Open Collections

UBC Theses and Dissertations

UBC Theses Logo

UBC Theses and Dissertations

Wave loads and motions of long structures in directional seas 1985

You don't seem to have a PDF reader installed, try download the pdf

Item Metadata

Download

Media
UBC_1985_A7 N86.pdf
UBC_1985_A7 N86.pdf [ 3.95MB ]
Metadata
JSON: 1.0062835.json
JSON-LD: 1.0062835+ld.json
RDF/XML (Pretty): 1.0062835.xml
RDF/JSON: 1.0062835+rdf.json
Turtle: 1.0062835+rdf-turtle.txt
N-Triples: 1.0062835+rdf-ntriples.txt
Citation
1.0062835.ris

Full Text

WAVE LOADS AND MOTIONS OF LONG STRUCTURES IN DIRECTIONAL SEAS by OKEY U. NWOGU B.A.Sc, U n i v e r s i t y of Ottawa, 1983 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF APPLIED SCIENCE i n FACULTY OF GRADUATE STUDIES Department of C i v i l E n g i n e e r i n g We accept t h i s t h e s i s as conforming to the r e q u i r e d standard THE UNIVERSITY OF BRITISH COLUMBIA J u l y 1985 © OKEY U. NWOGU, 1985 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 the requirements f o r an advanced degree at the THE UNIVERSITY OF BRITISH COLUMBIA, I agree t h a t the L i b r a r y s h a l l make i t f r e e l y a v a i l a b l e f o r r e f e r e n c e and study. I f u r t h e r agree that p e r m i s s i o n f o r e x t e n s i v e copying of t h i s t h e s i s f o r s c h o l a r l y purposes may be granted by the Head of my Department or by h i s or her r e p r e s e n t a t i v e s . I t i s understood that copying or p u b l i c a t i o n of t h i s t h e s i s f o r f i n a n c i a l gain s h a l l not be allowed without my w r i t t e n p e r m i s s i o n . Department of C i v i l E n g i n e e r i n g THE UNIVERSITY OF BRITISH COLUMBIA 2075 Wesbrook P l a c e Vancouver, Canada V6T 1W5 Date: J u l y 1985 ABSTRACT The e f f e c t s of wave d i r e c t i o n a l i t y on the l o a d s and mo t i o n s of l o n g s t r u c t u r e s i s i n v e s t i g a t e d i n t h i s t h e s i s . A n u m e r i c a l method based on Green's theorem i s d e v e l o p e d t o compute the e x c i t i n g f o r c e s and hydrodynamic c o e f f i c i e n t s due t o the i n t e r a c t i o n of a r e g u l a r o b l i q u e wave t r a i n w i t h an i n f i n i t e l y l o n g , semi-immersed f l o a t i n g c y l i n d e r of a r b i t r a r y shape. Comparisons a r e made w i t h p r e v i o u s r e s u l t s o b t a i n e d u s i n g o t h e r s o l u t i o n t e c h n i q u e s . The r e s u l t s o b t a i n e d from t h e s o l u t i o n of the o b l i q u e wave d i f f r a c t i o n problem a r e used t o dete r m i n e the t r a n s f e r f u n c t i o n s and response a m p l i t u d e o p e r a t o r s f o r a s t r u c t u r e of f i n i t e l e n g t h and hence t h e l o a d s and a m p l i t u d e s of moti o n of the s t r u c t u r e i n s h o r t - c r e s t e d s e a s . The wave l o a d s and body motions i n s h o r t - c r e s t e d seas a r e compared t o c o r r e s p o n d i n g r e s u l t s f o r l o n g - c r e s t e d s e a s . T h i s i s e x p r e s s e d as a d i r e c t i o n a l l y a v e r a g e d , f r e q u e n c y dependent r e d u c t i o n f a c t o r f o r the wave l o a d s and a response r a t i o f o r the body m o t i o n s . N u m e r i c a l r e s u l t s a r e p r e s e n t e d f o r the f o r c e r e d u c t i o n f a c t o r and response r a t i o of a l o n g f l o a t i n g box s u b j e c t t o a d i r e c t i o n a l wave spectrum w i t h a c o s i n e power t y p e energy s p r e a d i n g f u n c t i o n . A p p l i c a t i o n s of th e r e s u l t s of t h e p r e s e n t p r o c e d u r e i n c l u d e such l o n g s t r u c t u r e s as f l o a t i n g b r i d g e s and b r e a k w a t e r s . i i Table of Contents ABSTRACT . i i LIST OF TABLES v LIST OF FIGURES . . v i NOMENCLATURE v i i i ACKNOWLEDGEMENTS x i i 1 . INTRODUCTION 1 1 . 1 GENERAL 1 1.2 LITERATURE SURVEY 3 1.2.1 DIFFRACTION THEORY 3 1.2.2 EFFECTS OF DIRECTIONAL WAVES 5 1.3 DESCRIPTION OF METHOD 8 2. DIFFRACTION THEORY 11 2.1 INTRODUCTION 11 2.2 THEORETICAL FORMULATION 13 2.2.1 WAVE DIFFRACTION PROBLEM 13 2.2.2 FORCED MOTION PROBLEM 17 2.3 GREEN'S FUNCTION SOLUTION 19 2.4 EXCITING FORCES, ADDED MASSES AND DAMPING COEFFICIENTS 21 2.5 EQUATIONS OF MOTION 25 2.6 REFLECTION AND TRANSMISSION COEFFICIENTS 28 2.7 NUMERICAL PROCEDURE 30 2.8 EFFECT OF FINITE STRUCTURE LENGTH 35 3. EFFECTS OF DIRECTIONAL WAVES 39 3.1 REPRESENTATION OF DIRECTIONAL SEAS 39 3.2 RESPONSE TO DIRECTIONAL WAVES 44 4. RESULTS AND DISCUSSION 48 i i i 4.1 EXCITING FORCES, ADDED MASS AND DAMPING COEFFICIENTS 48 4.2 MOTIONS OF AN UNRESTRAINED BODY 52 4.3 EFFECTS OF DIRECTIONAL WAVES 53 5. CONCLUSIONS AND RECOMMENDATIONS 57 5.1 CONCLUSIONS 57 5.2 RECOMMENDATIONS FOR FURTHER STUDY 59 BIBLIOGRAPHY 61 APPENDIX I .65 i v LIST OF TABLES Table page 1. Comparison of the sway added mass and damping c o e f f i c i e n t s of a s e m i - c i r c u l a r c y l i n d e r (d/a=») o b t a i n e d i n the present study with the r e s u l t s of GAR ( G a r r i s o n , 1 984) 68 2. Comparison of the heave added mass and damping c o e f f i c i e n t s of a s e m i - c i r c u l a r c y l i n d e r (d/a=°°) ob t a i n e d i n the present study with the r e s u l t s of B&U ( B o l t o n and U r s e l l , 1 973) 69 3. Comparison of the sway e x c i t i n g f o r c e c o e f f i c i e n t and wave amplitude r a t i o of a s e m i - c i r c u l a r c y l i n d e r (d/a=°°) o b t a i n e d i n the prese n t study with the r e s u l t s of GAR ( G a r r i s o n , 1 984) 70 4. Comparison of the heave e x c i t i n g f o r c e c o e f f i c i e n t and wave amplitude r a t i o of a s e m i - c i r c u l a r c y l i n d e r (d/a=») o b t a i n e d i n the prese n t study with the r e s u l t s of B&U (Bolton and U r s e l l , 1973) 71 v LIST OF FIGURES F i g u r e page 1. D e f i n i t i o n sketch f o r a r e c t a n g u l a r c y l i n d e r 72 2. D e f i n i t i o n sketch f o r f l o a t i n g c y l i n d e r showing component motions . 73 3. Sketch of c l o s e d s u r f a c e 73 4. Sketch showing r e l a t i o n s h i p between x, £, and 74 5. A t y p i c a l boundary element mesh f o r a r e c t a n g u l a r c y l i n d e r (b/a=1 ,d/a=2) 74 6. Square of r e d u c t i o n f a c t o r r f o r d i f f e r e n t v a l u e s of 0 75 7. Sketch of a d i r e c t i o n a l wave spectrum 75 8. D i r e c t i o n a l spreading f u n c t i o n f o r d i f f e r e n t v a l u e s of the parameter s.. 76 9. Sway e x c i t i n g f o r c e c o e f f i c i e n t f o r a r e c t a n g u l a r c y l i n d e r (b/a=1,d/a=2) 76 10. Heave e x c i t i n g f o r c e c o e f f i c i e n t f o r a r e c t a n g u l a r c y l i n d e r (b/a=1 ,d/a=2) 77 11. R o l l e x c i t i n g moment c o e f f i c i e n t f o r a r e c t a n g u l a r c y l i n d e r (b/a=1,d/a=2) 77 12. R e f l e c t i o n c o e f f i c i e n t f o r a r e c t a n g u l a r c y l i n d e r (b/a=1 ,d/a=2) ....78 13. Sway e x c i t i n g f o r c e c o e f f i c i e n t f o r a r e c t a n g u l a r c y l i n d e r (b/a=0.265,d/a=») 78 14. Heave e x c i t i n g force, c o e f f i c i e n t f o r a r e c t a n g u l a r c y l i n d e r (b/a=0 . 265 ,d/a=») 79 15. R o l l e x c i t i n g moment c o e f f i c i e n t f o r a r e c t a n g u l a r c y l i n d e r (b/a=0 .265,d/a==>) 79 16. Sway added mass c o e f f i c i e n t f o r a r e c t a n g u l a r c y l i n d e r (b/a=0.265,d/a=») 80 17. Sway damping c o e f f i c i e n t f o r a r e c t a n g u l a r c y l i n d e r (b/a=0.265,d/a=») 80 18. Heave added mass c o e f f i c i e n t f o r a r e c t a n g u l a r c y l i n d e r (b/a=0.265,d/a=») 81 v i 19. Heave damping c o e f f i c i e n t f o r a r e c t a n g u l a r c y l i n d e r (b/a=0.265,d/a=») 81 20. R o l l added mass c o e f f i c i e n t f o r a r e c t a n g u l a r c y l i n d e r (b/a=0.265,d/a=«) 82 21. R o l l damping c o e f f i c i e n t f o r a r e c t a n g u l a r c y l i n d e r (b/a=0.265,d/a=») 82 22. Sway response amplitude operator f o r a long f l o a t i n g box (a=7.5m,b=3m,l=75m,d=12m) 83 23. Heave response amplitude operator f o r a long f l o a t i n g box (a=7 . 5m, b=3m, l = 75m,d= 1 2m) 83 24. R o l l response amplitude operator f o r a long f l o a t i n g box (a=7.5m,b=3m,l=75m,d=l2m) 84 25. Force and moment r e d u c t i o n f a c t o r s f o r a long f l o a t i n g box (a=7 . 5m, b=3m, l=75m,d= 1 2m) 84 26. Force r e d u c t i o n f a c t o r s f o r a long f l o a t i n g box (a=7.5m,b=3m,l=75m,d=l2m) i n normal and o b l i q u e mean seas 86 27. Response r a t i o s f o r a long f l o a t i n g box (a=7.5m,b=3m,l = 75m,d=1 2m) 87 v i i NOMENCLATURE a = h a l f beam of c y l i n d e r a. . = matrix c o e f f i c i e n t A = d i s p l a c e d volume per u n i t l e n g t h A 0 = complex amplitude of v e l o c i t y p o t e n t i a l A^j = complex wave amplitude b = d r a f t of c y l i n d e r b. . = matrix c o e f f i c i e n t 13 B = beam of c y l i n d e r c ^ j = h y d r o s t a t i c s t i f f n e s s matrix c o e f f i c i e n t Cj = e x c i t i n g f o r c e c o e f f i c i e n t C ( s ) , C ' ( s ) = n o r m a l i z i n g c o e f f i c i e n t s f o r d i r e c t i o n a l s p reading f u n c t i o n s d = water depth f = c i r c u l a r frequency f ^ k * = c o e f f i c i e n t d e f i n e d i n eqn. (2.92) F j = e x c i t i n g f o r c e F.. = f o r c e i n the i t h d i r e c t i o n due to the j t h ^ mode of motion of c y l i n d e r g = g r a v i t a t i o n a l a c c e l e r a t i o n G(fa>,0) = d i r e c t i o n a l spreading f u n c t i o n G(x;£) = Green's f u n c t i o n ; \ H = i n c i d e n t wave he i g h t Hj = system response f u n c t i o n i = v/(-D I 0 = p o l a r mass moment of i n e r t i a about the y a x i s per u n i t l e n g t h k = i n c i d e n t wavenumber K = Keulegan-Carpenter number v i i i K D,K_ = r e f l e c t i o n and t r a n s m i s s i o n c o e f f i c i e n t s H 1 K 0,K, = m o d i f i e d B e s s e l f u n c t i o n s of o r d e r s z e r o and one 1* = l e n g t h of s t r u c t u r e L = i n c i d e n t wavelength m = mass per u n i t l e n g t h of c y l i n d e r m^j = mass matrix c o e f f i c i e n t N number of segments on S B+Sp+S R n = u n i t normal v e c t o r d i r e c t e d out of f l u i d r e g i o n n . n = d i r e c t i o n c o s i n e s of n p = pressure q(kl,/3) = f a c t o r d e f i n e d i n eqn. (2.103) r = d i s t a n c e between x and J[ Ty = r a d i u s of g y r a t i o n of c y l i n d e r