UBC Theses and Dissertations

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UBC Theses and Dissertations

A method for drag reduction on bluff bodies Lesage, François 1983

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A METHOD FOR DRAG REDUCTION ON BLUFF BODIES by FRANCOIS LESAGE B.Sc.A., Universite Laval, 1979 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF APPLIED SCIENCE i n THE FACULTY OF GRADUATE STUDIES Department Of Mechanical Engineering We accept t h i s thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA A p r i l 1983 © Francois Lesage, 1983 I n p r e s e n t i n g t h i s t h e s i s i n p a r t i a l f u l f i l m e n t o f t h e r e q u i r e m e n t s f o r an 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 a nd 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 D e p a r t m e n t o r by h i s o r h e r r e p r e s e n t a t i v e s . I t i s u n d e r s t o o d t h a t c o p y i n g o r p u b l i c a t i o n o f t h i s t h e s i s f o r f i n a n c i a l g a i n s h a l l n o t be a l l o w e d w i t h o u t my w r i t t e n p e r m i s s i o n . 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 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 2075 Wesbrook P l a c e V a n c o u v e r , Canada V6T 1W5 D a t e : A p r i l 25, 1983 i i A b s t r a c t T h i s t h e s i s p r e s e n t s an 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 on t h e d r a g o f t y p i c a l t w o - d i m e n s i o n a l b l u f f b o d i e s w i t h a s m a l l c i r c u l a r c y l i n d e r ( h e r e c a l l e d a " r o d " ) p l a c e d u p s t r e a m on t h e s t a g n a t i o n l i n e . I n most c a s e s , t h e i n t e r a c t i o n was b e n e f i c i a l a n d t h e d r a g o f t h e o v e r a l l s y s t e m was r e d u c e d . F l u c t u a t i n g s i d e f o r c e due t o v o r t e x s h e d d i n g f r o m t h e m a i n body was a l s o i n v e s t i g a t e d f o r v a r i o u s r o d p o s i t i o n s a n d d i a m e t e r s . Two t y p i c a l b l u f f b o d i e s were i n v e s t i g a t e d : a c i r c u l a r c y l i n d e r a n d a f l a t p l a t e . The f r o n t r o d d i a m e t e r (d) v a r i e d f r o m 0.17D t o 0.5D where D i s t h e f r o n t a l w i d t h o f t h e m a i n c i r c u l a r c y l i n d e r o r f l a t p l a t e . R e y n o l d s numbers d e f i n e d u s i n g d i m e n s i o n D were i n t h e r a n g e o f 1x10" t o 7 x 1 0 " , a n d t h e l o n g i t u d i n a l s p a c i n g L f r o m t h e r o d c e n t r e t o t h e m a i n body was 0.4 < L/D < 7.0. F o r b o t h t h e f l a t p l a t e a n d t h e c i r c u l a r c y l i n d e r , t h e m e a s u r e d o v e r a l l d r a g c o e f f i c i e n t a t v a r i o u s L/D showed a d i s c o n t i n u o u s "jump" a t some c r i t i c a l s p a c i n g . T h i s c h a n g e c o r r e s p o n d s t o t h e e l i m i n a t i o n o f t h e u s u a l s i n g l e s t a g n a t i o n p o i n t on t h e b l u f f body c e n t r e l i n e and t h e a p p e a r a n c e o f two s t a g n a t i o n p o i n t s s y m m e t r i c a l l y p l a c e d , c l o s e t o t h e l a t e r a l e d g e s o f t h e b o d y . F o r t h e f l a t p l a t e , t h e optimum c o n f i g u r a t i o n was f o u n d t o be d/D = 0.33 a t a s p a c i n g L/D o f 1.81, a n d t h e o v e r a l l d r a g r e d u c t i o n ( b a s e d on t h e d r a g o f t h e p l a t e a l o n e ) was 36%. The d r a g r e d u c t i o n was due t o a c h a n g e i n f r o n t p r e s s u r e o n l y , t h e b a s e p r e s s u r e r e m a i n i n g e s s e n t i a l l y c o n s t a n t a n d i n d e p e n d e n t o f L/D. No R e y n o l d s number d e p e n d e n c e was f o u n d , a s e x p e c t e d . F o r t h e c i r c u l a r c y l i n d e r , t h e b e s t o v e r a l l d r a g r e d u c t i o n was f o u n d t o be 58% f o r t h e c o n f i g u r a t i o n d/D = 0.33 a n d L/D = 1.73. P a r t o f t h e d r a g r e d u c t i o n on t h e c y l i n d e r was due t o t h e r o d wake b e i n g t u r b u l e n t w h i c h made t h e f l o w o v e r t h e m a i n c y l i n d e r c r i t i c a l , w i t h a l o w e r v a l u e o f b a s e p r e s s u r e c o e f f i c i e n t . The s e c o n d e f f e c t was a s u d d e n c h a n g e i n t h e f r o n t p r e s s u r e d i s t r i b u t i o n s i m i l a r t o t h a t o b s e r v e d on t h e f l a t p l a t e . U n l i k e t h e f l a t p l a t e , h o w e v e r , t h e f l o w o v e r t h e c i r c u l a r c y l i n d e r was s e n s i t i v e t o R e y n o l d s number, w i t h d e c r e a s i n g CD f o r i n c r e a s i n g Re. The f l u c t u a t i n g s i d e f o r c e ( C I ' ) on t h e c y l i n d e r due t o v o r t e x s h e d d i n g was a l s o i n v e s t i g a t e d . The v a r i a t i o n o f C I ' w i t h s p a c i n g f o l l o w e d d i f f e r e n t t r e n d s d e p e n d i n g on t h e r o d s i z e a n d R e y n o l d s number b u t C I ' n e v e r i n c r e a s e d s i g n i f i c a n t l y w i t h t h e u s e o f t h e f r o n t r o d a n d was r e d u c e d i n most c a s e s . The r o d d/D = 0.17 seemed t o be more s u i t a b l e f o r minimum f l u c t u a t i n g s i d e f o r c e . The p o s i t i o n f o r C l ' minimum was f o u n d t o d e p e n d upon t h e p a r t i c u l a r c o n f i g u r a t i o n . i v T a b l e o f C o n t e n t s A b s t r a c t i i L i s t o f T a b l e s v i L i s t o f F i g u r e s v i i N o m e n c l a t u r e x A c k n o w l e d g e m e n t s x i i C h a p t e r I INTRODUCTION 1 1.1 M e t h o d s Of R e d u c i n g F l u i d F o r c e s 1 1.2 B l u f f Body I n t e r a c t i o n 3 1 . 3 O b j e c t i v e 4 C h a p t e r I I DESCRIPTION OF APPARATUS AND EXPERIMENTS 6 2.1 Smoke T u n n e l 6 2.2 W i n d T u n n e l 6 2.3 W i n d T u n n e l B a l a n c e 7 2.4 M o d e l s 7 2.5 P r e s s u r e D i s t r i b u t i o n M e a s u r e m e n t s 9 2.6 Mean D r a g M e a s u r e m e n t s 10 2.7 F l u c t u a t i n g S i d e F o r c e M e a s u r e m e n t s 11 2.8 D a t a A c q u i s i t i o n 13 C h a p t e r I I I RESULTS AND DISCUSSION I : FLAT PLATE ..14 3.1 P r e s s u r e D i s t r i b u t i o n On F l a t P l a t e 14 3.2 P r e s s u r e D i s t r i b u t i o n On F l a t P l a t e W i t h F r o n t Rod 1 5 3.3 F l o w Regime A ...16 3 . 4 F l o w Regime B 18 3.5 D r a g ; 20 3.6 E f f e c t Of Yaw 21 3.7 E f f e c t Of R e y n o l d s Number 22 3.8 P o t e n t i a l F l o w M o d e l 22 C h a p t e r I V RESULTS AND DISCUSSION I I : CIRCULAR CYLINDER 27 4.1 P r e s s u r e D i s t r i b u t i o n On C i r c u l a r C y l i n d e r 27 4.1.1 B a c k g r o u n d 27 4.1.2 R e s u l t s 28 4.2 P r e s s u r e D i s t r i b u t i o n On C i r c u l a r C y l i n d e r W i t h F r o n t Rod 30 4.3 F l o w Regime A 30 4.4 F l o w Regime B 32 4.5 D r a g 32 4.6 E f f e c t Of R e y n o l d s Number 34 4.7 Optimum C o n f i g u r a t i o n 34 4.8 E f f e c t Of Yaw 35 4.9 F l u c t u a t i n g S i d e F o r c e 36 4.10 S i d e F o r c e On M o d e l W i t h F r o n t Rod 36 4.10.1 F r o n t Rod d/D = 0.17 37 V 4.10.2 F r o n t Rod d/D = 0.33 38 4.10.3 F r o n t Rod d/D = 0.50 39 4.10.4 G e n e r a l R emarks 40 C h a p t e r V CONCLUSIONS 41 5.1 C o n c l u s i o n s 41 5.2 A r e a s Of F u r t h e r Work 43 BIBLIOGRAPHY 45 APPENDIX A - PRESSURE TAP LOCATION FOR SIDE FORCE MEASUREMENT 89 v i L i s t o f T a b l e s I . C a l c u l a t e d wake v e l o c i t y d e f i c i t ( U 0 ) a n d c h a r a c t e r i s t i c wake w i d t h ( l 0 ) 47 v i i L i s t o f F i g u r e s 1. C o o r d i n a t e s y s t e m a nd s y m b o l s f o r : ( a ) f l a t p l a t e ; (b) c i r c u l a r c y l i n d e r 48 2. Smoke t u n n e l 48 3. O u t l i n e o f t h e U.B.C. a e r o n a u t i c a l w i n d t u n n e l 49 4. S k e t c h o f a t y p i c a l m o d e l i n s i d e t h e w i n d t u n n e l 50 5. End p l a t e s 51 6. C r o s s s e c t i o n o f m o d e l s s h o w i n g p r e s s u r e t a p l o c a t i o n 52 7. B a l a n c e c a l i b r a t i o n c u r v e 53 8. S k e t c h o f m a n i f o l d f o r p n e u m a t i c a v e r a g i n g 53 9. A p p a r a t u s f o r c a l i b r a t i o n o f t h e s i d e f o r c e m e a s u r i n g s y s t e m a g a i n s t f r e q u e n c y d i s t o r t i o n 54 10. T r a n s f e r f u n c t i o n f o r t h e s i d e f o r c e m e a s u r i n g s y s t e m . The s o l i d l i n e was u s e d t o c o r r e c t d a t a 55 11. P r e s s u r e d i s t r i b u t i o n on f l a t p l a t e (Re=4.0x10") 56 12. P r e s s u r e d i s t r i b u t i o n on f l a t p l a t e w i t h f r o n t r o d d/D = 0.33 ( R e = 4 . 0 x 1 0 " ) ; F l o w r e g i m e A 57 13. P r e s s u r e d i s t r i b u t i o n on f l a t p l a t e w i t h f r o n t r o d d/D = 0.33 ( R e = 4 . 0 x l 0 4 ) ; F l o w r e g i m e B 58 14. P r e s s u r e d i s t r i b u t i o n on f l a t p l a t e w i t h f r o n t r o d d/D = 0.17 ( R e = 4 . 0 x 1 0 " ) ; F l o w r e g i m e A 59 15. P r e s s u r e d i s t r i b u t i o n on f l a t p l a t e w i t h f r o n t r o d d/D = 0.17 ( R e = 4 . 0 x 1 0 " ) ; F l o w r e g i m e B 60 16. P r e s s u r e d i s t r i b u t i o n on t h e f r o n t r o d d/D = 0.33 a t two d i f f e r e n t s p a c i n g s (Re = 5.Ox 1 0 4 ) ; L/D = 3.42 ( r e g i m e A) a n d L/D = 1.42 ( r e g i m e B) 61 17. V e l o c i t y d e f i c i t ( U 0 ) v e r s u s s p a c i n g ( L / d ) f o r f l o w r e g i m e A 62 18. B a s e p r e s s u r e f o r r e g i m e s A a n d B 63 v i i i 19. F l o w v i s u a l i s a t i o n a t Re = 5 X 1 0 3 f o r f r o n t r o d d/D = 0 . 2 1 ; (a) L/D = 1.36 ( r e g i m e A); (b) L/D = 0.79 ( r e g i m e B) 64 20. D r a g c o e f f i c i e n t o f f l a t p l a t e w i t h r o d d/D = 0.33 ( R e = 4 . 0 x l 0 " ) 65 2 1 . D r a g c o e f f i c i e n t o f f l a t p l a t e w i t h r o d d/D = 0.17 (Re=4.0xl0«) 66 22. E f f e c t o f yaw on t h e o v e r a l l d r a g ( p l a t e a n d r o d ) f o r r o d d/D = 0.33 a t L/D = 1.42 (Re = 5.Ox 1 0 4 ) 67 23. S e p a r a t e d f l o w p a s t a n o r m a l f l a t p l a t e f r o m wake s o u r c e m o d e l 67 24. S t r e a m l i n e s o v e r p o t e n t i a l model u s i n g c o n d i t i o n f o r s t a t i o n a r y p a i r o f v o r t i c e s ; Cpb = - 1 . 2 4 68 25. T y p i c a l d r a g c o e f f i c i e n t v e r s u s R e y n o l d s number f r o m A c h e n b a c h 68 26. P r e s s u r e d i s t r i b u t i o n a r o u n d a c i r c u l a r c y l i n d e r a t v a r i o u s R e y n o l d s n u m b ers; s o l i d l i n e i s f r o m ESDU f o r s u b c r i t i c a l r a n g e 69 27. P r e s s u r e d i s t r i b u t i o n a r o u n d a c i r c u l a r c y l i n d e r a t Re = 1.0x10" w i t h f r o n t r o d d/D = 0.17. (a) r e g i m e A; (b) r e g i m e B 70 28. P r e s s u r e d i s t r i b u t i o n a r o u n d a c i r c u l a r c y l i n d e r a t Re = 3.3x10" w i t h f r o n t r o d d/D = 0.17. (a ) r e g i m e A; ( b ) r e g i m e B 71 29. P r e s s u r e d i s t r i b u t i o n a r o u n d a c i r c u l a r c y l i n d e r a t Re = 6.5x10" w i t h f r o n t r o d d/D = 0.17. (a) r e g i m e A; (b) r e g i m e B 72 30. P r e s s u r e d i s t r i b u t i o n a r o u n d a c i r c u l a r c y l i n d e r a t Re = 1.0x10" w i t h f r o n t r o d d/D = 0.33. (a) r e g i m e A; (b) r e g i m e B 73 31. P r e s s u r e d i s t r i b u t i o n a r o u n d a c i r c u l a r c y l i n d e r a t Re = 3.3x10" w i t h f r o n t r o d d/D = 0.33. ( a ) r e g i m e A; ( b ) r e g i m e B 74 32. P r e s s u r e d i s t r i b u t i o n a r o u n d a c i r c u l a r c y l i n d e r a t Re = 6.5x10" w i t h f r o n t r o d d/D = 0.33. (a) r e g i m e A; (b) r e g i m e B 75 33. P r e s s u r e d i s t r i b u t i o n a r o u n d a c i r c u l a r c y l i n d e r a t Re = 1.0x10" w i t h f r o n t r o d d/D = 0.50. (a) r e g i m e A; (b) r e g i m e B 76 i x 34.. P r e s s u r e d i s t r i b u t i o n a r o u n d a c i r c u l a r c y l i n d e r a t Re = 3.3x10" w i t h f r o n t r o d d/D = 0.50. (a ) r e g i m e A; ( b ) r e g i m e B 77 35. P r e s s u r e d i s t r i b u t i o n a r o u n d a c i r c u l a r c y l i n d e r a t Re = 6.5x10" w i t h f r o n t r o d d/D = 0.50. (a) r e g i m e A; (b) r e g i m e B 78 36. O v e r a l l d r a g c o e f f i c i e n t f o r c y l i n d e r w i t h f r o n t r o d d/D = 0.17 79 37. O v e r a l l d r a g c o e f f i c i e n t f o r c y l i n d e r w i t h f r o n t r o d d/D = 0.33 80 38. O v e r a l l d r a g c o e f f i c i e n t f o r c y l i n d e r w i t h f r o n t r o d d/D = 0.50 81 39. Rod s i z e f o r minimum d r a g ( c i r c u l a r c y l i n d e r ) 82 40. C r i t i c a l s p a c i n g r a n g e s f o r d i f f e r e n t r o d s i z e s a n d R e y n o l d s numbers ( c i r c u l a r c y l i n d e r ) 83 41 . F l u c t u a t i n g s i d e f o r c e on a c i r c u l a r c y l i n d e r ; ( a ) S t r o u h a l number v e r s u s R e y n o l d s number; (b) i n t e n s i t y C l ' v e r s u s R e y n o l d s number 84 42. F l u c t u a t i n g s i d e f o r c e on a c i r c u l a r c y l i n d e r w i t h r o d d/D = 0.17; C l ' f i s f r o m F i g . 4 1 ( b ) 85 43. F l u c t u a t i n g s i d e f o r c e on a c i r c u l a r c y l i n d e r w i t h r o d d/D = 0.33 86 44. F l u c t u a t i n g s i d e f o r c e on a c i r c u l a r c y l i n d e r w i t h r o d d/D = 0.50 .86 45. S t r o u h a l number on a c i r c u l a r c y l i n d e r w i t h f r o n t r o d d/D = 0.17 87 46. S t r o u h a l number on a c i r c u l a r c y l i n d e r w i t h f r o n t r o d d/D = 0.33 (Re = 3 . 3 x 1 0 " ) ; p e r c e n t a g e s shown r e p r e s e n t p e r c e n t a g e o f ( C l 1 ) 2 due t o t h a t f r e q u e n c y .87 47. S t r o u h a l number on a c i r c u l a r c y l i n d e r w i t h f r o n t r o d d/D = 0.50 (Re = 3 . 3 x 1 0 " ) ; p e r c e n t a g e s shown r e p r e s e n t p e r c e n t a g e o f ( C l 1 ) 2 due t o t h a t f r e q u e n c y .88 X N o m e n c l a t u r e CD d r a g c o e f f i c i e n t = ( d r a g ) / ( 0 . 5 p U , 2 H D ) C I ' f l u c t u a t i n g s i d e f o r c e c o e f f i c i e n t = (rms o f s i d e f o r c e f l u c t u a t i o n ) / ( 0 . 5 p U , 2 H D ) Cp p r e s s u r e c o e f f i c i e n t = ( p - p , ) / ( 0 . 