UBC Theses and Dissertations

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UBC Theses and Dissertations

A mathematical model for vortex-induced oscillation Lee, Francis Ngai-Ho 1974

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A MATHEMATICAL MODEL FOR VORTEX-INDUCED OSCILLATION BY FRANCIS NGAI-HO LEE B . A . S c . , 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 , 1971 A THESIS SUBMITTED I N PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF APPLIED SCIENCE 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 . THE UNIVERSITY OF B R I T I S H COLUMBIA SEPTEMBER,1974 In presenting t h i s t h e s i s in p a r t i a l f u l f i 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 reference and study. I f u r t h e r agree that permission for extensive 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 representatives. It 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 permission. Department of MgCt/Atf/CAL The U n i v e r s i t y of B r i t i s h Columbia Vancouver 8, Canada Date i A b s t r a c t The 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 e x h i b i t i n g v o r t e x -i n d u c e d o s c i l l a t i o n i s m o d e l l e d by 2 p o t e n t i a l v o r t i c e s i n a 2-d i m e n s i o n a l , i n v i s c i d a nd i r r o t a t i o n a l f l o w . The l i f t on t h e c y l i n d e r i s o b t a i n e d f r o m t h e g e n e r a l f o r m o f t h e B l a s i u s e q u a t i o n . P r e s s u r e d i s t r i b u t i o n i s o b t a i n e d f r o m t h e p r e s s u r e a q u a t i o n i n a m o v i n g f r a m e o f r e f e r e n c e . The l i f t e x p r e s s i o n i s c o u p l e d t o t h e d y n a m i c e q u a t i o n o f t h e c y l i n d e r . The p h a s e and a m p l i t u d e o f o s c i l l a t i o n a r e d e t e r m i n e d by t h e method o f e q u i v a l e n t l i n e a r i z a t i o n . A r e l a t i o n s h i p b e t w e e n a m p l i t u d e o f o s c i l l a t i o n and s t r e n g t h o f t h e v o r t i c e s i s p r o p o s e d . B o o t mean s q u a r e 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 S t r o u h a l f r e q u e n c y on t h e s u r f a c e o f t h e o s c i l l a t i n g c y l i n d e r i s d e t e r m i n e d . i i P a^e I . I n t r o d u c t i o n . 1 • I I . F o r m u l a t i o n o f t h e K a t h e m a t i c a l M o d e l . 6 i ) E x p e r i m e n t a l O b s e r v a t i o n s 6 i i ) D e r i v a t i o n o f t h e C o m p l e x P o t e n t i a l 7 i i i ) L i f t and D r a g E q u a t i o n s 10 i v ) P r e s s u r e E q u a t i o n 17 v) Dynamic E q u a t i o n G o v e r n i n g t h e S p r i n g - C y l i n d e r S y s t e m 22 I I I . M e thod o f S o l u t i o n . 25 i ) P r e s s u r e L o a d i n g on t h e C y l i n d e r 25 i i ) The D y n a m i c E q u a t i o n 26 i i i ) P r o p o s e d R e l a t i o n s h i p B e t w een A m p l i t u d e and C i r c u l a t i o n 27 i v ) N u m e r i c a l S o l u t i o n 28 I V . A n a l y s i s o f R e s u l t s . 30 V. F u t u r e R e s e a r c h and C o n c l u d i n g Remarks. 35 R e f e r e n c e s 36 A p p e n d i x I . M e t h o d o f E q u i v a l e n t L i n e a r i z a t i o n . 39 A p p e n d i x I I . C o m p u t e r P r o g r a m S o u r c e L i s t i n g . 41 I l l L I S T OF FIGURES Page 50 F i g u r e 1. S i n g l e - v o r t e x model and p r e s s u r e d i s t r i b u t i o n . F i g u r e 2. T w o - v o r t e x m o d e l and p r e s s u r e d i s t r i b u t i o n 51 F i g u r e 3. V o r t e x - i n d u c e d o s c i l l a t i o n c h a r a c t e r i s t i c s o f c i r c u l a r c y l i n d e r 52 F i g u r e 4. G e n e r a l s e t - u p f o r t h e p r e s e n t m o d e l 53 F i g u r e 5. C o n t o u r s o f i n t e g r a t i o n 54 F i g u r e 6. N o t a t i o n s u s e d i n t h e p r e s s u r e e q u a t i o n 55 F i g u r e 7. P r o p o s e d r e l a t i o n s h i p b e t w e e n oi0 and Y 56 F i g u r e 8. Phase and a m p l i t u d e o b t a i n e d f r o m t h e p r e s e n t m o d e l F i g u r e 9. C p r m s a t S t r o u h a l f r e q u e n c y w i t h V=.963, Y=.45 , £ =98° a n d 55 F i g u r e 10. C o m p a r i s i o n o f Cp p r e d i c t e d by t h e p r e s e n t model w i t h a r b i t r a r y v a l u e s f o r t h e p a r a m e t e r s a n d m e a s u r e d v a l u e s 59 57 58 ACKNOWLEDGEMENT The a u t h o r w o u l d l i k e t o t h a n k D r . G. V. P a r k i n s o n f o r h i s a d v i c e a n d g u i d a n c e i n t h e c o u r s e o f t h i s r e s e a r c h . The c o m p u t i n g f a c i l i t i e s o f t h e C o m p u t i n g C e n t e r o f t h e U n i v e r s i t y o f B r i t i s h C o l u m b i a were u s e d t o do t h e c a l c u l a t i o n s i n v o l v e d i n t h i s r e a s e a r c h . L I S T O F S Y M B O L S P o s i t i o n s o f v o r t i c e s i n m o v i n g f r a m e r e f e r e n c 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 L i f t c o e f f i c i e n t D r a g C o m p l e x p o t e n t i a l , = d/2a V o r t i c i t y s h e d d i n g r a t e L i f t C o n s t a n t s = c/2a S t r o u h a l n u m b e r F r e e s t r e a m v e l o c i t y C y l i n d e r v e l o c i t y C o m p l e x v e l o c i t y =tW2aw„ C y l i n d e r c o m p l e x v e l o c i t y N o n - d i m e n s i o n a l c y l i n d e r d i s p l a c e m e n t = y / 2 a N o n - d i m e n s i o n a l a m p l i t u d e = y / 2 a C o m p l e x f i e l d v a r i a b l e R a d i u s o f c y l i n d e r P o s i t i o n s o f v o r t i c e s i n s t a t i o n a r y f r a m e r e f e r e n c e E x t e r n a l f o r c i n g f u n c t i o n V o r t e x f o r m a t i o n f r e q u e n c y v i f j # f 2 » £ 3 » f 4 A n g l e s ( s e e f i g u r e 6 ) t T i m e m H a s s o f c y l i n d e r p e r u n i t l e n g t h y f i C y l i n d e r d i s p l a c e m e n t y c A m p l i t u d e o f o s c i l l a t i o n / D a m p i n g 0 A n g l e **Vr N a t u r a l f r e q u e n c y o f s p r i n g - c y l i n d e r s y s t e m «*V V o r t e x f o r m a t i o n f r e g u e n c y i n r a d i a n s ^ V e l o c i t y p o t e n t i a l ^ S t r e a m f u n c t i o n ^ M a s s p a r a m e t e r =2a^«/m > . N o n - d i m e n s i o n a l t i m e s t a J ^ 2 P h a s e a n g l e C i r c u l a t i o n s c(0 N o n - d i m e n s i o n a l c i r c u l a t i o n « T5 / * ^ A •A* N o n - d i m e n s i o n a l f r e q u e n c y *u)f/u>„ F l u i d d e n s i t y 1 1 IHiI2^^S.ii°B. The phenomenon o f w i n d - i n d u c e d t r a n s v e r s e o s c i l l a t i o n o f s i n g l e b l u f f - s h a p e d s t r u c t u r e s c a n be d i v i d e d i n t o two m a j o r c a t e g o r i e s a c c o r d i n g t o t h e ways e n e r g y i s e x t r a c t e d f r o m t h e f l o w f i e l d . G a l l o p i n g o s c i l l a t i o n i s t h e t y p e o f v i b r a t i o n c a u s e d by f o r c e s a r i s i n g f r o m t h e s h a p e o f t h e s t r u c t u r e i n t h e s e p a r a t e d f l o w . W i t h a s m a l l t r a n s v e r s e d i s t u r b a n c e v e l o c i t y g i v e n t o t h e s t r u c t u r e , t h e two s h e a r l a y e r s s e p a r a t i n g f r o m t h e s u r f a c e o f t h e body i n t e r a c t w i t h t h e b o d y i t s e l f , g e n e r a t i n g a f o r c e i n t h e same d i r e c t i o n a s t h e d i s t u r b a n c e . S m a l l d i s t u r b a n c e s t h e r e f o r e c a n grow i n 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 s . The o t h e r t y p e o f o s c i l l a t i o n i s t h e v o r t e x -i n d u c e d o s c i l l a t i o n . The s h e a r l a y e r s c o m i n g o f f f r o m t h e s u r f a c e o f t h e s t r u c t u r e a t t h e s e p a r a t i o n p o i n t s a r e u n s t a b l e t o s m a l l d i s t u r b a n c e s i n t h e f l o w f i e l d . T h e y t e n d t o r o l l up i n t o l a r g e d i s c r e t e v o r t i c e s . T h e s e l a r g e d i s c r e t e v o r t i c e s a r e c r e a t e d a l t e r n a t e l y f r o m b o t h s h e a r l a y e r s , f o r m i n g t h e w e l l -known Karman v o r t e x s t r e e t . T h e s e v o r t i c e s i n d u c e a l t e r n a t i n g p r e s s u r e l o a d i n g on t h e s u r f a