UBC Theses and Dissertations

UBC Theses Logo

UBC Theses and Dissertations

On the unsteady aerodynamics of stationary elliptic cylinders during organised wake condition Dikshit, Ajai Kumar 1970

Your browser doesn't seem to have a PDF viewer, please download the PDF to view this item.

Notice for Google Chrome users:
If you are having trouble viewing or searching the PDF with Google Chrome, please download it here instead.

Item Metadata

Download

Media
831-UBC_1970_A7 D53.pdf [ 16.03MB ]
Metadata
JSON: 831-1.0102215.json
JSON-LD: 831-1.0102215-ld.json
RDF/XML (Pretty): 831-1.0102215-rdf.xml
RDF/JSON: 831-1.0102215-rdf.json
Turtle: 831-1.0102215-turtle.txt
N-Triples: 831-1.0102215-rdf-ntriples.txt
Original Record: 831-1.0102215-source.json
Full Text
831-1.0102215-fulltext.txt
Citation
831-1.0102215.ris

Full Text

ON THE UNSTEADY A E R O D Y N A M I C S OF S T A T I O N A R Y E L L I P T I C C Y L I N D E R S DURING O R G A N I S E D WAKE C O N D I T I O N by A O A I KUMAR D I K S H I T B . S c , G o r a k h p u r U n i v e r s i t y , U .P., I n d i a B . T e c h . , I n d i a n I n s t i t u t e o f T e c h n o l o g y , M a d r a s , I n d i a A T H E S I S S U B M I T T E D IN P A R T I A L F U L F I L L M E N T OF TH E R E Q U I R E M E N T S FOR T H E DE G R E E OF M . A . S c . i n t h e D e p a r t m e n t o f M e c h a n i c a l E n g i n e e r i n g We a c c e p t t h i s t h e s i s a s c o n f o r m i n g t o t h e r e q u i r e d s t a n d a r d T H E U N I V E R S I T Y OF B R I T I S H C OLUMBIA J u l y , 1 9 7 0 In p r e s e n t i n g t h i s t h e s i s in p a r t i a l f u l f i l l m e n t of the requirements f o r an advanced degree at the U n i v e r s i t y of B r i t i s h Columbia, I agree that the L i b r a r y s h a l l make i t f r e e l y a v a i l a b l e f o r r e f e r e n c e and study. I f u r t h e r agree that permission f o r e x t e n s i v e copying of t h i s t h e s i s f o r s c h o l a r l y purposes may be granted by the Head of my Department or by h i s r e p r e s e n t a t i v e s . I t i s understood that copying or p u b l i c a t i o n of t h i s t h e s i s f o r f i n a n c i a l gain s h a l l not be allowed without my w r i t t e n p e r m i s s i o n . A.K. D i k s h i t Department of Mechanical E n g i n e e r i n g The U n i v e r s i t y of B r i t i s h Columbia Vancouver 8, B.C., Canada Date /rf 7^ ABSTRACT The a e r o d y n a m i c s o f a s e t o f t w o - d i m e n s i o n a l , s t a t i o n a r y e l l i p t i c c y l i n d e r s w i t h e c c e n t r i c i t y o f 0.44, 0.92 and 0.98 i s s t u d i e d e x p e r i m e n t a l l y d u r i n g t h e o r g a n i s e d wake c o n d i t i o n ( R = 2 x l 0 ^ - 1 0 5 ) . T h e r e s u l t s i n d i c a t e t h e e f f e c t o f e c c e n t r i c i t y and a n g l e o f a t t a c k on mean and u n s t e a d y p r e s s u r e c o e f f i c i e n t s , S t r o u h a l number and wake g e o m e t r y , t h e t h r e e i m p o r t a n t p a r a m e t e r s i n t h e a e r o e l a s t i c i n s t a b i l -i t y s t u d y . In c o n j u n c t i o n w i t h t h e a v a i l a b l e l i t e r a t u r e , t h e i n f o r m a t i o n p r e s e n t e d h e r e a t t e m p t s a t p r o v i d i n g b e t t e r u n d e r s t a n d i n g o f t h e b l u f f b o dy a e r o d y n a m i c s i n t h e r e g i o n b e t w e e n t h e two e x t r e m e c a s e s o f c i r c u l a r c y l i n d e r (e=0) a n d f l a t p i a t e (e=°°) . T h e c y l i n d e r e c c e n t r i c i t y and a t t i t u d e has c o n -s i d e r a b l e e f f e c t on 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 v a r i a -t i o n o f t h e S t r o u h a l number i n t h e r a n g e 0 . 0 3 7 - 0 . 2 7 4 . A l t h o u g h b a s i n g S t r o u h a l number on p r o j e c t e d d i m e n s i o n o r wake w i d t h r e d u c e s i t s d e p e n d e n c e on a n g l e o f a t t a c k , t h e u s e o f t r a n s v e r s e d i s t a n c e b e t w e e n t h e s e p a r a t i o n p o i n t s p r o v i d e s , i n a d d i t i o n , a s m o o t h e r t r a n s i t i o n e v e n a t h i g h e r e c c e n t r i c i t i e s and s m a l l e r a n g l e s o f a t t a c k . T h e m e a s u r e m e n t o f f l u c t u a t i n g p r e s s u r e i n t h e n a r r o w b a n d a r o u n d t h e S t r o u h a l f r e q u e n c y s h o w e d s u b s t a n -t i a l d e p e n d e n c e on t h e R e y n o l d s n u m b e r a t z e r o a n g l e o f a t t a c k , p a r t i c u l a r l y f o r t h e e l l i p s e s o f h i g h e c c e n t r i c i t y C o n s i d e r a b l e p h a s e d i f f e r e n c e may e x i s t b e t w e e n t h e p r e s -s u r e s i g n a l s , h o w e v e r , i t h a s n e g l i g i b l e e f f e c t on t h e u n s t e a d y l i f t . T h e t h e s i s a l s o p r e s e n t s a n a l y t i c a l r e s u l t s on t h e l o c a t i o n o f s h e a r l a y e r s e p a r a t i o n as o b t a i n e d u s i n g G o r t l e r ' s s e r i e s i n c o n j u n c t i o n w i t h mean p r e s s u r e p r o f i l e T h e a n a l y t i c a l d a t a c o m p a r e d f a v o u r a b l y w i t h t h e r e s u l t s o f f 1 o w v i s u a ! i s a t i o n t h r o u g h S c h l i e r e n t e c h n i q u e . T h e h i g h s p e e d m o v i e o f t h e n e a r ' w a k e r e g i o n g a v e p r e l i m i n a r y i n f o r m a t i o n a b o u t t h e l o c a t i o n o f t h e f i r s t v o r t e x a n d t h e o s c i l l a t i o n s o f t h e s e p a r a t i n g s h e a r l a y e r s . T h e r i s e a n d d e c a y o f t h e u n s t e a d y p r e s s u r e i n t h e v o r t e x f o r m a t i o n r e g i o n a p p e a r e d t o s u b s t a n t i a t e t h e v i s u a l o b s e r v a t i o n s . I n g e n e r a l , t h e w a k e g e o m e t r y r a t i o v a r i e d a r o u n d t h e Ka>man s t a b i l i t y v a l u e o f 0 . 2 8 1 . T h e c o r r e l a t i o n o f s p a n w i s e r e s u l t s c l e a r l y e m p h a -s i z e d t h e t h r e e d i m e n s i o n a l c h a r a c t e r o f t h e u n s t e a d y a e r o d y n a m i c s . I n g e n e r a l , t h e s p a n w i s e v a r i a t i o n o f f l u c t u a t i n g p r e s s u r e i s e n h a n c e d w i t h i n c r e a s e i n a n g l e o f a t t a c k . T h e p h a s e d a t a s u g g e s t i n c l i n a t i o n o f t h e v o r t e x l i n e a t t h e m o d e l t o be < 1 1 ° . I n c r e a s e i n a n g l e o f a t t a c k r e s u l t s i n i m p r o v e d a l i g n m e n t o f t h e v o r t e x l i n e w i t h t h e c y l i n d e r a x i s . i v TABLE OF CONTENTS C h a p t e r Page 1 I n t r o d u c t i o n . . . 1 2 P u r p o s e and Scope of the I n v e s t i g a t i o n . . 6 3 M o d e l s and S u p p o r t i n g System 8 4 T e s t P r o c e d u r e s 13 4 . 1 I n s t r u m e n t a t i o n and C a l i b r a t i o n . . . 13 4 . 2 B a l a n c e Measurements 17 4 . 3 Mean S t a t i c P r e s s u r e on Model S u r f a c e 18 4 . 4 V o r t e x S h e d d i n g F r e q u e n c y 19 4 . 5 F l u c t u a t i n g P r e s s u r e on Model S u r f a c e 19 4 . 6 Wake S u r v e y 22 5 T e s t R e s u l t s and D i s c u s s i o n 26 5 . 1 S t e a d y A e r o d y n a m i c C o e f f i c i e n t s . . 26 5 . 2 Mean S t a t i c P r e s s u r e D i s t r i b u t i o n . . 31 5 . 3 L o c a t i o n of S e p a r a t i o n U s i n g G o r t l e r ' s Method . 38 5 . 4 S e p a r a t i o n S tudy U s i n g F l o w -V i s u a l i s a t i o n 41 5 . 5 S t r o u h a l Number 51 5 . 6 F l u c t u a t i n g P r e s s u r e D i s t r i b u t i o n . . 53 5 . 7 F l u c t u a t i n g L i f t 75 5 . 8 Spanwise E f f e c t s 78 5 . 9 Wake Geometry 90 V C h a p t e r Page 6 C o n c l u d i n g R e m a r k s 122 7 Recommendat ions f o r F u t u r e Work 126 B i b l i o g r a p h y 129 v i LIST OF TABLES T a b l e Page 3 - 1 Model d a t a 8 5 - 1 C o m p a r i s o n of a n a l y t i c a l l y d e t e r m i n e d s e p a r a t i o n p o i n t s and t h o s e o b t a i n e d u s i n g f l o w v i s u a l i s a t i o n 49 5 - 2 P o s i t i o n s o f the p r e s s u r e t a p s used i n the s t u d y o f f l u c t u a t i n g p r e s s u r e dependency on the R e y n o l d s number 59 5 - 3 A v e r a g e i n c l i n a t i o n of v o r t e x l i n e w i t h r e s p e c t to c y l i n d e r a x i s 91 5 - 4 V a r i a t i o n of l o n g i t u d i n a l v o r t e x s p a c i n g and v o r t e x v e l o c i t y w i t h e c c e n t r i c i t y and a n g l e o f a t t a c k 99 5 - 5 V a r i a t i o n of l a t e r a l s p a c i n g w i t h down-s t r e a m c o o r d i n a t e f o r s e v e r a l b l u f f b o d i e s 108 5 - 6 Dependence o f l a t e r a l v o r t e x s p a c i n g and wake geometry r a t i o on e c c e n t r i c i t y and ang l e of a t t a c k I l l v i i LIST OF FIGURES F i g u r e Page 3 - 1 C o n s t r u c t i o n a l d e t a i l s of a t y p i c a l model 9 3 - 2 N o t a t i o n f o r model and wake geometry . . . . 11 3 - 3 S c h e m a t i c d i a g r a m of the low speed wind t u n n e l used i n t e s t programme 12 4 - 1 C a l i b r a t i o n p l o t f o r B a r o c e l p r e s s u r e t r a n s d u c e r w i t h damping b o t t l e 15 4 - 2 D i s c probe and c a l i b r a t i o n p l o t s 16 4 - 3 F l u c t u a t i n g p r e s s u r e m e a s u r i n g s e t - u p . . . . 21 4 - 4 I n s t r u m e n t a t i o n f o r measurement of phase . . 23 4 - 5 S c h e m a t i c d i a g r a m of i n s t r u m e n t a t i o n used i n wake s u r v e y 24 5 - 1 V a r i a t i o n of mean a e r o d y n a m i c c o e f f i c i e n t s w i t h a n g l e of a t t a c k ( s y m b o l s w i t h s l a s h i n d i c a t e p r e s s u r e i n t e g r a t e d v a l u e s ) : (a) L i f t 27 (b) Drag . 28 (c ) P i t c h i n g moment about m i d - c h o r d . . . . 29 5 - 2 M i d s p a n d i s t r i b u t i o n of mean p r e s s u r e c o e f f i c i e n t : (a) a = 0 32 (b) a = 30° 33 (c ) a = 60° 3 4 (d) a = 90° 3 5 5 - 3 V a r i a t i o n o f base p r e s s u r e w i t h a n g l e of a t t a c k 37 5 - 4 Dependence of minimum p r e s s u r e l o c a t i o n on c y l i n d e r a t t i t u d e and e c c e n t r i c i t y 39 v i i i F i g u r e Page 5 - 5 S c h l i e r e n p h o t o g r a p h of f l o w around a c i r c u l a r c y l i n d e r , R = 1 7 , 0 0 0 43 5 - 6 S c h l i e r e n s t u d y of l a m i n a r s e p a r a t i o n on e l 1 i p t i c c y l i n d e r s : (a) e = 0 . 4 4 44 (b) e = 0 . 6 0 45 ( c ) e = 0 . 8 0 '. 46 (d) e = 0 . 9 2 47 (e) e = 0 . 9 8 48 5 - 7 V a r i a t i o n of S t r o u h a l number w i t h a n g l e of a t t a c k and e c c e n t r i c i t y as based o n : (a) m i n o r a x i s o f e l l i p s e 54 (b) p r o j e c t e d d i m e n s i o n 55 (c ) R o s h k o ' s c r i t e r i o n . c r o b (d) t r a n s v e r s e d i s t a n c e between s e p a r a t i o n p o i n t s 57 5 - 8 S t r o u h a l number as a f u n c t i o n of e c c e n -t r i c i t y o f the e l l i p s e a t ex t reme a t t i t u d e s 58 5 - 9 V a r i a t i o n of f l u c t u a t i n g p r e s s u r e c o e f f i c -i e n t w i t h R e y n o l d s number: (a) e = 0 . 4 4 60 (b) e = 0 . 9 2 61 (c ) e = 0 . 9 8 62 5 - 1 0 E f f e c t of the R e y n o l d s number on a e r o -dynamics of e l l i p t i c c y l i n d e r a t a = 0 : (a) u n s t e a d y p r e s s u r e 64 (b) mean base p r e s s u r e 65 (c ) c o r r e s p o n d e n c e between u n s t e a d y p r e s s u r e and base p r e s s u r e 66 5 - 1 1 D i s t r i b u t i o n of midspan f l u c t u a t i n g p r e s s u r e c o e f f i c i e n t : (a) a = 0 • • 68 (b) a = 30° 68 i x F i g u r e Page (c ) a = 60° 69 (d) a = 90° 69 5 - 1 2 The v a r i a t i o n of maximum f l u c t u a t i n g p r e s -s u r e w i t h e c c e n t r i c i t y and a n g l e o f a t t a c k 70 5 - 1 3 A m p l i t u d e m o d u l a t i o n of p r e s s u r e s i g n a l s on the s u r f a c e of the model : (a) e = 0 . 4 4 72 (b) e = 0 . 9 2 73 (c ) e = 0 . 9 8 74 5 - 1 4 M i d s p a n c i r c u m f e r e n t i a l phase s h i f t between p r e s s u r e s i g n a l s on the s u r f a c e of the model : (a ) a = 0 76 (b) a = 90° 77 5 - 1 5 V a r i a t i o n of f l u c t u a t i n g l i f t c o e f f i c i e n t w i t h a n g l e of a t t a c k 79 5 - 1 6 R e p r e s e n t a t i v e s p a n w i s e v a r i a t i o n of mean p r e s s u r e c o e f f i c i e n t 81 5 - 1 7 Spanwise d i s t r i b u t i o n of f l u c t u a t i n g p r e s -ure c o e f f i c i e n t : (a) e- = 0 . 4 4 82 (b) e = 0 . 9 2 83 (c ) e = 0 . 9 8 84 5 - 1 8 Phase d i f f e r e n c e between p r e s s u r e s i g n a l s i n s p a n w i s e d i r e c t i o n : (a) e = 0 . 4 4 85 (b) e = 0 . 9 2 86 (c ) e = 0 . 9 8 87 5 - 1 9 S t r e a m w i s e v a r i a t i o n of l o n g i t u d i n a l v o r t e x s p a c i n g and v o r t e x v e l o c i t y : (a) e = 0 . 4 4 93 (b) e = 0 . 8 0 94 X F i g u r e Page (c ) e = 0 . 9 2 95 (d) e = 0 . 9 8 96 5 - 2 0 Dependence of l o n g i t u d i n a l s p a c i n g on e c c e n t r i c i t y a t ex t reme v a l u e s of a n g l e of a t t a c k 98 5 - 2 1 L a t e r a l v a r i a t i o n of the f l u c t u a t i n g p r e s s u r e i n the wake of e l l i p t i c c y 1 i n d e r s : (a) e = 0 . 4 4 101 (b) e = 0 . 8 0 102 (c ) e = 0 . 9 2 103 (d) e = 0 . 9 8 104 5 - 2 2 L a t e r a l p o s i t i o n of v o r t e x c o r e s b e h i n d e l l i p t i c c y l i n d e r s 105 5 - 2 3 V a r i a t i o n o f l a t e r a l v o r t e x s p a c i n g w i t h downstream c o o r d i n a t e and a n g l e of a t t a c k 107 5 - 2 4 Wake geometry r a t i o f o r e l l i p t i c c y l i n -d e r s as a f u n c t i o n of downstream d i s t a n c e 109 5 - 2 5 V a r i a t i o n of S t r o u h a l number , d rag c o e f f i c i e n t and near i n f i n i t y v a l u e s of wake p a r a m e t e r s w i t h c y l i n d e r a t t i t u d e : (a ) e = 0 . 4 4 , 0 . 6 0 , 0 . 8 0 1 1 2 (b) e = 0 . 9 2 , 0 . 9 8 1 1 3 5 - 2 6 V a r i a t i o n of peak f l u c t u a t i n g p r e s s u r e i n the wake: (a) e = 0 . 4 4 , 0 . 9 8 1 1 5 (b) e = 0 . 9 2 , 0 . 8 0 1 1 6 5 - 2 7 V i s u a l s t u d y of near wake s h o w i n g : (a) p o s i t i o n o f the f i r s t v o r t e x as a f f e c t e d by a n g l e of a t t a c k 1 1 7 - 1 1 8 (b) e f f e c t of e c c e n t r i c i t y 119 ( c j u n s t e a d y c h a r a c t e r of s h e a r l a y e r . . . 120 L I S T OF SYMBOLS mean d r a g c o e f f i c i e n t , p r e s s u r e d r a g / { ( 1 / 2 ) pVj 2 a c } mean l i f t c o e f f i c i e n t , 1 i f t / { ( 1 / 2 ) p V ^ 2 2 a c } mean p i t c h i n g m o ment c o e f f i c i e n t , m o m e n t a b o u t m i d - c h o r d / { ( l / 2 ) p V o o 2 4 a 2 c } mean s t a t i c p r e s s u r e c o e f f i c i e n t , ( p - p ) / { ( 1 / 2 ) p V o o 2 } a m p l i t u d e o f f l u c t u a t i n g s e c t i o n a l l i f t c o e f f i c i e n t , b a s e d on rms v a l u e f l u c t u a t i n g p r e s s u r e c o e f f i c i e n t , p ' / U l / 2 ) p V £ } d i a m e t e r o f c i r c u l a r c y l i n d e r o r l e n g t h o f s q u a r e c y l i n d e r s i d e s t r e a m f u n c t i o n , \p (X ,Y) = v F ( £ , n ) r a t i o o f t h e maximum f l u c t u a t i n g p r e s s u r e ( a v e r a g e a m p l i t u d e ) a t a t r a n s v e r s e s t a t i o n i n t h e w a k e t o t h e c o r r e s p o n d i n g v a l u e on t h e s u r f a c e o f t h e m o d e l f o r a g i v e n a n g l e o f a t t a c k , p,',/(P ' ) m a v  rw m'max l o n g i t u d i n a l s p a c i n g b e t w e e n v o r t i c e s t h e n u m b e r o f p e d e s t a l s a t w h i c h c o r r e l a t i o n o u t p u t r e a c h e s t h e m aximum v a l u e R e y n o l d s n u m b e r , V o o 2 a / v p e r c e n t a g e c i r c u m f e r e n c e , m e a s u r e d a n t i -c l o c k w i s e f r o m t a p 0 S t r o u h a l n u m b e r s b a s e d on R o s h k o ' s c r i t e r i o n 2 1/2 ( f W / [ V ^ { l - ( C p ) b > ] , s p a c i n g b e t w e e n s e p a r a -t i o n p o i n t s ( f h ' / [ V 2 { l - ( C p ) b } ] 1 / 2 ) , m i n o r a x i s ( f 2 b / V o o ) a n d p r o j e c t e d h e i g h t o f t h e m o d e l ( f h / V ) , r e s p e c t i v e l y x i i o u t e r p o t e n t i a l f l o w v e l o c i t y p a r a l l e l to c i r c u m f e r e n c e of e l l i p s e a t the edge o f boundary l a y e r s t r e a m w i s e v o r t e x v e l o c i t y f r e e s t r e a m v e l o c i t y l a t e r a l s p a c i n g between v o r t i c e s d i s t a n c e measured a l o n g c i r c u m f e r e n c e o f e l l i p s e f rom f r o n t s t a g n a t i o n p o i n t d i s t a n c e f rom c i r c u m f e r e n c e of e l l i p s e a l o n g the normal s e m i - m a j o r a x i s of model s e m i - m i n o r a x i s of model l e n g t h of model P 1/2 e c c e n t r i c i t y of e l l i p t i c c y l i n d e r , [ l - ( b / a ) ] f r e q u e n c y of v o r t e x s h e d d i n g , S t r o u h a l f r e q u e n c y 2 2 2 2 1/2 p r o j e c t e d h e i g h t of m o d e l , 2 (a s i n a+b cos a ) t r a n s v e r s e d i s t a n c e between the s e p a r a t i o n p o i n t s mean p r e s s u r e a m p l i t u d e o f the f l u c t u a t i n g p r e s s u r e s i g n a l u n s t e a d y p r e s s u r e ( a v e r a g e a m p l i t u d e based on the rms v a l u e ) about mean mean s t a t i c p r e s s u r e i n the f r e e s t r e a m c i r c u m f e r e n t i a l l o c a t i o n of minimum p r e s s u r e f rom f r o n t s t a g n a t i o n e x p r e s s e d as a f r a c t i o n o f s e c t i o , n a l p e r i m e t e r t i m e d e l a y s e t t i n g i n c o m p u t i n g c r o s s c o r r e l -a t i o n f u n c t i o n r e f e r e n c e c o o r d i n a t e sys tem w i t h o r i g i n on the a x i s of a model a t the c e n t r a l s e c t i o n , F i g u r e 3 - 2 x i i i a e v P T ( X ) S u b s c r i p t s b 1 , u m max w a n g l e of a t t a c k i n c l i n a t i o n of the v o r t e x l i n e to z - a x i s o f the model k i n e m a t i c v i s c o s i t y of a i r a i r d e n s i t y s h e a r s t r e s s a t the s u r f a c e of e l l i p s e phase a n g l e between f l u c t u a t i n g p r e s s u r e s i g n a l s base v a l u e of p a r a m e t e r v a l u e of p a r a m e t e r on l o w e r and upper s u r f a c e s , r e s p e c t i v e l y p a r a m e t e r v a l u e on model s u r f a c e maximum v a l u e of p a r a m e t e r p a r a m e t e r v a l u e i n wake ACKNOWLEDGEMENT T h e 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 a p p r e c i a -t i o n f o r t h e g u i d a n c e a n d a n d a s s i s t a n c e g i v e n b y D r . V - J . M o d i t h r o u g h o u t t h e p r o g r a m . H i s h e l p d u r i n g S c h l i e r e n p h o t o g r a p h y a n d i n t h e p r e p a r a t i o n o f t h e t h e s i s d e s e r v e s s p e c i a l m e n t i o n . T h a n k s a r e a l s o d u e 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 i a n s E d A b e 1 1 , F r e d K n o w l e s , J o h n H o a r a n d P h i l H u r r e n f o r t h e i r i n v a l u a b l e a n d s k i l l e d c o n t r i b u t i o n i n t h e p r e p a r a t i o n o f t h e e x p e r i m e n t a l f a c i l i t i e s . S i n c e r e a p p r e c i a t i o n s a r e c o n v e y e d t o D r . Z. R o t e m f o r t h e u s e o f h i s s e m i - f o c u s i n g S c h l i e r e n s y s t e m a n d Mr. F.W. A n d e r s o n f o r h i s a s s i s t a n c e d u r i n g t h i s p a r t o f t h e e x p e r i m e n t . F i n a n c i a l s u p p o r t was r e c e i v e d 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 C a n a d a , G r a n t A - 2 1 8 1 . 