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UBC Theses and Dissertations

The effect of twist on flexible hydrofoils Milne, John Gordon 1974

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THE EFFECT OF TWIST ON FLEXIBLE HYDROFOILS by JOHN GORDON MILNE B.A. Sc . , University of B r i t i sh Columbia, 1972 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF APPLIED SCIENCE in the Department of C i v i l Engineering We accept th is thesis as conforming to the required standard Marchi 1974 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 an 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 make i t f r e e l y a v a i l a b l e f o r r e f e r e n c e and s t u d y . I f u r t h e r 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 by t h e Head o f my Depar tment o r by 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 be a l l o w e d w i t h o u t my w r i t t e n p e r m i s s i o n . Depar tment o f C i v i l E n g i n e e r i n g The U n i v e r s i t y o f B r i t i s h C o l u m b i a V a n c o u v e r 8, B . C . , Canada Date M a r c h ' 1 9 7 4 i i ABSTRACT F l e x i b l e low a s p e c t r a t i o h y d r o f o i l s were t e s t e d t o i n v e s t i g a t e t h e e f f e c t o f t w i s t on t h e l i f t and d r a g c h a r a c t e r i s t i c s and i n p a r t i c u l a r , t h e h i g h i n d u c e d d r a g f o r c e . F l e x i b l e models w i t h s t i f f e n e r s a t v a r i o u s p o s i t i o n s and r i g i d mode ls w i t h a s e t t w i s t were t e s t e d i n a w a t e r f l u m e and the l i f t and d r a g f o r c e s were measured o v e r a r a n g e o f a n g l e s o f a t t a c k . I t was c o n f i r m e d t h a t t w i s t i n g can be used as an a l t e r n -a t i v e t o t a p e r i n g t o p r o d u c e e l l i p t i c l o a d i n g . T w i s t i n g o f t a p e r e d w i n g s was shown t o be b e n e f i c i a l i n t h e c a s e s t e s t e d and t h e b e n e f i t s a r e i n c o n t r a d i c t i o n t o t h e s i m p l e t h e o r y o f e l l i p t i c l o a d i n g . I t was f ound t h a t t w i s t i n g i s more b e n e f i c i a l t h a n t h e s i m p l e t h e o r y shows and t h a t t h e t w i s t e d w i n g i s l e s s s e n s i t i v e t o changes i n a n g l e o f a t t a c k . i i i ACKNOWLEDGEMENTS The a u t h o r w i s h e s t o t h a n k h i s s u p e r v i s o r , D r . M .C . Q u i c k , f o r h i s a s s i s t a n c e and e n c o u r a g e m e n t t h r o u g h o u t t h e c o u r s e o f t h i s p r o j e c t . The a u t h o r a l s o w i s h e s t o t h a n k t h e V a n c o u v e r P u b l i c A q u a r i u m f o r t h e i r h e l p w h i c h a l l o w e d t h e a u t h o r t o o b t a i n measurements o f t h e k i l l e r w h a l e f l u k e . F i n a l l y , t h e a u t h o r w i s h e s t o t h a n k t h e N a t i o n a l R e s e a r c h C o u n c i l o f Canada f o r t h e i r f i n a n c i a l s u p p o r t . i v TABLE OF CONTENTS Chapter Page LIST OF FIGURES vi LIST OF PLATES v i i i 1. INTRODUCTION 1 2. THEORY OF LIFT AND DRAG ON AEROFOILS 6 2.1 Lift and Drag 6 2.2 Circulation 8 2.3 Vortex System 10 2.4 Induced Drag 12 2.5 Expression for Induced Velocity 14 2.6 Expressions for Lift and Induced Drag 16 2.7 El l iptic Loading 19 2.8 The Effect of Twist 19 2.9 The Effect of Yawing 20 3. APPARATUS AND INSTRUMENTATION 21 3.1 Water Flume 21 3.2 Strain Gauge Balance . 24 3.3 Models 27 3.4 Electronic Equipment 29 4. EXPERIMENTAL PROCEDURE 33 4.1 Calibration Methods 33 4.2 Test Procedure 33 4.3 Errors 36 V Chapter Page 5. DISCUSSION OF RESULTS 38 SUMMARY OF CONCLUSIONS 50 LIST OF REFERENCES 51 APPENDIX A - TABLE OF RESULTS 52 APPENDIX B - METHOD OF SOLUTION 57 APPENDIX C - MOLDING AND CASTING 61 v i L I S T OF FIGURES F i g u r e Page 1 . D e f i n i t i o n D iagram f o r L i f t , D r a g , A n g l e o f A t t a c k and C i r c u l a t i o n 7 2 . P l o t o f L i f t t o Drag R a t i o V e r s u s A n g l e o f A t t a c k f o r Unsymmet r i c A e r o f o i l s o f A s p e c t R a t i o , A = 1 , 3 , 6 9 3 . H o r s e s h o e V o r t e x S y s t e m 11 4 . D e f i n i t i o n D iagram f o r I n d u c e d V e l o c i t y and Reduced A n g l e o f A t t a c k 13 5 . D e f i n i t i o n D iagram f o r I n t e g r a t i o n o f K i n S p a n w i s e D i r e c t i o n 15 6 . S c h e m a t i c L a y o u t o f E x p e r i m e n t a l Equ ipment 23 7 . S t r a i n Gauge B a l a n c e 26 8 . D i m e n s i o n s o f t h e T h r e e M o d e l s 32 9 . P l o t o f L i f t t o Drag R a t i o V e r s u s A n g l e o f A t t a c k f o r t h e T h r e e Epoxy A e r o f o i l s 39 1 0 . P l o t o f L i f t t o Drag R a t i o V e r s u s A n g l e o f A t t a c k f o r t h e A luminum A e r o f o i l a t V a r i o u s A n g l e s o f T w i s t t o Reduce A n g l e o f A t t a c k 41 v i i F i g u r e Page 1 1 . P l o t o f L i f t t o Drag R a t i o V e r s u s A n g l e o f A t t a c k f o r t h e A luminum A e r o f o i l a t V a r i o u s A n g l e s o f T w i s t t o I n c r e a s e t h e A n g l e o f A t t a c k 43 1 2 . P l o t o f L i f t t o Drag R a t i o V e r s u s A n g l e o f A t t a c k f o r t h e Whale F l u k e M o d e l s 44 1 3 . P l o t o f L i f t t o Drag R a t i o V e r s u s A n g l e o f Yaw f o r t h e A luminum A e r o f o i l 49 v i i i L I S T OF PLATES P l a t e Page 1 . O v e r a l l V iew o f Flume •. 22 2 . B a l a n c e and R e c o r d i n g Equ ipment 25 3 . Epoxy A e r o f o i l Model 28 4 . Whale F l u k e Model . 31 5 . S t r a i n Gauge B a l a n c e 34 6 . Sample O u t p u t f rom C h a r t R e c o r d e r 37 1 CHAPTER 1 INTRODUCTION When a f l u i d f l o w s p a s t o b j e c t s o f d i f f e r e n t s h a p e s i t can p r o d u c e a v a r i e t y o f e f f e c t s . I t may p r o d u c e a s t e a d y f o r c e i n t h e d i r e c t i o n o f m o t i o n o r i t may p r o d u c e l a r g e l a t e r a l components i n a d d i t i o n t o t h e f o r c e i n t h e d i r e c t i o n o f m o t i o n . Some s h a p e s o f o b j e c t a r e s u b j e c t t o o s c i l l a t i n g l a t e r a l o r i n - l i n e f o r c e s and a f u r t h e r c a t e g o r y e x p e r i e n c e i n s t a b i l i t i e s te rmed g a l l o p i n g o r d i v e r g e n c e . T h i s s t u d y c o n c e n t r a t e s on a p a r t i c u l a r g roup o f shapes w h i c h a r e c l a s s i f i e d as f l e x i b l e l ow a s p e c t a e r o f o i l s . I nduced d r a g i s an i m p o r t a n t f a c t o r w i t h s u c h a e r o f o i l s and t h i s s t u d y s e t s o u t t o i n v e s t i g a t e w h e t h e r c o n t r o l l e d s p a n w i s e t w i s t i n g o f t h e a e r o f o i l can be used t o r e d u c e t h e i n d u c e d d r a g . T y p i c a l l y , a e r o f o i l s a r e o b j e c t s w h i c h p r o d u c e a h i g h l i f t component and a r e l a t i v e l y s m a l l d r a g component . Low a s p e c t r a t i o , w h i c h means s h o r t span compared w i t h c h o r d , t e n d s t o p r o d u c e h i g h e r d r a g components b e c a u s e o f w i n g t i p l o s s e s , r e f e r r e d t o as i n d u c e d d r a g . To r e d u c e t h e i n d u c e d d r a g t o a minimum f o r a g i v e n l i f t , a e r o f o i l s a r e u s u a l l y t a p e r e d t o a p p r o x i m a t e t h e op t imum, so c a l l e d , e l l i p t i c l o a d i n g . Such m i n i m i z a t i o n can a l s o be a c h i e v e d by r e d u c i n g t h e t h i c k n e s s t o c h o r d r a t i o and by t w i s t i n g t h e a e r o f o i l . T h i s t h e s i s i s c o n c e r n e d w i t h t w i s t i n g o f t h e a e r o f o i l and p a r t i c u l a r l y w i t h a f l e x i b l e a e r o f o i l i n w h i c h t h e amount o f t w i s t i s a f u n c t i o n o f t h e l i f t . 2 In t h r e e d i m e n s i o n s t h e d r a g o f an a e r o f o i l i s made up o f t h r e e s e p a r a t e p a r t s : f r i c t i o n d r a g , f o r m d r a g and i n d u c e d d r a g . The f r i c t i o n d r a g o r s k i n f r i c t i o n i s d e p e n d e n t on t h e smoothness o f t h e s u r f a c e o f t h e w i n g . The f o r m d r a g o r eddy r e s i s t a n c e depends on t h e shape o f t h e w i n g and i s g r e a t e r f o r t h i c k e r p r o f i l e s . The f r i c t i o n d r a g and fo rm d r a g a r e g e n e r a l l y lumped t o g e t h e r and c a l l e d t h e p r o f i l e d r a g w h i c h depends m a i n l y on t h e shape o f t h e s e c t i o n . In two d i m e n s i o n a l t h e o r y p r o f i l e d r a g makes up t h e t o t a l d r a g o f t h e a e r o -f o i l . I n t h r e e d i m e n s i o n s t h e i n d u c e d downf low n e a r t h e w i n g i n c r e a s e s t h e a n g l e o f a t t a c k and t h e r e b y r o t a t e s t h e two d i m e n s i o n a l l i f t away f r o m t h e p e r p e n d i c u l a r . The h o r i z o n t a l component o f l i f t r e s u l t i n g f rom t h i s r o t a t i o n i s c a l l e d t h e i n d u c e d d r a g . The i n d u c e d d r a g may a l s o be t h o u g h t o f as t h e work done on t h e f l u i d t o i n c r e a s e t h e l e n g t h o f t h e t r a i l i n g v o r t e x p a t t e r n , w h i c h c o n s i s t s o f a s t r o n g v o r t e x o r i g i n a t i n g f rom e a c h w i n g t i p . The k i n e t i c e n e r g y o f t h e t i p v o r t i c e s and t h u s t h e i n d u c e d d r a g a r e a l m o s t i n d e p e n d e n t o f s p a n . T h i s t h e s i s w i l l c o n c e n t r a t e on r e d u c i n g t h e i n d u c e d d r a g and t h e r e b y m a k i n g a s i g n i f i c a n t r e d u c t i o n i n t o t a l d r a g . I n a d d i t i o n t h i s t h e s i s w i l l c o n c e n t r a t e on low s p e e d a e r o f o i l s w h i c h a r e i d e n t i c a l w i t h h y d r o f o i l s i f c o m p r e s s i b i l i t y a n d . c a v i t a t i o n can be i g n o r e d . H y d r o f o i l s a r e , o f n e c e s s i t y , s h o r t and s t u b b y due t o t h e h i g h e r s t r e s s e s p r o d u c e d by w a t e r l o a d i n g . H y d r o f o i l s , t h e n , must have l o w a s p e c t r a t i o t o c o u n t e r a c t t h e h i g h e r s t r u c t u r a l l o a d i n g . The a s p e c t r a t i o has a g r e a t e f f e c t upon t h e l i f t t o d r a g r a t i o and g e n e r a l l y a l ow a s p e c t r a t i o o r s h o r t a e r o f o i l i s l e s s e f f i c i e n t . A good d e m o n s t r a t i o n o f t h i s comes f r o m a c o m p a r i s o n o f t h e a l b a t r o s s and t h e s p a r r o w ; the a l b a t r o s s h a v i n g a h i g h a s p e c t r a t i o , i s a v e r y 3 e f f i c i e n t g l i d i n