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T h e p r o b l e m of f i n d i n g the n a t u r e of a s y m p t o t i c e x p a n s i o n s of N a v i e r - S t o k e s s o l u t i o n s w h i c h a r e u n i f o r m l y v a l i d f o r h i g h R e y n o l d s n u m b e r s i s c o n s i d e r a b l y m o r e d i f f i c u l t t h a n the c o r r e s p o n d i n g p r o b l e m f o r l o w R e y n o l d s n u m b e r s . W e s h a l l h e r e n e g l e c t q u e s t i o n s of s t a b i l i t y ( i n the c l a s s i c a l s e n s e of t h i s t e r m ) a n d c o n s i d e r o n l y s t a t i o n a r y s o l u t i o n s . E v e n f o r this r e s t r i c t e d c a s e m a n y u n s o l v e d f u n d a m e n t a l p r o b l e m s s t i l l r e m a i n . One e x p e c t s , of c o u r s e , that P r a n t d l ' s b o u n d a r y - l a y e r t h e o r y w i l l p l a y a n i m p o r t a n t r o l e . T h e a p p l i c a t i o n of this t h e o r y p r e s u p p o s e s a k n o w l e d g e of the c o r r e s p o n d i n g i n v i s c i d f l o w w h i c h i n p r i n c i p l e m a y b e o b t a i n e d b y a p p l y i n g the f i r s t p r i n c i p a l l i m i t , i . e . the o u t e r or E u l e r l i m i t to the e x a c t s o l u t i o n ( s e e C h a p t e r I of P a r t I ) . H o w e v e r , this o u t e r l i m i t is r a r e l y k n o w n . A p p l y i n g the E u l e r l i m i t to the N a v i e r - S t o k e s e q u a t i o n s one o b t a i n s the E u l e r e q u a t i o n s ( w h i c h h e r e m e a n s the e q u a t i o n s f o r n o n - v i s c o u s , but p o s s i b l y r o t a t i o n a l f l o w ) . It i s to be e x p e c t e d that the E u l e r l i m i t of the N a v i e r - S t o k e s s o l u t i o n s h o u l d b e a s o l u t i o n of the E u l e r e q u a t i o n s . T h e r e m a y b e s o m e doubt a b o u t the u n i q u e n e s s of the N a v i e r - S t o k e s s o l u t i o n s f o r h i g h R e ( a d o u b t w h i c h w e s h a l l d i s r e g a r d h e r e ) but t h e r e i s no d o u b t that that, f o r m o s t i n t e r e s t i n g c o n f i g u r a t i o n s , the E u l e r s o l u t i o n i s not u n i q u e . T h e q u e s t i o n is t h e n w h i c h E u l e r s o l u t i o n f o r a g i v e n c o n f i g u r a t i o n is the E u l e r l i m i t of the N a v i e r - S t o k e s s o l u t i o n ( w e s h a l l a s s u m e t h i s l i m i t to e x i s t ) . In s o m e c a s e s the a n s w e r is s i m p l e : F o r f l o w p a s t a s e m i - i n f i n i t e b o d y , s a y a p a r a b o l a it s e e m s p l a u s i b l e that the l i m i t i n g s o l u t i o n i s p o t e n t i a l f l o w p a s t t h i s b o d y . P r a n d t l
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s o r i g i n a l p a p e r on b o u n d a r y - l a y e r t h e o r y ( P r a n d t l (1904) ) c o n t a i n e d a n i n t e r e s t i n g r e s u l t w h i c h f o r a l o n g t i m e w a s d i s r e g a r d e d ( o r f o r g o t t e n ) . F o r s t e a d y f l o w i n s i d e a finite c o n t a i n e r ( w i t h p r e s c r i b e d t a n g e n t i a l v e l o c i t y at the w a l l ) the f l o w f i e l d i n the l i m i t of z e r o v i s c o s i t y i s m a d e u p of
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w h i c h a r e m a x i m a l r e g i o n s of c l o s e d n e s t e d s t r e a m l i n e s . P r a n d t l
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s r e s u l t is that v o r t i c i t y is c o n s t a n t in e a c h s u c h v o r t e x . It i s then n a t u r a l to t r y to f i n d the v a l u e of this c o n s t a n t b y a c o n s i s t e n c y a r g u m e n t . L e t us a s s u m e f o r i n s t a n c e t h e r e i s o n l y one v o r t e x a n d that the