1 . INTRODUCTION The p r o b l e m of s e l f - i n d u c e d o s c i l l a t i o n s of e l a s -t i c a l l y mounted b l u f f b o d i e s , when exposed to a f l u i d s t r e a m , i s not new. S e v e r a l i n s t a n c e s of v i b r a t i o n and s u b s e q u e n t breakdown of smoke s t a c k s , t r a n s m i s s i o n l i n e s , a i r c r a f t w i n g s , b r i d g e s e t c . , have been r e p o r t e d . The n a t u r e of the f o r c i n g f u n c t i o n , S t r o u h a l number and wake geometry r e p r e s e n t t h r e e i m p o r t a n t p a r a m e t e r s i n an a e r o e l a s t i c i n s t a b i l i t y s t u d y . The d e t e r m i n a t i o n of the c o r r e s p o n d i n g i n f o r m a t i o n f o r a s e t of two d i m e n s i o n a l , s t a t i o n a r y , e l 1 i p t i c c y l i n d e r s i n the R e y n o l d s number range 4 5 of 2x10 - 1 0 forms the s u b j e c t of t h i s i n v e s t i g a t i o n . In g e n e r a l , the e l a s t i c a l l y s u p p o r t e d b l u f f b o d i e s e x h i b i t two d i s t i n c t forms o f a e r o d y n a m i c a l l y i n d u c e d v i b r a t i o n t r a n s v e r s e t o the f l o w d i r e c t i o n . In the f i r s t t y p e , commonly known as v o r t e x r e s o n a n c e , the p e r i o d i c i t y o f the o r g a n i s e d wake due to v o r t e x s h e d d i n g c o i n c i d e s w i t h the n a t u r a l f r e q u e n c y of the s t r u c t u r e . T h i s r e p r e -s e n t s f o r c e d v i b r a t i o n s i n c e the s u s t a i n i n g a l t e r n a t i n g f o r c e e x i s t s i n d e p e n d e n t of the m o t i o n and p e r s i s t s even when the m o t i o n i s s t o p p e d . A l t h o u g h any b l u f f body of a r b i t r a r y c r o s s - s e c t i o n , when s u i t a b l y m o u n t e d , would e x h i b i t v o r t e x e x c i t e d o s c i l l a t i o n , the a v a i l a b l e l i t e r a -t u r e i s l a r g e l y c o n f i n e d to such s t u d i e s on c i r c u l a r 2 c y l i n d e r s because of the g e o m e t r i c s i m p l i c i t y o f the s e c t i o n . The second form of i n s t a b i l i t y , r e f e r r e d to as g a l l o p i n g , r e p r e s e n t s an i m p o r t a n t t ype of s e l f - e x c i t e d v i b r a t i o n . The f l u i d f o r c e s wh ich c r e a t e a c o n d i t i o n of i n s t a b i l i t y a r i s e due to the f a c t t h a t the c r o s s - s e c t i o n of the body i s a e r o d y n a m i c a l l y u n s t a b l e to s m a l l d i s t u r -b a n c e s . These f o r c e s r e s u l t i n o s c i l l a t i o n s wh ich grow i n a m p l i t u d e u n t i l the energy e x t r a c t e d f rom the f l u i d s t r e a m b a l a n c e s t h a t d i s s i p a t e d t h r o u g h v a r i o u s forms of d a m p i n g . G a l l o p i n g o s c i l l a t i o n s a re r e f e r r e d to as " s e l f -e x c i t e d " because the f l u i d f o r c e s t h a t s u s t a i n the m o t i o n a re c r e a t e d and c o n t r o l l e d by the m o t i o n , i t s e l f , and i f the m o t i o n s t o p s , the u n s t e a d y f o r c e s d i s a p p e a r . The main f e a t u r e s of g a l l o p i n g a re t h a t the v i b r a t i o n can o c c u r i n a s i n g l e degree of f reedom and the s t e a d y - s t a t e a m p l i t u d e tends to i n c r e a s e w i t h i n c r e a s i n g wind v e l o c i t y . S t r o u h a l ^ was the f i r s t to c o r r e l a t e the p e r i o d i c v o r t e x s h e d d i n g w i t h the d i a m e t e r of a c i r c u l a r c y l i n d e r and f l u i d v e l o c i t y . T h i s was f o l l o w e d by the c l a s s i c a l * •* 2 3 s t u d y of Von Karman , wake a n a l y s i s by H e i s e n b e r g and 4 e x p e r i m e n t s on wake geometry by Benard . Ever s i n c e , academic and p r a c t i c a l i n t e r e s t i n the v o r t e x s h e d d i n g phenomenon has r e s u l t e d i n many t h e o r e t i c a l and e x p e r i m e n -t a l i n v e s t i g a t i o n s by R o s h k o ^ - ^ , Kovas zn ay ^ , Rosenhead 1 ? 1 1 14 "] jr p i ' , S c h a e f e r and E s k i n a z i ; G e r r a r d " and o t h e r s . 3 R o s e n h e a d 2 2 , W i l l e 2 3 ' 2 4 , M a r r i s 2 5 , M o r k o v i n 2 6 and S l a t e r 2 7 have r e v i e w e d t h i s l i t e r a t u r e a t some l e n g t h . I t would b e , t h e r e f o r e , s u f f i c i e n t to emphas i ze o n l y the i m p o r t a n t f e a t u r e s o f the a v a i l a b l e i n f o r m a t i o n r e l e v e n t to the p r e s e n t i n v e s t i g a t i o n . Among the numerous p a p e r s w r i t t e n on the a s p e c t s of f l u i d m e c h a n i c s and dynamics of b l u f f b o d i e s , o n l y a few a r e c o n c e r n e d w i t h the a c t u a l measurements of u n s t e a d y p r e s s u r e s . M c G r e g o r 2 8 and G e r r a r d ^ 5 measured f l u c t u a t i n g f o r c e s on 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 s u s i n g a c o n d e n s e r m i c r o p h o n e sys tem b u i l t i n t o the m o d e l . The p r e s s u r e d i s -29 t r i b u t i o n was o b t a i n e d by t u r n i n g the c y l i n d e r . Keefe c a r r i e d out f l u c t u a t i n g f o r c e measurements w i t h the h e l p of a c a r e f u l l y d e s i g n e d s t r a i n - g a u g e t r a n s d u c e r wh ich a l s o 30 a c t e d as a t e s t m o d e l . Mo lyneux has a l s o d e s c r i b e d a low f r e q u e n c y s t r a i n - g a u g e t ype t r a n s d u c e r mounted i n s i d e the model to measure p r e s s u r e s on o s c i l l a t i n g w i n g s . More 31 32 33 r e c e n t l y , Humphreys , Fung , Grove e t a l . , B i s h o p and 34 35 Hassan , and F e r g u s o n and P a r k i n s o n measured u n s t e a d y l o a d i n g on f i x e d c i r c u l a r c y l i n d e r o v e r d i f f e r e n t ranges o f the R e y n o l d s number. The s t u d y o f t h r e e - d i m e n s i o n a l s t r u c t u r e o f the w a k e , t h r o u g h f l o w - v i s u a l i s a t i o n and/or c o r r e l a t i o n of a e r o d y n a m i c c o e f f i c i e n t s , a s s o c i a t e d w i t h c i r c u l a r c y l i n d e r has been r e p o r t e d by s e v e r a l i n v e s t i g a t o r s ' ' In g e n e r a l , t w o - d i m e n s i o n a l c h a r a c t e r of the f l o w was 4 m a i n t a i n e d o n l y o v e r two t o t h r e e d i a m e t e r s i n t h e n e i g h -b o u r h o o d o f t h e c e n t r a l s e c t i o n . B o t h s h a r p e d g e d s e p a r a t i o n and o s c i l l a t i o n o f t h e model l e d t o i m p r o v e d c o r r e 1 a t i on . So f a r as t h e e f f e c t o f t u r b u l e n c e on t h e u n s t e a d y a e r o d y n a m i c s o f 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 s c o n c e r n e d , 43 t h e i n v e s t i g a t i o n c a r r i e d o u t by S u r r y a p p e a r s t o be t h e m o s t s i g n i f i c a n t o n e . He o b s e r v e d t h a t mean d r a g c o e f f i c -i e n t and v o r t e x s h e d d i n g f r e q u e n c y a r e c o n s i s t e n t w i t h an e f f e c t i v e i n c r e a s e i n t h e R e y n o l d s number w h i c h now d e p e n d s b o t h on i n t e n s i t y and s c a l e o f t h e t u r b u l e n c e . The f l u c t u -a t i n g l i f t i s d o m i n a t e d by b r o a d e n e d S t r o u h a l peak w h i l e t h e u n s t e a d y d r a g i s p r i m a r i l y due t o s t r e a m w i s e t u r b u l e n c e . In g e n e r a l , t h e p r e s e n c e o f t u r b u l e n c e r e s u l t e d i n t h e r e d u c t i o n o f s p a n w i s e c o r r e l a t i o n l e n g t h . The a v a i l a b l e i n f o r m a t i o n c o n c e r n i n g b l u f f b o d y i n t e r a c t i o n w i t h t h e s e p a r a t e d f l o w o f s t a b l e v o r t e x t y p e i s n o t l i m i t e d t o t h e c y l i n d e r s o f c i r c u l a r c r o s s - s e c t i o n . I n v e s t i g a t i o n s w i t h s q u a r e , r e c t a n g u l a r , t r i a n g u l a r and h e x a g o n a l c y l i n d e r s , S t r u c t u r a l H and a n g l e s e c t i o n s , as 44-56 w e l l as s e v e r a l i r r e g u l a r g e o m e t r i e s a r e a l s o r e p o r t e d B u t i t m u s t be e m p h a s i z e d t h a t t h e b u l k o f t h e l i t e r a t u r e i s i n d e e d d e v o t e d t o t h e c i r c u l a r g e o m e t r y . T h i s p o i n t i s w e l l e m p h a s i z e d by t h e f a c t t h a t t h e p r e v i o u s work on e l l i p t i c a l c y l i n d e r s seems t o be l i m i t e d t o r a t h e r p r e -4 9 l i m i n a r y u n s t e a d y p r e s s u r e m e a s u r e m e n t s by Modi and H e i n e , 50 Strouhal number study by Schramm , together with f l u c t u -a t i n g l o a d i n g and vortex shedding frequency study (e = 0.60 ,0.80) by Modi and W i l a n d 5 7 . A comment concerning the r e l a t e d but somewhat mod i f i e d problem i n v o l v i n g c a l c u l a t i o n of the boundary l a y e r s e p a r a t i o n would be a p p r o p r i a t e here. Although Hiemenz^ 8. 59 6 0 Chiu and Achenbach have provided experimental data f o r two-dimensional 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 s , the major re s e a r c h e f f o r t s i n t h i s area have been a n a l y t i c in c h a r a c t e r 61 S c h l i c h t i n g has presented an e x c e l l e n t review of t h i s l i t e r a t u r e . The a e r o e l a s t i c i n s t a b i l i t y of b l u f f bodies has been under i n v e s t i g a t i o n in t h i s department s i n c e 1958. The review of the progress made has been reported in two survey 6 2 63 papers ' . The i n v e s t i g a t i o n d e s c r i b e d here forms the par t of t h i s c o n t i n u i n g programme and intends to study, e x p e r i m e n t a l l y , the e f f e c t of e c c e n t r i c i t y of the c y l i n d r i -cal b l u f f body on the fundamental parameters l i s t e d b e f o r e . 6 2 . PURPOSE AND SCOPE OF THE INVESTIGATION A s y s t e m a t i c s t u d y of the a e r o e l a s t i c i n s t a b i l i t y o f b l u f f b o d i e s , wh ich has been p r o g r e s s i n g i n t h i s d e p a r t m e n t f o r o v e r a d e c a d e , has c o n t r i b u t e d to our u n d e r -s t a n d i n g of w ind e f f e c t s on b u i l d i n g s and s t r u c t u r e s . The i n v e s t i g a t i o n r e p o r t e d here forms a p a r t o f t h i s b r o a d -based programme. The s i g n i f i c a n c e of e l l i p t i c geometry i n such a s t u d y becomes a p p a r e n t when one r e c o g n i s e s the f a c t t h a t a c i r c u l a r c y l i n d e r , i n yawed c o n d i t i o n , p r e s e n t s e l l i p t i c c r o s s - s e c t i o n a l o n g the r e l a t i v e w i n d . F u r t h e r m o r e , an e l l i p t i c s e c t i o n r e p r e s e n t s a more g e n e r a l c o n f i g r a t i o n p e r m i t t i n g r e a l i z a t i o n of a wide range of g e o m e t r i c a l s h a p e s , f rom c i r c u l a r c y l i nder (e = 0) to f l a t p l a t e (e=°°), by a s y s t e m a t i c v a r i a t i o n of e c c e n t r i c i t y . The p r o j e c t aims a t s t u d y i n g the a e r o d y n a m i c s o f s t a t i o n a r y , t w o - d i m e n s i o n a l , e l l i p t i c c y l i n d r i c a l models of e c c e n t r i c i t i e s 0 . 4 4 , 0 . 9 2 and 0 . 9 8 . In p a r t i c u l a r , i t p r e s e n t s e x p e r i m e n t a l r e s u l t s o n : ( i ) mean l i f t , d rag and p i t c h i n g moment; ( i i ) v a r i a t i o n of S t r o u h a l number w i t h R e y n o l d s number ; ( i i i ) mean and f l u c t u a t i n g s t a t i c p r e s s u r e d i s t r i b u t i o n s ; ( i v ) f l u c t u a t i n g l i f t ; (v ) wake geomet ry ; 7 as f u n c t i o n s of a n g l e of a t t a c k and e c c e n t r i c i t y . The growth and decay of the f l u c t u a t i n g p r e s s u r e i n the wake are s t u d i e d i n some d e t a i l to o b t a i n i n f o r m a t i o n c o n c e r n -i n g the p o s i t i o n of the f i r s t v o r t e x . In most c a s e s the 4 5 R e y n o l d s number range was c o n f i n e d t o 2 x 1 0 - 1 0 . When-e v e r a v a i l a b l e , r e s u l t s f o r f l a t p l a t e , c i r c u l a r c y l i n d e r 6 4 and W i l a n d ' s d a t a on e l l i p t i c c y l i n d e r s ( e = 0 . 6 0 , 0 . 8 0 ) a re i n c l u d e d to p e r m i t c o m p a r i s o n and e s t a b l i s h t r e n d . The f l o w f i e l d was a l s o examined v i s u a l l y t h r o u g h S c h l i e r e n t e c h n i q u e i n c o n j u n c t i o n w i t h h i g h speed p h o t o -g raphy to p r o v i d e , a t l e a s t , q u a l i t a t i v e u n d e r s t a n d i n g of the complex phenomenon. The p o s i t i o n of the s e p a r a t i n g s h e a r l a y e r as g i v e n by f l o w v i s u a l i z a t i o n i s compared w i t h the a n a l y t i c a l r e s u l t s o b t a i n e d u s i n g G o r t l e r s e r i e s s o l u t i o n ^ a p p r o a c h . S i n c e the i n f l u e n c e of w ind t u n n e l w a l l s on the a e r o d y n a m i c c o e f f i c i e n t s i s not w e l l e s t a b l i s h e d , the r e s u l t s p r e s e n t e d are u n c o r r e c t e d f o r t h a t e f f e c t . 3 . MODELS AND SUPPORTING SYSTEM 8 Th ree e l l i p t i c c y l i n d r i c a l m o d e l s , each 27 i n c h e s l o n g , were d e s i g n e d to span the w ind t u n n e l c r o s s - s e c t i o n thus a p p r o x i m a t i n g the t w o - d i m e n s i o n a l f l o w c o n d i t i o n . The c o n s t r u c t i o n a l d e t a i l s of a t y p i c a l model a re shown i n F i g u r e 3 - 1 and p h y s i c a l p a r a m e t e r s are l i s t e d i n the t a b l e b e l o w . As W i l a n d ' s d a t a have been quoted e x t e n s i v e l y t h r o u g h o u t the t h e s i s f o r c o m p a r i s o n , the g e o m e t r i c a l f e a t u r e s o f h i s models a r e a l s o i n c l u d e d . T a b l e 3 - 1 Model d a t a e a/b In . / I n . M a t e r i a l Wei ght l b . Number of B u l k h e a d s S k i n T h i c k n e s s i n . 0 . 4 4 5 . 0 / 4 . 5 P1 e x i g 1 a s 1 .81 7 0 .020 * 0 . 6 0 5 . 0 / 4 . 0 A l u m i num 4 . 2 0 7 0 .020 * 0 . 8 0 5 . 0 / 3 . 0 P1 e x i g 1 a s 1 .45 7 0 . 0 2 0 0 . 9 2 5 . 0 / 2 . 0 P1 e x i g 1 a s 1 .38 7 0 . 0 1 5 0 . 9 8 5 . 0 / 1 . 0 P1 e x i g 1 a s 2 . 0 0 Modu1ar -C o n s t r u c t i o n * W i l a n d ' s models A l t h o u g h the p r e s e n t s e t of e x p e r i m e n t s employ models under s t a t i o n a r y c o n d i t i o n , the dynamic t e s t s p l a n n e d f o r f u t u r e r e q u i r e d c a r e f u l c o n s i d e r a t i o n of w e i g h t and s t r e n g t h . The model d e s i g n p e r m i t t e d them to be mounted on the wind t u n n e l b a l a n c e or s u p p o r t e d by the e x i s t i n g a i r b e a r i n g sys tem f o r measurement under v i b r a t i n g c o n d i t i o n . The p ronounced s l e n d e r n e s s of e l l i p s e w i t h e=0.98 r e q u i r e d s l i g h t l y d i f f e r -e n t c o n s t r u c t i o n a l t e c h n i q u e . Here p l e x i g l a s modules were a c c u r a t e l y f i n i s h e d and bonded t o g e t h e r . 10 The c e n t r a l b u l k h e a d of each model c a r r i e d 32 p r e s -s u r e t a p s ( d i a . = 0 . 0 2 5 i n . ) e q u a l l y s p a c e d around the c i r c u m -f e r e n c e . In a d d i t i o n the model was p r o v i d e d w i t h 8 p r e s s u r e t a p s i n s p a n w i s e d i r e c t i o n as i n d i c a t e d i n F i g u r e 3 - 2 . The p r e s s u r e t a p s a r e c o n n e c t e d to 5 f t . l o n g " I n t r a m e d i a c " p o l y e t h e l e n e t u b i n g s of i n s i d e d i a m e t e r 0 . 0 6 6 i n . wh ich a r e b r o u g h t out f rom the bot tom end of the models ( F i gu re 3 - 1 ) . The models were t e s t e d i n a low s p e e d , low t u r b u -l e n c e , r e t u r n t y p e wind t u n n e l where t h e a i r speed can be v a r i e d f rom 4 - 1 5 0 f t . / s e c . w i t h a t u r b u l e n c e l e v e l l e s s than 0 . 1 % . A B e t z m i c r o m a n o m e t e r , w i t h an a c c u r a c y of 0 . 2 mm. of w a t e r , i s used to measure 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 s e c t i o n of 7 :1 r a t i o . 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 the above p r e s s u r e d i f f e r e n t i a l . The r e c t a n g u l a r c r o s s - s e c t i o n , 36 i n . x 27 i n . , i s p r o v i d e d w i t h 45° c o r n e r f i l l e t s w h i c h v a r y f rom 6 i n . x 6 i n . to 4 . 7 5 i n . x 4 . 7 5 i n . t o p a r t l y compensate f o r the boundary l a y e r g r o w t h . 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 the t e s t s e c t i o n i s l e s s than 0 . 2 5 % . The t u n n e l i s powered by a 15 ho rsepower d i r e c t c u r r e n t motor 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 a W a r d - L e o n a r d sys tem of speed c o n t r o l . F i g u r e 3 - 3 shows the o u t l i n e of the t u n n e l . F i g u r e 3 - 2 N o t a t i o n f o r model and wake g e o m e t r y Turning vanes LO CO LO Figure 3-3 Schematic diagram of the low speed wind tunnel used in t e s t programme 13 4 . TEST PROCEDURES 4 . 1 I n s t r u m e n t a t i o n and C a l i b r a t i o n Most o f the i n s t r u m e n t a t i o n used f o r the e x p e r i m e n -t a l programme, e . g . , wind t u n n e l b a l a n c e , m a n o m e t e r s , f i l t e r s , rms m e t e r , e t c . , c o n s t i t u t e s t a n d a r d equ ipment i n any a e r o d y n a m i c s l a b o r a t o r y and hence needs no e l a b o r a t i o n . However , a b r i e f a c c o u n t of the u n s t e a d y p r e s s u r e and wake geometry measurement p r o c e d u r e would be a p p r o p r i a t e . In 1963 D a t a m e t r i c s I n c . of W a l t h a m , M a s s a c h u s e t t s d e v e l o p e d a p r e s s u r e t r a n s d u c e r c a l l e d B a r o c e l M o d u l a r P r e s s u r e T r a n s d u c i n g s y s t e m . I t i s a h i g h p r e c i s i o n s e n s i -t i v e i n s t r u m e n t c o n s i s t i n g of a s t a b l e c a p a c i t i v e v o l t a g e d i v i d e r w i t h a s t a i n l e s s s t e e l d i a p h r a g m s e p a r a t i n g the two p r e s s u r e c h a m b e r s . When exposed to a p r e s s u r e s i g n a l , the d i a p h r a g m d e f l e c t s a t t a i n i n g a v o l t a g e l e v e l d e t e r m i n e d by i t s r e l a t i v e p o s i t i o n between the p l a t e s . Wi th B a r o c e l a p p r o p r i a t e l y a r r a n g e d i n a b r i d g e c i r c u i t , the o u t p u t v o l t a g e i s d e t e r m i n e d by the r a t i o of c a p a c i t a n c e of the d i a p h r a g m to each of the s t a t i o n a r y e l e c t r o d e s . I t can now be read on any of the 8 s e n s i t i v i t y ranges (5V d . c . f u l l s c a l e ) w h i c h have a l i n e a r i t y of ±0.1% and s t a b i l i t y o f ±0.1% o v e r ±15°F a m b i e n t t e m p e r a t u r e c h a n g e s . The p r e s t r e s s e d d i a p h r a g m has a n a t u r a l f r e q u e n c y o f 2500 cps and t h e t r a n s i e n t r e s p o n s e of the p r e s s u r e s e n s o r head i s l e s s t h a n 2 ms. t o a s t e p p r e s s u r e i n p u t . From e x p e r i m e n t t h e H e l m h o l t z r e s o n a t o r f r e q u e n c y of the c a v i t y and c o n n e c -t i o n on one s i d e of d i a p h r a g m was found to be a p p r o x i m a t e l y 210 c p s . The B a r o c e l i s a c c u r a t e l y c a l i b r a t e d f o r s t e a d y p r e s s u r e s . However , f o r f l u c t u a t i n g s i g n a l s t r a n s m i t t e d t h r o u g h r e l a t i v e l y l o n g , s m a l l d i a m e t e r tubes c o n s i d e r a b l e a t t e n u a t i o n would o c c u r . T h e r e f o r e , the o u t p u t e l e c t r i c a l s i g n a l r e q u i r e d c a l i b r a t i o n a g a i n s t known i n p u t f l u c t u a t i n g p r e s s u r e a t the model s u r f a c e . T h i s was a c h i e v e d u s i n g t h e c a l i b r a t i o n s y s t e m d e v e l o p e d by W i l a n d . F i g u r e 4 - 1 shows the c a l i b r a t i o n p l o t f o r the t r a n s d u c e r i n terms of a t t e n u a t i o n as a f u n c t i o n of s i g n a l f r e q u e n c y . The wake s u r v e y was c a r r i e d out u s i n g a d i s c probe 6 6 c o n s t r u c t e d by F e r g u s o n i n a c c o r d a n c e w i t h the d e s i g n d i s c u s s e d i n d e t a i l by B r y e r e t a l . I t was mounted on a 1 i n . h y p o d e r m i c n e e d l e wh ich i n t u r n was c o n n e c t e d to a 14 i n . l o n g , 1/4 i n . d i a m e t e r s t i n g . The mean p r e s s u r e c a l i b r a t i o n r e s u l t s f o r the probe are shown i n F i g u r e 4 - 2 . The measurements i n d i c a t e the probe to be r e l a t i v e l y i n s e n s i t i v e to a p i t c h of ±5° and Yaw of ± 2 0 ° . The wake t r a v e r s i n g gear d e s i g n e d by F e r g u s o n 6 6 was used to p o s i t i o n the probe a t a d e s i r e d l o c a t i o n i n the wind t u n n e l t e s t s e c t i o n . The a c c u r a c y i n p o s i t i o n i n g the probe was approx i m a t e l y 0 . 0 2 i n . O u t p u t / I n p u t 0 . 5 0 . 4 0 . 3 0 . 2 0 . 1 0 0 1 Inner d i a m e t e r of t u b i n g (d j) = 0 . 0 6 6 L e n g t h of t u b i n g (Jfc) = 5 ' 10 2 0 3 0 S i g n a l f r e q u e n c y , H z . 4 0 5 0 F i g u r e 4 - 1 C a l i b r a t i o n p l o t f o r B a r o c e l p r e s s u r e t r a n s d u c e r w i t h damping b o t t l e 17 4 . 