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Forces on a cylinder due to waves and a colinear current Buckingham, William Richard 1982

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FORCES ON A CYLINDER DUE TO WAVES AND A COLINEAR CURRENT by WILLIAM RICHARD BUCKINGHAM B.Sc., A c a d i a U n i v e r s i t y , 1973 M.Sc., 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 , 1976 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER•OF APPLIED SCIENCE i n THE FACULTY OF GRADUATE STUDIES Department Of C i v i 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 BRITISH COLUMBIA March 1982 © W i l l i a m R i c h a r d Buckingham, 1982 In p r e s e n t i n g t h i s t h e s i s i n p a r t i a l f u l f i l m e n t . of t h e r e q u i r e m e n t s f o r an a d v a n c e d d e g r e e a t t h e U n i v e r s i t y o f B r i t i s h C o l u m b i a , I a g r e e t h a t t h e L i b r a r y s h a l l make i t f r e e l y a v a i l a b l e f o r r e f e r e n c e and s t u d y . I f u r t h e r a g r e e t h a t p e r m i s s i o n f o r e x t e n s i v e c o p y i n g o f t h i s t h e s i s f o r s c h o l a r l y p u r p o s e s may be g r a n t e d by t h e Head of my D e partment or by h i s o r h e r r e p r e s e n t a t i v e s . I t i s u n d e r s t o o d t h a t c o p y i n g o r p u b l i c a t i o n o f t h i s t h e s i s f o r f i n a n c i a l g a i n s h a l l n o t be a l l o w e d w i t h o u t my w r i t t e n p e r m i s s i o n . D e partment of C i v i l E n g i n e e r i n g The U n i v e r s i t y of B r i t i s h C o l u m b i a 2075 Wesbrook P l a c e V a n c o u v e r , Canada V6T 1W5 D a t e : March 26, 1982 i i A b s t r a c t A s e r i e s of l a b o r a t o r y e x p e r i m e n t s was c o n d u c t e d t o examine t h e o s c i l l a t o r y f o r c e s " and t h e wave runup on a v e r t i c a l , c i r c u l a r , s u r f a c e p i e r c i n g , r i g i d c y l i n d e r . i n t h e p r e s e n c e of b o t h waves and a c o l i n e a r c u r r e n t . I t was f o u n d t h a t a c u r r e n t w h i c h r a n o p p o s i t e t o t h e d i r e c t i o n o f wave p r o p a g a t i o n r e d u c e d t h e o s c i l l a t o r y f o r c e and t h e runup on t h e c y l i n d e r . The r e s u l t s f o r a c u r r e n t r u n n i n g i n t h e same d i r e c t i o n as t h e waves were more s c a t t e r e d , w i t h some c a s e s i n d i c a t i n g an i n c r e a s e i n f o r c e w h i l e o t h e r s a d e c r e a s e . The runup, however, i n c r e a s e d i n a l l c a s e s . An i n n o v a t i v e n u m e r i c a l t e c h n i q u e w h i c h i s c u r r e n t l y under d e v e l o p m e n t was a p p l i e d t o t h i s p r o b l e m . The l o a d s on. t h e c y l i n d e r were o b t a i n e d by a t i m e s t e p p i n g p r o c e d u r e i n w h i c h t h e f l o w a t e a c h t i m e s t e p was c a l c u l a t e d by an i n t e g r a l e q u a t i o n method b a s e d on G r e e n ' s t h e o r e m . The g e n e r a l r e s u l t s of t h e n u m e r i c a l method a g r e e d q u i t e w e l l w i t h t h e e x p e r i m e n t a l o b s e r v a t i o n s , w i t h i n t h e c o n s t r a i n t s of some s i m p l i f y i n g a s s u m p t i o n s . i i i TABLE OF CONTENTS Page Abstract i i List of Figures v Acknowledgement v i CHAPTER 1 INTRODUCTION 1 CHAPTER 2 FUNDAMENTAL FLOWS 4 2.1 Steady Viscous Flow Past a Circular Cylinder 4 2.2 Accelerating Flow of an Inviscid Fluid Past a Circular Cylinder 5 2.3 Oscillating Flow of a Real Fluid Past a Circular Cylinder 6 2.4 Linear Wave Theory 8 2.5 Diffraction Theory 9 2.6 Wave and Current Interaction 11 CHAPTER 3 THE PRESENT INVESTIGATION 15 3.1 Objectives 15 3.2 Inertial Force Prediction 18 CHAPTER 4 THEORETICAL CONSIDERATIONS 21 4.1 Governing Equations 21 4.2 Numerical Solution 24 4.2.1 Theory 24 4.2.2 Results 29 CHAPTER 5 EXPERIMENTAL APPARATUS 31 5.1 The Wave Tank and Moving Support Carriage 31 5.2 The Test Cylinders 32 i v CHAPTER 6 EXPERIMENTAL PROCEDURE 35 6.1 Wave Basin R e f l e c t i o n C o e f f i c i e n t 35 6.2 Force Measurements 37 6.3 Runup Measurements 38 CHAPTER 7 EXPERIMENTAL RESULTS AND DISCUSSION 39 7.1 Beat Phenomenon 39 7.2 I n - l i n e Force 42 7.2.1 Description of Results 42 7.2.2 Comparison With Predicted I n e r t i a Force and Numerical Solution 44 7.3 Runup 45 7.4 Flow V i s u a l i z a t i o n 48 CHAPTER 8 CONCLUSIONS 50 BIBLIOGRAPHY 52 V LIST OF FIGURES P a 9 e 1. Wave and c y l i n d e r d e f i n i t i o n sketch 55 2. Experimental waves categorized by depth parameter (kd) 56 3. Experimental waves categorized by d i f f r a c t i o n parameter and Keulegan Carpenter number 57 4. I n e r t i a l force p r e d i c t i o n 58 5. Integration surfaces f o r numerical method 59 6. Free surface elevation and current v e l o c i t y evolution 60 7. Free surface elevation r e s u l t s from numerical method 61 8. Force r e s u l t s from numerical method 62 9. General view of wave basin 63 10. Test c y l i n d e r and support rod 64 11. Gain-frequency response curve of e l e c t r o n i c f i l t e r 65 12. Block diagram of datalogging system 66 13. Moveable support carriage and datalogging e l e c t r o n i c s 67 14. Test c y l i n d e r moving toward wave source 68 15. Beat phenomenon 69 16. Average force versus current v e l o c i t y f o r small c y l i n d e r .... 70 17. Maximum force versus current v e l o c i t y f o r small c y l i n d e r .... 71 18. Average force versus current v e l o c i t y f o r large c y l i n d e r .... 72 19. Maximum force versus current v e l o c i t y f o r large c y l i n d e r .... 73 20. Maximum runup (front face) on small c y l i n d e r 74 21. Maximum runup (front face) on large c y l i n d e r 75 22. Local maximum runup (rear face) on large c y l i n d e r 76 2 3. Flow separation around c y l i n d e r 77 24. Breaking wave on side of c y l i n d e r 78 v i Acknowledgement The work d e s c r i b e d i n t h i s t h e s i s was c a r r i e d o u t i n t h e h y d r a u l i c s l a b o r a t o r y o f t h e D e p a r t m e n t of C i v i l E n g i n e e r i n g of t h e 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 w h i l e t h e a u t h o r was i n r e c e i p t o f a r e s e a r c h a s s i s t a n t s h i p from t h e N a t u r a l S c i e n c e and E n g i n e e r i n g R e s e a r c h C o u n c i l o f Canada. The r e s e a r c h was s u p e r v i s e d by D r . M. de S t . Q. I s a a c s o n whose s u p p o r t and a d v i c e has been g r e a t l y a p p r e c i a t e d . The a u t h o r would a l s o l i k e t o e x p r e s s h i s t h a n k s t o K u r t N e i l s o n , Don Smythe, Andy Chan, and Thorn G a r d e f o r t h e i r i n v a l u a b l e a s s i s t a n c e d u r i n g t h i s p r o j e c t . F i n a l l y , t h e a u t h o r w i s h e s t o t h a nk h i s w i f e , J o a n i e , f o r h e r encouragement and l o v e . 1 1. INTRODUCTION The e v e r i n c r e a s i n g w o r l d w i d e demand f o r e n e r g y has r e s u l t e d i n d r a m a t i c a d v a n c e s i n o f f s h o r e t e c h n o l o g y a s i t r e l a t e s t o t h e p e t r o l e u m i n d u s t r y . The i n i t i a l a c t i v i t y i n o f f s h o r e a r e a s i n v o l v e d l a n d - b a s e d e x p l o r a t i o n and p r o d u c t i o n s y s t e m s , w i t h m i n o r m o d i f i c a t i o n s t o a l l o w f o r t h e s h a l l o w w a t e r i n w h i c h t h e y were s i t u a t e d . As t h e c o n t i n u i n g s e a r c h f o r s u b s e a h y d r o c a r b o n s p r o g r e s s e d i n t o d e e p e r water a n d more h a r s h e n v i r o n m e n t s , i t became i n c r e a s i n g l y e v i d e n t t h a t a b e t t e r u n d e r s t a n d i n g of t h e m e c h a n i c s of t h e f l u i d f o r c e s on v a r i o u s o f f s h o r e s t r u c t u r e s was needed i n o r d e r t o p r o d u c e s a f e a n d e c o n o m i c a l d e s i g n s . A c c o r d i n g l y , t h e r e h a s r e c e n t l y been a keen i n t e r e s t i n r e s e a r c h on s e v e r a l a s p e c t s o f t h i s p r o b l e m . The f o l l o w i n g t h e s i s i s c o n c e r n e d p r i m a r i l y w i t h c e r t a i n h y d r o d y n a m i c a l a s p e c t s o f a v e r t i c a l s u r f a c e - p i e r c i n g c i r c u l a r c y l i n d e r . T h i s t y p e o f c y l i n d e r i s a f u n d a m e n t a l s t r u c t u r a l e l e m e n t f o u n d i n most o f f s h o r e s t r u c t u r e s i n t h e f o r m o f s u p p o r t l e g s and b r a c e s , s u b m a r i n e r i s e r s , c a b l e s , p i p e l i n e s , and s t o r a g e t a n k s t o name a few. Sometimes even t h e e n t i r e s t r u c t u r e i t s e l f , a s a m o n o l i t h , t a k e s on t h e shape o f a v e r t i c a l c i r c u l a r c y l i n d e r . C e r t a i n a s p e c t s of f l o w a r o u n d a c i r c u l a r s e c t i o n have been e x t e n s i v e l y e x amined by a number o f i n v e s t i g a t o r s . Good r e v i e w s of some of t h e f u n d a m e n t a l t y p e s o f f l o w a r e g i v e n i n S a r p k a y a and I s a a c s o n ( 1 9 8 1 ) , a n d B e r g e r a n d W i l l e ( 1 9 7 2 ) . The f o r c e s and r u n u p on a l a r g e s u r f a c e - p i e r c i n g c i r c u l a r c y l i n d e r due t o 2 i n c i d e n t wave a c t i o n have been w e l l e s t a b l i s h e d (MacCamy and F u c h s ( 1 9 5 4 ) ) . A l s o , t h e f o r c e s a c t i n g on s u c h a c y l i n d e r w h i l e s u b j e c t e d t o a s t e a d y c u r r e n t o n l y have been w e l l s t u d i e d and documented ( B a t c h e l o r ( 1 9 6 7 ) ) . S t a n s b y e t a l (1981) r e p o r t e d on t h e i r s t u d y o f t h e combined f o r c e s on a c y l i n d e r due t o t h e p r e s e n c e o f an o s c i l l a t o r y f l o w and a s t e a d y c r o s s - c u r r e n t . T h i s t h e s i s c o n s i d e r s , t h r o u g h a s e r i e s o f e x p e r i m e n t s , t h e m o d i f i c a t i o n o f w a v e - i n d u c e d i n - l i n e ( i e . p e r p e n d i c u l a r t o the wave c r e s t s ) o s c i l l a t o r y f o r c e s and r u n u p on a s u r f a c e - p i e r c i n g r i g i d v e r t i c a l c i r c u l a r c y l i n d e r due t o t h e p r e s e n c e o f a c o l i n e a r s t e a d y s t a t e c u r r e n t . The i m p e t u s f o r t h i s s t u d y a r i s e s f r o m t h e o b s e r v a t i o n t h a t i n n a t u r e , t h e r e v e r y s e l d o m e x i s t s a wave f i e l d w i t h o u t t h e p r e s e n c e o f a s i g n i f i c a n t o c e a n c u r r e n t a s w e l l . A l s o , when an o f f s h o r e r i g , f o r example, i s towed t o i t s o p e r a t i o n s i t e , i t e f f e c t i v e l y e x p e r i e n c e s a " c u r r e n t " e q u a l and o p p o s i t e t o i t s own v e l o c i t y . I f , on i t s j o u r n e y , i t e n c o u n t e r s a wave f i e l d , t h e n i t comes under t h e combined i n f l u e n c e of wave and c u r r e n t a t t h e same t i m e . A l t h o u g h a r i g o r o u s m a t h e m a t i c a l a n a l y s i s o f t h i s phenomenon i s beyond t h e scope of t h i s p r e s e n t t h e s i s , t h e f o l l o w i n g p r e s e n t a t i o n and d i s c u s s i o n of e x p e r i m e n t a l r e s u l t s a t t e m p t s t o d e l i n e a t e t h e b a s i c t r e n d s and c h a r a c t e r i z e t h e f u n d a m e n t a l a s p e c t s of t h i s complex and p e r t i n e n t p r o b l e m . The n o t a t i o n used t h r o u g h o u t t h i s t h e s i s f o l l o w s f a i r l y s t a n d a r d c o n v e n t i o n and i s i n d i c a t e d i n F i g u r e 1. The b a s i c p r o b l e m c o n s i s t s of a s t e a d y c u r r e n t and waves o f c o n s t a n t f o r m e n c o u n t e r i n g a v e r t i c a l r i g i d 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 i n 3 water whose s t i l l depth i s d. The t o t a l water column moves w i t h a v e l o c i t y U, r e p r e s e n t i n g a u n i f o r m c u r r e n t whose d i r e c t i o n i s c o l i n e a r w i t h the wave o r t h o g o n a l s . The waves a r e two-d i m e n s i o n a l w i t h a t r o u g h - t o - c r e s t h e i g h t H, p e r i o d T, l e n g t h L, and c e l e r i t y c r e l a t i v e t o a r e f e r e n c e frame moving a t v e l o c i t y U ( F i g u r e l a ) . R e l a t i v e t o a f i x e d r e f e r e n c e frame and i n t h e p resence of the c u r r e n t ( F i g u r e 1 b ) , the p e r i o d and c e l e r i t y a r e d e s i g n a t e d T and c r e s p e c t i v e l y . The f i x e d c o o r d i n a t e system i s d e f i n e d w i t h i t s o r i g i n a t the s t i l l water s u r f a c e and the c y l i n d e r a x i s , w i t h z measured v e r t i c a l l y upwards and x measured h o r i z o n t a l l y i n the same d i r e c t i o n as the wave p r o p a g a t i o n . The f r e e s u r f a c e e l e v a t i o n i s denoted by r\. Time t i s d e f i n e d as z e r o when a wave c r e s t passes th e c y l i n d e r a x i s . The f o r c e s c o n s i d e r e d i n t h i s t h e s i s a r e the i n - l i n e f o r c e s on the c y l i n d e r , t h a t i s , the f o r c e s a c t i n g i n a d i r e c t i o n c o l i n e a r w i t h t h e c u r r e n t v e l o c i t y and the wave p r o p a g a t i o n . The runup, R, i s c o n s i d e r e d m o s t l y on the s e c t i o n of the c y l i n d e r d i r e c t l y f a c i n g the oncoming waves where i t i s e x p e cted t o be a t a maximum. Some c o n s i d e r a t i o n i s a l s o g i v e n t o the s e c t i o n of the c y l i n d e r d i r e c t l y o p p o s i t e , where a l o c a l maximum i s e x p e c t e d t o o c c u r . 