UBC Theses and Dissertations

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UBC Theses and Dissertations

A mathematical model for airfoils with spoilers or split flaps Yeung, William Wai-Hung 1985

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A M a t h e m a t i c a l Model f o r A i r f o i l s w i t h S p o i l e r s or S p l i t F l a p s by W i l l i a m Wai-Hung Yeung B.A.Sc. The U n i v e r s i t y of B r i t i s h C o l u m b i a , 1983 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF APPLIED SCIENCE i n THE FACULTY OF GRADUATE STUDIES Department of M e c h a n i c a l E n g i n e e r i n g We a c c e p t the t h e s i s as c o n f o r m i n g to the r e q u i r e d s t a n d a r d THE UNIVERSITY OF BRITISH COLUMBIA August 1985 © W i l l i a m Wai-Hung Yeung, 1985 I n p r e s e n t i n g t h i s t h e s i s i n p a r t i a l f u l f i l m e n t o f t h e r e q u i r e m e n t s f o r an advanced degree a t t h e U n i v e r s i t y o f B r i t i s h C o l u m b i a , I agree t h a t t h e L i b r a r y s h a l l make i t f r e e l y a v a i l a b l e f o r r e f e r e n c e and s t u d y . I f u r t h e r agree t h a t p e r m i s s i o n f o r e x t e n s i v e c o p y i n g o f t h i s t h e s i s f o r s c h o l a r l y purposes may be g r a n t e d by t h e head o f my department o r by h i s o r h e r r e p r e s e n t a t i v e s . I t i s u n d e r s t o o d t h a t c o p y i n g o r p u b l i c a t i o n o f t h i s t h e s i s f o r f i n a n c i a l g a i n s h a l l n o t be a l l o w e d w i t h o u t my w r i t t e n p e r m i s s i o n . Department o f Mechanical F n g i n e e r i n g The U n i v e r s i t y o f B r i t i s h Columbia 1956 Main Mall Vancouver, Canada V6T 1Y3 D a t e August 28. 1985 r\T?_c ti / o i \ i i A b s t r a c t A f l o w model f o r a Joukowsky a i r f o i l w i t h an i n c l i n e d s p o i l e r or s p l i t f l a p i s c o n s t r u c t e d based on the e a r l y work by P a r k i n s o n and J a n d a l i . No r e s t r i c t i o n i s imposed on the a i r f o i l camber, t h e i n c l i n a t i o n and l e n g t h of the s p o i l e r or s p l i t f l a p , and the an g l e of i n c i d e n c e . The f l o w i s assumed t o be s t e a d y , t w o - d i m e n s i o n a l , i n v i s c i d and i n c o m p r e s s i b l e . A sequence of c o n f o r m a l t r a n s f o r m a t i o n s i s de v e l o p e d t o deform t h e contour of the a i r f o i l and the s p o i l e r ( s p l i t f l a p ) onto the c i r c u m f e r e n c e of the u n i t c i r c l e over which the f l o w problem i s s o l v e d . The p a r t i a l l y s e p a r a t e d f l o w r e g i o n b e h i n d t h e s e b l u f f b o d i e s i s s i m u l a t e d by s u p e r i m p o s i n g s u i t a b l e s i n g u l a r i t i e s i n the t r a n s f o r m p l a n e . The t r a i l i n g edge, the t i p of the s p o i l e r ( f l a p ) a r e made c r i t i c a l p o i n t s i n the mappings so t h a t K u t t a c o n d i t i o n s a r e s a t i s f i e d t h e r e . The p r e s s u r e s a t t h e s e c r i t i c a l p o i n t s a r e matched t o the p r e s s u r e i n s i d e the wake, the o n l y e m p i r i c a l i n p u t t o the model. Some s t u d i e s of an a d d i t i o n a l boundary c o n d i t i o n f o r s o l v i n g the f l o w problem were c a r r i e d out w i t h c o n s i d e r a b l e s u c c e s s . The ch o r d w i s e p r e s s u r e d i s t r i b u t i o n s and the o v e r a l l l i f t f o r c e v a r i a t i o n s a r e compared w i t h e x p e r i m e n t s . Good agreement i n g e n e r a l i s a c h i e v e d . The model can be extended r e a d i l y t o a i r f o i l s of a r b i t r a r y p r o f i l e w i t h t h e a p p l i c a t i o n of the Theodorsen t r a n s f o r m a t i o n . i i i T a b l e of C o n t e n t s 1. INTRODUCTION 1 2. HISTORICAL SURVEY 5 3. THEORY 7 3.1 K i n e m a t i c s 11 3.1.1 Conformal T r a n s f o r m a t i o n s f o r a S p o i l e r 11 3.1.2 Conformal T r a n s f o r m a t i o n s f o r a S p l i t F l a p 22 3.2 Dynamics 29 3.2.1 M a t h e m a t i c a l Flow Model 30 3.2.2 Boundary C o n d i t i o n s 33 3.2.3 1-source Models 34 3.2.4 A d d i t i o n a l Boundary C o n d i t i o n s 36 3.2.5 Method of S o l u t i o n and C a l c u l a t i o n 42 4. EXPERIMENTS 48 5. RESULTS and COMPARISONS 53 6. CONCLUSIONS and RECOMMENDATIONS 82 REFERENCES 85 dz APPENDIX A. C o n t i n u i t y of | — | a t G 86 dz APPENDIX B. C o n t i n u i t y of j — | a t • 88 APPENDIX C. L o c a t i o n s of t h e T r a i l i n g Edge 90 APPENDIX D. D e r i v a t i o n of E q u a t i o n s (4) & (5) from J a n d a l i ' s T h e s i s 92 i v APPENDIX E. D e r i v a t i o n of E q u a t i o n (5) 95 APPENDIX F. D e r i v a t i o n of E q u a t i o n (35) 97 APPENDIX G. E v a l u a t i o n s of f , i", f 2 & £2 a t C r i t i c a l P o i n t s 100 V LIST OF FIGURES 1 E x p e r i m e n t a l C_ V a r i a t i o n of NACA 23012 P r o f i l e 4 2 S t r e a m l i n e P a t t e r n s around A i r f o i l w i t h S p o i l e r / S p l i t F l a p 8 3 S e p a r a t i o n Bubble i n f r o n t of S p o i l e r ( J a n d a l i , U n p u b l i s h e d Data) 10 4 Complex Tr a n s f o r m P l a n e s ( S p o i l e r ) 12,16,19 5 Complex Tr a n s f o r m P l a n e s ( S p l i t F l a p ) 24,25,26 6 L o c a t i o n s of S i n g u l a r i t i e s 31 7 C i r c u l a t i o n around A i r f o i l 38 8 Contour D e f o r m a t i o n 39 9 C C o n t r i b u t i o n from the Wake ( S p l i t - F l a p ) 46 L 10 C p b V a r i a t i o n s of 20% & 30% c S p l i t F l a p s 54 11 Comparison on C^ D i s t r i b u t i o n s from 1-source Models and E x p e r i m e n t s 56 12 Comparison on C^ D i s t r i b u t i o n s from Eqns. (35) & (38) and E x p e r i m e n t s 58 13 Comparison on C^ D i s t r i b u t i o n s from Eqns. (38) & (39) and E x p e r i m e n t s 60 14 C D i s t r i b u t i o n s a t D i f f e r e n t 6 (20% c S p l i t P F l a p ) 62 15 C V a r i a t i o n over a a t D i f f e r e n t 8 (20% c S p l i t Li F l a p ) 63 16 C D i s t r i b u t i o n s a t D i f f e r e n t 8 (30% c S p l i t F l a p ) .. 64 P 17 C_ V a r i a t i o n over a a t D i f f e r e n t 8 (30% c S p l i t L v i F l a p ) 65 18 C D i s t r i b u t i o n s ( S p o i l e r , e/c = 50%, P h/c = 5 %, 6 = 45°) 66 19 C D i s t r i b u t i o n s ( S p o i l e r , e/c = 70%, P h/c = 5 %, 6 = 45°) 67 20 C D i s t r i b u t i o n s ( S p o i l e r , e/c = 90%, P h/c = 5 %, 6 = 45°) 68 21 C V a r i a t i o n over a ( S p o i l e r , L e/c = 50%, h/c = 5%, 6 = 45°) 69 22 C V a r i a t i o n over a ( S p o i l e r , L e/c = 70%, h/c = 5%, 6 = 45°) 70 23 C V a r i a t i o n over a ( S p o i l e r , L e/c = 90%, h/c = 5%, 6 = 45°) 71 24 C D i s t r i b u t i o n s ( S p o i l e r , e/c = 50%, P h/c = 10%, a = 6°) 72 25 Cp D i s t r i b u t i o n s ( S p o i l e r , e/c = 70%, h/c = 10%, a = 6°) 73 26 C D i s t r i b u t i o n s ( S p o i l e r , e/c = 90%, P h/c = 10%, a = 6°) 74 27 C D i s t r i b u t i o n s ( S p o i l e r , e/c = 50%, P h/c = 10%, a = 12°) 75 28 C D i s t r i b u t i o n s ( S p o i l e r , e/c = 70%, P h/c = 10%, a = 12°) 76 29 C D i s t r i b u t i o n s ( S p o i l e r , e/c = 90%, P h/c = 10%, a = 12°) 77 30 C V a r i a t i o n s over a ( S p o i l e r , e/c = 50%, , h/c = 10%) 78 v i i 31 C V a r i a t i o n s over a ( S p o i l e r , e/c = 70%, L h/c = 10%) 79 32 C V a r i a t i o n s over a ( S p o i l e r , e/c = 90%, h/c = 10%) 80 F1 T r a n s f o r m p l a n e s Z and $ 98 G1 D e f i n i t i o n s of V and 6 102 G2 D e f i n i t i o n s of A, v, <f>, m, TJ and R 103 v i i i LIST OF TAELES 1 S p o i l e r 14 2 C o - o r d i n a t e s of P o i n t s B, C, D, E, A, G, F i n D i f f e r e n t P l a n e s f o r S p o i l e r s 17 3 S p l i t F l a p 27 4 C o - o r d i n a t e s of P o i n t s B, C, D, E, A, G, F i n D i f f e r e n t P l a n e s f o r S p l i t F l a p s 28 o i x ACKNOWLEDGEMENT I would l i k e t o t a k e t h i s o p p o r t u n i t y t o thank P r o f e s s o r G. V. P a r k i n s o n f o r h i s e n t h u s i a s t i c and i n v a l u a b l e guidance d u r i n g the co u r s e of t h i s r e s e a r c h . H i s support and s u g g e s t i o n s were much a p p r e c i a t e d . Mr. T. Y. Lu, graduate s t u d e n t , d e s i g n e d and c a r r i e d out the wind t u n n e l measurements on s p l i t f l a p s . Mr. E. A b e l of the M e c h a n i c a l E n g i n e e r i n g Machine Shop b u i l t the p a r t s f o r the a u t h o r ' s e x p e r i m e n t s on s p o i l e r s . T h e i r h e l p i s g r a t e f u l l y acknowledged. The a u t h o r would a l s o l i k e t o thank the Computing C e n t e r of the U n i v e r s i t y of B r i t i s h Columbia f o r the use of t h e i r f a c i l i t i e s . 1 1. INTRODUCTION T h e o r e t i c a l as w e l l as e x p e r i m e n t a l i n v e s t i g a t i o n s of b l u f f body aerodynamics, e s p e c i a l l y p a r t i a l l y s e p a r a t e d f l o w s a s s o c i a t e d w i t h a i r f o i l s equipped w i t h s p o i l e r s , have been conducted i n the Department of M e c h a n i c a l E n g i n e e r i n g a t the U n i v e r s i t y of B r i t i s h Columbia f o r some y e a r s . The m o t i v a t i o n b e h i n d t h i s s tudy i s t h a t s p o i l e r s have been w i d e l y used on a i r c r a f t wings as a d e v i c e f o r l a n d i n g and maneuvering p u r p o s e s . They de c r e a s e the l i f t by r e d u c i n g c i r c u l a t i o n , and a l s o f u n c t i o n as a i r b r a k e s t o i n c r e a s e s t r o n g l y the d r a g , t h u s r e d u c i n g c o n s i d e r a b l y the speed and g e n e r a t i n g a s t e e p e r g l i d e a n g l e . They can a l s o s e r v e f o r r o l l c o n t r o l i f d e f l e c t e d a s y m m e t r i c a l l y . The c o u n t e r p a r t of a s p o i l e r , a s p l i t f l a p , has been s t u d i e d r e c e n t l y f o r one of i t s f e a t u r e s of p r o d u c i n g h i g h l i f t a t lower a n g l e s of i n c i d e n c e . W i t h the s p l i t f l a p d e p l o y e d , the maximum l i f t c o e f f i c i e n t would be i n c r e a s e d so t h a t the t a k e - o f f d i s t a n c e of an a i r c r a f t may be s h o r t e n e d i f the l i f t - t o - d r a g r a t i o i s kept s u f f i c i e n t l y h i g h . A l t h o u g h m u l t i - s l o t t e d f l a p s a r e used i n a i r c r a f t wings i n r e c e n t y e a r s f o r v a r i o u s r e a s o n s , a somewhat unexpected r e s u l t [1] has been ob s e r v e d t h a t when based on the extended c h o r d a t a Reynolds number (Re) 6 x 10 6 or h i g h e r , the maximum a t t a i n a b l e C i s L i n e a r l y the same f o r s p l i t , s l o t t e d , and d o u b l e - s l o t t e d f l a p s . 2 In o t h e r words, f o r p r o d u c i n g the l i f t f o r c e , the s p l i t f l a p may be j u s t as e f f i c i e n t as o t h e r t y p e s of f l a p . Both s p o i l e r s and s p l i t f l a p s have been employed, f o r i n s t a n c e , on the B-52 s t r a t o f o r t r e s s and o t h e r r e c e n t o p e r a t i o n a l a i r p l a n e s , and t h e y c o n t i n u e t o p l a y an i m p o r t a n t r o l e i n the d e s i g n of a i r p l a n e s . In p r a c t i c e , wing f l a p s and s p o i l e r s as used on wings, a r e t h r e e d i m e n s i o n a l d e v i c e s , l i m i t e d i n span and sometimes r a k e - l i k e i n shape, u s u a l l y i n t e r r u p t e d by the f u s e l a g e and i n t e r f e r e d w i t h by n a c e l l e s and p r o p e l l e r s l i p s t r e a m s . However, the two d i m e n s i o n a l aerodynamic c h a r a c t e r i s t i c s a r e the b a s i s from which more c o m p l i c a t e d c o n f i g u r a t i o n s can be u n d e r s t o o d . The s i m p l e two d i m e n s i o n a l wake s i n g u l a r i t y model f o r b l u f f body f l o w s d e v e l o p e d by P a r k i n s o n and J a n d a l i [2,3] p r o v i d e s the b a s i c i n f o r m a t i o n f o r the f l o w a s s o c i a t e d w i t h a i r f o i l normal s p o i l e r s . In a d d i t i o n , i t s r e l i a b l i t y , m a t h e m a t i c a l e l e g a n c e and f l e x i b i l i t y a l l o w the model t o be used f o r a c c u r a t e p r e d i c t i o n of the l o a d i n g on any a i r f o i l - s h a p e d p r o f i l e equipped w i t h a s p o i l e r or s p l i t f l a p . C o s t l y and t e d i o u s wind t u n n e l t e s t s f o r p r e s s u r e d i s t r i b u t i o n and l i f t v a r i a t i o n may then be reduced t o a c e r t a i n e x t e n t . A s p l i t f l a p i s formed by d e p l o y i n g the r e a r p o r t i o n of t h e lower s u r f a c e of the a i r f o i l about th e f o r w a r d edge of the f l a p . T h e r e f o r e , the l e n g t h of the f l a p i s e q u a l t o the d i s t a n c e between i t s h i n g e and the t r a i l i n g edge. Whereas f o r a s p o i l e r , i t i s l o c a t e d on the upper s u r f a c e and u s u a l l y 3 p o s i t i o n e d somewhere between the mid-chord and the t r a i l i n g edge, i n o r d e r t o a c h i e v e the d e s i r e d performance. I t s h e i g h t o f t e n e x t e n d s from 5 t o 10 % of the c h o r d . In p r a c t i c e , the a n g l e of d e f l e c t i o n , 8, f o r both d e v i c e s ranges from 0° t o 60°, a l t h o u g h no such r e s t r i c t i o n i s imposed on 6 i n the p r e s e n t t h e o r y . F i g . 