2 B a l a n c e Measurements For f o r c e and moment m e a s u r e m e n t s , the models were mounted on the w ind t u n n e l b a l a n c e t h r o u g h a b r a c k e t . To e n s u r e n e a r l y t w o - d i m e n s i o n a l f l o w c o n d i t i o n , the c l e a r -ance between the t u n n e l c e i l i n g and the model end was k e p t to a minimum ( - 1 / 3 2 i n . ) . H a v i n g s e t the model a t a p r e -s e l e c t e d v a l u e of a n g l e of a t t a c k , the wind t u n n e l speed was b r o u g h t t o the d e s i r e d v a l u e and the l i f t , d rag and p i t c h i n g moment r e s u l t s r e c o r d e d . The p r o c e d u r e was r e -p e a t e d a t s e v e r a l w ind speeds to s t u d y R e y n o l d s number d e p e n d e n c y . The a n g l e of a t t a c k was v a r i e d s y s t e m a t i c a l l y o v e r the range 0 -90° i n the s t e p of 5° e x c e p t near the ends of the i n t e r v a l where f i n e r changes were r e q u i r e d t o e s t a b -l i s h t r e n d . As i n d i c a t e d by P a n k h u r s t and H o l d e r 6 8 the i n t e r -f e r e n c e f rom wind t u n n e l w a l l s d u r i n g s t e a d y f l o w c o n d i t i o n s may be s u b d i v i d e d i n t o : ( i ) S o l i d b l o c k a g e ; ( i i ) Wake b l o c k a g e ; ( i i i ) L i f t e f f e c t ; ( i v ) I n t e r f e r e n c e due to s t a t i c p r e s s u r e g r a d i e n t ; (v) W a l l - b o u n d a r y l a y e r i n t e r f e r e n c e . In case of an e l l i p t i c c r o s s - s e c t i o n m o d e l , s y m m e t r i c -a l l y p l a c e d i n a f l o w f i e l d , ( i i i ) does not e x i s t . For the wind t u n n e l a t the U n i v e r s i t y of B r i t i s h C o l u m b i a , the w a l l boundary l a y e r t h i c k n e s s i n the t e s t s e c t i o n i s r e l a t i v e l y s m a l l p a r t l y due to the f i l l e t e d c o r n e r s w h i c h compensate f o r boundary l a y e r g r o w t h . F u r t h e r m o r e , the p r e s s u r e i n t e -g r a t e d and b a l a n c e measured a e r o d y n a m i c c o e f f i c i e n t s showed good c o r r e l a t i o n ( F i g u r e 5 - 1 ) . T h u s , (v) a p p e a r s to be n e g l i g i b l e . S i m i l a r l y , f o r the t u n n e l ( i v ) was found to be of m i n o r s i g n i f i c a n c e (<1%). T h e r e f o r e s o l i d and wake b l o c k a g e r e p r e s e n t majo r w a l l i n t e r f e r e n c e on model a e r o d y n a m i c s . A p p r o x i m a t e c o r r e c t i o n to the d rag c o e f f i c i e n t 69 was o b t a i n e d u s i n g the method d e s c r i b e d by W h i t b r e a d I t amounts t o 13%, 14%, and 17% f o r e l l i p s e s of e = 0 . 4 4 , 0 . 9 2 and 0 . 9 8 , r e s p e c t i v e l y , a t a = 9 0 ° . At a=0 i t i s 1 2 % , 6% and 1% f o r e = 0 . 4 4 , 0 . 9 2 , 0 . 9 8 r e s p e c t i v e l y . For the fi 4 e l l i p s e s t e s t e d by W i l a n d ( e = 0 . 6 0 , 0 . 8 0 ) t h e s e c o r r e c t i o n s were i n the range of 8 - 1 2 % . As s u g g e s t e d by W h i t b r e a d the same c o r r e c t i o n can be a p p l i e d to the l i f t and moment c o e f f i c i e n t s . 4 . 3 Mean S t a t i c P r e s s u r e on Model S u r f a c e The mean p r e s s u r e d i s t r i b u t i o n was o b t a i n e d u s i n g a Lambrecht manometer . Hypodermic n e e d l e s were i n t r o d u c e d i n the t u b i n g c o n n e c t i n g the p r e s s u r e tap to the manometer t o m i n i m i z e o s c i l l a t i o n of the a l c o h o l column caused by the f l u c t u a t i n g component of the s t a t i c p r e s s u r e . The p r e s s u r e on the model was measured r e l a t i v e to the t o t a l head i n the s e t t l i n g s e c t i o n of the wind t u n n e l to f a c i l i -t a t e r e d u c t i o n of the d a t a . T h i s e l i m i n a t e d the e f f e c t of a t m o s p h e r i c p r e s s u r e changes d u r i n g the t e s t s and p r o -duced a p r e s s u r e d i f f e r e n t i a l w h i c h was a l w a y s p o s i t i v e . The t e s t s were c o n d u c t e d o v e r a range of wind speed and a n g l e of a t t a c k . 4 . 4 V o r t e x S h e d d i n g F r e q u e n c y The S t r o u h a l f r e q u e n c y was d e t e r m i n e d by d i s p l a y -i n g a f l u c t u a t i n g p r e s s u r e s i g n a l on a s t o r a g e o s c i l l o s c o p e . A band pass f i l t e r was i n t r o d u c e d i n the c i r c u i t to e l i m -i n a t e e x t r a n e o u s n o i s e . The models were t e s t e d f o r v o r t e x s h e d d i n g f r e q u e n c y v a r i a t i o n w i t h a n g l e of a t t a c k and R e y n o l d s number. In g e n e r a l , the s i g n a l s were s t r o n g e r f o r e l l i p s e s a t h i g h e r a n g l e s o f a t t a c k . Wi th the s l e n d e r e l l i p s e of e = 0 . 9 8 , the s y s t e m f a i l e d to r e c o r d any s i g n a l f o r a < 1 2 ° , p r o b a b l y , due to poor r e s p o n s e c h a r a c t e r i s t i c s of the p r e s s u r e t r a n s d u c e r at h i g h f r e q u e n c y . The f l o w v i s u a l i s a t i o n s t u d y i n d i c a t e d the p r e s e n c e of w e l l d e f i n e d Karman v o r t e x s t r e e t , even a t a = 0° , f o r the same e l l i p s e . 4 . 5 F l u c t u a t i n g P r e s s u r e on Model S u r f a c e F l u c t u a t i n g p r e s s u r e , r e l a t i v e to the mean s t a t i c p r e s s u r e a t the tap i n q u e s t i o n , was measured u s i n g the 20 instrumentation set-up shown in Figure 4-3. Due to large seemingly random amplitude modulations of flu c t u a t i n g pressure, i t was necessary to present the results as time-average values. Under certain conditions i t was necessary to average results over several minutes to obtain repro-ducible r e s u l t s . This was accomplished through the use of true rms voltmeter which converts the flu c t u a t i n g pressure signal into an equivalent D.C. output l e v e l . An external r-c damping c i r c u i t was used to reduce the variation of the signal caused by amplitude modulations. The f i n a l steady d.c. level was measured on a vaccum tube voltmeter. Because of the extremely high input resistance of the l a t t e r no measurable voltage drop occurred over the damping r e s i s t o r s . The extent of the amplitude modulation was obtained by determining the average peak signal over more than 2,000 cycles as recorded on a storage oscilloscope. The band pass f i l t e r attenuation was determined for each wind speed and f i l t e r cut-off frequency settings by using a sinusoidal signal from a low frequency function generator. The phase difference between the flu c t u a t i n g pres-sure at the model taps was obtained using function correlator manufactured by Princeton Applied Research Corporation. For cross co r r e l a t i o n mode the phase is given by a simple formu1 a , F i g u r e 4-3 F l u c t u a t i n g p r e s s u r e m e a s u r i n g s e t up <f> = 3 . 6 N t f ( 4 . 1 ) The e f f e c t of any phase change i n h e r e n t to the i n s t r u m e n -t a t i o n was n u l l i f i e d by m e a s u r i n g a l l phase d a t a w i t h r e -s p e c t t o r e f e r e n c e t a p s a t 90° to the wind d i r e c t i o n . The s c h e m a t i c d i a g r a m of the phase m e a s u r i n g s y s t e m i s shown i n F i g u r e 4 - 4 . 4 . 6 Wake Survey The d e t e r m i n a t i o n of wake geometry was a c c o m p l i s h e d by e x a m i n i n g the f l u c t u a t i n g p r e s s u r e f i e l d , a s s o c i a t e d w i t h the v o r t i c e s shed f rom the m o d e l , u s i n g the i n s t r u m e n -t a t i o n l a y o u t shown i n F i g u r e 4 - 5 . T r a v e r s i n g the d i s c probe l a t e r a l to the wake a t v a r i o u s downstream s t a t i o n s and r e c o r d i n g the rms v a l u e of the p r e s s u r e s i g n a l gave a s e t of c u r v e s each h a v i n g two peaks near the v o r t e x c e n t r e l i n e s . The d i s t a n c e between t h e s e m a x i m a , at each y -s t a t i o n , was t a k e n t o be a measure of the l a t e r a l s p a c i n g between the two rows of v o r t i c e s . I t was c o n v e n i e n t to p r e s e n t the r e s u l t s of l a t e r a l t r a v e r s e as a r a t i o of p robe to model tap rms s i g n a l s . The s e l e c t i o n of the model tap f o r the r a t i o was somewhat a r b i t r a r y , but r e p r e s e n t e d the p o s i t i o n on the model h a v i n g a near maximum v a l u e of the u n s t e a d y p r e s s u r e . V a r i a t i o n of the peak p r e s s u r e v a l u e s a l o n g the y - c o o r d i n a t e gave an i n d i c a t i o n as to the p o s i t i o n 23 Barocel Damping bottles Signal conditioner Power supply Filter Correlator Barocel Signal conditioner Filter Oscilloscope F i g u r e 4-4 I n s t r u m e n t a t i o n f o r measurement of phase O c Probe 24 Barocel Damping bottle Damping bottle Signal conditioner Power supply Correlator Barocel Signal conditioner Oscilloscope VJ rms voltmeter . rc damping unit V T V M F i g u r e 4 - 5 S c h e m a t i c d i a g r a m of i n s t r u m e n t a t i o n used i n wake s u r v e y 25 of the f i r s t v o r t e x f o r m a t i o n and i t s decay i n downstream d i r e c t i o n . The l o n g i t u d i n a l s p a c i n g between the c o n s e c u t i v e v o r t i c e s was o b t a i n e d t h r o u g h r e c o g n i t i o n of the f a c t t h a t the d i s t a n c e c o r r e s p o n d s to a 360° phase d i f f e r e n c e between the f l u c t u a t i n g p r e s s u r e s i g n a l s a s s o c i a t e d w i t h them. U s i n g a p r e s s u r e t a p on the model as r e f e r e n c e , the d i s c probe was moved downstream near the c e n t r e l i n e of a v o r t e x row. The phase d a t a , o b t a i n e d u s i n g the c o r r e l a t o r , was r e c o r d e d as a f u n c t i o n of downstream c o o r d i n a t e ( y ) . The p r o c e s s was c o n t i n u e d u n t i l l i m i t e d by the t r a v e l o f the t r a v e r s i n g gear (40 i n . ) . The p l o t of phase v s . y - c o o r d i n a t e gave c o n t i n u o u s v a r i a t i o n of t h e l o n g i t u d i n a l s p a c i n g t h r o u g h the r e l a t i o n L = (360°) ( 4 . 2 ) The S t r o u h a l f r e q u e n c y b e i n g known, the v o r t e x v e l o c i t y i n the wake was a l s o d e t e r m i n e d . The wake t r a v e r s e m e a s u r e -ments were c o n f i n e d to the midspan of the c y l i n d e r . 26 5 . TEST RESULTS AND DISCUSSION 5 . 1 S t e a d y A e r o d y n a m i c C o e f f i c i e n t s The mean l i f t , d rag and p i t c h i n g moment e x p e r i e n c e d by e l l i p t i c c y l i n d e r s were measured o v e r a range of R e y n o l d s number and a n g l e of a t t a c k u s i n g the A e r o l a b s i x component s t r a i n - g a u g e b a l a n c e . The r e s u l t s a re p r e s e n t e d i n F i g u r e 5 - 1 . I t i s a p p a r e n t t h a t ove r the w i n d - s p e e d range i n v e s -t i g a t e d , the f o r c e and moment c o e f f i c i e n t s a re e s s e n t i a l l y i n d e p e n d e n t of the R e y n o l d s number , e x c e p t f o r the few i s o l a t e d d i s c r e p a n c i e s . P e r t i n e n t r e s u l t s f o r f l a t p l a t e 7 ^ 1 , 5 64 c i r c u l a r and e l l i p t i c c y l i n d e r s as q u o t e d i n l i t e r a t u r e a r e a l s o i n c l u d e d . The p r e s s u r e i n t e g r a t e d r e s u l t s , a v a i l -a b l e f o r a=0, 3 0 ° , 60° and 9 0 ° , a re i n d i c a t e d f o r c o m p a r i s o n . T h e i r c l o s e agreement w i t h the b a l a n c e d a t a s u g g e s t s e s s e n -t i a l l y t w o - d i m e n s i o n a l f l o w c o n d i t i o n . A degree of s i m i l a r i t y between the l i f t and moment p l o t s t o g e t h e r w i t h the p r e s e n c e of hump near a - 1 0 ° f o r s l e n d e r e l l i p s e s i s of i n t e r e s t . The sudden drop i n t h e s e a e r o d y n a m i c c o e f f i c i e n t s a l o n g the l e e w a r d s i d e of the hump may be a t t r i b u t e d to the s h i f t i n top s u r f a c e s e p a r a t i o n p o i n t f rom the v i c i n i t y of the t r a i l i n g end to the l e a d i n g e d g e . T h i s b e h a v i o r was c o n f i r m e d d u r i n g the f l o w v i s u a l i s a t i o n . In g e n e r a l , l i f t at^d p i t c h i n g moment i n c r e a s e w i t h e c c e n t r i c i t y and a n g l e of a t t a c k r e a c h i n g t h e i r peak v a l u e s 0 o A 6 A V • • A N A Q O A A A A O V • 8 A v 6 8 0 0 0) , > 1 0 2 0 0 0 V 0 . 4 4 _ J O 1 3 6 0 0 0 ) 6 8 0 0 0 0 . 6 0 ) r e f > 6 8 0 0 0 0 . 8 0 / 64* A 3 4 0 0 0 ) o 6 8 0 0 0 [ 0 « 9 2 • 1 0 2 0 0 0 ) v 4 8 0 0 0 ) O 5 9 0 0 0 ) 0 . 9 8 o 6 8 0 0 0 ) V — 1 5 7 0 0 0 ^ 0 0 / r e f . 7 0 8 A a A 8 A • \ I o V ° A O A A 10 2 0 3 0 4 0 cr v 0 - L 5 0 6 0 L. 7 0 80 9 0 F i g u r e 5 - 1 V a r i a t i o n o f mean a e r o d y n a m i c c o e f f i c i e n t s w i t h a n g l e of a t t a c k ^ ( s y m b o l s w i t h s l a s h i n d i c a t e p r e s s u r e i n t e g r a t e d v a l u e s ) : (a ) L i f t . ^ IR v 68000 ) > 102000 ) O 136000 ) - - 6 8 0 0 0 — 6 8 0 0 0 A 3 4 0 0 0 ) o 6 8 0 0 0 ) • 1 0 2 0 0 0 ) • 4 8 0 0 0 ) O 5 9 0 0 0 ) © 6 8 0 0 0 ) — 15 7 0 0 0 ei 0.44 0.6 0) 0 . 8 0 / 0.92 0.98 ref.70 * ref. 64 60 70 80 90 Fi g u r e 5 - 1 V a r i a t i o n of mean aerodynamic c o e f f i c i e n t s with angle of a t t a c k (symbols with s l a s h i n d i c a t e p r e s s u r e i n t e g r a t e d v a l u e s ) : (b) Drag ^ CO co v 68000 > 102000 O 136000 68000 68000 A 34000 O 68000, • 102000J • 48000] • 59000 • 68000 0.44 0-60 0«80 0-92 0.98 ref . 64 $ o 8 g 9 §§ » % S A '89 3 5 4 0 a 5 0 60 7 0 F i g u r e 5 - 1 V a r i a t i o n of mean a e r o d y n a m i c c o e f f i c i e n t s w i t h a n g l e o f ( s y m b o l s w i t h s l a s h i n d i c a t e p r e s s u r e i n t e g r a t e d v a l u e s ) ( c ) P i t c h i n g moment about m i d - c h o r d 8 0 a t t a c k 30 a r o u n d 4 0 ° - 5 5 ° . T h e maximum v a l u e o f t h e mean l i f t f o r t h e s l e n d e r e l l i p s e o f e = 0 . 9 8 was f o u n d t o be 1.0 w h i c h c o m p a r e s w i t h t h e f l a t p l a t e v a l u e o f 1.2. I t i s o f i n t e r e s t t o n o t e t h a t a c o r r e s p o n d i n g t o t h e maximum l i f t i n c r e a s e s w i t h d e c r e a s i n g e c c e n t r i c i t y ( a ) C , = 4 0 ° , 5 5 ° f o r e=°°, 0.44 max r e s p e c t i v e l y ) . On t h e o t h e r h a n d , | C M - 4 5 ° a n d i s 1 "max r e l a t i v e l y u n a f f e c t e d b y t h e c y l i n d e r e c c e n t r i c i t y . B e y o n d t h i s , l i f t a n d moment g r a d u a l l y d e c r e a s e a n d v a n i s h c o m p l e t e l y a t a = 9 0 ° d u e t o s y m m e t r y . As e x p e c t e d , d r a g c o e f f i c i e n t i n c r e a s e s w i t h b l u f f -n e s s i n t h e a n g l e o f a t t a c k r a n g e o f 0 - 4 5 ° . H o w e v e r , i n g e n e r a l , t h e t r e n d i s r e v e r s e d b e y o n d t h i s p o i n t . T h e maximum d r a g c o e f f i c i e n t o f 2.3 f o r e = 0 . 9 8 i s s l i g h t l y h i g h e r t h a n t h e f l a t p l a t e v a l u e o f 2.12 r e p o r t e d b y F a g e a n d J o h a n s e n 7 ^ , p r o b a b l y d u e t o w a l l c o n f i n e m e n t . U s i n g t h e m e a s u r e d b a s e p r e s s u r e v a l u e f o r t h e c y l i n d e r w i t h e = 0 . 9 8 a t a = 9 0 ° , R o s h k o ' s ^ n o t c h e d h o d r o g r a p h s o l u t i o n p r e -d i c t s a v a l u e o f C D = 2 . 6 2 , w h i c h i s s o m e w h a t h i g h e r t h a n t h e b a l a n c e d a t a . T h e a t t i t u d e o f a r o u n d 4 5 ° a p p e a r s t o p l a y an i m p o r t a n t r o l e i n t h e s t e a d y s t a t e a e r o d y n a m i c s o f t h e e l l i p t i c c y l i n d e r s . B e s i d e s a t t a i n i n g t h e s t a l l a c c o m p a n i e d by t h e maximum r a t e o f i n c r e a s e i n d r a g , t h e e l l i p t i c m o d e l s , i r r e s p e c t i v e o f t h e i r e c c e n t r i c i t i e s , o f f e r a p p r o x i m a t e l y t h e same r e s i s t a n c e as t h e c i r c u m s c r i b i n g c i r c u l a r c y 1 i n d e r . For a g i v e n e l l i p s e , v a r i a t i o n of the d rag c o e f f i c -i e n t w i t h a n g l e of a t t a c k can be r e d u c e d by b a s i n g i t on the p r o j e c t e d h e i g h t . However , the d rag c o e f f i c i e n t does not become c o m p l e t e l y i n d e p e n d e n t of the model o r i e n t a t i o n . I t may be e m p h a s i z e d t h a t such a s y s t e m a t i c s t u d y of l i f t , d rag and p i t c h i n g moment c o e f f i c i e n t s as a f u n c -t i o n o f a n g l e o f a t t a c k and e c c e n t r i c i t y of the e l l i p s e , though r o u t i n e , has not been r e c o r d e d i n l i t e r a t u r e . Only a v a i l a b l e d a t a a re f o r e l l i p s e s at z e r o a n g l e of 71 a t t a c k as r e p o r t e d by Hoerner . They compare q u i t e w e l l w i t h the c o r r e s p o n d i n g r e s u l t s i n F i g u r e 5 - l ( b ) . V a r i o u s o t h e r a e r o d y n a m i c p a r a m e t e r s such as f l u c u -a t i n g p r e s s u r e , S t r o u h a l number , wake g e o m e t r y , e t c . , were f o u n d t o be s u b s t a n t i a l l y i n d e p e n d e n t of the R e y n o l d s number. T h e r e f o r e , the r e s u l t s p r e s e n t e d h e n c e f o r t h are f o r R=68,000 u n l e s s o t h e r w i s e s p e c i f i e d . 5 . 2 Mean S t a t i c P r e s s u r e D i s t r i b u t i o n The mean p r e s s u r e on the s u r f a c e of the e l l i p t i c c y l i n d e r was found to be e s s e n t i a l l y i n d e p e n d e n t of the 4 5 R e y n o l d s number o v e r the range 2x10 - 1 0 . M i d s p a n p r e s s u r e d i s t r i b u t i o n s f o r the e l l i p s e s a re shown i n F i g u r e 5 - 2 . The p l o t s not o n l y p roved u s e f u l i n the a n a l y t i c a l d e t e r m i n a t i o n of the s e p a r a t i o n p o i n t s , but a l s o p r o v i d e d d a t a on l o c a t i o n of f r o n t s t a g n a t i o n and base p r e s s u r e c o e f f i c i e n t n e c e s s a r y i n e v a l u a t i n g wind t u n n e l w a l l c o r r e c t i o n s and 36 s e p a r a t e d s h e a r l a y e r v e l o c i t y . They a l s o s e r v e d as a check on b a l a n c e d a t a . As the f l o w moves downstream f rom the s t a g n a t i o n p o i n t a l o n g the c o n t o u r of an e l l i p s e , the p r e s s u r e i n c r e a s e s n e g a t i v e l y , a t t a i n i n g a peak v a l u e near s e p a r a t i o n p o i n t s . Beyond t h i s , i n the w a k e , the mean p r e s s u r e r e m a i n s a p p r o x -i m a t e l y u n i f o r m . As can be e x p e c t e d , the p r e s s u r e d i s t r i -b u t i o n i s s u b s t a n t i a l l y a f f e c t e d by the e c c e n t r i c i t y and a n g l e of a t t a c k of the e l l i p s e . However , i t i s of i n t e r e s t t o no te t h a t some d e f i n i t e t r e n d can be d i s c e r n e d . In g e n e r a l , the mean p r e s s u r e i n c r e a s e s n e g a t i v e l y w i t h i n c r e a s e i n a n g l e of a t t a c k . For a=0, the s t e e p e r change i n p r e s s u r e , i n the i m m e d i a t e v i c i n i t y of the s t a g n a -t i o n p o i n t , w i t h i n c r e a s e i n e c c e n t r i c i t y may be a t t r i b u t e d t o the h i g h e r c u r v a t u r e of the n o s e . T h i s l e a d s to d e -c r e a s i n g p r e s s u r e w i t h e c c e n t r i c i t y , h o w e v e r , the t r e n d i s r e v e r s e d f o l l o w i n g the s e p a r a t i o n as shown i n F i g u r e 5 - 2 ( a ) . The b e h a v i o u r , though not q u i t e w e l l d e f i n e d , a p p e a r s t o undergo c o m p l e t e r e v e r s a l a t h i g h e r a n g l e s of a t t a c k ( F i g u r e s 5 - 2 c , d ) . The t r a n s i t i o n i n the t r e n d seems to o c c u r a round 3 0 ° , where no w e l l d e f i n e d p a t t e r n can be e s t a b l i s h e d . The v a r i a t i o n of base p r e s s u r e w i t h a n g l e of a t t a c k and e c c e n t r i c i t y i s shown i n F i g u r e 5 - 3 . The base p r e s s u r e i n c r e a s e s w i t h r i s e i n e c c e n t r i c i t y a t a=0, w i t h the minimum 72 v a l u e of - 1 . 0 7 f o r e=0 . However , t h i s t r e n d b e g i n s to change a t h i g h e r a n g l e s of a t t a c k , and at a = 90° , the base F i g u r e 5-3 V a r i a t i o n o f base p r e s s u r e w i t h a n g l e o f a t t a c k to p r e s s u r e i s more n e g a t i v e f o r the e l l i p s e s of h i g h e r e c c e n -t r i c i t i e s . The f l a t p l a t e d a t a shown i n the d i a g r a m do not c o n f o r m to t h i s t r e n d , p r o b a b l y , due to the d i f f e r e n c e i n the b l o c k a g e r a t i o s f o r the two s e t s of e x p e r i m e n t s . The s i m i l a r i t y between the base p r e s s u r e and the d rag c o e f f i c i e n t c u r v e s h o u l d a l s o be r e c o g n i s e d . In F i g u r e 5 - 4 i s shown the dependence of the minimum p r e s s u r e p o i n t s ( a p p r o x i m a t e l o c a t i o n s of the s e p a r a t i o n p o i n t s ) on c y l i n d e r e c c e n t r i c i t y and a n g l e of a t t a c k . For a=0, the s e p a r a t i o n appears to o c c u r e a r l y f o r s l e n d e r e l l i p s e s , p r o b a b l y , due t o s m a l l r a d i u s o f c u r v a t u r e of the l e a d i n g e d g e . I t i s e v i d e n t t h a t the minimum p r e s s u r e p o i n t on the top s u r f a c e moves f o r w a r d , w h i l e t h a t on the l o w e r s u r f a c e moves r e a r w a r d s w i t h i n c r e a s e i n a n g l e of a t t a c k , h o w e v e r , t h i s t r e n d r e v e r s e s a t h i g h e r a t t i t u d e ( > 4 0 ° ) . The e f f e c t of i n c r e a s e i n e c c e n t r i c i t y a t a=90° i s t o s h i f t t h e s e p o i n t s d o w n s t r e a m . 5 . 