4 2 . FUNDAMENTAL FLOWS A b r i e f d i s c u s s i o n of r e l a t e d fundamental flows past a r i g i d c y l i n d e r w i l l p r o v i d e a u s e f u l background r e f e r e n c e f o r the d i s c u s s i o n of the present i n v e s t i g a t i o n . The flows mentioned here have been e x t e n s i v e l y researched by v a r i o u s people and many good r e f e r e n c e s may be found i n the l i t e r a t u r e . 2 . 1 STEADY VISCOUS FLOW PAST A CIRCULAR CYLINDER A r e a l v i s c o u s f l u i d i n a s t a t e of steady flow past a r i g i d c i r c u l a r c y l i n d e r tends to shed eddies or v o r t i c e s from a l t e r n a t i n g s i d e s of the body, which are then c a r r i e d downstream. T h i s a l t e r n a t i n g flow s e p a r a t i o n i n t r o d u c e s a f l u c t u a t i n g component to the drag f o r c e . The frequency, f , with which a p a i r of a l t e r n a t e eddies are shed i s dependent upon the diameter, D, of the c y l i n d e r , the flow v e l o c i t y , U, the f l u i d d e n s i t y p , and the f l u i d v i s c o s i t y u . T h i s r e l a t i o n s h i p may be expressed as: f D S = — ^ = f (Re) ( 2 , 1 ) U where S i s the S t r o u h a l number and Re i s the Reynolds number (Re=pUD/y). The value of S has been found e x p e r i m e n t a l l y to have . a reasonably constant value of 0 . 2 over the s u b c r i t i c a l range, l 0 3 < R e < l 0 5 . The "steady" component of the drag f o r c e i s g e n e r a l l y 5 e x p r e s s e d a s F n = Jg p D C d U 2 where C d i s t h e d r a g c o e f f i c i e n t w h i c h depends on t h e body shape and t h e R e y n o l d s number. 2 . 2 ACCELERATING FLOW OF AN INVISCID FLUID PAST A CIRCULAR  CYLINDER An i n v i s c i d f l u i d f l o w i n g p a s t a b l u f f body e x p e r i e n c e s no f l o w s e p a r a t i o n and hence t h e body i s n o t s u b j e c t e d t o a d r a g f o r c e . I f t h e f l u i d i s a c c e l e r a t i n g , however, t h e body w i l l e x p e r i e n c e a h y d r o d y n a m i c f o r c e a c t i n g i n t h e d i r e c t i o n o f t h e a c c e l e r a t i o n . T h i s f o r c e i s c a l l e d t h e " i n e r t i a f o r c e " and may be e x p r e s s e d a s : FI =V V - o f < 2' 2 ) where V i s t h e volume of t h e immersed body, and t h u s p V i s t h e mass of t h e d i s p l a c e d f l u i d . The p a r a m e t e r C m i s t h e i n e r t i a c o e f f i c i e n t . I t i s c o n v e n i e n t t o r e g a r d t h i s i n e r t i a l f o r c e a s b e i n g t h e sum of two d i s t i n c t components. The f i r s t component i s d i r e c t l y due t o t h e f o r c e f i e l d w h i c h c a u s e s t h e f l u i d t o a c c e l e r a t e w i t h t i m e a t a s p e c i f i c p o i n t i n an a n a l o g o u s f a s h i o n t o A r c h i m e d e ' s P r i n c i p l e . . T h i s f o r c e i s c a l l e d t h e F r o u d e -K r y l o v f o r c e : F k - pV (2.3) The s e c o n d component of t h e i n e r t i a l f o r c e i s t h a t f o r c e , F^ , r e s u l t i n g f r o m t h e d e v i a t i o n o f t h e f l u i d p a t h a r o u n d t h e body. 6 C o m b i n i n g t h e above e x p r e s s i o n s and r e - a r r a n g i n g , we h a v e : The t e r m ( C m - l ) p V i s o f t e n c a l l e d t h e "added mass" and i s t h e e f f e c t i v e mass a s s o c i a t e d w i t h t h i s component o f t h e i n e r t i a l f o r c e . 2.3 OSCILLATING FLOW OF A REAL FLUID PAST A CIRCULAR CYLINDER When c o n s i d e r i n g a r e a l ( v i s c o u s ) f l u i d a c c e l e r a t i n g p a s t a c i r c u l a r c y l i n d e r , t h e p r o b l e m becomes s u b s t a n t i a l l y more c o m p l i c a t e d t h a n t h e above two flows.. Not o n l y must eddy s h e d d i n g be a c c o u n t e d f o r , but t h e e v e r - c h a n g i n g f l o w p a t t e r n a r o u n d t h e body must a l s o be c o n s i d e r e d . The most w i d e l y a c c e p t e d a p p r o a c h t o t h i s p r o b l e m has been t h r o u g h a s e m i -e m p i r i c a l f o r m u l a f i r s t d e v e l o p e d by M o r i s o n , O ' B r i e n , J o h n s o n , and S c h a a f (1950) and commonly r e f e r r e d t o as t h e M o r i s o n e q u a t i o n . T h i s f o r m u l a i s b a s e d on t h e two " r e f e r e n c e " f l o w s d i s c u s s e d a b o v e . The h y p o t h e s i s p u t f o r w a r d by M o r i s o n e t a l i s t h a t t h e t o t a l i n - l i n e f o r c e on t h e immersed body may be c a l c u l a t e d by s i m p l y summing t h e a p p r o p r i a t e a n a l o g i e s t o t h e above two f l o w s . T h a t i s , t o t a l i n - l i n e f o r c e = d r a g f o r c e + i n e r t i a f o r c e . F o r a c i r c u l a r s e c t i o n of d i a m e t e r D i n a t w o - d i m e n s i o n a l f l o w , t h e i n - l i n e f o r c e p e r u n i t l e n g t h would t h u s be: (2.4) F = \ pDC . u | u | + p TT D - du (2.5) 4 Sri d t 7 where t h e f i r s t t e r m i s t h e d r a g f o r c e a n d t h e s e c o n d i s t h e i n e r t i a f o r c e ( u | u | r e p l a c e s u 2 t o a c c o u n t f o r f l o w r e v e r s a l ) . I t must be remembered t h a t t h i s f o r m u l a i s b a s e d on a n a l o g y o n l y a n d s o t h e v a l u e s o f a n d n e e d n o t n e c e s s a r i l y be t h e same a s f o r t h e two r e f e r e n c e f l o w s . I n d e e d , t h e s e v a l u e s must be o b t a i n e d e x p e r i m e n t a l l y f o r t h e p a r t i c u l a r f l o w i n q u e s t i o n . I m p o r t a n t f e a t u r e s o f an o s c i l l a t o r y f l o w p a s t a s t a t i o n a r y c y l i n d e r d e p e n d s t r o n g l y on t h e a m p l i t u d e o f t h e w a t e r p a r t i c l e e x c u r s i o n s r e l a t i v e t o t h e d i a m e t e r o f t h e c y l i n d e r . F o r i n s t a n c e , i f t h e p a r t i c l e e x c u r s i o n s a r e l a r g e c o m p a r e d t o t h e c y l i n d e r d i a m e t e r , t h e n t h e r e i s t h e o p p o r t u n i t y f o r many e d d i e s t o be s h e d b e f o r e t h e f l o w r e v e r s e s . A f t e r r e v e r s i n g , what had been t h e d o w n s t r e a m v o r t e x s t r e e t now becomes p a r t o f t h e . u p s t r e a m i n c i d e n t f l o w , c o m p l e t e w i t h s w i r l i n g e d d i e s t o c o m p l i c a t e t h e r e t u r n f l o w p a s t t h e c y l i n d e r . On t h e o t h e r h a n d , i f t h e p a r t i c l e e x c u r s i o n s a r e s h o r t c o m p a r e d t o t h e c y l i n d e r d i a m e t e r , o n l y a few, o r p e r h a p s no e d d i e s w i l l h a ve e n o u g h t i m e t o d e v e l o p on t h e d o w n s t r e a m s i d e b e f o r e t h e f l o w r e v e r s e s . K e u l e g a n a n d C a r p e n t e r ( 1 9 5 8 ) made a p a r t i c u l a r l y i m p o r t a n t c o n t r i b u t i o n t o t h e u n d e r s t a n d i n g o f t h i s f l o w . They e m p l o y e d t h e p a r a m e t e r u mT/D t o r e p r e s e n t t h e r e l a t i v e p a r t i c l e o r b i t s i z e . T h i s p a r a m e t e r h a s come t o be t e r m e d t h e K e u l e g a n C a r p e n t e r number: K = _^L- = - i ^ _ ( 2 . 6 ) where u m i s t h e semi p e a k - t o - p e a k a m p l i t u d e o f t h e o s c i l l a t o r y f l o w v e l o c i t y a n d x i s t h e c o r r e s p o n d i n g w a t e r p a r t i c l e 8 d i splacement. 2.4- LINEAR WAVE THEORY I t i s u s e f u l t o assemble the r e s u l t s of l i n e a r wave t h e c i r as we s h a l l r e f e r t o them throughout t h i s t h e s i s . The c l a s s i c a l development of t h i s t h e o r y i s w e l l known and the reader i s r e f e r r e d t o S t o k e s (1847) and Lamb (1945) f o r f u r t h e r d e t a i l . The s a l i e n t r e s u l t s of t h i s t h e o r y a r e as f o l l o w s : cz = = -3- tanh (kd) ( 2 . 7 ) T k n = 4j- cos(kx - wt) (2.8) " = ¥ - s ? n h i ^ P Z " < 2- 9 > where c i s the wave c e l e r i t y , n i s the f r e e s u r f a c e e l e v a t i o n , and u and w a r e the h o r i z o n t a l and v e r t i c a l components of the water p a r t i c l e v e l o c i t y r e s p e c t i v e l y . A l s o , k=2ir/L i s the wave number andw=2Tr/T i s the a n g u l a r wave f r e q u e n c y . C l e a r l y , the water p a r t i c l e s f o l l o w e l l i p t i c a l o r b i t s whose e c c e n t r i c i t y depends on the v a l u e of kd which i s i n d i c a t i v e of the water depth t o wave l e n g t h r a t i o . Large v a l u e s of kd i n d i c a t e "deep water" waves which cause c i r c u l a r water p a r t i c l e o r b i t s whose magnitude d i m i n i s h e s e x p o n e n t i a l l y w i t h depth u n t i l a t a depth of a p p r o x i m a t e l y L/2 the p a r t i c l e motion i s c o n s i d e r e d n e g l i g i b l e . In o t h e r words the waves do not " f e e l " the bottom. 9 C o n v e r s e l y , " i n t e r m e d i a t e d e p t h " and " s h a l l o w w a t e r " waves w i t h s m a l l e r v a l u e s o f kd p r o d u c e f l a t t e r w a ter p a r t i c l e o r b i t s w i t h more h o r i z o n t a l t h a n v e r t i c a l movement. A s t a n d a r d c l a s s i f i c a t i o n f o r t h i s r e l a t i o n s h i p i s r o u g h l y : d e ep water waves kd > 3 . i n t e r m e d i a t e d e p t h waves 0.3 < kd < 3 s h a l l o w water waves kd < 0.3 The p r e s e n t i n v e s t i g a t i o n c o n s i d e r s waves whose v a l u e o f kd r a n g e s from 1.5 t o 6.0. In o t h e r words, most d a t a s e t s i n v o l v e d e ep water waves w h i l e a few i n v o l v e i n t e r m e d i a t e d e p t h waves. 2.5 DIFFRACTION THEORY The M o r i s o n e q u a t i o n d i s c u s s e d i n s e c t i o n 2.3 may be a p p l i e d t o i n c i d e n t waves e n c o u n t e r i n g a v e r t i c a l c y l i n d e r whose d i a m e t e r D i s s m a l l compared t o t h e wave l e n g t h L. In t h i s c a s e , t h e c y l i n d e r does n o t s i g n i f i c a n t l y a f f e c t t h e wave t r a i n . L a r g e r b o d i e s , however, do c a u s e a change i n t h e k i n e m a t i c s o f t h e u n d i s t u r b e d f l o w i n t h e i n c i d e n t wave d i r e c t i o n . The r a t i o D/L ( o r e q u i v a l e n t l y ka = 2iTa/L=TrD/L, where a i s t h e r a d i u s of t h e c y l i n d e r ) i s commonly u s e d t o gauge t h e i m p o r t a n c e of t h i s e f f e c t and i s c a l l e d t h e d i f f r a c t i o n p a r a m e t e r . The wave s c a t t e r i n g t h a t o c c u r s when t h e i n c i d e n t wave e n c o u n t e r s t h e body i s g e n e r a l l y c o n s i d e r e d s i g n i f i c a n t when t h e body s p a n s more t h a n a b o u t one f i f t h o f t h e i n c i d e n t wave l e n g t h (D/L>0.2). I n e r t i a f o r c e s g e n e r a l l y p l a y a more i m p o r t a n t r o l e i n t h i s c a s e 10 as compared t o t h e s m a l l e r b o d i e s where f l o w s e p a r a t i o n and d r a g e f f e c t s a r e d o m i n a n t . The wave f o r c e s on b o d i e s i n t h e d i f f r a c t i o n r e g i m e a r e g e n e r a l l y l e s s t h a n t h o s e t h a t would be c a l c u l a t e d u s i n g t h e M o r i s o n e q u a t i o n and n e g l e c t i n g d i f f r a c t i o n s i n c e t h e v a r i a t i o n i n wave k i n e m a t i c s w i t h r e s p e c t t o t h e x-a x i s a l o n g t h e d i a m e t e r o f t h e c y l i n d e r c a n no l o n g e r be n e g l e c t e d . F o r c o n v e n i e n c e , t h e i n e r t i a component of t h e M o r i s o n e q u a t i o n ( F = p 1 ^ C m-g^) i s s t i l l u s e d , b u t w i t h an " e f f e c t i v e i n e r t i a c o e f f i c i e n t " w h i c h g e n e r a l l y d e c r e a s e s as t h e w a v e l e n g t h i s r e d u c e d . In t h e d i f f r a c t i o n r e g i m e , when D/L i s s u f f i c i e n t l y l a r g e , t h e water p a r t i c l e e x c u r s i o n s may be s m a l l enough compared t o t h e d i a m e t e r o f t h e c y l i n d e r s u c h t h a t t h e e f f e c t s of eddy s h e d d i n g a r e m i n i m a l o r a t w o r s t l o c a l i z e d , and hence t h e f l u i d may be c o n s i d e r e d i n v i s c i d and t h u s t h e f l o w i r r o t a t i o n a l . Thus t h e p r o b l e m i s g e n e r a l l y s o l v e d u s i n g p o t e n t i a l f l o w t h e o r y , w i t h t h e v e l o c i t y p o t e n t i a l d e s c r i b i n g t h e f l o w b e i n g c o m p r i s e d o f a p o t e n t i a l due t o t h e i n c i d e n t , u n d i s t u r b e d wave t r a i n and a p o t e n t i a l due t o t h e s c a t t e r e d i n c i d e n t waves. MacCamy and F u c h s (1954) have d e v e l o p e d t h i s t h e o r y f o r a v e r t i c a l c i r c u l a r c y l i n d e r e x t e n d i n g from t h e ocean f l o o r and p i e r c i n g t h e water s u r f a c e . 