S1 i s o f p a r t i c u l a r s i g n i f i c a n c e asit seems t o cor respond t o th eof tenobse rveda x i s y m m e t r i cexplos ivebreakdown. The p hy s i ca ls i g n i f i c a n c e of t h eh i g h e r

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s i n g u l a r i t i e s i s n o t q u i t e c l e a r a t p r e s e n t . A second mpor tan tv a l u e of t h e swirl paramete r i s So w h i c h s e p a r a t e s f l o w whichdecay s smo oth ly f rom vor tex f low .whi ch even t ua l ly b reak s down.

T y p i c a lv o r t e x f l o w s( i n i t i a l l y u n i f o r m a x i a l f l o w , e a d i n ge d g e v o r t e x , r a i l i n g v o r t e x ) are computed fo r a w i d e v a r i e t y o fi n i t i a l swir l p a ra m e te r s u s in g a method o f we igh ted res idu a lsw h ic hexpresses th e v e l o c i t y and c i r c u l a t i o n a p p r o x i m a t i o n s nterms o fe x p o n e n t i a l s . The e f f e c t s of e x t e r n a la x i a lv e l o c i t ya n d c i r c u l a t i o n g r a d i e n t sa r e n v e s t i g a t e d .

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TABLE OF CONTENTSPage

I . I N T R O D U C T I O N . . . . . . . . . . . . . . . . . . . . 11.1 P h y s i c sfhe Pr'oblem . . . . . . . . . . . 11.2 S w i r l Parameter . . . . . . . . . . . . . . . 81 . 3 Numerical Approach . . . . . . . . . . . . . 91.4 Overview . . . . . . . . . . . . . . . . . . 11

I1. COMPUTATIONAL METHOD . . . . . . . . . . . . . . . 132.1 Governing Equat ions . . . . . . . . . . . . . 132.2 I n t e g r a l R e l a t i o n s . . . . . . . . . . . . . 1 52.3 ApproximatingndWeighting F u n c t i o n s . . . . 1 62.4 S y s t emfO r d i n a r yD i f f e r en t i a lq u a t i o n s . . 172.5 I n i t i a l P r o f i l e s . . . . . . . . . . . . . . 1 82.6 Accuracynd Convergence . . . . . . . . . . 1 8

I11. SINGULARITIES AND CRITICAL SWIRLARAMETERS . . . 213.1 S i n g u l a r i t i e s fh e Governing System . . . . 213.2 S w i r l Parameter . . . . . . . . . . . . . . . 233.3 S i n g u l a r i t i e s o r n i t i a l l y Uniform

A x i a l Flow . . . . . . . . . . . . . . . . . 233.4 Behavior of A x i a l Der iva t ives . . . . . . . . 253.5 E f f e c tf V e l o c i t y P r o f i l e . . . . . . . . . 253.6 Assessment of Breakdown Behavior . . . . . . 27

I V . VORTEX COMPUTATIONSORCONSTANT EXTERNAL A X I A LVELOCITY AND C I R C U L A T I O N . . . . . . . . . . . . . 294.1 I n i t i a l P r o f i l e s a n d Distance t o F a i l u r e . . 29

Uniform i n i t i a l a x i a l f low . . . . . . . . 29Leadingdge vortex . . . . . . . . . . . . . . 3 1T r a i l i n g v o r t e x . . . . . . . . . . . . . . 31P o s s i b i l i t y o f b r e a k d o w n - s t a b l eso1utJons of t y p e 2.. 3. 4 . . . . . . . . . . . 33

V

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4.2 V e l o c i t y on t h e Axis . . . . . . . . . . . .Be ha v i o r of i n i t i a l l y uniform a x i a l f lowso f y p e 1-4 . . . . . . . . . . . . . . . .I n i t i a l l yu n i f o r m a x i a l f l owof ype 1 . .

343436

Leadingdge vor t ex o fy p e 1 . . . . . . . 37T r a i l i n g vor t ex of t y p e 1 . . . . . . . . . 384.3 V e l o c i t y P r o f i l e s . . . . . . . . . . . . . . 38

V . EFFECTS O F EXTERNAL V E L O C I T Y AND C I R C U L A T I O NDISTRIBUTION . . . . . . . . . . . . . . . . . . . 445 . 1 Type 1 Vortex Flow . . . . . . . . . . . . . 445.2. Type 2 Vortex Flow . . . . . . . . . . . . . 485.3 Type 3 Vor tex Flow . . . . . . . . . . . . . 495.4 Type 4 Vor tex Flow . . . . . . . . . . . . . 495.5 Avoidance of S i n g u l a r i t i e s . . . . . . . . . 5 1

V I . CONCLUSIONS . . . . . . . . . . . . . . . . . . . 55A P P E N D I X . . . . . . . . . . . . . . . . . . . . . 59REFERENCES . . . . . . . . . . . . . . . . . . . . 6 3

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LIST OF SYMBOLS

a paramete r inh ee l o c i t yp p r o x i m a t i o nnbn p ar am e te r i n t h e c i r c u l a t i o n a p p r o x i m a t i o n

[ - 1[ - 1

Fk c o e f f i c i e n t snh er d i n a r yi f f e r e n t i a l e q u a -A- t i o n sd e f i n e dnhe Appendix) [ - 1n Ik

e = 2.71828183f k (Y) w e i g h t i n gu n c t io n sgk ( Y ) w e i g h t i n gu n c t io n sH = V R (nondimensionalarameter)j nk = wr-circulation

z e r o s o f h e Besse l f u n c t i o n J1

K = WR n o n d i m e n s i o n a li r c u l a t i o nn o n d i m e n s i o n a le x t e r n a lc i r c u l a t i o n

Kei initial KeN number of p a r a m e t e r snh ee l o c i t yn dc i r c u l a t i o n a p p r o x i m a t i o n s

r-I[ - I1- 11-11- 1

[ - 1

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PPePOP'POpmrrRC

RCReS

U

ue'b,U

pres sureexternal static pressurefree-stream total pressure= p/ph2/2) nondimensional pressurenondimensional external static pressurenondimensional free-stream total pressurenondimensional free-strean static pressureradial coordinatevortex core radius= ,/iG r/rc nondimensiond. radial coordinatenondimensional vortex core radius= I&,rc/V - core Reynolds numberswirl parameter S = wC/u, = WC/U,or S = (dW/dR )=RC/Uaxx-dependent swirl parameters function ofvelocity profile developmentvalues of S at which singularities existinitial swirl parametercritical swirl parameter value (velocity profile-dependent) separating vortex flow which decayssmoothly (S


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ve external radial velocityV =6 v/la, nondimensional radial velocityW swirlelocitycircumferentialirection)WC swirl elocity t ore adius rcW = w/% nondimensionalwirlelocityWC = wc/l&)X axialoordinateX = x/rcondimensionalxialoordinateXf ondimensional osition f omputational ailureX0 x value at which velocity gradient is appliedY = R 2 / 2 nondimensionaladialoordinatea exponent n elocity nd irculation pproximationsY exponent n he irculation rofilerl = e Cry

V kinematiciscosityP densityk weightingunctionxponent$(Y) complete set of linearly independent functions

t d/dYSubscriptsk integer, 1 k N + 1n integer, 1 5 n 5 N

rL/T1[-Ir m 1r L/T 1[-I[-I[ L I[ - I[-I[- I

r -1[ - I[-I

[-I

axnhexis

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I. I N T R O D U C T I O N

1.1 Physics.of he.ProblemThe p r e s e n t r e p o r t p r e s e n t s and d i s c u s s e s n u m e r i c a l

s o l u t i o n s t o a s p e c i f i c p a r a b o l i c s u b s e t - a p p l i c a b l e t o q u a s i -c y l i n d r i c a l v o r t e x f l o w s - o f h eax isymrne t r ic ncompress ib leN a v i e r - S t o k e se q u a t io n s .T h i ss u b s e tc a nb eu s e d as a r e a s o n a b l ea p p r o x i m a t i o nf o r many v o r t e x f l o w s o c c u r r i n g n e c h n o l o g i c a la p p l i c a t i o n s and i n n a t u r e (w ing v o r t i c e s , r o t a t i n g u b e s ,s w i r l i n gp i p e l o w ,d u s td e v i l s , o r n a d o s ) .S o l u t i o n so f h e .q u a s i - c y l i n d r ic a l v o r t e x e q u a t i o n s a re t h e r e f o r e o f some i n t e r e s ti n t h e i r own r i g h t , b u t o f e v e n g r e a t e r i n t e r e s t a r e p e r h a p s t h ec o n d i t i o n su n d e rw h i c hs o l u t i o n s c a n n o tb eo b ta in e d . ns u c hc a s e s h ep h y s i c so f h ep r o b l e m a re d i f f e r e n t t h a n assum ed i nt h e q u a s i - c y l i n d r i c a l a p p r o x i m a t i o n , and t h i s subset becomesin ad eq ua te . When t h i sh a p p e n s , h ep h y s i c a lc o u n t e r p a r to f h ef lo w u n d e rc o n s id e ra t io n most l i k e l y e x p e r i e n c e s a r a p id e x p a n -s i o n o r c o n t r a c t i o n o f t h e c o r e t o v i o l a t e t h e c o n d i t i o n o fq u a s i - c y l i n d r i c i t y .

O b s e r v a t i o n so fv o r t e x f l o w s a t h igh s w i r l do nde ed showr a p i de x p a n s i o n sa n d / o rc o n t r a c t i o n su n d e rc e r t a i nc o n d i t i o n s .The ob se rv ed c l os e l y- re la te d phenomena havebeen lumped und ert h eh e a d in g s o f ' v o r t e xb r e a k d o w n 'a n d v o r t e xburs t ing ' , ands e v e r a le x p l a n a t i o n sh a v eb e e na d v a n c e d .U n t i l e c e n t l y ap e c u l i a r i t yo f v o r t e x f l o w s - t h e i r e x t r e m e s e n s i t i v i t y t o p r o b ei n s e r t i o n - h a s p r e v e n t e d h e g a t h e r i n go f r e l i a b l e e x p e r i m e n t a ld a t a o v e r i f y h e s ee x p l a n a t i o n s . Only now, w i t h h ea d v e n to fp robe less ve loc i tym e a s u re m e n tu s in g h e a s e rD o p p le ra n e m o m e te r ,can r e l i a b l e d a t a b e c o l l e c t e d , a nd t h e o r i e s be checked,anda s s i g n e d h e i rp r o p e rp l a c e n h eo v e r a l l l o wp i c t u r e . I tt u r n s o u tr e a l l y i nphenomenau n d e r h e

- once more - t h a t t h e c o n f l i c t i n g t h e o r i e s a re n o tc o n f l i c t a f t e r a l l , b u t t h a t t h e y d e s c r i b e d i f f e r e n twhich a re nd eed o b s e r v e d t o o f t e n happen more o r l e s ssame c i rcumstances .

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I n orde r t o p u t t h e r e s u l t s of th i s s t u d y i n t o t h e p r o p e rp h y s i c a lp e r s p e c t i v e , an i n t e r p r e t a t i o n of ' vor texbreakdown 'o b s e r v a t i o n s w i l l f i r s t be made p a r t l y b a s e d on r e c e n t l a se ranemometer s t u d i e s which have l e a d t o some unexpectedobserva-t i o n s . The c o n c l u s i o n s contras t sOme:-~terpre ta t iOps .till to bef w d . n %h+ cwmn%literature.

