On the calculation of ducted propeller performance in axisymmetric flows

O N T H E C A L C U L A T I O N O F

D U C T E D P R O P E L L E R

P E R F O R M A N C E I N

A X I S Y M M E T R I C F L O W S

C A L C U L A T I O N OF D U C T E D P R O P E L L E R P E R F O R M A N C E IN

A X I S Y M M E T R I C FLOWS

BIBLIOTHEEK T U Delft P 1734 3404

O N T H E C A L C U L A T I O N O F

D U C T E D P R O P E L L E R

P E R F O R M A N C E I N

A X I S Y M M E T R I C F L O W S

T E R V E R K R I J G I N G V A N D E G R A A D V A N D O C T O R I N D E

T E C H N I S C H E W E T E N S C H A P P E N A A N D E T E C H N I S C H E

H O G E S C H O O L D E L F T OP G E Z A G V A N D E R E C T O R

M A G N I F I C U S , P R O F . IR. B. P. T H . V E L T M A N

V O O R E E N C O M M I S S I E A A N G E W E Z E N D O O R H E T C O L L E G E

V A N D E K A N E N T E V E R D E D I G E N O P D I N S D A G

1 4 J U N I 1983 T E 14.00UUR

J O S É A L B E R T O C A I A D O F A L C Â O D E C A M P O S

P R O E F S C H R I F T

D O O R

E N G E N H E I R O M E C Á N I C O

G E B O R E N T E L I S S A B O N

H. V E E N M A N E N Z O N E N B.V. - W A G E N I N G E N

Dit proefschrift is goedgekeurd

door de promotoren

Prof. Dr. Ir. J. D. van Manen

Prof. Dr. Ir. P. J. Zandbergen

CONTENTS

1. INTRODUCTION 1

2. ANALYSIS OF THE FLOW PAST A PROPELLER DUCT 6

2.1. I n t r o d u c t o r y remarks 6 2.2. P o t e n t i a l f l o w a n a l y s i s 9 2.2.1. F o r m u l a t i o n o f t h e p r o b l e m and boundary c o n d i t i o n s 9 2.2.2. N u m e r i c a l s o l u t i o n 15 2.2.3. C a l c u l a t i o n o f t h e d u c t c i r c u l a t i o n . F i r s t i n v i s c i d a p p r o x i m a t i o n and t h e K u t t a c o n d i t i o n 21 2.3. C a l c u l a t i o n o f t h e d u c t v i s c o u s l a y e r s 25 2.4. V i s c o u s - i n v i s c i d c o u p l i n g 28 2.5. R e s u l t s i n u n i f o r m f l o w and c o m p a r i s o n w i t h e x p e r i m e n t 30 2.6. C a l c u l a t i o n o f t h e d u c t s t e a d y l o a d f o r a d u c t w i t h p r o p e l l e r 38 2.6.1. P r o p e l l e r model and p r o p e l l e r i n d u c e d v e l o c i t i e s on t h e d u c t 38 2.6.2. Remarks on v i s c o u s e f f e c t s on t h e d u c t f o r a d u c t e d p r o p e l l e r 49 2.7. N u m e r i c a l r e s u l t s and c o m p a r i s o n w i t h e x p e r i m e n t 51

3. DUCTED PROPELLER IN AXISYMMETRIC SHEAR FLOW 70

3.1. I n t r o d u c t i o n 70 3.2. G o v e r n i n g e q u a t i o n s 73 3.3. I t e r a t i v e s o l u t i o n by a d i s c r e t e v o r t e x s h e e t method 82 3.3.1. V o r t e x s h e e t a p p r o x i m a t i o n t o t h e v o r t i c i t y i n t h e f l o w 82 3.3.2. F i r s t a p p r o x i m a t i o n t o t h e a c t u a t o r d i s k v o r t e x s h e e t s and t h e f l o w s t r e a m s u r f a c e s 85 3.3.3. C a l c u l a t i o n o f t h e f l o w s t r e a m s u r f a c e s and d i s c r e t i z a t i o n o f t h e v o r t e x s h e e t s 87 3.3.4. C a l c u l a t i o n o f t h e s t r e n g t h o f t h e v o r t e x s h e e t s 95 3.3.5. I t e r a t i v e p r o c e d u r e 97 3.4. N u m e r i c a l r e s u l t s and c o m p a r i s o n w i t h e x p e r i m e n t 98

4. INTERACTION STUDIES BETWEEN A DUCTED PROPELLER AND THE STERN FOR AX I SYMMETRIC FLOWS

4.1. I n t r o d u c t i o n

4.2. A p p l i c a t i o n t o t h e c a l c u l a t i o n o f t h e s t e r n f l o w f o r an a x i s y m m e t r i c body

4.3. I n t e r a c t i o n between a d u c t e d p r o p e l l e r and t h e s t e r n 4.4. D i s c u s s i o n o f t h e r e s u l t s

5. DUCTED PROPELLER DESIGN 5.1. I n t r o d u c t i o n 5.2. D e s i g n p r o c e d u r e 5.3. P r o p e l l e r i n d u c e d v e l o c i t i e s 5.4. Duct t h r u s t and d u c t i n d u c e d v e l o c i t i e s 5.5. The d e s i g n w i t h t h e i n d u c t i o n f a c t o r method 5.6. R e s u l t s and d i s c u s s i o n 6. CONCLUSIONS A p p e n d i x 1 A p p e n d i x 2 R e f e r e n c e s N o m e n c l a t u r e Summary S a m e n v a t t i n g Acknowledgement C u r r i c u l u m v i t a e 114 114 119 126 130 139 139 140 142 145 147 152 161 163 167 168 175 188 190 192 193

1. I n t r o d u c t i o n

c a v i t a t i o n on p r o p e l l e r s , as i n t h e c a s e o f pump j e t s . In r e c e n t y e a r s , w i t h t h e permanent i n c r e a s e o f power i n s t a l l e d on s h i p s , h i g h l e v e l s o f v i b r a t i o n i n d u c e d on t h e h u l l may o c c u r p r i m a r i l y due t o u n s t e a d y c a v i t a t i o n phenomena on t h e p r o p e l l e r b l a d e s o p e r a t i n g i n t h e h i g h l y n o n - u n i f o r m s h i p ' s wake. In c e r t a i n c a s e s , as s u g g e s t e d f o r example by O o s t e r v e l d (1971), a p p l i c a t i o n o f a n o n - a x i s y m m e t r i c d u c t may l e a d t o a d e c r e a s e o f t h e e x t e n t o f c a v i t a t i o n on t h e p r o p e l l e r and r e d u c e i t s i n d u c e d v i b r a t i o n and r a d i a t e d n o i s e l e v e l s . In v i e w o f an i n c r e a s i n g number o f a p p l i c a t i o n s o f t h e d u c t e d p r o p e l l e r , the development o f t h e o r e t i c a l models f o r d e s c r i b i n g i t s h y d r o d y n a m i c a l p e r f o r m a n c e has r e c e i v e d a t t e n t i o n o f many a u t h o r s .

E a r l y t h e o r e t i c a l work on d u c t e d p r o p e l l e r s a i m i n g a t t h e e v a l u a t i o n o f i t s p e r f o r m a n c e and t h e s e t up o f adequate d e s i g n methods were c o n c e n t r a -t e d on -t h e u n i f o r m f l o w c a s e . In s u c h i d e a l i z e d c o n d i -t i o n s -t h e h y d r o d y n a m i c p r o b l e m f o r t h e d u c t e d p r o p e l l e r c o n f i g u r a t i o n p r e s e n t s i t s e l f as an i n t e r f e r e n c e p r o b l e m between p r o p e l l e r and d u c t . Under t h e a s s u m p t i o n s o f i d e a l f l u i d and f l o w i r r o t a t i o n a l i t y the f l o w p a s t p r o p e l l e r and d u c t may be r e p r e s e n t e d by s i n g u l a r i t y d i s t r i b u t i o n s on t h e d u c t and p r o p e l l e r s u r f a c e s and t h e c o r r e s p o n d i n g t r a i l i n g v o r t e x s h e e t s r e p r e s e n t i n g t h e i r wakes.

the d i s t u r b a n c e s i n t r o d u c e d by p r o p e l l e r and d u c t on t h e u n d i s t u r b e d f l o w i n a r e f e r e n c e frame r o t a t i n g w i t h t h e p r o p e l l e r , t h e s i n g u l a r i t i e s and t h e p o t e n t i a l f l o w boundary c o n d i t i o n s c a n be t r a n s f e r r e d t o s p e c i f i c r e f e r e n c e s u r f a c e s : a c y l i n d r i c a l s u r f a c e f o r t h e d u c t and a s e t o f h e l i c o i d a l v o r t e x s h e e t s f o r t h e p r o p e l l e r . The b a s i c i d e a s f o r t h e a n a l y s i s o f t h e s t e a d y p e r f o r m a n c e o f t h e d u c -t e d p r o p e l l e r i n a x i a l and u n i f o r m f l o w , were a l r e a d y c o n -t a i n e d i n -t h e work o f Dickmann and W e i s s i n g e r ( 1 9 5 5 ) . They r e p r e s e n t e d t h e d u c t w i t h an e l l i p t i c a l d i s t r i b u t i o n o f r i n g v o r t i c e s on a c y l i n d e r o f c o n s t a n t r a d i u s and t h e p r o p e l l e r was modeled by an a c t u a t o r d i s k o f c o n s t a n t l o a d w i t h i t s c o r r e s -pondent s l i p s t r e a m r i n g v o r t i c i t y .