11 2.6 WAVE AND CURRENT INTERACTION In d e v e l o p i n g a wave t h e o r y s u c h a s t h a t p r e s e n t e d i n s e c t i o n 2.4 , i t i s u s u a l t o c o n s i d e r a r e f e r e n c e frame ( x ' , z ) w i t h t h e same v e l o c i t y a s t h e waves. T h i s p r o v i d e s a u n i q u e s o l u t i o n f o r t h e wave f l o w . In o r d e r t o f u l l y e s t a b l i s h t h e f l o w w i t h r e s p e c t t o t h e f i x e d o r s t a t i o n a r y r e f e r e n c e frame ( x , z ) , an a s s u m p t i o n must be made r e g a r d i n g t h e a b s e n c e o f an u n d e r l y i n g o r s u p e r p o s e d c u r r e n t . T h i s a s s u m p t i o n i s o f t e n t a k e n t o be t h e c o n d i t i o n t h a t t h e t i m e a v e r a g e d h o r i z o n t a l water p a r t i c l e v e l o c i t y a t any one l o c a t i o n i s z e r o . The h o r i z o n t a l p a r t i c l e v e l o c i t y u' i n t h e moving r e f e r e n c e f r a m e , i n terms o f i t s c o u n t e r p a r t u i n t h e f i x e d r e f e r e n c e frame and t h e wave c e l e r i t y , c, a l s o i n t h e f i x e d r e f e r e n c e frame, may t h e n be d e n o t e d u'=u-c. I f we c o n s i d e r a s t e a d y , u n i f o r m , and c o l i n e a r u n d e r l y i n g c u r r e n t U, u' w i l l r e m a i n unchanged and may now be e x p r e s s e d a s : u ' = u + U - c (2.11) c where c c i s t h e wave c e l e r i t y i n t h e p r e s e n c e o f a c u r r e n t and i n t h e f i x e d r e f e r e n c e f r a m e . Note t h a t u i s s t i l l j u s t t h e o s c i l l a t o r y component of t h e w a t e r p a r t i c l e s i n t h e f i x e d r e f e r e n c e frame due t o t h e wave a c t i o n . I f we once a g a i n assume t h a t t h e t i m e a v e r a g e o f t h e s e o s c i l l a t i o n s i s z e r o (u=0), t h e n we have t h e r e l a t i o n : c c = c + U (2.12) w i t h c d e n o t i n g t h e wave c e l e r i t y i n t h e a b s e n c e o f a c u r r e n t and b e i n g d e t e r m i n e d by t h e d i s p e r s i o n r e l a t i o n , w h i c h i n t h e 12 c a s e of l i n e a r wave t h e o r y i s : c 2 = g/k t a n h ( k d ) a s m e n t i o n e d e a r l i e r i n s e c t i o n 2.4 ( e q u a t i o n 2 . 7 ) . S i n c e t h e wave l e n g t h L and wave number k a r e c l e a r l y unchanged by a change i n r e f e r e n c e f r ame, i t f o l l o w s f r o m e q u a t i o n (2.12) t h a t t h e a n g u l a r wave f r e q u e n c y , u c , i n t h e p r e s e n c e o f a c u r r e n t U w i l l be g i v e n a s : a) = u + kU (2.13) o r , i n t h e c a s e of l i n e a r t h e o r y : o j c = (gk t a n M k d ) ) ^ + kU. (2.14) The e f f e c t s o f c u r r e n t s on t h e o v e r a l l l o a d i n g of a s t r u c t u r e i n a wave f i e l d have been summarized by Hogben and S t a n d i n g (1975) who i d e n t i f i e d t h r e e d i s t i n c t e f f e c t s . The f i r s t e f f e c t i s t h a t t h e wave s p e e d i s a l t e r e d due t o t h e f a c t t h a t t h e waves p r o p a g a t e o v e r a moving r a t h e r t h a n a s t a t i o n a r y f l u i d , a s d i s c u s s e d a b o v e . A s e c o n d e f f e c t i s due t o t h e r e s u l t i n g change t h a t a c u r r e n t p r o d u c e s upon t h e f l u i d p a r t i c l e v e l o c i t y a s s o c i a t e d w i t h o s c i l l a t i n g wave a c t i o n . I f t h e c u r r e n t were not p r e s e n t , t h e n t h e e x p e c t e d f l u i d p a r t i c l e v e l o c i t y would be a s i m p l e h a r m o n i c , a s : u = u m c o s ( u i t ) . (2.15) The f o r c e on t h e c y l i n d e r would be w h o l l y i n e r t i a l p r o v i d e d t h e r e was no f l o w s e p a r a t i o n , t h a t i s , t h e K e u l e g a n C a r p e n t e r number, K, had a v a l u e o f l e s s t h a n 1 o r 2 ( B i d d e ( 1 9 7 0 ) , S a r p k a y a and I s a a c s o n ( 1 9 8 1 ) ) . I f K were s l i g h t l y l a r g e r ( f r o m 5 t o 15) t h e r e would be some f l o w s e p a r a t i o n , but i t would l i k e l y 13 be a s y m m e t r i c i n n a t u r e w i t h a v o r t e x b e i n g s h e d from o n l y one s i d e o f t h e c y l i n d e r . Above K=15, r e g u l a r p a i r s of e d d i e s would be s h e d , f o r m i n g a downstream v o r t e x s t r e e t . W i t h t h e a d d i t i o n o f a c o l i n e a r c u r r e n t , however, t h e p a r t i c l e v e l o c i t y must now t a k e on t h e f o r m : u c = U + u m c o s ( w t ) (2.16) and so t h e p a r t i c l e o r b i t s can no l o n g e r be c l o s e d e l l i p s e s but w i l l r a t h e r be c y c l o i d i c i n n a t u r e w i t h t h e p a r t i c l e t r a v e l l i n g f u r t h e r i n t h e downstream d i r e c t i o n t h a n i n t h e u p s t r e a m d i r e c t i o n . T h i s means t h a t t h e p a r t i c l e v e l o c i t y w i l l a l s o be g r e a t e r i n t h e downstream t h a n i n t h e u p s t r e a m d i r e c t i o n and t h u s t h e eddy s h e d d i n g phenomenon w i l l be b i a s e d w i t h more e d d i e s s h e d on t h e c u r r e n t downstream s i d e o f t h e c y l i n d e r . In f a c t , i f t h e body/wave . r e l a t i o n l i e s i n t h e d i f f r a c t i o n r e g i m e , t h e n a l t h o u g h w i t h no c u r r e n t p r e s e n t t h e r e would be o n l y n e g l i g i b l e ( i f any) f l o w s e p a r a t i o n and h e n c e no d r a g , t h e s u p e r p o s i t i o n o f even s m a l l c u r r e n t s may i n d u c e eddy s h e d d i n g and . t h u s s i g n i f i c a n t d r a g f o r c e s i n one d i r e c t i o n . S i n c e d r a g f o r c e s a r e p r o p o r t i o n a l t o t h e s q u a r e of t h e f l u i d v e l o c i t y and v e l o c i t y g e n e r a l l y d e c r e a s e s o n l y s l o w l y w i t h d e p t h , a c o m p a r a t i v e l y s m a l l c u r r e n t a c t i n g on a s u r f a c e p i e r c i n g c y l i n d e r can c l e a r l y i n c r e a s e t h e d r a g a p p r e c i a b l y . The t h i r d e f f e c t i d e n t i f i e d by Hogben and S t a n s b y i s t h a t t h e i n c i d e n t c u r r e n t w i l l c a u s e t h e c y l i n d e r i t s e l f t o g e n e r a t e waves. A body i n a u n i f o r m c u r r e n t c a u s e s a s t a t i o n a r y wave p a t t e r n t o form on t h e f r e e s u r f a c e i n t h e same f a s h i o n t h a t a s h i p moving t h r o u g h s t i l l w a t e r w i l l c r e a t e waves t h a t a p p e a r 14 s t a t i o n a r y when v i e w e d from t h e s h i p . A c o r r e s p o n d i n g n e t f o r c e , c a l l e d t h e "wave-making r e s i s t a n c e " , i s f e l t by t h e body. Hogben (1974) examined t h i s e f f e c t i n r e l a t i o n t o l a r g e o f f s h o r e s t r u c t u r e s and c o n c l u d e d t h a t i t was much s m a l l e r t h a n t h e e f f e c t of d r a g . A l t h o u g h an a t t e m p t a t a r i g o r o u s m a t h e m a t i c a l c l o s e d form s o l u t i o n o f t h e w a v e - c u r r e n t i n t e r a c t i o n p r o b l e m i s beyond t h e scope of t h i s t h e s i s , t h e g o v e r n i n g e q u a t i o n s and i n h e r e n t d i f f i c u l t i e s a r e d i s c u s s e d i n C h a p t e r 3, and a p o s s i b l e n u m e r i c a l a p p r o a c h t o t h e p r o b l e m i s examined i n C h a p t e r 4. 15 3. THE PRESENT INVESTIGATION 3.1 OBJECTIVES The main t h r u s t of t h i s t h e s i s i s b a s e d upon a s e r i e s o f e x p e r i m e n t s w h i c h a r e d e s c r i b e d i n d e t a i l i n C h a p t e r 6. The p r i m a r y emphasis o f t h e e x p e r i m e n t s was on t h e i n - l i n e f o r c e s o n a v e r t i c a l c i r c u l a r s u r f a c e - p i e r c i n g c y l i n d e r under t h e combined i n f l u e n c e of c o l i n e a r waves and c u r r e n t , w h i l e a s e c o n d a r y s e t o f e x p e r i m e n t s c o n c e r n e d t h e runup on t h e same c y l i n d e r under s i m i l a r c o n d i t i o n s . A l s o , a n u m e r i c a l p r o c e d u r e c u r r e n t l y under d e v e l o p m e n t by I s a a c s o n ( s e e I s a a c s o n (1981)) was u s e d t o compare th e g e n e r a l t r e n d s o f t h e i n - l i n e f o r c e c h a r a c t e r i s t i c s . F o r c o n v e n i e n c e and s i m p l i c i t y d u r i n g t h e e x p e r i m e n t s , t h e w a t e r c u r r e n t was s i m u l a t e d by moving t h e c y l i n d e r a t a s t e a d y s p e e d t h r o u g h t h e w a t e r , r a t h e r t h a n h a v i n g t h e water f l o w p a s t t h e c y l i n d e r . I t c a n be e a s i l y shown t h a t t h i s change i n r e f e r e n c e f r a m e s does n o t a f f e c t t h e f l u i d v e l o c i t i e s r e l a t i v e t o t h e c y l i n d e r and hence t h e f l o w p a t t e r n s a r e i d e n t i c a l i n b o t h c a s e s . The r e l e v a n c e and r e l a t i o n s h i p s o f t h e v a r i o u s p a r a m e t e r s i n v o l v e d i n t h e s t a t e d p r o b l e m becomes more a p p a r e n t i f we c o n s i d e r a d i m e n s i o n a l a n a l y s i s o f t h e p r o b l e m . The maximum h o r i z o n t a l f o r c e , F, on t h e c y l i n d e r may be w r i t t e n i n a g e n e r a l f o r m a s : F = f ( a , d , g , H , k , p,U) (3.1) where f s i g n i f i e s a f u n c t i o n a l r e l a t i o n s h i p and a i s t h e 16 c y l i n d e r r a d i u s , d t h e s t i l l w a t er d e p t h , g t h e g r a v i t a t i o n a l c o n s t a n t , H t h e wave h e i g h t , k ( = 2TT/L) t h e wave number, p t h e f l u i d d e n s i t y , and U t h e c u r r e n t v e l o c i t y ( t h e e f f e c t o f v i s c o s i t y o r R e y n o l d ' s number i s n o t i n c l u d e d h e r e ) . The v a l u e s o f d, g, H, and k a r e s u f f i c i e n t t o d e f i n e t h e i n c i d e n t wave t r a i n w h i l e t h e wave p e r i o d , T, may be e x p r e s s e d i n terms of t h e s e by a c h o s e n wave t h e o r y . The m a g n i t u d e o f U t o g e t h e r w i t h a d i r e c t i o n s i g n s u f f i c i e n t l y d e f i n e s t h e s t e a d y s t a t e , v e r t i c a l l y homogeneous, i n c i d e n t c u r r e n t . A d i m e n s i o n a l a n a l y s i s t h e n g i v e s : — = f (ka, kd, 4- , — ) (3.2) h pgHa2 d c where ka i s a d i f f r a c t i o n p a r a m e t e r , kd a wave d e p t h p a r a m e t e r , H/d a n o n l i n e a r i t y p a r a m e t e r , and U/c a c u r r e n t v e l o c i t y p a r a m e t e r ( n o t e t h a t c=L/T i s t h e wave c e l e r i t y i n t h e a b s e n c e of a c u r r e n t ) . T h i s d e s i g n a t i o n o f n o n - d i m e n s i o n a l p a r a m e t e r s i s n ot u n i q u e . F o r example, r a t h e r t h a n use t h e c u r r e n t v e l o c i t y t o wave c e l e r i t y r a t i o U/c, we c o u l d a s e a s i l y use t h e F r o u d e number w h i c h i s e s s e n t i a l l y a r a t i o o f i n e r t i a f o r c e t o g r a v i t y f o r c e and i s u s u a l l y e x p r e s s e d a s : F r = ~^=r ( 3 . 3 ) where D i s t h e d i a m e t e r o f t h e c y l i n d e r (=2a). The F r o u d e number i s o f t e n u s e d i n h y d r o d y n a m i c p r o b l e m s i n w h i c h a f l u i d f r e e s u r f a c e p l a y s an e s s e n t i a l r o l e and t h u s g r a v i t y f o r c e s must be t a k e n i n t o a c c o u n t . C o n s t a n c y o f t h i s number i n p h y s i c a l model p r o b l e m s e n s u r e s dynamic s i m i l a r i t y between model 17 and p r o t o t y p e . The p r e s e n t e x p e r i m e n t s a t t e m p t t o c o v e r a s u i t a b l e r a n g e o f h y d r o d y n a m i c c o n d i t i o n s u n d e r w h i c h t o examine t h e w a v e / c u r r e n t e f f e c t s on a v e r t i c a l c y l i n d e r . The i n c i d e n t waves a r e m o s t l y d e e p - w a t e r waves w i t h some s e t s i n t h e i n t e r m e d i a t e d e p t h r a n g e , a s shown i n F i g u r e 2 w h i c h i n d i c a t e s t h e r a n g e o f e x p e r i m e n t a l v a l u e s of kd from 1.5 t o 6.0. The K e u l e g a n -C a r p e n t e r number K r a n g e d f r o m a p p r o x i m a t e l y 0.1 t o 0.6 and t h e d i f f r a c t i o n p a r a m e t e r ka c o v e r e d t h e range 0.5 t o 3.0, a s i l l u s t r a t e d i n F i g u r e 3. The R e y n o l d s number r e l a t i n g t o t h e f l o w of t h e c u r r e n t , U, p a s t t h e c y l i n d e r r a n g e d from 0 t o 8 x 1 0 s , and t h e c o r r e s p o n d i n g eddy s h e d d i n g f r e q u e n c y , f , went from 0 t o a p p r o x i m a t e l y 0.3 h e r t z . The r a t i o o f t h e c u r r e n t v e l o c i t y t o t h e wave c e l e r i t y , U/c, r a n g e d i n e a c h d a t a s e t from -0.2 t o +0.2 w h i l e t h e v a l u e o f t h e F r o u d e number, U/i'gD, r a n g e d from -0.19 t o +0.19 t h r o u g h o u t t h e e x p e r i m e n t . Waves e n c o u n t e r i n g a r i g i d body w i l l r e s u l t i n w a t e r " r u s h i n g up" t h e s i d e o f t h e body t o a h e i g h t g r e a t e r t h a n t h a t o f t h e i n c i d e n t wave c r e s t . The h e i g h t t o w h i c h t h e water r i s e s a b ove t h e s t i l l w a t er l e v e l i s c a l l e d t h e r u n u p and depends l a r g e l y upon t h e shape and o r i e n t a t i o n o f t h e body. In t h e s i m p l e s t c a s e o f a f l a t v e r t i c a l w a l l , t h e i n c i d e n t wave i s c o m p l e t e l y r e f l e c t e d and a s t a n d i n g wave r e s u l t s w i t h a runup o f H, where H/2 i s t h e semi p e a k - t o - p e a k a m p l i t u d e o f t h e i n c i d e n t wave. Wave r u n u p on v e r t i c a l c i r c u l a r c y l i n d e r s has r e c e n t l y been a s u b j e c t of i n t e r e s t , p a r t i c u l a r l y w i t h r e g a r d s t o t h e d e s i g n of s t r u c t u r e s w h i c h must n o t s u f f e r o v e r t o p p i n g . S e v e r a l 18 e x p e r i m e n t a l s t u d i e s have been c a r r i e d o u t on s u c h c y l i n d e r s under v a r y i n g wave c o n d i t i o n s ( e g . C h a k r a b a r t i and Tam ( 1 9 7 5 ) , I s a a c s o n ( 1 9 7 7 ) ) , but none w i t h an i n t e r a c t i n g c u r r e n t and wave f i e l d . The p r e s e n t s t u d y e x a m i n e s r u n u p u n d e r t h e same c o n d i t i o n s a s i n d i c a t e d f o r t h e f o r c e measurements, t h e runup measurements b e i n g made p r i m a r i l y a t t h e f r o n t o f t h e c y l i n d e r ( f a c i n g t h e oncoming w a v e s ) , w h i c h i s where t h e maximum o c c u r s , w i t h a l s o a few measurements a t t h e r e a r o f t h e c y l i n d e r where t h e r e i s a l o c a l maximum. F o r wave-body i n t e r a c t i o n s i n t h e d i f f r a c t i o n r e g i m e w i t h no c u r r e n t , t h e runup p r o b l e m has been s o l v e d f o r a r b i t r a r y d e p t h s u s i n g l i n e a r p o t e n t i a l f l o w t h e o r y by MacCamy and F u c h s ( 1 9 5 4 ) . 