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R . & M . No . 3 4 2 1

t.mim,,BEDFORD.

M I N I S T R Y O F A V I A T I O N

AERONAUTICAL RESEARCH COUNCILREPORTS AND MEMORANDA

B oun dary-Layer Separation inSupersonic P rop e l lingN ozz lesBy M. V. HERBERT and R. J. HERD

LONDONt HER MAJESTY'S STATIONERY OFFICE1966

PRICE 1 8S. 0d. ]NET

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B oun dary-L ayer Separation inSup ersonic Pro pel l ing N ozzlesB y M . V . H E R B E R T a n d R . J . H E R D

COMMUNICATED BY THE DEPUTY CONTROLLER AIRCRAFT (RESEARCH AND DEVELOPMENT),MINISTRY OF AVIATION

Reports and Memoranda No. 3421"August, i964

Summary.A c o m p a r i s o n h a s b e e n m a d e b e t w e e n t h e p r e s s u r e r i s e a t s e p a r a t i o n i n c o n v e r g e n t - d i v e r g e n t n o z z le s a n dt h a t i n o t h e r m o d e l s w i t h s u p e r s o n i c f l o w a n d a t u r b u l e n t b o u n d a r y l a y e r . I n t h e c a s e o f n o zz l e s w i t h u n i f o r mdive rgence , c lose s imi l a r i t y i s found to t he cha rac t e r i s t i c s o f sepa ra t ion induced on su r faces wi th i n i t i a l l y ze rop r e s s u r e g r a d i e n t . W h e r e e x c e p t io n s t o t h i s g e n e r a l a g r e e m e n t o c c u r , t h r e e s p e c i a l c a te g o r i es o f n o z z le d a t ac a n b e d i s t in g u i s h e d . T h e f i rs t , i n v o l v in g a c h a n g e o f b e h a v i o u r i n t h e v i c i n i t y o f s e p a r at i o n , i s a t t r ib u t a b l et o t h e e x i s te n c e o f a l a m i n a r b o u n d a r y l a y e r o r o f o n e i n a s t a t e o f t ra n s i t io n . A c r i t ic a l v a l u e o f R e y n o l d sn u m b e r a t s e p a ra t i o n , b a s e d o n e q u i v a l e n t f l at - p l a te l e n g t h , i s f o u n d t o b e a r o u n d 0 . 7 m i l li o n . T h e s e c o n d f o r mo f a b n o r m a l i t y c o r r e s p o n d s t o a n u n u s u a l l y l a r g e a m o u n t o f p r e s s u r e r i s e i n t h e r e g i o n b e t w e e n s e p a r a t i o na n d n o z z l e o u t l e t ; t h i s c a n o c c u r i n n o z z le s o f l o w d i v e r g e n c e a n g le w i t h e i t h e r t u r b u l e n t o r l a m i n a r s e p a r a t i o n ,t h e e f f e c t b e i n g m o s t p r o n o u n c e d w i t h t h e l a tt e r. T h i r d c o m e s t h e c a s e o f a s h o c k . s y s t e m i n c l o s e p r o x i m i t yt o n o z z l e o u t le t , w h e r e t h e f u l l i n te r a c t i o n p r e s s u r e r i s e w i t h a t u r b u l e n t b o u f i d a r y l a y e r is u n a b l e t o d e v e l o p .

Section1.2.

L I S T O F C O N T E N T S

I n t r o d u c t i o nT h e G e n e r a l P ic t u re2 .1 S e p a r a t i o n c r k e r i a

2 .1 .1 T u r b u l e n t f l o w2 . 1. 2 A g r e e m e n t w i t h e x p e r i m e n t2 . 1. 3 L a m i n a r f l o w

2 .2 E f f e ct o f R e y n o l d s n u m b e r. R e p l ac e s N . G . T . E . R e p o r t N o . R . 2 6 0 - - A . R . C . 2 6 1 02 .

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Sect ion3.4 .

L I S T O F C O N T E N T S - - c o n t i n u e dR e g i m e s o f N o z z l e O p e r a t i o nO v e r e x p a n d e d N o z z l es4 .1 P rev iou s co r re la t ions4 . 2 R e y n o l d s n u m b e r c h a n g e a l o n g a n o z z l e

5 . N o z z l e S e p a r a t i o n D a t a : Q u i e s c e n t A i r5 .1 T h e ' k i n k p o i n t '5 . 2 C o m m e n t s

5 .2 .1 Sepa ra t ion c lose to nozz le ou t le t6 . S e p a r a t i o n w i t h E x t e r n a l F l o w

6 .1 L a m i n a r o r t u r b u l e n t ?7 . E f f e c t o f D i v e r g e n c e A n g l e

7 .1 T h e m i x i n g r e g i o n7 .2 S e p a r a t i o n n e a r t h e t h r o a t o f a n o z z l e

8 . S e p a r a t i o n i n C o n t o u r e d N o z z l e s9 . Conc lus ionsA c k n o w l e d g e m e n t sR e f e r e n c e sT a b l e l - - N o z z l e t h r o a t R e y n o l d s n u m b e r s ( c o n v e n t io n a l s e p a ra t io n d a t a )A p p e n d i c e s I t o I I II l l u s t r a t i o n s - - F i g s . 1 t o 3 2D e t a c h a b l e A b s t r a c t C a r d s

Ap p e n d i xI . N o t a t i o n

I I . D e f i n i t i o n s/

I I I .

L I S T O F A P P E N D I C E S

M o m e n t u m t h i c k n e s s o f a l a m i n ar b o u n d a r y l a y e r

F z g z i F eI .2 .3.4.5.6.

L I S T O F I L L U S T R A T I O N SS e p a r a ti o n p r e s s u r e r a t i o - - IS e p a r a ti o n p r e s s u re r a t i o - - I IS e p a r a ti o n p r e ss u r e r a t i o - - I I IN o z z l e f l o w w i t h s e p a r a t i o nN o z z l e s e p a r a t i o n p r e s s u r e sR e y n o l d s n u m b e r v a r i a t i o n a l o n g a n o z z l e

2

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F@ye7 .8 .9 .

10 .11 .12 .13 .14 .15 .16 .17 .18 .19 .20 .21 .22 .23 .24 .25 .26 .27 .28.29 .30.31 .32 .

L I S T O F I L L U S T R A T I O N S - - c o n t i n u e dC o n v e n t i o n a l n o z z l e s e p a r a t i o n d a t a - - IC o n v e n t i o n a l n o z z l e s e p a r a t i o n d a t a - - I IC o n v e n t i o n a l n o z z l e s e p a r a t i o n d a t a - - I I IC o n v e n t i o n a l n o z z l e s e p a r a t i o n d a t a - - I VT u r b u l e n t b o u n d a r y - l a y e r ' k in k p o in t s 'U n c o n v e n t i o n a l n o z z l e s e p a r a t i o n d a t aN o z z l e p r e s s u r e d i s t r i b u t i o n f r o m R e f . 5 0 - - IN o z z l e p r e s s u r e d i s t r i b u t i o n f r o m R e f . 5 0 - - I IN o z z l e p r e s s u r e d i s t r i b u t i o n f r o m R e f . 5 0 - - I I IN o z z l e p r e s s u r e d i s t r i b u t i o n f r o m R e f . 5 1 - - IN o z z l e p r e s s u r e d i s t r ib u t i o n f r o m R e /: . 5 1 - - I IS e p a ra t io n R e y n o l d s n u m b e r s o f s o m e d a t aT h r o a t R e y n o l d s n u m b e r s o f s o m e s e p a ra t io n d a taT u r b u l e n t n o z z l e s e p a r a t i o n d a t a a t l o w a p p l i e d p r e s s u r e r a t i oTh e e f f ec t o f d iv e r g en ce an g leP r e s s u r e d i s t r i b u t i o n d o w n K u h n s 5 n o z z leP r e s s u r e d i s t r i b u t i o n d o w n K u h n s 1 0 n o z z leF r e e j e t b o u n d a r i e s a f t e r l a m i n a r s e p a r a t i o nF r e e j e t b o u n d a r i e s a f t e r t u r b u l e n t s e p a r a t i o nM o d i f i c a t i o n t o j e t b o u n d a r y c a u s e d b y p r e s s u r e r i s e i n m i x i n g r e g i o nPr es su r e d i s t r ib u t io n s in sev e r a l 1 0 co n id a l ,n o zz le s' T u l i p ' n o z z le p r e s s u r e d i s t r i b u t i o n fr o m R e f . 5 1 - - I' T u l i p ' n o z z l e p r e s s u r e d i s t r i b u t i o n f r o m R e f . 5 1 - - I IT u r b u l e n t s e p a r a t i o n d a t a f o r c o n t o u r e d n o z z l e s - - IT u r b u l e n t s e p a ra t io n d a t a f o r c o n t o u r e d n o z z l e s - - I IE f f e c t o f m e a n w a l l a n g l e o n p e r f o r m a n c e O f c o n t o u r e d n o z z le s ( f r o m R e f . 4 2 )

1 . I n t r o d u c t i o n .T h e p e r f o r m a n c e o f a s u p e r s o n i c i n te r n a l - e x p a n s i o n p r o p e l l i n g n o z z le o p e r a t i n g a t a p r e s s u r e r a ti o

b e l o w t h a t r e q u i r e d t o p r o d u c e c o r r e c t e x p a n s io n d e p e n d s , a s 'i s w e l l k n o w n , u p o n t h e o c c u r r e n c e o ff l o w s e p a ra t i o n w i t h i n t h e n o z z le . T h i s t a k e s p l a ce w h e n t h e o b l i q u e s h o ck , f o r m e d t o r e c o m p r e s s t h eo v e r e x p a n d e d j e t to t h e b a c k p r e s s u r e o f t h e e x h a u s t s y s t e m , m o v e s i n s i d e t h e w a l ls o f t h e n o z z l e .I n i n v i s c i d f l o w , t h e s h o c k w o u l d r e m a i n a t o u t l e t d o w n t o a m u c h l o w e r p r e s s u r e r a t i o , a n d w o u l do n l y e n t e r t h e n o z z l e a f t e r i n c r ea s i n g i n s t r e n g t h t o t h e f o r m o f a p l a n e s h o c k .' H o w e v e r , a b o u n d a r y

3(91949) A 2

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layer is unable to withstand the pressure rise associated with so strong a shock at outlet, and conse-quently in practice the shock starts to move inside while still oblique in form.

Shock-induced boundaryZlayer separation in other supersonic flow systems has been studied quiteextensively, and sufficient knowledge is available to estimate the quantities there associated with it.Several authors have applied the results of this work to separation from the walls of convergent-divergent propelling nozzles. They have found that experimental measurements of the pressure risein nozzles exhausting into quiescent air agree fairly well with appropriate values for other systemswith tu rbul ent boundary layers. This suggests that no great differences are introduced by the factorsof pressure gradient and wall divergence which exist in a nozzle.

However, some recent tests, carried out partly in the presence of an external stream, have indicatedpatterns of separation different from those obtained previously. There is in consequence need for afresh appraisal of the factors upon whic h separation in a supersonic nozzle depends, in order todetermine by what agency the observed difference in results is produced.

2 . Th e G en er a l P i c t u r e .A large body of experimental data exists on boundary-layer separation due both to obstacles and

incident shocks, in supersonic flow with initially zero pressure gradient. The cases of forward-facingsteps, curved surfaces, and compression corners, wedges or ramps, have been quite comprehensivelystudied two-di mensionall y for both laminar and t urbu lent flow, as have incident shocks on flat plates.Limited evidence is available for separation at axisymmetric versions of steps, compression cornersand curved surfaces, and for the interaction of a plane shock with a cylindrical body. It seems that thebehaviour of two-dimensional and axisymmetric obstacles is qualitatively similar. Various theoreticalapproaches have been tried in order to describe the mechanism and occurrence of separation,generally including some factor for whic h an empirical value has to be taken.

Two points of immediate interest to us stand out when surveying this field of work (Refs. 1 to 21).First: in every experimental model studied, re-attachme nt of the boundar y layer follows quite soonafter separation. Second: the early part of the pressure rise (associated with separation) has featuresclosely similar between all models, while the later part (associated with re-attachment) has not.