c e o f t h e s t r u c t u r e . When t h e v o r t e x f o r m a t i o n f r e q u e n c y i s i n t h e n e i g h b o r h o o d o f any o f t h e n a t u r a l f r e q u e n c i e s o f a l i g h t l y - d a m p e d s t r u c t u r e , t h e s t r u c t u r e w o u l d be e x c i t e d i n t o r e s o n a n t o s c i l l a t i o n . The s u b j e c t o f v o r t e x - i n d u c e d o s c i l l a t i o n h a s b e e n u n d e r i n v e s t i g a t i o n s i n c e 1878 by d i f f e r e n t r e s e a r c h e r s i n d i f f e r e n t p a r t s o f t h e w o r l d . A t r e m e n d o u s amount o f d a t a h a s been a c c u m u l a t e d . I t h a s been known t h a t v o r t e x - i n d u c e d o s c i l l a t i o n 2 i s n o t a s i m p l e c a u s e - a n d - e f f e c t p h e n o m e n o n . T h e r e i s a n i n t e r d e p e n d e n c e b e t w e e n t h e f o r c e t h a t c a u s e s t h e s t r u c t u r e t o o s c i l l a t e a n d t h e o s c i l l a t i o n . N a u d a s c h e r ( 1 ) a n d T o e b e s ( 2 ) h a v e r e c o g n i z e d t h i s f a c t a n d t e r m e d i t f l u i d - e l a s t i c i n t e r a c t i o n . H o w e v e r , w i t h o u t s o m e k i n d o f a n a l y t i c m o d e l , t h e i n t e r p l a y b e t w e e n t h e d i f f e r e n t p h y s i c a l q u a n t i t i e s i n v o l v e d i n / t h e p h e n o m e n o n c a n n o t b e a d e q u a t e l y e s t a b l i s h e d . E n g i n e e r i n g , a t t e m p t s e i t h e r t o m a k e u s e o f o r t o e l i m i n a t e t h e p h e n o m e n a o f v o r t e x - i n d u c e d o s c i l l a t i o n w o u l d , b e n e f i t f r o m a s u c c e s s f u l m a t h e m a t i c a l d e s c r i p t i o n o f t h e p h e n o m e n a . F u r t h e r m o r e , a d e e p e r i n s i g h t c o u l d b e g a i n e d i n t o t h e , p h y s i c s o f t h e f l o w . • • \ • T h e s e a r c h f o r a p r o p e r m a t h e m a t i c a l m o d e l h a s a l r e a d y b e g u n . A s i n e x p e r i m e n t a l w o r k , t h e m o s t w i d e l y s t u d i e d b l u f f s h a p e i s t h e c i r c u l a r c y l i n d e r . S i n c e a l l N e w t o n i a n f l u i d f l o w p h e n o m e n a a r e g o v e r n e d b y t h e N a v i e r - S t o k e s e q u a t i o n s , a n o b v i o u s a p p r o a c h t o t h i s p r o b l e m i s t o s o l v e t h e f u l l N a v i e r -S t o k e s e q u a t i o n s f o r t h e f l o w f i e l d i n t h e " p r e s e n c e o f t h e o s c i l l a t i n g c i r c u l a r c y l i n d e r . B e c a u s e o f t h e c o m p l e x i t y o f t h i s s y s t e m o f e q u a t i o n s , n o s u c h s o l u t i o n h a s b e e n f o u n d . W i t h s o m e s i m p l i f i c a t i o n , J o r d a n a n d F r o m m ( 6 ) h a v e d e v e l o p e d c o m p u t e r p r o g r a m s t o s o l v e t h e N a v i e r - S t o k e s e q u a t i o n s f o r t h e t i m e -d e p e n d e n t , v i s c o u s a n d i n c o m p r e s s i b l e f l o w p a s t 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 . T h e y o b t a i n e d r e s u l t s f o r l i f t , d r a g , t o r q u e a n a p r e s s u r e d i s t r i b u t i o n o n t h e c y l i n d e r a t 3 R e y n o l d s n u m b e r s j ( R n ) o f 1 0 0 , 4 0 0 a n d 1 0 0 0 . R e s u l t s f o r t h e f i r s t t w o a r e i n g o o d a g r e e m e n t w i t h e x p e r i m e n t a l r e s u l t s , w h i l e f o r t h e t h i r d o n e , s i g n i f i c a n t d i s c r e p a n c i e s w e r e f o u n d b e c a u s e o f t h e d i f f i c u l t y e n c o u n t e r e d i n m o d e l l i n g t h e t h i n b o u n d a r y l a y e r u p s t r e a m o f t h e s e p a r a t i o n p o i n t , a n d t h e 3 - d i m e n s i o n a l n a t u r e o f t h e f l o w . B e c a u s e o f t h e c o m p l e x i t y o f t h e N a v i e r - S t o k e s e g u a t i o n s , s i m p l e r m o d e l s w o u l d s e e m t o b e a b e t t e r a p p r o a c h f r o m a n a n a l y t i c p o i n t o f v i e w . P o t e n t i a l f l o w m o d e l s a r e a p p e a l i n g i n t h i s r e s p e c t b e c a u s e o f t h e s i m p l e r b u t e l e g a n t m a t h e m a t i c a l \ t h e o r y t h a t h a s b e e n d e v e l o p e d . I f o n e l o o k s a t a s e p a r a t e d f l o w , t h e i m p o r t a n t e n t i t i e s i n v o l v e d a r e t h e t h i n s h e a r l a y e r s a n d t h e d i s c r e t e v o r t i c e s , w i t h t h e s e i n m i n d , G e r r a r d (7), A b e r n a t h y a n d K r o n a u e r (8) a n d o t h e r s h a v e s t u d i e d t w o -d i m e n s i o n a l p o t e n t i a l m o d e l s o f ^ 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 i n u n i f o r m f l o w . I n t h e i r m o d e l s , t h e t h i n s h e a r l a y e r s a r e m o d e l l e d a s s h e e t s o f d i s c r e t e p o i n t v o r t i c e s . T h e y h a v e s h o w n , b y u s e o f l e n g t h y c o m p u t e r p r o g r a m s , t h a t t h e v o r t e x s h e e t s d o r o l l u p t o f o r m l a r g e c l u s t e r s o f v o r t i c e s a t t h e o b s e r v e d p o s i t i o n s o f a c t u a l d i s c r e t e v o r t i c e s b e h i n d t h e c i r c u l a r c y l i n d e r w i t h c i r c u l a t i o n s t r e n g t h s t h a t a g r e e w i t h e x p e r i m e n t a l o n e s . T h e y a l s o o b t a i n e d v a l u e s f o r t h e f o r c e s o n t h e s t a t i o n a r y c y l i n d e r i n g o o d a g r e e m e n t w i t h t h e m e a s u r e d o n e s O n e d i s a p p o i n t i n g d r a w b a c k o f t h e s e m o d e l s i s t h a t i m p o r t a n t - a n a l y t i c r e l a t i o n s h i p s b e t w e e n t h e p r i n c i p a l q u a n t i t i e s a r e m a s k e d i n f i n e n u m e r i c a l d e t a i l s . T h e y d o n o t t h r o w m u c h l i g h t o n t h e p h y s i c s o f t h e p h e n o m e n a . Y e t s i m p l e r i m o d e l s a r e r e q u i r e d . A s i n g l e - v o r t e x m o d e l ( s e e f i g u r e 1) w a s / p r o p o s e d b y E t k i n , a n d M c G r e g o r ( 9 ) u s e d i t t o o b t a i n p r e s s u r e d i s t r i b u t i o n s o n t h e s u r f a c e o f a s t a t i o n a r y c y l i n d e r . A s s h o w n 4 i n f i g u r e 1, t h e r o o t mean s q u a r e (RMS) o f 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 a t t h e S t r o u h a l f r e q u e n c y o b t a i n e d f r o m t h i s m o d e l a g r e e s f a i r l y w e l l w i t h t h e ones he m e a s u r e d o v e r t h e f r o n t p o r t i o n o f t h e c y l i n d e r . A l o g i c a l e x t e n s i o n o f t h i s m o d e l w o u l d be t o have two s u c h p o t e n t i a l v o r t i c e s l o c a t e d on e i t h e r s i d e o f t h e wake c e n t e r l i n e (see f i g u r e 2 ) . Madderom (10) a p p l i e d t h i s m o d e l t c o b t a i n t h e RMS o f 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 a t t h e S t r o u h a l f r e q u e n c y 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 . A s c a n be s e e n f r o m f i g u r e 2, t h e a g r e e m e n t i s b e t t e r t h a n f o r E t k i n ' s s i n g l e - v o r t e x m o d e l . I n t h e s t a t i o n a r y - c y l i n d e r m o d e l s m e n t i o n e d a b o v e , c n l y t h e f l o w f i e l d n e e d s t o be m o d e l l e d . F o r v o r t e x - i n d u c e d o s c i l l a t i o n of a b l u f f c y l i n d e r , t h i s i s o n l y h a l f o f t h e p r o b l e m . The d y n a m i c s o f t h e o s c i l l a t i n g c y l i n d e r must a l s o be d e a l t w i t h . S e v e r a l d y n a m i c m o d e l s f o r t h e 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 h ave been p r o p o s e d . T h e s e m o d e l s o f t e n lump t h e e f f e c t o f t h e wake i n t o some k i n d o f o s c i l l a t o r w h i c h g e n e r a t e s t h e l i f t on t h e c y l i n d e r . A p r o m i s i n g a n d more s u c c e s s f u l one was p r o p o s e d by H a r t l e n and C u r r i e ( 1 1 ) . I n t h e i r m o d e l , l i f t i s g o v e r n e d by a n o n - l i n e a r d i f f e r e n t i a l e q u a t i o n c o u p l e d t o t h e v e l o c i t y o f o s c i l l a t i o n l i n e a r l y . T h i s m o d el was a b l e t o p r e d i c t most o f t h e c h a r a c t e r i s t i c s o f v o r t e x - i n d u c e d o s c i l l a t i o n f a i r l y w e l l . None o f t h e m o d e l s m e n t i o n e d s o f a r t a k e b o t h t h e d y n a m i c s of t h e f l o w f i e l d and t h e o s c i l l a t i n g body i n t o a c c o u n t . I t i s 5 fe l t that perhaps a better understanding of vortex-induced osc i l la t ion can be gained i f we couple Madderom's flow f i e ld model, because of i t s a b i l i t y to predict the pressure distribution in the static case and i t s r e a l i s t i c s implici ty and analytic t rac tab i l i t y , to the dynamic equation governing the motion of the c ircular cylinder. It i s to this end this research has been aimed. 