5 p U , 2 ) Cpb b a s e p r e s s u r e c o e f f i c i e n t D f r o n t a l w i d t h o f t h e m a i n body d r o d d i a m e t e r F ( z ) c o m p l e x p o t e n t i a l i n t h e z - p l a n e f f r e q u e n c y o f v o r t e x s h e d d i n g (Hz) H t e s t s e c t i o n h e i g h t h p l a t e w i d t h ( p o t e n t i a l f l o w m o d e l ) L l o n g i t u d i n a l s p a c i n g f r o m t h e r o d c e n t e r t o t h e m a i n body 1 0 c h a r a c t e r i s t i c wake h a l f - w i d t h p p r e s s u r e on model s u r f a c e p, s t a t i c p r e s s u r e Q s t r e n g t h o f s o u r c e s R c i r c l e r a d i u s i n $ - p l a n e Re R e y n o l d s number S S t r o u h a l number U f r e e s t r e a m v e l o c i t y i n t h e z - p l a n e U 0 c h a r a c t e r i s t i c wake v e l o c i t y d e f i c i t U, f r e e s t r e a m v e l o c i t y V f r e e s t r e a m v e l o c i t y i n t h e $ - p l a n e w ( z ) c o m p l e x v e l o c i t y i n t h e z - p l a n e x d i s t a n c e d e f i n i n g p o s i t i o n on t h e s u r f a c e o f t h e f l a t p l a t e x i z c o m p l e x v a r i a b l e d e f i n i n g t h e f l a t p l a t e p l a n e 6 a n g u l a r p o s i t i o n o f s o u r c e T c i r c u l a t i o n s t r e n g t h p d e n s i t y o f a i r 4> a n g l e d e f i n i n g p o s i t i o n on t h e s u r f a c e o f t h e c i r c u l a r c y l i n d e r ii s t r e a m f u n c t i o n $ c o m p l e x v a r i a b l e d e f i n i n g t h e c i r c l e p l a n e So p o s i t i o n o f u p p e r v o r t e x i n t h e $ - p l a n e $"0 p o s i t i o n o f l o w e r v o r t e x i n t h e $ - p l a n e A c k n o w l e d g e m e n t The a u t h o r w i s h e s t o e x p r e s s h i s s i n c e r e g r a t i t u d e t o P r o f e s s o r I . S . G a r t s h o r e f o r h i s e n c o u r a g e m e n t and v a l u a b l e d i r e c t i o n t h r o u g h o u t t h i s s t u d y . T h a n k s a r e a l s o due t o 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 f o r t h e u s e o f t h e i r f a c i l i t i e s , a n d t o t h e t e c h n i c a l s t a f f members f o r t h e i r v a l u a b l e h e l p d u r i n g t h e c o n s t r u c t i o n o f t h e w i n d t u n n e l m o d e l s a n d d u r i n g t h e e x p e r i m e n t s . S p e c i a l t h a n k s t o my f e l l o w g r a d u a t e s t u d e n t s f o r t h e i r h e l p f u l a d v i c e s a nd t o my w i f e , C h a n t a l , f o r p l a y i n g t h e r o l e o f a s t u d e n t ' s w i f e s o w e l l . S u p p o r t f o r t h i s r e s e a r c h f r o m 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 i s g r a t e f u l l y a c k n o w l e d g e d . C o m p u t i n g f a c i l i t i e s were p r o v i d e d by t h e U n i v e r s i t y C o m p u t i n g C e n t e r . 1 I . INTRODUCTION The p u r p o s e o f t h i s work i s t o i n v e s t i g a t e a new method o f r e d u c i n g mean d r a g a n d u n s t e a d y s i d e f o r c e on a c y l i n d e r i n a u n i f o r m f l o w . 1.1 M e t h o d s Of R e d u c i n g F l u i d F o r c e s D i f f e r e n t m e t h o d s have been d e v e l o p e d t o r e d u c e b o t h d r a g a n d v o r t e x s h e d d i n g s i d e f o r c e on a b l u f f b o d y . R e d u c i n g t h e s e f l u i d d y n a m i c f o r c e s i s o f g r e a t i m p o r t a n c e s i n c e i t means p o s s i b l y r e d u c i n g t h e r e q u i r e d s t r e n g t h o f a s t r u c t u r e , i t s w e i g h t a n d t h e r e f o r e i t s c o s t . F o r a b l u f f b o d y , t h e d r a g i s 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 t h e f l o w l a r g e l y due t o b o u n d a r y l a y e r s e p a r a t i o n . The s e p a r a t e d r e g i o n , on t h e b a c k o f t h e b o d y , i s a r e g i o n o f low p r e s s u r e w h e r e a s t h e f r o n t o f t h e body i s s u b j e c t e d t o a h i g h e r p r e s s u r e . T h i s d i f f e r e n c e c r e a t e s a s t e a d y f o r c e a n d t h e s t r u c t u r e h a s t o be d e s i g n e d t o w i t h s t a n d t h i s l o a d i n g . A l t h o u g h t h e r e a r e u n s t e a d y c o m p o n e n t s t o t h e d r a g , t h e s e a r e u s u a l l y s m a l l c o m p a r e d t o t h e t i m e a v e r a g e d v a l u e a n d a r e n o t c o n s i d e r e d h e r e . The f l u c t u a t i n g s i d e f o r c e i s due t o a l t e r n a t e s h e d d i n g o f v o r t i c e s f r o m e a c h s i d e o f t h e c y l i n d e r . E a c h t i m e a v o r t e x i s s h e d f r o m t h e c y l i n d e r , t h e l o c a l p r e s s u r e d i s t r i b u t i o n i s a l t e r e d , c r e a t i n g a t i m e - v a r y i n g f o r c e on t h e c y l i n d e r , a t t h e f r e q u e n c y o f v o r t e x s h e d d i n g . I f t h e c y l i n d e r i s r i g i d , t h e f l u i d d y n a m i c f o r c e s a r e m a i n l y a f u n c t i o n o f t h e sh a p e o f t h e bod y . B u t i f t h e 2 c y l i n d e r d e f l e c t s , an i n t e r a c t i o n o f t h e d i s p l a c e m e n t w i t h t h e f l u i d f o r c e s may r e s u l t w h i c h i n t u r n c a n c a u s e s t r u c t u r a l v i b r a t i o n s . The i m p o r t a n t p a r a m e t e r s i n t h i s c a s e a r e t h e shape o f t h e body , t h e n a t u r a l f r e q u e n c y a n d t h e d a m p i n g o f t h e s t r u c t u r e . I n many c a s e s , - t h e f l u c t u a t i n g f o r c e due t o v o r t e x s h e d d i n g i s more i m p o r t a n t i n t h e d e s i g n o f a s t r u c t u r e t h a n t h e d r a g . I m p o r t a n t p r o b l e m s a r e a s s o c i a t e d w i t h t h i s f l u c t u a t i n g f o r c e s u c h a s m a t e r i a l f a t i g u e a n d s y n c h r o n i s a t i o n o f v o r t e x s h e d d i n g f r e q u e n c y w i t h t h e n a t u r a l f r e q u e n c y o f t h e s t r u c t u r e . T h e r e a r e b a s i c a l l y two methods o f r e d u c i n g t h e f l u i d d y n a m i c f o r c e s a n d v i b r a t i o n s on a c y l i n d e r : (1) - A l t e r i n g t h e sh a p e o f t h e body t o r e d u c e t h e a m p l i t u d e a n d c h a n g e t h e f r e q u e n c y o f t h e f o r c e a p p l i e d t o t h e b o d y . (2) - A l t e r i n g t h e n a t u r a l f r e q u e n c y o r d a m p i n g o f t h e s t r u c t u r e t o r e d u c e t h e a m p l i t u d e a n d c h a n g e t h e f r e q u e n c y o f o s c i l l a t i o n . A good r e v i e w o f a v a i l a b l e m e t h o d s was made by E v e r y e t a l . ( 1 ) . The f i r s t c l a s s o f methods i n c l u d e s h e l i c a l s t r a k e s , f a i r i n g s , s p l i t t e r p l a t e s , f l a g s , p e r f o r a t e d s h r o u d s a n d o t h e r s . T h e s e methods a r e b a s e d on an a l t e r a t i o n o f t h e shap e o f t h e s t r u c t u r e i n o r d e r t o c h a n g e t h e b o u n d a r y l a y e r s e p a r a t i o n p o i n t s , p r e v e n t o r r e d u c e t h e f o r m a t i o n o f v o r t i c e s a n d p r e v e n t t h e c o r r e l a t i o n o f v o r t e x s h e d d i n g 3 a l o n g t h e l e n g t h o f t h e s t r u c t u r e . Some o f t h e s e d e v i c e s r e d u c e v o r t e x s h e d d i n g b u t i n c r e a s e d r a g ( h e l i c a l s t r a k e s ) ; some r e d u c e b o t h b u t a r e e x p e n s i v e ( f a i r i n g s ) o r n o t v e r y e f f i c i e n t ( s p l i t t e r p l a t e s ) ; some a r e u n i d i r e c t i o n a l ( f a i r i n g s ) , o t h e r s a r e o m n i d i r e c t i o n a l ( h e l i c a l s t r a k e s ) . The s e c o n d c l a s s o f methods i n c l u d e s t h e u s e o f d a m p e r s , s t i f f e n e r s and d i f f e r e n t m a t e r i a l s i n o r d e r t o c h a n g e t h e n a t u r a l f r e q u e n c y a n d d a m p i n g o f t h e s t r u c t u r e . 1.2 B l u f f Body I n t e r a c t i o n I t i s known t h a t two b l u f f b o d i e s p l a c e d i n l i n e i n a u n i f o r m s t r e a m c a n l e a d , i n some c a s e s , t o a t o t a l d r a g s i g n i f i c a n t l y l o w e r t h a n t h a t o f e i t h e r body a l o n e . M o r e l & Bohn (2) i n v e s t i g a t e d t h e f l o w o v e r two d i s k s o f u n e q u a l d i a m e t e r p l a c e d i n tandem a nd R o s h k o & K o e n i g (3) i n v e s t i g a t e d t h e f l o w o v e r a f l a t f a c e d c i r c u l a r c y l i n d e r p r e c e d e d by a c o n c e n t r i c c i r c u l a r d i s k . They b o t h f o u n d a r e m a r k a b l e d r a g r e d u c t i o n f r o m t h e v a l u e o f t h e r e f e r e n c e b o d y . M o r e l & Bohn showed a d r a g r e d u c t i o n o f 8 1 % i n t h e optimum c a s e . T h i s i d e a i s u s e d i n t h i s work t o d e s i g n a d e v i c e t h a t c o u l d be u s e d on t w o - d i m e n s i o n a l s t r u c t u r e s t o r e d u c e d r a g a n d v o r t e x s h e d d i n g s i d e f o r c e . T h r o u g h w i n d t u n n e l e x p e r i m e n t s , t h e f l o w o v e r t y p i c a l b l u f f b o d i e s w i t h a s m a l l c i r c u l a r c y l i n d e r p l a c e d u p s t e a m on t h e s t a g n a t i o n l i n e i s i n v e s t i g a t e d . Two t y p i c a l b l u f f b o d i e s a r e s t u d i e d : a f l a t p l a t e a n d a c i r c u l a r c y l i n d e r . 4 I g a r a s h i (4) h a s a l r e a d y i n v e s t i g a t e d t h e c h a r a c t e r i s t i c s o f t h e f l o w o v e r two c i r c u l a r c y l i n d e r s w i t h r a t i o d l / d 2 = 0.68, t h e s m a l l e r c y l i n d e r b e i n g d o w n s t r e a m . The e x i s t e n c e o f d i f f e r e n t f l o w p a t t e r n s v a r y i n g w i t h s p a c i n g a n d R e y n o l d s number was shown. Z d r a v k o v i c h & P r i d d e n ( 5) i n v e s t i g a t e d t h e i n t e r a c t i o n b e t w e e n two c i r c u l a r c y l i n d e r s o f t h e same s i z e p l a c e d i n tandem. H i w a d a e t a l . ( 6 ) l o o k e d a t t h e f l u i d f l o w a n d h e a t t r a n s f e r a r o u n d two c i r c u l a r c y l i n d e r s o f d i f f e r e n t d i a m e t e r s i n tandem. The s m a l l e r c y l i n d e r was p l a c e d u p s t r e a m a n d u s e d a s a t u r b u l e n c e g e n e r a t o r . 1.3 O b j e c t i v e I n a l l p r e v i o u s w o r k s on t h e i n t e r a c t i o n o f two-d i m e n s i o n a l b l u f f b o d i e s , d i f f e r e n t f l o w p a t t e r n s were i d e n t i f i e d a t d i f f e r e n t s p a c i n g r a t i o s . H o wever, no work h a s y e t been done t o i d e n t i f y t h e optimum s p a c i n g f o r l e a s t o v e r a l l d r a g o r f o r l e a s t v o r t e x s h e d d i n g s i d e f o r c e . T h i s i s t h e o b j e c t i v e o f t h i s w o r k . I n t h e f o l l o w i n g e x p e r i m e n t s , t h e f r o n t c i r c u l a r c y l i n d e r , f r o m now on c a l l e d t h e f r o n t r o d , i s r e l a t i v e l y s m a l l c o m p a r e d t o t h e m a i n body. I t s d i a m e t e r i s v a r i e d f r o m 0.17 D t o 0.50 D, where D i s t h e f r o n t a l w i d t h o f t h e m a i n body. S i n c e t h e f r o n t r o d i s u s e d a s a method f o r d r a g o r v o r t e x s h e d d i n g r e d u c t i o n on a b l u f f b o d y , i t s s i z e must be k e p t s m a l l t o make i t c o s t c o m p e t i t i v e w i t h o t h e r m e t h o d s o f f o r c e r e d u c t i o n . 5 E x p e r i m e n t s a r e c a r r i e d o u t , f i r s t l y u s i n g a f l a t p l a t e a s t h e m a i n body a n d s e c o n d l y u s i n g a c i r c u l a r c y l i n d e r . The r e s u l t s f o r t h e f l a t p l a t e s h o u l d be i n d e p e n d e n t o f R e y n o l d s number and t h e r e i s no s i g n i f i c a n t s i d e f o r c e p r e s e n t . The p a r a m e t e r s v a r i e d a r e t h e s p a c i n g L, t h e r o d d i a m e t e r d a n d t h e R e y n o l d s number Re. The f l a t p l a t e c o o r d i n a t e s y s t e m a n d s y m b o l s a r e d e f i n e d i n F i g . 1 ( a ) . The e x p e r i m e n t s u s i n g a c i r c u l a r c y l i n d e r a s t h e m a i n body a r e c a r r i e d o u t v a r y i n g t h e same p a r a m e t e r s , b u t t h i s t i m e t h e f l u c t u a t i n g s i d e f o r c e i s m e a s u r e d a n d t h e R e y n o l d s number i s e x p e c t e d t o be an i m p o r t a n t p a r a m e t e r . The c i r c u l a r c y l i n d e r c o o r d i n a t e s y s t e m a n d s y m b o l s a r e d e f i n e d i n F i g . 1 ( b ) . 6 I I . DESCRIPTION OF APPARATUS AND EXPERIMENTS 2.1 Smoke T u n n e l A smoke t u n n e l ( F i g . 2 ) was u s e d f o r f l o w v i s u a l i s a t i o n . I t i s an E l e k t r o n open c i r c u i t w i n d t u n n e l , w i t h a two-d i m e n s i o n a l t e s t s e c t i o n . I t s t e s t s e c t i o n i s 558 mm l o n g w i t h a c r o s s s e c t i o n o f 29 mm by 330 mm. The t w o -d i m e n s i o n a l model was p l a c e d 150 mm d o w n s t r e a m o f t h e end o f t h e c o n t r a c t i o n . Smoke was i n j e c t e d a h e a d o f t h e model by a s e r i e s o f 29 h o r i z o n t a l n o z z l e s . The smoke, f o l l o w i n g t h e a i r f l o w , made t h e f l o w p a t t e r n v i s i b l e . The smoke was g e n e r a t e d by b u r n i n g p i p e t o b a c c o a n d i t s f l o w r a t e c o u l d be a d j u s t e d . The maximum R e y n o l d s number b a s e d on t h e m o d e l c r o s s s e c t i o n was 5 X 1 0 3 . P i c t u r e s o f d i f f e r e n t a r r a n g e m e n t s were t a k e n . 2.2 W ind T u n n e l The q u a n t i t a t i v e e x p e r i m e n t s were c o n d u c t e d i n t h e U.B.C. l o w s p e e d , low t u r b u l e n c e , c l o s e d r e t u r n t y p e w i n d t u n n e l i n w h i c h t h e v e l o c i t y c a n be v a r i e d f r o m 0 t o 46 m/s w i t h an i n h e r e n t u n d i s t u r b e d t u r b u l e n c e l e v e l o f l e s s t h a n 0.1%. The s p a t i a l v a r i a t i o n o f mean v e l o c i t y i n t h e t e s t s e c t i o n i s l e s s t h a n 0.25%. T h r e e s c r e e n s s m o o t h t h e f l o w a t t h e e n t r a n c e o f t h e s e t t l i n g chamber a n d a 7:1 c o n t r a c t i o n a c c e l e r a t e s t h e f l o w , i m p r o v i n g i t s u n i f o r m i t y a s i t r e a c h e s t h e t e s t s e c t i o n . The t e s t s e c t i o n i s 2.74 m l o n g w i t h a c r o s s - s e c t i o n o f 7 914 mm by 686 mm. F o u r f i l l e t s d e c r e a s i n g f r o m 152 mm a t t h e u p s t r e a m t o 121 mm a t t h e d o w n s t r e a m end o f f s e t t h e e f f e c t o f b o u n d a r y l a y e r g r o w t h i n t h e t e s t s e c t i o n . The t u n n e l i s p o w e r e d by a 15 HP d i r e c t c u r r e n t m o t o r d r i v i n g a c o m m e r c i a l a x i a l f l o w f a n w i t h T h y r i s t o r s p e e d c o n t r o l . The p r e s s u r e d i f f e r e n t i a l a c r o s s t h e c o n t r a c t i o n i s m e a s u r e d on a B e t z m i c r o m a n o m e t e r w h i c h c a n be r e a d t o 0.02 mm o f w a t e r . The t e s t s e c t i o n v e l o c i t y i s c a l i b r a t e d a g a i n s t t h e a b o v e p r e s s u r e d i f f e r e n c e . F i g . 3 shows an o u t l i n e o f t h e t u n n e l . 2.3 Wind T u n n e l B a l a n c e F o r c e m e a s u r e m e n t s on m o d e l s were t a k e n on an A e r o l a b s t r a i n gauge b a l a n c e . F o r t h e p u r p o s e o f t h i s e x p e r i m e n t , o n l y t h e mean d r a g was r e a d f r o m t h e b a l a n c e . The f r e q u e n c y r e s p o n s e o f t h e b a l a n c e i s much t o o l o w t o a l l o w i t s u s e i n t h e measurement o f f l u c t u a t i n g s i d e f o r c e . 