The second point may be put another way; in the words of Mager 14, the boundary layer does not'know' what combination of circumstances creates the pressure rise leading to separation; it onlyknows what pressure rise is required at given conditions of Mach number and Reynolds number tocause it to separate. In this context Chapman et a l 1 use the term 'free interaction', with relation toregions of flow 'which are free from direct influences of downstream geometry', and i p s o f a c t o'independent of the mode of inducing separation'. Experimental results from several sources showthat, at least as far as their respective separation points, an incident shock and all the forms ofobstacle mentioned above are indeed 'free interactions' in this sense. But, once separated, the effectsof geometry--either physical in the case of obstacles, or that imposed by mainstream requirementssuch as shock reflection--which the boundary layer must negotiate during the process of re-attach-ment, put the latter phase outside the category of a free interaction, and similarity of the variousmodels ceases to be found.

In passing, it may be noted tha t Mager 14 had earlier coined the phrase 'free shock-separation 'with reference to 'that type of separation where the flow downstre am of the separation region is free

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t o a d j u s t t o a n y d i r e c ti o n t h a t m a y r e s u l t f r o m t h e s h o c k - b o u n d a r y - l a y e r i n t e r a c ti o n p r o c e s s '. T h i sc o n d i t i o n s e e m s t o b e r e s t r i c te d t o c a s es i n w h i c h r e - a t t a c h m e n t e i t h e r d o e s n o t o c c u r o r o n l y d o e ss o s o m e c o n s i d e r a b l e d i s t a n c e a f te r s e p a r a t i o n . O v e r e x p a n d e d n o z z l e s a n d ' s u f f i c i e n t ly l a r g e ' s t e p sa r e c i te d a s p o s s i b l e e x a m p l e s , w h i l e o t h e r s y s t e m s s u c h a s i n c i d e n t s h o c k s ( w i t h e a r l y r e - a t t a c h m e n t )a r e t e r m e d ' r e s t r i c t e d s h o c k - s e p a r a t i o n s ' . M a g e r n o t e s t h a t ' a t le a s t i n s o m e c a se s , t h e i n t e r a c t i o np r o c e ss o f t h e r e s t r i c t e d s h o c k - s e p a r a t e d b o u n d a r y l a y e r s t a rt s a s a f r e e s h o c k - s e p a r a t i o n ', w h i c h i sa n o t h e r w a y o f s a y in g t h a t u p t o t h e s e p a r a t i o n p o i n t a ll m o d e l s b e h a v e s i m i la r l y. A l t h o u g h t h e r ei s n o b a s i c c o n f l ic t o f i d ea s , s o m e c o n f u s i o n c o u l d a r i se b e t w e e n t h i s t e r m i n o l o g y a n d t h a t o fC h a p m a n e t a l q u o t e d a b o v e , w i t h r e g a r d t o u s e o f t h e w o r d ' f r e e ' a n d w h a t i t i s t a k e n t o im p l y .T h e t w o d e f i n i t i o n s a r e s t r ic t l y i n c o m p a t i b l e , t o t h e e x t e n t t h a t M a g e r s p e c i f ic a l l y r e f e r s t o c o n -d i t i o n s ' d o w n s t r e a m o f t h e s e p a r a t i o n r e g i o n ' a s d e c i d i n g w h e t h e r t h e i n t e r a c t io n a s a w h o l e i s' f r e e ' o r ' r e s t ri c t e d ' ( a c c o r d in g t o w h i c h t h e g r e a t m a j o r i t y o f t h e c a s e s i n v e s t ig a t e d m u s t b e r e g a r d e da s n o t fr e e ), w h e r e a s C h a p m a n e t a l a s c ri b e t h e w o r d t o th a t p a r t o f a f l o w s y s t e m w h i c h i s in d e p e n -d e n t o f th e d o w n s t r e a m c o n d i t i o n s ( w h e n p a r t o f th e p r o c e s s i n a ll m o d e l s c a n g e n e r a l ly b e c a ll e df r e e ) . W h e r e u s e d i n t h e p r e s e n t p a p e r , i t w i l l h a v e t h e l a t t e r m e a n i n g .

N o w t h e f o l l o w i n g s t a ti o n s in t h e f l o w m a y b e r e c o g n i s e d :1 N t h e u n d i s t u r b e d s t r e a m j u s t a h e a d o f t h e i n it ia l c o m p r e s s i o n .s ~ t h e p o i n t o f a c t u a l b o u n d a r y - l a y e r s e p a r a t io n ( u p s t r e a m e n d o f t h e ' b u b b l e ' o r r e c i r c u la t i o n ,

u s u a l l y d e t e r m i n e d b y o p t i c a l o r o i l - fi l m m e t h o d s ) .2 ~ a p o i n t o f h i g h e r p r e s s u r e w h i c h c a n b e r e g a r d e d , r a t h e r a r b i t ra r i l y , a s a d i v i s i o n b e t w e e n

t h e p r o c e s s es o f s e p a r a t io n a n d r e - a t t a c h m e n t . T h i s is d e f i n e d a c c o r d i n g to s o m e c h a r a c t e r -i s ti c o f t h e p r e s s u r e d i s t r i b u t i o n c u r v e , v a r i o u s ly t e r m e d t h e ' p l a t e a u ' i n l a m i n a r s e p a r a t i o n,a n d ' f i r s t p e a k ' ( f o r f o r w a r d - f a c i n g s t e p s) o r ' i n f l e c t i o n p o i n t ' ( f o r w e d g e s o r i n c i d e n ts h o c k s ) in t u r b u l e n t f l o w . I n t h e l a m i n a r c a s e , t h i s p l a t e a u o f p r e s s u r e e x t e n d s o v e r a na p p r e c ia b l e d i s t a n c e a h e a d o f t h e s u r f a c e d i s c o n t i n u i t y o r p o i n t o f s h o c k im p i n g e m e n t .W i t h a tu r b u l e n t b o u n d a r y l a y er , t h e s e l e c t e d f e a tu r e o f t h e p r e s s u r e c u r v e i s a l o ca l o ne ,a n d o c c ur s a t th e m a x i m u m b u b b l e w i d t h b e h i n d a n i n c i d e n t s h o c k - in d u c e d s e p a ra t io n e ,c o r r e s p o n d i n g t o t h e p o i n t o f s h o c k im p i n g e m e n t , a t o r v e r y s l i g h tl y a h e a d o f t h e s u r f a c ed i s c o n t i n u i t y o n a w e d g e o r r a m p , w h e r e t h e s e p a r a t i o n b u b b l e a g a i n h a s i ts g r e a te s t w i d t h ,a n d i n t h e c a s e o f a f o r w a r d - f a c i n g s te p a b o u t 6 0 p e r c e n t o f t h e d i s ta n c e a l o n g t h e s e p a r a t e do r r e c i r c u l a t i o n r e g i o n , i. e . m e a s u r i n g f r o m t h e s e p a r a t i o n p o i n t t o w a r d s t h e s t e p .

r ,-, t h e p o i n t o f r e - a t t a c h m e n t o r e n d o f t h e s e p a r a te d r e gi o n , o c c u r r i n g a t t h e e d g e o f a f o r w a r d -f a c i n g s te p , a n d i n t h e c a se s o f w e d g e s a n d i n c i d e n t s h o c ks s o m e d i s ta n c e d o w n s t r e a m o ft h e s u r f a c e d i s c o n t i n u i t y o r p o i n t o f s h o c k i m p i n g e m e n t .

3 ,,~ w h e re p a r a l l e l f l o w i s f i n a l l y r e s t o r ed an d p r e s s u re r i s e ceas e s .I n t h e l i t e r a t u r e i t s e e m s t o b e g e n e r a l l y e s t a b l i s h e d t h a t a l l t h e m o d e l s i n v e s t i g a t e d e x h i b i t p r e s s u r e -r is e c h a r a c te r is t ic s w h i c h c o r r e s p o n d c l os e ly t o o n e a n o t h e r u n d e r s i m i la r c o n d i t i o n s o f M a c h a n dR e y n o l d s n u m b e r s a s f a r a s s t a t i o n 2 f o r l a m i n a r f l o w , a n d a t l e a s t u p t o s t a t i o n s f o r t u r b u l e n t .T h u s f a r t h e y c o n f o r m w i t h t h e c o n c e p t o f a f r e e i n t e r a c t i o n i n t h e s e n s e u s e d b y C h a p m a n e t a l 1 .

e I t shou ld be no ted tha t s t a t ion 2 as here def ined fo r inc iden t shock in terac t ions i s n o t the same as thep o i n t o f 'k i n k p re s s u re ' u s ed b y G ad d , H o l d e r an d R eg an8. Th e l a t t e r occurs cons iderab ly nearer the separa t ionpoin t , an d i s no t a lways easy to id en t i fy in da ta f rom o ther sources .

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A d i f f i c u lt y a p p e a r s , h o w e v e r , i n r e s p e c t o f t h e f u r t h e r p r e s s u r e r i se f r o m s t o 2 in t u r b u l e n t f l o w ,w h e r e t h e d e t a i l e d p r e s s u r e d i s t r i b u t i o n s f r o m v a r i o u s m o d e l s a r e n o l o n g e r i d e n t i c a l . C h a p m a ne t a l ~ c o n c l u d e t h a t i n g e n e r a l t h i s r e g i o n m u s t b e r e g a r d e d a s o u t s id e t h e c a t e g o r y o f a fr e e i n t e r -a c t i o n . N e v e r t h e l e s s , i n s o m e i n s ta n c e s s i m i l a r i t y o f p r e s s u r e d i s t r i b u t i o n h a s b e e n o b s e r v e d b e t w e e nd i f f e r e n t g e o m e t r i e s o f m o d e l o v e r a c o n s i d e r a b l e p a r t o f t h e r i s e f r o m s t o 2 . T w o s t r i k i n g e x a m p l e sa r e av a i l ab l e , b o t h a t M s ~ 3 :

( i) F r o m R e f . 4 ( F ig . 1 9 o f B o g d o n o f f a n d K e p l e r , o r F i g . 2 6 o f B o g d o n o f f ) : t h i s c o m p a r e s t h ep r e s s u r e d i s t r i b u t i o n f o r a s t r o n g i n c i d e n t s h o c k ( 1 4 d e f l e c ti o n ) w i t h t h a t f o r a s t e p(h e i g h t h = 1 . 8 3 1 ) .

( ii) F r o m R e f . 1 ( F i g . 3 9 b ) : a c o m p a r i s o n i s g i v e n o f t h e p r e s s u r e d i s t r i b u t i o n s f o r a s t e p( e s t i m a t e d h = 2 . 1 81) a n d a 2 5 c o m p r e s s i o n c o r n e r .

B o t h t h e s e p i c t u r e s s h o w s i m i l a r f e a t u r e s :( a) T h e r e i s a la c k o f d e p e n d e n c e o n g e o m e t r y o f m o d e l n o t o n l y u p t o t h e s e p a r a t io n p o i n t b u t

a s f a r b e y o n d a s t h e ' i n f l e c t i o n p o i n t ' i n t h e c u r v e s f o r i n c i d e n t s h o c k a n d c o m p r e s s i o nc o r n e r .

( b) T h e s e i n f l e c t i o n p o i n t s o c c u r a t p r e s s u r e l e v el s s l i g h t l y b e l o w ' f i r st p e a k ' f o r t h e s t e p i nb o t h s e t s o f e x p e r i m e n t s . ( A s im i l a r o b s e r v a t i o n i s m a d e b y L a n g e T .)

( c) O n t h e w h o l e , v a l u e s o f p r e s s u r e r i se a n d i n t e r a c t i o n l e n g t h ( w h e r e l i s m e a s u r e d f r o ms t a t i o n 1 ) a r e in f a i r ly g o o d q u a n t i t a t i v e a g r e e m e n t b e t w e e n t h e d i f f e r e n t t e s ts , a s s h o w n i nt h e f o l l o w i n g t a b l e .