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c o n t a i n e r i s s i m p l y c o n n e c t e d . F o r s m a l l v a l u e s of v i s c o s i t y the i n v i s c i d v o r t e x w i l l d e t e r m i n e a b o u n d a r y - l a y e r at the w a l l . F o r s t e a d y f l o w the b o u n d a r y l a y e r m u s t o b v i o u s l y b e p e r i o d i c ( m e a n i n g s i m p l y o n e - v a l u e d ) . It t u r n s out, h o w e v e r , that t h e r e i s o n l y one v a l u e of the v o r t i c i t y f o r w h i c h the b o u n d a r y - l a y e r e q u a t i o n s h a v e a s t a t i o n a r y s o l u t i o n ( t h e r e m a y b e no s u c h v a l u e w h i c h w o u l d m e a n that the a s s u m p t i o n of one v o r t e x i n the l i m i t is w r o n g ) . N o w c o n s i d e r f l o w p a s t a finite b o d y . One w o u l d hope to b e a b l e to a p p l y a c o n s i s t e n c y a r g u m e n t to s h o w that o n l y one of the E u l e r s o l u t i o n s p a s t the b o d y is c o n s i s t e n t w i t h the e x i s t e n c e of a b o u n d a r y - l a y e r . H e r e , h o w e v e r , one e n c o u n t e r s t w o k i n d s of d i f f i c u l t i e s . ( F o r the f o l l o w i n g , cf. L a g e r S t r o m (1964), p p . 106-107 a n d 131-133).
T h e f i r s t d i f f i c u l t y is that the p o s s i b l e E u l e r s o l u t i o n f o r f l o w p a s t a b o d y a r e i n s u f f i c i e n t l y k n o w n . T h e r e is the c l a s s i c a l p o t e n t i a l s o l u t i o n w h i c h , h o w e v e r , s e e m s to b e a n u n l i k e l y c a n d i d a t e e x c e p t p o s s i b l y f o r b o d i e s w i t h t r a i l i n g e d g e c u s p s (cf. C h a p t e r I , S e c t i o n 2 . 5 c ) . If the l i m i t i n g f l o w i s not p o t e n t i a l then the t w o s t r e a m l i n e s c o m i n g f r o m the f o r w a r d s t a g n a t i o n point ( w e c o n s i d e r a finite t w o - d i m e n s i o n a l b o d y of s i m p l e s h a p e , f o r i n s t a n c e a c i r c u l a r c y l i n d e r ) e a c h l e a v e s the b o d y at a p o i n t w h i c h w e s h a l l c a l l a point of d e t a c h m e n t . A c l a s s i c a l e x a m p l e of s u c h a s o l u t i o n is the K i r c h h o f f s o l u t i o n f o r w h i c h the d e t a c h e d s t r e a m l i n e s continue to d o w n s t r e a m infinity a n d b e t w e e n t h e m e n c l o s e a w a k e w h e r e the v e l o c i t y is z e r o . F o r a g i v e n c o n f i g u r a t i o n the p o s i t i o n of the p o i n t s of d e t a c h m e n t m a y v a r y w i t h i n a c e r t a i n r a n g e . M a n y w r i t e r s on the s u b j e c t h a v e a d v o c a t e d the K i r c h h o f f f l o w a s the c o r r e c t l i m i t i n g f l o w ; h o w e v e r , the a r g u m e n t s g i v e n in f a v o r of this h y p o t h e s i s a p p e a r to be r a t h e r i r r e l e v a n t . O n the o t h e r h a n d the a r g u m e n t s g i v e n a g a i n s t it a l s o a p p e a r i r r e l e v a n t . It s e e m s that the o r i g i n a l a r g u m e n t in f a v o r of the K i r c h h o f f s o l u t i o n w a s s i m p l y that the p o t e n t i a l - f l o w s o l u t i o n s e e m e d u n l i k e l y to b e the c o r r e c t l i m i t i n g f l o w . T h i s a r g u m e n t , h o w e v e r , n e g l e c t s the o t h e r p o s s i b l e E u l e r s o l u t i o n s . T o o u r k n o w l e d g e t h e r e h a s b e e n no s y s t e m a t i c s t u d y of the p o s s i b l e E u l e r s o l u t i o n s f o r a g i v e n c o n f i g u r a t i o n . T h i s is the f i r s t d i f f i c u l t y m e n t i o n e d a b o v e . I n g e n e r a l , it
s e e m s that the E u l e r e q u a t i o n s h a v e b e e n u n j u s t l y n e g l e c t e d . A s y s t e m a t i c i n v e s t i g a t i o n of t h e s e e q u a t i o n s m a y p o s s i b l y
s o o n r e s o l v e the f i r s t d i f f i c u l t y . ( K a p l u n o b t a i n e d v a r i o u s r e s u l t s a b o u t E u l e r f l o w at l a r g e d i s t a n c e s f r o m a finite b o d y . T h e e x i s t i n g m a n u s c r i p t s a r e , u n f o r t u n a t e l y , t o o i n c o m p l e t e to be e d i t e d ) .