T h e o r i e s i n t r o d u c e d s u b s e q u e n t l y by Ordway e t a l (1960),and Morgan (1961) made use o f l i n e a r i z e d r i n g a i r f o i l t h e o r y , o r i g i n a l l y d e v e l o p e d by

W e i s s i n g e r (1955,1957), and i n c l u d e d a l i f t i n g l i n e model f o r t h e p r o p e l l e r . Ordway e t a l c o n s i d e r e d a l i g h t l y l o a d e d p r o p e l l e r , w h i l e Morgan u s e d t h e i n d u c t i o n f a c t o r method as i n t r o d u c e d by L e r b s (1952) f o r m o d e r a t e l y l o a d e d p r o p e l l e r s . These t h e o r i e s c o u l d be a p p l i e d t o t h e e v a l u a t i o n o f b o t h s t e a d y and n o n - s t e a d y l o a d i n g on a r o t a t i o n a l symmetric d u c t under t h e i n f l u e n c e o f the p r o p e l l e r . D i s c u s s i o n o f n o n - s t e a d y d u c t p e r f o r m a n c e i s o u t s i d e the s c o p e o f t h e p r e s e n t work. F o r an a c c o u n t on t h e l i n e a r i z e d t h e o r i e s m e n t i o n e d above we r e f e r t o t h e r e v i e w work o f W e i s s i n g e r and Maas (1968). A f u n d a m e n t a l r e s u l t w h i c h f o l l o w e d from t h o s e i n v e s t i g a t i o n s and which i s o f i m p o r t a n c e f o r t h e e v a l u a t i o n o f t h e d u c t ' s s t e a d y p e r f o r m a n c e , i s t h e e q u i v a l e n c e between t h e time a v e r a g e d f l o w f i e l d i n d u c e d by a r o t a t i n g s e t o f bound r a d i a l l i f t i n g l i n e s t o g e t h e r w i t h t h e i r t r a i l i n g v o r t e x s h e e t s and t h e a x i s y m m e t r i c f l o w i n d u c e d by an i n f i n i t e l y b l a d e number p r o p e l l e r model, t h e a c t u a t o r d i s k .

T h i s e q u i v a l e n c e was d e m o n s t r a t e d by Hough and Ordway (1965) i n t h e s t r i c t l y l i n e a r i z e d c a s e o f a l i g h t l y l o a d e d p r o p e l l e r f o r w h i c h t h e

h e l i c o i d a l v o r t e x s h e e t s a r e assumed t o have a c o n s t a n t p i t c h d e t e r m i n e d by t h e u n d i s t u r b e d advance and r o t a t i o n a l v e l o c i t i e s .

In t h e c a s e o f t h e m o d e r a t e l y l o a d e d p r o p e l l e r l i f t i n g l i n e model t h e e q u i v a l e n c e does n o t h o l d . However, t h e a c t u a t o r d i s k model has been used

as an a p p r o x i m a t i o n t o s t u d y n o n - l i n e a r e f f e c t s o f c o n t r a c t i o n and p i t c h v a r i a t i o n o f t h e h e l i c o i d a l v o r t e x l i n e s t a k i n g p l a c e i n t h e s l i p s t r e a m o f m o d e r a t e l y and h e a v i l y l o a d e d p r o p e l l e r s as c o n s i d e r e d by Van Gent (1976).

S t u d i e s o f t h e e f f e c t s o f s l i p s t r e a m c o n t r a c t i o n on d u c t p e r f o r m a n c e i n d u c t e d p r o p e l l e r a p p l i c a t i o n s u s i n g s i m p l i f i e d a c t u a t o r d i s k models have been c a r r i e d o u t by C h a p l i n (1964) and Van G u n s t e r e n (1973).

A p a r t from such r e f i n e m e n t s o f t h e p r o p e l l e r models, improvement o f t h e r e p r e s e n t a t i o n o f t h e d u c t has been a c h i e v e d by a p p l i c a t i o n o f s u r f a c e v o r t i c i t y models which t a k e i n t o a c c o u n t i n a r a t h e r a c c u r a t e way,the d u c t ' s geometry. T h i s has been done by L e w i s and Ryan ( 1 9 7 1 ) .

In g e n e r a l t h e d u c t e d p r o p e l l e r o p e r a t e s i n t h e s h i p ' s wake i n t h e p r o x i m i t y o f t h e h u l l and t h e w a t e r s u r f a c e s . The f a c t t h a t t h e p r o p u l s o r works i n a f l o w r e g i o n w i t h h i g h c o n c e n t r a -t i o n o f v o r -t i c i -t y b a s i c a l l y i n v a l i d a -t e s -t h e a s s u m p -t i o n o f p o -t e n -t i a l f l o w w h i c h u n d e r l i e s u n i f o r m f l o w t h e o r i e s . N e v e r t h e l e s s , t h e s u c c e s s e n j o y e d by t h e d e s i g n o f wake a d a p t e d p r o p e l l e r s b a s e d on t h e i n d u c t i o n f a c t o r method o f L e r b s (1952) has e n s u r e d t h e a p p l i c a t i o n o f p o t e n t i a l f l o w t h e o r i e s i n a s l i g h t l y m o d i f i e d form t o t h e g e n e r a l n o n - u n i f o r m f l o w c a s e . The m o d i f i c a t i o n s i n t r o d u c e d i n t h e t h e o r y o f wake a d a p t i o n c o n s i s t i n c o n s i d e r i n g t h e f l o w p e r t u r b a t i o n s i n d u c e d by t h e bound and t r a i l i n g v o r t i c i t y t o be added t o t h e u n d i s t u r b e d l o c a l i n f l o w v e l o c i t i e s t o t h e p r o p e l l e r d i s k assumed t o v a r y w i t h t h e r a d i a l c o o r d i n a t e . In a d d i t i o n t h e p i t c h o f t h e h e l i c o i d a l v o r t e x l i n e s assumed c o n s t a n t i n a x i a l d i r e c t i o n i s d e t e r m i n e d a t t h e p r o p e l l e r p l a n e by t h e l o c a l t o t a l v e l o c i t i e s . The l o c a l i n f l o w v e l o c i t i e s i n t o t h e p r o p e l l e r i . e . t h e t o t a l v e l o c i t i e s minus t h e p r o p e l l e r p e r t u r b a t i o n s , a r e known as e f f e c t i v e v e l o c i t i e s and i t s knowledge i s c o n s i d e r e d i n d i s p e n s a b l e i n wake a d a p t e d p r o p e l l e r d e s i g n . They d i f f e r from t h e n o m i n a l v e l o c i t i e s w h i c h o c c u r b e h i n d the s h i p ' s h u l l i n t h e a b s e n c e o f t h e o p e r a t i n g p r o p e l l e r . Such d i f f e r e n c e i s r e g a r d e d as a c o n s e q u e n c e o f t h e p r o p e l l e r - h u l l i n t e r a c t i o n phenomena. The p r e v i o u s c o n s i d e r a t i o n s i l l u s t r a t e some o f t h e p r o b l e m s i n v o l v e d i n t h e a p p l i c a t i o n o f t h e a v a i l a b l e t h e o r i e s and p o i n t o u t t h e need o f c o n s i d e r i n g t h e p r o b l e m o f d u c t e d p r o p e l l e r h u l l i n t e r a c t i o n . S p e c i f i c a l l y i t i s t h o u g h t n e c e s s a r y t o a s c e r t a i n t o what e x t e n t some

3

o f t h e most r e l e v a n t e f f e c t s o f t h e i n t e r a c t i o n phenomena may i n f l u e n c e t h e d e t a i l e d p e r f o r m a n c e o f t h e p r o p u l s o r .

Recent s t u d i e s by Huang e t a l (1976, 1977), S c h e t z and F a v i n ( 1 9 7 9 ) , on t h e i n t e r a c t i o n between a c o n v e n t i o n a l p r o p e l l e r and t h e s t e r n have been c o n -c e n t r a t e d on a x i s y m m e t r i -c b o d i e s . In v i e w o f t h e -c o n s i d e r a b l e s i m p l i -c i t y o f t h e s t e r n f l o w when compared w i t h t h e s i t u a t i o n b e h i n d t h e s h i p , o f f e r e d by t h e f l o w axisymmetry, t h e s e s t u d i e s employ models w h i c h attempt c o m p l e t e p r e -d i c t i o n s o f t h e f l o w f i e l -d a r o u n -d t h e s t e r n w i t h t h e p r o p e l l e r i n o p e r a t i o n .