Values of P / P 1

Separat ion Inflect ion(s) (2) Fi rs t peak. ( 2 )

"Values of 1/81

Separat ion Inflect ion(s) (2) Fi rs t peak(2 )

Inc iden t shock : Ref . 4Step l Ref. 4Step : Ref . 1Compress ion corner : Ref . 1

2 . 1 0

2"06

2 ' 5 5

2 . 3 4

2 .672 .43

2~

2

5

5 ~

m

77

I t i s u n f o r t u n a t e l y t r u e t h a t i n g e n e r a l n o u n i q u e r e l a ti o n f o r P ~ / P 1 e x i st s f o r th e c a s e o f a t u r b u l e n tb o u n d a r y l a y e r . S o f a r a s t h e p r e s e n t p a p e r i s c o n c e r n e d , h o w e v e r , w h e r e t h e o b j e c t i s m a i n l y t om a k e a c o m p a r i s o n w i t h d a t a o n s e p a r a t i o n i n n o z z l e s , w e a r e o n l y s e e k i n g a c o n d i t i o n w h i c h f a i r l yd e s c r ib e s t h e e n d o f t h e s e p a r a t i o n p ro c e s s a n d t h e b e g i n n i n g o f r e - a t t a c h m e n t i n o t h e r m o d e l s , a n df o r t h i s a r e a s o n a b l e b u t a p p r o x i m a t e e s t i m a t e o f P 2 / P 1 w i l l s u f f i ce . T h i s i s e s p ec i a l l y t h e ca s e s i n cet h e p r e s s u r e r is e t o s t a t io n s, w h i c h i s u n i q u e a n d c a p a b l e o f f a i r l y c e r t a i n d e t e r m i n a t i o n , r e p r e s e n t s

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at least 80 per cent of the whole rise between 1 and 2. Of the various geometries of model studiedin the literature, the most comprehensive data for P2/P1 as here defined come from forward-facingsteps. This is perhaps not surprising as their 'first peak' is the feature most readily identifiableamong the turbulen t pressure distributions for any of the models. The value of P2/P1 at 'first peak'for certain types of step is considered near enough to that at 'inflection' for incident shocks andwedges to justify us, for the present purpose, in treating such step data as typifying the end of theseparation process in all models.

There remains the complication of step height, which at times has been found to exert a quitepronounced effect. It seems to be generally agreed1, 4, n that whe n step height is small the pressurerise to first peak--but not that to the separation point--diminishes as step height is reduced. Asimilar effect has been noted 4 in the case of weak inc ident shocks, and it is thoug ht probable t hatsmall-angle wedges behave in like manner.

Bogdonoff and Kepler ~ show curves of P~/P1 for varying step height which level off when h/31reaches a value around 1.5 to 2. They did not, however, test much above 2. Love 9 did, and statesthat no effect of height was observed when h/3 1 >1 2, although his upper limit is not given. On theother hand, Chapman et aP in their Fig. 46 present data which show a considerable furth er increasein P2/P1 as h/31 rises from 2 to 6. This effect is apparently related to Mach number : departure fr omthe general trend shown by the data of Ref. 4 (for h/3~ ~ 2) and Ref. 9 only becomes pronouncedas M 1 increases above 2.5. It is advisable, therefore , wh en considering the use of step data astypifying P ~ / P 1 for different geometric models in turbulent flow, to restrict the presumption ofsimilarity to values of h/31 around 2.

The first peak data available in this category cover the range of Mach number 1.4 to 6, as shownin Fig. 2b. They are derived from three sources:

(i) Loveg: as already mentioned, these results relate to steps with h /31 between 2 and someunkn own higher value such that no effect of step he ight was produced.

(ii) Sterrett and Emery11:h/31 is estimated to be within the range 1.6 to 3.0.(iii) Bogdonoff4: only the results for h/S 1 > 1.5 are included.

2.1. Separation Criteria.Many relations, some wholly empirical and some based on theoretical considerations, are given

in the literature for the ratiosPs P~ and P22PI' Ps PI'

and kindred functions associated with the mainstream flow, such as Mach number ratio and shockdeflection angle, are also employed to relate conditions at stations 1, s and 2. All analytical trea tmentsassume two-dimensional flow.

2.1.1. Turbulent f/ ow .- -Ma geP a seems to have been the first to derive the relationMs K'/2M1where K is some constant.

( 1 )

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I n o r d e r t o c o n v e r t t h i s in t o a s im p l e e x p r e s s i o n f o r p r e s s u r e r a t io , h e t h e n u s e s a li n e a r i s e d a p p r o x i -m a t i o n t o t h e o b l i q u e s h o c k , a n d o b t a i n s

(1 - K ) 2 M I ~P s - - 1 + _ (2 )P1 + ~_~_I i1o r

w h e r e

1 - KCp, s - - _ (2a)+ r - 2 - -1 M ?P - P~c ~ -

Y_ P iM l~2T h e v a l u e s u g g e s t e d f o r K i n R e f . 1 3 is 0 -4 9 4 , g i v i n g M s / M 1 = 0 . 7 0 3 . I n a la t e r p ap e r 14, M ag e rc h a n g e s t o K = 0 . 5 5 o n t h e g r o u n d s t h a t a b e t t e r f i t i s o b t a i n e d t o e x p e r i m e n t a l m e a s u r e m e n t s o ft h e s e p a r a t i o n p o i n t ~. T h i s w o u l d g i v e M d M ~ = 0 . 742 .

N u m e r i c a l c o m p a r i s o n o f e q u a t i o n s ( 1 ) a n d ( 2) , u s i n g c o r r e c t o b l i q u e s h o c k r e l a t i o n s i n t h ef o r m e r , d i s c l o s e s a g r o s s d i s c r e p a n c y . A t M ~ = 2 , e q u a t i o n ( 1 ) y i e l d s a v a l u e o f P s / P I 2 4 p e r c e n ta b o v e t h a t g i v e n b y e q u a t i o n ( 2 ), t a k i n g K = 0 . 5 5 , w h i l e a t M 1 = 4 t h e f i g u r e h a s r i s e n to 5 0 p e rc e n t. T h i s w a s p o i n t e d o u t b y R c s h o t k o a n d T u c k e r ~5 w h o , n o d o u b t c o r r e ct ly , a t t r i b u t e t h ed i s c r e p a n c y t o M a g e r ' s u s e o f a s h o c k - w a v e a p p r o x i m a t i o n . I t s e e m s , t h e r e f o r e , t h a t e q u a t i o n s ( 2)a n d ( 2 a ) s h o u l d n o t b e u s e d .

W h a t i s h a r d t o u n d e r s t a n d is t h a t M a g e r i n h i s la t e r p a p e r d o e s n o t h i n g t o a n s w e r t h i s c r i ti c is mo r t o c o r r e c t t h e s i t u a t i o n . H e r e i t e r a t e s b o t h e q u a t i o n s ( 1 ) a n d ( 2 ) , a n d t h e n p r o c e e d s t o d e v e l o pa n e x p r e s s i o n f o r P ~ / P s i n t e r m s o f M s a n d A 2, f o r t h e c a s e o f h i s ' f r e e s h o c k - s e p a r a t i o n ' m o d e li n w h i c h r e - a t t a c h m e n t d o e s n o t o c c u r ( see Sec t i o n 2 ) . T h i s g i v es h i m

0" 328 ~,KM~2,~~p8 I + . , _ i ( 3)1 + ~ -- 2 K M I 2

B y r e g a r d i n g t h e f l o w o u t s i d e t h e b o u n d a r y l a y e r a s a d j u s t i n g i t s e lf t o t h e c o n d i t i o n s o f t h e ' j e t- l i k e 's e p a r a t e d b o u n d a r y l a y e r a t s t a t i o n 2 , h e i s a b l e t o i n v o k e c o n d i t i o n s a c r o s s a t w o - d i m e n s i o n a lm a i n s t r e a m s h o c k t o c o n n e c t M ~ w i t h ~ t2 . F o r t h i s s h o c k h e i n tr o d u c e s a f u r t h e r a p p r o x i m a t er e l a t i o n

A - - (M 12 - 1)~/~

a n d s u g g e s t s it s u s e i n c o n j u n c t i o n w i t h e q u a t i o n s ( 2) a n d ( 3) , g iv i n g th e c o m b i n e d e x p r e s s i o n ,

I , ! 1 t 1 1 + 0 1 ,,1 1 ,1 , +R e s h o t k o a n d T u c k e r 1'~ g i v e a g e n e r a l t h e o r e t i c a l a n a l y si s f o r t h e ' c h a n g e o f b o u n d a r y - l a y e r -

t h i c k n e s s p a r a m e t e r s a n d f o r m f a c t o r c a u s e d b y a d i s c o n t i n u i t y i n t h e a b s e n c e o f f r i c t i o n e f f e c t s ' .

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S i n c e f ri c t i o n e ff e c ts a r e c o n s i d e r e d n e g l i g i b le c o m p a r e d w i t h t h o s e o f p r e s s u r e g r a d i e n t d u r i n gs e p a r a t io n , t h e s e a u t h o r s c o n c l u d e ' t h e f o r m o f t h e r e s u l t s u g g e s t s t h a t t h e M a c h n u m b e r r a t i oa c r o ss t h e s h o c k is a c h a r a c t e r is t i c p a r a m e t e r f o r d e f i n i n g s h o c k - i n d u c e d s e p a r a t i o n ' . I t s h o u l d b en o t e d , a s M a g e r 14 p o i n t s o u t , t h a t t h e y d o n o t s p e c i f ic a l l y d i s t i n g u i s h b e t w e e n s t a ti o n s s a n d 2 i nt h e i r a n a l ys i s, a n d i t i s p r o b a b l y r e a s o n a b l e t o r e g a r d i t a s a p p l y i n g s t r i c tl y o n l y t o w h a t w e h a v ea g r e e d t o c a ll ' f r e e i n t e r a c t i o n s ' . I n t u r b u l e n t f l o w , t h is c o v e r s s b u t n o t n e c e s s a r i ly 2 . R e s h o t k o a n dT u c k e r d o i n f a c t q u o t e d i f f e r e n t v a l u e s f o r t h e r a t i o M 2 / M 1 a s b e t w e e n ' f i rs t p e a k ' d a t a f o r f o r w a r d -f a c i n g s te p s ( 0 . 7 6 2 ) a n d ' i n f le c t i o n p o i n t ' d a t a f o r w e d g e s ( 0 . 8 1 ) .

L o v e 9 s u g g e s t s t h a t a v a l u e o f M s / M 1 a r o u n d 0 . 8 5 i s r e q u i r e d t o f i t e x p e r i m e n t a l d a t a, w h i c hw o u l d g i v e K = 0 . 7 2 2 . T h i s f i g u r e i s m u c h h i g h e r t h a n e i t h e r o f t h o s e t a k e n b y M a g e r la,14 .H o w e v e r , s i n c e P2/P1 > PJP1, i t i s n e c e s s a r y t o h a v e M~/ M 1 < Ms /M 1 , a n d L o v e s s h o w s a v e r yf a i r c o r r e l a t i o n o f e x p e r i m e n t a l ' f ir s t p e a k ' d a t a fo r s te p s u p t o M t = 3 u s i n g t h e v a l u e M2/M 1 =0 . 7 6 2 c o p i e d f r o m R e s h o t k o a n d T u c k e r 15. M o r e s e p a r a t i o n p o i n t d a t a h a v e r e c e n t l y b e c o m e a v a i l -a b le , c o l l e c t e d i n F i g . 2a , c o v e r i n g M a c h n u m b e r s b e t w e e n 1 - 5 a n d 6 . A fa i r a v e r a g e fi t o v e r t h ew h o l e r a n g e o f M 1 i s g i v e n b y M d M 1 = 0 - 82 ( K = 0 - 67 ) .

S o f ar as t h e v a l id i t y o f e i t h e r M a c h n u m b e r r a t io t o c o r r e l a t e s e p a r a t i o n p e r f o r m a n c e i s c o n c e r n e d ,i t s e e m s t h a t w e s h o u l d a c c e p t M , / M 1 a s b e i n g a p a r a m e t e r w h i c h a n a ly s i s s u g g e st s t o b e s i g n i f i c a n ti n t h e m e c h a n i s m o f s e p a r a t io n , a n d a s h a v i n g a t l e a st in t h ' e o ry a n u n i q u e v a l u e , w h i l e t h e u s e o fM 2 / M 1 m a y b e j u s t i f i e d b y e x p e r i m e n t a l d a t a fo r a n y p a r t i c u l a r g e o m e t r i c a r r a n g e m e n t , t h e v a l u e si n t h is c a s e b e i n g n o t n e c e s s a ri l y i n d e p e n d e n t o f g e o m e t r y .