3.2 INERTIAL FORCE PREDICTION I t • i s an i n s t r u c t i v e e x e r c i s e t o d e t e r m i n e what we would e x p e c t o u r e x p e r i m e n t a l r e s u l t s t o be i f we c o n s i d e r e d t h e f l u i d p a r t i c l e f l o w t o be w h o l l y i n e r t i a l , t h a t i s , w i t h no f l o w s e p a r a t i o n and t h u s no d r a g f o r c e . C o n s i d e r a r i g i d c y l i n d e r i n a wave and c u r r e n t f i e l d as shown i n F i g u r e 1b. L e t t h e s u b s c r i p t c d e n o t e p a r a m e t e r s i n t h e p r e s e n c e o f an u n d e r l y i n g c u r r e n t ( i e . i n our e x p e r i m e n t s , t h i s would r e f e r t o t h e moving r e f e r e n c e frame o f t h e c y l i n d e r ) . U n s u b s c r i p t e d v a r i a b l e s w i l l be t a k e n as n o t b e i n g under t h e i n f l u e n c e o f an u n d e r l y i n g c u r r e n t ( i e . i n our e x p e r i m e n t s , t h i s would r e f e r t o t h e f i x e d r e f e r e n c e frame of t h e wave b a s i n ) . The v a r i a b l e s u n a f f e c t e d by 0 19 t h e p r e s e n c e of t h e c u r r e n t a r e H, D, d, L, and k, w h i l e t h o s e a f f e c t e d a r e T, u, and c . From s e c t i o n 2.5, we see t h a t t h e t o t a l i n e r t i a l f o r c e on t h e c y l i n d e r i s : F t = J ^ i C f ° ^ £ d z (3.4) r I c 4 Sn J _ d 9 t Now, t h e p a r t i c l e v e l o c i t y u i s g i v e n on t h e b a s i s o f l i n e a r wave t h e o r y by: " c ' U + ' f 3 S n i l $ ) * , ) ) c " ( k ( x - C c t » ( 3 - 5 ) where t h e a m p l i t u d e o f t h e o s c i l l a t o r y component, -rrH cosh (k (d + z)) T sinh (kd) d o e s n ot depend on t h e m a g n i t u d e o f t h e c u r r e n t U. D i f f e r e n t i a t i o n of u c w i t h r e s p e c t t o t i m e g i v e s : R e c a l l i n g f r o m s e c t i o n 2.6 t h a t c c=c+U, we c a n w r i t e e q u a t i o n (3.6) a s : - I - . 0 t tt ) k c c.«h j ^ j " ^ » , 1 n ( k (x-c et» (3.7) Hence t h e maximum p a r t i c l e a c c e l e r a t i o n i s : , 5tJc x _ N . _ U _ x k r TTH cosh (k (d + z))  K 9t 'max 1 1 c ; K C T sinh (kd) I 1 + ~ c " ' '"Ii' max < 3 - 8 ' Thus i t f o l l o w s t h a t t h e maximum i n e r t i a l f o r c e i n t h i s c a s e may 2 0 be e x p r e s s e d a s : ( F i c ) m a v = ( 1 + £ ) (FT) - ( 3 . 9 ) max c 1 max where ( F T ) r e f e r s , of c o u r s e , t o t h e maximum i n e r t i a l f o r c e I max ' ' i n t h e a b s e n c e o f a c u r r e n t . I f we p l o t t h i s r e l a t i o n s h i p w i t h ( F i P ) a s t h e o r d i n a t e and U/c t h e a b s c i s s a , t h e r e s u l t w i l l c max be a s t r a i g h t l i n e w i t h s l o p e ( F _ ) and U/c=0 i n t e r c e p t a t 1 max (F ) =(F ) . F i g u r e 4 shows s u c h a p l o t i n a form l c max 1 max c o m p a t i b l e w i t h t h a t i n w h i c h t h e e x p e r i m e n t a l d a t a a r e p l o t t e d . T h a t i s , (F ) n o n - d i m e n s i o n a l i z e d i n t h e fo r m : I c max <Flc> max 7T-ispgHa' i s p l o t t e d a s a f u n c t i o n o f U/c and t h e r e s u l t i s a f a m i l y of c u r v e s e a c h w i t h s l o p e and U/c=0 i n t e r c e p t o f (F _ ) I max JjpgHa2 E a c h c u r v e c o r r e s p o n d s t o a d i f f e r e n t v a l u e of ka, s i n c e Fj i s a f u n c t i o n o f ka a l s o . 21 4. THEORETICAL CONSIDERATIONS 4.1 GOVERNING EQUATIONS The g o v e r n i n g e q u a t i o n s f o r our p r o b l e m a r e p r e s e n t e d h e r e u s i n g t h e n o t a t i o n as i n d i c a t e d i n F i g u r e 1. The s e a b e d i s assumed h o r i z o n t a l a l o n g t h e p l a n e z = - d . The f l u i d i s assumed i n c o m p r e s s i b l e and i n v i s c i d , a nd t h e f l o w i r r o t a t i o n a l . The f l u i d m o t i o n i s d e s c r i b e d by a v e l o c i t y p o t e n t i a l <{> w h i c h s a t i s f i e s t h e L a p l a c e e q u a t i o n w i t h i n t h e f l u i d r e g i o n : V2<(> = 0 (4.1) and i s s u b j e c t t o t h e f o l l o w i n g boundary c o n d i t i o n s : • | i = 0 at z = -d (4.2) 3z | i = 0 on S K (4.3) 3n b |£- |5-n = 0 o n S f ( 4 . 4 ) 3n dt z f |£ + gn + h(V<i>)2 = 0 on S f (4.5) Here g i s t h e g r a v i t a t i o n a l c o n s t a n t , n d e n o t e s d i s t a n c e i n t h e d i r e c t i o n of t h e u n i t n o r m a l v e c t o r n d i r e c t e d o u t w a r d from t h e f l u i d r e g i o n , i s t h e immersed body s u r f a c e , S f i s t h e f r e e s u r f a c e (z=n) and n i s t h e d i r e c t i o n c o s i n e o f n w i t h r e s p e c t z — t o t h e z d i r e c t i o n . E q u a t i o n s 4.2 and 4.3 c o r r e s p o n d t o t h e k i n e m a t i c b o u n d a r y c o n d i t i o n s a t t h e s e a b e d and body s u r f a c e r e s p e c t i v e l y w h i l e e q u a t i o n s 4.4 and 4.5 r e p r e s e n t t h e k i n e m a t i c and dynamic f r e e s u r f a c e b o u n d a r y c o n d i t i o n s r e s p e c t i v e l y . The form of t h e k i n e m a t i c f r e e s u r f a c e b o u n d a r y c o n d i t i o n g i v e n i n 2 2 e q u a t i o n 4.4 i s a c c o u n t e d f o r i n I s a a c s o n ( 1 9 8 1 ) . In a d d i t i o n , i t i s w e l l known t h a t f o r an e s t a b l i s h e d wave m o t i o n a r a d i a t i o n c o n d i t i o n , r e q u i r i n g t h a t t h e s c a t t e r e d wave f i e l d c o r r e s p o n d s t o o u t w a r d r a t h e r t h a n i n w a r d t r a v e l l i n g waves, i s needed t o e n s u r e a u n i q u e s o l u t i o n t o t h e p r o b l e m ( e g . S a r p k a y a and I s a a c s o n ( 1 9 8 1 ) ) . A s o l u t i o n t o t h i s p r o b l e m f o r t h e c a s e of waves o n l y , w i t h no c u r r e n t , may be a c c o m p l i s h e d u s i n g a p e r t u r b a t i o n p r o c e d u r e s u c h as was o r i g i n a l l y u s e d by S t o k e s ( 1 8 4 7 ) . In t h i s method, i t i s assumed t h a t t h e v a r i a b l e s d e s c r i b i n g t h e f l o w may be d e v e l o p e d as a power s e r i e s i n t e r m s o f a p e r t u r b a t i o n p a r a m e t e r w h i c h i s s m a l l . T h a t i s , <j> and a s s o c i a t e d v a r i a b l e s ( n , u , w , . . . ) , may be w r i t t e n i n t h e f o r m : <t> = e<J>i + e 2 $ 2 + .... (4.6) i n w h i c h e i s t h e p e r t u r b a t i o n p a r a m e t e r w h i c h i s on t h e o r d e r o f , s a y , H/L. E a c h s u b s c r i p t e d v a r i a b l e a p p e a r i n g i n t h e d i f f e r e n t s e r i e s i s t a k e n as h a v i n g t h e same o r d e r o f m a g n i t u d e and _ t h u s e a c h a d d i t i o n a l t e r m i n t h e s e r i e s r e p r e s e n t s a q u a n t i t y s m a l l e r t h a n t h e p r e c e d i n g one by a f a c t o r o f o r d e r e. By s u b s t i t u t i n g t h e p e r t u r b a t i o n e x p a n s i o n i n t o t h e g o v e r n i n g e q u a t i o n s and c o l l e c t i n g t e r m s o f l i k e o r d e r s o f m a g n i t u d e we o b t a i n a new s e t of e q u a t i o n s . T h e s e a r e s t r a i g h t f o r w a r d e x c e p t f o r t h e f r e e s u r f a c e boundary c o n d i t i o n e q u a t i o n s w h i c h c o n t a i n n o n l i n e a r t e r m s and a p p l y a t t h e unknown e l e v a t i o n z=n. I f t h e p r o b l e m i s c o n s i d e r e d t o a f i r s t a p p r o x i m a t i o n ( i e . s a v i n g o n l y f i r s t o r d e r e t e r m s and n e g l e c t i n g h i g h e r o r d e r e ' s ) t h e n t h e i n c i d e n t waves a r e t h o s e r e s u l t i n g f r o m l i n e a r wave t h e o r y as 23 p r e s e n t e d i n s e c t i o n 2.4. The s o l u t i o n of t h i s w a v e - s t r u c t u r e i n t e r a c t i o n p r o b l e m i s w e l l e s t a b l i s h e d and was f i r s t p r e s e n t e d by MacCamy and F u c h s ( 1 9 5 4 ) . In a s i m i l a r f a s h i o n , one may c o n s i d e r a s t e a d y , u n i f o r m f l o w p a s t a c y l i n d e r , w i t h no i n c i d e n t waves. H e r e , a p e r t u r b a t i o n p a r a m e t e r a on t h e o r d e r o f , s a y , U 2/gD, may be c h o s e n and t h e same p r o c e d u r e c a r r i e d o u t t o p r o v i d e a f i r s t a p p r o x i m a t i o n s o l u t i o n t o t h e p r o b l e m . I f we a t t e m p t t h e same p r o c e d u r e once a g a i n , o n l y t h i s t i m e , w i t h a c o m b i n a t i o n of waves and an u n d e r l y i n g c u r r e n t U, we q u i c k l y run j r n t o d i f f i c u l t i e s . The v e l o c i t y p o t e n t i a l must now be expanded i n t h e form: <j> = U + + e 2 $ 2 + ««• • (4.7) and t h e f r e e s u r f a c e b o u n d a r y c o n d i t i o n s w i l l c o n t a i n n o n l i n e a r p r o d u c t t e r m s even i n t h e f i r s t o r d e r o f e . A l s o , i f we combine w i t h t h i s a p e r t u r b a t i o n power s e r i e s i n a f o r t h e c u r r e n t v e l o c i t y , t h e n t h e p r o b l e m q u i c k l y e s c a l a t e s beyond t h e r e a c h o f s i m p l e a n a l y t i c a l t e c h n i q u e s . F o r t h i s r e a s o n t h e r e i s no w e l l e s t a b l i s h e d c l o s e d - f o r m s o l u t i o n f o r t h e w a v e - c u r r e n t i n t e r a c t i o n p r o b l e m . 24 4.2 NUMERICAL SOLUTION 4.2.1 T h e o r y An a l t e r n a t e a p p r o a c h t o s o l v i n g t h i s p r o b l e m i n an a n a l y t i c a l f a s h i o n i s t o use n u m e r i c a l p r o c e d u r e s . One s u e p r o c e d u r e i s examined h e r e . I t i s b a s e d on a method developed by I s a a c s o n (1981) w h i c h c a l c u l a t e s t h e i n t e r a c t i o n of steep ( n o n l i n e a r ) o c e a n waves w i t h l a r g e o f f s h o r e s t r u c t u r e s o f a r b i t r a r y s h a p e . In t h i s method t h e wave d i f f r a c t i o n and c u r r e n t i n t e r a c t i o n a r e t r e a t e d as a t r a n s i e n t p r o b l e m . The known i n i t i a l c o n d i t i o n s c o r r e s p o n d t o s t i l l w ater i n t h e v i c i n i t y o f t h e c y l i n d e r w i t h a p r e s c r i b e d i n c i d e n t wave form a p p r o a c h i n g t h e c y l i n d e r and a u n i f o r m c u r r e n t i n c r e a s i n g w i t h t i m e from z e r o t o a s p e c i f i e d m a g n i t u d e . The d e v e l o p m e n t o f t h e f l o w c a n t h e n be o b t a i n e d by a t i m e s t e p p i n g p r o c e d u r e i n w h i c h t h e v e l o c i t y p o t e n t i a l o f t h e f l o w a t any one i n s t a n t i s o b t a i n e d by an i n t e g r a l e q u a t i o n method b a s e d on Green's t h e o r e m . The g o v e r n i n g e q u a t i o n s a r e as i n d i c a t e d i n s e c t i o n 4.1. A b o u n d a r y i n t e g r a l method i n v o l v i n g a G r e e n ' s f u n c t i o n i s u s e d as t h e b a s i s f o r a n u m e r i c a l e v a l u a t i o n of <f>. I f we d e n o t e t h e p o i n t x = ( x , y , z ) as l y i n g on t h e f l u i d b o u n d a r y ( a p p r o a c h e d from w i t h i n t h e f l u i d r e g i o n c o n t a i n e d by t h e c l o s e d s u r f a c e S ) , t h e n t h e v e l o c i t y p o t e n t i a l <J>(x) a t t h i s p o i n t may be e x p r e s s e d a s : = 2T I s { G { ^ W ~ " • ( i , t o ( i ' i , } d s (4-8) 25 where H r e p r e s e n t s a p o i n t (?,n,C) on t h e s u r f a c e S o v e r w h i c h t h e i n t e g r a t i o n i s p e r f o r m e d . A l s o , G i s a s u i t a b l e G r e e n ' s f u n c t i o n and n i s measured from t h e p o i n t K . As shown i n F i g u r e 5, t h e i n t e g r a t e d s u r f a c e S c o m p r i s e s t h e immersed s u r f a c e o f t h e body, , t h e i n s t a n t a n e o u s f r e e s u r f a c e , S f , a v e r t i c a l " f a r f i e l d " c o n t r o l s u r f a c e , S c , and t n e s e a b e d . I n a c t u a l p r a c t i c e , i t i s f o u n d t o be more e f f i c i e n t t o r e p l a c e t h e s e a b e d w i t h a r e f l e c t i o n of what i s above i t . A G r e e n ' s f u n c t i o n w h i c h a c c o u n t s f o r t h i s symmetry i s u s e d , namely: G = 1/r + 1 / r ' where r i s . t h e d i s t a n c e between t h e p o i n t s x and £ and r ' i s d i s t a n c e between x and t h e p o i n t £' w h i c h i s t h e r e f l e c t i o n o f i n t h e s e a b e d . The i n i t i a l c o n d i t i o n s f o r t h e p r o b l e m a r e c h o s e n s u c h t h a t t h e r e i s s t i l l w a t er i n t h e v i c i n i t y o f t h e c y l i n d e r and a p r o g r e s s i v e wave t r a i n e m a n a t i n g f r o m t h e c o n t r o l s u r f a c e S t o w a r d t h e body. The c o n t r o l s u r f a c e i s c h o s e n t o b e s u f f i c i e n t l y removed f r o m t h e body s u c h t h a t s c a t t e r e d waves w h i c h r e s u l t f r o m t h e i n t e r a c t i o n between t h e i n c i d e n t waves and t h e body w i l l n o t r e a c h t h e c o n t r o l s u r f a c e w i t h i n t h e t i m e - s p a n o f t h e a n a l y s i s . Thus, t h e v a l u e s o f 4> and H on t h e c o n t r o l s u r f a c e a r e known a t a l l t i m e s o f t h e a n a l y s i s , b e i n g due t o t h e i n c i d e n t wave a s p r e s c r i b e d by t h e c h o s e n r e p r e s e n t a t i v e wave t h e o r y . E q u a t i o n 4.8 may now be s e p a r a t e d i n t o a s u r f a c e i n t e g r a l o v e r S c w h i c h i s known a t any t i m e , and a s u r f a c e i n t e g r a l o v e r 26 t h e r e m a i n i n g s u r f a c e , S b + S f , w h i c h c o n t a i n s t h e bou n d a r y f u n c t i o n s <j> and | | t o be d e t e r m i n e d . The i n t e g r a l e q u a t i o n o v e r t h e s u r f a c e S b + S f i s s u b j e c t t o t h e v a r i o u s b o u n d a r y c o n d i t i o n s d i s c u s s e d i n t h e p r e v i o u s s e c t i o n . The b o u n d a r y c o n d i t i o n on th e immersed s u r f a c e o f t h e c y l i n d e r , S f a, i s s i m p l y g i v e n by e q u a t i o n 4 . 