6 1 1 ZSOSl^ iSU 2! t h e M a t h e m a t i c a l M o d e l i ) E x p e r i m e n t a l O b s e r v a t i o n s A v e r y c o m p r e h e n s i v e e x p e r i m e n t a l s t u d y o f v o r t e x - i n d u c e d o s c i l l a t i o n i n b o t h c i r c u l a r a n d D - s e c t i o n c y l i n d e r s w a s c a r r i e d o u t i n t h i s l a b o r a t o r y b y C . C . F e n g (3) a t R e y n o l d s n u m b e r s i n \ t h e n e i g h b o r h o o d o f 2 x 1 0 * . H i s f i n d i n g s s h o w t h a t v o r t e x -i n d u c 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 i s s i n u s o i d a l i n t i m e t s o t h a t t h e d i s p l a c e m e n t y c c a n b e e x p r e s s e d b y ; ' \ t w h a r e jft i s t h e a m p l i t u d e , u)„ t h e n a t u r a l f r e q u e n c y o f t h e s p r i n g - c y l i n d e r s y s t e m h e u s e d , <g t h e p h a s e a n g l e m e a s u r e d r e l a t i v e t o e x c i t a t i o n . H e f o u n d h y s t e r e s i s l o o p s e x i s t i n b o t h a m p l i t u d e a n d p h a s e . A s s h o w n i n t h e 7 v s V c u r v e i n f i g u r e 3 , w h e r e ? s Jju/Za. i s t h e n o n - d i m e n s i o n a l a m p l i t u d e a n d V»tf©ykm)„ i s t h e n o n - d i m e n s i o n a l v e l o c i t y , U0 b e i n g t h e f r e e s t r e a m v e l o c i t y , s t a r t i n g f r o m A , i f w e i n c r e a s e t h e f l o w v e l o c i t y v , a m p l i t u d e i n c r e a s e s a c c o r d i n g l y t o B . H o w e v e r , f u r t h e r i n c r e a s e i n v e l o c i t y b e y o n d B w o u l d r e s u l t i n a s u d d e n d r o p i n a m p l i t u d e , t o C . T h e a m p l i t u d e w i l l t h e n d i m i n i s h s l o w l y t o D . I f w e s t a r t f r o m D a n d d e c r e a s e t h e f l o w v e l o c i t y , a m p l i t u d e w i l l i n c r e a s e s l o w l y , t h r o u g h C t o E . F u r t h e r d e c r e a s e w i l l m a k e t h e a m p l i t u d e j u m p f r o m E t o F . I t w i l l t h e n d e c r e a s e w i t h v e l o c i t y t o A . 7 P h a s e m e a s u r e m e n t s show t h e same k i n d o f jump a s shown i n f i g u r e 3. I t i s w e l l known t h a t v o r t e x f o r m a t i o n f r e q u e n c y -f y i n t h e wake o f a b l u f f body i s g o v e r n e d by t h e S t r o u h a l r e l a t i o n s h i p 2 ^ C2) where S i s t h e S t r o u h a l number and a i s t h e r a d i u s o f t h e c y l i n d e r . I f we l o o k a t t h e r a t i o **>v/t&* i n f i g u r e 3, where co„z:zr,j-^ , a s a f u n c t i o n o f v e l o c i t y , s t a r t i n g a t t h e n e i g h b o r h o o d o f V=.8, i n s t e a d o f f o l l o w i n g t h e S t r o u h a l r e l a t i o n s h i p , t h e v o r t e x f o r m a t i o n f r e q u e n c y i s t h e same a s t h e n a t u r a l f r e q u e n c y o f t h e c y l i n d e r - s p r i n g s y s t e m f o r a r a n g e o f v e l o c i t i e s up t o V=1.1. I t l o o k s a s t h o u g h t h e v o r t e x f o r m a t i o n f r e q u e n c y i s l o c k e d i n t o t h e n a t u r a l f r e q u e n c y o f t h e s y s t e m . T h i s r a n g e o f v e l o c i t y i s known a s t h e l o c k - i n r a n g e . The f r e q u e n c y r e v e r t s b a c k t o t h e S t r o u h a l f r e q u e n c y b e y o n d V=1.1 i i ) D e r i v a t i o n o f t h e C o m p l e x P o t e n t i a l W i t h t h e s e t - u p shown i n f i g u r e 4, we have i n t h e c o m p l e x Z - p l a n e , a c i r c u l a r c y l i n d e r o f r a d i u s a, f r e e t o move i n t h e y-d i r e c t i o n i n a u n i f o r m , 2 - d i m e n s i o n a l , i n v i s c i d and i r r o t a t i o n a l f l o w w i t h f r e e s t r e a m v e l o c i t y U 0 . L o c a t e d d o w n s t r e a m o f t h e 8 c y l i n d e r a r e t w o p o t e n t i a l v o r t i c e s a t p o i n t s A , + i d . T h e s t r e n g t h s o f t h e s e v o r t i c e s a r e ' . v e n b y Ji a r , c-»s uj^t > a t a n d =rj ( M - N CJ>&'uo*t> "i a t |T , if a n d N a r e p o s i t i v e c o n s t a n t s s u c h t h a t M>N a n : M+N=1. T ; I S t h e e f f e c t i v e c i r c u l a t i o n . I n t h e i r l a b o r a t o r y s t u d i e s , F e n g (3) , K o o p m a n n (4) a n d G e r r a r d (5) a l l o b s e r v e d t h a t w h e n t h e c y l i n d e r w a s s t a t i o n a r y o r o s c i l l a t i n g w i t h l o w a m p l i t u d e , t h e v o r t e x f i l a m e n t c o m i n g o f f t h e s u r f a c e o f t h e c y l i n d e r w a s i n c l i n e d a t a n a n g l e t o t h e a x i s o f t h e c y l i n d e r . W h e n t h e c y l i n d e r w a s e x h i b i t i n g m a x i m u m a m p l i t u d e o f o s c i l l a t i o n , t h e v o r t e x f i l a m e n t w a s a l i g n e d p a r a l l e l t o t h e a x i s o f t h e c y l i n d e r . T h i s i s a 3-d i m e n s i o n a l e f f e c t w h i c h a 2 - d i m e n s i o n a l m o d e l l i k e t h i s i s u n a b l e t o i n c l u d e . T h e r e f o r e i t i s a s s u m e d t h a t f 6 = K C Y ) a n d t h e f o r m o f t h e f u n c t i o n r e m a i n s t o b e d e t e r m i n e d . T h e c o m p l e x p o t e n t i a l F ( Z ) f o r t h e c y l i n d e r - v o r t e x s y s t e m a s s h o w n i n f i g u r e 4 w i t h </c=o i n a f l o w w i t h a r b i t r a r y c o m p l e x v e l o c i t y ( 0 » + i O e ) i n t h e c o m p l e x p l a n e i s g i v e n b y w h e r e cf> i s t h e v e l o c i t y p o t e n t i a l a n d 0 i s t h e s t r e a m f u n c t i o n . N o w i f w e i m p o s e a c o m p l e x v e l o c i t y - ( U 6 + i U c ) o n t h e w h o l e s y s t e m , i . e . s t o p p i n g t h e o n - c o m i n g f l o w a n d s e t t i n g t h e 9 c y l i n d e r - v o r t e x s y s t e m i n t o m o t i o n w i t h c o m p l e x v e l o c i t y • * ( U 0 + i U g ) , t h e c o m p l e x p o t e n t i a l b e c o m e s F i n a l l y t h e t w o v o r t i c e s a r e g i v e n a m o t i o n t h a t w o u l d s t o p t h e m f r o m m o v i n g i n t h e y - d i r e c t i o n . T h i s w i l l s h o w u p i n t h e p o s i t i o n o f t h e v o r t i c e s r e l a t i v e t o t h e c y l i n d e r . T h e c o m p l e x p o t e n t i a l b e c o m e s i n w h i c h t \ n * &* Jmtyt, J & i s t n e d i s p l a c e m e n t o f t h e c y l i n d e r w i t h r e s p e c t t o t h e f i x e d f r a m e o f r e f e r e n c e Z = ( x + i y ) . N e g l e c t i n g s o m e c o n s t a n t s , F ( Z ) c a n b e s i m p l i f i e d t o , - ) 1 § [A C2-A)- JU a - f!) ] 1 0 i i i ) L i f t a n d D r a g E q u a t i o n B y p o t e n t i a l f l o w t h e o r y , t h e r c s c i e x v e l o c i t y ( U - i V c ) c a n b e o b t a i n e d b y d i f f e r e n t i a t i n g h e c o m p l e x p o t e n t i a l w i t h r e s p e c t t o t h e f i e l d v a r i a b l e Z . T h e r e f o r e -t I 2* c ^ j (4) f r o m t h e g e n e r a l f o r m o f t h e B l a s i u s e q u a t i o n , ( s e e 1 2 , p a g e 2 5 5 ) t h e l i f t L a n d d r a g D o n a m o v i n g c y l i n d e r a r e g i v e n b y O-a = { iff (gfif» - f?c*Wf c c w h e r e i s t h e f l u i d d e n s i t y , V t h e t o t a l c i r c u l a t i o n i n s i d e t h e c y l i n d e r , W = - 0 6 - i U c , t h e c o m p l e x v e l o c i t y o f t h e c y l i n d e r , a n d A t t h e c r o s s - s e c t i o n a l a r e a o f t h e c y l i n d e r . T h e C a t t h e b o t t o m o f t h e i n t e g r a l s i g n d e n o t e s a c o n t