g b i r d , w h i l e t h e h i g h l y m a n o e u v e r a b l e s p a r r o w , h a v i n g a l ow a s p e c t r a t i o , must c o n t i n u a l l y b e a t i t s w i n g s t o s t a y i n t h e a i r . S i n c e t h e i n d u c e d d r a g i s a l m o s t i n d e p e n d e n t o f span i t i s f a r more i m p o r t a n t f o r s h o r t spans t h a n l o n g ones as i t w i l l have a p r o p o r t i o n a l l y g r e a t e r e f f e c t f o r t h e s h o r t e r s p a n . A r e d u c t i o n i n t h e i n d u c e d d r a g , w h i c h c o n s t i t u t e s a l a r g e p a r t o f t h e t o t a l d r a g f o r h y d r o f o i l s ( s h o r t a e r o f o i l s ) , w i l l t h e n be q u i t e b e n e f i c i a l . The minimum p o s s i b l e i n d u c e d d r a g f o r c e f o r a g i v e n l i f t i s o b t a i n e d by e l l i p t i c l o a d i n g , t h e m a j o r a x i s o f t h e e l l i p s e b e i n g t h e span and t h e m i n o r a x i s b e i n g t h e m a g n i t u d e o f t h e l i f t f o r c e . E l l i p t i c l o a d i n g c a n be a c h i e v e d by a w i n g t h a t i s e l l i p t i c i n p l a n f o r m o r i t can be b r o u g h t a b o u t by t w i s t i n g a r e c t a n g u l a r a e r o f o i l a l o n g i t s l e n g t h t o d e c r e a s e t h e a n g l e o f a t t a c k t o w a r d s t h e t i p . In e f f e c t t h e t w i s t d e c r e a s e s t h e s t r e n g t h o f t h e t r a i l i n g v o r t i c e s and s o d e c r e a s e s t h e k i n e t i c e n e r g y o r work done on t h e f l u i d by t h e s e v o r t i c e s w i t h t h e r e s u l t t h a t t h e i n d u c e d d r a g f o r c e i s . r e d u c e d . The use o f t w i s t i s n o t new and has been used i n n a t u r e f o r m i l l i o n s o f y e a r s t o h e l p c o n t r o l b i r d s i n f l i g h t . In 1903 t h e W r i g h t B r o t h e r s , c o p y i n g t h e f l i g h t o f b i r d s , a d a p t e d t w i s t as a method o f o b t a i n i n g l a t e r a l c o n t r o l o v e r t h e i r p l a n e w h i c h made t h e f i r s t r e c o g n i z e d powered f l i g h t . T h i s m e t h o d , c a l l e d w i n g w a r p i n g , c o n s i s t e d o f p r o v i d i n g t h e p i l o t w i t h c o n t r o l o f f l e x i b l e w i n g s w h i c h c o u l d be t w i s t e d t o i n c r e a s e o r d i m i n i s h t h e a n g l e o f a t t a c k a t t h e w i n g t i p . I n t h i s way t h e w i n g on one s i d e c o u l d be made t o p r o d u c e a h i g h e r o r l o w e r l i f t t h a n t h e w i n g on t h e o t h e r s i d e and t h e l i f t 4 d i f f e r e n t i a l w o u l d c a u s e t h e p l a n e t o r o l l a b o u t i t s l o n g i t u d i n a l a x i s . In l a t e r a e r o n a u t i c a l work w i n g w a r p i n g was r e p l a c e d by a i l e r o n s b e c a u s e o f t h e i r p r a c t i c a l i t y and b e c a u s e i t was r e c o g n i z e d t h a t t w i s t -i n g c a u s e d f l u t t e r and c o n t r o l r e v e r s a l , so t h a t t w i s t i n g o f l o n g a e r o f o i l s came t o be r e g a r d e d as h a z a r d o u s . T w i s t i n g o f s h o r t a e r o -f o i l s o r h y d r o f o i l s , h o w e v e r , may be q u i t e b e n e f i c i a l . P r e v i o u s l y a c o m p a r i s o n o f t h e a l b a t r o s s and t h e s p a r r o w showed t h e s p a r r o w t o be a r a t h e r i n e f f i c i e n t c r e a t u r e i n t e rms o f l i f t t o d r a g r a t i o . However , even f o r t h e s p a r r o w , w h i c h r e q u i r e s m a n o e u v e r a b i l i t y , and so i s b e s e t w i t h l o w a s p e c t r a t i o w i n g s , n a t u r e seems t o do i t s u t m o s t t o i n c r e a s e t h e e f f i c i e n c y . An e x a m i n a t i o n o f s u c h a c r e a t u r e as t h e s p a r r o w s h o u l d g i v e us some i d e a s t o w a r d r e d u c i n g t h e i n d u c e d d r a g f o r c e . The c r e a t u r e c h o s e n f o r s t u d y was t h e k i l l e r w h a l e , l a r g e s t member o f t h e d o l p h i n f a m i l y . The t h r u s t u n i t o r t a i l f l u k e s a r e s i m i l a r t o t h e w i n g s o f t h e s p a r r o w , b u t t h e ma in r e a s o n t h e k i l l e r w h a l e was c h o s e n was t h a t i t c o u l d be r e a d i l y o b s e r v e d and p h o t o g r a p h e d f r o m c l o s e q u a r t e r s a t t h e V a n c o u v e r P u b l i c A q u a r i u m . The l e a d i n g edge o f t h e f l u k e i s v e r y s t i f f so t h a t t h e c e n t r e o f t w i s t i s n e a r t o t h e f o r w a r d e d g e . The swept back shape means t h a t t h e c e n t r e o f l i f t t e n d s t o move a f t , f u r t h e r away f r o m t h e r o o t . A p a r t f r o m t h e l e a d i n g e d g e , t h e f l u k e i s q u i t e f l e x i b l e and t h i s , combined w i t h the e f f e c t o f t h e c e n t r e o f t w i s t b e i n g ahead o f t h e c e n t r e o f p r e s s u r e , b r i n g s a b o u t a r e d u c t i o n i n t h e a n g l e o f a t t a c k . 5 The k i l l e r w h a l e f l u k e has a r a t h e r d i f f e r e n t s e c t i o n f r o m t h e t y p i c a l a e r o f o i l s h a p e , b e i n g much t h i c k e r f o r t h e same l e n g t h . A model o f t h i s shape was i n v e s t i g a t e d a s p a r t o f t h e s t u d y . The e f f e c t o f y a w i n g t h e a e r o f o i l f o r w a r d and b a c k w a r d was a l s o i n v e s t i g a t e d t o d e t e r m i n e i t s i n f l u e n c e on t h e d r a g f o r c e . In summary, t h e n , t h e p u r p o s e o f t h i s t h e s i s i s t o s t u d y t w i s t i n g as a method o f i m p r o v i n g t h e e f f i c i e n c y o f h y d r o f o i l s by r e d u c i n g t h e i n d u c e d d r a g . The i n d u c e d d r a g c a n n o t be d e c r e a s e d w i t h o u t r e d u c i n g t h e l i f t f o r c e , so t h e p r o b l e m i s t o f i n d t h e optimum a n g l e o f t w i s t f o r w h i c h t h e a d v a n t a g e o f d e c r e a s i n g t h e d r a g i s n o t o u t w e i g h e d by t h e d i s a d v a n t a g e o f d e c r e a s i n g t h e l i f t . 6 CHAPTER 2 THEORY 2.1 L i f t and Drag When a body i s s u b j e c t t o a f l u i d f l o w , t h e r e s u l t a n t f o r c e i t e x p e r i e n c e s i s a r e s i s t a n c e w h i c h g e n e r a l l y l i e s a l o n g t h e d i r e c t i o n o f m o t i o n i n t h e same d i r e c t i o n as . t h e f l u i d m o t i o n . Some b o d i e s e x p e r i e n c e a r e s u l t a n t f o r c e w h i c h l i e s a t some a n g l e t o t h e d i r e c t i o n o f m o t i o n and a e r o f o i l s a r e t h e c l a s s o f body f o r w h i c h t h e r e s u l t a n t f o r c e i s v e r y n e a r l y p e r p e n d i c u l a r t o t h e d i r e c t i o n o f m o t i o n . L i f t i s t h e t e r m u s e d t o d e s c r i b e t h e component o f t h e r e s u l t a n t f o r c e p e r p e n d i c u l a r t o t h e d i r e c t i o n o f m o t i o n and d r a g i s t h e t e r m d e s c r i b i n g t h e r e s i s t a n c e f o r c e a l o n g t h e d i r e c t i o n o f m o t i o n . F i g u r e 1 i s a d e f i n i t i o n d i a g r a m f o r l i f t and d r a g . L i f t i s t h e u s e f u l compon-e n t w h i c h c a r r i e s t h e a e r o p l a n e w h i l e d r a g i s t h e f o r c e w h i c h must be overcome by t h r u s t . S y m m e t r i c a e r o f o i l s , w h i c h a r e d e a l t w i t h e x c l u s i v e l y i n t h i s p a p e r , have z e r o l i f t f o r c e when t h e a n g l e o f a t t a c k i s z e r o . The a n g l e o f a t t a c k i s t h e a n g l e between t h e d i r e c t i o n o f m o t i o n o f t h e f l u i d and t h e . c h o r d , t h e c h o r d b e i n g t h e l i n e drawn between t h e s h a r p - t r a i l i n g edge and t h e c e n t r e o f c u r v a t u r e o f t h e n o s e . The q u a l i t y o r e f f i c i e n c y o f an a e r o f o i l i s measured by t h e r a t i o o f l i f t t o d r a g . P r a c t i c a l a e r o f o i l s have a h i g h l i f t t o d r a g F I G U R E I 7 L i f t S t a r t ing Vortex D E F I N I T I O N D I A G R A M F O R L I F T , D R A G , A N G L E O F A T T A C K A N D C I R C U L A T I O N . 8 r a t i o and t h e s e a r e t h e ones f o r w h i c h t h e n e t r e s u l t a n t f o r c e i s v e r y n e a r l y p e r p e n d i c u l a r t o t h e d i r e c t i o n o f m o t i o n . The l i f t t o d r a g r a t i o depends t o a l a r g e e x t e n t on t h e a n g l e o f a t t a c k and t h e a s p e c t r a t i o . The a s p e c t r a t i o may be d e f i n e d as t h e r a t i o o f t h e span s q u a r e d t o t h e a r e a o f t h e a e r o f o i l , t h i s may be r e d u c e d t o t h e r a t i o o f span t o c h o r d l e n g t h f o r a r e c t a n g u l a r a e r o f o i l . F i g u r e 2 shows a p l o t o f l i f t t o d r a g r a t i o v e r s u s a n g l e o f a t t a c k f o r u n s y m m e t r i c a e r o f o i l s o f v a r i o u s a s p e c t r a t i o s and d e m o n s t r a t e s c l e a r l y t h e dependence o f l i f t t o d r a g r a t i o on a n g l e o f a t t a c k and a s p e c t r a t i o . 2.2 C i r c u l a t i o n The f o r m a t i o n o f l i f t can be e x p l a i n e d i n t e rms o f c i r c u l a t i o n t h e o r y . When a body i s s u b j e c t t o l i f t t h e n i t must be s u b j e c t t o a c e r t a i n p o s i t i v e p r e s s u r e on t h e b o t t o m and a c e r t a i n s u c t i o n p r e s s u r e on t h e t o p . From B e r n o u l l i ' s e q u a t i o n we can s t a t e t h a t t h e v e l o c i t y above the body must be g r e a t e r t h a n t h e v e l o c i t y be low t h e body . T h i s c o n d i t i o n can be o b t a i n e d by s u p e r i m p o s i n g on t h e f l o w p a s t t h e b o d y , a c i r c u l a t i o n a r o u n d i t , w h i c h f l o w s so as t o i n c r e a s e t h e f l u i d s t r e a m v e l o c i t y on t o p o f t h e body and d e c r e a s e i t on t h e b o t t o m o f t h e body . So t h a t i f a body i s s t a t i o n a r y and t h e f l u i d f l o w i s t o t h e r i g h t t h e n t o d e v e l o p a l i f t f o r c e t h e c i r c u l a t i o n must be c l o c k -w i s e a r o u n d t h e body . When an a e r o f o i l s t a r t s f r o m r e s t a v o r t e x i s fo rmed a t t h e s h a r p r e a r e d g e . T h i s v o r t e x , c a l l e d t h e s t a r t i n g v o r t e x , i s a l w a y s FIGURE 2 9 25 4 -2 0 2 4 6 8 10 12 14 16 Angle of Att a c k ( degrees) PLOT OF LIFT TO DRAG RATIO VERSUS ANGLE OF ATTACK FOR UNSYMMETRIC AEROFOILS OF ASPECT RATIO,A= 1,3,6 . ( B A S E D ON DATA FROM = THEORY OF WING S E C T I O N S , ABBOT AND VON DOENHOFF ). 