1 from [4] shows e x p e r i m e n t a l l y how C L i v a r i e s w i t h a, a n g l e of i n c i d e n c e , as 6 i s i n c r e a s e d from 0° t o 60° f o r the NACA 23012 p r o f i l e f i t t e d w i t h a s p l i t f l a p of 20 % c h o r d . I t i s c l e a r from F i g . 1 t h a t d e f l e c t i n g the s p l i t f l a p i s e q u i v a l e n t t o a l t e r i n g the e f f e c t i v e camber of the a i r f o i l , or more p r e c i s e l y , t o d e c r e a s i n g the a n g l e of z e r o l i f t , but w i t h o u t c o n s i d e r a b l y d i s t o r t i n g the l i f t c u r v e s l o p e . The o p p o s i t e e f f e c t t a k e s p l a c e i f a s p o i l e r i s e r e c t e d . D e p l o y i n g t h e s p l i t f l a p or s p o i l e r p r o v i d e s the s i m p l e s t method o f a c c o m p l i s h i n g the d e s i r a b l e l i f t c o e f f i c i e n t t h r o u g h the temporary a l t e r a t i o n of t h e a i r f o i l geometry. F i g . 1 E x p e r i m e n t a l C V a r i a t i o n of NACA 23012 P r o f i l e 5 2 . HISTORICAL SURVEY An e a r l y t h e o r e t i c a l study of p o t e n t i a l f l o w around an a i r f o i l w i t h a s p l i t f l a p and c i r c u l a t i o n c o n t r o l by s u c t i o n , was c o nducted by Whitehead, Cheers and Mandl [5] a t NAE. The co m b i n a t i o n of the a i r f o i l and the s p l i t f l a p i s i d e a l i z e d by a s t r a i g h t - l i n e c o n f i g u r a t i o n w i t h s u c t i o n produced by a s i n k l o c a t e d on t h e upper s u r f a c e of the f l a p i n o r d e r t o s i m u l a t e the e f f e c t of the s e p a r a t e d wake. The e f f e c t s from the l e a d i n g edge s e p a r a t i o n b u b b l e , the f i n i t e t h i c k n e s s of the a i r f o i l and the wind t u n n e l c o n s t r a i n t t e n d t o widen the d i s c r e p a n c y between the e x p e r i m e n t a l and t h e o r e t i c a l l i f t v a r i a t i o n over a range of a n g l e s of i n c i d e n c e . B e s i d e s , no i n f o r m a t i o n c o n c e r n i n g t h e l o a d i n g over the a i r f o i l i s a v a i l a b l e . Woods [6] d e a l t w i t h a s i m i l a r problem i n the form of a l i n e a r p e r t u r b a t i o n f r e e s t r e a m l i n e p o t e n t i a l t h e o r y . As p o i n t e d out i n P a r k i n s o n [ 7 ] , Woods' t h e o r y i s awkward t o a p p l y because i t r e q u i r e s two e m p i r i c a l d e t a i l s of the s e p a r a t e d f l o w , and does not account f o r a i r f o i l t h i c k n e s s , which may be a s i g n i f i c a n t c o n t r i b u t i o n . A comparsion between the t h e o r e t i c a l p r e s s u r e d i s t r i b u t i o n s by Woods and P a r k i n s o n & J a n d a l i can be found i n r e f e r e n c e [ 8 ] . The p r e s e n t work f o l l o w s c l o s e l y J a n d a l i ' s [3] approach but has been . extended t o d e a l w i t h a r b i t r a r i l y i n c l i n e d 6 s p o i l e r s or s p l i t f l a p s . The f l o w i s a l s o assumed t o be i n v i s c i d but the t h e o r y i s e x a c t i n the sense t h a t no l i n e a r i z a t i o n t e c h n i q u e i s a p p l i e d . That i s , the a i r f o i l i s of f i n i t e t h i c k n e s s and camber. T h e r e f o r e , the r e s u l t i n g p r e s s u r e l o a d i n g over the a i r f o i l does not p o s s e s s a l e a d i n g edge s i n g u l a r i t y . The o n l y e m p i r i c a l i n p u t i s the p r e s s u r e i n s i d e the wake beh i n d the s p o i l e r or s p l i t f l a p because t h e r e i s not any t h e o r y t o p r e d i c t i t s v a l u e s a t v a r i o u s g e o m e t r i c a l c o n f i g u r a t i o n s . 7 3. THEORY The work d e s c r i b e d h e r e i n c o n c e r n s o n l y the Joukowsky a i r f o i l f a m i l y . However, i t can be r e a d i l y extended t o o t h e r a i r f o i l shapes by a p p l y i n g the Theodorsen t r a n s f o r m a t i o n t o the p h y s i c a l p l a n e where the a i r f o i l i s l o c a t e d . In f a c t , J a n d a l i [3] a c c o m p l i s h e d t h i s e x t e n s i o n t o a C l a r k - Y a i r f o i l w i t h a normal s p o i l e r and a c h i e v e d s a t i s f a c t o r y r e s u l t s . The p r e s e n t t h e o r y d e a l s w i t h the k i n e m a t i c s as w e l l as the dynamics of the f l u i d f l o w around an a i r f o i l w i t h an i n c l i n e d s p o i l e r or s p l i t f l a p . I t i s assumed t h a t the f l o w i s everywhere s t e a d y , two d i m e n s i o n a l , i n v i s c i d and i n c o m p r e s s i b l e except i n the r e g i o n where a broad wake, caused by f l o w s e p a r a t i o n , i s l o c a t e d ( F i g s . 2a,b). T h e r e f o r e , the p o t e n t i a l f l o w t h e o r y w i t h the t e c h n i q u e of c o n f o r m a l t r a n s f o r m a t i o n can be used t o d e a l w i t h the k i n e m a t i c s . The d y n a m i c a l i n p u t i s t h r o u g h the p r e s c r i b e d boundary c o n d i t i o n s w i t h an e m p i r i c a l parameter, which i s the c o n s t a n t p r e s s u r e i n the wake behind the s p o i l e r o r s p l i t f l a p . Shear l a y e r s emanating from the p o i n t s of s e p a r a t i o n ( i . e . the t r a i l i n g edge, and the t i p . of the s p o i l e r or s p l i t f l a p ) a r e m o d e l l e d as s e p a r a t i n g s t r e a m l i n e s of i n i t i a l l y c o n s t a n t p r e s s u r e . B e f o r e s e p a r a t i o n t a k e s p l a c e , i t i s r e a s o n a b l e t o assume, p a r t l y as a m a t t e r of o b s e r v a t i o n and p a r t l y from F i g . 2 S t r e a m l i n e P a t t e r n s around A i r f o i l w i t h S p o i l e r / S p l i t F l a p 9 m a t h e m a t i c a l a n a l y s i s , t h a t the v i s c o u s boundary l a y e r i s t h i n over an a i r f o i l s u r f a c e . T h e r e f o r e , the f l o w can be t r e a t e d as i r r o t a t i o n a l and B e r n o u l l i ' s p r i n c i p l e i s a p p l i c a b l e t o c a l c u l a t e the p r e s s u r e v a r i a t i o n a l o n g a s t r e a m l i n e of i n t e r e s t , which i s the a i r f o i l p r o f i l e as c o n s i d e r e d h e r e . However, i n s e r t i n g a s p o i l e r or s p l i t f l a p w i l l d e f i n i t e l y slow down the f l u i d p a r t i c l e s a p p r o a c h i n g i t , and t h u s a l t e r the t h i c k n e s s of the boundary l a y e r ahead of t h i s o b s t a c l e . In f a c t , i n the case of an i n c l i n e d s p o i l e r , the s t r o n g adverse p r e s s u r e g r a d i e n t over the upper s u r f a c e and near the l e a d i n g edge of the a i r f o i l enhances t h i s d e c e l e r a t i o n of f l u i d f l o w so t h a t an e a r l y f l o w s e p a r a t i o n , i n c l u d i n g some backward f l o w , t a k e s p l a c e upstream of the s p o i l e r . T h i s s e p a r a t i o n of f l o w f o l l o w e d by reattachment over t h e s p o i l e r s u r f a c e , forms a c l o s e d bubble of c i r c u l a t o r y a i r a t a f a i r l y c o n s t a n t p r e s s u r e . F i g . 3 i s a f l o w v i s u a l i z a t i o n of t h e bubble t a k e n i n a smoke t u n n e l a t low Reynolds number. As t h e Reynolds number i s i n c r e a s e d , s t r o n g e r f l o w o u t s i d e the boundary l a y e r r e s i s t s the backward f l o w from s t a r t i n g f u r t h e r upstream of the s p o i l e r , and thus causes the s i z e of the bubble t o d i m i n i s h . On the o t h e r hand, the o c c u r r e n c e of the bubble i s l e s s pronounced i n f r o n t of the s p l i t f l a p because the p r e s s u r e g r a d i e n t i s i n g e n e r a l l e s s prominent over the l o w e r s u r f a c e . T h e r e f o r e , the s e p a r a t i o n bubble i s s m a l l e r and t h e t h e o r e t i c a l p r e d i c t i o n of the p r e s s u r e near i t i s r e a s o n a b l y good. N e v e r t h e l e s s , s i n c e v i s c o u s e f f e c t s dominate i n t h i s r e g i o n , t h e p o t e n t i a l f l o w t h e o r y p r e s e n t e d here i s , F i g . 3 S e p a r a t i o n Bubble i n f r o n t of S p o i l e r ( J a n d a l U n p u b l i s h e d Data) 11 i n g e n e r a l , e x p e c t e d t o p r e d i c t l e s s s a t i s f a c t o r y r e s u l t s . E v i d e n c e i s found i n b o t h J a n d a l i [3] and the p r e s e n t work. No attempt has been made i n the t h e o r y t o model the presence of the bubble i n e i t h e r c a s e . 3.1 K i n e m a t i c s S o l v i n g the f l o w problem of any b l u f f body s e c t i o n d i r e c t l y i n the p h y s i c a l p l a n e i s not always easy. More o f t e n , the degree of c o m p l e x i t y of the geometry can be s u b s t a n t i a l l y reduced i f t h e method of c o n f o r m a l t r a n s f o r m a t i o n i s u t i l i z e d . P a r k i n s o n and J a n d a l i [2] s u c c e s s f u l l y found the mappings f o r a number of b l u f f body shapes. In J a n d a l i [ 3 ] , c o n f o r m a l maps were used t o deform t h e c o n t o u r of an a i r f o i l w i t h a normal s p o i l e r onto the c i r c u m f e r e n c e of a u n i t c i r c l e over which the f l o w problem was s o l v e d . For an a i r f o i l w i t h an i n c l i n e d s p o i l e r or s p l i t f l a p , the use of c o n f o r m a l mapping i s i n e v i t a b l e because of the n o n - t r i v i a l geometry i n v o l v e d . J a n d a l i ' s method i s used here but the sequence of mappings i s more i n t r i c a t e . 3.1.1 C o n f o r m a l T r a n s f o r m a t i o n s f o r an I n c l i n e d S p o i l e r A c i r c l e of r a d i u s R c e n t e r e d a t t 0 (~e,n) i n t h e t - p l a n e ( F i g . 4b) w i l l become a Joukowsky a i r f o i l i n the z - p l a n e ( F i g . 4a) when g i v e n the well-known Joukowsky t r a n s f o r m a t i o n : z = t + 1/t ... (1) The camber and the t h i c k n e s s of t h e d e r i v e d a i r f o i l a r e F i g . 4 Complex Transform P l a n e s ( S p o i l e r ) 13 r e l a t e d t o the l o c a t i o n of t 0 . The magnitude of R can be c a l c u l a t e d by the formula An i n c l i n e d s p o i l e r which i s r e p r e s e n t e d by the fence BCD i n the z-plane i s mapped from a s t r a i g h t l i n e segment BCD emanating from p o i n t B i n the t - p l a n e . The i n c l i n a t i o n 6 90°), which i s the measure of the a n g l e of i n t e r s e c t i o n between BC and the tangent l i n e a t B, s t a y s c o n s t a n t under the f o r e m e n t i o n e d c o n f o r m a l t r a n s f o r m a t i o n . In o t h e r words, the s p o i l e r i n the p h y s i c a l p l a n e z i s a l s o making the same a n g l e 6 w i t h the s u r f a c e of the a i r f o i l . However, s i n c e (1) i s not a l i n e a r t r a n s f o r m a t i o n , the s p o i l e r BCD becomes a s l i g h t l y c u r v e d segment which i s not i n a p p r o p r i a t e f o r p r a c t i c a l s p o i l e r h e i g h t s . N e v e r t h e l e s s , i t s h o u l d be noted t h a t the s p o i l e r ( s p l i t f l a p ) t i p d e f l e c t i o n a n g l e s w i l l be from 0.5° t o 1.0° l e s s than the r o o t v a l u e s of 6 used throughout the t h e s i s . The chordwise l o c a t i o n e and the h e i g h t ft of the s p o i l e r i n the z-plane a r e r e l a t e d t o the a n g u l a r v a r i a b l e 0 O, h e i g h t ft and i n c l i n a t i o n 6 i n t h e t - p l a n e . T a b l e 1 p r o v i d e s some v a l u e s of e/c, 0 O, 8, & and n used i n the c a l c u l a t i o n s . A c o u n t e r - c l o c k w i s e r o t a t i o n t h r ough 7 combines w i t h t r a n s l a t i o n of A t o produce the t r a n s f o r m a t i o n R = / (1 + e ) 2 + u2 ( l a ) s = ( t - A) e i7 (2) where 7 = TT/2 - 0 O - 8 (2a) and A = t 0 + R cos 8 e i ( 0 o + M (2b) T a b l e 1: SpoiIer e/c do .90 32.25° .70 61.25° .50 86.25" ft/c = 5 % , 6 = 45°, n = 1.5 e/c h I .90 .2013 5.9817 6.9252 .70 .1284 9.4638 10.4268 .50 &/c = 10 e/c .1183 %, 6 = 30°, r n 10.2855 3 I T 11.2513 J? .90 .4832 2.3162 2.0201 .70 .2616 4.3887 3.2539 .50 fc/c = 10 e/c .2366 %, 6 = 60°, r n 4.8730 1 1 3.5380 V .90 .4329 2. 