E a r l i e r e x p e r i m e n t a l a n d t h e o r e t i c a l work i n the are.a o fc o n c e n t r a t e dv o r t e x f l o w s h as been eviewed in H a l l ( 1 9 6 6 a ) , a n da v e r yc o m p r e h e n s i v eb i b l i o g r ap h y o 1967 i s g i v e n i n T i m( 1 9 6 7 ) . A r e v i e wo fvo rte x breakdown ha sr e c e n t l y been made by

H a l l ( 1 9 7 2 ) , and a c o m p re h e n s iv e e v i e wo fconf inedv o r t e x lo w si s due t o . Lewellen ( 1 9 7 1 ) . O b s e rv a t io n so fw in gvor tex f lowsand v o r t e x b reakd ow n, e s p e c i a l l y on d e l t a w i n g s , a re numerous- r e f e r e n c e sc a n be found i n th e reviews c i t e d . W ith v e r y fewe x c e p t i o n s e s p e c i a l l yH u m e l1965, McCormick e t a l . 1968)t h e s es t u d i e s a re q u a l i t a t i v e . I t i s e x p e c t e d h a t r e l i a b l eq u a n t i t a t i v e d a t a f o r wing v o r t ex f lows w i l l become ava i lab lei n t h e n e a r - f u t u r e w i t h the a d v e n to f t w o - and hree-d imensionaland rapids c a n n i n g a s e rD o p p l e ra n e m o m e t e r s .

Much us ef u l nf or m at io n on t h e phenomenon ofv o r t e xbreakdown has come from t h e s t u d y of s w i r l i n g f l o w n p i p e s ( i np a r t i c u l a r H a r v e y 1 9 6 2 , Sarpkaya1971a, 1 9 7 1 b , O r l o f f 1 9 7 1 ,Orlo f f and Bossel 1 9 7 1 ) . T h e dye tud ieso fHarvey andSarpkaya a re q u a l i t a t i v e o n l y , and t h e c o n c l u s i o n so f theseau thorshav e been somewhat re vi se d b y t h e q u a n t i t a t i v e o p t i c a lve loc i tymeasurementso fOrloffand Bossel . More i n fo rm a t io n o nthe vortex breakdown phenomenon has come from t h e s tudyoff lowi n a s t a t i o n a r y c y l i n d e r w i t h a r o t a t i n g l i d (M axworthy 1 9 6 7 ,Vogel 1 9 6 1 ) . I n the c a r e f u l and comprehensive work ofV o g e l q u a n t i t a t i v e d a t a on vortex breakdown a re ob ta ined f romt r ace r s tu d ie s .S w i r l i n g l o w su n d e r g o i n g a r a p i de x p a n s i o nhavebeenstudiedbyG or e and Ranz ( 1 9 6 4 ) , NissanandBresan( 1 9 6 1 ) , P o t t e r e t a l . ( 1 9 5 8 ) , So ( 1 9 6 7 ) , Vonnegut 1954) ndo the r s .

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S e v e r a lc o n f l i c t i n g e x p l a n a t i o n s h a v e b e e n a d v a n c e d t os u d d e n e x p a n s i o no fv o r t e x co res , most commonly

as ' v o r t exb r e a k d o w n ' : t h e s p i r a l i n s t a b i l i t y t h e o r y( 1 9 6 2 , 1 9 6 5 ) , h ehy d ra u l i c jump ana logyofBenjamin

o r , i n a m i ld erf o r m , h ec o n c e p to f a s t a t i o n a r y wave1 9 6 7 , L e i b o v i c h1 9 6 8 ) ,a n df i n a l l y h ee x p l a n a t i o n onof s t a t io n a r y c o n t i n u o u s s o l u t i o n s o f h e N a v i e r - S t o k e s

f o r s w i r l i n gf l o w , o r a p r o p e rs u b s e t h e r e o f V a i s e yand Fe j e r 1 9 6 6 , Bossel 1 9 6 7 , 1 9 6 9 , O r l o f f 1 9 7 1 ,

an d Bossel 1 9 7 1 , Torranceand Kopecky 1 9 7 1 ) . On t h eh a n d , - t h e f a i l u r e o f h e q u a s i - c y l i n d r i c a l subset o f t h e

eq u a t i o n s ( i . e . t h e a i l u r eo f h eq u a s i - c y l i n d r i -l ap p r o x i m a t i o n i n r e g i o n s o f r a p i d l y e x p a n d i n g o r c o n t r a c t i n g

h asb e e nu s e db yG a r t s h o r e 1 9 6 3 ) , H a l l (1966b1,( 1 9 6 7 , 1 9 7 1 ) , Mager ( 1971) t o p r e d i c t h e p o s i t i o no f

k e l y break dow nand t h e b e h a v i o r of f l o w p r eced i n g it.Most e x p e r i m e n t a lo b s e r v a t i o n s seem t o s u p p o r t h e v i e w

t t h e su dd ene x p a n s i o no fv o r t ex cores i s m o s to f t ena nphenomenon and n o t t h e r e s u l t o f s p i r a l i n s t a b i l i t y .

wever, ac co rd in g ou n p u b l i s h e dexper iments (Ludwieg 1 9 7 1 ,a t e c o m m u n i c a t i o n ) , s p i r a l i n s t a b i l i t y may i nd e edp l a y a

r o l e n h e b re ak do wno f d e l t a w ing v o r t i c e s .O t h e rh a v e b e e n d e n t i f ie d b y S a r p k a y a ( 1 9 7 1 a , b ) , b u t t h e

m in an t r o l e i n w ha t w e know as t h e b r eak d o w n ' - p r o ces sa p p e a r sb e a s u d d e na x i s y m m e t r i ce x p a n s i o no f h e core . I t i s f o r

axisymmetric e q u a t i o n s a re s t u d i e d i n t h eB e f o r ee n t e r i n g n t o h ea n a l y s i s , it w i l l b e

t o p i e c e t o g e t h e r a c o m p r e h e n s i v ep i c t u r eo f h ea x i s y m - -' v o r t e x b r e a k d o w n ' - p r o c e s sb a s e d o n h e most r e c e n ti v e d a t a o b t a i n e d t h r o u g h o p t i c a l v e . l o c i t y m e a s u r e m e n t s

O r l o f f (1 9 7 1 ) an dOr lo f f nd Bossel ( 1 9 7 1 ) . It i s f o u n d h a te e x p l a n a t i o n s a n d h e o r i e s p r o p o s e d so f a r a l l h a v e h e i r

p l a c e a n d a re n o t e a l l yc o m p e t i t i v e . The p r o p e r a n g el i c a t io n w i l l b e i d e n t i f i e d .

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C on sid er f i r s t a v o r t e x f lo w w i t h more o r l e s s c y l i n d r i c a le x t e r n a ls t r e a m s u r f a c e s a n d a h i g h ' swir l ' (some r a t i o of ar e p r e s e n t a t i v ec i r c u m f e r e n t i a l ( s w i r l ) v e l o c i t y t o a r e p r e s e n t a -t i v e a x i a lv e l o c i t y ) . An a d v e r s ep r e s s u r eg r a d i e n t i ssuperimposed on thea x i sb y h ep r e s e n c eo fa na x i s y m m e t r i cobs tac le (O r lo f fan dBo s s e l 1 9 7 1 ) . The f l o w observed i s t h e n asshown in Fig. 1.1. The flow initially approaches the obstacle in aq u a s i - c y l i n d r i c a l m anner. The v e l o c i t yo n h ea x i s decreasesu n t i l a s ta g n a t i o n p o i n t i s reached on th e a x i s a n d f l o w has t omove outward rom t h e a x i sn e a r t h e s t a g n a t i o np o i n t . nk e e p i n gw i t h t r a d i t i o n a l n o t a t i o n , t h i s p a r t i c u l a r p r o c e s s w i l l be c a l l e d' v o r t e xb r e a k d o w n ' n t h e f o l l o w i n g .

There i s r e v e r s e d u p s t r e a m )a x i a l l o w on t he downstreams i d e of t h e s t a g n a t i o np o i n t . T h i s r e v e r s e d lo w i s p a r t ofthe f lo w i n t h e ( ' f o r c e d ' ) v o r t e xb u b b l e ' between s t a g n a t i o np o i n t and o b s t a c l e , a c l o s e d r e g i o n w i t h p r a c t i c a l l y no i n t e r -change of f l u i d w i t h t h e u r ro u n d in g lo w . The e x t e r n a l l o wp a s s e so v e r t h e bubbleandover t h e r e a r of t h e o b s t a c l e . A sit p a s s e so v e r t h e r e a r it undergoes a 'v o r t ex jump' i n t h esense of t h e hydraul ic jump analogyofBenjamin ( 1 9 6 2 ) , w i t h as u b s t a n t i a l l o s s of a x i a l momentum an da n a t t e n d a n te x p a n s i o nof stream s u r f a c e s n e a r t h e a x i s .

I n F i g . 1 . 2 t h e s t a g n a t i o n p o i n t , a n d t h e r a p ide x p a n s i o n .n e a r h ea x i sa r ec a u s e d b y t h e e f f e c t of a n e x t e r n a lp r e s s u r eg ra d ie n ts u c h as producedby a f lo wdivergence . With a p roperf a v o r a b l e p r e s s u r e g r a d i e n t t h e bubble w i l l c o n t r a c t d ow ns tre amand may e v e w b e more o r l e s s c lo s e d . F lo w shaving uch(seemingly) c l o s e d ' f r e e ' v o r t e xb u b b l e sc a n e i t h e r be g e n e r a t e db y p ro p e rs h a p i n go fa no u t e rs u r f a c ea s nF i g . 1 . 2 ( B o s s e land Orloff 1 9 7 1 , u n p u b l i s h e d )o r by a s e l f - i n d u c e dp r o c e s sb e tw e e nvor texb ub bl e and e x t e r n a l f l o w f i e l d a s i n the v o r t e xtubelows fHarvey ( 1 9 6 2 ) andSarpkaya ( 1 9 7 1 a , b ) . Sarpkayahas evensu cceed ed i n g e n e r a t i n g s e v e r a l v o r t e x b u b b l e s i n arow. J u s t d ow n st re amo f a v o r t e x b u b b l e a vor tex jump m a y

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o c c u r , b u t does n o t a p p e a r t o o c c u r i n many casesbubb le i s smooth or repeated .Fo l lowingBen jamin

where the( 1 9 6 7 ) and

Le ibov ich (1968) t h e s ev o r t e xb u b b l e sc a nb ev i e w e d as s o l i t a r ywaves. H o w e v e r , t h e i rd e s c r i p t i o na p p e a r s t o b e more s t r a i g h t -fo rw ard and a c c e s s i b l e t o n u m e r i c a l q u a n t i t a t i v e a n a l y s i s i fd i r e c t u s e i s made of t h e N a v i e r - S t o k e s e q u a t i o n s o r a p r o p e rs u b s e t o f t h e s e e q u a t i o n s .

F i g . 1.3 shows a n o t h e r . p o s s i b i l i t y w hic happears t o be.t y p i c a lo fd e l t a w i n gvor tex lowsand some t o r n a d o s . I n h i scase t h e v o r t e x b u b b le fo I lo w in g h e v o r t e x break dow n i s n o t ,closeddownstream. A t some d is ta n ce f rom th e breakdown p o i n tt h e l o wagain becomesmore o r l e s s c y l i n d r i c a l . no t h e r l o w s(on de l taw i n g s , f o re x a m p l e ) a s t a g n a t i o np o i n to c c a s i o n a l l y

d o e s n o t seem t o e x i s t : t h e r e i s o n l y a s w e l l i n g of t h e c o r e w i t ha n a t t e n d a n td e c r e a s eo fv e l o c i t i e s n h ec o r e .T h i ss i t u a t i o ni s d e p i c t e d nF i g . 1.4; t h e term ' v o r t e xb u r s t i n g 's h o u l dp r o b a b l yb er e s e r v e dfo r it .