o u r i n t e g r a l a l o n g t h e s u r f a c e o f t h e c y l i n d e r . We w i l l n o w c o n s i d e r t h e t w o 11 i n t e g r a l s I , a n d I ; . F r o m e q u a t i o n (4) , w e o b t a i n e d C^f a n d _ n K r J. L _ 7 r — i - r i - 7 5 rf* B y u s i n g t h e m e t h o d o f r e s i d u e s , t h i s i n t e g r a l c a n b e e a s i l y e v a l u a t e d . T h e f i n a l r e s u l t i s S e p a r a t i n g t h i s i n t o t h e r e a l a n d i m a g i n a r y p a r t s , w e o b t a i n cr '' is/ 4 12 /Al« _ J c / ; where iAi and IBI a r e t h e a b s o l u t e v a l u e s o f A and B. How we t u r n o u r a t t e n t i o n t o I z i n e q u a t i o n ( 5 ) . The i n t e g r a l t h a t we h a v e t o e v a l u a t e i s The f i r s t t e r m c a n be e v a l u a t e d r e a d i l y by t h e method o f r e s i d u e s and However, t h e s e c o n d t e r m i n v o l v e s , i n s t e a d o f a s i n g u l a r i t y , two 13 b r a n c h p o i n t s a t Z = A a n d Z = . S i n c e Z = A i s o u t s i d e t h e c y l i n d e r , i t d o e s n o t h a v e a n y c o n t r i b u t i o n . T h e o n l y b r a n c h p o i n t w e h a v e t o c o n s i d e r i s z = < a V ^ , t h e i n v e r s e p o i n t o f A . T a k i n g a b r a n c h c u t a s s h o w n i n f i g u r e ( 5 A ) , w e d e f o r m t h e c o n t o u r o f i n t e g r a t i o n a s i n d i c a t e d b y t h e a r r o w s . B y C a u c h y ' s / t h e o r e m , \ J - f - s + S =V J-*t -t***-, u^r* I n t h e l i m i t £ - * o $ 1 ' i z S- J - - { f * f-~ f ] : <i> F i r s t c o n s i d e r t h e l i n e i n t e g r a l f r o m A-+t> . L e t \ < * ' . *. t i t * -* e ' V f A l o n g t h e p a t h , T = c o n s t a n t = iru a n d ^ v a r i e s f r o m ^ . ^ f c - ^ a t 4 t o - £ » o a t ^ . « t e = e < & y . 14 A l o n g t h e p a t h t -?u , -5» c o n s t a n t s t a s we c h o s e i t t o be s o when we c h o s e t h e p a t h o f i n t e g r a t i o n . V v a r i e s f r o m -v=vu a t t t o r = -2n +ru a t u . a ! ? - - ^ e c V ^ . A s ^ln^ - ^ e o ana 5 - ? © . B u t - 5 - * * f a s t e r t h a n l n ^ —9\oo . T h e r e f o r e we c a n s a y t h e i n t e g r a l i s z e r o . A l o n g t h e p a t h u - r r f = c o n s t a n t= -zn + v„ and ^  v a r i e s f r o m z e r o a t «. to (n-'-jfj) a t r . Ui^eCr^^ . P u t t i n g t h e r e s u l t s b a c k i n t o e q u a t i o n ( 9 ) , we o b t a i n Ml F o r t h e t h i r d t e r m i n e q u a t i o n ( 8 ) , t h e b r a n c h p o i n t o f i n t e r e s t i s z= *>/g . He c h o o s e o u r b r a n c h c u t a s shown i n f i g u r e ( 5 B ) . F o l l o w i n g t h e same p r o c e d u r e , t h e i n t e g r a l i s f o u n d 15 to be and equation (8) becomes This expression can be separated into i t s real and imaginary parts. The remaining two terms in equation (5) can be evaluated quite readily. 16 W i t h t h e r e s u l t s i n e q u a t i o n s ( 7 ) , (10) , ( 1 1 ) , and (12) s u b s t i t u t e d back i n t o e q u a t i o n ( 5 ^ o b t a i n /Z4l [Uclc\cyc + c{f]-2UtC(icjd)] + Tuft C(cz+tf-e{i) ? IB I4-  7 1 ( f y c ' - * 1 ) * * 4 < * V J We d e f i n e a l i f t c o e f f i c i e n t Cu i n t h e c o n v e n t i o n a l way and C L = L CL . F u r t h e r m o r e , a l l q u a n t i t i e s i n v o l v e d a r e n o n -d i m e n s i o n a l i z e d a c c o r d i n g t o t t s . f o l l o w i n g scheme: (1) C i r c u l a t i o n i s n o n - l i m e n s i o n a l i z e d w i t h U 0 a (2) L e n g t h i s n o n - d m e n s i o n a l i z e d w i t h 2a. (3) Time i s n o n - d i m e i s i c n a l i z e d w i t h co^ . (4) F r e q u e n c y i s n o n - d m e n s i o n a l i z e d w i t h u?n . (5) V e l o c i t y i s n o n - d i m e n s i o n a l i z e d w i t h 2a£c7„ . 17 T h e f o l l o w i n g e x p r e s s i o n f o r Cu i s o b t a i n e d : a <*Z + <*l <Y+") _ ^ f 2YCYrH)?+ \/ Cfx- (Y+ - 2CYrH) [Ct/-Yj*+ p l 2 K P Y _ / 7 w h e r e Y s y t y £ a # H°ei/za. * P * c / z « . » \A*4>/2«<J, S * 4 * ^ i s t h e S t r o u h a l n u m b e r a n d . D o t r e p r e s e n t s d i f f e r e n t i a t i o n w i t h r e s p e c t t o t h e d i m e n s i o n l e s s t i m e *2r»-^u>^. i v ) P r e s s u r e E q u a t i o n I n a m o v i n g f r a m e o f r e f e r e n c e , t h e p r e s s u r e P a t a n y p o i n t N i n t h e f i e l d i s g i v e n b y ( s e e 1 2 p a g e 2 5 2 ) - . j W h e r e i s t h e p r e s s u r e a t i n f i n i t y , 4> t h e v e l o c i t y p o t e n t i a l 18 a s d e f i n e d i n e q u a t i o n ( 3 ) , oi t h e a n g l e shown i n f i g u r e 6. A p r e s s u r e c o e f f i c i e n t c a n be d e f i n e d i n t h e c o n v e n t i o n a l way, i . e . On t h e s u r f a c e o f t h e c y l i n d e r , Z= a.case+Ca.sth s> a n a (p \ s o b t a i n e d t h e r e f r o m t h e r e a l p a r t o f e q u a t i o n (3) where ^ , f y , f 2 , f 3 and f^. a r e shown i n f i g u r e 6. • -} 19 * I -tU, +L(JC f \ ~ " " W h e r e p r i m e i n d i c a t e s d i f f e r e n t i a t i n g w i t h r e s p e c t t c t i m e t . W i t h s o m e l e n g t h y a n d t e d i o u s a l g e b r a , t h i s e x p r e s s i o n c a n b e s i m p l i f i e d t o t h e f o l l o w i n g : \ 1 w h e r e s t f & L l ! ^ f - t-i-i'' s^t/t'** C W J * 7 i I - -w i t h : # = r ^ - i Q 4 - T W -_Cf(±¥c) 21 and - ^ f C -eft / { 1 ^ H o w e v e r , we a r e o n l y i n t e r e s t e d i n t h e f l u c t u a t i n g d p a t t h e S t r o u h a l f r e q u e n c y . A f t e r d r o p p i n g a l l t h e t e r m s t h a t have no S t r o u h a l f r e q u e n c y c o m p o n e n t s , we a r r i v e a t - 2A ^ _ „. .. _ 3 c»0 F o l l o w i n g t h e scheme t h a t was u s e d t o n o n - d i m e n s i o n a l i z e CL , we c h a n g e Cp i n t o a d i m e n s i o n l e s s f o r m . T hus J]L - <K it f M £4X3 SltL 22 r r> Tr £££Jl——if [z*>4fess* rjjr •> -**4<#-Y)V j f ^ ? p 4- ( (//-Y)%fzJ - 4 [COOP/>-+ Sth& c#-r))J v) Dynamic E q u a t i o n G o v e r n i n g t h e S p r i n g - C y l i n d e r S y s t e m The t r a n s v e r s e m o t i o n o f a r i g i d 2 - d i m e n s i o n a l c y l i n d e r , w i t h v i s c o u s - t y p e d a m p i n g , mounted on l i n e a r s p r i n g s i s g o v e r n e d by t h e d i f f e r e n t i a l e q u a t i o n 23 where m i s t h e mass p e r u n i t l e n g t h , i s a m e a s u r e o f t h e d a m p i n g o f t h e s y s t e m e x p r e s s e d i n f r a c t i o n s o f t h e c r i t i c a l d a m p i n g , t h e l e v e l o f d a m p i n g a b o v e w h i c h no o s c i l l a t i o n c a n t a k e p l a c e when t h e s y s t e m i s n o t f o r c e d e x t e r n a l l y , to* i s t h e n a t u r a l f r e q u e n c y o f t h e s y s t e m and i s t h e e x t e r n a l t r a n s v e r s e f o r c e a p p l i e d on t h e s y s t e m . I n t h e c a s e we a r e c o n s i d e r i n g , F j = L A n o n - d i m e n s i o n a l f o r m o f t h i s e q u a t i o n i s o b t a i n e d b y d i v i d i n g t h r u w i t h 2a*Mto n l . rhus where =• L&£-. E a c h d o t r e p r e s e n t s d i f f e r e n t i a t i o n w i t h r e s p e c t t o t h e n o n - d i m e n s i o n a l t i m e o n c e . cu i s a s d e f i n e d i n e q u a t i o n (11) . S u b s t i t u t i n g e q u a t i o n (14) i n t o e q u a t i o n ( 1 8 ) , ana a f t e r r e a r r a n g i n g some t e r m s we o b t a i n Y (HZTL*)* Y - 7 K * f 4 ^ < ^ 4 v^CYtH)z tp7l J - Y ) T h i s e q u a t i o n g o v e r n s t h e d y n a m i c s o f t h e s p r i n g - c y l i n d e r s y s t e f o r a g i v e n s e t o f p a r a m e t e r s . 2 5 I I I M e t h o d o f S o l u t i o n W i t h t h e e g u a t i o n s d e v e l o p e d i n I I , i t i s p o s s i b l e t o g o a h e a d a n d s o l v e f o r t h e p r e s s u r e l o a d i n g a n d p h a s e a n d a m p l i t u d e o f o s c i l l a t i o n f o r a l l v e l o c i t i e s . H o w e v e r , m o s t o f t h e c h a r a c t e r i s t i c s o f v o r t e x - i n d u c e d o s c i l l a t i o n a r e o b s e r v e d m t h e l o c k - i n r a n g e w h e n uJ^au)^, W e w i l l t h e r e f o r e c o n s i d e r o n l y \ : h e c a s e i n t h e l o c k - i n r a n g e . 