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Steady and nonsteady potential flow methods for airfoils with spoilers Brown, Geoffrey Phillip 1971

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S T E A D Y A N D N O N S T E A D Y P O T E N T I A L F L O W M E T H O D S F O R A I R F O I L S W I T H S P O I L E R S B Y G E O F F R E Y P . BROWN B . E . , U n i v e r s i t y o f S y d n e y , I967 A T H E S I S S U B M I T T E D 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 T H E D E G R E E O F D O C T O R O F P H I L O S O P H Y i n t h e D e p a r t m e n t o f 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 t h i s t h e s i s a s c o n f o r m i n g t o t h e r e q u i r e d s t a n d a r d 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 J u l y , 1971 In 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 a n a d v a n c e d d e g r e e 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 a g r e e t h a t t h e L i b r a r y s h a l l m a k e 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 a n d S t u d y . I f u r t h e r a g r e e 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 p u r p o s e s may be g r a n t e d b y t h e H e a d o f my D e p a r t m e n t o r b y h i s 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 b e 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 . D e p a r t m e n t o f The 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 V a n c o u v e r 8, C a n a d a i A B S T R A C T In p a r t I a l i n e a r i z e d c a v i t y p o t e n t i a l f l o w t h e o r y i s dev-eloped t o s o l v e both the steady s t a t e a i r f o i l s p o i l e r problem and the t r a n s i e n t loads on an a i r f o i l d u r i n g and a f t e r s p o i l e r a c t u a t i o n . The t h e o r y i s a l s o a p p l i c a b l e t o the case of an a i r f o i l w i t h a s p o i -l e r and a f l a p . The the o r y uses conformal t r a n s f o r m a t i o n s t o map t h a t p a r t of the a i r f o i l exposed t o the fl o w onto the upper h a l f of a u n i t c i r c l e . The complete flow f i e l d about the a i r f o i l maps i n t o the upper h a l f plane e x t e r i o r t o t h i s u n i t c i r c l e . A l t hough no l i m i t a t i o n s are imposed i n the paper upon the s p o i -l e r h e i g h t , angle, or l o c a t i o n , good agreement w i t h experiment would not be expected i n such a l i n e a r i z e d t h e o r y f o r ve r y l a r g e s p o i l e r s . S p o i l e r h e i g h t s up to 10% of the a i r f o i l chord are c o n s i d e r e d , and the t h e o r y shows good agreement with experiment. A t h e o r y f o r the steady s t a t e a i r f o i l s p o i l e r problem f o r a s o l i d a i r f o i l , and an a i r f o i l w i t h a s l o t t e d f l a p i s developed i n p a r t I I . An exact p o t e n t i a l f r e e s t r e a m l i n e t h e o r y u s i n g the s u r f a c e s i n g u l a r i t y technique i s used i n t h i s work. The wake Is contained between two f r e e s t r e a m l i n e s . F o l l o w i n g J a n d a l i ' s technique ( 1 ) , the wake f l o w i s c r e a t e d by p o s i t i o n i n g sources on the a i r f o i l s u r -f a c e i n t h a t r e g i o n exposed t o the wake. The a c t u a l f l o w i n the wake r e g i o n i s i g n o r e d , and the base p r e s s u r e i s taken t o be constant a t the e x p e r i m e n t a l v a l u e . The theory agrees w e l l w i t h the r e s u l t s ob-t a i n e d by J a n d a l i . 11 TABLE OF CONTENTS Page I INTRODUCTION 1 PART I A LINEARIZED CAVITY POTENTIAL FLOW THEORY FOR THE STEADY STATE AND TRANSIENT AIRFOIL SPOILER PROBLEM 4 I I THE ACCELERATION POTENTIAL 5 I I I STEADY THEORY 10 3.1 F o r m u l a t i o n of the Problem 10 3 .2 T r a n s f o r m a t i o n s 10 3.3 Boundary C o n d i t i o n s 14 3«4 Mathematical Flow Model 15 3 . 4 . 1 I n c i d e n c e Case 16 3 . 4 . 2 Camber Case 17 / 3 . 4 . 3 T h i c k n e s s Case 19 3 . 4 . 4 S p o i l e r Case 21 3 . 4 . 5 F l a p Case 23 3.5 - Method of S o l u t i o n 23 IV NONSTEADY THEORY 32 4 . 1 F o r m u l a t i o n of the Problem 32 4 . 2 Blowing Theory 35 4 . 2 . 1 Boundary C o n d i t i o n s 35 4 . 2 . 2 T r a n s f o r m a t i o n s 35 4 . 2 . 3 Method of S o l u t i o n 36 4 . 3 U n i t Step A c t u a t i o n 44 4 . 4 F i n i t e Time A c t u a t i o n 46 V EXPERIMENTS 48 VI RESULTS AND COMPARISONS 52 6 . 1 Steady Theory 52 6 . 2 Nonsteady Theory 68 i n PART I I AN EXACT FREE STREAMLINE POTENTIAL FLOW THEORY FOR THE STEADY STATE AIRFOIL SPOILER AND SPOILER PLUS SLOTTED FLAP PROBLEM , ° * VII SOLID AIRFOIL WITH A SPOILER 8 3 7 .1 S u r f a c e S i n g u l a r i t y Theory 83 7.2 F o r m u l a t i o n of the Problem 83 7.3 Boundary C o n d i t i o n s 85 7 .4 1-Source Model 86 7.5 2-Source Model 89 V I I I AIRFOIL WITH A SLOTTED FLAP AND A SPOILER 92 8 . 1 F o r m u l a t i o n of the Problem 92 8.2 Boundary C o n d i t i o n s 92 8 . 3 1-Source Model 92 8 . 4 2-Source Model 95 IX RESULTS AND COMPARISONS 98 9 . 1 S o l i d A i r f o i l w i t h a S p o i l e r 98 9.2 A i r f o i l w i t h a S l o t t e d F l a p and a S p o i l e r 104 * * * * * X CONCLUSION 111 i v LIST OF FIGURES Page 1 A i r f o i l i n z-plane 11 2 Complex Transform Planes 12 3 A i r f o i l i n the z 1 - p l a n e 34 4 14$ T h i c k C l a r k Y A i r f o i l 49 5 14$ T h i c k C l a r k Y A i r f o i l w i t h 32.5$ F l a p 50 6 L i f t C o e f f i c i e n t f o r B a s i c C l a r k Y A i r f o i l 53 ? L i f t C o e f f i c i e n t f o r C l a r k Y A i r f o i l w i t h S p o i l e r 55 8 L i f t C o e f f i c i e n t f o r C l a r k Y A i r f o i l w i t h S p o i l e r 56 9 L i f t C o e f f i c i e n t f o r C l a r k Y A i r f o i l w i t h S p o i l e r 57 10 L i f t C o e f f i c i e n t f o r C l a r k Y A i r f o i l w i t h S p o i l e r 58 11 L i f t C o e f f i c i e n t f o r C l a r k Y A i r f o i l w i t h S p o i l e r 60 12 L i f t C o e f f i c i e n t f o r C l a r k Y A i r f o i l w i t h S p o i l e r 6 l 13 L i f t C o e f f i c i e n t f o r C l a r k Y A i r f o i l w i t h S p o i l e r 62 14 L i f t C o e f f i c i e n t f o r C l a r k Y A i r f o i l w i t h S p o i l e r 63 15 L i f t C o e f f i c i e n t f o r C l a r k Y A i r f o i l w i t h S p o i l e r 64 16 L i f t C o e f f i c i e n t f o r C l a r k Y A i r f o i l w i t h S p o i l e r 65 17 L i f t C o e f f i c i e n t f o r C l a r k Y A i r f o i l w i t h S p o i l e r 66 18 L i f t C o e f f i c i e n t f o r C l a r k Y A i r f o i l w i t h S p o i l e r 67 19 L i f t C o e f f i c i e n t f o r C l a r k Y A i r f o i l w i t h S p o i l e r and F l a p 69 20 L i f t C o e f f i c i e n t f o r C l a r k Y A i r f o i l w i t h S p o i l e r and F l a p 70 21 P r e s s u r e D i s t r i b u t i o n f o r C l a r k Y A i r f o i l w i t h S p o i l e r 71 22 P r e s s u r e D i s t r i b u t i o n f o r C l a r k Y A i r f o i l w i t h S p o i l e r 72 23 Blowing Theory S o l u t i o n 74 24 Blowing Theory S o l u t i o n 75 25 U n i t Step S p o i l e r A c t u a t i o n S o l u t i o n 76 26 U n i t Step and F i n i t e Time S p o i l e r A c t u a t i o n S o l u t i o n s 77 27 U n i t Step and F i n i t e Time S p o i l e r A c t u a t i o n S o l u t i o n s 78 V 28 U n i t Step and F i n i t e Time S p o i l e r A c t u a t i o n S o l u t i o n s 80 29 U n i t Step and F i n i t e Time S p o i l e r A c t u a t i o n S o l u t i o n s 81 30 A i r f o i l i n the z-plane 84 31 A i r f o i l i n the z-plane 93 32 P r e s s u r e D i s t r i b u t i o n f o r C l a r k Y A i r f o i l w i t h S p o i l e r 100 33 P r e s s u r e D i s t r i b u t i o n f o r C l a r k Y A i r f o i l w i t h S p o i l e r 101 34 P r e s s u r e D i s t r i b u t i o n f o r C l a r k Y A i r f o i l w i t h S p o i l e r 102 35 P r e s s u r e D i s t r i b u t i o n f o r C l a r k Y A i r f o i l w i t h S p o i l e r 103 36 P r e s s u r e D i s t r i b u t i o n f o r C l a r k Y A i r f o i l w i t h and without S p o i l e r 105 37 P r e s s u r e D i s t r i b u t i o n f o r NACA 23012 A i r f o i l and S l o t t e d F l a p without S p o i l e r 106 38 P r e s s u r e D i s t r i b u t i o n f o r NACA 23012 A i r f o i l and S l o t t e d F l a p w i t h S p o i l e r 108 39 P r e s s u r e D i s t r i b u t i o n f o r NACA 23012 A i r f o i l and S l o t t e d F l a p w i t h and without S p o i l e r 109 ACKNOWLEDGEMENTS I would l i k e t o thank Dr. G.V. Pa r k i n s o n f o r h i s a b l e a s s i s t -ance d u r i n g the course of t h i s r e s e a r c h and f o r h i s g e n e r a l g u i d -ance d u r i n g ray time i n graduate s c h o o l , I would a l s o l i k e t o thank Dr. I.S. G a r t s h o r e f o r h i s suggestions d u r i n g the e a r l y stages of t h i s work and Dr. L. Mysak f o r h i s a d v i c e c o n c e r n i n g the mathemat-i c s . S p e c i a l thanks go t o the many from whom I borrowed r e f e r e n c e •books and t o Mr. E. A b e l l who b u i l t the experimental equipment used. T h i s work was supported by the U n i v e r s i t y of B r i t i s h Columbia, the N a t i o n a l Research C o u n c i l of Canada and the Defense Research Board of Canada. v i i LIST OF MAIN SYMBOLS-PART I a a' - i ¥ - J F 1 c St - a c c e l e r a t i o n v e c t o r a x - x-component of a c c e l e r a t i o n ay - y-component of a c c e l e r a t i o n b - a/ B e - cons t a n t i n c a v i t y t e r m i n a t i o n term c - a i r f o i l chord C^ - f l a p chord CD - cons t a n t i n l e a d i n g edge term C,' - constant from a i r f o i l term G^ 1 - constant from s p o i l e r term C L - l i f t c o e f f i c i e n t C t s - q u a s i - s t e a d y l i f t c o e f f i c i e n t C,^  - f i n a l l i f t c o e f f i c i e n t C p - p r e s s u r e c o e f f i c i e n t D0 - co n s t a n t e - s u b s c r i p t i n d i c a t i n g e r e c t i o n F - complex a c c e l e r a t i o n f u n c t i o n F - (*-=l,2,3) complex f u n c t i o n s of F-tH F-„ - complex i n c i d e n c e f u n c t i o n Fe - complex camber f u n c t i o n F + - complex f l a p f u n c t i o n F s - complex s p o i l e r f u n c t i o n F t - complex t h i c k n e s s f u n c t i o n v i i i F w - nonsteady complex f u n c t i o n h - s p o i l e r h e i g h t I - a c t u a t i o n response f u n c t i o n i - i p i n p h y s i c a l plane or i t s transforms j - ^T? i n phase plane or i t s transforms J„ - nonsteady a c c e l e r a t i o n term F o u r i e r s e r i e s c o e f f i c i e n t K - c a v i t a t i o n number k - yW-C I - c a v i t y l e n g t h M„ - camber F o u r i e r s e r i e s c o e f f i c i e n t N* - t h i c k n e s s F o u r i e r s e r i e s c o e f f i c i e n t P - s t a t i c p r e s s u r e Poo - u n d i s t u r b e d f r e e stream s t a t i c p r e s s u r e Pc - p r e s s u r e i n s i d e the c a v i t y p - s t a t i c p r e s s u r e p e r t u r b a t i o n Q - frequency response f u n c t i o n Tq" - v e l o c i t y v e c t o r q e - magnitude of v e l o c i t y on the c a v i t y R - r e a l p a r t of Q S - imaginary p a r t of Q s - s p o i l e r p o s i t i o n s' - d i s t a n c e moved i n chords t - time U - f r e e stream v e l o c i t y u - p e r t u r b a t i o n v e l o c i t y i n the x d i r e c t i o n v - p e r t u r b a t i o n v e l o c i t y i n the y d i r e c t i o n v„ - amplitude of v e l o c i t y \£ - amplitude of v e l o c i t y w - complex v e l o c i t y i x V/ - a c t u a t i o n response f u n c t i o n x,y- c o o r d i n a t e system i n z-plane z - complex v a r i a b l e d e s c r i b i n g a i r f o i l plane z ! - complex v a r i a b l e d e s c r i b i n g z 1 - p l a n e - angle of i n c i d e n c e S - s p o i l e r angle t o a i r f o i l s u r f a c e 8 - complex v a r i a b l e d e s c r i b i n g 6-plane §^ - p o i n t a t i n f i n i t y i n §-pane % - f l a p angle © - a n g u l a r measurement i n 1»-plane ©„ - a n g u l a r p o s i t i o n of l e a d i n g edge i n S-plane &, - a n g u l a r p o s i t i o n of s p o i l e r base i n §-plane &x - a n g u l a r p o s i t i o n of f l a p hinge p o i n t i n §-plane V - complex v a r i a b l e d e s c r i b i n g V-plane ^ - d e n s i t y of f l o w *f - a c c e l e r a t i o n p o t e n t i a l 'Y' - a c c e l e r a t i o n stream f u n c t i o n w - frequency of blowing LIST OF MAIN SYMBOLS-PART I I element of m a t r i x A element of m a t r i x ATI element of m a t r i x A element of m a t r i x B TI element of m a t r i x B element of m a t r i x B T* a i r f o i l chord base p r e s s u r e c o e f f i c i e n t p r e s s u r e c o e f f i c i e n t a t the i t h c o n t r o l p o i n t s p o i l e r h e i g h t symbol r e p r e s e n t i n g i t h element on the f o i l or f l a p symbol r e p r e s e n t i n g j t h element on the f o i l or f l a p symbol r e p r e s e n t i n g k t h element on the main f o i l sumbol r e p r e s e n t i n g 1 t h element on the f l a p s u b s c r i p t i n d i c a t i n g normal d i r e c t i o n number of l a s t element b e f o r e the s p o i l e r t i p number of f i r s t element a f t e r the s p o i l e r t i p number of l a s t element b e f o r e the f o i l t r a i l i n g edge number of f i r s t element a f t e r the f o i l t r a i l i n g edge number of l a s t element b e f o r e the f l a p t r a i l i n g edge number of f i r s t element a f t e r the f l a p t r a i l i n g edge s p o i l e r p o s i t i o n v e l o c i t y on the i t h c o n t r o l p o i n t s u b s c r i p t i n d i c a t i n g t a n g e n t i a l d i r e c t i o n v e l o c i t y due t o source s angle of Inci d e n c e c i r c u l a t i o n about the a i r f o i l s p o i l e r angle t o a i r f o i l s u r f a c e (without s u b s c r i p t ) x i *l - f l a p angle X - source strength - v e l o c i t y p o t e n t i a l - strength of the source on the J t h element 1. SECTION 1 INTRODUCTION T h e o r e t i c a l i n v e s t i g a t i o n of a i r f o i l s p o i l e r aerodynamics i s an a r e a of c o n t i n u i n g importance. The t r a n s i e n t loads of s p o i l e r a c t -u a t i o n and the s p o i l e r p l u s f l a p combination have p r a c t i c a l a p p l i c a -t i o n i n the r e c e n t i n t e r e s t i n h i g h l i f t d e v i c e s . S p o i l e r s are used f o r r o l l c o n t r o l a t low speeds when a c t u a t e d a s y m m e t r i c a l l y , or f o r l i f t and d r a g c o n t r o l i f a c t u a t e d s y m m e t r i c a l l y . 