On t h e o t h e r hand, w i t h t h e advent o f L a s e r - D o p p l e r anemometry, t h e measurement o f t h e v e l o c i t y f i e l d i n t h e c l o s e v i c i n i t y o f t h e o p e r a t i n g p r o p e l l e r , (Huang e t a l , 1976, 1977), has e n a b l e d t h e d e t a i l e d c o m p a r i s o n w i t h t h e t h e o r e t i c a l p r e d i c t i o n s .

A t t e m p t s t o a d r e s s t h e p r o b l e m from a d i r e c t n u m e r i c a l s o l u t i o n o f t h e f u l l N a v i e r - S t o k e s e q u a t i o n s have been u n d e r t a k e n by S c h e t z and F a v i n (1979). From t h e c o m p a r i s o n s w i t h e x p e r i m e n t a l d a t a t h e a u t h o r s r e c o g n i z e d some o f t h e s h o r t comings o f t h e t u r b u l e n c e model employed and t h e need o f i t s r e f i n e m e n t i n o r d e r t o improve t h e t h e o r e t i c a l r e s u l t s . The a p p r o a c h o f Huang e t a l (1976, 1977), makes use o f a c a l c u l a t i o n method o f t h e v i s c o u s f l o w on the s t e r n r e g i o n b a s e d on boundary l a y e r t h e o r y b r o u g h t i n t o i n t e r -a c t i o n w i t h t h e o u t e r p o t e n t i -a l f l o w i n -an i t e r -a t i v e scheme. The i n f l u e n c e o f t h e p r o p e l l e r i n t h e boundary l a y e r i s e x e r t e d t h r o u g h a m o d i f i c a t i o n o f t h e e x t e r n a l p o t e n t i a l f l o w . A n o v e l f e a t u r e o f t h e a p p r o a c h i s , however, t h e c a l c u l a t i o n o f t h e f l o w f i e l d i n the c l o s e v i c i n i t y o f t h e p r o p e l l e r by an i n v i s c i d r o t a t i o n a l f l o w model b a s e d on t h e E u l e r ' s e q u a t i o n s o f m o t i o n . R a t h e r a c c u r a t e p r e d i c t i o n s o f t h e t o t a l v e l o c i t i e s ahead o f t h e p r o p e l l e r a r e o b t a i n e d w i t h s u c h model.

The need f o r an e l u c i d a t i o n o f some o f t h e f u n d a m e n t a l a s p e c t s o f d u c t e d p r o p e l l e r h u l l i n t e r a c t i o n and t h e development o f c a l c u l a t i o n t e c h -n i q u e s a p p r o p r i a t e t o the -n o -n - u -n i f o r m f l o w s i t u a t i o -n whe-n t h e v o r t i c i t y o f t h e i n c o m i n g f l o w i s t a k e n i n t o c o n s i d e r a t i o n has m o t i v a t e d t h e p r e s e n t i n v e s t i g a t i o n . The b a s i c a p p r o a c h p u r s u e d i n t h i s s t u d y assumes t h a t t h e i n t e r a c t i o n f l o w between d u c t e d p r o p e l l e r and h u l l w h i c h u l t i m a t e l y d e t e r m i n e s t h e

4

p e r f o r m a n c e o f d u c t and p r o p e l l e r i s i n v i s c l d i n n a t u r e and t h e r e f o r e may a d e q u a t e l y be t r e a t e d by t h e c o n s i d e r a t i o n o f t h e E u l e r ' s e q u a t i o n s o f m o t i o n . A l t h o u g h t h e i n v i s c i d a n a l y s i s might be t h e o r e t i c a l l y j u s t i f i c a b l e o r e x p e r i m e n t a l l y v a l i d a t e d when d e a l i n g w i t h t h e g r o s s e f f e c t s o f t h e i n t e r -a c t i o n p r o b l e m s t o t h e d u c t e d p r o p e l l e r , v i s c o u s e f f e c t s i n t h e bound-ary l a y e r s on t h e v a r i o u s components o f t h e d u c t e d p r o p e l l e r s y s t e m may be o f p r i m a r y i m p o r t a n c e i n d e t e r m i n i n g t h e o v e r a l l f o r c e s a c t i n g on t h e s y s t e m . V i s c o u s e f f e c t s on p r o p e l l e r b l a d e s and t h e i r i n f l u e n c e on p r o p e l l e r c h a r a c t e r i s t i c s have been i n v e s t i g a t e d f o r y e a r s . On t h e o t h e r hand, t h e i n f l u e n c e o f v i s c o s i t y on t h e d u c t p e r f o r m a n c e has r e c e i v e d much l e s s a t t e n t i o n i n s p i t e o f b e i n g a s o u r c e o f s e r i o u s s c a l e e f f e c t s on

f u l l - s c a l e p r e d i c t i o n s when s e p a r a t i o n f l o w phenomena o c c u r on model s c a l e .

T h e r e f o r e , t h e s e c o n d c h a p t e r i s c o n c e r n e d w i t h t h e a n a l y s i s o f t h e v i s c o u s f l o w p a s t an a x i s y m m e t r i c d u c t e i t h e r i n u n i f o r m a x i a l f l o w o r when r e g a r d e d as b e i n g a p a r t o f t h e d u c t e d p r o p e l l e r . In t h e t h i r d c h a p t e r , t h e f l o w p a s t an a n n u l a r a e r o f o i l and a d u c t e d p r o p e l l e r i n a x i s y m m e t r i c s h e a r f l o w i s c o n s i d e r e d and a p p r o x i m a t e n u m e r i c a l s o l u t i o n s o f t h e E u l e r ' s e q u a t i o n by a d i s c r e t e v o r t e x method a r e g i v e n . C h a p t e r f o u r d e a l s w i t h t h e a p p l i c a t i o n s o f t h e methods d e v e l o p e d i n c h a p t e r two t o t h e i n t e r a c t i o n p r o b l e m o f a d u c t e d p r o p e l l e r b e h i n d a r e v o l u t i o n body. In c h a p t e r f i v e some c o n s i d e r a t i o n s on t h e d e s i g n o f d u c t e d p r o p e l l e r s a r e g i v e n .

The r e s u l t s o f t h e b a s i c f l o w models i n t h e f i r s t and second c h a p t e r s a r e v e r i f i e d by c o r r e l a t i o n w i t h e x p e r i m e n t .

2. A n a l y s i s o f t h e f l o w p a s t a p r o p e l l e r d u c t

2.1. INTRODUCTORY REMARKS

F o r t h e c a l c u l a t i o n o f d u c t p e r f o r m a n c e i t i s n e c e s s a r y t o have a method f o r t h e e v a l u a t i o n o f t h e p r e s s u r e d i s t r i b u t i o n and f o r e s t i m a t i n g i t s f r i c t i o n a l d r a g . T h i s i s n o r m a l l y a c c o m p l i s h e d by combined p o t e n t i a l f l o w and boundary l a y e r c a l c u l a t i o n methods.

In g e n e r a l , i n d u c t e d p r o p e l l e r a p p l i c a t i o n s , f o r d u c t s w i t h s h a r p t r a i l i n g edges o r w i t h s m a l l r a d i u s o f c u r v a t u r e a t t h e t r a i l i n g edge and o p e r a t i n g n e a r d e s i g n c o n d i t i o n s , a p o t e n t i a l f l o w c a l c u l a t i o n i g n o r i n g t h e p r e s e n c e o f t h e boundary l a y e r a l r e a d y g i v e s r e l i a b l e v a l u e s o f t h e o v e r a l l f o r c e s a c t i n g on t h e d u c t p r o v i d e d t h a t f l o w s e p a r a t i o n o c c u r s from t h e d u c t ' s s u r f a c e o n l y i n t h e v i c i n i t y o f t h e t r a i l i n g edge. In t h i s c a s e , t h e v i s c o u s d r a g i s r a t h e r s m a l l when compared w i t h t h e a x i a l f o r c e a c t i n g on t h e d u c t . The p o t e n t i a l f l o w s o l u t i o n o b t a i n e d as a f i r s t a p p r o x i m a t i o n by d i s r e g a r d i n g t h e p r e s e n c e o f t h e boundary l a y e r and wake, o r f i r s t i n v i s c i d a p p r o x i m a t i o n , i s assumed t o s a t i s f y t h e K u t t a - J o u k o w s k y c o n d i t i o n f o r t h e f l o w at t h e t r a i l i n g edge. The a p p l i c a t i o n o f t h e K u t t a - J o u k o w s k y c o n d i t i o n f o r t h e c a l c u l a t i o n of t h e p o t e n t i a l f l o w on a p r o f i l e i s n o t a t r i v i a l m a t t e r and i t s v a r i o u s i n t e r p r e t a t i o n s and c o r r e s p o n d i n g i m p l e m e n t a t i o n s may b e a r c o n s i d e r a b l e i n f l u e n c e on t h e s o l u t i o n . G o s t e l o w (1974) , g i v e s a r e v i e w o f t h e a p p l i c a t i o n o f t r a i l i n g edge c o n d i t i o n s on t w o - d i m e n s i o n a l and t u r b o m a c h i n e r y b l a d e s e c t i o n s . F o r p r o f i l e s w i t h b l u n t t r a i l i n g edges, a t r a i l i n g edge c o n d i t i o n e q u i v a l e n t t o t h e c l a s s i -c a l K u t t a - J o u k o w s k y -c o n d i t i o n -cannot be f o r m u l a t e d , and any h y p o t h e t i -c a l f i r s t i n v i s c i d a p p r o x i m a t i o n b a s e d on an a r b i t r a r y v a l u e o f c i r c u l a t i o n i s d e p r i v e d from p h y s i c a l meaning. C l e a r l y , f o r t h e s o l u t i o n o f t h i s p r o b l e m , v i s c o u s e f f e c t s have t o be c o n s i d e r e d . In r e l a t i o n t o p r o p e l l e r d u c t s t h i s s u b j e c t w i l l be c o n s i d e r e d more e x t e n s i v e l y l a t e r i n t h i s C h a p t e r .