F i n a l l y , i f t h e s w i r l i s v e r yh i g h n a v o r t e x , h e v o r t e xassumes a dec idedly d i f f e r e n tc o l u m n a r c h a r a c t e rw h e r e f l o wd ep en de nc e on a x i a l p o s i t i o n has e s s e n t i a l l y d i s a p p e a r e d , andt h eT a y l o r o l u m n ' t r e t c h e s a rd o w n s t r e a m F i g . 1 . 5 ) . Th i ss t r u c t u r e i s founddownstream of a vo r t ex jump or mm edia te lydownstreamof t h e f l o w e n t r a n c e i f t h e i n i t i a l s w i r l i s v e r yh i g h . I n k e e p in gw i th c c e p te dn o ta t io n Be n ja m in 1 9 6 2 ) t h i sf low i s termed ' s u b c r i t i c a l 'w h i l e h ef l o w u p s t r e a m o f a v o r t e xjump would b e su p er c r i t i ca l ' . S ta t i o na ry waves may occ ur i n' s u b c r i t i c a l ' l o w ,w h i l e n s u p e r c r i t l c a l ' flow t h ea x i a l l o w .i s t o o f a s t , sw eepinganywavesdownstream).

Summarizing thee x p e r i m e n t a lo b s e r v a t i o n s , it i s obv ioust h a t c l e a r d i s t i n c t i o n s e x i s t betw een vortexbreakdown'and' v o r t e x u m p ' , s u p e r c r i t i c a l ' n d ' s u b c r i t i c a l ' l o w , v o r t e xb re a k d o w n 'a n d v o r t e xburs t ' , ' ax i symmetr icb re a k d o w n 'an dn o n a x i s y m m e t r i c n s t a b i l i t i e s . For c o n v e n ie n c e , ' w eh a v e l s o

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- -___cvo r t ex b reakdow' f o r c e d ' vortex bub o l wU P

Fig . 1.1. Vortexbreakdown,causedbypresence of an o b s t a c l e :on t h e a x i s .

' v o r t ex jump' f r e e ' v o r t exubble (may o r may n o tF ig . 1 . 2 ' F r e e ' vo r te x b reakdown wi t h c lo sed 'b u b b i e .

v o r t e x bF i g . ,1.3 'Free ' vo r t ex breakdown w i t h ' open 'bubble ." "Fig . 1.4 Expansion of v o r t e x core; no breakdown with f r e es t a g n a t i o n p o i n t .

o ccu r )

F i g . 1 .5 S u b c r i t i c a l T a y l o r column) b eh av i o r .

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d i s t i n g u i s h e db e t w e e n f r e e 'a n d f o r c e d 'v o r t e xb u b b l e s ,a l t h o u g h h e ya p p e a r t o b e t h e r e s u l t o f t h e same b a s i c .m e c h a -nism. In this report, the term 'vortex breakdown' will be used todesignate conditions here the quasi-cylindrical solutions becomesingular and the corresponding computations fail. This includes rapidcore expansion, as in the vicinity of the first stagnation point, andrapid core contraction.

wltn a c c u r a t e o p t i c a l v e l o c i ty m e a s u r e m e n t s a h e a d o f ,a round ,beh ind ,a nd i n h e v o r t e x b u b b le now a v a i l a b l e ( O r l o f f1 9 7 1 , O r l o f f a n dBosse l 1 9 7 1 ) , i t has become poss ib le t o t e s tt h ea c c u r a c y and a p p l i c a b i l i t y o f n u m e r i c a l a p p r o a c h e s o h es o l u t i o no f t h e ' vo r tex b reakdown 'problem. I t i s f o u n d t h a tt h e q u a s i - c y l i n d r i c a la p p r o x i m a t i o n c a n be c o r r e c t l y a p p l i e du p st re a m t o a d i s t a n c eo f h eo r d e ro f h eb u b b l ed i a m e t e ra h e a dof t h es t a g n a t i o np o i n t F i g s . 1 . 1 - 1 . 3 ) . Flow i n h en e i g h b o r -hood of t h e s t a g n a t i o n p o i n t and i n andaround theb u b b lem u s tbe t r e a t e d by u s i n g e i t h e r h e f u l l N a v i e r - S t o k e s e q u a t i o n s o ra p ro p e ra p p ro x im a te u b s e t F ig s . 1 . 1 - 1 . 3 ) . It w a s shown i nBo s s e l ( 1 9 6 7 ) t h a t h ep r o p e r s u b s e t i s t h ee q u a t i o n o ri n v i s c i d o t a t i n g l o w .N u m e r i c a l o l u t i o n s show e x c e l l e n tq u a n t i t a t i v e a g r e e m e n t w i t h t h e m e a s u r e d v e l o c i t y p r o f i l e s w i t han i n a c c u r a t e d e s c r i p t i o n of t h e s w i r l v e l o c i t y p r o f i l e i n s i d eth eb u b b l e a s t h eo n l ye x c e p t i o n (Orloff andBosse l 1 9 7 1 ) . F o ra c c u r a t e d e s c r i p t i o n of t h ev e r ys l o wf l o w n s i d e h e b u b b l e ,v i s c o u s terms should be t a k e n n t oa c c o u n t ; how ever t h e e x a c tb u b b l es h a p ea n d h ee x t e r n a lf l o wa r eq u i t eo b v i o u s l yd e t e r -mined a l m o s t o l e l y by t h e n v i s c i d l o w f i e l d . The i n v i s c i de q u a t i o n of r o t a t i n g l o wc a n be a p p l i e d o com pute ' f r e e ' and' f o r c e d 'v o r t e xb u b b l e se q u a l l y w e l l (Bosse l 1 9 6 7 , 1 9 6 9 ,O r lo f f1 9 7 1 , O r lo f f a n dBosse l 1 9 7 1 ) .

A c o n t i n u o u s s o l u t i o n i s n o t p o s s i b l e i n a r eg io nw h e re av o r t e x jump o c c u r s ; h e f l o w o r c e 'c o n c e p t so fB e n j a m i n ( 1 9 6 2 )must hen be a p p l i e d o o i n two c o n j u g a t e l o w s . n h e

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o b s e rv a t io n so fO r l o f fa n d Bossel (1971) the v o r t e x jump o c c u r sover a d is tanceo fa p p ro x im a te lyo n ebubb le d ia m e te ra n d i smarked b y v e r y s t r o n g t u r b u l e n c e .1.2 Swirl Parameter

The axisymmetric behavior of a v o r t e x flow i s very much afu n c t io no f i t s s w i r l . A s w i r l p a ra m e te rcan be d e f i n e d ast h e r a t i o o f s o m e r e p re s e n ta t iv e( c i r c u m f e r e n t i a l ) swirl v e l o c i t yt o a r e p r e s e n t a t i v ea x i a lv e l o c i t y (see Sec. 111). F o rze roswirl , t h e f l o w r e d u c e s o a j e t o r wake, dependingon whethera v e l o c i t ye x c e s s or d e f i c i t e x i s t s on t h e a x i s . T h i s j e t orwake v e l o c i t y p r o f i l e w i l l u l t i m a t e l y d e c a y t o th e freestreamv e l o c i t y as t h e f l o w proceedsdownstream. The same i s t r u e o rsmall and medium amounts of s w i r l , u n t i l th e swirl v e l o c i t y anda x i a lv e l o c i t y reach comparablemagnitudes. A t h i g h e r swirlp a r a m e t e r s , the a s y m p to t i ca p p ro a c h t o freestream c o n d i t i o n si s r e p l a c e d b y d e c e l e r a t i o n of t h e flowon th e a x i s u n t i l as t a g n a t io np o i n ta n ds u b s e q u e n t b r e a k d o w nb u b b le ' , or a t l e a s ta s w e l l i n g of t h e core appear.. A l l o t h e r c o n d i t i o n sb e i n ge q u a l , t he re i s o b v i o u s l yo n e p a r t i c u l a r swirl p a r a m e t e rv a l u ew h i c h d i v i d e s a v o r t e x flow which d e c a y s i n a wake-like mannerf romano ther which ' b r e a k s d o w n ' .Th i sd iv id ing swirl p a ra m e te ri s d e s i g n a t e d So i n t h e f o l l o w i n g ; it i s a f u n c t i o no f h ev o r t e x v e l o c i t y p r o f i l e s .

Axisymmetr icvortexbreakdown i s a c o m p l e t e l y symmetricphenomenon wi t h . i t s o r i g i n a t t h e v a r ya x i s ( S a r p k a y a 1971a,b,O r l o f f 1 9 7 1 ) . Thus c o n d i t i o n s on the a x i sm u s tb e of v i t a lim p o rt a nc e i n i t s i n c e p t i o n . np a r t i c u l a r ,v i s c o s i t y estab-l i s h e s r i g i d r o t a t i o n a t a n d n e a r t h e a x i s . For a r i g i d l yr o t a t i n g i n v i s c i d c y l i n d r i c a l s l u g of f l u i d of r a d i u s r c ,a x i a l v e l o c i t y uax,and swirl v e l o c i t y wc a t r c , a swirl parame-t e r c an be d e f i n e db y S = wc/uax, and t w o i m p o r t a n t t h e o r e t i c a lr e s u l t s follow:

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As the s w i r l i s i n c r e a s e d a n d crosses So = fi frombelow, a n i n i t i a l d e c e l e r a t i o n o f the core elemellt w i l lampl i fyand ead t o s t a g n a t i o n (Bossel 1 9 6 8 ) .As t h e swirl i s i n c r e a s e d f u r t h e ra n d crosses S1 = j11/2(= 3.8317/2 = 1.9159)* rom below, t h e c ha ra c t e r oft h es o l u t i o nc h a n g e s r o m s u p e r c r i t ic a l ' w h i c hc a n n o ts u p p o r t t a n d in gw a v e s ) t o ' Subcr i t ica l ' (which ans u p p o r t t a n d in gw a v e s ) F r a e n k e l1 9 5 6 ) . As S r e a c h e st h e v a l u e s jln/2 cor respond ing t o h i g h e rz e r o so f t h eBeesel f u n c t i o n J1, more waves a p p e a r .

It c a nb ee x p e c t e d t h a t the c r i t i c a l s w i r l v a l u e s So, S1,S 2 , . . . of a v i s c o u sv o r t e x w i l l be a f u n c t i o no f h ev o r t e xv e l o c i t y p r o f i l e s and w i l l b e s o m e w h a t d i f f e r e n t f ro m those f o rt h e i g i d l y o t a t i n g n v i s c i dc y l i n d r i c a ls l u g . O f p a r t i c u l a rs i g n i f i c a n c ef o r v o r t e x f lows a r e So (d iv id ingwake- type rombreakdown behavior)a nd S1 (beyond S1 s u b s t a n t i a lu p s t r e a mi n f l u e n c e i s p o s s i b l e ) .1 .3NumericalApproach

La m in a r n c o m p re s s ib l es teadya x i s y m m e t r iovor texf lo w sa r e descr ibed by t h e c o r r e s p o n d i n g form o f h eN a v ie r -S to k e se q u a t io n s .T h e s ee q u a t i o n s can be approximatedby a p a r a b o l i cv i s c o u s s e t , a n a lo g o u s t o t h e b o u n d a r y a y e re q u a t i o n s , nr e g i o n s where t h e stream s u r f a c ea n g l er e m a i n s s m a l l ( q u a s i -c y l i n d r i c a l v o r t e x f l o w ) , andby the i n v i s c i d e q u a t i o n s o fr o t a t i n g f l o w a t and n e a r t h e a x i s , and where stream s u r f a c ea n g l e s become l a r g e( e x p a n s i o n or c o n t r a c t i o no f the core)(Bossel 1 9 6 9 ) .

Some s o l u t i o n s of t h e i n v i s c i d s e t p e r t a i n i n g t o v o r t e xf lo ws a t h i gh swir l have b e e np r e s e n t e d n Bossel ( 1967 , 19691 ,C h o w ( 19691 , O r l o f f and Bossel .(197.1): here w e. shall now di.scuss*

j l l i s t h e f i r s t z e r o of the Bessel f u n c t i o n J1.'