T h e r e i s a s e c o n d d i f f i c u l t y w h i c h p r e s e n t s a r a t h e r f o r m i d a b l e o b s t a c l e f o r the s t u d y of f l o w at h i g h R e y n o l d s n u m b e r s . It s e e m s c l e a r that the c l a s s i c a l b o u n d a r y - l a y e r t h e o r y , w h i l e a p p l i c a b l e a l o n g the b o d y b e t w e e n the f o r w a r d s t a g n a t i o n p o i n t a n d the p o i n t s of d e t a c h m e n t is i n s u f f i c i e n t to h a n d l e o t h e r r e g i o n s of the f l o w f i e l d . F o r e x a m p l e , one a r g u m e n t a d v a n c e d a g a i n s t the K i r c h h o f f f l o w i s the f o l o w i n g : If one a p p l i e s P r a n d t l ' s t h e o r y to the d i s c o n t i n u i t y b e t w e e n the w a k e a n d the p o t e n t i a l f l o w one o b t a i n s a s h e a r l a y e r w h o s e t h i c k n e s s d o w n s t r e a m g r o w s a s V v x / U . H o w e v e r , the t h i c k n e s s of the w a k e r e g i o n i t s e l f g r o w s p a r a b o l i c a l l y w i t h x. H e n c e , f o r ν s u f f i c i e n t l y s m a l l the s h e a r l a y e r s do not m e e t . O n e then a r r i v e s at the i m p l a u s i b l e c o n c l u s i o n that the w a k e i s i n f i n i t e l y l o n g at a finite R e y n o l d s n u m b e r . ( H e r e w a k e m e a n s the w a k e b u b b l e i . e . the r e g i o n of c l o s e d s t r e a m l i n e s b e h i n d the b o d y ) . T h e w e a k n e s s i n t h i s a r g u m e n t i s , of c o u r s e , that it p r e s u p p o s e s that s t a n d a r d
b o u n d a r y - l a y e r t h e o r y s h o u l d b e a p p l i c a b l e a l o n g the i n f i n i t e l y l o n g w a k e b o u n d a r y . A l l the a r g u m e n t s h o w s is that o u r
p e r t u r b a t i o n t e c h n i q u e s m a y b e h i g h l y i n a d e q u a t e . T h e r e i s , of c o u r s e , no a p r i o r i d i f f i c u l t y in v i s u a l i z i n g a finite but l a r g e v i s c o u s w a k e b u b b l e w h i c h , a s R e t e n d s to i n f i n i t y , t e n d s to the K i r c h h o f f d e a d w a t e r r e g i o n . O n l y d e t a i l e d a n d d e l i c a t e m a t h e m a t i c a l c o n s i d e r a t i o n s c o u l d r u l e t h i s out. I n f a c t K a p l u n s t u d i e d the c a s e of a w a k e b u b b l e of l e n g t h R e and t h i c k n e s s V R e . H e s e e m e d to a r r i v e at the c o n c l u s i o n that at l a r g e d i s t a n c e s f r o m the b o d y the w a k e b u b b l e t e n d s to a K i r c h h o f f d e a d w a t e r r e g i o n ( t h i s w o u l d not e x c l u d e finite v o r t i c e s n e a r the b o d y ) . U n f o r t u n a t e l y , h e r e a g a i n , the w o r k is t o o i n c o m p l e t e to be e d i t e d . A n o t h e r e x a m p l e of the d i f f i c u l t y is the f o l l o w i n g : P r o p o n e n t s of the K i r c h h o f f f l o w h a v e t r i e d to d e t e r m i n e the p o s i t i o n of the b r e a k - a w a y p o i n t b y a c o n s i s t e n c y a r g u m e n t . O n e c h o o s e s t h i s p o s i t i o n s u c h that the r e s u l t i n g p o t e n t i a l f l o w o u t s i d e the w a k e g i v e s r i s e to a b o u n d a r y - l a y e r f o r w h i c h the s k i n f r i c t i o n v a n i s h e s at the point of d e t a c h m e n t . T h i s c o n s i s t e n c y a r g u m e n t p r e s u p p o s e s that the p o i n t of v a n i s h i n g s k i n f r i c t i o n i s i d e n t i c a l w i t h the point of d e t a c h m e n t . T h i s is often t a c i t l y a s s u m e d w h e n one r e f e r s t o e i t h e r p o i n t a s the " s e p a r a t i o n p o i n t " . H o w e v e r , the t w o p o i n t s a r e c e r t a i n l y c o n c e p t u a l l y d i f f e r e n t a n d no g o o d a r g u m e n t s h a v e e v e r b e e n a d v a n c e d w h y t h e y s h o u l d c o i n c i d e .