A n a l t e r n a t i v e a n d w h o l l y e m p i r i c a l r e la t i o n is o f f e r e d b y L o v e s t o d e s c r i b e m o r e c l o s e l y h i s' f i rs t p e a k ' d a t a o n f o r w a r d - f a c i n g s t e p s o v e r t h e r a n g e 1 " 4 < M 1 < 3 . 5 , n a m e l y

3 . 2C P '2 = 8 + ( M 1 - 1 ) ~ " ( 6 )T h i s r a n g e i s e x t e n d e d u p t o M 1 = 6 - 3 b y S t e r r e t t a n d E m e r y n , w h o s h o w t h a t e q u a t i o n ( 6 ) i s i na p p r o x i m a t e a g r e e m e n t w i t h t h e i r d a t a u p t o M 1 - - 5 a n d a n i n c r e a s i n g l y p o o r f i t t h e r e a f t e r . T h e yg i v e a f u r t h e r e m p i r i c a l e q u a t i o n t o f i t t h e i r o w n d a t a , b u t i t c a n n o t b e s e n s i b l y a p p l i e d w h e nM 1 < 3 . 5 . U s e o f a n y c o n s t a n t v a l u e f o r M 2 / M 1 o v e r t h e w h o l e r a n g e i s a g a i n f o u n d t o b e n o tc o m p l e t e l y s a t i s fa c t o r y ]- .

A n o t h e r s e m i - t h e o r e t i c a l e x p r e s s i o n f o r PJP1 i s d e v e l o p e d b y G a d d l L T h i s i s e x p l i c it l y g i v e n f o rt h e c a s e o f a n i n c i d e n t s h o c k i n t w o - d i m e n s i o n a l f l o w , b u t i n a n e a r li e r v e r s i o n o f th e s a m e w o r k +s i m i l a r i t y is s u g g e s t e d t o t h e c a s e o f a c o m p r e s s i o n c o r n e r . A c c o r d i n g t o p r e v i o u s d i s c u s s i o n , i ts h o u l d i n fa c t b e a p p l i c ab l e t o a n y ' f re e i n t e r a c t i o n ' . G a d d c o n s i d e r s a o n e - s e v e n t h p o w e r v e l o c i t yp r o f i l e as h a v i n g a ' s h o u l d e r ' a t s o m e c h a r a c t e r i st i c f r a c t i o n Y o f t h e f r e e - s t r e a m v e l o c i ty , a n d , a g a i nt a k i n g f r i c t io n f o r ce s as n e g li g i b le b y c o m p a r i s o n w i t h t h o s e d u e t o t h e p r e s s u r e g r a d i e n t , h e

o T he mea n l ine in F ig . 2a f it s M s/M 1 -- 0- 825 ve ry closely ov er the rang e 1" 85 < M 1 < 3" 6 ; a t valuesof 11/I < 1 .85 the e f fec t ive Ma ch n um be r ra t io drops sharp ly (0-79 a t M1 = 1"5) , and a t h igh va lues of M 1i t f al l s aga in mor e gent ly (0" 81 a t M 1 = 6) . Hen ce the accuracy in prac t ice of th is form of c r i te r ion , pa r t icu la r lya t l ow M a c h n um b e r , i s on ly a pp r ox im a te .

"~ T he va lue of M2/iVI 1 required to f i t the da ta for forward- fac ing s teps , co l lec ted in F ig . 2b , r i ses f rom0-762 a t M 1 < 3 to 0-78 a t M1 = 4 and the n fa ll s r apid ly to about 0" 74 a t M~ = 6 .

J; A.R .C. 15 543. Janu ary, 1953.

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e v a l u a te s t h e p r e s s u r e i n c r e a s e r e q u i r e d t o b r i n g t h e f l u i d o n t h i s s t r e a m l i n e t o r e s t i s e n tr o p i c al l y .T h i s p r e s s u r e r is e h e a r g u e s is a p p r o x i m a t e l y e q u a l t o t h a t a t t h e s e p a r a t i o n p o i n t , w h e n c e

P1 1 + ~ 0 7~ ~)2)M1~ (7)

G a d d i n it ia l ly r e c o m m e n d s 17 tak ing Y as 0 . 6 , bu t th i s i s l a te r r ev i s ed s to 0 . 54 ; w i t h th e l a t t e r va lueof J equ a t ion (7) i s show n i s to g ive qu i te g ood f i t to exp er im en t .

A m odi f ica t ion to eq ua t ion (7) i s p ropose d b y A rens an d Sp ieg le r =2,2a fo r the case in wh ich thec h a r a c t e r i s ti c s t r e a m l i n e i s in i t ia l l y su p e r s o n i c ( w h i c h , f o r y = 1 . 4 , w o r k s o u t t o b e w h e nM 1 > 2 . 0 8 ) , w h e n t h e y t h e n a s s u m e a n o r m a l s h o c k f o l l o w e d b y i s e n t r o p i c s u b s o n ic s t a g n a ti o n .T h i s l e a d s t o t h e r e l a t i o n

P~P1 { 1

( '+ Y - @ - ( 1 - J 2 ) M I ~ } { M ~ [ ( y + I ) J 2 ( Y - 1 ) 2 ]7 ~ - 1 ] ; ~ }

1 ' y - - l ) " ( 8 )

G o o d a g r e e m e n t i s c l a i m e d ~2 b e t w e e n e x p e r i m e n t a l d a t a ( f o r st e p s, c o m p r e s s i o n s u r f a c e s a n di n c i d e n t - s h o c k m o d e l s ) a n d e q u a t i o n ( 8) u s i n g t h e v a l u e J = 0 . 5 6 .

Var ious o ther w ho l ly e m pi r ica l r e la t ions a re ava i l ab le . C ooke ~1 sugges t s tha t the da ta o f C hap m ane t a l 1 a r e r e a so n a b l y w e l l d e s c r i b e d b y t h e e q u a t i o nP" - 0. 079 M12 + 1.5 2 0- 22p l - - ( 9 )

T h i s d o e s n o t s a t i s f a c t o r i l y f i t t h e d a t a o f S t e r r e t t a n d E m e r y 11 a t h i g h e r M a c h n u m b e r s , a n dsho u ld no t be used w he n M1 > 4 . F o r r e la t ing s ta t ions s and 2 , C ooke ~1 no tes th a t va r iousc o m p r e s s i o n - c o r n e r a n d i n c i d e n t - s h o c k d a t a f i t t h e f o r m u l a

- 1 - i s ( lO )P~ P1 "G u m a n 2, r e f e r r i n g s i m p l y to ' p e a k ' p r e s s u r e , w i t h o u t s p e c i f y i n g t h e i n t e n d e d m e a n i n g i n r e l a t i o nt o d i f f e r e n t g e o m e t r i c a r ra n g e m e n t s , p r o p o s e s

C p , 2 = 1"54 Cp,~ (11)w h i c h , i n c o n j u n c t i o n w i t h e q u a t i o n ( 2 ) a n d K = 0 . 5 5 , i s s a i d t o g i v e ' f a i r t o g o o d a g r e e m e n t 'w i t h t h e d a t a o f C h a p m a n e t a l 1 .

2.1.2. A g r e e m e n t w i th e x p e r i m e n t . - - A t th i s s t age i t i s wor th s i f t ing equa t ions (1 ) to (11 )i n t h e l i g h t o f e x p e r i m e n t a l e v i d e n c e , t a k e n f r o m F i g s. 2 a a n d b . I n o r d e r t o a c h i e v e g r e a t e rs ens i t iv i ty , r ec ip roca l p res su re r a t io s w i l l be used in fu tu re : tha t i s to s ay P 1 / P ~ , e t c . r a t h e r t h a nt h e f o r m P . ; / P 1 , e t c. w h i c h a p p e a r s i n t h e p r e c e d i n g e q u a t i o n s . T h e v a l u e Y = 1 . 4 i s t a k e nt h r o u g h o u t .

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O n F i g . 1 r e la t i n g t o P1/Ps a r e s h o w n t h e f o l l o w i n g c u r v e s :( a) E x p e r i m e n t a l m e a n l i ne f r o m F i g . 2 a( b ) E q u a t i o n ( 1) w i t h K = 0 . 5 5 ( a f t e r M a g e r 1~)( c ) E q u a t i o n ( 1) w i t h K = 0 . 6 7( d ) E q u a t i o n ( 2) w i t h K = 0 ' 5 5 {s a id b y M a g e r t o b e a s u f f i c ie n t a p p r o x i m a t i o n t o (b ) }( e ) E q u a t i o n ( 7 ) w i t h J = 0 . 6( f) E q u a t i o n ( 7 ) w i t h J = 0 . 5 6( g ) E q u a t i o n ( 8 ) w i t h J = 0 - 6( h ) E q u a t i o n ( 8 ) w i t h J = 0 . 5 6 .

I t c a n b e s e e n t h a t c u r v e s ( b ) , ( e ) a n d ( g ) a l l o v e r e s t i m a t e t h e v a l u e o f Ps /P1 . C u r v e s ( f ) a n d ( h )s h o w t h a t w h e n J = 0 . 5 6 , t h e r e i s l it tl e d if f e r en c e b e t w e e n e q u a t i o n s ( 7 ) a n d ( 8 ), a n d t h e s i m p l e rf o r m o f e q u a t i o n ( 7) m a y b e p r e f e r r e d . Q u i t e g o o d a g r e e m e n t w i t h t h e e x p e r i m e n t a l d a t a is g i v e nb y c u r v e s ( c ), ( f) a n d ( h ) , e x c e p t t h a t n o n e o f t h e e q u a t i o n s f it s s a ti s f a c t o r il y w h e n M 1 < 1 . 7 .

A n e x p e r i m e n t a l c u r v e o f Ps/P2 f o r f o r w a r d - f a c i n g s te p s h a s b e e n o b t a i n e d b y c o m b i n i n g t h ed a t a o f F i g s . 2 a a n d b . F i g . 2 c t h e n c o m p a r e s t h e f o l l o w i n g c u r v e s f o r P.~/P2:( i) E x p e r i m e n t a l m e a n l in e d e r i v e d a s a b o v e( j) E q u a t i o n ( 3 ) f o r K = 0 . 5 5 ( a f t e r M a g e r ~4)( k ) E q u a t i o n ( 3 ) f o r K = 0 " 6 7( 1 ) E q u a t i o n ( 1 0 )

( m ) E q u a t i o n ( 1 1 ) i n c o n j u n c t i o n w i t h t h e e x p e r i m e n t a l c u r v e o f P1/P~ a s u s e d i n F i g . 1 .E q u a t i o n ( 3) w a s e v a l u a t e d u s i n g c o r r e c t s h o c k d e f l e ct i o n re l a ti o n s f o r ~ , r a t h e r t h a n ' M a g e r ' sa p p r o x i m a t i o n i n e q u a t i o n ( 4 ). F i g . 2 c s h o w s t h a t n o n e o f t h e r e l a t i o n s g i v i n g c u r v e s ( j ) t o ( m ) i sv e r y s a t i s f a c t o r y . E q u a t i o n ( 3 ) i s i n s e n s i t i v e t o t h e v a l u e o f K , a n d e v i d e n t ly m u c h u n d e r e s t im a t e sthe r . a t io P~/Ps. A sl ig h t i m p r o v e m e n t w o u l d b e o b t a i n e d b y u s e o f t h e e x p r e s s io n

P--~ = 1 .0 5 + 0" 05 M ~. (12)P sF i n a l l y , F i g . 3 g iv e s c u r v e s f o r P1/P~ a s f o l l o w s :

( n ) E x p e r i m e n t a l m e a n l in e f o r f o r w a r d - f a c i n g s t e p s f r o m F i g . 2 b( o ) E q u a t i o n ( 5 ) f o r K = 0 - 5 5 ( a f t e r M a g e P 4 )( p ) E q u a t i o n ( 6 )(q ) M2 /M~ = 0 . 7 6 2( r) E q u a t i o n ( 7 ) w i t h J = 0 . 5 6 i n c o n j u n c t i o n w i t h e q u a t i o n (1 2 )( s) E q u a t i o n ( 8 ) w i t h J = 0 . 5 6 i n c o n j u n c t i o n w i t h e q u a t i o n ( 1 2) .

A l s o e v a l u a t e d , b u t n o t s h o w n i n t h e f i g u r e , w a s( t) e q u a t i o n ( 1 ) w i t h K = 0 . 6 7 i n c o n j u n c t i o n w i t h e q u a t i o n (1 2 ) .

C u r v e ( o ) b e a r s l i t t l e r e s e m b l a n c e t o t h e e x p e r i m e n t a l l i n e ; t h e r e i s n o p a r t i c u l a r r e a s o n w h y i ts h o u l d , s i n c e i t i s m a d e u p o f t w o e x p r e s s i o n s a t le a s t o n e o f w h i c h { c u r v e ( d ) o f F ig . 1 } h a s b e e n

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f o u n d t o b e i n a c c u r a t e . C u r v e ( p ) f r o m t h e e m p i r i c a l e q u a t i o n ( 6 ) gi v es t h e b e s t f it s o l o n g a sM 1 ~< 4 - 5 , a s i t w a s d e s i g n e d t o d o . F o r t h e c o m p l e t e r a n g e o f M 1 u p t o 6 , t h e r e i s li t tl e t o c h o o s eb e t w e e n c u r v e s ( q ), ( r) , (s ) a n d ( t) , a n y o f w h i c h a g r e e w i t h t h e e x p e r i m e n t a l l i n e a b o u t a s w e l l a si t s p e c u l i a r f o r m w i l l a l l o w .