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a m ethod o f s o l v in g th e p a r a b o l i c v i s c o u s s e t a n d g iv e corres-ponding r e s u l t s f o r d i f f e r e n t . s w i r 1 p a r a m e t e r s , i n i t i a l v e l o c i t yp r o f i l e s , a nd e x t e r n a l p r e s s u r e and c i r c u l a t i o n g r a d i e n t s .S i n c e h e r e g i o n s of v a l i d i t y of th e two s e t s p a r t i a l l y o v e r l a pon and n e a r t h e a x i s , it s h o u l d b e p o s s i b l e t o conf i rm some oft h e e a r l i e r r e s u l t s . The p a r a b o l i cs y s t e m h a s beenusedbefo reinn u m e r i c a lc o m p u ta t io n sb yd i f f e r e n tm e t h o d s (Gartshore 1963,Ha l l1965 , Bossel 1 9 6 7 , Mager 1 9 7 1 ) . Gar tshore and Mager eachused a momentum-integralapproachandencountered a s i n g u l a r i t yw h i c h h e y i n k e d o t h e v o r t e x breakdown phenomenon and t o th ec r i t i c a l s w i r l parameter3.8317/2of low i n i n i t i a l l y r i g i dr o t a t i o n . The p r e s e n t e s u l t s e i n f o r c e t h i s view. Beyondt h i s c r i t i c a l s w i r l r a t i o , Bossel (1967) and Mager ( 1971) a l soo b ta in e df l o w s which c o n t r a s t e d n b e h a v i o r w i t h thosebelowt h e c r i t i c a l swir l r a t i o .

T h e num erica l method N-parameter exponent ia l se r i e si n t e g r a l m e t h o d ' " ) o b e u s e d f o r t h e p r e s e n t s t u d y hasp r e v i o u s l yb e e no u t l i n e d n Bossel (1970a, 1 9 7 1 ) . A r e l a t e dmethod N-parameterpower s e r i e s i n t e g r a lm e t h o d ' ) h as e a r l i e rbeen u s e d s u c c e s s f u l l y o c a l c u l a t e v i s c o u s v o r t e x f l o w s(Bo s s e l 1 9 6 7 ) . The methods were i n s p i r e d by th e s u c c e s so ft h e Dorodni tsynmethodfor t h e c a l c u l a t i o n o fb o u n d a r y a y e r s(Dorodni tsyn 1 9 6 2 , B e t h e l 1 9 6 8 ) , e s p e c i a l l y by i t s a s p e c t so fspee d nd ccur acy. However, t h e Dorodni tsynapproach w i t h i t suse of Y (U) i n p l a c e of U(Y) canno t be d i r e c t l y a p p l i e d t ov o r t e x f l o w s where a x i a l v e l o c i t y p r o f i l e s o f t e n e x h i b i t o v e r -shoo t and f l o w r e v e r s a l . The p re s e n ta p p ro a c hc i r c u m v e n t s t h i sproblem . The ,soundne ssof t h e e x p o n e n t i a l s e r i e s i n t e g r a lmethod hasb e e nd e m o n s t r a t e d n i t s a p p l i c a t i o n o n c o m p r e s s i -b l e andcompress ib leboundary ayer lows Bosse l 1 9 7 0 a , b ,Mitra andBossel 1 9 7 1 ) .* The term ' i n t e g r a lm e t h o d ' i s used here t oa v o i d t h e cumber-some (b u t more a cc u ra te ) terms 'me thodo fweigh ted es idua ls 'o r 'me thod of i n t e g r a l r e l a t i o n s ' .

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There a re good reasonsfo rc h o o s i n ga nN - p a r a m e t e r n t e g r a la f i n i t e d i f f e r e n c e method f o r a s tu d ysuch as t h e

on e. Many of t h e o p e nq u e s t i o n s n h i sc a s eo n l ya q u a l i t a t i v e a n s w e r , w hi chcanbeobta ined a t minimal

c o s tb y a o ne o r tw o p a ra m e te rs o l u t i o n . fa c c u r a c ys d e s i r e d , t h e number o fp a ra m e te r s i s i n c r e a s e dw i t h o u ta n y

ange i n t h e p rogram: R es u l t s o f h e same d e t a i l and a c cu ra c ys f o r f i n i t e d i f f e r e n c e m e th od sc a n h e nbeobta ined . The

s o l u t i o n o f p a r t i a l d i f f e r e n t i a lt o s o l u t i o n o f a s e t of o r d i n a r y d i f f e r e n t i a l equa-

o n s . Well-known s ta b le n te g ra t io ns c h e m e s ,s u c h as t h emethod, an be a p p l i e d .T h e r e . i s little chanceofa t r u e s i n g u l a r i t y o f t h e s e t o f e q u a t i o n s t o be

w i th a s i n g u l a r i t y o f a f i n i t e d i f f e r e n c e scheme r e p l a c i n gt. No i t e r a t i o n s a r e n e c e s s a r y .F i n a l l y , e l o c i t i e s , stream

p r e s s u r e ,b o u n d a r y a y e r h i c k n e s s ,s h e a r , e t c . , c anl becomputedsimplyand d i r e c t l y f r o m a n a l y t i c a l e x p r e s s i o n s

p a r a m e t e r s n h e v e l o c i t y a p p r o x i m a t i o n s .. 4 Overview

The fo l lowing S e c t i o n I1 w i l l f i r s t des c r i be the computa-o n a l m ethod nd g iv e a nexample f o r i t s accuracy .andS i n g u l a r i t i e so f h ee q u a t i o n sa r e n v e s t i g a t e d i n

111, and a s w i r l p a ra m e te r i s i n t r o d u c e d .S e c t i o n I Vt i g a t e s , f o r c o n s t a n t e x t e r n a l a x i a l v e l o c i t y and c i r c u l a -

v o r t ex f l o w s which a r e o f p a r t i c u l a r i n t e r e s t t od y n a m i c i s t : (1) v o r t e x l o ww i t h n i t i a lu n i f o r ma x i a l

(2) v o r t e x l o ww h e r e h ev e l o c i t y on t h e a x i s i sh i g h e r h a n h e f r e e s t r e a m v e l o c i t y ( l e a d i n g e d g e, and ( 3 ) v o r t e x f l o w w h e r e h ev e l o c i t y on t h e a x i s i s

lower t h a n h e f r e e s t r e a m v e l o c i t y ( t r a i l i n g v o r t e x ) .v e l o c i t y p r o f i l e s and v e l o c i t y d i s t r i b u t i o n s on

e a x i s as a f u n c t i o no fd i s t a n c e are g i v en . The e f f e c t s ofo r n e g a t i v e g r a d i e n t s i n t h e e x t e r n a l v e l o c i t y and

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c i r c u l a t i o nd i s t r i b u t i o n are s t u d i e d nS e c t i o n V . Somec o n c lu s io n s a re drawn,a n d r e s u l t s are s um ma rize d in S e c t i o n V I .

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11. C O M P U T A T I O N A L METHOD

I n d e v e l o p i n g a m e th o d o f w e ig h te d r e s id u a l s f o r a prob le mi n t w o c o o r d i n a t e s , t h e d e p e n d e n t v a r i a b l e s n h e p a r t i a ld i f f e r e n t i a l e q u a t i o n s a r e r e p l a c e d b y a p p r o x i m a t i n g u n c t i o n s ,

a s e t of w e i g h t i n g f u n c t i o n s ,a n d h e n f o r m a l l yi n t e g r a t e d w i t h r e s p e c t t o o n ec o o r d i n a t e . There remains a s e tof 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 n t h e o t h e r c o o r d i n a t e o rt h e p a r a m e t e r s n t h e a p p r o x i m a t i n ge x p r e s s i o n sfo r t h e d e p en d e n t

a r i a b l e s . The major t a s k i nd e v e l o p i n g the i n t e g r a l , method i st h e c o r r e c t . d e v e l o p m e n t of t h e c o e f f i c i e n t s i n t h e s e t o fo r d i n a r y- n o t a d i f f i c u l t t a s k , b u t o ne r e q u i r i n gc a re fu lb o o k k e e p in g . The FORMAC* method hasbeenused i n a caseofcomparablecomplexi ty t o r e l e g a t e t h i s t a s k t o t h e computer(M i t ra 1 9 7 0 ) . The o r d i n a r yd i f f e r e n t i a le q u a t i o n sa r es o l v e d f o rt h e paramete rsbyany of a number of s t a n d a r d methods . Knowledgeof t h e p a r a m e t e r s a t each s t e p p e r m i t s c a l c u l a t i o n of th ed e p e n -

. .

2 . 1

v a r i a b l e s a n d of a number of r e l a t e d v a r i a b l e s o f i n t e r e s t .GoverningEquat ionsThe i n d e p e n d e n t and d e p e n d e n t v a r i a b l e sa r e n t r o d u c e d n2 . 1 . The d i m e n s i o n a le q u a t i o n s o r n c o m p r e s s i b l ev i s c o u s

v o r t e x l o w a r e (Bo s s e l 1 9 6 9 ) :

* FORmula W i p u l a t i o n o nComputer, a method developed by IBMfo r n o n -n u m e r i c a l m a n ip u la t io n s . It u s e s PL-1 o r FORTRAN.1 3


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rFig. 2.1 Coordinate system and velocities.

with the boundary conditions:at r = 0: v =at r + 03: u +

NondimensionalX = x/ru = u/u,

C

where the core

0, w = o , u = u (free)ue(x) , wr = k +- ke(x)ax

variables areY = R / 2 =

defined as follows:Re (r /2rc2)2 p = P/(PU, /2)2

define H = VR and K = WR (circulation). These quantities areintroduced into the conservation equations. The pressureiseliminated by cross-differentiation, and two equations remainfor U and K:

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where

aax ayU K ) + [HK - 2Y + 2 = 0Kax + q + y2 a 2 (Y'2 H U - 2 Y E ) = O

Y

0The b o u n d a ryco n d it io n s now become:a t Y = ' O : H = 0 , K = 0 , U = U ( f r e e )a t Y -f a: U -f U e ( X ) , p7R = K -f Ke ( X )a x

The p r e s s u r e a t any p o i n t i n t h e f low i s g iv e n b y

where - - u e 2 = P m + l - U e 2'e - '02.2 I n t e q r a lR e l a t i o n s

A s a f i r s t s t e p i n d e v e lo p in g t h e computa t iona lmethod ,in t e g ra l r e l a t i o n s a r e d e r iv e d f ro m t h e two momentum e q u a t io n s(2.2) . I no r d e r t o o b t a i n a s u f f i c i e n t number o fe q u a t i o n s o rd e t e r m i n a t i o n of he unknown pa ra m et er s n t h e a x i a l v e l o c i t ya n d c i r c u l a t i o n a p p r o x i m a t i o n s t o b e n t r o d u c e db e l o w , h ee q u a t i o n s a r e m u l t i p l i e d by members of two s e t s o fw e ig h t in gf u n c t i o n sgk(Y) and fk Y ) , r e s p e c t i v e l y , and t h e n i n t e g r a t e df o r m a l l y n t h e Y -d i r e c t io n ro m z e r o t o n f i n i t y . To f a c i l i t a t et h e i n t e g r a t i o n , w e r e q u i r e g k (0) f k ( 0 ) = f i n i t e , and gk ( m ) ,f k ( " ) = 0 . A f t e r n t e g r a t i o n by p a r t s , e a r r a n g e m e n t , ands i m p l i f i c a t i o n :

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dX JgkUK d Y - \ gktHK dY - (2gkt'Y + 4 g k ' ) K dY = 0I0 0 . - 00 0

00- , ( f K I I Y 2 + 4 Y f i + 2fk)HU dY0

0

2.3ApproximatingandWeight ingFunctionsT h e i n t e g r a t i o n i s c o m p le t e dw i th h efo l lowingc h o ic e s f o r

w e i g h t i n gf u n c ti o n s and a x i a l v e l o c i t y a n d c i r c u l a t i o n a p p r o x i -mat ions :