K a p l u n w a s f u l l y c o n s c i o u s of the v a r i o u s d i f f i c u l t i e s r e f e r r e d to a b o v e . A s i n d i c a t e d he m a d e m a n y m o r e o r l e s s i n c o m p l e t e s t u d i e s of s e v e r a l a s p e c t s of the p r o b l e m . H e
F L U I D M E C H A N I C S A N D S I N G U L A R P E R T U R B A T I O N S
e v e n t u a l l y d e c i d e d to m a k e a v e r y d e t a i l e d s t u d y of one p a r t i c u l a r a s p e c t , n a m e l y the b e h a v i o r of the b o u n d a r y - l a y e r s o l u t i o n at the point of v a n i s h i n g s k i n f r i c t i o n ( h e r e r e f e r r e d to a s s e p a r a t i o n p o i n t ) . H e r e a l i z e d that t h i s i s o n l y one d e t a i l of the p r o b l e m of h i g h R e y n o l d s n u m b e r f l o w . H o w e v e r , he c o n s i d e r e d it n e c e s s a r y that t h i s d e t a i l be c o m p l e t e l y c l e a r e d u p . H i s s t a r t i n g p o i n t w a s a n a r t i c l e b y G o l d s t e i n (1948) in w h i c h it i s p r o p o s e d that a b o u n d a r y - l a y e r s o l u t i o n e x h i b i t s a s i n g u l a r b e h a v i o r at the s e p a r a t i o n p o i n t . S o m e t i m e b e f o r e 1956 K a p l u n b e g a n a d e t a i l e d i n v e s t i g a t i o n of this p r o b l e m . H e d i s c o v e r e d v e r y e a r l y the e s s e n t i a l s of the r e s u l t s p r e s e n t e d h e r e in C h a p t e r I . In that c h a p t e r a n i n i t i a l v a l u e p r o b l e m is p o s e d . A n i n i t i a l p r o f i l e i s p r e s c r i b e d s u c h that the c o r r e s p o n d i n g s k i n f r i c t i o n is s m a l l a n d its v a l u e i s c o n s i d e r e d a s a s m a l l p a r a m e t e r . P e r t u r b a t i o n m e t h o d s m a y t h e n b e a p p l i e d t o f i n d the f l o w d o w n s t r e a m of the i n i t i a l p r o f i l e . It is f o u n d that u n d e r c e r t a i n i n i t i a l c o n d i t i o n s s e p a r a t i o n o c c u r s a n d it is p o s s i b l e to s t u d y the s e p a r a t i o n p r o f i l e i n a n a l y t i c d e t a i l . It i s i n d e e d h i g h l y s i n g u l a r . K a p l u n w a s c o n v i n c e d that the p r o b l e m s t u d i e d i n C h a p t e r I , a l t h o u g h in a p p e a r a n c e s p e c i a l , w a s s u f f i c i e n t l y r e p r e s e n t a t i v e in the s e n s e that its s o l u t i o n s h o w s the e s s e n t i a l n a t u r e of the s i n g u l a r i t y at a n a c t u a l s e p a r a t i o n p r o f i l e . A l l s u b s e q u e n t w o r k of his b o r e out t h i s c o n j e c t u r e .