A c l a s s i c a l a p p r o a c h c o n s i s t s i n s o l v i n g t h e p o t e n t i a l f l o w and t h e boundary l a y e r p r o b l e m s i t e r a t i v e l y . On e a c h i t e r a t i o n s t e p t h e p o t e n t i a l f l o w s o l u t i o n p r o v i d e s t h e r e q u i r e d boundary c o n d i t i o n s f o r t h e boundary l a y e r c a l c u l a t i o n s and these, i n t u r n , f u r n i s h t h e r e q u i r e d boundary c o n d i t i o n s f o r t h e p o t e n t i a l f l o w p r o b l e m .

One o f t h e methods o f c o u p l i n g t h e two s o l u t i o n s t h r o u g h t h e i r c o r r e s p o n d e n t boundary c o n d i t i o n s c o n s i s t s i n d i v i d i n g t h e f l o w f i e l d i n t o two w e l l - d e f i n e d r e g i o n s : an o u t e r i n v i s c i d p o t e n t i a l f l o w r e g i o n and a v o r t i c a l r e g i o n d o m i n a t e d by t h e e f f e c t o f v i s c o s i t y . These two f l o w r e g i o n s are s e p a r a t e d by some boundary s u r f a c e where t h e s o l u t i o n s o f t h e two f l o w p r o b l e m s s h o u l d be matched by p r o p e r s p e c i f i c a t i o n o f t h e r e s p e c t i v e boundary c o n d i t i o n s . Such an a p p r o a c h r e q u i r e s an a d e q u a t e d e f i n i t i o n o f t h e boundary s u r f a c e w h i c h n a t u r a l l y c o u l d be t a k e n as t h e s u r f a c e s p e c i f i e d by t h e boundary l a y e r and t h e wake t h i c k n e s s e s and s h o u l d be d e t e r m i n e d as a p a r t o f t h e s o l u t i o n , (Rom, 1977).

Such a p r o c e d u r e i s , however, d i s a d v a n t a g e o u s from t h e p o i n t o f v i e w o f c o m p u t a t i o n a l e f f i c i e n c y , s i n c e t h e i n v i s c i d p a r t o f t h e c o m p u t a t i o n has t o be p e r f o r m e d w i t h boundary c o n d i t i o n s p r e s c r i b e d on a s u r f a c e c h a n g i n g i t s p o s i t i o n d u r i n g t h e i t e r a t i o n p r o c e s s . An a l t e r n a t i v e a p p r o a c h , which c i r c u m v e n t s t h i s p r o b l e m , c o n s i s t s i n t r a n s f e r r i n g t h e m a t c h i n g c o n d i t i o n s t o t h e body's s u r f a c e by assuming t h a t t h e o u t e r i n v i s c i d f l o w may be c o n t i n u e d down t o t h e b o d y ' s s u r f a c e . F o r t h i n s h e a r l a y e r s t h e t r u n c a t i o n e r r o r o f t h e T a y l o r e x p a n s i o n about t h e p o i n t s on t h e o r i g i n a l m a t c h i n g s u r f a c e i s i n g e n e r a l s m a l l e x c e p t n e a r s e p a r a t i o n . L i g h t h i l l , (1958), showed t h a t such a c o u p l i n g p r o c e d u r e between the two f l o w s may be f o r m u l a t e d i n terms o f an e q u i v a l e n t s o u r c e d i s t r i b u -t i o n on -t h e body's s u r f a c e .

The s o l u t i o n o f t h e v i s c o u s f l o w p r o b l e m a t h i g h R e y n o l d s numbers by s o l v i n g i t e r a t i v e l y t h e o u t e r i n v i s c i d f l o w and t h e boundary l a y e r forms t h e c l a s s i c a l "weak i n t e r a c t i o n " t h e o r y .

One o f t h e main d i f f i c u l t i e s e n c o u n t e r e d i n t h e a p p l i c a t i o n o f t h e t h e o r y l i e s on t h e f a c t t h a t t h e p r o c e d u r e may b r e a k down i n r e g i o n s o f " s t r o n g i n t e r a c t i o n " o f t h e boundary l a y e r and t h e i n v i s c i d f l o w such as n e a r a s e p a r a t i o n p o i n t o r a t t h e t r a i l i n g edge r e g i o n .

In f a c t , i t might be i m p o s s i b l e t o c o n t i n u e t h e boundary l a y e r c a l c u l a -t i o n s beyond s e p a r a -t i o n , u s i n g d i r e c -t me-thods i . e . w i -t h p r e s c r i b e d p r e s s u r e

d i s t r i b u t i o n a t t h e edge o f t h e l a y e r .

R i g o r o u s a n a l y s i s o f t h e l o c a l f l o w i n a r e g i o n o f s t r o n g v i s c o u s -i n v -i s c -i d -i n t e r a c t -i o n f o r l a m -i n a r f l o w s a t t h e t r a -i l -i n g edge o f c u s p e d and wedged a i r f o i l shapes b a s e d on a s y m p t o t i c t h e o r y , h a v e been g i v e n by v a r i o u s a u t h o r s : R i l e y and S t e w a r t s o n , (1969), Brown and S t e w a r t s o n , (1970), Chow and M e l n i k , (1976). E s t i m a t e s f o r t h e c o r r e c t i o n due t o t h e e f f e c t o f v i s c o s i t y on t h e c i r c u l a t i o n as d e t e r m i n e d by t h e K u t t a c o n d i t i o n have been g i v e n i n t h o s e a n a l y s e s . A new method o f g e n e r a l a p p l i c a t i o n i n r e g i o n s o f s t r o n g i n t e r a c t i o n , has been r e c e n t l y p r o p o s e d by Veldman, ( 1 9 7 9 ) , (1981).

The a n a l y s i s b a s e d on t h e "weak i n t e r a c t i o n " t h e o r y i s c l a s s i c a l ( s e e T h w a i t e s , 1960), and i n many, c a s e s i s c a p a b l e o f p r o v i d i n g p r e d i c t i o n s o f s e c t i o n l i f t and d r a g f o r c e s w i t h e n g i n e e r i n g a c c u r a c y .

In t h e a p p l i c a t i o n t o p r o p e l l e r d u c t s one s u c h a method has been c o n s i d e r e d . In t h i s r e s p e c t t h e f o l l o w i n g remarks s h o u l d be made: - L a m i n a r s e p a r a t i o n phenomena o c c u r s f r e q u e n t l y on p r o p e l l e r d u c t s a t model s c a l e , Dyne ( 1 9 7 7 ) . T h e r e f o r e t h e t r e a t m e n t o f l a m i n a r s e p a r a t i o n b u b b l e s i n t h e c a l c u l a t i o n method was f e l t n e c e s -s a r y . - Due t o o p e r a t i o n r e q u i r e m e n t s p r o p e l l e r d u c t s have, r a t h e r o f t e n , c o n s i d e r a b l y t h i c k round t r a i l i n g e d g e s . A c c o r d i n g l y , t h e e f f e c t s o f t r a i l i n g edge b l u n t n e s s had t o be c o n s i d e r e d . T h i s c h a p t e r i s d i v i d e d i n t o t h r e e p a r t s . In t h e f i r s t p a r t t h e p o t e n t i a l f l o w a n a l y s i s i s g i v e n and n u m e r i c a l r e s u l t s i n u n i f o r m f l o w a r e p r e s e n t e d and compared w i t h e x p e r i m e n t a l d a t a . In t h e second p a r t e x t e n s i o n o f t h e method t o i n c l u d e t h e e f f e c t o f v i s c o u s l a y e r s i s g i v e n and e x p e r i m e n t a l v e r i f i c a t i o n i s s u p p l i e d . F i n a l l y , i n t h e t h i r d p a r t t h e c a s e o f t h e d u c t w i t h p r o p e l l e r i s examined i n u n i f o r m f l o w by means o f p o t e n t i a l t h e o r y . F o r t h a t p u r p o s e an a c t u a t o r d i s k r e p r e s e n t a t i o n o f t h e p r o p e l l e r i s u s e d . The r e s u l t s a r e compared w i t h e x p e r i m e n t a l d a t a . An attempt i s made t o i n c l u d e v i s c o u s e f f e c t s i n t h e d u c t a n a l y s i s i n t h e p r e s e n c e o f t h e p r o p e l l e r . 8