3 L o c a t i o n o f S e p a r a t i o n U s i n g G o r t l e r ' s Method The l o c a t i o n of s h e a r l a y e r s e p a r a t i o n as i n d i c a t e d by minimum p r e s s u r e p o i n t s h o u l d be c o n s i d e r e d o n l y q u a l i t a t i v e . Numerous a n a l y t i c a l methods have been 73 r e p o r t e d , w h i c h a p p r o x i m a t e l y p r e d i c t the mean p o s i t i o n 7 4 of s e p a r a t i o n , e . g . , B l a s i u s s e r i e s method , P o h l h a u s e n ' s a p p r o a c h , S c h l i c h t i n g and U l r i c h ' s e x t e n s i o n of P o h l h a u s e n ' s t e c h n i q u e to the two d i m e n s i o n a l c a s e , e t c . * re f. 64 0 30 Q O 60 90 F i g u r e 5-4 Dependence of minimum pressure l o c a t i o n on c y l i n d e r a t t i t u d e and £ e c c e n t r i c i ty However , a r e l a t i v e l y r e c e n t method by G S r t l e r a p p e a r s to be q u i t e p r o m i s i n g . As a g a i n s t the a p p a r e n t l i m i t a t i o n s (use of o n l y f o u r t h degree p o l y n o m i a l by P o h l h a u s e n , r e s t r i c t e d a p p l i c a t i o n of the B l a s i u s s e r i e s to c y l i n d e r s where f l o w s t a r t s w i t h a s t a g n a t i o n p o i n t , m o s t l y f l a t 7 7 78 p l a t e r e l e v a n c e of H o w a r t h ' s and T a n i ' s a p p r o a c h e s , and of c o u r s e , poor c o n v e r g e n c e c h a r a c t e r i s t i c s ) o f the methods m e n t i o n e d a b o v e , G o r t l e r ' s a p p r o a c h r e p r e s e n t s g e n e r a l i s a t i o n of the a v a i l a b l e s e r i e s s o l u t i o n t e c h n i q u e , p r o m i s e s b e t t e r c o n v e r g e n c e and i s a p p l i c a b l e to the b o d i e s i r r e s p e c t i v e of the l e a d i n g edge c o n d i t i o n . T h i s s e r i e s a p p r o a c h , w h i c h does not i n v o l v e any p r e f e r e n t i a l c h o i c e o f c o o r d i n a t e s , s o l v e s the d i m e n s i o n l e s s form of the boundary l a y e r e q u a t i o n F + B ( 5 ) [ 1 - ( F ) 2 ]+ FF = 2£(F F R - F R F ) ( 5 . 1 ) w h e r e , £ o I H X , Y ) = v r n r F U.n) 3(X) = 2 ( U ( X ) } 2 U ' (X ) X / U(X)dX o As i n f l u e n c e of the body geometry a p p e a r s i n the e q u a t i o n t h r o u g h B , G b r t l e r r e f e r r e d to i t as the " P r i n c i p a l F u n c t i o n . " U s i n g measured p r e s s u r e d i s t r i b u t i o n , the p o t e n t i a l f l o w f i e l d was r e p r e s e n t e d by a s e r i e s l e a d i n g to the d e t e r m i n a t i o n of the c o e f f i c i e n t i n the p r i n c i p a l f u n c t i o n e x p a n s i o n . F u r t h e r m o r e , r e p r e s e n t i n g d i m e n s i o n l e s s s t r e a m f u n c t i o n F ( £ , n ) i n terms of u n i v e r s a l f u n c t i o n a l c o e f f i c i e n t s l e a d to the s h e a r s t r e s s e x p r e s s i o n 2 T ( X ) = F n n U , 0 ) ( 5 . 2 ) / 2 l n n The s e p a r a t i o n r e s u l t s o b t a i n e d u s i n g G o r t l e r s e r i e s a p p r o a c h are shown i n T a b l e 5 - 1 . The method a p p e a r s to work q u i t e w e l l f o r the e l l i p s e of e=0.44 ove r the e n t i r e range of a n g l e s of a t t a c k . However , f o r h i g h e r e c c e n t r i c i t y e l l i p s e s , i t c o u l d be a p p l i e d o n l y a t a=0. The f a i l u r e of the method at h i g h e r a t t i t u d e s may be a t t r i b u t e d to the poor c o n v e r g e n c e of the p o l y n o m i a l r e p r e s e n t i n g p o t e n t i a l v e l o c i t y d i s t r i b u t i o n . 5 . 4 S e p a r a t i o n Study U s i n g F l o w - V i s u a l i s a t i o n To b e t t e r a p p r e c i a t e the c h a r a c t e r of s e p a r a t i o n , i t s v i s u a l s t u d y u s i n g s e m i - f o c u s s i n g S c h l i e r e n s y s t e m 7 ^ ' 8 ^ was u n d e r t a k e n . An open c i r c u i t w ind t u n n e l w i t h 6 i n . x 12 i n . t e s t s e c t i o n w i t h a maximum speed of 20 f p s was used f o r the p u r p o s e . The models used were so p r o p o r t i o n e d as to p r o v i d e the b l o c k a g e c o n d i t i o n i d e n t i c a l to t h a t o b s e r v e d d u r i n g the t e s t i n the main t u n n e l . The R e y n o l d s number d u r i n g t h i s s t u d y was t y p i c a l l y 2 0 , 0 0 0 . The r e q u i r e d d e n s i t y g r a d i e n t was o b t a i n e d t h r o u g h s t r e a k -l i n e s o f low d e n s i t y a i r i n t r o d u c e d by p l a c i n g a h e a t e d probe (N ich rome V w i r e s of 0 . 0 1 0 i n . d i a m e t e r ) u p s t r e a m of the m o d e l . The p o s i t i o n of the probe was a d j u s t e d w i t h r e s p e c t to the model such t h a t the s t r e a k l i n e s were c l o s e t o the r e g i o n of s e p a r a t i n g s h e a r l a y e r s . The f l o w p a t t e r n s were p h o t o g r a p h e d u s i n g 35 mm. s t i l l ( A s h a h i P e n t a x SP) and 16 mm. h i g h speed movie (HYCAM K30S1) c a m e r a s . To p r o v i d e some i n d i c a t i o n as t o the p r o p e r w o r k i n g of the i n s t r u m e n t a t i o n , the w e l l known case of f l o w around c i r c u l a r c y l i n d e r was p h o t o g r a p h e d ( F i g u r e 5 - 5 ) . The s e p a r a t i o n p o i n t was found t o be a t a p p r o x i m a t e l y 8 0 ° , w h i c h i s i n f a i r agreement w i t h the a c c e p t e d v a l u e of 82° as o b t a i n e d by H i e m e n z ^ . The S c h l i e r e n p h o t o g r a p h s s h o w i n g the c h a r a c t e r of s e p a r a t i o n as a f u n c t i o n of c y l i n -der e c c e n t r i c i t y and a n g l e of a t t a c k a re shown i n F i g u r e 5 - 6 . The mean l o c a t i o n s o f the s e p a r a t i n g s h e a r l a y e r s as o b t a i n e d t h r o u g h f l o w v i s u a l i s a t i o n a re compared w i t h the c o r r e s p o n d i n g a n a l y t i c a l d a t a and e x p e r i m e n t a l l y o b t a i n e d minimum p r e s s u r e p o i n t s i n T a b l e 5 - 1 . P e r t i n e n t r e s u l t s f rom l i t e r a t u r e a re a l s o i n c l u d e d f o r c o m p a r i s o n . F i g u r e 5 - 5 S c h l i e r e n photograph of f l o w around a c i r c u l a c y l i n d e r , R = 1 7 , 0 0 0 o a = 0 s = 0 . 2 1 o a = 60 S U = 0.19>; S| =0.24 44 o a= 90 S=0.23 Figure 5-6 Schl ieren study of laminar separation on e l l i p t i c cyl inders : (a) e = 0.44 Figure 5-6 Schlieren study of laminar separation on e l l i p t i c cyl inders: (b) e = 0.60 Figure 5-6 Schlieren study of laminar separation on e l l i p t i c cy l inders : (c) e = 0.80 47 Figure 5-6 Schl ieren study of laminar separation on e l l i p t i c cy l inders : (d) e = 0.92 o a = o o a =30 Su=0.05;Sj=0.44 a=60 SU=0.12;S| =0.35 o a= 90 S = 0.22 Figure 5-6 Schlieren study of laminar separation on e l l i p t i c cyl inders: (e) e = 0.98 Table 5-1 Comparison of a n a l y t i c a l l y determined s e p a r a t i o n p o i n t s and those obtained e x p e r i m e n t a l l y e a° P o s i t i o n of minimum pressure point A n a l y t i c a l l y determined s e p a r a t i o n values Experimental Values Si Su Si Su Si 0 0 *0.19 **0.20 *0.23 ***0 .23 §0.23 tO.30 *0.23 0.44 0 0.19 0.23 0.21 30 60 0.17 0.20 0.20 0. 20 0.19 0.23 0.(24 0.24 0.21 0.19 0.21 0.24 90 0.19 0.25 0.23 0.60 0 0.20 0.24 0.22 30 60 0.14 0.16 0.22 0.22 0.16 0.19 0.29 0.25 90 0.19 0.22 0.80 0 0.16 0.20 0.22 30 60 0.11 0.16 0.28 0.25 0.12 0.16 0.31 0.27 90 0. 22 0.23 tO.87 0 0.36 0.92 0 0.16 0.20 0.18 30 60 0.08 0.14 0.32 0.28 0.08 0.15 0.33 0.31 90 0.22 0.23 AO.94 0 0.19 V0.29 0.28 0.97 0 0.42 0.98 0 0.16 0 .21 ( ? ) 30 60 0.03 0.11 0.44 0 .34 0.05 0.12 0.44 0.35 90 0. 23 0.22 tO.998 0 0.46 *Hiemenz 5 8, * * F 1 a c h s b a r t 7 2 , ***Poh1hausen 7 5, § G o r t l e r 6 5 , t S c h l i c h t i n g and U l r i c h 82 81 ASchubauer , VMeksyn In g e n e r a l , the agreement between the a n a l y t i c a l and e x p e r i m e n t a l d a t a a p p e a r s to be q u i t e a c c e p t a b l e . As can be e x p e c t e d , the s e p a r a t i o n o c c u r s downstream of the minimum p r e s s u r e p o s i t i o n s , h o w e v e r , the d i f f e r e n c e i n t h e i r l o c a t i o n s a p p e a r s to d i m i n i s h w i t h i n c r e a s e i n e c c e n t r i c i t y . Based on the t r e n d , s u g g e s t e d by the r e -s u l t s , c o n c e r n i n g the dependence of s h e a r l a y e r s e p a r a t i o n on the c y l i n d e r e c c e n t r i c i t y and a n g l e o f a t t a c k , the f o l l o w i n g o b s e r v a t i o n s can be made: ( i ) A t a=0 and e i n the range of 0 - 0 . 8 0 , the l o c a t i o n o f the s e p a r a t i o n (s ) i s s u b s t a n t i a l l y i n d e p e n d e n t of c y l i n d e r e c c e n t r i c i t y , v a r i a t i o n s b e i n g l e s s than 5% around the c i r c u l a r c y l i n d e r v a l u e . How-e v e r , f o r s l e n d e r e l l i p s e s w i t h e g r e a t e r than 0 . 8 0 , the s e p a r a t i o n p o i n t s move c o n s i d e r a b l y f o r w a r d . T h i s i s i n agreement w i t h the o b s e r v a t i o n c o n c e r n i n g movement of the minimum p r e s s u r e p o i n t s made e a r l i e r . I t may be p o i n t e d out t h a t the 7 fi a n a l y t i c a l r e s u l t s of S c h l i c h t i n g and U l r i c h 8 2 and S c h u b a u e r do not c o n f i r m to t h i s t r e n d , ( i i ) As s u g g e s t e d by the e x p e r i m e n t a l r e s u l t s , the e f f e c t of i n c r e a s e i n a n g l e of a t t a c k , up to a c r i t i c a l v a l u e , i s to move the l o w e r s e p a r a t i o n p o i n t r e a r w a r d s and upper s e p a r a t i o n p o i n t f o r -w a r d . However , any f u r t h e r i n c r e a s e i n a beyond the c r i t i c a l v a l u e r e v e r s e s t h i s t r e n d . The c r i t i c a l a t t i t u d e f o r a l l e l l i p s e s was o b s e r v e d to be i n the range a = 3 0 ° - 6 0 ° . ( i i i ) A t a=30° , 6 0 ° , the s e p a r a t i o n of s h e a r l a y e r f rom the l o w e r s u r f a c e tends t o move d o w n s t r e a m , w h i l e t h a t f rom the upper s u r f a c e s h i f t s f o r w a r d w i t h i n c r e a s e i n e c c e n t r i c i t y , ( i v ) In g e n e r a l , a t a=90° the l o c a t i o n of s e p a r a t i o n p o i n t s i s r e l a t i v e l y i n d e p e n d e n t of e c c e n t r i c i t y . 5 . 5 S t r o u h a l Number The v o r t e x s h e d d i n g f r e q u e n c y was measured i n the 4 5 R e y n o l d s number range of 2x10 - 1 0 w i t h the s y s t e m a t i c v a r i a t i o n i n a n g l e of a t t a c k . I t was o b s e r v e d to v a r y l i n e a r l y w i t h i n c r e a s i n g w ind speed thus i n d i c a t i n g i n d e -pendence o f the S t r o u h a l number o v e r the R e y n o l d s number range i n v e s t i g a t e d . F i g u r e 5 - 7 ( a ) shows the v a r i a t i o n of S t r o u h a l number , based on m i n o r a x i s of the m o d e l , as a f u n c t i o n of a n g l e of a t t a c k . In g e n e r a l , d e c r e a s e s w i t h i n c r e a s e i n a n g l e of a t t a c k , the drop b e i n g more p r o n o u n c e d i n the range a=0 -50° f o r the e l l i p s e s ' o f h i g h e c c e n t r i c i t i e s . I t may be a t t r i b u t e d to the f a c t t h a t h i g h e c c e n t r i c i t y e l l i p -ses at low a n g l e s of a t t a c k e s s e n t i a l l y behave as s l e n d e r b o d i e s s h e d d i n g v o r t i c e s a t h i g h f r e q u e n c y . However , a t 52 h i g h e r a n g l e s o f a t t a c k , b l u f f b o d y c h a r a c t e r p r e d o m i n a t e s l e a d i n g t o t h e a l m o s t u n i f o r m S t r o u h a l n u m b e r . T h i s , t o g e t h e r w i t h i n c r e a s e i n t h e S t r o u h a l n u m b e r w i t h r e -d u c t i o n i n e c c e n t r i c i t y a t h i g h e r a n g l e s o f a t t a c k ( a _ > 2 0 ° ) , s u g g e s t s t h a t b a s i n g t h e d i m e n s i o n l e s s f r e q u e n c y p a r a m e t e r on p r o j e c t e d h e i g h t h may m i n i m i z e i t s d e p e n d e n c e on b l u f f n e s s . I t i s a p p a r e n t f r o m F i g u r e 5 - 7 ( b ) t h a t , i n m o s t c a s e s , t h e S t r o u h a l n u m b e r i s c o n f i n e d t o t h e r a n g e 0 . 1 8 - 0 . 2 2 i r r e s p e c t i v e o f t h e e c c e n t r i c i t y a n d a n g l e o f a t t a c k . I t may be p o i n t e d o u t t h a t t h e t e s t r e s u l t s , b e i n g u n c o r r e c t e d f o r b l o c k a g e , a r e s o m e w h a t h i g h e r t h a n t h e n e a r l y u n c o n f i n e d f l a t p l a t e v a l u e s a s g i v e n b y F a g e a n d J o h a n s e n 7 ^ . F o r t h e s l e n d e r e l l i p s e o f e = 0 . 9 8 a t s m a l l a n g l e s o f a t t a c k ( a < 1 2 ° ) no r e l i a b l e S t r o u h a l d a t a c o u l d be o b t a i n e d d u e t o m i n u t e p r e s s u r e f l u c t u a t i o n s a n d p o o r r e s p o n s e c h a r a c t e r i s t i c s o f t h e i n s t r u m e n t a t i o n i n t h e h i g h e r f r e q u e n c y r a n g e . I t was d e c i d e d t o e x p l o r e a p p l i c a b i l i t y o f R o s h k o ' s p u n i v e r s a l S t r o u h a l n u m b e r , S^, b a s e d on wake w i d t h a n d s h e a r l a y e r s e p a r a t i o n v e l o c i t y . T h e v a r i a t i o n o f S^, b a s e d on m e a s u r e d m i n i m u m b a s e p r e s s u r e , w a k e g e o m e t r y a n d s h e d d i n g f r e q u e n c y , w i t h a n g l e o f a t t a c k a n d e c c e n t r i c i t y i s s h o w n i n F i g u r e 5 - 7 ( c ) . I n g e n e r a l , t h e r e s u l t s c o l l a p s e d a r o u n d t h e u n i v e r s a l v a l u e o f 0 . 1 6 4 e x c e p t f o r t h e s l e n d e r e l l i p s e s o f e = 0 . 9 2 , 0 . 9 8 a t s m a l l a n g l e s o f a t t a c k . H e r e , a s p o i n t e d o u t by R o s h k o , t h e t h e o r y d o e s n o t a p p l y d i r e c t l y 53 p r o b a b l y d u e t o t h e d i f f e r e n c e i n t h e h i s t o r y o f t h e b o u n d a r y l a y e r d e v e l o p m e n t . T h e d e p e n d e n c e o f S t r o u h a l n u m b e r on c y l i n d e r b l u f f n e s s c a n be m i n i m i z e d f u r t h e r by r e l a t i n g i t t o 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 c o n d i t i o n s . F i g u r e 5 - 7 ( d ) s h o w s t h e v a r i a t i o n o f S<. b a s e d on t r a n s v e r s e d i s t a n c e b e t w e e n s e p a r a t i o n p o i n t s a n d v e l o c i t y a t t h e o u t e r e d g e of t h e b o u n d a r y l a y e r a r o u n d t h e s e p a r a t i o n p o i n t . A l t h o u g h t h e p o s s i b i l i t y o f c o l l a p s i n g t h e d a t a on a s i n g l e c u r v e a p p e a r s t o be r e m o t e , t h i s m a n n e r o f r e p r e s e n t a t i o n r e d u c e s t h e e x c u r s i o n o f d a t a w i t h b l u f f n e s s , e v e n a t h i g h e c c e n t r i c i t i e s , a s c o m p a r e d t o R o s h k o ' s a p p r o a c h . T h e S t r o u h a l n u m b e r r e s u l t s c a n be s u m m a r i s e d c o n -v e n i e n t l y , a s i n d i c a t e d i n F i g u r e 5 - 8 . F o r c l a r i t y o n l y r e s u l t s c o r r e s p o n d i n g t o t h e e x t r e m e a t t i t u d e s o f t h e m o d e l ( a = 0 , 9 0 ° ) a r e i n c l u d e d . I t i s a p p a r e n t t h a t up t o e = 0 . 8 0 t h e c r i t e r i o n b a s e d on p r o j e c t e d h e i g h t f a r e s q u i t e w e l l . H o w e v e r , f o r l a r g e r e c c e n t r i c i t y c y l i n d e r s , S t r o u h a l n u m b e r b a s e d on s e p a r a t i o n p o i n t s s h o w s s m o o t h e r t r e n d a t a=0 a n d i s r e l a t i v e l y u n i f o r m a t a = 9 0 ° . 5.6 F l u c t u a t i n g P r e s s u r e D i s t r i b u t i o n M i d s p a n f l u c t u a t i n g p r e s s u r e m e a s u r e m e n t s w e r e made f o r a = 0 , 3 0 ° , 6 0 ° a n d 9 0 ° T o d e t e r m i n e t h e d e p e n d e n c e o f f l u c t u a t i n g p r e s s u r e on t h e R e y n o l d s n u m b e r , t h e l a t t e r was 4 5 v a r i e d f r o m 2 . 5 x 1 0 -10 . T o b e t t e r r e f l e c t t h e c h a r a c t e r 0 . 0 4 F i g u r e 5-7 V a r i a t i o n of Strouhal number with angle of a t t a c k and e c c e n t r i c i t y as „ based on: (a) minor axis of e l l i p s e F i g u r e 5-7 V a r i a t i o n of Strouhal number with angle of a t t a c k and e c c e n t r i c i t y as ^ based on: (b) p r o j e c t e d dimension cn a F i g u r e 5-7 V a r i a t i o n of Strouhal number with angle of a t t a c k and e c c e n t r i c i t y as w based on: (c) Roshko's c r i t e r i o n <^  F i g u r e 5-7 V a r i a t i o n o f S t r o u h a l n u m b e r w i t h a n g l e o f a t t a c k a n d e c c e n t r i c i t y a s b a s e d o n : ( d ) t r a n s v e r s e d i s t a n c e b e t w e e n s e p a r a t i o n p o i n t s 0 . 2 6 0 . 2 2 S 2 b ' S h ' S s 0.1 8 0 . 1 4 0 . 1 0 0.0 6 b a s e d on v e l o c i t y at the m i n i m u m pressure p o i n t a n a t r a n s v e r s e d i s ta nce b e t w e e n s e p a r a t i o n p o i n t s , S S b a s e d on p r o j e c t e d d i m e n s i o n of the m o d e l n o r m a l to f l o w d i r e c t i o n , b a s e d on minor a x i s of the e l l i p s e , 0 . 0 2 0 0.1 0 . 2 0 . 3 0.4 0 . 5 0 . 6 0 . 7 0 . 8 0 - 9 1.0 A F i g u r e 5-8 S t r o u h a l n u m b e r a s a f u n c t i o n o f eccentricity o f t h e e l l i p s e a t e x t r e m e a t t i t u d e s CO o f t h e f l o w f o u r t a p s w e r e u s e d , two i n t h e l a m i n a r a n d two i n t h e s e p a r a t e d f l o w r e g i o n s . T h e R e y n o l d s n u m b e r d e p e n d e n c y was c o n v e n i e n t l y e x p r e s s e d a s a p e r c e n t a g e v a r i a t i o n o f t h e f l u c t u a t i n g p r e s s u r e a b o u t a mean v a l u e a t a t a p i n q u e s t i o n . T h e s e l e c t i o n o f t h e p o i n t s d e p e n -d e d on 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 a n d h e n c e v a r i e d w i t h c y l i n d e r e c c e n t r i c i t y a n d a t t i t u d e . T h e p r e s s u r e t a p s u s e d a r e t a b u l a t e d b e l o w a n d t h e R e y n o l d s n u m b e r d e p e n d e n c y o f u n s t e a d y p r e s s u r e i s i n d i c a t e d i n F i g u r e 5 - 9 . T a b l e 5-2 P o s i t i o n s o f t h e p r e s s u r e t a p s u s e d i n t h e s t u d y o f f l u c t u a t i n g p r e s s u r e d e p e n d e n c y on t h e R e y n o l d s n u m b e r 0 ° 3 0 ° 6 0 ° 9 0 ° 0 .44 * L - 5 ,1 1 * S - 2 1 ,28 L - 8 , 1 2 S-20 ,31 L - l 1 ,16 S-20 ,28 L - 1 3 ,14 S-3 ,4 0.92 L - 5 ,11 S-22 ,26 L - 7 ,1 3 S-0 ,4 L - 1 0 , 1 8 S-2 ,24 L - l l ,21 S-6 ,26 0 . 9 8 no w e l 1 d e -f i n e d s i g n a l L - 1 9 ,20 S-4 ,5 L - 1 8 , 1 9 S-4 ,5 L - 1 2 ,20 S-5 ,27 * L a n d S r e f e r t o t h e t a p s l o c a t e d i n l a m i n a r a n d s e p a r -a t e d r e g i o n s , r e s p e c t i v e l y . No s i g n i f i c a n t R e y n o l d s n u m b e r d e p e n d e n c y was n o t i c e d e x c e p t f o r t h e e l l i p s e s a t z e r o a n g l e o f a t t a c k . M o d i a n d 5 7 W i l a n d a l s o o b s e r v e d s i m i l a r b e h a v i o r d u r i n g t h e t e s t w i t h e l l i p t i c c y l i n d e r s o f e = 0 . 6 0 , 0 . 8 0 . F o r c o m p a r i s o n F i g u r e 5 - 9 V a r i a t i o n o f f l u c t u a t i n g p r e s s u r e c o e f f i c i e n t w i t h R e y n o l d s number : (a) e = 0 . 4 4 cn o F i g u r e 5-9 V a r i a t i o n o f f l u c t u a t i n g p r e s s u r e c o e f f i c i e n t with R e y n o l d s n u m b e r : ( b ) e = 0.92 63 t h e s e r e s u l t s a r e s u m m a r i z e d i n F i g u r e 5 - 1 0 ( a ) . T e s t d a t a 1 5 on c i r c u l a r c y l i n d e r s a s o b t a i n e d by G e r r a r d a r e a l s o i n c l u d e d . T h e i n f l u e n c e o f t h e R e y n o l d s n u m b e r a p p e a r s t o be c o n f i n e d t o t h e r a n g e 3 - 7 x l 0 4 a n d i n c r e a s e s w i t h e c c e n -t r i c i t y . H o w e v e r , f o r c i r c u l a r c y l i n d e r s , G e r r a r d o b s e r v e d t h e R e y n o l d s n u m b e r e f f e c t t o e x t e n d a t l e a s t o v e r t h e r a n g e 2 , 0 0 0 - 1 8 0 , 0 0 0 i n v e s t i g a t e d b y h i m . No c o n c l u s i v e e x p l a n a t i o n c o n c e r n i n g t h i s p h e n o m e n o n i s a v a i l a b l e , b u t i t i s l i k e l y t o be a s s o c i a t e d w i t h t h e m e a s u r e d b a s e p r e s -s u r e v a r i a t i o n s ( F i g u r e 5 - 1 0 b ) , w h i c h i n t u r n may r e f l e c t c h a n g e s i n s e p a r a t i o n c o n d i t i o n . I n g e n e r a l , f o r t h e c y l i n d e r a t t i t u d e a n d t h e R e y n o l d s n u m b e r c o n d i t i o n s u n d e r i n v e s t i g a t i o n , a d e c r e a s e i n b a s e p r e s s u r e a p p e a r s t o be a c c o m p a n i e d b y an i n c r e a s e i n f l u c t u a t i n g p r e s s u r e o v e r t h e e n t i r e r a n g e o f c y l i n d e r e c c e n -t r i c i t y . F o r c l a r i t y t h i s i s s h o w n s e p a r a t e l y i n F i g u r e 5 - 1 0 ( c ) f o r a f e w r e p r e s e n t a t i v e c a s e s . How t h i s w o u l d a f f e c t mean l o c a t i o n a n d u n s t e a d y m o v e m e n t o f t h e s e p a r a t i o n p o i n t s n e e d s t o be e x p l o r e d . T h e m e a s u r e m e n t o f i n s t a n t a n e o u s p r e s s u r e p r o f i l e s , s h e a r s t r e s s a n d / o r f l o w v i s u a l i s a t i o n s t u d i e s c o u l d t h r o w some l i g h t on t h e e x c u r s i o n o f s e p a r a t i o n p o i n t s . As c a n be e x p e c t e d , w i t h i n c r e a s e i n a n g l e o f a t t a c k t h e m o v e m e n t o f s e p a r a t i o n p o i n t s i s m i n i m i s e d l e a d i n g t o a r e d u c e d d e p e n -d e n c y o f f l u c t u a t i n g p r e s s u r e on t h e R e y n o l d s n u m b e r . F l u c t u a t i n g p r e s s u r e d i s t r i b u t i o n a t t h e m i d s p a n s t a t i o n was m e a s u r e d f o r R = 6 8 , 0 0 0 a n d c y l i n d e r a t t i t u d e F i g u r e 5 - 1 0 E f f e c t o f t h e R e y n o l d s n u m b e r on a e r o d y n a m i c s o f e l l i p t i c c y l i n d e r a t a = 0: ( a ) u n s t e a d y p r e s s u r e CT) - 0 . 