3 . An e x p l i c i t t i m e - s t e p p i n g p r o c e d u r e i s u s e d t o t r e a t t h e f r e e s u r f a c e ( S f ) bou n d a r y c o n d i t i o n s . The f r e e s u r f a c e S.,^ and t h e v e l o c i t y p o t e n t i a l * , , b o t h a t f(t+A=t) J * rt+At t i m e t + A t , a r e e x p r e s s e d e x p l i c i t l y i n terms of t h e s o l u t i o n up t o t i m e t . T h a t i s , W= f ( t' f c-At ) ° n S f ( t + A t ) ( 4 ' 9 ) where f ( ) i n d i c a t e s a known q u a n t i t y a n d t h e s u b s c r i p t s i n d i c a t e t h e t i m e a t w h i c h t h e c o r r e s p o n d i n g q u a n t i t i e s a r e c o n s i d e r e d . U s i n g a two s t e p A d a m s - B a s h f o r t h method, t h e f r e e s u r f a c e b o u n d a r y c o n d i t i o n s , e q u a t i o n s 4 . 4 and 4 . 5 , a r e e x p r e s s e d a s : w • \ + f % > t - { i S u . . o ) Z" z * t + A t = • t " x[ 3 { g n + ^ «w2>t - ^ + ^ <7*)2}t_At] ( 4 . 1 D Note t h a t t h e r i g h t hand s i d e s c o n t a i n q u a n t i t i e s a t t i m e s t and t - A t , w h i c h a r e known f r o m t h e p r e v i o u s t i m e i t e r a t i o n , a s p r e s c r i b e d i n e q u a t i o n 4 . 9 . Thus t h e i n t e g r a l e q u a t i o n o v e r t h e s u r f a c e +S f may be s o l v e d n u m e r i c a l l y t o o b t a i n t h e d i s t r i b u t i o n s o f <f> and | | o v e r +S f a t t i m e t + A t . Hence t i m e can be a d v a n c e d s t e p by s t e p t o d e s c r i b e t h e e v o l u t i o n o f t h e f l o w . The f o r c e s on t h e c y l i n d e r a r e of i n t e r e s t i n t h i s s t u d y 27 and t h e s e may be d e t e r m i n e d i n terms o f t h e v e l o c i t y p o t e n t i a l . The p r e s s u r e d i s t r i b u t i o n o v e r t h e body s u r f a c e i s f i r s t d e t e r m i n e d by a p p l y i n g t h e u n s t e a d y B e r n o u l l i e q u a t i o n : P = -p{gz + |f + MV*) 2} (4.12) where p i s t h e f l u i d d e n s i t y . The f o r c e v e c t o r F i s t h e n e x p r e s s e d by i n t e g r a t i n g t h e p r e s s u r e o v e r t h e body s u r f a c e : I = / s PH d s (4.13) The r e p r e s e n t a t i o n o f t h e i n c i d e n t wave may be o f s e v e r a l d i f f e r e n t f o r m s , but we s h a l l c o n s i d e r h e r e o n l y a s m a l l a m p l i t u d e s i n u s o i d a l wave a s d e s c r i b e d by l i n e a r wave t h e o r y . T h i s wave form c o r r e s p o n d s t o a wave g r o u p w i t h a l e a d i n g edge p r o p a g a t i n g o v e r s t i l l w a t e r . a n d i n i t i a l l y u p s t r e a m o f t h e body. As w i t h t h e e x p r e s s i o n i n s e c t i o n 2.4, t h e f r e e s u r f a c e e l e v a t i o n c a n be d e n o t e d a s : n = A ( x ) f cos{k(x-ct) } (4.14) where A ( x ) i s t h e a m p l i t u d e o f t h e g r o u p ' s e n v e l o p e , x = < ( x - c G t ) , K (<k) i s t h e wave number o f t h e e n v e l o p e w h i c h s h o u l d be s u f f i c i e n t l y s m a l l t o p r o v i d e f o r a slow v a r i a t i o n o f A ( x ) , k i s th e i n c i d e n t wave number, c i s t h e wave s p e e d , and cQ i s t h e g r o u p v e l o c i t y . I f t h e body l i e s w i t h i n |x|<a, t h e n t h e c o n d i t i o n A ( x ) = 0 s h o u l d e x i s t f o r x>-a, t=0. T h i s e n s u r e s an i n i t i a l c o n d i t i o n of s t i l l w ater i n t h e v i c i n i t y o f t h e c y l i n d e r . A s i m p l e form o f A ( x ) w h i c h i s employed i n t h i s s t u d y i s : r 1 for x + <a < - I T A (X) = < h{l - cos (x + <a) } for - I T < x + *a < 0 (4.15) 0 for x + K a > 0 28 and t h i s wave p r o f i l e i s i l l u s t r a t e d i n F i g u r e 6a. The i n t r o d u c t i o n o f a c o l i n e a r c u r r e n t , U, p e r f o r c e c h a n g e s the above wave r e p r e s e n t a t i o n s l i g h t l y . The c u r r e n t i t s e l f i s imposed i n a p r o g r e s s i v e f a s h i o n , s t a r t i n g f r o m z e r o and i n c r e a s i n g s m o o t h l y t o t h e s t e a d y s t a t e c o n d i t i o n i n t h e same f a s h i o n and w i t h i n t h e same t i m e as t h e wave t a k e s t o b u i l d up t o i t s s t e a d y form ( s e e F i g u r e 6 b ) . The f r e e s u r f a c e i s t h e same as b e f o r e e x c e p t t h e wave c e l e r i t y c i s now c ( r e c a l l c c=c+U i n s e c t i o n 2.6) and t h u s : n = A ( x)j cos{k(x-c ct) } (4.16) S i m i l a r l y , c Q i s now a l s o a f u n c t i o n of c Q r a t h e r t h a n c . As w e l l , we n o t e t h a t t h e v e l o c i t y p o t e n t i a l <J> , f o r m e r l y e x p r e s s e d a s : <J>- = f (x-ct) (4.17) i s now e x p r e s s e d a s : 4) = f (x-c ct) + ux ( 4 . 1 8 ) i n t h e p r e s e n c e of a c u r r e n t . D u r i n g t h e i n i t i a l c a l c u l a t i o n s b e f o r e t h e f l o w has become f u l l y d e v e l o p e d , t h e v a l u e o f c c c h a n g e s from t i m e s t e p t o t i m e s t e p c o r r e s p o n d i n g t o t h e t i m e e v o l u t i o n o f t h e c u r r e n t v e l o c i t y U. 29 4.2.2 R e s u l t s U s i n g a n u m e r i c a l p r o c e d u r e b a s e d on t h e above method w i t h t h e s u r f a c e s S. , S , and S, d i s c r e t i z e d i n t o f i n i t e numbers o f b e f e l e m e n t s , s e v e r a l s e t s of r e s u l t s were g e n e r a t e d f o r v a r y i n g c o n d i t i o n s of wave, c u r r e n t , and c y l i n d e r s i z e . The c y l i n d e r c i r c u m f e r e n c e was d i v i d e d i n t o e i g h t e q u a l s t r a i g h t - l i n e segments and t h e h e i g h t of t h e c y l i n d e r i n t o f i v e s egments. Thus t h e body s u r f a c e c o n s i s t e d o f f o r t y segments. b S i m i l a r l y , t h e f r e e s u r f a c e S f c o n s i s t e d o f s i x t y segments and t h e c o n t r o l s u r f a c e S c c o n s i s t e d o f n i n e t y - s i x segments. The t i m e s t e p i n t e r v a l was c h o s e n t o be o n e - f i f t i e t h of t h e wave p e r i o d . The g r o w t h l e n g t h s of t h e i n c i d e n t wave and c u r r e n t were c h o s e n s u c h t h a t t h e y were b o t h f u l l y d e v e l o p e d by a p p r o x i m a t e l y one and a h a l f wave p e r i o d s . F i g u r e s 7 and 8 i l l u s t r a t e some r e p r e s e n t a t i v e r e s u l t s f o r t h e c o n d i t i o n s ka=1.0 and kd=3.0. E a c h t r a c e r e p r e s e n t s t h e e f f e c t of d i f f e r e n t c u r r e n t s , w i t h U/c h a v i n g v a l u e s o f -0.05, -0.02, 0.0, 0.02, and 0.05. F i g u r e 7 shows t h e non-d i m e n s i o n a l i z e d f r e e s u r f a c e e l e v a t i o n n/H (where H i s t h e p r e s c r i b e d c r e s t t o t r o u g h wave h e i g h t ) a t t h e c y l i n d e r ' s c e n t r e as a f u n c t i o n of n o n - d i m e n s i o n a l i z e d t i m e t / T (where t = e l a p s e d t i m e i n s e c o n d s and T = wave p e r i o d ) . I t c a n be seen t h a t t h e wave form i s f u l l y d e v e l o p e d w i t h i n one and a q u a r t e r wave-l e n g t h s a t w h i c h p o i n t n/H=-0.5. The e f f e c t o f t h e c u r r e n t on t h e f r e e s u r f a c e e l e v a t i o n t r a c e s i s e v i d e n t i n t h e o b s e r v e d wave p e r i o d s a t t h e c y l i n d e r c e n t r e . The wave p e r i o d d e c r e a s e s w i t h i n c r e a s i n g p o s i t i v e c u r r e n t , as e x p e c t e d . C o n v e r s e l y , t h e 30 wave p e r i o d i n c r e a s e s w i t h i n c r e a s i n g n e g a t i v e c u r r e n t . F i g u r e 8 shows t h e n o n - d i m e n s i o n a l i z e d f o r c e , , r — as a f u n c t i o n of >5pgHaz t / T . As w i t h t h e s u r f a c e e l e v a t i o n , we e x p e c t t h e c u r r e n t / w a v e f l o w t o be f u l l y d e v e l o p e d by a t l e a s t t/T=1.3. I t i s s u s p e c t e d t h a t due t o t h e n e c e s s a r y t r a n s i e n t c o n d i t i o n w i t h r e g a r d t o t h e d u r a t i o n of t h e c a l c u l a t i o n s , t h e n u m e r i c a l method becomes u n s t a b l e and g i v e s u n r e l i a b l e r e s u l t s a f t e r t h e n e x t c r e s t ( i e . a r o u n d t / T = 1 . 8 ) . At any r a t e , t h e g e n e r a l t r e n d a t t/T=1.3 i s c l e a r l y i d e n t i f i a b l e , w i t h t h e f o r c e b e i n g d i r e c t l y p r o p o r t i o n a l t o t h e c u r r e n t . T h a t i s , t h e g r e a t e r t h e p o s i t i v e c u r r e n t , t h e g r e a t e r t h e t o t a l f o r c e and c o n v e r s e l y , t h e more n e g a t i v e t h e c u r r e n t , t h e l e s s t h e f o r c e on t h e c y l i n d e r . The v a l u e of t h e f o r c e peak a t t/T=1.3 f o r U/c=0.0 i s -4.22 w h i c h i s i n c l o s e a c c o r d w i t h t h e c a l c u l a t i o n s b a s e d on t h e MacCamy-Fuchs f o r m u l a t i o n f o r t h e same s i t u a t i o n o f -4.28. I t must be remembered when r e v i e w i n g t h e r e s u l t s o f t h i s n u m e r i c a l p r o c e d u r e t h a t i t i s s u b j e c t t o t h e a s s u m p t i o n of an i n v i s c i d f l u i d and as s u c h , d o e s n o t t a k e i n t o a c c o u n t any e f f e c t o f c u r r e n t i n d u c e d d r a g . 31 5. EXPERIMENTAL APPARATUS 5.1 THE WAVE TANK AND MOVING SUPPORT CARRIAGE The e x p e r i m e n t s were p e r f o r m e d i n a r e c t a n g u l a r wave b a s i n 13.7 m e t r e s l o n g , 4.9 m e t r e s w i d e , and 0.9 m e t r e s deep. The waves were g e n e r a t e d a t one end of t h e t a n k by a p a d d l e whicn e x t e n d e d f r o m one s i d e o f t h e t a n k t o t h e o t h e r . T h i s p a d d l e was h i n g e d a t t h e b a s i n f l o o r and was d r i v e n by a v a r i a b l e - s p e e d e l e c t r i c motor t h r o u g h a c r a n k of a d j u s t a b l e s t r o k e . A wave f i l t e r c o n s i s t i n g of two l a y e r s o f w i r e mesh was l o c a t e d d i r e c t l y i n f r o n t of t h e wave p a d d l e i n o r d e r t o a b s o r b any h i g h - f r e q u e n c y i r r e g u l a r i t i e s i n t h e g e n e r a t e d waves. S i d e - w a l l wave r e f l e c t i o n s i n t h e b a s i n were r e d u c e d by t h e u s e o f s t r i p s o f a r t i f i c i a l h a i r m a t t i n g hung a l o n g t h e s i d e s o f t h e b a s i n , e x t e n d i n g t h r o u g h t h e s u r f a c e of t h e water and a n g l e d a t a b o u t t h i r t y d e g r e e s away from t h e v e r t i c a l w a l l s . Wave r e f l e c t i o n s f r o m t h e end of t h e b a s i n o p p o s i t e t h e wave p a d d l e were r e d u c e d by t h e use o f an a r t i f i c i a l h a i r m a t t i n g b e ach w i t h a c o n s t a n t s l o p e of 1:10. R e f l e c t i o n c o e f f i c i e n t s f o r t h e b a s i n were d e t e r m i n e d by two d i f f e r e n t methods ( s e e s e c t i o n 6.1) and t h e r e s u l t s i n d i c a t e d t h a t f o r t h e p a r t i c u l a r r ange of wave p e r i o d s and h e i g h t s of i n t e r e s t i n t h e e x p e r i m e n t a l s e r i e s , t h e r e f l e c t i o n c o e f f i c i e n t s were l e s s t h a n e i g h t p e r c e n t , w i t h t h e h i g h e r c o e f f i c i e n t s b e i n g g e n e r a l l y a s s o c i a t e d w i t h t h e l o n g e r p e r i o d waves. A c a r r i a g e p l a t f o r m s p a n n i n g t h e w i d t h o f t h e b a s i n was mounted on r a i l s w h ich r a n t h e e n t i r e l e n g t h o f t h e b a s i n . The 32 c a r r i a g e w h e e l s were d r i v e n by a v a r i a b l e s p e e d e l e c t r i c motor and t h u s i t was p o s s i b l e t o make t h e c a r r i a g e t r a v e l t h e l e n g t h of t h e b a s i n a t any d e s i r e d s p e e d from a l o w e r t h r e s h o l d o f a p p r o x i m a t e l y e i g h t c e n t i m e t r e s p e r s e c o n d (cm/sec) t o a maximum o f a r o u n d f i f t y cm/sec i n e i t h e r d i r e c t i o n . A m o u n t i n g f a c i l i t y f o r t h e t e s t c y l i n d e r s was c o n s t r u c t e d on t h e u n d e r s i d e of t h e c a r r i a g e i n t h e m i d d l e o f t h e s p a n . The g e n e r a l c o n f i g u r a t i o n o f t h e wave b a s i n and i t s components i s shown i n F i g u r e 9. 5.2 THE TEST CYLINDERS The h o l l o w aluminum c y l i n d e r s o f d i f f e r e n t d i a m e t e r s were c o n s t r u c t e d t o a c t as t h e model s t r u c t u r e s . One c y l i n d e r was t w e n t y c e n t i m e t r e s i n d i a m e t e r w h i l e t h e o t h e r was f o r t y c e n t i m e t r e s i n d i a m e t e r . An i n s t r u m e n t e d v e r t i c a l r o d was f i t t e d down t h r o u g h a h o l e i n t h e t o p o f t h e c y l i n d e r and was b o l t e d t o t h e i n s i d e o f t h e c y l i n d e r b a s e . T h i s r o d was c o n n e c t e d a t i t s upper end t o t h c c a r r i a g e frame and formed t h e s u p p o r t f o r t h e c y l i n d e r s u c h t h a t t h e b a s e o f t h e c y l i n d e r r e m a i n e d a s h o r t d i s t a n c e above t h e f l o o r of t h e b a s i n . In o r d e r t o measure t h e f o r c e e x e r t e d on t h e c y l i n d e r , t h e s u p p o r t r o d was i n s t r u m e n t e d w i t h two s t r a i n gauges i n l i n e w i t h e a c h o t h e r , one mounted ne a r t h e t o p and one n e a r t h e b o t t o m of t h e r o d . A d i a g r a m of a t e s t c y l i n d e r and t h e s u p p o r t r o d i s shown i n F i g u r e 10. The c y l i n d e r s , r o d , and c a r r i a g e were d e s i g n e d t o e n s u r e t h a t 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 was h i g h enough. 