2.4 M o d e l s E a c h m o d e l c o n s i s t e d o f two p a r t s : t h e d o w n s t r e a m m a i n b l u f f body a nd t h e u p s t r e a m r o d . The two m a i n b l u f f body m o d e l s u s e d (a f l a t p l a t e a n d a c i r c u l a r c y l i n d e r ) h a d same f r o n t a l w i d t h o f 38.1 mm. T h r e e d i f f e r e n t d i a m e t e r s o f f r o n t r o d were u s e d : 6.35, 12.7 a n d 19.1 mm, g i v i n g n o n -d i m e n s i o n a l i s e d r o d s i z e s d/D o f 0.17, 0.33 a n d 0.50. The m a i n body was f i x e d t o a s t a n d w h i l e t h e f r o n t r o d was f i x e d t o a s u p p o r t w h i c h was m o v a b l e on t h e s t a n d s o 8 t h a t t h e a x i s o f t h e two c y l i n d e r s w o u l d s t a y i n t h e d i r e c t i o n o f t h e f r e e s t r e a m . The m o d e l s v e r t i c a l l y s p a n n e d a l l t h e way a c r o s s t h e w i n d t u n n e l and e x t e n d e d o u t s i d e . The two p a r t s were c o n n e c t e d t o g e t h e r on t h e b a l a n c e so t h a t t h e d r a g m e a s u r e d was t h e t o t a l f o r c e on t h e two p a r t s ( F i g . 4)... End p l a t e s were mounted on e a c h m o d e l a t a d i s t a n c e o f 38 mm f r o m t h e w i n d t u n n e l r o o f a n d f l o o r , s o t h a t t h e y were j u s t o u t s i d e t h e t u n n e l w a l l b o u n d a r y l a y e r s . S t a n s b y (7) showed t h a t t h e end e f f e c t s on a two-d i m e n s i o n a l c i r c u l a r c y l i n d e r w i t h o u t e n d p l a t e s a l t e r e d t h e t r u e b a s e p r e s s u r e o v e r t h e e n t i r e l e n g t h o f t h e m o d e l , e v e n when t h e a s p e c t r a t i o was a s h i g h a s 20, I t was shown t h a t t h i s c h a n g e i n b a s e p r e s s u r e c o u l d be r e c t i f i e d by t h e u s e o f end p l a t e s . L e e (8), d o i n g r e l a t e d work on s q u a r e s e c t i o n s , c o n c l u d e d t h a t i n t h e c a s e o f t w o - d i m e n s i o n a l m o d e l s w i t h s h a r p c o r n e r s , t h e u t i l i s a t i o n o f end p l a t e s was o f l i t t l e u s e . I n t h i s w o r k , b o t h c i r c u l a r c y l i n d e r a n d s h a r p c o r n e r m o d e l s were u s e d . S i n c e end p l a t e s seem t o be c o m p u l s o r y f o r t h e c i r c u l a r c y l i n d e r , end p l a t e s were u s e d f o r c o n s i s t e n c y on a l l m o d e l s i n c l u d i n g t h e f l a t p l a t e . T h e r e was a s e c o n d r e a s o n f o r u s i n g e n d p l a t e s ; b e c a u s e a b a l a n c e was u s e d t o m e asure t h e d r a g , a gap was l e f t a r o u n d t h e m o d el where i t p a s s e s t h r o u g h t h e w i n d t u n n e l r o o f a n d f l o o r ; f r o m t h i s gap, a i r c o u l d be s u c k e d i n o r o u t 9 due t o t h e p r e s s u r e d i f f e r e n c e a n d t h e p r e s s u r e d i s t r i b u t i o n on t h e model c o u l d be a f f e c t e d . End p l a t e s were a s s u m e d t o m i n i m i z e t h i s e f f e c t . The e nd p l a t e s were d e s i g n e d a c c o r d i n g t o S t a n s b y ' s r e c o m m e n d a t i o n s ( F i g . 5 ) . The f r o n t r o d e n d p l a t e s were o f v a r i o u s s i z e s t o s u i t d i f f e r e n t r o d s p a c i n g s a n d a v o i d i n t e r f e r e n c e . 2.5 P r e s s u r e D i s t r i b u t i o n M e a s u r e m e n t s A l l m o d e l s were f i t t e d w i t h p r e s s u r e t a p s a t t h e i r m i d -s p a n p o s i t i o n . F o r t h e f l a t p l a t e , s e v e n p r e s s u r e t a p s were d i s t r i b u t e d on t h e f r o n t a n d one p r e s s u r e t a p was l o c a t e d a t t h e b a c k . The p r e s s u r e a t t h e bac k was assumed c o n s t a n t a c r o s s t h e w i d t h o f t h e p l a t e , an a s s u m p t i o n t h a t was v e r i f i e d e x p e r i m e n t a l l y . F o r t h e c i r c u l a r c y l i n d e r , p r e s s u r e t a p s were l o c a t e d a t 20° i n t e r v a l s a l l a r o u n d , e x c e p t n e a r t h e b a s e where t h e p r e s s u r e i s e s s e n t i a l l y c o n s t a n t ( s e e F i g . 2 9 ) . One p r e s s u r e t a p was l o c a t e d a t m i d -s p a n on t h e 12.7 mm d i a m e t e r f r o n t r o d , a n d p r e s s u r e m e a s u r e m e n t s a t v a r i o u s a n g l e s were made by r o t a t i n g t h e r o d . F o r t h e o t h e r two r o d s , 6.35 a n d 19.1 mm d i a m e t e r , t h e p r e s s u r e d i s t r i b u t i o n was n o t m e a s u r e d . F i g . 6 shows t h e l o c a t i o n o f p r e s s u r e t a p s on t h e m o d e l s . P r e s s u r e m e a s u r e m e n t s were made u s i n g a S e t r a 237 d i f f e r e n t i a l p r e s s u r e t r a n s d u c e r i n a s c a n i v a l v e . The p r e s s u r e t r a n s d u c e r was c a l i b r a t e d a g a i n s t t h e w i n d t u n n e l B e t z manometer and t h e c a l i b r a t i o n was f r e q u e n t l y c h e c k e d . The p r e s s u r e t a p s were c o n n e c t e d t o t h e s c a n i v a l v e t h r o u g h 1 10 m o f t y g o n t u b i n g w i t h i n t e r n a l d i a m e t e r o f 1.68 mm. The r e f e r e n c e p r e s s u r e was t h e s t a t i c p r e s s u r e i n t h e t e s t s e c t i o n . 2.6 Mean D r a g M e a s u r e m e n t s The o v e r a l l d r a g o f t h e two p a r t s o f t h e m o d e l , f r o n t r o d a n d m a i n c y l i n d e r , was m e a s u r e d by t h e b a l a n c e . A d r a g c o e f f i c i e n t was c a l c u l a t e d f r o m t h e m e a s u r e d f o r c e , t h e f r o n t a l a r e a o f t h e m a i n c y l i n d e r a n d t h e d y n a m i c p r e s s u r e o f t h e f l o w . The f r o n t a l a r e a was t a k e n a s t h e h e i g h t o f t h e t e s t s e c t i o n m u l t i p l i e d by t h e d i m e n s i o n D o f t h e m o d e l . An e r r o r i n t h e m e a s u r e d d r a g c o e f f i c i e n t was i n t r o d u c e d due t o end e f f e c t s a n d f r i c t i o n on t h e end p l a t e s . T h i s e r r o r i n d r a g was assumed t o be p r o p o r t i o n a l t o t h e t o t a l d r a g on t h e m o d e l . A c a l i b r a t i o n was made be t w e e n t h e b a l a n c e r e a d i n g ( t o t a l f o r c e on t h e m o d e l ) and t h e f o r c e m e a s u r e d by i n t e g r a t i o n o f t h e p r e s s u r e d i s t r i b u t i o n ( f o r c e on t h e m i d - s p a n s e c t i o n o f t h e m o d e l ) . F i g . 7 shows t h e c a l i b r a t i o n c u r v e , u s i n g d i f f e r e n t m o d e l s , f r o m w h i c h a c a l i b r a t i o n c o n s t a n t o f 1.048 was d e d u c e d ; a l l l a t e r b a l a n c e d r a g c o e f f i c i e n t s were c o r r e c t e d i n t h i s way: C D ( t r u e ) = 1.048 C D ( b a l a n c e ) 11 2.7 F l u c t u a t i n g S i d e F o r c e M e a s u r e m e n t s S i d e f o r c e m e a s u r e m e n t s were o n l y t a k e n on t h e c i r c u l a r c y l i n d e r . A d i f f e r e n t a r r a n g e m e n t o f p r e s s u r e t a p s was u s e d ; e i g h t p r e s s u r e t a p s were l o c a t e d on one s i d e o f t h e c y l i n d e r i n s u c h a way t h a t t h e i r a v e r a g e p r e s s u r e was p r o p o r t i o n a l t o t h e s i d e f o r c e . D e t a i l s o f t h e c a l c u l a t i o n s o f t h e p r e s s u r e t a p l o c a t i o n a r e shown i n A p p e n d i x A. The a v e r a g i n g o f t h e p r e s s u r e was made u s i n g t h e t e c h n i q u e f o r p n e u m a t i c a v e r a g i n g a s d e s c r i b e d by S t a t h o p o u l o s ( 9 ) . The e i g h t p r e s s u r e t u b e s were c o n n e c t e d t o a s p e c i a l s y m m e t r i c a l m a n i f o l d d e s i g n e d t o h a v e minimum p o s s i b l e i n t e r n a l v o l u m e . The m a n i f o l d h a d a s i n g l e o u t p u t t u b e w h i c h was e x p e c t e d t o g i v e a measurement o f t h e a v e r a g e o f t h e e i g h t i n p u t p r e s s u r e s . S t a t h o p o u l o s showed t h a t t h i s s y s t e m w o r k s w e l l a n d h a s a g o o d f r e q u e n c y r e s p o n s e , a t l e a s t a t low f r e q u e n c i e s ( b e l o w a b o u t 60 Hz d e p e n d i n g on t u b i n g c o n n e c t i o n s u s e d ) . F i g . 8 shows a s k e t c h o f t h e mani f o l d . B e c a u s e t h e t u b i n g s y s t e m , t h e m a n i f o l d and t h e t r a n s d u c e r c a v i t y i n t r o d u c e f r e q u e n c y d e p e n d e n t d i s t o r t i o n o f t h e p r e s s u r e f l u c t u a t i o n s , t h e s i d e f o r c e m e a s u r i n g s y s t e m was c a l i b r a t e d . A s c h e m a t i c r e p r e s e n t a t i o n o f t h e c a l i b r a t i o n a p p a r a t u s i s shown i n F i g . 9 . A f l u c t u a t i n g p r e s s u r e was c r e a t e d i n t h e c y l i n d e r by a f l u c t u a t i n g d i a p h r a g m . The d i a p h r a g m was a c t i v a t e d by a v i b r a t o r c o n n e c t e d t o a f r e q u e n c y g e n e r a t o r . N i n e p r e s s u r e t a p s were f i t t e d t o t h e c y l i n d e r h e a d so t h a t one c o u l d be u s e d a s a 12 r e f e r e n c e w h i l e t h e o t h e r e i g h t were c o n n e c t e d t o t h e m a n i f o l d . The r e f e r e n c e p r e s s u r e was m e a s u r e d by t h e s c a n i v a l v e t h r o u g h a s h o r t l e n g t h o f t u b i n g (50 mm). The o u t p u t p r e s s u r e o f t h e s y s t e m t o be c a l i b r a t e d was a l s o m e a s u r e d by t h e s c a n i v a l v e on a s e c o n d c h a n n e l . The a m p l i t u d e r a t i o s (RMS/RMS r ef ) were p l o t t e d a g a i n s t f r e q u e n c y f o r f r e q u e n c i e s b e t w e e n 8 Hz and 240 Hz. The r e s u l t , c a l l e d t h e t r a n s f e r f u n c t i o n , i s shown i n F i g . 1 0 . F o u r c a s e s a r e shown where e i g h t , s i x , f o u r a n d two m a n i f o l d i n p u t s were c o n n e c t e d t o t h e c y l i n d e r h e a d , t h e o t h e r i n p u t s b e i n g l e f t open t o a m b i e n t p r e s s u r e . The r e f e r e n c e v a l u e was t a k e n a s t h e a v e r a g e f l u c t u a t i o n o f t h e e i g h t m a n i f o l d i n p u t s , i f no d i s t o r t i o n due t o f r e q u e n c y e x i s t e d . I n t h e c a s e o f f o u r i n p u t s l e f t o p e n , f o r e x a m p l e , t h e r e f e r e n c e v a l u e was h a l f o f t h e a m p l i t u d e r e a d a t t h e c y l i n d e r r e f e r e n c e t a p . T h e r e i s g ood a g r e e m e n t b e t w e e n t h e f o u r d i f f e r e n t c o n f i g u r a t i o n s e v e n a t h i g h f r e q u e n c i e s , w h i c h shows t h a t t h e a v e r a g i n g m e thod w o r k s w e l l e v e n a t f r e q u e n c i e s a s h i g h a s 240 Hz. The o n l y i n c o n s i s t e n c y i n t h e r e s u l t s i s a r o u n d t h e r e s o n a n t peak (~60 Hz) and c o u l d be a s o u r c e o f e r r o r . The s o l i d c u r v e i n F i g . 1 0 i s u s e d i n C h a p t e r I V t o c o r r e c t f l u c t u a t i n g s i d e f o r c e r e a d i n g s f o r f r e q u e n c y d i s t o r t i o n . A S p e c t r a s c o p e I I f r e q u e n c y a n a l y s e r was u s e d d u r i n g t h e e x p e r i m e n t t o m e a s u r e t h e f r e q u e n c y s p e c t r u m and a s a 1 3 r e s u l t t h e d o m i n a n t f r e q u e n c y o f v o r t e x s h e d d i n g . 2.8 D a t a A c q u i s i t i o n S i g n a l s f r o m b o t h t h e b a l a n c e a n d t h e p r e s s u r e t r a n s d u c e r were d i g i t i z e d by a NEFF D a t a A c q u i s i t i o n S y s t e m . D i g i t a l s i g n a l s were t h e n p r o c e s s e d by a PDP-11/34 c o m p u t e r . The s a m p l i n g r a t e was s e t t o 400 Hz and s a m p l i n g t i m e was 10 s e c o n d s . From t h e s e 4000 r e a d i n g s , t h e a v e r a g e a n d t h e RMS v a l u e o f t h e f l u c t u a t i o n were c a l c u l a t e d . The p r o g r a m w o u l d a l s o t r a n s f o r m t h e s e v a l u e s i n t o c o e f f i c i e n t f o r m a s f o l l o w s : Cp = (p - p , ) / ( 0 . 5 p U , 2 ) CD = ( D r a g ) / ( 0 . 5 p U 1 2 D H ) p - P i i s g i v e n by t h e p r e s s u r e t r a n s d u c e r 0.5pU, 2 i s g i v e n by t h e B e t z manometer D r a g i s g i v e n by t h e b a l a n c e DH i s g i v e n by 38.1 mm x 685.8 mm ( m o d e l f r o n t a l w i d t h m u l t i p l i e d by w i n d t u n n e l h e i g h t ) 1 4 I I I . RESULTS AND DISCUSSION I ; FLAT PLATE T e s t s were c o n d u c t e d on t h e f l a t p l a t e w i t h two s i z e s o f f r o n t r o d , d/D = 0.17 a n d d/D = 0.33, a t t h r e e R e y n o l d s numbers ( 2 . 5 x 1 0 " , 4 x 1 0 " and 5x10") b a s e d on D. No R e y n o l d s number d e p e n d e n c e was f o u n d , a s e x p e c t e d . S i m i l a r t e s t s were c o n d u c t e d i n t h e smoke t u n n e l f o r v i s u a l i s a t i o n p u r p o s e s a t a R e y n o l d s number o f 5 x 1 0 3 . 3.1 P r e s s u r e D i s t r i b u t i o n On F l a t P l a t e I n t h e w i n d t u n n e l , t h e p l a t e was a l i g n e d p e r p e n d i c u l a r t o t h e a i r s t r e a m by r o t a t i n g t h e model u n t i l t h e p r e s s u r e d i s t r i b u t i o n on t h e f r o n t f a c e was s y m m e t r i c a l . The mean p r e s s u r e d i s t r i b u t i o n on t h e f l a t p l a t e i s shown i n F i g . 1 1 . T h e s e r e s u l t s , a s w e l l a s a l l l a t e r v a l u e s o f Cp and CD, h a v e been c o r r e c t e d f o r w i n d t u n n e l b l o c k a g e u s i n g M a s k e l l ' s m e thod (10) w i t h t h e b l o c k a g e b e i n g 4.1%. T h i s p r e s s u r e d i s t r i b u t i o n i s c o m p a r e d w i t h F a g e & J o h a n s e n ' s r e s u l t s ( 1 1 ) , c o r r e c t e d f o r b l o c k a g e , a nd P a r k i n s o n ' s wake s o u r c e m o d e l ( 1 2 ) . N u m e r i c a l i n t e g r a t i o n o f t h e p r e s s u r e d i s t r i b u t i o n g i v e s a v a l u e o f d r a g c o e f f i c i e n t CD = 1.94 a s c o m p a r e d w i t h t h e c a l i b r a t e d b a l a n c e r e a d i n g CD = 1.95 a n d w i t h Fage & J o h a n s e n ' s v a l u e o f CD = 1.90. T h e r e i s a d i s c r e p a n c y b e t w e e n Fage & J o h a n s e n ' s r e s u l t s a n d t h e c u r r e n t r e s u l t s , m a i n l y i n t h e b a s e p r e s s u r e v a l u e . T h i s i s p r o b a b l y due t o a d i f f e r e n c e i n t h i c k n e s s t o w i d t h r a t i o . The m o d e l u s e d i n t h i s e x p e r i m e n t had a t h i c k n e s s t o w i d t h r a t i o o f 1/12 w h i l e F a g e 1 5 & J o h a n s e n ' s m odel h a d s h a r p e d g e s . E n g i n e e r i n g S c i e n c e D a t a (1 3 ) shows t h a t t h e e f f e c t o f a t h i c k n e s s t o w i d t h r a t i o o f 1/12 i s a b o u t t h e same a s o b s e r v e d h e r e . 