8 - 0 . 6 r - 0 . 4 J -- 0 . 2 0 0 G e r r a rid 17 R A 3 4 0 0 0 v 4 0 0 0 0 o 6 8 0 0 0 • 1 0 2 0 0 0 a = o • o 0 . 2 0 .4 8 i i W i l a n d 6 4 0 .6 0.8 J a 6 F i g u r e 5 - 1 0 E f f e c t o f t h e R e y n o l d s n u m b e r on a e r o d y n a m i c s o f e l l i p t i c c y l i n d e r a t a = 0: ( b ) mean b a s e p r e s s u r e 66 - 1 . 0 -1 .4 |ND_ O) o - 1 . 8 - 1 . 0 - 1 . 2 — I e = 0 . 9 2 a = o e= o . 4 4 a=o o - 1 . 4 - 0 . 2 - 0 . 4 - 0 ' . 6 4.3 tap in l a m i n a r r e g i o n t a p in separated r e g i o n o ( C p ) b G = 0 , G e r r a r d 15, 17 o t a p 22 t a p 5 t a p 21 t a p 11 t a p at 120° p o s i t i o n tap at 6 0 ° p o s i t i o n 4.4 •0-35 - 0 - 5 5 (C , - - 0 . 7 5 - 0 . 9 0 - 1 . 0 0 - 1 . 1 0 - - 0 . 8 0 - 0 . 9 0 F i g u r e 5 - 1 0 E f f e c t o f t h e R e y n o l d s n u m b e r on a e r o d y n a m i c s o f e l l i p t i c c y l i n d e r a t a = 0: ( c ) c o r r e s p o n -d e n c e b e t w e e n u n s t e a d y p r e s s u r e a n d b a s e p r e s s u r e 67 o f 0, 3 0 ° , 6 0 ° a n d 9 0 ° . 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 u r e 5 - 1 1 . B a s e d on t h i s d a t a f o l l o w i n g r e m a r k s c a n be m a d e : ( i ) T h e maximum f l u c t u a t i n g p r e s s u r e o c c u r s n e a r t h e r e g i o n w h e r e t h e mean p r e s s u r e a p p r o a c h e s t h e m i n i m u m v a l u e , i . e . , c l o s e t o t h e s e p a r a t i o n p o i n t s . On t h e o t h e r h a n d , t h e r e a r e two p o s i t i o n s , a p p r o x i m a t e l y 1 8 0 ° a p a r t , w h e r e t h e f l u c t u a t i n g p r e s s u r e t e n d s t o v a n i s h . One w o u l d e x p e c t t h i s d u e t o c a n c e l l a t i o n o f p r e s s u r e s w h i c h a r e 1 8 0 ° o u t o f p h a s e . A s c a n be e x p e c t e d t h i s e f f e c t i s l e s s c o m p l e t e a t t h e r e a r o f t h e c y l i n d e r , p r o b a b l y d u e t o i r r e g u l a r i t i e s i n t h e w a k e . ( i i ) F o r a=0 a n d 3 0 ° , t h e l e v e l o f f l u c t u a t i n g p r e s -s u r e a p p e a r s t o i n c r e a s e w i t h b l u f f n e s s . H o w e v e r , a t h i g h e r a n g l e s o f a t t a c k , t h e t r e n d i s n o t w e l l e s t a b l i s h e d b e c a u s e o f t h e r e l a t i v e l y s m a l l e r v a r i a t i o n s i n C-, w i t h e c c e n t r i c i t y , ( i i i ) I n g e n e r a l , f o r a g i v e n e l l i p s e , t h e maximum f l u c t u a t i n g p r e s s u r e i n c r e a s e s w i t h i n c r e a s e i n a n g l e o f a t t a c k ( F i g u r e 5 - 1 2 ) . T h e f l u c t u a t i n g p r e s s u r e c o e f f i c i e n t c a n ' a t t a i n a v a l u e a s h i g h a s 0.8 ( e = 0 . 6 0 , a = 9 0 ° ) . ( i v ) I t i s i n t e r e s t i n g t o n o t e t h a t t h e f l u c t u a t i n g p r e s s u r e i n c r e a s e s a s t h e mean p r e s s u r e d e c r e a s e s . T h u s a c u r v e p r e s e n t i n g v a r i a t i o n o f t h e mean F i g u r e 5-12 T h e v a r i a t i o n of-maximum f l u c t u a t i n g p r e s s u r e w i t h e c c e n t r i c i t y a n d a n g l e o f a t t a c k o p r e s s u r e on t h e s u r f a c e o f a c y l i n d e r f o l l o w s t h e t r e n d s i m i l a r t o t h a t o f C - , . P T h e f l u c t u a t i n g p r e s s u r e s i g n a l s on t h e m o d e l s u r -f a c e s h o w e d r a n d o m a m p l i t u d e m o d u l a t i o n , w h i c h may be a t t r i b u t e d t o t h e g e n e r a l i n s t a b i l i t y o f t h e s e p a r a t e d s h e a r l a y e r a n d a s s o c i a t e d v o r t i c e s . F i g u r e 5-13 s h o w s e x t e n t o f t h e m o d u l a t i o n , e x p r e s s e d a s a r a t i o o f t h e maximum t o a v e r a g e a m p l i t u d e , on t h e m o d e l s u r f a c e . T h e maximum v a l u e was o b t a i n e d b y m o n i t o r i n g t h e s i g n a l o v e r a p e r i o d o f t h r e e m i n u t e s . I n s p i t e o f t h e a p p a r e n t s c a t t e r , a p e a k v a l u e a t t a i n e d by t h e r a t i o n e a r t h e m i n i m u m f l u c -t u a t i n g p r e s s u r e p o i n t ( d i a m e t r i c a l l y o p p o s i t e t o 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 q u i t e d i s t i n c t . T h e r e s u l t s a l s o s u g g e s t a t r e n d t o w a r d s r e d u c t i o n o f m o d u l a t i o n i n l a m i n a r r e g i o n . A 4- A k i • 4. • * 1 5 , 28,66 ,83 A s s u g g e s t e d b y s e v e r a l i n v e s t i g a t o r s t h e a m p l i t u d e m o d u l a t i o n w e r e f o u n d t o be i n p h a s e a r o u n d t h e m o d e l . 1 5 G e r r a r d was p r o b a b l y t h e f i r s t o n e t o m e a s u r e t h e 1 8 0 ° p h a s e c h a r a c t e r o f t h e p r e s s u r e s i g n a l s on t h e two s i d e s o f a c i r c u l a r c y l i n d e r . H o w e v e r , c o n t r a r y t o h i s o b s e r v a t i o n , M o d i a n d W i 1 a n d 5 7 f o u n d c o n s i d e r a b l e p h a s e d i f f e r e n c e b e t w e e n a d j a c e n t p r e s s u r e s i g n a l s , p r o b a b l y , d u e t o t h e a d j u s t m e n t o f t h e f l o w f i e l d a r o u n d t h e m o d e l f o l l o w i n g t h e s h e d d i n g o f v o r t e x a n d f o r m a t i o n o f t h e n e x t 72 a = 9 0 3 2 1 0 4 o „ o o o o o o 0 o 0 o O O ° o 0 0 o 0 0 o 0 o O o H h a = 6 0 | "a. 3 X D o 3 4 ~o o E 3 ~5 2 e- i 0 4 3 2 1 0 o o o o 0 o 0 0 0 0 o o o o 0 o o 0 o -I h ° o o o o o rt o ° ° o o ° oo o o « ~ o 0 o o o ° a=3o o o ~ ° oo H h 1 h 0 o o O o 0 ° 0 o o 0 0 o o 0 o 0 o o o o a= o o o o o o o ° 0 0 04 1 ' I L 0 4 8 12 1 6 2 0 2 4 2 8 Tap n u m b e r F i g u r e 5 - 1 3 A m p l i t u d e m o d u l a t i o n o f p r e s s u r e s i g n a l s o n t h e s u r f a c e o f t h e m o d e l : ( a ) e = 0.44 3 2 1 0 4 4> Too o o O o e=o.92 Q = 9 0 ° 0 o ° ° ° o ° ° o o ° o o o o ° o ° o o o o + 1 1 r-a = 6 0 I a * 2 a . E t 1 c o - 0 o E o o r, O O O •o o o o 0 o o 0 0 o % o O o o ° o o o 0 o •\ r-Q = 3 0 c CL E < 2 1 0 4 3 2 1 t 0 o 0 o 0 o o U O o 0 0 o u o o o o °o ^ . ^ o 0 n o o 0 -I 1 h o o o o ° ~ o o Q o 4 H 1 h a =0 o o o o o o O o o 0 0 o o o o 0 o 0 o o o o 0 o 0 o | o F i g u r e 5 - 1 3 A m p l i t u d e m o d u l a t i o n o f p r e s s u r e s i g n a l s t h e s u r f a c e o f t h e m o d e l : ( b ) e = 0.92 I 1 1 1 ! I I L 1 0 4 8 12 16 20 24 28 0 Tap number F i g u r e 5 - 1 3 A m p l i t u d e m o d u l a t i o n o f p r e s s u r e s i g n a l s o n t h e s u r f a c e o f t h e m o d e l : ( c ) e = 0 . 9 8 v o r t e x c o r e . H e n c e , a s y s t e m a t i c s t u d y o f p h a s e b e t w e e n u n s t e a d y p r e s s u r e s as a f u n c t i o n o f e c c e n t r i c i t y was u n d e r -t a k e n . T h e m e a s u r e m e n t s w e r e c o n f i n e d t o t h e e x t r e m e a t t i t u d e s , a = 0 , 9 0 ° . Two p r e s s u r e t a p s a t 9 0 ° p o s i t i o n t o t h e f l o w d i r e c t i o n w e r e c h o s e n , e a c h s e r v i n g as a r e f e r -e n c e f o r t a p s on i t s s i d e . T h i s l e d t o t w o s e t s o f d a t a , w h i c h , t h e o r e t i c a l l y , s h o u l d be i d e n t i c a l b u t ' s h o w e d some s c a t t e r p e r h a p s d u e t o m i n o r d i s c r e p a n c y i n t h e f l o w s y m m e t r y . 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 u r e 5 - 1 4 . F o r a = 0 , t h e p h a s e c h a n g e i n g e n e r a l was f o u n d t o i n c r e a s e w i t h i n c r e a s e i n e c c e n t r i c i t y a n d r e a c h e d a v a l u e as h i g h as - 6 5 ° f o r e = 0 . 9 2 . T h e e f f e c t o f e c c e n t r i c i t y a p p e a r s t o be m o r e p r o n o u n c e d i n a n d n e a r t h e s e p a r a t e d f l o w r e g i o n as c o m p a r e d t o t h e l e a d i n g e d g e . I t i s o f i n t e r e s t t o n o t e t h a t f o r c y l i n d e r s a t a = 0 , t h e s i g n a l s f r o m u p s t r e a m a n d d o w n s t r e a m p r e s s u r e t a p s a r e l a g g i n g a n d l e a d i n g , r e s p e c -t i v e l y , w i t h r e s p e c t t o t h e r e f e r e n c e l o c a t e d a t 9 0 ° p o s i t i o n ( t a p s 0 a n d 1 6 ) . On t h e o t h e r h a n d , f o r a = 9 0 ° a l m o s t a l l t h e p r e s s u r e s i g n a l s l a g t h e r e f e r e n c e ( t a p s 8 a n d 2 4 ) . R e l a t i v e l y l a r g e s c a t t e r i n r e s u l t s n e a r p r e s s u r e c a n c e l l a t i o n r e g i o n s may be p a r t i a l l y a t t r i b u t e d t o r a t h e r w e a k s i g n a l s , c o n s i d e r a b l y d i s t o r t e d by t h e n o i s e . 5.7 F l u c t u a t i n g L i f t T h e u n s t e a d y l i f t e x p e r i e n c e d by t h e e l l i p t i c c y l i n d e r s was d e t e r m i n e d t h r o u g h t h e i n t e g r a t i o n o f p r e s -31 0 1 1 7 1 6 15 Tap n u m b e r F i g u r e 5 - 1 4 M i d s p a n c i r c u m f e r e n t i a l p h a s e s h i f t b e t w e e n p r e s s u r e s i g n a l s o n t h e s u r f a c e o f t h e m o d e l : ( a ) a = 0 6 0 4 0 U 2 0 0 8 •©-G-«-0-r e f . 6 4 -0-043-0-24 O ) O J c j ® e 0 . 4 4 0 - 6 0 ) 0 . 8 0 i 0 - 9 2 0 . 9 8 Tap 8 as r e f e r e n c e Tap 2 4 as r e f e r e n c e 31 F i g u r e 2 3 3 0 2 9 4 2 8 5 2 7 6 7 8 9 2 6 2 5 2 4 2 3 Tap n u m b e r 10 2 2 11 21 12 2 0 13 19 14 1 8 15 1 7 0 5 - 1 4 M i d s p a n c i r c u m f e r e n t i a l p h a s e s h i f t b e t w e e n p r e s s u r e s i g n a l s o n t h e s u r f a c e o f t h e m o d e l : ( b ) a = 9 0 ° s u r e d i s t r i b u t i o n p l o t s p r e s e n t e d e a r l i e r . T h e v a r i a t i o n o f t h e f l u c t u a t i n g l i f t c o e f f i c i e n t , C - , , b a s e d on t h e m a j o r a x i s o f t h e e l l i p s e , w i t h a n g l e o f a t t a c k i s s h o w n i n F i g u r e 5 - 1 5 . I n t e g r a t e d v a l u e s a c c o u n t i n g f o r t h e c i r c u m f e r e n t i a l p h a s e a r e a l s o i n c l u d e d . E x c e p t f o r e = 0 . 4 4 , t h e e l l i p t i c c y l i n d e r s , i n g e n e r a l , s h o w t h e f l u c t u a t i n g l i f t t o i n c r e a s e w i t h t h e a n g l e o f a t t a c k u n t i l t h e c r i t i c a l a t t i t u d e o f - 7 0 ° - 8 0 ° i s a t t a i n e d . B e y o n d t h i s t h e f l u c t u a t i n g l i f t t e n d s t o d i m i n -i s h . F o r e = 0 . 4 4 , C r , | was f o u n d t o be 0 . 5 8 , as c o m p a r e L 1 max v 42 t o t h e c i r c u l a r c y l i n d e r v a l u e o f 0 . 4 5 . T h e s u b s t a n t i a l m a g n i t u d e o f f l u c t u a t i n g l i f t f o r e = 0 . 5 0 , 0.80 ( C [ ' l m a x ~ 1 a n d 0 . 7 5 , r e s p e c t i v e l y ^ 7 ) w o u l d make t h e m s u s c e p t i b l e t o l a r g e a m p l i t u d e o s c i l l a t i o n d u r i n g v o r t e x r e s o n a n c e . I t i s o f i n t e r e s t t o n o t e t h a t t h e c i r c u m f e r e n t i a l p h a s e h a s n e g l i g i b l e e f f e c t ( < 8 % a t a=0 a n d 9 0 ° ) on t h e f l u c t u a t i n g l i f t . T h i s may be a t t r i b u t e d t o t h e f a c t t h a t , i n g e n e r a l , l a r g e p r e s s u r e f l u c t u a t i o n s a r e a s s o c i a t e d w i t h s m a l l p h a s e d i f f e r e n c e s a n d v i c e v e r s a . H e n c e , t h e c o n t r i b u t i o n t o t h e f l u c t u a t i n g l i f t i s h a r d l y s i g n i f i c a n t . 5.8 S p a n w i s e E f f e c t s A s p o i n t e d o u t by V i c k e r y , P r e n d e r g a s t a n d 42 F e n g , t h e s p a n w i s e c o r r e l a t i o n o f t h e f l o w p a s t t h e c i r c -u l a r c y l i n d e r e x i s t s o n l y f o r a f i n i t e l e n g t h . F o r s t a t i o n a r y m o d e l s t h i s w a s , t y p i c a l l y , two t o t h r e e c y l i n d e r e = o.6o* 0 30 60 „o 90 F i g u r e 5 - 1 5 V a r i a t i o n o f f l u c t u a t i n g l i f t c o e f f i c i e n t w i t h a n g l e o f a t t a c k ^ d i a m e t e r s w i t h i n c r e a s e d t w o - d i m e n s i o n a l i t y d u r i n g m o d e l o s c i l l a t i o n . I n g e n e r a l , t h e c o r r e l a t i o n l e n g t h i m p r o v e s f o r s h a r p - e d g e d b o d i e s , a n d i s o f t h e o r d e r o f f i v e t o s i x 51 d i a m e t e r s f o r s t a t i o n a r y s q u a r e member i n a s m o o t h s t r e a m N o r m a l l y t u r b u l e n c e a f f e c t s t h e s p a n w i s e c o r r e l a t i o n a d v e r s e l y . W i t h t h i s b a c k g r o u n d , i t was d e c i d e d t o u n d e r t a k e a p r e l i m i n a r y s t u d y o f t h e e f f e c t o f e c c e n t r i c i t y a n d a n g l e o f a t t a c k o f an e l l i p t i c c y l i n d e r on t h e s p a n w i s e c h a r a c t e r o f t h e f l o w . T h e e x p e r i m e n t a l p r o g r a m u s e d t h r e e m o d e l s ( e = 0 . 4 4 , 0 . 9 2 , 0 . 9 8 ) a t a n g l e s o f a t t a c k o f 3 0 ° , 6 0 ° a n d 9 0 ° . E a c h m o d e l was p r o v i d e d w i t h a s e t o f n i n e , e q u a l l y s p a c e d p r e s s u r e t a p s a l o n g i t s l e a d i n g e d g e ( F i g u r e 3 - 2 ) . T h e m o d e l s p a n n e d t h e t e s t s e c t i o n a l m o s t c o m p l e t e l y , w i t h no g a p a t t h e b o t t o m a n d t h e t o p g a p l e s s t h a n 0 . 0 1 5 i n . T h e d i s t r i b u t i o n o f t h e mean p r e s s u r e a s s h o w n i n F i g u r e 5-16 s u g g e s t s n e g l i g i b l e s p a n w i s e e f f e c t . T h e r e s u l t s f o r a=0 a r e o m i t t e d a s no s p a n w i s e d e p e n d e n c y was n o t i c e d . On t h e o t h e r h a n d , u n s t e a d y p r e s s u r e a n d p h a s e s h o w e d m a r k e d v a r i a t i o n w i t h s p a n w i s e l o c a t i o n . T h e s e r e s u l t s a r e p r e -42 s e n t e d i n F i g u r e 5-17 a n d 5 - 1 8 . F e n g ' s r e s u l t s f o r c i r c u l a r c y l i n d e r a r e a l s o i n c l u d e d f o r c o m p a r i s o n . I n g e n e r a l , t h e v a r i a t i o n o f f l u c t u a t i n g p r e s s u r e i n s p a n w i s e d i r e c t i o n i n c r e a s e s w i t h i n c r e a s e i n a n g l e o f a t t a c k , how-e v e r , t h e f o r m s o f t h e p l o t s r e m a i n e s s e n t i a l l y s i m i l a r . A d e g r e e o f d i s t o r t e d s y m m e t r y a b o u t t h e m i d s p a n s e c t i o n 1.0 0.5 I-! j e = 0.92 S p a n w i s e t a p s a t l e a d i n g e d g e i?--e=0.44 "oo -v — T • • T 30 60° o 90 ± 4 4 3 2 1 0 -1 -2 -3 S p a n w i s e t a p n u m b e r F i g u r e 5-16 R e p r e s e n t a t i v e s p a n w i s e v a r i a t i o n o f m e a n p r e s s u r e c o e f f i c i e n t -4 F i g u r e 5- 1 7 S p a n w i s e d i s t r i b u t i o n o f f l u c t u a t i n g p r e s s u r e c o e f f i c i e n t : ( a ) e = 0.44 CO 0.6 0.5 0.4 0.3 0.1 0 oo S p a n w i s e taps a t l e a d i n g e d g e e=o.92 a e 0 o * 30 * 60° B 90° J*J A — A " A . s • 1 — o —e-i e i e i 1 1 1 0 - 1 S p a n w i s e tap n u m b e r - 2 -3 - 4 F i g u r e 5-17 Spanwise d i s t r i b u t i o n of f l u c t u a t i n g p r e s s u r e c o e f f i c i e n t : (b) e = 0 S p a n w i s e tap number F i g u r e 5 - 1 7 S p a n w i s e d i s t r i b u t i o n o f f l u c t u a t i n g p r e s s u r e c o e f f i c i e n t : ( c ) e = 0. 9 8 CO -P. 4 3 2 1 0 -1 -2 -3 -4 S p a n w i s e tap number F i g u r e 5 - 1 8 P h a s e d i f f e r e n c e b e t w e e n p r e s s u r e s i g n a l s i n s p a n w i s e d i r e c t i o n : ( a ) e = 0.44 S p a n w i s e tap n u m b e r F i g u r e 5 - 1 8 P h a s e d i f f e r e n c e b e t w e e n p r e s s u r e s i g n a l s i n s p a n w i s e d i r e c t i o n : ( b ) e = 0.92 S p a n w i s e taps at t r a i l i n g e d g e 4 3 2 1 0 -1 -2 -3 -4 S p a n w i s e tap n u m b e r I e = 0.98 9 - 4 - 3 - 2 01 il F i g u r e 5 - 1 8 P h a s e d i f f e r e n c e b e t w e e n p r e s s u r e s i g n a l s i n s p a n w i s e d i r e c t i o n : ( c ) e = 0.98 c a n a l s o be d i s c e r n e d i n m o s t o f t h e c a s e s i n v e s t i g a t e d . T h e maximum f l u c t u a t i n g p r e s s u r e i n v a r i a b l y o c c u r r e d away f r o m t h e m i d s p a n . S i m i l a r b e h a v i o r was a l s o o b s e r v e d b y 15 49 G e r r a r d a n d F e n g f o r c i r c u l a r c y l i n d e r . T h e s p a n w i s e v a r i a t i o n s w e r e f o u n d t o be a s l a r g e as 8 0 % ( e = 0 . 4 4 , a = 6 0 ° ) o f t h e r e f e r e n c e m i d s p a n v a l u e . T h e v a r i a t i o n o f p h a s e b e t w e e n u n s t e a d y p r e s s u r e s i g n a l s a l o n g t h e s p a n i s s h o w n i n F i g u r e 5 - 1 8 . N o r m a l l y t h e r e s u l t s w e r e o b t a i n e d f o r t h e l e a d i n g e d g e as w e l l as t h e t r a i l i n g e d g e by r o t a t i n g t h e m o d e l t h r o u g h 1 8 0 ° . F o r a p a r t i c u l a r c a s e o f e = 0 . 9 8 , t h e e x p e r i m e n t was r e p e a t e d f o r R = 3 3 , 0 0 0 . I n g e n e r a l , t h e r e s u l t s d i d n o t s h ow a n y w e l l d e -f i n e d t r e n d s as t o t h e e f f e c t o f c y l i n d e r e c c e n t r i c i t y a n d a n g l e o f a t t a c k on t h e s p a n w i s e v a r i a t i o n o f t h e p h a s e . H o w e v e r , i n m o s t c a s e s t h e t w o - d i m e n s i o n a l c h a r a c t e r was m a i n t a i n e d o n l y o v e r a s h o r t d i s t a n c e a r o u n d t h e m i d s p a n s e c t i o n w i t h t h e s m a l l a s p e c t r a t i o e f f e c t b e c o m i n g q u i t e a p p a r e n t t o w a r d s t h e e n d s . T h e p l o t s s u g g e s t o d d o r e v e n f u n c t i o n t y p e o f s y m m e t r y a b o u t t h e m i d s p a n , h o w e v e r , d u e t o l i m i t e d n u m b e r o f s p a n w i s e s t a t i o n s a n d i n h e r e n t l y u n s t e a d y c h a r a c t e r o f t h e f l o w , no d e f i n i t e c o n c l u s i o n s c a n be d r a w n . As t i m e h i s t o r y o f - t h e p h a s e i s a s s o c i a t e d w i t h t h s p a t i a l l o c a t i o n o f t h e v o r t e x s h e e t , t h e a v e r a g e i n c l i n -a t i o n o f t h e c o r e l i n e w i t h r e s p e c t t o t h e c y l i n d e r a x i s c a n be c a l c u l a t e d . T h e i n c l i n a t i o n 0, b a s e d on t h e s t r a i g h t l i n e v a r i a t i o n o f t h e p h a s e n e a r m i d s p a n s t a t i o n , i s s h o w n on t h e d i a g r a m . T h e r e i s an i n d i c a t i o n t h a t a t h i g h v a l u e s o f a , t h e v o r t e x l i n e i n c l i n a t i o n d i m i n i s h e s s u g g e s t i n g i m p r o v e d a l i g n m e n t w i t h t h e c y l i n d e r a x i s . I t i s s i g n i f i c a n t 51 t o p o i n t o u t t h a t V i c k e r y f o u n d i m p r o v e m e n t i n c o r r e l a t i o n l e n g t h f o r s q u a r e c y l i n d e r ( 5 . 6 D) c o m p a r e d . t o t h e c i r c u l a r 36 37 c y l i n d e r r e s u l t s ( 3 . 5 D) o f P r e n d e r g a s t a n d e l B a r o u d i 1 8 G e r r a r d ' s r e s u l t s on t h r e e - d i m e n s i o n a l c h a r a c t e r 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 a t R = 2 x l 0 4 s h o u l d be m e n t i o n e d h e r e . T h e h o t w i r e m e a s u r e m e n t s made t h r e e d i a m e t e r s d o w n s t r e a m a n d o v e r a s m a l l a x i a l d i s t a n c e ( t e s t l e n g t h = 9 i n . , m o d e l l e n g t h = 20 i n . ) s h o w e d v o r t e x l i n e s t o be a l m o s t s t r a i g h t a n d p a r a l l e l t o t h e c y l i n d e r a x i s . T h i s b e h a v i o r was c o n f i r m e d t h r o u g h f l o w v i s u a l i s a t i o n u s i n g t i t a n i u m t e t r a c h l o r i d e . T h e r a n d o m f l u c t u a t i o n s i n t h e v o r t e x l i n e i n c l i n a t i o n , w h i c h was a l w a y s p r e s e n t , was f o u n d t o be a s l a r g e a s ± 1 5 ° , h o w e v e r , i t o c c u r r e d a t a f r e q u e n c y an o r d e r o f m a g n i t u d e b e l o w t h e f u n d a m e n t a l f r e -42 q u e n c y . On t h e o t h e r h a n d , F e n g ' s r e s u l t s w i t h c i r c u l a r c y l i n d e r a t t h e same R e y n o l d s n u m b e r a n d p r e s e n t d a t a w i t h e l l i p t i c c y l i n d e r s a t R = 6 8 , 0 0 0 s h o w e d t h e p r e s e n c e o f v o r t e x l i n e i n c l i n a t i o n e v e n when b a s e d on a v e r a g e p h a s e r e s u l t s a t t h e f u n d a m e n t a l f r e q u e n c y . A Gomment c o n c e r n i n g t h e e f f e c t o f c y l i n d e r m o t i o n 40 41 39 42 a s r e p o r t e d by K o o p m a n n ' , T o e b e s a n d F e n g w o u l d be a p p r o p r i a t e . F o r l o w a m p l i t u d e ( < < T 0 % o f t h e c y l i n d e r d i a m e t e r ) f o r c e d a n d w i n d i n d u c e d o s c i l l a t i o n s a t l o w R e y n o l d s n u m b e r (1 0 0 - 3 0 0 ) , K o o p m a n n 4 0 r e p o r t e d v o r t e x l i n e i n c l i n a t i o n o f 1 7 ° . H o w e v e r , l a r g e a m p l i t u d e o s c i l l a t i o n s t e n d e d t o p r o m o t e a l i g n m e n t o f v o r t e x l i n e s w i t h c y l i n d e r a x i s s u g g e s t i n g i m p r o v e d t w o - d i m e n s i o n a l i t y o f t h e n e a r w a k e f l o w . Some o f t h e r e s u l t s on t h r e e - d i m e n -s i o n a l i t y o f t h e w a k e a r e s u m m a r i s e d i n T a b l e 5 - 3 . 