1) P r e s s u r e L o a d i n g o n t h e C y l i n d e r F o r t h e d y n a m i c c a s e b e i n g c o n s i d e r e d , we a r e i n t e r e s t e d i n t h a c o m p o n e n t o f f l u c t u a t i n g p r e s s u r e a t t h e f u n d a m e n t a l S t r o u h a l f r e g u e n c y . T h e o r e t i c a l l y , e q u a t i o n ( 1 6 ) c a n b e d e c o m p o s e d i n t o i t s F o u r i e r s e r i e s a n d t h e f u n d a m e n t a l f r e q u e n c y c a n b e o b t a i n e d . H o w e v e r , t h i s s e e m s t o b e t e c t e d i o u s m a t h e m a t i c a l l y . I n s t e a d i t i s d o n e n u m e r i c a l l y o n t h e c o m p u t e r . U s i n g e q u a t i o n ( 1 6 ) , w e c a n o b t a i n t h e F o u r i e r c o e f f i c i e n t s o f t h e f u n d a m e n t a l f r e q u e n c y b y i n t e g r a t i o n , i . e . T h e r o o t m e a n s g u a r e v a l u e o f t h i s f l u c t u a t i n g p r e s s u r e i s g i v e n b y 26 i i ) The Dynamic Equation In equation (19), the right hand side i s multiplied by the mass parameter ^ • This parameter i s proportional to the ratio of the mass of air and the mass of the cylinder and i t s magnitude is of the order of oC/o~*J. Therefore this equation can be solved by the small parameter approximation method in non-linear analysis. This method, the method of equivalent linearization, i s explained in Appendix I. The method yields r Sin ( t f i ) \ and where and £ i s as shown in equation (19). In the case of steady state oscillation. otf s 0 xs (.2.1 ) L2. Z.) 27 m ) P r o p o s e d R e l a t i o n s h i p Between A m p l i t u d e a n d C i r c u l a t i o n As m e n t i o n e d i n I I s e c t i o n v o r t i c e s m t h e wake a r e o r g a n i z e d by t h e a m p l i t u d e o f o s c i l l a t i o n . The p h a s e between t h e f o r m a t i o n o f v o r t i c e s a l o n g t h e s p a n w i s e d i r e c t i o n i s r e d u c e d t o z e r o as a m p l i t u d e i n c r e a s e s t o i t s maximum. So t h e r e i s a r e l a t i o n s h i p b e t w e e n a m p l i t u d e and c i r c u l a t i o n w h i c h i s t o be i n v e s t i g a t e d i n t h i s m o d e l . I n t h e c o u r s e o f t h i s r e s e a r c h , v a r i o u s r e l a t i o n s h i p s between c i r c u l a t i o n and a m p l i t u d e have been l o o k e d a t . O b v i o u s o n e s a r e t h e l i n e a r a n d q u a d r a t i c f u n c t i o n s . I t was f o u n d t h a t t h e y o n l y p r o d u c e d good r e s u l t s i n p a r t s o f t h e l o c k - i n r a n g e . Tha q u e s t i o n , t h e n , i s what k i n d o f r e l a t i o n s h i p b e t w e e n a m p l i t u d e and c i r c u l a t i o n t h e model w o u l d r e q u i r e f o r i t t o work i n t h e l o c k - i n r e g i o n . By n a k i n g ^ and & t h e unknowns i n t h e s y s t e m o f e q u a t i o n s (21) and ( 2 2 ) , t h e s e c o n d p r o g r a m i n A p p e n d i x I I was m o d i f i e d t o s o l v e f o r t h e s e t ^ o q u a n t i t i e s w i t h i n p u t p a r a m e t e r s , M , ^ , P , i-l a n d v g i v e n . G u i d e d by F e n g " s a m p l i t u d e m e a s u r e m e n t s , a f u n c t i o n i s d e s i g n e d r e l a t i n g oi„ a n d Y. T h i s f u n c t i o n i s g i v e n by < = -36 e - 7 ? i i l s ~ T - / *7}\ /-If i'f^TJ - ,.,4]' 28 iv) Numerical S o l u t i o n s (1) Pressure Equation The i n t e g r a t i o n as i n d i c a t e d by equations (20A) and (20B) i s c a r r i e d out n u m e r i c a l l y on the IB" 360-70 computer at the computing c e n t e r of UBC. The program, shown i n Appendix I I , uses a l i b r a r y r o u t i n e SQUAN'K to perform the i n t e g r a t i o n . T h i s r o u t i n e i s based on Simpson's method of d i v i d i n g the i n t e r v a l of i n t e g r a t i o n i n t o a number of d i v i s i o n s . The area under the curve i n each of those d i v i s i o n s i s c a l c u l a t e d by assuming t h a t the curve can be approximated by a q u a d r a t i c . The accuracy of the value f o r the i n t e g r a l depends on the number of d i v i s i o n s . The r o u t i n e keeps on d i v i d i n g the i n t e r v a l i n t o more and more d i v i s i o n s and comparing the r e s u l t with the previous one. When the d i f f e r e n c e i s s m a l l e r than an amount 6 , s p e c i f i e d ty the user, the i t e r a t i o n w i l l stop and the f i n a l r e s u l t i s ob t a i n e d . In t h i s case, 4 was s p e c i f i e d to be .0001. (2) Dynamic Equations The dynamics of the c y l i n d e r - s p r i n g system are governed by equations (21) , (22), and (23). These form a system of t h r e e equations i n three unknowns Y , and . Again t h i s system i s s o l v e d by using the f a c i l i t i e s a t the computing cent e r i n 29 UBC. A l i b r a r y r o u t i n e NONLIN i s u s e d t o s o l v e t h e s y s t e m o f n o n - l m e a r e q u a t i o n s . I h i s r o u t i n e i s b a s e d on t h e Newton-Raphson method o f i t e r a t i o n as f o u n d i n (14). The f i r s t non-l i n e a r e q u a t i o n i s e x p r e s s e d i n a T a y l o r s e r i e s a b o u t t h e i n i t i a l g u e s s e s s u p p l i e d by t h e u s e r . K e e p i n g o n l y t h e l i n e a r t e r m s , t h e r e s u l t i n g s e r i e s i s e q u a t e d t o z e r o and i s s o l v e d f o r cna v a r i a b l e , s a y xk, m t e r m s o f t h e o t h e r r e m a i n i n g v a r i a b l e s . I n t h e s e c o n d n o n - l i n e a r e q u a t i o n , t h e same p r o c e d u r e i s a p p l i e d e x c e p t x, i s r e p l a c e d by t h e v a l u e o b t a i n e d e a r l i e r . T h i s p r o c e s s i s r e p e a t e d u n t i l m t h e l a s t e q u a t i o n o n l y one v a r i a b l e ^ n i s l e f t . Xy, i s t h e n s o l v e d f o r and b a c k s u b s t i t u t e d i n t o t h e f i r s t e q u a t i o n i n p l a c e o f t h e g u e s s u s e d . An i m p r o v e d X , i s t h u s o b t a i n e d . Then X , and X* , a r e u s e d m t h e s e c o n d e q u a t i o n t o o b t a i n a n i m p r o v e d X 2 • T n e p r o c e s s g o e s on u n t i l t h e i m p r o v e d s o l u t i o n s a r e c a l c u l a t e d t o w i t h i n t h e number o f s i g n i f i c a n t f i g u r e s s p e c i f i e d by t h e u s e r . I n t h i s c a s e , i t i s s e t e q u a l t o a . 30 I V . A a a l _ y s i s o f R e s u l t s ^ A l t h o u g h t h e e q u a t i o n s o b t a i n e d a r e c o m p l e x , t h e c o n t r i b u t i o n s o f e a c h o f tr.e q u a n t i t i e s a r e c l e a r l y i s o l a t e d . The v a l u e s f o r P and H d e f i n i n g t h e p o s i t i o n o f t h e v o r t i c e s a r e b a s e d on t h e e x p e r i m e n t a l v a l u e s o b t a i n e d by F e r g u s o n (13) m t h e R e y n o l d s number r a r g e 1.5 - U . 1 x 1 0 4 . He d e t e r m i n e d t h e p o s i t i o n o f t h e f i r s t s i g n a l i n t h e wake i n phase w i t h 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 on t h e s u r f a c e o f t h e c y l i n d e r t o be a t H=.47 and P=1. f o r t h e o s c i l l a t i n g c y l i n d e r a s c o m p a r e d w i t h H>.5 and P=1. f o r t h e p r e s e n t m o d e l . F u r t h e r m o r e , he f o u n d t h a t t h i s p o s i t i o n d i d n o t c h a n g e s i g n i f i c a n t l y w i t h w i n d s p e e d . The v a l u e s f o r M and r« a r e n o t b a s e d on any e x p e r i m e n t a l v a l u e . T h e y a r e c h o s e n f r o m p h y s i c a l a r g u m e n t s . I n ftadderom's m o d e l , he c h o s e t h e v a l u e s t o be .5 t o g i v e good d i s t r i b u t i o n . T h i s means t h a t b e t w e e n t h e f o r m a t i o n o f t h e v o r t i c e s i n e i t h e r t h e u p p e r o r l o w e r row, t h e r e e x i s t s a t i m e i n w h i c h t h e r e i s no v o r t i c i t y a t a l l . P h y s i c a l l y t h i s i s n o t q u i t e c o r r e c t a s v o r t i c i t y i s g e n e r a t e d a l l t h e t i m e a t t h e s e p a r a t i o n p o i n t and s w e p t d o w n s t r e a m i n t h e t h i n s h e a r l a y e r b e f o r e i t r o l l s up t o f o r m t h e d i s c r e t e v o r t i c e s . Thus t n e r e s h o u l d be some c i r c u l a t i o n a l l t h e t i m e i n t h e wake and t h i s i s m o d e l l e d by M b e i n g g r e a t e r t h a n N. I t i s f o u n d t h a t B=. 