C h a p t e r I f o r m s a s e l f - c o n t a i n e d i n v e s t i g a t i o n of a c e r t a i n p e r t u r b a t i o n p r o b l e m . A t the s a m e t i m e it w a s m e a n t a s a p r e l i m i n a r y e x p l o r a t i o n b e f o r e t a c k l i n g the r e a l s e p a r a t i o n p r o b l e m . I n s u b s e q u e n t w o r k K a p l u n s t u d i e d the a c t u a l s e p a r a t i o n p r o f i l e a n d the r e s t r i c t i o n s i m p o s e d u p o n it b y the d e m a n d that it s h o u l d h a v e a n u p s t r e a m c o n t i n u a t i o n , a n d he a l s o s t u d i e d the p o s s i b i l i t y of c o n t i n u i n g it d o w n s t r e a m . T h e e s s e n c e of this is c o n t a i n e d in C h a p t e r I I I . C h a p t e r I I i s a p r e l i m i n a r y s t u d y to C h a p t e r I I I a n d a l s o i n v e s t i g a t e s s o m e s i d e i s s u e s . It d u p l i c a t e s s o m e of the m a t e r i a l i n C h a p t e r I I I . It h a s b e e n r e t a i n e d h e r e b e c a u s e it c o n t a i n s m a n y i n t e r e s t i n g i d e a s a n d r e s u l t s . H o w e v e r , it s e e m s l e s s i m p o r t a n t t h a n the o t h e r t w o c h a p t e r s a n d i s not n e c e s s a r y f o r u n d e r s t a n d i n g C h a p t e r I I I . O n e of the e s s e n t i a l r e s u l t s of C h a p t e r I I I i s a d e t a i l e d d e s c r i p t i o n of the s e p a r a t i o n p r o f i l e , e s p e c i a l l y the o c c u r r e n c e of l o g a r i t h m i c t e r m s . I - D e e C h a n g c o l l a b o r a t e d w i t h K a p l u n on p a r t of this c h a p t e r . A c o m p a r i s o n w i t h the r e s u l t s in, s a y , S t e w a r t s o n (1958) h a s not b e e n m a d e h e r e s i n c e K a p l u n a r r i v e d at his b a s i c r e s u l t s quite i n d e p e n d e n t l y a n d none of his m a n u s c r i p t s c o n t a i n s u c h a c o m p a r i s o n .
S o m e a u t h o r s h a v e a s s u m e d that the o u t e r f l o w a n d the b o u n d a r y - l a y e r i n t e r a c t i n s u c h a w a y that the p r e s s u r e d i s t r i b u t i o n a l o n g the b o d y t a k e s on the h i g h l y s p e c i a l f o r m n e e d e d to a v o i d s p e a r a t i o n . K a p l u n d i s m i s s e d t h i s p o s s i b i l i t y and in f a c t a s s u m e d that a c o n s t a n t p r e s s u r e g r a d i e n t w o u l d be r e p r e s e n t a t i v e . H e a l s o f o u n d it u n l i k e l y that the s e p a r a t i o n p r o f i l e e x h i b i t e d w h a t he c a l l e d h i g h l y o s c i l l a t o r y b e h a v i o r . H e p r o p o s e d to s e t t l e this doubt b y n u m e r i c a l c o m p u t a t i o n a n d d i d c o n s i d e r a b l e w o r k on n u m e r i c a l m e t h o d s f o r s o l v i n g the b o u n d a r y - l a y e r e q u a t i o n s . T h i s w o r k is too i n c o m p l e t e t o b e i n c l u d e d h e r e .
D u r i n g the l a s t y e a r s of his l i f e , s e v e r e l y h a m p e r e d b y i l l h e a l t h , S a u l K a p l u n w o r k e d on a p r o j e c t e d p u b l i c a t i o n of t h r e e p a p e r s on s e p a r a t i o n . T h e f i r s t one w a s m e a n t to c o r r e s p o n d to C h a p t e r I , the s e c o n d one to C h a p t e r I I I and the t h i r d one w a s to c o n t a i n n u m e r i c a l r e s u l t s . S o m e t i m e in 1963 he b e g a n a n e x t e n s i v e r e w r i t e of the o l d d r a f t f o r C h a p t e r I I I . W h e n he s u c c u m b e d to a h e a r t a t t a c k in F e b r u a r y , 1964, he h a d a l m o s t f i n i s h e d w h a t i s h e r e c a l l e d S e c t i o n 2.3e. U p to this p o i n t C h a p t e r I I I u s e s the r e v i s e d d r a f t , the r e s t i s t a k e n f r o m a m u c h o l d e r d r a f t .