10 a s s o c i a t e d w i t h t h e same p a r t i c l e s o f f l u i d s o i t b r e a k s away and i s washed downst ream as t h e a e r o f o i l c o n t i n u e s i n m o t i o n . S i n c e t h e c i r c u l a t i o n a r o u n d any l a r g e c o n t o u r w h i c h i n c l u d e s t h e a e r o f o i l and t h e s t a r t i n g v o r t e x must r e m a i n z e r o , t h e n t h e g e n e r a t i o n o f c i r c u l a t i o n can be e x p l a i n e d by t h e f a c t t h a t t h e r e must be a v o r t e x a r o u n d t h e w i n g e q u a l and o p p o s i t e t o t h e s t a r t i n g v o r t e x . T h i s i s n o t t o say t h a t o t h e r v o r t i c e s a r e n o t shed as t h e a e r o f o i l moves a l o n g , i n f a c t v o r t i c e s a r e s h e d c o n t i n u o u s l y f r o m t h e t r a i l i n g edge t o m a i n t a i n t h e c i r c u l a t i o n w h i c h w o u l d o t h e r w i s e d e c a y . F i g u r e 1 shows t h e c i r c u l a t i o n a r o u n d an a e r o f o i l and t h e s t a r t i n g v o r t e x . 2 . 3 V o r t e x S y s t e m The c i r c u l a t i o n moves w i t h t h e a e r o f o i l and i s made up o f a s e r i e s o f l i n e v o r t i c e s a l o n g t h e s p a n . These a r e c a l l e d t h e bound v o r t i c e s . S i n c e a l i n e a r v o r t e x c a n n o t t e r m i n a t e i n t h e i n t e r i o r o f a f l u i d , t h e n t h e bound v o r t i c e s c a n n o t t e r m i n a t e a t t h e w i n g t i p s b u t must c o n t i n u e as f r e e v o r t i c e s . These f r e e v o r t i c e s , c a l l e d t h e t r a i l i n g v o r t i c e s , a r e t u r n e d p e r p e n d i c u l a r t o t h e w i n g and a r e f o r c e d downst ream w i t h t h e f l u i d f l o w . G e n e r a l l y t h e s e v o r t i c e s a r e shed a l o n g t h e l e n g t h o f t h e w i n g as a s h e e t , however a s i m p l e r s i t u a t i o n i s when t h e c i r c u l a t i o n K a r o u n d t h e w i n g has c o n s t a n t v a l u e and the t r a i l i n g v o r t i c e s s p r i n g o n l y f r o m t h e w i n g t i p s . The bound v o r t e x i s now a s i n g l e l i n e v o r t e x and t h e t r a i l i n g v o r t i c e s a r e a l s o l i n e v o r t i c e s o r i g i n a t i n g f r o m each w i n g t i p . T h i s s i t u a t i o n , c a l l e d t h e h o r s e s h o e v o r t e x s y s t e m , i s shown i n F i g u r e 3 . FIGURE 3 11 H O R S E S H O E VORTEX S Y S T E M 12 The open end o f t h e v o r t e x r e c t a n g l e i s c l o s e d by t h e s t a r t i n g v o r t e x w h i c h r e m a i n s a t t h e s t a r t i n g p o i n t w h i l e t h e t r a i l i n g v o r t i c e s c o n -t i n u a l l y i n c r e a s e i n l e n g t h . The f o r m a t i o n o f t h e t r a i l i n g v o r t i c e s can a l s o be e x p l a i n e d i n t e rms o f s p a n w i s e f l o w s . Due t o t h e h i g h e r p r e s s u r e b e l o w t h e w i n g t h a n above i t t h e r e i s a t e n d e n c y f o r f l u i d t o f l o w f r o m t h e b o t t o m t o t h e t o p a r o u n d t h e w i n g t i p s . T h i s c a u s e s an o u t w a r d f l o w t o w a r d t h e w i n g t i p a l o n g t h e b o t t o m o f t h e span and a f l o w t o w a r d t h e c e n t r e on t h e t o p . The f l o w w h i c h r o l l s a r o u n d t h e w i n g t i p s f r o m b o t t o m t o t o p fo rms t h e two d i s t i n c t t r a i l i n g v o r t i c e s . 2 . 4 I n d u c e d Drag The t o t a l d r a g o f an a e r o f o i l can be d i v i d e d i n t o t h r e e s e p a r a t e p a r t s : f r i c t i o n d r a g , f o r m d r a g and i n d u c e d d r a g . The f r i c t i o n d r a g depends upon t h e smoothness o f t h e s e c t i o n and t h e f o r m d r a g depends upon t h e shape o f t h e p r o f i l e , b e i n g g r e a t e r f o r t h i c k e r s e c t i o n s . The f r i c t i o n d r a g and f o r m d r a g a r e lumped t o g e t h e r and c a l l e d t h e p r o f i l e d r a g s i n c e t h e y depend m a i n l y on t h e shape o f t h e s e c t i o n . The i n d u c e d d r a g can be e x p l a i n e d i n t e rms o f t h e i n d u c e d v e l o c i t y w h i c h i s due t o t h e i n f l u e n c e o f t h e t r a i l i n g v o r t i c e s and i s d i r e c t e d g e n e r a l l y downwards. T h i s normal v e l o c i t y , d e n o t e d w, i s s m a l l compared t o t h e v e l o c i t y V o f t h e a e r o f o i l . The e f f e c t o f t h e i n d u c e d v e l o c i t y i s e q u i v a l e n t t o a r e d u c t i o n i n t h e a n g l e o f a t t a c k as shown i n F i g u r e 4 ( f r o m G l a u e r t ( 3 ) ) . FIGURE 4 13 DEFINITION DIAGRAM FOR INDUCED VELOCITY AND REDUCED ANGLE OF A T T A C K . ( FROM G L A U E R T ) 14 Due t o t h e i n d u c e d v e l o c i t y t h e o r i g i n a l a n g l e o f a t t a c k a i s now r e d u c e d by t h e amount w / V so t h a t t h e new a n g l e o f i n c i d e n c e i s aQ = a - w / V . The l i f t f o r c e i s now t i l t e d b a c k w a r d s a t an a n g l e w / V and so t h e r e i s a component i n t h e d r a g d i r e c t i o n e q u a l t o w / V t i m e s t h e o r i g i n a l l i f t f o r c e ( t h e two d i m e n s i o n a l l i f t ) . T h i s component o f t h e l i f t i s c a l l e d t h e i n d u c e d d r a g f o r c e . The i n d u c e d d r a g may a l s o be t h o u g h t o f as the , ;work done on t h e f l u i d t o i n c r e a s e t h e l e n g t h o f t h e t r a i l i n g v o r t e x p a t t e r n . The te rm " t i p l o s s " r e f e r s t o t h e i n d u c e d d r a g f o r c e and o r i g i n a t e s f rom t h e s p a n w i s e f l o w e x p l a n a t i o n where t h e f l o w a r o u n d t h e w i n g t i p f r o m h i g h t o l o w p r e s s u r e was t h e c a u s e o f t h e t r a i l i n g v o r t i c e s . 2 . 5 E x p r e s s i o n f o r I nduced V e l o c i t y In t h i s s e c t i o n t h e v a l u e f o r t h e i n d u c e d v e l o c i t y a d j a c e n t t o a v o r t e x w i l l be computed . P r a n d t l has shown t h a t t h e normal downward v e l o c i t y i n d u c e d a d i s t a n c e h away f r o m a l i n e a r v o r t e x o f s t r e n g t h K i s g i v e n by - " ( 1 ) G e n e r a l l y t h e c i r c u l a t i o n K v a r i e s a c r o s s t h e span and t h e e x p r e s s i o n f o r w becomes more c o m p l i c a t e d . The above r e s u l t ca i i be a p p l i e d t o a s t r i p dy o f t h e v o r t e x band as i n F i g u r e 5 . Between y and y + dy a l o n g t h e span t h e c i r c u l a t i o n d e c r e a s e s by ^ dy and a t r a i l i n g v o r t e x o f t h i s s t r e n g t h s p r i n g s f r o m e l e m e n t d y . The normal i n d u c e d F I G U R E 5 15 D E F I N I T I O N D I A G R A M F O R I N T E G R A T I O N O F K I N S P A N W I S E D I R E C T I O N . ( F R O M G L A U E R T ) 16 v e l o c i t y a t any p o i n t y^ must be o b t a i n e d f r o m t h e sum o f t h e t r a i l -i n g v o r t i c e s o r i g i n a t i n g a l o n g t h e l e n g t h o f t h e w i n g . So t h a t f o r a h a l f span s f dy /•s dy * w ( y j = (2 ) J _ s 4 T r (y 1 - y ) 2 . 6 E x p r e s s i o n s f o r L i f t and I n d u c e d Drag Most o f t h e m a t e r i a l f o r t h i s s e c t i o n o r i g i n a t e s f rom G l a u e r t ( 3 ) . The l i f t c o e f f i c i e n t i s g i v e n by k L = a o a o ( 3 ) where a Q i s t h e s l o p e o f t h e c u r v e o f l i f t c o e f f i c i e n t v e r s u s a n g l e o f a t t a c k f o r t h e a e r o f o i l i n two d i m e n s i o n a l m o t i o n , w h i c h may be t a k e n e q u a l t o ir i n most c a s e s . The c i r c u l a t i o n K a r o u n d t h e a e r o f o i l i s K = k L c V = a Q c (Va - w) (4) w h e r e - c i s t h e c h o r d l e n g t h . Once t h e c i r c u l a t i o n K and t h e normal i n d u c e d v e l o c i t y w a r e known t h e n t h e l i f t L and i n d u c e d d r a g D c a n be f o u n d f r o m 17 D = / U * (5 ) s p w K dy (6) - s i n w h i c h p i s t h e f l u i d d e n s i t y . F o r p u r p o s e o f s o l u t i o n we c a n assume K c a n be e x p r e s s e d by t h e F o u r i e r s e r i e s 4 s V Z A s i n ne (7) n=l where 0 i s t h e c i r c u l a r c o - o r d i n a t e w h i c h r u n s f r o m 0 t o IT ( - s t o +s) a l o n g t h e a e r o f o i l , s o t h a t y i s r e l a t e d t o 6 by y = - s cos 0 . S u b s t i t u t i n g t h e F o u r i e r e x p r e s s i o n f o r K i n t o t h e e x p r e s s i o n f o r w we g e t ., f[ I n A cos n0 w(e,) = I - — D de (8) J0 c o s 0 - c o s e 1 fir c o s n0 d0 and s i n c e s i n n6 0 c o s 6 - c o s = TT 1 s i n e 1 t h e n w(e.,) = V E n A n s i n n0 1 s i n e (9) 1 S i n c e K = 4s V £ A n s i n n6 = a Q c ( V a - w) 18 s i n n6 Then 4s V I A s i n n0 = a c ( V a - VEn A, n (10) s i n 6 o r Z A s i n ne (nu + s i n 6) = u a s i n 0 ( I D f o r t h e g e n e r a l p o i n t 6 , where T h i s e q u a t i o n can now be used t o f i n d t h e v a l u e s o f t h e c o e f f i c i e n t s A , so t h a t t h e c i r c u l a t i o n K and hence t h e v a l u e s o f L and D c a n be n f o u n d . The l i f t and i n d u c e d d r a g c a n be e x p r e s s e d i n te rms o f t h e c o e f f i c i e n t s L - s p V K dy 2 2 = 2T\ S^ p r A (12) D = p w K dy - s (13) F o r t h e c a s e o f t h e t w i s t e d a e r o f o i l t h e a n g l e o f a t t a c k a may be e x p r e s s e d as a = a - e c o s 6 where a i s t h e a n g l e o f 19 a t t a c k o f t h e c e n t r e ( r o o t ) o f t h e a e r o f o i l and. e i s t h e a n g u l a r t w i s t f r o m t h e c e n t r e t o t h e t i p . T h i s v a l u e f o r a may be s u b s t i t u t e d i n t o t h e c o e f f i c i e n t e q u a t i o n and e a c h c o e f f i c i e n t i s t h e n d e t e r m i n e d i n two p a r t s , one p r o p o r t i o n a l t o a and one p r o p o r t i o n a l t o e . The method o f s o l u t i o n i s d i s c u s s e d i n A p p e n d i x B. 2 . 7 E l l i p t i c L o a d i n g When a w i n g has an e l l i p t i c l o a d d i s t r i b u t i o n t h e n t h e i n d u c e d d r a g w i l l be a minimum f o r a g i v e n l i f t . The m a j o r a x i s o f t h e e l l i p s e w i l l be t h e span and t h e m i n o r a x i s t h e m a g n i t u d e o f t h e c i r c u l a t i o n w h i c h i s p r o p o r t i o n a l t o t h e l i f t f o r c e . The e l l i p t i c l o a d d i s t r i b u t i o n can be a c h i e v e d by u s i n g an e l l i p t i c w i n g o r by t w i s t i n g a r e c t a n g u l a r w i n g so t h a t t h e a n g l e o f a t t a c k d e c r e a s e s t o w a r d s t h e t i p . W i th e l l i p t i c l o a d i n g t h e i n d u c e d v e l o c i t y has a c o n s t a n t v a l u e a l o n g t h e s p a n . 