about the y a x i s r(kl,/3) = r e d u c t i o n f a c t o r r' = d i s t a n c e between x and Rp = f o r c e r e d u c t i o n f a c t o r Rj^ = response r a t i o s = c o s i n e power of spreading f u n c t i o n S ( C J ) = s p e c t r a l energy d e n s i t y S(co,/3) = d i r e c t i o n a l wave spectrum S D = immersed body s u r f a c e S D = seabed S F = f r e e s u r f a c e S _ = r a d i a t i o n s u r f a c e S N = waterplane area moment of i n e r t i a about the x a x i s per u n i t l e n g t h t = time ix T = wave p e r i o d u = f l u i d v e l o c i t y v e c t o r U = wind speed U"m = maximum p a r t i c l e v e l o c i t y V = d i s p l a c e d volume of c y l i n d e r V = normal v e l o c i t y of body x = h o r i z o n t a l c o o r d i n a t e normal to c y l i n d e r a x i s x = v e c t o r of point (x,z) x^ = c e n t r o i d of the waterplane l i n e measured from the c e n t r e of g r a v i t y X R = x c o o r d i n a t e of the r a d i a t i o n s u r f a c e y = h o r i z o n t a l c o o r d i n a t e p a r a l l e l to c y l i n d e r a x i s z = v e r t i c a l c o o r d i n a t e measured upwards from the s t i l l water l e v e l z B = z c o o r d i n a t e of the c e n t r e of buoyancy Z Q = Z c o o r d i n a t e of the ce n t r e of g r a v i t y = response amplitude operator (5 = angle of in c i d e n c e measured from the p o s i t i v e x a x i s P0 = p r i n c i p a l d i r e c t i o n of wave propaga t i o n 77 = water s u r f a c e e l e v a t i o n measured from the s t i l l water l e v e l 77^ = asymptotic wave amplitude 7 j R , 7 j T = r e f l e c t e d and t r a n s m i t t e d wave amplitudes 5.. = Kronecker d e l t a f u n c t i o n 13 A = phase angle 7 = angle between x-jj_ and n; a l s o E u l e r ' s c o n s t a n t 7' = angle between x-£' and n' X^j = damping c o e f f i c i e n t x u = nondimensional frequency parameter (see eqn. 2.14) u- • = added mass c o e f f i c i e n t 1D v = nondimensional frequency parameter (see eqn. 2.14) 6 = angle of i n c i d e n c e measured from p r i n c i p a l wave d i r e c t i o n p = d e n s i t y of f l u i d 4> = v e l o c i t y p o t e n t i a l 0^ = complex v e l o c i t y p o t e n t i a l s cj = wave angular frequency £ = v e c t o r of p o i n t (£,$) on f l u i d boundary £' = v e c t o r of p o i n t U,-($+2d)] = nondimensional amplitude of body motion Hj = displacement or r o t a t i o n of body 5^ = complex wave amplitude r a t i o x i ACKNOWLEDGEMENTS The author wishes t o express h i s immense g r a t i t u d e to Dr. M i c h a e l de S t . Q. Isaacson f o r h i s guidance and ad v i c e throughout the p r e p a r a t i o n of t h i s t h e s i s . F i n a n c i a l support i n the form of a r e s e a r c h a s s i s t a n t s h i p from the N a t u r a l S c i e n c e s and E n g i n e e r i n g Research C o u n c i l of Canada i s g r a t e f u l l y acknowledged. x i i 1. INTRODUCTION 1.1 GENERAL W i t h t h e g r o w t h i n t h e d e v e l o p m e n t o f o f f s h o r e r e s o u r c e s , t h e r e h as been a n e e d f o r t h e s a f e a n d ec o n o m i c d e s i g n o f 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 . An i m p o r t a n t a s p e c t i n t h e d e s i g n o f t h e s e s t r u c t u r e s i n v o l v e s t h e d e t e r m i n a t i o n o f b o t h t h e e x c i t i n g f o r c e s due t o wave i n t e r a c t i o n w i t h a f i x e d body and t h e r e s p o n s e o f t h e s t r u c t u r e . The s t r u c t u r e s h o u l d be d e s i g n e d n o t o n l y t o w i t h s t a n d t h e t h e l o a d s f r o m t h e complex o c e a n e n v i r o n m e n t , b u t i n a d d i t i o n i t s m o t i o n s g e n e r a l l y have t o be w i t h i n a c c e p t a b l e l i m i t s . The t r a d i t i o n a l a p p r o a c h t o t h e d e s i g n o f o f f s h o r e s t r u c t u r e s o f t e n assumes t h e i n c i d e n t wave f i e l d t o be u n i d i r e c t i o n a l o r l o n g - c r e s t e d . R e a l s e a s a r e , however, b o t h random a n d m u l t i - d i r e c t i o n a l , i . e . t h e waves n o t o n l y have d i f f e r e n t a m p l i t u d e s and f r e q u e n c i e s but a l s o may a p p r o a c h a s t r u c t u r e f r o m d i f f e r e n t d i r e c t i o n s . T h i s p r o p e r t y i s a l s o s ometimes r e f e r r e d t o a s wave s h o r t - c r e s t e d n e s s . The d i r e c t i o n a l i t y o f t h e waves c a n s i g n i f i c a n t l y i n f l u e n c e t h e l o a d s and m o t i o n s e x p e r i e n c e d by t h e s t r u c t u r e . The use o f d i r e c t i o n a l s p e c t r a i n wave f o r c e c a l c u l a t i o n s o f t e n l e a d s t o a r e d u c t i o n i n t h e computed f o r c e s , compared t o t h e c a s e o f l o n g - c r e s t e d waves. T h i s c o u l d l e a d t o s i g n i f i c a n t s a v i n g s i n c o n s t r u c t i o n c o s t s . I t c o u l d a l s o a f f e c t d e c i s i o n s a s t o whether d e s i g n s a r e a c c e p t e d o r r e j e c t e d i n f e a s i b i l i t y s t u d i e s . W i t h t h e r e c e n t 1 2 developments i n methods of determining d i r e c t i o n a l wave s p e c t r a (Borgman (1969), Mitsuyasu et al (1975), Leblanc and Middleton(1982)) and the b u i l d i n g of l a b o r a t o r y wave basins capable of g e n e r a t i n g d i r e c t i o n a l waves, the use of d i r e c t i o n a l s p e c t r a models i s soon becoming an e s t a b l i s h e d p a r t of the o f f s h o r e design p r o c e s s . When a wave t r a i n i s i n c i d e n t upon an i n f i n i t e l y long semi-immersed s t r u c t u r e , the s t r u c t u r e responds i n three degrees of freedom : heave ( v e r t i c a l motion), sway (beamwise motion), and r o l l (angular motion about the l o n g i t u d i n a l a x i s ) . There are not only e x c i t i n g f o r c e s due to the presence of the waves but a l s o hydrodynamic f o r c e s a s s o c i a t e d with the response of the s t r u c t u r e . For slender s t r u c t u r e s , the presence of the body does not s i g n i f i c a n t l y a f f e c t the i n c i d e n t wave kinematics and Morison's equation (Morison et al ,1950) i s o f t e n used to estimate the e x c i t i n g f o r c e s . I f the s t r u c t u r e i s l a r g e enough to d i f f r a c t the i n c i d e n t wave f i e l d , flow s e p a r a t i o n e f f e c t s are o f t e n n e g l e c t e d and the problem i s s o l v e d u s i n g p o t e n t i a l flow theory (Kellogg,1929). The complete problem i s n o n l i n e a r and i s u s u a l l y l i n e a r i z e d by assuming a s m a l l amplitude wave t r a i n . A numerical method based on Green's theorem i s used i n t h i s t h e s i s to s o l v e f o r the e x c i t i n g f o r c e s and hydrodynamic c o e f f i c i e n t s of an i n f i n i t e semi-immersed c y l i n d e r of a r b i t r a r y shape i n o b l i q u e seas. The r e s u l t s are f i r s t extended to s t r u c t u r e s of f i n i t e l e n g t h and then to 3 d i r e c t i o n a l seas using the transfer function approach. The wave loads and motions of the structure in d i r e c t i o n a l seas are compared with those of long-crested waves. The applications of the re s u l t s of t h i s thesis include such long structures as f l o a t i n g breakwaters, f l o a t i n g bridges and pipelines. It could also be used in the study of ship motions where Korvin-Kroukovsky's (1955) s t r i p theory i s often used to reduce the three-dimensional problem to a two-dimensional one. 1 . 2 L I T E R A T U R E S U R V E Y 1 . 2 . 1 D I F F R A C T I O N T H E O R Y A number of authors ( U r s e l l (1949), MacCamy (1964), Kim (1965), Bai (1972), Ijima et al (1976)) have treated the two-dimensional wave-structure interaction problem. Much less work has however been reported for the case of obliquely incident waves. Previous studies of oblique wave-structure interaction include those conducted by Black and Mei (1970), Bai (1975), Leonard et al (1983) for f i n i t e water depth, and by Garrison (1969), Bolton and U r s e l l (1973), and Garrison (1984) for i n f i n i t e depth. Garrison (1969) used a Green's function procedure to compute the ex c i t i n g forces, added mass and damping c o e f f i c i e n t s , and r e f l e c t i o n and transmission c o e f f i c i e n t s for a shallow draft cylinder f l o a t i n g at 4 the f r e e s u r f a c e . The method i n v o l v e s e x p r e s s i n g the p o t e n t i a l at any p o i n t i n the f l u i d r e g i o n i n terms of a continuous d i s t r i b u t i o n of sources a l o n g the body s u r f a c e . The Green's f u n c t i o n r e p r e s e n t s a p o i n t source of u n i t s t r e n g t h . The boundary c o n d i t i o n on the body s u r f a c e r e s u l t s i n an i n t e g r a l equation which can be s o l v e d n u m e r i c a l l y to o b t a i n the source s t r e n g t h s and hence the v e l o c i t y p o t e n t i a l . G a r r i s o n (1984) extended t h i s approach t o c y l i n d e r s of a r b i t r a r y shape. Bolton and U r s e l l (1973) used a m u l t i p o l e method to s o l v e the problem a s s o c i a t e d with a c i r c u l a r c y l i n d e r o s c i l l a t i n g i n heave with the amplitude of motion v a r y i n g s i n u s o i d a l l y along the l e n g t h of the c y l i n d e r . The Haskind r e l a t i o n s were then used t o r e l a t e t h i s r a d i a t i o n problem to the wave d i f f r a c t i o n problem. Black and Mei (1970) used a v a r i a t i o n a l technique based on Schwinger's v a r i a t i o n a l p r i n c i p l e t o o b t a i n the f a r f i e l d s o l u t i o n of the problem. Bai (1975) a l s o used a v a r i a t i o n a l technique to s o l v e f o r the e x c i t i n g f o r c e s and r e f l e c t i o n and t r a n s m i s s i o n c o e f f i c i e n t s i n water of f i n i t e depth. The method i n v o l v e s e x p r e s s i n g the governing d i f f e r e n t i a l e q u a t i o n as the minimum of some f u n c t i o n a l . The f l u i d domain i s d i v i d e d i n t o subregions and a set of i n t e r p o l a t i o n f u n c t i o n s with nodal v a r i a b l e s i s used to d e f i n e the v e l o c i t y p o t e n t i a l over the domain. M i n i m i s i n g the f u n c t i o n a l with r e s p e c t to the nodal v a r i a b l e s y i e l d s a set of l i n e a r equations 5 w h i c h c a n be s o l v e d t o g i v e t h e p o t e n t i a l f i e l d . The v a r i a t i o n a l a p p r o a c h l e a d s t o a s y s t e m o f e q u a t i o n s much l a r g e r t h a n t h a t o f t h e i n t e g r a l e q u a t i o n method. The m a t r i x i s however s y m m e t r i c and banded and can be s o l v e d u s i n g e f f i c i e n t t e c h n i q u e s . L e o n a r d et al (1983) u s e d an a p p r o a c h s i m i l a r t o t h a t o f B a i (1975) i n s t u d y i n g t h e c a s e of m u l t i p l e c y l i n d e r s . 