-OkYgk(Y) = e k = 1, 2 , . . ., N-a.-Y k = l , 2 , . . . , N + 1k ( Y ) = e K

U ( X , Y ) = (1- e -aY) Nue ( X ) + 2 n(X)e + uaX ( X ) e-aY[ n = lK X , Y ) = W (X,Y) = (1 - e'uy) Ke (X) + 1 (X)e 'nayN[ n = l n 1 ( 2 . 4 )The e x t e r n a l .v e l o c i t y and c i r c u l a t i o n U e ( X ) and Ke ( X ) are

p r e s c r i b e d : h e a , (X) , n ( X ) and th e v e l o c i t y on t h e a x i s U a x ( X )are f r e ep a r a m e t e r s . Major r e a s o n s f o r t h e s e choices (Bossel19704 a re t h e e x p o n e n t i a l c h a r a c t e r o$ t h e f l o w s , ease of

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analytical integration, the fact that the approximations satisfythe boundary conditions, and the observation that the approxi-mations satisfy Weierstrass' approximation theorem after acoordinate transformationof the semi-infinite region 0 5 Y i 00into 1 - q > 0, with = e aY -2.'4 Svstem of Ordinarv.Differentia1 Eauations

Introduction of the 'approximating expressions and theweighting functions (2.4) into equation (2.3) results in thefollowing set of ordinary differential equations for Uaxnd theparameters .an(X) and bn(X)

N N

2 nAn ,k + 'nBn ,k + eaxCk = -6 e k - ieEk - Fkn=l n=lk = l , 2 , . . , N + 1

N N1 n%,k + 1 nEn,kn=l -+ caxek = -fieDk - keEk - Fk-k = 1, 2, . . ., N

The coefficients are given in the Appendix. Only a small partof the coefficient calculation has toe done at each step: themajor part is done only oncet the beginning of the computation.The system ( 2 . 5 ) of (2N f 1) first order ordinary differentialequations is first solved (by Gaussian elimination) for thederivative vector (dan/dX: dbn/dX; dUa,/dx). A standard methodof integration then produces the an, bn,nd U (the Runge-Kutta method has been sed in the present work). Axial velocityU(X,Y), circulation K(X,Y) , and swirl velocity W ( X , Y ) followfrom relations (2.4) .

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2 . 5 I n i t i a lP r o f i l e sI n i t i a l p r o f i l e p a r a m e t e r s a n , b n , and Uax f o r t h e i n i t i a l

a x i a l v e l o c i t y and c i r c u l a t i o n p r o f i l e s ( 2 . 4 ) are r e q u i r e d os t a r t t h ec a l c u l a t i o n . n h ep r e s e n ts t u d y ,s i m p l ee x p o n e n t i a lp r o f i l e s h avebeenused hroughout , i . e . ,

and (Y) = K e ( l - e 0 Y ) o r K e ( l . - e 2 a Y )K i n i t i a lExponents a = 1, andak = k , k = 1, 2 , . . . N + 1were u s ed i na l l o f t h e c a lc u la t io n s .

I n h e c a s e of a r b i t r a r y p r o f i l e s , t h e p a r a m e t e r s a n , b n ,and Uax a re o b t a i n e da so u t l i n e d nB o s s e l (1 9 7 0 a o r b). Theprocedures f o r t h e a x i a l v e l o c i t y p r o f i l e U ( Y ) and t h e c i r c u l a -t i o np r o f i l e K(Y) are i d e n t i c a l .T h u s , i f U ( Y ) i s the( a n a l y t i c a l o r a b u l a t e d ) p r o f i l e o be approximated,UN(Y;an)t h e p r o f i l ea p p r o x i m a t i o n ( w i t h unknown parametersa n ) ,a n df k ( Y ) a c o n v e n i e n tw e i g h t i n gf u n c t i o n , h e n th e N unknownp a ra m e te r s a a re obta ined rom t h e N e q u a t i o n sn

k = 1, 2 , . . ., N2 . 6 AccuracyndConvergence

R e s u l t so f th e presen t me thod were compared w i t h p r e v i o u scomputa t ions Bosse l 1 9 6 7 ) u s i n gp o l y n o m i a l s n Y i n th e approx i -m a t in g u n c t io n s .F or N = 2 t h e agreement was wi th in 3% i n t h ev e l o c i t yp r o f i l e s . n cases where theycanbecompared , thecomputa t ionsa pp ea r t o a g r e e w e l l w i t h h o s e of H a l l ( 1966b)( o n l y a , q u a l i t a t i v e c o m p a r i s o nc a nbe made,however , s ince H a l l

p r e s c r i b e s h eo u t e rs t r e a ms u r f a c es h a p e ) . C r i t i c a l runshave

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b e e n e p e a t e dw i t hd i f f e r e n to r d e r s N o fap p r o x i m a t i o n .F i g u r e s2 .2 and2.3 show t h e v e l o c i t y on t h e a x i s , . swir l p a r am e t e r , an dp r o f i l e d e v e l o p m e n t f o r a breakdown case ( t y p e b ) and N = 1,2 , 3 , 4 , and 5. There i s obvious onvergence a s N i n c r e a s e s .Cases l i k e h e s e . h a v e y p i c a l r u n n i n g times o n h e IBM 360/75of5 t o 1 5 s e c o n d sf o r N = 1 and 2 , and 1 0 t o 30 s e c o n d s o r N = 3t o 5 , A l l c a s e sd i s c u s s e d l a t e r were r u nw i t h N = 3,whicha p pe a re d t o b e a n e f f i c i e n t co mprom isebetw een t h e c o n f l i c t i n gdemandsof h ighaccu r acya n d l o w computing t ime.

O1/.6"ox 1.4 N=I 354 2

0' 1 1 I I I I I0 0.01 0.02 0.03 0.04 0.05 0.06 0.07X

F i g , 2.2 Development of v e l o c i t yo n h ea x i s a n d local s w i r lp a r a m e t e r S ( X ) as a f u n c t i o n o f number ofparamete r su s e d , for i n i t i a l s w i r l p a r a m e t e r Si = 1.425.

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4Si 1.425 I 4

3R

2

0

I N=5b&tween---and- 1

t 0 0.04,0 02 0.4 0.6 0.8 1.oU

3R

2

1

0 0 02 0.4 0.6 0.8 1.oWFig: 2.3 V e lo c i typ ro f i l ed e v e l o p m e n t as a f u n c t i o n of numberof paramete rs used i n th e p r o f i l e a p p r o x i m a t i o n .

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III. SINGULARITIES AND CRITICAL SWIRLPARAMETERS

3 .1S in g u la r i t i e so f h eG o v e r n i n gS y s t e mThe system of 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 ( 2 .5 ) becomes

s i n g u l a r when th e d e t e r m i n a n t o f c o e f f i c i e n t s . f o r th e unknownsa n ? bn*and Uax v a n i s h e s , and th e i g h t -h a n ds i d e . r e m a $ n snonzero. The s p e c i a l case when t h e r igh t-han d s ide v a n i s h e sa l s o h a s some s i g n i f i c a n c e and i s c o n s i d e r e d l a t e r .

. A t a p a r t i c u l a r a x i a l p o s i t i o n X t h e p r o f i l e s h a p e s( r e l a t i o n s 2 . 4 ) are determined by t h e l o c a l an (Xo), bn(Xo) , and

'ax (Xo). Without loss i n g e n e r a l i t y w e s h a l l assume Ue = 1 andk e e pon ly Ke as a f ree p a ra m e te r . T h i s amounts t o i n v e s t i g a t i n ga g i v e n a x i a l v e l o c i t y p r o f i l e and a g i v e n c i r c u l a t i o n p r o f i l eshape a t d i f f e r e n tv a l u e s of e x t e r n a l c i r c u l a t i o n Ke. Thec o n d i t i o n f o r v a n i s h i n g o f t h e d e t e r m i n a n t of c o e f f i c i e n t s o nt h e l e f t - h a n d s i de of system (2.5) i s r e p r e s e n t e d b y a po lynomia le x p r e s s i o n of th e form

f 2 nGn (K e 1 = 0 (3 1)n=Ow i t h N r o o t s f o r c h a r a c t e r i s t i c e x t e r n a l c i r c u l a t i o n v a l u e s(K e 1 . The G n a r en u m e r i c a l o e f f i c i e n t s .T h u s , t h e r e a re N

d i s t i n c t v a l u e s , o f Ke2 which l e a d t o s i n g u l a r i t i e s , and f o rN -+ Q) t h e f o l l o w i n g r e s u l t s o b t a i n :

2

For a g i v e n a x i a l v e l o c i t y p r o f i l e and g i v e ns h a p e of t h ec i r c u l a t i o np r o f i l e , h ee q u a t i o n s of q u a s i - c y l i n d r i c a lv i s c o u si n c o m p r e s s i b l ev o r t e x l o w ( 2 . 1 ) have a c o u n t a b l y n f i n i t e s e tof s i n g u l a r i t i e s f o r d i s c r e t ev a Z u e s of e x t e r n a l c i r c u l a t i o n ,1% I

It shou ld be s t r e s s e d t h a t t h e s i n g u l a r i t i e s a re a f u n c t i o nof t h e l o c a l p r o f i l e s o n l y a nd n de pe nd en t of l o c a ld i e n t s of e x t e r n a l a x i a l v e l o c i t y and c i r c u l a t i o n .

a x i a l g r a -However, t h e

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p r o f i l e d e v e l o p m e n t t s e l f is, of course ,d e p en d e n t on e x t e r n a lg r a d i e n t s .

The e f f e c t o f g r a d i e n t s o f e x t e r n a l a x i a l v e l o c i t y Ue andc i r c u l a t i o n Ke ( i . e . dUe/dX and dKe/dX)becomes ob vi ou s i f t h es p e c i a l case i s c o n s i d e r e dw h e r e h er ig h t -h a n d s ide o fe q u a t i o n s(2.5) v a n i s h e s o g e t h e r w i t h h e d e t e r m i n a n t o f c o e f f i c i e n t s .System ( 2 . 5 ) i s t h e nnons ingu la r . The r igh t -han d s i d e i s al i n e a r f u n c t i o n o f t h e g r a d i e n t s d U e / d X and d K e / d X and thep a ra m e te r s a and Uax i n t h e a x i a l v e l o c i t y and c i r c u l a t i o np r o f i l e s .F o re a c hp r o f i l ec o m b i n a t i o n ( i . e . U e , K e , a n , b n ,'axg r a d i e n t s d U e / d X and dKe/dX e x i s t s f o r which th e r i g h t - h a n ds i de vanishesand he ys tem ( 2 . 5 ) i s n o n s i n g u l a r , i . e . :

n ' b n ') a p a r t i c u l a rc o m b in a t io n Of e x t e r n a l a x i a l and c i r c u l a t i o n

A s i n g u z a r i t yc a nb ea v o i d e d by a p p l i c a t i o n of a p p r o p r i a t ee x t e r n a la x i a i !a n dc i r c u l a t i o ng r a d i e n t s .

T h i s r e s u l t i s s u pp o rt ed by t h e r e s u l t s of computa t ionsf o r d i f f e r e n t e x t e r n a l a x i a l . and c i r c u l a t i o n g r a d i e n t s d e s c r i b e di n t h i s r e p o r t .

The two resultsd e r i v e da b o v ea r e n d e p e n d e n to f h ep a r t i c u l a ra p p r o x im a t io n and w e ig h t in g u n c t io n sc h o s e n ( i . e .e x p r e s s i o n s ( 2 . 4 ) ) s ince an ycon t inuous unc t ion U ( Y ) o r K(Y)c anbe ep resen ted by an i n f i n i t e se r i e s of l i n e a r l y n d e p e n d e n tf u n c t i o n s { $ n ( ~ ) lorming a complete s e t .