2 . 8 The E f f e c t o f T w i s t F o r a r i g i d u n t w i s t e d r e c t a n g u l a r a e r o f o i l ( w h i c h i s n o t l o a d e d e l l i p t i c a l l y ) t h e i n d u c e d v e l o c i t y t e n d s t o i n c r e a s e t o w a r d s t h e t i p . By t w i s t i n g t h e a e r o f o i l a l o n g i t s l e n g t h by j u s t t h e r i g h t amount t o d e c r e a s e t h e i n d u c e d v e l o c i t y t o w a r d s t h e t i p so t h a t i t has c o n s t a n t v a l u e a l o n g t h e s p a n we o b t a i n e l l i p t i c l o a d i n g . I f t h e w i n g i s d i v i d e d i n t o a number o f segments a c r o s s t h e span t h e n t h e i n d u c e d d r a g i s p r o p o r t i o n a l t o t h e sum o f t h e i n d i v i d u a l i n d u c e d v e l o c i t i e s o f e a c h segment s q u a r e d . C l e a r l y t h e 20 minimum p o s s i b l e i n d u c e d d r a g w i l l be o b t a i n e d when t h e i n d u c e d v e l o c i t y o f each segment i s t h e same o r when t h e i n d u c e d v e l o c i t y has c o n s t a n t v a l u e a c r o s s t h e s p a n . T h i s means t h a t i f t h e a e r o f o i l i s t w i s t e d t o o much so t h a t t h e n e t r e s u l t i s t h a t t h e i n d u c e d v e l o c i t y d e c r e a s e s t o w a r d s t h e t i p t h e n t h e e f f e c t w i l l be d e t r i m e n t a l and t h e a e r o f o i l w i l l be l e s s e f f i c i e n t . F o r a r i g i d r e c t a n g u l a r a e r o f o i l t w i s t e d a g i v e n amount t h e r e i s o n l y one a n g l e o f a t t a c k f o r w h i c h e l l i p t i c l o a d i n g can o c c u r . A f l e x i b l e r e c t a n g u l a r a e r o f o i l may, h o w e v e r , be so d e s i g n e d t h a t i t t w i s t s j u s t t h e r i g h t amount t o p r o d u c e e l l i p t i c l o a d i n g a t a l l a n g l e s o f a t t a c k . One way t o do t h i s i s t o p l a c e s t i f f e n e r s t o w a r d s t h e l e a d i n g e d g e . T h i s has t h e e f f e c t o f mov ing t h e c e n t r e o f t w i s t f o r w a r d and i f i t moves f a r enough f o r w a r d t o be ahead o f t h e c e n t r e o f p r e s s u r e t h e n the s e c t i o n w i l l t w i s t t o r e d u c e t h e a n g l e o f a t t a c k t o w a r d s t h e t i p . In t h i s p r o j e c t t e s t s were made on mode ls w i t h s t i f f e n e r s i n b o t h l e a d i n g and t r a i l i n g edges so t h a t t h e e f f e c t o f s t i f f e n e r s c o u l d be c l e a r l y shown. 2 . 9 E f f e c t o f Yawing When t h e a e r o f o i l i s yawed f o r w a r d t h e n t h e f l o w has a component d i r e c t e d a l o n g t h e span t o w a r d s t h e c e n t r e , t h i s t e n d s t o oppose t h e normal s p a n w i s e f l o w s on t h e h i g h p r e s s u r e s i d e f rom t h e c e n t r e t o t h e t i p . Thus t h e v o r t e x r o l l i n g a r o u n d t h e w i n g t i p and s o t h e i n d u c e d d r a g f o r c e a r e r e d u c e d . 21 CHAPTER 3 APPARATUS AND INSTRUMENTATION 3.1 Water Flume A l l t e s t s were p e r f o r m e d i n 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 ' s 75 f o o t w a t e r f l u m e i n w h i c h v e l o c i t y c a n be v a r i e d up t o .3 f t / s e c i n t h e 2 3 . 5 x 30 i n c h s e c t i o n . F o r a l l t h e t e s t s e x c e p t t h e y a w i n g t e s t ; w a t e r v e l o c i t y and d e p t h were h e l d c o n s t a n t a t 2 . 4 f t / s e c and 19 i n . r e s p e c t i v e l y . These c o n d i t i o n s were o b t a i n e d by a 6 i n . w e i r a t t he end o f t h e f l u m e and by h a v i n g b o t h v a l v e s o f t h e 6 c f s pump f u l l y o p e n . The y a w i n g t e s t was p e r f o r m e d a f t e r t h e o t h e r t e s t s and a f t e r some m o d i f i c a t i o n s had been made t o t h e f l u m e . F o r t h e s e t e s t s w a t e r v e l o c i t y was 3 . 5 f t / s e c and dep th 18 i n , t h e s e c o n d i t i o n s were o b t a i n e d by a 12 i n . w e i r a t t h e end o f t h e f l u m e and by u s i n g t h e na r row 12 i n . x 24 i n . s e c t i o n . W a t e r e n t e r s t h e f l u m e t h r o u g h a s t i l l i n g t a n k w h i c h c o n t a i n s two s e t s o f b a f f l e s and an i n l e t d e f l e c t o r t o r e d u c e t u r b u l e n c e . F u r t h e r s m o o t h i n g was o b t a i n e d by a s c r e e n p l a c e d 15 f t u p s t r e a m o f t h e s t r a i n gauge b a l a n c e , w h i c h was i t s e l f l o c a t e d 28 f t f r o m t h e e n t r a n c e . P l a t e 1 shows an o v e r a l l v i e w o f t h e f l u m e and F i g u r e 6 shows a s c h e m a t i c l a y o u t o f t h e e q u i p m e n t . 22 P l a t e 1 O v e r a l l V iew o f Flume FIGURE 6 23 I Ur^ Rigid Mounting From S t r a i n Gauges — S t r a i n Gauge B a l a n c e Water Surface F l o w Model B A L A N C I N G V O L T - C H A R T U N I T M E T E R R E C O R D E R P.O W E R S U P P L Y SCHEMATIC LAYOUT OF EXPERIMENTAL EQUIPMENT 24 3 . 2 S t r a i n Gauge B a l a n c e ' F o r c e measurements were t a k e n on a s t r a i n gauge b a l a n c e s y s t e m w h i c h was d e s i g n e d f o r t h i s p r o j e c t . T h i s c o n s i s t e d o f a 26 i n . x 1 . 625 i n , r o u n d s t e e l s h a f t m a c h i n e d t o 0 . 7 5 i n . r o u n d a l o n g most o f i t s l e n g t h . Two m a c h i n e d r e c t a n g u l a r s e c t i o n s , 9 i n . a p a r t , c o n t a i n e d s t r a i n gauges on e a c h f a c e and t h e s e were c o v e r e d by a p l e x i g l a s s s h i e l d t o e l i m i n a t e w a t e r s p l a s h . The l o w e r end o f t h e s h a f t was t h r e a d e d t o a c c e p t a b o l t and a p l a t e w h i c h c o u l d be l o c k e d i n p l a c e . The model was f a s t e n e d t o t h e l o w e r end o f t h i s c a n t i l e v e r and t h e s y s t e m was c o n n e c t e d t o t h e u p p e r p l a t e o f t h e b a l a n c e by b o l t s p a s s i n g t h r o u g h s l o t s so t h a t t h e model c o u l d be r o t a t e d t h r o u g h 30 d e g r e e s . The s h a f t was s e c u r e d t o a r i g i d m o u n t i n g a c r o s s t h e t o p o f t h e f l u m e . A s l i d i n g m o u n t i n g p e r m i t t e d t h e d e p t h o f i m m e r s i o n t o be v a r i e d . The b a l a n c e s y s t e m i s d e s i g n e d t o s u p p o r t a model i n t h e w a t e r f l u m e , a d j u s t i t s a n g l e o f a t t a c k o v e r a r a n g e o f ± 15 d e g r e e s and measure t h e l o c a t i o n and m a g n i t u d e o f t h e l i f t and d r a g f o r c e s on t h e m o d e l . The a n g u l a r p o s i t i o n o f t h e model i s i n d i c a t e d by g r a d u -a t i o n m a r k i n g s on t h e l o w e r p l a t e measured r e l a t i v e t o a m a r k i n g on t h e upper p l a t e . The f o r c e components a r e measured t h r o u g h s t r a i n gauge p a i r s and t h e o u t p u t i s r e c o r d e d on a p p r o p r i a t e e l e c t r o n i c e q u i p m e n t . The s y s t e m was d e s i g n e d t o r e c o r d a maximum l i f t f o r c e o f 10 l b s and a maximum d r a g o f 1 .5 l b s . P l a t e 2 shows t h e b a l a n c e s e t - u p and t h e r e c o r d i n g e q u i p m e n t and F i g u r e 7 shows t h e d i m e n s i o n s o f t h e b a l a n c e . P l a t e 2 B a l a n c e and R e c o r d i n g Equ ipment FIGURE 7 Q. IVe V DiQ. l5/e" Did. Centre Line - Stra i n Gauges T 3/ 1 1 '* Centre Line - Strain Gauges r i l 1 1 i I :: • 5 / , 6 VIEW IN DRAG DIRECTION i ; i • • J o © © © . VIEW IN LIFT DIRECTION / 2 THE STRAIN GAUGE B A L A N C E 27 3 . 3 Mode ls T h e r e were b a s i c a l l y two t y p e s o f m o d e l s u s e d i n t h e p r o j e c t ; t h e s e were t h e a e r o f o i l m o d e l s and t h e w h a l e f l u k e m o d e l s . The a e r o -f o i l mode ls were a l l r e c t a n g u l a r and s y m m e t r i c and had a t y p i c a l a e r o -f o i l shape h a v i n g t h e maximum t h i c k n e s s a b o u t 40% o f t h e c h o r d l e n g t h f r o m t h e l e a d i n g e d g e . The t h r e e e p o x y a e r o f o i l mode ls were c a s t f r o m t h e same s i l a s t i c r u b b e r m o l d and were 17 i n . x 4 . 7 5 i n . and 0 . 1 8 8 i n . a t t h e i r maximum t h i c k n e s s . I t was d e c i d e d t o c a s t t h e s e mode ls o f epoxy s i n c e t h i s a l l o w e d a c e r t a i n amount o f f l e x i b i l i t y , and a l s o a l l o w e d b a r s t o be p l a c e d w i t h i n t h e l i q u i d b e f o r e t h e epoxy s e t . The f l e x i b i l i t y o f t h e s e mode ls was such t h a t t h e t i p d e f l e c t e d 1 i n . o v e r an e f f e c t i v e span o f 15 i n . when a 1 l b l o a d was a p p l i e d a t t h e t i p . S i l a s t i c r u b b e r was c h o s e n f o r t h e mo ld m a t e r i a l s i n c e t h i s has a l o w s h r i n k a g e c o e f f i c i e n t , and i s q u i t e f l e x i b l e when s e t (more i n f o r m a t i o n on m o l d i n g and c a s t i n g i s g i v e n i n A p p e n d i x C ) . The b a r s , p l a c e d i n t h e model t o v a r y t h e c e n t e r o f t w i s t , were 0 . 0 9 3 8 i n . m i l d s t e e l w e l d i n g r o d s . The f i r s t model had two b a r s p l a c e d i n t h e l e a d i n g e d g e , 0 . 3 i n . a n d 0 . 9 i n . f r o m t h e e d g e . T h i s model i s shown i n P l a t e 3 . The s e c o n d epoxy model had no b a r s and t h e t h i r d had 2 b a r s p l a c e d a t t h e t r a i l i n g e d g e , 0 . 4 i n . a n d 0 . 8 i n . f r o m t h e e d g e . The a luminum a e r o f o i l model was 18 i n . x 5 . 5 i n . and 0 . 