3.2 P r e s s u r e D i s t r i b u t i o n On F l a t P l a t e W i t h F r o n t Rod The mean p r e s s u r e d i s t r i b u t i o n on t h e f l a t p l a t e when t h e f r o n t r o d d/D = 0.33 was l o c a t e d b e t w e e n L/D = 7.17 and L/D = 1.97 i s shown i n F i g . 1 2 . F i g . 1 3 shows t h e p r e s s u r e d i s t r i b u t i o n on t h e same m o d e l , when t h e r o d was l o c a t e d b e t w e e n L/D = 1.81 a n d L/D = 0.84. Two d i f f e r e n t f l o w r e g i m e s a r e o b s e r v e d . The f i r s t , r e g i m e A, i s o b s e r v e d f o r <» > l / D > 1.97 a n d t h e s e c o n d , r e g i m e B, f o r 1.97 > L/D > 0.84. The f l o w a t s p a c i n g L/D = 1.97 was f o u n d t o be b i s t a b l e , s w i t c h i n g f r o m one r e g i m e t o t h e o t h e r . No e x p e r i m e n t s were done w i t h L/D l e s s t h a n 0.84. S i m i l a r r e s u l t s w e r e o b t a i n e d when t h e r o d d/D = 0.17 was u s e d . P r e s s u r e d i s t r i b u t i o n s a r e shown i n F i g . 1 4 f o r r e g i m e A a n d i n F i g . 1 5 f o r r e g i m e B. The p r e s s u r e d i s t r i b u t i o n on t h e f r o n t r o d d/D = 0.33 a t two d i f f e r e n t s p a c i n g s h a s been m e a s u r e d and t h e r e s u l t s a r e shown i n F i g . 1 6 . The p r e s s u r e d i s t r i b u t i o n a t s p a c i n g L/D = 3.42 ( f l o w r e g i m e A) i s v e r y s i m i l a r t o a s u b c r i t i c a l p r e s s u r e d i s t r i b u t i o n a r o u n d a c i r c u l a r c y l i n d e r i n a u n i f o r m f l o w . A t s p a c i n g L/D = 1.42 ( f l o w r e g i m e B) t h e s t r o n g i n t e r a c t i o n o f t h e two b o d i e s makes t h e r o d b a s e p r e s s u r e h a v e a c o n s t a n t a n d h i g h e r v a l u e t h a n t h e p r e v i o u s c a s e . 1 6 3.3 F l o w Regime A The c h a r a c t e r i s t i c s o f t h i s f l o w r e g i m e ( F i g s . 1 2 & 14) a r e t h a t t h e p r e s s u r e on t h e f r o n t o f t h e p l a t e d e c r e a s e s w i t h d e c r e a s i n g r o d s p a c i n g a n d t h a t t h e b a s e p r e s s u r e r e m a i n s c o n s t a n t , i n d e p e n d e n t o f t h e r o d p o s i t i o n . The d e c r e a s i n g f r o n t p r e s s u r e w i t h d e c r e a s i n g s p a c i n g c a n be e x p l a i n e d by t h e f a c t t h a t i t i s n o t a u n i f o r m f l o w t h a t r e a c h e s t h e p l a t e , b u t a f l o w w i t h t h e t y p i c a l mean v e l o c i t y p r o f i l e o f a wake b e h i n d a c i r c u l a r c y l i n d e r . The p l a n e wake b e h i n d a c i r c u l a r c y l i n d e r i n a u n i f o r m s t r e a m h a s been c a r e f u l l y s t u d i e d by Townsend ( 1 4 ) . The mean v e l o c i t y p r o f i l e i n t h e wake i s c h a r a c t e r i s e d by a v e l o c i t y d e f i c i t a t t h e c e n t e r , U 0 , d e c r e a s i n g w i t h i n c r e a s i n g d o w n s t r e a m d i s t a n c e a nd by a c h a r a c t e r i s t i c wake w i d t h , 1 0 , i n c r e a s i n g w i t h d o w n s t r e a m d i s t a n c e . F o r a s m a l l d e f i c i t wake, s i m i l a r i t y a r g u m e n t s showed t h a t U 0o: x " V 2 a n d l 0 o c x 1 / 2 , when x i s m e a s u r e d i n t h e s t r e a m w i s e d i r e c t i o n f r o m an a p p r o p r i a t e o r i g i n . I f t h e p l a t e d o e s n o t h a v e a s t r o n g i n t e r a c t i o n w i t h t h e r o d wake, t h e s t a g n a t i o n p r e s s u r e a t t h e c e n t r e o f t h e p l a t e c a n be c o n s i d e r e d a s a m e a s u r e o f t h e t o t a l h e a d a t t h e c e n t r e o f t h e wake. The s t a t i c p r e s s u r e c a n be assumed t o be c o n s t a n t a c r o s s t h e f l o w a nd t h e v e l o c i t y d e f i c i t o f t h e wake c a n be f o u n d f r o m t h e m e a s u r e d s t a g n a t i o n p r e s s u r e a t t h e c e n t r e o f t h e p l a t e , p ( c e n t r e ) : 17 C p ( c e n t r e ) = ( p ( c e n t r e ) - p,)/(0.5pU, 2) = ( 0 . 5 p U ( c e n t r e ) 2 ) / ( 0 . 5 p U , 2 ) = ( U ( c e n t r e ) / U , ) 2 U 0 = u, - U ( c e n t r e ) = (1 - V C p ( c e n t r e ) ) U , The v a l u e s o f v e l o c i t y d e f i c i t on t h e c e n t r e l i n e , a s c a l c u l a t e d (1 - \ / C p ( c e n t r e ) ) U , , a r e t a b u l a t e d f o r b o t h s i z e s o f r o d i n T a b l e I , a n d s h o u l d be p r o p o r t i o n a l t o x ~ 1 / 2 . F i g . 1 7 shows Uo/U, v e r s u s L / d , d i s t a n c e n o n -d i m e n s o n a l i s e d w i t h r e s p e c t t o t h e r o d s i z e . The c u r v e U 0 / U i = ( L / d ) ' 1 / 2 i s a l s o shown a n d f o l l o w s t h e d a t a p o i n t s w e l l . T h i s r e s u l t i s i n g o o d a g r e e m e n t w i t h Townsend who showed t h a t on t h e c e n t r e l i n e o f a wake b e h i n d a c i r c u l a r c y l i n d e r : ( U o / U , ) ( L / d ) 1 / 2 = 1 T h i s l a s t f i g u r e w e l l s u p p o r t s t h e e x p l a n a t i o n t h a t t h e f r o n t s t a g n a t i o n p r e s s u r e c o e f f i c i e n t i s l e s s t h a n one due t o t h e v e l o c i t y d e f i c i t o f t h e wake o f t h e r o d . The s e c o n d c h a r a c t e r i s t i c o f r e g i m e A, w h i c h i s a l s o a c h a r a c t e r i s t i c o f r e g i m e B, i s a b a s e p r e s s u r e e s s e n t i a l l y i n d e p e n d e n t of r o d p o s i t i o n . F i g . 1 8 shows t h e b a s e p r e s s u r e c o e f f i c i e n t v e r s u s s p a c i n g f o r t h e two s i z e s o f r o d . T h e r e i s some s c a t t e r b u t t h e b a s e p r e s s u r e c o e f f i c i e n t i s f a i r l y 18 c o n s t a n t a r o u n d -1.24. T u r b u l e n c e i n t h e f l o w d o e s n o t seem t o be an i m p o r t a n t f a c t o r g o v e r n i n g t h e b a s e p r e s s u r e v a l u e , p r o b a b l y b e c a u s e o f t h e f i x e d s e p a r a t i o n p o i n t s and no p o s s i b i l i t y o f r e a t t a c h m e n t . The wake v e l o c i t y p r o f i l e d o e s n o t seem t o a f f e c t t h e b a s e p r e s s u r e e i t h e r . I f t h e v e l o c i t y p r o f i l e i n t h e wake o f t h e f r o n t r o d i s c o n s i s t e n t w i t h Townsend's r e s u l t s a s i t seems t o be f r o m F i g . 1 7 , t h e n t h e c h a r a c t e r i s t i c wake w i d t h , 1 0 , w h i c h w o u l d e x i s t i n t h e a b s e n c e o f t h e p l a t e , c a n be e v a l u a t e d . Townsend showed t h a t : 1 0 = 0 . 4 ( L d ) 1 / 2 ( h a l f w i d t h t o h a l f d e p t h ) From t h e l a s t e q u a t i o n , v a l u e s o f 1 0 / D were c a l c u l a t e d a n d a r e shown i n T a b l e I . 1 0 / D i s a l w a y s s m a l l e r t h a n 0 . 5 o r 1 0 a l w a y s s m a l l e r t h a n t h e p l a t e h a l f - w i d t h , e x c e p t when t h e r o d i s f a r a n d h a s a weak e f f e c t . T h e r e f o r e , t h e v e l o c i t y a t t h e edge o f t h e p l a t e i s e x p e c t e d t o k e e p a c o n s t a n t v a l u e s o t h a t t h e b a s e p r e s s u r e i s a l s o c o n s t a n t . 3.4 F l o w Regime B T h i s f l o w r e g i m e i s c o m p l e t e l y d i f f e r e n t f r o m t h e t y p i c a l f l o w o v e r a f l a t p l a t e a l o n e . The two b o d i e s h a v e a s t r o n g i n f l u e n c e on e a c h o t h e r , s u b s t a n t i a l l y a f f e c t i n g t h e f l o w f i e l d . T h i s c h a n g e i n f l o w p a t t e r n b e g i n s a t one t y p i c a l s p a c i n g , t h a t we w i l l c a l l t h e c r i t i c a l s p a c i n g . F l o w v i s u a l i s a t i o n p i c t u r e s ( F i g . 1 9 ) show a c h a n g e i n f l o w p a t t e r n o c c u r r i n g b e t w e e n L/D = 1.36 a n d L/D = 0.79. 19 The p a t t e r n i n F i g . 1 9 ( a ) shows a t y p i c a l wake b e h i n d a c i r c u l a r c y l i n d e r i n t h e r e g i o n b e t w e e n t h e r o d and t h e p l a t e f o r L/D = 1.36. A d i f f e r e n t f l o w p a t t e r n i s shown i n F i g . 1 9 ( b ) f o r L/D = 0.79. A c l o s e d r e g i o n b e t w e e n t h e p l a t e a nd t h e r o d i s f o r m e d where r e c i r c u l a t i o n o c c u r s . The two s e p a r a t e d s h e a r l a y e r s f r o m t h e r o d r e a c h t h e p l a t e v e r y c l o s e t o t h e e d g e s o f t h e p l a t e . B e c a u s e o f t h e s t r o n g b l o c k a g e a n d t h e t w o - d i m e n s i o n a l t e s t s e c t i o n o f t h e smoke t u n n e l , t h e v i s u a l i s a t i o n r e s u l t s show a l o w e r v a l u e o f c r i t i c a l s p a c i n g t h a n t h a t shown i n t h e w i n d t u n n e l . N e v e r t h e l e s s , t h e f l o w v i s u a l i s a t i o n s a r e h e l p f u l i n u n d e r s t a n d i n g t h e d i f f e r e n t r e g i m e s . I n r e g i m e B ( F i g s . 1 3 & 1 5 ) , i n s t e a d o f one s t a g n a t i o n p o i n t a t t h e c e n t r e , t h e p r e s s u r e d i s t r i b u t i o n s show two s y m m e t r i c a l n e a r - s t a g n a t i o n p o i n t s c l o s e t o t h e e d g e s o f t h e p l a t e . They r e p r e s e n t t h e r e a t t a c h m e n t o f t h e s e p a r a t e d s h e a r l a y e r s o f t h e r o d o n t o t h e p l a t e . T h e i r v a l u e i s l e s s t h a n Cp = 1 b e c a u s e o f l o s s e s i n t h e s h e a r l a y e r s a n d u n s t e a d i n e s s . The e x a c t l o c a t i o n a nd p r e s s u r e o f t h e s e n e a r -s t a g n a t i o n p o i n t s c o u l d n o t be m e a s u r e d b e c a u s e t h e y were l o c a t e d b e t w e e n t h e edge o f t h e p l a t e a n d t h e f i r s t p r e s s u r e t a p . B u t f r o m t h e p r e s s u r e d i s t r i b u t i o n s i n r e g i m e B, t h e r e must be a p o i n t o f maximum p r e s s u r e b e t w e e n t h e edge o f t h e p l a t e ( p r e s s u r e g i v e n by t h e b a s e p r e s s u r e m e a s u r e m e n t ) a n d t h e f i r s t p r e s s u r e t a p . The p r e s s u r e d i s t r i b u t i o n on t h e c e n t r a l p o r t i o n o f t h e 20 p l a t e , s a y 0.3 < x/D < 0.7, i s f a i r l y c o n s t a n t . F o r t h e r o d d/D = 0.33 a t s p a c i n g L/D = 1.42, t h i s c o n s t a n t v a l u e i s a p p r o x i m a t e l y Cp = -0.10 a n d i s t h e same a s t h e f r o n t r o d b a s e p r e s s u r e a t t h e same s p a c i n g ( F i g . 1 6 ) . T h i s s u p p o r t s t h e i d e a t h a t a c l o s e d r e g i o n i s f o r m e d b e t w e e n t h e r o d and t h e p l a t e , where t h e mean v e l o c i t y i s s m a l l a n d where t h e p r e s s u r e i s e s s e n t i a l l y c o n s t a n t . The c h a n g e i n p r e s s u r e d i s t r i b u t i o n w i t h s p a c i n g f o r r e g i m e B, i s an i n c r e a s i n g f r o n t p r e s s u r e w i t h d e c r e a s i n g s p a c i n g . The f a c t t h a t t h e c l o s e d r e g i o n g e t s s m a l l e r w i t h s m a l l e r s p a c i n g s u g g e s t s t h a t l e s s d i s s i p a t i o n due t o t u r b u l e n c e o c c u r s a t s m a l l e r s p a c i n g a n d h e n c e an i n c r e a s e i n f r o n t p r e s s u r e . As a l r e a d y p o i n t e d o u t , t h e p l a t e b a s e p r e s s u r e i s f a i r l y c o n s t a n t w h a t e v e r t h e p o s i t i o n o f t h e r o d ( F i g . 1 8 ) . 3.5 D r a g F i g . 2 0 a nd 21 show d r a g c o e f f i c i e n t v e r s u s r o d s p a c i n g f o r t h e two r o d s , d/D = 0.33 and d/D = 0.17 r e s p e c t i v e l y . E a c h g r a p h shows t h e o v e r a l l d r a g c o e f f i c i e n t m e a s u r e d by t h e b a l a n c e a nd t h e d r a g c o e f f i c i e n t o f t h e p l a t e o n l y , c a l c u l a t e d by n u m e r i c a l i n t e g r a t i o n o f t h e p r e s s u r e d i s t r i b u t i o n . The d r a g c o e f f i c i e n t o f t h e r o d i s g i v e n by t h e d i f f e r e n c e b e t w e e n t h e two c u r v e s . A minimum d r a g c o e f f i c i e n t i s r e a c h e d a t t h e c r i t i c a l s p a c i n g . F o r t h e r o d d/D = 0.33 t h e minimum CD i s 1.25 o c c u r r i n g a t L/D = 1.81, a s c o m p a r e d t o 1.94 f o r t h e p l a t e a l o n e ; t h i s i s a d r a g r e d u c t i o n o f 36%. F o r t h e r o d 21 d/D = 0 . 1 7 a maximum d r a g r e d u c t i o n o f 24% i s a c h i e v e d a t t h e c r i t i c a l s p a c i n g , L/D = 1.64. The c h a n g e i n f l o w p a t t e r n i s r e p r e s e n t e d by a d o t t e d l i n e b e c a u s e t h e d r a g c o e f f i c i e n t d r a s t i c a l l y d r o p s when t h e f l o w p a t t e r n s w i t c h e s f r o m r e g i m e A t o r e g i m e B. A t t h i s p o i n t t h e f l o w was f o u n d t o be b i s t a b l e . The d r a g o f t h e r o d k e e p s a f a i r l y c o n s t a n t v a l u e i n e a c h r e g i m e a n d i s a l w a y s s m a l l e r t h a n t h e d r a g o f t h e same r o d i n a u n i f o r m f l o w . The o v e r a l l d r a g c o e f f i c i e n t d e c r e a s e s w i t h d e c r e a s i n g s p a c i n g i n r e g i m e A a n d i n c r e a s e s w i t h d e c r e a s i n g s p a c i n g i n r e g i m e B. 3.6 E f f e c t Of Yaw S i n c e t h e d r a g r e d u c t i o n d e v i c e i n v e s t i g a t e d i s u n i d i r e c t i o n a l a n d , i n p r a c t i c a l c a s e s , t h e s t r u c t u r e may n o t be p e r f e c t l y a l i g n e d w i t h t h e f l o w , i t i s o f i n t e r e s t t o s t u d y t h e e f f e c t o f s m a l l yaw a n g l e on t h e d r a g . The o v e r a l l d r a g c o e f f i c i e n t v e r s u s yaw i s shown i n F i g . 2 2 when t h e r o d d/D = 0.33 was l o c a t e d a t s p a c i n g L/D = 1.42 ( r e g i m e B ) . The m e a s u r e d d r a g c o e f f i c i e n t i s d i v i d e d by C D r e ^ w h i c h i s t h e d r a g c o e f f i c i e n t o f t h e p l a t e a l o n e a t z e r o yaw. The d r a g r e d u c t i o n i s 34% a t 0° and i s o n l y 17% a t 12°. The e f f e c t o f yaw i s t h e r e f o r e i m p o r t a n t a n d t h e d e v i c e s h o u l d be a l i g n e d w i t h t h e f l o w a s much a s p o s s i b l e i f a l a r g e d r a g r e d u c t i o n i s t o be a c h i e v e d . 22 3.7 E f f e c t Of R e y n o l d s Number As e x p e c t e d , R e y n o l d s number d i d n o t h a v e much e f f e c t on p r e s s u r e c o e f f i c i e n t b e c a u s e t h e s e p a r a t i o n p o i n t s a r e f i x e d a t t h e e d g e s o f t h e p l a t e . The r e s u l t s a t t h e t h r e e d i f f e r e n t s R e y n o l d s n u mbers, 2 . 5 x 1 0 " , 4 x 1 0 " and 5 x 1 0 " , were e x a c t l y t h e same w i t h i n t h e a c c u r a c y o f m e a s u r e m e n t s . 