It becomes evident f rom the a n a l y s i s h a t h e ( swir l -d e p e n d e n t ) s i n g u l a r i t i e s o f t h e q u a s i - c y l i n d r i c a lv o r t e xe q u a t i o n sa r e d i f f e r e n t i n c h a r a c t e r f r o m h e s e p a ra t i o n s i n g u la r i ty o f h eb o u n d a r y a y e re q u a t i o n s . np a r t i c u l a r , i n c o n t r a s t o b ou ndaryl a y e r s o l u t i o n s , t h e r e e x i s tm e a n i n g f u l v o r t e x s o l u t i o n s f o rswir l v a l u e sb e t w e e n h es i n g u l a rv a l ue s , and e s p e c i a l l y f o r aswirl v a l u eg r e a t e r h a n t h a t o f h e f i r s ts i n g u l a r i t y . F u l lv o r t e xc o m p u t a t i o n s (Secs. I V and V ) haveshown,however , tha tt h e s e s o l u t i o n s w i l l n o r m a l l y o n l y p e r s i s t f o r a f i n i t e d i s t a n c e ,b e f o r e h e y a re d r i v e n t o t h e n e x t s i n g u l a r i t y .

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3 .2 Sw ir l a ram ete rsI n c o m p a r i n g d . i f f e r e n t v o r t e x f l o w s , Ke i s n o t a v e ry

meaningfulparameter .Since he xisymm etricbreakdownbehavioro fvor texf lo w s i s e v i d e n t l y c o n t r o l l e d by c o n d i t i o n s a t t h everyc o re Sa rp k a y a 1 9 7 1 a , b , O r lo f f a n dBosse l 1 9 7 1 ) , a s w i r lparameter i s in t roducedwhich i s based on corec o n d i t i o n s , i . e .t h e swir l v e l o c i t y p r o f i l e g r a d i e n t on t h e a x i s , t h e a x i a lv e l o c i t y on t h e a x i s , and t h e c o r e r a d i u s Rw max where t h es w i r l v e l o c i t y W r e a c h e s i t s maximum value:

T h i s d e f i n i t i o n of t h e s w i r l p a r a m e t e r r e d u c e s o h e f a m i l i a rone S = RRc/U = Wc/U f o r a c y l i n d e ro f f l u i d o f r a d i u s Rc i nr i g i d o t a t i o n R , w i t hu n i f o r ma x i a lv e l o c i t y U . I n v i s c i ds o l u t i o n s o rs u c h l o w s F r a e n k e l1 9 5 6 )h a v ec r i t i c a lv a l u e sof t h e s w i r l p a ra m e te r a t j ln /2 = 1.9159,3.5078,5.0867,6 . 6 6 1 8 , 8.2353, ..., where j ln a r ez e r o s of t h e Bessel f u n c t i o n3 . 3S i n g u l a r i t i e s o r n i t i a l l y Uniform A x i a l Flow

F i g u r e 3 . 1 p r e s e n t s h e f i r s t f i v e s i n g u l a r i t i e s f o r t h ev o r t e x p r o f i l e s

U ( Y ) = 1 = c o n s t a n t

as a fu n c t io n o f s w i r l p a ra m e te ran dorder of approximat ionused .T h i s l o wa p p r o x i m a t e s r i g i d o t a t i o n n h ec o r e . The s i n g u l a r -i t i e s were o b t a i n e d b yc o m p u t i n g h ed e r i v a t i v e s in, n , fiaxo v e r a wide angeof Ke. Each add i t iona la p p r o x i m a t i n g t e r maccoun ts for a new s i n g u l a r i t y , a s e x p e c t e d ,w h i l e h e o c a t i o nof p r e v i o u s l y c o m p u t e d s i n g u l a r i t i e s i s o n l y s l i g h t l y a f f e c t e d .

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10 -8 -

6 -sait

4 -

O ss

O s4a

-0-0 s,

' t -0-0-0-0 s,s," " - x " " - n - " " ~ " " - ~&

rgid rotationiqvisciiF i g .3 .1S i n g u l a r s w i r l p a r a m e t e rv a l u e s o r U(Y) = 1, andK ( Y ) = WR = S (l-e'Y)/O .792 as a f u n c t i o n of o r d e rof pproximat ion N. ( I n v i s c i dv a l u e sa r e o r i g i db o d y r o t a t i o n . )

Fig. 3.20 1 2 3 4 S s 6 7 8 0 10 11

Ma g n i tu d eofg rad ien t of v e l o c i t y on t h e axis asf u n c t i o no f swir l p a ra m e te r .U n i fo rmax ia l lowU ( Y ) = 1, and K ( Y ) = WR = S ( l - e - Y ) / 0 . 7 9 2 .

2 4


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Likewis.e , t h e u s e of d i f f e r e n t OL o r w e i g h t i n g f u n c t i o n e x p o n e n t sk has o n l ym i n i m a le f f e c t o n the l o c a t i o n of th e lower o r d e r 's i n g u l a r i t i e s . np a r t i c u l a r , h es i n g u l a r i t y of g r e a t e s ti n t e r e s t , S1, r e m a i n sp r a c t i c a l l yc o n s t a n t .S i n c e t he re i sample evidence t h a t t h i s f i r s t s i n g u l a r i t y i s a s s o c i a t e d w i t ha x i s y q tm e t r i cvor texb re a k d o w n ,c o m p u ta t io n sfa i l i .ng a t s i n g u l a r -i t i e s , and i n p a r t i c u l a r a t S1, w i l l a l s o be s a i d t o 'b reak do wn "i n t h e f o l l o w i n g work. The c r i t i c a l s w i r l v a l u e s j1J2 asfound from t h e h e o r y of i n v i s c i d r i g i d r o t a t i o n ( F r a e n k e l 1 9 5 6 )' a re a l s o shown inF i g .3 . 1 . T h i s f i g u r e a l s o g i v e s h e swirlparamete r So d i v i d i n g f l o w whichdecayssmooth ly type l a )from f l o w which i s a b r u p t l y decelera ted on t h e a x i s and ' b reaksdown' because S1 is r e a c h e d ty p e l b ) . The t h e o r e t i c a l( i n v i s c i d )v a l u e (Bossel 1968) fo ru n i f o r m f l o w n n i t i a l l yr i g i d r o t a t i o n i s So = fi. The v i scousv a l u e o r So i s foundfrom a f u l lv o r t e xc o m p u t a t i o n (see Sec. I V ) . There i s s t r i k i n gagreementbetween t h e v i s c o us and i n v i s c i d v a l u e s f o r So, S1's 2 , ... .

'*3.4 ~Behavior of A x i a l D e r i v a t i v e s

A t each S i n g u l a r i t y , a l l a x i a l d e r i v a t i v e s jump t o i n f i i l i t yi n m ag ni tu de n d r e v e r s es i g n s . T h i s r e s u l t s nc o n t r a s t i n gb e h a v i o r f o r f l o w s s e p a r a t e d by a s i n g u l a r i t y . The a x i a ld e r i v a t i v e of t h e v e l o c i t y on t h e a x i s f o r t h e v o r t e x w i t h uni-form a x i a lf l o w ' i s p l o t t e d n F i g . 3.2 a s a f u n c ti o no f s w i r lp a ra m e te r S. T h e s i n g u l a r i t i e s a re v e r y v i d e n t . N ote e s p e c i a l -l y t h a t t h e . s i g n of th e a x i a l g r a d i e n t r e v e r s e s a t eachs i n g u l a r i t y . np a r t i c u l a r ' , o r a swir l va luebe low S1, thea x i a l f l o w i s dece le ra ted i n t h e c o r e , w h i l e f o r a swir l v a l u eS1 < S < S 2 t h e a x i a l f l o w i s a c c e l e r a t e d n t h e i n n e rc o r e .3.5 E f f e c t of V e l o c i t y P r o f i l e

The c r i t i c a l s w i r l p a ra m e te r s So, S1, S 2 , ....are f u n c t i o n s25


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of the a n , bn ' U a x r and Ue and a re t he r e fo re p ro f i l e -d e p e n d e n t .The examplegiven , which approximates r i g i d r o t a t i o n 4 n i t si n n e rc o r e ,a p p e a r s t o c o n f i r m h e e a r l i e r c o n t e n t i o n (Bossel1 9 6 7 , 1968) t h a t t h eb e h a v i o ro fv i s c o u sv o r t e xf l o w s i s governed

, m a i n l yb y n v i s c i dc o r ep r o p e r t i e s . T h e e f f e c t ofnonuniformi n ' i t i a l a x i a l f l o w on the c r i t i c a l s w i r l p a r a m e t e r s i s showni n F i g . 3 . 3 f o r t h e f a m i ly o fp r o f i l e s

K ( Y ) = WR = K e ( l - e-aY)Flows with Ua x < 1 a r eo f t h e t r a i l i n g v o r t e x t y p e , w i t h av e l o c i t y d e f i c i t on t h e a x i s , w h i l e f l o w s w i t h U > 1 approx i -m a t e h e e a d i n ge d g ev o r t e x w i t h a v e l o c i t y e x c e s s on the a x i s .ax.

F i g u r e 3 . 3 shows the samev a r i o u sl o w s . The magnitudes

q u a l i t a t i v e b e h a v i o r f o r t h eof t h e c r i t i c a l swir l v a l u e sf o r

141 %-\

Fig. 3 . 3

10 -s -

8 -

6 -

4 -

2 - """""E f f e c t of v e l o c i t y p r o f i l e on the c r i t i c a l swir lv a l u e s . U ( Y ) = 1 + e - y ( U a x - l ) , K ( Y ) = S (1-e-y) /O .792.

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t h en o n u n i f o r mprof i le s show a p r o f i l e e f f e c t : h e y a r ec o n s i s t e n t l y h i g h e r f o r t h e p r o f i l e w i t h i n i t i a l a x i a l v e l o c i t yd e f i c i t , and c o n s i s t e n t l y l o w e r f o r t h e p r o f i l e w i t h a x i a l e x c e s snear he axis. T h i s e s u l t i s t o beexpected. The av era geveloci tyneart h e axis i s most ce r t a in ly igh er h a n n h e r a i l i n g o r t e x a s edue t o t h e e f f e c t o f f as te r f low i n a neighborhood of th e axis. I n h e c a s eof the ead ingedgevor tex , lu idsurrounding he axis is slower hanUax and thea v e r a g ev e l o c i t yn e a r h e exis would be l e s s . If propera v e r a g eve loc i t i e s were used in he ca l c u l a t io n of S , h e i n e s ofc r i t i c a l s w i r l parameters would be more n ea rl y ho riz on ta l .

S i n g u l a r i t y a n a l y s e s c a n be p er fo rm e d f o r a n yg i v e na r b i -t r a r y e x p e r i m e n t a l )a x i a l a n d s w i r l v e l o c i t yp r o f i l ec o m b i n a t i o ni n o r d e r t o o b t a i n i n f o r m a t i o n on l ik e ly f l o w b e h a v i o r , i np a r t i c u l a r h ep r o x i m i t yo f S1. T h e c o e f f i c i e n t s a n , bn, andUax i n an N-th o r d e rp r o f i l ea p p r o x i m a t i o n ( 2 . 4 ) a r e f i r s tde te rminedby hemethod of B o s s e l (1970a). The c r i t i c a l s w i r lv a l u e s a re t h e n e i t h e r o b t a i n e d by d e t e r m i n i n g t h e c o e f f i c i e n t sGn i n h e c h a r a c t e r i s t i c e q u a t i o n ( 3 .1 ) a n ds o l v i n g o r h ef i r s t N c r i t i c a l Ke ( o r S o ) , o r by d e t e r m i n i n g h eg r a d i e n t sd/dX o f h ean and bn from ystem ( 2 . 5 ) as a func t iono f Ke ( o rs ) .3.6 Ass essm ent f Breakdown Be ha vio r

The r e s u l t s o f t h i s s e c t i o n o f f e r s e v e r a l p o s s i b i l i t i e so f a s s e s s i n g h e p r o b a b l eax is ym me tr ic b reakdown beh av io r ofv o r t e xv e l o c i t yp r o f i l e s . n h eo r d e ro f n c r e a s i n ga c c u r a c y :

(1) Comparison f t h e swir l v a l u e

w i t h t h e i n v i s c i d c r i t i c a l v a l u e s f o r r i g i d , r o t a t i o n , i np a r t i c u l a r So = fi ( d i v i d i n gs t a g n a t i n g (S > So) from

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wake-type vo r t ex lo w ( S < So)) and S1 = 1 . 9 1 5 9( d i v i d i n gs u p e r c r i t i c a l ( S < S1) f r o m s u b c r i t i c a l

v o r t e xf l o w (S > S1), w i th a r e v e r s a l of b e ha v io r a tsl)

( 2 ) Comparisonof t h e computed s w i r l v a l u ew i t h h e e s u l t sof F i g . 3 . 3 .( 3 ) A p p r o x i m a t i o no f h ev e lo c i typ ro f i l e s b yparameters

a n t b n t U e t U a X t Ke u s in g h e m ethod in B o s s e l (1970a)f o r n i t i a lp r o f i l ea p p r o x i m a t i o n .U s i n g h ea p p r o a c hof t h e p r e s e n t s e c t i o n , h e a x i a l g r a d i e n t s a r e t h e ncomputedand t h e a c t u a l s i n g u l a r i t i e s d e t e r m i n e d b yv a r y i n g Ke o r t h e swir l S .