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 G r e e n ' s s e c o n d i d e n t i t y i s u s e d i n t h i s t h e s i s t o s o l v e t h e wave d i f f r a c t i o n p r o b l e m . The a p p r o a c h has p r e v i o u s l y been u s e d by I j i m a et al (1976) and F i n n i g a n and Yammamoto (1979) f o r t w o - d i m e n s i o n a l wave p r o b l e m s and by I s a a c s o n (1981) f o r n o n l i n e a r w a v e - s t r u c t u r e i n t e r a c t i o n . The p r e s e n t method a v o i d s t h e c o m p l e x i t y o f d e r i v i n g a G r e e n ' s f u n c t i o n w h i c h h a s t o s a t i s f y 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 i n w a t e r o f f i n i t e d e p t h . The r e s u l t s o f t h e p r e s e n t p r o c e d u r e a r e compared w i t h t h o s e o f B a i (1975) f o r f i n i t e w a t e r d e p t h , a s w e l l as B o l t o n and U r s e l l (1973) and G a r r i s o n (1984) f o r i n f i n i t e w a t e r d e p t h . 1 . 2 . 2 EFFECTS OF DIRECTIONAL WAVES P r e v i o u s s t u d i e s o f t h e l o a d i n g and r e s p o n s e o f s t r u c t u r e s i n d i r e c t i o n a l s e a s a r e few and w i d e l y s c a t t e r e d i n t h e l i t e r a t u r e . T h e r e have been two g e n e r a l a p p r o a c h e s u s e d t o d e t e r m i n e t h e r e s p o n s e o f s t r u c t u r e s i n s h o r t - c r e s t e d 6 seas. The more common approach i s the frequency domain approach where l i n e a r t h e o r i e s are used to determine t r a n s f e r f u n c t i o n s which r e l a t e the i n c i d e n t wave s p e c t r a to the response s p e c t r a . Time domain s i m u l a t i o n s are o f t e n used when the wave-structure i n t e r a c t i o n process i s of a n o n l i n e a r nature. Time domain d e s c r i p t i o n of d i r e c t i o n a l seas i n v o l v e e i t h e r the d i g i t a l f i l t e r i n g of white n o i s e or Fast F o u r i e r Transform (FFT) tec h n i q u e s . The time domain a n a l y s i s i s however g e n e r a l l y more expensive than the frequency domain approach. Huntington and Thompson (1976) computed the wave loads on a l a r g e v e r t i c a l c y l i n d e r i n s h o r t - c r e s t e d seas. L i n e a r d i f f r a c t i o n theory was used to determine the t r a n s f e r f u n c t i o n s . The t h e o r e t i c a l r e s u l t s were found to be i n good agreement with experimental measurements. Dean (1977) proposed a h y b r i d method of computing the wave loads on o f f s h o r e s t r u c t u r e s which i n c o r p o r a t e s both the n o n l i n e a r i t y and d i r e c t i o n a l i t y of the waves. A l i n e a r i z e d form of Morison's equation was used to determine . the e f f e c t of d i r e c t i o n a l waves. Force r e d u c t i o n f a c t o r s were presented f o r the c o s i n e power spreading f u n c t i o n . B a t t j e s (1982) s t u d i e d the e f f e c t s of d i r e c t i o n a l waves on the loads on a long s t r u c t u r e . Reduction f a c t o r s were presented f o r a v e r t i c a l w a l l occupying the 7 e n t i r e water depth and a p i p e l i n e f o r the c o s i n e power type d i r e c t i o n a l spreading f u n c t i o n . D a l l i n g a et al (1984) i n v e s t i g a t e d the e f f e c t s of d i r e c t i o n a l s preading on the loads and motions of a barge used f o r the t r a n s p o r t of a jackup p l a t f o r m . L i n e a r d i f f r a c t i o n theory was used t o o b t a i n the t r a n s f e r f u n c t i o n s . Bryden and Greated (1984) and Lambrakos (1982) both s t u d i e d the response of long s l e n d e r f l e x i b l e h o r i z o n t a l c y l i n d e r s i n d i r e c t i o n a l seas. Lambrakos (1982) used a f i n i t e number of wave f r e q u e n c i e s and d i r e c t i o n s to d e s c r i b e the sea s u r f a c e . The wave loads were determined from Mori son's equation and the response of the s t r u c t u r e was o b t a i n e d by s o l v i n g the d i f f e r e n t i a l e quation of motion using a f i n i t e d i f f e r e n c e scheme. Hackley (1979) and Shinozuka et al (1979) used a time domain approach to s i m u l a t e the l o a d i n g and response of s t r u c t u r e s i n s h o r t - c r e s t e d s e a s . The Fast F o u r i e r Transform technique was used to determine the water p a r t i c l e v e l o c i t i e s and a c c e l e r a t i o n s f o r use i n Morison's e q u a t i o n . Shinozuka et al (1979) found a r e d u c t i o n i n the i n l i n e response i n s h o r t - c r e s t e d seas compared to l o n g - c r e s t e d seas. There was a l s o a s i g n i f i c a n t t r a n s v e r s e response. G e o r g i a d i s (1984) used a Monte C a r l o s i m u l a t i o n to determine the a p p r o p r i a t e nodal f o r c e s on s t r u c t u r e s i n s h o r t - c r e s t e d seas. The response of the s t r u c t u r e was 8 t h e n e v a l u a t e d u s i n g a d e t e r m i n i s t i c a n a l y s i s . 1.3 DESCRIPTION OF METHOD The a n a l y s i s of the dynamic response of l o n g s t r u c t u r e s i n d i r e c t i o n a l seas can be d i v i d e d i n t o two p a r t s . The f i r s t p a r t i n v o l v e s s o l v i n g t he problem of the d i f f r a c t i o n of a r e g u l a r o b l i q u e wave t r a i n by an i n f i n i t e semi-immersed c y l i n d e r . An i n t e g r a l e q u a t i o n method based on Green's second i d e n t i t y i s used t o compute the e x c i t i n g f o r c e s and hydrodynamic c o e f f i c i e n t s . The f l u i d motion i s d e s c r i b e d i n terms of a v e l o c i t y p o t e n t i a l which c o n s i s t s of components due t o t h e i n c i d e n t wave, d i f f r a c t e d wave, and f o r c e d waves f o r each mode of motion of the c y l i n d e r . Green's second i d e n t i t y i s used t o r e l a t e t he v a l u e s of the unknown v e l o c i t y p o t e n t i a l s and t h e i r normal d e r i v a t i v e s on a boundary t o the Green's f u n c t i o n and i t s normal d e r i v a t i v e s . The boundary c o n s i s t s of t he immersed body s u r f a c e , f r e e s u r f a c e and r a d i a t i o n s u r f a c e . The Green's f u n c t i o n o n l y has t o s a t i s f y the g o v e r n i n g d i f f e r e n t i a l e q u a t i o n which i s t h e t w o - d i m e n s i o n a l m o d i f i e d H e l m h o l t z e q u a t i o n . The boundary i s d i v i d e d i n t o a f i n i t e number of segments. A p p l i c a t i o n of the v a r i o u s boundary c o n d i t i o n s on the v a r i o u s s u r f a c e s y i e l d s a s e t of a l g e b r a i c e q u a t i o n s which can be s o l v e d t o o b t a i n t he v e l o c i t y p o t e n t i a l s . B e r n o u l l i ' s e q u a t i o n i s then used t o compute the p r e s s u r e s and hence the e x c i t i n g f o r c e s and hydrodynamic 9 f o r c e s due to the motions of the c y l i n d e r . The hydrodynamic f o r c e s can be expressed i n terms of components i n phase with the body a c c e l e r a t i o n and v e l o c i t y . These are r e f e r r e d to as the added mass and damping c o e f f i c i e n t s r e s p e c t i v e l y . The r e f l e c t i o n and t r a n s m i s s i o n c o e f f i c i e n t s are determined by e v a l u a t i n g the asymptotic wave amplitudes at the r a d i a t i o n s u r f a c e . B e r n o u l l i ' s equation i s used to r e l a t e the water s u r f a c e e l e v a t i o n t o the v e l o c i t y p o t e n t i a l with the pr e s s u r e set to zero at the f r e e s u r f a c e . The added mass and damping c o e f f i c i e n t s are then combined with the mass or moment of i n e r t i a of the body and the h y d r o s t a t i c s t i f f n e s s c o e f f i c i e n t s to o b t a i n three coupled l i n e a r equations of motion f o r the body. The equations of motion are then s o l v e d to o b t a i n the amplitudes of body motion per u n i t wave amplitude o f t e n r e f e r r e d to as the response amplitude o p e r a t o r . For a r i g i d s t r u c t u r e of f i n i t e l e n g t h , the two-dimensional f o r c e s are i n t e g r a t e d along the body a x i s to o b t a i n the t o t a l wave loads on the s t r u c t u r e . The second p a r t of the a n a l y s i s i n v o l v e s extending the r e s u l t s f o r a r e g u l a r o b l i q u e wave t r a i n to random m u l t i - d i r e c t i o n a l seas using the l i n e a r t r a n s f e r f u n c t i o n approach. The s h o r t - c r e s t e d sea s u r f a c e i s d e s c r i b e d i n terms of a d i r e c t i o n a l wave spectrum. The d i r e c t i o n a l wave spectrum can be expressed as the product of the c o n v e n t i o n a l one-dimensional frequency spectrum and a d i r e c t i o n a l s p r e a d i n g f u n c t i o n . A c o s i n e power spreading f u n c t i o n which 1 0 i s independent of frequency i s used i n t h i s study. The e x c i t i n g f o r c e and body response s p e c t r a are ob t a i n e d by m u l t i p l y i n g the i n c i d e n t wave spectrum with the a p p r o p r i a t e t r a n s f e r f u n c t i o n or response amplitude o p e r a t o r . The e f f e c t s of wave d i r e c t i o n a l i t y i s expressed as a d i r e c t i o n a l l y averaged, frequency dependent r e d u c t i o n f a c t o r to be a p p l i e d t o the one-dimensional f o r c e spectrum. The mean square v a l u e s of the response i n s h o r t - c r e s t e d seas are a l s o compared to co r r e s p o n d i n g r e s u l t s f o r l o n g - c r e s t e d seas. 2 . D I F F R A C T I O N T H E O R Y 2 . 