S u m m i n g u p , o f t h e s e m i - t h e o r e t i c a l re l a t i o n s g i v e n :e q u a t i o n s ( 2 ) , ( 3 ) a n d ( 5 ) s h o u l d n o t b e u s e d ;e q u a t i o n ( 1) w i t h K = 0 . 6 7 , a n de q u a t i o n s ( 7 ) a n d ( 8 ) w i t h J = 0 . 5 6 a r e s a t i s fa c t o r y i n p r e d i c t i n g Ps/P1 ;

w h i l e o f t h e e m p i r i c a l e x p r e s s io n s :e q u a t i o n ( 6 ) a n d t h e r e l a t i o n M 2 / M 1 = 0 - 7 6 2 a r e b o t h o f s o m e v a l u e i n d e t e r m i n i n g P2/P1 f o r

f o r w a r d - f a c i n g s t e p s ;e q u a t i o n s ( 1 0 ) a n d ( 1 1) a re n o t v e r y s a t i s f a c t o r y , a n de q u a t i o n ( 1 2 ) i s s l i g h t l y b e t t e r i n g i v i n g P J P s .

2 .1 .3 .a n a l y t ic a l t r e a t m e n t t h a n i s t h e t u r b u l e n t .

G a d d 17 d e r i v e s t h e f o l l o w i n g e x p r e s s i o n f o r a n i n c i d e n t s h o c k

w h e r e

L a m in a r f l o w . - - T h e c a s e o f a l a m i n a r b o u n d a r y l a y e r i s m o r e s u s c e p t i b l e t o e x a c t{ I 1 f ,j } l /20 6 3 6 t a n l { ( )C1~ ~. = 1 . 5 6 Z 1/2

M 11 + 0 . 6 93 ( y - 1 ) M 1 ~

z = [ ( M 1

a n d m o r e g e n e r a l l y 19 f o r s e p a r a t i o n i n a n y t w o - d i m e n s i o n a l s u p e r s o n i c f l o wCp, 8 = 1 . 1 3 Z .

A n o t h e r a n d l e s s r i g o r o u s a n a l y s i s b y H a k k i n e n et a110 g i v e s t h e a l m o s t i d e n t i c a l r e l a t i o nCj,,s = 1 . 1 5 Z

w h i l e t h e s a m e a u t h o r s p r e s e n t a n a r g u m e n t s u g g e s t i n g t h a tC p 2~/2 < C' i~s < 1 "9

o n t h e b a s is o f w h i c h t h e y p r o p o s e

(13)

Cp, 2 = 1 " 9 Z .T h e i r o w n d a t a f o r i n c i d e n t s h o c k s a t M 1 --~ 2 a g r e e q u i t e w e l l w i t h t h e i r r e l a t i o n s f o r b o t h Cp,a n d C p , 2 . O t h e r a n d m o r e c o m p r e h e n s i v e d a ta u p t o M 1 = 3 . 5 f r o m t h e w o r k o f C h a p m a n et al 1h a v e b e e n s h o w n t o f i t

C ~ .~ = 0 .9 3 Z ( d u e t o G ad d 19)Cp, ~ = 1 . 8 2 Z ( d u e t o S t e r r e t t a n d E m e r y 11 ) .

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Gu man 2 suggestsCp , 2 = 2.2Z

as fitting the same data.Taking the average of all experimentally supported values except Gum an's , we get

1.04Cp, 8 = [(M12- 1)Rex]it 4 (14)an d 1.86

Cp ,~ = [(M12_ 1 ) R e x ] l l ~ . (15)It is worth noting that equations (13) and (14) give coefficients of Z which agree quite closely

(within + 6 per cent in the range 2 < M 1 < 5, taking values of y between 1 .2 and 1.4), so that therather tedious computation involved in the use of equation (13) can be avoided.

2.2. E f f e c t o f R e y n o l d s N u m b e r .In the foregoing review of separation criteria, equations (1) to (11) for a turbulent boundary layer

are all independent of Reynolds number, and equations (13) to (15) for a laminar b oundary layer allhave a dependence on R e x -~ /~ .General analyses of the effect of Reynolds number have been attempted by several workers.The earliest is that given by Donaldson and Lange6 for the case of shock-i nduced separation, whoadvance the suggestion that the 'critical' pressure coefficient (defined as being based on the pressurerise across a shock wave which just causes separation of the boundary layer) is proportional to theskin-friction coefficient. Hence

Cp,orit OCR e x - l / ~ in laminar floworC~,cri~ oc Rex- l /5 in turbulent flow.

These expressions are apparently intended to take full account of Mach nu mber as well as Reynoldsnumber. The same authors cite work by Stewartson ~ as predicting that for laminar flow

C p oc R e x - 2 / s .Apart fro m those relations given above, there seems to be general unanim ity in favour of the squareroot of the skin-friction coefficient, with some additional dependence on Mach n umber. This leadsto expressions, for constant Mach number, of the form

C p o: R e x - l l ~ in laminar flowo r C p o c R e x - l / l o in turbulent flow.It will be observed that the laminar result is in agreement with equations (13) to (15). Chapmane t a P give their derivation of the fo rm

oc clas applying generally to any free interaction, which therefore covers both CrJ ~ and Cp , 2 in laminarflow and Cp,8 in turbulent. The y add a rather tentative suggestion of (3/1 z - 1) -1t4 for the furt herMach number dependence, which is also in line with equations (13) to (15). A later analysis byErdos and Pallone16 produces the same indices of - 1/4 and - 1/10 for Reynolds number.

K. Stewartson. On the interaction between shock waves and boundary layers. Proc. Cambridge Phil . Soc. ,Vol. 47, Part 3, p.545, July, 1951.

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A great deal of the available experimental evidence covers too narrow a range of Reynolds numberand exhibits too much scatter for any certain correlation to be made. In this category for laminarflow come the data from Refs. 6, 7, 10 and 11. Donaldson and Lange 6 claim that the data theysho wsupport their predicted dependence on R e x - i/ 2, but the correlation is unconvincing. All that canreally be said of the laminar data published by Hakkinen et al 1 and Sterrett and Emeryn is that abetter fit is given by the index - 1/4 than by 0. The most comprehensive correlations are thosepresented by Chapman et al ~, and these leave little doubt that R e x -~14 is at least very close to thetruth for both Cp, s and C~, ~ in laminar interactions.

Turning to turbulent separation, data on the effect of Reynolds number can be found in Refs.1 to 3, 6 to 9, and 11. Once again Donaldson and Lange 6 claim to find agreement between theirprediction of R e x -~ /a and a collection of experimental data at varying Mach number, of which theirown forms the whole of the high Reynolds number end. However, the tren d shown is more obviouslywith Mach number than with Reynolds number, and the validity of their own data is in any casecalled in question by LangC. Other collections7, 9, ~1 of data relating to Cp, ~ create a fairly clearimpression that a large dependence on Mach number exists but none at all on Reynolds number.

Chapman e t a l i publish data in the form Co,.~ versus R e x , and show a quite good and distinctcorrelation of this quanti ty with R e x - i ll for forward-facing steps and compression corners. W hen,however, they come to plot Cp. ~ the picture changes somewhat, a fact which they associate with itsexclusion from the category of a free interaction. The effects they observe in the latter case are avariation of the Reynolds number index with Mach number and type of model, both above andbelow the value -1/10, with zero in several cases. Similarly Kuehn~,a shows that the overallpressure ratio for incipient separation at compression corners, curved surfaces and incident shocks,does not depend in a consistent manner on Reynolds number, but is again subject to changes withMach number and geometry. In general the trend is towards zero dependence when the Machnumber is low and the Reynolds number high, with a quite large effect being experienced whenthe Mach number is high and the Reynolds number low.

All this is rather a nuisance if one is seeking a general criterion for use in other circumstances.One may note, however, that there is no disagreement withCI~, ~ ~: R e x

for a tu rbu len t bounda ry layer at M i ~< 2, and that no lower index than - 1/10 is found up toM i = 2.5. This is quite trifling, and for approximate estimates with in that range of M 1 no significanterror should result from neglecting the effect altogether.

3 . R eg i m es o f N o z z l e O p er a ti o n .As the applied pressure ratio of a convergent-divergent nozzle with internal expansion is reduced

from above its design pressure ratio towards unity, it is well known that four flow regimes occursuccessively:

(i) The flow is everywhere attached to the walls, when the nozzle is said to be running full.Either an expansion fan or an oblique shock is located at the nozzle outlet, dependingupon whether the internal flow is under- or overexpanded. Transition from this regimeto the next occurs at the 'kink point', at which the oblique shock commences to move insidethe nozzle.

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(ii) The flow separates from the walls inside the divergent portion of the nozzle, as a result ofinteraction between the internal oblique shock and the boundary layer.

(iii) Separation takes place close to the nozzle throat, the accompanying compression tendingin form towards a normal shock, followed by subsonic diffusing flow. The change fromregime (ii) is of course gradual, but various approximate estimates have been made of thecondi tion at which a difference in flow model must be in trodu ced. Arens and Spiegler 22, 2ahave recently suggested for this case M 1 < 1.13, while an earlier paper =4 proposed taking anormal shock at M 1 = 1.15 as the 'limiting condit ion eventua lly reached as the shocksystem moves upstream in any nozzle' (see also Section 5).

(iv) The nozzle ceases to be choked, and runs with subsonic flow throughout.In the present work we are concerned primarily with regime (ii), in which a parallel may be foundto other models with separation in supersonic flow.

4. Overexpanded Nozzles .Two salient differences set nozzles apart from the other models described in Section 2.

1. The fluid is expanding, and the pressure gradient ahead of the interaction is not zero.2. Re-at tachment generally does not occur, and the separation bubble is extended into a'mixing region' co ntinuing to the nozzle outlet (station b, Fig. 4), wit h ambient air entering

to take part in the recirculatory flow system set up.Some further pressure rise is known to take place along the wall in this mixing region,

and the amo unt may be expected to depend on the angle and Mach num ber of the separatedmain jet downstream of the interaction, on the wall angle, and on the level of Reynoldsnumber controlling the mixing processes in the shear layer. This further rise in the mixingregion has been called 'pressure recovery', having in mind some loose analogy with thebehaviour of diffusing subsonic flow, and the terminology is retained here for convenience.In amount it is usually found to be small.

If, in the absence of data, the possible effect of a favourable pressure gradient in increasing theallowable shock strength be neglected, then there seems little doubt that the pressure rise in anozzle should follow that in other models so far as they remain free interaction s--i.e, up to station sin turbulent flow and station 2 in laminar. Beyond s, in the case of a turbulent boundary layer, theexact similarity in behaviour breaks down; but among various geometries of model some approximatecorrespondence was noted previously (Section 2) in the additional pressure rise to station 2, wh ichmay be regarded as the boundary dividing separation from re-attachment. It therefore seems quitelikely that the pressure rise within the interaction region of a nozzle, which is experienced as theseparation bubble develops, may also show some agreement as far as station 2. This it will be thepurpose to demonstrate.

With subsequent and dissimilar pressure rises around re-attachment the present paper is happilyunconcerned; instead there is the largely unkn own quantit y of the mixing pressure recovery. In thefirst instance, lacking any systematic measurements or method of theoretical prediction, noquantitative allowance for this final phase of pressure rise in a nozzle can be introduced. But wherethe amount of recovery must be small, as for instance in nozzles of rather wide divergence angle orwith separation fairly close to the outlet, one can reasonably look for agreement between the overallpressure rise from 1 to b (Figs. 4 and 5) and those cri teria which effectively describe behav iour upto station 2.

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I n c e r t a i n sp e c i a l ca s es t o b e d i s c u s s e d l a t e r, t h e p r e s s u r e r e c o v e r y c a n a s s u m e m u c h g r e a t e ri m p o r t a n c e , a n d t h e s e e x c e p t i o n s to t h e g e n e r a l t r e a t m e n t w i ll b e n o t e d a s th e y a r is e .