1979 4.8253 .70 .2668 3.6177 7.3304 .50 .2418 4.0015 8.0026 15 T h i s mapping r e s u l t s i n b r i n g i n g the l i n e segment BCD i n the t - p l a n e onto the h o r i z o n t a l a x i s i n the s - p l a n e . And the c i r c l e i s c e n t e r e d a t s = -R cos 8 i , F i g 4c. Wit h the K a r m a n - T r e f f t z t r a n s f o r m a t i o n s = i R s i n 8 c o t (co/2) ... (3) the c o n t o u r i n the s-plane becomes a deg e n e r a t e p o l y g o n i n the co-plane as shown i n F i g . 4d. The two c i r c u l a r a r c s DEAG and GFB become two i n f i n i t e v e r t i c a l l i n e s t o the l e f t and the r i g h t of the o r i g i n of the co-plane, r e s p e c t i v e l y , whereas the s t r a i g h t l i n e segment BCD i s a v e r t i c a l ray s t a r t i n g from the p o i n t a) = h i . The f l o w o u t s i d e the a i r f o i l i n t h e z-plane i s mapped onto the i n t e r i o r of the polygon' bounded by GBDG. co = 0 co r r e s p o n d s t o z = °°. T a b l e 2 summarizes the c o - o r d i n a t e s of the p o i n t s of i n t e r e s t i n the t , s and co-planes. Note t h a t n = 2 (1 - 8/TT) . . . (3a) a n d h • l n "I 2 5 l " 6 + * ] ... (3b) n where 1 £ n < 2. The i n t e r i o r of the po l y g o n i s mapped o n t o t h e upper h a l f p l a n e of X ( F i g . 4e) by t h e S c h w a r z - C h r i s t o f f e l t r a n s f o r m a t i o n 3^ = k X (X + n ) " 1 (X - 2 + n ) " 1 ... (4) dX which can be i n t e g r a t e d t o co = - ? ( 2 - n ) + i h - J {n l n ( - + 1) + (2-n) l n ( - ^ - - 1} 2 2 n 2-n (5) T a b l e 2 : C o - o r d i n a t e s of p o i n t s B , C , D, E , A . G , F In d i f f e r e n t p l a n e s f o r spoilers. P o i n t t - p l a n e s - p l a n e o - p l a n e B i 0 o t o + R e R s i n 6 n n / 2 + i - o r + i ~ C 10o - i y t o + R e + ft e R s i n 6 + ft h i D i 0 o to + R e R s i n 6 ( 2 - n ) n 7 2 + i ~ or + i -E 1 'T (1 - A)e 2 c o t "1 [ (A - 1)e / ( R s i n 6 ) ] A 1 ( 0 o + 6 ) t o - R e - ( R + R cos 6 ) i - ( 2 - n ) n / 2 G i ( 0 o + 2 6 ) to + R e -R s i n 6 n n / 2 - i-> or - ( 2 - n ) n / 2 - i ~ F i ( 0 o + 6 ) t o + R e (R - R cos 6 ) i n n / 2 18 The s c a l e f a c t o r k and t h e i n t e g r a t i o n c o n s t a n t a re e v a l u a t e d by s e t t i n g to = h i f o r X = 0 7T co = n- + i» f o r X = -n~ co = => i f o r X = -n* , as shown i n appendix E. Le t X = H + irj c o r r e s p o n d t o co = 0 or z = °°. Then X and rj are oo g i v e n by e 2 h = [ ( I + i ) 2 + ( S ) 2 ] n / 2 [ ( J _ _ 1 ) a + ( J L ) 2 ] ( 2 - n ) / 2 n n 2-n 2-n (6) 7r(2-n) = n t a n - M r 2 - ) + (2-n) t a n " M r - 2 - - > ; • • • <7> X+n J-2+n When n=1 or 6 = 90°, (6) & (7) can be s i m p l i f i e d t o T = 0 and TJ = / ( e 2 h - D . For 1 < n < 2, % and 7? have t o be found n u m e r i c a l l y . See T a b l e 1 f o r a summary of v a l u e s of T, ri, n and f i . The t r a n s f o r m a t i o n ( 8 ) , which i s a c o m b i n a t i o n of s h i f t i n g and s c a l i n g , X = X + n T ... (8) ens u r e s t h a t 7v = i i s e q u i v a l e n t t o z = », see F i g . 4 f . Here, Oo r\ i s t a k e n t o be non-zero. F i n a l l y , a b i l i n e a r t r a n s f o r m a t i o n and a r o t a t i o n , (9) F i g . 4 Complex Transform P l a n e s ( S p o i l e r ) 20 5 - e~ i a°(i±|) ... (9) map the f l o w on the upper h a l f p l a n e of X t o the e x t e r i o r of the u n i t c i r c l e i n the $-plane. Angle a0, which i s d e f i n e d by ( 1 3 ) , i s chosen so t h a t the u n i f o r m f l o w a t i n f i n i t y i s p a r a l l e l t o the r e a l a x i s i n the $-plane, F i g . 4g. A f t e r a sequence of 6 mappings, the c o n t o u r of the a i r f o i l w i t h the s p o i l e r becomes the p e r i m e t e r of the u n i t c i r c l e , F i g . 4 g . The d i s t r i b u t i o n of p o i n t s B, C, D, E, e t c . , can be o b t a i n e d once the f o r m u l a f o r a 0 i s e s t a b l i s h e d . dz The combined t r a n s f o r m a t i o n d e r i v a t i v e — can be o b t a i n e d d$ by the c h a i n r u l e : I dz dz dt ds do; dX d*X d$ " dt ds doj dX d* d$ and reduces t o dz t 2 - 1 _ i 7 r .~ s i n 6 -,u>..r X i , n ( i - 7 v ) 2 i a 0 - = — e [ - 1 R - j - c s c 2 ( - ) ] [ ( x + e (10) (10) has e s s e n t i a l l y two s i m p l e z e r o s and two s i m p l e p o l e s i n the r e g i o n c o r r e s p o n d i n g t o the f l o w f i e l d and the a i r f o i l boundary : a) s i m p l e z e r o a t the t r a i l i n g edge E, t = 1, b) s i m p l e z e r o a t the t i p of t h e s p o i l e r C, X = 0, c) s i m p l e p o l e a t the s p o i l e r base B, X = -n, and d) s i m p l e p o l e a t the s p o i l e r base D, X = 2-n. 21 dz The p o i n t s a t which — has s i m p l e z e r o s a r e c a l l e d the c r i t i c a l p o i n t s . Because of the d o u b l i n g of the a n g l e s at c r i t i c a l p o i n t s , the s t a g n a t i o n s t r e a m l i n e s l e a v i n g E and C i n the S-plane become the t a n g e n t i a l s e p a r a t i o n s t r e a m l i n e s a t the c o r r e s p o n d i n g p o i n t s i n the z - p l a n e . The p o i n t s a t which 4^ " has s i m p l e z e r o s w i l l be s t a g n a t i o n p o i n t s i n the z - p l a n e . dz Only p o i n t B i s of i n t e r e s t because p o i n t D i s w i t h i n the wake i n w h i c h t h e f l o w i s i g n o r e d . However, t h e p o i n t G on the a i r f o i l s u r f a c e , co = - ( 2 - n ) ~ - i» or n~ - i°° , and the p o i n t a t i n f i n i t y , "X = i , a r e not s i m p l e z e r o s of (10) but removable s i n g u l a r i t i e s . The p r o o f s a r e g i v e n i n a p p e n d i c e s A and B, r e s p e c t i v e l y . I L e t Ue l a and V be the v e l o c i t i e s of the f l o w a t i n f i n i t y i n t h e z and ^ - p l a n e s . They a r e r e l a t e d t h rough U e " i a > V ( ^ ) | ... (11) dz 1 Z =co From e q u a t i o n s (11) and (B4,appendix B ) , one can deduce t h a t ¥ . ,dz U 'dS 1 z=°° _ R s i n 6 / [ (T+n) 2 + T ? 2][ (T-2+n) 2 + rj2 ] n / ( I 2 + V) (12) And a 0 = ^ + a + y + 0, - 62 - 03 ... (13) where 0, = t a n " 1 ( | ) (13a) 62 = t a n ' 1 ( r ^ - ) ... (13b) and 6 3 = t a n " 1 ( F f r r ; ) 22 (13c) The a n g u l a r p o s i t i o n s of the s i m p l e z e r o s i n the $-plane a r e g i v e n by The e q u a t i o n s used t o determine the p o i n t E i n the "K p l a n e , "X E , a r e g i v e n i n appendix C. E q u a t i o n s (14) and (15) s h o u l d reduce t o the c o r r e s p o n d i n g ones g i v e n i n J a n d a l i [3] when 5 = 90° and n = 1 s i n c e both J a n d a l i ' s and the c u r r e n t mapping sequences deform the a i r f o i l c o n t o u r onto the c i r c u m f e r e n c e of the u n i t c i r c l e . The d e t a i l s of the d e r i v a t i o n a r e g i v e n i n appendix D. 3.1.2 Conformal T r a n s f o r m a t i o n f o r I n c l i n e d S p l i t F l a p s The mappings f o r an i n c l i n e d s p l i t f l a p a r e s i m i l a r t o t h o s e f o r an i n c l i n e d s p o i l e r . F i g s . 5 a-g d e p i c t t h e sequence i n v o l v e d . The e q u a t i o n s of T r a n s f o r m a t i o n s a r e : (14) 6„ = - cos [ ( 1 - * E ) / ( - 1 + 7 ; 2 ) ] - a0 (15) z = t + 1/t (16) s = ( t - A) e -17 (17) s = I R s i n 5 c o t (—) (18) nir , ., i f i / X = + l h - - {n l n ( - -2 2 n 1) + (2-n) l n ( 2-n X + U (19) 23 x = -\ + TJ r ... (20) 5 - e ~ i a ° ( ^ | ) ... (21) l ~7v where A = t 0 + R cos 6 e 7 = \ - 0O - 6 - i ( 0 o + 5 ) and e q u a t i o n s ( 3 a ) , ( 3 b ) , ( 6 ) , (7) a p p l y w i t h X = -T+ir}. A l s o 00 §?- = i i i i e ^ [ . i R s i ^ _ 5 c s c 2 ( s ) ] I _ ^ L ^ ] a L i i n a e i a 0 d$ t 2 2 2 (X-n)(X+2-n) 2 i (22) a 0 = | + a - 7 + 0, - 02 ~ 0 3 (23) where 0, = TT - t a n ' 1 ( ^ ) ... (23a) 02 = v - t a n - 1 ( ^ r ) . .. (23b) 'T+n' JL lT-2+n and 03 = t a n - 1 U L-) ... (23c) The a n g u l a r p o s i t i o n s of the s i m p l e z e r o s i n the $-plane a r e g i v e n by 0 C = -it + 2 t a n " 1 ( ^ ) - a 0 ... (24) 0 E = c o s - H d - T g J / d + T : 2 ) ] - a 0 ... (25) where "K i s d e f i n e d i n appendix C. V a l u e s of e/c, 60, 6, R, T and rjf a r e g i v e n i n T a b l e 3. T a b l e 4 g i v e s the l o c a t i o n s of p o i n t s B, C, D, E, A, G and F. © @ B fit B . D D hi .(2-n) | (d) 5 Complex Transform P l a n e s ( S p l i t F l a p ) to 01 F i g . 5 Complex Transform P l a n e s ( S p l i t F l a p ) T a b l e 3: Spl i t FI ay e/c = .80, d0 = 51.25 6 h T V 10° .7647 0.6769 0.3865 30° .7146 1.5339 1 .5341 45° .6842 1.6851 2.5233 60° .6591 1 .4212 4.3691 e/c = .70, 0 O = 63.75° 1 6 h T 1 V 10° .9861 0.5139 0.3461 30° .9459 1 .1403 1.2768 45° .9257 1.2278 2.0250 60° .9056 1.0208 2.6849 T a b l e 4 : C o - o r d i n a t e s of p o i n t s B, C, D, E, A, G. F i n d i f f e r e n t p l a n e s f o r spli t f1aps. P o i n t t - p i a n e s-plane o-piane B - 100 t o + R e R s i n 6 -nir / 2 + i ~ or + 1-C "100 1 T t o + R e + h e R s i n 6 + n h 1 D - 100 t o + R e R s i n 6 ( 2 - n ) n / 2 + 1~ or + 1~ E 1 (1 - A)e 2 c o t - 1 [ ( A - 1)e /(R s i n 6 ) ] A -1 ( 0 o + 6 ) t o - R e (R + R cos 6 )1 ( 2 - n ) i r / 2 G -1 ( 0 o + 2 6 ) t o + R e -R s1n 6 -nn / 2 - 1- or ( 2 - n ) n / 2 - i ~ F -1 ( 0 o + 6 ) t o + R e - (R. - R cos 6 )1 -n7f/2 29 3.2 Dynamics In the p h y s i c a l p l a n e z the u n i f o r m f l o w , when a p p r o a c h i n g the a i r f o i l , s e p a r a t e s from the t r a i l i n g edge and the t i p of t h e s p o i l e r or s p l i t f l a p . A h i g h l y t u r b u l e n t broad wake, s i t u a t e d b e h i n d the s p o i l e r or s p l i t f l a p and bounded by the s e p a r a t i n g shear l a y e r s emanating from the p o i n t s of s e p a r a t i o n , i s formed and found, e x p e r i m e n t a l l y , t o be n e a r l y at c o n s t a n t p r e s s u r e . Owing t o the c o m p l e x i t y of the wake dynamics, i n c l u d i n g the f o r m a t i o n of Karman v o r t e x s t r e e t s , and t o the i n s u f f i c i e n t u n d e r s t a n d i n g of the i n t e r a c t i o n between the wake and the shear l a y e r s , no t h e o r i e s a r e a v a i l a b l e t o c o r r e c t l y p r e d i c t the p r e s s u r e i n s i d e the wake. A good r e v i e w of the work done i n t h i s f i e l d can be found i n Chang [ 9 ] . As a r e s u l t , t h i s base p r e s s u r e i s t r e a t e d as a c o n s t a n t v a l u e and p r o v i d e d e m p i r i c a l l y , as done i n [ 2 , 3 ] . I t i s e x p r e s s e d i n a d i m e n s i o n l e s s form, c p f c = <P„ - p.)/(i»o»> where p^ i s t h e p r e s s u r e i n s i d e the wake, and p i s t h e upstream u n d i s t u r b e d p r e s s u r e . Oo The f l o w e x t e r i o r t o the a i r f o i l i s t r a n s f o r m e d t o t h a t o u t s i d e of the u n i t c i r c l e i n the $-plane. I t i s m o d e l l e d by s u p e r i m p o s i n g on the b a s i c f l o w p a s t the t r a n s f o r m c i r c l e , s u i t a b l e s i n g u l a r i t i e s i n s i d e the a p p r o p r i a t e r e g i o n t o s i m u l a t e t h e wake and i t s e f f e c t s on the o u t e r f l o w . The s e m i - i n f i n i t e wake thus c r e a t e d i s e n c l o s e d by two 30 n o n - i n t e r s e c t i n g s t r e a m l i n e s , see F i g . 28 of [ 3 ] , which, when viewed from the p h y s i c a l p l a n e , i s assumed t o r e p r e s e n t the a pproximate shapes of the s e p a r a t e d shear l a y e r s or the t i m e - a v e r a g e d b o u n d a r i e s of the wake bu b b l e . These sou r c e s i n g u l a r i t i e s , however, have t o be l o c a t e d on the c i r c u m f e r e n c e , i n the wake r e g i o n , t o s a t i s f y t h e s e p a r a t i o n p r e s s u r e boundary c o n d i t i o n s . 3.2.1 M a t h e m a t i c a l Flow Model F i g s . 6a,b d e p i c t the r e p r e s e n t a t i o n of the r e q u i r e d s i n g u l a r i t i e s i n the $-plane. They must l i e w i t h i n a r c EC. I f the c a s e of a s p o i l e r i s c o n s i d e r e d , then F i g . 6a shows t h a t p o i n t C w i l l r e p r e s e n t the s p o i l e r t i p /and p o i n t E, the t r a i l i n g edge. For the case of a s p l i t f l a p , p o i n t E becomes the t r a i l i n g edge and p o i n t C i s t h e t i p of the f l a p , as shown i n F i g . 6b. The b a s i c f l o w p a s t a c i r c l e i s the u s u a l c o m b i n a t i o n of the u n i f o r m f l o w a t z e r o i n c i d e n c e w i t h r e s p e c t t o the r e a l a x i s , and a d o u b l e t of s u i t a b l e s t r e n g t h a t t h e o r i g i n . S i n c e an a i r f o i l i s a l i f t i n g body of non-zero c i r c u l a t i o n around i t , a v o r t e x l o c a t e d a t the o r i g i n i s needed. In o r d e r t o s i m u l a t e f l o w s e p a r a t i o n a t p o i n t s E and C, e i t h e r one or two s o u r c e s a r e added on t h e a r c EC. Both p o s s i b i l i t i e s have been e x p l o r e d and w i l l be r e f e r r e d t o as 1-source and 2-source models r e s p e c t i v e l y . 