5.9 Wake G e o m e t r y T h e r e a p p e a r s t o be r a t h e r s c a n t i n f o r m a t i o n c o n -c e r n i n g t h e o r g a n i s e d w a k e a s s o c i a t e d w i t h e l l i p t i c c y l i n d e r . 5 7 M o d i a n d W i l a n d p r e s e n t e d p r e l i m i n a r y r e s u l t s on i t s n e a r i n f i n i t y c h a r a c t e r , i . e . , u n i f o r m v a l u e s o f v o r t e x s p a c i n g s a t t a i n e d f i v e t o e i g h t ' d i a m e t e r s ' ( m a j o r a x i s ) d o w n s t r e a m , f o r c y l i n d e r s o f e c c e n t r i c i t y 0 . 5 0 a n d 0 . 8 0 . H o w e v e r , no a t t e m p t was made t o e x p l o r e t h e v a r i a t i o n o f w a k e g e o m e t r y a s a f u n c t i o n o f d o w n s t r e a m c o o r d i n a t e . T h i s s e c t i o n d i s -c u s s e s t h e r e s u l t s o f a d e t a i l e d i n v e s t i g a t i o n on t h e w a k e o f c y l i n d e r s w i t h e = 0 . 4 4 , 0 . 8 0 , 0.92 a n d 0 . 9 8 . T o s t u d y t h e d e p e n d e n c e o f v a r i o u s w a k e p a r a m e t e r s on t h e a t t i t u d e o f t h e m o d e l s , t h e e x p e r i m e n t s w e r e c o n d u c t e d a t a = 0 , 3 0 ° , 6 0 ° a n d 9 0 ° . S i n c e t h e R e y n o l d s n u m b e r e f f e c t i n t h e r a n g e 4 5 5 7 3 x 1 0 -10 i s r e p o r t e d t o be n e g l i g i b l e , a l l t h e t e s t s w e r e c o n f i n e d t o R = 6 8 , 0 0 0 . T h e r e s u l t s on p h a s e v a r i a t i o n i n t h e d o w n s t r e a m d i r e c t i o n s h o w e d i t t o a t t a i n a c o n s t a n t v a l u e s u g g e s t i n g T a b l e 5-3 A v e r a g e i n c l i n a t i o n o f v o r t e x l i n e w i t h r e s p e c t t o c y l i n d e r a x i s E c c e n t r i c i t y C o n d i t i on R T e c h n i q u e I n c l i n a t i o n I n v e s t i g a t o r e = 0 s t a t i o n a r y 50 p h a s e m e a s u r e m e n t u s i n g h o t - w i r e a l i g n e d w i t h t h e a x i s f o r a t l e a s t 30 d i a -m e t e r s d o w n s t r e a m K o v a s z n a y 1 1 , 1 9 4 9 s t a t i o n a r y <60 75 a l u m i n a f l a k e s u s -p e n s i o n i n w a t e r 0 1 0 ° T a n e d a8 4 , 1 9 5 2 s t a t i o n a r y $ 1 0 0 $ 8 0 h o l l o w c y l i n d e r f i l l e d w i t h d y e , i n w a t e r 0, i n s t i l l w a t e r 0, i n d i s t u r b e d w a t e r P h i 1 1 i p s 8 5 , 1 9 5 6 s t a t i o n a r y 117 190 r e d d y e l e a k e d t h r o u g h s m a l 1 h o l e s a l i g n e d p a r a l l e l t o c y l i n d e r a x i s 0 < 1 0 ° , w a v y H a m a 8 6 , 1 9 5 7 s t a t i o n a r y 1 0 5 - 3 x l 0 7 u s e o f f i n e s i l k t h r e a d s i n w i n d -t u n n e l w a v e p a t t e r n w i t h c e l l s i z e b e t w e e n 1.4D a n d 1.7D 31 H u m p h r e y s , 1 9 6 0 s t a t i o n a r y v - ? 0 0 - 1 4 0 h y d r o g e n b u b b l e f l o w v i s u a l i z -a t i o n 0 - 3 0 ° A b e r n a t h y ^ ^ , 1 9 6 4 - s t a t i o n a r y 85 2 x l 0 4 h o t - w i r e a n e m o m e t r y 1 4 ° a t 1 7 . 2 d i a -m e t e r s d o w n s t r e a m 0 a t 3 d i a m e t e r s d o w n s t r e a m 1 8 G e r r a r d , 1 9 6 6 s t a t i o n a r y v i b r a t i n g : w i n d i n -d u c e d f o r c e d 2 0 0 2 0 0 s m o k e v i s u a l i s a t i o n 1 7 . 5 ° a t t h e c y l i n -d e r a n d 3 3 ° a t 14 d i a m e t e r s d o w n s t r e a m } -41 K o o p m a n n , 1 9 6 7 s t a t i o n a r y w i n d i n d u c e d 1 7 6 0 0 2 0 4 0 0 1 7 0 0 0 2 0 7 0 0 2 6 3 0 0 m e a s u r e m e n t o f p h a s e b e t w e e n p r e s s u r e s i g n a l s a t t h e c y l i n d e r s u r f a c e go 7 . 1 ° 7 . 8 ° 0 go F e n g 4 2 s 1 9 6 8 e = 0 . 4 4 a=0 a = 3 0 ° a = 6 0 ° a = 9 0 ° s t a t i o n a r y 6 8 0 0 0 w eak s i g n a l , a b s e n c e o f w e l 1 d e f i n e d p h a s e 5 ° 0 - 5 ° 1° P r e s e n t e x p e r i m e n t e = 0 . 9 2 a = 0 a = 3 0 ° a = 6 0 ° a = 9 0 ° s t a t i o n a r y 6 8 0 0 0 m e a s u r e m e n t o f p h a s e b e t w e e n p r e s s u r e s i g n a l s a t t h e c y l i n d e r s u r f a c e w e a k s i g n a l . a b s e n c e o f w e l l d e f i n e d p h a s e 1 1 ° 1 0 ° 1 0 ° P r e s e n t e x p e r i m e n t e = 0 . 9 8 a = 3 0 ° a = 6 0 ° a = 9 0 ° s t a t i o n a r y 6 8 0 0 0 7° 0 2 ° P r e s e n t e x p e r i m e n t _ 92 an u n i f o r m v o r t e x v e l o c i t y a n d l o n g i t u d i n a l s p a c i n g . T h e l i m i t e d r e a c h o f t h e t r a v e r s i n g g e a r a n d p o o r s t r e n g t h o f t h e f l u c t u a t i n g p r e s s u r e s i g n a l r e s t r i c t e d t h e l o n g i t u d i n a l m e a s u r e m e n t s t o e i g h t d i a m e t e r s . F o r t u n a t e l y , t h i s d o e s 8 7 n o t r e p r e s e n t a s e r i o u s l i m i t a t i o n a s F r i m b e r g e r ' s e x p e r i -m e n t s w i t h c i r c u l a r c y l i n d e r s , i n a i r a n d w a t e r , s h o w e d t h e w a k e g e o m e t r y t o a t t a i n u n i f o r m d i m e n s i o n s a r o u n d t h e s ame d i s t a n c e d o w n s t r e a m . T h e v a r i a t i o n o f l o n g i t u d i n a l s p a c i n g b e t w e e n v o r t i c e s o b t a i n e d u s i n g e q u a t i o n ( 4 . 2 ) i s p r e s e n t e d i n F i g u r e 5 - 1 9 . T h e p l o t s s h o w a r a p i d i n c r e a s e i n s p a c i n g f o r t h e d i s t a n c e o f two t o t h r e e d i a m e t e r s b e h i n d t h e c y l i n d e r , g r a d u a l l y a p p r o a c h i n g a c o n s t a n t v a l u e by s e v e n d i a m e t e r s . I n g e n e r a l , a l o w e r a n g l e o f a t t a c k i s a c c o m p a n i e d by e a r l y a t t a i n m e n t o f u n i f o r m s p a c i n g . T h e v o r t e x s p a c i n g s h o w e d i n c r e a s e w i t h i n c r e a s e i n a n g l e o f a t t a c k , t h e l a r g e r c h a n g e , a s e x p e c t e d , b e i n g a s s o c i a t e d w i t h h i g h e c c e n t r i c i t y e l l i p s e s . T h e v o r t e x v e l o c i t y was d e d u c e d f r o m t h e r e l a t i o n , V y = f L ( 5 . 3 ) A l t h o u g h , V y s h o w s a t r e n d s i m i l a r t o t h e l o n g i t u d i n a l s p a c i n g , i t i s r e l a t i v e l y l e s s d e p e n d e n t on t h e c y l i n d e r a t t i t u d e a n d e c c e n t r i c i t y . I n g e n e r a l , t h e e f f e c t o f i n c r e a s e i n a n g l e o f a t t a c k a n d e c c e n t r i c i t y i s t o d e l a y t h e e s t a b l i s h m e n t o f n e a r i n f i n i t y v a l u e s d o w n s t r e a m . i g u r e 5 - 1 9 S t r e a m w i s e v a r i a t i o n o f l o n g i t u d i n a l v o r t e x s p a c i n g a n d v o r t e x v e l o c i t y : ( a ) e = 0.44 co F i g u r e 5 - 1 9 S t r e a m w i s e v a r i a t i o n o f l o n g i t u d i n a l v o r t e x s p a c i n g a n d v o r t e x v e l o c i t y : ( b ) e = 0.80 97 F i g u r e 5 - 2 0 s u m m a r i s e s t h e d e p e n d e n c e o f l o n g i t u d i n a l s p a c -i n g a t t w o e x t r e m e a t t i t u d e s o f t h e m o d e l s . A s c a n be e x p e c t e d , t h e e f f e c t o f e c c e n t r i c i t y a t a = 9 0 ° i s r e l a t i v e l y s m a l l . On t h e o t h e r h a n d , t h e p l o t s a t a=0 s h ow s i g n i f i -c a n t d e p e n d e n c e on e c c e n t r i c i t y . A t a g i v e n s t a t i o n , t h e l o n g i t u d i n a l s p a c i n g i n c r e a s e s as t h e c y l i n d e r b e c o m e s m o r e s l e n d e r , w h i c h a l s o p r o m o t e s e a r l y a t t a i n m e n t o f i t s u n i f o r m v a l u e . F o r t h e e l l i p s e s o f e = 0 . 9 2 a n d 0 . 4 4 , t h e n e a r i n f i n i t y v a l u e s w e r e r e a c h e d a p p r o x i m a t e l y t h r e e a n d f i v e d i a m e t e r s d o w n s t r e a m , r e s p e c t i v e l y . I t may be m e n t i o n e d h e r e t h a t b a s i n g d o w n s t r e a m d i s t a n c e a n d l o n g i -t u d i n a l s p a c i n g on m i n o r a x i s s h o w e d r e d u c t i o n i n T a t t e r ' s v a r i a t i o n w i t h e a t a = 0 . I n p a r t i c u l a r t h e n e a r i n f i n i t y v a l u e o f t h e p a r a m e t e r was f o u n d t o be -4 e x c e p t f o r e = 0 . 9 2 i n w h i c h c a s e i t was 3 . 2 . F o r o t h e r a n g l e s o f a t t a c k t h e v a l u e s r a n g e d b e t w e e n 4.0 a n d 5 . 5 . T a b l e 5-4 p r e s e n t s t h e r e s u l t s on u n i f o r m v a l u e s a t t a i n e d by l o n g i t u d i n a l s p a c i n g a n d v o r t e x v e l o c i t y a s a f u n c t i o n o f c y l i n d e r a t t i t u d e a n d e c c e n t r i c i t y . I t i s a p p a r e n t t h a t l o n g i t u d i n a l s p a c i n g d i m i n i s h e s w i t h i n c r e a s e i n e c c e n t r i c i t y a t s m a l l a n g l e s o f a t t a c k . H o w e v e r , a t a = 6 0 ° a n d 9 0 ° , no d e f i n i t e t r e n d c a n be o b s e r v e d a l t h o u g h , i n g e n e r a l , t h e v a r i a t i o n i s r e l a t i v e l y s m a l l . T h e v a r i a t i o n o f v o r t e x v e l o c i t y d o e s n o t s e e m t o f o l l o w a n y s y s t e m a t i c b e h a v i o r a n d f o r t h e c a s e s i n v e s t i g a t e d i t s n o n - d i m e n s i o n a l v a l u e was c o n f i n e d t o t h e r a n g e 0 . 7 7 - 0 . 9 7 . F i g u r e 5 - 2 0 D e p e n d e n c e o f l o n g i t u d i n a l s p a c i n g o n e c c e n t r i c i t y a t e x t r e m e v a l u e s o f a n g l e o f a t t a c k T a b l e 5-4 V a r i a t i o n o f l o n g i t u d i n a l v o r t e x s p a c i n g a n d v o r t e x v e l o c i t y w i t h e c c e n t r i c i t y a n d a n g l e o f a t t a c k e a = 0 = 30° a=60° a= 90° (L/2a) (1-/23^ 0* 4.27 0.80 0.44 4.1 0.95 4.2 0.92 4.8 0.96 5.0 0.97 0.60t 3.2 0.86 3.9 0.93 4.4 0 .94 4.6 0.93 0.80 2.4 0.89 3.3 0.90 4.7 0 .90 5 .1 0.90 0.92 1 .3 0.86 3.0 0.97 4.7 0.92 4.9 0 .96 0. 98 - - 2.8 0.96 4.5 0.95 4.8 0.90 A OO - - 2.8 0.84 4.5 0 .76 5.2 0.77 * F a g e a n d J o h a n s e n , t M o d i a n d W i l a n d , A F a g e a n d J o h a n s e n M O IX ) 1 00 T h e l a t e r a l v a r i a t i o n s o f t h e f l u c t u a t i n g p r e s s u r e a m p l i t u d e i n t h e w a k e a t v a r i o u s d o w n s t r e a m s t a t i o n s i s p r e s e n t e d i n F i g u r e 5 - 2 1 . As e x p e c t e d , t h e u n s t e a d y p r e s -s u r e d i s t r i b u t i o n s a r e s i m i l a r on b o t h s i d e s o f t h e w a k e f o r s y m m e t r i c a l l y o r i e n t e d m o d e l s . T h e wake c e n t r e l i n e , t h u s , c o i n c i d e s w i t h t h e y - a x i s . H o w e v e r , f o r o t h e r o r i e n -t a t i o n s , t h e w a k e i s u n s y m m e t r i c a l w i t h t h e h i g h e r p e a k v a l u e on t h e s i d e f o r w h i c h t h e s e p a r a t i o n o f t h e s h e a r l a y e r i s m o r e r e a r w a r d . S i m i l a r b e h a v i o r was a l s o o b s e r v e d 27 by S l a t e r i n h i s s t u d y w i t h s t r u c t u r a l a n g l e s e c t i o n . As t h e v o r t i c i t y r a t e a s s o c i a t e d w i t h t h e u p p e r a n d l o w e r 8 8 s u r f a c e s o f a b o d y i s t h e same , t h e p r e s e n c e o f h i g h e r p r e s s u r e p e a k on o n e s i d e may be e x p l a i n e d b y r e l a t i v e l y s h o r t e r d i s t a n c e a n d h e n c e r e d u c e d d i s s i p a t i o n s u f f e r e d b y t h e c o r r e s p o n d i n g v o r t e x . F i g u r e 5-22 s h o w s t h e p o s i t i o n o f t h e w a k e o b t a i n e d by c o n s i d e r i n g , a p p r o x i m a t e l y , t h e l o c a t i o n o f p e a k f l u c t u a t i o n a s t h e p o s i t i o n o f t h e v o r t e x 89 c o r e . H o w e v e r , as p o i n t e d o u t by H o o k e r , t h e maximum v e l o c i t y f l u c t u a t i o n s ( a n d h e n c e p r e s s u r e v a r i a t i o n s ) do n o t o c c u r a l o n g t h e p a t h o f v o r t e x c e n t r e s a s some e x p e r i -m e n t e r s h a v e a s s e r t e d b u t r a t h e r d e v e l o p i n t h e n e i g h b o u r -h o o d o f t h e c o r e f a r t h e s t f r o m t h e s t r e e t c e n t r e l i n e . T h u s , t h e d i s t a n c e b e t w e e n t h e p e a k s a s p r e s e n t e d h e r e w o u l d o v e r e s t i m a t e t h e wake w i d t h b y an a m o u n t e q u a l t o t h e d i a m e t e r o f t h e v o r t e x c o r e . B a s e d on S c h a e f e r a n d 1 4 E s k i n a z i ' s m a t h e m a t i c a l m o d e l f o r v o r t e x s t r e e t i n 0.5 2.0 _ _.A A.-ArA--A-A--A.. A-. A ^ ^ A A A -\ 60 / - - ^ ^ A-' A "A'--A-•-•D-- "13--- * - - „ - - • • " : . J 2 - - E > - - I J - - L T J3-r>-n. 1.0 --O A - -•s ^ ^ "V * O 30 . A j& A - -A - - - -A - - . A _ -A- ""A--, ,A' 2.0 <3V.--•la*--. . " A . 6f 1.0 - T - - J 7 - - ^ - °V\ a=o A ' 0rt -0o6 -0.4 -0.2 0.2 0.4 0.6 °x/2a F i g u r e 5-21 L a t e r a l v a r i a t i o n o f t h e f l u c t u a t i n g p r e s s u r e i n t h e w a k e o f e l l i p t i c c y l i n d e r s : ( a 1 e = n . 4 4 _ _g g ^ ^ 0 3.0 \ Ji A 2.0 V \ \ 30 \ \ LO 0.5 6.0 \ IS 4.0 a=o / / f / / f / 2.0 1.0 • A' -0.2 o x/2a a 2 0.4 0.6 1 L a t e r a l v a r i a t i o n o f t h e f l u c t u a t i n g p r e s s u r e i n t h e w a k e o f e l l i p t i c c y l i n d e r s : ( b ) e = 0. -0.4 -0.2 0 x / 2 a 0.2 0.4 0.6 F i g u r e 5-21 L a t e r a l v a r i a t i o n o f t h e f l u c t u a t i n g p r e s s u r e i n t h e w a k e o f e l l i p t i c c y l i n d e r s : ( c ) e = 0.92 F i g u r e 5-21 L a t e r a l v a r i a t i o n o f t h e f l u c t u a t i n g p r e s s u r e i n 106 v i s c o u s f l u i d s h o w e d c o r r e c t i o n i n t h e wake w i d t h t o be n e g l i g i b l e ( < 1 % ) a t t h e m o d e l a n d l e s s t h a n 5% e i g h t d i a m e t e r s d o w n s t r e a m . I t may be p o i n t e d o u t , h o w e v e r , t h a t t h e c o r r e c t i o n w o u l d f a i l f a r d o w n s t r e a m a s i t d o e s n o t a c c o u n t f o r t u r b u l e n c e o r w a k e i n s t a b i l i t y w h i c h w o u l d i n f l u e n c e t h e p o s i t i o n a n d g r o w t h o f t h e v o r t e x c o r e s . T h e v a r i a t i o n o f t h e l a t e r a l s p a c 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 i s p r e s e n t e d i n F i g u r e 5-23 f o r v a r i o u s e l l i p s e s a s a f u n c t i o n o f a n g l e o f a t t a c k . I t i s a p p a r e n t t h a t b o t h t h e m o d e l g e o m e t r y a n d a t t i t u d e h a v e s u b s t a n t i a l e f f e c t o n t h e s p a c i n g b e t w e e n t h e v o r t e x r o w s . I n g e n e r a l , t h e s p a c i n g i n c r e a s e d w i t h d o w n s t r e a m c o o r d i n a t e e x c e p t f o r t h e s h o r t f o r m a t i o n r e g i o n , a n d i n some c a s e s f a i l e d t o a t t a i n t h e u n i f o r m v a l u e w i t h i n t h e l i m i t o f t h e t r a v e r s e . A s p o i n t e d o u t by s e v e r a l i n v e s t i g a t o r s ^ , v o r t i c e s do n o t f l o w d o w n s t r e a m i n d e f i n i t e l y i n p a r a l l e l r o w s b u t a l w a y s move away f r o m t h e c e n t r e l i n e w i t h i n c r e a s -i n g y e v e n when an i n t e r m e d i a t e s e q u e n c e o f v o r t i c e s h a v e some u n i f o r m i t y o f c o n f i g u r a t i o n . T h i s p o i n t i s w e l l 70 e m p h a s i z e d b y t h e w a k e w i d t h r e s u l t s f o r f l a t p l a t e , 91 c i r c u l a r a n d e l l i p t i c c y l i n d e r s p r e s e n t e d i n T a b l e 5 - 5 . F i g u r e 5 - 2 3 V a r i a t i o n o f l a t e r a l v o r t e x s p a c i n g w i t h d o w n -s t r e a m c o o r d i n a t e a n d a n g l e o f a t t a c k 108 Table 5-5 V a r i a t i o n of l a t e r a l spacing with downstream co o r d i n a t e f o r s e v e r a l b l u f f bodies c i r c u l a r c y l i n d e r , ^ e = 0,R = 4 ,900 e l l i p t i c c y l i n d e r e = 0.92 ,cc = 90° ,R=68,000 f l a t p l a t e * e=~,a=90°,R=31,000 y/2a w/2a y/2a w/2a y/2a w/2a 3.22 1 .06 1 .0 0.72 5.0 1 . 30 4.60 1.19 3.0 1 .20 10.0 2 .00 7.00 1 .34 5.0 1 .20 20.0 2.75 8.80 1 .40 7.0 1 .40 t T y l e r , *Fage and Johansen L o n g i t u d i n a l and l a t e r a l spacing r e s u l t s may be com-bined to o b t a i n c l a s s i c a l wake geometry parameter, W/L ( F i g u r e 5-24). With i n c r e a s e i n downstream c o o r d i n a t e , i n g e n e r a l , there i s a drop in the value up t o , approximately, 2-4 diameters downstream f o l l o w e d by a region of constant value and f i n a l l y a tendency towards r i s e i n W/L. In p a r t i c u l a r f o r e=0.98, the r a t i o ranged between 0.4 to 0.6 at y/2a=1.0, diminished to -0.3 in the v i c i n i t y of y/2a=4.0 and showed upward trend t h e r e a f t e r . This c o r r e s -ponds reasonably well with Karman's o b s e r v a t i o n of W/L -0.36 behind the body approaching the value 0.281 f a r t h e r downstream. A r i s e in the wake geometry r a t i o when f a r downstream i s a l s o r e p o r t e d by s e v e r a l i n v e s t i g a t o r s 7 1 ^ 1 0 9 F i g u r e 5 - 2 4 Wake g e o m e t r y r a t i o f o r e l l i p t i c c y l i n d e r s a s a f u n c t i o n o f d o w n s t r e a m d i s t a n c e i n t h e f l a t p l a t e a n d c i r c u l a r c y l i n d e r s t u d i e s . I n f a c t F a g e a n d J o h a n s e n 7 ^ , i n t h e i r e x p e r i m e n t w i t h f l a t p l a t e n o r m a l t o t h e f l o w , f o u n d W/L t o i n c r e a s e by m o r e t h a n 1 0 0 % o v e r t h e d o w n s t r e a m d i s t a n c e o f 5-20 p l a t e w i d t h s . I t i s o f i n t e r e s t t o n o t e t h a t t h e l o c a t i o n o f m i n i m u m w a k e g e o m e t r y r a t i o s e e m s t o move d o w n s t r e a m w i t h i n c r e a s e i n e c c e n t r i c i t y o f an e l l i p s e . T h e w a k e w i d t h d i d n o t s h o w a n y w e l l d e f i n e d d e p e n d e n c e on c y l i n d e r a t t i t u d e . T h e n e a r i n f i n i t y v a l u e s f o r t h e w a k e w i d t h a n d g e o m e t r y r a t i o a s a f f e c t e d by c y l i n d e r e c c e n t r i c i t y a n d a t t i t u d e a r e s u m m a r i s e d i n T a b l e 5 - 6 . *The q u o t e d f i g u r e s o c c u r a r o u n d s i x t o s e v e n d i a m e t e r s d o w n s t r e a m . F o r ct=0, t h e w a k e w i d t h d i m i n i s h e s w i t h i n c r e a s e i n e c c e n t r i c i t y , h o w e v e r , t h e w a k e g e o m e t r y r a t i o i n c r e a s e s s u g g e s t i n g g r e a t e r r e d u c t i o n i n l o n g i t u d i n a l s p a c i n g . E s s e n t i a l l y t h e same b e h a v i o r was o b s e r v e d a t a = 3 0 ° . No c l e a r p a t t e r n c a n be d i s c e r n e d a t a = 6 0 ° , h o w e v e r , a t h i g h e r a t t i t u d e , t h e r e a p p e a r s t o be a c o m p l e t e r e v e r s a l i n t h e t r e n d o f ( W / ^ a ) ^ o b s e r v e d e a r l i e r . F u r t h e r m o r e , t h e w a k e g e o m e t r y r a t i o r e m a i n s r e a s o n a b l y c l o s e t o K a r m a n ' s s t a b i l i t y v a l u e o f 0 . 2 8 1 . I t w o u l d be a p p r o p r i a t e t o e m p h a s i z e a d e g r e e o f s i m i l a r i t y i n t h e v a r i a t i o n o f ( h / L ) ^ , ( W / L j ^ . S ^ a n d ( 1 / C J J ) w i t h m o d e l o r i e n t a t i o n . T h i s i s s h o w n i n F i g u r e 5 - 2 5 . I n g e n e r a l , t h e y a p p e a r t o a t t a i n t h e p e a k v a l u e a t a = 0 , d i m i n s h w i t h i n c r e a s e i n a n g l e o f a t t a c k t o r e a c h T a b l e 5-6 D e p e n d e n c e o f l a t e r a l v o r t e x s p a c i n g a n d w a k e g e o m e t r y r a t i o on e c c e n t r i c i t y a n d a n g l e o f a t t a c k e a = 0 a = 3 0 ° a = 6 0 ° a = 9 0 ° ( w / 2 a ) x ' ' oo ( w / 2 a ) x ' 00 ( w / 2 a ) * ' ' 00 ( W / L ) . ( w / 2 a ) r o ( W / L ) . 0* 1 . 3 - 1 . 8 0 . 2 4 - 0 . 2 8 0.44 1.12 0.27 1.10 0.26 1 .28 0.27 1.14 0.23 0 . 6 0 t 1 .0 0.31 1 .04 0.27 1 .14 0 .26 1.16 0.25 0 . 8 0 0.90 0.38 0.96 0.29 1.12 0.24 1 .20 0.24 0.92 0.62 0.48 0.96 0.32 1 .25 0.27 1 . 32 0.27 0 . 9 8 A - - 0.78 0.28 1 .25 0 . 2 8 1 .45 0.30 i i oo 1 .65 0.32 * S c h a e ' f e r a n d E s k i n a z i , +Modi a n d W i l a n d , A F a g e a n d J o h a n s e n 0.25—.1.5 —iCU5 112 (-) 0.5 H 0 . 2 5 0.15 - Jo —'0.05 —10-31 —f0.29 0.27 0.25 —\ 0.25 —i 1.5 - i 0^40 Huo H 0.20 -105 -"0.20 030 F i g u r e 5 - 2 5 V a r i a t i o n o f S t r o u h a l n u m b e r , d r a g c o e f f i c i e n t a n d n e a r i n f i n i t y v a l u e s o f w a k e p a r a m e t e r s w i t h c y l i n d e r a t t i t u d e : ( a ) e = 0 . 4 4 , 0 . 6 0 , 0 . 