2.2. POTENTIAL FLOW ANALYSIS 2.2.1. F o r m u l a t i o n o f t h e p r o b l e m and boundary c o n d i t i o n s We c o n s i d e r t h e f l o w o f an i n v i s c i d and i n c o m p r e s s i b l e f l u i d p a s t a d u c t i n an o n s e t f l o w . The d i s t u r b a n c e p o t e n t i a l s a t i s f i e s L a p l a c e e q u a t i o n

V

<f> = 0 . (2-1)

and t h e boundary c o n d i t i o n a t t h e impermeable d u c t s u r f a c e D

m

<f>

s u b j e c t t o t h e c o r r e s p o n d e n t boundary c o n d i t i o n s and i s assumed t o be known f o r the p r e s e n t p u r p o s e s . In t h e l i f t i n g c a s e we a r e c o n s i d e r i n g , t h e r e i s c i r c u l a t i o n a r o u n d a c o n t o u r e n c i r c l i n g a s e c t i o n o f t h e d u c t and t h e p o t e n t i a l i s d i s c o n t i n u o u s on a s u r f a c e W e x t e n d i n g from a l i n e on t h e d u c t ' s s u r f a c e t o i n f i n i t y ( F i g . 2 - 1 ) . 9

The d i s t u r b a n c e p o t e n t i a l at a p o i n t P o u t s i d e t h e s u r f a c e s D and W i s g i v e n i n terms o f i t s boundary v a l u e s (Lamb, 1952), by

( P ) =

• A ƒ ƒ ( Z ^ L

^ D+W

3 n

' i d s +

i ƒ ƒ (c

R 4 7 1

D+W

(2-6)

where (J) , T T1— and d> , T T — denote the values of the p o t e n t i a l and i t s normal dn on

d e r i v a t i v e s , r e s p e c t i v e l y on t h e o u t e r and i n n e r s i d e s o f t h e s u r f a c e s D and W, andR=|Rj i s t h e d i s t a n c e between t h e f i e l d p o i n t P and t h e p o i n t on t h e s u r f a c e where t h e i n t e g r a l i n (2-6) i s b e i n g e v a l u a t e d . E q u a t i o n (2-6) g i v e s t h e r e p r e s e n t a t i o n o f t h e p o t e n t i a l i n terms o f a s o u r c e d i s t r i b u t i o n on t h e s u r f a c e w i t h s t r e n g t h a e q u a l t o t h e d i s c o n t i -n u i t y i -n t h e -n o r m a l d e r i v a t i v e

3<j)

3 n

3<j>

3 n

and a d i p o l e d i s t r i b u t i o n w i t h axes d i r e c t e d a l o n g t h e normal t o t h e s u r f a c e and w i t h i t s s t r e n g t h y e q u a l t o t h e d i s c o n t i n u i t y o f t h e p o t e n t i a l

(2-8)

W i t h eq. (2-7) and (2-8), eq. (2-6) w r i t e s

D+W D+W

(2-9)

Uo

Fig. 2.1. Schematic representation of the flow past a propeller

duct section.

By assuming c o n t i n u i t y o f t h e p o t e n t i a l on t h e s u r f a c e , w e o b t a i n a r e p r e s e n t a t i o n f o r t h e p o t e n t i a l i n t e r m s s o l e l y o f t h e s o u r c e d i s t r i b u t i o n and by assuming c o n t i n u i t y o f t h e n o r m a l d e r i v a t i v e s on t h e s u r f a c e we o b t a i n a r e p r e s e n t a t i o n i n terms o f a d i p o l e d i s t r i b u t i o n o n l y . In t h e l i f t i n g c a s e , f o r which t h e p o t e n t i a l i s d i s c o n t i n u o u s t h e l a t t e r r e p r e s e n -t a -t i o n o r a c o m b i n a -t i o n o f -t h e -two as i n (2-6) has -t o be a d o p -t e d . As s t a t e d i n t h e p r e v i o u s s e c t i o n , i n t h e i n t e r a c t i o n between v i s c o u s and i n v i s c i d f l o w r e g i o n s , t h e d i s p l a c e m e n t e f f e c t s due t o t h e boundary l a y e r and wake can be r e p r e s e n t e d by a s o u r c e d i s t r i b u t i o n on t h e s u r f a c e s D and W i n t h e manner s u g g e s t e d by L i g h t h i l l (1958). The Neumann boundary c o n d i t i o n (2-3) s h o u l d t h e n be a p p l i e d on t h e s u r f a c e B d i s p l a c e d from t h e o r i g i n a l s u r f a c e s D and W by t h e boundary l a y e r and wake d i s p l a c e m e n t t h i c k -n e s s .

However, as remarked b e f o r e , t h e boundary c o n d i t i o n on B can be r e p l a c e d by t h e Neumann boundary c o n d i t i o n on t h e o r i g i n a l s u r f a c e D+W s p e c i f y i n g t h e v a l u e o f t h e normal d e r i v a t i v e f o r t h e o u t e r p o t e n t i a l a t t h e s u r f a c e . The normal d i s c o n t i n u i t y i s e q u a l t o t h e s o u r c e s t r e n g t h on t h e s u r f a c e From (2-7) we o b t a i n | i _ + O = 0 on D , (2-11) s i n c e d> and i t s d e r i v a t i v e s a r e c o n t i n u o u s on D. o U s i n g G r e e n ' s theorem, we c o n c l u d e f r o m (2-11) t h a t t h e p o t e n t i a l i s c o n s t a n t i n s i d e D: $• + $ = C , (2-12) where C i s an a r b i t r a r y c o n s t a n t . E q u a t i o n (2-12) c o n s t i t u t e s a D i r i c h l e t boundary c o n d i t i o n f o r t h e i n n e r p o t e n t i a l w h i c h may be employed i n s t e a d o f t h e Neumann c o n d i t i o n ( 2 - 1 0 ) .

As i n t h e p r e s e n t a n a l y s i s t h e s o u r c e d i s t r i b u t i o n i s o n l y u s e d t o r e p r e s e n t boundary l a y e r and wake d i s p l a c e m e n t t h i c k n e s s e f f e c t s , i t s s t r e n g t h f o l l o w s from t h e s h e a r l a y e r f l o w s o l u t i o n s . In s u c h c a s e , a p p l i c a -t i o n o f -t h e b o u n d a r y c o n d i -t i o n s (2-10) o r (2-12) l e a d s -t o i n -t e g r a l e q u a -t i o n s f o r t h e d i p o l e d i s t r i b u t i o n . A p p l y i n g t h e Neumann c o n d i t i o n ( 2 - 1 0 ) , a F r e d h o l m i n t e g r a l e q u a t i o n of t h e f i r s t k i n d i s d e r i v e d 9 (J) ^ D + W P ( q ) ^ ) d S =

~ 3 ^ H

4 V

7

^

I

where p d e n o t e s t h e p o i n t where t h e boundary c o n d i t i o n i s s t a t e d and t h e terms i n t h e s o u r c e d i s t r i b u t i o n have been p l a c e d on t h e r i g h t - h a n d s i d e t o emphasize t h e f a c t t h a t t h e y a r e c o n s i d e r e d t o be known. A l t e r n a t i v e l y , we may d i f f e r e n t i a t e e q u a t i o n (2-12) a l o n g any d i r e c t i o n t a n g e n t t o the s u r f a c e r e q u i r i n g t h e v e l o c i t y component i n t h a t d i r e c t i o n on i t s i n n e r s i d e t o v a n i s h . Such c o n d i t i o n i s e x p r e s s e d by t h e f o l l o w i n g v e c t o r e q u a t i o n n x V (<J> +<J>_) = 0 . (2-14) — p p Y o

By u s i n g a well-known e q u i v a l e n c e between t h e p o t e n t i a l due t o a d i p o l e s h e e t and a v o r t e x s h e e t , i t i s u s e f u l t o show t h a t e q u a t i o n (2-14) l e a d s t o t h e s u r f a c e v o r t i c i t y f o r m u l a t i o n o f t h e p o t e n t i a l f l o w problem. To do t h i s we f i r s t e v a l u a t e t h e v e l o c i t y i n d u c e d by t h e d i p o l e d i s t r i b u t i o n w h i c h , i n any c a s e , i s r e q u i r e d f o r a p p l i c a t i o n o f e i t h e r boundary c o n d i t i o n (2-13) o r ( 2 - 1 4 ) . We have