6 and N=.4 works w e l l f o r t h i s m o d e l . I t m i g h t be w o r t h w h i l e t o p o i n t out t h a t tr.e e l e m e n t a r y v o r t i c e s t h a t G e r r a r d (5) u s e d i n h i s model o f t h e wake h a r t mean s t r e n g t h s o f a b o u t .682 and t h e s e o s c i l l a t e a t t h e S t r o u . i a l f r e q u e n c y w i t h an a m p l i t u d e o f a b o u t 31 .502. However, one must remember t h i s i s a t o t a l l y d i f f e r e n t m o d e l . The v a l u e s f o r and ^ a r e . 0 0 1 C 3 and .00257 r e s p e c t i v e l y . T h e s e a r e f r o m t h e c y l i n d e r - s p r i n g s y s t e m u s e d by F e n g . The s t r e n g t h o f c i r c u l a t i o n o(t r a n g e s f r o m .22 t o .58 i n the l o c k - m r e g i o n . A l t h o u g h i t i s d i f f i c u l t t c cor&rrent cn t h e r e s u l t s f r o m a p o t e n t i a l f l o w model i n w h i c h t h e r e i s no t o t a l head l o s s i n t h e wake, i t i s i n t e r e s t i n g t c compare t h i s r a n g e o f v a l u e s w i t h t h e one c a l c u l a t e d f r o m t h e d a t a o b t a i n e d by Fage a n l J o h a n s e n (12) a t a R e y n o l d s number o f 3 x 1 0 * . As shown i n t h e d i a g r a m b e l o w , t h e r a t e v o r t i c i t y i s s h e d i s g i v e n by The a v e r a g e K i s .9 and t h e a v e r a g e c o n v e c t i o n s p e e d \£ i s . 757U 0 f r o m x=0 t o x-.796 ( 2 a ) . Now o n l y an amount £ o f t h e 1? t o t a l v o r t i c i t y s h e d shows up i n t h e d i s c r e t e v e r t e x . T h e r e f o r e £tT0z^K . et was e s t i m a t e d t o be .5. I f b i s t h e v o r t e x s p a c i n g , . T h i s g i v e s where t h e v a l u e b/a=8.51 i s f r o m t h e same p a p e r As shown m f i g u r e 8, t h i s m o d e l i s c a p a b l e o f g e n e r a t i n g t h e jump c o n d i t i o n i n a m p l i t u d e and p h a s e o b s e r v e d i n e x p e r i m e n t s . One t h i n g s h o u l d be p o i n t e d i s t h a t t h e b r a n c h o f c u r v e b e t w e e n B and C i s u n s t a b l e . S t e a d y s t a t e o s c i l l a t i o n i n t h i s r e g i o n i s i m p o s s i b l e a c c o r d i n g t o n o n - l i n e a r a n a l y s i s . T h e r e i s no s u r p r i s e t h a t t h e m o d e l g i v e s a m p l i t u d e s o f o s c i l l a t i o n t h a t f o l l o w t h e e x p e r i m e n t a l o n e s c l o s e l y i n t h e w h o l e l o c k - i n r e g i o n a s i t i s d e s i g n e d t o do s o . A l t h o u g h t h e p h a s e a n g l e b e t w e e n e x c i t a t i o n and o s c i l l a t i o n does n o t a g r e e v e r y w e l l n u m e r i c a l l y w i t h t h e o n e s o b t a i n e d by F e n g , t h i s m o d e l shows t h e c o r r e c t t r e n d a t t h e f i r s t h a l f o f t h e l o c k - i n r a n g e . The v a l u e s i n b r a n c h e-d a r e h i g h e r t h a n t h o s e i n a-b. T h i s i s n o t t h e t r e n d o b s e r v e d i n t h e l a b o r a t o r y . The r a n g e o f v a l u e s f o r <£. i s n o t a s g r e a t as t h e one shown i n f i g u r e 3. I n t h e m o d e l , t h e v a l u e f o r i s a b o u t c o n s t a n t a t 100° a s c o m p a r e d t o t h e v a r i a t i o n f r o m z e r o t o 190°. A l l t h e i m p o r t a n t jumps a r e o b s e r v e d a t 100° i n t h e model w h i l e 33 i t i s a t -100° m t h e me a s u r e d c a s e . I t i s more i n t e r e s t i n g t o e x a m i n e t h e r e l a t i o n s h i p b e t w e e n cAB and Y . As shewn i n f i g u r e 7, ot^and y a r e e s s e n t i a l l y r e l a t e d by two l i n e a r e q u a t i o n s j o i n e d t o g e t h e r a t one en d . T h i s seems t o i n d i c a t e t h a t t h e wake v o r t i c e s a r e o r g a n i z e d , o r d i s o r g a n i z e d , i n a l i n e a r f a s h i o n d e p e n d i n g on wheher you go up or down a l o n g t h e c u r v e s . T h i s means t h a t t h e v o r t e x f i l a m e n t s a r e a l i g n e d f r o m t h e i r t i l t i n g p o s i t i o n t o t h e a x i s c f t h e c y l i n d e r g r a d u a l l y i n a l i n e a r manner. However, Feng and Koopmann b o t h r e p o r t e d a s u d d e n c h a n g e i n a l i g n m e n t w h i c h would mean a ' t h r e s h o l d ' a m p l i t u d e i s r e g u i r e d t o a l i g n t h e v o r t i c e s . W i t h a l l t h e p a r a m e t e r s t h u s d e t e r m i n e d , t h e p r e s s u r e d i s t r i b u t i o n on t h e s u r f a c e o f t h e c y l i n d e r i s o b t a i n e d f o r V=. 9 6 3 . A t t h i s v e l o c i t y , Y =.U5, $ = 9 8 ° , and ri.e = . t l 5 5 . The r e s u l t i s shown i n f i g u r e 9. T h i s model i s a b l e t o p r e d i c t t h e d r a m a t i c r i s e i n p r e s s u r e c o e f f i c i e n t o b s e r v e d i n v o r t e x - i n d u c e d o s c i l l a t i o n . The m a g n i t u d e o f t h e RKS v a l u e a t t h e S t r o u h a l f r e q u e n c y i s m t h e same o r d e r a s t h o s e m e a s u r e d by F e n g . However, t h e ' v a l l e y ' a t &-9oa d e v i a t e s f r o m m e a s u r e d v a l u e s . The c a u s e o f t h i s v a l l e y c a n e a s i l y be i d e n t i f i e d t o be t h e 4ucsinl&/i/<> t e r m i n e q u a t i o n ( 1 6 ) . A t t e m p t s have been made t o see w h e t h e r t h i s m o d e l i s a b l e t o p r e d i c t t h e r i g h t k i n d o f p r e s s u r e d i s t r i b u t i o n . The b e s t r e s u l t a s f a r as t h e v a l l e y i s c o n c e r n e d i s o b t a i n e d w i t h a r b i t r a r y v a l u e s f o r t h e p a r a m e t e r s as shown i n f i g u r e 10. The f r o n t h a l f i s i n good a g r e e m e n t w i t h t h o s e m e a s u r e d by Feng. The b a c k h a l f i s more t h a n t w i c e a s h i a h 3 4 d S the measured values in some places. This i s to fce expected because in a potential model, there is no total head loss across the shear layers. What happens in the real separated flow is part of the flow energy i s used m the formation of the shear layers. The shear layers r o l l up to form the wake vortices alternately. These vortices account partly for the low base pressure on the cylinder. 3 5 7 Suture R e s e a r c h and C o n c l u d i n g Remarks The p o t e n t i a l m o d el f o r v o r t e x - i n d u c e d o s c i l l a t i o n p r e s e n t e d h e r e i s a b l e t o g i v e t h e jump c o n d i t i o n i n b o t h a m p l i t u d e and p h a s e o f o s c i l l a t i o n a s o b s e r v e d i n t h e l a b o r a t o r y . I t h a s d e m o n s t r a t e d i t s a b i l i t y t o i s o l a t e t h e e f f e c t s o f d i f f e r e n t q u a n t i t i e s i n v o l v e d i n t h e phenomenon. I t j u s t i f i e s f u t u r e r e s e a r c h t o e l u c i d a t e t h e phenomenon by t h i s a p p r o a c h . Much i m p r o v e m e n t o f t h e m o d e l i s r e q u i r e d t o make i t s a t i s f a c t o r y . T h i s c a n p r o b a b l y be a c h i e v e d by i n c o r p o r a t i n g t h e f a c t t h a t c i r c u l a t i o n d e p e n d s on a m p l i t u d e i n t o t h e c h a r a c t e r i s t i c s o f t h e v o r t i c e s , a n d r e l a t i n g t h e p h a s e o f o s c i l l a t i o n t o some p h y s i c a l q u a n t i t i e s i n t h e wake. 36 REFERENCES (1 ) N a u d a s c h e r , E. , *From F l o w I n s t a b i l i t y t o F l o w - I n d u c e d E x c i t a t i o n 1 , P r o c . AS^E, Hy 4, J u l y 1 9 6 7 , 15-40. (2) 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 ' , P r o c . I n t . Res. Sem., Wind 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 , O t t a w a , 1967. (3) F e n g , C.C., 'The Measurement o f V o i 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 and O s c i l l a t i n g C i r c u l a r a n d E-s e c t i o n C y l i n d e r s ' , F.A.Sc. T h e s i s , U.B.C., 1968. (4) Koopniann, G.H., 'The v o r t e x Wake 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 1 , J F P , 28 , 3, 1969, 50 1-12. (5) G e r r a r d , J . H . , 'The 3 - 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 K , 25 , 1966, 143-164. (6) J o r d a n , S. K. , and Fromm, J . E . , ' O s c i l l a t i n g D r a g , L i f t , and T o r q u e on a C i r c u l a r C y l i n d e r i n a U n i f o r m F l o w ' , P h y s . F l u i d s , J5 , 3 , M a r c h , 1972, 371- 6. (7) 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 ', P h i l . T r a n . Roy. S o c , A, 2 6_1 J a n . , 1967, 137- 162. 