1 8 8 i n . a t i t s maximum t h i c k n e s s and i t c o u l d be t w i s t e d t o i t s y i e l d p o i n t s o t h a t i t w o u l d r e t a i n a l o c k e d i n t w i s t a l o n g i t s l e n g t h . T h i s model was t e s t e d w i t h f i v e d i f f e r e n t a n g l e s o f t w i s t : no t w i s t , 1 d e g r e e , 1 1 / 2 d e g r e e s , 2 d e g r e e s and 5 d e g r e e s , and was l a t e r used i n t h e c a n t i n g t e s t . I P l a t e 3 Epoxy A e r o f o i l Model 29 The w h a l e f l u k e mode ls were 5 / 8 s c a l e s y m m e t r i c r e d u c t i o n s o f t h e k i l l e r w h a l e S k a n a ' s s t a r b o a r d t a i l f l u k e . Measurements were made on t h e w h a l e i n t h e p o o l o f t h e V a n c o u v e r P u b l i c A q u a r i u m . The w h a l e i s t r a i n e d t o h o l d h e r f l u k e s o u t o f t h e w a t e r , so t h a t i t was p o s s i b l e t o s t a n d on t h e p o o l - s i d e p l a t f o r m and t a k e measurements on t h e f l u k e s d i r e c t l y . An a c c u r a t e p l a n f o r m was o b t a i n e d by t r a c i n g t h e shape o n t o a l a r g e p i e c e o f c a r d b o a r d and t h i c k n e s s p r o f i l e s were o b t a i n e d by t h e a d r o i t use o f a s e t o f l a r g e c a l i p e r s and a t a p e m e a s u r e . The f l u k e i s s l i g h t l y u n s y m m e t r i c i n s e c t i o n b u t t h i s was n e g l e c t e d i n the mode ls b e c a u s e i t was f e l t t h a t f o r p u r p o s e s o f t h i s p r o j e c t s y m m e t r i c mode ls w o u l d s u f f i c e . These mode ls were 10 i n . x 15 i n . o v e r a l l and 1 .7 i n . a t t h e i r maximum t h i c k n e s s . A hand made wooden model was c o n s t r u c t e d f r o m t h e f i e l d measurements and t h i s was used t o make a m o l d o f s i l a s t i c r u b b e r t h a t w o u l d be used t o c a s t an e p o x y model w i t h an a luminum c o r e w h i c h c o u l d h o l d a l o c k e d i n t w i s t . The wooden model i s shown i n P l a t e 4 . A l l t h e m o d e l s had r e c t a n g u l a r end s e c t i o n s d r i l l e d w i t h f o u r h o l e s t o f i t t h e f l a n g e c o n n e c t i o n on t h e l o w e r p l a t e o f t h e b a l a n c e . The a luminum model was l a t e r m o d i f i e d by r o u n d i n g o f f t h e end s e c t i o n s , and m a r k i n g i t o f f i n a s c a l e o f d e g r e e s , so t h a t i t c o u l d be yawed back and f o r t h v e r t i c a l l y i n t h e f l u m e . The d i m e n s i o n s o f t h e t h r e e m o d e l s a r e shown i n F i g u r e 8 . 3 . 4 E l e c t r o n i c Equ ipment F o l l o w i n g i s a l i s t o f e l e c t r o n i c a p p a r a t u s used i n t h e e x p e r i m e n t a l w o r k : V o l t m e t e r s : H e w l e t t P a c k a r d , 419 A D .C . N u l l V o l t m e t e r Dana D i g i t a l V o l t m e t e r , Model 5330 Power S u p p l y : H a r r i s o n 620 4B D .C . Power S u p p l y C h a r t R e c o r d e r : H o n e y w e l l E l e c t r o n i k 19 R e c o r d e r A l s o a f i v e c h a n n e l B a l a n c i n g U n i t d e s i g n e d and b u i l t i n t h e C i E n g i n e e r i n g D e p a r t m e n t . P l a t e 4 Whale F l u k e Model DIMENSIONS OF THE THREE MODELS r\3 33 CHAPTER 4 EXPERIMENTAL PROCEDURE 4.1 C a l i b r a t i o n Methods The s t r a i n gauge b a l a n c e was s e t up on a b e n c h , and an e x t e n s i o n was b o l t e d t o t h e v e r t i c a l s h a f t , s o t h a t known l o a d s c o u l d be a p p l i e d by w i r e s i n t h e h o r i z o n t a l l i f t and d r a g d i r e c t i o n s . The c a l i b r a t i o n was c o m p l i c a t e d by t h e f a c t t h a t t h e c r o s s - s e c t i o n i n t h e d r a g d i r e c t i o n was much s m a l l e r t h a n t h e c r o s s - s e c t i o n i n t h e l i f t d i r e c t i o n , so t h a t when pu re d r a g was a p p l i e d t h e l i f t gauges showed a r e a d i n g . These l i f t gauge c o r r e c t i o n s were l i n e a r , s o t h e y c o u l d e a s i l y be e l i m i n a t e d f o r t h e r a n g e o f d r a g f o r c e s i n v o l v e d . The c a l i b r a t i o n c u r v e s were f e d i n t o t h e c o m p u t e r , and a s i m p l e p rogram d e v i s e d , t h a t w o u l d i n t a k e t h e f o u r v o l t a g e r e a d i n g s f r o m t h e f o u r s t r a i n gauge p a i r s , and o u t p u t t h e l i f t and d r a g f o r c e s , and t h e i r r e s p e c t i v e d i s t a n c e t o t h e c e n t e r o f e f f o r t measured f r o m t h e l o w e r s t r a i n gauge s e c t i o n . P l a t e 5 shows t h e s t r a i n gauge b a l a n c e mounted on t h e bench f o r c a l i b r a t i o n . 4 . 2 T e s t P r o c e d u r e L i f t and d r a g measurements were made on t h e t h r e e epoxy m o d e l s , t he a luminum model a t v a r i o u s a n g l e s o f t w i s t and t h e two w h a l e f l u k e m o d e l s . The t o p end o f e a c h model was a t t a c h e d t o t h e l o w e r 34 Plate 5 Stra in Gauge Balance 35 p l a t e o f t h e b a l a n c e , and t h e b a l a n c e was r a i s e d t o a h e i g h t s u c h t h a t t h e l o w e r p l a t e was a b o u t 3 i n c h e s above t h e w a t e r s u r f a c e . The d e p t h o f i m m e r s i o n was a b o u t 15 i n c h e s f o r a l l t h e model s e c t i o n s . Due t o t h e symmetry o f t h e model s e c t i o n a t z e r o a n g l e o f a t t a c k , no l i f t f o r c e was p r o d u c e d , and t h e p o s i t i o n o f minimum l i f t was u s e d t o e s t a b l i s h t h e b a l a n c e p o s i t i o n f o r z e r o a n g l e o f a t t a c k . B e f o r e e a c h t e s t on t h e a e r o f o i l s e c t i o n s , t h e f o u r b a l a n c e b r i d g e c i r -c u i t s were s e t t o t h e n u l l p o s i t i o n w i t h no w a t e r i n t h e f l u m e , l a t e r b e c a u s e o f t h e l a r g e buoyancy e f f e c t o f t h e w h a l e f l u k e s e c t i o n s , t h e b r i d g e was b a l a n c e d w i t h t h e f l u m e f u l l o f w a t e r . I m m e d i a t e l y a f t e r e a c h t e s t t h e z e r o r e a d i n g s were c h e c k e d t o e n s u r e t h a t no d r i f t had o c c u r r e d d u r i n g t h e t e s t . A t e a c h a n g l e o f a t t a c k t h e f o u r v o l t a g e s were r e c o r d e d i n s u c c e s s i o n , each one b e i n g a l l o w e d t o run f o r two m i n u t e s t o a l l o w t h e t u r b u l e n c e e f f e c t s t o e v e n o u t . The a n g l e o f a t t a c k was g e n e r a l l y i n c r e a s e d by e i t h e r 1 , 2 o r 3 d e g r e e s d e p e n d i n g on t h e r a n g e and t h e s e n s i t i v i t y . The epoxy a e r o f o i l s e c t i o n s were r a t h e r f l e x i b l e , and c o u l d o n l y be t e s t e d up t o an a n g l e o f a t t a c k o f 8 d e g r e e s , a t w h i c h t h e y showed r a t h e r l a r g e d e f l e c t i o n s i n t h e l i f t d i r e c t i o n . The more s t i f f a luminum and w h a l e f l u k e mode ls were t e s t e d up t o t h e f u l l r a n g e o f 15 d e g r e e s . The a luminum model was t w i s t e d u n i f o r m l y so as t o d e c r e a s e t h e a n g l e o f a t t a c k a t t h e t i p f o r p o s i t i v e a n g l e s o f a t t a c k , and t h i s was t e s t e d a t b o t h p o s i t i v e and n e g a t i v e a n g l e s , so t h a t t h e a d v a n t a g e o u s p o s i t i v e p o s i t i o n c o u l d be compared w i t h t h e d i s a d v a n t a g e o u s n e g a t i v e p o s i t i o n . The a luminum a e r o f o i l model was a l s o used f o r t h e y a w i n g t e s t where t h e a n g l e o f a t t a c k was h e l d c o n s t a n t a t - 8 d e g r e e s and t h e model 36 yawed back and f o r t h f r o m t h e v e r t i c a l p o s i t i o n t h r o u g h a r a n g e o f a n g l e s f r o m 3 t o 13 d e g r e e s . 4 . 3 E r r o r s D u r i n g t h e c a l i b r a t i o n p r o c e d u r e t h e s i g n a l f r o m t h e r e c o r d e r was s t a b l e and p r o d u c e d a s t r a i g h t l i n e on t h e c h a r t . Measurements made i n the f l u m e , b e c a u s e o f t h e low f r e q u e n c y o s c i l l a t i o n s , showed a r a n g e o f v a l u e s and t h e c h a r t i s as shown i n P l a t e 6 . The a c t u a l r e a d i n g was o b t a i n e d by a v e r a g i n g t h e v a r i a t i o n s by eye and t h i s was deemed a r e a s o n a b l e m e t h o d . The e r r o r i n t h e c a l i b r a t i o n f a c t o r s o b t a i n e d f r o m t h e s p r e a d o f r e a d i n g s was a b o u t ± 5 mv f o r b o t h t h e l i f t and d r a g g a u g e s . The e r r o r i n t h e v o l t a g e r e a d i n g s f r o m t h e f l u m e were ± 20 mv f o r t h e d r a g gauges and ± 35 mv f o r t h e l i f t gauges ( a v e r a g e d o v e r t h e r a n g e o f a n g l e s o f a t t a c k ) . The t o t a l e r r o r i n t h e d r a g r e a d i n g was ± 25 mv w h i c h c o r r e s p o n d s t o a d r a g f o r c e o f ± 0 . 0 2 l b . The t o t a l e r r o r i n t h e l i f t r e a d i n g was ± 40 mv w h i c h c o r r e s p o n d s t o a l i f t f o r c e o f ± 0 . 2 0 l b . The a v e r a g e e r r o r i n t h e l i f t t o d r a g r a t i o was a b o u t ± 0 . 8 . The e r r o r i n r e a d i n g t h e a n g l e o f a t t a c k f r o m t h e g r a d u a t e d m a r k i n g s on t h e b a l a n c e was ± 0 . 2 5 d e g r e e s . The e r r o r i n c u r r e d f r o m s e t t i n g t h e z e r o a n g l e o f a t t a c k p o s i t i o n was e s t i m a t e d t o be ± 0 . 2 5 d e g r e e s . The e r r o r i n t h e a n g l e o f t w i s t measurements was j u d g e d t o be o f t h e o r d e r o f ± 0 . 2 5 d e g r e e s . Plate 6 Sample Output from Chart Recorder 38 CHAPTER 5 DISCUSSION OF RESULTS I n t h i s s e c t i o n t h e r e s u l t s f r o n v t h e v a r i o u s t e s t s w i l l be d i s c u s s e d . The r e s u l t s a r e p r e s e n t e d i n t a b u l a r f o r m i n A p p e n d i x A . In t h i s s e c t i o n t h e r e s u l t s a r e p r e s e n t e d i n g r a p h i c a l f o r m , g e n e r a l l y as a p l o t o f l i f t t o d r a g r a t i o v e r s u s a n g l e o f a t t a c k . The a d v a n t a g e s and d i s a d v a n t a g e s o f t h e v a r i o u s mode ls a r e i l l u s t r a t e d on t h e g r a p h s and t h e s e w i l l now be d i s c u s s e d i n d e t a i l . I t s h o u l d be made c l e a r a t t h e o u t s e t t h a t t h e l i f t t o d r a g r a t i o i s i n t e rms o f t h e t o t a l d r a g and n o t t h e i n d u c e d d r a g w h i c h i s o n l y a p o r t i o n o f t h e t o t a l d r a g . I n t h e f i r s t s e r i e s o f t e s t s f l e x i b l e a e r o f o i l s were t e s t e d t o d e t e r m i n e t h e i n f l u e n c e o f t w i s t on t h e i r l i f t t o d r a g r a t i o s . F i g u r e 9 shows t h e r e s u l t s f r o m t h e t e s t s on t h e t h r e e epoxy a e r o f o i l s . The model w i t h t h e l e a d i n g edge s t i f f e n e r s has a g r e a t e r l i f t t o d r a g r a t i o f o r a l l a n g l e s o f a t t a c k t h a n t h e model w i t h t h e t r a i l i n g edge s t i f f e n e r s . The model w i t h no s t i f f e n e r s ,has a g r e a t e r l i f t t o d r a g r a t i o t h a n t h e model w i t h l e a d i n g edge s t i f f e n e r s a t s m a l l a n g l e s o f a t t a c k - - a r o u n d 1 o r 2 d e g r e e s . The model w i t h l e a d i n g edge s t i f f e n e r s has t h e h i g h e r l i f t t o d r a g r a t i o f o r a l l a n g l e s o f a t t a c k g r e a t e r t h a n 2 d e g r e e s . In t h i s t e s t i t was d i s a p p o i n t i n g t h a t t h e model w i t h l e a d i n g edge s t i f f e n e r s d i d no t show g r e a t e r e f f i c i e n c y o v e r t h e f u l l r a n g e o f a n g l e s o f a t t a c k . A t l ow a n g l e s o f a t t a c k t h e P L O T O F L I F T T O D R A G R A T I O V E R S U S A N G L E O F A T T A C K F O R T H E T H R E E E P O X Y A E R O F O I L S . 