3 3 t o a v o i d s i g n i f i c a n t dynamic a m p l i f i c a t i o n a t t h e e x p e r i m e n t a l wave f r e q u e n c i e s . A l s o , t h e s y s t e m was r i g i d enough t o p r o d u c e o n l y s m a l l d e f l e c t i o n s u nder t e s t c o n d i t i o n s so t h a t t h e r e were n e g l i g i b l e a l t e r a t i o n s t o t h e h y d r o d y n a m i c l o a d i n g on t h e c y l i n d e r . The s t r a i n g auges w h i c h were mounted on t h e s u p p o r t r o d were i n c o r p o r a t e d i n t o a W h e a t s t o n e b r i d g e c i r c u i t i n s u c h a f a s h i o n so as t o p r o d u c e measurements o f t o t a l f o r c e a c t i n g on t h e c y l i n d e r . The r o d was o r i e n t e d i n t h e wave b a s i n so t h a t o n l y t h e i n - l i n e f o r c e s ( p e r p e n d i c u l a r t o t h e wave c r e s t s ) were mea s u r e d . The i n s t r u m e n t e d r o d was c a l i b r a t e d f o r f o r c e by s e c u r i n g i t i n a h o r i z o n t a l p o s i t i o n and h a n g i n g a s e r i e s of known w e i g h t s from i t . The o u t p u t from t h e Wh e a t s t o n e b r i d g e c i r c u i t was p a s s e d t h r o u g h an e l e c t r o n i c l o w - p a s s f i l t e r . T h i s was n e c e s s a r y i n o r d e r t o a t t e n u a t e t h e h i g h f r e q u e n c y n o i s e g e n e r a t e d by t h e v i b r a t i o n s of t h e c y l i n d e r due t o t h e movement of t h e s u p p o r t c a r r i a g e a l o n g t h e r a i l s . The f i l t e r u s e d was a f i v e h e r t z f o u r t h o r d e r l o w - p a s s a c t i v e f i l t e r w i t h 24 d B / o c t a v e a t t e n u a t i o n . The g a i n - f r e q u e n c y r e s p o n s e c u r v e i s shown i n F i g u r e 11. T h i s was q u i t e e f f e c t i v e i n f i l t e r i n g o u t t h e unwanted v i b r a t i o n s w i t h o u t a l t e r i n g t h e o u t p u t a t t h e i n c i d e n t wave f r e q u e n c i e s . The f i l t e r d i d i n t r o d u c e a p h a s e s h i f t t o t h e f o r c e s i - g n a l (due t o t h e f i n i t e amount of t i m e r e q u i r e d f o r t h e f i l t e r t o a c t ) , but a s t h e phase r e l a t i o n was n o t b e i n g s t u d i e d , t h i s was o f no c o n s e q u e n c e . A f t e r b e i n g p a s s e d t h r o u g h t h e low-p a s s f i l t e r , t h e f o r c e s i g n a l was d i s p l a y e d on an o s c i l l o s c o p e f o r v i e w i n g p u r p o s e s and t h e n r e c o r d e d on an u l t r a v i o l e t c h a r t 34 r e c o r d e r . The c h a r a c t e r i s t i c s o f t h e i n c i d e n t waves were measured u s i n g a v a r i a b l e c a p a c i t a n c e wave p r o b e w h i c h was mounted on t h e s u p p o r t c a r r i a g e and whose o u t p u t was a l s o r e c o r d e d on t h e u l t r a v i o l e t c h a r t r e c o r d e r . A s c h e m a t i c o f t h i s d a t a l o g g i n g s y s t e m i s i l l u s t r a t e d i n F i g u r e 12. A l s o , F i g u r e 13 shows t h e m o v e a b l e s u p p o r t c a r r i a g e w i t h t h e s p e e d c o n t r o l box on t h e r i g h t . The e l e c t r o n i c s i n t h e p h o t o g r a p h a r e , from l e f t t o r i g h t , t h e W h e a t s t o n e b r i d g e c i r c u i t , t h e o s c i l l o s c o p e , and t h e u l t r a v i o l e t c h a r t r e c o r d e r w i t h t h e wave p r o b e power s u p p l y mounted on t o p . The e l e c t r o n i c f i l t e r i s out o f s i g h t b e h i n d ' t h e W h e a t s t o n e b r i d g e . .The t e s t c y l i n d e r s were f i t t e d w i t h v e r t i c a l s c a l e s b o t h f o r e and a f t upon w h i c h t h e water l e v e l c o u l d be e a s i l y r e a d by eye f r o m t h e s i d e o f t h e t a n k . T h e s e s c a l e s were u s e d t o e s t i m a t e t h e r u n u p on t h e l e a d i n g and t r a i l i n g f a c e s o f t h e c y l i n d e r w a l l s under t h e v a r y i n g c o n d i t i o n s of waves and c u r r e n t . One o f t h e s c a l e s c a n be seen on t h e f r o n t of a c y l i n d e r i n F i g u r e 14. A l s o t o be n o t i c e d i n t h i s p h o t o g r a p h i s t h e wave p r o b e mounted on t h e c a r r i a g e t o t h e r i g h t o f t h e c y l i n d e r . 3 5 6. EXPERIMENTAL PROCEDURE 6.1 WAVE BASIN REFLECTION COEFFICIENT The f i r s t step i n the experimental procedure was to determine the wave c h a r a c t e r i s t i c s of the wave b a s i n . The i d e a l basin would have wave gen e r a t i o n at one end and the r e s u l t i n g p r o g r e s s i v e wave would t r a v e l along the l e n g t h of the b a s i n with no s i d e - w a l l r e f l e c t i o n s and no end-wall r e f l e c t i o n . In the r e a l case, however, the wave energy was not completely d i s s i p a t e d by the mat beach at the end of the basin and a. c e r t a i n amount of wave energy was r e f l e c t e d back toward the wave generator. T h i s r e s u l t e d i n a p a r t i a l standing wave, or i n other words, a s l i g h t l y v a r y i n g wave height p r o f i l e along the l e n g t h of the b a s i n . A r e f l e c t i o n c o e f f i c i e n t , K , i s d e f i n e d as the r a t i o of the r e f l e c t e d wave height to the i n c i d e n t wave he i g h t . The r e f l e c t i o n c o e f f i c i e n t i s expected to change somewhat with wave le n g t h , the c o e f f i c i e n t being l a r g e r f o r longer waves. A s e r i e s of measurements were conducted i n the wave bas i n to determine the' r e f l e c t i o n c o e f f i c i e n t f o r the p a r t i c u l a r geometry and composition of the energy d i s s i p a t i n g beach that was used. For these measurements, the t e s t c y l i n d e r was removed from the tank and only the wave probe was l e f t p r o t r u d i n g i n t o the water. Two d i f f e r e n t methods were employed to determine K r. In the f i r s t method a " t r a n s i e n t " wave t r a i n was used. That i s , the water s u r f a c e was allowed to become f l a t calm before the wave paddle was a c t i v a t e d . The paddle was then a c t i v a t e d f o r a short p e r i o d of time causing a f i n i t e number of 36 waves (on the order of 10) to progress past the wave probe (situated near the paddle end of the basin), r e f l e c t off the beach, and progress past the wave probe in the opposite d i r e c t i o n . The t o t a l wave action was recorded on the u l t r a v i o l e t chart recorder and the height of the refl e c t e d wave was compared to the height of the incident wave. This s e q u e n c e was repeated for four d i f f e r e n t wave lengths which covered t h e range of wavelengths of interest in the overall experiment. By this method, i t was determined that the c o e f f i c i e n t of r e f l e c t i o n was consistently no greater than eight percent. Another method for determining'K was also used as a check on the f i r s t r e s u l t s . In this method, the wave paddle was set continuously in motion and the maximum and minimum wave heights were measured throughout the length of the basin. In other words, the magnitude of the modulation was measured. Using a method described by Sarpkaya and Isaacson (1981), i t can be shown that: H - "H ".' „ max min r H + H . max min where H and H . are the maximum and minimum wave heights max min 3 respectively. The measured values of H and H were applied max min to this formula and i t was found that the results of thi s method were consistent with the "transient wave" method. It was concluded thar the c o e f f i c i e n t of r e f l e c t i o n from the hair-mat beach was no greater than eight percent throughout the major set of experiments. 37 6.2 FORCE MEASUREMENTS The main o b j e c t i v e s of t h e p r i m a r y e x p e r i m e n t s were t o q u a n t i f y t h e m o d i f i c a t i o n of t h e o s c i l l a t o r y wave f o r c e s and t h e run u p on t h e two t e s t c y l i n d e r s u nder t h e i n f l u e n c e of a c o l i n e a r c u r r e n t . The c u r r e n t was s i m u l a t e d by moving the c y l i n d e r t h r o u g h t h e w a t e r a t a c o n s t a n t v e l o c i t y . T h i s was a c c o m p l i s h e d by a t t a c h i n g t h e c y l i n d e r t o t h e i n s t r u m e n t e d s u p p o r t on t h e u n d e r s i d e of t h e m o b i l e c a r r i a g e . The o s c i l l a t i n g f o r c e o f a s p e c i f i c wave t r a i n on t h e s t a t i o n a r y c y l i n d e r was measured, t h e i n s t r u m e n t e d c y l i n d e r s u p p o r t r o d h a v i n g f i r s t been c a l i b r a t e d f o r f o r c e by s u p p o r t i n g i t i n a h o r i z o n t a l f a s h i o n and h a n g i n g measured w e i g h t s f r o m i t . A f t e r t h e m a g n i t u d e o f t h e o s c i l l a t i n g wave f o r c e on t h e s t a t i o n a r y c y l i n d e r had been d e t e r m i n e d , t h e c y l i n d e r was moved t h r o u g h t h e wat e r a t a s p e c i f i c v e l o c i t y and t h e o s c i l l a t i n g p a r t of t h e t o t a l f o r c e was measured. The " c u r r e n t " v e l o c i t y was chosen s u c h t h a t t h e r a t i o o f t h i s v e l o c i t y t o t h e wave c e l e r i t y (U/c) was 0 . 1 and 0.2 i n b o t h t h e p o s i t i v e and n e g a t i v e d i r e c t i o n s . The c h a r a c t e r i s t i c s o f t h e waves t h a t i m p i n g e d upon t h e c y l i n d e r s were c h o s e n i n t h e f o l l o w i n g manner. A c o n v e n i e n t measure o f t h e i m p o r t a n c e of wave d i f f r a c t i o n a r o u n d a body i s t h e " d i f f r a c t i o n p a r a m e t e r " which i s d e f i n e d a s t h e body s i z e t o wave l e n g t h r a t i o , D / L (where D i s t h e c h a r a c t e r i s t i c d i m e n s i o n o f t h e body -- t h e d i a m e t e r i n t h e c a s e o f a v e r t i c a l c i r c u l a r c y l i n d e r — and L i s t h e wave l e n g t h ) . I f t h e r a d i u s o f t h e t e s t c y l i n d e r i s d e n o t e d a and t h e wave number i s d e n o t e d k, t h e n ka ( = TTD/L) i s s i m p l y an a l t e r n a t i v e e x p r e s s i o n f o r t h e 3 8 d i f f r a c t i o n parameter. The ex p e r i m e n t s were d e s i g n e d t o c o v e r the range k a = 0 . 5 t o k a = 3 . 0 a t 0 . 5 i n t e r v a l s , w i t h the s m a l l c y l i n d e r spanning k a = 0 . 5 t o 2 . 0 and the l a r g e c y l i n d e r spanning k a = l . 5 t o 3 . 0 . Three wave h e i g h t s were used f o r each v a l u e of ka and the s t i l l water depth was kept c o n s t a n t throughout the s e r i e s of ex p e r i m e n t s a t t h i r t y c e n t i m e t r e s . 6 . 3 RUNUP MEASUREMENTS The s e r i e s of runup e x p e r i m e n t s were conducted i n the same manner as the f o r c e e x p e r i m e n t s except t h a t the runup was e s t i m a t e d v i s u a l l y a g a i n s t c a l i b r a t e d markings on the l e a d i n g and t r a i l i n g f a c e s of the c y l i n d e r s . 39 7. EXPERIMENTAL RESULTS AND DISCUSSION 7.1 BEAT PHENOMENON The f i r s t observation that was made at the outset of the experimental runs was that the magnitude of the runup and the amplitude of the i n - l i n e o s c i l l a t i n g wave force on the cylinder did not remain constant when the cylinder was moving through the water ( i e . in the presence of a simulated current). In fact, the runup and the amplitude of the force o s c i l l a t i o n s appeared to fluctuate in a regular fashion. This observation was not f e l t to be simply a manifestation of the wave r e f l e c t i o n c h a r a c t e r i s t i c s of the basin for two reasons. F i r s t l y , the magnitude of the r e f l e c t i o n c o e f f i c i e n t for the basin was shown in section 6.1 to be less than eight .percent, whereas the magnitude of the effect just described was substantially larger. Secondly, i f i t were due to basin r e f l e c t i o n s , then the direc t i o n of the cylinder motion should have no influence on the frequency of thi s modulation e f f e c t . This was not so. The frequency of the modulation was higher with the cylinder moving into the waves than with the cylinder moving in the same direc t i o n as the waves. Furthermore, changes in towing speed gave r i s e to envelope maxima at d i f f e r e n t locations in the wave basin which should not occur on account of wave r e f l e c t i o n s within the basin. The observation can be explained, however, as a "beat phenomenon" due to the superposition of two wave motions having only s l i g h t l y d i f f e r e n t frequencies. When a progressive wave encounters the motionless cylinder, a certain portion i s 40 r e f l e c t e d or s c a t t e r e d i n a r a d i a l f a s h i o n w i t h the same freque n c y as the i n c i d e n t wave. Thus a t the f r o n t of the c y l i n d e r t h e r e e x i s t s a p a r t i a l s t a n d i n g wave, which becomes more and more " p a r t i a l " w i t h d i s t a n c e from the c y l i n d e r due t o the a t t e n u a t i o n of the r e f l e c t e d wave i n h e r e n t i n i t s r a d i a l p r o p a g a t i o n . Now, i