3.8 P o t e n t i a l F l o w M o d e l A t w o - 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 p o t e n t i a l f l o w m o d e l i s p r e s e n t e d h e r e a n d was d e v e l o p e d a s a n u m e r i c a l e x p e r i m e n t . I t r e p r e s e n t s t h e f l o w o v e r t h e f l a t p l a t e i n r e g i m e B. The b l o c k a g e due t o t h e r o d was assumed t o be o f l i t t l e i m p o r t a n c e , t h e m a i n e f f e c t o f t h e r o d b e i n g t o f i x t h e u p s t r e a m s t a g n a t i o n p o i n t a n d c r e a t e a wake i n d e p e n d a n t o f r o d s h a p e o r s i z e . I n t h i s m o d e l , t h e n o r m a l f l a t p l a t e i s mapped f r o m a c i r c l e by t h e J o u k o w s k i t r a n s f o r m a t i o n . The r e p r e s e n t a t i o n o f t h e s e p a r a t e d f l o w b e h i n d t h e p l a t e i s o b t a i n e d u s i n g P a r k i n s o n ' s wake s o u r c e m o d e l ( 1 2 ) . The s e p a r a t e d f l o w c r e a t e d by t h e r o d u p s t r e a m t h e f l a t p l a t e i s r e p r e s e n t e d by a s y m m e t r i c a l p a i r o f v o r t i c e s . C o n s i d e r t h e wake s o u r c e m o d e l f o r a n o r m a l f l a t p l a t e a s shown i n F i g . 2 3 . The c o m p l e x p o t e n t i a l o f t h e r e s u l t i n g f l o w i n t h e $ - p l a n e i s : F(0 = V(?+| ).+Q{1n(^Rei6)+ln(^-Re-',6)-lnU 23 a n d t h e c o m p l e x v e l o c i t y i s : •«> • ^ • s t b + ^  - & The J o u k o w s k i t r a n s f o r m a t i o n , w h i c h p r e s e r v e s v e l o c i t y a t i n f i n i t y , t r a n s f o r m s t h e c i r c l e i n t o a n o r m a l f l a t p l a t e : z = $-R 2/S where R = h/4 The b o u n d a r y c o n d i t i o n s a r e : (1) s t a g n a t i o n p o i n t s i n t h e $ - p l a n e a t t h e c r i t i c a l p o i n t s o f t h e t r a n s f o r m a t i o n so t h a t t h e a n g l e s a r e d o u b l e d t h e r e a n d t h e f l o w l e a v e s t h e p l a t e t a n g e n t i a l l y : w($) = 0 a t J = ±iR (2) t h e b a s e p r e s s u r e i s s p e c i f i e d . A t t h e c r i t i c a l p o i n t s , w ( z ) i s f i n i t e a n d : w ( z ) = U(1 - C p b ) V 2 a t z = ±h/2 From t h e s e two c o n d i t i o n s , Q a n d 6 a r e d e t e r m i n e d . Now t o a d d a s y m m e t r i c a l p a i r o f v o r t i c e s i n f r o n t o f t h e c i r c l e ( a t $ 0 a n d $ 0 ) i n o r d e r t o c r e a t e a b u b b l e , we a l s o h a v e t o a d d i m a g e s i n s i d e t h e c i r c l e t o s a t i s f y t h e c i r c l e b o u n d a r y c o n d i t i o n s . The l o c a t i o n ( $ 0 ) a n d s t r e n g t h ( D . o f t h e p a i r o f v o r t i c e s w i l l d e t e r m i n e t h e s i z e o f t h e b u b b l e . The c o m p l e x p o t e n t i a l becomes: 24 2 F ( ? ) = )+?{ ln ( ; -Re l 5 )+ ln(c -Re' l 6)-1n?}+ g { l n ( r ; - c 0 ) + l n ( c - R 2 / ^ ) - 1 n ( S - " 0 ) - l n ( c - R 2 / C 0 ) } and the velocity p o t e n t i a l : -<<> • v ( . - 5 + ^ ^ - D , i r r 1 l + l _ l -i with again the Joukowski transformation and the conditions: (1) w(I) = 0 at 5 = ±iR (2) w(z) = U(1 - Cpb) 1 7 2 at z = h/2 and extra conditions to determine the location and strength of v o r t i c e s . A condition determining both the location and the strength of the vortices at the same time was used here. The condition is that each vortex be stationary, which means that the ve l o c i t y at i t s location due to a l l s i n g u l a r i t i e s except i t s e l f , i s zero: T^—\ l— ~ ZT^ = ° at c = 5 4 ^ e-R K0 c-R /Co 0 From these conditions, we can determine a locus of vortex positions where Q, T and 5 can also be determined. Fig.24 shows the streamlines of the flow over the f l a t plate with the front stagnation point at L/D = 0.87. The base pressure was specified as Cpb = -1.24 as found experimentally in a previous section. 25 I t i s n o t i c e d t h a t t h e s i z e o f t h e b u b b l e seems t o be t o o l a r g e a n d , a s a r e s u l t , t h e s t r e a m l i n e s go o v e r t h e p l a t e q u i t e s m o o t h l y , c r e a t i n g a much n a r r o w e r wake t h a n e x p e c t e d . E v e n t h o u g h t h e s t r e a m l i n e s l o o k v e r y smooth when p a s s i n g o v e r t h e p l a t e , t h e s t r e a m l i n e \p = 0 l e a v e s t h e p l a t e t a n g e n t i a l l y a n d c u r v e s d o w n s t r e a m v e r y r a p i d l y . T h i s i s a r a t h e r p o o r s i m u l a t i o n o f t h e s t r e a m l i n e s i f we compare t h i s w i t h t h e f l o w v i s u a l i s a t i o n p i c t u r e o f F i g . 1 9 ( b ) . T h i s r e s u l t c o u l d be i m p r o v e d by c h o o s i n g a v o r t e x s t r e n g t h a n d p o s i t i o n t h a t w o u l d s a t i s f y t h e f i r s t two b o u n d a r y c o n d i t i o n s a n d a l s o g i v e a g o o d s t r e a m l i n e r e p r e s e n t a t i o n . One m i g h t be i n t e r e s t e d i n e v a l u a t i n g t h e d r a g f r o m t h e p r e s e n t m o d e l . The r e a l f l o w i s o b v i o u s l y n o t p o t e n t i a l i n s i d e t h e b u b b l e a n d t h e p r e s s u r e must be p r e s c r i b e d e m p i r i c a l l y . From t h e e x p e r i m e n t , t h e p r e s s u r e i n s i d e t h e b u b b l e i s f a i r l y c o n s t a n t , i t w o u l d t h e r e f o r e make s e n s e t o use a c o n s t a n t v a l u e o f p r e s s u r e . I f t h e p r e s s u r e i s s p e c i f i e d i n s i d e t h e b u b b l e , a n d s i n c e t h e b a s e p r e s s u r e i s a l s o s p e c i f i e d , o n l y t h e p r e s s u r e n e a r t h e e d g e s o f t h e p l a t e c a n be v a r i e d by t h e m o d e l , w h i c h w o u l d n o t c h a n g e t h e t o t a l d r a g s i g n i f i c a n t l y . I n c o n c l u s i o n , t h e model i s n o t u s e f u l t o p r e d i c t t h e d r a g a n d , i n i t s p r e s e n t f o r m , d o e s n o t g i v e r e a l i s t i c s t r e a m l i n e s s h a p e . A b e t t e r c o n d i t i o n f o r T a n d $ 0 c o u l d i m p r o v e t h e m o d e l a s f a r a s s t r e a m l i n e s a r e c o n c e r n e d . B u t t h i s " b e t t e r " c o n f i g u r a t i o n w o u l d h a v e t o be f o u n d by t r i a l 2 6 a n d e r r o r . T h i s m o d e l shows t h e l i m i t a t i o n s o f p o t e n t i a l f l o w i n r e p r e s e n t i n g s e p a r a t e d f l o w b u t i t may be u s e f u l i n r e p r e s e n t i n g how much a c h a n g e i n p a r a m e t e r s , s u c h a s t h e b a s e p r e s s u r e , c a n a f f e c t t h e o u t e r f l o w . 27 I V . RESULTS AND DISCUSSION I I ; CIRCULAR CYLINDER The f l a t p l a t e was r e p l a c e d by a c i r c u l a r c y l i n d e r o f t h e same f r o n t a l w i d t h , 38.1 mm, and t h e same type's o f m e a s u r e m e n t s were c a r r i e d o u t . I n t h i s c a s e , u n l i k e t h e f l a t p l a t e , t h e R e y n o l d s number was e x p e c t e d t o be an i m p o r t a n t p a r a m e t e r g o v e r n i n g t h e f l o w . The f l u c t u a t i n g s i d e f o r c e , s i g n i f i c a n t i n t h i s c a s e , was a l s o m e a s u r e d . E x p e r i m e n t s were c a r r i e d o u t i n t h e w i n d t u n n e l f o r R e y n o l d s numbers i n t h e r a n g e 1.0x10" t o 7.0x10". S i m i l a r e x p e r i m e n t s were done i n t h e smoke t u n n e l a t a R e y n o l d s number o f 5 x 1 0 3 . 4.1 P r e s s u r e D i s t r i b u t i o n On C i r c u l a r C y l i n d e r 4.1.1 B a c k g r o u n d I n t h e r a n g e o f R e y n o l d s number o f i n t e r e s t f o r a e r o d y n a m i c i s t s , b e t w e e n 1 0 3 a n d 1 0 7 , t h e f l o w o v e r a c i r c u l a r c y l i n d e r i s s u b j e c t e d t o d i f f e r e n t f l o w p a t t e r n s . F i g . 2 5 shows t h e d r a g c o e f f i c i e n t , CD, i n t h i s r a n g e o f Re, t a k e n f r o m A c h e n b a c h ( 1 5 ) . F o u r r a n g e s a r e s p e c i f i e d : s u b c r i t i c a l , c r i t i c a l , s u p e r c r i t i c a l a n d t r a n s c r i t i c a l . I n t h e s u b c r i t i c a l r a n g e , t h e f l o w i s n o t i n f l u e n c e d by t h e s u r f a c e r o u g h n e s s and t h e d r a g c o e f f i c i e n t i s e s s e n t i a l l y c o n s t a n t . A t h i g h e r Re, CD s u d d e n l y d r o p s . T h i s i s due t o t h e l a m i n a r b o u n d a r y l a y e r s e p a r a t i o n a n d t h e t u r b u l e n t r e a t t a c h m e n t . T u r b u l e n t m i x i n g r e - e n e r g i z e s t h e d e c e l e r a t i n g b o u n d a r y l a y e r , m a k i n g t h e f i n a l s e p a r a t i o n o c c u r f u r t h e r d o w n s t r e a m . I t r e s u l t s i n a n a r r o w e r wake 28 ( l o w p r e s s u r e r e g i o n ) c o r r e s p o n d i n g t o a f a l l i n d r a g c o e f f i c i e n t . E x c e e d i n g t h e R e y n o l d s number o f CD minimum, t h e d r a g c o e f f i c i e n t i n c r e a s e s a g a i n i n t h e s u p e r c r i t i c a l r a n g e a n d r e a c h e s a n e a r l y c o n s t a n t v a l u e i n t h e t r a n s c r i t i c a l r a n g e . N e a r t h e c r i t i c a l Re, a t w h i c h CD d r o p s s h a r p l y , i t i s known t h a t t h e f l o w p a t t e r n i s s t r o n g l y d e p e n d e n t on t h e t u r b u l e n c e l e v e l o f t h e a p p r o a c h i n g s t r e a m and on t h e r o u g h n e s s o f t h e s u r f a c e . A c h e n b a c h showed t h a t t h e e f f e c t o f t u r b u l e n c e was m a i n l y t o s h i f t t h e c u r v e CD v e r s u s Re t o t h e l e f t so t h a t t h e f l o w becomes c r i t i c a l a t a l o w e r Re. 4.1.2 R e s u l t s M e a s u r e d p r e s s u r e d i s t r i b u t i o n s a r o u n d t h e c i r c u l a r c y l i n d e r a l o n e a t d i f f e r e n t R e y n o l d s numbers i n t h e s u b c r i t i c a l r a n g e a r e shown i n F i g . 2 6 . The s o l i d l i n e r e p r e s e n t s t h e p r e s s u r e d i s t r i b u t i o n a r o u n d a c i r c u l a r c y l i n d e r i n t h e s u b c r i t i c a l r a n g e , a s shown i n E n g i n e e r i n g S c i e n c e D a t a ( 1 6 ) . The d a t a p o i n t s do n o t f o l l o w e x a c t l y t h e s o l i d l i n e , f i r s t , b e c a u s e t h e y h a v e n o t been c o r r e c t e d f o r b l o c k a g e a n d s e c o n d , b e c a u s e t h e y r e p r e s e n t p r e s s u r e d i s t r i b u t i o n s a t v a r i o u s R e y n o l d s numbers. D e s p i t e some d i s c r e p a n c i e s , t h e d a t a p o i n t s a r e i n g ood a g r e e m e n t w i t h t h e s o l i d l i n e . The u n c o r r e c t e d v a l u e o f d r a g c o e f f i c i e n t f o r 3.3x10" < Re < 7.1x10", a s m e a s u r e d by t h e b a l a n c e , was f o u n d t o be c o n s t a n t w i t h CD = 1.19. T h i s v a l u e w i l l be u s e d l a t e r a s t h e r e f e r e n c e v a l u e . 29 As m e n t i o n e d a b o v e , t h e d a t a p o i n t s h a v e n o t been c o r r e c t e d f o r b l o c k a g e a s i s t h e c a s e f o r a l l t h e r e s u l t s i n t h i s c h a p t e r . I t h a s been shown r e c e n t l y by West & A p e l t (17) t h a t when two w e l l known methods f o r b l o c k a g e c o r r e c t i o n , one o f them b e i n g M a s k e l l ' s ( 1 0 ) , a r e a p p l i e d t o a c i r c u l a r c y l i n d e r t h e c o r r e c t e d v a l u e s o f d r a g c o e f f i c i e n t a r e n o t any c l o s e r t o t h e v a l u e s f r e e o f b l o c k a g e e f f e c t s . On t h e o t h e r h a n d , t h e c o r r e c t e d v a l u e f o r b a s e p r e s s u r e u s i n g M a s k e l l ' s method i s v e r y c l o s e t o t h e e x p e c t e d v a l u e . The s u g g e s t e d e x p l a n a t i o n was t h a t t h e s h a p e o f t h e p r e s s u r e d i s t r i b u t i o n , s u c h a s t h e l o c a t i o n o f t h e s e p a r a t i o n p o i n t s , i s c h a n g e d by t h e b l o c k a g e . As a r e s u l t t h e c o r r e c t i o n , w h i c h i s made o n l y by r e s i z i n g t h e p r e s s u r e d i s t r i b u t i o n c u r v e w i t h o u t c h a n g i n g i t s s h a p e , i n t r o d u c e s an e r r o r . T h e r e f o r e t h e v a l u e s o f d r a g a n d p r e s s u r e c o e f f i c i e n t i n t h i s c h a p t e r were n o t c o r r e c t e d f o r b l o c k a g e e v e n t h o u g h i t ha s been a common p r a c t i c e i n p r e v i o u s w o r k s . One v a l u e o f b a s e p r e s s u r e i s c o r r e c t e d h e r e by M a s k e l l ' s method i n o r d e r t o compare i t w i t h o t h e r q u o t e d v a l u e s . F o r t h e c i r c u l a r c y l i n d e r a t Re = 6.5x10", t h e c o r r e c t e d b a s e p r e s s u r e c o e f f i c i e n t i s -1.30 a s c o m p a r e d w i t h ESDU's v a l u e (16) o f -1.23 f o r t h i s r a n g e o f R e y n o l d s n u m b e r s . 30 4.2 P r e s s u r e D i s t r i b u t i o n On C i r c u l a r C y l i n d e r W i t h F r o n t  Rod The p r e s s u r e d i s t r i b u t i o n on t h e c y l i n d e r was m e a s u r e d when a r o d was l o c a t e d u p s t r e a m a t v a r i o u s s p a c i n g s . T h r e e s i z e s o f f r o n t r o d were u s e d , d/D = 0.17, d/D = 0.33 a n d d/D = 0.50. The e x p e r i m e n t s were done a t t h r e e d i f f e r e n t R e y n o l d s numbers, 1.0x10", 3.3x10" a n d 6.5x10". The n i n e r e s u l t i n g g r a p h s a r e p r e s e n t e d i n F i g s . 2 7 t o 35. F o r c l a r i t y , e a c h g r a p h i s p r e s e n t e d i n two p a r t s , ( a ) and ( b ) . The two f l o w r e g i m e s , A and B d e s c r i b e d i n t h e p r e v i o u s c h a p t e r , s t i l l o c c u r r e d f o r t h e c i r c u l a r c y l i n d e r . F o r e a c h g r a p h , ( a ) r e p r e s e n t s r e g i m e A w h i l e (b) r e p r e s e n t s r e g i m e B. The n i n e g r a p h s h a v e t h e same g e n e r a l c h a r a c t e r i s t i c s w h i c h w i l l be d i s c u s s e d i n t h e f o l l o w i n g s e c t i o n s . I n most c a s e s , t h e c h a n g e s i n p r e s s u r e d i s t r i b u t i o n a r e p r o g r e s s i v e w i t h r o d s p a c i n g e x c e p t f o r t h e l o w R e y n o l d s number, 1.0x10", where t h e i n a c c u r a c y o f t h e p r e s s u r e m e a s u r e m e n t s makes t h e r e s u l t s show some s c a t t e r . 4.3 F l o w Regime A The e f f e c t o f t h e f r o n t r o d on t h e c y l i n d e r f r o n t p r e s s u r e d i s t r i b u t i o n i s s i m i l a r t o t h a t d e s c r i b e d f o r t h e f l a t p l a t e . The f r o n t s t a g n a t i o n p r e s s u r e c o e f f i c i e n t i s l e s s t h a n one a n d i s d e c r e a s i n g w i t h d e c r e a s i n g r o d s p a c i n g s . T h i s i s due t o t h e v e l o c i t y d e f i c i t o f t h e f r o n t r o d wake a n d i t w o u l d be p o s s i b l e t o p r e s e n t a s i m i l a r g r a p h o f UQ / U, v e r s u s L / d a s f o r t h e f l a t p l a t e . 