W

P

" * * > ? 5 - < i >

= i ¥ / /

^ ) v

[ n ^ . v ( | ) ] d s (2-15)

12

S i n c e 1 ^ V . ' • 7 we o b t a i n R R Vp[ nq. Vq( i ) ] = ( nq. Vp) ^ + nqx ( Vpx - 3 ) . F o r an a r b i t r a r y v e c t o r A t h e f o l l o w i n g r e l a t i o n h o l d s n x ( V x A ) = ( n x V ) x A - ( n . V ) A - n ( V . A ) and (2-16) becomes R To i n t e g r a t e (2-15) by p a r t s we n o t e t h a t (2-16) V p [ nq. Vq( | ) ] = - ( ^ x V q) x ^ n g V q ( i ) . (2-17) R 5 5: y ( n x V J x - T = ( n x V ) x ( y - ^ - ) - ( n x V y ) x - ~ - (2-18) ~ q q R3 - q q R3 - q R3 and the i n t e g r a l i n (2-15) w r i t e s 1 5 + -r-

// (n x V u x - 7 d S +

c o n t r i b u t i o n t o t h e i n d u c e d v e l o c i t y by t h e boundary edges o f t h e s u r f a c e D+W. In t h e p r e s e n t c a s e such edges a r e i n e x i s t e n t and t h e i n t e g r a l v a n i s h e s .

When i d e n t i f y i n g t h e s t r e n g t h o f t h e v o r t e x s h e e t y as

1 = - n x V y (2-21)

the second i n t e g r a l e x p r e s s e s t h e f a m i l i a r r e s u l t o b t a i n a b l e from B i o t -S a v a r t law. F i n a l l y , t h e l a s t i n t e g r a l i n (2-19) v a n i s h e s i n view o f t h e f a c t t h a t 2 1 V (—) i s z e r o e v e r y w h e r e e x c e p t when t h e p o i n t P c o i n c i d e s w i t h t h e p o i n t q. R W i t h t h e p r e v i o u s r e s u l t ( 2 - 1 9 ) , (2-20) and ( 2 - 2 1 ) , e q u a t i o n (2-14) y i e l d s t h e f o l l o w i n g i n t e g r a l e q u a t i o n f o r t h e s u r f a c e v o r t i c i t y d i s t r i b u t i o n R (2-22)

D+W "

R"

D+W

-0 -p f o l l o w i n g i n t e g r a l e q u a t i o n

- | Y (S ) + $s y ( s1) k ( s , s ' ) d s ' = f ( s ) (2-23)

The k e r n e l k ( s , s ' ) r e p r e s e n t s t h e i n d u c e d v e l o c i t y t a n g e n t t o t h e c o n t o u r a t the p o i n t s due t o a r i n g v o r t e x l o c a t e d a t s=s' and i s g i v e n by

k ( s , s ' ) = ( x - x1 ; r , r ' ) ~ - ( x - x ' ; r , r ' ) | | (2-24)

where u and v a r e t h e a x i a l and r a d i a l v e l o c i t i e s i n d u c e d a t t h e p o i n t

Y Y

d s

ds

y y 0 a

c o m p l e t e e l l i p t i c i n t e g r a l s by Kiichemann and Weber, (1953).

We n o t e t h a t , a s u r f a c e v o r t i c i t y f o r m u l a t i o n c o u l d be u s e d i n combi-n a t i o combi-n w i t h t h e Neumacombi-ncombi-n boucombi-ndary c o combi-n d i t i o combi-n l e a d i combi-n g t o a F r e d h o l m i combi-n t e g r a l e q u a t i o n o f t h e f i r s t k i n d . E q u a t i o n s o f s e c o n d k i n d a r e , however, a d v a n t a -geous from t h e p o i n t o f v i e w o f t h e i r n u m e r i c a l s o l u t i o n . 2.2.2. N u m e r i c a l s o l u t i o n N u m e r i c a l s o l u t i o n p r o c e d u r e s f o r i n t e g r a l e q u a t i o n s o f t h e t y p e o f e q u a t i o n (2-23) o f t h e l a s t s e c t i o n e i t h e r , f o r t w o - d i m e n s i o n a l o r axisymme-t r i c p o axisymme-t e n axisymme-t i a l f l o w p r o b l e m s , h a v e been g i v e n by many a u axisymme-t h o r s . The g r e a t m a j o r i t y o f t h e s o l u t i o n t e c h n i q u e s employs a c o l l o c a t i o n method. In the c o l l o c a t i o n method t h e e q u a t i o n i s o n l y e x a c t l y s a t i s f i e d a t a d i s c r e t e s e t o f p o i n t s and t h e number o f p o i n t s i s c h o s e n e q u a l t o t h e number o f k n o t s employed i n t h e n u m e r i c a l q u a d r a t u r e u s e d t o a p p r o x i m a t e t h e i n t e g r a l . F o r l i n e a r i n t e g r a l e q u a t i o n s t h i s p r o c e d u r e l e a d s t o a l i n e a r s y s t e m o f e q u a t i o n s h a v i n g as unknowns the v a l u e s o f t h e f u n c t i o n a t t h e k n o t l o c a t i o n s . 15

Two b a s i c a p p r o a c h e s have been f o l l o w e d : A p p l i c a t i o n o f a t r a n s f o r m a t i o n t o t h e i n t e g r a t i o n v a r i a b l e , p r i o r t o the a p p l i c a t i o n o f t h e c o l l o c a t i o n method,or d i r e c t s o l u t i o n o f t h e o r i g i n a l e q u a t i o n by c o l l o c a t i o n h a v i n g t h e a r c l e n g t h as i n d e p e n d e n t v a r i a b l e . The o r i g i n a l v e r s i o n o f t h e s u r f a c e v o r t i c i t y method, d e v e l o p e d by M a r t e n s e n ( 1 9 5 9 ) , f o r t w o - d i m e n s i o n a l a i r f o i l s i s o l a t e d o r i n c a s c a d e , b e l o n g s t o t h e f i r s t c a t e g o r y . The method was s u b s e q u e n t l y d e v e l o p e d by J a c o b and R i e g e l s (1963), W i l k i n s o n (1967), and o t h e r s and a p p l i e d t o p r o p e l -l e r d u c t s by L e w i s and Ryan (1971).

A l l t h e s e methods make use o f a t r a n s f o r m a t i o n o f t h e a r c l e n g t h i n t o t h e p o l a r a n g l e w i t h r e s p e c t t o a p o i n t i n s i d e t h e c o n t o u r and employed t r a p e z o i d a l i n t e g r a t i o n t o a p p r o x i m a t e t h e i n t e g r a l .

The b a s i c d i f f e r e n c e s between t h e v a r i o u s methods r e g a r d t h e d i s t r i -b u t i o n o f c o l l o c a t i o n p o i n t s and t h e n u m e r i c a l p r o c e d u r e s u s e d t o e v a l u a t e t h e g e o m e t r i c a l p a r a m e t e r s o f t h e c o n t o u r . As remarked by W i l k i n s o n ( 1 9 6 7 ) , o r i g i n a l l y recommended p i v o t a l p o i n t d i s t r i b u t i o n s , ( J a c o b and R i e g e l s , 1963), i n c l u d e d t h e t r a i l i n g edge as a c o l l o c a t i o n p o i n t and gave u n r e l i a b l e r e s u l t s f o r p r o f i l e s w i t h s h a r p t r a i -l i n g e d g e s . The p o t e n t i a -l f -l o w p r o b -l e m i s n o t u n i q u e -l y d e t e r m i n e d by t h e s a t i s f a c t i o n o f t h e k i n e m a t i c a l boundary c o n d i t i o n on t h e c o n t o u r and t h e c i r c u l a t i o n must be g i v e n t o s p e c i f y t h e s o l u t i o n . The main d i f f i c u l t y a r o s e i n t h e a p p l i c a t i o n o f t h e K u t t a c o n d i t i o n i n t h e t r a n s f o r m e d v a r i a b l e . Due t o t h e p r o p e r t i e s o f t h e t r a n s f o r m a t i o n a t t h e t r a i l i n g edge t h e implemen-t a implemen-t i o n o f implemen-t h e K u implemen-t implemen-t a c o n d i implemen-t i o n i n implemen-t h e implemen-t r a n s f o r m e d v a r i a b l e d i d n o implemen-t i m p l y z e r o l o a d i n g a t a s h a r p t r a i l i n g edge. To m i n i m i z e the e r r o r s i n t r o d u c e d by t h e t r a p e z o i d a l i n t e g r a t i o n s a "back d i a g o n a l c o r r e c t i o n " was n o r m a l l y a p p l i e d t o t h e o r i g i n a l m a t r i x w h i c h r e n d e r e d i t s i n g u l a r . W i l k i n s o n showed t h a t t h e s y s t e m o f e q u a t i o n s a f t e r t h e a p p l i c a t i o n o f t h e K u t t a c o n d i t i o n became i l l - c o n d i t i o n e d a t s m a l l e r t r a i l i n g edge r a d i i . The p r o b l e m has been c i r c u m v e n t e d by a l t e r n a t i v e i m p l e m e n t a t i o n s o f t h e K u t t a c o n d i t i o n , ( W i l k i n s o n , 1967), and by d i f f e r e n t c h o i c e s o f p i v o t a l p o i n t l o c a t i o n s , ( L e w i s and Ryan, 1971).