1 I N T R O D U C T I O N B e f o r e t r e a t i n g t h e p r o b l e m of t h e dynamic r e s p o n s e o f l o n g s t r u c t u r e s i n m u l t i - d i r e c t i o n a l s e a s , we s h a l l f i r s t c o n s i d e r t h e i n t e r a c t i o n o f a r e g u l a r o b l i q u e wave t r a i n w i t h an i n f i n i t e s emi-immersed h o r i z o n t a l c y l i n d e r of a r b i t r a r y s h a p e . The c y l i n d e r i s c o n s i d e r e d l a r g e enough so a s t o d i f f r a c t t h e i n c i d e n t f l o w f i e l d . Flow s e p a r a t i o n e f f e c t s a r e assumed n e g l i g i b l e and t h e e f f e c t s of v i s c o s i t y a r e assumed c o n f i n e d t o a t h i n b o u n d a r y l a y e r on t h e body s u r f a c e . The f l u i d f l o w c a n t h u s be c o n s i d e r e d t o be i r r o t a t i o n a l and t h e p r o b l e m 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 . An i n d i c a t i o n o f t h e i m p o r t a n c e of f l o w s e p a r a t i o n e f f e c t s i s t h e K e u l e g a n - C a r p e n t e r number, R. The K e u l e g a n - C a r p e n t e r number i s d e f i n e d as t h e r a t i o o f t h e a m p l i t u d e o f f l u i d m o t i o n t o a t y p i c a l d i m e n s i o n o f t h e body, t h a t i s K = U mT/B (2.1) where U m i s t h e maximum p a r t i c l e v e l o c i t y , T i s t h e wave p e r i o d and B i s a t y p i c a l d i m e n s i o n of t h e body. F o r t h e r a n g e o f f r e q u e n c i e s u s e d i n t h i s s t u d y , K w i l l u s u a l l y be l e s s t h a n two and f l o w s e p a r a t i o n s h o u l d n o t o c c u r ( s e e S a r p k a y a and I s a a c s o n , 1 9 8 1 ) . F o r r e c t a n g u l a r s e c t i o n c y l i n d e r s w h i c h a r e u s e d i n t h i s s t u d y , v o r t i c e s a r e u s u a l l y formed a t t h e s h a r p 11 12 c o r n e r s . V a r i o u s authors (Bearman et al (1979), Mogridge and Jamieson (1976)) have however found good agreement between p o t e n t i a l flow theory and experimental r e s u l t s f o r such c y l i n d e r s when f i x e d d e s p i t e the formation of the v o r t i c e s . For f l o a t i n g c y l i n d e r s , the r o l l amplitude of motion i s s i g n i f i c a n t l y a f f e c t e d by v i s c o u s damping p a r t i c u l a r l y near the resonance frequency and an e m p i r i c a l v i s c o u s damping c o e f f i c i e n t should be i n c l u d e d i n the equations of motion. I t i s u s u a l l y convenient to separate the wave-structure i n t e r a c t i o n problem f o r f l o a t i n g bodies i n t o two p a r t s : (1) e x c i t i n g f o r c e s due to wave d i f f r a c t i o n by a f i x e d c y l i n d e r , and (2) hydrodynamic f o r c e s a s s o c i a t e d with an i n f i n i t e c y l i n d e r o s c i l l a t i n g i n heave, sway and r o l l i n an otherwise s t i l l water expressed i n terms of added mass and damping c o e f f i c i e n t s . The wave h e i g h t and o s c i l l a t o r y motions of the c y l i n d e r are assumed small so that the complete problem of wave i n t e r a c t i o n with a f l o a t i n g c y l i n d e r can be r e p r e s e n t e d by a l i n e a r s u p e r p o s i t i o n of the d i f f r a c t i o n and f o r c e d motion problems. The c y l i n d e r i s assumed f l e x i b l e with i t s amplitude of o s c i l l a t i o n p e r i o d i c along the a x i s of the c y l i n d e r , so the t h r e e - d i m e n s i o n a l problem can be reduced t o a two-dimensional one. Even though the numerical r e s u l t s are o b t a i n e d f o r an i n f i n i t e c y l i n d e r , they are extended to s t r u c t u r e s of f i n i t e l e n g t h by i n t e g r a t i n g along the body a x i s , i g n o r i n g end e f f e c t s . For non-uniform bodies such as s h i p s , Korvin-Kroukovsky's (1955) s t r i p theory can be used 13 w i t h t h e t w o - d i m e n s i o n a l r e s u l t s , F o r head seas (wave c r e s t s normal t o the c y l i n d e r a x i s ) , t h e wavelength a l o n g t h e l e n g t h of the c y l i n d e r becomes of the same o r d e r of magnitude as a t y p i c a l c r o s s s e c t i o n a l d i m e n s i o n and the p r o c e d u r e i s no l o n g e r a p p l i c a b l e . A t h r e e - d i m e n s i o n a l model wh i c h c o n s i d e r s end e f f e c t s would have t o be used as the i n c i d e n t wave d i r e c t i o n moves s u b s t a n t i a l l y away from the beam d i r e c t i o n . 2.2 THEORETICAL FORMULATION 2.2.1 WAVE DIFFRACTION PROBLEM A r e g u l a r s m a l l a m p l i t u d e wave t r a i n of h e i g h t H and a n g u l a r f r e q u e n c y co i s o b l i q u e l y . i n c i d e n t upon an i n f i n i t e l y l o n g f i x e d h o r i z o n t a l c y l i n d e r . The waves p r o p a g a t e i n water of depth d i n a d i r e c t i o n making an a n g l e 0 w i t h the x a x i s (see F i g . 1). The c o o r d i n a t e system i s r i g h t handed w i t h z measured upwards from the s t i l l water l e v e l and the x-y p l a n e h o r i z o n t a l . The y a x i s i s p a r a l l e l t o the a x i s of the i n f i n i t e c y l i n d e r . The o r i g i n of the ( x , y , z ) c o o r d i n a t e system i s a t the s t i l l water l e v e l v e r t i c a l l y above or below the c e n t r e of g r a v i t y . The f l u i d i s assumed t o be i n v i s c i d and i n c o m p r e s s i b l e and the 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 may t h e r e f o r e be d e s c r i b e d i n terms of a v e l o c i t y p o t e n t i a l d e f i n e d by u = V # ( x , y , z , t ) (2.2) 1 4 where u i s the f l u i d v e l o c i t y v e c t o r and # must s a t i s f y the Laplace equation V2<I>(x,y,z,t) = 0 (2.3) w i t h i n the f l u i d r e g i o n . The wave height i s assumed s u f f i c i e n t l y small so that l i n e a r wave theory i s a p p l i c a b l e and consequently $ i s s u b j e c t to the usual l i n e a r i z e d boundary c o n d i t i o n s . On the f r e e s u r f a c e , the dynamic pr e s s u r e i s given by the B e r n o u l l i equation | | + grj + ^ ( V # ) 2 = R (2.4) where g i s the g r a v i t a t i o n a l a c c e l e r a t i o n and R i s the B e r n o u l l i constant s et equal to zero f o r convenience. The kinematic f r e e s u r f a c e boundary c o n d i t i o n r e q u i r e s that the normal v e l o c i t y of the f r e e s u r f a c e e l e v a t i o n be equal to the normal v e l o c i t y of a f l u i d p a r t i c l e at the f r e e s u r f a c e . T h i s can be expressed as H = & + H i 2 + ! f ! ? < 2 - 5 ) Eqns. (2.4) and (2.5) are l i n e a r i z e d by n e g l e c t i n g the f l u i d v e l o c i t y square term i n eqn. (2.4) and the wave steepness terms i n eqn. (2.5), and by a p p l y i n g the c o n d i t i o n s at the s t i l l water l e v e l z=0 r a t h e r than at the instantaneous water su r f a c e e l e v a t i o n z=7j. The two equations can then be combined to give the l i n e a r i z e d f r e e s u r f a c e boundary c o n d i t i o n 15 | | - = 0 at z = 0 (2.6) fo r simple harmonic motion. The immersed body s u r f a c e i s assumed impermeable and hence the normal v e l o c i t y of the f l u i d on the body s u r f a c e , S D must equal zero a | | = 0 on S B (2.7) where n i s a d i r e c t i o n normal to the body s u r f a c e d i r e c t e d i n t o the body. The seabed i s assumed h o r i z o n t a l and impermeable g i v i n g | | = 0 at z=-d (2.8) In a d d i t i o n to the above boundary condtions, $ has to s a t i s f y a r a d i a t i o n c o n d i t i o n at the f a r f i e l d t o ensure a unique s o l u t i o n . I t i s convenient to assume the v e l o c i t y p o t e n t i a l to be of the form * = # 0 + *a (2.9) where 4>0 and 4>j, are the v e l o c i t y p o t e n t i a l s f o r the i n c i d e n t and d i f f r a c t e d waves r e s p e c t i v e l y . r The i n c i d e n t wave p o t e n t i a l i s given by l i n e a r wave theory as • „ ( , . y . « , t ) - R e t ^ g H c ° ^ d f " x exp{i (kxcos0+kysin/3-o>t)} ] (2.10) where k i s the wave number which i s r e l a t e d to the angular frequency u by the d i s p e r s i o n r e l a t i o n 1 6 k tanh(kd) = £p (2.11) The r a d i a t i o n c o n d i t i o n which ensures that the d i f f r a c t e d waves are t r a v e l l i n g away from the c y l i n d e r i s given by j-— + ikcos/3 = 0 at x = ±» (2.12) In a numerical approximation, the i n f i n i t e boundary i s t r u n c a t e d at a f i n i t e d i s t a n c e , X from the o r i g i n where the evanescent modes due to the presence of the the body are assumed to have decayed s u f f i c i e n t l y . An approximate a n a l y s i s to f i n d the optimum d i s t a n c e X R at which the r a d i a t i o n c o n d i t i o n i s a p p l i e d i s given i n appendix I. The f l u i d motion i s c o n s i d e r e d p e r i o d i c i n time as w e l l as along the a x i s of the c y l i n d e r . A nondimensional p o t e n t i a l , <t> can thus be d e f i n e d by „ * ( x , y , z , t ) = R e [ ^ i r g ^ > ( x , z ) e x p { i (kysin/3 - cot)}] (2.13) I t i s a l s o convenient to nondimensionalize the v a r i a b l e s u sing the h a l f beam of the c y l i n d e r , a. x' = x/a, z' = z/a, y' = y/a, k' = ka d' = d/a, u = ^g 3-, v = kasin/3 (2.14) For convenience, the primes have been dropped from the v a r i a b l e s and i t i s understood t h a t the v a r i a b l e s are now nondimensional. Dimensional v a r i a b l e s w i l l h e n c e f o r t h be b a r r e d where necessary f o r c l a r i t y . 17 The boundary value problem f o r the d i f f r a c t e d p o t e n t i a l can now be s t a t e d i n nondimensional form as V 2 0 « - v24>k = 0 i n the f l u i d (2.15a) -g-pj— = u<t>* at z=0 (2. 1 5b) •g-̂ - = 0 at z=-d (2.15c) •g^- = ikcos/3tf>a at x = ± X R (2.15d) d(f>n d<t>0 1TT = -cTfr o n S B (2.l5e) where * ° • C ° c S s h U d f ) ] exp(ikxcos^) (2.16) The t h r e e - d i m e n s i o n a l Laplace equation (2.3) has now been reduced to the two-dimensional m o d i f i e d Helmholtz equation (2.15a). 2.2.2 FORCED MOTION PROBLEM Consider an i n f i n i t e l y long c y l i n d e r o s c i l l a t i n g i n heave, sway and r o l l as shown i n F i g . 