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 shows t h a t the f l o w i n the r e g i o n between a s p o i l e r and the t r a i l i n g edge i s se p a r a t e d . S i n c e no s a t -i s f a c t o r y method e x i s t s t o p r e d i c t the base p r e s s u r e t h e o r e t i c a l l y , a t h e o r e t i c a l model w i l l r e q u i r e a t l e a s t one e m p i r i c a l parameter. Some pro g r e s s has been made i n the attempt t o s o l v e the a i r -f o i l s p o i l e r problem. J a n d a l l (1) used an exact f r e e s t r e a m l i n e pot-e n t i a l f l o w t h e o r y t o s o l v e f o r l i f t and pre s s u r e on a s o l i d a i r f o i l w i t h a f i x e d normal s p o i l e r . Both J a n d a l i ' s 1- and 2 - s o u r c e models ar e c a l c u l a t e d s i m i l a r l y , and s i n c e they depend on conformal mappings of the Theodorsen type, they cannot be used t o s o l v e the s p o i l e r p l u s f l a p case. The 2 - s o u r c e model which g i v e s good agreement w i t h experiment r e q u i r e s both base p r e s s u r e and zero l i f t angle i n p u t . A l i n e a r i z e d f r e e s t r e a m l i n e p o t e n t i a l f l o w t h e o r y has been developed by Woods ( 2 ) , who gave e x p r e s s i o n s t o c a l c u l a t e i n c r e m e n t a l p r e s s u r e s , l i f t , drag and i n c r e m e n t a l p i t c h i n g moment as a f u n c t i o n of a i r f o i l i n c i d e n c e , s p o i l e r h e i g h t , angle t o the a i r f o i l s u r f a c e and chordwise p o s i t i o n . Woods c o n s i d e r s the a i r f o i l t h i c k n e s s o n l y In d e t e r m i n i n g the s p o i l e r angle, and drops i t from the t h e o r y as a sec -ond o r d e r term. Barnes ( 3 ) used the r e s u l t s of wind t u n n e l experiments on two a i r f o i l s t o d e v i s e an e m p i r i c a l m o d i f i c a t i o n t o Woods' 2 . f o r normal s p o i l e r s . Barnes used the boundary l a y e r t h i c k n e s s on the b a s i c a i r f o i l t o determine an e f f e c t i v e s p o i l e r h e i g h t t h a t gave him good agreement w i t h experiment f o r normal s p o i l e r s . From t h i s he was a b l e t o d e v e l o p an e m p i r i c a l r e l a t i o n s h i p t h a t he s t a t e s i s a p p l i c -a b l e t o most a i r f o i l s . The t h e o r i e s presented so f a r have been f r e e s t r e a m l i n e separ-a t e d p o t e n t i a l f l o w t h e o r i e s . The l i n e a r i z e d c a v i t y 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 i n p a r t I of the c u r r e n t work was developed by P a r k i n {k), who c o n s i d e r e d the h y d r o f o i l case of s e p a r a t i o n from the l e a d i n g edge. F a b u l a ( 5 ) extended t h i s work t o the g e n e r a l case of s e p a r a t i o n from a p o i n t on the upper s u r f a c e , t h i s case b e i n g a p p l i c -a b l e t o a h y d r o f o i l w i t h blowing or a s t e p on the upper s u r f a c e . The c u r r e n t t h e o r y , presented i n p a r t I, extends t h i s r e s u l t t o the a i r -f o i l problem, and r e l a t e s the s e p a r a t i o n t o a s p o i l e r h e i g h t and angle, The s p o i l e r p l u s f l a p s o l u t i o n i s a l s o o b t a i n e d . A new type of f u n c -t i o n xtfith a s i n g u l a r i t y a t the p o i n t c o r r e s p o n d i n g t o the l e a d i n g edge has been used t o s o l v e f o r the t h i c k n e s s s o l u t i o n . T h i s t h e o r y i s developed i n p a r t I and a p p l i e d t o a lk% t h i c k C l a r k Y a i r f o i l . The t h e o r y f o r the a i r f o i l s p o i l e r problem i s then extended t o s o l v e f o r the t r a n s i e n t l o ads on an a i r f o i l d u r i n g and a f t e r s p o i l e r a c t -u a t i o n . The l i n e a r i z e d t h e o r y of p a r t I i s mainly u s e f u l f o r the p r e -d i c t i o n of t o t a l f o r c e s and moments on the a i r f o i l . I n p a r t I I of the c u r r e n t work the steady s t a t e s p o i l e r problem i s once a g a i n t r e a t e d , t h i s time u s i n g an e n t i r e l y d i f f e r e n t t e c h -n i q u e . The exact s u r f a c e s i n g u l a r i t y p o t e n t i a l f l o w t h e o r y developed by Smith ( 6 ) i s combined w i t h the p r i n c i p l e s i n h e r e n t i n J a n d a l i ' s t h e o r y ( l ) . T h i s type of t h e o r y -is mainly u s e f u l f o r the p r e d i c t i o n of s u r f a c e p r e s s u r e d i s t r i b u t i o n . The s o l i d a i r f o i l w i t h a s p o i l e r , and the case of a s p o i l e r p l u s a s l o t t e d f l a p , are t r e a t e d . F o r the s o l i d a i r f o i l case the t h e o r y i s a p p l i e d t o a 14$ t h i c k C l a r k Y a i r f o i l . F o r the s p o i l e r p l u s s l o t t e d f l a p case the th e o r y i s a p p l i e d t o an NACA 23012 a i r f o i l w i t h a 25.66$ s l o t t e d f l a p . F o r the t h e o r i e s developed i n "both p a r t s of the c u r r e n t work the s p o i l e r h e i g h t , l o c a t i o n and angle are u n r e s t r i c t e d . The l i n e a r i z e d t h e o r y developed i n p a r t I i s bounded i n i t s agreement w i t h ex periment by the u s u a l r e s t r i c t i o n s f o r p e r t u r b a t i o n t h e o r i e s . PART I A LINEARIZED CAVITY POTENTIAL FLOW THEORY FOR THE STEADY STATE AND TRANSIENT AIRFOIL SPOILER PROBLEM 5 SECTION 2 THE ACCELERATION POTENTIAL C o n s i d e r an i r r o t a t i o n a l f l o w of an i n c o m p r e s s i b l e i n v i s c i d f l u i d i n which an a i r f o i l of chord c i s immersed. At p o i n t s f a r from the a i r f o i l the v e l o c i t y f i e l d c o n s i s t s of a c o n s t a n t f r e e s t -ream v e l o c i t y U i n the p o s i t i v e x d i r e c t i o n . C o n s i d e r the a i r f o i l t o be p o s i t i o n e d i n the x-y c o o r d i n a t e system such t h a t the l e a d i n g edge i s f i x e d a t the o r i g i n . Suppose t h a t t h e r e i s a f u l l y developed c l o s e d c a v i t y of l e n g t h Z, s p r i n g i n g from an a i r f o i l s p o i l e r on the upper s u r f a c e and from the t r a i l i n g edge. I f the u n d i s t u r b e d f r e e stream s t a t i c p r e s s u r e i s denoted by P^, and the c o n s t a n t p r e s s u r e i n s i d e the c a v i t y by Pe , a c a v i t a t i o n number f o r the f l o w , d e s i g n a t e d by K, i s d e f i n e d by the r e l a t i o n s h i p : K - 3sz3 where ^ i s the d e n s i t y of the f l o w . T h i s type of f l o w model, o r i g i n -a l l y developed f o r c a v i t y flows i n l i q u i d s , i s a l s o u s e f u l f o r se-p a r a t e d a i r f l o w s , where the base p r e s s u r e i s c o n s t a n t i n the wake j u s t behind the g e n e r a t i n g body, and the base p r e s s u r e c o e f f i c i e n t , as u s u a l l y d e f i n e d , i s I n the s p e c i a l case of steady f l o w the c a v i t y s u r f a c e v e l o c i t y mag-n i t u d e i s a constant q c . The s t e a d y - s t a t e B e r n o u l l i e q u a t i o n can be used to r e l a t e q & t o the c a v i t a t i o n number, w i t h the r e s u l t t h a t 6. T h i s q u a n t i t y o ^ i s sometimes used as the fundamental r e f e r e n c e speed i n p l a c e of the more commonly used f r e e stream v e l o c i t y U. When the a c c e l e r a t i o n p o t e n t i a l i s used t o s o l v e the t h i n a i r f o i l problem pro-posed, both systems reduce t o the same l i n e a r i z e d e q u a t i o n s . In the u s u a l l i n e a r i z e d a i r f o i l problem without c a v i t y , U i s the o n l y char-a c t e r i s t i c v e l o c i t y , and i t i s convenient t o r e t a i n i t here as the r e f e r e n c e v e l o c i t y . Thus a t any p o i n t (x,y) i n the nonsteady f l o w around the a i r f o i l the v e l o c i t y v e c t o r (T can be expressed i n terms of i t s x and y-components as , The d i m e n s i o n l e s s q u a n t i t i e s u ( x , y , t ) and v ( x , y , t ) are components of the s m a l l d i s t u r b a n c e v e l o c i t y i n the x and y d i r e c t i o n s r e s p e c t i v e l y . Both components u and v d i s a p p e a r at upstream i n f i n i t y . Correspond-i n g t o these v e l o c i t y p e r t u r b a t i o n s , a s m a l l d i s t u r b a n c e i n the f i e l d of s t a t i c p r e s s u r e P ( x , y , t ) can be d e f i n e d by p u t t i n g T h i s d i s t u r b a n c e p r e s s u r e a l s o d i s a p p e a r s at i n f i n i t y . To f i r s t o r d e r terms i n the s m a l l p e r t u r b a t i o n s , E u l e r ' s equat-i o n s of motion may be w r i t t e n as and (1) U dt 3x y • ?. The a c c e l e r a t i o n components a* and a y o f the a c c e l e r a t i o n v e c t o r "a(x,y,t) can be expressed as the g r a d i e n t of a s c a l a r : a"(x,y,t) - [«x,ay} - Vcp(x,y,t) . The s c a l a r q?(x,y,t) i s the; a c c e l e r a t i o n p o t e n t i a l and i t s r e l a t i o n -s h i p w i t h the p e r t u r b a t i o n pressure i s where <p has been d e f i n e d t o be z e r o on the c a v i t y . I n terms of the s t a t i c p r e s s u r e t h i s becomes In an i n c o m p r e s s i b l e f l u i d the equation of c o n t i n u i t y , w i l l a l s o h o l d . I f t h i s e quation i s combined w i t h equations ( 1 ) , the d i v e r g e n c e of a g r a d i e n t b e i n g z e r o g i v e s V * y = o a t every i n s t a n t . Hence <p i s a harmonic f u n c t i o n s , and a harmonic conjugate f can be d e f i n e d by means of the Cauchy-Rlemann equations as f o l l o w s : he. « m Q X £x ^y and ey <§x I n t r o d u c i n g the complex v a r i a b l e , the complex a c c e l e r a t i o n p o t e n t i a l which i s an a n a l y t i c f u n c t i o n of z a t every i n s t a n t can be w r i t t e n as The conjugate f u n c t i o n -^(x.y.t) i s the a c c e l e r a t i o n stream f u n c t i o n The a n a l y t i c i t y of F guarantees t h a t the complex a c c e l e r a t i o n , olz ' w i l l a l s o be an a n a l y t i c f u n c t i o n of z, The p r e s s u r e c o e f f i c i e n t takes i t s customary d e f i n i t i o n : i ? L l x • I n t r o d u c i n g the a c c e l e r a t i o n p o t e n t i a l t h i s becomes C p «-Xf-K (2) U s i n g the E u l e r and Cauchy-Riemann equations the x and y-com-ponents of p e r t u r b a t i o n v e l o c i t y can be expressed by the l i n e a r f i r s t o r d e r d i f f e r e n t i a l e q u a t i ons, + U dx u dx and a t u a x . F o r the s p e c i a l case of steady flow these equations reduce t o : u = u>+K (3) i and where the c o n d i t i o n s a t i n f i n i t y have been used t o determine the c o n s t a n t s of i n t e g r a t i o n . A p p l i c a t i o n of B e r n o u l l i ' s e q u a t i o n b e t -ween i n f i n i t y and a p o i n t on the a i r f o i l g i v e s the l i n e a r i z e d p r e s -sure c o e f f i c i e n t on the a i r f o i l as Cp » - l m . S u b s t i t u t i n g e q u a t i o n (3) Into t h i s r e s u l t shows c o n s i s t e n c y w i t h e q u a t i o n ( 2 ) 0 Equations (3) and ( M are both c o n s i s t e n t w i t h the r e s u l t o b t a i n e d u s i n g the c a v i t y v e l o c i t y magnitude q c as the fund-amental r e f e r e n c e v e l o c i t y . I t should be r e c o g n i z e d however, t h a t the v e l o c i t y v i n equation (k) has been n o n d i m e n s i o n a l l z e d by r e f -erence v e l o c i t y U r a t h e r t h a n q f e. 10. SECTION 3 STEADY THEORY 3.1 F o r m u l a t i o n of the Problem The f u l l y developed c l o s e d c a v i t y f l o w model proposed f o r the s o l u t i o n t o the a i r f o i l s p o i l e r problem has many f e a t u r e s i n common w i t h the u s u a l t h e o r y f o r t h i n a i r f o i l s . In f a c t , one can use the same techn i q u e s here t h a t have proven so u s e f u l i n s o l v i n g the t h i n a i r f o i l t h e o r y . In the present t h e o r y the c a v i t y - f o i l system must be regarded as b e i n g a t h i n body, and f o r t h i s reason the c a v i t y term-i n a t i o n i s accompanied by a s i n g u l a r i t y . In p r a c t i c e the f l o w down-stream of the a i r f o i l i s not a p o t e n t i a l f l o w but a h i g h l y d i s s i p a t -i v e f l o w , and the presence of the s i n g u l a r i t y i s an attempt t o approx-imate t h i s v e r y c o m p l i c a t e d wake f l o w i n the s i m p l e s t a n a l y t i c a l way. The important r e s u l t of such a procedure Is t h a t i n the neighbourhood of the a i r f o i l , the flow appears t o be w e l l r e p r e s e n t e d . 3.2 T r a n s f o r m a t i o n s The a i r f o i l of chord c Is l o c a t e d i n the z-plane w i t h the l e a d -i n g edge p o s i t i o n e d a t the o r i g i n . The u n d i s t u r b e d f l o w i s In the pos-i t i v e x d i r e c t i o n and the a i r f o i l i s i n c l i n e d a t a. s m a l l angle oL t o t h i s f l o w . The a i r f o i l s p o i l e r of h e i g h t h i s p o s i t i o n e d a t x =• s . The s p o i l e r angle t o the chord l i n e i s denoted by S. The a i r f o i l con-f i g u r a t i o n i s shown i n f i g u r e ( 1 ) . The a i r f o i l can a l s o have a f l a p of chord c u a t an angle t o the chord l i n e of the a i r f o i l , not shown i n f i g u r e ( 1 ) . The p h y s i c a l r e p r e s e n t a t i o n of the a i r f o i l i n the z-plane i s shown i n f i g u r e ( 2 ) . In order t o apply the methods of t h i n F - X G U R E ( 1 ) ; A I R F O I L I N Z - P L A N E Z-(Jane S S*W C I • z* - planet -1 V- plane t a b §-plane e i o« F I G U R E ( 2 ) : C O M P L E X T R A N S F O R M P L A N E S a i r f o i l t h e o r y i t i s h e l p f u l to t r a n s f o r m the z-plane I n t o a more u s e f u l p l a n e . F i r s t c o n s i d e r the conformal t r a n s f o r m a t i o n which maps the z-plane i n t o the z'-plane. In the z'-plane the a i r f o i l chord, becomes equal t o g^: and the c a v i t y t e r m i n a t i o n p o i n t has been mapped t o z' = oo. The p o i n t a t i n f i n i t y has been mapped t o the p o i n t z ' = - l . The conformal t r a n s f o r m a t i o n v » «JF , a - J U T ' Cm maps the e n t i r e z 1 - p l a n e i n t o the upper h a l f of the i ) - p l a n e . The f o i l o c c u p i e s the s l i t - l ^ V ^ b on the r e a l a x i s of the V - p l a n e . The p o i n t a t i n f i n i t y has been mapped t o + i a . The c a v i t y extends a l o n g V > b and V<-1 on the r e a l a x i s . The v a l u e of the con s t a n t b i s The f o i l i s then mapped from the V-plane on t o the upper h a l f of the u n i t c i r c l e i n the S-plane by means of the Joukowski t r a n s f o r m a t i o n , The l e a d i n g edge of the a i r f o i l corresponds t o the p o i n t %=.e°f the s p o i l e r base t o the p o i n t Sae 4*^, the f l a p hinge p o i n t t o the p o i n t ? e e ° S the t r a i l i n g edge t o the p o i n t S = - l and the s p o i l e r t i p to the p o i n t § = 1 . The c a v i t y extends a l o n g the r e a l a x i s S>1 and § < - l . 14. U s i n g the mapping f u n c t i o n s the a n g u l a r l o c a t i o n s of the c r i t i c a l p o i n t s i n the g-plane can be determined as e 0 - c o s " ' ( ^ ) t 0, » cos"'{r?b[aj^ + 4r]} and where the i n v e r s e c o s i n e s are taken between 0 and i r . The p o i n t a t i n f i n i t y i n the S-plane i s the f i e l d p o i n t S i g i v e n by The complex t r a n s f o r m planes are shown i n f i g u r e ( 2 ) . The complex a c c e l e r a t i o n p o t e n t i a l s i n the v a r i o u s t r a n s f o r m planes are i n v a r i a n t a t c o r r e s p o n d i n g p o i n t s and so the a c c e l e r a t i o n s d i f f e r o n l y by the d e r i v a t i v e of the mapping f u n c t i o n s , and thus 3 . 3 Boundary C o n d i t i o n s . The steady s t a t e boundary c o n d i t i o n s on the a i r f o i l are as f o l -lows: ( i ) <p«R I .F»o on the c a v i t y , £ » x > ( s + h ) , y=0*and £ > x > c , y=0" . 15, ( i i ) K u t t a c o n d i t i o n s , tp continuous, at the s p o i l e r t i p x=(s+h), y=0*and a t the t r a i l i n g edge of the a i r f o i l x=c, y=0" . ( i i i ) Normal boundary c o n d i t i o n of no f l o w through the a i r -f o i l s u r f a c e . Hence i f the a i r f o i l s u r f a c e i s denoted t>y C x,y ( x ) ] , the c o n d i t i o n becomes, u s i n g equation ( 4 ) ( i v ) The c o n d i t i o n F=-£ f o r the p o i n t a t i n f i n i t y , (v) The body c a v i t y system must be e q u i v a l e n t t o a c l o s e d body. Hence f o r a c l o s e d wake Tm.<£ w (z) d z = 0. I n the above boundary c o n d i t i o n s y=0 r e f e r s t o the upper s u r f a c e of the s l i t and y=Cf r e f e r s t o the lower s u r f a c e of the s l i t . In cond-i t i o n ( v ) , w(z) i s the complex v e l o c i t y and equations ( 3 ) and ( 4 ) combine t o g i v e Boundary c o n d i t i o n (v) can then be r e w r i t t e n Irrx£-R*>o»z = 0 . 3 . 