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I V . .VORTEX COMPUTATIONS FOR CONSTANT EXTERNAL AXIAL VELOCITYAND CIRCULATION

T h i s s e c t i o n p r e s e n t s r e s u l t s o f v o r t e x c o m p u t a t i o n s f o rc o n s t a n te x t e r n a la x i a lv e l o c i t y and c i r c u l a t i o n .T h r e ed i f f e r e n t i n i t i a l v e l o c i t y p r o f i l e s were used i n t h i s . a n d i n t h ef o l l o w i n g s e c t i o n ,c o r r e s p o n d i n g t o i n i t i a l l y . u n i f o r m a x i a l f l o wand tof l o w so f e a d i n ge dg e and t r a i l i n g v o r t e x y p e , r e s p e c -t i v e l y . D i a g r a m so f d i s t a n c e Xf t o f a i l u r e of t h ec o m p u t a t i o na s f u n c t i o n o f i n i t i a l s w i r l p a ra m e te r w i l l f i r s t b e p r e s e n t e df o r h e s e h r e e l o w y p e s . The d ev elo pm en t of t h ev e l o c i t y o nt h e a x i s and o f t h e v e l o c i t y p r o f i l e s w i l l b e d i s c u s s e d i n somed e t a i l . The r e s u l t sc o n f i rm h es i n g u l a rb e h a v i o rp r e d i c t e d nSec. 111.

4 . 1 ___n i t i a l- P r o f i l e s '. ~. and D i s t a n c e t o F a i l u r eUni form { n i t i a Z ax iaZ fZou

The case w h e r e h e n i t i a l a x i a l v e l o c i t y p r o f i l e i s uniformw h i l e h e s w i r l v e l o c i t y p r o f i l e i s l i n e a r w i t h R i n t h e v i c i n i t yof t h e a x i s p r o v i d e s a good t e s t o f t h e a s s e r t i o n t h a t v o r t e xb e h a v io r i s g o ve rn e d by t h e n v i s c i d p r o p e r t i e s o f t h e r i g i d l yr o t a t i n g c o r e a s c o n d i t i o n s on t h e a x i s a r e h e n d e n t i c a l i nb o t hc a s e s . n i t i a lp r o f i l e s B u r g e r s 1 9 4 0 ) f o r h i sc a s e were

U ( Y ) = 1 = c o n s t a n tK (Y> = W ( Y ) R = K~ (1- e-Yy)

Most c o m p u ta t io n sp re s e n te dh e re were o b t a i n e d f o r y = 1.y = 2 was u se d o t e s t s c a l i n g e f f e c t s on t h e r e s u l t s o f t h ecomputa t ion . The d i f f e r e n c e s were found t o be minimal . The r e l a -t i o n sh i p b e t w e e n e x t e rn a l c i r c u l a t io n and i n i t i a l s w i r l c a n becomputed from t h e n i t i a l p r o f i l e s . For y = 1

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(dW/dR) Ra x c -s = - . .792 K /U = .792 Ke%X e . a x

and f o r y = 2S = 1 . 1 2 1 Ke/Uax = 1 . 1 2 1 Ke

F i g u r e 4 . 1 p r e s e n t s , a s a f u n c t i o n of i n i t i a l swir l p a ra -meter a n d f o r N = 3 , t h e d i s t a n c e o f s u c c e s s f u lc o m p u t a t i o nb e f o r e a i l u r eo c c u r r e d a t X f . T h i s p l o t shows s e v e r a l d i s t i n c tr e g i o n s s e p a ra t e d by s i n g u l a r p o i n t s i n a c c o r d a n c e w i t h t h er e s u l t so f Sec. 111. C o n t i n u o u s o l u t i o n sc o u l do n l y be foundi n r e g i o n l a , c o r r e s p o n d i n g o a n n i t i a l s w i r l p a ra m e te rS 5 fi. T h e c h a r a c t e r so f h es o l u t i o n s n h ed i f f e r e n t e g i o n sw i l l be d i s c u s s e d more f u l l y n h e n e x ts e c t i o n . A s p o i n t e do u t i n Sec. 111, t h ec o m p u t a t i o n w i t h N = 3 o n l yp i c k s u p t h ef i r s t h r e es i n g u l a r i t i e s . The r e s u l t s a r e t h e r e f o r eo n l ya c c u r a t e f o r S 5 S2..

00yo0

000

/0

0 0.02 0.04 0.06.08 0.0 0.12 0.14 0.16 0.18 0.20xfs w i r l p a r a m e t e r . n i t i a l l yu n i f o r m a x i a l f l o w .Open c i r c l e s d e n o t ef a i l u r eo fc o m p u t a t i o n .

Fi.g. 4 . 1 D i s t a n c e t o f a i l u r e ( a t X f ) a s f u n c t i o n of i n i t i a l

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L e a d i n g e d g e v o r t e xExper imen ta l ( H u m m e l ' 1965)a n d t h e o r e t i c a l ( H a l l 1 9 6 1 )

r e s u l t s o r e a d i n ge d g ev o r t e x l o w s a sg e n e r a t e d a t t h el e a d in ge d g e s of d e l t a w i n g s , f o r e x a m p l e ) all show a l a r g ee x c e s s o fa x i a lv e l o c i t yn e a r h e a x i s o v e r h e f r e e s t r e a ma x i a lv e l o c i t y . It i s o f some i n t e r e s t o see w hat e f f e c t , fa n y ,t h i sv e l o c i t ye x c e s sh a s on t h ev o r t e xb e h a v i o r . The e f f e c t o ft h e n o n u n if o r m s h a p e o f h e a x i al v e lo c i t y p r o f i l e on t h ec r i t i c a l s w i r l v a lu e sh a sb e e nd i s c u s s e d i n Sec. 111.

The i n i t i a l v e l o c i t y p r o f i l e s i n v e s t i g a t e d w e r e( i . e . uaX = 2 . 0 , ue = 1.0)

Th e r e l a t i o n s h ip b e t w e e n Ke and S f o r h e s e p r o f i l e s i ss = . 7 9 2 Ke/Uax = .792 K e / 2 = . 3 9 6 Ke

T h i sc h o ic e i s n o t o o goodan a p p r o x i m a t i o n o h ee x p e r i m e n t a land t h e o r e t i c a l p r o f i l e s of H u m m e l and H a l l , b u t i t s h o u l ds e rv et o d e m o n s t r a t e h e e f fe c t o f h ig h e r a x i a l c o r e v e lo c i t y .

Dis tances t o f a i l u r e of t h e computa tion a t X f for N = 3are shown i n F i g . 4 . 2 a s a f u n c t i o n o f n i t i a l s w i r l p a ra m e te r .The behavior i s q u a l i t a t i v e l y h e same a s f o r h e c a s e o f n i t i a lu n i fo rma x i a l l o w F i g u r e 4 . 1 ) . The c r i t i c a l s w i r l v a l u e sa r enow somewhat d i f f e r e n t from thep r e v i o u sca se . They havebeenmore a c c u r a t e l yd e t e r m i n e d ,w i t h N = 5 , i n Sec. 111.T ' r a i Zing v o r t e x

The t r a i l i n g v o r t e x , a s foundd ow ns tre am o f a s t r a i g h twing ,h as a w a k e - l i k e a x i a l v e l o c i t y d i s t r i b u t i o n w i t h a c o r ev e l o c i t ys m a l l e r h a n h e f r e e s t r e a m v e l o c i t y( B a t c h e l o r 1 9 6 4 ,McCormick et a l . 1 9 6 8 ) . T h i sv e l o c i t y e t a r d a t i o nn e a r h ea x i s canbee xp ,e ct ed t o l e a d t o e a r l i e r v o r t e x s t a g n a t i o n and

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- @$"-

"- # 0(. 0 --5 - --/#--0""!+p?"- " -""" - 0 " -S p L "_"""" -"""./-"" 0 "_"" 8"""""0"-

0 breakdown-stableS r " ~ " 0 - /1b -+a ---- n-SO- - ~ ." "-0' I I I 1 I 1 I I I 10 0.02 0.04 0.06 0.08 0.10 0.12 0.14 0.16 0.18 0.20

. ". . . .

Fig . 4 . 2

breakdown

XfD i s t a n c e o a i l u r e ( a t X f ) a s u n c t i o no f n i t i a ls w i r l p a r a m e t e r . n i t i a lp r o f i l e so f e a d i n ge d g evo r t ex yp e . Open c i r c l e s d e n o t e a i l u r eo fcomputa t ion .

The i n i t i a l p r o f i l e chosen was( i . e . uax = 0 . 5 , ue = 1 . 0 )

For t he s e p r o f i l e s , e x t e r n a l c i r c u l a t i o n and s w i r l v a l u e s a r er e l a t e d by

S = . 792 Ke/Uax = . 7 9 2 Ke/0.5 = 1.584 Ke

T h i sc h o i c ea pp ea rs o come f a i r l y c l o s e t o a c t u a l t r a i l i n gvo rt ex lo w s (McCormick e t a l . 1 9 6 8 ) .D i s t a n c e s t o f a i l u r e of t h e c o m p u ta t io n a t X f f o r N = 3 a r e

shown i n F i g . 4 . 3 a s a f u n c t i o n o f h e n i t i a l s w i r l p a ra m e te r .Again the b e h a v i o ro f h e s o l u t i o n s i s q u a l i t a t i v e l y t h e same a sf o r h e t w o p r e v i o u s cases. T h e a c t u a l c r i t i c a l s w i r l p a ra m e te r

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1 I I Io 0.02.04 006 0.m 0.10 0.12 a14 0.16 0.m azox1

3 D i s t a n c e o a i l u r e a t X f ) a s f u n c t i o no f n i t i a ls w i r l p a r a m e t e r . n i t i a lp r o f i l e so f r a i l i n gv o r t e xtype. Open c i r c l e s d e n o t e a i l u r eo fc o m p u t a t i o n .F i g . 4 .

v a l u e s d i f f e r som ewhat fro m th e o th e r two c a s e s a nd a g a in a r en o ta sa c c u r a t ea s h o s ef o u n d nS e c . I11 f o r N = 5 , e s p e c i a l l yfor S > S2 .PossibiZity of b r e a k d o w n - s t a b l e s o l u t i o n s of t y p e 2, 3, 4, . . .