2. Each mode of motion i s p e r i o d i c i n time as w e l l as along the a x i s of the c y l i n d e r . The displacement or r o t a t i o n i n the kth mode i s given by E k ( y , t ) = ReU kexp{iUy-cot)}] { J I \'2} (2.17) where £^ * s the complex amplitude of o s c i l l a t i o n of the c y l i n d e r with k=1,2,3 corres p o n d i n g to the sway, heave and r o l l modes r e s p e c t i v e l y . Throughout the f o l l o w i n g 18 development, the upper terms i n the c u r l y b r a c k e t s apply with each other, and s e p a r a t e l y the lower terms apply with each other. (•, and £ 2 have been nondimensionalized by a, while i - 3 corresponds to the r o l l angle i n r a d i a n s . The v e l o c i t y of the body s u r f a c e i n the d i r e c t i o n n i s given by 3 _, 3 V n = Z | f n k = R e [ 1 - i " a £ k n k e x p { i ( v y - u t ) } ] (2.18) k=1 k=1 where n1 = n x ) n~ = n„ } (2.19) n 3 = ( z - e ) n x - x n z ) and n , n are the d i r e c t i o n c o s i n e s of the u n i t normal v e c t o r n on the immersed body s u r f a c e and (0,e) denotes the p o i n t about which the r o l l motion i s p r e s c r i b e d . The normal v e l o c i t y of the f l u i d on the immersed body s u r f a c e must equal the normal v e l o c i t y of the body y i e l d i n g ^ = V on S. (2.20) 3n n B T h i s boundary c o n d i t i o n i s s a t i s f i e d at the e q u i l i b r i u m p o s i t i o n of the body r a t h e r than at the instantaneous p o s i t i o n of the body. From equations (2.18) and (2.20), the f o r c e d motion p o t e n t i a l f o r the kth mode of motion can be expressed as = R e [ - i c j a 2 ^ . ^ . exp(iUy-cot)} ] (2.21) 19 The l i n e a r i z e d boundary c o n d i t i o n on the body s u r f a c e can thus be expressed as 3 0 v The boundary value problem f o r the f o r c e d motion p o t e n t i a l s 0j c(k=1,2,3) i s hence governed by eqns. (2.l5a-d) and eqn. (2.22). 2.3 GREEN'S FUNCTION SOLUTION A boundary i n t e g r a l method i n v o l v i n g Green's i d e n t i t y i s used as the b a s i s f o r the numerical e v a l u a t i o n of the p o t e n t i a l s <j>^ (k= 1 ,2, 3,4). The second form of Green's theorem may be a p p l i e d over a c l o s e d s u r f a c e S c o n t a i n i n g the f l u i d r egion i n order to r e l a t e the va l u e s of the p o t e n t i a l <f>(x) i n the f l u i d r e g i o n to the boundary values of the p o t e n t i a l </>(Jj.) and i t s normal d e r i v a t i v e 3t/>(JL)/9n. T h i s can be expressed as where G(x;£) i s an a p p r o p r i a t e Green's f u n c t i o n , x denotes the p o i n t (x,z) being c o n s i d e r e d and £ denotes the p o i n t (£/$) over which the i n t e g r a t i o n i s performed. The c l o s e d s u r f a c e S comprises the immersed body s u r f a c e S f i, the mean f r e e s u r f a c e S„, the r a d i a t i o n s u r f a c e S_., and the seabed S_. F R D as shown i n F i g . 3. When the i n t e r i o r p o i n t x approaches the boundary from w i t h i n , eqn. (2.23) reduces to the f o l l o w i n g i n t e g r a l k=1,2,3 (2.22) * < i > = ^ f U ( i ) f § ( x ; i ) " i $( J L)G ( x ; I)]dS (2.23) 20 equation • (£> - i Si^V^liV " U{i)G{*''i)]dS (2'24) S The Green's f u n c t i o n which s a t i s f i e s the m o d i f i e d Helmholtz equation (2.15a) i n an unbounded f l u i d and i s s i n g u l a r at the p o i n t x=i. i s given by G(x;£) = -K0(vr) (2.25) where K 0 i s the mod i f i e d B e s s e l f u n c t i o n of order zero and r i s the d i s t a n c e between the p o i n t s x and £ r = |1 - x| = [ U - x ) 2 + ( S - Z ) 2 ] 1 / 2 (2.26) The f u n c t i o n K 0 ( x ) -In x as x —s» 0. The Green's f u n c t i o n which s a t i s f i e s the two-dimensional Laplace equation G ( x ; l ) = In r (2.27) i s thus obtained as 0 —> 0°. Since the seabed i s assumed h o r i z o n t a l , i t i s com p u t a t i o n a l l y more e f f i c i e n t to exclude the seabed from S and an a l t e r n a t i v e Green's f u n c t i o n which takes i n t o account symmetry about the seabed can be d e f i n e d G(x;£) = -[K 0(;/r) +K 0(*»r')] (2.28) where r' i s the d i s t a n c e between the p o i n t s x and £' = (£/~($ +2d)) which i s the r e f l e c t i o n of £ about the seabed: r « = |£' - x| = [ U - x ) 2 + ($ + 2 d + z ) 2 ] l / 2 (2.29) If the depth v a r i a t i o n s are s i g n i f i c a n t , the seabed would have to be i n c l u d e d i n S and the Green's f u n c t i o n given by eqn. (2.25) used i n s t e a d . 21 The i n t e g r a l e quation (2.24) can now be eva l u a t e d n u m e r i c a l l y to give the p o t e n t i a l </> at any p o i n t i n the f l u i d and hence provide the s o l u t i o n to the boundary value problem. 2 . 4 EXCITING FORCES, ADDED MASSES AND DAMPING COEFFICIENTS Once the v e l o c i t y p o t e n t i a l i s o b t a i n e d , the hydrodynamic pressure can be computed from the l i n e a r i z e d B e r n o u l l i equation p = -p|| = iw/o$ (2.30) The f o r c e s and moments per u n i t l e n g t h are determined by i n t e g r a t i n g the hydrodynamic pressure over the immersed body su r f a c e S„. a The e x c i t i n g f o r c e per u n i t l e n g t h which i s due to the i n c i d e n t and s c a t t e r e d waves and i s p r o p o r t i o n a l t o the wave height i s given by F j = H M f * n j d s { i - 3 , 2I (2-31) B where Fj(j=1,2) denotes the sway and heave f o r c e r e s p e c t i v e l y while F 3 denotes the r o l l moment. S u b s t i t u t i o n of equations (2.9) and (2.13) i n t o eqn. (2.31) y i e l d s F j ( y , t ) = pg§{f[2}Re[;(0 o+ t>„)n : ].exp{i(vy-cjt)}dS] |1:3' 2|(2.32) S B The d i m e n s i o n l e s s e x c i t i n g f o r c e amplitude i s given by 22 F . ( y , t ) Cj = — = J(*o+*«>nj dS Ull'2} (2-33> ^•pgHa s B The e x c i t i n g f o r c e c o u l d a l t e r n a t i v e l y be d e f i n e d by F • (y, t ) r~7"{i} = l C j l c o s ( , y - W t + A . ) {1l3'2} (2.34) 2PgHa where the phase angle Aj i s d e f i n e d by Aj = t a n " 1 Im(C..) Re(Cj) (2.35) There are a l s o hydrodynamic f o r c e s a s s o c i a t e d with the motions of the c y l i n d e r which are p r o p o r t i o n a l to the amplitude of c y l i n d e r motion. The i t h component of the f o r c e due to the j t h component of motion can be expressed as F i j = * j n i d s ji=3'2j 3-1'2'3 ( 2- 3 6 ) S B ' S u b s t i t u t i o n of the equation f o r the f o r c e d motion p o t e n t i a l s (2.21) i n t o eqn. (2.36) y i e l d s Fiy" p c j 2 ^ | ^ a } R e [ / « ;.n iexp{i(vy-wt)}dS] S B ) i = 3 ' 2 | 3=1,2,3 (2.37) T h i s f o r c e can a l s o be expressed i n terms of two components; one component i n phase with the a c c e l e r a t i o n and the other i n phase with the v e l o c i t y F i j * - - O i j S j - X i j E j 3 " 1 ' 2 ' 3 ( 2 ' 3 8 ) where p^j and X^j are the added mass and damping 23 c o e f f i c i e n t s r e s p e c t i v e l y . S u b s t i t u t i o n of eqn. (2.17) i n t o eqn. (2.38) g i v e s F i j = { * } R e [ { c j 2 M i j * j + i a , X i j ̂  j ) e x P { 1 (f'Y-^t)} ] jj=3'2j i=1,2,3 (2.39) Comparing eqn. (2.37) with eqn (2.39) g i v e s the nondimensional added mass and damping c o e f f i c i e n t s as M i j m = Re[/ 4>.n. dS] (2.40) pa S B 3 — m = Im[J 4>.n. dS] (2.41) pwa s B where the constant m i s given as 2 f o r ( i , j ) = (1,1) and (2,2) m = {3 f o r ( i , j ) = (1,3) and (3,1) (2.42) 4 f o r ( i , j ) = (3,3) The Haskind (1953) r e l a t i o n s ( a l s o see Newman,1962) provide an a l t e r n a t e way of c a l c u l a t i n g the e x c i t i n g f o r c e s . A p p l y i n g Green's theorem to the d i f f r a c t i o n p o t e n t i a l g i v e s ( 0 j - 5 r r " *«-9n ) d S = 0 3 = 1/2,3 (2.43) S u b s t i t u t i n g the above e x p r e s s i o n i n t o eqn. (2.32) giv e s / 9#- 90K (0O-3W1 + * j 7 J n - ) d s { 2 ' 4 4 ) A p p l y i n g the boundary c o n d i t i o n given by eqn. (2.l5e) e l i m i n a t e s the d i f f r a c t i o n p o t e n t i a l from the ex p r e s s i o n f o r the e x c i t i n g f o r c e 24 / d<j> • 30o U O - S T P " 0 j ? T r ) d S (2.45) S B There i s a l s o a d i r e c t r e l a t i o n between the damping c o e f f i c i e n t and the amplitude of the waves generated by an o s c i l l a t i n g c y l i n d e r symmetrical about x=0. An amplitude r a t i o |$^| can be d e f i n e d as the r a t i o of the wave amplitude at |x|=°° t o the amplitude of o s c i l l a t i o n of the c y l i n d e r , that i s where |7j ^ | i s the amplitude of the r a d i a t e d waves at | x | =°° f o r the i t h mode of o s c i l l a t i o n of the c y l i n d e r . By equating the work done i n o s c i l l a t i n g the c y l i n d e r to the energy f l u x r a d i a t e d a c r o s s a c o n t r o l s u r f a c e at i n f i n i t y , i t can be shown that (see Newman,1977) f o r the o b l i q u e case. The e x c i t i n g f o r c e can a l s o be r e l a t e d to the amplitude r a t i o by e v a l u a t i n g the i n t e g r a l i n eqn. (2.45) at the negative r a d i a t i o n s u r f a c e (x=-X R). The i n t e g r a l does not va n i s h s i n c e the i n c i d e n t wave p o t e n t i a l does not s a t i s f y the r a d i a t i o n c o n d i t i o n . Since the f o r c e d motion p o t e n t i a l i s p r o p o r t i o n a l to the square root of the energy f l u x , eqn. (2.45) can be i n t e g r a t e d over the depth at x=-X R to g i v e the e x c i t i n g f o r c e c o e f f i c i e n t as 25 Equations (2.47) and (2.48) can be combined to p r o v i d e a d i r e c t r e l a t i o n between the e x c i t i n g f o r c e c o e f f i c i e n t s and the damping c o e f f i c i e n t s X. . I c i l = f — m <1 + 2 k d )tanhkd c o s / 3 ] 1 / 2 (2.49) 1 pcua sinh2kd Equations (2.47)-(2.49) p r o v i d e a u s e f u l check on the numerical r e s u l t s o b t a i n e d . 2.5 EQUATIONS OF MOTION . The dynamic response of the c y l i n d e r due to the e x c i t i n g waves can now be o b t a i n e d by s o l v i n g the equations of motion. The equations of motion are of the form 3 2 [-co 2 (m. .