4 Mathematical Flow Model F o l l o w i n g the methods of t h i n a i r f o i l t h e o r y we determine a s e t of f u n c t i o n s i n the g - p l a n e t h a t s a t i s f y the boundary c o n d i t i o n s 16. (I) through ( v ) . The problem can be s p l i t i n t o the d e t e r m i n a t i o n of i n c i d e n c e , camber, t h i c k n e s s , s p o i l e r and f l a p s o l u t i o n s and then superposed as i n standard l i n e a r i z e d t h e o r i e s . The next s e c t i o n s are then concerned w i t h d e t e r m i n i n g mathematical f u n c t i o n s w i t h the de-s i r e d c h a r a c t e r i s t i c s f o r these i n d i v i d u a l c a s e s . 3.4 .1 I n c i d e n c e Case C o n s i d e r the complex f u n c t i o n F, i s p u r e l y imaginary on the c a v i t y where § i s r e a l and iSl > I . T h i s s a t i s f i e s c o n d i t i o n ( i ) . On the u n i t c i r c l e S-e*"0, Ft(e**e) • CoCcose^cos© . At the s p o i l e r t i p and t r a i l i n g edge e-*>o,ir r e s p e c t i v e l y and F, i s c l e a r l y c ontinuous, hence s a t i s f y i n g c o n d i t i o n ( i i ) . A c o n s t a n t term does not v i o l a t e c o n d i t i o n s ( i ) or ( i i ) and i s an a c c e p t a b l e f u n c t i o n . I t was p r e v i o u s l y d e c i d e d t h a t t'he c a v i t y t e r m i n a t i o n must be a s i n g -u l a r p o i n t t o account f o r the b r a n c h i n g of the f r e e s t r e a m l i n e s a t t h i s p o i n t . In the ^ - p l a n e t h i s p o i n t Is l o c a t e d a t i n f i n i t y so t h a t the p o l e t h e r e i s simply g i v e n by <.§. For the u n i t c i r c l e t o remain a s t r e a m l i n e a simple pole of o p p o s i t e s i g n must be added t o the i n -v e r s e p o i n t of the u n i t c i r c l e . A complex f u n c t i o n term g i v i n g the net c o n t r i b u t i o n of the s i n g u l a r i t y at the c a v i t y t e r m i n a t i o n can 17. then be w r i t t e n as F a i s p u r e l y imaginary on the c a v i t y where § i s r e a l and hence s a t -i s f i e s c o n d i t i o n ( l ) . On the u n i t c i r c l e S»e f c S, F/e 1®) = - z B o S m ® . The f u n c t i o n c l e a r l y s a t i s f i e s the K u t t a c o n d i t i o n . I t i s t o be r e -membered t h a t boundary c o n d i t i o n s ( i i i ) through (v) are y e t to be s a t i s f i e d . The f u n c t i o n s i n c l u d e d i n the i n c i d e n c e case are then, PinC^ - i C . C f e ^ l + S e k ' i ] «• *H * ^BM"k). (6) On the j f o i l the a c c e l e r a t i o n p o t e n t i a l becomes f - ^ S^SS . (?) 3.4.2 Camber Case C o n s i d e r the complex f u n c t i o n e w - - ; . 2 & ( 8 ) c 1 ^ > where the M„ are r e a l c o n s t a n t s . T h i s f u n c t i o n i s p u r e l y imaginary on the c a v i t y where S i s r e a l and so s a t i s f i e s c o n d i t i o n ( i ) . On the u n i t c i r c l e 1 8 . oo — 2L M„ Sivn n o - -L 2. M n c o s n o Once a g a i n the f u n c t i o n c l e a r l y s a t i s f i e s the K u t t a c o n d i t i o n s . Bound ar y c o n d i t i o n ( i i i ) can now be used t o s o l v e f o r the unknown M n. F o r the camber case f*y? as a f u n c t i o n of © i s a continuous curve t h a t oix can be r e p r e s e n t e d by i t s known F o u r i e r c o s i n e s e r i e s as f o l l o w s : ¥° + l M * c o s * e , (9) olx z • where f i r . M ^ - ~ J ^ ' c o s n e d © . o The stream f u n c t i o n c o n d i t i o n then becomes » _ M°-% M „ c o s n e . I f the term -£!• i s Included i n the con s t a n t terms of the i n c i d e n c e case i t i s apparent t h a t the camber f u n c t i o n of eq u a t i o n (8) s a t -i s f i e s c o n d i t i o n s ( i ) , ( i i ) and ( i i i ) . Next i t i s t o be demonstrated t h a t the constant terms of the stream f u n c t i o n a l s o s a t i s f y c o n d i t i o n ( i i i ) . In the i n c i d e n c e case of a f l a t p l a t e a t angle of a t t a c k oc. dx ' and c o n d i t i o n ( i i i ) g i v e s 19. Hence the c o n s t a n t p a r t s of the stream f u n c t i o n from the i n c i d e n c e case and from the camber case are as f o l l o w s : or 0o = * - M « - C 0 . (10) Hence the camber f u n c t i o n i s and on the f o i l the a c c e l e r a t i o n p o t e n t i a l becomes <pc - - X M„sinne . (11) 3.4.3 T h i c k n e s s Case F o r the t h i c k n e s s s o l u t i o n c o n s i d e r the complex f u n c t i o n where the N n are r e a l c o n s t a n t s . On the c a v i t y where § i s r e a l and |S|>1, t h i s f u n c t i o n i s p u r e l y imaginary and t h e r e f o r e s a t i s f i e s c o n d i t i o n ( i ) . On the u n i t c i r c l e 2 0 . I t can be seen t h a t F + a l s o c l e a r l y s a t i s f i e s c o n d i t i o n ( i i ) . The unknown N„ can be s o l v e d f o r i d e n t i c a l l y as In the camber s o l u t i o n , through boundary c o n d i t i o n ( i i i ) . In the t h i c k n e s s s o l u t i o n <tl* as dx a f u n c t i o n of o i s a curved f u n c t i o n w i t h a d i s c o n t i n u i t y a t the p o i n t c o r r e s p o n d i n g t o the l e a d i n g edge of the a i r f o i l . F o r the t h i c k -ness f u n c t i o n assumed t o be a p p l i c a b l e t o the s o l u t i o n of t h i s pro-blem the f o l l o w i n g r e l a t i o n must be t r u e : dx ~ x(cose0-cose") I t i s apparent t h a t t h i s r e l a t i o n reduces t o the s o l u t i o n of a Fou-r i e r s e r i e s as i n the camber s o l u t i o n . The unknown N H are then g i v e n by N H = $1 j**(cos®.-cos©") cos n© de , n » i o and N„ = %\ S}2\cose0-cose)cle . The t h i c k n e s s complex f u n c t i o n i s then and on the f o i l the a c c e l e r a t i o n p o t e n t i a l i s a(cose0-cose) . I t was found n e c e s s a r y t o use a t h i c k n e s s f u n c t i o n t h a t has a s i n g -u l a r i t y a t the p o i n t c o r r e s p o n d i n g t o the l e a d i n g edge. An attempt t o use a f u n c t i o n i d e n t i c a l t o the camber case was not s u c c e s s f u l . I t was found t h a t the F o u r i e r c o e f f i c i e n t s would not converge. 3 . 4 . 4 S p o i l e r Case The base of the s p o i l e r maps i n t o the p o i n t §»e ,' e\ i n the S-pl a n e . At p o i n t s on the c i r c l e when S passes through e4"^, the r e i s a s t e p change i n v, and t h e r e f o r e through equation ( 4 ) , a s t e p change i n •y. The l o g a r i t h m i c f u n c t i o n i s an a n a l y t i c f u n c t i o n which w i l l p r o v i d e such a jump. I t i s a l s o r e q u i r e d t h a t the imaginary p a r t of the f u n c t i o n be cons t a n t over a p p r o p r i a t e p o r t i o n s of the c i r c l e . C o n s i d e r the f u n c t i o n The imaginary p a r t of t h i s f u n c t i o n i n the r e q u i r e d range of - i r > e » o i s I f t h i s f u n c t i o n i s combined w i t h one of the same type as F from the i n c i d e n c e case, the r e s u l t i n g f u n c t i o n i s am (14) On the c i r c l e where § • e , t h i s becomes 2 2 . stn le-e.| -^j (• O -Tor T T > © > - © , i s m f f f o r S \ > © > 0 . Boundary c o n d i t i o n ( i i i ) then becomes o o n +he f o i l ( i r r\ S o n t n e s p o i l e r ( e ,> -e >• o") The s p o i l e r angle £ has not been r e s t r i c t e d t o a s m a l l angle and i t i s r e a l i s e d t h a t t h i s contravenes one of the b a s i c assumptions of l i n e a r i z e d t h e o r y . In most p r a c t i c a l c o n f i g u r a t i o n s the s p o i l e r h e i -ght i s o n l y a s m a l l percentage of the chord. I t i s t h e r e f o r e o p t i -m i s t i c a l l y assumed t h a t the l i n e a r i z e d f l o w has not been too severe-l y d i s t u r b e d . Both Woods ( 2 ) and Barnes ( 3 ) have had some success w i t h normal s p o i l e r s u s i n g l i n e a r i z e d t h e o r i e s . I t now remains t o demonstrate t h a t the s p o i l e r f u n c t i o n F $ s a t i s f i e s c o n d i t i o n ( i ) and ( i i ) . The f i r s t p a r t of t h i s f u n c t i o n , h a v ing been drawn from the i n c i d e n c e case, has a l r e a d y been shown t o s a t i s f y the c o n d i t i o n s . On the c a v i t y i n the S-plane where § i s r e a l and |§I>1 , the l o g a r i t h m term i s p u r e l y imaginary and hence s a t i s f i e s c o n d i t i o n ( i ) . In the e q u a t i o n f o r F $ ( e < e ) , i t can be seen t h a t <p-*>o as © - • O j t r and hence the f u n c t i o n s a t i s f i e s the K u t t a c o n d i -t i o n s . The s p o i l e r f u n c t i o n i s then and the a c c e l e r a t i o n p o t e n t i a l on the f o i l i s 23. TT Lcoso0-cose Umo+o. j j 3 . 4 . 5 F l a p Case The f l a p hinge p o i n t i n the S-plane corresponds to the p o i n t $ » e * d * I t can e a s i l y be r e c o g n i z e d t h a t t h i s case i s i d e n t i c a l t o the s p o i l e r case, except t h a t the f l a p i s r e s t r i c t e d t o s m a l l a n g l e s . The v e l o c i t y c o n d i t i o n s ( i i i ) on the s u r f a c e f o r t h i s case are r o on +hc fo«l ( e x >e>o ) V a i I on +Vie -flap (tr> e>0^) , The complex f u n c t i o n f o r the f l a p case i s then and on the f o i l the a c c e l e r a t i o n p o t e n t i a l i s „ »r tociJi"- + i - t - ' f f l ] ] ( 1 7 , V U0S© 0 -C0S0 Uln®±©«.jJ . I t has been shown i n the s p o i l e r case t h a t these f u n c t i o n s making up F^ s a t i s f y the boundary c o n d i t i o n s ( i ) and ( i i ) . 3.5 Method of S o l u t i o n The f u n c t i o n s developed i n the p r e v i o u s paragraphs t o s o l v e the i n d i v i d u a l cases of i n c i d e n c e , camber, t h i c k n e s s , s p o i l e r and f l a p s o l u t i o n s were shown to s a t i s f y boundary c o n d i t i o n s ( i ) through ( i i i ) . I t remains then t h a t boundary c o n d i t i o n s ( i v ) and (v) be 24. s a t i s f i e d . C o n d i t i o n ( i v ) , the c o n d i t i o n a t i n f i n i t y can be express-ed as 'F-CSi")- FctSi>* 5<$> « - J < (18) where § i t the p o i n t a t i n f i n i t y i n the S-plane, i s g i v e n by equat-i o n (5)» The l a s t f o u r terms on the l e f t hand s i d e of equation (18) are known f u n c t i o n s ; the unknown c o n s t a n t s are c o n t a i n e d i n F i n ( § ^ ) . The e q u a t i o n can then be w r i t t e n as (19) where and u s i n g equations (10) and (6), . F l n( g^) can be w r i t t e n as In t h i s e q u a t i o n put and E q u a t i o n (19) can then be expressed as 2 5 . C e x , - - E - i . The r e a l and imaginary p a r t s of t h i s e quation f u r n i s h two equations i n the two unknowns C 0 and B 0 . The v a l u e s of these c o n s t a n t s are B . - R u r J m . E - f r - % 0 - I m . X , g l E f - ^ J ^ . A , (20) R U . I m X t - Iw^. x, R|. X v and C . - R I . E - B 0 R l . X i - 4 : (21) RI.X% A l l the f u n c t i o n c o n s t a n t s have been determined. The remaining unknowns a r e the c a v i t y number K and the c a v i t y l e n g t h 2. There i s not c u r r e n t l y a the o r y t h a t w i l l c o r r e c t l y p r e d i c t the base pr e s s u r e , and a t l e a s t t h i s parameter w i l l be an e m p i r i c a l i n p u t . I t turns out however, t h a t t h i s i s the onl y e m p i r i c a l Input needed s i n c e the cav-i t y l e n g t h Z can be r e l a t e d t o K through boundary c o n d i t i o n ( v ) . C o n d i t i o n (v) p h y s i c a l l y means t h a t , I f the body c a v i t y system i s t o be a c l o s e d body, then the sum of the sources i n s i d e a contour i n c l u d i n g the b o d y - c a v i t y system must be z e r o . M a t h e m a t i c a l l y i t was found t h a t c o n d i t i o n (v) c o u l d be expressed as Im.j* Rz)dz » 0 . From the mapping f u n c t i o n s , p o i n t s i n the f»-plane are r e l a t e d t o the z-plane by (22) 26. S i n c e any contour of i n t e g r a t i o n i s s u i t a b l e , one can be chosen such t h a t |z|^£ . The i n t e g r a l can then be s o l v e d u s i n g a Laurent expan-s i o n . F o r lzl^£ equation (22) becomes § . a 0 + + , where 0 s 4 + -=i5- «. and The c o e f f i c i e n t s of -k i n the f o l l o w i n g terms are given: 0 r i^n _§ - _ e -1 *a ,L a , "aoe* 0 ^! a,e-*9»,i J S Although the s o l u t i o n appears s t r a i g h t forward, such Is not the case and an i t e r a t i v e s o l u t i o n Is n e c e s s a r y . The angles i n the §-plane, 0 O , © , and © x , the p o i n t s c o r r e s p o n d i n g t o the l e a d i n g edge, the spo-i l e r base and the f l a p hinge p o i n t on the u n i t c i r c l e r e s p e c t i v e l y , are complex f u n c t i o n s of Z, The c o e f f i c i e n t of 4% can be w r i t t e n as • H c ^ ) - ( S T ^ i l * t Hat*"*} -gpfeaep From e q u a t i o n ( 2 0 ) and ( 2 1 ) B 0 and G„ can be w r i t t e n as B 0 = * l X , [ I ^ . E - ( * - ^ ] - I W . ) J , W . E + K. J ^ X ,  and C o S B R l X f c C l m . E - («- x*)3 " Xrw.A„RLE + K X ~ v X , . . RlX^Im.X.-Xm.X^Rl.X, * RIX iTm.X,-lm.X%Rl.X, Now Xrv».£ F(*)dz a 2TT Rl. £cOcffiCien+ o f } } and hence the c l o s u r e c o n d i t i o n becomes RI.£coeff«cten+ o f ^.^ * 0 • P u t t i n g the v a l u e s of B 0 and C e i n the above e x p r e s s i o n f o r the coe-f f i c i e n t of g i v e s 28. + L^ca.-e**) " " C o . - e - ^ ) l " " ( o c - e - ^ l 4- * «i.x,.xm.x,-irr».xtRs.x, Up.e^O ( a o e-* e2.0 v - i RU.ImXi-Xm.X.RI.X* a » V J . * ^ Hence w i t h K g i v e n from experiment the c o r r e c t v a l u e of the cavity-l e n g t h 6, can be determined by p l o t t i n g a graph of K v»s £, or by i t e r a t i v e l y changing the v a l u e of £ i n equation ( 2 3 ) . The r e q u i r e d r e a l p a r t s of t h i s e q u a t i o n c o u l d be expressed by a l g e b r a i c a l l y s p l -i t t i n g each term i n t o i t s r e a l and Imaginary p a r t s . However i t i s d e s i r a b l e t o r e t a i n the c o n c i s e form of equation ( 2 3 ) . T h i s completes the problem f o r m u l a t i o n f o r the steady t h e o r y . I t remains t o determine the pr e s s u r e and l i f t c o e f f i c i e n t s . Using e q u a t i o n (2) and c o l l e c t i n g the v a l u e s of a c c e l e r a t i o n p o t e n t i a l from equations ( 7 ) , (11) , ( 1 3 ) . (15) and (17) , the p r e s s u r e coe-f f i c i e n t , as a f u n c t i o n of angular p o s i t i o n on the u n i t c i r c l e i n the <?-plane, can be w r i t t e n as 2 9 . where * - K . * * ^ * ^ ] d g f e « « - 8 „ - o ( 2 4 ) ^ L s in J T 1S1„ t ^ v j -2 M * sin *e - fXs.nne C o n s i d e r a t i o n of the t r a n s f o r m a t i o n s l e a d s to the r e s u l t t h a t on the a i r f o i l a n g u l a r p o i n t s on the u n i t c i r c l e i n the S-plane, and p o i n t s on the a i r f o i l i n the z-plane are r e l a t e d by x . i L ^ - ' - ^ r (25) I t f o l l o w s immediately t h a t equations ( 2 ) , (24) and (25) are used, t o r e l a t e the p r e s s u r e c o e f f i c i e n t t o p o i n t s on the a i r f o i l . The l i f t c o e f f i c i e n t can be determined by the use of the B l a s -l u s e q u a t i o n , D-iL - i|U^w*Cx"^dz , (26) where the contour e n c l o s e s the body and c a v i t y system. To the f i r s t o r d e r i n the d i m e n s i o n l e s s p e r t u r b a t i o n v e l o c i t i e s w and e q u a t i o n (26) can be w r i t t e n as D-i-L = ( u - W ) d z . (2?) 1 U s i n g e q u a t i o n s (3) and (4) and n o n d i m e n s i o n a l i z i n g the l i f t and drag, e q u a t i o n (27) can be expressed as 3 0 . C p - ^ C U F ( x )dz . ( 2 8 ) F ( z ) can be expanded i n a Laurent s e r i e s f o r a contour x such t h a t |zl*>£. E q u a t i o n (28) then a l l o w s the l i f t c o e f f i c i e n t t o be w r i t t e n as C t = ^Im.