31 U n l i k e t h e f l a t p l a t e , t h e c i r c u l a r c y l i n d e r b a s e p r e s s u r e i s a f f e c t e d by t h e p o s i t i o n o f t h e r o d . The r o d c r e a t e s t u r b u l e n c e i n t h e s t r e a m r e a c h i n g t h e c y l i n d e r . As m e n t i o n e d i n s e c t i o n 4.1, t u r b u l e n c e makes t h e s e p a r a t i o n p o i n t s o c c u r f u r t h e r d o w n s t r e a m on t h e s u r f a c e o f t h e c y l i n d e r , p e r m i t t i n g a more c o m p l e t e p r e s s u r e r e c o v e r y . The p r o x i m i t y o f t h e r o d i n c r e a s e s t h e t u r b u l e n c e l e v e l a n d t h e r e f o r e t h e b a s e p r e s s u r e . F o r e x a m p l e , a t Re = 3.3x10" when t h e r o d i s a b s e n t ( F i g . 2 6 ) , t h e b a s e p r e s s u r e , 180° f r o m t h e f r o n t s t a g n a t i o n p o i n t , i s Cpb = - 1 . 3 2 . By b r i n g i n g t h e r o d d/D = 0.17 c l o s e r t o t h e c y l i n d e r , t h e b a s e p r e s s u r e i n c r e a s e s ( F i g . 2 8 ( a ) ) . A t a c e r t a i n s p a c i n g , t h e b a s e p r e s s u r e becomes a l m o s t c o n s t a n t w i t h d e c r e a s i n g s p a c i n g a n d r e m a i n s c o n s t a n t a t Cpb = -0.83 u n t i l t h e c h a n g e i n f l o w r e g i m e . The s e p a r a t i o n p o i n t s a r e t h e r e f o r e e x p e c t e d t o have been s h i f t e d d o w n s t r e a m by b r i n g i n g t h e r o d c l o s e r . The s o l i d l i n e i n F i g . 2 8 ( a ) r e p r e s e n t s t h e p r e s s u r e d i s t r i b u t i o n f o r c r i t i c a l f l o w a r o u n d a c i r c u l a r c y l i n d e r , t a k e n f r o m E n g i n e e r i n g S c i e n c e D a t a ( 1 6 ) . Good a g r e e m e n t i s f o u n d b e t w e e n t h e sh a p e o f t h e s o l i d c u r v e a n d t h e p r e s e n t d a t a a t s m a l l s p a c i n g s (L/D < 2 . 4 ) , e x c e p t f o r t h e f r o n t s t a g n a t i o n p o i n t . T h i s s u g g e s t s t h a t t h e e f f e c t o f t u r b u l e n c e i s t o make t h e f l o w c r i t i c a l . The t r e n d s o f t h e g r o u p o f c u r v e s a r e , i n r e g i m e A, t o w a r d a l a r g e r b a s e p r e s s u r e v a l u e f o r l a r g e r r o d d i a m e t e r s a n d l a r g e r R e y n o l d s n u m b ers. 32 4.4 F l o w Regime B A d i f f e r e n t f l o w r e g i m e o c c u r s b e c a u s e o f t h e s t r o n g i n t e r a c t i o n b e t w e e n t h e r o d and t h e c y l i n d e r a t s m a l l e r s p a c i n g s . A g a i n , l i k e t h e f l a t p l a t e , two s y m m e t r i c a l s t a g n a t i o n p o i n t s a r e f o r m e d a s w e l l a s a c l o s e d r e g i o n of r o u g h l y c o n s t a n t p r e s s u r e . The m a i n d i f f e r e n c e w i t h t h e f l a t p l a t e r e s u l t s i s t h a t t h e b a s e p r e s s u r e i s i n f l u e n c e d by t h e r o d p o s i t i o n . I t seems t h a t t h e s e p a r a t i o n o c c u r s e a r l i e r when t h e r o d g e t s c l o s e r i n r e g i m e B b e c a u s e t h e b a s e p r e s s u r e d e c r e a s e s w i t h d e c r e a s i n g s p a c i n g . The t r e n d s o f t h i s g r o u p o f c u r v e s a r e s i m i l a r t o t h o s e o f r e g i m e A, t o w a r d a l a r g e r b a s e p r e s s u r e f o r l a r g e r r o d d i a m e t e r s a n d l a r g e r R e y n o l d s n u m b ers. 4.5 D r a g O n l y t h e t o t a l d r a g o f t h e two p a r t s o f the- m o d e l was l o o k e d a t i n t h e c a s e o f t h e c i r c u l a r c y l i n d e r , s i n c e i t was known f r o m t h e f l a t p l a t e r e s u l t s ( s e c t i o n 3.5) t h a t t h e d r a g o f t h e r o d i t s e l f was n o t i m p o r t a n t . I t s v a l u e was a l m o s t c o n s t a n t i n e a c h f l o w r e g i m e a n d a l w a y s s m a l l e r t h a n t h e d r a g o f t h e same r o d i n a u n i f o r m f l o w . T h e r e f o r e , o n l y t h e b a l a n c e m e a s u r e m e n t s were p l o t t e d . F i g . 3 6 shows t h e o v e r a l l d r a g a t Re = 3.3x10" and 6.5x10" f o r t h e f r o n t r o d d/D = 0.17. F i g s . 3 7 a n d 38 shows s i m i l a r r e s u l t s f o r d/D = 0.33 a n d d/D = 0.50 r e s p e c t i v e l y . The t h i r d R e y n o l d s number, 1.0x10", i s n o t r e p r e s e n t e d h e r e b e c a u s e b a l a n c e r e a d i n g s were v e r y i n a c c u r a t e a s t h e f o r c e was v e r y s m a l l . 33 The s h a p e o f e a c h p l o t i s s i m i l a r t o t h e f l a t p l a t e r e s u l t s . I n r e g i m e A, t h e o v e r a l l d r a g d e c r e a s e s w i t h d e c r e a s i n g s p a c i n g . Then t h e r e i s a s u dden jump i n t h e c u r v e a t t h e c r i t i c a l s p a c i n g c o r r e s p o n d i n g t o a c h a n g e i n f l o w p a t t e r n . The minimum d r a g i s o b t a i n e d a t t h a t c r i t i c a l s p a c i n g . T h e n , i f t h e r o d i s b r o u g h t c l o s e r i n r e g i m e B, t h e d r a g i n c r e a s e s a g a i n . The m a i n d i f f e r e n c e b e t w e e n t h e f l a t p l a t e a n d t h e c i r c u l a r c y l i n d e r r e s u l t s i s t h a t t h e r e i s a l r e a d y an i m p o r t a n t d r a g r e d u c t i o n f o r t h e c i r c u l a r c y l i n d e r when t h e r o d i s q u i t e d i s t a n t , s a y L/D = 7, w h e r e a s t h e r e was no s i g n i f i c a n t d r a g r e d u c t i o n a t t h a t s p a c i n g f o r t h e f l a t p l a t e . The d r a g r e d u c t i o n on t h e c y l i n d e r i n r e g i m e A i s due t o t h e t u r b u l e n c e c r e a t e d by t h e r o d ; t h i s makes t h e f l o w e n t e r t h e c r i t i c a l r e g i o n , d e c r e a s i n g i t s d r a g . An e x t r a r e d u c t i o n i s o b t a i n e d a t t h e c r i t i c a l s p a c i n g when t h e f l o w s w i t c h e s t o r e g i m e B. T h i s e x t r a d r a g r e d u c t i o n i s due t o a c h a n g e i n f r o n t p r e s s u r e o n l y , t h e b a s e p r e s s u r e b e i n g e s s e n t i a l l y t h e same a s t h a t j u s t b e f o r e t h e c r i t i c a l s p a c i n g . F o r e x a m p l e , i n t h e c a s e o f d/D = 0.33 a t Re = 6.5x10" ( F i g . 3 7 ) , t h e d r a g r e d u c t i o n j u s t b e f o r e t h e c r i t i c a l s p a c i n g ( r e g i m e A) i s a l r e a d y 34% (CD = 0.78) a n d a t t h e c r i t i c a l s p a c i n g i t i s 58% (CD = 0 . 5 0 ) . The d i f f e r e n c e , 24%, i s m a i n l y due t o a c h a n g e i n t h e c y l i n d e r f r o n t p r e s s u r e . 34 4.6 E f f e c t Of R e y n o l d s Number I n t h e r a n g e o f R e y n o l d s numbers s t u d i e d , t h e m a i n e f f e c t o f i n c r e a s i n g Re was t o l o w e r t h e o v e r a l l d r a g , s h i f t i n g t h e c u r v e o f d r a g v e r s u s s p a c i n g downwards ( F i g s . 3 6 t o 3 8 ) . T h i s r e s u l t c a n 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 y n o l d s number a r o u n d t h e c y l i n d e r i s i n c r e a s e d , b r i n g i n g t h e f l o w c l o s e r t o t h e c r i t i c a l r a n g e , i m p r o v i n g t h e b a s e p r e s s u r e r e c o v e r y a n d r e d u c i n g t h e d r a g c o e f f i c i e n t . A n o t h e r p o s s i b l e e x p l a n a t i o n i s an i n c r e a s e i n t u r b u l e n c e r e a c h i n g t h e c y l i n d e r due t o a h i g h e r R e y n o l d s number a r o u n d t h e f r o n t r o d . 4.7 Optimum C o n f i g u r a t i o n I t h a s been shown t h a t a t any R e y n o l d s number t h e r e i s a r o d p o s i t i o n g i v i n g a minimum d r a g , t h i s p o s i t i o n c o r r e s p o n d i n g t o t h e c r i t i c a l s p a c i n g . The i m p o r t a n t p a r a m e t e r s g o v e r n i n g t h e optimum c o n f i g u r a t i o n a r e : (1) - s i z e o f r o d d/D (2) - s p a c i n g L/D (3) - R e y n o l d s number Re F i g . 3 9 shows t h e e f f e c t o f r o d s i z e on t h e minimum d r a g f o r two R e y n o l d s n u m b e r s . A t Re = 6.5x10", a d r a g r e d u c t i o n o f 5 8 % i s a c h i e v e d w i t h a r e l a t i v e l y s m a l l r o d , d/D = 0.33. A t l o w e r R e y n o l d s n u m b e r s , t h e d r a g r e d u c t i o n i s l e s s a n d t h e s i z e o f t h e r o d must be l a r g e r f o r maximum d r a g r e d u c t i o n . A d r a g r e d u c t i o n o f 44% i s a c h i e v e d a t Re = 3.3x10" w i t h t h e r o d d/D = 0.50 b u t a l o w e r d r a g c o u l d p r o b a b l y be a c h i e v e d w i t h an e v e n l a r g e r r o d . 35 F i g . 4 0 shows t h e c r i t i c a l s p a c i n g f o r d i f f e r e n t r o d s i z e s a n d R e y n o l d s numbers. The c r i t i c a l s p a c i n g i s a l w a y s i n t h e r a n g e 1.2 < L/D < 2.4. I t was assumed t h a t t h e p o s i t i o n o f t h e c r i t i c a l s p a c i n g was b e t w e e n t h e l a s t v a l u e i n r e g i m e A and t h e f i r s t v a l u e i n r e g i m e B. The r e s u l t s show some s c a t t e r b u t t h e t r e n d i s t o w a r d a l a r g e r c r i t i c a l s p a c i n g f o r l a r g e r r o d d i a m e t e r s a n d f o r l a r g e r R e y n o l d s n u m b e r s . The R e y n o l d s number had a s t r o n g e f f e c t on t h e minimum d r a g ( F i g . 3 7 t o 3 9 ) . U n f o r t u n a t e l y , f o r many p r a c t i c a l a p p l i c a t i o n s t h e R e y n o l d s number w i l l be h i g h e r t h a n t h e maximum v a l u e i n v e s t i g a t e d h e r e ( t h e maximum v a l u e o f t h e e x p e r i m e n t , 6.5x10", was f i x e d by l i m i t s o f t h e i n s t r u m e n t a t i o n ) . F o r h i g h e r R e y n o l d s numbers i t i s e x p e c t e d t h a t t h e minimum d r a g c o e f f i c i e n t w i l l be e v e n l o w e r . On t h e o t h e r h a n d , t h e r e f e r e n c e d r a g c o e f f i c i e n t ( c y l i n d e r i n u n i f o r m f l o w ) w i l l be l o w e r b e c a u s e t h e f l o w may be i n t h e c r i t i c a l r a n g e . T h e r e f o r e t h e d r a g r e d u c t i o n c o u l d be c o n s e q u e n t l y l o w e r t h a n f o r t h e r a n g e o f R e y n o l d s numbers s t u d i e d h e r e . No c h a n g e i n t r e n d i s a n t i c i p a t e d a t h i g h R e y n o l d s number h o w e v e r . 4.8 E f f e c t Of Yaw The e f f e c t o f s m a l l yaw on t h e d r a g was l o o k e d a t b u t t h e r e s u l t s were v e r y i n a c c u r a t e b e c a u s e t h e m o d el was n o t d e s i g n e d f o r t h a t p u r p o s e . From t h e r e s u l t s g a t h e r e d , t h e t r e n d o f t h e d r a g w i t h yaw seems t o be s i m i l a r t o t h a t o f 36 t h e f l a t p l a t e . 4.9 F l u c t u a t i n g S i d e F o r c e The f l u c t u a t i n g s i d e f o r c e due t o v o r t e x s h e d d i n g on a c i r c u l a r c y l i n d e r a l o n e h a s i t s f r e q u e n c y a n d i n t e n s i t y c h a n g i n g w i t h R e y n o l d s number. However i n t h e r a n g e o f R e y n o l d s numbers i n v e s t i g a t e d t h e n o n - d i m e n s i o n a l f r e q u e n c y , t h e S t r o u h a l number S = ( f D ) / U 1 , i s c o n s t a n t . T h i s was v e r i f i e d e x p e r i m e n t a l l y a nd t h e r e s u l t s a r e p r e s e n t e d i n F i g . 4 1 ( a ) , g i v i n g S = 0.195. T h i s r e s u l t w e l l a g r e e s w i t h t h e v a l u e o f S = 0.20 g e n e r a l l y a c c e p t e d . The i n t e n s i t y C l ' , g i v e n by t h e RMS v a l u e o f t h e f l u c t u a t i n g f o r c e , was m e a s u r e d by t h e method d e s c r i b e d i n C h a p t e r I I . The r e s u l t s ( F i g 4 1 ( b ) ) a r e shown i n two s e t s o f d a t a p o i n t s : one i s c o r r e c t e d f o r f r e q u e n c y d i s t o r t i o n by t h e t r a n s f e r f u n c t i o n ( F i g . 1 0 ) , t h e s e c o n d i s l e f t u n c o r r e c t e d . O t h e r c u r v e s by K e e f e (18) a n d G e r r a r d ( 1 9 ) a r e a l s o shown. None o f t h e r e s u l t s seem t o a g r e e , p r o b a b l y b e c a u s e o f d i f f e r e n t t u r b u l e n c e l e v e l s . I t h a s been shown by G e r r a r d (20) t h a t i n t h i s r a n g e o f R e y n o l d s numbers t h e o s c i l l a t i n g s i d e f o r c e i s r e m a r k a b l y s e n s i t i v e t o t u r b u l e n c e l e v e l . 4.10 S i d e F o r c e On M o d e l W i t h F r o n t Rod The f l u c t u a t i n g s i d e f o r c e was i n v e s t i g a t e d t o make s u r e t h a t no p r o b l e m was c r e a t e d by t h e f r o n t r o d . F i g . 4 2 shows t h e f l u c t u a t i n g s i d e f o r c e on t h e c y l i n d e r w i t h t h e r o d d/D = 0.17 a t d i f f e r e n t s p a c i n g s . S i m i l a r r e s u l t s f o r 37 d/D = 0.33 and d/D = 0.50 a r e shown i n F i g s . 4 3 a n d 44 r e s p e c t i v e l y . The m e a s u r e d v a l u e s o f C l ' were n o t a s a c c u r a t e a s t h e o t h e r r e s u l t s b e c a u s e o f t h e method o f m e a s u r e m e n t . T h e r e f o r e t h e s e v a l u e s s h o u l d n o t be c o n s i d e r e d a s d e f i n i t i v e . F o r t h e s m a l l r o d (d/D = 0.17) t h e r e s u l t a t Re = 1.0x10" shows c o n s i d e r a b l e s c a t t e r a n d i s d i f f i c u l t t o i n t e r p r e t . The r e a s o n i s p r o b a b l y a s a m p l i n g t i m e t o o s m a l l (10 s e c . ) f o r t h e l o w f r e q u e n c y i n v o l v e d ( ~ 2 0 H z ) . T h i s r e s u l t i s t h e r e f o r e d i s c a r d e d a s w e l l a s r e s u l t s a t t h e same Re f o r t h e o t h e r r o d d i a m e t e r s . The b e h a v i o r s o f t h e s i x r e m a i n i n g c u r v e s ( F i g s . 4 2 t o 44) a r e d i f f e r e n t b u t t h e r e a r e some common c h a r a c t e r i s t i c s t o a l l o f them. I n a l l c a s e s , t h e r e a r e two f l o w r e g i m e s o b s e r v e d , c o r r e s p o n d i n g t o r e g i m e s A a n d B d e s c r i b e d p r e v i o u s l y . Regime A c o r r e s p o n d s t o s p a c i n g s where t h e s i d e f o r c e i s a l m o s t c o n s t a n t . Regime B c o r r e s p o n d s t o s p a c i n g s where t h e s i d e f o r c e v a r i e s r a p i d l y . 4.10.1 F r o n t Rod d/D = 0.17 The two s e t s o f d a t a p o i n t s o f i n t e r e s t i n F i g . 4 2 (Re = 3.3x10" a n d 6.5x10") have t h e same sh a p e w i t h a minimum s i d e f o r c e n e a r t h e c r i t i c a l s p a c i n g . The minimum v a l u e o f C l ' was a l w a y s s m a l l e r t h a n t h e r e f e r e n c e a t t h e same Re ( t h e r e f e r e n c e i s C l ' f o r t h e c y l i n d e r a l o n e ) ; t h e maximum r e d u c t i o n o f C l ' i s 67% f o r Re = 6.5x10" a n d 25% f o r Re = 3.3x10". Any s p a c i n g i n r e g i m e A gave a s i d e f o r c e r e d u c t i o n . I n r e g i m e B, where t h e s i d e f o r c e v a r i e d 38 r a p i d l y , t h e maximum s i d e f o r c e was n o t s i g n i f i c a n t l y l a r g e r t h a n t h e r e f e r e n c e . F i g . 4 5 shows t h e e f f e c t o f r o d s p a c i n g on S t r o u h a l number f o r t h e r o d d/D = 0.17; t h e S t r o u h a l number i n c r e a s e s w i t h R e y n o l d s number. 4.10.2 F r o n t Rod d/D = 0.33 F i g . 