f the c y l i n d e r i s g i v e n m o t i o n , say, i n t o the incoming waves, i t e f f e c t i v e l y moves i n t o i t s own s t a n d i n g wave. From the v i e w p o i n t of the c y l i n d e r , the i n c i d e n t wave now has a s l i g h t l y h i g h e r f r e q u e n c y than when t h e c y l i n d e r was m o t i o n l e s s , w h i l e the r e f l e c t e d wave (moving i n the same d i r e c t i o n as the c y l i n d e r ) has a s l i g h t l y lower f r e q u e n c y . The net r e s u l t i s t h a t the c y l i n d e r sees modulated wave h e i g h t and f o r c e f l u c t u a t i o n s . I f we examine t h i s phenomenon i n an a n a l y t i c a l but n o n - r i g o r o u s f a s h i o n , we can l e t the i n c i d e n t s u r f a c e e l e v a t i o n be: nj = ^ - c o s ( k ( x - ( c + U ) t ) ) and the r e f l e c t e d s u r f a c e e l e v a t i o n be: where H i s t h e c r e s t t o t r o u g h wave h e i g h t and U i s the speed of the c y l i n d e r . C o n s i d e r i n g the frame of r e f e r e n c e of the moving c y l i n d e r and s u p e r p o s i n g the s u r f a c e e l e v a t i o n s of the i n c i d e n t and r e f l e c t e d waves we g e t : n = n. + n = Hcos(wt ) c o s ( k u t ) . t i r Now i f we c o n s i d e r the c y l i n d e r ( c u r r e n t ) v e l o c i t y t o be s m a l l such t h a t the r a t i o of the c y l i n d e r v e l o c i t y t o the • wave c e l e r i t y i s : U/c = E and e « 1 , 41 t h e n we have t h e t o t a l r e s u l t a n t s u r f a c e e l e v a t i o n r e p r e s e n t e d a s : n t = H c o s C w t ) c o s ( e u ) t ) w h i c h i s t h e p r o d u c t of a q u i c k l y v a r y i n g c o s i n e and a s l o w l y v a r y i n g c o s i n e . The r e s u l t o f t h i s i s a wave f o r m t h a t has a " c a r r i e r " f r e q u e n c y o f co arid a " g r o u p " o r m o d u l a t i n g f r e q u e n c y of etc and r e s e m b l e s t h e wave i n F i g u r e 15. The p e r i o d o f t h e " c a r r i e r " wave, T . w i l l be t h e p e r i o d o f t h e i n c i d e n t wave, carrier — , w h i l e t h e p e r i o d o f t h e " g r o u p " wave, T , w i l l be — . to group eco The r a t i o : T group _ 2Tr/ea) _ _1 _ £ T . ~ 2 T T / O O e ~ U carrier g i v e s an i n d i c a t i o n o f t h e p e r i o d of t h e g r o u p wave i n terms of c y c l e s o f t h e c a r r i e r wave. The v a l u e s o f U/c u s e d i n t h e s e r i e s o f e x p e r i m e n t s were +0.2 and +0.1, so t h e r e s p e c t i v e g r o u p maximums c o u l d be e x p e c t e d t o o c c u r a t e v e r y f i f t h o r t e n t h wave. T h i s was, i n f a c t , a p p r o x i m a t e l y what was o b s e r v e d . 42 7.2 IN-LINE FORCE 7.2.1 D e s c r i p t i o n Of R e s u l t s The i n - l i n e f o r c e measurements a s r e c o r d e d on t h e u l t r a -v i o l e t c h a r t r e c o r d e r were v e r y s t e a d y when t h e c y l i n d e r was not moving. W i t h t h e c y l i n d e r i n m o t i o n , however, t h e r e c o r d e d s i g n a l t e n d e d t o f l u c t u a t e s u b s t a n t i a l l y , m a i n l y due t o t h e b e a t phenomenon as m e n t i o n e d a b o v e . O t h e r a n o m a l i e s were u n d o u b t e d l y i n t r o d u c e d f r o m v a r i o u s s o u r c e s o f e x p e r i m e n t a l e r r o r s u c h as wave r e f l e c t i o n and n o i s e due t o t h e c a r r i a g e movement. Two q u a n t i t i e s were measured and r e t a i n e d from t h e f o r c e r e c o r d s . One was t h e a v e r a g e of t h e semi p e a k - t o - p e a k f l u c t u a t i o n s . The a v e r a g i n g p r o c e d u r e was a c c o m p l i s h e d i n a v i s u a l manner r a t h e r t h a n d i g i t a l f o r c o n v e n i e n c e . The o t h e r r e t a i n e d q u a n t i t y was t h e maximum semi p e a k - t o - p e a k f l u c t u a t i o n f o r e a c h r u n . These f o r c e s were n o n - d i m e n s i o n a l i z e d i n t h e form %pqfta2 and p l o t t e d a g a i n s t t h e n o n - d i m e n s i o n a l i z e d c u r r e n t v e l o c i t y U/c a s shown i n F i g u r e s 16 t o 19. A l s o r e p r e s e n t e d on t h e x - a x i s of t h e s e p l o t s a r e t h e a p p r o p r i a t e v a l u e s o f t h e F r o u d e number ( F r =7 g^' )• T n e f i r s t s e t o f f i g u r e s (16 t o 17) show t h e r e s u l t s f o r t h e s m a l l e r of t h e two t e s t c y l i n d e r s , w i t h F i g u r e 16 r e p r e s e n t i n g t h e a v e r a g e semi p e a k - t o - p e a k f o r c e and F i g u r e 17 r e p r e s e n t i n g the maximum semi p e a k - t o - p e a k f o r c e . In e a c h f i g u r e t h e r e a r e t h r e e c u r v e s f o r e a c h v a l u e o f ka ( f r o m ka=0.5 t o ka=2.0) w i t h e a c h c u r v e r e p r e s e n t i n g t h e r e s u l t from a p a r t i c u l a r wave h e i g h t . The wave h e i g h t s a r e d e s i g n a t e d H1 f o r t h e s m a l l e s t t o H3 f o r 4 3 t h e l a r g e s t . The p o i n t s c o r r e s p o n d i n g t o U/c=0 (Fr=0) r e p r e s e n t t h e "no c u r r e n t " s i t u a t i o n w h i l e U/c>0 (Fr>0) r e p r e s e n t s c u r r e n t s i n t h e same d i r e c t i o n as t h e waves ( o r t h e c y l i n d e r m o t i o n o p p o s i t e t o t h e wave d i r e c t i o n ) and U/c<0 (Fr<0) r e p r e s e n t s a c u r r e n t o p p o s i t e t o t h e wave d i r e c t i o n . As e x p e c t e d , t h e c u r v e s rank t h e m s e l v e s i n o r d e r of ka, w i t h the l o n g e r waves ( s m a l l e r ka) p r o d u c i n g t h e h i g h e r t o t a l f o r c e s on t h e c y l i n d e r . T h i s r a n k i n g a p p e a r s t o be n o n - l i n e a r , as t h e d i f f e r e n c e i n f o r c e between s u c c e s s i v e l y l a r g e r v a l u e s of ka becomes l e s s and l e s s . The r e l a t i o n s h i p between t h e wave h e i g h t and t h e n o n - d i m e n s i o n a l i z e d f o r c e f o r a p a r t i c u l a r ka i s not o b v i o u s from t h e s e r e s u l t s . A l t h o u g h i n some i n s t a n c e s t h e r e a p p e a r s t o be a p a t t e r n , i t i s f a r from c o n s i s t e n t , p e r h a p s due t o t h e e x p e r i m e n t a l e r r o r l i m i t s b e i n g o f t h e same m a g n i t u d e as t h e n o n l i n e a r o r s e c o n d o r d e r wave h e i g h t e f f e c t s . T h e r e i s some c o n s i s t e n c y , however, i n t h e o b s e r v a t i o n t h a t t h e f o r c e v a l u e s a t t h e extreme n e g a t i v e v a l u e s of U/c ( t h a t i s , when t h e c u r r e n t i s o p p o s i t e t h e d i r e c t i o n of wave p r o p a g a t i o n w i t h t h e m a g n i t u d e o f U/c=-0.2) a r e g e n e r a l l y t h e minimum f o r c e s on t h e c u r v e . The a v e r a g e f o r c e v a l u e a t U/c=0 a p p e a r s t o be t h e maximum f o r t h e c u r v e s when ka=1.0, 1.5, and 2.0. The a v e r a g e f o r c e c u r v e s f o r ka=0.5 a p p e a r anomalous when compared w i t h t h o s e of l a r g e r k a. The d i s t i n g u i s h i n g f e a t u r e of t h e ka=0.5 c u r v e s i s t h e l o c a l minimum f o r c e a t U/c, w i t h l a r g e r f o r c e s a t b o t h U/c=0.1 and U/c=-0.1, and f i n a l l y t h e s m a l l e s t f o r c e s a t U/c=0.2 and -0.2. A l t h o u g h t h e maximum f o r c e c u r v e s a g r e e more o f t e n t h a n not w i t h a v e r a g e f o r c e c u r v e s h a v i n g a minimum a t 4 4 U/c=-0.2, i n g e n e r a l t h e i r s h a p e s a r e d i f f e r e n t , b e i n g more l i k e t h e a v e r a g e f o r c e c u r v e f o r ka=0.5 and r e s e m b l i n g an "M" w i t h a l o c a l minimum a t U/c=0. The r e s u l t s f o r t h e l a r g e r o f t h e two t e s t c y l i n d e r s a r e shown i n F i g u r e s 18 and 19 u s i n g t h e same f o r m a t a s t h o s e f o r t h e s m a l l e r c y l i n d e r . Once a g a i n t h e c u r v e s rank t h e m s e l v e s as e x p e c t e d a c c o r d i n g t o t h e v a l u e of k a . The d i f f e r e n c e i n t h e c u r v e s becomes q u i t e s m a l l a t t h e h i g h e r ka v a l u e s o f 2.5 and 3.0 and w i t h o u t undue s c a t t e r i n g o f p o i n t s , t h e c u r v e s o v e r l a p one a n o t h e r . In a l l c a s e s , t h e minimum f o r c e o c c u r s a t U/c=-0.2 whereas t h e maximum o c c u r s a t t i m e s a t U/c=0.0, a t o t h e r t i m e s a t U/c=+0.2, and even a c o u p l e of t i m e s a t U/c=-0.1. 7.2.2 C o m p a r i s o n W i t h P r e d i c t e d I n e r t i a F o r c e And  N u m e r i c a l S o l u t i o n S e c t i o n 3.2 d e s c r i b e d what t h e above m e n t i o n e d c u r v e s might l o o k l i k e i f t h e f l o w a r o u n d t h e c y l i n d e r had been p u r e l y i n e r t i a l . A l t h o u g h t h e measured c u r v e s do not g r e a t l y r e s e m b l e t h e p r e d i c t e d c u r v e s , t h e r e a r e some c l e a r s i m i l a r i t i e s . The p r e d i c t e d c u r v e s have p o s i t i v e s l o p e s and t h e measured c u r v e s c l e a r l y e x h i b i t t h i s t e n d e n c y w i t h t h e minimum f o r c e s g e n e r a l l y o c c u r i n g a t U/c=-0.2. The p r e d i c t e d c u r v e s a l s o i n d i c a t e a d e c r e a s e i n s l o p e as ka i n c r e a s e s . A l t h o u g h t h e s c a t t e r o f t h e p o i n t s o f t h e measured c u r v e s i s f a i r l y l a r g e , t h e r e i s a vague t e n d e n c y f o r t h i s t o be so w i t h t h e measured c u r v e s . The d a t a 45 s e t most c l o s e l y r e s e m b l i n g t h e p r e d i c t e d i n e r t i a f o r c e c u r v e s i s t h a t o f t h e maximum semi p e a k - t o - p e a k f o r c e on t h e l a r g e c y l i n d e r . In g e n e r a l , t h e p r e d i c t e d c u r v e s t e n d t o match t h e m easured c u r v e s b e t t e r a s ka i n c r e a s e s . The l a r g e s t d i f f e r e n c e between t h e p r e d i c t e d and measured c u r v e s i s t h a t t h e p r e d i c t e d c u r v e s a r e a l l l i n e a r i n n a t u r e , whereas t h e m e asured c u r v e s t e n d t o c u r v e downward somewhat, p a r t i c u l a r l y when U/c i s p o s i t i v e . To t h e same e x t e n t t h a t t h e p r e d i c t e d i n e r t i a l f o r c e s r e s e m b l e t h e measured f o r c e s , so t o o do t h e f o r c e s d e r i v e d from t h e n u m e r i c a l p r o c e d u r e d e s c r i b e d i n s e c t i o n 4.2. The n u m e r i c a l p r o c e d u r e r e s u l t e d i n a t r e n d t o w a r d l a r g e r f o r c e s where t h e c u r r e n t was i n c r e a s e d i n t h e p o s i t i v e d i r e c t i o n . T h i s r e l a t i o n s h i p was not l i n e a r as w i t h t h e p r e d i c t e d i n e r t i a l f o r c e s , however, w i t h t h e f o r c e s i n c r e a s i n g a t a g r e a t e r r a t e w i t h i n c r e a s i n g p o s i t i v e c u r r e n t s . 7.3 RUNUP The r e s u l t s o f t h e runup e x p e r i m e n t s a r e i l l u s t r a t e d i n F i g u r e s 20 t o 22. The f o r m a t i s s i m i l a r t o t h e f o r c e p l o t s e x c e p t t h e y - a x i s i s t h e d i m e n s i o n l e s s p a r a m e t e r : R _ Runup  H Incident Wave Height The f i r s t s e t o f f i g u r e s ( F i g u r e s 20 and 21) i l l u s t r a t e t h e r u n u p a t t h e p o i n t on t h e c y l i n d e r w a l l where t h e i n c i d e n t wave 46 f i r s t s t r i k e s . T h i s i s t h e p o s i t i o n o f maximum runup on t h e c y l i n d e r . E a c h c u r v e r e p r e s e n t s t h e r e s u l t s f o r a p a r t i c u l a r i n c i d e n t wave h e i g h t , a g a i n d e n o t e d H1 f o r t h e s m a l l e s t h e i g h t up t o H3 f o r t h e l a r g e s t h e i g h t . The most s a l i e n t f e a t u r e o f t h e s e p l o t s i s t h a t a l l t h e c u r v e s have a p o s i t i v e s l o p e . The d i f f e r e n t wave h e i g h t s f o r e a c h v a l u e o f ka p r o d u c e d s i m i l a r r e s u l t s w i t h a t r e n d f o r t h e l a r g e r wave h e i g h t s t o p r o d u c e h i g h e r r e l a t i v e r unup (R/H). T h e r e i s no c l e a r l y r e c o g n i z a b l e p a t t e r n r e l a t i v e t o c h a n g i n g v a l u e s of ka, however. The i n c r e a s e d r u n u p w i t h p o s i t i v e U/c i s q u i t e u n d e r s t a n d a b l e s i n c e t h e i n c o m i n g waves would s u p e r p o s e on t h e s t a n d i n g "bow wave". A l s o , t h e i n c o m i n g wave c r e s t s would be moving f a s t e r r e l a t i v e t o t h e c y l i n d e r and would have more momentum t o c a r r y t h e c r e s t h i g h e r up t h e w a l l t h a n i n t h e s t a t i o n a r y c a s e . F o r U/c l e s s t h a n z e r o , a l t h o u g h t h e s m a l l s t e r n wave t e n d e d t o c a u s e an i n c r e a s e i n ru n u p (by s u p e r p o s i t i o n ) , t h e wave c r e s t s a p p r o a c h e d t h e c y l i n d e r more s l o w l y and t h u s had l e s s e n e r g y t o r i d e up t h e w a l l of t h e c y l i n d e r . The t o t a l e f f e c t was t o d e c r e a s e t h e run u p f o r U/c<0. A n o t h e r f a c t o r w h i c h may have a f f e c t e d t h e run u p when U/c<0 was t h e o b s e r v e d e d d y - s h e d d i n g b e h i n d t h e c y l i n d e r as i t was moved t h r o u g h t h e w a t e r . The s w i r l i n g m o t i o n o f t h e e d d i e s c a u s e d t h e water p a r t i c l e v e l o c i t i e s a s s o c i a t e d w i t h t h e wave p r o p a g a t i o n t o l o s e some of t h e i r c o h e r e n c y and t h u s d e c r e a s e d t h e wave h e i g h t n e a r t h e c y l i n d e r . A l t h o u g h t h i s was not q u a n t i f i e d , i t was o b s e r v e d t h a t t h e wave h e i g h t d i d d e c r e a s e s l i g h t l y a s t h e wave t r a v e l l e d o v e r t h e wake o f t h e c y l i n d e r . 