31 32 From Milne-Thomson's C i r c l e Theorem, w i t h a source p l a c e d o u t s i d e a s o l i d c i r c u l a r boundary, an image source and a s i n k a r e r e q u i r e d i n s i d e the boundary so t h a t the c i r c u l a r shape of the s t r e a m l i n e i s p r e s e r v e d . In the l i m i t i n g case when the added source i s on the p e r i m e t e r , the image source w i l l move t o the same l o c a t i o n as the added s o u r c e , and the s i n k w i l l s t i l l be a t the o r i g i n . I t i s the reason why two double s o u r c e s on a r c EC and a double s i n k a t the o r i g i n a r e shown i n F i g s . 6a,b f o r the 2-source model. The complex p o t e n t i a l f o r the 2-source model i n the S-plane i s F($) = V ($ + j) + ^ l n $ + ) ^ l n ($ - l n ($ - $ 2) -7T 7T 2 7T dF The complex v e l o c i t y , W($) = — , becomes w(5) - v (i - I , ) + + 2_L _ L _ + 2± _J (Qi + 9^) 1 (27) V i s d e f i n e d i n e q u a t i o n ( 1 2 ) . Q 1 f Q 2, T, $, and S 2 a r e s t r e n g t h s of the s o u r c e s and c i r c u l a t i o n , and t h e a n g u l a r p o s i t i o n s of the s o u r c e s , r e s p e c t i v e l y . S i n c e the s o u r c e s a r e on the s u r f a c e , one can w r i t e i 0 , i 0 i y i02 S = e , Si = e , $ 2 = e 33 E q u a t i o n (27) can then be s i m p l i f i e d t o W(S) e " l g r „ . ,6-8, t .e-6,,. ~ ^ p - = -^-[-4 s i n 0 - 2 7 + q i c o t ( - ^ — L ) + q 2 c o U - y - 2 - ) ] (28) where q, = , q 2 = ^ , 7 -The complex v e l o c i t y i n the z-plane i s W(z) = W(5)/(||) ... (29) The q u a n t i t i e s q 1 r q 2 , 7, 6, and 62 w i l l be determined t h r o u g h s u i t a b l e boundary c o n d i t i o n s . For t h e 1-source model, o n l y q,, 7 and 5, w i l l be of non-zero v a l u e s . The number of unknowns then reduces from 5 t o 3. I 3.2.2 Boundary C o n d i t i o n s There a re t o t a l l y 5 unknowns f o r t h e 2-source model so t h a t we have t o seek the same number of independent boundary c o n d i t i o n s . In the sequence of t r a n s f o r m a t i o n s , sharp edges such as dz p o i n t s E and C a r e made c r i t i c a l p o i n t s a t which — = 0. A c c o r d i n g t o t h e e q u a t i o n ( 2 9 ) , W(z) a t these p o i n t s w i l l be i n f i n i t e i f W($) i s non-zero, a c a s e c o n s i d e r e d t o be u n r e a l i s t i c , s i n c e the p r e s s u r e w i l l a l s o be i n f i n i t e . As a r e s u l t , t he f i r s t two boundary c o n d i t i o n s w i l l c o r r e s p o n d t o s e t t i n g W($) = 0 a t the s e p o i n t s i n the $-plane. These c o n d i t i o n s a re u s u a l l y r e f e r r e d t o a s K u t t a c o n d i t i o n s , and are e q u i v a l e n t t o h a v i n g the f l o w l e a v e t h e t r a i l i n g edge and 34 the t i p of the s p o i l e r or s p l i t f l a p t a n g e n t i a l l y . T h e r e f o r e , we a r r i v e a t 8 — 8 8~~ 8 g, c o U - y - 1 ) + q 2 c o t f - j - 2 - ) - 4 s i n « - 2 7 = 0 a t 8 = 0_ ... (30) E a t 0 = # c ... (31) The p r e s s u r e a t the p o i n t s of s e p a r a t i o n w i l l be - matched t o t h a t of the wake through B e r n o u l l i ' s e q u a t i o n : B + M s l l i . f a + S i . . . ( 3 2 , p 2 P 2 T h e r e f o r e , two a d d i t i o n a l boundary c o n d i t i o n s a r e e v o l v e d : P E - P ; _ . H t z i | 2 . P E ' \ ••• <33) and PC " P=° _ ,W(z) x ^2 u C -b C p r = 1 P U 2 = 1 " ' U I* = C p ... (34) W (z) However, the l i m i t i n g v a l u e s of | — — | a t E and C must be o b t a i n e d by u s i n g l ' H o p i t a l ' s r u l e s i n c e W(z) i s i n an i n d e t e r m i n a t e form, see a p p e n d i c e s F & G f o r d e t a i l s . 3.2.3 1-source Models There a r e two d i s t i n c t 1-source models, depending on the c h o i c e of t h e boundary c o n d i t i o n s . These models a r e o b t a i n e d by s e t t i n g e i t h e r q, or q 2 t o z e r o . To be c o n s i s t e n t , q 2 i s e l i m i n a t e d f o r the f o l l o w i n g d i s c u s s i o n . The unknowns a r e then q!, 8! and 7. 35 The f i r s t model c o r r e s p o n d s t o s a t i s f y i n g the K u t t a c o n d i t i o n b o t h a t the c r i t i c a l p o i n t s E and C, and t o matching the p r e s s u r e a t p o i n t E t o t h a t of the wake. The o t h e r model i s o b t a i n e d i n e x a c t l y the same way but w i t h the p r e s s u r e a t C, i n s t e a d of E, e q u a l t o t h a t of the wake. M a t h e m a t i c a l l y , they can be w r i t t e n as f i r s t 1-source model: and w($) = 0 a t 5 = $ E and $ c C = C ; Pb PE second 1-source model: W(S) = 0 a t $ = $ E and $ c and C = C . Pb PC These models a r e , however, c o n s i d e r e d t o be u n r e a l i s t i c because, i n g e n e r a l , the p r e s s u r e a t E i s not the same as t h a t at C i n each model. T h i s d i s c o n t i n u i t y of p r e s s u r e i s u n d e s i r a b l e s i n c e e x p e r i m e n t a l r e s u l t s do not s u p p o r t t h i s phenomenon. N e v e r t h e l e s s , t h e s e models a r e s i m p l e i n the sense t h a t o n l y a s i n g l e e m p i r i c a l i n p u t , the base p r e s s u r e , i s r e q u i r e d . I t w i l l be seen l a t e r on t h a t t h e s e models p l a y a s i g n i f i c a n t r o l e i n c o n s t r u c t i n g a 5 ^ boundary c o n d i t i o n f o r the 2-source model. 36 3.2.4 A d d i t i o n a l Boundary C o n d i t i o n s C o n t r i v i n g a 5 f c^ boundary c o n d i t i o n , which i s a d m i s s i b l e b o t h p h y s i c a l l y and m a t h e m a t i c a l l y , i s not a t r i v i a l t a s k . More o f t e n , more e m p i r i c a l i n f o r m a t i o n may be i n v o l v e d , as seen i n J a n d a l i ' s d i s c u s s i o n [ 3 ] , However, as shown i n [ 1 0 ] , the c r i t e r i o n of s t r e a m l i n e s p o s s e s s i n g f i n i t e c u r v a t u r e a t s e p a r a t i o n p r o v e s t o s u c c e s s f u l l y model the f l o w around a c i r c u l a r c y l i n d e r a t Re = 2 ( 1 0 ) 6 . C o n s e q u e n t l y , the e m p i r i c a l s p e c i f i c a t i o n of the a n g l e of s e p a r a t i o n i s e l i m i n a t e d . T h i s c r i t e r i o n sounds p r o m i s i n g f o r s u r f a c e s of c o n t i n u o u s c u r v a t u r e , such as t h e p o r t i o n near the t r a i l i n g edge of an a i r f o i l . As i n d i c a t e d by Woods[6]-, i t a l s o l e a d s t o the consequence of f i n i t e p r e s s u r e g r a d i e n t a t s e p a r a t i o n . A l t h o u g h the e x p e r i m e n t a l support of t h i s c r i t e r i o n i s not o b v i o u s f o r s p o i l e r s , t h e r e i s a f a i r amount of d a t a which s u b s t a n t i a t e s t h i s i d e a even though they a r e c o n cerned w i t h a i r f o i l s o t h e r than the Joukowsky f a m i l y . The m a t h e m a t i c a l e q u a t i o n , which d e s c r i b e s t h e c r i t e r i o n of f i n i t e p r e s s u r e g r a d i e n t , i s f 2 f ? - t\ f3 = 0 ... (35) where f , = | W ( J ) | E , f 2 = |§|l E ... (36a,b) and ( )' " f i • The d e r i v a t i o n of (35) and t h e f u r t h e r s i m p l i f i c a t i o n of (36a,b) can be found i n appendices F and G. Note t h a t the n o n - d i m e n s i o n a l c i r c u l a t i o n unknown, 7, does not appear i n 37 (35) s i n c e o n l y the d e r i v a t i v e s of f , a r e i n v o l v e d . Another p l a u s i b l e boundary c o n d i t i o n can be s e t up t o ap p r o x i m a t e the c i r c u l a t i o n a c r o s s the wake. L e t T be the t o t a l c i r c u l a t i o n around the c o n t o u r C e n c l o s i n g the a i r f o i l w i t h a s p o i l e r or a s p l i t f l a p , see F i g . 7. T h i s c o n t o u r i s then shrunk on t o the boundary of t h e a i r f o i l and s p l i t i n t o two p a r t s : one r u n n i n g from the t i p of t h e s p o i l e r ( t h e t r a i l i n g edge) t o the t r a i l i n g edge (the t i p of the f l a p ) i n c l u d i n g the p o r t i o n exposed t o the wake, and t h e o t h e r one g o i n g around the l e f t - o v e r p o r t i o n , F i g . 8. S i n c e the t o t a l head i s s u b s t a n t i a l l y reduced t h e r e , e x p e r i m e n t s show t h a t the ti m e - a v e r a g e d v e l o c i t y i n the wake i s near z e r o . I f the c i r c u l a t i o n i n the wake i s d e f i n e d as ( i n the z- p l a n e ) T = / v • ds ... (37) wake w a k e s t t h e n , s i n c e |v| i s s m a l l , t o the 1 a p p r o x i m a t i o n , ^wake ^ S i n c e c i r c u l a t i o n i s i n v a r i a n t under t r a n s f o r m a t i o n s , e q u a t i o n (37) i n the $-plane can be r e w r i t t e n as r = / |W($)| d0 = 0 ... (38) w a k e wake where W($) i s g i v e n by ( 2 8 ) . A f t e r c a r r y i n g out the i n t e g r a t i o n , the g e n e r a l e x p r e s s i o n i s [4cos 9 - 2 7 0 + 2 q. In I s i n A 1 I + 2 q 2 In | s i n A 2 | ] f l 1 = 0 8 6 where A, = ( — j - 1 ) c F i g . 7 c i r c u l a t i o n around A i r f o i l OJ CD r - r wetted s u r f a c e s wake wetted s u r f a c e s exposed t o wake r - r wake wake F i g . 8 Contour D e f o r m a t i o n 40 A , - < ^ > 0, = 8Q 02 = &E f o r s p o i l e r s , 0n = 0E 02 = 0c f ° r s p l i t f l a p s . F i n a l l y , another boundary c o n d i t i o n based on the l a s t one i s d e v e l o p e d because i t g i v e s the b e s t r e s u l t when compared w i t h e x p e r i m e n t s . I n s t e a d of e q u a t i n g f k e t o z e r o , i t i s changed t o a number, which depends upon the s o l u t i o n of the two 1-source models. L e t T and T be the v a l u e s of the c i r c u l a t i o n a c r o s s the wake o b t a i n e d by the two 1-source models by ( 3 7 ) . Then r . = ( r + r )/2 ... (39) wake w1 w2 t h 1 i s the m o d i f i e d 5 boundary c o n d i t i o n , where r k e i s d e f i n e d i n ( 3 8 ) . E q u a t i o n (39) seems a r t i f i c i a l and not p h y s i c a l l y o b v i o u s because i t i n v o l v e s the two u n r e a l i s t i c 1-source models. However, i t s v a l i d i t y s u r f a c e s i n t h e l i g h t of an example, the f l o w around a c i r c u l a r c y l i n d e r . Because of the g e o m e t r i c symmetry, the s o u r c e s ' s t r e n g t h s must be i d e n t i c a l and t h e i r l o c a t i o n s a r e a t 0 = ± 5 , , when the wake sour c e model i s used t o d e a l w i t h t h i s problem [ 2 ] . T h e r e f o r e , the c o r r e s p o n d i n g " 1 - s o u r c e " models w i l l be no d i f f e r e n t from the 2-source model s i n c e the p r e s s u r e s a t the p o i n t s of s e p a r a t i o n a r e a u t o m a t i c a l l y matched i n a l l c a s e s . C o n s e q u e n t l y , the s o l u t i o n s of t h e unknowns a r e the same and so a r e the v a l u e s of T , T and T . . E q u a t i o n (39) then reduces t o an w, w 2 wake 41 i d e n t i t y . In o t h e r words, no ambiguous and a d d i t i o n a l r e s t r i c t i o n i s imposed on the model. In f a c t , the s i m i l a r d e d u c t i o n h o l d s f o r f l o w s around any s y m m e t r i c a l b o d i e s a t z e r o i n c i d e n c e . Of c o u r s e , the above argument shows t h a t e q u a t i o n (39) i s o n l y a n e c e s s a r y c o n d i t i o n f o r b o d i e s around which T = r ^ , = r , = 0 . S i m i l a r d e d u c t i o n s , however, wetted wake cannot be reached f o r b o d i e s of a r b i t r a r y shapes at non-zero i n c i d e n c e . For asymmetric b o d i e s such as the one c o n s i d e r e d i n t h i s work, the non-zero r i g h t - h a n d s i d e of (39) i s used t o account f o r the c i r c u l a t i o n a c r o s s the wake, which i s not n e c e s s a r i l y z e r o i n g e n e r a l . 