37 (8) A b e r n a t h y , F.F. And K r o n a u e r , R.E., 'The F o r m a t i o n o f V o r t e x S t r e e t s ' , JFh, JJj r 1962, 1-20. (9) 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 on a C i r c u l a r C y l i n d e r i n a U n i f o r m S t r e a m ' , UTIAS TN 14, 1957. (10) B a d d e r o m , P., 'Two V o r t e x P o t e n t i a l M odel f o r T u r b u l e n t F l o w P a s t C i r c u l a r C y l i n d e r s ' , UBC ME E e p t . U n p u b l i s h e d N o t e , A p r i l , 1966. (11) H a r t l e n , R.T. And C u r r i e , I . G . , ' l i f t - C s c i l l a t o r M o d e l o f V o r t e x - I n d u c e d V i b r a t i o n ' , P r o c . ASCE, EM 5, O c t . 1970, 5 7 7 - 5 9 1 . (12) M i l n e - T h o m s o n , L.M., ' T h e o r e t i c a l H y d r o d y n a m i c s ' , 5 t h E d . , M a c M i l l a n , 1968. (13) F e r g u s o n , N., 'The M easurement o f Wake and 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 and 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 * , M.A.Sc. T h e s i s , UBC, 1965. (14) F a g e , A. And J o h a n s e n , ? . C , 'The 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 . 7, V o l . 5, #28, Feb. 1928, 4 17-4 41. 38 (15) K r y l o f f , N. , and B o g o l i u b o f f, N. r ' I n t r o d u c t i o n t o Non-L i n e a r M e c h a n i c s ' , P r i n c e t o n U n i v e r s i t y P r e s s , 1947. (16) B r o w n , K.M., 'The S o l u t i o n o f S i m u l t a n e o u s N o n l i n e a r E q u a t i o n s ' , P r o c . ACM N a t i o n a l M e e t i n g , 1967, 110-114. 39 APPENDIX I U§£hod o f E q u i v a l e n t L i n e a r i z a t i o n We c o n s i d e r n o n - l i n e a r d i f f e r e n t i a l e q u a t i o n s o f t h e f orm U ) where >j i s a s m a l l p a r a m e t e r . The e x t e r n a l f o r c i n g f u n c t i o n -^(svtjY, Y ) c a n be r e p l a c e d by an e q u i v a l e n t l i n e a r one e w i t h an a c c u r a c y t o w i t h i n t h e o r d e r o f 7il ( s e e r e f e r e n c e 15). A s s u m i n g Y = Y $>in ( - ' v c - r , we p u t C o m p a r i n g t h i s w i t h t h e e x t e r n a l f o r c i n g f u n c t i o n f , we o b t a i n — w r 2 T L \< - - =r^~ \ -$C*>t, Y ; Y ) Sin (-at-r 5 ) » ( ( A t - t i ) T h u s e g u a t i o n (1) becomes Y > Y + 0 + £ ) Y = O 40 From t h e t h e o r y o f l i n e a r o s c i l l a t i o n , we g e t oLt, F o r t h e l o c k - m range,-^-=1. 41 APPENDIX I I The following program calculates C at Strouhal frequency R E A L M , N C0MM0N/FUN/THETA,Xl,X2,ALFA.P,M,N,S,V,HiPl S= 2 P-3 = 3 „ H 1 5 Q RCAD(5,10) X1,X2,ALFA,M,N,P,H,V 10 FORMATClOFtO'.S) D 0-12 V-I-I-= 1,-3-7 THETA=PI/lfl', T H ETAsTHCTA*(FLOAT(II>*1,) F-X-T-F-R-N-AL CPS-J-N CPRM8S=SQUANK(CPSIN.,0,6,28318/.0001,TOL#FIFTH) EXTERNAL CPCOS CP R-M SG-= vS 0 LU N K-(-^P -Cn5- # - r 0,6,2 8-3 l-fl-#40-0-04-»~T-0 F-I-F-T-H-) CPFUNsSRRT((CPRMSS**2+CPRMSC**2)/2,/PI **2) A N G I E s ( T H E T A / P I ) * 1 8 0 ' , WRITE(6,124) ANGLE ,CPFUN tCPRMSS,CPRMSS,CPRMSC 124 F0RMATC7G15U) 123 CONTINUC S-T-^ P END IN EFFECT* ID,CBCDIC,SOURCE,NOLIST,NODECK,LOAD,NOMAP 4-N EFFECT* NAME = MAIN , LINECNT = 51 CS* SOURCE STATEMCNTS s 20 # PROGRAM S I Z E s 762 CS* NO DIAGNOSTICS GENERATED •WA-J+J-R M I N A L S Y S T C M F O R T R A N G C 4 1 3 3 6 ) C P C O S F U N C T I O N C P C O S ( T A U ) C n M M O N / F U N / T H E T A , X l , X 2 , A L F A . P . M , N , S » V , H # P I R E A L M , N Y-^X4-*S4-N (TAIJ + X g ) Y D O T = X l * C 0 S ( T A U + X 2 ) P S I 1 = A T A N ? ( H - Y , P ) P-S-I-2-s A T-A N 2_(L-H-Y-^P-5 S T H s S I N ( T H E T A ) C r H s C O S C T H E T A ) S^1-*S4^P-S-J4-J C S l s C O S f P S H ) S S 2 a S l N ( P S I 2 ) GS^S&OS-C-P^J-?-) -D 1 = P * P + C H - Y D * ( H - Y D D 2 = P * P + C H + Y ) * C H + Y ) S ^ P l = S O R T ( D l ) S O D ? = S Q R T ( 0 2 ) P 3 I s A T A N 2 ( X l + H , P ) P 4 4 4 - ^ A W - * l ( , S * S I M ( P F t l s A T A N 2 ( S T H * S S l / C 2 „ * S Q D l ) , C T H - C S 1 / ( 2 ' , * S O D 1 ) ) I F C F l l . L T ' . O ' . ) GO TO 1 2 5 I F ( T H E T A , G T . r . 5 * P I ) CO TO 1 3 4 F 1 = F 1 1 , GO TO 1 2 8 -4-25 I F ( T H F T A . L T ' . P H I ) GO TO 1 2 6 1 3 6 F 1 = F 1 1 + 2 . * P I GO TO 1 2 8 -4-2-6 F 1 = F U ; 1 2 8 F 2 2 = A T A N 2 ( S T H + 2 . * Y * 2 . * H «CTH«2.*P) I F ( F ? 2 , L T ' . 0 ' . ) GO TO 1 4 0 F-2^F-2-2 • GO TO U l H O F 2 = 2 . * P I + F 2 2 •4-4-1 F 3 3 = A T A N 2 ( S T H + 2 , * Y + 2 ' , * H , C T H ~ 2 ' . * P ) I F ( F 3 3 , L T ' . 0 ' . ) GO TO 1 5 0 F 3 = F 3 3 GO TO 154 1 5 0 F 3 = 2 . * P I + F 3 3 1 5 1 F 4 4 = A T A N 2 ( S T H ^ S S 2 / ( 2 , * S Q D 2 ) , C T H - C S 2 / ( 2 , * S 0 D 2 ) ) IF(F/W !«LX0 r , . ) GO TO 1 6 5 I F ( T H E T A ' . G T , r . 5 * P I ) GO TO 1 7 6 F 4 = F 4 4 GO TO 1-&0 1 6 5 C O N T I N U E 1 7 6 F 4 = F 4 4 + 2 ' , * P I — 1 8 0 P F 1 0 T e ( ( ( C S 1 * P . » S S I » ( H - Y ) ) a Y O O T ) * ( G T H - G - S 1 / ( 2 ' . »S&D 1 ) ) - ( S T H - S S 1 / ( ? « * S 1 0 0 1 ) ) •* ( ( C S 1 * ( Y - P ) - S S 1 * P ) * Y 0 0 T ) ) / ( « , * V * D l * * l ' . 5 * ( ( C T H - C S i / ( 2 , * S O O l ) ) 2 * * 2 + ( S T H - S S l / C ? . * S Q D l ) ) * * 2 ) ) OF -40T^ - ( -< -C~T44 - tS^ -^^ 2 ^ ( H ^ Y - ) - ) ^ Y - O ^ ^ ^ W ^ S a ^ ^ ^ ^ O t V 1 2 ) ) * ( C S ? * ( H + Y ) - S S 2 * P ) * Y D 0 T ) / C 4 , * V * D 2 * * l ' . 5 * ( CCT 2 H - C S 2 / ( 2 . * S Q D 2 ) ) * * 2 + ( S T H - S S ? / ( 2 ' . * S 0 D 2 ) ) * * 2 ) ) D F 2 D l a Y 0 0 T » ( C T H " 2 U P ) / ( V * ( ( C T H " 2 , * P ) » * ? * ( S T H » 2 > » Y * 2 > » H ) » « 2 ) ) D F 3 D T = Y D 0 T * ( C T H - 2 . * P ) / ( V * ( ( C T H - 2 , * P ) * * 2 + C S T H + 2 . * Y + 2 ' . * H ) * * 2 ) ) A U F A 1 = A L F A * ( M + N * S I N ( T A U ) ) / W ^ t ^ - W W W f ) R M I N A L S Y S T F M F O R T R A N G ( 4 1 3 3 6 ) C P C O S 4 3 A L i n O T = A L F A A N * C O S C T A U ) Al 2 D 0 T = - A L F A * N * C 0 S t T A U ) AKOIJ = A L F A l * ( l ' , - 4 , * D l ) / ( 2 , *P T * ( T.+ '4 , * D 1 - 4 , * ( C T H * P + S T H * ( H - Y ) ) ) 5 - A U F A 1 . - / I . v r ^ V - ( - ^ - ^ - I - 4 M 4 - ^ ^ C P = - A L 1 0 0 1 * S * ( F 1 - F 2 ) - S * A L 2 D 0 T * ( F 3 * F 4 ) - A L F A1 / P I * ( O F 1 D T - 0 F 2 D T ) - A I . F A2 1 / P I A (DF3DT«DF4DT) -4 .*YDOT * S I N ( 2 ' . * T H E T A ) / V * 2 ' . * A K O U * ( * 2 . * S T H M , * Y D O T 2_* n-T-H-/-V-)--A ICO U*-A K-OU—Y-/-(-V~*-V4-*S~T-H CPCOSsCP*COS(TAU) R E T U R N I N E F F E C T * ! D , r B C D I C , S O U R C E , N O L 1 S T , N O D E C K , L O A D , N O M A P I N C F F E C T * NAMT = C P C O S , L I N E C N T = 5 7 TC-5* S W M - S T ^ W i C - H W - s ^6-T-P-R-OGRAM S I Z E — « 2-7-24-I C S * NO D I A G N O S T I C S G E N E R A T E D C P C O S \ G I N S ON * S O U R C F * C A U S E S A R E T U R N TO M T S » R M I N A T E D [TRMINAL SYSTEM FORTRAN GC41336) OPSIN j 44 ; ALlDOT=AlFAAN*rOS(TAtO > A|.2DOT = -Al FA*N*COS(TAU) , AKOU=ALTAI* C l ' . - a , *D1) / (2 , *P T * (1'.+4 . *u 1 f 4 , * (CTH*PtSTH* (H-Y) ) ) )«ALFA [ (1 , *-&2-)/(3,«PI»(l,wr.« 02*4'. *(CTH* p^-jMMM^-y-)^-) ; CP=-AL100T*S*(Fl-F2)-S*AL2D0T*(F3^Fa)"AUFAi/PI*(DFlDT^DF2DT)-ALFA2 1 l/PI*(nF3DT-0F«nT)-.«'.*Y00T*STN(2'.*THETA)/V + 2'.*AK0U*(»2'.*STH'.a,*Y00T 2 * GT-H/V-) - AKOIMAK-O U_*-YV-(-V-* V-)-*$T H ' CPSIN=CP*SIN(TAU) RETURN FWO 5 IN EFFECT* 10,CBCDIC,SOURCE,NOLI ST,NODFCK,LOAD,NOMAP 5 IN EFFECT* NAME = CPSlN , LINECNT = 57 -I-C-S* SOUR-G-E—ST-A-T-E-MEN-T-S-s 58-,P-RQGR AM S-I-Z-E—a 2-7-2-S-' I C S * NO DIAGNOSTICS GFNERATED i CPSIN \ \ S Y S T C M F O R T R A N G C 4 1 3 3 6 ) C P S I N 45 F U N C T I O N C P S I N ( T A U ) R E A L M f N C 0 M M 0 N / F U N / T H E T A , X 1 , X 2 , A L F A , P , M , N , S . V , H # P I Y _ x X 4 ^ g i M ( T A l H X ? ) Y 0 0 T = X 1 * C 0 S ( T A U + X 2 ) P 9 I 1 = A T A N 2 ( H - Y f P ) -P-S-I-?