40 l i f t f o r c e s p r o d u c e d a r e s m a l l and so t h e t w i s t i n g e f f e c t o f t h e s t i f f e n e r s i s s m a l l , and one w o u l d e x p e c t t h e l e a d i n g edge c a s e t o a t l e a s t match t h e p e r f o r m a n c e o f t h e model w i t h no s t i f f e n e r s . T h i s may have been due t o a shape d e f e c t i n o n e i o r b o t h o f t h e two m o d e l s , as t h e e p o x y had a t e n d e n c y t o warp a f t e r i t was removed f rom t h e m o l d . I t was v e r y i n t e r e s t i n g t o n o t e t h a t t h e model w i t h l e a d i n g edge s t i f f e n e r s h e l d i t s peak l i f t t o d r a g r a t i o o v e r a w i d e r a n g e o f a n g l e s o f a t t a c k , i t was t h u s q u i t e e f f i c i e n t o v e r a w i d e range and l e s s s e n s i t i v e t o changes i n a n g l e o f a t t a c k . In t h e s e c o n d s e r i e s o f t e s t s a r i g i d a luminum a e r o f o i l was used w h i c h c o u l d be t w i s t e d and made t o h o l d a g i v e n t w i s t f o r e a c h t e s t . F i g u r e 10 shows t h e r e s u l t s f r o m t e s t i n g t h e a luminum a e r o f o i l i n w h i c h t h e model was t w i s t e d t o ' r e d u c e t h e a n g l e o f a t t a c k a t t h e t i p . The model had a peak l i f t t o | d r a g r a t i o o f 1 0 . 2 b e f o r e i t was t w i s t e d , b u t w i t h an a n g l e o f t w i s t o f 1 d e g r e e t h e maximum r a t i o was 1 2 . 4 and a t 1 .5 d e g r e e s i t was 9 . 8 w h i l e a t 5 d e g r e e s i t was r e d u c e d t o a b o u t 8 . 0 . These peak l i f t t o d r a g r a t i o s o c c u r r e d a t h i g h e r a n g l e s o f a t t a c k f o r t h e h i g h e r a n g l e s o f t w i s t . The u n t w i s t e d model p r o v e d t o be t h e most e f f i c i e n t up t o an a n g l e o f a t t a c k o f 4 d e g r e e s , t h e n t h e model w i t h 1 d e g r e e o f t w i s t was t h e most e f f i c i e n t up t o an a n g l e o f a t t a c k o f a b o u t 11 d e g r e e s , t h e n t h e model w i t h 5 d e g r e e s o f t w i s t was most e f f i c i e n t . I t i s u n f o r t u n -a t e t h a t t h e c u r v e f o r 1 .5 d e g r e e s o f t w i s t d i d n o t c r o s s t h e 1 d e g r e e o f t w i s t c u r v e a t a b o u t 9 d e g r e e s . However t h e r e s u l t s f rom t h e 2 d e g r e e s o f t w i s t c a s e show i t t o be t h e most e f f i c i e n t f rom 10 41 FIGURE 10 0 2 4 6 8 10 12 A n g l e of A t t a c k ( d e g r e e s ) PLOT OF LIFT TO DRAG RATIO VERSUS ANGLE OF ATTACK FOR THE ALUMINUM AEROFOIL AT VARIOUS ANGLES OF TWIST TO REDUCE ANGLE OF ATTACK . 42 t o 12 d e g r e e s . T h i s means t h a t as t h e a n g l e o f a t t a c k i s i n c r e a s e d t h e h i g h e r a n g l e s o f t w i s t a r e t h e most e f f i c i e n t . T h i s f o l l o w s f r o m what was s a i d e a r l i e r a b o u t t h e t w i s t i n g e f f e c t b e i n g s m a l l f o r t h e f l e x i b l e a e r o f o i l w i t h l e a d i n g edge s t i f f e n e r s when t h e a n g l e s o f a t t a c k a r e s m a l l . As t h e a n g l e o f a t t a c k i n c r e a s e s t h e l i f t i n c r e a s e s and so t h e s e c t i o n t w i s t s t o g r e a t e r a n g l e s . F i g u r e 10 t h u s r e p r e s e n t s an e n v e l o p e o f t h e b e s t l i f t t o d r a g r a t i o t h a t c a n be o b t a i n e d f o r each a n g l e o f a t t a c k . The c u r v e drawn t h r o u g h t h e peaks o f e a c h o f t h e i n d i v i d u a l t w i s t c u r v e s s h o u l d r e p r e s e n t t h e b e s t t h a t c a n be done a t any g i v e n a n g l e o f a t t a c k , i t s h o u l d be i d e n t i c a l t o t h e c u r v e f o r a p r o p e r l y d e s i g n e d f l e x i b l e a e r o f o i l t h a t t w i s t s j u s t t h e r i g h t amount t o o b t a i n e l l i p t i c l o a d i n g a t a l l a n g l e s o f a t t a c k . A n o t h e r p o i n t t o 'note f r o m F i g u r e 10 i s t h a t t h e peak o f t h e c u r v e f l a t t e n s o u t f o r i n c r e a s i n g a n g l e s o f t w i s t . The r e s u l t s f r o m t h e s e t e s t s v e r i f y t h a t t h e peak o f t h e c u r v e f o r t h e u n t w i s t e d epoxy model i s t o o h i g h , s i n c e i t s h o u l d be a b o u t t h e same as t h e peaks o b t a i n e d i n t h e a luminum a e r o f o i l t e s t s . F i g u r e 11 shows t h e r e s u l t s f r o m t h e a luminum a e r o f o i l s e r i e s o f t e s t s i n w h i c h t h e model was t w i s t e d i n t h e o p p o s i t e s e n s e t o i n c r e a s e r a t h e r t h a n d e c r e a s e t h e a n g l e o f a t t a c k a t t he t i p . A l l o f t h e s e t e s t s show t h e d e t r i m e n t a l e f f e c t o f i n c r e a s i n g t h e l i f t a t t h e w i n g t i p , a l l o f t h e mode ls have a l o w e r l i f t t o d r a g r a t i o t h a n t h e no t w i s t c a s e . I n t h e n e x t s e r i e s o f t e s t s t h e w h a l e f l u k e mode ls were t e s t e d t o i n v e s t i g a t e t h e i n f l u e n c e o f t w i s t and shape on t h e l i f t t o d r a g r a t i o . F i g u r e 12 shows t h e r e s u l t s f r o m t h e w h a l e f l u k e F I G U R E II 43 2 8 6 4 ^^ ^^ ^ c Twisted to of a t t a c k c i n c r e a s e a ]t Tip 1 -ig le 0 2 4 6 8 10 12 A n g l e of A t t a c k ( d e g r e e s ) P L O T O F L I F T T O D R A G R A T I O V E R S U S A N G L E O F A T T A C K F O R T H E A L U M I N U M A E R O F O I L A T V A R I O U S A N G L E S O F T W I S T T O I N C R E A S E A N G L E O F A T T A C K . PLOT OF LIFT TO DRAG RATIO VERSUS ANGLE OF ATTACK FOR THE WHALE FLUKE MODELS . N 45 s e r i e s o f t e s t s . The u n t w i s t e d wooden model had a peak l i f t t o d r a g r a t i o o f 8 . 5 and t h e epoxy t w i s t e d model had a maximum r a t i o o f 9 . 5 . The t w i s t e d model ( w i t h an a n g l e o f t w i s t o f a b o u t 1 d e g r e e ) was more e f f i c i e n t o v e r a l a r g e r a n g e o f a n g l e s o f a t t a c k , h a v i n g h i g h e r l i f t t o d r a g r a t i o s o v e r t h e r a n g e 0 t o 10 d e g r e e s . P r e s u m a b l y , as b e f o r e , h i g h e r a n g l e s o f t w i s t w o u l d have been more e f f i c i e n t a t t h e h i g h e r a n g l e s o f a t t a c k . The peaks o f b o t h c u r v e s a r e s l i g h t l y f l a t t e r t h a n t h e peak f o r t h e u n t w i s t e d a luminum a e r o f o i l and t h i s i s q u i t e i n t e r e s t -i n g as a f l a t t e n i n g o u t was a l s o n o t i c e d f o r t h e t w i s t e d a e r o f o i l s . On t h e o t h e r hand when t h e epoxy model was t w i s t e d so t h a t t h e a n g l e o f a t t a c k i n c r e a s e d a t t h e t i p , t h e l i f t t o d r a g r a t i o s f e l l l o w e r t h a n t h e u n t w i s t e d c a s e f o r a l l a n g l e s o f a t t a c k . The e p o x y a e r o f o i l w i t h l e a d i n g edge s t i f f e n e r s seems t o v e r i f y t h e e l l i p t i c l o a d i n g t h e o r y p u t f o r w a r d e a r l i e r . E l l i p t i c l o a d i n g p r o d u c e s opt imum l i f t t o d r a g r a t i o s and c a n be o b t a i n e d by e i t h e r an e l l i p t i c w i n g o r a r e c t a n g u l a r w i n g t w i s t e d t o r e d u c e t h e a n g l e o f a t t a c k a t t h e t i p . The e l l i p t i c w i n g w i l l be e f f i c i e n t a t a l l a n g l e s * o f a t t a c k b u t t h e r e c t a n g u l a r w i n g w i t h a f i x e d a n g l e o f t w i s t can o n l y match i t s p e r f o r m a n c e a t one s p e c i f i c a n g l e o f a t t a c k d e p e n d i n g on t h e a n g l e o f t w i s t . T h i s shows up q u i t e c l e a r l y i n F i g u r e 10 as e a c h i n d i v i d u a l a n g l e o f t w i s t has i t s own p a r t i c u l a r r a n g e o f a n g l e s o f a t t a c k f o r w h i c h i t i s more e f f i c i e n t t h a n t h e u n t w i s t e d c a s e . The f l e x i b l e r e c t a n g u l a r w i n g d e s i g n e d t o t w i s t a t j u s t t h e r i g h t a n g l e f o r each a n g l e o f a t t a c k w i l l p r o d u c e e l l i p t i c l o a d -i n g a t a l l a n g l e s o f a t t a c k . The f l e x i b l e a e r o f o i l c a n be d e s i g n e d i f r o m t h e e n v e l o p e c u r v e o b t a i n e d f r o m a s e r i e s o f t e s t s s u c h as t h o s e ! I 46 o f F i g u r e 10. I f a l a r g e number o f t h e s e t e s t s were p e r f o r m e d , i n c r e m e n t i n g t h e a n g l e o f t w i s t a s m a l l amount e a c h t i m e , t h e n a s e r i e s o f c u r v e s w o u l d be o b t a i n e d f o r w h i c h t h e r a n g e o f a n g l e s o f a t t a c k i n w h i c h t h a t p a r t i c u l a r t w i s t was most e f f i c i e n t w o u l d be s m a l l . An a c c u r a t e e n v e l o p e c u r v e c o u l d t h e n be drawn t h r o u g h t h e peak r a t i o s f o r e a c h t w i s t and f rom t h i s c u r v e t h e t w i s t r e q u i r e d f o r t h e f l e x i b l e a e r o f o i l a t any p a r t i c u l a r a n g l e o f a t t a c k c o u l d be r e a d o f f . The e p o x y a e r o f o i l w i t h l e a d i n g edge s t i f f e n e r s h e l d i t s peak l i f t t o d r a g r a t i o o v e r a w i d e r a n g e and t h i s must be due t o t h e i n f l u e n c e o f e l l i p t i c l o a d i n g . U n f o r t u n a t e l y t h i s a e r o f o i l was n o t d e s i g n e d c o r r e c t l y and s o i t d i d n o t match t h e p e r f o r m a n c e g i v e n by t h e e n v e l o p e c u r v e t h r o u g h t h e peaks o f t h e i n d i v i d u a l c u r v e s o f F i g u r e 10. The t w i s t e d w h a l e f l u k e epoxy model was more e f f i c i e n t t h a n t h e u n t w i s t e d model o v e r a l a r g e r a n g e o f a n g l e s o f a t t a c k . The t w i s t e d model was n o t f l e x i b l e so e l l i p t i c l o a d i n g c o u l d n o t be t a k i n g p l a c e o v e r t h e e n t i r e r a n g e , p e r h a p s t h e g r e a t e r e f f i c i e n c y was b r o u g h t a b o u t by a c o m b i n a t i o n o f t h e s m a l l a n g l e o f t w i s t and t h e u n u s u a l shape o f t h e w h a l e f l u k e . I t i s c l e a r t h a t t h e u n u s u a l shape o f t h i s s e c t i