1 3.2.5 Method of S o l u t i o n and C a l c u l a t i o n 42 The 3 unknowns a s s o c i a t e d w i t h each of the 1-source models, q,, 6, and 7, a r e determined by s o l v i n g e q u a t i o n s ( 3 0 ) , (31) and one of (33) and ( 3 4 ) , depending upon which model i s c o n s i d e r e d . These e q u a t i o n s can be s i m p l i f i e d t o one i n v o l v i n g 6, a l o n e because 7 can be e l i m i n a t e d between (30) and ( 3 1 ) , and q, appears o n l y l i n e a r l y i n a l l e q u a t i o n s . However, the reduced s i n g l e e g u a t i o n must be s o l v e d n u m e r i c a l l y because of i t s complex form. S e v e r a l computer s u b r o u t i n e s a r e a v a i l a b l e i n the computer system at UBC. The most r o b u s t one, NDINVT, which i s w r i t t e n t o l o c a t e a r o o t of N n o n - l i n e a r s i m u l t a n e o u s e q u a t i o n s by u s i n g the g e n e r a l i z e d s e c a n t method, i s invoked t o s o l v e f o r 6,. An i n i t i a l e s t i m a t e of t h e unknown, which must be p r o v i d e d by the u s e r , can be of any number chosen between the v a l u e s of 8 and 6 , where the E C s o u r c e i s assumed t o l i e . I f the i t e r a t i v e p r o c e s s does not l e a d t o convergence a f t e r 30 i t e r a t i o n s , a d i f f e r e n t c h o i c e of the i n i t i a l e s t i m a t e s h o u l d be c o n s i d e r e d . M u l t i p l e r o o t s w i t h i n the s p e c i f i e d domain a r e not l i k e l y and have been checked e i t h e r by u s i n g a n o t h e r s u b r o u t i n e t o s e a r c h f o r m u l t i p l e r o o t s or by s k e t c h i n g the c u r v e r e p r e s e n t i n g t h i s s i n g l e e q u a t i o n . A l t h o u g h a s o l u t i o n i s not g u a r a n t e e d , e x p e r i e n c e shows t h a t i t does e x i s t and can be found w i t h o u t much d i f f i c u l t y . A f t e r hy i s o b t a i n e d , q, can be computed by s u b s t i t u t i n g 6, i n t o e q u a t i o n (33) or ( 3 4 ) , a g a i n depending upon which model i s used. 7 i s then c a l c u l a t e d by 43 s u b s t i t u t i n g qy and 6, i n t o (30) or ( 3 1 ) . The s o l u t i o n s from 1-source models are e s s e n t i a l because they s e r v e as good e s t i m a t e s of the unknowns and p r o v i d e the r i g h t hand s i d e of e q u a t i o n (39) f o r the 2-source model which i s c o n s i d e r e d t o be more i n t e r e s t i n g and r e a l i s t i c . For t h e case of the 2-source model, e q u a t i o n s ( 3 0 ) , ( 3 1 ) , ( 3 3 ) , (34) and one of ( 3 5 ) , (38) and (39) a r e used t o determine the 5 unknowns q,, 6 1 f q 2 , 6 2 and 7. A g a i n , s i n c e 7, q,, q 2 appear l i n e a r l y i n t h i s s e t of e q u a t i o n s , i t i s p o s s i b l e t o e l i m i n a t e them and a r r i v e a t 2 e q u a t i o n s i n v o l v i n g 6, and 6 2. F u r t h e r s i m p l i f i c a t i o n , however, i s i m p o s s i b l e because t h e s e two e q u a t i o n s a r e r a t h e r c o m p l i c a t e d . T h e r e f o r e , NDINVT i s used t o s o l v e f o r 8, and 5 2 n u m e r i c a l l y . N e v e r t h e l e s s , both 8^ and 6 2 appear i n the arguments of the t r a n s c e n d e n t a l f u n c t i o n s l i k e c o s e c a n t ( e s c ) , c o t a n g e n t ( c o t ) , and t h e i r p r o d u c t s whose p e r i o d i c i t i e s and s i n g u l a r i t i e s o f t e n p r e v e n t the i t e r a t i v e p r o c e s s from c o n v e r g i n g . A l s o , s o l u t i o n s do not always e x i s t i f (35) or (38) i s used as the 5 ^ e q u a t i o n u n l e s s the e x p e r i m e n t a l base p r e s s u r e i s a l t e r e d . The above pr o c e d u r e a l s o does not h e l p i n g i v i n g any c l u e t o the e x i s t e n c e of the s o l u t i o n s . C o n s e q u e n t l y , a d i f f e r e n t s t r a t e g y towards l o c a t i n g the s o l u t i o n s i s d e v i s e d and d e s c r i b e d h e r e i n . Even though they do not c o n s t i t u t e a complete s e t of e q u a t i o n s u n i q u e l y d e t e r m i n i n g t h e 5 unknowns, ( 3 0 ) , ( 3 1 ) , (33) and (34) a r e the 44 f undamental e q u a t i o n s i n the model i n the sense t h a t any s e t of a c c e p t a b l e s o l u t i o n s must s a t i s f y them, n s e t s of " s o l u t i o n s " of q,, 6,, q 2 and 7, however, can be o b t a i n e d by s o l v i n g t h e s e e q u a t i o n s n u m e r i c a l l y by NDINVT i f the p o s i t i o n of q 2 i s s e t t o ( 6r - 6e ) i 5 2 = 6E + n+1 ' 1 = 1 ' 2 - - " n where 8 < 8 i s assumed h e r e . E C These n s e t s of " s o l u t i o n s " a r e then s u b s t i t u t e d i n t o w h i c h e v e r i s chosen as t h e 5*"^  e q u a t i o n i n o r d e r t o e v a l u a t e t h e r e s i d u e , which i s e x a c t l y z e r o i f the c o r r e s p o n d i n g s e t of t h t h s o l u t i o n s s a t i s f i e s a l l 5 e q u a t i o n s . I f the i and the i+1 s e t s of s o l u t i o n s cause a change of s i g n of the r e s i d u e , then the e x i s t e n c e of a s e t of s o l u t i o n s i s g u a r a n t e e d . More a c c u r a t e v a l u e s of t h i s s e t of s o l u t i o n s can be found n u m e r i c a l l y by s o l v i n g a l l 5 e q u a t i o n s u s i n g NDINVT w i t h the i * " * 1 or i + l * " * 1 s e t of s o l u t i o n s as the i n i t i a l g u e s s e s . Very o f t e n , o n l y a few i t e r a t i o n s a r e r e q u i r e d b e f o r e convergence i s a r r i v e d a t because the i n i t i a l e s t i m a t e s a r e r e a s o n a b l y c l o s e t o the r o o t s . The p o s s i b i l i t y of the e x i s t e n c e of m u l t i p l e r o o t s w i t h i n a r c EC has t o be i n v e s t i g a t e d by u s i n g d i f f e r e n t i n i t i a l e s t i m a t e s of q,, 6,, 7 and q 2 a t each p r e s c r i b e d v a l u e of 8 2 m a i n l y because no computer r o u t i n e s a r e a v a i l a b l e f o r d e t e c t i n g m u l t i p l e r o o t s of n (n>l) n o n - l i n e a r s i m u l t a n e o u s e q u a t i o n s . I f the r e s i d u e does not change s i g n f o r d i f f e r e n t v a l u e s of 5 2 and no m u l t i p l e r o o t s e x i s t , then i t can be c o n c l u d e d t h a t no s o l u t i o n s can s a t i s f y t hese 5 45 e q u a t i o n s . T h i s i s what i s o f t e n e n c o u n t e r e d when (35) or (38) i s used as the 5*"*1 boundary c o n d i t i o n . 6, and 62 are i n t e r c h a n g e a b l e i n the above scheme of s e a r c h i n g f o r r o o t s s i n c e no p a r t i c u l a r c o n d i t i o n s a r e imposed t o d i s t i n g u i s h one from the o t h e r . However, because t h e r e i s no o b v i o u s method of d e t e r m i n i n g t h e upper and lower bounds of q 1 f q 2 or 7, u s i n g any one of them t o f u n c t i o n as 6, or 62 i s out of the q u e s t i o n . A l s o , i f 6,(6 2) i s s p e c i f i e d , the e q u a t i o n s a r e much more easy t o s o l v e s i n c e o n l y 6,(62) appears n o n - l i n e a r l y . Once the s o l u t i o n s f o r the above unknowns a r e o b t a i n e d , the p r e s s u r e on the a i r f o i l s u r f a c e can b e / c a l c u l a t e d by C p = 1 - |W($)/(dz/d$)| 2 dz where W(£) i s d e f i n e d by (28) and — i s g i v e n by e i t h e r (10) or ( 2 2 ) . I f and C a r e d e s i g n a t e d as the lo w e r and upper s u r f a c e p r e s s u r e c o e f f i c i e n t s r e s p e c t i v e l y , then i n t e g r a t i n g ( 0 ^ ^ - ) w i t h r e s p e c t t o t h e n o n - d i m e n s i o n a l q u a n t i t y ( x ' / c ) , where x' i s i n the d i r e c t i o n of the f l o w a t i n f i n i t y and c i s t h e c h o r d l e n g t h of the a i r f o i l , t he o v e r a l l l i f t c o e f f i c i e n t i s g i v e n by Cosot, c_ = ; ( c - c ) d ( x ' / c ) L c p i pu A c c o r d i n g t o F i g . 9, o n l y p a r t of the p r e s s u r e i n s i d e the wake c o n t r i b u t e s t o the l i f t s i n c e the p r e s s u r e i s assumed t o u •=0 F i g . 9 C C o n t r i b u t i o n from the Wake ( S p l i t F l a p ) 47* be the same on the p o r t i o n of the a i r f o i l and the f l a p both exposed t o the wake. In o t h e r words, 1 x> c; = - ; ( c - c . ) d x ' = o L c J . pa p f s i n c e C = C r The same r e s u l t h o l d s f o r s p o i l e r s , pa pf In the c a l c u l a t i o n s , t 0 i s (-0.085,0.05). The p r e s s u r e c o e f f i c i e n t i s e v a l u a t e d a t 98 p o i n t s on the a i r f o i l s u r f a c e and 10 p o i n t s on the s p o i l e r or s p l i t f l a p f a c i n g the upstream f l o w . The p r e s s u r e i n t h e wake i s assumed c o n s t a n t and e q u a l t o t h a t from e x p e r i m e n t s . A computer s u b r o u t i n e , QINT4P, i s used t o i n t e g r a t e the v a l u e s because t h i s i s the o n l y a v a i l a b l e program f o r i n t e g r a t i o n of a s e t of u n e q u a l l y spaced / d a t a p o i n t s . The method i s t h a t f o r each i n t e r v a l , x. t o x. . c l i + 1 , a c u b i c i n t e r p o l a t i o n - p o l y n o m i a l based on the f o u r d i s t i n c t p o i n t s x j _ - | ' xi» x i + i a n c ^ x i + 2 * S ^ n t e 9 r a t e c ' * However, over the p o r t i o n where i s c o n s t a n t , t h i s program i s not used because m e a n i n g l e s s r e s u l t s a r e o b t a i n e d i n t r y i n g t o f i t a s t r a i g h t l i n e w i t h a c u b i c s p l i n e . T h i s p a r t of C i s s e t t o CpAx', where Ax' i s the l e n g t h of the p o r t i o n . 48 4. EXPERIMENTS There a r e two purposes i n p e r f o r m i n g e x p e r i m e n t s i n t h i s work. The f i r s t i s t o measure the base p r e s s u r e v a l u e i n s i d e the wake beh i n d the s p o i l e r or s p l i t f l a p because i t i s the r e q u i r e d e m p i r i c a l i n p u t t o the t h e o r y d e s c r i b e d i n the p r e v i o u s s e c t i o n s . The second i s t o make comparisons between the t h e o r e t i c a l and e x p e r i m e n t a l p r e s s u r e l o a d i n g and the o v e r a l l l i f t f o r c e on the a i r f o i l a t d i f f e r e n t a n g l e s of a t t a c k and f o r the v a r i o u s c o n f i g u r a t i o n s i n v o l v e d . I Two s e r i e s of e x p e r i m e n t s were c a r r i e d out : one i n v o l v i n g the a i r f o i l and s p o i l e r s , and the o t h e r w i t h the a i r f o i l and s p l i t f l a p s . They were conducted i n the s m a l l low speed a e r o n a u t i c a l wind t u n n e l i n the Department of M e c h a n i c a l E n g i n e e r i n g a t the U n i v e r s i t y of B r i t i s h C o lumbia. I t has a t e s t s e c t i o n of 27 i n c h h e i g h t and 36 i n c h w i d t h . The t u n n e l p o s s e s s e s good f l o w u n i f o r m i t y and a t u r b u l e n c e l e v e l of l e s s than 0.1 p e r c e n t over i t s speed range. The Joukowsky a i r f o i l of 27 i n c h span, 12.08 i n c h c h o r d , 11 % t h i c k n e s s and 2.4% camber, i s the same one used by J a n d a l i . I t was mounted v e r t i c a l l y and spanned the t e s t s e c t i o n , w i t h s m a l l c l e a r a n c e s a t the c e i l i n g and t h e f l o o r . The a i r f o i l was a t t a c h e d t o a s i x component p y r a m i d a l b a l a n c e s i t u a t e d beneath the t e s t s e c t i o n of the t u n n e l , a t the q u a r t e r c h o r d p o s i t i o n . 2 f o r c e and 1 moment components were measured over a wide range of 49 a n g l e s of a t t a c k . The a i r f o i l was o r i g i n a l l y d e s i g n e d f o r J a n d a l i ' s e x p e r i m e n t s on normal upper s u r f a c e s p o i l e r s . The d e t a i l e d d e s c r i p t i o n of i t can be found i n h i s t h e s i s [ 3 ] , N e v e r t h e l e s s , t h e r e i s a p o i n t worth n o t i n g . S i n c e the Joukowsky p r o f i l e was s t r u c t u r a l l y weak near the cusped t r a i l i n g edge, the upper s u r f a c e i n t h i s p o r t i o n was t h i c k e n e d t o g i v e an a p p r o x i m a t e l y c o n s t a n t t h i c k n e s s of 1/8 i n c h . The whole p r o f i l e i s shown i n F i g . 9 i n [ 3 ] . T h i s m o d i f i e d p o r t i o n does not i n f l u e n c e the p r e s s u r e measurements f o r the upper s u r f a c e s p o i l e r e x p e r i m e n t s because i t i s c o m p l e t e l y embedded i n the wake and has no e f f e c t on the o u t e r f l o w . However, u n c e r t a i n t y i n p r e s s u r e measurements may r e s u l t from c l e a n a i r f o i l e x p e r i m e n t s and those w i t h the lower s u r f a c e s p l i t f l a p d e f l e c t e d . In f a c t , F i g . 