+fi..) - ico\.. + c. .]S. = F . ( y , t ) i = 1,2,3 (2.50) j_1  1J 1 J • lJ 1 J J A where m^j and c ^ j are the mass and h y d r o s t a t i c s t i f f n e s s matrix c o e f f i c i e n t s r e s p e c t i v e l y . A d d i t i o n a l f o r c e s due to moorings or v i s c o u s damping may be i n c l u d e d i n eqn. (2.50) i f p r e s e n t . I t should be noted that f o r the case of r o l l motion, n o n l i n e a r v i s c o u s damping i s important p a r t i c u l a r l y near the resonance frequency and would have to be i n c l u d e d in p r a c t i c a l a p p l i c a t i o n s . I t was assumed i n d e r i v i n g the added mass and damping c o e f f i c i e n t s that the c y l i n d e r was f l e x i b l e with i t s amplitude of motion v a r y i n g s i n u s o i d a l l y along the l e n g t h of the c y l i n d e r as w e l l as i n time. The term sin/3 can be thought of as the r a t i o of the i n c i d e n t wave l e n g t h to the 26 the wave l e n g t h along the a x i s of the c y l i n d e r . A r i g i d c y l i n d e r has an i n f i n i t e wavelength along the a x i s of the c y l i n d e r and hence corresponds to a f l e x i b l e c y l i n d e r w i t h 0=0°. The components of the mass matrix are given as m. 1D m -mz, 0 m 0 -mz. (2.51) 0 'G ~ where m i s the mass per u n i t l e n g t h of the body, z^ i s the z c o o r d i n a t e of the c e n t r e of g r a v i t y and I 0 i s the p o l a r mass moment of i n e r t i a about the y a x i s per u n i t l e n g t h . I 0 may be expressed as I 0 = m(r 2 + z*) • (2.52) y G where r ^ i s the r a d i u s of g y r a t i o n of the body about the y a x i s . The h y d r o s t a t i c s t i f f n e s s matrix i s determined by c a l c u l a t i n g the f o r c e s r e q u i r e d to r e s t o r e the body to i t s e q u i l i b r i u m p o s i t i o n f o r s m a l l amplitude d i s p l a c e m e n t s . The s t i f f n e s s matrix components are given as 0 0 0 c. . = 1D 0 C 2 2  C 2 3 0 C 2 3 C 3 3 (2.53) where ' c 2 2 = pgB c 2 3 = pgBx f (2.54a) (2.54b) 27 = pgA[(S,,/A) + z f i - z Q ] (2.54c) where B i s the beam of the c y l i n d e r , x^ i s the c e n t r o i d of the waterplane l i n e and i s equal to zero f o r bodies symmetrical about x=0, z f i i s the z c o o r d i n a t e of the c e n t r e of buoyancy, A i s the d i s p l a c e d volume per u n i t l e n g t h , and i s the waterplane area moment of i n e r t i a about the x a x i s per u n i t l e n g t h , t h a t i s S t a t i c s t a b i l i t y i n r o l l r e q u i r e s t h a t the c o e f f i c i e n t c 3 3 be p o s i t i v e . From eqn. (2.54c), i t i s evident t h a t the metacentre ( S ^ / A ) + z f i has to be l o c a t e d higher than the c e n t r e of g r a v i t y z^ f o r the f l o a t i n g body to be s t a b l e . The equations of motion (2.50) can now be s o l v e d to o b t a i n the complex amplitudes of o s c i l l a t i o n , £j f o r any given wave frequency and d i r e c t i o n u s i n g a complex matrix i n v e r s i o n technique. The amplitude of body motion i s o f t e n d e s c r i b e d i n terms of the response amplitude o p e r a t o r d e f i n e d as The response, amplitude operator r e p r e s e n t s the amplitude of body motion due to a u n i t amplitude wave of frequency co, t r a v e l l i n g i n d i r e c t i o n /3. = / x 2dx = B 3/12 B (2.55) Z.(u,/3) = (2.56) 28 2.6 REFLECTION AND TRANSMISSION COEFFICIENTS Another two q u a n t i t i e s of p h y s i c a l i n t e r e s t e s p e c i a l l y f o r such s t r u c t u r e s as f l o a t i n g breakwaters are the r e f l e c t i o n and t r a n s m i s s i o n c o e f f i c i e n t s . The c o e f f i c i e n t s are o b t a i n e d by e v a l u a t i n g the component wave amplitudes at the r a d i a t i o n s u r f a c e s ( x = ± X R ) . There are c o n t r i b u t i o n s to t h i s asymptotic wave amplitude from: (1) the o s c i l l a t i o n s of the c y l i n d e r i n i t s three modes, and (2) the r e f l e c t i o n and t r a n s m i s s i o n of the i n c i d e n t wave by a f i x e d body. From B e r n o u l l i ' s equation, the wave amplitude i s r e l a t e d to the v e l o c i t y p o t e n t i a l by * = "g- f f ^ ' Y ' O ' t ) < 2- 5 7> S u b s t i t u t i n g the equation f o r the f o r c e d motion p o t e n t i a l s (2.21) i n t o eqn. (2.57) y i e l d s the asymptotic wave amplitude f o r each mode of motion T J. = Ret ^-a2^i<t>i(x.r0) e x p { i U y - o t ) } ] (2.58) The wave amplitude r a t i o p r e v i o u s l y d e f i n e d by eqn. (2.46) i s now given as l* i l = TaTtl = ^ i ^ i ( x ' ° > I { 2 - 5 9 ) e v a l u a t e d at x=±X R. At the r a d i a t i o n s u r f a c e , the evanescent modes are assumed to have decayed s u f f i c i e n t l y (see appendix I ) and the v e l o c i t y p o t e n t i a l s are of the form 0(x,z) = A o c o s h [ k ( z + d ) ] e x p ( ± i k x c o s g ) a t x = ± X R (2.60) cosh(kd) K 29 where A 0 i s the complex amplitude of the p o t e n t i a l at z=0. Given that J c o s h 2 [ k ( z + d ) ] d z = sinh(2kd) + 2kd ( 2 # g l ) -d 4 K the c o e f f i c i e n t A 0 can be obtained by a p p l y i n g the o r t h o g o n a l i t y c o n d i t i o n of the h y p e r b o l i c c o s i n e f u n c t i o n and i s given f o r the j t h p o t e n t i a l as A ° i = sinhtSkdi^Zkd exp(±ikxcos/5) /«.cosh[ k (z+d) ]dz -d J at x=+XR (2.62) The wave amplitude r a t i o can thus be e v a l u a t e d as ISjl = ^ I A O J I (2.63) f o r each mode of motion. The r e f l e c t i o n and t r a n s m i s s i o n c o e f f i c i e n t s due to the presence of a f i x e d body are obta i n e d i n a s i m i l a r manner. The r e f l e c t i o n c o e f f i c i e n t can be obtained by e v a l u a t i n g the asymptotic wave amplitude of the s c a t t e r e d waves at the negative r a d i a t i o n s u r f a c e (x=-X R). S u b s t i t u t i o n of the form of the d i f f r a c t e d wave p o t e n t i a l given by eqn. (2.13) i n t o eqn. (2.57) y i e l d s T J r = Re[.§0,(-X ,0)_ e x p { i U y - u t ) } ] (2.64) The r e f l e c t i o n c o e f f i c i e n t 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 amplitude to the i n c i d e n t wave amplitude and i s thus given by I V I K R = - i - ^ - = | 0 „ ( - X R f O ) | (2.65) H/ 2 30 The t r a n s m i s s i o n c o e f f i c i e n t i s due to the asymptotic wave amplitude of the i n c i d e n t and s c a t t e r e d waves at the p o s i t i v e r a d i a t i o n s u r f a c e (x=X R) and i s s i m i l a r l y given by K T = |# 0(X R,0) + * , ( X R f 0 ) | (2.66) The e x p r e s s i o n s on the r i g h t s i d e of eqns. (2.65) and (2.66) are e v a l u a t e d using eqn. (2.62). A p p l y i n g c o n s e r v a t i o n of energy p r i n c i p l e s , remembering that the energy i n a wave i s p r o p o r t i o n a l to the square of the wave amplitude, the r e f l e c t i o n and t r a n s m i s s i o n c o e f f i c i e n t s are r e l a t e d by K R + K 2 = 1 (2.67) A f t e r o b t a i n i n g the amplitudes of body motion by s o l v i n g the equations of motion, the r e f l e c t i o n and t r a n s m i s s i o n c o e f f i c i e n t s f o r a f r e e l y f l o a t i n g body are determined r e s p e c t i v e l y as 3 K D = |0,(-X O,O) + Z $.Z.(u,/3)| (2.68a) R j=1 3 D 3 K T = |0 O(X R,O) + 0,(X R,O) + Z S j Z j ( w , 0 ) | (2.68b) 2 .7 NUMERICAL PROCEDURE In order to ev a l u a t e the i n t e g r a l equation (2.24), the boundary i s d i v i d e d i n t o N segments with the value of <t> or 90/3n c o n s i d e r e d constant over each segment and equal to the value at the midpoint of the segment. Eqn. (2.24) can be r e p l a c e d by the summation equation 31 1  N ar 9 0 i 0 k(x.) = i 2 { * k ( £ j U I g ^ i ^ ^ d S - ^ / G ( x . ; x . ) d S } j j k=1,2,3,4 (2.69) where the summation i n eqn. (2.69) i s performed i n a counter c l o c k w i s e manner around the boundary. Eqn. (2.69) can be r e w r i t t e n as N m b<j>[k) • 5 1 { ( a i j + 6 i j ) 0 j + hijJn3 } = 0 k=1,2,3,4 (2.70) where S^j i s the Kronecker d e l t a f u n c t i o n given by (1 i - j 6^ = { (2.71) The c o e f f i c i e n t s a ^ j and b ^ j are d e f i n e d as a i j - ij InlKot^r.j) + K o U r l ^ J d S (2.72) b. • = 4 S^oivr. •) + .KoUr! -)]dS (2.73) r . . and r ! . are given as 13 13 y r-j = [ ( X j - x . ) 2 + ( 2 j - z . ) 2 ] l / 2 (2.74) r l j = [ ( X j - x ^ 2 + ( z j + 2 d + z . ) 2 ] l / 2 (2.75) x^ and X j are e v a l u a t e d at the midpoint of each segment. The gr a d i e n t 9G/9n may be expressed as ! § < £ i ? £ j > - H c o s 7 + I f ' COST' (2.76) where 7 and 7' are as shown i n F i g . 4 and correspond to the angles between n. and r=x.-x., and between n'. and r'=x'.-x. -3 3 - i ' -D -3 - 1 r e s p e c t i v e l y , that i s 32 n.•(x .-x. ) C O S T = ^ — ~ 3 ~* (2.77) COST' = =3—71 - 1 (2.78) where n. = n i + n k ~1 X ~ 2 ~ (2.79) n'. = n i - n k —3 x— z— and X j i s t h e p o i n t ( x j ( z j + 2 d ) ) . The u n i t normal v e c t o r n i s g i v e n by n = | | i - | | k (2.80a) The above e x p r e s s i o n can be ap p r o x i m a t e d as Az • _ Ax , 2S i 715" - (2.80b) The d e r i v a t i v e of the Green's f u n c t i o n i s g i v e n by IfKodr) = - J » K , (vr) ( 2 . 8 1 ) where K , i s t h e m o d i f i e d B e s s e l f u n c t i o n of o r d e r one. When i * j , the i n t e g r a l s i n eqns. ( 2 . 7 2 ) and ( 2 . 7 3 ) a r e appr o x i m a t e d by e v a l u a t i n g t h e Green's f u n c t i o n and i t s normal d e r i v a t i v e a t the m i d p o i n t of each segment. The c o e f f i c i e n t a ^ j i s thus g i v e n as K,Ur..) a . . = - v- (x .-x. )n„ + (z .-z. )n lAS. 13 Trr.j 3 i x ] i ' z' 3 K , Ur! .) - v [ (x .-x. )n + ( z . + 2d+z.)n ]AS . i * j (2.82a) Trr!. D i x 3 1 2 3 i ] S u b s t i t u t i n g t h e a p p r o x i m a t i o n f o r the d i r e c t i o n c o s i n e s g i v e n i n eqn. (2.80b), a ^ j becomes 33 K , ( v r . .) a . . = - v [ (x .-x . ) Az . - ( z : - z . ) A x . ] *D ,rr. . D I D D 1 D ID K 1 ( T I • ) - v (x .-x. )Az . - ( z - + 2d+z. )Ax .] i # j (2.82b) 7rr!^j D 1 D D 1 D where A z . = z ... - z . D D + 1 D A X j = x j + 1 - x. ASj = [ ( A Z j ) 2 + ( A X j ) 2 ] l / 2 The c o e f f i c i e n t b.^ i s g i v e n a s b i : j = " ^ [ K o d ^ r . j ) + K 0 ( » r J j ) ] A S j i * j (2.83) When i = j , t h e i n t e g r a l s i n e g n s . (2.72) and (2.73) become s i n g u l a r . E v a l u a t i n g t h e n o n s i n g u l a r components K 0('i>r') and 9K 0(j>r' )/3n a s b e f o r e and u s i n g t h e a s y m p t o t i c f o r m u l a f o r KQ{vr) ( s e e Abramowitz and S t e g u n , l 9 6 4 ) K 0(*>r) » -{lnUr/2) + 7 } as vr^O (2.84) where 7 i s E u l e r ' s c o n s t a n t , t h e d i a g o n a l c o e f f i c i e n t s a r e g i v e n as a.. = £ K , [ 2 v ( z . + d ) ] A x . (2.85) 1 1 7T 1 1 AS- i>AS. b H = - y l l l n - ^ - i + 7 ~ 1 - K 0{2i»(z.+d)}] (2.86) F o r 0=0° , t h e p r o b l e m r e d u c e s t o t h e t y p i c a l two d i m e n s i o n a l one. U s i n g t h e G r e e n ' s f u n c t i o n g i v e n i n eqn. (2.27) w i t h symmetry a b o u t t h e s e a b e d t a k e n i n t o a c c o u n t , t h e c o e f f i c i e n t s f o r 0=0° a r e g i v e n by 34 1 i j [ (XyX.)Az. - (2 j+2d+z i)AXj] i * j (2.87) b t j = l ( l n r . j + In r l ^ A S j i * j (2.88) For i=j Ax. a i i = 2*U*+d) ( 2 ' 8 9 ) AS. AS. b.. = — i t l n - T i - 1 + In 2(z.