N o b r e a k d o w n - s t a b l es o l u t i o n so f y p e 2 , 3 , o r 4 w e r efound i n t h e p r e s e n t i n v e s t i g a t i o n f o r c o n s t a n t e x t e r n a l a x i a lv e l o c i t y and c i r c u l a t i o n . T h i s a g r e e sw i t h h e r e s u l t s o f t h es i n g u l a r i t y a n a l y s i s i n Sec'. I11 where it became e v i d e n t t h a to n l y s p e c i f i c e x t e r n a l a x i a l v e l o c i t y a n d c i r c u l a t i o n g r a d i e n t sc anp re v e n tf lo w s of t h e s e ' t y p e sf r o m a p p r o ac h i n g a s i n g u l a r i t y .It w i l l a g a i nb e shown i n Sec. V i n s e v e r a l exam ples t h a ta p p l i c a t i o n o f e x t e r n a l a x i a l v e l o c i t y o r c i r c u l a t i o n g r a d i e n t sc a n h a v e b e n e f i c i a l e f f e c t s on f lowd ev el op me nt f o r t h e s e t y p e s .There i s some othere x p e r i m e n t a la n da n a l y t i c a le v i d e n c e(D o n a ld s o na n dSu l l i v a n 1 9 6 0 ) s u g g e s t in gb re a k d o w n -s t a b l es o lu -t i o n s o f h e m u l t i - l a y e r e d y p e .

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4 . 2 V e l o c i t y on theA x i sThe behavioro fvor texf low i s d i f f e r e n t f o r e a c h o f t h e

r e g i o n s l a , lb, 2 , 3 , ... . The m o s t m p o r t a n t n d i c a to ro fv o r t e x b e h a v i o r i s a p l o t of v e l o c i t y on t h e a x i s Uax vs.d i s t a n c e X. The deve lopmento f h ev e lo c i ty on th e a x i s w i l lnow b e d i s c u s s e d f o r f l o w s w i t h c o n s t a n t e x t e r n a l a x i a l v e l o c i t ya n d c i r c u l a t i o n .B e f o r ed e s c r i b i n g ng r e a t e rd e t a i l l o w so ft y p e 1 h a v i n g d i f f e r e n t i n i t i a l p r o f i l e s , b e h a v i o r o f i n i t i a l l yu n i f o r m a x i a l f l o w so f h e f i r s t f i v e y p e s w i l l be d i s c u s s e d .A l l r e s u l t s were computed with N = 3 .Behavior of initially uniform axial f l o w s of t y p e s 1-4

F i g u r e 4 . 4 p r e s e n t s th e d e v e lo p m e n to fv e l oc i ty on t h e a x i si n t h e f i v e e g i o n s l a , lb , 2 , 3 , and 4 . Each of thec u r v e s i sr e p r e s e n t a t i v eo fo t h e r l o w s n h e same reg ion .Fo r l o w so fa g i v e n y p e , h ec h a r a c t e ro f h e s ec u r v e s ,a n d of t h e v e l o c i t yp r o f i l e s , e m a i n s t h e same. Type la , heo n l yb r e a k d o w n - s t a b l e4 r

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-2F i g . 4 . 4 Developmento f v e l o c i t y on t h ea x i s o rd i f f e r e n tf l o w y p e s . n i t i a l l y n i f o r m x i a l l o w . Open

c i r c l e s d e n o t e f a i l u r e p o i n t s .

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f l o w ,d e v e lo p s a re du cc v e l o c i t y on th ea x i s w h ic h e m a in sa p p r o x i m a t e l y c o n s t a n t f o r a c o n s i d e r a b l e d i s t a n c e b e f o r e i ta g a i na p p r o a c h e s h e freestream v a lu e . Type l b f l o wh asgradients (dUax/dX) which area l w a y ss t e e p e r h a na n y y p e ag r a d i e n t s .T h e s eg r a d i e n t s n c r e a s e , and t h e o m p u t a ti o nt e r m i n a t e sd u e t o e x c e s s i v e g r a d i e n t s , w i t h Uax i n h e n e i g h b o r -hood of 0 . 5 . A t t h e same t i m e , t h es t r e a ms u r f a c ea n g l e ss u d d e n l y n c r e a s e o a r g e p o s i t i v e v a l u e s , n d i c a t i n g a ne x p l o s i v eb e h a v i o ro f h e core . Gra d ien ts become la r g e r , andt h e c a l c u l a t i o n b r e a k s down in c re a s in g ly s o o n e r , as t h e c r i t i c a ls w i r l v a l u e S1 i s approached.

For s w i r l v a l u e s g r e a t e r h a n S1, s o l u t i o n s of t y p e 2 a r eo b ta in e d ,w h ic h now show a n a c c e l e r a t i o n of t h e v e l o c i t y on t h ea x i s .G r a d ie n ts (dUax/dX) a r ev e r y a r g e o r S n e a r S1, b u td e c r e a s e as S i s i n c r e a s e d , e a d i n g o o n g e rc a l c u l a t i o n sb e f o r ef a i l u r eo c c u r s .F a i l u r e of t y p e 2 i s anmplos ive ne : t reams u r f a c e a n g le s assum e l a rg e n e g a t i v e v a l u e s v e r y q u i c k l y n h eim me diate v i c i n i t y o f h e a i l u r ep o i n t .G r a d i e n t sd e c r e a s ef u r t h e rw i t h c o r r e s p o n d i n g l y o n g e r r e s u l t i n gc o m p u t a t i o n s ,u n t i l t h e v i c i n i t y o f S2 i s r e a c h e d ,w h e reaga in no s o l u t i o n sca n b eo b t a i n e d .

For s w i r l S > S2, t h e l o w i s a g ai n of d i f f e r e n t y p e 3 .The v e l o c i ty on th e a x i s now a g a i nd e . c e l e r a t e s ,e v e n t u a l l yr e a ch i n g a s t a g n a t i o np o i n t on t h ea x i s . Downstream of t h i sp o i n tr e v e r s e df l o wd e v e l o p s o nand near t h e a x i s , g r a d i e n t ss t e e p e n and c om pu ta tio n f a i l s w i t h e x p l o s i v e b e h a v i o r o f t h eo u t e rc o r e . As t h e s w i r l i s f u r t h e r n c r e a s e d ,g r a d i e n t sa r ea g a i n r e d u c e d ,a n d h e c o m p u t a t i o n s f a i l f a r t h e r- d o w n s t r e a mu n t i l h e v i c i n i t y o f S3 i s r e a c h e d . N o s o l u t i o n sc a nbe ob-t a i n e d i n t h e immediate neighborhoodof t h i s p o i n t .

For s w i r l S . S3, th ef lowh a so n c ea g a inc h a n g e d i t st y p e 4 f l o w ) . The v e l o c i t y on t h ea x i sa g a i n e n d su n t i l g r a d i e n t s a g a i n s t e e p e n r a p i d l y , l e a d i n g . t o

b e h a v i o r t ot o n c r e a s e ,

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l a r g e n e g a t i v e stream su r f a c e a n g l e s i n - t h e o u t e r core r e g i o n s( im p lo s iv eb e h a v io r ) ,a n d t h e c o m p u t a t i o nf a i l s .

As S i s i n c r e a s e d , more and more s i n g u l a r i t i e s a re encoun-t e r e d . The v e l o c i t y on t h e a x i s a l t e r n a t e s b e t w e e n a d e c e l e r a t i n gten den cy fo r odd type low)andanacce le ra t ing endency ( f o reven type f l o w ) .I n i t i a l l y uniform axial f l o w of t y p e 1

Pro b a b ly h eo n ly v o r t e x f lo w o fm a j o rp h y s i c a l s ig n i f i -cance i s t y p e 1 flo w. The develop ment of i t s v e l o c i t y o n thea x i s and i t s s w i r l paramete r S (X ) w i l l now b e g i ve n n mored e t a i l f o r th e v o r t e x w i t h i n i t i a l ly u n i f o r m a x i a l f l o w .

F i g u r e 4.5 p r e s e n t s r e s u l t s f o r d i f f e r e n t c h o i c e s o f n i t i a lswirl . p a r a m e te r Si. It a l s o g i v e s the s w i r l parameter S (X) f o rtwo n eighb or ing low s of t y p e l a and lb . A swi r l parameterva lueofapprox imate ly Si = 1 . 4 s e p a r a t e s t h e s mo oth lydecayingv o r t e x l o w s t y p e a : Si 5 1 . 4 ) f romvor tex lows which

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e v e n t u a l l ybr e ak dowr- ( ty p e b : Si 2 1 . 4 ) . N o t e t h a t thet h e o r e t i c a l i n v i s c i d v a l u e f o r t h e i n i t i a l s w i r l v a l u e e a d i n gt o s t a g n a t i o n i s Si > fi = 1 . 4 1 f o r r i g i d r o t a t i o n ( B o s s e l 1 9 6 8 ) .

Type l a f l o w a d j u s t s t o a n a l m o s t c o n s t a n t v a l u e o f a x i a lv e l o c i t y and th e r e a f t e rb e h a v e s n a wake-likemanner. Thes w i r l paramet e r S X ) g e n e r a l l yd e c r e a s e s . T h e s w i r l stillc o u n t e r a c t s h en o r m a ls p r e a d i n go f h ew a k e ' a n d h ea t t e n d a n td i m i n i s h i n g of t h e a x i a l v e l o c i t y d e f i c i t ; b u t e v e n t u a l l y t h ea x i a l v e l o c i t y d e f i c i t a g a i n d e c r e a s e s , a n d t h e a x i a l and s w i r lv e l o c i t y p r o f i l e s d e c a y t o g e t h e r .

F l o w s o f t y p e l b i n i t i a l l y b e h a v e much. l i k e t h o s e o f y p el a , e x c e p t t h a t t h e i r r a t e o fd e c r e a s e of U ax i s s t e e p e r , a n dt h e s w i r l paramet e r S X ) i n c r e a s e s .E v e n t u a l l y h e s w i r l e f f e c t soverwhelm the r e s t o r i n g e n d e n c i e s o f t h e w ake, t h ed r o p o fv e l o c i t y on t h ea x i ss t e e p e n s ra p i d l y , and t hecomput a t i onf a i l s . A s an i n i t i a l s w i r l paramet e rva l ueo f Si 'I 1 . 8 i sa p p r o a c h e d , h e o m p u t a t i o n a i l s n c r e a s i n g l y o o n e r . N os o l u t i o n sc a n b e o b t a i n e d n e a r h i s p o i n t , whose t h e o r e t i c a li n v i s c i d v a l u e i s Si = 3.8317/2 = 1.9159 f o r r i g i d r o t a t i o n( F r a e n k e l1 9 5 6 ) .L e a d i n ge d g ev o r t e x of t y p e 1

The developmento f h eve l oc i t y on t h e a x i s f o r d i f f e r e n ti n i t i a l s wi r l paramet e r s Si f o r a v o rt e xof ead i ngedge ype( i n i t i a l l y Uax = 2, Ue = 1) i s g i ven i n F i g . 4 . 6 . The s w i r lparameter S ( X ) i s a l s o shown f o r two ne ighbor ing lows of typ el a and l b . The beh avio r i s q u a l i t a t i v e l ya g a i n t h e same a s f o rt h ev o r t e xw i t h n i t i a l l yu n i f o r ma x i a lv e l o c i t y . A s w i r lparamet e rva l ueo f Si = 1.1 d i v i d e s thesmooth lydecaying ypel a f r o m h e b r e a k d ow n y p e b .Th i sva l ueo f Si i s somewhatlower t h a n h e h e o r e t i c a l v a l u e So = Jz f o r i n v i s c i d f l o w i nr i g . i d r o t a t i o n a ndappears t o r e f l e c t t h e i n f l u e n c e o f t h ep r o f i l eshape . One would e x p e c t h a t h e e g i o n sa d j a c e n t ot h e a x i s , w h e r e h e a x i a l v e l o c i t y i s lower,wouldhave some

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XFig . 4 . 6 Developmento f v e l o c i t y on t h ea x i s a s a fu n c t io no fi n i t i a l s w i r l . I n i t i a l f lowof ead inge d g evor textype. Open c i r c l e s d e n o t e a i l u r e of t h e computa t ions .

e f f e c t on t h e f l o w b e h a v i o ra n dw o u l d h e r e f o r e e n d ob r i n gdown t h e c r i t i c a l v a l u e of Si.T r a i l i n g v o r t e x of t y p e 1

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vortex singularities, numberical solution, and a xi symmetric vortex breakdown

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