10 i n J a n d a l i ' s t h e s i s p r o v i d e s some e v i d e n c e . I t would be p r e f e r a b l e f o r the s p l i t f l a p e x p e r i m e n t s t o have t h i s m o d i f i e d p o r t i o n l o c a t e d on the lower s u r f a c e of the a i r f o i l so t h a t i t i s exposed t o the wake. The i n f o r m a t i o n of the p r e s s u r e v a r i a t i o n a l o n g t h i s p o r t i o n of the upper s u r f a c e i s c r u c i a l t o t h e c r i t e r i o n of the f i n i t e p r e s s u r e g r a d i e n t a t s e p a r a t i o n d i s c u s s e d i n 3.2.4. D u r i n g the c o u r s e of e x p e r i m e n t s , end p l a t e s were used f o r s u p p o r t i n g the s p o i l e r or s p l i t f l a p but t h e y do not t o u c h e i t h e r the r o o f or the f l o o r . These p l a t e s were d e s i g n e d t o a l l o w the s p o i l e r or s p l i t f l a p t o be l o c a t e d a t v a r i o u s p o s i t i o n s and a n g l e s of i n c l i n a t i o n . The s p o i l e r s of h e i g h t 5% 50 and 10% c h o r d c o u l d be mounted a t d i s t a n c e s of 50%, 70%, and 90% c h o r d p o s i t i o n s from the l e a d i n g edge of the a i r f o i l . The 5% c h o r d s p o i l e r can o n l y make an i n c l i n a t i o n of 45° whereas the 10% one i s a l l o w e d t o d e f l e c t a t 30° and 60° w i t h r e s p e c t t o t h e l o c a l upper s u r f a c e of the a i r f o i l . The two s p l i t f l a p s used have l e n g t h s of 20% and 30% c h o r d . T h e i r l o c a t i o n s , measured from the t r a i l i n g edge, a r e e x a c t l y e q u a l t o t h e i r l e n g t h s . The a n g l e s of i n c l i n a t i o n a r e 10°, 30°, 45°, and 60°. The s m a l l gap between the s p o i l e r or f l a p and the a i r f o i l s u r f a c e i s unvented by s e a l i n g i t w i t h masking tapes i n the wake. Owing t o the s m a l l s i z e s of s p o i l e r s used, p r e s s u r e measurements were made o n l y on the wette d s u r f a c e of each f l a p . They were measured by t a p i n g p r e s s u r e t a p s over the s u r f a c e so t h a t the tubes were exposed t o the o u t e r f l o w . The p r e s s u r e s on the s u r f a c e of the a i r f o i l , i n c l u d i n g the p o r t i o n w i t h i n the wake, however, were measured by u s i n g the b u i l t - i n p r e s s u r e t a p s i n s i d e the Joukowsky a i r f o i l . A l l p r e s s u r e taps were c o n n e c t e d t o a 48 p o r t s c a n i v a l v e , a manually s c a n n i n g p r e s s u r e t r a n s d u c e r . A S e t r a 237 d i f f e r e n t i a l p r e s s u r e t r a n s d u c e r , a HP 6204B D.C. power s u p p l y , a S o l a r t r o n JM 1860 time domain a n a l y s e r and a F l u k e 8000A d i g i t a l m u l t i m e t e r were used f o r d a t a r e c o r d i n g . Because of the l i m i t a t i o n i n t i m e , no d a t a a c q u i s i t i o n system c o n t r o l l e d by a m i c r o p r o c e s s o r was s e t up . 51 B e s i d e s s u p p o r t i n g t h e a i r f o i l , t he b a l a n c e was used t o measure not o n l y the l i f t , but the drag and p i t c h i n g moment, which were needed f o r the wind t u n n e l w a l l c o r r e c t i o n s . The 1 c wake b l o c k a g e term = , as suggested by J a n d a l i , was employed f o r g i v i n g a b e t t e r c o l l a p s e f o r the d a t a , c i s the a i r f o i l c h o r d l e n g t h and H i s the e f f e c t i v e t e s t s e c t i o n w i d t h , d e f i n e d as the r a t i o of the t u n n e l c r o s s - s e c t i o n a l a r e a t o i t s h e i g h t . i s d e f i n e d as the d i f f e r e n c e between the drag c o e f f i c i e n t s w i t h and w i t h o u t the s p o i l e r or f l a p d e f l e c t e d . The c o r r e c t e d C i s g i v e n by LJ C = C T [ 1 - 2 e - a ] - T 2! C_ + 4 C ] (40) where b l o c k a g e f a c t o r e = e + e } and s o i l d b l o c k a g e f a c t o r e i s as d e f i n e d i n [ 1 3 ] , 5 7T 2 C ° = l 8 ( i ) 2 ' C w = q u a r t e r c h o r d moment. The f o r m u l a f o r c o r r e c t i n g the p r e s s u r e c o e f f i c i e n t i s 1 - Cp _ C L 1 - C C P T l T where ( ) T denotes a v a l u e measured i n the t u n n e l . The c o r r e c t e d C^ v a l u e s a r e then i n t e g r a t e d t o g i v e C L by u s i n g the computer s u b r o u t i n e QINT4P d e s c r i b e d i n t h e p r e v i o u s s e c t i o n . T h i s reduced C i n s t e a d of the one c a l c u l a t e d from the b a l a n c e d a t a w i l l be compared t o t h e t h e o r e t i c a l p r e d i c t i o n . The t e s t R e y n o l d s number i s 3 ( 1 0 ) 5 . 52 I t i s j u s t a matter of i n t e r e s t t o mention t h a t when the s p o i l e r was l o c a t e d a t 50% or 70% c h o r d p o s i t i o n when r u n n i n g the e x p e r i m e n t s , the a i r f o i l was b u f f e t i n g a p p r e c i a b l y . T h i s phenomenon i s a r e s u l t of h a v i n g a l a r g e p o r t i o n of t h e a i r f o i l s u r f a c e exposed t o the wake. The c o r r e s p o n d i n g p r e s s u r e measurements s h o u l d not be i n f l u e n c e d t o a g r e a t e x t e n t . 53 5. RESULTS AND COMPARISONS To ensure t h a t a l l p r e s s u r e t a p s on the a i r f o i l were f u n c t i o n i n g p r o p e r l y , an experiment on the c l e a n a i r f o i l w i t h o u t end p l a t e s was performed. The r e s u l t s c l o s e l y agree w i t h those o b t a i n e d by J a n d a l i , which can be found i n [3] and w i l l not be r e p e a t e d h e r e . By e x a m i n i n g F i g . 10 i n [ 3 ] , i t i s c l e a r t h a t t h e r e i s a l i t t l e d i s c r e p a n c y between the t h e o r e t i c a l and e x p e r i m e n t a l p r e s s u r e d i s t r i b u t i o n s a t the l e a d i n g edge a t the same a n g l e of a t t a c k . S i m i l a r l y , d e v i a t i o n s of the t h e o r e t i c a l and e x p e r i m e n t a l r e s u l t s a r e observed i n the v a r i a t i o n of l i f t w i t h the angle of i n c i d e n c e i n F i g . 11 i n [ 3 ] . A l l the s e are e x p l a i n e d by J a n d a l i i n terms of the p r e s e n c e of the boundary l a y e r v o r t i c i t y . The measurements of base p r e s s u r e i n the wake e x h i b i t some i n t e r e s t i n g t r e n d s when p l o t t e d a g a i n s t 5, a t d i f f e r e n t v a l u e s of a f o r s p l i t f l a p c a s e s i n F i g s . 10a,b. A l t h o u g h t h e r e i s no e x i s t i n g t h e o r y f o r p r e d i c t i n g C p f c ' * s p l a u s i b l e t o c o n s i d e r an e m p i r i c a l f o r m u l a of the f o l l o w i n g form C , = A + B 5 + C 5 2 + D S 3 ... (41) pb where A, B, C, and D a r e f u n c t i o n s of a, f l a p l e n g t h , and o t h e r p a r a m e t e r s . Of c o u r s e , more measurements i n v a r i o u s g e o m e t r i c c o n f i g u r a t i o n s must be p e r f o r m e d i n o r d e r t o f i n d 54 - . 2 -.44--6f -.8* - . 2 °pb -.6f 1 0 3 0 - + -4 5 —+— 6 0 — h -(a) 2 0 % c F l a p (b) 3 0 V . c F l a p 90 — i oC o 8' o •o 4 0 " - 4 8' — O 4 - 4 * F i g . 1 0 c , V a r i a t i o n s of 20% & 30% c S p l i t F l a p s pb 55 out the c o e f f i c i e n t s of ( 4 1 ) . However, a f o r m u l a f o r C , would pb be v a l u a b l e t o the t h e o r y d e s c r i b e d above. Tarn Doo [11] conducted some thorough e x p e r i m e n t a l i n v e s t i g a t i o n s on C ^ f o r both v e n t e d and unvented normal s p o i l e r s on 2D and 3D wings. I n t e r e s t i n g r e s u l t s can be found i n h i s t h e s i s . The purpose of t h i s work, however, i s not on the s t u d y of the v a r i a t i o n of the base p r e s s u r e . C o n s e q u e n t l y , no f u r t h e r work was c a r r i e d o u t . For the sake of s i m p l i c i t y , the t h e o r e t i c a l C d i s t r i b u t i o n s c o r r e s p o n d i n g t o the 1-source models were c a l c u l a t e d once the e x p e r i m e n t a l C ^ v a l u e s were a v a i l a b l e . For s p l i t f l a p s , i t i s found i n g e n e r a l t h a t the e x p e r i m e n t a l I C d i s t r i b u t i o n over the upper s u r f a c e of the a i r f o i l l i e s P w i t h i n the two t h e o r e t i c a l c u r v e s . More p r e c i s e l y , the 1-source model w i t h C , = C a t the t r a i l i n g edge (T.E.) Pb P o v e r e s t i m a t e s the C d i s t r i b u t i o n , and t h e model w i t h C , = C P Pb p at the t i p of the f l a p (T.F.) u n d e r e s t i m a t e s i t . Over the lower s u r f a c e , however, the p r e d i c t i o n c o r r e s p o n d i n g t o C ^ = C _ f i t s w e l l t o e x p e r i m e n t s , whereas t h e o t h e r p T.F. t h e o r e t i c a l c u r v e shows a backward s h i f t of t h e f r o n t a l s t a g n a t i o n p o i n t so t h a t a d i s c r e p a n c y r e s u l t s . F i g . 11 i s a t y p i c a l r e s u l t as d e s c r i b e d above. The t h e o r e t i c a l C^ ^ Cpb = C _ ) near the l e a d i n g edge i s f i n i t e , about -8.0, t o o p T.E. l a r g e t o be shown i n the diagram. A l t h o u g h each of t h e s e models f a i l s t o match the p r e s s u r e a t t h e t r a i l i n g edge t o t h a t of the t i p of the f l a p , t hey p r o v i d e the a e r o d y n a m i c i s t w i t h a f a i r e s t i m a t e of the o v e r a l l p r e s s u r e d i s t r i b u t i o n . F i g . 11 Comparison on C D i s t r i b u t i o n s from 1-source Models and Experiments 57 Very s i m i l a r r e s u l t s were o b t a i n e d f o r s p o i l e r s so t h a t the above d e s c r i p t i o n i s a p p l i c a b l e except t h a t the f l a p i s r e p l a c e d by the s p o i l e r . In o r d e r t o improve the agreement between the t h e o r y and e x p e r i m e n t s , the c r i t e r i o n of f i n i t e c u r v a t u r e of the s e p a r a t i n g s t r e a m l i n e s or t h a t of the f i n i t e p r e s s u r e g r a d i e n t at s e p a r a t i o n f o r s p l i t f l a p s was i n v e s t i g a t e d . I t i s found i n g e n e r a l t h a t t o have i t s a t i s f i e d , one of t h e s o u r c e s i n the 2-source model must l i e o u t s i d e t h e s i m u l a t e d wake i f the e x p e r i m e n t a l C i s a p p l i e d . T h i s i s not d e s i r a b l e because when e v a l u a t i n g the p r e s s u r e a l o n g the a i r f o i l s u r f a c e , a s i n g u l a r b e h a v i o r of C would be o b t a i n e d a t the l o c a t i o n P / where the s o u r c e i s s i t u a t e d . In o r d e r t o f o r c e the source t o l o c a t e i n the wake r e g i o n , the i n p u t base p r e s s u r e needs t o be a l t e r e d . Very o f t e n , i t i s more p o s i t i v e than the e m p i r i c a l v a l u e . For i n s t a n c e , f o r the case of the 20% c h o r d s p l i t f l a p d e f l e c t e d a t 30° a t an a n g l e of a t t a c k of 4°, t h e r e q u i r e d C ^ i s -.10 compared t o -.54 from t h e e x p e r i m e n t . S i n c e t h e r e i s no method of p r e d i c t i n g t h i s m o d i f i e d v a l u e e x c e p t by t r i a l and e r r o r , the s t u d y of t h i s c r i t e r i o n was not p u r s u e d f u r t h e r even though the c o r r e s p o n d i n g 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 r e s t of t h e a i r f o i l s u r f a c e i s not f a r from the e x p e r i m e n t a l r e s u l t as d e p i c t e d i n F i g . 12. S i m i l a r r e s u l t s have been found f o r s p o i l e r s , which seems u n d e r s t a n d a b l e s i n c e not s u f f i c i e n t e x p e r i m e n t a l d a t a s u p p o r t the b e h a v i o r of f i n i t e p r e s s u r e g r a d i e n t a t s e p a r a t i o n . F i g . 12 Comparison on C D i s t r i b u t i o n s from Eqns. (35) k (38) and Experiments en oo 59 The boundary c o n d i t i o n of z e r o c i r c u l a t i o n a c r o s s the wake was then e x p l o r e d . O v e r a l l agreement on the c h o r d w i s e p r e s s u r e d i s t r i b u t i o n i s improved as can be seen i n F i g . 12. B e s i d e s , f o r a good number of c a s e s , the e x p e r i m e n t a l C ^ i s a c c e p t e d by t h i s 5t^1 boundary c o n d i t i o n i n the sense t h a t both s o u r c e s a r e l y i n g w i t h i n the wake. Near the t r a i l i n g edge, the p r e s s u r e g r a d i e n t i s r a t h e r s t e e p when compared w i t h t h a t o b t a i n e d by a p p l y i n g the f i n i t e p r e s s u r e g r a d i e n t c o n d i t i o n . The p r e s s u r e near the l e a d i n g edge i s o v e r e s t i m a t e d by the t h e o r y , a s i m i l a r r e s u l t r e s e m b l i n g t h a t of the c l e a n a i r f o i l . The problem of a c c e p t i n g the e x p e r i m e n t a l C ^ v a l u e s comes up a g a i n as the a n g l e of d e f l e c t i o n i n c r e a s e s . For l example, F i g . 13 shows the case of the 20 % c h o r d f l a p a t 6 = 6 0 ° , a = 0 ° . To have r = 0 s a t i s f i e d , the e x p e r i m e n t a l C , wake pb must be changed from - .67 t o - . 78 . The r e s u l t i n g i s not a c c u r a t e on both upper and lower s u r f a c e s . B e s i d e s , the p r e s s u r e a t the t i p of the f l a p i s matched a b r u p t l y t o t h a t at the t r a i l i n g edge. I t s u g g e s t s t h a t e q u a t i o n (38) s h o u l d be used w i t h r e s e r v a t i o n . C o n s i d e r a b l e improvement i s a c h i e v e d i f the boundary c o n d i t i o n (39) i s u t i l i z e d . The e x p e r i m e n t a l C , can be used pb w i t h o u t a l t e r a t i o n and the a b r u p t jump of the C^ no l o n g e r p r e v a i l s . A l t h o u g h the l e a d i n g edge C^ v a l u e i s s l i g h t l y o v e r e s t i m a t e d by the t h e o r y , as shown i n F i g . 13, the o v e r a l l agreement between the t h e o r y and e x p e r i m e n t s i s r e m a r k a b l e . As a consequence, e q u a t i o n (39) i s chosen over the o t h e r s as the F i g . 13 Comparison on C D i s t r i b u t i o n s from Eqns. (38) & (39) and Experiments 61 boundary c o n d i t i o n t o determine the C and C v a l u e s f o r both . P L • s p l i t f l a p s and s p o i l e r s . F i g . 