47 A few r u n u p measurements were a l s o made on t h e s i d e o f t h e c y l i n d e r d i r e c t l y o p p o s i t e t h e oncoming i n c i d e n t waves. T h i s i s a r e g i o n o f l o c a l maximium. The r e s u l t s of t h e s e measurements a r e shown i n F i g u r e 22. I t i s n o t i c e d t h a t t h e s e p l o t s e x h i b i t t h e same b a s i c c h a r a c t e r i s t i c s a s t h e p r e v i o u s ones i n w h i c h t h e runup was measured a t t h e f r o n t o f t h e c y l i n d e r , i n t h a t t h e c u r v e s have p o s i t i v e s l o p e s . T h e r e a l s o a p p e a r s t o be a r e l a t i o n s h i p amongst t h e d i f f e r e n t wave h e i g h t s whereby a t U/c=0 t h e d i m e n s i o n l e s s runup R/H i n c r e a s e s w i t h i n c r e a s i n g wave h e i g h t H. However t h i s r e l a t i o n s h i p c h a n g e s somewhat w i t h t h e p r e s e n c e o f a c u r r e n t , e s p e c i a l l y when U/c>0. A t U/c=0.2, t h e r e l a t i o n s h i p i s r e v e r s e d w i t h t h e s m a l l e r wave h e i g h t s p r o d u c i n g l a r g e r v a l u e s of R/H. T h i s i s n o t s u r p r i s i n g s i n c e , i n t h e l i m i t o f no waves, t h e r e w o u l d s t i l l be a r u n u p due t o t h e " s t e r n wave" o f t h e c y l i n d e r moving t h r o u g h t h e w a t e r . The r u n u p f o r a s t a t i o n a r y v e r t i c a l c i r c u l a r c y l i n d e r has been p r e d i c t e d by MacCamy and F u c h s (1954) u s i n g p o t e n t i a l f l o w t h e o r y . The p r e d i c t e d r e s u l t s a r e d i s p l a y e d a s l a r g e s q u a r e symbols on t h e runup p l o t s . As c a n be s e e n , t h e e x p e r i m e n t a l r u n u p r e s u l t s f o r U/c=0 a g r e e v e r y w e l l w i t h t h e p r e d i c t e d v a l u e s . 48 7.4 FLOW VISUALIZATION A s e r i e s of p h o t o g r a p h s were t a k e n t o i l l u s t r a t e c e r t a i n f e a t u r e s of t h e f l o w under v a r i o u s c o n d i t i o n s . F i g u r e 14 shows t h e i n c i d e n t waves coming from t h e l e f t and t h e c y l i n d e r moving to w a r d t h e l e f t . In t h i s s i t u a t i o n U/c i s p o s i t i v e . The f e a t u r e t o n o t i c e h e r e i s t h e p r e s e n c e of s i g n i f i c a n t l y l a r g e waves b e i n g r e f l e c t e d away from t h e c y l i n d e r i n a r a d i a l f a s h i o n . As m e n t i o n e d i n s e c t i o n 7.1, t h e a r e a d i r e c t l y i n f r o n t o f t h e c y l i n d e r r e s e m b l e s a p a r t i a l s t a n d i n g wave f i e l d , w i t h t h e c y l i n d e r moving s l o w l y t h r o u g h i t . The s i g n i f i c a n c e o f f l o w s e p a r a t i o n and t h e r e s u l t i n g d r a g f o r c e i s shown i n p l a n view i n F i g u r e 23 w h i c h has t h e i n c i d e n t waves a p p r o a c h i n g t h e c y l i n d e r from t h e t o p o f t h e p i c t u r e and t h e c y l i n d e r moving i n t h e same d i r e c t i o n t o w a r d t h e b o t t o m of t h e p i c t u r e . E d d i e s a r e c l e a r l y b e i n g s h e d i n t h e wake o f t h e moving c y l i n d e r and t h e i n c o m i n g waves must c r o s s t h i s a r e a of l o c a l t u r b u l e n c e . I t was o b s e r v e d t h a t t h e wave h e i g h t s i n t h i s r e g i o n were s l i g h t l y d e c r e a s e d . Some of t h e i n - l i n e f o r c e r e c o r d s showed t h e i n f l u e n c e of t h e eddy s h e d d i n g i n t h e form of o s c i l l a t i o n s a t t w i c e t h e eddy s h e d d i n g f r e q u e n c y , f , as e x p e c t e d f o r t h e i n - l i n e d i r e c t i o n . Thus, t h e s e r e c o r d s c o n s i s t e d o f a s u p e r p o s t i o n of t h r e e f r e q u e n c i e s , t h e o b s e r v e d wave f r e q u e n c y a t t h e c y l i n d e r , t h e m o d u l a t i n g " g r o u p wave" o r b e a t f r e q u e n c y , and t w i c e t h e eddy s h e d d i n g f r e q u e n c y . F i g u r e 24 i l l u s t r a t e s t h e o c c u r e n c e of a phenomenon t h a t o c c u r r e d q u i t e r e g u l a r l y when U/c had a l a r g e n e g a t i v e v a l u e . In t h i s c a s e t h e c y l i n d e r was t r a v e l l i n g i n t h e same d i r e c t i o n 4 9 a s t h e w a v e s . A s t h e i n c o m i n g w a v e c r e s t e n c o u n t e r e d t h e c y l i n d e r a n d m o v e d a r o u n d t o t h e s i d e , i t b e c a m e s e v e r e l y n o n l i n e a r a n d t u r n e d i n t o a b r e a k i n g w a v e i n t h e n e a r v i c i n i t y o f t h e c y l i n d e r . O n e p o s s i b l e r e a s o n f o r t h i s o c c u r e n c e i s t h a t a s t h e w a t e r p a r t i c l e s i n t h e u n d e r l y i n g c u r r e n t a c c e l e r a t e t o g e t a r o u n d t h e s i d e s o f t h e c y l i n d e r , t h e p a r t o f t h e w a v e c l o s e t o t h e c y l i n d e r w a l l " s e e s " a n i n c r e a s e d a d v e r s e c u r r e n t . T h i s s i t u a t i o n o f a w a v e f i e l d e n c o u n t e r i n g a n a d v e r s e c u r r e n t c a u s e s t h e w a v e s t o s t e e p e n s i g n i f i c a n t l y . I f t h i s e f f e c t i s l a r g e e n o u g h , t h e w a v e s w i l l b r e a k . I n t h e c a s e o f t h e c y l i n d e r , t h i s i s p u r e l y a l o c a l e f f e c t a r o u n d i t s s i d e s . 50 8. CONCLUSIONS The r e s u l t s o f t h e e x p e r i m e n t s c o n d u c t e d on a s u r f a c e -p i e r c i n g v e r t i c a l c i r c u l a r c y l i n d e r i n t h e p r e s e n c e o f a c o l i n e a r wave and c u r r e n t f i e l d i n d i c a t e t h a t t h e p r e s e n c e o f a c u r r e n t does i n d e e d i n f l u e n c e t h e o s c i l l a t o r y wave f o r c e on t h e c y l i n d e r . As w e l l , t h e r u n u p on t h e c y l i n d e r i s a p p r e c i a b l y a f f e c t e d . The e x a c t r e l a t i o n s h i p between t h e i n - l i n e o s c i l l a t o r y f o r c e s and t h e c u r r e n t m a g n i t u d e i s n o t d e f i n i t i v e l y o b v i o u s , and c a n n o t be t o t a l l y p r e d i c t e d as an i n e r t i a l f o r c e . A l t h o u g h t h e f o r c e d e c r e a s e s w i t h U/c<0 when compared t o U/c=0, th e f o r c e i n t h e r e g i m e U/c>0 does not a l w a y s i n c r e a s e a s would be e x p e c t e d when c o n s i d e r i n g t h e i n e r t i a f o r c e s o n l y . I n d e e d , t h e f o r c e o f t e n a p p e a r s t o " r o l l o f f " o r d e c r e a s e f o r l a r g e v a l u e s of p o s i t i v e U/c. The e x a c t r o l e t h a t eddy s h e d d i n g ( i n th e p r e s e n c e of a c u r r e n t ) p l a y s w i t h r e s p e c t t o t h e o s c i l l a t o r y wave f o r c e i s n o t t o t a l l y c l e a r a p a r t f r o m t h e low f r e q u e n c y ( 2 f e ) f l u c t u a t i o n s t h a t a r e s u p e r p o s e d on t h e i n c i d e n t wave f r e q u e n c y . I t i s n o t a b l e , however, t h a t when t h e i n c i d e n t wave c r o s s e s t h e t u r b u l e n t wake o f t h e c y l i n d e r , i t s h e i g h t i s s l i g h t l y d e c r e a s e d , p e r h a p s due t o a s m a l l l o s s of c o h e r e n c y i n th e d i r e c t e d e n e r g y of t h e water p a r t i c l e s i n v o l v e d i n t h e wave m o t i o n . The o t h e r n o t a b l e o b s e r v a t i o n i n t h e e x p e r i m e n t s was th e b e a t phenomenon r e s u l t i n g f r o m t h e c y l i n d e r moving s l o w l y t h r o u g h i t s own p a r t i a l s t a n d i n g wave. The runup o b s e r v a t i o n s i n d i c a t e d a more s t r a i g h t f o r w a r d l i n e a r e f f e c t due t o t h e a d v e n t o f a c u r r e n t . B a s i c a l l y , t h e runup was d i r e c t l y p r o p o r t i o n a l t o t h e c u r r e n t v e l o c i t y so t h a t 51 t h e runup d e c r e a s e d when U/c was made more n e g a t i v e and i n c r e a s e d when U/c i n c r e a s e d i n t h e p o s i t i v e s e n s e . The a p p l i c a t i o n of t h e n u m e r i c a l method, a l t h o u g h i t i s s t i l l i n a d e v e l o p m e n t a l s t a g e , p r o v e d i n s t r u c t i v e . The r e s u l t i n g t r e n d s c o r r e l a t e d r e a s o n a b l y w e l l w i t h o b s e r v a t i o n w i t h i n t h e l i m i t s of t h e i m p l i e d a s s u m p t i o n s o f t h e n u m e r i c a l p r o c e d u r e . I t i s c l e a r t h a t t h i s method h o l d s much p r o m i s e f o r s u c c e s s f u l a p p l i c a t i o n t o d i f f i c u l t a nd complex p r o b l e m s i n t h e f u t u r e . 52 BIBLIOGRAPHY 1. B a t c h e l o r , G. K. 1967. An I n t r o d u c t i o n t o F l u i d Dynamics Cambridge U n i v . P r e s s , B e n t l e y House, L o n d o n . 2. B e r g e r , E. and W i l l e , R. 1972. P e r i o d i c F l o w Phenomena. A n n u a l Review o f F l u i d M e c h a n i c s , V o l . 4. 3. B i d d e , D. D. 1970. Wave F o r c e s On a C i r c u l a r P i l e Due t o Eddy S h e d d i n g . Ph.D. t h e s i s , U n i v e r s i t y o f C a l i f o r n i a , B e r k e l e y , T e c h . R e p o r t No. HEL 9-16. 4. C h a k r a b a r t i , S. K. and Tarn, W. A. 1975. Wave H e i g h t D i s t r i b u t i o n A r o u n d V e r t i c a l C y l i n d e r . J . Waterways H a r b o u r s and C o a s t a l Eng. D i v . , ASCE, V o l . 101, No. WW2, pp. 225-230. 5. Hogben, N. 1974. Wave R e s i s t a n c e o f S u r f a c e P i e r c i n g V e r t i c a l C y l i n d e r s i n U n i f o r m C u r r e n t s . N a t i o n a l P h y s i c a l L a b o r a t o r y , S h i p D i v . , London, R e p o r t No. 183. 6. Hogben, N. and S t a n d i n g R. G. 1975. E x p e r i e n c e i n Computing Wave Loads on L a r g e B o d i e s . P r o c . O f f s h o r e T e c h . C o n f . , H o u s t o n , Paper No. OTC 2189, V o l . I I , pp. 413-431. 7. I s a a c s o n , M. de S t . Q. 1977. S h a l l o w Wave D i f f r a c t i o n A r o u n d L a r g e C y l i n d e r . J . Waterway P o r t C o a s t a l and Ocean D i v . , ASCE, V o l . 103, No. WW1, pp. 69~82. 8. I s a a c s o n , M. de S t . Q. 1981. N o n l i n e a r Wave F o r c e s on L a r g e O f f s h o r e S t r u c t u r e s . C o a s t a l / O c e a n E n g i n e e r i n g R e p o r t , D epartment o f C i v i l E n g i n e e r i n g , 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 , V a n c o u v e r , B.C. 9. K e u l e g a n , G. H. and C a r p e n t e r , L . H. 1958. F o r c e s on C y l i n d e r s and P l a t e s i n an O s c i l l a t i n g F l u i d . J . Res. Na t . B u r e a u o f S t a n d a r d s , V o l . 60, No. 5, pp. 423-430. 10. Lamb, H. 1945. H y d r o d y n a m i c s . 6 t h ed., D o v e r , New Y o r k . 11. MacCamy, R. C. and F u c h s , R. A. 1954. Wave F o r c e s on P i l e s : A D i f f r a c t i o n T h e o r y . U.S. . Army C o r p s o f E n g i n e e r s , Beach E r o s i o n B o a r d , T e c h . Memo. No. 69. 12. M o r i s o n , J . R., O ' B r i e n , M. P., J o h n s o n , J . W., and S c h a a f , S. A. 1950. The F o r c e E x e r t e d by S u r f a c e Waves on P i l e s . P e t r o l . T r a n s . , AIME, V o l . 189, pp. 149-154. 53 13. S a r p k a y a , T. and I s a a c s o n , M. de S t . Q. 1981. M e c h a n i c s of Wave F o r c e s on O f f s h o r e S t r u c t u r e s . Van N o s t r a n d R e i n h o l d , New Y o r k . 14. S t a n s b y , P. K., E l - K h a i r y , N., and B u l l o c k , G. N. 1981. F o r c e s on a C y l i n d e r i n O s c i l l a t o r y Flow w i t h a C r o s s C u r r e n t . S u b m i t t e d t o ASCE. 15. S t o k e s , G.G. 1847. On t h e T h e o r y o f O s c i l l a t o r y Waves. T r a n s . Camb. P h i l . S o c , V o l . 8, pp. 441-455. 54 55 ( a ) incident wave direction ( b ) wave speed = cc= c + U Figure 1. Wave and cylinder definition sketch a) no underlying current b) underlying current, c. . ^. EXPERIMENTAL WAVES to kd = 6.0 SHALL* WATER WAVES > W INTERMEDIATE DEPTH WAVES DEEP WATER WAVES 1 1 0 1 2 3 k d Figure 2. Experimental waves categorized by depth parameter (kd). Figure 3. Experimental waves c a t e g o r i z e d by d i f f r a c t i o n parameter and Keulegan Carpenter number. 58 0 + U/c Figure 4. I n e r t i a l force p r e d i c t i o n . 59 Sf z=-d Figure 5. Integration surfaces for numerical method. 60 t / T Figure 6. Free surface e l e v a t i o n and current v e l o c i t y e v o l u t i o n . 63 Figure 9. General view of wave basin. 64 rigid s u p p o r f test cy linder strain gauges o 1 1 ° o 1 o 1 o o protective s leeve instrumented bar Figure 10. Test cylinder and support rod. Figure 11. Gain-frequency response curve of•electrbnic f i l t e r . power supply o o o wave probe X T o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o Wheatstone Bridge test cylinder 0 OOD o v w o © e ° © • 8 ultra-violet recorder 0> low-pass filter oscilloscope Figure 12. Block diagram of datalogging system. 67 Figure 13. Moveable support carriage and datalogging electronics. Figure 14. Test cylinder moving toward wave source. group wave Figure 15. Beat phenomenon. Figure 16. Average forc e versus current v e l o c i t y f o r s m a l l c y l i n d e r . Figure 17. Maximum force versus current velocity for small cylinder. 72 73 Figure 19. Maximum force versus current velocity for large cylinder. 74 75 Figure 2 2 . Local maximum runup (rear face) on large cylinder. 77 Figure 23. Flow separation around cylinder Figure 24. Breaking wave on side of cylinder. 

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