+d)] (2.90) With the c o e f f i c i e n t s a ^ j and b ^ j now known, eqn. (2.70) provides N equations r e l a t i n g the valu e s of <$> and 9</>/9n over S +S+S_. The v a r i o u s boundary c o n d i t i o n s around S +S +S_ provide the remaining N equations needed to solve f o r 4> and d<j>/dn. S u b s t i t u t i o n of the v a r i o u s boundary condition's given i n eqn. (2.15) i n t o eqn. (2.70) y i e l d s N 1 2 N2 , , x Z ( a . . + 6. •+^ra-b. . ) * • + S ( a - . + S.-U- + N 3  W2« (k) N 4 (k) Z ( a i i + 6..+2gab i.)^ K' + I ( a . . + 6 i , m c o s 0 b . . i > ^ , u j=N2+1 1 3 3 3 3 j=N3+1 3 -1 J 3 N (k) N 2 (k) I ( a . .+.6. .+ikcos0b. ,)0- ; = - Z b..t\K> j=N4+1 J J J J j=N1+TJ J f o r i=1,....,N; k=1,2,3,4 (2.91) (k) where f . i s d e f i n e d as 35 f (k) k=1,2,3 k=4 (2.92) The e x p r e s s i o n s on the r i g h t - h a n d s i d e of the above equation are given as a / 4 ) c o s h [ k ( z .+d) ] Az. TTn^ = i kcos/3 C o s h ( k d ) exp( ikcos/3x .) s i n h [ k ( z .+d)] k c o s h U d ) Ax. exp(ikcos/3x . ) — . 3 AS (2.93) and AZj/ASj -AXj/ASj Az (z . - e ) — : 3 AS Ax . + x . — 1 3 AS . D k=1 k=2 k = 3 (2.94) F i g . 5 shows a t y p i c a l d i s c r e t i z e d boundary with the c o n s t a n t s N1, N2, N3 and N4 shown. Eqn. (2.91) y i e l d s N (k) equations f o r N unknown <f>\ (k=1,2,3,4) v a l u e s which can be s o l v e d using a matrix i n v e r s i o n technique to obtain the unknown v e l o c i t y p o t e n t i a l s on the boundary. The e x c i t i n g f o r c e s , added mass and damping c o e f f i c i e n t s , and r e f l e c t i o n and t r a n s m i s s i o n c o e f f i c i e n t s can now be determined using the e x p r e s s i o n s given i n the p r e c e d i n g s e c t i o n s . 2.8 EFFECT OF FINITE STRUCTURE LENGTH Let us now c o n s i d e r the f o r c e s and response of a r i g i d s t r u c t u r e of f i n i t e l ength, 1. The l e n g t h of the s t r u c t u r e i s assumed to be much gr e a t e r than the i n c i d e n t wavelength. 36 The f o r c e per u n i t l e n g t h i s given by eqn. (2.32) as F j ( y , t ) = pq%{l2}Re[C.(to,ti)exp{i(1,y-tot)}] ^]ll'2\ (2.95) where j = 1,2,3 corresponds to the sway, heave and r o l l e x c i t i n g f o r c e s (or moment). The t o t a l f o r c e on the s t r u c t u r e i s obtained by i n t e g r a t i n g two-dimensional f o r c e along i t s l e n g t h , i g n o r i n g end e f f e c t s 1/2 F . ( t ) = / F . ( y , t ) d y (2.96a) 3 -1/2 3 S u b s t i t u t i o n of eqn. (2.95) i n t o eqn. (2.96a) y i e l d s p , a , 2sin(- k-isin^) ?\ F . ( t ) = pg§l \ l 2 C- ?- e x p ( - i u t ) \ ]Z' \ (2.96b) J ^ a -1 klsin/J 13--* J f o r p*0°. The above ex p r e s s i o n can be thought of as the product of the f o r c e per u n i t l e n g t h , the le n g t h of the s t r u c t u r e , and a f a c t o r r(kl,/3) d e f i n e d as r(kl,/3) = 2sin(^is i n / 3 ) ^ (1*0° klsin/3 1 0=0° (2.97) The f a c t o r r ( k l , 0 ) can be c o n s i d e r e d to be a r e d u c t i o n of the l o a d per u n i t l e n g t h due to the f i n i t e l e n g t h of the s t r u c t u r e f o r a given angle of approach, or due to the ob l i q u e n e s s of the waves f o r a given s t r u c t u r e l e n g t h . F i g . 6 shows a p l o t of r 2 a g a i n s t k l f o r /3=0°, 15°, 30° and 60°. The separate i n f l u e n c e s of k l and /3 on the load per u n i t l e n g t h can be seen. The f a c t o r r(kl,/3) has an o s c i l l a t o r y behavior at l a r g e v a l u e s of k l with an i n f i n i t e number of zeros g i v e n by 37 ^ s i n j 3 = n it n=1,2,... (2.98) For an i n f i n i t e span s t r u c t u r e , the t o t a l l o a d per u n i t l e n g t h tends to zero. The maximum t o t a l l o a d occurs on a span of le n g t h 1 = L/2sin/3 (2.99) where L=27r/k i s the wavelength. T h i s maximum f o r c e i s F j ( t ) = P g j ^ C j U ^ T f g 2 ^ exp(-icot) {]ll'2} (2.100) The motions of a r i g i d c y l i n d e r of f i n i t e l e n g t h i n obli q u e seas can be d e s c r i b e d i n terms of s i x degrees of freedom. In a d d i t i o n to the sway, heave and r o l l modes present i n beam seas, the c y l i n d e r can a l s o surge, yaw and p i t c h c o rresponding to the t r a n s l a t i o n a l motion along the y a x i s and r o t a t i o n a l motions about the z and x axes r e s p e c t i v e l y . The added mass and damping c o e f f i c i e n t d e r i v e d i n s e c t i o n 2.4 corresponds to the o s c i l l a t i o n s of a f l e x i b l e c y l i n d e r with a s i n u s o i d a l v a r i a t i o n of the amplitude of motion along the le n g t h of the c y l i n d e r . The added mass and damping c o e f f i c i e n t s of a r i g i d c y l i n d e r correspond to the case of beam seas (0=0°). The hydrodynamic c o e f f i c i e n t s f o r the sway, heave and r o l l motions of a f i n i t e l e ngth s t r u c t u r e are obtained by m u l t i p l y i n g the s e c t i o n a l c o e f f i c i e n t s with the l e n g t h of the s t r u c t u r e . The e x c i t i n g f o r c e s and hydrodynamic c o e f f i c i e n t s f o r the p i t c h and yaw motions can be obtained from the s e c t i o n a l c o e f f i c i e n t s f o r the heave and sway motions u s i n g a s t r i p theory approach 38 d e s c r i b e d i n Bhattacharyya (1978). The yaw and p i t c h e x c i t i n g e x c i t i n g moment c o e f f i c i e n t s are given as 1/2 F j + 3 ( t ) = J . y F . ( y , t ) d y -1/2 j=1,2 (2.101) where j=4,5 corresponds to the yaw and p i t c h modes r e s p e c t i v e l y . S u b s t i t u t i o n of e x p r e s s i o n f o r the two-dimensional f o r c e s (2.95) i n t o eqn. (2.101) y i e l d s F j + 3 ( t ) = pg5al2Cj-q(kl,/3)exp(-icjt) j = 1,2 (2.102) where q ( k l , 0 ) i s d e f i n e d as q = 2i (klsin/3) 2 [ ^ s i n j 3 c o s ( ^ s i n / 3 ) s i n ( ^ s i n / 3 ) ] 0*0' (2.103) 0 = 0 ' 3. EFFECTS OF DIRECTIONAL WAVES 3.1 REPRESENTATION OF DIRECTIONAL SEAS Before proceeding to determine the response of s t r u c t u r e s i n d i r e c t i o n a l seas, we s h a l l f i r s t present a mathematical r e p r e s e n t a t i o n of d i r e c t i o n a l seas. The p r e c e d i n g chapter d e a l t with the e x c i t i n g forces and response of a s t r u c t u r e s u b j e c t to regular u n i d i r e c t i o n a l waves. Ocean waves however e x h i b i t a wave pa t t e r n which i s h i g h l y complex and i r r e g u l a r . T h i s complex sea surface i s o f t e n modelled by a l i n e a r s u p e r p o s i t i o n of l o n g - c r e s t e d waves of a l l p o s s i b l e f r e q u e n c i e s approaching a p o i n t from a l l d i r e c t i o n s . The sea s u r f a c e e l e v a t i o n i s assumed to be a zero mean, s t a t i o n a r y , e r g o d i c random Gaussian p r o c e s s . The assumption of a Gaussian process i m p l i e s symmetry about the s t i l l water l e v e l which i s only r e a l i s t i c f o r smal l amplitude waves. A l o n g - c r e s t e d wave t r a i n t r a v e l l i n g at angle 0 r e l a t i v e to the p o s i t i v e x a x i s may be rep r e s e n t e d by r?(x,y,t) = Re [A exp{ i (kxcos0+kysin/3-cot)} ] (3.1) where A i s the complex wave amplitude w i t h a random phase, k i s the wave number r e l a t e d to the frequency co by the l i n e a r d i s p e r s i o n r e l a t i o n (eqn. 2.11). A random sea surface can be c o n s i d e r e d to be a d i s c r e t e sum of l i n e a r waves of d i f f e r e n t f r e q u e n c i e s and d i r e c t i o n s 77 = Re[ZZ A i^exp{ i (k^cos/3 j+k^sin/3^-co^ )} ] (3.2) 39 40 where k^ denotes the wave number of the i - t h wave component t r a v e l l i n g i n d i r e c t i o n /3j, co^ i t s frequency and A^j i t s amplitude. I f we l e t the t o t a l number of harmonics tend to i n f i n i t y while the d i f f e r e n c e between adjacent f r e q u e n c i e s and d i r e c t i o n s tends to zero, the summation i n eqn. (3.2) can be r e p l a c e d by an i n t e g r a l over a continuous range of fre q u e n c i e s and d i r e c t i o n s r?(x,y,t) = R e [ J/exp{ i (kxcos/3+kysin/3-a>t) }dA(co, 0) ] (3.3) where dA rep r e s e n t s the d i f f e r e n t i a l wave amplitude i n the two-dimensional (co,/3) space bounded by ico,co+dco) and (/3,/3+d|3). The mean square value of the water s u r f a c e e l e v a t i o n i s given by T77 = i//dA(w,/3)dA*(u,/5) = U S(co,/3)dcod/J (3.4) - T T O where dA*(co,/3) i s the complex conjugate of dA(to,/3) and S(co,/3) i s a d i r e c t i o n a l wave spectrum. Since the average energy d e n s i t y i n the waves i s p r o p o r t i o n a l to the square of the wave amplitude, the product S ( C J , j3)dcod/3 can be c o n s i d e r e d to be the c o n t r i b u t i o n to the t o t a l mean energy d e n s i t y due to waves with f r e q u e n c i e s between co and co+dco, t r a v e l l i n g i n d i r e c t i o n s between /3 and /3+dj3. A sketch of a t y p i c a l d i r e c t i o n a l wave spectrum i s shown i n F i g . 7. The one-dimensional spectrum, S(co) can be obtained by i n t e g r a t i n g the d i r e c t i o n a l wave spectrum over a l l d i r e c t i o n s 41 7T S(w) = / S(cj,/3)d0 (3.5) The one-dimensional wave spectrum can be determined from measurements of the f r e e s u r f a c e e l e v a t i o n at a s i n g l e p o i n t i n space; f o r i n s t a n c e by r e c o r d i n g the motions of a heaving buoy. In order to o b t a i n i n f o r m a t i o n about the d i r e c t i o n a l i t y of the waves, one has to r e s o r t to more complicated t echniques. The most common methods f o r e v a l u a t i n g d i r e c t i o n a l wave s p e c t r a i n c l u d e 1. a n a l y s i s of the water s u r f a c e e l e v a t i o n and the h o r i z o n t a l o r b i t a l v e l o c i t i e s at an o b s e r v a t i o n p o i n t (e.g. F o r r i s t a l l et a l (1978), Sand (1980)). 2. a n a l y s i s of the measurements of the water s u r f a c e e l e v a t i o n , s l o p e and c u r v a t u r e from the motions of a f l o a t i n g buoy (e.g. Longuet-Higgins et al (1961), Cartwright and Smith (1964), Mitsuyasu et al (1975)). 3. a n a l y s i s of the measurements of the water s u r f a c e e l e v a t i o n from an array of guages (e.g. Borgman (1969), Panicker (1971), Davis and Regier (1977)). 4. by means of stereophotographs (e.g. Cot6 et al (1960), Holthujsen (1981)). I t i s o f t e n convenient to express the d i r e c t i o n a l wave spectrum i n terms of an energy spreading f u n c t i o n a p p l i e d to the one-dimensional spectrum S(u,0) = S(u)G(u,0) (3.6) where G(co,|3) i s a d i r e c t i o n a l spreading f u n c t i o n . 