8 0 113 0.15 0 20 40 60 80 '00 -10.22—.3.0 H2 .0 - i0 .32 «0.20- 1.0 -40.30 0.2 8 H0.18 0.16 0.30—12.0 -i0.50 0.25—h.5 -^0.20- fl.O -A 0.40 0.30 0.15 - J 0 . 5 - ' 0.20 ± F i g u r e 5 - 2 5 V a r i a t i o n o f S t r o u h a l n u m b e r , d r a g c o e f f i c i e n t a n d n e a r i n f i n i t y v a l u e s o f wa k e p a r a m e t e r s w i t h c y l i n d e r a t t i t u d e : ( b ) e = 0 . 9 2 , 0 . 9 8 114 a m i n i m u m a r o u n d 5 0 ° f o l l o w e d by a s m a l l r i s e o v e r t h e r e m a i n i n g r a n g e . A p p a r e n t l y , t h i s s u g g e s t s s i g n i f i c a n t r e l a t i o n s h i p b e t w e e n t h e w a k e g e o m e t r y a n d b o d y a e r o -d y n a m i c s . T h e u s e o f u n i v e r s a l n u m b e r s i n s u c h a s t u d y b y s e v e r a l a u t h o r s may be a t t r i b u t e d t o t h e s i m i l a r o b s e r v a t i o n o f w a k e - b o d y i n t e r a c t i o n . T h e r e s u l t s o f l a t e r a l t r a v e r s e a l s o p r o v i d e i n f o r m a t i o n a b o u t v a r i a t i o n o f p e a k f l u c t u a t i n g p r e s s u r e w i t h d o w n s t r e a m c o o r d i n a t e ( F i g u r e 5 - 2 6 ) . I t a p p e a r s t h a t p e a k p r e s s u r e a m p l i t u d e a t t a i n s a maximum v a l u e a r o u n d o n e d i a m e t e r d o w n s t r e a m . T h e p r e s s u r e d e c a y t h a t f o l l o w s c o m p a r e s f a v o u r a b l y w i t h t h e a n a l y t i c a l r e s u l t s o b t a i n e d u s i n g v o r t e x s t r e e t m o d e l o f S c h a e f e r a n d E s k i n a z i ^ 4 . 66 27 F e r g u s o n ( c i r c u l a r c y l i n d e r ) a n d S l a t e r ( s t r u c t u r a l a n g l e s e c t i o n ) a l s o o b s e r v e d s i m i l a r d e c a y o f u n s t e a d y p r e s s u r e i n t h e w a k e . H o w e v e r , a s y s t e m a t i c s t u d y o f i t s r i s e a n d d e c a y i n t h e d o w n s t r e a m d i r e c t i o n a n d as a f u n c t i o n o f c y l i n d e r e c c e n t r i c i t y a n d a t t i t u d e d o e s n o t s e e m t o be r e p o r t e d i n l i t e r a t u r e . I n g e n e r a l , t h e i n c r e a s e i n a n g l e o f a t t a c k t e n d s t o move p r e s s u r e p e a k u p s t r e a m . T h i s may s u g g e s t c l o s e r f o r m a t i o n o f t h e f i r s t v o r t e x a t h i g h e r a n g l e s o f a t t a c k . T h e b e h a v i o r was c o n f i r m e d t h r o u g h f l o w v i s u a l i s a t i o n . F i g u r e 5-27 s h o w s t y p i c a l f r a m e s f r o m t h e h i g h s p e e d 16 mm. m o v i e o f t h e n e a r w a k e r e g i o n . I t p r o v i d e s some i n f o r m a t i o n c o n c e r n i n g t h e p o s i t i o n o f t h e f i r s t 115 F i g u r e 5 - 2 6 V a r i a t i o n o f p e a k f l u c t u a t i n g p r e s s u r e i n t h e w a k e : ( a ) e = 0 . 44 , 0 . 9 8 F i g u r e 5-26 V a r i a t i o n o f p e a k f l u c t u a t i n g p r e s s u r e i n t h e w a k e : ( b ) e = 0 . 9 2 , 0 . 8 0 F i g u r e 5-27 V i s u a l s t u d y o f n e a r w a k e s h o w i n g : ( a ) p o s i t i o n o f t h e f i r s t v o r t e x as a f f e c t e d by a n g l e o f a t t a c k e=o.60 F i g u r e 5-27 V i s u a l s t u d y o f n e a r wake s h o w i n g : ( a ) p o s i t i o n o f t h e f i r s t v o r t e x a s a f f e c t e d b y a n g l e o f a t t a c k 119 Figure 5-27 Visual study of near wake showing: (b) effect of eccentricity Figure 5-27 Visual study of near wake showing: (c) unsteady character of shear layer v o r t e x a n d h e n c e t h e v o r t e x f o r m a t i o n l e n g t h ' a s a f f e c t e d b y c y l i n d e r e c c e n t r i c i t y a n d a n g l e o f a t t a c k . S e v e r a l a s p e c t s o f i n t e r e s t b e c o m e a p p a r e n t : ( i ) T h e p h o t o g r a p h s s u g g e s t t h a t t h e f i r s t v o r t e x i s l o c a t e d a r o u n d 0 . 7 5 - 1 . 5 d i a m e t e r s d o w n s t r e a m f r o m t h e c y l i n d e r a x i s . T h i s i s i n a g r e e m e n t w i t h t h e l o c a t i o n o f p r e s s u r e p e a k s i n F i g u r e 5 - 2 6 . ( i i ) A c o m p a r i s o n o f v o r t e x p o s i t i o n s a t a=0 a n d 9 0 ° f o r a s e t o f e l l i p s e s s h o w s f o r w a r d m o v e m e n t o f t h e f i r s t v o r t e x a t h i g h e r a n g l e s o f a t t a c k . T h i s , i n g e n e r a l , c o n f i r m s t h e t r e n d i n d i c a t e d by p r e s s u r e ' p e a k s i n t h e w a k e . On t h e o t h e r h a n d , no c o n c l u s i v e r e m a r k s c a n be made a b o u t t h e e f f e c t o f e c c e n t r i c i t y a t t h e same a n g l e o f a t t a c k , ( i i i ) T h e u n s t e a d y c h a r a c t e r o f t h e p o s i t i o n o f s e p a r -a t i o n i s q u i t e e v i d e n t . F o r c i r c u l a r c y l i n d e r i t was o b s e r v e d t o b e , a p p r o x i m a t e l y , o v e r t h e r a n g e o f 7 0 ° t o 8 8 ° . T h i s m o v e m e n t i s a c c o m p a n i e d w i t h t h e p e r i o d i c c h a n g e i n t h e i n c l i n a t i o n o f s e p a r -a t i n g s h e a r l a y e r . T h e s e o b s e r v a t i o n s on t h e n e a r w a k e s h o u l d be c o n s i d e r e d p r e l i m i n a r y a n d h e n c e o n l y i n d i c a t i v e o f q u a l i t a t i v e f e a t u r e s o f t h e p h e n o m e n o n . 122 6. C O N C L U D I N G REMARKS B a s e d on t h e e x p e r i m e n t a l r e s u l t s t h e f o l l o w i n g g e n e r a l r e m a r k s c a n be made c o n c e r n i n g t h e u n s t e a d y a e r o -d y n a m i c s o f t h e s t a t i o n a r y e l l i p t i c c y l i n d e r s : ( i ) T h e mean l i f t , d r a g a n d p i t c h i n g m o ment show c o n -s i d e r a b l e d e p e n d e n c e on a n g l e o f a t t a c k a n d e c c e n t r i c i t y o f an e l l i p s e . H o w e v e r , t h e y a r e s u b s t a n t i a l l y i n d e p e n d e n t o f t h e R e y n o l d s n u m b e r 4 5 i n t h e r a n g e 2 x 1 0 - 1 0 . T h e b a l a n c e r e s u l t s com-p a r e f a v o u r a b l y w i t h t h e p r e s s u r e i n t e g r a t e d s t e a d y a e r o d y n a m i c c o e f f i c i e n t s t h u s i n d i c a t i n g t w o - d i m e n s i o n a l c h a r a c t e r o f t h e mean f l o w . T h e p l o t s s u g g e s t t h a t t h e e l l i p s e w i t h e c c e n t r i c i t y o f 0 . 9 8 i s s u s c e p t i b l e t o g a l l o p i n g t y p e o f i n s t a b i l i t y i n t h e v i c i n i t y o f a = 1 0 ° . ( i i ) E c c e n t r i c i t y a n d t h e a t t i t u d e o f t h e m o d e l a f f e c t mean p r e s s u r e d i s t r i b u t i o n on i t s s u r f a c e s u b s t a n -t i a l l y . T h e l o c a t i o n o f m i n i m u m p r e s s u r e on t h e t o p s u r f a c e m o v e s f o r w a r d w h i l e t h a t on t h e l o w e r s u r f a c e m o v e s r e a r w a r d s w i t h i n c r e a s e i n a n g l e o f a t t a c k . H o w e v e r , t h i s t r e n d r e v e r s e s a t a r o u n d 4 0 ° . T h e s t u d y o f s e p a r a t i n g s h e a r l a y e r t h r o u g h f l o w v i s u a l i s a t i o n u s i n g S c h l i e r e n t e c h n i q u e 123 c o n f i r m e d t h i s t r e n d . T h e a g r e e m e n t b e t w e e n t h e a n a l y t i c a l d a t a on t h e s e p a r a t i o n p o i n t s , o b t a i n e d u s i n g G o r t l e r ' s s e r i e s s o l u t i o n , a n d t h e e x p e r i m e n t a l v a l u e s a p p e a r s t o be q u i t e a c c e p t a b l e . ( i i i ) T h e S t r o u h a l f r e q u e n c y i n c r e a s e s l i n e a r l y w i t h w i n d s p e e d i n t h e R e y n o l d s n u m b e r r a n g e o f 4 5 2 x 1 0 -10 . S t r o u h a l n u m b e r e x h i b i t s c o n s i d e r a b l e d e p e n d e n c e on t h e c y l i n d e r e c c e n t r i c i t y a n d a n g l e o f a t t a c k , r a n g i n g f r o m 0 . 0 3 7 ( e = 0 . 9 8 , a = 9 0 ° ) t o 0 . 2 7 4 ( e = 0 . 9 2 , a = 0 ) . T h e v a r i a t i o n o f S t r o u h a l n u m b e r c a n be r e d u c e d by b a s i n g i t on t h e p r o j e c -t e d d i m e n s i o n , w a k e w i d t h o r t r a n s v e r s e d i s t a n c e b e t w e e n s e p a r a t i o n - p o i n t s . R o s h k o ' s c r i t e r i o n p r o v i d e s s a t i s f a c t o r y c o l l a p s e a r o u n d t h e v a l u e o f 0 . 1 6 4 e x c e p t f o r t h e s l e n d e r e l l i p s e s a t s m a l l a n g l e s o f a t t a c k . H o w e v e r , S<. h a s t h e a d v a n t a g e o f o f f e r i n g t h e s m o o t h e r t r e n d e v e n a t l o w a n g l e s o f a t t a c k f o r e l l i p s e s o f h i g h e c c e n t r i c i t y , ( i v ) T h e m i d s p a n f l u c t u a t i n g p r e s s u r e d i s t r i b u t i o n i s e s s e n t i a l l y i n d e p e n d e n t o f t h e R e y n o l d s n u m b e r e x c e p t f o r t h e e l l i p s e s a t a=0 i n t h e r a n g e 2 . 5 x 4 5 10 -10 . I n g e n e r a l , f o r a g i v e n e l l i p s e , t h e m a ximum f l u c t u a t i n g p r e s s u r e c o e f f i c i e n t i n c r e a s e s w i t h i n c r e a s e i n a n g l e o f a t t a c k . T h e p r e s s u r e s i g n a l s a r e a l w a y s a m p l i t u d e m o d u l a t e d a l t h o u g h 124 t h e e x t e n t o f m o d u l a t i o n i s l e s s i n t h e l a m i n a r r e g i o n . C o n s i d e r a b l e p h a s e d i f f e r e n c e c a n e x i s t e v e n b e t w e e n t h e p r e s s u r e s i g n a l s on t h e same s i d e o f t h e m o d e l . F o r t h e s l e n d e r e l l i p s e o f e = 0 . 9 2 , a t cc=0, t h e p h a s e c h a n g e a s l a r g e a s 6 5 ° was r e c o r d e d , ( v ) I n g e n e r a l , t h e f l u c t u a t i n g l i f t c o e f f i c i e n t i n c r e a s e s w i t h i n c r e a s e i n a n g l e o f a t t a c k r e a c h -i n g m a ximum v a l u e a r o u n d 7 0 ° - 8 0 ° , b e y o n d w h i c h i t t e n d s t o d r o p . H i g h e s t f l u c t u a t i n g l i f t r e c o r d e d was - 1 . 0 f o r e = 0 . 6 0 . T h e p h a s e b e t w e e n t h e p r e s -s u r e s i g n a l s h a s n e g l i g i b l e e f f e c t on u n s t e a d y l i f t d a t a . ( v i ) T h e u n s t e a d y a e r o d y n a m i c s o f e l l i p t i c c y l i n d e r s e x h i b i t s t h r e e d i m e n s i o n a l c h a r a c t e r , f l u c t u a t i n g p r e s s u r e a n d p h a s e b e i n g s u b s t a n t i a l l y d e p e n d e n t on s p a n w i s e l o c a t i o n . A v e r a g e i n c l i n a t i o n o f v o r t e x l i n e s was f o u n d t o be a r o u n d 0 - 1 1 ° w i t h i m p r o v e d a l i g n m e n t w i t h t h e c y l i n d e r a x i s a t h i g h e r a n g l e s o f a t t a c k . ( v i i ) B o t h l o n g i t u d i n a l a n d l a t e r a l s p a c i n g s b e t w e e n t h e v o r t i c e s a r e s u b s t a n t i a l l y i n d e p e n d e n t - o f t h e 4 5 R e y n o l d s n u m b e r i n t h e r a n g e 3 x 1 0 -10 . L o n g i t u d i n a l s p a c i n g b e t w e e n v o r t i c e s i n c r e a s e s r a p i d l y b e h i n d t h e m o d e l a n d a t t a i n s a c o n s t a n t v a l u e a r o u n d s e v e n d i a m e t e r s ( l i m i t o f t h e t r a v e r -s i n g g e a r ) d o w n s t r e a m . 125 I t i n c r e a s e s w i t h i n c r e a s e i n a n g l e o f a t t a c k , t h e l a r g e r c h a n g e b e i n g a s s o c i a t e d w i t h h i g h e c c e n t r i c i t y e l l i p s e s . T h e v o r t e x v e l o c i t y h a s r e l a t i v e l y s m a l l d e p e n d e n c e on a t t i t u d e a n d e c c e n t r i c i t y o f a n e l l i p s e . T h e l a t e r a l s p a c i n g c o n t i n u e s t o i n c r e a s e w i t h d o w n s t r e a m c o o r d i n a t e , f o r some c a s e s e v e n b e y o n d t h e l i m i t o f t r a v e r s e . A t s m a l l a n g l e s o f a t t a c k , t h e e f f e c t o f i n c r e a s e i n e c c e n t r i c i t y i s t o d i m i n i s h t h e w a k e w i d t h . H o w e v e r , a t a = 9 0 ° , t h e t r e n d u n d e r g o e s a c o m p l e t e r e v e r s a l . A c e r t a i n a m o u n t o f s i m i l a r i t y c a n be d i s -c e r n e d b e t w e e n t h e v a r i a t i o n o f ( h / L ) , ( b / a ) , % ' ' 00 ' oo S h a n d 1 / C D w i t h a n g l e o f a t t a c k . T h i s e m p h a s i z e s h t h e i n t e r a c t i o n b e t w e e n w a k e g e o m e t r y a n d b o d y a e r o d y n a m i c s . I n g e n e r a l , t h e w a k e g e o m e t r y r a t i o r e m a i n s c l o s e t o K a r m a n s t a b i l i t y v a l u e o f 0 . 2 8 1 . ( v i i i ) T h e r i s e a n d d e c a y o f f l u c t u a t i n g p r e s s u r e i n t h e w a k e s u g g e s t s f o r m a t i o n o f t h e f i r s t v o r t e x t o o c c u r i n t h e v i c i n i t y o f o n e d i a m e t e r d o w n s t r e a m . T h e f l o w v i s u a l i s a t i o n c o n f i r m s t h i s o b s e r v a t i o n , ( i x ) T h e h i g h s p e e d p h o t o g r a p h y i n d i c a t e s , q u a l i t a t i v e l y , u n s t e a d y c h a r a c t e r o f p o s i t i o n a n d i n c l i n a t i o n o f t h e s e p a r a t i n g s h e a r l a y e r s . 7. RECOMMENDATIONS FOR F U T U R E WORK T h e c o n c e p t o f a s y s t e m a t i c c h a n g e o f e c c e n t r i c i t y t o c o v e r a w i d e r a n g e o f g e o m e t r i c a l s h a p e s , t h o u g h a t t r a c ^ t i v e , d o e s n o t s e e m t o h a v e r e c e i v e d s u f f i c i e n t a t t e n t i o n i n t h e s t u d y o f b l u f f b o d y a e r o d y n a m i c s . T h r o u g h t h e i n v e s t i g a t i o n p r e s e n t e d h e r e come t o l i g h t s e v e r a l a r e a s w h e r e a v a i l a b l e i n f o r m a t i o n i s l i m i t e d t h u s s u g g e s t i n g p o s s i b l e c o u r s e f o r f u t u r e r e s e a r c h . Some o f t h e m o r e i m p o r t a n t p r o b l e m s a r e l i s t e d b e l o w : ( i ) I t w o u l d be u s e f u l t o e s t a b l i s h t h e l i m i t o f a p p l i c a b i l i t y o f t h e G o r t l e r ' s s e r i e s a p p r o a c h f o r d e t e r m i n i n g s e p a r a t i o n p o i n t s , w h i c h , a s p o i n t e d o u t i n t h e t e x t , c o u l d be u s e d o n l y i n t h e a n a l y s i s o f c y l i n d e r s a t z e r o a n g l e o f a t t a c k , p a r t i c u l a r l y f o r h i g h e c c e n t r i c i t y e l l i p -s e s . T h e p r o b a b l e l i m i t i n g f a c t o r i s l i k e l y t o be s t e e p p r e s s u r e g r a d i e n t s on t h e s u r f a c e o f t h e m o d e l a t h i g h e r a n g l e s o f a t t a c k . T h e u s e o f m o r e t e r m s i n E, s e r i e s may e x t e n d t h e l i m i t o f a p p l i c a b i l i t y o f t h i s a p p r o a c h . T h e s t u d y w o u l d r e q u i r e , o b v i o u s l y , m e a s u r e m e n t o f mean p r e s s u r e d i s t r i b u t i o n a t f i n e r i n t e r v a l o f a t t i t u d e . ( i t i ) T h e d e p e n d e n c e o f f l u c t u a t i n g p r e s s u r e on t h e R e y n o l d s n u m b e r a t a=0 s h o u l d be e x p l o r e d f u r t h e r A l t h o u g h t h e p r e l i m i n a r y r e s u l t s s u g g e s t c o r r e l -a t i o n b e t w e e n t h i s b e h a v i o u r a n d b a s e p r e s s u r e , a d e t a i l e d i n v e s t i g a t i o n i s n e c e s s a r y t o p r o v e i t c o n c l u s i v e l y . A p p a r e n t l y , t h e p h e n o m e n o n i s l i k e l y t o be a s s o c i a t e d w i t h c h a n g e s i n s e p a r a t i o n a n d n e a r w a k e c o n d i t i o n . T h e m e a s u r e m e n t o f i n s t a n t a n e o u s f l u c t u a t i n g p r e s s u r e o r s h e a r s t r e s s w i t h a v i e w t o ' e s t a b l i s h e x c u r s i o n o f t h e s e p a r -a t i o n p o i n t s , a n d t h e l e n g t h o f f o r m a t i o n r e g i o n 21 a s d i s c u s s e d b y G e r r a r d may t h r o w some l i g h t on t h i s u n e x p l a i n e d b e h a v i o u r . F l o w v i s u a l i s a t i o n s t u d i e s may a l s o p r o v e t o be o f some u s e . ( i i i ) A s y s t e m a t i c s t u d y o f t h e t h r e e d i m e n s i o n a l c h a r a c t e r o f f l o w a r o u n d e l l i p t i c c y l i n d e r s a s a f u n c t i o n o f e n d c o n d i t i o n s , e c c e n t r i c i t y a n d a t t i t u d e s h o u l d p r o v i d e i n f o r m a t i o n o f f u n d a m e n t a l i n t e r e s t . T h e i n v e s t i g a t i o n o f p r e s s u r e c o r r e l -a t i o n l e n g t h i n s p a n w i s e d i r e c t i o n a n d i n c l i n a t i o n o f v o r t e x f i l a m e n t w o u l d s u p p l e m e n t t h e e x i s t i n g i n f o r m a t i o n on c i r c u l a r c y l i n d e r p r e s e n t e d b y 36 37 39 40 4 P r e n d e r g a s t , e l B a r o u d i , T o e b e s , K o o p m a n n ' e t a l . ( i v ) A s y s t e m a t i c s t u d y o f w a k e a s s o c i a t e d w i t h c i r c u l a r a n d e l l i p t i c a l c y l i n d e r s w i t h p a r t i c u l a r e m p h a s i s on f a r d o w n s t r e a m b e h a v i o u r w o u l d be r e l e v e n t . 1 28 ( v ) T h e a e r o d y n a m i c s a n d d y n a m i c s o f e l l i p t i c c y l i n d e r s i n v o r t e x i n d u c e d a n d g a l l o p i n g mode o f o s c i l l a t i o n s a p p e a r t o be t h e l o g i c a l e x t e n -s i o n o f t h i s s t u d y . 129 B I B L I O G R A P H Y 1. S t r o u h a l , V . , " U b e r e i n e b e s o n d e r e A r t d e r T o n e r r e g u n g , " W i e d . A n n . P h y s i k u . Chem. , N e u e F o l g e , V o l . V , 1 8 7 8 , p p . 2 1 6 - 2 5 1 . 2. K a r m a n , T h . V o n . ," F l u'gg i g k e j . t u . L u f t w i d e r s t a n d ," P h y s . 1., V o l . 1 3 , 1 9 1 1 , p . 4 9 ; " U b e r d e n M e c h a n i s m u s d e s F l u s s i g k e i t s - u n d L u f t w i d e r s t a n d e s ," P h y s i k . I. , V o l . 1 3 , 1 9 1 2 , p p . 4 9 - 5 9 ; " U b e r d e n M e c h a n i s m u s d e s W i d e r s t a n d s , d e n e i n b e w e g t e r K o r p e r i n e i n e r F l u ' s s i g k e i t e r f a h r t , " G o t t i n g e r N a c h r i c h t e n , M a t h . P h y s i k . K l . , 1 9 1 1 , p p . 5 0 9 -5 1 9 ; a n d , 1 9 1 2 , p p . 5 4 7 - 5 5 6 . 3. H e i s e n b e r g , W. V o n . , " D i e A b s o l u t e n D i m e n s i o n e n d e r K a r m a n s c h e n W i r b e l b e w e g u n g ," P h y s i k . I. , V o l . 23 , 1 9 2 2 , p p . 3 6 3 - 3 6 6 . 4. B g n a r d . H., " H y d r o d y n a m i q u e e x p e r i m e n t a l e - f o r m a t i o n de c e n t r e s de g i r a t i o n a l ' a r r i e r e d ' u n o b s t a c l e e n m o u v e -m e n t i " C o m p t e s R e n d u s ( A c a d e m i e d e s S c i e n c e s ) , V o l . 1 4 7 , 1 9 0 8 , pp~. 8 3 9 - 8 4 2 ; "TTydrodynami q u e e x p e r i m e n t a l e - s u r ^ l e s l o i s rje l a f r e q u e n c e d e s t o u r b i l l o n s a t t e r n e s d e t a c h e s d e r r i e r e un o b s t a c l e , " C o m p t e s R e n d u s , V o l . 1 8 2 , 1 9 2 6 , p p . 1 3 7 5 - 1 3 7 7 ; " H y d r o d y n a m i q u e e x p e > i m e n t a l e - s u r 1' i n e x a t i t u d e , p o u r l e s l i q u i d e s r e e l s , d e s l o i s t h e o r i q u e s d e K a r m a n r e l a t i v e s a l a s t a b i l i t e d e s t o u r b i l l o n s a l t e r n e s , " C o m p t e s R e n d u s , V o l . 1 8 2 , 1 9 2 6 , p p . 1 5 2 3 -1 5 2 5 . 5. R o s h k o , A . , "On t h e D r a g a n d S h e d d i n g F r e q u e n c y 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 , " NACA, T e c h . N o t e 3 1 6 9 , 1 9 5 3 . 6. R o s h k o , A . , "On t h e D e v e l o p m e n t o f T u r b u l e n t Wakes F r o m V o r t e x S t r e e t s , " N ACA, T e c h . N o t e 291 3, 1 9 5 4 . 7. R o s h k o , A . , " A New H o d o g r a p h f o r F r e e - S t r e a m l i n e T h e o r y , " NACA, T e c h . N o t e 3 1 6 8 , J u l y 1 9 5 4 . 8. R o s h k o , A . , "On t h e Wake a n d D r a g o f B l u f f B o d i e s , " J . A e r o n a u t i c a l S c i e n c e s , V o l . 2 2 , No. 2, F e b r u a r y 1 9 5 5 , p p . 1 2 4 - 1 3 3 . 9. R o s h k o , A . , " E x p e r i m e n t s on 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 a t V e r y H i g h R e y n o l d s N u m b e r , " J . F l u i d M e c h . , V o l . 1 0 , 1961 , p p . 3 4 5 - 3 5 6 . 130 1 0 . R o s h k o , A . , " T r a n s i t i o n i n I n c o m p r e s s i b l e N e a r - W a k e s , " T h e P h y s i c s o f F l u i d s S u p p l e m e n t , V o l . 1 0 , No. 9, 1 9 6 7 , p p . S 1 8 1 - S 1 8 3 . 1 1 . K o v a s z n a y , L . S . G . , " H o t - W i r e I n v e s t i g a t i o n o f t h e Wake B e h i n d C y l i n d e r s a t Low R e y n o l d s N u m b e r s , " P r o c . R o y . S o c . o f L o n d o n , S e r i e s A, V o l . 1 9 8 , 1 9 4 9 , p p . 1 7 4 - 1 9 0 . 1 2 . R o s e n h e a d , L . , " T h e K a r m a n S t r e e t o f V o r t i c e s i n a C h a n n e l o f F i n i t e B r e a d t h , " P h i l . T r a n s . , S e r i e s A , V o l . 2 2 8 , J u n e 1 9 2 9 , p p . 2 7 5 - 3 2 9 . 1 3 . R o s e n h e a d , L . a n d S c h w a b e , M., "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 F l o w B e h i n d C i r c u l a r C y l i n d e r s i n C h a n n e l s o f D i f f e r e n t B r e a d t h s , " P r o c . R o y . S o c . o f  L o n d o n , S e r i e s A , V o l . 1 29 , 1 9 3 0 , p p . 