A l t h o u g h t h e s e t y p e o f methods may r e q u i r e a r a t h e r s m a l l number o f k n o t s f o r an "optimum" t r a n s f o r m a t i o n , t h e main drawback l i e s i n t h e s e l e c -t i o n o f an adequa-te -t r a n s f o r m a -t i o n .

Methods o f t h e s e c o n d c a t e g o r y employ t h e a r c l e n g t h as i n d e p e n d e n t v a r i a b l e . They may d i f f e r on the o r d e r o f a p p r o x i m a t i o n used t o d i s c r e t i z e t h e p r o f i l e c o n t o u r and t h e s u r f a c e s i n g u l a r i t y d i s t r i b u t i o n . F i r s t o r d e r a p p r o x i m a t i o n s d i s c r e t i z e the c o n t o u r by s t r a i g h t e l e m e n t s and assume a c o n -s t a n t -s i n g u l a r i t y d i -s t r i b u t i o n on each e l e m e n t .

H i g h e r o r d e r a p p r o x i m a t i o n s , d i s c r e t i z i n g t h e c o n t o u r and t h e s i n g u l a -r i t y d i s t -r i b u t i o n s by p o l y n o m i a l s o f h i g h e -r d e g -r e e , can -r e d u c e s u b s t a n t i a l l y the c o m p u t a t i o n time by r e d u c i n g t h e number o f e l e m e n t s r e q u i r e d f o r a p r e s c r i b e d a c c u r a c y at c o s t s o f a d d i t i o n a l a n a l y t i c work. The e f f e c t i v e n e s s o f s e v e r a l s e c o n d o r d e r methods r e l y i n g on v a r i o u s f o r m u l a t i o n s o f t h e p o t e n t i a l f l o w p r o b l e m i n c o m b i n a t i o n w i t h d i f f e r e n t n u m e r i c a l a p p r o x i m a t i o n s o f t h e c o n t o u r and s i n g u l a r i t y d i s t r i b u t i o n s , h a s been r e c e n t l y a s s e s s e d by L a b r u j e r e (1979), f o r t w o - d i m e n s i o n a l f l o w s on p r o f i l e s . A l t h o u g h s u r f a c e v o r t i c i t y f o r m u l a t i o n s were c o n t e m p l a t e d , t h e y have been o n l y examined i n c o m b i n a t i o n w i t h t h e Neumann b o u n d a r y c o n d i t i o n and a boundary c o n d i t i o n i n t e r m s o f t h e s t r e a m f u n c t i o n . In t h i s r e s p e c t , w e w i l l be l i m i t e d t o a c o m p a r i s o n w i t h methods o f t h e f i r s t c a t e g o r y o r e x p e r i m e n t a l r e s u l t s .

The s e c o n d o r d e r method p r o p o s e d by Hess (1973, 1974), i n c o n n e c t i o n w i t h h i s s u r f a c e s o u r c e p a n e l method has been a d o p t e d i n t h e p r e s e n t n u m e r i -c a l s o l u t i o n .

The d u c t i s p a n e l l e d by e l e m e n t s o f p a r a b o l i c shape where v o r t i c i t y i s d i s t r i b u t e d a c c o r d i n g t o a p o l y n o m i a l f u n c t i o n . C o n s t a n t , l i n e a r and p a r a b o -l i c f u n c t i o n s have been u s e d .

The v o r t i c i t y d i s t r i b u t i o n i s expanded about a c o n t r o l p o i n t c h o s e n as t h e mid p o i n t o f t h e element i n t h e form

Y j U ) = Y <

Y ^ C

yfh

where Y j ^ > Y j an<i Y j a r e , r e p e c t i v e l y t h e s t r e n g t h o f t h e v o r t e x s h e e t ,

i t s f i r s t and h a l f t h e s e c o n d d e r i v a t i v e s e v a l u a t e d a t t h e c o n t r o l p o i n t j

X

Fig. 2.2. D e f i n i t i o n of a parabolic panel on the duct's section contour.

and E, i s t h e a r c l e n g t h on t h e element measured from t h e c o n t r o l p o i n t , F i g . 2.2.

The v a l u e s o f t h e f i r s t and s e c o n d d e r i v a t i v e s a t t h e c o n t r o l p o i n t s a r e o b t a i n e d by a d i v i d e d d i f f e r e n c e scheme, as g i v e n by Hess and M a r t i n (1974)

We n o t e t h a t when u s i n g s u c h scheme, d i s c o n t i n u i t i e s i n t h e v o r t i c i t y d i s t r i b u t i o n a r e i n t r o d u c e d a t t h e j u n c t i o n p o i n t s between e l e m e n t s .

The p a r a m e t r i c e q u a t i o n s o f t h e a r c element admit a s i m i l a r e x p a n s i o n

2

x _.(£;) = x_. +cosct_.5 - C j sina_.5 (2-30)

- 2

(^) = r_. +sina_.£ + c_. c o s o u ? (2-31)

where ( x ^ , r ^ ) a r e t h e c o o r d i n a t e s o f t h e c o n t r o l p o i n t , a. i s t h e s l o p e o f t h e c h o r d on t h e element and c . i s t h e element c u r v a t u r e .

J

The boundary c o n d i t i o n o f z e r o v e l o c i t y t a n g e n t t o t h e c o n t o u r on t h e i n n e r s i d e a p p l i e d a t N c o n t r o l p o i n t s l e a d s t o t h e l i n e a r s y s t e m o f e q u a t i o n s N ( 0 ) I k . • Y j = f , i = 1, ( 1 ) ,N (2-32) j = l 3 3 The m a t r i x o f i n f l u e n c e c o e f f i c i e n t s k.. i s o n l y a f u n c t i o n o f t h e d u c t ' s geometry and i s g i v e n by k. . = h&. • + c o s a . X . . + s i n a . Y . . , i = l , ( l ) , N , j = 1 , ( 1 ) , N (2-33) 6 i s t h e K r o n e c k e r d e l t a 6. . = 1 f o r i = j and 6. .=0 f o r The m a t r i x X. . and Y . a r e a x i a l and r a d i a l i n d u c e d v e l o c i t y m a t r i c e s , and f o r t h e i r e v a l u a t i o n we r e f e r t o Hess and M a r t i n (1974).

The r i g h t - h a n d s i d e i n e q u a t i o n (2-32) i s o b t a i n e d by e v a l u a t i n g e q u a t i o n (2-25) a t t h e c o n t r o l p o i n t s and i n c l u d e s t h e u n d i s t u r b e d f l o w and t h e d i s t u r b a n c e v e l o c i t y c a u s e d by t h e boundary l a y e r and wake d i s p l a c e m e n t t h i c k n e s s e s . The d e t e r m i n a t i o n o f t h e s e v e l o c i t y f i e l d s i s t h e s u b j e c t o f t h e s u b s e q u e n t s e c t i o n s .

The c o m p u t a t i o n a l a d v a n t a g e s o f a c c o u n t i n g f o r t h e i n f l u e n c e o f t h e boundary l a y e r on t h e p o t e n t i a l f l o w by an a d d i t i o n a l d i s t u r b a n c e v e l o c i t y t o t h e b a s i c o n s e t f l o w , become e v i d e n t from t h e form o f e q u a t i o n ( 2 - 3 2 ) . The m a t r i x o f i n f l u e n c e c o e f f i c i e n t s k _ w h i c h depends o n l y on t h e geometry o f t h e d u c t , does n o t need t o be changed i n t h e v i s c o u s - i n v i s c i d i t e r a t i o n p r o c e s s . As a c o n s e q u e n c e o f t h e n o n - u n i q u e n e s s o f t h e p o t e n t i a l f l o w p r o b l e m e i g e n s o l u t i o n s o f e q u a t i o n (2-25) r e p r e s e n t i n g c i r c u l a t o r y f l o w s may be added t o a p a r t i c u l a r s o l u t i o n w i t h o u t a f f e c t i n g t h e boundary c o n d i t i o n on t h e s u r f a c e . 19

S t r i c t l y s p e a k i n g a t i n c r e a s i n g number o f e l e m e n t s t h e m a t r i x k ^ becomes i l l - c o n d i t i o n e d . C l e a r l y any s o l u t i o n o f t h e system ( 2 - 3 2 ) , i f made p o s s i b l e t h r o u g h the d i s c r e t i z a t i o n , does n o t , i n g e n e r a l , s a t i s f y t h e K u t t a - J o u k o w s k y c o n d i t i o n o f smooth f l o w a t t h e t r a i l i n g edge.