14 shows the v a r i a t i o n of C a t 6 = 10°, 30° and 60° * P at a = 4° f o r the 20 % c h o r d f l a p . The i n t e g r a t e d C i s L i compared w i t h e x p e r i m e n t s and shown i n F i g . 15. F i g s . 16 and 17 a r e the C and C„ v a r i a t i o n s f o r the 30 % c h o r d f l a p , p L r e s p e c t i v e l y . F i g s . 18-20 show the d i s t r i b u t i o n s f o r the s p o i l e r c a s e s . The s p o i l e r i s l o c a t e d a t e/c = 50 %, 70 % and 90 % c h o r d d i s t a n c e s from the l e a d i n g edge. The s p o i l e r h e i g h t i s 5 % c h o r d , the a n g l e of d e f l e c t i o n i s 45° and the a n g l e s of a t t a c k a r e 6° and 12°. The corresponding^ 1 C c u r v e s a r e shown L i i n F i g s . 21-23. The C d i s t r i b u t i o n s on the a i r f o i l w i t h 10 % c h o r d P s p o i l e r a re d e p i c t e d i n F i g s . 24-26 a t 6 = 30° and 60° and a t a = 6°. F i g s . 27-29 show the C d i s t r i b u t i o n s a t a = 12°. P F i g s . 30-32 a r e the l i f t c o e f f i c i e n t s of the a i r f o i l when the s p o i l e r i s l o c a t e d a t 50 %, 70 % and 90 % cho r d p o s i t i o n s . The t h e o r y p r e d i c t s the base of the s p o i l e r as a s t a g n a t i o n p o i n t . T h e r e f o r e , t h e c o r r e s p o n d i n g v a l u e i s 1.0. However, exp e r i m e n t s show t h a t f l o w s e p a r a t i o n t a k e s p l a c e t o form a s e p a r a t i o n bubble b e f o r e the f l u i d p a r t i c l e s r e a c h t h i s l o c a t i o n . The presence of the bubble i s not so c l e a r i n f r o n t of the s p l i t f l a p but r a t h e r o b v i o u s upstream of the s p o i l e r . As a r e s u l t , d i f f e r e n c e between t h e o r e t i c a l F i g . 14 C D i s t r i b u t i o n s a t D i f f e r e n t 8 ( 2 0 % c S p l i t F l a p ) 63 F i g . 1 6 C D i s t r i b u t i o n s P a t D i f f e r e n t 6 (30% c S p l i t F l a p ) 65 —I 1 1 1 1 - 8 - 4 0 ec* A 8 F i g ; 17 C, V a r i a t i o n s over oc a t D i f f e r e n t 6 S p o i l e r e/c - 50% Theory 45' h/c • 5% Expt. A O 12 6 10 L a as F i q . 18 C D i s t r i b u t i o n s ( s p o i l e r , e/c P 50%, h/c - 5 %, 6 - 45°) F i g . 19 C D i s t r i b u t i o n s ( s p o i l e r , e/c = 70%, n / c = 5 %, 69 70 71 S p o i l e r a = 6° F i g . 25 C D i s t r i b u t i o n s ( s p o i l e r , e/c = 70%, h/c = 10%, a * 6°) CO S p o i l e r a = 6* e/c = 90% h/c = 10% Theory i f -E x p t A O 8° 30 60 F i g . 26 c D i s t r i b u t i o n s ( s p o i l e r , e/c • 90%, h/c » P F i g . 27 C D i s t r i b u t i o n s ( s p o i l e r , e/c = 50%, h/c = 10%, o = 12°) F i q . 28 C D i s t r i b u t i o n s ( s p o i l e r , e/c - 70%, h/c - 10%, o - 12°) P --4 cn F i g . 29 C D i s t r i b u t i o n s ( s p o i l e r , e/c = 90%, h/c = 10%, o « 12°) Fig. 30 C. Variations over cC Fig. 31 C. Variations over eC Fig. 32 C. Variations over oC 8 1 and e x p e r i m e n t a l v a l u e s i n t h i s neighbourhood i s i n e v i t a b l e . Only w i t h i n a s m a l l p o s i t i v e range of a do the e x p e r i m e n t a l C c u r v e s behave l i n e a r l y f o r s p l i t f l a p s , as L i shown i n F i g s . 15 and 17. T h i s i s p r o b a b l y due t o the e a r l y f l o w s e p a r a t i o n t a k i n g p l a c e near the l e a d i n g edge a t h i g h e r a n g l e s of i n c i d e n c e . T h e r e f o r e , the o v e r a l l c i r c u l a t i o n g e n e r a t e d i s reduced and so i s the l i f t f o r c e . On the o t h e r hand, the n o n - l i n e a r i t y of the C c u r v e s f o r s p o i l e r s i s the L i consequence of t h e s e p a r a t i o n bubble e x p l a i n e d above. In g e n e r a l , the agreement between t h e o r y and e x p e r i m e n t s i s the best f o r s p o i l e r c a s e s , f a i r f o r the c l e a n a i r f o i l and a c c e p t a b l e f o r s p l i t f l a p s . T h i s may be due t o the f a c t t h a t the i n v i s c i d f l o w assumption o v e r e s t i m a t e s the c i r c u l a t i o n around a c l e a n a i r f o i l . E r e c t i n g the s p o i l e r , w hich h e l p s reduce the o v e r a l l c i r c u l a t i o n , t h e r e f o r e , produces b e t t e r agreement. To t h e c o n t r a r y , d e f l e c t i n g the s p l i t f l a p , which a s s i s t s i n e n h a n c i n g the c o r r e s p o n d i n g c i r c u l a t i o n , widens the a l r e a d y e x i s t e d d i s c r e p a n c y i n t h e c l e a n a i r f o i l c a s e . 82 6. CONCLUSIONS AND RECOMMENDATIONS A sequence of c o n f o r m a l t r a n s f o r m a t i o n s has been found u s e f u l f o r deforming the c o n t o u r of an a i r f o i l w i t h an i n c l i n e d s p o i l e r or s p l i t f l a p onto the p e r i m e t e r of the u n i t c i r c l e o v e r which the f l o w problem i s s o l v e d . The wake sour c e model o r i g i n a l l y d e v e l o p e d by P a r k i n s o n and J a n d a l i was a p p l i e d t o d e a l w i t h the c u r r e n t p a r t i a l l y s e p a r a t e d f l o w s i t u a t i o n . The 1-source models p r o v i d e a f i r s t a p p r o x i m a t i o n of the I C d i s t r i b u t i o n i n the sense t h a t the e x p e r i m e n t a l C P P d i s t r i b u t i o n u s u a l l y l i e s w i t h i n the t h e o r e t i c a l c u r v e s . For t h e 2-source model, the c r i t e r i o n of the f i n i t e c u r v a t u r e of the s e p a r a t i n g s t r e a m l i n e s or t h a t of the f i n i t e p r e s s u r e g r a d i e n t a t s e p a r a t i o n r e q u i r e s a m o d i f i c a t i o n of the e x p e r i m e n t a l C ^ v a l u e s . However, s i n c e the a i r f o i l used i n e x p e r i m e n t s has an a r t i f i c i a l l y t h i c k e n e d t r a i l i n g edge exposed t o t h e o u t e r f l o w f o r the case of s p l i t f l a p s , the m o d i f i e d s e c t i o n may have some i n f l u e n c e on the measurements i n the neighbourhood. T h e r e f o r e , f u r t h e r work s h o u l d be c o n s i d e r e d b e f o r e the v a l i d i t y of the c r i t e r i o n can be p i n - p o i n t e d . The boundary c o n d i t i o n , ^ ^ = 0, must be used w i t h c a u t i o n because the problem of a c c e p t i n g the C ^ comes up when 83 the a n g l e of d e f l e c t i o n 6 i n c r e a s e s . I n v e s t i g a t i o n s show t h a t the range i n which t h i s c r i t e r i o n works w e l l i s 0 < 6 < 45° w i t h the s p o i l e r or s p l i t f l a p l o c a t e d no l e s s than 80 % c h o r d d i s t a n c e from the l e a d i n g edge. The m o d i f i e d v e r s i o n of F w k e = n* e q u a t i o n (39) , works w e l l f o r most c a s e s even though i t s p h y s i c a l s i g n i f i c a n c e i s not o b v i o u s . T h i s c o n d i t i o n a l l o w s the use of the e m p i r i c a l C , w i t h the two s o u r c e s l o c a t e d w i t h i n the s i m u l a t e d wake pb r e g i o n . A l t h o u g h the p r e s s u r e near the l e a d i n g edge i s o f t e n o v e r e s t i m a t e d by the t h e o r y , the o v e r a l l p r e d i c t e d d i s t r i b u t i o n i s remarkably c l o s e t o the e x p e r i m e n t a l r e s u l t s . Upstream of the s p o i l e r where the s e p a r a t i o n bubble i s l o c a t e d , the d i s c r e p a n c y of the d i s t r i b u t i o n s from the t h e o r y and the e x p e r i m e n t s s t i l l p r e v a i l s as o b t a i n e d p r e v i o u s l y by J a n d a l i [ 3 ] . T h i s r e g i o n i n which v i s c o u s e f f e c t s dominate can o n l y be coped w i t h i f some mechanism of d i s s i p a t i o n i s i n t r o d u c e d i n the model. F u r t h e r work s h o u l d be i n t e r e s t i n g and c h a l l e n g i n g . To the c o n t r a r y , the e f f e c t due t o the bubble i s not a pronounced one f o r the case of s p l i t f l a p s because the p r e s s u r e g r a d i e n t upstream of i t i s l e s s p r o minent. There i s an i n t e r e s t i n g t r e n d of the base p r e s s u r e f o r s p l i t f l a p s when p l o t t e d a g a i n s t the a n g l e of d e f l e c t i o n , see (41) and F i g s . 10a,b. T h i s t r e n d has not been d i s c u s s e d here f o r s p o i l e r s o n l y because of the l a c k of d a t a . However, i t would be p r e f e r a b l e t o c a r r y out more e x p e r i m e n t s t o c o n s t r u c t 84 a f o r m u l a f o r C r o r both i n c l i n e d s p o i l e r s and s p l i t f l a p s . F i n a l l y , the t h e o r y d e s c r i b e d above can be r e a d i l y extended t o d e a l w i t h any a r b i t r a r y t h i c k a i r f o i l p r o f i l e w i t h the h e l p of the Theodorsen t r a n s f o r m a t i o n . By u t i l i z i n g the W i l l i a m s t r a n s f o r m a t i o n s [ 1 2 ] , i t would be i n t e r e s t i n g t o a p p l y the p r e s e n t work t o m u l t i - e l e m e n t a i r f o i l s . 85 R e f e r e n c e s 1. McCormick, B.W.Jr. - Aerodynamics of V/STOL F l i g h t , Academic P r e s s , 1967. 2. P a r k i n s o n , G.V. and J a n d a l i , T. - A Wake Source Model f o r B l u f f Body P o t e n t i a l Flow, JFM, V o l . 40, No.3, pp577-594, Feb., 1970. 3. J a n d a l i , T. - A P o t e n t i a l Flow Theory f o r A i r f o i l S p o i l e r s , Ph.D T h e s i s , U n i v e r s i t y of B r i t i s h C olumbia, 1 970. 4. S c h l i c h t i n g , H . and T r u c k e n b r o d t , E. - Aerodynamics of the A i r p l a n e , M c G r a w - H i l l , 1979. 5. Whitehead, L.G., Cheer s , F., and Mandl, P. - Flow about A i r f o i l s w i t h S p l i t F l a p s w i t h A p p l i c a t i o n t o C i r c u l a t i o n C o n t r o l by S u c t i o n , NAE L a b o r a t o r y Report LR-226, A p r i l , 1958. 6. Woods, L.C. - Theory of Subsonic P l a n e Flow, Cambridge U n i v e r s i t y P r e s s , 1961. 7. P a r k i n s o n , G.V. - C l o s e E n c o u n t e r s w i t h S e p a r a t e d F l o w s , The W. Rupert T u r n b u l l L e c t u r e a t the CASI Annual G e n e r a l M e e t i n g i n Ottawa on May 2, 1979, CASJ, V o l . 25, No.3, T h i r d Q u a r t e r , 1979. 8. J a n d a l i , T. and P a r k i n s o n , G.V. - A P o t e n t i a l Flow Theory f o r A i r f o i l S p o i l e r s , CASI T r a n s . , V o l . 3 , No.1, p p l - 7 , March, 1970. 9. Chang, P.K. - S e p a r a t i o n of Flow, Pergamon P r e s s , 1970 10. P a r k i n s o n , G.V. and Yeung, W.W.H. - Improvements i n Wake S i n g u l a r i t y P o t e n t i a l Flow Models, P r o c e e d i n g s of the 10th CANCAM, London, O n t a r i o , June, 1985. 11. Tam Doo, P. - A P r e d i c t i o n Method For S p o i l e r Performance, Ph.D. T h e s i s , U n i v e r s i t y of B r i t i s h C o l u m b i a , 1977. 12. W i l l i a m s , B.R. - An E x a c t T e s t Case f o r the P l a n e P o t e n t i a l Flow About Two A d j a c e n t L i f t i n g A e r o f o i l s , ARC RM No. 3717, 1971. 13. Pope, A. and Harper, J . J . - Low-Speed Wind Tunnel t e s t i n g , John W i l e y and Sons, 1966. 8 6 Appendix A • dz . C o n t i n u i t y of l ^ l a t G dz . . . I t i s shown here t h a t | — | i s f i n i t e a t the p o i n t G on the a i r f o i l s u r f a c e . U s i n g the t r i g o n o m e t r i c i d e n t i t y and e q u a t i o n ( 3 ) , one g e t s e s c 2 ( - ) = 1 + c o t 2 ( - ) = 1 + [ - — 5 l 2 I R s i n 8 j ( R s i n 6 - s ) ( R s i n 6 + s) R 2 s i n 26 Near p o i n t G, s — > -R s i n 6, and X, ~K — > ±°° from F i g . 4d. T h e r e f o r e , i t i s adequate t o examine the p r o d u c t | ( s + R s i n 6)X| ... (A1) dz i n t h e e q u a t i o n f o r |—|, see ( 1 0 ) . U s i n g (3) t o r e w r i t e (s + R s i n 6 ) , one g e t s (s + R s i n 6) = R s i n 5 (1 + i c o t (^)) I f to = to + i to , where to and to a r e r e a l , i t can be shown x y x y t h a t .to. cos X c o s h Y - i s i n X s i n h Y c o t ( — ) = —: ~ , „ , : ~ : r ~ ... (A2) 2 s i n X c o s h Y + i cos X s m h Y 87 where X = to /2, Y = to /2. x y With some m a n i p u l a t i o n , i ^ ,OK . s i n h Y + cosh Y 1 + 1 C O t ( - ) = / . 2 <* • V.2 -r ( A 3 ) 1 2 1 / s i n 2 X + s i n h 2 Y In o r d e r t o be p r e c i s e , l e t us approach p o i n t G from p o i n t A i n the co-plane. T h e r e f o r e , u s i n g ( 5 ) , w x = " §(2-n), u = h - ^{n l n ( - + 1) + (2-n) l n ( r ^ - 1} y 2 n 2-n - h - In X, s i n c e X — > °° o r , e e /X. Cl) /2 The numerator of (A3) i s j u s t e ¥ , whereas i t s denominator approaches -^ e " y ^ 2 near p o i n t G s i n c e s i n h (~^) dominates. T h e r e f o r e , (A1) can be w r i t t e n as |(s + R s i n 8) X | oC e ^ X = e^, some f i n i t e v a l u e . C o n s e q u e n t l y , p o i n t G i s not a s i n g u l a r i t y . Q.E.D. Appendix B dz C o n t i n u i t y of | — | a t » The p o i n t a t i n f i n i t y , r e p r e s e n t e d by z = °°, t = °°, ~K i , or co = 0, i s shown here t o be a removable s i n g u l a r i t y dz dS* From e q u a t i o n ( 1 0 ) , — c s c 2 ( ^ ) ( i - T C ) 2 . S i n c e esc x — > ^ as x —> 0, i t i s easy t o see t h a t oC 4 ( 1 - ^ ) 2 ... (B1) dS co J U s i n g e q u a t i o n ( 5 ) , one can examine the n a t u r e of the p o i n t = oo by w r i t i n g CO — CO- = ^ [ { n ln(£»+ 1) + (2-n) ln(^r» - 1 ) } -{n l n ( - + 1) + (2-n) l n ( r ^ - 1 ) } ] n 2-n i r ,X + n. . . , .X -2+n.. = " I [ n l n ( x ^ > + ( 2 " n ) l n ( x ^ ) ] - - i [ n l n ( 1 + ^ f f ) + (2-n) I n d + * - ^ ) ] As co — > co — > 0, X — > X = X + V i r OO OO 2 Xoo+n Xo.