1 1 5 - 1 3 5 . 1 4 . S c h a e f e r , J.W. a n d E s k i n a z i , S., "An A n a l y s i s o f t h e V o r t e x S t r e e t G e n e r a t e d i n a V i s c o u s F l u i d , " J . F l u i d  M e c h . , V o l . 6 , 1 9 5 9 , p p . 241 - 2 6 0 . 1 5 . G e r r a r d , J . H . , " A n 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 M e c h . , V o l . 1 1 , 1961 , p p . 2 4 4 - 2 5 6 . 1 6 . G e r r a r d , J . H . , " T h e C a l c u l a t i o n o f t h e F l u c t u a t i n g L i f t on a C i r c u l a r C y l i n d e r a n d i t s A p p l i c a t i o n t o t h e D e t e r m i n a t i o n o f A e o l i a n T o n e I n t e n s i t y , " AGARD R e p . No. 4 6 3 , A p r i l , 1 9 6 3 . 1 7 . G e r r a r d , J . H . , "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 N u m b e r R a n g e 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 M e c h . , V o l . 2 2 , P a r t I , 1 9 6 5 , p p . 187-1 9 6 . 1 8 . G e r r a r d , J . H . , " T h e T h r e e - D i m e n s i o n a l S t r u c t u r e o f t h e Wake o f a C i r c u l a r C y l i n d e r , " J . F l u i d M e c h . , V o l . 2 5 , P a r t I , 1 9 6 6 , p p . 1 43-1 6 4 . 1 9 . G e r r a r d , J . H . , " N u m e r i c a l C o m p u t a t i o n o f t h e M a g n i t u d e a n d F r e q u e n c y o f t h e L i f t on a C i r c u l a r C y l i n d e r , " T r a n s . R o y a l S o c i e t y o f L o n d o n . , S e r i e s A , V o l . 2 6 1 , J a n u a r y 1 9 6 7 , p p . 1 3 7 - 1 6 2 . 2 0 . G e r r a r d , J . H . , " 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 S e p a r a t e d B o u n d a r y L a y e r U n d e r g o i n g T r a n s i t i o n t o T u r b u l e n c e , " T h e P h y s i c s o f F l u i d s S u p p l e m e n t , V o l . 1 0 , No. 9, 1 9 6 7 , p p . S 9 8 - S 1 0 0 . 131 2 1 . G e r r a r d , J . H . , " T h e M e c h a n i c s o f t h e F o r m a t i o n R e g i o n o f V o r t i c e s B e h i n d B l u f f B o d i e s , " J . F l u i d M e c h . , V o l . 2 5 , P a r t 2, 1 9 6 8 , p p . 4 0 1 - 4 1 3 . 2 2 . R o s e n h e a d , L . , " V o r t e x S y s t e m s i n W a k e s , " A d v . In  A p p l i e d M e c h a n i c s , V o l . 3, 1 9 5 3 , p p . 1 8 5 - 1 9 5 . o 2 3 . W i l l e , R., " U b e r S t r b ' m u n g s e r s c h e i n u n g e n im U b e r g a n g -s g e b i e t v o n g e o r d n e t e r z u u n g e o r d n e t e r B e w e g u n g , " J a h r b . S c h i f f b a u t e c h n . G e s . , V o l . 46 , 1 952 , p p . 1 76-1 8 7 . 2 4 . W i l l e , R., " K a r m a n V o r t e x S t r e e t s ," A d v . i n A p p l i e d  M e c h a n i c s , V o l . 6 , 1 9 6 0 , p p . 2 7 3 - 2 8 7 . 2 5 . M a r r i s , A.W., "A R e v i e w on V o r t e x S t r e e t s , P e r i o d i c W a k e s , a n d I n d u c e d V i b r a t i o n P h e n o m e n a , " J . B a s i c E n g . , A S M E , V o l . 8 6 , 1 9 6 4 , p p . 1 8 5 - 1 9 6 . 2 6 . M o r k o v i n , M.V., " F l o w A r o u n d C i r c u l a r C y l i n d e r - A K a l e i d o s c o p e o f C h a l l e n g i n g F l u i d P h e n o m e n a , " Sym. on F u l l y S e p a r a t e d F l o w s , A S M E , 1 9 6 4 , p p . 1 0 2 - 1 1 8 . 2 7 . S l a t e r , J . E . , " A e r o e l a s t i c I n s t a b i l i t y o f A S t r u c t u r a l A n g l e S e c t i o n , " 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 , P h . D . T h e s i s , M a r c h 1 9 6 9 . 2 8 . M c G r e g o r , D.M., "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 P r e s s u r e s on a C i r c u l a r C y l i n d e r i n a F l u i d - S t r e a m ," U n i v e r s i t y o f T o r o n t o I n s t , o f A e r o -p h y s i c s , T e c h . N o t e 1 4 , 1 9 5 7 . 2 9 . K e e f e , R . T . , "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 , a n d o f t h e A s s o c i a t e d S o u n d F i e l d , " U n i v e r s i t y o f T o r o n t o I n s t , o f A e r o p h y s i c s , R e p o r t 76 , 1 961 . 3 0 . M o l i n e u x , W.G., " M e a s u r e m e n t o f t h e A e r o d y n a m i c F o r c e s on O s c i l l a t i n g A i r f o i l s , " AGARD R e p o r t 3 5 , 1 9 5 6 . 3 1 . H u m p h r e y s , J . S . , "On a C i r c u l a r C y l i n d e r i n a S t e a d y W i n d a t T r a n s i t i o n R e y n o l d s N u m b e r s , " J . F l u i d M e c h . , V o l . 9, 1 9 6 0 , p p . 6 0 3 - 6 1 2 . 3 2 . F u n g , Y . C . , " F l u c t u a t i n g L i f t a n d D r a g A c t i n g on a C y l i n d e r i n a F l o w a t S u p e r c r i t i c a l R e y n o l d s N u m b e r , " J . A e r o s p a c e S c i e n c e s , V o l . 2 7 , No. 1 1 , N o v e m b e r 1 9 6 0 , p p . 8 0 1 - 8 1 4 . 132 3 3 . G r o v e , A . S . , S h a i r , F . H . , P e t e r s o n , E . E . a n d A c r i v o s , A . , "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 S t e a d y S e p a r a t e d 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 M e c h . , V o l . 1 9 , P a r t 1, 1 9 6 4 , p p . 6 0 - 8 0 . 3 4 . B i s h o p , R . E . D . , a n d H a s s a n , A . Y . , " T h e L i f t a n d D r a g F o r c e s on a C i r c u l a r C y l i n d e r i n a F l o w i n g F l u i d , " P r o c . R o y . S o c . o f L o n d o n , S e r i e s A , V o l . 277 , 1 9 6 4 , p p . 3 2 - 5 0 . 3 5 . F e r g u s o n , N. a n d P a r k i n s o n , G.V., " S u r f a c e a n d Wake F l o w P h e n o m e n a o f t h e V o r t e x - E x c i t e d O s c i l l a t i o n o f a C i r c u l a r C y l i n d e r , " ASME V i b r a t i o n C o n f e r e n c e , P a p e r 6 7 - V i b r . - 3 1 , 1 9 6 7 . 3 6 . P r e n d e r g a s t , V . , " M e a s u r e m e n t o f T w o - P o i n t C o r r e l a t i o n s o f t h e S u r f a c e P r e s s u r e on a C i r c u l a r C y l i n d e r , " U n i v e r s i t y o f T o r o n t o I n s t , o f A e r o p h y s i c s , T e c h . N o t e 2 3 , 1 9 5 8 . 3 7 . EL B a r o u d i , M.Y., " M e a s u r e m e n t o f T w o - P o i n t C o r r e l -a t i o n s ^ o f V e l o c i t y N e a r 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 a K a r m a n V o r t e x S t r e e t , " U n i v e r s i t y o f T o r o n t o I n s t . o f A e r o p h y s i c s , T e c h . N o t e 3 1 , 1 9 6 0 . 3 8 . M a t t i n g l y , G . E . , "An E x p e r i m e n t a l S t u d y o f t h e T h r e e D i m e n s i o n a l i t y o f t h e 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 , " U n i v e r s i t y o f M a r y l a n d , T e c h . N o t e B N - 2 9 5 , 1 9 6 2 . 3 9 . T o e b e s , G.H., " F l u i d e l a s t i c F e a t u r e s o f F l o w A r o u n d C y l i n d e r s , " I n t . R e s e a r c h S e m i n a r : W i n d E f f e c t s on  B l d g s . a n d S t r u c t u r e s , NRC, O t t a w a , V o l . 2, S e p t e m b e r 1 9 6 7 , p p . 3 2 3 - 3 4 1 . 4 0 . K o o p m a n n , G . H . , "On t h e W i n d I n d u c e d V i b r a t i o n s o f C i r c u l a r C y l i n d e r s , " C a t h o l i c U n i v e r s i t y , M . A . S c . T h e s i s , M a r c h 1 9 6 7 . 4 1 . K o o p m a n n , G . H . , " T h e V o r t e x W a k es o f V i b r a t i n g C y l i n d e r s a t Low R e y n o l d s N u m b e r s , " J . F l u i d M e c h . , V o l . 2 8 , 1 9 6 7 , p p . 5 0 1 - 5 1 2 . 4 2 . F e n g , C . C . , " T h e M e a s u r e m e n t o f V o r t e x I n d u c e d E f f e c t s i n F l o w P a s t S t a t i o n a r y a n d O s c i l l a t i n g C y l i n d e r a n d D - S e c t i o n C y l i n d e r s , " 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 , M . A . S c . T h e s i s , O c t o b e r 1 9 6 8 . 4 3 . S u r r y , D., " T h e E f f e c t o f H i g h I n t e n s i t y T u r b u l e n c e on t h e A e r o d y n a m i c s o f a R i g i d C i r c u l a r C y l i n d e r a t S u b c r i t i c a l R e y n o l d s N u m b e r , " U n i v e r s i t y o f T o r o n t o  I n s t , f o r A e r o s p a c e S t u d i e s , R e p o r t 1 4 2 , O c t o b e r 1 9 6 9 . 133 4 4 . P a r k i n s o n , G.V. a n d B r o o k s , N . P . H . , "On t h e A e r o e l a s t i c I n s t a b i l i t y o f B l u f f C y l i n d e r s , " J . A p p l . M e c h . , A S M E , V o l . 8 3 , No. 2, J u n e 1 9 6 1 , p p . 2 5 2 - 2 5 8 . 4 5 . T o e b e s , G.H. a n d E a g l e s o n , P . S . , " H y d r o e l a s t i c V i b r a t i o n s o f F l a t P l a t e s R e l a t e d t o T r a i l i n g E d g e G e o m e t r y , " J . B a s i c E n g n g . , A S M E , V o l . 8 3 , No. 4, D e c e m b e r 1 9 6 1 , p p . 671 -678". 4 6 . E a g l e s o n , P . S . , N o u t s c p o u 1 o s , G.K. a n d D a i l y , J.W., " T h e N a t u r e o f S e l f - E x c i t a t i o n i n t h e F l o w - I n d u c e d V i b r a t i o n o f F l a t P l a t e s , " J . B a s i c E n g n g . , A S M E , V o l . 8 6 , No. 3, S e p t e m b e r 1 9 6 4 , p p . 5 9 9 - 6 0 6 . 4 7 . P a r k i n s o n , G.V. a n d S m i t h J . D . , " T h e S q u a r e P r i s m a s an A e r o e l a s t i c N o n - L i n e a r O s c i l l a t o r , " Q u a r t . J . M e c h . a n d A p p l . M a t h . , V o l . 1 7 , P a r t 2, May 1 9 6 4 , p p . 2 2 5 -239 . 4 8 . W a r d l a w , R . L . a n d D a v e n p o r t , A . G . , "Some E x p e r i m e n t s on t h e F l u c t u a t i n g F o r c e s on F l a t P l a t e s i n T u r b u l e n t F l o w , " 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 C a n a d a , A e r o n a u -t i c a l R e p o r t L R - 4 1 6 , D e c e m b e r 1 9 6 4 . 4 9 . M o d i , V . J . a n d H e i n e , W. , "On t h e P r e s s u r e F l u c t u a t i o n s a n d Wake G e o m e t r y A s s o c i a t e d w i t h S e v e r a l B l u f f B o d i e s , " J S M E P r o c . 1 5 t h J a p a n N a t . C o n g . A p p l . M e c h . , 1 9 6 5 , p p . 7 - 1 8 . 5 0 . S c h r a m m , W., " W i r b e l f r e q u e n z m e s s u n g e n an u m s t r o m t e n B a u t e i l e n , " I f L - M i t t . , V o l . 5, No. 8, 1 9 6 6 , p p . SOS-S I S . 5 1 . V i e k e r y , B . J . , " F l u c t u a t i n g L i f t a n d D r a g on a L o n g C y l i n d e r o f S q u a r e C r o s s - S e c t i o n i n a S m o o t h a n d i n a T u r b u l e n t S t r e a m , " J . F l u i d M e c h . , V o l . 2 5 , P a r t 3, 1 9 6 6 , p p . 4 8 1 - 4 9 4 . 5 2 . P a r k i n s o n , G.V. a n d S a n t o s h a m , T . V . , " C y l i n d e r s o f R e c t a n g u l a r S e c t i o n a s A e r o e l a s t i c N o n l i n e a r O s c i l l a t o r s , " ASME V i b . C o n f . , P a p e r No. 6 7 - V I B R - 5 0 , M a r c h 1 9 6 7 . 5 3 . W a r d l a w , R . L . , " A e r o d y n a m i c a l l y E x c i t e d V i b r a t i o n s o f a 3 - i n c h x 3 - i n c h A l u m i n u m A n g l e i n S t e a d y F l o w , " 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 C a n a d a , A e r o n a u t i c a l R e p o r t No. L R - 4 8 2 , J u n e 1 967. 5 4 . M o d i , V . J . a n d S l a t e r , J . E . , "On t h e A e r o e l a s t i c I n s t a b i l i t y o f A S t r u c t u r a l A n g l e , " J S M E S e m i - I n t e r -n a t i o n a l Sym. , S e p t e m b e r 1 9 6 7 , p p . 2 0 7 - 2 1 4. 134 5 5 . K o s k o , E . , " T h e F r e q u e n c y S p e c t r u m o f a S t r u c t u r a l M e m b e r i n C o u p l e d F l e x u r a l - T o r s i ona.l V i b r a t i o n , " J . S o u n d a n d V i b r a t i o n , V o l . 7, No. 2, 1 9 6 8 , p p . 1 4 3 -1 5 5 . 5 6 . P e a k e , D . J . , " T h r e e - D i m e n s i o n a l F l o w S e p a r a t i o n s on U p s w e p t R e a r F u s e l a g e s , " C a n a d i a n A e r o n a u t i c s a n d  S p a c e J o u r . , V o l . 1 5 , No. 1 0 , D e c e m b e r 1 9 6 9 , p p . 3 9 9 -4 0 8 . 5 7 . M o d i , V . J . a n d W i l a n d , E . , " U n s t e a d y A e r o d y n a m i c s o f S t a t i o n a r y E l l i p t i c C y l i n d e r s i n S u b c r i t i c a l F l o w , " C A S I / A I A A S u b s o n i c A e r o - a n d - H y d r o - D y n a m i c s M e e t i n g , P a p e r No. 6 9 - 7 4 5 , J u l y 1 9 6 9 ; I n P r e s s A I A A J o u r . 5 8 . H i e m e n z , K., " D i e G r e n z s c h i c h t an e i n e m i n d e n g l e i c h -f o r m i g e n F l u'ss i g k e i t s s t r o m e i n g e t a u c h t e n g e r a d e n K r e i s z y 1 i n d e r ," D i s s e r t a t i o n G b t t i n q e n , 1 9 1 1 . 5 9 . C h i u , W.S., " T h e B o u n d a r y L a y e r F o r m a t i o n ' a n d V o r t e x S h e d d i n g on Y a w e d C y l i n d e r s , " W a s h i n g t o n S t a t e U n i v e r -s i t y , C o l 1 e g e o f E n g n g . , B u i 1 e t i n 299 , 1 9 6 6 . 6 0 . A c h e n b a c h , A . , " D i s t r i b u t i o n o f L o c a l P r e s s u r e a n d S k i n F r i c 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 C r o s s -F l o w up t o R e = 5 x l 0 6 , " J . F l u i d M e c h . , V o l . 3 4 , P a r t 4, 1 9 6 8 , p p . 6 2 5 - 6 3 9 . 6 1 . S c h l i c h t i n g , H., " B o u n d a r y - L a y e r T h e o r y , " M c G r a w H i l l  B o o k C o . , New Y o r k , 1 9 6 8 , C h a p t e r s IX a n d T~. 6 2 . P a r k i n s o n , G.V., " A e r o e l a s t i c G a l 1 o p i n g i n One D e g r e e o f F r e e d o m , " P r o c . F i r s t I n t . C o n f . on W i n d E f f e c t s  on B l d g s . a n d S t r u c t s . , N PL . , L o n d o n , V o l . I I , 1 9 6 5 , p p . 5 8 1 - 6 0 9 . 6 3 . P a r k i n s o n , G.V. a n d M o d i , V . J . , " R e c e n t R e s e a r c h on W i n d E f f e c t s on B l u f f T w o - D i m e n s i o n a l B o d i e s , " I n t . R e s e a r c h S e m i n a r : W i n d E f f e c t s on B l d g s . a n d S t r u c t s . , NRC, O t t a w a , V o l . 1, S e p t e m b e r 1 9 6 7 , p p . 4 8 5 - 5 1 3 . 6 4 . W i l a n d , E . , " U n s t e a d y A e r o d y n a m i c s o f S t a t i o n a r y E l l i p t i c C y l i n d e r s i n S u b c r i t i c a l F l o w , " 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 , M . A . S c . T h e s i s , A p r i l 1 9 6 8 . 6 5 . G b r t l e r , H., "A New S e r i e s f o r t h e C a l c u l a t i o n o f S t e a d y L a m i n a r B o u n d a r y L a y e r F l o w s , " J . o f M a t h s . a n d M e c h a n i c s , V o l . 6, No. 1 , 1 9 5 7 , p p . 1 - 6 6 . 6 6 . F e r g u s o n , N., " T h e M e a s u r e m e n t o f Wake a n d S u r f a c e E f f e c t s i n t h e S u b c r i t i c a l 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 a t R e s t a n d i n V o r t e x E x c i t e d O s c i l l a t i o n s , " 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 , M . A . S c . T h e s i s , 1 9 6 5 . 135 6 7 . B r y e r , D.W., W a l s h e , D.E. a n d G a r n e r , H.C., " P r e s s u r e P r o b e s S e l e c t e d f o r T h r e e - D i m e n s i o n a l F l o w M e a s u r e m e n t , " A e r o n a u t i c a l R e s e a r c h C o u n c i l , R. a n d M. No. 3 0 3 7 , 1 9 5 8 . 6 8 . P a n k h u r s t , R.C. a n d H o l d e r , D.W., " W i n d T u n n e l T e c h n i q u e s , " P i t m a n a n d S o n s L i m i t e d , L o n d o n , 1 9 5 2 , C h a p t e r 8. 6 9 . W h i t b r e a d , R . E . , " M o d e l S i m u l a t i o n o f W i n d E f f e c t s on S t r u c t u r e s , " P r o c . F i r s t I n t . C o n f . on W i n d E f f e c t s  on B l d g s . a n d S t r u c t s . , N P L , L o n d o n , V o l . 1, 1 9 6 5 , p p . 2 8 3 - 3 0 6 . 7 0 . F a g e , A. a n d J o h a n s e n , F . C . , " 0 n 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 . o f L o n d o n , S e r i e s A , V o l . 1 1 6 , 1 9 2 7 , p p . 1 7 0 - 1 9 7 . 7 1 . H o e r n e r , S . F . , F l u i d - D y n a m i c D r a g , 1 9 6 5 , C h a p t e r 3, p . 11 . 7 2 . F l a c h s b a r t , 0 . , " W i n d d r u c k a u f G a s b e h a l t e r , " R e p , o f  t h e A e r o d y n . V e r s u c h s a n s t a l t i n G b t t i n g e n , IV S e r i e s , 1 9 3 2 , p p . 1 3 4 - 1 3 8 . 7 3 . R o s e n h e a d , L . , L a m i n a r B o u n d a r y L a y e r s , C l a r e n d o n P r e s s , O x f o r d , 1 9 6 3 , p p . 2 5 8 - 3 4 8 . 7 4 . B l a s i u s , H., " G r e n z s c h i c h t e n i n F l i i s s i g k e i t e n m i t k l e i n e r R e i b u n g , " Z. M a t h . u . P h y s . , V o l . 56 , 1 9 2 7 , pp . 1-37 . 7 5 . P o h l h a u s e n , K., " Z u r n a h e r u n g s w e i s e n I n t e g r a t i o n d e r D i f f e r e n t i a l g l e i c h u n q d e r l a m i n a r e n R e i b u n g s s c h i c h t , " ZAMM 1 , 1921 , p p . 2 5 2 - 2 6 8 . 7 6 . S c h l i c h t i n g , H., a n d U l r i c h , A . , " Z u r B e r e c h u n g d e s U m s c h l a g e s l a m i n a r - t u r b u l e n t , " J a h r b u c h d . d t . L u f t f a h r t f o r s c h u n g , V o l . I , 1 9 4 2 , p p . 8 - 3 5 . 7 7 . H o w a r t h , L . , "On t h e S o l u t i o n o f t h e L a m i n a r B o u n d a r y L a y e r E q u a t i o n s , " P r o c . R o y . S o c . o f L o n d o n , S e r i e s A , V o l . 1 6 4 , 1 9 3 8 , p p . 5 4 7 - 5 7 9 . 7 8 . T a n i , I . , "On t h e S o l u t i o n o f t h e L a m i n a r B o u n d a r y L a y e r E q u a t i o n s , " J . P h y s . S o c . o f J a p a n , V o l . 4, 1 9 4 9 , p p . 1 4 9 - 1 5 4 . 7 9 . R o t e m , Z., H a u p t m a n n , E . G . , a n d C l a a s s e n , L . , " S e m i -f o c u s i n g C o l o r S c h l i e r e n S y s t e m f o r U s e i n F l u i d M e c h a n i c s a n d H e a t T r a n s f e r , " A p p l i e d O p t i c s , V o l . 8, No. 1 1 , N o v e m b e r 1 9 6 9 , p p . 2 3 2 6 - 2 3 2 8 . 136 8 0 . R o t e m , Z . , a n d C l a a s s e n , L . , " N a t u r a l C o n v e c t i o n a b o v e U n c o n f i n e d H o r i z o n t a l S u r f a c e , " J . F l u i d M e c h . , V o l . 3 8 , P a r t I , 1 9 6 9 , p p . 1 7 3 - 1 9 2 . 8 1 . M e k s y n , D., New M e t h o d s i n L a m i n a r B o u n d a r y - L a y e r  T h e o r y , P e r g a m o n P r e s s , L o n d o n , 1 9 6 1 , C h a p t e r 1 1 . 8 2 . S c h u b a u e r , G.B., " A i r f l o w i n a S e p a r a t i n g L a m i n a r B o u n d a r y L a y e r , " NACA, R e p . No. 5 2 7 , 1 9 3 5 . 8 3 . H e i n e , W., "On t h e 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 V o r t e x E x c i t e d P r e s s u r e F l u c t u a t i o n s , " 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 , M . A . S c . T h e s i s , 1 9 6 4 . 8 4 . T a n e d a , S . , " 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 W a k es B e h i n d C y l i n d e r s a n d P l a t e s a t Low R e y n o l d s N u m b e r , " J . P h y s i c a l S o c i e t y o f J a p a n , V o l . 1 1 , No. 3, M a r c h 1 9 5 6 . 8 5 . A b e r n a t h y , F . H . , C o m m u n i c a t i o n a t IUTAM C o n f e r e n c e  on C o n c e n t r a t e d V o r t e x M o t i o n s , A n n A r b o r , 1 9 6 4 . 8 6 . Hama, F.R., " T h r e e D i m e n s i o n a l V o r t e x P a t t e r n B e h i n d a C i r c u l a r C y l i n d e r , " J . A e r o s p a c e S c i e n c e s , V o l . 2 4 , No. 2, F e b r u a r y 1 9 5 7 , p p . 1 5 6 - 1 5 8 . 8 7 . F r i m b e r g e r , R. , " E x p e r i m e n t e l l e U n t e r s u c h u n g e n an K a r m a n s c h e n W i r b e l s t r a B e n ," Z. F l u g w i s s . , V o l . 5, 1 9 5 7 , p p . 3 5 5 - 3 5 9 . 8 8 . F a g e , A. a n d J o h a n s e n , F . C . , " T h e S t r u c t u r e o f V o r t e x S h e e t s , " P h i l . Mag. , S e r i e s 7, V o l . 5, No. 28 , 1 9 2 8 , p p . 3 1 7 - 4 4 0 . 8 9 . H o o k e r , S . G . , "On t h e A c t i o n o f V i s c o s i t y i n I n c r e a s -i n g t h e S p a c i n g R a t i o o f a V o r t e x S t r e e t , " P r o c . R o y . S o c . o f L o n d o n , S e r i e s A , V o l . 1 5 4 , 1 9 3 6 , p p . 6 7 - 8 9 . 9 0 . W i l l e , R. a n d T i m m e , A . , " U b e r d e s V e r h a l t e n v o n W i r b e l s t r a B e n , J a h r b u c h d e r S c h i f f b a u t e c h n . G e s e l l s -c h a f t , " V o l . 51 , 1 9 5 7 , p p . 2 1 5 - 2 2 1 . 9 1 . T y l e r , E . , " V o r t e x F o r m a t i o n B e h i n d O b s t a c l e s o f V a r i o u s S e c t i o n s , " P h i l . M a g . , S e r i e s 7, V o l . 1 1 , No. 72 , 1931 , p p . 8 4 9 - 8 9 0 . 9 2 . B l o o r , M.S., " T h e T r a n s i t i o n t o T u r b u l e n c e i n t h e Wake o f a C i r c u l a r C y l i n d e r , " J . F l u i d M e c h . , V o l . 1 9 , Par,t I I , 1 9 6 4 , p p . 2 9 0 - 3 0 4 . 

Cite

Citation Scheme:

        

Citations by CSL (citeproc-js)

Usage Statistics

Share

Embed

Customize your widget with the following options, then copy and paste the code below into the HTML of your page to embed this item in your website.
                        
                            <div id="ubcOpenCollectionsWidgetDisplay">
                            <script id="ubcOpenCollectionsWidget"
                            src="{[{embed.src}]}"
                            data-item="{[{embed.item}]}"
                            data-collection="{[{embed.collection}]}"
                            data-metadata="{[{embed.showMetadata}]}"
                            data-width="{[{embed.width}]}"
                            data-media="{[{embed.selectedMedia}]}"
                            async >
                            </script>
                            </div>
                        
                    
IIIF logo Our image viewer uses the IIIF 2.0 standard. To load this item in other compatible viewers, use this url:
https://iiif.library.ubc.ca/presentation/dsp.831.1-0102215/manifest

Comment

Related Items