4 .1 . Previous Correlations.A r e c e n t c o n t r i b u t i o n b y A r e n s a n d S p i e g l e r 22 b e c a m e a v a i l a b le a s t h e p r e s e n t p a p e r w a s i n c o u r s e

o f p r e p a r a t i o n . T h i s c o n s t i t u t e s a n a t t e m p t , a l t h o u g h n o t t h e f i rs t , t o a p p l y t h e r e s u l ts o f w o r k o no t h e r s e p a r a t i o n m o d e l s t o t h e c a s e o f a n o v e r e x p a n d e d n o z z l e , i g n o r i n g in t h e s a m e w a y a s o u r se l v e st h e e f fe c t s o f p r e s s u r e g r a d i e n t . I t is m e n t i o n e d f i rs t b e c a u s e A r e n s ' p i c t u r e i s s i m i l a r to t h a t s h o w ni n F i g . 4 ; h e d i v i d e s a n o z z l e in t o ' t h e i n it i al r e g i o n o f c o m p r e s s i o n t e r m i n a t i n g a t th e p o i n t o fs e p a r a ti o n a n d t h e s u b s e q u e n t m i x i n g r e g i o n w h e r e i n t h e s e p a r a t e d f l o w m i x e s w i t h e n t r a i n e d a i rf r o m a t m o s p h e r e ' . O t h e r i n t e r a c t i o n m o d e l s a r e a ls o r e f e r r e d t o a s c o m p r i s i n g z o n e s o f s e p a r a t i o na n d m i x i n g , w h e r e i n th a t c a s e ' m i x i n g i s t e r m i n a t e d b y f u r t h e r c o m p r e s s i o n a s t h e f l o w r e - a t t a c h e s ' .

F r o m t h e a b o v e d e s c r i p t i o n it a p p e a r s t h a t A r e n s a n d S p i e g l e r r e g a r d t h e w h o l e p r e s s u r e r i s e i n an o z z l e b e y o n d t h e s e p a r a t i o n p o i n t , i . e . t h e r i s e f r o m s t o 2 a s W e l l a s t h a t f r o m 2 t o b, a s p a r t o ft h e ' m i x i n g ' , w h i c h t h e y t h e n p r o c e e d t o i gn o r e% A c o m p a r i s o n is p r e s e n t e d b e t w e e n t h e o v e r a llp r e s s u r e r i s e t o n o z z l e o u t l e t f r o m t h e d a t a o f n u m e r o u s w o r k e r s a n d t h e r e l a ti o n s f o r Ps/P1 i nt u r b u l e n t f l o w g i v e n b y e q u a t i o n s ( 7) a n d ( 8 ), t a k in g G a d d ' s o r i g in a l v a l u e o f J = 0 . 6 i n b o t h .R a t h e r s u r p r i s i n g ly , i n v i e w o f t h e a s s u m p t i o n s n o t e d a b o v e , fa i r ly g o o d a g r e e m e n t w i t h t h e m a s so f n o z z l e d a t a i s s h o w n . A c o n t r i b u t o r y r e a s o n f o r t h i s e v i d e n t l y a r is e s f r o m t h e t e n d e n c y , i l l u s t r a t e di n F i g . 1 , o f e q u a t i o n s ( 7 ) a n d ( 8) w i t h t h i s v a l u e o f J t o o v e r e s t i m a t e t h e r a t i o P J P 1 . T h e r e i sc e r t a i n l y n o i n h e r e n t r e a s o n w h y a n y v a li d r e l a ti o n f o r Ps/P1 s h o u l d a g r e e w i t h t h e o v e r a l l p r e s s u r er i se o c c u r r i n g i n a n o v e r e x p a n d e d n o z z l e . I n t h is c o n n e c t i o n i t m a y b e n o t e d t h a t A r e n s a n dS p i e g l e r 22 s h o w o n t h e s a m e g r a p h a c u r v e l a b e l l e d ' s e p a r a t i o n p r e s s u r e r a t i o ' f r o m R e f . 1 1, w h i c hl ie s w e l l a w a y f r o m t h e n o z z l e d a t a i n t h e d i r e c t i o n w h i c h t h e d i f f e r e n c e b e t w e e n s t a ti o n s s a n d 2w o u l d l e a d o n e t o e x p e c t . T h e a u t h o r s o f R e f . 2 2 s e e m t o r e c o g n i s e t h e f o r t u i t o u s a n d e m p i r i c a ln a t u r e o f t h e a g r e e m e n t w h i c h t h e y p r e s e n t , a s t h e y g o o n t o s u g g e s t th a t a m o r e a c c u r a t e v a l u e o fJ f o r u s e in e q u a t i o n ( 8 ) i s a r o u n d 0 . 5 6 .

M u c h e a r l ie r L e F u r 25 a p p l i e d t h e r e l a t io n M J M 1 = 0 . 7 6 2 f r o m R e f s. 9 a n d 1 5 w i t h f a i r s u c c e s st o s o m e r a t h e r l im i t e d e x p e r i m e n t a l s e p a r a t io n d a t a f r o m c o n v e r g e n t - d i v e r g e n t n o z z l e s, a s s u m i n gP 2 = P ~ .

O t h e r a t t e m p t s a t c o r r e l a ti o n o f n o z z l e d a t a h a v e b e e n e s s e n t i a ll y e m p i r ic a l , a n d t w o s e ts o fw o r k e rs 2 4 , 26 h a v e f o u n d a c o n v e n i e n t c r i t e r i o n t o b e c o n s t a n t f l o w d e f l e c t i o n a n g l e ( ;~ ) c o r r e s p o n d i n gt o t h e s e p a r a t i o n s h o c k , a g a i n a s s u m i n g P 2 = P b . A f a i rl y w i d e s u r v e y o f d a t a 24 o n s e p a r a t i o n i nn o z z l e s w i t h o b v i o u s l y t u r b u l e n t b o u n d a r y l a y e rs s u g g e s ts a v a l u e o f ;~ = 13.

L o v e 9 g i v es c u r v e s o f ) t c o r r e s p o n d i n g t o M 2 / M 1 = 0 . 7 6 2 , w h i c h s h o w a s u b s t a n t ia l l y c o n s t a n tv a l u e ( 13 + 1 ) f o r M 1 > 1 . 9 w h e n 7 = 1 . 4 t , b u t t h e r e i s a p r o n o u n c e d e f f e c t o f la r g e c h a n g e s i n7, ~ i n c r e a s i n g w i t h r e d u c t i o n o f 7 .

~ In ano ther ve r s ion o f the same pape r 2a, Arens speci fica l ly neg lec t s ' the smal l p ressure r i se occu r r ing int h e m i x i n g r eg io n b e t w ee n t h e s ep a r a ti o n p o i n t an d t h e n o zz l e ex i t p l an e ' .

-~ Ra ther m ore genera l ly , r e f erence to shock tab les (7 = 1 -4 ) shows tha t fo r any value o f Mac h n um be rra t io in th i s ne igh bourh ood , the cor r espo nd ing def lec t ion ang le ~ is near ly cons tan t once the in it i a l Ma chnu m be r i s f a i rly h igh , so tha t so me theore t ica l suppor t can be fo und fo r the eff icacy o f such a cr i te rion .

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T h i s p r e d i c t e d e f f e c t o f y o n s e p a r a t i o n i s o f c o n s i d e r a b l e i m p o r t a n c e , a n d i t is d i f fi c u l t t o o b t a i na n y v e r y c o n c l u s i v e p i c t u r e f r o m t h e e v i d e n c e s o f a r a v a i la b l e . I n g e n e r a l , t h e d a t a o f R e f s. 1 to 1 2a r e a ll f o r Y = 1 . 4 , a n d a l t h o u g h t h e v a r i o u s r e l a t i o n s q u o t e d i n p r e c e d i n g s e c t i o n s a re f u n c t i o n so f y , it s e f f e c t c a n n o t b e c h e c k e d f r o m c o n v e n t i o n a l s e p a r a t i o n m o d e l s . A si n g l e e x p e r i m e n t i s c it e db y L o v e 9 i n w h i c h h e l i u m ( y = 1 - 66 7 ) w a s u s e d i n a t w o - d i m e n s i o n a l n o z z l e w i t h s e p a r a t i o n a tM 1 = 3 " 48 , y i e l d i n g a v a l u e o f C p , 2 i n g o o d a g r e e m e n t w i t h t h a t p r e d i c t e d b y t h e r e l a t io nM 2 / M 1 = 0 . 7 6 2 - - a g r e e m e n t a t l e a st a s c lo s e a s g i v e n b y L o v e ' s o w n d a t a f o r 7 = 1 . 4 t o t h e s a m er e la t io n . T o t h a t e x t e n t p r e d i c t i o n i s a p p a r e n t l y b o r n e o u t .

A l l o t h e r s e p a r a t i o n d a t a i n w h i c h y d e p a r t s a p p r e c i a b l y f r o m 1 . 4 r e l at e t o r o c k e t n oz z le s w i t hl o w e r v a l u e s o f y . S u m m e r f i e l d et a126 r e p r o d u c e t h e r e s u l t s o f F o s t e r a n d C o w l e s 3a, f o r a x i s y m m e t r i cn o z z l e s w i t h M I > 3 a n d y = 1 . 2 3 , w h i c h t h e y s u g g e s t s h o w s o m e v a l u e o f ;~ a r o u n d 1 6 , b u t t h ec o r r e l a ti o n is p o o r. W h e n c o m p a r e d w i t h c o l le c t io n s o f o t h e r n o z z l e s e p a r a ti o n d a t a f o r w h i c hy = 1 . 4 , i t is f o u n d 24, 27 t h a t t h e r e s u l t s o f F o s t e r a n d C o w l e s d o n o t d i f f e r a t a l l s i g n i f i c a n t l y f r o mt h e g e n e r a l p a t t e r n .

O t h e r d a t a c i t e d b y S u m m e r f i e l d et a l 2G a r e t h o s e o f M c K e n n e y 34 o n t w o - d i m e n s i o n a l n o z z l es ,w i t h M 1 i n th e r a n g e 3 t o 3 . 4 a n d 7 = 1 . 4 . T h e s e r e s u lt s f o r m a m u c h b e t t e r c u rv e , g i v i n g v a l u e so f ;~ b e t w e e n 1 4 } a n d 1 3 , w h i c h a g r e e q u i t e w e l l w i t h t h o s e c o r r e s p o n d i n g t o t h e r e l a t i o nM 2 / M 1 = O . 7 6 2 a t t h i s 7 .

M a g e r 2v u s e s h i s e x p r e s s i o n f o r P 2 / P 1 { o b t a i n e d b y c o m b i n i n g e q u a t i o n s ( 2 ) a n d ( 3 ) , w i t hK = 0 . 5 5 } t o c o r r e l a te a v a r i e t y o f n o z z l e d a t a i n c l u d i n g h i s o w n , b u t t h e f o r m o f p r e s e n t a t i o n i si n s u f f i c i e n t l y s e n s i t iv e t o g i v e a n y t r u e i n d i c a t i o n o f it s m e r i t s i n th e r e g i o n w h e r e m o s t d a t a e x i s t( M 1 < 3 . 5 ) . A t h i g h e r M a c h n u m b e r , s o m e f e w d a t a e f o r l o w 7 d o i n d e e d f a l l a r o u n d M a g e r ' sc o m p u t e d c u r v e s f o r 7 = 1 . 2 a n d 1 . 2 5 a n d w e l l a w a y f r o m t h a t fo r 7 = 1 . 4 , b u t i t i s n o t y e t c l e arw h e t h e r t h i s d e v i a t i o n is c a u s e d b y v a r i a ti o n o f 7 o r b y t h e u s e o f a f a u l t y e q u a t i o n (see S e c t i o n2 . 1 . 2 an d F i g . 3 ) .