[coeffic»e«+ of -k]f (29) and the t o t a l drag c o e f f i c i e n t on the'body c a v i t y system as CP»-3r Rl. [ c o e f f i c i e n t - o f £ } . (30) The Rl£coeff (fc)} was shown t o be z e r o i n the c o n s i d e r a t i o n of the c l o s u r e c o n d i t i o n and the drag c o e f f i c i e n t i s z e r o as expected i n such a p o t e n t i a l t h e o r y . The drag on the a i r f o i l can s t i l l be worked out t h e o r e t i c a l l y . The drag on the a i r f o i l i s balanced e q u a l l y by the f o r c e on the s i n g u l a r i t y a t the c a v i t y t e r m i n a t i o n p o i n t . In d e t e r m i n i n g the drag on the a i r f o i l t h i s c a v i t y t e r m i n a t i o n p o i n t o n l y need be c o n s i d e r e d . The drag p r e d i c t e d i s u n r e a l i s t i c a l l y h i g h due t o the t h e o r y b e i n g unable t o model the s e p a r a t i o n bubble i n -f r o n t of the s p o i l e r e v i d e n t i n the r e a l flow. The drag theory t h e r e -f o r e w i l l not be pursued f u r t h e r . U s i n g the r e s u l t s j u s t determined f o r the c a v i t y c l o s u r e con-d i t i o n the l i f t c o e f f i c i e n t can be w r i t t e n as CL , 4$ I. .[«. B>.0 + <y - i tc. + 2 W S & S $ 3 1 . The B l a s i u s e q u a t i o n f o r the p i t c h i n g moment cou l d e q u a l l y be a p p l i e d t o determine the p i t c h i n g moment c o e f f i c i e n t of the a i r f o i l . 32. SECTION 4 NONSTEADY THEORY 4 . 1 F o r m u l a t i o n of the Problem The nonsteady problem t o be c o n s i d e r e d i s the case of s p o i l e r a c t u a t i o n on a f i x e d a i r f o i l i n i n i t i a l l y steady f l o w . Nonsteady a i r -f o i l motions w i t h f i x e d s p o i l e r angles have not been c o n s i d e r e d . Such cases are a much s i m p l e r a p p l i c a t i o n of t h i s t h e o r y and they have many f e a t u r e s i n common w i t h e x i s t i n g r e s u l t s as g i v e n by Par-k i n ( 4 ) . The f o l l o w i n g theory i s developed f o r the case of zer o c a v i t a -t i o n number. The reasons behind t h i s m e r i t c a r e f u l c o n s i d e r a t i o n . F i r s t c o n s i d e r an a i r f o i l w ithout a s p o i l e r i n steady f l o w . I f a s p o i l e r i s a c t u a t e d on the upper s u r f a c e , the f l o w i s going t o be d i s t u r b e d . Even a d e t a i l e d e x p e r i m e n t a l t a b u l a t i o n of how the c a v i t y number v a r i e s w i t h the v a r i a b l e s of time and s p o i l e r angle w i l l not enable a s o l u t i o n t o be fo r m u l a t e d . I t must be remembered t h a t the c a v i t y l e n g t h i s r e l a t e d t o the c a v i t y number due t o the f a c t t h a t t h e r e can be no drag on the b o d y - c a v i t y system. I t w i l l be r e c a l l e d t h a t t h i s was expressed by boundary c o n d i t i o n (v) of the steady s t a t e . Hence, knowing the c a v i t y number as a f u n c t i o n of time, the c a v i t y l e n g t h can be c a l c u l a t e d . S i n c e the t r a n s f o r m a t i o n s of s e c t i o n ( 3 . 2 ) depend upon c a v i t y l e n g t h , the mapping f u n c t i o n w i l l change as cav-i t y number changes w i t h time. M o d i f i c a t i o n to the c u r r e n t theory would be n e c e s s a r y t o e f f e c t a s o l u t i o n i n such a s i t u a t i o n . The onl y s o l u t i o n s would appear t o be, to assume that as soon as the s p o i l e r s t a r t s t o move the c a v i t y number assumes i t s f i n a l steady s t a t e v a l u e , o r assume t h a t the c a v i t y number i s at a l l times z e r o . The f i r s t s o l u t i o n has obvious l i m i t a t i o n s and the second s o l u t i o n a l -33. though not p h y s i c a l l y a t t a i n a b l e does have m e r i t . The average p r e s -sure on the r e a r p a r t of the upper s u r f a c e of an a i r f o i l i n most c o n f i g u r a t i o n s i s v e r y c l o s e t o z e r o . Hence d u r i n g the i n i t i a l p a r t of the s p o i l e r a c t u a t i o n the c a v i t a t i o n number i s c l o s e t o z e r o . The z e r o c a v i t a t i o n number s o l u t i o n i s c o m p a r a t i v e l y s i m p l e r mathemati-c a l l y and i t s complete l i n e a r i t y permits easy comparisons to e x i s t -i n g nonsteady t h i n a i r f o i l t h eory problems such as change of angle of a t t a c k . At z e r o c a v i t a t i o n number the f l o w resembles a Helmholtz flow -w i t h the c a v i t y behind the body extending to i n f i n i t y . In t h i s case the c a v i t y p r essure i s equal to the u n d i s t u r b e d f r e e scream s t a t i c p r e s s u r e . The f o i l Is p o s i t i o n e d as d e s c r i b e d i n s e c t i o n ( 3.2). The a i r f o i l c o n f i g u r a t i o n i s shown i n f i g u r e ( 3 ) . In s o l v i n g t h i s problem once a g a i n l i n e a r a i r f o i l techniques are employed. The f i r s t problem f o r which a s o l u t i o n i s r e q u i r e d i s the u n i t s t e p s p o i l e r a c t u a t i o n . To achieve t h i s s o l u t i o n a case t h a t i s not p h y s i c a l l y a t t a i n a b l e must be c o n s i d e r e d . The s t e p bound-a r y c o n d i t i o n on the s p o i l e r i s a s t e p i n the y-component of v e l o c i t y g i v e n by v=sin£ over t h a t p o r t i o n of the x - a x i s t h a t corresponds t o the s p o i l e r . Such a s t e p change i n v e l o c i t y can be achieved by S O l -Vujt v i n g f o r the case of v»v„e over t h i s r e g i o n and then i n t e g r a t i n g over a l l f r e q u e n c i e s . Looking more c l o s e l y a t t h i s problem of a s i n -u s o i d a l v e l o c i t y however i t can be seen t h a t i t i s not p h y s i c a l l y p o s s i b l e . Over the n e g a t i v e p o r t i o n of the v e l o c i t y c y c l e when phy-s i c a l l y the p o r t i o n of the x - a x i s c o r r e s p o n d i n g t o the s p o i l e r has a s u c t i o n on i t , the a i r f o i l could not p o s s i b l y support a c a v i t y . There are however, no mathematical l i m i t a t i o n s and the c a v i t y i s j u s t c o n s i d e r e d to e x i s t . T h i s p o i n t i s e a s i e r to understand i f the s i n u s o i d a l v e l o c i t y i s considered to be a d i s t u r b a n c e on the e x i s t -i n g steady s t a t e s o l u t i o n f o r some s p o i l e r a n g l e . In such a case the r-plane F I G U R E , (3 ) : ' , , A I R F O I L ' I N T H E Z ' - P L A N E 35. c a v i t y does a l r e a d y e x i s t . T h i s case of a s i n u s o i d a l v e l o c i t y over the s p o i l e r r e g i o n h e n c e f o r t h w i l l be r e f e r r e d t o as blowing t h e o r y . 4.2 Blowing Theory In c o n s i d e r i n g t h i s problem o n l y the f l a t p l a t e s o l u t i o n of zero i n c i d e n c e i s c o n s i d e r e d . T h i s s e c t i o n i s the e q u i v a l e n t of the s p o i l e r case i n the steady s o l u t i o n . The remaining e x i s t i n g steady s t a t e s o l u t i o n s of i n c i d e n c e , camber, t h i c k n e s s , and f l a p f o r K=0 are f u l l y a d d i t i v e t o t h i s nonsteady s o l u t i o n . 4.2.1 Boundary C o n d i t i o n The boundary c o n d i t i o n s f o r t h i s nonsteady blowing problem are as f o l l o w s : (i ) < f=0 on the c a v i t y , x»(s+h), y=0* and x»c, y=CT. (11) K u t t a c o n d i t i o n s , ( f continuous, a t the s p o i l e r t i p x=(s+h), y=0* and a t the t r a i l i n g edge of the a i r f o i l x=c, y=0~. ( i i i ) TO on the f o i l 0*x*ss, y-0* and 0«x«c, y=0~. ^v ee* u , t on the s p o i l e r s*x«(s+h), y=0"*". ( i v ) F=0 f o r the p o i n t a t i n f i n i t y , z = - © o . (v) r=0 on the f o i l 0«x«ss, y=0"*" and 0«*x«c, y=0" . • I T - \ L=-£/u/oe0 on the s p o i l e r s*x«(s+h), y=0 . In the above boundary c o n d i t i o n s 4.2.2 T r a n s f o r m a t i o n s The p h y s i c a l plane f o r t h i s case i s the z-plane of f i g u r e (2) w i t h £ = o o 0 The t r a n s f o r m a t i o n s are obtained from the steady s t a t e t r a n s f o r m a t i o n s g i v e n i n s e c t i o n (3*2), w i t h the z'-plane omitted. The t r a n s f o r m a t i o n s are 36. V s» a'J~z , a' = ^jp and These t r a n s f o r m a t i o n s are shown In f i g u r e ( 2 ) . P o i n t s on the a i r f o i l i n the z-plane are r e l a t e d t o the c o r r e s p o n d i n g p o i n t s on the §-plane "by the r e l a t i o n : £ = ( - i r ) £cos©-co5©j*^ (32) 4 . 2 . 3 Method of S o l u t i o n U s i n g boundary c o n d i t i o n (v) and the d e f i n i t i o n of the a c c e l -e r a t i o n stream f u n c t i o n from s e c t i o n ( 1 ) , the stream f u n c t i o n can be w r i t t e n as f = o on + n e -foil dx i =»-^ /AV^ e^ rU,t on Wie s p o i l e r . I n t e g r a t i o n of t h i s e q u a t i o n g i v e s f = C » e i U > o n H , e f o i l I » ""J> M ' v «» e x + ^ ' e o w +«e. s p o i l e r , where the 'constants* of i n t e g r a t i o n have been assumed to be harmon-i z e f u n c t i o n of t . S u b s t i t u t i o n of x from e q u a t i o n (32) i n t o these r e l a t i o n s enables 'f to be expressed as 37. r « C » c 4 U , i o n + U e - f o » l (33) L » - j / A V e e < ' L A T T V i c o s i o ; c o s d ] + C , e < o « 4We s p o i l * where C, =Cr<j/n&( )*"(|+cos *"e a). The s o l u t i o n t o the problem c l o s e l y f o l l o w s the techniques of the steady s t a t e problem. Complex f u n c t -i o n s must be determined t o s a t i s f y the boundary c o n d i t i o n s developed. The c o n s t a n t terms of e q u a t i o n (33) can be s a t i s f i e d by f u n c t i o n s g i v e n i n the s p o i l e r case s o l u t i o n as f o l l o w s : (34) The p a r t of e q u a t i o n (33) t h a t i s a f u n c t i o n of e i s s i m i l a r t o a camber type problem and once again a F o u r i e r s e r i e s complex f u n c t i o n i s r e q u i r e d . Suppose the f u n c t i o n of e i n equations (33) i s express-* ed as A F o u r i e r c o s i n e s e r i e s can be determined as f o l l o w s : oo X X,cosne =af(e) , i t where X * % f ( o ) C O S n O d © , 3 8 . S i n c e t h i s F o u r i e r c o s i n e s e r i e s has t o be the imaginary p a r t of any f u n c t i o n determined, i t f o l l o w s t h a t the r e q u i r e d complex f u n c t i o n can be w r i t t e n as Hence combining e x p r e s s i o n (3*+) w i t h t h i s f u n c t i o n a l l o w s the t o t a l nonsteady complex f u n c t i o n to be w r i t t e n as I t remains t o demonstrate t h a t these f u n c t i o n s s a t i s f y a l l the bound-a r y c o n d i t i o n s . C o n d i t i o n s ( i ) and ( i i ) are i d e n t i c a l c o n d i t i o n s t o the steady s t a t e s o l u t i o n and s i n c e the f u n c t i o n s are of the steady s t a t e type these c o n d i t i o n s are f u l l y s a t i s f i e d . Through the mapping f u n c t i o n s i t can be shown t h a t ISi approaches i n f i n i t y as 1*1 approach-es i n f i n i t y . Hence the complex f u n c t i o n s i n e q u a t i o n ( 3 5 ) d i s a p p e a r a t i n f i n i t y and c o n d i t i o n ( i v ) Is s a t i s f i e d . Boundary c o n d i t i o n (v) was used t o determine the nonsteady complex f u n c t i o n s and i s i n h e r -e n t l y s a t i s f i e d . The v e l o c i t y boundary c o n d i t i o n ( i i i ) now must be s a t i s f i e d . U s i n g the d e f i n i t i o n of the stream f u n c t i o n from s e c t i o n ( 1 ) , e q u a t i o n (1) can be w r i t t e n as dx v a x ( 3 6 ) 39. where the nonsteady flox* quantities have been written as ip-.tfoe^^ y a ^ e ^ * and v»\£e i U , t» In equation (35) these values enable the non-steady complex function to be expressed as F„(g),L(f„C§) + i^.(S)3e*tt't. (37) Since V 0 , the v e l o c i t y amplitude about the x-axis of a given point i n the flow f i e l d , must vanish at i n f i n i t y , the i n t e g r a l of equation (36) can be written as V . . - e ^ " £ ^ e ^ * d j ( 3 8 , where x re f e r s to any point on the f o i l or s p o i l e r region and $ i s a dummy variable of in t e g r a t i o n . At points on the f o i l V£=0 and at points on the s p o i l e r region Vo»v0 . Consideration of the leading edge and the s p o i l e r base i n equation (38) w i l l r e s u l t i n two equ-ations i n the two unknowns C, and C%. I t should be noted that con-s i d e r a t i o n of any general points r e s u l t s i n the same equations, but mathematical s i m p l i c i t y makes the leading edge and the s p o i l e r base the desired choice. The equations are and Integrating these equations by parts and changing the variable of integ r a t i o n gives 4 0 . — C O « £> and (39) where J has been r e p l a c e d by - 5 ' . S u b s t i t u t i n g the v a l u e of oj,0 from e q u a t i o n (37) i n t o the f i r s t of equations (39) r e s u l t s i n the equa-t i o n " o © o S u b s t i t u t i n g ^ i n t o the second of equations (39) g i v e s o I n the above equations and T 3 =» R| 41. The s o l u t i o n t o the two simultaneous equations f o r C, and C x i s and C%* v0Cl The v a l u e s of C,' and C,J are and ( W where I t can be seen t h a t G,» and C* are v e r y complicated e x p r e s s i o n s t h a t cannot be s o l v e d a n a l y t i c a l l y , . The n u m e r i c a l s o l u t i o n of C,' and C* as a f u n c t i o n of /A . i n v o l v e s l a r g e numbers of c a l c u l a t i o n s and nec-e s s i t a t e s the use of a computer. In s o l v i n g the e x p r e s s i o n s f o r C,' and , 5' o b v i o u s l y cannot be extended to i n f i n i t y . F o r t u n a t e l y t h i s does not l i m i t the s o l u t i o n . A c l o s e r look at the Integrands of the e x p r e s s i o n s In equations (40) helps to demonstrate t h i s p o i n t . These Integrands a l l approach z e r o as .j' approaches i n f i n i t y . I t i s not t h i s f a c t however, t h a t f a c i l i t a t e s the s o l u t i o n of these I n t e g r a l s . 42. I t i s found t h a t when £' i s g r e a t e r than a few chords, the int e g r a n d s are v e r y s l o w l y v a r y i n g f u n c t i o n s of . The important aspect of the i n t e g r a l s then becomes the f a c t t h a t the i n t e g r a n d must be t r u n c a t e d at the end of a complete c y c l e . T h i s means t h a t has t o be an ex-a c t even m u l t i p l e of TT. Using t h i s technique i t i s found t h a t 10 ch-ords g i v e s a h i g h degree of acc u r a c y . The v a l u e w i l l a c t u a l l y f l u c t -uate around 10 chords as i s kept as an exact even m u l t i p l e of T . In the i n t e g r a l s i n v o l v i n g T% , T 3 , and T A the i n t e g r a n d i s i n f i n i t e a t the lower l i m i t . Care must be taken i n n u m e r i c a l l y d e t e r m i n i n g the Cauchy P r i n c i p a l v a l u e of these i n t e g r a l s . From eq u a t i o n (37) the a c c e l e r a t i o n p o t e n t i a l ^ f o r p o i n t s on the f o i l can be w r i t t e n as Us i n g e q u a t i o n (2) and i n t e g r a t i n g the p r e s s u r e , the l i f t c o e f f i c i e n t can be w r i t t e n as S u b s t i t u t i n g the e x p r e s s i o n f o r x from equation (32) g i v e s where C, i s the amplitude of the unsteady l i f t c o e f f i c i e n t , and Cu=Cujer o P u t t i n g the e x p r e s s i o n f o r *f>0 i n t h i s e q u a t i o n and i n t e g r a t -i n g g i v e s where 0, * Tr[(^-') f cC i e.CoS i e«-S^ l©.II -r (•-tf*)[.Si*e i-© l«*0] ; 4 3 . t \ . = TT [ ( ^ i ) E 2 - ( Q , - T r ) C o S J L e o - S«n X © , ] + C»-fc)'*)Ls»n©, - ( A - * " ) COS © J ] } and D, » t C ^ J ^ + i s i n ^ - | ( ^ ) 0 - b , X ) [ s . - » © , ^ 5 i n 30,] ^ i C - H C e . + is.nz©,] . Suppose the q u a s i - s t e a d y l i f t c o e f f i c i e n t C u $ i s d e f i n e d t o be the v a l u e of the unsteady l i f t c o e f f i c i e n t for/*.