4 6 shows t h e e f f e c t o f r o d s p a c i n g on S t r o u h a l number f o r Re = 3.3x10". T h e r e was n o t a w e l l d e f i n e d d o m i n a n t f r e q u e n c y f o r Re = 6.5x10". F o r e a c h c o n f i g u r a t i o n t e s t e d a t Re = 3.3x10", t h e s p e c t r a l d e n s i t y o f t h e f l u c t u a t i o n showed two f r e q u e n c y p e a k s c o r r e s p o n d i n g t o t h e S t r o u h a l numbers o f F i g . 4 6 . The t o t a l a r e a u n d e r t h e s p e c t r a l d e n s i t y c u r v e b e i n g p r o p o r t i o n a l t o ( C l ' ) 2 , i t was assumed t h a t t h e h e i g h t o f e a c h f r e q u e n c y p eak was an e s t i m a t e o f t h e c o n t r i b u t i o n o f t h a t f r e q u e n c y t o ( C l ' ) 2 . The c o n t r i b u t i o n s t o ( C l ' ) 2 i n p e r c e n t a g e a r e shown i n F i g . 4 6 . The two f r e q u e n c i e s were assumed t o be t h e v o r t e x s h e d d i n g f r e q u e n c i e s o f b o t h b o d i e s , t h e c i r c u l a r c y l i n d e r p r o d u c i n g t h e low f r e q u e n c y a n d t h e f r o n t r o d p r o d u c i n g t h e h i g h f r e q u e n c y . The v a l u e s o f C l ' were c o r r e c t e d f o r f r e q u e n c y d i s t o r t i o n u s i n g t h e t r a n s f e r f u n c t i o n ( F i g . 1 0 ) a nd c o n s i d e r i n g t h e c o n t r i b u t i o n o f e a c h o f t h e two f r e q u e n c i e s t o C l ' . The r e s u l t i s p r e s e n t e d i n F i g . 4 3 f o r Re = 3.3x10". F o r Re = 6.5x10" i t was n o t p o s s i b l e t o c o r r e c t t h e d a t a b e c a u s e no d o m i n a n t f r e q u e n c i e s were n o t i c e a b l e ; t h e s e 39 r e s u l t s a r e p l o t t e d a s i f t h e f r e q u e n c y r e s p o n s e was f l a t . The t r e n d s o f t h e two c u r v e s ( F i g . 4 3 ) d i f f e r s i g n i f i c a n t l y . T h i s may be due t o one c u r v e n o t b e i n g c o r r e c t e d . A n o t h e r p o s s i b l e c a u s e o f t h e d i f f e r e n c e i s t h e p r e s e n c e o f two d o m i n a n t f r e q u e n c i e s i n one c a s e . N e v e r t h e l e s s , t h e s i d e f o r c e i s r e l a t i v e l y l o w , n o t s i g n i f i c a n t l y h i g h e r t h a n t h e r e f e r e n c e . 4.10.3 F r o n t Rod d/D = 0.50 F i g . 4 7 shows t h e e f f e c t o f r o d s p a c i n g on S t r o u h a l number a t Re = 3.3x10". Two d o m i n a n t f r e q u e n c i e s were n o t i c e d a g a i n , b u t t h i s t i m e t h e i n t e n s i t y due t o t h e h i g h f r e q u e n c y ( f r o n t r o d ) was d o m i n a n t a s shown by t h e p e r c e n t a g e v a l u e s . A g a i n a t Re = 6.5x10" t h e r e was no d o m i n a n t f r e q u e n c y n o t i c e a b l e . The v a l u e s o f C l ' ( F i g . 4 4 ) a t Re = 3.3x10" were c o r r e c t e d f o r f r e q u e n c y d i s t o r t i o n b u t n o t t h e v a l u e s a t Re = 6.5x10". A g a i n , t h e t r e n d s o f t h e two c u r v e s d i f f e r s i g n i f i c a n t l y f o r t h e same r e a s o n s a s m e n t i o n e d i n s e c t i o n 4.10.2. The maximum C l ' , o b t a i n e d h e r e i n r e g i m e A, i s a b o u t 50% more t h a n t h e r e f e r e n c e b u t t h e f r e q u e n c y i s a l s o h i g h e r . A low C l ' i s u s e f u l i n p r a c t i c e b u t a h i g h f r e q u e n c y i s a l s o o f p r a c t i c a l u s e s i n c e i t c a n p r e v e n t s y n c h r o n i s a t i o n o f t h e s h e d d i n g f r e q u e n c y w i t h t h e n a t u r a l f r e q u e n c y o f t h e s t r u c t u r e . 40 4.10.4 G e n e r a l Remarks The s i d e f o r c e r e s u l t s show d i f f e r e n t t r e n d s v a r y i n g w i t h Re a n d r o d s i z e . They a l s o a r e , i n some c a s e s , t h e s u p e r p o s i t i o n o f r e s u l t s h a p p e n i n g a t d i f f e r e n t f r e q u e n c i e s . I t i s t h e r e f o r e d i f f i c u l t t o draw g e n e r a l c o n c l u s i o n s on t h e e f f e c t o f t h e f r o n t r o d on t h e f l u c t u a t i n g s i d e f o r c e . I n a l l c a s e s , t h e s h e d d i n g f r e q u e n c y was i n c r e a s e d f r o m t h a t o f t h e c y l i n d e r a l o n e . An i n c r e a s e i n f r e q u e n c y o r t h e s u p p r e s s i o n o f t h e d o m i n a n t f r e q u e n c y i s s u i t a b l e f o r p r a c t i c a l p u r p o s e s b e c a u s e i t p r e v e n t s t h e s i d e f o r c e f r o m m a t c h i n g t h e n a t u r a l f r e q u e n c y o f t h e s t r u c t u r e . I f t h e d e v i c e i n v e s t i g a t e d h e r e ( f r o n t r o d ) i s u s e d p r i m a r i l y a s a d r a g r e d u c t i o n d e v i c e , i t s h o u l d n o t i n c r e a s e t h e f l u c t u a t i n g s i d e f o r c e i f t h e s p a c i n g i s p r o p e r l y c h o s e n a n d may i n f a c t d e c r e a s e i t somewhat. I f t h e d e v i c e i s t o be u s e d p r i m a r i l y a s a f l u c t u a t i n g s i d e f o r c e r e d u c t i o n d e v i c e , f u r t h e r i n v e s t i g a t i o n s h o u l d be u n d e r t a k e n t o f i n d t h e o p t i m u m c o n f i g u r a t i o n . From t h e p r e s e n t r e s u l t s , i t seems t h a t t h e s m a l l r o d d/D = 0.17 u s e d i n r e g i m e A w o u l d be t h e most e f f i c i e n t . 41 V. CONCLUSIONS 5.1 C o n c l u s i o n s The o b j e c t i v e o f t h i s p r o j e c t was t o i n v e s t i g a t e t h e e f f e c t o f p l a c i n g a s m a l l r o d i n f r o n t o f a b l u f f body f o r f l u i d f o r c e r e d u c t i o n . Two b l u f f b o d i e s were s t u d i e d : a f l a t p l a t e a n d a c i r c u l a r c y l i n d e r . The c o n c l u s i o n s drawn a r e : (1) - E x p e r i m e n t s on t h e two t y p e s o f b l u f f body w i t h d i f f e r e n t s i z e s o f f r o n t r o d a n d d i f f e r e n t R e y n o l d s numbers a l w a y s showed, a t some s p a c i n g , a s i g n i f i c a n t d r a g r e d u c t i o n f r o m t h a t o f t h e s i n g l e b o d y . F o r t h e f l a t p l a t e , a d r a g r e d u c t i o n o f 36% was f o u n d f o r r o d d/D = 0.33 a t s p a c i n g L/D = 1.81 and t h e r e s u l t s were i n d e p e n d e n t o f R e y n o l d s number i n t h e r a n g e s t u d i e d . F o r t h e c i r c u l a r c y l i n d e r a d r a g r e d u c t i o n o f 58% was f o u n d f o r r o d d/D = 0.33 a t s p a c i n g L/D = 1.73 and Re = 6.5x10". (2) - I n a l l c a s e s , two f l o w r e g i m e s were o b s e r v e d , d e p e n d i n g on t h e r o d s p a c i n g . The s w i t c h i n f l o w r e g i m e o c c u r r e d a t a " c r i t i c a l " s p a c i n g , d e p e n d i n g on t h e r o d d i a m e t e r a n d t h e R e y n o l d s number. The minimum d r a g a l w a y s o c c u r r e d a r o u n d t h i s s p a c i n g . Regime A was o b s e r v e d f o r s p a c i n g s l a r g e r t h a n c r i t i c a l w h i l e r e g i m e B was o b s e r v e d f o r s p a c i n g s s m a l l e r t h a n c r i t i c a l . T y p i c a l v a l u e s o f t h e c r i t i c a l s p a c i n g were a r o u n d L/D = 2. (3) - Regime A was o b s e r v e d f o r l a r g e r o d s p a c i n g s . The e f f e c t o f t h e r o d c o n s i s t e d f i r s t , i n r e d u c i n g t h e r e a r body s t a g n a t i o n p r e s s u r e by s l o w i n g down (on a v e r a g e ) t h e 42 f l u i d r e a c h i n g i t a n d s e c o n d l y , i n c r e a t i n g t u r b u l e n c e w h i c h p e r m i t t e d , i n t h e c a s e o f t h e c i r c u l a r c y l i n d e r , a b e t t e r b a s e p r e s s u r e r e c o v e r y . The o v e r a l l e f f e c t o f d e c r e a s i n g r o d s p a c i n g i n t h i s r e g i m e was t o r e d u c e d r a g , e s p e c i a l l y f o r t h e c i r c u l a r c y l i n d e r where t h e d r a g r e d u c t i o n was s i g n i f i c a n t m a i n l y due t o t u r b u l e n c e . (4) - F o r s m a l l e r s p a c i n g s t h a n c r i t i c a l , a d i f f e r e n t f l o w r e g i m e ( r e g i m e B) was o b s e r v e d ; t h e s t r o n g i n t e r a c t i o n b e t w e e n t h e two b o d i e s c r e a t e d a c l o s e d r e g i o n i n t h e gap where t h e p r e s s u r e was e s s e n t i a l l y c o n s t a n t . The s w i t c h made t h e d r a g d r o p s h a r p l y . F o r s m a l l e r s p a c i n g s t h e d r a g i n c r e a s e d back a g a i n . (5) - The d r a g o f t h e f r o n t r o d i t s e l f was a l w a y s s m a l l e r t h a n t h e d r a g o f t h e same r o d i n a u n i f o r m f l o w . (6) - I n t h e c a s e o f t h e f l a t p l a t e , t h e b a s e p r e s s u r e was e s s e n t i a l l y c o n s t a n t f o r a l l r o d s i z e s a n d p o s i t i o n s ( r e g i m e A a n d B ) . (7) - I n t h e c a s e o f t h e c i r c u l a r c y l i n d e r , i n c r e a s i n g R e y n o l d s number r e d u c e d t h e o v e r a l l d r a g due t o an i n c r e a s e i n b a s e p r e s s u r e . (8) - An optimum r o d s i z e was d e t e r m i n e d b u t i t c h a n g e d w i t h Re ( c i r c u l a r c y l i n d e r o n l y ) ; a l a r g e r r o d d i a m e t e r was r e q u i r e d f o r l o w e r Re. (9) - Yaw was f o u n d t o be i m p o r t a n t f o r d r a g r e d u c t i o n . A yaw o f 5° c a n c e l s 50% o f t h e f l a t p l a t e d r a g r e d u c t i o n i n t h e tandem c o n f i g u r a t i o n . ( 1 0 ) - The f l u c t u a t i n g s i d e f o r c e ( c i r c u l a r c y l i n d e r 43 o n l y ) was n o t s i g n i f i c a n t l y i n c r e a s e d by t h e use o f t h e f r o n t r o d a n d was r e d u c e d i n most c a s e s . A s m a l l r o d s u c h a s d/D = 0.17 seemed t o be more e f f e c t i v e f o r C l ' r e d u c t i o n t h a n a l a r g e r r o d s u c h a s d/D = 0.50. The s i d e f o r c e r e d u c t i o n was b e t t e r a t l a r g e r s p a c i n g s ( r e g i m e A ) . The s i d e f o r c e f r e q u e n c y was i n c r e a s e d by p l a c i n g a r o d u p s t r e a m a n d , f o r l a r g e r o d s , t h e f r e q u e n c y p r o d u c e d by t h e r o d i t s e l f ( h i g h f r e q u e n c y ) became v e r y i m p o r t a n t on t h e c y l i n d e r . The optimum c o n f i g u r a t i o n f o r l e a s t s i d e f o r c e ( p o s i t i o n and s i z e o f t h e f r o n t r o d ) was n o t p r e c i s e l y d e t e r m i n e d i n t h i s w o r k . 5.2 A r e a s Of F u r t h e r Work (1) - A more e x t e n s i v e s t u d y o f t h e f l u c t u a t i n g s i d e f o r c e on t h e c i r c u l a r c y l i n d e r w o u l d p r e c i s e l y d e t e r m i n e t h e o p timum c o n f i g u r a t i o n f o r a f l u c t u a t i n g s i d e f o r c e r e d u c t i o n d e v i c e . (2) - Many a p p l i c a t i o n s o f t h i s f l u i d f o r c e r e d u c t i o n d e v i c e a r e a t h i g h e r R e y n o l d s numbers t h a n t h e maximum s t u d i e d h e r e ( 7 . 0 x 1 0 " ) . I t w o u l d t h e r e f o r e be u s e f u l t o c a r r y o u t s i m i l a r e x p e r i m e n t s a t h i g h e r Re, i f t h e e x p e r i m e n t a l a p p a r a t u s p e r m i t s . (3) - S i m i l a r e x p e r i m e n t s s h o u l d be c a r r i e d o u t on d i f f e r e n t s h a p e s o f b l u f f b o d y , s u c h a s a s q u a r e c y l i n d e r . (4) - The e f f e c t o f t u r b u l e n t f l o w i n s t e a d o f smooth f l o w s h o u l d be t e s t e d . (5) - F u r t h e r work c o u l d i n c l u d e t e s t i n g a b l u f f body m o d e l w i t h two r o d s , one a t t h e f r o n t and t h e s e c o n d a t t h e 44 b a c k . A p r e l i m i n a r y s t u d y o f t h i s model showed t h a t a d d i n g a r o d a t t h e back ( i n t h e wake) o f t h e p r e s e n t m o d e l ( w i t h f r o n t r o d ) d i d n o t c h a n g e s u b s t a n t i a l l y t h e o v e r a l l d r a g ; t h e d r a g r e d u c t i o n when t h e two r o d s were l o c a t e d s y m m e t r i c a l l y a t t h e c r i t i c a l s p a c i n g was a b o u t t h e same a s w i t h t h e f r o n t r o d o n l y . I f t h i s d e v i c e i s s u c c e s s f u l , i t w o u l d be a b i d i r e c t i o n a l d e v i c e a n d w o u l d f i n d many p r a c t i c a l a p p l i c a t i o n s . 45 BIBLIOGRAPHY 1. E v e r y , M.J., K i n g , R., Weaver, D.S., " V o r t e x - E x c i t e d V i b r a t i o n s o f C y l i n d e r s a n d C a b l e s a n d T h e i r S u p p r e s s i o n " , Ocean Engng, V o l . 9 , No.2, p p . 1 3 5 - 1 5 7 , 1982. 2. M o r e l , T., Bohn, M., "Fl o w O v e r Two C i r c u l a r D i s k s i n Tandem", J . F l u i d s E n g n g , V o l . 1 0 2 , M a r c h 1980. 3. R o s h k o , A. and K o e n i g , K., " I n t e r a c t i o n E f f e c t s on t h e D r a g o f B l u f f B o d i e s i n Tandem", P r o c e e d i n g s o f t h e Symposium on A e r o d y n a m i c D r a g M e c h a n i s m s , Ed.G. S o v r a n , T. M o r e l a n d W.T. Mason, J r . , P l e n u m P r e s s , New Y o r k , 1978. 4. I g a r a s h i , T . , " C h a r a c t e r i s t i c s o f a F l o w A r o u n d Two C i r c u l a r C y l i n d e r s o f D i f f e r e n t D i a m e t e r s A r r a n g e d i n Tandem", B u l l . J . S . M . E . , V o l . 2 5 , No.201, M a r c h 1982. 5. Z d r a v k o v i c h , M.M. a n d P r i d d e n , D.L., " I n t e r f e r e n c e B e t w een Two C i r c u l a r C y l i n d e r s ; S e r i e s o f U n e x p e c t e d D i s c o n t i n u i t i e s " , J . I n d . A e r o . , 2 ( 1 9 7 7 ) 255-270 6. H i w a d a , M., T a g u s h i , T., M a b u s h i , I . , Kumana, M., " F l u i d F l o w a n d H e a t T r a n s f e r A r o u n d Two C i r c u l a r C y l i n d e r s o f D i f f e r e n t D i a m e t e r s i n C r o s s F l o w " , B u l l . J . S . M . E . , V o l . 2 2 , No.167 ( 1 9 7 9 ) , p . 7 1 5 . 7. S t a n s b y , P.K., "The E f f e c t o f End P l a t e s on t h e B a s e P r e s s u r e C o e f f i c i e n t o f a C i r c u l a r C y l i n d e r " , A e r o . J . , J a n u a r y 1974, pp.36-37 8. L e e , B.E., "The S u s c e p t i b i l i t y o f T e s t s on Two-D i m e n s i o n a l B l u f f B o d i e s t o I n c i d e n t F l o w V a r i a t i o n " , J . I n d . A e r o . , 2 ( 1 9 7 7 ) 133-148 9. S t a t h o p o u l o s , T., " T e c h n i q u e o f P n e u m a t i c a l l y A v e r a g i n g P r e s s u r e s " , R e s e a r c h R e p o r t BLWT-2-1975, F a c u l t y o f E n g i n e e r i n g S c i e n c e , U n i v e r s i t y o f W e s t e r n O n t a r i o , L o n d o n , C a n a d a . 10. M a s k e l l , E.C., "A T h e o r y o f B l o c k a g e E f f e c t s on B l u f f B o d i e s a n d S t a l l e d W i n g s i n a C l o s e d Wind T u n n e l " , A e r o . R e s . C o u n . R&M 3400, 1965 11. F a g e , A. a n d J o h a n s e n , F.C.,."On t h e F l o w o f A i r B e h i n d an I n c l i n e d F l a t P l a t e o f I n f i n i t e S p a n " , P r o c . R o y . S o c . L o n d . , A 116, 170, 1927. 