4 .2 . R e y n o l d s N u m b e r C l ia n g e a lo n g a N o z z l e .I n e x p e r i m e n t s r e l a t i n g t o s e p a r a t i o n i n f l o w w i t h i n i t i a l l y z e r o p r e s s u r e g r a d i e n t , t h e l o c al s t a t e

o f t h e b o u n d a r y l a y e r i s u s u a l l y r e p r e s e n t e d b y t h e R e y n o l d s n u m b e r b a s e d o n d i s t a n c e f r o m t h el e a d i n g e d g e o f t h e s u r f a c e. W h e n c o n s i d e r i n g a n oz z l e w i t h f i n it e p r e s s u re g r a d i e n t a n d w a l ld i v e r g e n c e , i t i s n e c e s s a r y t o d e t e r m i n e a n ' e q u i v a l e n t f l a t - p l a t e l e n g t h ' J~. S t r a t f o r d a n d B e a v e r s e sp r o p o s e a m e t h o d f o r c a l c u l a t in g t h is l e n g t h f o r a x i s y m m e t r i c f l o w w i t h a t u r b u l e n t b o u n d a r y l a y e r,w h i c h y i e l d s t h e f o l l o w i n g r e l a ti o n f o r 7 = 1 . 4

J o l+0-2M JF o r t w o - d i m e n s i o n a l f l o w w i t h a t u r b u l e n t b o u n d a r y l a y e r, th e e x p r e s s io n r e d u c e s t o

X = [ 1 + 0 " 2 M 2 7 ~ ( x M- ] J o I 1 + 6 7 " 2M Z l d x . ( 1 7 )* Produc ed b y Elko and Stary , c i t ed in R e 32 , o r ig inal no t avai lab le .t Fo r def in i t ion see Ap pendix I I .

r17

( 9 1 9 4 9 )

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C o m p a r a b l e r e l a ti o n s fo r a l a m i n a r b o u n d a r y l a y e r a r e d e v e l o p e d i n A p p e n d i x I I I , g i v in g f o ra x i s y m m e t r i c f l o w

- - M ~ 0 1 + 0 - 2 M ~ M y 2 d x (18)a n d f o r t w o - d i m e n s i o n a l f l o w

X = [ 1 + 0 " 2 M ~ ] 4 1 ( ~ F M - 1 4~7.2M2 M dx (19)J M 3 0 L 1 + I n e a c h ca s e t h e t r e a t m e n t a s s u m e s t h a t y = 1 . 4 .

I t c a n b e s h o w n ( e. g. A p p e n d i x I V o f R e f. 5 1) t h a t t h e e f fe c t u p o n c o n d i t i o n s f a r t h e r d o w n an o z z l e o f i g n o r i n g t h e b o u n d a r y - l a y e r t h i c k n e s s a t t h e t h r o a t i s g e n e r a l l y s m a l l . C o n s e q u e n t l y , i t i sa s s u m e d t h r o u g h o u t t h i s w o r k t h a t t h e t h r o a t b o u n d a r y - l a y e r th i c k n e s s is z e ro .

F i g . 6 g iv e s t w o e x a m p l e s o f t h e v a r i a ti o n o f R e y n o l d s n u m b e r d o w n t h e d i v e r g e n t p a r t o f an o z z l e , b a s e d u p o n t h e e q u i v a l e n t f l a t - p l a t e l e n g t h a s a b o v e (Rex) . F o r c o n v e n i e n c e t h e t h r o a tR e y n o l d s n u m b e r (Re*) h a s b e e n u s e d a s r e fe r e n c e q u a n t i t y , a n d t h e d i ff e r e n ce b e t w e e n l a m i n a ra n d t u r b u l e n t b o u n d a r y l a y e r s i s s h o w n .

5. No zzle Separation Da ta: Quiescent Air.T h e f o r m o f p r e s e n t a t i o n a d o p t e d r e l a te s th e o v e r a l l s e p a r a t i o n p r e s s u r e r a t i o Pt/Po ( d e n o t e d b y k ;

see F i g s . 4 a n d 5 ) t o n o z z l e a p p l i e d p r e s s u r e r a t i o ( A . P . R . = P#Po). K n o w l e d g e o f t h e c o m p l e tep r e s s u r e r i se f r o m s t a t i o n 1 t o b i s w h a t a p o w e r - p l a n t d e s i g n e r r e q u i r e s , a n d i s a ll t h a t t h e r e l a t iv e l yc r u d e i n s t r u m e n t a t i o n i n g e n e r a l u s e p e r m i t s t o b e m e a s u r e d . I n c o m p a r i n g n o z z le s e p a r a ti o n d a t ai n t h e s e t e r m s w i t h t h e r e s u l t s f r o m o t h e r m o d e l s , i t m u s t b e r e m e m b e r e d f r o m S e c t i o n 4 t h a t n os p e c i a l a l l o w a n c e c a n a t p r e s e n t b e m a d e f o r t h e p r e s s u r e r i s e f r o m s t a t i o n 2 t o b o c c u r r i n g i n an o z z l e . W i t h t h i s l i m i t a t i o n in m i n d , i t m a y s t il l b e i n s t r u c t i v e t o e x a m i n e h o w v a l u e s o f t h e r a t i oP1/P2 o b t a i n e d a c c o r d i n g t o t h e e x p r e s s i o n s g i v e n i n S e c t i o n 2 . 1 a g r e e w i t h t h e q u a n t i t y k a s a b o v e .T h e r e is o f c o u r se a d i r e c t c o n n e c t i o n b e t w e e n M 1 a n d P#P1 ( = A . P . R . /k ) , w h i le th e a s s u m p t i o no f P~ = P o e n a b l e s a n y r e l a t i o n in t e r m s o f M 1 , M 2 o r ;~ t o b e t r a n s l a t e d i n t o o n e b e t w e e n k a n dA . P . R .

F i g s . 7 t o 1 0 s h o w a c o l l e c t io n o f d a t a o b t a i n e d f r o m n o z z l e s w i t h c o n s t a n t d i v e r g e n c e h a l f - a n g l e si n t h e r a n g e 1 0 to 3 0 , t a k e n f r o m R e f s . 2 7 a n d 3 2 to 4 2 . I n m o s t c a s e s v a l u e s o f k a re q u o t e dd i r e c t l y , o r p r e s s u r e d i s t r i b u t i o n s a r e a v a i l a b l e f r o m w h i c h t o o b t a i n i t . R e f . 3 4 , h o w e v e r , g i v e ss e p a r a t i o n a r e a r a t i o s w h i c h h a v e b e e n c o n v e r t e d t o p r e s s u r e r a t io s b y m e a n s o f i s e n t r o p i c r e l a t io n sf o r 7' = 1 . 4 ; a n y e r r o r s o i n t r o d u c e d s h o u l d b e w e l l w i t h i n t h e g e n e r a l s p r e a d o f r e s u l ts .

O n e p o i n t w h i c h s h o u l d b e n o t e d c o n c e r n s t h e v a l u e s o f y, q u o t e d f o r th e d a t a i n F i g s . 7 t o 1 0 , an u m b e r o f w h i c h a r e a r o u n d 1 . 2 . T h e s e v a l u es r el a te to r o c k e t - m o t o r e x h a u s t a n d a r e , so f a r asc a n b e a s c e r t a i n e d , f o r t h e i n i ti a l c o n d i t i o n a t e n t r y t o t h e n o z z l e . S i n c e t h e e f f e c t o f ~ i s t o r e c e i v es o m e a t t e n t i o n in t h e f o l l o w i n g p ag e s , it c o u l d w e l l be a s k e d w h e t h e r 7 m a y n o t h a v e r i s e n m u c hn e a r e r t o 1 . 4 d u r i n g t h e e x p a n s i o n p r o c e s s p r e c e d i n g s e p a ra t io n . T h i s , h o w e v e r , s e e m s n o t t o b et h e c a s e : a f a i r ly t y p i c a l e x a m p l e t a k e n f o r t h e p r o p e l l a n t m i x t u r e o f R e f . 38 s u g g e s t s t h a t o v e r a ne x p a n s i o n p r e s s u r e r a t io o f 1 0 0 0 t h e r e l e v a n t ~, i n c r e as e s b y l e s s t h a n 7 p e r c e n t . O n e s h o u l dt h e r e f o r e b e j u s t i f i e d i n r e g a r d i n g t h e q u o t e d v a l u e s o f 7 a s r e p r e s e n t a t i v e o f t h e w h o l e f l o w s e q u e n c eo f e x p a n s io n a n d s u b s e q u e n t s h o c k c o m p r e s s i o n o c c u r r i n g in t h e n o z z l e.

1- Th e au thors are inde b ted to the R ocket Propu l s ion Es tab l i shment , Wes tco t t , fo r th is in fo rm at ion .18

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A c e r t a i n a m o u n t o f t h e r a t h e r w i d e s c a t t e r a p p e a r i n g i n F i g s . 7 t o 1 0 is n o d o u b t c a u s e d b yf a i lu r e to a c c o u n t p r o p e r l y f o r t h e p r e s s u r e r i se o c c u r r i n g i n t h e m i x i n g r e g io n d o w n s t r e a m o f t h ei n t e r a c t i o n (see S e c t i o n 5 .2 ). P r e s s u r e d i s t r ib u t i o n s p u b l i s h e d b y m a n y a u t h o r s s h o w s o m e v a r i a t i o ni n t h e e x t e n t o f t h i s r is e , w h i c h m i g h t d e p e n d u p o n d i v e r g e n c e a n g le a n d l e n g t h o f m i x i n g r e g io n ,i .e . u p o n a r e a r a t io o r d e s i g n p r e s s u r e r a ti o . W i t h t h i s e x c e p ti o n , n o d e p e n d e n c e o n n o z z l e D . P . R .w o u l d b e e x p e c t e d , a s m a y b e r e a li s ed b y c o n s i d e r i n g F i g . 5. W i t h a t u r b u l e n t b o u n d a r y l a y e r t h ei n i t i a l p r e s s u r e r i s e i s s t e e p , a n d w e r e i t n o t f o r t h e s u b s e q u e n t ' r e c o v e r y ' a n i d e a l i s e d p r e s s u r ep a t t e r n w o u l d b e a v e r t ic a l li n e a t s e p a r a t i o n , f o l l o w e d b y a h o r i z o n t a l li n e c o n t i n u i n g t o o u t l e t .W i t h s u c h a m o d e l a c h a n g e i n D . P . R . o n l y a ff e c ts t h e e x t e n t o f t h e h o r i z o n t a l li n e, s o th a t t h ep r e s s u r e P 1 a n d h e n c e k c a n b e s e e n t o d e p e n d o n l y o n A . P . R .

A m o n g t h e e x p e r i m e n t a l s c a t t e r n o s i g n i f ic a n t i n f l u e n c e c a n b e f o u n d o f a n y o f t h e f o l l o w i n gp r o p e r t i e s :

( i ) N o z z l e D . P . R .( ii ) W a l l d i v e r g e n c e i n t h e r a n g e 1 0 to 3 0 h a l f - a n g l e .

( ii i) A x i s y m m e t r i c or t w o - d i m e n s i o n a l f o r m ( F i g . 7 ).( iv ) 7 i n th e r a n g e 1 . 2 t o 1 . 4 ( F i g s . 7 a n d 1 0, n o t i n g t h a t s o l i d s y m b o l s a r e u s e d t o d e n o t e

v a l u e s o f 7 n e a r 1 . 2 ) .C o m m e n t s m a d e a b o v e c o v e r t h e f i rs t o f t h e s e o b s e r v a ti o n s . T h e t h i r d i s al so n o t r e m a r k a b l e , s i n ceb o u n d a r y - l a y e r s e p a r a ti o n c a n b e r e g a r d e d a s l o c a ll y t w o - d i m e n s i o n a l , w h a t e v e r t h e g e o m e t r y o ft h e s u r f a c e a s a w h o l e . A s i m i l a r c o r r e s p o n d e n c e w a s n o t e d i n S e c t i o n 2 b e t w e e n t h e b e h a v i o u r o ft w o - d i m e n s i o n a l a n d a x i a l l y - s y m m e t r i c v e r s io n s o f o t h e r s e p a r a t io n m o d e l s. I t i s th e f o u r t h p o i n tt h a t m a y e x c i t e s o m e s u r p r i s e , c a l l in g f o r f u r t h e r a t t e n t i o n .

T h e s u r p r is e i s l a r g e ly o c c a s io n e d b e c a u s e w e h a v e b e e n c o n d i t i o n e d b y t h e f o r m o f t h e v a r i o u ss e p a r a t i o n c r i t e r i a g i v e n i n S e c t i o n 2 .1 .1 * t o e x p e c t a d e p e n d e n c e o n 7 . U p o n r e f l e c t i o n , t h e a b s e n c eo f a n y d i s t i n g u i s h a b l e e f f e c t a m o n g s t t h e d a t a d o e s n o t i n i ts e l f s e e m i n c r e d i b l e . S o f a r a s t h eb o u n d a r y l a y e r c a n b e c o n s i d e r e d i n c o m p r e s s i b le , t h e r e w i l l b e n o i n f l u e n c e o f 7 o ll s e p a ra t i o n ;t h i s c a n o n l y a r is e t h r o u g h t h e t r a n s f o r m a t i o n n e c e s s a r y t o ta k e a c c o u n t o f c o m p r e s s i b il i ty , a n d i t isu n l i k e l y t h a t w e n e e d l o o k f o r th e e f fe c t t o b e l a r ge i n m a g n i t u d e . T h a t n o n e a p p a r e n t l y e x is ts i s af a c t w h i c h r e f le c t s m o s t l y u p o n t h e v a l i d it y o f t h e c r it e ri a .