•*•<>, then, s i n c e C.'sl and C^ «=o -for ^-•-o. The r a t i o of the nonsteady l i f t c o e f f i c i e n t t o the q u a s i - s t e a d y l i f t c o e f f i c i e n t can be w r i t t e n as where The l i f t c o e f f i c i e n t i s then expressed as T h i s l i f t c o e f f i c i e n t must now be used t o determine the u n i t s t e p s p o i l e r a c t u a t i o n problem. 44. 4.3 U n i t Step A c t u a t i o n The complete l i n e a r i t y of the present system f o r K=0 makes i t p o s s i b l e t o use the methods of s u p e r p o s i t i o n t o o b t a i n the t r a n s i e n t s o l u t i o n t o the s p o i l e r a c t u a t i o n problem. The second term of equ-a t i o n (42) r e p r e s e n t s the c o n t r i b u t i o n of the apparent mass term and w i l l be d i s c a r d e d s i n c e i t has no c o n t r i b u t i o n t o the s o l u t i o n . B e f o r e p r o c e e d i n g w i t h the problem, a c l o s e r look a t Q(/*) Is war-r a n t e d . The n u m e r i c a l s o l u t i o n of Q has l i m i t a t i o n s f o r l a r g e v a l u e s of jx. F o r t u n a t e l y such l a r g e v a l u e s of/*. have l i t t l e e f f e c t on the s o l u t i o n t o the problem. Q^ /n) i s shown i n s e c t i o n (6) f o r d i f f e r e n t s p o i l e r p o s i t i o n s and s i z e s . I t can be seen t h a t as /*. gets l a r g e r , Q tends t o the Imaginary a x i s and the r a t e of change of the r e a l p a r t of Q w i t h jx i s v e r y s m a l l . T h i s c o n t r i b u t e s t o the s o l u t i o n converg-i n g r a p i d l y as yu. i n c r e a s e s . A f u r t h e r d i s c u s s i o n of t h i s p o i n t w i l l be g i v e n a f t e r the s o l u t i o n has been developed. A f u r t h e r p o i n t to n o t i c e i s t h a t a s / t approaches i n f i n i t y , Q becomes asymptotic to the imaginary a x i s . T h i s means t h a t the r e a l p a r t of the l i f t approaches z e r o as jx approaches I n f i n i t y . P h y s i c a l l y i t can be argued t h a t the blowing and s u c k i n g c y c l e s occur so r a p i d l y t h a t the wake c i r c u l a t i o n c a n c e l s the l i f t more e f f e c t i v e l y f o r these s h o r t e r wavelengths. E q u a t i o n (42) can be w r i t t e n as C t = vt>,Q</>0, (43) where v-Voe^^ and the apparent mass term has been dropped. The v e l -o c i t y boundary c o n d i t i o n on the s p o i l e r r e g i o n f o r the u n i t s t e p s p o i l e r a c t u a t i o n problem i s as f o l l o w s : r o , t * o v » 1 4 5 . T h i s can be expressed as s i n S . K t ) . I f v0 i s put e q u a l t o f=JFjbr and e q u a t i o n (43) i s i n t e g r a t e d over a l l f r e q u e n c i e s , the t r a n s i e n t l i f t c o e f f i c i e n t becomes f e d Suppose Ut » s'c where s* i s the d i s t a n c e moved i n chords, then t h i s e q u a t i o n can be w r i t t e n as Cu * C^ lCsO, (44) where C. , i s the f i n a l steady s t a t e v a l u e of the l i f t c o e f f i c i e n t T and ICsO - & F }$^T QCIOdk. (45) The frequency parameter k i s g i v e n by The technique f o r s o l v i n g t h i s i n t e g r a l i s g i v e n by B i s p l i n g h o f f ( 8 ) , Suppose where R and S are both r e a l f u n c t i o n s of k. The i n t e g r a l of equation (45) becomes I(s') = ^ p ^ p J i i . ' dtk . (46) 46. R i s o n l y known n u m e r i c a l l y and t h i s i n t e g r a l must a l s o be so l v e d n u m e r i c a l l y . I t w i l l be r e c a l l e d t h a t as /*. and hence k, i n c r e a s e s i n s i z e , R(k) and the r a t e of change of R(k) approach z e r o . T h i s and the k f a c t o r i n the denominator make the i n t e g r a n d v e r y r a p i d -l y approach z e r o . Once a g a i n i t i s advantageous t o i n t e g r a t e over a l e n g t h c o r r e s p o n d i n g t o an even m u l t i p l e of i r . I t i s not neces s a r y to take k any g r e a t e r than 25 to g i v e v e r y good accurac y f o r t h i s i n t e g r a l . The s o l u t i o n s t o t h i s problem are g i v e n g r a p h i c a l l y f o r d i f f e r e n t a i r f o i l c o n f i g u r a t i o n s i n s e c t i o n (6). 4.4 F i n i t e Time A c t u a t i o n In p r a c t i c a l a p p l i c a t i o n s the s p o i l e r a c t u a t i o n takes a f i n i t e p e r i o d of time. Once a g a i n the complete l i n e a r i t y of the z e r o c a v i t y number s o l u t i o n a l l o w s t h i s problem t o be e a s i l y s o l v e d . The u n i t s t e p i n t e g r a l i n eq u a t i o n (46) i s e n t i r e l y independent of the s p o i -l e r a n g l e . T h i s means t h a t nonsteady s p o i l e r a c t u a t i o n s o l u t i o n s can be superposed. F o r example, suppose a t t=0 the s p o i l e r angle jumps £E and a f t e r t=dfc the s p o i l e r angle a g a i n jumps AS. At t=A"b the d i s -tance t r a v e l l e d , As ', i s U&fc and the t o t a l s p o i l e r angle i s 2 A £ . A f t e r time t=t e=NAt, d u r i n g which time the f o i l has t r a v e l l e d S1 = Sg=NAs\ the s p o i l e r angle has r i s e n t o i t s e r e c t i o n angle U s i n g e q u a t i o n (44) and summing up each s m a l l s t e p g i v e s A f t e r time t> t e , d u r i n g which time the f o i l has t r a v e l l e d a d i s t a n c e (s'-s'e) w i t h i t s s p o i l e r f u l l y e r e c t e d , the l i f t c o e f f i c i e n t i s Now A S « I and s i n AS can be r e p l a c e d by T h i s g i v e s the l i f t c o e f f i c i e n t Cu = CU4 WCs') where WCsO -^{i.Cs') + XN+l(s'-s4)] . S t r i c t l y speaking i n t h i s s o l u t i o n C u^ should e q u a l D,S and not D,stn£ . However s i n c e i t i s the response f u n c t i o n W(s') t h a t i s the important p a r t of t h i s s o l u t i o n i t i s a l l o w a b l e , f o r c o n f o r m i t y purposes, t o w r i t e the s o l u t i o n as gi v e n i n equa t i o n ( 4 4 ) . Once N i s chosen, the s o l u t i o n i s a f u n c t i o n of the t o t a l d i s t a n c e t r a v e l l e d and the d i s -ance t r a v e l l e d d u r i n g s p o i l e r e r e c t i o n . T y p i c a l l y N=20 shows very good r e s u l t s . T h i s s o l u t i o n f o r d i f f e r e n t a i r f o i l c o n f i g u r a t i o n s i s a l s o shown g r a p h i c a l l y i n s e c t i o n ( 6 ) . 4 7 . 48. SECTION 5 EXPERIMENTS E x p e r i m e n t a l measurements f o r l i f t , drag and p i t c h i n g moment were taken f o r the \k% t h i c k C l a r k Y a i r f o i l , shown g r a p h i c a l l y i n f i g u r e ( 4 ) . The a i r f o i l was c o n s t r u c t e d of wood and had a 14 i n c h chord. Each end of the a i r f o i l had 1/8 i n c h s t e e l - p l a t e s attached t o a l l o w spanwise s p o i l e r s of h e i g h t s $% and 1 0 $ chord to be mounted a t s p o i l e r angles of 1 5 , 3 ° » 4 5 , 6 0 , 7 5 , and 9 0 degrees t o the a i r -f o i l s u r f a c e a t v a r y i n g p o s i t i o n s on the a i r f o i l . Measurements were taken f o r s p o i l e r p o s i t i o n s of 5 0 and 1/0% chord. The a i r f o i l itfas mounted at the mid-chord p o s i t i o n on a s i x -component s t r a i n gauge balance s y s t e m . The l i f t , d r a g and p i t c h i n g moment were measured over a wide range of incidence„ The gap be-tween the s p o i l e r and the a i r f o i l s u r f a c e was s e a l e d w i t h masking tape f o r each c o n f i g u r a t i o n . The base p r e s s u r e i n the wake r e g i o n was measured by t a p i n g a t h i n tube to the a i r f o i l i n the wake r e -g i o n . T h i s tube was connected t o an a l c o h o l manometer t o g e t h e r w i t h a tube l e a d i n g t o a s t a t i c probe measuring the upstream u n d i s t u r b e d 5 s t a t i c p r e s s u r e . The t e s t Reynolds number was 4 x 1 0 . I d e n t i c a l measurements were taken on a lk% t h i c k C l a r k Y a i r -f o i l w i t h a 3 2 . 5 $ f l a p . T h i s a i r f o i l and f l a p combination i s shown g r a p h i c a l l y i n f i g u r e ( 5 ) » F o r these measurements the a i r f o i l was mounted at the ^ - c h o r d p o i n t . The gap on the lower s u r f a c e between the main f o i l and the f l a p was s e a l e d w i t h masking tape. A l l measurements were made i n the low speed wind, t u n n e l of the Mechanical E n g i n e e r i n g .Department of the U n i v e r s i t y of B r i t i s h C o l -umbia. T h i s t u n n e l has a t e s t s e c t i o n o f . 3 f t . by 2\ f t . over a l e n g t h of 8 2 / 3 f t . The t u n n e l produces a v e r y uniform flow, with a F I G U R E ( 4 ) : lk°/o T H I C K C L A R K Y A I R F O I L G U R L ( 5 ) : iW/o T H I C K C L A R K Y A I R F O I L W I T H 32.5°/o F L A P 51. t u r b u l e n c e l e v e l of l e s s than 0.1$ over a wind speed range of 0-150 f p s . The wind t u n n e l w a l l c o r r e c t i o n technique employed was the same as t h a t employed by J a n d a l i ( 1 ) . T h i s method uses the c o r r e c t -i o n s e s t a b l i s h e d by Pope and Harper (9)» w i t h a wake blockage term of i ( c / H ) C p i n s t e a d of ^(c/H)C^. J a n d a l i found t h a t measurements on a i r -f o i l s of v a r y i n g chord l e n g t h s c o l l a p s e d more s u i t a b l y u s i n g these c o r r e c t i o n s . There e x i s t s some c o n t r o v e r s y over the techniques emp-loyed f o r c o r r e c t i n g the wake p r e s s u r e c o e f f i c i e n t . B l u f f body and s t a l l e d a i r f o i l t echniques such as those presented by M a s k e l l (10) are not s t r i c t l y a p p l i c a b l e . To overcome t h i s problem, base pressure measurements were taken over a range of i n c i d e n c e s on a i r f o i l s of ch-ords 9 , 14, 19 and 24 Inches f o r normal s p o i l e r s of 5$ and 10$ h e i g h t s l o c a t e d a t bot h the 50$ and ?0$ chord p o s i t i o n s . These measurements were p l o t t e d and i n t e r p o l a t e d back t o zer o chord ( o r i n f i n i t e stream). The base p r e s s u r e c o e f f i c i e n t f o r the remaining s p o i l e r a n g l e s , f o r which measurements were taken on the 14 i n c h chord a i r f o i l , were then c o r r e c t e d i n the same r e s p e c t i v e r a t i o . I t i s r e a l i z e d t h a t , as the s p o i l e r angle changes, the wake c h a r a c t e r i s t i c s a l s o change s l i g h t l y . T h i s i n t u r n would a f f e c t the c o r r e c t i o n r a t i o s s l i g h t l y . T h i s t e c h -nique does however, g i v e r e a s o n a b l y r e a l i s t i c r e s u l t s and was used i n the absence of a b e t t e r t e c h n i q u e . At low a i r f o i l i n c i d e n c e s and s m a l l s p o i l e r a ngles the p o s s i b -i l i t y of f l o w reattachment o c c u r s . In such a case the t h e o r i e s deve-loped i n s e c t i o n s (3) and (4) are not a p p l i c a b l e . To ensure t h a t measurements were not taken f o r such cases t u f t s of c o t t o n were a t t -ached t o the a i r f o i l s u r f a c e i n the wake r e g i o n . O b s e r v a t i o n of these t u f t s i n a l l a i r f o i l c o n f i g u r a t i o n s was c a r e f u l l y c a r r i e d out. The lower s u r f a c e of the C l a r k Y a i r f o i l Is f l a t , and t h i s base i s used as a r e f e r e n c e f o r i n c i d e n c e r a t h e r than the u s u a l chord l i n e . 52. SECTION 6 RESULTS AND COMPARISONS The f i r s t p a r t of t h i s s e c t i o n shows the r e s u l t s from the steady l i n e a r i z e d c a v i t y f l o w t h e o r y developed i n s e c t i o n ( 3 ) . The l i f t c o e f f i c i e n t s over a wide range of i n c i d e n c e , obtained f o r nor-mal s p o i l e r s , are compared to those obtained t h e o r e t i c a l l y by Woods (2) and J a n d a l i ( 1 ) . The t h e o r e t i c a l l i f t c o e f f i c i e n t s as f u n c t i o n s of i n c i d e n c e f o r s e v e r a l s p o i l e r a n g l e s , i n c l u d i n g the normal s p o i -l e r s , are compared w i t h experiment„ Comparisons between theory and experiment are a l s o presented showing the v a r i a t i o n of l i f t c o e f -f i c i e n t w i t h s p o i l e r angle f o r a g i v e n a i r f o i l i n c i d e n c e . Examples of the p r e s s u r e c o e f f i c i e n t p r e d i c t e d by the l i n e a r i z e d t h e o r y com-p l e t e the steady s t a t e r e s u l t s p r e s e n t e d . The l a t t e r p a r t of t h i s s e c t i o n c o n t a i n s the r e s u l t s from the unsteady l i n e a r i z e d c a v i t y f l o w t h e o r y developed i n s e c t i o n (4) t o s o l v e the s p o i l e r a c t u a t i o n problem. I n the blowing case t h e o r e t i c a l r e s u l t s are g i v e n showing the v a r i a t i o n w i t h frequency of the r a t i o of l i f t c o e f f i c i e n t t o q u a s i - s t e a d y l i f t c o e f f i c i e n t ; - The r a t i o of l i f t c o e f f i c i e n t t o f i n a l l i f t c o e f f i c i e n t as a f u n c t i o n of d i s t a n c e moved i n chords i s presented t h e o r e t i c a l l y f o r the u n i t s t e p and f i n -i t e time s p o i l e r a c t u a t i o n problems» 6 . 1 Steady Theory In f i g u r e (6) the ex p e r i m e n t a l l i f t c o e f f i c i e n t and the l i f t c o e f f i c i e n t c a l c u l a t e d u s i n g standard l i n e a r i z e d t echniques are p l o t -t e d as a f u n c t i o n of i n c i d e n c e f o r the t h i c k C l a r k Y a i r f o i l . As i s expected the agreement around the zer o l i f t angle of i n c i d e n c e Is good, the agreement becoming worse as the i n c i d e n c e i n c r e a s e s . Around the z e r o l i f t i n c i d e n c e the v o r t i c i t y d i s s i p a t i o n i n the boundary 5 3 . 