12. P a r k i n s o n , G.V., J a n d a l i , T., "A Wake S o u r c e M o d e l f o r B l u f f Body P o t e n t i a l F l o w " , J . F l u i d Mech. ( 1 9 7 0 ) , V o l . 4 0 , p p . 5 7 7 - 5 9 4 . 46 13. E n g i n e e r i n g S c i e n c e D a t a I t e m Number 71016, J u l y 1973, F i g u r e 19, E n g i n e e r i n g S c i e n c e D a t a U n i t s L i m i t e d , L o n d o n . 14. Townsend, A.A., 1956, The S t r u c t u r e o f T u r b u l e n t S h e a r  F l o w , C a m b r i d g e U n i v e r s i t y P r e s s , L o n d o n . 15. A c h e n b a c h , E., " F l o w P a s t Rough C y l i n d e r s a t H i g h Re", J . F l u i d Mech., V o l . 4 6 , No.2 ( 1 9 7 1 - 3 ) , p . 3 2 1 . 16. E n g i n e e r i n g S c i e n c e D a t a I t e m Number 70 0 1 3 , F i g u r e 12, E n g i n e e r i n g S c i e n c e D a t a L t d , L o n d o n , 1970. 17. West, G.S. and A p e l t , C . J . , "The E f f e c t o f T u n n e l B l o c k a g e and A s p e c t R a t i o on t h e Mean F l o w P a s t a C i r c u l a r C y l i n d e r w i t h R e y n o l d s Numbers b e t w e e n 10" a n d 1 0 5 " , J . F l u i d Mech ( 1 9 8 2 ) , V o l . 1 1 4 , pp.361-377 18. K e e f e , R.T., 1961, "An I n v e s t i g a t i o n o f t h e F l u c t u a t i n g F o r c e s A c t i n g on a S t a t i o n a r y C i r c u l a r C y l i n d e r i n a S u b s o n i c S t r e a m and t h e A s s o c i a t e d Sound F i e l d " , U n i v . T o r o n t o I n s t . A e r o p h y s . Rep. No.76. 19. G e r r a r d , J.H., 1961, "An 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 o f t h e O s c i l l a t i n g L i f t a n d D r a g o f a C i r c u l a r C y l i n d e r S h e d d i n g T u r b u l e n t V o r t i c e s " , J . F l u i d Mech., 11, 244. 20. G e r r a r d , J.H., 1965, "A D i s t u r b a n c e S e n s i t i v e R e y n o l d s Number Range o f t h e F l o w P a s t a C i r c u l a r C y l i n d e r " , J . F l u i d Mech., V o l . 2 2 , p . 1 8 7 . L/D d/D C p ( c e n t r e ) U 0 / U l L/d 1 0 /D 7 .00 0.17 .721 .151 4 1 . 2 .436 2 .32 0.17 .548 .260 13 .6 .251 1.98 0.17 .516 .280 11 .6 .232 1.81 0.17 .540 .290 10 .6 .221 7 .17 0.33 .621 .212 21 . 7 .615 3 .15 0.33 .467 .317 9 .5 .408 2 .78 0.33 .431 .343 8 .4 .383 2.44 0 .33 .401 .367 7 .4 .359 2 .13 0.33 .384 .380 6 .5 .335 T a b l e I - C a l c u l a t e d wake v e l o c i t y d e f i c i t ( U n ) and c h a r a c t e r i s t i c wake w i d t h ( l n ) F i g u r e 2 - Smoke t u n n e l T u r n i n g vanes Figure 3 - Outline of the U.B.C. aeronautical wind tunnel 50 Pressure tapl location Circular Cylinder S.S /- y y y y/ y vs y y y s s s s .?-\ Wind Tunnel Floor Stand End plates y y y y y y y y y ^ W i n d T u n n e l R o o f y y ^y s s s s s s y y y y y r Front Rod y y y y y y y y y y y yy -Movable Support ////A'///////' I Balance F i g u r e 4 - S k e t c h o f a t y p i c a l m o d e l i n s i d e t h e w i n d t u n n e l 51 FRONT ROD (2 sizes of end plates depending on spacing) F i g u r e 5 - End p l a t e s 52 6.35 3.2 r FRONT RODS F i g u r e 6 - C r o s s s e c t i o n o f m o d e l s s h o w i n g p r e s s u r e t a p l o c a t i o n 53 2.4 I 1 i 1 1 1 1 1 1 1 1 r 12.7 mm F i g u r e 8 - S k e t c h o f m a n i f o l d f o r p n e u m a t i c a v e r a g i n g 54 F r e q u e n c y G e n e r a t o r V i b r a t o r V o l t m e t e r -O s c i l l o s c o p e S c a n i v a l v e & T r a n s d u c e r 914 mm l o n g Tyqon t u b i n g Channe l 1 C h a n n e l 52 mm d i a . C y l i n d e r I |o M a n i f o l d 9 - 9 1 4 mm l o n g Tygon t u b i n g F i g u r e 9 - A p p a r a t u s f o r c a l i b r a t i o n o f t h e s i d e f o r c e m e a s u r i n g s y s t e m a g a i n s t f r e q u e n c y d i s t o r t i o n F i g u r e 10 - T r a n s f e r f u n c t i o n f o r t h e s i d e f o r c e m e a s u r i n g s y s t e m . ^The s o l i d l i n e was u s e d t o c o r r e c t d a t a 56 F i g u r e 11 - P r e s s u r e d i s t r i b u t i o n on f l a t p l a t e ( R e = 4 . 0 x l O a ) 57 0.6 0.4 0 .2 0 .0 -•0.2 " •0.4 --0 .6 --0 .8 - 1 . 0 -11 -1 .2 " n 1 r.—i 1 1 1 1 1 1 1 1 1 r i i i r + t x t X • X L/D -CD 7.17 -+ 3 .15 -2.78 -X 2.13 -1.97 — _1 I I I I I J l_ J I I L_ I 1 I I L_ 0 .0 0.2 0.4 0 .6 0 .8 1.0 x/D F i g u r e 12 - P r e s s u r e d i s t r i b u t i o n on f l a t p l a t e w i t h f r o n t r o d d/D = 0.33 (Re=4.Ox 1 0 0 ) ; F l o w r e g i m e A 58 1.0 i 1 1 1 — i — i — i ' — i 1 1 — 1 — 1 — 1 1 1 1 1 ' r 0 .0 X + • X ID X o x • X 8 0.4 L/D 0.6 - e> 1.81 - + 1.47 0.8 - 1.29 - X 1.16 1.0 -a 0.84 l ' 2 l X X + 0 I I t I I I I I I I I I 1 J 1 1 L 0.2 0.4 0 .6 0 .8 1.0 x/D F i g u r e 13 - P r e s s u r e d i s t r i b u t i o n on f l a t p l a t e w i t h f r o n t r o d d/D = 0.33 ( R e = 4 . 0 x 1 0 " ) ; F l o w regime'B 59 1.0 I 1 1 1 1 1 i i 1 1 1 1 1 1 1 1 1 1 1 r 0 . 8 -0 .6 -0 .4 -0 .2 -0 . 0 -•0.2 ' •0.8 CD i J 1 I L 0 . 0 0 .2 * - L/D - CD 7.00 - + 2.32 - 1.98 X 1.81 CD X J I I 1 1 I I 1 I i I I I L 0.4 0 .6 0 .8 1 .0 x/D F i g u r e 14 - P r e s s u r e d i s t r i b u t i o n on f l a t p l a t e w i t h f r o n t r o d d/D = 0.17 ( R e = 4 . 0 x 1 0 * ) ; F l o w r e g i m e A 60 1. 0 i 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 i r 0.8 r 0.6 h 0.4 0.2 0 .0 o X 6 8 9 S cb 8 L/D 1.64 + 1.30 «> 1.13 X 0.97 CD 0 .83 CD + CD J I I I I I I I I 1 1 1 1 L . J I I 1_ 0 .0 0 .2 0.4 0 .6 0 .8 1.0 x/D F i g u r e 15 - P r e s s u r e d i s t r i b u t i o n on f l a t p l a t e w i t h f r o n t r o d d/D = 0.17 ( R e = 4 . 0 x 1 0 " ) ; F l o w r e g i m e B F i g u r e 16 - P r e s s u r e d i s t r i b u t i o n on t h e f r o n t r o d d/D = 0.33 a t two d i f f e r e n t s p a c i n g s (Re=5.Ox 1 0 * ) ; L/D = 3.42 ( r e g i m e A) and L/D = 1.42 ( r e g i m e B) F i g u r e 17 - V e l o c i t y d e f i c i t ( U 0 ) v e r s u s s p a c i n g ( L / d ) f o r f l o w r e g i m e A - 0 . 6 Cp (ba se ) - 0 . 8 - 1 . 0 - 1 . 2 - 1 . 4 - 1 . 6 i r i r ,9 4+ 0 + ° CD ° J I L I I L i 1 r d/D + 0.17 * — CD 0 .33 -P l a t e a l o n e CD l I I I I I L 0 .0 1.0 2.0 3 .0 4 . 0 5 .0 6 .0 7 .0 8 . 0 L/D F i g u r e 18 - Base p r e s s u r e f o r r e g i m e s A a n d B 64 F i g u r e 19 - F l o w v i s u a l i s a t i o n a t Re = 5 X 1 0 3 f o r f r o n t r o d d/D = 0.21; ( a ) L/D = 1.36 ( r e g i m e A ) ; ( b ) L/D = 0.79 ( r e g i m e B) 2.4 CD 2 .0 1.6 1.2 0 .8 0.4 0 .0 i r i 1 1 r i 1 r P l a t e a l o n e + + + ++ / + + 0 CD CD CD CD J L J I L CD P l a t e o n l y + O v e r a l l J L CD J I I L 0 .0 1.0 Z.O 3.0 4 . 0 5.0 6 .0 7 .0 8 . 0 L/D F i g u r e 20 - Drag c o e f f i c i e n t o f f l a t p l a t e w i t h r o d d/D = 0 .33 (Re^ .Ox lO 1 * ) en ui 2.4 CD 2.0 1.6 1.2 0.8 0.4 0 .0 i r i r P l a t e a l o n e i 1 1 1 r © P l a t e o n l y + O v e r a l l + O i i I I I I I 1 L J I I L 0 0 1.0 2.0 3 .0 4 . 0 5.0 6 .0 7 .0 8 . 0 L/D F i g u r e 21 - D r a g c o e f f i c i e n t o f f l a t p l a t e w i t h r o d d/D = 0.17 ( R e = 4 . 0 x l 0 " ) 67 F i g u r e 23 - S e p a r a t e d f l o w p a s t a n o r m a l f l a t p l a t e f r o m wake s o u r c e m o d e l 68 F i g u r e 24 - S t r e a m l i n e s o v e r p o t e n t i a l m o d e l u s i n g c o n d i t i o n f o r s t a t i o n a r y p a i r o f v o r t i c e s ; Cpb =- 1.24 CD S u b c r i t i c a l . Q - i ' i c a l i S u p e r c r i t i c a l ) T r a n s c f i t i c a l Re c r i t Re F i g u r e 25 - T y p i c a l d r a g c o e f f i c i e n t v e r s u s R e y n o l d s number f r o m A c h e n b a c h 69 F i g u r e 26 - P r e s s u r e d i s t r i b u t i o n a r o u n d a c i r c u l a r c y l i n d e r a t v a r i o u s R e y n o l d s n u m b e rs; s o l i d l i n e i s f r o m ESDU f o r s u b c r i t i c a l r a n g e - 1 . 8 - i i i _ i i i i i i t i i i i - 1 . 8 _ i i i i i u _l I I I l _ L_ 60 120 180 60 120 180 F i g u r e 30 - P r e s s u r e d i s t r i b u t i o n a r o u n d a c i r c u l a r c y l i n d e r a t Re = 1.0x10" w i t h f r o n t r o d d/D = 0.33. (a) r e g i m e A; (b) r e g i m e B (a) 1.0 Cp 0.6 -I 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 r L/D e 1.73 + 2.06 « 2.24 X 2.61 EJ 2.94 -1.4 -1.8 I i i J i i 1—i L . 0 60 i i i i i i i_ 120 180 Cp 1.0 0.6 -0.6 -1.0 -1.4 (b) - i 1 1 1 1 1 1 1 1 1 1 1 i 1 1 r -L/D 1.02 -+ 1.10 -1.45 -X 1.60 --1 al i i i ' i i i i i i i i i i — i — i — i — 0 60 120 180 F i g u r e 31 - P r e s s u r e d i s t r i b u t i o n a r o u n d a c i r c u l a r c y l i n d e r a t Re = 3.3x10" w i t h f r o n t r o d d/D = 0.33. ( a ) r e g i m e A; (b) r e g i m e B 1.0 Cp 0.6 (a) - i 1 1 1 1 1 1 1 1 r — i 1 1 i i r L/D 0 1.91 t 2.06 • 2.40 X 2.94 6.74 -1.8 I I I I I 1 1 1 i L 1 1— 60 120 180 Cp 1.0 0.6 •1.4 •1.8 (b) - i 1 1 1 1 1 1 r L/D 0 1.02 -+ 1.10 -» 1.45 -X 1.60 -a 1.73 -• I I I I I I 1—I 1 1 1 1 1-60 120 180 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 a r o u n d a c i r c u l a r c y l i n d e r a t Re = 6.5x10" w i t h f r o n t r o d d/D = 0.33. (a) r e g i m e A; (b) r e g i m e B (a) ~i i 1 1 1 1 1 1 1 1 r L/D 2.26 + 2.56 © 2.74 X 6.56 -I—1 1—I—I 1 I I I I I I 120 180 (b) 1.0 Cp 0.6 0.2 -0.2 -0.6 -1.0 i i 1 1 1 1 1 r -1.4 -1.8 I—i— i L. 0 - i — i — i i _ 60 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 a r o u n d a c i r c u l a r c y l i n d e r a t Re = 1.0x10" w i t h f r o n t r o d d/D = 0.50. (a) r e g i m e A; (b) r e g i m e B (a) 1.0 -i 1 1 1 1 1 1 1 1 1 1 1 i i r Cp 0 .6 A-- 1 . 4 -1 .8 L/D e 1.90 + 2.06 « 2.26 X 2.74 EJ 6.56 I I I I 1 1— J I — _ J — 1 _ 60 120 180 (b) Cp 1.0 C.6 0 .2 -1 .4 •1.8 -I 1 1 1 1 1 1 1 1 1 1 1 1 1 1 r-L/D -o 1.08 -+ 1.23 -o 1.42 -X 1.55 -I l l 1 I I 1 1 1 1 _ L . 60 120 180 F i g u r e 34 - P r e s s u r e d i s t r i b u t i o n a r o u n d a c i r c u l a r c y l i n d e r a t Re = 3.3x10" w i t h f r o n t r o d d/D = 0.50. (a) r e g i m e A; (b) r e g i m e B (a) 1.0 - i 1 1 1 1 1 1 1 1 1 1 1 r Cp 0.6 - 1 . 8 L/D 1.72 + 2.06 o 2.26 X 2.74 B 6.56 _J I L. ' I I I I I ,J I l _ 60 120 180 (b) 1.0 Cp 0.6L 0.2 - 1 . 4 i 1 1 r n 1 1 1 1 1 1 1 1 1 1 r L/D e 0.63 + 1.08 o 1.23 X 1.42 El 1.55 1.81—i L. I I I I 1_ 60 120 180 Figure 35 - Pressure d i s t r i b u t i o n around a c i r c u l a r cylinder at Re = 6.5x10" with front rod d/D = 0.50. (a) regime A; (b) regime B 1.0 0 .8 0 .6 1 1 1 r — r - t i i i i i i i i i i i i i i i i i i i i i C y l i n d e r a l o n e + \ . O ^ CD \ ' 0 O © ° ~~~ i i i i I I Re + 3 . 3 x l 0 4 0 6 . 5 x l 0 4 1 i i i 0 i i i 'O.O 1.0 2.0 3.0 4 . 0 5 .0 6 .0 7 .0 L/D F i g u r e 37 - O v e r a l l d r a g c o e f f i c i e n t f o r c y l i n d e r w i t h f r o n t r o d d/D = 0 . 3 3 F i g u r e 38 - O v e r a l l d r a g c o e f f i c i e n t f o r c y l i n d e r w i t h f r o n t r o d d/D = 0 . 5 0 0.6 d/D 0.5 0.2 0.1 1 1 1 1 i 1 1 1 1 1 1 1 1 1 Re - <W—«vf ©—® — - © 6 . 5 x l 0 4 -- + 3 . 3 x l 0 4 -— l . O x l O 4 — — • CD O -—m\—o -1 1 I l l 1 1 1 1 1 1 i i .0 0 .4 0 .8 1.2 1.6 2 .0 2.4 2 .8 L / D c r i t . F i g u r e 40 - C r i t i c a l s p a c i n g r anges f o r d i f f e r e n t r od s i z e s and Reyno l d s numbers ( c i r c u l a r c y l i n d e r ) 84 0.3 "i i i i i i i i i i i i i r 0.2 r- <3 ® 0 © B C D © O © ID S ffi B ffl O.i h 0 .0 J I I 1 I I i i ' I I i I i i 0 1 6 7 8 R e ( x l 0 " 4 ) (a ) F i g u r e 41 - F l u c t u a t i n g s i d e f o r c e on a c i r c u l a r c y l i n d e r ; ( a ) S t r o u h a l number v e r s u s R e y n o l d s number; (b) i n t e n s i t y C l 1 v e r s u s R e y n o l d s number 85 F i g u r e 42 - F l u c t u a t i n g s i d e f o r c e on a c i r c u l a r c y l i n d e r w i t h r o d d/D = 0.17; C l ' r e f > i s f r o m F i g . 4 1 ( b ) 86 i r i r i i r 0.6 c i ' 0.4 0.2 0.0. 0 o , ^ 0 Re C 1 ' r e f . -3.3xl04 0.36 — + 6.5xl04 1.36 -0 + -J L J L i I I 6. 7. L/D Figure 43 - Fluctuating side force on a c i r c u l a r cylinder with rod d/D = 0.33 Cl 0.6 0.4 0.2 0.0' i 1 r J L 0. 1. J I I 1 2. O H Re Cl ' f 3.3xl04 0.36 + 6.5xl04 1.36 1 1 1 1 1 1 1 1 3. 4. 5. 6. L/D Figure 44 - Fluctuating side force on a c i r c u l a r cylinder with rod d/D = 0.50 87 0.3 0.2 0.1 0.0 1 1 1 1 I - 4 + 4 + 4 4 + . 4 1 I I I 1 ' o 4 C5 --Re -- + 6.5xl04 -- CD 3.3xi04 -1 1 l l i 1 i i i i I 0. 1. 6. 7. L/D Figure 45 - Strouhal number on a c i r c u l a r cylinder with front rod d/D = 0.17 0.6 1 • l I 1 I —1 r — i 1 1 1 • 1 1 0 ^ 22% 34% cK25% 38% ; --/ / / / / -® 0 / / o 0 X --0 1 J 1 J I 1 i i i i i i i 0.4 0.2 0.0 0. 3. 6. 7. L/D Figure 46 - Strouhal number on a c i r c u l a r cylinder'with front rod d/D = 0.33 (Re = 3 . 3 x 1 0 " ) ; percentages shown represent percentage of ( C l 1 ) 2 due to that frequency 88 i i r "i 1 1 r 0.4 • a 69% o 79% '80% / 0.2 CD (D CD o — o — © a CD 56% J 0.0 J L J 1 I I 0. 1 3. 4. 5. 6. 7. L/D F i g u r e 47 - S t r o u h a l number on a c i r c u l a r c y l i n d e r w i t h f r o n t r o d d/D = 0.50 (Re = 3 . 3 x 1 0 " ) ; p e r c e n t a g e s shown r e p r e s e n t p e r c e n t a g e o f ( C l ' ) 2 due t o t h a t f r e q u e n c y 89 APPENDIX A ~ PRESSURE TAP LOCATION FOR SIDE FORCE MEASUREMENT C o n s i d e r a c i r c u l a r c y l i n d e r s u b m i t t e d t o a p r e s s u r e d i s t r i b u t i o n : Now d i v i d e t h e Y a x i s i n t o e i g h t e l e m e n t s a n d c o n s i d e r e a c h . e l e m e n t a s b e i n g s u b j e c t e d t o a u n i f o r m p r e s s u r e e q u a l t o t h e a c t u a l p r e s s u r e a t t h e c e n t r e o f t h e e l e m e n t . The u n i f o r m l y d i s t r i b u t e d p r e s s u r e c a n be r e p l a c e d by a p o i n t f o r c e F: F = pAA = p A y / s i n 8 We a r e i n t e r e s t e d o n l y i n t h e s i d e f o r c e , o r t h e Y component o f F: Fy = F s i n e = pAy where Fy i s t h e s i d e f o r c e on t h e e l e m e n t c o n s i d e r e d . The t o t a l s i d e f o r c e on t h e . c y l i n d e r i s t h e sum o v e r a l l e l e m e n t s : F y T = F y x + FY 2 + """ + f Y q = p 1 A y 1 + P 2 A y 2 + --- + P 8 A y g I f t h e e l e m e n t s h a v e t h e same l e n g t h , t h e n : F y T = ^ + p 2 + - — + P 8 > A y = ( P l + p ? + — + r ^ ) x 8Ay = ( a v e r a g e p r e s s u r e ) x ( d i a m e t e r ) 90 I f t h e e l e m e n t s a r e o f e q u a l l e n g t h , o n l y t h e a v e r a g e p r e s s u r e h a s t o be m e a s u r e d t o o b t a i n t h e s i d e f o r c e . F o r e l e m e n t s o f t h e same l e n g t h , t h i s i s t h e l o c a t i o n o f p r e s s u r e t a p s : C l ' i s c a l c u l a t e d i n t h e f o l l o w i n g way: C l ' = 2 x m a n i f o l d p r e s s u r e ( R M S ) x D 0.5PU, 2 x D The f a c t o r 2 i s t h e r e t o i n c l u d e t h e s i d e f o r c e f r o m t h e o t h e r s i d e o f t h e c y l i n d e r . By d o i n g t h a t i t i s assumed t h a t t h e f l u c t u a t i n g s i d e f o r c e i s o u t o f p h a s e by 180° f r o m one s i d e t o t h e o t h e r . T h i s a s s u m p t i o n was n o t v e r i f i e d h e r e h o w e v e r , b u t , i f a n y t h i n g , C l ' w i l l be o v e r e s t i m a t e d . 

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