-2+n , co 1 -iXg,  o r , (- — ) — > 2  X - X ' (X^+n)(X^-2+n) ... (B2) From ( 8 ) , X - X = rj(TC-i) ... (B3) OO 89 Combining ( 1 0 ) , ( B 1 ) , (B2) and ( B 3 ) , i t can be shown t h a t dz _ -(Xpo+n) (\g,-2+n)R s i n 6 i i / / , . d$ TJ X CO where v// = ^ + a 0 ~ 7 dz I t i s e a s i l y seen t h a t — i s a complex number of f i n i t e d$ magnitude. T h e r e f o r e , the pro o f i s c o m p l e t e d . A more s i m p l e p r o o f suggested by P r o f . P a r k i n s o n i s t o d7v a p p l y l ' H o p i t a l r u l e t o the r i g h t hand s i d e of ( B 1 ) . Si n c e — dz i s f i n i t e a t z , — must a l s o be so. oo a s Q.E.D. 90 Appendix C L o c a t i o n s of the T r a i l i n g Edge The l o c a t i o n of the p o i n t E i n the t - p l a n e can be e x p r e s s e d as t = 1 or t = t 0 + R e ^ ... (C1) where tft = t a n _ 1 ( ~ j ) and R i s d e f i n e d i n ( l a ) . Combining (C1),(2) and ( 2 b ) , f o r the case of an i n c l i n e d s p o i l e r , i n t h e s - p l a n e , s = R cos(7-tf>) + i R [sin(7~ti>) - cos 8 ] , ) i n the co-plane, u> = -~{2-n) + iw E Z Ey 1 x 2 + Y 2 where u = In { g 2 + y 2 ) ... (C2) and X = cos<7-0) - s i n 6 Y = s i n ( 7 - 0 ) - cos 8 Z = c o s ( 7 ~ 0 ) + s i n 8 In t h e X-plane, X i s g i v e n i m p l i c i t l y by E o) = h - h n l n ( - E + 1) + (2-n) l n ( - ^ E - 1)} ... (C3) Ey 2 n 2-n T h e r e f o r e , i n the 7v-plane, X = (X - T-)/r} ... (C4) E E For t h e case of a s p l i t f l a p , u s i n g (C1) and (17) i n t h e s - p l a n e , s = R cos(7+#) + i R [ c o s 6 - s i n ( c * + 7 ) ] , 91 7T i n t h e co-plane, co = -(2-n) + ico E l Ey 1 X 2 + Y 2 where u = -- In { g 2 + y 2 } ... (C5) and X = cos(7+#) - s i n 8 Y = - sin(7+<>) + cos 8 Z = cos(7+0) + s i n 8 In t h e X-plane, X i s g i v e n i m p l i c i t l y by E co = h - U n l n ( - E - 1) + (2-n) l n ( - ^ E + 1 ) } ... (C6) Ey 2 n 2-n T h e r e f o r e , i n the "X-plane, "X = (X + T£)/r\ ... (C7) E E I 92 Appendix D D e r i v a t i o n of E q u a t i o n s (4) & (5) from J a n d a l i ' s T h e s i s J a n d a l i ' s e q u a t i o n s (4) & (5) (from h i s t h e s i s ) , w i l l be r e - d e r i v e d i n t h i s appendix from the g e n e r a l e x p r e s s i o n s d e v e l o p e d i n t h i s t h e s i s f o r an i n c l i n e d s p o i l e r . F o r a normal s p o i l e r , 5 = 90° and n=1, (6) and (7) p r o v i d e 1 = 0 and V = e -1 ... (D1) I From (13a, b, c ) , 01 = \ , 0 2 = t a n - ' ( i j ) f fl3 = t a n " 1 ( 7 ^ ) = w - 0 2 A l s o from ( 2 a ) , 7 = ^ - 0 o - ^ = - 0 o it it Hence, (13) g i v e s a0 = — + a - 80 + — - 62 - ( T ~ 0 2 ) = a - 0 O From e q u a t i o n ( 1 4 ) , t h e l o c a t i o n of p o i n t C on the u n i t c i r c l e i s t h e n 6C = w ~ 2 ( 2 } " a ° = 0 O - a whic h i s e x a c t l y the same as e q u a t i o n (4) i n J a n d a l i ' s t h e s i s . From ( 3 b ) , one g e t s 93 J a n d a l i [2] d e f i n e s £ = ^ , which i s e q u i v a l e n t t o £ = R1 n T h e r e f o r e , e h = (D2) From appendix C, f o r normal s p o i l e r , i t i s e a s i l y d e r i v e d from (C1) t h a t 1 i n r 1 ~ C O S (6q + <I>) 2 X l1 + cos (60 + <t>)n 0 r , _ ~2uy = 2 cos(gc.+?)  e i + c o s ( e 0 + ^ ) 1 + e y = 1 + C O S ( 0 o + c6) I t i s a l s o o b v i o u s from (C2) t h a t (D3a) (D3b) 2 = ! + e 2 ( c y - h ) E (D4) and "KE = \E/fj ... (D5) U s i n g (D1), (D3a,b), (D4) and (D5), m a n i p u l a t i o n , one g e t s 1 - _ 1 - e2"Y - 2 e " 2 h 1 + H ' i + e " 2 w y cos ( e 0 + ^ ) - [ i+cos(e 0+c6) ] ( | 7 T ) 2 cos {«.+•) { 7 ^ 7 ? ) - < ^ j > 2 and a f t e r some 2 c o s ( g 0 + t6) + ( 1-k) (1+k) (D6) 1 £ 2 + 1 where k = — * S u b s t i t u t i n g (D6) i n t o (15) and r e p l a c i n g 4>0 by 0o + 0f J a n d a l i ' s e q u a t i o n (5) i s r e c o v e r e d . Appendix E D e r i v a t i o n of E q u a t i o n (5) E q u a t i o n (5) w i l l be d e r i v e d by i n t e g r a t i n g ( 4 ) . I n t e g r a t i n g ( 4 ) , CJ = | [ n In (X+n) + (2-n) In (X-2+n)] + C ... (E1 ) where C i s the i n t e g r a t i o n c o n s t a n t . Note t h a t b oth k and can be complex so k = k, + k 2 i and C = C, + C 2 i I where k 1 f k 2 , C 1 f C 2 are r e a l numbers. When X = -n• or -n-e as e —> 0, u —> n7r/2 + °° i S u b s t i t u t i n g i n t o (E1) n7r/2+=°i = (k,+ k 2 i ) [ n In (-e) + (2-n) In (-2-e) ] /2 + C, + C 2 i = (k,+ k 2 i ) [ n In (e) + (2-n) In (2+e) + n j r i / 2 + (2-n)7ri/2 ] /2 + C, + C 2 i T a k i n g the r e a l p a r t of b o t h s i d e s nir = k, [ n I n c + (2-n) In (2+e) ] - k 27r + 2 C, As e •—> 0, In c — > - ». T h e r e f o r e , k, must be 0. nit = - k 2 i r + 2 C, ... (E2) 96 When X = - n + or -n + e as e — > 0, u — > °° i S u b s t i t u t i n g i n t o (E1) »i = (k,+ k 2 i ) [ n In (e) + (2-n) In (-2+e) ] /2 + C, + C 2 i = (k,+ k 2 i ) [ n In (e) + (2-n) In (2e) + ( 2 - n ) i r i / 2 ]/2 + C, + C 2 i T a k i n g the r e a l p a r t of bo t h s i d e s 0 = 2 k, [ n In e + (2-n) I n (2-e) ] - k 2 ( 2 - n ) 7 r / 2 + 4 C, S i n c e k, = 0, 4 C, = k 2 (2-n)ir ... (E3) S u b s t i t u t i n g (E3) i n t o ( E 2 ) , I k 2 = -2 and 1 C, = - ( 2 - n ) 7 r / 2 When X = 0 and CJ = h i , (E1) becomes, a f t e r e q u a t i n g the im a g i n a r y p a r t s of both s i d e s , C 2 = h + n In n + (2-n) In (2-n) S u b s t i t u t i n g k,, k 2, C 1 f C 2 i n t o ( E 1 ) , one o b t a i n s e q u a t i o n (5) . 97 Appendix F D e r i v a t i o n of E q u a t i o n (35) E q u a t i o n (35) f o r f i n i t e p r e s s u r e g r a d i e n t a t s e p a r a t i o n i s d e r i v e d h e r e . The p r e s s u r e a l o n g the a i r f o i l s u r f a c e can be e x p r e s s e d most c o n v e n i e n t l y i n terms of I f s i s measured from the f o r w a r d s t a g n a t i o n p o i n t a l o n g l the a i r f o i l , see F i g . F 1 ( a ) , then where 8 i s t h e a n g u l a r v a r i a b l e i n t h e $-plane, F i g . F 1 ( b ) , and k = / I - C , , some non-zero number. pb S i n c e dz = ds e l e and d$ = i e ^ d f l , .dz. ds 'd$' = ~d8 because s i n c r e a s e s w i t h d e c r e a s i n g 6. A l s o |w(z)| = | w ( s ) | / | | | | - - f , / f a T h e r e f o r e , dC^, _k d .,,/ v , ,, dz, _k d ,fi« ,, . 98 (a) i F 1 T r a n s f o r m p l a n e s Z and $ 99 dC S i n c e I 2 L = 0, "r-P|_ i s f i n i t e o n l y i f E U S E 3S<f?l« - 9(»>IB - 0 But q(6) = ( f 2 t\ ~ f , f j ) / f | A p p l y i n g l ' H o p i t a l ' s r u l e t w i c e t o q(6) s i n c e both the numerator and denominator are z e r o s of o r d e r 2, then t\ f f - f , f j = 0 a t 6 = 6E ) 100 Appendix G E v a l u a t i o n s of t\, f " , f 2 & f 2 a t C r i t i c a l P o i n t s R e c a l l the f o l l o w i n g d e f i n i t i o n s : f1 = |W($)| and f 2 = |dz/d$| S i n c e both f , and f 2 a r e made equ a l t o z e r o a t p o i n t s E and C, e q u a t i o n s (33) and (34) must be e v a l u a t e d by L ' H o p i t a l ' s r u l e . T h e r e f o r e , a t t h e s e c r i t i c a l p o i n t s , |W(z)| = | j , where ( )' = |^ T h i s appendix i s devoted t o f i n d i n g f , , f " , f 2 and f 2 a t the c r i t i c a l p o i n t s E and C f o r b o t h s p o i l e r s and s p l i t f l a p s . f , i s a s i m p l e f u n c t i o n of 6 a l o n e . I t i s r a t h e r s t r a i g h t s t nd f o r w a r d t o compute i t s 1 and 2 d e r i v a t i v e s . f\(6) = -|[4 cos 6 + Sb- c s c M ^ 1 ) + f- c s c M ^ ) ] e- «1 2 e- «i 2 e- 6 2 2 On the o t h e r hand, f 2 i s a more c o m p l i c a t e d f u n c t i o n i n v o l v i n g more than one v a r i a b l e . T h e r e f o r e , i t i s c o n v e n i e n t t o reduce i t t o a f u n c t i o n of V and d so t h a t i t s d e r i v a t i v e s can be o b t a i n e d r e l a t i v e l y e a s i l y . A c c o r d i n g t o F i g s . G l ( a ) 101 and G 1 ( b ) , t = t 0 + R e 1 ^ if? , and $ = e ... (G2) I t can be shown t h a t , u s i n g F i g . G2, (G1 ) (G3) M t - O U + D I y/ A B | t | 2 E ... where A = 2 - 2 cos {V+<t>) B = 1 + m2 + 2m cos (7jf-A) E = 1 + T? 2 + 2TJ C O S ("&-»>) <}> = t a n ' M - r 1 - ) A = t a n - M r ^ ) v = TT - t a n ' M " ) 1 + e 1 - e e mR = / M 2 + (1 " e)2 TJR = / e 2 + y 2 y and R i s g i v e n by ( 1 a ) . S u b s t i t u t i n g ( G l ) t o ( 1 7 ) , the f o l l o w i n g e x p r e s s i o n r e s u l t s . s = R [ cos ( # - 7 ) + i { s i n C & - 7 ) + cos 6 }] (G4) U s i n g (G4) and appendix A, i t can be shown t h a t | c s c 2 ( | ) | = / C(2) D(S) ... (G5) where C(S) = 1 + s i n ( S - 7 - 6 ) D (7J ) = 1 + s i n (2 - 7+6) S u b s t i t u t i n g (G2) t o (21) and making use of ( 2 0 ) , i t f o l l o w s t h a t X = - t a n ( 6 * a ° ) ... (G6) 102 G1 D e f i n i t i o n s of If and 6 © G 2 D e f i n i t i o n s of A, v, 4>, m, rj and R o to 104 and X = - T - rj t a n ( — ^ ) ... (G7) As a r e s u l t , f o r the s p l i t f l a p s , f2 — K L(V) h(6) ... (G8) where K = R TJ/(2 s i n 6) L ( ^ = ^ t 2 | ^ C ( y ) D ( y ) ' a S d e f i n e d i n ( G 3 a n d G 5 ) ' h(e) - Vwnfr • ' " ' ' H 8 1 ' ' G ( . ) - - X H(0) = n - X and 1(0) = n - 2 - X An i m p l i c i t r e l a t i o n between # and 6 can be found i n the f o l l o w i n g way. U s i n g ( A 2 ) , e q u a t i o n (18) i s s i m p l i f i e d t o I • r , c r 1 ~ i tan X tanh Y n . s = l R s i n 8 — v ^ . . — — — ] ... (G9) tan X + I tanh Y E q u a t i n g the r e a l p a r t s of (G4) and (G9), when p o i n t E i s approached, \ • r s e c 2 X tanh Y., . cos ( V - 7 ) - s i n 8 [ t a n a x + t a n h 2 y ] ... (610) where 2X = (2-n) | 2Y = h - { n l n ( - - 1) + (2-n) l n ( l + -~)}/2 n 2-n T h e r e f o r e , e q u a t i o n s (G7) and (G10) can be used t o c a l c u l a t e — and i n d e r i v i n g i\ and f 2 s i n c e , from (G8), f 2 = K [ L'(V) hie) H + h'(e)) f3 = K [ L " ( 9 ) h ( 0 ) ( | | ) 2 + 2L' (Ti)h' ( 0 ) | | + L ' ( ? J ) h ( 0 ) 0 1 05 L ( S ) h " ( 0 ) ] Note t h a t the terms i n v o l v i n g L i V ) can be i g n o r e d s i n c e at the p o i n t E, # = -<j> and from (G3), L(-<t>) = 0. T a k i n g the d e r i v a t i v e of (10) w i t h r e s p e c t t o 0, a f t e r some m a n i p u l a t i o n , = — — 7 £ — r F(Y) h(0) ... (G11) d0 s i n (#-7) ( t a n 2 X - t a n h 2 Y ) s e c h 2 Y where F(Y) = and k = ( t a n 2 X + t a n h 2 Y ) 2  s i n 6 s e c 2 X 7? T a k i n g the d e r i v a t i v e of (G11) a g a i n , i t can be shown t h a t / 1 f . r - F' ( Y ) h 2 ( 0 ) 7 ? , . . d ? n , 5 F = s i n ( H ) { k [ F ( Y ) h ( 9 ) i ] - C O S ( ^ ) ( d V } - 9 t a n h V c o r * V i 2 V " h e r e F ' ( Y ) " (t.n*» + t a n h ' Y ) * ' 1 s e c f i ! Y * ( t a n ' X " t a n h ' Y ) » ( t a n 2 X + t a n h 2 Y ) + 2 s e c h 2 Y ( t a n 2 X - t a n h 2 Y ) ] Making use of (G3), (G5), i t can be shown t h a t r , / 7 M /B(9) C(V) D(V) L m = zm and at "& = -<p B' . C D' 2E L"(7j0 = /BCD { — + — + - ==r } BE CE DE E 2 where B' = -2 m s i n CB-A) C = cos (S-7-6) D' = cos (fl-7+8) E' = -2 77 s i n CB-v) 106 and h ' ( 0 ) ^ = ( F' G + F G' )(H I ) " 1 - F G G' ( H'2!.-1 + H " 1 I - 2 ) where F' = - F TC G' = T J F/2 H' = G' I ' = G' T h e r e f o r e , i\ and f 2 a t 8 = 8 can be c a l c u l a t e d . E Near p o i n t E, f o r s p o i l e r s , f 2 t a k e s the form s i m i l a r t o t h a t of (G8) w i t h X = T - n t a n ( ^ - ^ ) e x c e p t C = 1 - s i n (2+7+6) D = 1 - s i n (2+7-6) The r e l a t i o n between 2 and 8 i s . . „ r s e c 2 X tanh Y.. cos (2 + 7) = s i n 8 [ t a n 2 x + t a n h 2 y ] w i t h 2X = -(2-n) | 2Y = h - { n l n ( - + D + (2-n) l n ( ~ - - 1 }/2 n 2-n d2 k T h e r e f o r e , - = s i n ( y + < y ) F(Y) h<*> and f j | , = « L'(.)h(») g | , E E where k, F ( Y ) , h{8), K , L'(0) a r e the same as th o s e d e f i n e d i n above. f 2 can be c a l c u l a t e d by u s i n g t he s i m i l a r p r o c e d u r e 107 o u t l i n e d above. By examining e q u a t i o n s (10) and ( 2 2 ) , i t i s c l e a r t h a t f 2 i s z e r o a t p o i n t C because X — > 0 t h e r e . Z-ZIQ c a n e a s i l y be c a l c u l a t e d and i s e q u a l t o f , . _ I t l - 1 I R s i n 6 s e c M e + Q ° ) T s i n h M - n - 1 z | C " | t 2 | 4n(2-n) L f + 2 ) L s i n h ( 2 ) J s i n c e u>c = i h . So | s i n 2 ( ^ c ) | = s i n h 2 ( ^ ) For s p o i l e r s , t = t 0 + R e ld° + fi e + l T , but t = t 0 + R e + i e ° + fi e ~ 1 7 f o r s p l i t f l a p s . 

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