54. l a y e r i s a t a minimum and a c l o s e comparison betxveen t h e o r y and ex-periment i s expected. F i r s t l i f t c o e f f i c i e n t s over a range of i n c i d e n c e from the presen t t h e o r y w i l l be compared t o the r e s u l t s of ot h e r t h e o r i e s , and t o ex p e r i m e n t a l r e s u l t s , f o r the case of a normal s p o i l e r . A l l s p o i l e r h e i g h t s and p o s i t i o n s are g i v e n as a percentage of the a i r -f o i l chord. In f i g u r e (?) and (8) the l i f t c o e f f i c i e n t s f o r the t h r e e t h e o r i e s and the experimental l i f t c o e f f i c i e n t s are shown f o r a 10% normal s p o i l e r a t the 50 and 70% p o s i t i o n s r e s p e c t i v e l y . The co r r e s p o n d i n g comparisons f o r a 5% normal s p o i l e r are shown i n f i g -u r e s (9) and ( 1 0 ) . These f i g u r e s show the agreement between the th e o r y and experiment i s q u i t e good. The agreement w i t h experiment shown by J a n d a l i ' s t h e o r y i s v e r y good, but i t w i l l be r e c a l l e d the a d d i t i o n a l c o n d i t i o n of no l i f t i n c i d e n c e i s r e q u i r e d as an i n p u t i n t o t h i s t h e o r y . I n the c a l c u l a t i o n of Woods 1 t h e o r y the e m p i r i c a l e q u a t i o n f o r C P j r suggested by Barnes (3) was used. Although Woods' t h e o r y shows reas o n a b l e agreement w i t h experiment, i t does not seem t o p r e d i c t the c o r r e c t l i f t curve s l o p e . F o r the ex p e r i m e n t a l base p r e -ssure a p p l i c a b l e i n these cases, the c a v i t y l e n g t h i s q u i t e s h o r t . T h i s has the tendency of making the s o l u t i o n more h i g h l y dependent on the accurac y of the base p r e s s u r e c o e f f i c i e n t . The camber and t h i c k n e s s s o l u t i o n s which c o n s t i t u t e a c o n s i d e r a b l e p o r t i o n of the l i f t c o e f f i c i e n t , are dependent t o a v e r y s m a l l e x t e n t on the c a v i t y number. They have a v e r y s m a l l e f f e c t on the d e t e r m i n a t i o n of the c a v i t y l e n g t h , and en t e r p r i m a r i l y through s a t i s f y i n g the c o n d i t i o n at i n f i n i t y . T h i s f a c t reduces the s o l u t i o n dependence on the c a v i t y l e n g t h and hence the c a v i t y numbero The c o n t r i b u t i o n of the t h i c k n e s s s o l u t i o n i n c r e a s e s n e g a t i v e l y from z e r o as the s p o i l e r moves ahead ' F I G U R E ( 7 ) : LIFT COEFFICIENT FCR C L r R K Y A IK F O I L W I T H S P O T L h K 1.4 .. F I G U R E ( 9 ) : L I F T C O E F F I C I E N T FOR C L A R K Y A I R F O I L W I T H 3 R 0 I L E / Theory Woods — — Tandali F I G U R E ( 1 0 ) : L 1 ' ? T C O S J P T C T E N T FOR C L A R K Y A I R F O I L W I T H 59. from the t r a i l i n g edge. The c o n t r i b u t i o n of the camber s o l u t i o n de-c r e a s e s from a maximum as the s p o i l e r moves forward. F o r the cases presented of 50 and 70$ s p o i l e r p o s i t i o n s , the t h i c k n e s s c o n t r i b u -t i o n s i s the same order of magnitude as the camber c o n t r i b u t i o n . I n Woods' t h e o r y the t h i c k n e s s term i s d i s c a r d e d as a second order term. Comparisons between t h e o r e t i c a l and exp e r i m e n t a l r e s u l t s w i l l now be c o n s i d e r e d f o r s p o i l e r angles other than 90 degrees. In f i g -ures (11) and (12) the comparisons are shown f o r a s p o i l e r of angle 45 degrees p o s i t i o n e d a t the 50$ chord p o i n t , the s p o i l e r h e i g h t s b e i n g 10$ and 5$ r e s p e c t i v e l y . F i g u r e s (13) and (14) show a compar-i s o n of t h e o r e t i c a l and ex p e r i m e n t a l l i f t c o e f f i c i e n t s f o r s p o i l e r s l o c a t e d a t the 70$ chord p o i n t w i t h a s p o i l e r angle of 60 degrees and s p o i l e r h e i g h t s of 10$ and 5$ r e s p e c t i v e l y . F i g u r e s (15) and (16) show the c o r r e s p o n d i n g comparisons f o r a s p o i l e r angle of 30 degrees. These r e s u l t s w i l l now be presented f o r a f i x e d a i r f o i l i n c i d -ence, the s p o i l e r angle b e i n g the v a r i a b l e a g a i n s t which the l i f t c o e f f i c i e n t i s p l o t t e d . The s p o i l e r i s l o c a t e d a t the 70$ chord p o i n t . F i g u r e (17) i s a comparison i n l i f t c o e f f i c i e n t s between t h e o r y and experiment a t an a i r f o i l i n c i d e n c e of 6 degrees, and a s p o i l e r h e i g h t of 10$. F i g u r e (18) shows a cor r e s p o n d i n g comparison at an a i r f o i l i n c i d e n c e of 6 degrees and a s p o i l e r h e i g h t of 5$. As expected the l i f t c o e f f i c i e n t decreases as the s p o i l e r angle i n c r e -a s e s . The agreement between theory and experiment i s once agai n q u i t e good. I t should be r e c a l l e d t h a t t h i s graph of l i f t c o e f f i c i -ent v e r s e s s p o i l e r angle cannot be e x t r a p o l a t e d back t o zer o s p o i l e r a ngle, s i n c e the theory i s onl y a p p l i c a b l e as long as the c a v i t y e x i s t s , and i n such a case the c a v i t y p h y s i c a l l y c o u l d not e x i s t . s/c =0.5", h / c » 0 0 5 , S»4-5" o o Experiimen+ T h e o r y FICURF (12) : LIFT OOFFFICIKNT FOR CLARK Y AIRFOIL WITH Gi'OILl F I G U R E (13): L I F T COEFFICIENT FOR C L A R K Y A I R F O I L W I T H S P O I L E R 6 3 . 64, s/c 3 0.7 , h/c »o . I , S»30 o o E x p e r i m e n t Theory FIGURE (15) : LIFT COEFFICIENT FOR CLARK Y AIRFOIL WITH 8 F O I L sjc * O 7 , n/c » O.05" , S • 3 0 o — o Experiment T h e o r y F I G U R E ( 1 6 ) : L I F T C O E F F I C I E N T F O R C L A R K Y A I R F O I L W I T H S P O I L 6 6 . I.O .. O.g 0 . 4 O.X . . IS" 3o 4 r 6 0 "T5- 9 0 s/csO .7 j J i / c - O . I ,0^-4 ° o — o Expenrnein+ T h e o r y F I G U R E ( 1 7 ) : L I F T COEFFICIENT FOR CLARK Y AIRFOIL W I T H BROILER 6?. o.x .. — 1 1 1 1 1 •a 30 45" 60 If 9 0 S° S/c a O . T n/o " O . 05" ; e( s t° o o E x p e r i m e n t » T h e o r y FIG-URL' ' ( 1 8 ) : L I F T C O E F F I C I E N T ^ O R C L A R K Y A I R F O I L W I T H S P O I L E R 6 8 . Comparisons f o r the a i r f o i l w i t h s p o i l e r and f l a p 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 l i f t c o e f f i c i e n t s f o r a 10$ normal spo-i l e r a t the 70$ chord p o i n t w i l l now be g i v e n f o r d i f f e r e n t f l a p a n g l e s . F i g u r e (19) shows the comparison f o r a f l a p angle of 15 de-g r ees, and f i g u r e (20) i s a comparison f o r a f l a p angle of 30 deg-r e e s . T h i s r e s u l t i s c o n s i s t e n t w i t h simple l i n e a r f l a p t h e o r y , the t h e o r y over p r e d i c t i n g the l i f t as the f l a p angle gets l a r g e . To complete the r e s u l t s presented f o r the steady s t a t e s o l u t -i o n , some examples of the t h e o r e t i c a l p r e s s u r e c o e f f i c i e n t d i s t r i -b u t i o n f o r normal s p o i l e r s w i l l be g i v e n . F i g u r e (21) shows the theo-r e t i c a l p r e s s u r e c o e f f i c i e n t d i s t r i b u t i o n f o r a 10$ s p o i l e r l o c a t e d a t the 70$ chord p o s i t i o n w i t h an a i r f o i l I n cidence of 12 degrees. The t h e o r e t i c a l p r e s s u r e c o e f f i c i e n t d i s t r i b u t i o n f o r a 5$ s p o i l e r l o c a t e d at the 50$ chord p o i n t f o r an a i r f o i l i n c i d e n c e of 10 degre-es, i s g i v e n i n f i g u r e ( 2 2 ) . F o r comparison the r e s u l t s g i v e n by J a n d a l i (1) are shown on these graphs f o r the same a i r f o i l c o n f i g -u r a t i o n s o The s i n g u l a r i t i e s i n the p r e s s u r e d i s t r i b u t i o n s are i n h e r -ent i n l i n e a r i z e d t e c h n i q u e s , and the d i s t r i b u t i o n s g i v e b a s i c a l l y q u a l i t a t i v e i n f o r m a t i o n . 6 . 2 Nonsteady Theory In the c o n s i d e r a t i o n of the nonsteady s o l u t i o n s i t w i l l be r e -c a l l e d t h a t the s o l u t i o n i s independent of the s p o i l e r angle and t h a t the c a v i t y must a l r e a d y e x i s t . T h i s f a c t means t h a t the t h i c k -ness, camber, i n c i d e n c e , and f l a p s o l u t i o n s are not going to change d u r i n g f u r t h e r s p o i l e r a c t u a t i o n and t h e i r steady s t a t e s o l u t i o n s are f u l l y a d d i t i v e t o the nonsteady s p o i l e r s o l u t i o n . I t i s proposed t h e r e f o r e to p r e s e n t o n l y the r e s u l t s to the nonsteady s p o i l e r s o l -u t i o n . 6 9 . Lb 1 T h e o r y FIGVJ?B ( 1 9 ) : L I F T C O E F F I C I E N T FOR C L A R K Y A I R F O I L WITH S P O I L E R AI"]) F L A P 7 0 . c h o r d a 4.04-3 F I G U R E ( 2 1 ) : P R E S S U R E D I S T R I B U T I O N F O R C L A R K Y A I R F O I L W I T H S P O I L E R chorda 4-043 THeor -y F I G U R E ( 2 2 ) : P R E S S U R E D I S T R I B U T I O N F O R C L A R K Y A I R F O I L V / I T H S P O I L E R ro 73. Although they are of l i t t l e p h y s i c a l Importance and are prim-a r i l y an Intermediate r e s u l t , some blowing case s o l u t i o n s w i l l be pre s e n t e d . I t w i l l be r e c a l l e d from equation (42) t h a t the r a t i o of l i f t c o e f f i c i e n t t o q u a s i - s t e a d y l i f t c o e f f i c i e n t was expressed as a f u n c t i o n o f / * , the frequency of blowing. F i g u r e (23) g i v e s t h i s r a t i o f o r a s p o i l e r r e g i o n l e n g t h of 10$ p o s i t i o n e d a t the 70$ chord p o i n t . F i g u r e (24) g i v e s t h i s r a t i o f o r a s p o i l e r r e g i o n l e n g t h of 5$ p o s i t i o n e d at the 50$ chord p o i n t . I t can be seen t h a t these p l o t s a s y m p t o t i c a l l y approach the Imaginary a x i s , the r e a l p a r t of the l i f t c o e f f i c i e n t approaching z e r o . I t may be reas o n a b l e t o argue p h y s i c a l l y t h a t the blowing and s u c k i n g c y c l e s occur so q u i c k l y as the frequency gets l a r g e , t h a t the net a f f e c t approaches z e r o . The u n i t s t e p s p o i l e r a c t u a t i o n problem w i l l be c o n s i d e r e d next. The r e s u l t s f o r the s p o i l e r a c t u a t i o n problems are r e p o r t e d as the r a t i o of the ins t a n t a n e o u s l i f t c o e f f i c i e n t over the f i n a l steady s t a t e l i f t c o e f f i c i e n t , as a f u n c t i o n of the d i s t a n c e moved i n chords. Si n c e the s p o i l e r a c t u a t i o n problems are Independent of s p o i l e r a n g l e , the s o l u t i o n becomes a f u n c t i o n of s p o i l e r p o s i t i o n and hei g h t o n l y . F i g u r e (25) i s the s o l u t i o n f o r a 10$ and a 5$ s p o i l e r a t the 70$ chord p o i n t . The use of l i n e a r i z e d theory allowed the s u p e r p o s i t i o n of the u n i t s t e p s o l u t i o n s i n t o the f i n i t e time s p o i l e r a c t u a t i o n problem. T h i s a g a i n makes the s o l u t i o n independent of s p o i l e r a n g l e . These r e s u l t s a re a l s o r e p o r t e d as the r a t i o of the in s t a n t a n e o u s l i f t c o e f f i c i e n t t o the f i n a l steady s t a t e l i f t c o e f f i c i e n t as a f u n c t i o n of d i s t a n c e moved i n chords. The u n i t s t e p s p o i l e r a c t u a t i o n s o l u -t i o n i s i n c l u d e d w i t h the f i n i t e time s o l u t i o n s f o r purposes of com-p a r i s o n . T h e o r e t i c a l r e s u l t s are g i v e n f o r a c t u a t i o n d i s t a n c e s of 5. 10 and 15 chords. In f i g u r e s (26) and (27) the t h e o r e t i c a l s o l u t i o n s 74. F I G U R E '(2k) t B L O V / I - K G T H E O R Y S O L U T I O N o.s-i s / c a O . 7 K/C a O.O 5* h/CaO.I S*, chords F I G U R E ( 2 5 ) : U ? T I T S T E P S P O I L E R A C T U A T I O N S O L U T I O N s/c ssO.7 ^ h/c - O.I j/caO.5" , h/c-O.OS" 79. are g i v e n f o r 10$ s p o i l e r at the 70$ chord p o i n t and. a 5$ s p o i l e r at the 50$ chord p o i n t . In f i g u r e (28) the d i s t a n c e t r a v e l l e d f o r the l i f t c o e f f i c i e n t to f a l l t o 90$ of i t s f i n a l steady s t a t e v a l u e i s p l o t t e d a g a i n s t s p o i l e r p o s i t i o n f o r both 5$ and 10$ s p o i l e r s w i t h v a r y i n g e r e c t i o n d i s t a n c e s . F i g u r e (29) shows the d i s t a n c e t r a v e l l e d f o r the l i f t c o e f f -i c i e n t t o f a l l t o 90$ of i t s steady s t a t e v a l u e as a f u n c t i o n of e r e c t i o n d i s t a n c e f o r both 5$ and 10$ s p o i l e r s a t v a r i o u s s p o i l e r p o s i t i o n s . As the s p o i l e r e r e c t i o n time i n c r e a s e s the nonsteady s o l -u t i o n a s y m p t o t i c a l l y approaches the q u a s i - s t e a d y s o l u t i o n . For v e r y slow s p o i l e r a c t u a t i o n r a t e s only the q u a s i - s t e a d y s o l u t i o n i s nec-e s s a r y . However f o r f a s t e r r a t e s of a c t u a t i o n a f u l l nonsteady ana-l y s i s i s n e c e s s a r y . 15* o.i 0.3 0.5" — h -0.7 0.«5 s/c hjc » 0.05" . h/c a O.I M O U R E ( 2 8 ) : U N I T S T E P A N D F I N I T E T I N l E S P O I L E R A C T U A T I O N S O L U T I O I O I S " Spoiler Erco+ion Distance , chords n/c «OOzT F I G U R E ( 2 9 ) : U N I T S T E P A M ) F I N I T E T I M E S P O I L E R A C T U A T I O N S O L U T I O N S PART I I AN EXACT FREE STREAMLINE POTENTIAL FLOW THEORY FOR THE STEADY STATE AIRFOIL SPOILER AND SPOILER PLUS SLOTTED FLAP PROBLEMS 8 3 . SECTION 7 SOLID AIRFOIL WITH A SPOILER 7 . 1 S u r f a c e S i n g u l a r i t y Theory The s u r f a c e s i n g u l a r i t y t echnique employed i n the f o l l o w i n g problems was developed r e c e n t l y by A.M.O. Smith and h i s a s s o c i a t e s at the McDonnell-Douglas A i r c r a f t Company. S e v e r a l comprehensive pub-l i c a t i o n s ( 6 , 7 ) of t h i s theory e x i s t , and t h e r e f o r e i t w i l l not be re p e a t e d . A l t h o u g h Smith's th e o r y i s a p p l i c a b l e t o two and t h r e e d i m e n s i o n a l f l o w s , t h i s a n a l y s i s i s l i m i t e d t o a two d i m e n s i o n a l , i n c o m p r e s s i b l e , i n v i s c i d and i r r o t a t i o n a l f low f i e l d . 7 . 2 F o r m u l a t i o n of the Problem The a i r f o i l w ithout a f l a p i s p o s i t i o n e d i n the z-plane as d e s c r i b e d i n s e c t i o n ( 3 . 2 ) . T h i s c o n f i g u r a t i o n i s shown p i c t o r i a l l y i n f i g u r e ( 3 0 ) . In t h i s problem t h e r e i s a wake extending t o i n f i n -i t y . The wake i s bounded by f r e e s t r e a m l i n e s , o n e s p r i n g i n g from the s p o i l e r t i p and one from the t r a i l i n g edge. E x p e r i m e n t a l l y the base p r e s s u r e on the a i r f o i l i n t h i s wake r e g i o n takes a con s t a n t v a l u e . S i n c e no s a t i s f a c t o r y t h e o r y f o r p r e d i c t i n g the base p r e s s u r e c o r -r e c t l y has been d e v i s e d , a l l such t h e o r i e s r e q u i r e a t l e a s t one em-p i r i c a l parameter. There i s no advantage i n c o r r e c t l y m o d e l l i n g the con s t a n t v a l u e of t h i s base p r e s s u r e over the wake r e g i o n . I f the th e o r y c o r r e c t l y p r e d i c t s the s e p a r a t i o n p r e s s u r e c o e f f i c i e n t s on the f r e e s t r e a m l i n e s d e t a c h i n g from the s p o i l e r t i p and the t r a i l i n g edge, the p r e s s u r e on the a i r f o i l between the s t r e a m l i n e s i n s i d e the wake can be assumed c o n s t a n t , and equal t o the s e p a r a t i o n p r e s s u r e , r e g a r d l e s s of what type of fl o w i s i n t h i s r e g i o n . T h i s approach was used e f f e c t i v e l y by J a n d a l i ( 1 ) and Par k i n s o n ( 1 1 ) . T h e i r method of u s i n g sources on the body i n the wake r e g i o n t o c r e a t e the wake FIGURE (30) : AIRFOIL IN THE Z-PLANE oo 8 5 . i s a l s o pursued here. J a n d a l i c o n s i d e r s both 1 and 2-source models f o r a s o l i d a i r f o i l w i t h a normal s p o i l e r . The c u r r e n t t h e o r y i s not l i m i t e d t o a normal s p o i l e r . Use of the 2-source model a l l o w s the p r e s s u r e t o be s t i p u l a t e d on both s e p a r a t i n g f r e e s t r e a m l i n e s . The 1-source model allows the p r e s s u r e t o be s t i p u l a t e d on o n l y one of the s e p a r a t i n g f r e e s t r e a m l i n e s . The o t h e r v a l u e f l o a t s f r e e l y , but was shown by J a n d a l i t o be f i n i t e . The i t e r a t i v e method of p o s i t i o n -i n g the source i n the n u m e r i c a l technique employed i n the c u r r e n t t h e o r y makes i t extremely d i f f i c u l t t o c o n s i d e r the 2-source prob-lem. In the 2-source model th e r e are f i v e unknowns; the two source s t r e n g t h s , two source p o s i t i o n s and one unknown c i r c u l a t i o n . There a r e however o n l y f o u r c o n d i t i o n s ; the two K u t t a c o n d i t i o n s , and the s t i p u l a t i o n of the base p r e s s u r e a t the s p o i l e r t i p and at the t r a i l -i n g edge of the a i r f o i l . T h i s problem w i l l be d i s c u s s e d f u r t h e r when th e 2-source model i s t r e a t e d i n d e t a i l . 