=A T-A N H - A V P - 4 — S T H s S T N ( T H F T A ) C T H s C O S ( T H E T A ) - S S U S l N ( P S I l ) C S l s C n S ( P S I l ) S f l 2 s S l N ( P S I 2 ) -C-52sG-0S-(a$- I -2 - ) -D 1 = P * P + ( H - Y ) * (H-»Y) D 2 = P * P + C H + Y ) * ( H + Y ) - S W ^ S Q R T ( D l ) S O D 2 = S Q R T ( 0 ? ) PSI=ATAN2(X1+H,P) F l l s A T A N 2 ( 8 T H * S S l / ( 2 . * S Q 0 J ) . C T H » C S t / ( 2 , , * S 0 0 i ) ) I F C F l l . L T ' . O ' . ) GO TO 125 }. I F C T H E T A ' . G T ^ l ' . S a P I ) GO TO 136 ,' F l s F l l \ GO TO 128 ^ -W5 I F ( T H E T A . L T I P H I ) GO T O 126 , 136 F l = F l i + 2 „ * P I GO TO 128 4n2-6 F l - F l 1 128 F22=ATAN2(STH+2,*Y-2,*H ,CTH«.2>P) IFCF22,LT'.0'.) GO TO 140 F-2^F-^J GO TO 141 140 F 2 = 2 . * P I + F 2 2 -4-4-1 F32S = ATAN2(STH + ? ' , * Y + 2»*H, CTH-2'.+P) I F C F 3 3 . L T . 0 f . ) GO TO 1 5 0 F 3 = F 3 3 GO TO 151 1 5 0 F 3 = ? . * P I + F 3 3 1 5 1 F44=ATAN2(STH-8S2/(2,*S0D2).CTH-CS2/C2' *SQD2)) IF(F«4«LT.0 f.) 00 TO 1 6 5 I F ( T H E T A . G T . l ' . 5 * P I ) GO TO 1 7 6 F 4 = F 4 f l -GO TO 1-84 1 6 5 C O N T I N U F 1 7 6 F 4 = F 4 « + 2 , * P T 1 8 0 D F I D T B C ( ( C 8 1 » P - S 3 1 » ( H » Y ) ) * Y - O W - ^ - G T - H ~ £ S - l / (2'. * S Q D1 ) ) - (STH«3S 1 / (2*. * f l 1 0 0 1 ? ) * ( ( C 8 1 * ( Y - P ) - S S J * P ) * Y D 0 T ) ) / ( 4 t * V * D l * * l ' . 5 * ( ( C T H « C S l / ( 2 , * S Q D l ) ) 2 * * 2 + ( S T H - S S 1 / ( 2 . * S Q D 1 ) ) * * ? ) ) -D-F-40-T-=-(-{4^r-H^G-8P-AC-a-r^Q4>2-)-)-*-(-C-S-?-*P-+SS2*-fR+V-)-)-»-YDOT- (S - T-H~SS2/(2 1 2 ) > *(CS2*(H + Y)-SS2*P)*YD0T)/(4,*V*D2**l' ,5*((CT 2H-CS2/C2.*S0D2) )**2+(STH-S82/(2r.*SQD2))**2)) / • 0F2DTeYD0T»(CTH>»2^P)/(Vir((CTH».'>,»P)»»a*(aTH»2'.*Y-2l»H)»*?)) DF3DT = YD0T* (CTH-2.*P)/(V*((CTH"2,*P)**2+(STH+2.*Y+2'.*H)**2)) A L F A U A L F A * (M + N * S I N ( T A U ) ) -A4^F-A-2BALFA»(M-N*S-IN(TAU) ) 46 The following program calculates phase and amplitude of osc i l la t ion . REAL M/NN , DIMENSION I P 0 I N T ( 4 , 4 ) , I S U B ( 4 ) , C O E ( 4 , 4 ) , T E M P ( 4 ) , P A R T ( 4 ) , X ( 3 ) ' C 0 M M 0 N / 7 / X 1 , X 2 , X 3 , H , P , P I , S , V , M , N N , B E T A , R N , A L F A , H F MAXIT=20 ^ NUMSIG=4 101 R E A D C S , 1 0 ) M , N N , P , H , A L F A ' 10 FORMAT(15F8„5) g = p B E T A s . 0 0 1 0 3 \ R N a . 0 0 2 5 7 P I = 3 , 1 4 1 5 9 \ 1 R E A D ( S , 1 0 ) V \ IF ( V . E Q , 0 , ) GO TO 35 R F A D ( 5 , 1 0 ) X ( 1 ) , X ( 2 ) , X ( 3 ) , C 0 N T R 0 CALL NONLIN ( 4 , M A X I T , N U M S I G , I S I N G , X , I P 0 I N T , I S U B , C 0 E , T E M P , P A R T ) IF C M A X l T , E Q f 2 0 ) W R I T E ( 6 , 3 0 ) I F ( I S I N G , E Q . O ) W R T T E ( 6 , 3 0 ) WRITE ( 6 , 2 0 ) X , M A X I T , I S I N G , V , M , N N , P , H , A L F A 20 FORMATC'O X»'S ARE » ,3G15 t 8, l# I T = • , 1 3 , • S I N G ' , 1 2 , 6 F 1 0 , 3 ) 30 FORMAT (1X,»N0 C O N V E R G E N C E ' ) MAXIT=20 I F C C O N T R O ' . E O . l ) GO TO 101 GO TO 1 35 STOP ENO S IN E F F E C T * I D , C B C D I C , S O U R C E , N O L I S T , N O D E C K , L O A D , N O M A P S IN E F F E C T * NAME = MAIN , L INECNT = 57 T I C S * SOURCE STATEMENTS s 26 ,PROGRAM S I Z E a . 1078 T I C S * NO DIAGNOSTICS GENERATED N MAIN \ :RMINAL SYSTEM FORTRAN G ( 4 1 3 3 M F K 4 7 S U B R O U T I N E F K C N , X , Y , K ) C O M M O N / Z / X 1 , X 2 , X 3 , H , P , P I , S , V , M , N N , B E T A , R N , A L F A , HF D I M E N S I O N X ( N ) E X T E R N A L F 1 , F 2 A = - 1 0 8 9 , 8 i B = . 0 6 C = - 7 9 . 2 4 8 D = 1 . 0 7 X 1 = X C 1 ) X 2 = X ( 2 ) X 3 = X ( 3 ) R L I M = O s R L I M U = 3 , 1 4 1 5 9 * 2 , GO TO ( 1 0 , 2 0 , 3 0 ) , K 10 Y = S Q U A N K ( F 1 , R L I M , R L I M U , , 0 0 0 0 1 , T O L , F I F T H ) R E T U R N 2 0 Y = S Q U A N K ( F 2 , R L I M , R L I M U , . 0 0 0 0 1 , T O L , F I F T H ) R E T U R N 4 - 1 a 1 t « 5 9 * ( X ( l ) / ( X ( 3 ) - , 1 0 4 ) M . 2 5 ) * * 2 ) - . 0 7 5 * E X P ( M 7 3 2 . 8 7 5 7 * ( X ( l ) / ( X ( 3 ) - , 1 0 f l ) « * t 9 8 ) * * 2 ) R E T U R N END I N E F F E C T * I D , E B C D I C , S O U R C E , N O L I S T , N O D E C K , L O A D , N O M A P I N E F F E C T * NAME = F K , L I N E C N T = 5 7 C C S * S O U R C E S T A T E M E N T S = 2 1 , P R O G R A M S I Z E = 1 5 7 2 ICS* . NO D I A G N O S T I C S G E N E R A T E D F K 4 8 E R M I N A L S Y S T F M F O R T R A N G ( 4 1 3 3 6 ) F l F U N C T I O N F K T A O ) C O M M O N / Z / X 1 , X 2 , X 3 , H , P , P I , S , V , M , N N , B E T A , R N , A L F A , H F R E A L M , N N , N N = NN T A U = T A 0 - X 2 A L P A 1 = X 3 * ( M + N * S I N ( T A U ) ) A L F A 2 = X 3 * ( M - N * S I N ( T A U ) ) Y = X 1 * S I N ( T A U + X 2 D Y D 0 T = X 1 * C 0 S ( T A U + X 2 ) P S = P * P / H S = H * H Y S = Y * Y N A A s ( A L F A l ) * * 2 * ( Y - H ) / ( P I * ( l , - a . * ( C Y - H ) * * 2 + P S ) ) ) A B = ( A L F A 2 ) * * 2 * ( Y + H ) / ( P I * ( l , - 4 . * ( ( Y + H ) * * 2 + P S ) ) ) s A C = A L F A 1 * ( - 1 , + ( 2 , * Y D 0 T * C Y - H ) * P + C P S - ( Y " H ) * * 2 ) * V « 2 , * C Y * H ) * S G R T ( ( H - Y ) 1 * * 2 + P S ) * P * Y D 0 T ) / ( V * ( ( Y - H ) * * 2 + P S ) * * 2 * 4 . ) ) AD = A L F A 2 * ( l ' . - ( 2 , * Y D O T * P * ( Y + H ) + ( P S - ( Y + H ) * * 2 ) - 2 , * ( H + Y ) * S Q R T ( ( H " » Y ) * * 2 1 + P S ) * P * Y D 0 T ) / ( 4 . * V * ( ( Y + H ) * * 2 + P S ) * * 2 ) ) AE = A L F A 1 * A L F A 2 * Y * ( P S + Y S - H S ) / ( 2 , * P I * ( ( P S + Y S - H S ) * * 2 + 4 . * H S * P S ) ) A F = X 3 * N * C 0 S ( T A U ) * P I * S * P * ( S Q R T ( ( Y T ^ ) * * 2 + P S ) - , 5 ) / ( ( Y » H ) * * 2 + P S ) A G = X 3 * N * C 0 S ( T A U ) * P I * S * P * ( S Q R T ( (Y + H ) * * 2 + P S ) « * , 5 ) / ( (Y+H) * * 2 + P S ) F 1 = (AA,+ AB + AC + A D - A E - A F - A G ' 2 , * B E T A * Y D 0 T / ( R N * V * V ) ) * S I N ( T A U * X 2 ) R E T U R N END S I N E F F E C T * I D , E B C D I C , S O U R C E , N O L I S T , N O D E C K , L O A D , N O M A P S I N E F F E C T * NAME = F l , L I N E C N T = 5 7 T I C S * S O U R C E S T A T E M E N T S s 2 2 , P R O G R A M S I Z E s 1400 T I C S * . NO D I A G N O S T I C S G E N E R A T E D H F l 'ERMINAL SYSTEM FORTRAN 0 ( 4 1 3 3 6 ) F2 49 FUNCTION F 2 ( T A 0 ) REAL M , N N , N C 0 M M 0 N / 7 / X l , X 2 , X 3 , H , P , P I , S , V , M , N N , B E T A , R N , A L F A , H F N = NN T A U = T A 0 - X 2 Y = X 1 * S I N ( T A U + X 2 ) YD0T=X1*C0S (TAU+X2 ) PS=P*P HS=H*H A L F A 1 = X 3 * ( M + N * S I N ( T A U ) ) / A L F A 2 = X 3 * ( M - N * S I N ( T A U ) ) \ YS = Y*Y V A A = ( A L F A l ) * * 2 * ( Y - H ) / ( P I * ( l , - 4 , * ( ( Y - H ) * * 2 + P S ) ) ) A B = ( A L F A 2 ) * * 2 * ( Y + H ) / ( P I * ( 1 , - 4 . * ( ( Y + H ) * * 2 + P S ) ) ) ' AC=ALFA1*( -1,+(2,*YD0T*(Y-H)*P+(PS-(Y-H)**2)*V-2,*(Y"H)*SQRT((H«Y) 1 * * 2 + P S ) * P * Y D 0 T ) / ( V * ( ( Y ~ H ) * * 2 + P S ) * * 2 * 4 , ) ) AD = A L F A 2 * ( 1 . - ( 2 , * Y D O T * P * C Y + H ) + ( P S - ( Y + H ) * * 2 W , * ( H + Y ) * S Q R T ((H«Y)**2 i+PS)*P*YDQT)/ (4«*V*((Y+H)**2+PS)**2)) A E = A L F A 1 * A L F A 2 * Y * ( P S + Y S - H S ) / ( 2 , * P I * ( ( P S + Y S - H S ) * * 2 + 4 . * H S * P S ) ) AF = X 3 * N * C 0 S ( T A U ) * P I * S * P * ( S Q R T ( ( Y - H ) **2+PS ) - . 5 ) / ( ( Y-»H) * * 2 + P S ) A G = X 3 * N * C 0 S ( T A U ) * P I * S * P * ( S O R T ( ( Y + H ) * * 2 + P S ) - , 5 ) / ( ( Y + H ) * * 2 + P S ) F2=(AA+AB+AC+AD«AE-AF-AG-2,*BETA*YD0T/(RN*V*V) ) *COS (TAU + X2 ) RETURN END S IN E F F E C T * I D , F B C D I C , S O U R C E , N O L 1 S T , N O D E C K , L O A D , N O M A P S IN E F F E C T * NAME = F2 , L INECNT = 57 T I C S * SOURCE STATEMENTS = 22 ,PROGRAM S I Z E s T I C S * NO DIAGNOSTICS GENERATED N F2 1400 TS FLAGGED IN THE ABOVE C O M P I L A T I O N S , : G I N S 3 52 c i r c u l a r c y l i n d e r F i g u r e 4. G e n e r a l s e t - U p f o r t h e p r e s e n t m odel. N o t a t i o n s u s e d i n t h e p r e s s u r e e q u a t i o n 0 60 120 180 6' f i g u r e 10, C o m p a r i s i o n o f Cp r n 1 3 p r e d i c t e d by t h e p r e s e n t m o d e l w i t h a r b i t r a r y v a l u e s f o r t h e p a r a m e t e r s ^ and m e a s u r e d v a l u e s . - -

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