o n i m p r o v e s i t s p e r f o r m a n c e . The t e s t s on t h e a luminum a e r o f o i l showed t h a t a t s m a l l a n g l e s o f a t t a c k t h e c a s e o f no t w i s t was most e f f i c i e n t and t h i s was c o n s i s t e n t w i t h t h e t h e o r y . The w h a l e f l u k e , h o w e v e r , shows t h e t w i s t e d c a s e t o be more e f f i c i e n t even a t s m a l l a n g l e s o f a t t a c k and we s u r m i s e t h a t t h i s must be due t o t h e shape e f f e c t . T h i s seems t o c o n t r a d i c t t h e b a s i c t h e o r y o f e l l i p t i c l o a d i n g b e c a u s e we s h o u l d n o t e x p e c t t o do b e t t e r t h a n t h e u n t w i s t e d 47 c a s e a t s m a l l a n g l e s o f a t t a c k . T h i s i n d i c a t e s t h a t a more e x t e n s i v e s e r i e s o f t e s t s a r e r e q u i r e d t o i n v e s t i g a t e t h e shape e f f e c t , and d e f i n i t e c o n c l u s i o n s s h o u l d n o t be drawn h e r e . The most i n t e r e s t i n g r e s u l t o b t a i n e d f r o m t h e s e t e s t s i s t h a t t h e t a p e r e d w h a l e f l u k e b e n e f i t s f r o m t w i s t . O r i g i n a l l y i t was t h o u g h t t h a t t h e opt imum e l l i p t i c l o a d i n g c o u l d be b r o u g h t a b o u t by t a p e r i n g i n t h e fo rm o f an e l l i p s e o r by t w i s t i n g , and c o m b i n a t i o n s o f t h e two s h o u l d o n l y be u s e f u l when t h e amount o f t a p e r i n g i s s m a l l . The h i g h l y t a p e r e d w h a l e f l u k e , h o w e v e r , shows an i n c r e a s e i n e f f i c i e n c y when t w i s t e d t o r e d u c e t h e a n g l e o f a t t a c k . These r e s u l t s a r e b e t t e r t h a n i s i n d i c a t e d by t h e p r e d i c t i o n s o f t h e two d i m e n s i o n a l t h e o r y o f e l l i p t i c l o a d i n g . The ma in a d v a n t a g e o f t h e f l e x i b l e a e r o f o i l w h i c h p r o d u c e s e l l i p t i c l o a d i n g a t a l l a n g l e s o f a t t a c k i s i t s l a c k o f s e n s i t i v i t y t o w a r d changes i n a n g l e o f a t t a c k . The epoxy a e r o f o i l w i t h l e a d i n g edge s t i f f e n e r s t e s t e d h e r e h e l d i t s peak l i f t t o d r a g r a t i o o v e r q u i t e a w i d e r a n g e o f a n g l e s o f a t t a c k and we w o u l d e x p e c t a p r o p e r l y d e s i g n e d f l e x i b l e a e r o f o i l t o do much b e t t e r t h a n t h i s . T h i s k i n d o f s y s t e m i s v e r y f o r g i v i n g , i t i s l e s s d e m a n d i n g , and s m a l l c h a n g e s i n a n g l e o f a t t a c k p r o d u c e v e r y l i t t l e change i n t h e l i f t t o d r a g r a t i o . T h e r e i s no s h a r p d r o p o f f o r s t a l l i n t h e l i f t c u r v e and t h e f u l l r a n g e o f a n g l e s o f a t t a c k c a n be u t i l i z e d . T h i s i s i n ag reement w i t h many n a t u r a l s y s t e m s and we m i g h t s u r m i s e t h a t t h i s i s p a r t i c u l a r l y u s e f u l f o r b i r d s and swimming a n i m a l s w h i c h must f l a p t h e i r w i n g s o r f l u k e s f o r p r o p u l s i o n . The f l a p p i n g a c t i o n r o t a t e s t h e w i n g t h r o u g h q u i t e a r a n g e o f a n g l e s o f a t t a c k and t h e h i g h e r l i f t t o d r a g r a t i o 48 t h a t can be m a i n t a i n e d d u r i n g f l a p p i n g t h e b e t t e r . G l i d i n g b i r d s a r e a l s o s u b j e c t t o n a t u r a l changes i n a n g l e o f a t t a c k due t o t h e t u r b u l e n t c h a r a c t e r i s t i c s o f w i n d . Some c a l c u l a t i o n s were c o m p l e t e d a c c o r d i n g t o t h e t h e o r y o f G l a u e r t and some t y p i c a l r e s u l t s gave t h e l i f t as 1 .16 l b s and t h e i n d u c e d d r a g as 0 . 0 2 4 5 l b s as c o m p a r e d ' t o a measu red l i f t o f 2 . 4 7 l b s and a measured t o t a l d r a g o f 0 . 2 9 ! l b s , t h e s e r e s u l t s b e i n g f o r t h e u n t w i s t e d a luminum a e r o f o i l a t an a n g l e o f a t t a c k o f 6 d e g r e e s . F o r 1 d e g r e e o f t w i s t a t an a n g l e o f a t t a c k o f 6 d e g r e e s t h e computed l i f t was 1 . 0 8 l b s and t h e computed i n d u c e d d r a g was 0 .021 l b s w h i l e t h e measured l i f t was 2 . 0 8 l b s and t h e measured d r a g was 0 . 2 0 l b s . The computed v a l u e s f o r l i f t and i n d u c e d d r a g a r e much l o w e r t h a n t h e measured v a l u e s and t h i s must be due t o t h e t u r b u l e n t n a t u r e o f t h e f l o w i n t h e f l u m e . I t i s s i g n i f i c a n t t h a t t h e measured l i f t d r o p s o n l y 12% w h i l e t h e measured d r a g d r o p s 31%. T h i s 31% d r o p i i n d r a g c a n n o t be a d e q u a t e l y e x p l a i n e d by t h e u s u a l two d i m e n s i o n a l t h e o r y . P r e s u m a b l y t h e t h r e e d i m e n s i o n a l s p a n w i s e f l o w s p l a y a s i g n i f i c a n t r o l e and f u r t h e r s t u d y o f s u c h t h r e e d i m e n s i o n a l e f f e c t s w o u l d be w o r t h w h i l e . F i g u r e 13 shows t h e r e s u l t s f r o m t h e y a w i n g t e s t s and shows t h a t t h e r e i s v e r y l i t t l e d i f f e r e n c e i n p e r f o r m a n c e when t h e a e r o f o i l i s yawed f o r w a r d o r b a c k w a r d . These t e s t s show l i t t l e v a r i a t i o n o f l i f t t o d r a g r a t i o w i t h yaw b u t t h e r e s u l t s a r e n o t c o n c l u s i v e b e c a u s e i t i s p r o b a b l e t h a t w i n g t i p shape may have an o v e r - r i d i n g i n f l u e n c e on t h e r e s u l t s . 10 PLOT OF LIFT TO DRAG RATIO VERSUS ANGLE OF YAW FOR THE ALUMINUM AEROFOIL . SUMMARY OF CONCLUSIONS 50 I t has been c o n f i r m e d t h a t e l l i p t i c l o a d i n g c a n be o b t a i n e d by a f l e x i b l e a e r o f o i l w i t h s t i f f e n e r s i n t h e l e a d i n g e d g e , w h i c h w o u l d p e r m i t j u s t t h e r i g h t amount o f t w i s t a t e a c h a n g l e o f a t t a c k . T h i s c o r r e c t amount o f t w i s t c a n be o b t a i n e d by mak ing a s e r i e s o f t e s t s on r i g i d a e r o f o i l s i n w h i c h t h e t w i s t i s i n c r e m e n t e d a s m a l l amount f o r each t e s t . The e n v e l o p e o f t h e s e r i g i d , c o n t r o l l e d t w i s t , t e s t s g i v e s t h e d e s i r a b l e f l e x i b l e a e r o f o i l c h a r a c t e r i s t i c s . By t h e s i m p l e t h e o r y o f e l l i p t i c l o a d i n g , w i n g s w i t h c o n s i d e r a b l e t a p e r s h o u l d n o t b e n e f i t f r o m w a s h o u t t w i s t ( t w i s t t o r e d u c e t h e a n g l e o f a t t a c k t o w a r d s t h e t i p ) ; i n f a c t , r e v e r s e t w i s t s h o u l d make them a p p r o x i m a t e e l l i p t i c l o a d i n g . In t h i s s t u d y w a s h o u t t w i s t i n g o f t a p e r e d w i n g s has been shown t o be b e n e f i c i a l i n t h e c a s e s t e s t e d , and t h e s e b e n e f i t s a r e i n c o n t r a d i c t i o n t o t h e s i m p l e two d i m e n -s i o n a l t h e o r y o f e l l i p t i c l o a d i n g . F u r t h e r m o r e , t w i s t i n g i s more b e n e f i c i a l t h a n t h e s i m p l e t h e o r y s h o w s ; t h e r e d u c t i o n s i n i n d u c e d d r a g b e i n g much g r e a t e r t h a n t h e t h e o r y p r e d i c t s . As an a d d i t i o n a l i m p o r t a n t b e n e f i t t h e t w i s t e d w i n g i s l e s s s e n s i t i v e t o changes i n a n g l e o f a t t a c k , t h e r e i s no s h a r p d r o p o f f o r s t a l l , a r e s u l t w h i c h makes t h e t o t a l s y s t e m l e s s demanding and c o u l d be p a r t i c u l a r l y u s e f u l a t low s p e e d s , s a y d u r i n g commencement o f m o t i o n o f b i r d s and f i s h . L I S T OF REFERENCES A b b o t t , I . H . and von D o e n h o f f , A . E . T h e o r y o f Wing S e c t i o n s I n c l u d i n g a Summary o f A i r f o i l D a t a . New Y o r k , McGraw H i l l , 1949 . B a r r y , A . B . E n g i n e e r i n g M e a s u r e m e n t s . New Y o r k , W i l e y and S o n s , 1964 . G l a u e r t , H. The E l e m e n t s o f A e r o f o i l and A i r s c r e w T h e o r y . C a m b r i d g e , Cambr idge U n i v e r s i t y P r e s s , 1926 . Howard , F. and G u n s t o n , W. The C o n q u e s t o f t h e A i r . New Y o r k , Random H o u s e , 1 9 7 2 . P r a n d t l , L . and T i e t j e n s , O . G . F u n d a m e n t a l s o f H y d r o - and A e r o m e c h a n i c s . T r a n s ! . L . R o s e n h e a d . New Y o r k , McGraw H i l l , 1934 . P r a n d t l , L . and T i e t j e n s , O . G . A p p l i e d H y d r o - and A e r o m e c h a n i c s . T r a n s ! . J . P . Den H a r t o g . New Y o r k , Dover P u b l i c a t i o n s , 1957 . A n d e r s e n , H .T . The B i o l o g y o f M a r i n e Mammals. New Y o r k , A c a d e m i c P r e s s , 1969 . N o r r i s , K . S . W h a l e s , D o l p h i n s and P o r p o i s e s . B e r k e l y , U n i v e r s i t y o f C a l i f o r n i a P r e s s , 1966 . APPENDIX A TABLE OF RESULTS EPOXY AEROFOIL - STIFFENERS IN LEADING EDGE ANGLE L IFT L I F T DIST. DRAG DRAG DIST. L / D RATIO 1 T . 2 5 1 2 . 6 0 . 1 0 10 .1 1 3 . 0 2 1 . 2 3 1 6 . 7 0 . 1 2 1 0 . 7 1 0 . 5 3 1 . 8 9 1 1 . 2 0 . 1 6 1 0 . 4 1 1 . 9 4 2.01 1 2 . 5 0.21 1 0 . 6 9 . 8 5 2 .31 1 1 . 7 0 . 2 4 1 1 . 3 9 . 5 6 2 . 6 5 1 1 . 2 0 . 3 2 1 1 . 5 8 . 2 7 2 . 4 5 1 3 . 4 0.41 1 0 . 6 6 . 0 8 2 . 8 3 1 2 . 3 0 . 4 6 1 1 . 8 6 . 2 EPOXY AEROFOIL - NO STIFFENERS 1 1.41 1 1 . 7 0 . 0 9 1 0 . 4 1 6 . 5 2 1 .56 1 2 . 6 0 . 1 2 1 0 . 6 1 2 . 7 3 1 .96 1 1 . 7 0 . 1 7 11.1 1 1 . 7 4 2 . 3 5 1 1 . 3 0 . 2 3 1 1 . 3 1 0 . 0 5 2 . 2 5 1 2 . 8 0 . 2 7 1 2 . 6 8 . 2 6 2 . 6 0 1 2 . 0 0 . 3 7 1 1 . 8 7 . 0 EPOXY AEROFOIL - STIFFENERS IN TRAILING EDGE 1 0 . 8 3 1 4 . 4 0 . 0 8 8 . 2 1 0 . 0 2 1.01 15.1 0.11 9.1 9 . 0 3 1 . 5 3 1 1 . 7 0 . 1 6 9 . 7 9 . 7 4 1 .69 13.1 0 . 2 2 1 0 . 6 7 . 9 5 2 . 1 4 1 1 . 9 0 . 2 7 11.1 7 . 8 6 2 . 2 5 1 2 . 4 0 . 3 5 1 1 . 3 6 . 4 7 2 . 5 6 1 1 . 2 0.41 1 2 . 6 6 . 3 8 2 . 8 3 1 1 . 4 0 . 5 3 1 1 . 9 5 . 4 ALUMINUM AEROFOIL - 0 DEGREES OF TWIST 53 ANGLE L I F T L I F T DIST. DRAG DRAG DIST. L / D RATIO - 3 1 .43 14.1 0 . 1 4 1 2 . 6 1 0 . 2 - 6 2 . 4 7 1 2 . 7 0 . 2 9 1 3 . 7 8 . 5 - 9 3 . 2 8 1 2 . 5 0 . 5 4 1 3 . 3 6 . 0 ALUMINUM AEROFOIL - 1 DEGREE OF TWIST - 3 1 .44 1 5 . 2 0 . 1 6 1 1 . 6 8 . 7 - 6 2 . 5 0 1 2 . 7 0 . 3 2 1 4 . 0 7 . 8 - 9 3 . 3 7 1 2 . 4 0.61 1 3 . 5 5 . 5 -11 3 . 4 2 1 4 . 5 0 . 8 2 1 3 . 3 4 . 2 +3 0 . 3 8 2 . 6 0 . 0 5 9 . 