7 o 3 Boundary C o n d i t i o n s The boundary c o n d i t i o n s f o r the 1-source problem, i t e m i z e d i n d e t a i l , are as f o l l o w s : ( i ) K u t t a c o n d i t i o n a t the t r a i l i n g edge and s p o i l e r t i p . ( i i ) P r e s s u r e s t i p u l a t e d on the s e p a r a t i o n f r e e s t r e a m l i n e a t the s p o i l e r t i p . ( i i i ) S u r f a c e normal v e l o c i t y c o n d i t i o n of no f l o w through the s u r f a c e . A l t h o u g h Smith's t h e o r y i s an exact a n a l y t i c a l t h e o r y , n u m e r i c a l app-r o x i m a t i o n s are n e c e s s a r y i n o b t a i n i n g a s o l u t i o n . The a i r f o i l i s d i v i d e d i n t o s t r a i g h t l i n e elements the c e n t r e s of which are the c o n t r o l p o i n t s . The K u t t a c o n d i t i o n a t the t r a i l i n g edge i s then s a t -i s f i e d by matching the t a n g e n t i a l v e l o c i t y a t the l a s t c o n t r o l p o i n t 8 6 . on the upper s u r f a c e t o the t a n g e n t i a l v e l o c i t y a t the l a s t c o n t r o l p o i n t on the lower s u r f a c e . T h i s matching technique a p p l i e s i d e n t i -c a l l y i n 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 a t the s p o i l e r t i p . I t must be r e a l i z e d t h a t these c o n d i t i o n s are not a p p l i e d r i g h t a t the se-p a r a t i o n p o i n t , but at the l a s t c o n t r o l p o i n t s b e f o r e the s e p a r a t i o n p o i n t . T h i s i s i n h e r e n t i n the method and cannot be avoided. Hence i n s a t i s f y i n g c o n d i t i o n ( i i ) , the p r e s s u r e i s a c t u a l l y s t i p u l a t e d at the l a s t c o n t r o l p o i n t b e f o r e the s p o i l e r t i p . There are then t h r e e c o n d i t i o n s t o be s a t i s f i e d by t h r e e unknowns. The unknowns are; the source p o s i t i o n , the source s t r e n g t h and the c i r c u l a t i o n about the a i r f o i l . C o n d i t i o n ( i i i ) a l s o can o n l y be s a t i s f i e d a t the con-t r o l p o i n t s and t h i s i s done i n the u s u a l way of p u t t i n g a d i s t r i b u -t i o n of sources over the elements. F o r the 2-source model c o n d i t i o n s ( i ) and ( i i i ) remain unchanged and c o n d i t i o n ( i i ) becomes: ( i i ) P r e s s u r e s t i p u l a t e d on the s e p a r a t i o n f r e e s t r e a m l i n e a t the s p o i l e r t i p and a t the t r a i l i n g edge. 7 . 4 1-Source Model F i r s t the method of s o l v i n g t h i s problem f o r the 1-source model w i l l be d e s c r i b e d . Suppose t h a t a source i s p o s i t i o n e d somewhere on the a i r f o i l between the s p o i l e r and the t r a i l i n g edge. The s o l u t i o n can then be determined d i r e c t l y by 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 at the s p o i l e r t i p of the a i r f o i l , and by s t i p u l a t i n g the p r e s s u r e on the l a s t element b e f o r e the s p o i l e r t i p . The t r a i l i n g edge i s then observed t o see i f the K u t t a c o n d i t i o n there i s s a t i s f i e d . I f i t i s not s a t i s f i e d the source can be moved alo n g the s u r f a c e of the a i r -f o i l , and the s o l u t i o n determined a g a i n . T h i s procedure i s repeated u n t i l i t i s observed t h a t the K u t t a c o n d i t i o n i s s a t i s f i e d at the t r a i l i n g edge. When t h i s occurs the c o r r e c t s o l u t i o n has been d e t e r -mined. 8 7 . In o rder t o e f f e c t a s o l u t i o n a l a r g e c i r c u l a r f i l l e t i s p l a c -ed behind the a i r f o i l s p o i l e r . T h i s i s n e c e s s a r y s i n c e the s p o i l e r otherwise i s i n f i n i t e l y t h i n and not adaptable t o Smith's method. Such a m o d i f i c a t i o n t o the a i r f o i l and s p o i l e r i n the wake r e g i o n has no e f f e c t on the problem s o l u t i o n . M o d i f i c a t i o n t o the a i r f o i l s u r f a c e i n t h i s r e g i o n was a l s o employed by J a n d a l i . In p o s i t i o n i n g the source onset f l o w care must be taken t o ensure t h a t as the source moves al o n g the s u r f a c e , i t passes smoothly through the c o n t r o l p o i n t s , I t now remains t o adopt the above arguments i n t o a mathematical f o r m u l a t i o n of the problem. I n t h i s c o n f i g u r a t i o n t h e r e are t h r e e onset flows t o the a i r f o i l , namely the f r e e stream, the co n c e n t r a t e d source f l o w and the c i r c u l a t o r y f l o w . F o l l o w i n g Smith's technique the c i r c u l a t o r y f l o w Is c r e a t e d by d i s t r i b u t i n g v o r t i c i t y over the elements of the a i r f o i l . Suppose A^j , an element of m a t r i x A, i s the normal v e l o c i t y a t c o n t r o l p o i n t i due t o a u n i t source d i s t r i b u t i o n over element j , and , an element of mat r i x B, i s the t a n g e n t i a l v e l o c i t y . Then -B^ i s the normal v e l o c i t y a t c o n t r o l p o i n t i due t o a u n i t v o r t i c i t y d i s t r i b u t i o n over element j , and A^j i s the t a n -g e n t i a l v e l o c i t y . The normal v e l o c i t y i s defined as p o s i t i v e out-wards from the a i r f o i l and the t a n g e n t i a l v e l o c i t y i s d e f i n e d as p o s i t i v e i n a c l o c k w i s e d i r e c t i o n s t a r t i n g from the l e a d i n g edge. I f Vn<i and are the normal and t a n g e n t i a l v e l o c i t i e s r e s p e c t -i v e l y , due t o the co n c e n t r a t e d source of u n i t s t r e n g t h and and those due to a u n i t f r e e stream, the normal v e l o c i t y boundary c o n d i t i o n ( i i i ) can be w r i t t e n as where X i s the uniform v o r t i c i t y s t r e n g t h , X the co n c e n t r a t e d source s t r e n g t h and o? the s t r e n g t h of the source d i s t r i b u t i o n on the |+h 8 8 . element. Suppose t h a t s t a r t i n g from the l e a d i n g edge i n a c l o c k w i s e manner N, i s the number of the l a s t element b e f o r e the s p o i l e r t i p and the f i r s t element a f t e r the s p o i l e r t i p . Suppose a l s o t h a t N, and N 4 are r e s p e c t i v e l y the l a s t element b e f o r e and the f i r s t element a f t e r the t r a i l i n g edge. The K u t t a c o n d i t i o n at the s p o i l e r t i p then g i v e s the e q u a t i o n : A p p l i c a t i o n of boundary c o n d i t i o n (11) g i v e s the f i n a l e q u a t i o n : where C p i s the base p r e s s u r e , obtained e x p e r i m e n t a l l y , and e m p i r i -c a l l y e ntered i n t o the s o l u t i o n . These equations can be combined t o g i v e the equation: T h i s e s s e n t i a l l y r a i s e s the order of the m a t r i x A by two and can be s o l v e d d i r e c t l y . The K u t t a c o n d i t i o n a t the t r a i l i n g edge can then be expressed m a t h e m a t i c a l l y as The c o n c e n t r a t e d source must be i t e r a t i v e l y moved and the s o l u t i o n 89. r e c a l c u l a t e d u n t i l t h i s e q u a t i o n i s s a t i s f i e d . Once t h i s e q u a t i o n i s s a t i s f i e d the t a n g e n t i a l v e l o c i t y a t the <.+n c o n t r o l p o i n t , , i s g i v e n by The p r e s s u r e c o e f f i c i e n t a t the iA\\ c o n t r o l p o i n t denoted by C p i > i s Cp^ » i - TjL . . 7.5 2-Source Model The 2-source model i s s o l v e d i n an i d e n t i c a l manner. I t was p o i n t e d out i n s e c t i o n (7.2) t h a t i n t h i s problem t h e r e are f i v e unknowns w i t h o n l y f o u r c o n d i t i o n s . J a n d a l i (1) used the z e r o l i f t i n c i d e n c e as the e x t r a c o n d i t i o n . That c o n d i t i o n i s not adaptable to t h i s t h e o r y and some oth e r c o n d i t i o n i s n e c e s s a r y . J a n d a l i found t h a t the second source f e l l on the a i r f o i l between the f i r s t source and the t r a i l i n g edge and was weaker i n s t r e n g t h than the f i r s t sou-r c e . He found t h a t i f the second source was moved towards the t r a i l -i n g edge i t s s t r e n g t h approached z e r o , and the 2-source model appr-oached the 1-source model. In the c u r r e n t work i t i s not proposed t o s e a r c h f o r another c o n d i t i o n s u i t a b l e f o r t h i s theory, but merely to demonstrate how the boundary c o n d i t i o n s can be s a t i s f i e d , and how the n u m e r i c a l technique shows c o n s i s t e n c y w i t h J a n d a l i ' s r e s u l t s . The second source i n the 2-source model i s t h e r e f o r e merely f i x e d on the a i r f o i l s u r f a c e i n a p o s i t i o n t h a t makes the present t h e o r y agree w i t h J a n d a l i ' s 2-source model. T h i s c o n v e n i e n t l y p o s i t i o n s the second source between the f i r s t source and the t r a i l i n g edge of the a i r f o i l as i n J a n d a l i ' s t h e o r y . 90. Suppose V,M and V.+. are the normal and t a n g e n t i a l v e l o c i t i e s r e s -p e c t i v e l y due to the second source of u n i t s t r e n g t h . I f X f c i s the s t r -ength of the second source, and the f i r s t source s t r e n g t h and v e l -o c i t i e s are g i v e n the s u b s c r i p t 1, then development of t h i s problem as i n the 1-source problem g i v e s the equation: Once a g a i n t h i s e q u a t i o n must be s o l v e d and the K u t t a c o n d i t i o n at the t r a i l i n g edge observed. The f i r s t source, as i n the 1-source model must be i t e r a t i v e l y r e l o c a t e d u n t i l the K u t t a c o n d i t i o n at the t r a i l i n g edge i s s a t i s f i e d . S ince the p r e s s u r e , and hence the v e l o c i t y , has been s t i p u l a t e d on the l a s t c o n t r o l p o i n t b e f o r e the t r a i l i n g edge on the upper s u r f a c e of the a i r f o i l , t h i s K u t t a cond-i t i o n can be expressed m a t h e m a t i c a l l y as Once t h i s e q u a t i o n i s s a t i s f i e d the pressure c o e f f i c i e n t at the i+h c o n t r o l p o i n t i s where 91. The p r e s s u r e c o e f f i c i e n t can be n u m e r i c a l l y i n t e g r a t e d t o determine the l i f t c o e f f i c i e n t f o r any case d e s i r e d . T h i s completes the s o l -u t i o n t o the steady s t a t e a i r f o i l s p o i l e r problem f o r both 1 and 2-source models. R e s u l t s from the above theory f o r the 1 and 2-source models are presented i n s e c t i o n ( 9 K Some comment i s warranted on the n u m e r i c a l procedures. G e n e r a l -l y when g r a d i n g the elements and d e t e r m i n i n g element s i z e , the guide-l i n e s g i v e n by Smith (6) are s a t i s f a c t o r y . I t has been found necess-ary i n t h i s problem t o reduce the element s i z e on the underside of the a i r f o i l i n t h a t r e g i o n o p p o s i t e the concentrated sources which are s i t u a t e d on the upper s u r f a c e . R e l a t i v e l y speaking these sources are q u i t e s t r o n g and t h e element s i z e should be much l e s s than the d i s t a n c e between the elements and the source. A v a l u e of 110 e l e -ments was found t o g i v e good accuracy. On an I.B.M. 3°0/67 computer the i t e r a t i v e s o l u t i o n t o t h i s problem f o r 110 elements has a t y p i -c a l e x e c u t i o n time of 60 seconds f o r both 1 and 2-source models. T h i s time quoted I n c l u d e s s i x i t e r a t i o n s of r e l o c a t i n g the source p o s i t -i o n . 92. SECTION 8 AIRFOIL WITH A SLOTTED FLAP AND A SPOILER 8 . 1 F o r m u l a t i o n of the Problem The a i r f o i l i s p o s i t i o n e d i n the z-plane as d e s c r i b e d i n sec-t i o n ( 3 . 2 ) . The c o n f i g u r a t i o n i s shown p i c t o r l a l l y i n f i g u r e ( 3 1 ) . In t h i s case of a s l o t t e d f l a p , the problem i s e s s e n t i a l l y a two body problem where one of the bodies i s of the type d e s c r i b e d i n s e c t i o n ( 7 ) , and the o t h e r body, the f l a p , i s a b a s i c a i r f o i l . There i s one e x t r a unknown, the c i r c u l a t i o n about the f l a p , and one e x t r a boundary c o n d i t i o n , the K u t t a c o n d i t i o n at the f l a p t r a i l i n g edge. The c h o i c e s of an i t e r a t i v e or a d i r e c t s o l u t i o n , as presented by Smith (6) are open t o such a m u l t i p l e body problem. Both methods were t r i e d and the d i r e c t method was chosen as s i m p l e r and s h o r t e r i n com-p u t a t i o n time. 8 . 2 Boundary C o n d i t i o n s The boundary c o n d i t i o n s f o r the 1-source problem a r e : ( i ) K u t t a c o n d i t i o n s a t the main f o i l t r a i l i n g edge, f l a p t r a i l i n g edge and the s p o i l e r t i p . ( i i ) P r e s s u r e s t i p u l a t e d on the s e p a r a t i o n f r e e s t r e a m l i n e at the s p o i l e r t i p . ( i i i ) S u r f a c e normal v e l o c i t y c o n d i t i o n of no flow through the s u r f a c e . F o r the 2-source model boundary c o n d i t i o n ( i i ) becomes: ( i i ) P r e s s u r e s t i p u l a t e d on the s e p a r a t i o n f r e e s t r e a m l i n e s a t the s p o i l e r t i p and main f o i l t r a i l i n g edge. 8 . 3 1-Source Model The manner i n which t h i s problem Is s o l v e d f o l l o w s i d e n t i c a l l y z-plane F I G U R E (31): AIRFOIL I N T H E Z - P L A N E 94. the procedure d e s c r i b e d i n s e c t i o n s (7 .4 ) and ( 7 . 5 ) ' Suppose -BT'k i s the normal v e l o c i t y a t c o n t r o l p o i n t i , on the main f o i l or f l a p , due t o a u n i t v o r t i c i t y d i s t r i b u t i o n over element k, on the main f o i l , and A^ 'k i s the t a n g e n t i a l v e l o c i t y . S i m i l a r l y -Bjg and A^j are the v e l o c i t i e s due to a u n i t v o r t i c i t y d i s t r i b u t i o n over element Jl, on the f l a p . The normal v e l o c i t y boundary c o n d i t i o n ( i i i ) becomes where Jf,, i s the v o r t i c i t y s t r e n g t h on the main f o i l and tt, the v o r -t i c i t y s t r e n g t h on the f l a p . Suppose t h a t the elements are numbered from the l e a d i n g edge of the main f o i l c l o c k w i s e around the main f o i l , and then from the l e a d i n g edge of the f l a p c l o c k w i s e around the f l a p . A l l symbols and d e f i n i t i o n s g i v e n i n s e c t i o n (7) a p p l y d i r e c t l y t o t h i s s o l u t i o n . Suppose t h a t N , i s the number of the l a s t element b e f o r e the f l a p t r a i l i n g edge and NFC i s the f i r s t element a f t e r the f l a p t r a i l i n g edge. The K u t t a c o n d i t i o n a t the t r a i l i n g edge of the f l a p g i v e s the equation: The K u t t a c o n d i t i o n at the s p o i l e r t i p g i v e s the equation: A p p l i c a t i o n o f b o u n d a r y c o n d i t i o n ( i i ) g i v e s the f i n a l equation: * B«.i7 * + ^ »A*% * ^  - ^ - c P b T h e s e e q u a t i o n s c a n b e c o m b i n e d , t o g i v e t h e e q u a t i o n : 9 5 . -W .T I V K SE This equation can be solved d i r e c t l y . The Kutta condition at the t r a i l i n g edge of the main can be expressed as Once t h i s equation has been s a t i s f i e d the tangential v e l o c i t y at the -t+h control point i s and the pressure c o e f f i c i e n t i s 8 . 4 2-Source Model The discussion of the 2-source model given i n section ( 7 . 5 ) i s f u l l y applicable to t h i s s o l u t i o n . Development of the problem as given i n section ( 7 . 5 ) and ( 8 . 