T h r e e o f th e m o r e s a t is f a c t o ry re l a ti o n s ( S e c t io n 2 . 1 .2 ) h a v e b e e n u s e d , n a m e l y :( i) E q u a t i o n (1 ) w i t h K = 0 . 6 7 ] i n c o n j u n c t i o n w i t h e q u a t i o n ( 1 2 )( ii ) E q u a t i o n (7 ) w i t h J = 0 . 5 6

(iii) M 2 / M 1 = 0 . 7 6 2 .A s c a n b e s e e n f r o m F i g s . 7 a n d 8 , a n y o f t h e s e r e l a ti o n s w i t h 7 = 1 . 4 a d e q u a t e l y f i t t h e m a s s o fd a t a, s u g g e s t i n g t h a t t h e p r e s s u r e r is e a t s e p a r a ti o n i n o v e r e x p a n d e d n o z z le s o f u n i f o r m d i v e r g e n c eg e n e r a l ly fo l l o w s t h a t i n o t h e r m o d e l s u p t o s t a t io n 2 . I n o r d e r t o s h o w t h e a g r e e m e n t m o r e c l e a r ly ,t h e d a t a f o r 7 ~ 1 . 2 , w h i c h e x h i b i t m u c h t h e w o r s t s c a t t e r ( F i g . 1 0), h a v e b e e n o m i t t e d f r o m F i g . 8 .A l s o s h o w n o n F i g s . 7 a n d 8 i s t h e c u r v e o f h = 13 a s s u g g e s t e d i n R e f . 2 4 ; t h i s i s n o t a s a t i s f a c t o r yf i t a t v a l u e s o f A . P . R . b e l o w 4 o r a b o v e 6 0 .

See also Sect ion 4.1.19

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A n o t h e r l i n e o n F i g . 7 re l a t e s t o t h e l i m i t in g c o n d i t i o n o f a n o r m a l s h o c k r e a c h e d w h e n s e p a r a t i o no c c u r s c l o s e t o t h e n o z z l e t h r o a t (see S e c t i o n 3 ) . I n t e r s e c t i o n o f t h i s l in e w i t h t h e g r o u p o f th r e eo t h e r c u r v e s o c c u r s a r o u n d M 1 = 1 . 1 6 .

F i g . 9 s h o w s j u s t t h o s e d a t a f o r w h i c h 7 ~ 1 - 2 t a k e n f r o m F i g . 1 0, t o g e t h e r w i t h t h e s a m e t h r e er e l a ti o n s as a b o v e f o r t h e v a l u e 7 = 1 - 2 . T h e r e i s n o w a m u c h w i d e r s p r e a d b e t w e e n t h e t h r e ec r i te r i a t h e m s e l v e s t h a n w a s t h e c a s e w i t h y = 1 . 4 , a n d l i t tl e p r e t e n c e a t ' a g r e e m e n t w i t h t h e d a t a.O n e h a s t h e r e f o r e to c o n c l u d e t h a t n o n e o f th e f o r e g o i n g r e la t i o n s f o r a t u r b u l e n t b o u n d a r y l a y e rm a y p e r m i s s i b l y b e u s e d w i t h y o t h e r t h a n 1 - 4 .

I n s o m e a n a ly t ic a l t r e a t m e n t s - - f o r e x a m p l e t h a t o f R e f. 1 5 gi v in g c o n s ta n t M a c h n u m b e r r a t i o - -t h e q u a n t i t y 7 d o e s n o t e x p l i c it l y a p p e a r , a n d t h e r e i s n o r e a s o n t o a s s o c ia t e t h e v a l i d i ty o f t h i s f o r mo f c r i t e r io n w i t h p a r t i c u l a r v a l u e s o f y . H o w e v e r , t h e e m p i r i c a l c o n s t a n t c h o s e n t o g i v e a g r e e m e n tw i t h e x p e r i m e n t a l d a t a c o u l d v e r y w e l l v a r y w i t h 7 . T h e r e i s c e r t a i n ly n o j u s t i f i c a t io n f o r t a k i n gt h e c o n s t a n t 0 . 7 6 2 i n th e a b o v e e x a m p l e as a p p l y i n g b e y o n d t h e c o n d i t i o n o f y = 1 . 4 f o r w h i c h i tw a s d e r i v e d ; l i k e w i se t h e c o n s t a n t s K a n d J i n e q u a t i o n s ( 1) , (7 ) a n d ( 8) c o u l d a n d e v i d e n t l y s h o u l dd e p e n d o n ~ .

N o t e w a s m a d e i n S e c t i o n 4. 1 o f a s i n gl e e x p e r i m e n t a l v a l u e o b t a i n e d v e it h h e l i u m f o r w h i c hy = 1 . 6 6 7 , t h a t w a s s h o w n b y L o v e 9 t o s u p p o r t c l o s e ly t h e r e l a t i o n M J M 1 = 0 . 7 6 2 . T h i s p o i n ti s i n c l u d e d o n F i g . 7 , a n d a p p e a r s v e r y n e a r t o o t h e r d a t a f o r 7 = 1 . 2 3 a n d 1 . 4 o n t h e f r i n g e o ft h e s c a t t e r . I t s e e m s t h e n t h a t t h e a g r e e m e n t i n R e f . 9 is f o r t u i t o u s .

O n t h e e v i d e n c e o f F ig s . 7 a n d 1 0, w e m a y c o n c l u d e f a i r ly d e f i n i te l y t h a t t h i s f o r m o f p r e s s u r e - r i s ec h a r a c t e r i st i c is i n d e p e n d e n t o f y in t h e c a s e o f a t u r b u l e n t b o u n d a r y l a y e r .

I t i s w o r t h o b s e r v i n g t h a t t h e r e l a t io n s g i v e n f o r l a m i n a r s e p a r a t i o n a ls o c o n t a i n a n e f f e c t o f 7 .S o m e w h a t c u r i o u s l y , t h e t r e n d i s o p p o s i te in s e n s e t o t h a t n o t e d a b o v e f o r a t u r b u l e n t b o u n d a r yl a y e r . E x p r e s s i o n s o f t h e f o r m o f e q u a t i o n s ( 1) a n d ( 7 ) fo r t u r b u l e n t s e p a r a t i o n g i v e , as w e h a v e s e e n ,v a l u e s o f k d i m i n i s h i n g w i t h r e d u c t i o n o f y ; b u t , f o r a g iv e n v a l u e o f R e x , t h e t y p e o f e q u a t i o n ( 14 )f o r l a m i n a r s e p a r a t i o n p r o d u c e s a n i n c r e a s e i n k w i t h r e d u c t i o n o f y. I n t h i s c a s e c o n s t a n t s c a n b ed e r i v e d a n a l y t i ca l l y1, 19 w h i c h f a i r l y w e l l a g r e e w i t h t h e d a t a f o r 7 = 1 . 4 ( S e c t i o n 2 . 1 . 3 ) , a n d t h e r ei s n o o b v i o u s r e a s o n w h y t h e y s h o u l d b e in v a l i d fo r a n y o t h e r c o n d i t i o n . U n f o r t u n a t e l y t h e r e a ren o k n o w n d a t a f o r l a m i n a r s e p a r a t i o n a t v a l u e s o f y o t h e r t h a n 1 . 4 , s o t h e m a t t e r c a n n o t b e r e s o l v e d ;b u t o n p r e s e n t e v i d e n c e t h e r e i s a li k e l ih o o d t h a t t h e p r e s s u r e r is e o c c u r r i n g w i t h l a m i n a r s e p a r a t io n ,u n l ik e t h a t w i t h t u r b u l e n t , m a y v a r y w i t h 7 .

5 .1 . The 'K i nk P o i n t ' .T h e c o n d i t i o n a t w h i c h a n o b l i q u e s h o c k m o v e s f r o m t h e l i p in s i d e t h e n o z z l e i s t r a d i t i o n a l l y

k n o w n a s t h e ' k i n k p o i n t ' .I t s h o u l d o f c o u r s e b e a p p r e c i a t e d t h a t , e v e n w h e n t h e s h o c k i s n o m i n a l l y s ti ll a t t h e l i p , s o m e

c o m p r e s s i o n w i l l f e e d u p t h e b o u n d a r y l a y e r, r a is i n g th e w a l l p r e s s u r e i m m e d i a t e l y a h e a d o f t h eo u t l e t. A s t h e s h o c k s t r e n g t h e n s , t h i s z o n e o f c o m p r e s s i o n o r ' f o o t ' o f th e s h o c k w i ll s p r e a d f a r t h e ri n s i d e t h e n o z z l e , u n t i l e v e n t u a l l y t h e b o u n d a r y l a y e r j u s t s t a r t s t o s e p a r a t e a t o u t l e t ; a t t h i s c o n d i t i o no n l y t h e p r e s s u r e r i s e f r o m 1 t o s w i l l b e e x p e r i e n c e d o n t h e w a l l . F u r t h e r c h a n g e o f n o z z l e p r e s s u r er a t io w i l l b r i n g t h e s h o c k w h o l l y i n s i d e t h e n o z z l e , w i t h t h e f u l l p r e s s u r e r i s e f r o m 1 t o 2 .

S i n c e t h i s p r o c e s s o c c u r s g r a d u a l l y , t h e r e i s in r e a l i t y n o s i n g le ' p o i n t ' w h i c h d e f i n e s t h e d i f f e r e n c eb e t w e e n a n o z z l e r u n n i n g f u ll a n d o n e w i t h i n t e r n a l s e p a r a t i o n . H o w e v e r , a s a g e n e r a l g u i d e t o t h eo n s e t o f s e p a r a t i o n , i t is c o n v e n i e n t t o t a ke t h e c o n d i t i o n w h e r e t h e f u l l p r e s s u r e r i s e 1 t o 2 i s f i r s t

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a c h i e v e d . T h i s w e s h a l l t r e a t a s b e i n g t h e ' k i n k p o i n t ' . I t o c c u r s a t a v a l u e o f A . P . R . e q u a l t ok x n o z z l e D . P . R . ; a n d a t t h i s c o n d i t i o n t h e a s s u m p t i o n P 2 = P b is o b v i o u s l y c o r r e c t , si n c e t h e r ei s n o ' m i x i n g r e g i o n ' i n w h i c h p r e s s u r e r e c o v e r y c a n ta k e p la c e. T h e r e l a t io n b e t w e e n k i n k - p o i n tp r e s s u r e r a t io a n d n o z z l e D . P . R . f o r t u r b u l e n t f l o w m a y t h u s b e o b t a i n e d f r o m a n y o f t h e t h r e es a t i s f a c t o r y c r it e r i a g i v e n in F i g . 8 . T h i s i s p r e s e n t e d i n F i g . 1 1.

5 .2 . C o m m e n t s .D a t a c o l l e c t e d in F i g s . 7 t o 1 0 a r e f o r n o z z l e s w i t h c o n s t a n t d i v e r g e n c e h a l f - a n g l e s b e t w e e n 1 0 a n d

3 0 , a n d s u g g e s t a n e g l i g i b le i n f l u e n c e o f w a l l a n g l e o v e r t h i s r a n g e . A l t h o u g h o n e o f t h e s o u r c e s 35c l a i m s t o f i n d s o m e e f f e c t , i t s e e m s s m a l l e n o u g h t o b e l o s t a m o n g s t t h e g e n e r a l s c a t t e r . A r e n s a n dS p i e g l e r ~4 t e s t e d n o z z l e s w i t h 7 , 1 5 a n d 2 2 d i v e r g e n c e , a n d f o u n d n o t r e n d . T h e i r w o r k c o v e r e db o t h t w o - d i m e n s i o n a l a n d a x i s y m m e t r i c n o z z le s , a n d c o n f i r m s t h a t t h e s e p a r a t io n b e h a v i o u r i ss i m i l a r i n b o t h c a s e s .

D i v e r g e n c e a n g l e s o u t s i d e t h e a b o v e r a n g e h a v e n o t r e c e i v e d w i d e a t t e n t i o n , n o d o u b t b e c a u s et h e y w o u l d h av e l i t t le p r a c t i c