3 7 . 4 +6 1 . 1 5 1 0 . 8 0 . 0 9 1 1 . 6 1 2 . 3 +9 2 . 0 8 1 1 . 5 0 . 2 0 1 4 . 5 1 0 . 2 +11 2 . 6 2 1 1 . 4 0 . 3 3 1 3 . 8 8 . 0 +13 3 .21 1 2 . 0 0 . 4 9 14 .1 6 . 5 ALUMINUM AEROFOIL - 1 . 5 DEGREES OF TWIST - 3 2 . 0 5 1 2 . 5 0 . 2 0 1 3 . 0 1 0 . 2 - 6 2 . 8 4 13 .1 0 . 4 3 1 2 . 4 6 . 6 - 9 3 . 4 3 13.1 0 . 6 7 14 .1 5.1 -11 3 . 6 5 1 3 . 5 0 . 8 4 1 4 . 6 4 . 3 - 1 3 3 . 9 8 1 2 . 9 1 . 0 7 1 4 . 6 3 . 7 3 0 . 0 2 3 5 . 8 0 . 0 5 1 1 . 0 0 . 4 6 0 . 5 7 2 0 . 8 0 . 0 8 1 2 . 8 7 . 6 9 1 . 7 3 1 2 . 7 0 . 1 7 1 3 . 0 9 . 9 11 2 . 3 8 1 2 . 3 0 .31 1 2 . 2 7 . 6 13 3 . 1 3 1 1 . 7 0 . 4 5 1 3 . 6 7 . 0 ALUMINUM AEROFOIL - 2 DEGREES OF TWIST 54 ANGLE L I F T L I F T DIST. DRAG DRAG DIST. L / D RATIO - 3 1 .60 1 3 . 9 0 . 1 8 1 2 . 6 8 . 8 - 6 2 . 6 2 1 2 . 7 0 . 3 6 1 3 . 5 7 . 3 - 9 3 . 1 9 1 3 . 6 0 . 5 8 1 4 . 8 5 . 5 - 11 3 . 8 5 1 2 . 8 0 . 9 0 1 2 . 9 4 . 3 3 0 . 0 6 6 . 8 0 . 0 6 8 . 6 1.1 6 0 . 9 0 1 2 . 6 0 . 0 7 1 5 . 4 1 2 . 5 9 1 .82 1 2 . 4 0 . 2 0 1 2 . 6 8 . 9 11 2 . 4 3 1 2 . 5 0 . 2 8 1 5 . 3 8 . 7 13 2 . 8 4 1 3 . 0 0 . 4 5 1 4 . 4 6 . 3 ALUMINUM AEROFOIL - 5 DEGREES OF TWIST - 3 1 .93 1 5 . 4 0 . 2 6 1 4 . 2 7 . 4 - 6 2 . 8 6 1 4 . 4 0 .51 1 4 . 7 5 . 6 - 9 3 . 3 9 1 4 . 5 0 . 7 5 1 6 . 0 4 . 5 -11 3 . 7 8 1 3 . 7 0 . 9 8 15 .1 3 . 8 - 1 3 4 . 0 7 1 3 . 3 1 .17 1 5 . 5 3 . 5 3 0 . 4 2 1 3 . 4 0 . 0 7 9 . 4 5 . 8 6 0 . 2 8 1 5 . 8 0 . 0 8 8.1 3 . 6 9 1 . 2 0 1 3 . 3 0 . 1 5 9 . 8 8 . 2 11 2 . 1 3 11.1 0 . 2 0 1 2 . 6 1 0 . 4 13 2.61 1 1 . 6 0 . 3 3 1 2 . 6 7 . 9 15 3 . 3 4 1 1 . 3 0 . 5 3 1 1 . 9 6 . 3 WHALE FLUKE - 0 DEGREES OF TWIST ANGLE L I F T L I F T DIST. DRAG DRAG DIST. L / D RATIO 1 0 . 4 9 1 1 . 0 0 . 2 2 9 . 2 2 . 2 2 0 . 7 2 11 .8 0 . 2 5 8.1 2 . 9 3 0 . 8 8 1 3 . 2 0 .21 9 . 7 4 . 2 4 0 . 9 4 1 6 . 0 0 . 2 0 1 0 . 2 4 . 7 5 1 .38 13 .1 0 . 2 2 9 . 7 6 . 4 6 1 .66 1 2 . 4 0 . 2 2 10.1 7 . 5 7 1 . 74 1 3 . 9 0 . 2 4 1 0 . 5 7 . 4 8 2 . 0 9 13.1 0 . 2 8 8 . 9 7 . 6 10 2 . 6 7 1 2 . 2 0 . 3 0 1 0 . 6 8 . 9 12 3 . 1 4 1 1 . 9 0 . 3 7 9 . 9 8 . 5 15 3 . 5 3 1 2 . 4 0 . 4 7 1 0 . 5 7 . 6 WHALE FLUKE - 1 DEGREE OF TWIST - 2 0 . 5 7 1 7 . 3 0 . 1 8 1 0 . 2 3 . 2 - 4 1 .26 1 2 . 3 0 . 1 9 9 . 7 6 . 8 - 6 1 .69 1 2 . 9 0 . 1 9 1 0 . 7 9 .1 - 8 1.71 1 5 . 4 0 . 2 5 8 . 9 6 . 9 - 1 0 2 .51 1 2 . 7 0 . 2 9 9 . 7 8 . 8 - 1 2 2 . 5 2 1 4 . 5 0 . 3 3 1 0 . 3 7 . 5 2 0 . 1 7 2 3 . 0 0 . 2 4 1 0 . 5 0 . 7 4 0 . 9 5 1 0 . 3 0 . 2 6 1 1 . 9 3 . 6 6 1 .32 1 2 . 7 0 . 3 4 1 1 . 4 3 . 8 8 1 .70 1 3 . 7 0 . 4 3 1 1 . 5 4 . 0 10 2 . 1 0 1 4 . 0 0 . 5 2 1 2 . 3 4 .1 12 2 . 4 9 14.1 0 . 6 3 1 2 . 8 3 . 9 56 ALUMINUM AEROFOIL - CANTED FORWARD ANGLE* L I F T L I F T DIST. DRAG DRAG DIST. L / D RATIO 0 3 .11 1 4 . 3 0 .41 1 3 . 0 7 . 6 3 2 . 7 7 1 5 . 7 0 . 3 8 1 6 . 9 7 . 2 5 3 . 4 4 1 3 . 2 0 .41 1 4 . 9 8 . 3 8 3 . 4 9 1 3 . 0 0 . 4 2 1 5 . 2 8 . 2 10 3 . 3 4 1 3 . 4 0 . 4 0 1 6 . 5 8 . 3 13 3 . 6 2 1 2 . 3 0 . 4 0 1 7 . 9 9.1 ALUMINUM AEROFOIL - CANTED BACKWARD 3 2 . 8 5 1 4 . 4 0 . 3 7 1 3 . 4 7 . 7 5 3 . 4 4 1 2 . 7 0 . 3 9 1 1 . 7 8 . 8 8 3 . 3 0 1 3 . 2 0 . 3 9 1 0 . 7 8 . 5 10 3 . 3 0 13 .1 0 . 3 8 1 0 . 2 8 . 6 13 3.31 13.1 0 . 3 6 9 . 4 9 . 2 Note t h a t he re t h e a n g l e q u o t e d i s t h e a n g l e o f yaw and n o t t h e a n g l e o f a t t a c k . The a n g l e o f a t t a c k i s c o n s t a n t a t - 8 d e g r e e s . 57 APPENDIX B METHOD OF SOLUTION F o r p u r p o s e s o f s o l u t i o n t h e a luminum a e r o f o i l w i l l be t a k e n as h a l f a monoplane a e r o f o i l so t h a t i t i s t r e a t e d as b e i n g s y m m e t r i c a b o u t i t s m i d p o i n t h a v i n g an e f f e c t i v e h a l f span o f 15 i n c h e s . The a s p e c t r a t i o w i l l be t a k e n as 5 . 4 6 and t h e v a l u e o f 3 . 1 4 w i l l be used f o r a . The v a l u e s o f t h e c o n s t a n t t e rms w i l l be t a k e n a s : o a c a 3 . 1 4 o _ o _ 4 s 2 A 2 x 5 . 4 6 287 3 14 ya = . 287 x ^f^- x 6 .03 03 ye = = . 0 0 5 2TT S 2 p v 2 = 2 x 3 . 1 4 x ( y | ) 2 1 . 9 4 ( 2 . 4 ) 2 = 110 These v a l u e s c o r r e s p o n d t o an a n g l e o f a t t a c k o f 6° w i t h an a n g l e o f t w i s t o f 1 ° . As a f i r s t a p p r o x i m a t i o n t h e v a l u e s o f t h e f i r s t two c o e f f i c i e n t s , A-| and A g , w i l l be computed ( n o t e t h a t even numbered c o e f f i c i e n t s do n o t a p p e a r b e c a u s e t h e s i n e s e r i e s i s n o t s y m m e t r i c f o r t h e s e v a l u e s ) . The f u n d a m e n t a l e q u a t i o n t o s o l v e i s : £ A n s i n n8 (ny + s i n 8) = u s i n 6 (a-e c o s 6 ) and t h i s w i l l be d i v i d e d i n t o two p a r t s , each b e i n g s o l v e d s e p a r a t e l y . F i r s t p a r t : Z A n s i n n0 (ny + s i n 0) = ya s i n 6 A 1 s i n 6 (y + s i n 6) + A 3 s i n 36 (3y + s i n 6) = ya s i n 6 e = 4 5 , 90 . 707 A., ( . 5 7 5 + . 7 0 7 ) + . 707 A g O . 7 2 5 + . 707 ) = . 707 ya A ^ . 5 7 5 + 1) - A 3 ( l . 725 + 1 ) = ya S o l v i n g t h e s e two e q u a t i o n s s i m u l t a n e o u s l y f o r A-j and A^ A 1 = 0 . 7 0 4 ya = 0 .0211 A 3 = 0 . 0 4 0 ya = 0 . 0 0 1 2 S e c o n d p a r t : E A s i n n0 (ny + s i n 0) = - y e cos 6 s i n 6 59 A-j s i n e (y + s i n e ) + A 3 s i n 36 (3y + s i n e ) = - y e c o s e s i n e . 9 0 7 A 1 + 1 .72 A 3 = - . 5 y e 1 . 5 7 5 A 1 - 2 . 7 2 5 A 3 = 0 A ] = - . 2 6 4 y e = - 0 . 0 0 1 3 2 A 3 = - 0 . 1 5 2 y e = - 0 . 0 0 0 7 6 So t h a t f o r t h e u n t w i s t e d a e r o f o i l t h e c o n s t a n t s may be used as f o l l o w s : A 1 = .0211 A 3 = . 0 0 1 2 L i f t = ^ ( . 0 2 1 1 ) = 1 .16 l b I nduced Drag = ^ [ ( . 0 2 1 1 ) 2 + 3 ( . 0 0 1 2 ) 2 ] = 0 . 0 2 4 5 l b F o r t h e t w i s t e d a e r o f o i l t h e c o n s t a n t s become: A 1 = .0211 - . 00132 = . 0 1 9 8 A 0 = . 0 0 1 2 - . 00076 = . 0005 60 L i f t = 55 ( . 0198 ) = 1 . 0 8 l b I nduced Drag = 55 [ ( . 0 1 9 8 ) 2 + 3 ( . 0 0 0 5 ) 2 ] = .021 l b 61 APPENDIX C MOLDING AND CASTING The m a t e r i a l used f o r m o l d m a k i n g was Dow C o r n i n g S i l a s t i c B RTV Mo ldmak ing R u b b e r , t h i s was c h o s e n o v e r s u c h t h i n g s as P l a s t e r o f P a r i s o r F i b r e g l a s s b e c a u s e o f i t s l o w l i n e a r s h r i n k a g e and e a s e o f w o r k a b i l i t y . The m o l d fo rms were made f r o m p l e x i g l a s s , and g e n e r a l l y a l l o w e d a b o u t 0 . 2 5 i n - o f r u b b e r a t t h e maximum t h i c k n e s s . In t h e f i r s t a t t e m p t on t h e a e r o f o i l model t h e f o r m was s e t v e r t i c a l l y and a 5 t o 1 r a t i o between r u b b e r and c a t a l y s t (Dow C o r n i n g C a t a l y s t #1 ) u s e d . The i n s i d e s u r f a c e o f t h e fo rm and t h e s u r f a c e o f t h e model were c o a t e d w i t h mo ld r e l e a s e (Dow C o r n i n g 236 D i s p e r s i o n ) . The model was h e l d i n p l a c e i n s i d e t h e f o r m by s c r e w s t h r o u g h t h e f o r m a t t h e l o w e r c l o s e d end and by wedges a t t h e upper open e n d . T h i s a t t e m p t was u n s u c c e s s f u l b e c a u s e t h e low r u b b e r t o c a t a l y s t r a t i o s o a c c e l e r a t e d t h e w o r k i n g t i m e t h a t n o t enough t i m e was a l l o w e d f o r t h e low v i s c o s i t y r u b b e r t o p e n e t r a t e t h e n a r r o w 0 . 2 5 i n . n e c k s , a l s o i t was f o u n d t h a t t h e r u b b e r t e n d e d t o s t i c k q u i t e r e a d i l y t o t h e mo ld r e l e a s e c o a t e d s u r f a c e s o f t h e f o r m . In t h e s e c o n d a t t e m p t t h e f o r m was s e t h o r i z o n t a l l y and an e n d p l a t e a t t a c h e d w i t h a s l o t so t h a t t h e o r i g i n a l model c o u l d be s l i p p e d t h r o u g h and a t t a c h e d t o t h e o t h e r end o f t h e f o r m by t h e s c r e w s o n c e t h e f o r m was h a l f f u l l o f t h e r u b b e r . The u p p e r p l a t e o f t h e fo rm had been removed s o t h a t f i l l i n g c o u l d t a k e p l a c e o v e r a l a r g e a r e a , t h i s was r e p l a c e d when t h e model was i n p l a c e and t h e m o l d had been f i l l e d t o c a p a c i t y . T h i s a t t e m p t p r o d u c e d a v o i d b u t s i n c e t h e r e was a t h i n f i l m o f r u b b e r o v e r t h e model t h i s c o u l d be f i l l e d s e p a r a t e l y , and a r e a s o n a b l e mo ld was p r o d u c e d . In t h i s a t t e m p t no m o l d r e l e a s e was used on t h e p l e x i g l a s s f a c e s o f t h e f o r m , and i t a p p e a r e d t h a t t h i s was more s a t i s f a c t o r y t h a n when m o l d r e l e a s e had been u s e d . A l s o i n t h i s a t t e m p t a 20 t o 1 r a t i o o f r u b b e r t o c a t a l y s t had been used so t h a t t h e w o r k i n g t i m e was more t h a n d o u b l e d . The mo ld f r o m t h e w h a l e f l u k e was m a d e . i n t h e v e r t i c a l p o s i t i o n . I n t h i s c a s e t h e wooden model was a t t a c h e d t o a t o p p l a t e and t h e f o r m f i l l e d b e f o r e t h e model was i n s e r t e d . T h i s p r o v e d q u i t e s u c c e s s f u l and no v o i d s were o b s e r v e d . The o r i g i n a l mode ls were removed f r o m t h e mo lds by means o f a s l i t c u t i n t h e r u b b e r i n t h e l o n g d i r e c t i o n . The r u b b e r c o u l d t h e n be p e e l e d back and t h e model removed q u i t e r e a d i l y . When m o l d i n g f r o m wooden models i t i s i m p o r t a n t t o o b t a i n a good s e a l on t h e wood s u r f a c e , e i t h e r by p a i n t o r s h e l l a c , b e f o r e t h e m o l d r e l e a s e i s a p p l i e d . The m a t e r i a l used f o r c a s t i n g was 5V E p o x y , t h i s was c h o s e n b e c a u s e i t i n v o l v e d t h e m i x i n g o f a h a r d e n e r and b o n d i n g a g e n t so t h a t d i f f e r e n t m i x t u r e s w o u l d p r o d u c e d i f f e r e n t f l e x i b i l i t i e s o f t h e f i n a l p r o d u c t . I t was f o u n d t h a t a 1 t o 1 r a t i o p r o d u c e d t h e d e s i r e d f l e x i b i l i t y f o r t h i s p r o j e c t . F o r t h e c a s t i n g o p e r a t i o n t h e r u b b e r m o l d was e n c a s e d i n i t s o r i g i n a l p l e x i g l a s s f o r m t o g i v e i t t h e n e c e s s a r y r i g i d i t y . D u r i n g t h e p o u r i n g t h e f o r m was t i l t e d a b o u t 30 d e g r e e s f r o m t h e v e r t i c a l t o e l i m i n a t e t h e f o r m a t i o n o f b u b b l e s i n t h e m i x t u r e . The epoxy model was removed by o p e n i n g t h e s l i t used t o remove t h e o r i g i n a l m o d e l . 

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