3 ) leads to the equation The K u t t a c o n d i t i o n becomes Once t h i s e q u a t i o n i s s a t i s f i e d the pr e s s u r e c o e f f i c i e n t a t the -c+K c o n t r o l p o i n t i s where Once a g a i n t h i s p r e s s u r e c o e f f i c i e n t can be n u m e r i c a l l y i n t e g r a t e d t o determine the l i f t c o e f f i c i e n t . T h i s s o l v e s the problem of the steady s t a t e a i r f o i l w i t h a s l o t t e d f l a p and a s p o i l e r . In s o l v i n g t h i s problem care should be taken t o f o l l o w the g u i d e l i n e s g i v e n a t the end of s e c t i o n ( ? ) . The number of elements needed i s approximately 100 f o r the main f o i l and. 80 f o r the f l a p . With such a number of elem'ents the e x e c u t i o n time f o r t h i s s o l u t i o n i s a p p r o x imately k minutes f o r both 1 and 2-source models. T h i s time 97. i n c l u d e s s i x i t e r a t i v e changes In the source p o s i t i o n . 98. SECTION 9 RESULTS AND COMPARISONS In the f i r s t p a r t of t h i s s e c t i o n the t h e o r e t i c a l r e s u l t s f o r a s o l i d a i r f o i l w i t h a s p o i l e r are p r e s e n t e d . These r e s u l t s are com-pared w i t h t h e o r e t i c a l r e s u l t s obtained by J a n d a l l ( 1 ) . The l a t t e r p a r t of t h i s s e c t i o n shows the s o l u t i o n to the problem of an a i r f o i l w i t h a s p o i l e r and a s l o t t e d f l a p . E x p e r i m e n t a l or other t h e o r e t i c a l r e s u l t s are not a v a i l a b l e f o r t h i s case, and hence on l y the t h e o r e t -i c a l change i n the p r e s s u r e d i s t r i b u t i o n over the b a s i c f o i l and f l a p i s g i v e n . The second source i n the t h e o r i e s developed i n s e c t i o n s (7 .5) and ( 8 . 4 ) was a r b i t r a r i l y l o c a t e d between the f i r s t source and the t r a i l i n g edge. I t was d i s c u s s e d i n s e c t i o n (7 .5) t h a t some f u r t h e r c o n d i t i o n i s n e c e s s a r y to f i x the p o s i t i o n of t h i s second source. I f the c i r c u l a t i o n about the a i r f o i l i s n e g l e c t e d , one of the e x i s t -i n g c o n d i t i o n s could be used f o r a mathematical s o l u t i o n t o the pro-blem. However, such n e g l e c t of the c i r c u l a t i o n g i v e s an erroneous r e s u l t , v a r y i n g w i d e l y from experiment. I t i s t h e r e f o r e n e c e s s a r y to i n c l u d e the c i r c u l a t i o n , and look f o r another c o n d i t i o n . I t w i l l be r e c a l l e d from s e c t i o n (7 .5) t h a t t h i s second source i s much weak-er than the f i r s t source, and approaches z e r o s t r e n g t h as i t s l o c -a t i o n approaches the t r a i l i n g edge. The 2-source model then approach-es the 1-source model. The 2-source model agrees more c l o s e l y w i t h experiment and s a t i s f i e s a l l the boundary c o n d i t i o n s . The t h e o r e t -i c a l r e s u l t s presented here are intended t o demonstrate t h i s p o i n t and to show c o n s i s t e n c y w i t h J a n d a l i 1 s r e s u l t s . 9.1 S o l i d A i r f o i l w i t h a S p o i l e r The r e s u l t s g i v e n in. t h i s s e c t i o n were determined by J a n d a l l (1) 9 9 . f o r a 14$ C l a r k Y a i r f o i l shoism i n f i g u r e ( 3 0 ) . The s p o i l e r i s l o c -ated a t the 70% chord p o i n t . Since the theory presented by J a n d a l i i s l i m i t e d t o normal s p o i l e r s , t h e o r e t i c a l comparisons can o n l y be presented f o r t h i s case. F i r s t f i g u r e s are presented comparing the c u r r e n t r e s u l t s f o r normal s p o i l e r s w i t h those obtained t h e o r e t i c a -l l y by J a n d a l i . Some r e s u l t s f o r s p o i l e r s of v a r y i n g angles f o r the 1-source model then f o l l o w . A l l angles of i n c i d e n c e presented f o r the C l a r k Y are measured from the lower s u r f a c e . S p o i l e r h e i g h t s are g i v e n as a percentage of the a i r f o i l c hord. F i g u r e ( 32 ) shows a comparison between the p r e s s u r e d i s t r i b u -t i o n s f o r the J a n d a l i 1-source model and the present 1-source model f o r a 10% normal s p o i l e r a t an a i r f o i l i n c i d e n c e of 12 degrees. F i g -ure (33) shows the c o r r e s p o n d i n g comparison f o r the 2-source model. In these r e s u l t s the chord i s 4.043 as g i v e n by J a n d a l i . The agree-ment between t h e o r i e s i s seen t o be v e r y good. The agreement between t h e o r y and experiment was shown by J a n d a l i t o be q u i t e good except i n the r e g i o n i n f r o n t of the s p o i l e r , where t h e r e i s a c t u a l l y a r e g i o n of s e p a r a t e d f l o w . P o t e n t i a l t h e o r y i s not a b l e t o model such a f l o w . A n a l y t i c a l l y any such sharp concave c o m e r w i l l produce a s t a g n a t i o n p o i n t , and hence i n t h i s r e g i o n the t h e o r y d i v e r g e s from experiment. In the s u r f a c e s i n g u l a r i t y t h e o r y employed i t w i l l be r e c a l l e d t h a t the f l o w p r o p e r t i e s are c o n s i d e r e d a t c o n t r o l p o i n t s t h a t are s l i g h t l y removed from such sharp c o r n e r s , and t h e o r e t i c a l -l y the s t a g n a t i o n p o i n t w i l l never be reached. The trend towards a s t a g n a t i o n p o i n t however, i s s t r o n g l y e v i d e n t . F i g u r e (34) shows a comparison between the J a n d a l i 1-source model and the present 1-source model f o r a 5$ normal s p o i l e r at an a i r f o i l i n c i d e n c e of 10 degrees. The c o r r e s p o n d i n g comparison f o r the 2-source model i s pre-sented i n f i g u r e (35). Although o n l y r e p r e s e n t a t i v e r e s u l t s have s/css 0.7 ; W/caO.I chord>4-043 F I G U R E (32): P R E S S U R E D I S T R I B U T I O N F O R C L A R K Y A I R F O I L W I T H S P O I L E R £ o s/c at O.l , h/c » O . I S = 9 0 ° , a 1 2 * ckoroia4.04i T h e o r y z - s o u r c e T c m d a l i 2 - S O u r c e F I G U R E (33): P R E S S U R E D I S T R I B U T I O N F O R C L A R K Y A I R F O I L W I T H S P O I L E R s/ c= 0.7 , h/c • O.OS c W o r d »X.04-3 F I G U R E ( 3 4 ) : P R E S S U R E D I S T R I B U T I O N F O R C L A R K Y A I R F O I L V / I T H S P O I L E R £ s/c »0.7 , h/c »O.Of C h o r d » A.O AW Theory 2 - S o u r c e J a n d a l i 2.- S o u r c e -3 F I G U R E (35): P R E S S U R E D I S T R I B U T I O N F O R C L A R K Y A I R F O I L W I T H S P O I L E R 104. been presented i t can be seen t h a t the agreement i s e x c e l l e n t , and t h a t the two t h e o r i e s are c o n s i s t e n t . Next I t i s demonstrated how the t h e o r e t i c a l p r essure d i s t r i -b u t i o n changes as the s p o i l e r angle i s p r o g r e s s i v e l y i n c r e a s e d t o 90 degrees. In f i g u r e (36) the p r e s s u r e d i s t r i b u t i o n s f o r the 1-source model are presented f o r the C l a r k Y a i r f o i l w i t h a 10% s p o i -l e r a t the 10% chord p o i n t w i t h angles of 30» 6 0 , and 90 degrees, and these are compared to the t h e o r e t i c a l d i s t r i b u t i o n f o r the case of no s p o i l e r . The a i r f o i l i n c i d e n c e i n a l l cases i s 12 degrees. I t can be seen how the p r e s s u r e peak and the a r e a between the curve and the chord l i n e p r o g r e s s i v e l y decrease as the s p o i l e r angle i n -c r e a s e s . Such decreases s i g n i f y a p r o g r e s s i v e l o s s i n l i f t . The bas-i c a i r f o i l p r e s s u r e d i s t r i b u t i o n was c a l c u l a t e d u s i n g Smith's t h e o r y ( 6 ) , and i t can be once a g a i n seen t h a t the t h e o r y i s unable to pre-d i c t a s t a g n a t i o n p o i n t a t the f i n i t e angle t r a i l i n g edge. I t i s more n o t i c e a b l e f o r the cases w i t h i n c l i n e d s p o i l e r s t h a t , a l t h o u g h the p r e s s u r e d i s t r i b u t i o n i n f r o n t of the s p o i l e r tends towards a s t a g n a t i o n p o i n t , i t does not a c t u a l l y r each i t . The c o n s i d e r a t i o n of c o n t r o l p o i n t s around such a concave c o r n e r as the s p o i l e r base has the a f f e c t of rounding the c o r n e r and the t h e o r y i s not expected to model the s t a g n a t i o n p o i n t c o r r e c t l y . 9.2 A i r f o i l w i t h a Slotted. F l a p and a S p o i l e r The r e s u l t s presented In t h i s s e c t i o n are f o r the NACA 23012 a i r f o i l w i t h a 2 5 . 6 6 $ s l o t t e d f l a p shown i n f i g u r e ( 3 1 ) . The b a s i c 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 are shown i n f i g u r e (37) f o r t h i s a i r f o i l a t 8 degrees i n c i d e n c e and 20 degrees f l a p a n g l e . The e x p e r i m e n t a l r e s u l t s were obtained from r e f e r e n c e ( 1 2 ) . S i n c e no e x p e r i m e n t a l r e s u l t s are a v a i l a b l e f o r the s p o i l e r F L A P WITHOUT SPOILER 107. case o n l y r e p r e s e n t a t i v e examples of the t h e o r e t i c a l r e s u l t s are shown. I t w i l l be demonstrated how the present 2-source model i s s u p e r i o r t o the 1-source model. The pre s s u r e d i s t r i b u t i o n s , f o r the a i r f o i l w ith a s p o i l e r a t v a r y i n g angles w i l l be compared t o the b a s i c d i s t r i b u t i o n c a l c u l a t e d u s i n g Smith's t h e o r y (6). In the f o l -lowing r e s u l t s the s p o i l e r i s p o s i t i o n e d a t the 60% chord p o i n t . The a i r f o i l i n c i d e n c e i s 8 degrees and the a i r f o i l chord i s 4.0. The f l a p angle i s 20 degrees. The e m p i r i c a l i n p u t of base p r e s s u r e c o e f f i c i e n t i s s e t a t -1.0 In the absence of ex p e r i m e n t a l r e s u l t s . F i g u r e (38) i s a comparison between the present 1-and 2-source mod-e l s f o r a 10% normal s p o i l e r . T h i s comparison shows the 1-source model g i v i n g an u n r e a l i s t i c a l l y h i g h p r e s s u r e peak on the f l a p . I t w i l l be r e c a l l e d t h a t i n the 1-source model o n l y the K u t t a c o n d i t i o n i s s t i p u l a t e d a t the main f o i l t r a i l i n g edge, and the s e p a r a t i o n p r e s s u r e c o e f f i c i e n t , a l t h o u g h f i n i t e , determines i t s own v a l u e . In t h i s case i t determines a v e r y n e g a t i v e v a l u e of l e s s than -2. The main f o i l t r a i l i n g edge i s so c l o s e t o the f l a p t h a t t h i s h i g h neg-a t i v e v a l u e d i r e c t l y a f f e c t s the n e g a t i v e p r e s s u r e peak on the f l a p , making i t u n r e a l i s t i c a l l y h i g h . I t would be expected then t h a t i f the s e p a r a t i o n p r e s s u r e was s t i p u l a t e d a t the main f o i l t r a i l i n g edge, as i n the 2-source model, a d i r e c t e f f e c t would be n o t i c e d on the f l a p . T h i s p o i n t i s Indeed demonstrated i n f i g u r e (38), where i t can be seen t h a t the 2-source model g i v e s a much b e t t e r r e s u l t than the 1-source model. The second source i n t h i s case has been a r b i t r a r i l y l o c a t e d between the f i r s t source and the t r a i l i n g edge of the main f o i l . F o r t h i s reason the s o l u t i o n f o r the 2-source mod-e l w i l l not be pursued f u r t h e r . The e f f e c t of s p o i l e r angle i s shown i n f i g u r e (39). In t h i s f i g u r e the t h e o r e t i c a l r e s u l t f o r a b a s i c a i r f o i l w i t h a s l o t t e d s/c» o.d>, S» 90° 06 » g* ; *2=> 20° 2- Source / - Source F I G U R E ( 3 8 ) : P R E S S U R E D I S T R I B U T I O N F O R N A C A 2 3 0 1 2 A I R F O I L A N D S L O T T E D £ F L A P W I T H S P O I L E R co 109. 110. f l a p i s presented -with the 1-source model f o r a 10$ s p o i l e r at ang-l e s of 45 and 90 degrees. Once a g a i n the c h a r a c t e r i s t i c l o s s e s i n p r e s s u r e c o e f f i c i e n t can be observed as the s p o i l e r angle I n c r e a s e s . SECTION 10 CONCLUSION The l i f t g i v e n by the l i n e a r i z e d c a v i t y p o t e n t i a l t h e o r y dev-eloped i n p a r t I shows good agreement wi t h experiment. The pressure p r e d i c t e d by the t h e o r y g i v e s o n l y q u a l i t a t i v e i n f o r m a t i o n about the p r e s s u r e d i s t r i b u t i o n on the a i r f o i l . Such s i n g u l a r i t i e s i n the p r e s s u r e d i s t r i b u t i o n as g i v e n by the t h e o r y are i n h e r e n t i n l i n e a r -i z e d t e c h n i q u e s . Only t h e o r e t i c a l r e s u l t s f o r the s p o i l e r a c t u a t i o n problem have been pr e s e n t e d . E x p e r i m e n t a l measurement i s c u r r e n t l y b e i n g done i n an attempt t o v e r i f y these r e s u l t s . In p a r t I I the t h e o r y was shox-m t o g i v e p r e s s u r e d i s t r i b u t i o n s c o n s i s t e n t w i t h J a n d a l l r s t h e o r y (1) f o r the s o l i d a i r f o i l w i t h a normal s p o i l e r . J a n d a l i shox^ed t h a t t h i s r e s u l t was i n good agree-ment w i t h experiment except f o r a s m a l l r e g i o n i n f r o n t of the s p o i -l e r . I n t h i s r e g i o n the a c t u a l f l o w separates and the t h e o r e t i c a l p r e d i c t i o n of a s t a g n a t i o n p o i n t i s not i n agreement w i t h experiment. A l t h o u g h no e x p e r i m e n t a l or o t h e r t h e o r e t i c a l r e s u l t s are a v a i l a b l e f o r the case of a n g u l a r s p o i l e r s , the good agreement f o r the normal s p o i l e r case i s an i n d i c a t i o n t h a t these r e s u l t s most l i k e l y agree c l o s e l y w i t h experiment. F o r the case of an a i r f o i l x^ith a s p o i l e r and a s l o t t e d f l a p , e x p e r i m e n t a l r e s u l t s are not a v a i l a b l e . The 2-source model, which g i v e s much b e t t e r r e s u l t s f o r t h i s case than the 1-source model, i s p r o b a b l y i n q u i t e good agreement w i t h experiment. I n v e s t i g a t i o n i s n e c e s s a r y f o r a f u r t h e r c o n d i t i o n t o f i x the p o s i t i o n of the second s o u r c e . I t should a l s o be v e r i f i e d t h a t the t h e o r y does i n f a c t agree w i t h experiment. REFERENCES 1. J a n d a l i , T. 1970 "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 Columbia. 2 . Woods, L.C. 1953 "Theory f o r A i r f o i l S p o i l e r s " A.R.C. R&M 2 9 6 9 . 3. Barnes, C S . I965 "A Developed Theory of S p o i l e r s on A i r f o i l s " A.R.C. C P . 8 8 7 . 4 . P a r k i n , B.R. 1959 " L i n e a r i z e d Theory of C a v i t y Flow i n Two-Dimensions" RAND P - 1 7 4 5 . 5. F a b u l a , A.G. I 9 6 2 " T h i n - A i r f o i l Theory A p p l i e d t o H y d r o f o i l s w i t h a S i n g l e F i n i t e C a v i t y and A r b i t r a r y Free S t r e a m l i n e Detachment" J.F.M. pp. 227-240. 6. Hess, J.L.&Smith A.M.O. 1966 " C a l c u l a t i o n of P o t e n t i a l Flow about A r b i t r a r y B o dies" Prog, i n Aero. S c i e n c e s V o l . 8 , Per-gamon P r e s s , Oxford. 7 . G i e s i n g , J.P. I 9 6 5 " P o t e n t i a l Flow about Two-Dimenslonal A i r -f o i l s " Rept. LB 3 1 9 4 6 , Douglas A i r c r a f t Co. 8 . B i s p l i n g h o f f , R.L., A s h l e y , H.& Halfman, R.L. 1955 " A e r o e l a s t -i c i t y " Addison-Wesley P u b l i s h i n g Co. pp. 284 - 2 8 5 . 9 . Pope, A.& Harper, J . J . I 9 6 6 "Low-Speed Wind Tunnel T e s t i n g " John W i l e y & Sons. 1 0 . M a s k e l l , E . C I 9 6 3 "A Theory of Blockage E f f e c t s on B l u f f Bod-i e s and S t a l l e d Wings i n a C l o s e d Wind Tunnel" R&M 3400. 1 1 . P a r k i n s o n , G.V.& J a n d a l i , T. 1970 "A Wake Source Model f o r B l u f f Body P o t e n t i a l Flow" J.F.M. V o l . 40, pp. 5 7 7 - 5 9 4 . 1 2 . Wenzinger, C.J.& Delano, J.B. 1938 "Pressure D i s t r i b u t i o n over an NACA 23012 A i r f o i l w i t h a S l o t t e d and a P l a i n F l a p " T.R. No. 6 3 3 , NACA. 

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