Such c o n d i t i o n has t o be s p e c i f i e d i n a d d i t i o n t o the s y s t e m o b t a i n e d from t h e c o l l o c a t i o n method. In t h e l i t e r a t u r e v a r i o u s i m p l e m e n t a t i o n s o f t h e K u t t a c o n d i t i o n have been c o n s i d e r e d . A l s o t h e r e i s c o n s i d e r a b l e freedom i n t h e way t h e a d d i t i o n a l c o n d i t i o n s a r e c o u p l e d t o t h e e x i s t i n g system o f e q u a t i o n s .

R e g a r d i n g t h e form o f t h e K u t t a c o n d i t i o n , M a n g i e r and S m i t h ( 1 9 6 9 ) , showed t h a t , f o r s h a r p t r a i l i n g e d g e s , i n t w o - d i m e n s i o n a l i n v i s c i d f l o w , t h e s t a g n a t i o n s t r e a m l i n e s h o u l d l e a v e the t r a i l i n g edge a l o n g t h e b i s e c t o r t o t h e t r a i l i n g edge a n g l e . In f i r s t o r d e r methods t h i s c o n d i t i o n can be a p p r o x -i m a t e d , f o r example, by e q u a t -i n g t h e v o r t e x s t r e n g t h o f t h e f -i r s t and l a s t c o n t r o l p o i n t s , ( L e w i s and Ryan, 1971), o r by c o m p u t i n g t h e v e l o c i t y a t a p o i n t o u t s i d e t h e s u r f a c e c l o s e t o t h e t r a i l i n g edge and g i v i n g i t t h e d i r e c -t i o n o f -the b i s e c -t o r . T h i s a p p r o a c h may o b v i o u s l y be o f low a c c u r a c y i n -t h e c a s e o f l o a d e d t r a i l i n g e d g e s . In s e c o n d o r d e r methods e x t r a p o l a t e d forms o f t h e K u t t a c o n d i t i o n t o t h e t r a i l i n g edge i t s e l f become p o s s i b l e , which would improve t h e p r e d i c t i o n s f o r l o a d e d t r a i l i n g e d g e s . W i t h t h e i n c r e a s e o f t h e t r a i l i n g edge r a d i u s t h e s e forms o f K u t t a c o n d i t i o n becomes o f q u e s t i o n a b l e a p p l i c a t i o n i n v i e w o f t h e d i f f i c u l t y o f d e f i n i n g a s u i t a b l e " b i s e c t o r " t o t h e t r a i l i n g edge a n g l e . In s u c h c a s e , d i s c r e t i z a t i o n o f t h e t r a i l i n g edge i t s e l f and s p e c i f i c a t i o n o f s t a g n a t i o n p o i n t as done by M a r t e n s e n (1959) can be u s e d . W i t h r e s p e c t t o t h e c o u p l i n g t o t h e s y s t e m o f e q u a t i o n s v a r i o u s c h o i c e s a r e p o s s i b l e . S t r i c t l y s p e a k i n g , t h e system o f e q u a t i o n s h o u l d be r e n d e r e d s i n g u l a r p r i o r t o a d d i n g t h e K u t t a c o n d i t i o n . M a r t e n s e n (1959), makes the s y s -tem s i n g u l a r i n a l e a s t - s q u a r e s s e n s e . In t h e method o f L e w i s and Ryan ( 1 9 7 1 ) , t h e s o - c a l l e d "back d i a g o n a l " c o r r e c t i o n i s t h e most o b v i o u s c h o i c e o f ma-k i n g t h e e x i s t i n g system s i n g u l a r , s i n c e i t r e p l a c e s t h e c o u p l i n g c o e f f i c i e n t i n e a c h column most a f f e c t e d by t r a p e z o i d a l i n t e g r a t i o n e r r o r s by t h e one needed to'make t h e sum o f a l l column e l e m e n t s v a n i s h .

In methods w h i c h a c c u r a t e l y compute t h e c o u p l i n g c o e f f i c i e n t s s u c h c h o i c e i s n o t u n i q u e . In t h e p r e s e n t method t h e o v e r d e t e r m i n e d s y s t e m o f e q u a t i o n s which r e s u l t s from a d d i n g an a d d i t i o n a l e q u a t i o n e x p r e s s i n g t h e K u t t a c o n d i -t i o n , -t o -t h e o r i g i n a l s y s -t e m , i s s o l v e d by a l e a s -t - s q u a r e s me-thod. In -t h i s c a s e t h e K u t t a c o n d i t i o n i s s a t i s f i e d o n l y a p p r o x i m a t e l y w i t h t h e same d e g r e e o f a c c u r a c y as t h e e q u a t i o n s e x p r e s s i n g t h e boundary c o n d i t i o n on t h e c o n t r o l p o i n t s . S t r o n g e r i m p l e m e n t a t i o n s w h i c h r e s u l t from s a t i s f y i n g e x a c t l y t h e K u t t a c o n d i t i o n and o n l y a p p r o x i m a t e l y t h e o t h e r e q u a t i o n s have been c o n s i d e r -ed but d i d n o t i n t r o d u c e d i s c e r n a b l e changes i n t o t h e r e s u l t s .

2 . 2 . 3 . C a l c u l a t i o n of the duct c i r c u l a t i o n . F i r s t i n v i s c i d approximation

and t h e K u t t a c o n d i t i o n . In t h e v i s c o u s i n v i s c i d i t e r a t i o n p r o c e s s , t h e s o l u t i o n o f t h e p o t e n -t i a l f l o w p r o b l e m i s o b -t a i n e d by s p e c i f y i n g -t h e c i r c u l a -t i o n around -t h e d u c -t ' s s e c t i o n . The c i r c u l a t i o n i s d e f i n e d i n a c o n t o u r c o i n c i d i n g w i t h t h e o u t e r s i d e o f t h e d u c t ' s c o n t o u r . T = (5 Y ( s ) d s ( 2 - 3 4 ) and i s a p p r o x i m a t e d by N r = l g . y .0 ) ( 2 - 3 5 )

j = l

A s i m p l e r a p p r o a c h i s based on t h e a s s u m p t i o n t h a t f o r n o n - s e p a r a t e d

f l o w t h e p r e s s u r e g r a d i e n t normal t o t h e s t r e a m l i n e s at t h e t r a i l i n g edge l o c a t i o n i s r a t h e r s m a l l and c a n be n e g l e c t e d . T h i s l e a d s t o t h e e q u a l i t y o f t h e p r e s s u r e on t h e i n n e r and o u t e r s i d e s and c a n be e x p r e s s e d by C = C (2-37) ro u t ri n n P - PQ where C= — i s t h e p r e s s u r e c o e f f i c i e n t . E q u a t i o n (2-37) p r o v i d e s a good * 4 pUo a p p r o x i m a t i o n f o r a c u s p e d t r a i l i n g edge where t h e s t r e a m l i n e s l e a v e t h e t r a i l i n g edge p a r a l l e l t o i t . At i n c r e a s i n g t r a i l i n g edge a n g l e s t r e a m l i n e c u r v a t u r e e f f e c t s become i m p o r t a n t and c o n d i t i o n (2-37) i s i n p r i n c i p l e l e s s a c c u r a t e ( s e e T h w a i t e s , 1960). F o r s e p a r a t e d f l o w a t t h e t r a i l i n g e d g e , T h w a i t e s (1960) shows t h a t , under t h e a s s u m p t i o n s o f f i r s t o r d e r boundary l a y e r t h e o r y , a s i m i l a r c o n d i t i o n t o (2-37) h o l d s , p r o v i d e d t h a t t h e p r e s s u r e i s t a k e n a t t h e boun-d a r y l a y e r s e p a r a t i o n p o i n t s on t h e o u t e r anboun-d i n n e r s u r f a c e s : ( c

p

o u t

p ' i n n <

"

>

o

r ,. = r + p[ (c ) -(c ). 1 , (2-39)

where t h e r e l a x a t i o n f a c t o r f> has a v a l u e comprised between 0.1 and 0.3.

To s t a r t t h e i t e r a t i o n and i f t h e l o c a t i o n o f t h e s e p a r a t i o n p o i n t s a r e n o t known t h e f i r s t i n v i s c i d a p p r o x i m a t i o n i s c o n s i d e r e d t o c o n f o r m w i t h t h e c l a s s i c a l K u t t a c o n d i t i o n . F o r a s h a r p t r a i l i n g edge t h e o c c u r r e n c e o f