4 2 I t f o l l o w s from eqn. (3.5) t h a t G(w,0) must s a t i s f y 7T / G(u,0)d0 = 1 (3.7) -it V a r i o u s one-dimensional frequency s p e c t r a have been used d e s c r i b e ocean waves. The most commonly used ones i n c l u d e the B r e t s c h n e i d e r , Pierson-Moskowitz and JONSWAP s p e c t r a . These s p e c t r a are d e s c r i b e d i n d e t a i l i n Sarpkaya and Isaacson (1981) and hence are not given here. There have a l s o been s e v e r a l f o r m u l a t i o n s f o r G(CJ,0) proposed by v a r i o u s a u t h o r s . A few of the commonly used ones are o u t l i n e d below 1. Cosine-squared f o r m u l a t i o n S t . Denis and P i e r s o n (1953) proposed a spreading f u n c t i o n which i s independent of frequency ( | cos20 f o r I 0 I < T T / 2 G(0) = I * (3 . 8 1 ( 0 otherwise The spectrum i s c e n t r e d about 0=0°. 2 . Cosine-power f o r m u l a t i o n Longuet-Higgins et al (1961) proposed the f o l l o w i n g d i r e c t i o n a l spreading f u n c t i o n C(e) = C(s) c o s 2 s ( 0 ) (3.9) where 8 i s measured from the p r i n c i p a l d i r e c t i o n of wave pr o p a g a t i o n . C(s) i s a n o r m a l i z i n g c o e f f i c i e n t that ensures that eqn. (3.7) i s s a t i s f i e d and i s given by 43 C(s) = r ( s + ] ) (3.10) 2/TT r(s+^) r i s the gamma f u n c t i o n . F i g . 8 shows the d i r e c t i o n a l s preading f u n c t i o n f o r d i f f e r e n t v a l u e s of s. I t can be seen that s d e s c r i b e s the degree of spread about the p r i n c i p a l d i r e c t i o n with s — r e p r e s e n t i n g l o n g - c r e s t e d waves. On the b a s i s of t h e i r measurements f o r wind d r i v e n ocean waves, Mitsuyasu et al (1975) found the parameter s to depend on the dimensionless frequency ( 0 . 1 1 6 ( T ) ~ 2 * 5 f o r I>I s = { _ 5 _ v 5 m (3.11) ( 0.116(1) b ( T J / , b f o r T-<Im m m where T = dimensionless frequency = Uf/g T m = dimensionless modal frequency = Uf^/g U = wind speed at 19.5m above sea l e v e l Hasselmann et al (1980) on the b a s i s of the data obtained from the J o i n t North Sea Wave P r o j e c t (JONSWAP) found the parameter s to depend mainly on f / f m r a t h e r than I and proposed a d i f f e r e n t formula f o r s. Borgman (1969) used an a l t e r n a t i v e c o s i n e power f u n c t i o n given as • i r (s) c o s 2 s ( 0 ) f o r |0|<7r/2 G(6) ={ (3.12) otherwise The n o r m a l i z i n g c o e f f i c i e n t C'(s) i s given as C ( s ) = -1 r ( s + l> (3.13) H r(s+£) 44 3. SWOP f o r m u l a t i o n Cote et al (1960) proposed a d i r e c t i o n a l spreading f u n c t i o n which i s dependent on both frequency and d i r e c t i o n based on data obtained from the Stereo Wave Observation P r o j e c t (SWOP). 1 + acos20 + bcos40] for|0|<7r/2 (3.14) 0 otherwise where a = 0.50 + 0.82exp(-^u*) b = 0.32exp(-2^ a) co = nondimensional frequency = Uco/g 3.2 RESPONSE TO DIRECTIONAL WAVES The e x c i t i n g f o r c e on a r i g i d s t r u c t u r e of f i n i t e l e n g t h due to a re g u l a r o b l i q u e wave t r a i n of frequency co and d i r e c t i o n 0 can be expressed as F j ( t ) = Hj(a>,0)7?(t) (3.15) where Hj(co,j3) i s a complex-valued system response f u n c t i o n given by eqn. (2.96b) as Hj(co , / 3 ) = p q l { l ^ C j ( u , f i ) r ( k l , p ) {ill'2} (3*16) Since the wave-structure i n t e r a c t i o n process i s assumed l i n e a r , we expect the value of any f o r c e at a given wave frequency to be due to wave components at that same frequency but propagating from a l l p o s s i b l e d i r e c t i o n s . The f o r c e spectrum S„ (co) i s thus r e l a t e d to the i n c i d e n t wave spectrum S (co,0) by 45 S p (co) = / |H.(co,/3) | 2S (u,0)d0 (3.17) where jH^ (co, j3) | 2 i s the t r a n s f e r f u n c t i o n . For convenience, the s u b s c r i p t j w i l l h e n c e f o r t h be dropped and i t should be noted that a l l f o l l o w i n g e x p r e s s i o n s are v a l i d f o r j = 1,2,3. Since the water su r f a c e e l e v a t i o n i s assumed to be a Gaussian p r o c e s s , the f o r c e s w i l l possess a Gaussian p r o b a b i l i t y d i s t r i b u t i o n . Using the form of the d i r e c t i o n a l wave spectrum given in eqn. (3.6), eqn. (3.17) reduces to The f a c t o r i n the brackets re p r e s e n t s a frequency dependent, d i r e c t i o n a l l y averaged t r a n s f e r f u n c t i o n , d i s measured from the p r i n c i p a l wave d i r e c t i o n 0 O and i s thus r e l a t e d to 0 by The mean square value of the for c e can be obtained by i n t e g r a t i n g the f o r c e spectrum over the frequency co. The root mean square value (rms) of the f o r c e represents a c h a r a c t e r i s t i c f o r c e from which extreme value p r e d i c t i o n s are u s u a l l y made. The e f f e c t s of wave d i r e c t i o n a l i t y on the wave loads can be expressed as a f o r c e r e d u c t i o n f a c t o r d e f i n e d as the r a t i o of the frequency dependent, d i r e c t i o n a l l y averaged t r a n s f e r f u n c t i o n i n s h o r t - c r e s t e d seas to the t r a n s f e r f u n c t i o n f o r l o n g - c r e s t e d , normally i n c i d e n t waves (3.18) 6 = 0 - 0 O (3.19) 4 6 7T J | H ( C J , 0) | 2 G ( c o , 0 ) d0 R 2 = — (3.20) F | H ( c o , 0 ) | 2 A body response r a t i o R̂ ^ can a l s o d e f i n e d as the r a t i o of the rms v a l u e of t h e response i n s h o r t - c r e s t e d seas t o c o r r e s p o n d i n g r e s u l t s f o r l o n g - c r e s t e d s e a s , t h a t i s / /|Z(co ,0) | 2G(w , e)S r ?(co)d0dco R^ = °~* (3.21) 7| Z(w, 0) | 2S (u)dw 0 n where Z(to ,0) i s the r e s p o n s e a m p l i t u d e o p e r a t o r d e f i n e d p r e v i o u s l y i n eqn. ( 2 . 5 6 ) . The f i r s t example c o n s i d e r e d i s the wave f o r c e on an i n f i n i t e s i m a l segment of a s t r u c t u r e w i t h the s i n u s o i d a l v a r i a t i o n a l o n g the l e n g t h n e g l e c t e d . The h o r i z o n t a l f o r c e at any a n g l e 0 i s p r o p o r t i o n a l t o c o s 0 . The t r a n s f e r f u n c t i o n can thus be e x p r e s s e d as | H(w , 0 ) | 2 = |H(o>,0)|2 c o s 2 0 (3.22) The f r e q u e n c y independent c o s i n e - p o w e r d i r e c t i o n a l s p r e a d i n g f u n c t i o n g i v e n i n eqn. (3.12) i s used i n t h i s s t u d y . S u b s t i t u t i o n of the e x p r e s s i o n s f o r the t r a n s f e r f u n c t i o n (3.22) and s p r e a d i n g f u n c t i o n (3.12) i n t o eqn. (3.20) y i e l d s * / 2 2 s R 2 = C'(s) / c o s 2 0 cos s ( 0 ) d 0 (3.23) F - T T / 2 For the c a s e of o b l i q u e mean i n c i d e n c e , the d i r e c t i o n a l d i s t r i b u t i o n w i l l be c u t o f f t o ensure t h a t the waves approach t h e s t r u c t u r e from one s i d e o n l y . I f the p r i n c i p a l 47 d i r e c t i o n of wave propagation i s zero, eqn. (3.23) can be i n t e g r a t e d to gi v e For any given s t r u c t u r e of a r b i t r a r y shape and f i n i t e l e n g t h , the dependence on 0 i s no longer e x p l i c i t and eqn. (3.20) w i l l have t o be i n t e g r a t e d n u m e r i c a l l y to give the f o r c e r e d u c t i o n f a c t o r . S u b s t i t u t i o n of the expre s s i o n f o r the t r a n s f e r f u n c t i o n (3.16) i n t o eqn. (3.20) y i e l d s C (s) (3.24) C ( s + 1 ) C (s) * / 2 2s J |C .(w,0) 1 2 r 2 ( k l , / 3 ) c o s z s ( 0 ) d 0 - T T/2 3 (3.25) I C j ^ O ) ! 2 f o r the cosine-power type spreading f u n c t i o n . 4 . RESULTS AND DISCUSSION 4.1 EXCITING FORCES, ADDED MASS AND DAMPING COEFFICIENTS A computer program based on the procedure d e s c r i b e d i n the preceding s e c t i o n s was used to determine the e x c i t i n g f o r c e s , hydrodynamic c o e f f i c i e n t s , and r e f l e c t i o n and tra n s m i s s i o n c o e f f i c i e n t s f o r s e v e r a l t e s t cases i n order to compare the accuracy and e f f i c i e n c y of the present method with other s o l u t i o n techniques. The f i r s t case c o n s i d e r e d i s a r e c t a n g u l a r s e c t i o n c y l i n d e r with a d r a f t to half-beam (b/a) r a t i o of 1, i n water of f i n i t e depth (d/a=2). F i g s . 9-12 show a comparison of the computed e x c i t i n g f o r c e and r e f l e c t i o n c o e f f i c i e n t s with the r e s u l t s obtained by Bai (1975) using a f i n i t e element technique. The c o e f f i c i e n t s are p l o t t e d as a f u n c t i o n of the angle of i n c i d e n c e 0 f o r ka=0.1, 0.2 and 0.4. Bai's (1975) r e s u l t s are represented by the s o l i d and dashed curves while the present r e s u l t s are shown as p o i n t s . The d i s c r e t i z e d s u r f a c e had 40 node p o i n t s on the free s u r f a c e , 20 node p o i n t s on the r a d i a t i o n s u r f a c e and 16 node p o i n t s on the body s u r f a c e y i e l d i n g a matrix of dimension N=76. I t took approximately 3.0s on the Amdahl V8-II c e n t r a l processor under the Michigan Terminal System (MTS) to solve f o r the e x c i t i n g f o r c e c o e f f i c i e n t s f o r a given wavenumber and angle of i n c i d e n c e . Bai (1975) used an 88 element, 325 node f i n i t e element mesh with a CPU time of 12s on an IBM 370 computer. The present procedure i s thus r e l a t i v e l y q u i t e 48 49 e f f i c i e n t . The computed sway and heave e x c i t i n g f o r c e c o e f f i c i e n t s and the r e f l e c t i o n c o e f f i c i e n t agree q u i t e c l o s e l y with Ba i ' s (1975) r e s u l t s . The r o l l e x c i t i n g moment c o e f f i c i e n t was c o n s i s t e n t l y g r e a t e r than that presented by Bai (1975) with a maximum d i f f e r e n c e of about 7.5%. The use of a much l a r g e r set of node p o i n t s d i d not s i g n i f i c a n t l y change the present r e s u l t s . The d i f f e r e n c e i s expected to d i m i n i s h with the use of a f i n e r mesh i n Bai's computations. From F i g s . 9-11 i t can be seen t h a t the e x c i t i n g f o r c e c o e f f i c i e n t s decrease with i n c r e a s i n g angle of i n c i d e n c e v a n i s h i n g at 0=90°. The maximum f o r c e or moment occurs at 0=0°. The heave e x c i t i n g f o r c e c o e f f i c i e n t was f a i r l y constant up to c e r t a i n angle before d e c r e a s i n g to zero at 0=90°, while the sway and r o l l e x c i t i n g f o r c e (or moment) c o e f f i c i e n t s at, any angle 0 seemed t o be p r o p o r t i o n a l to cos/3 f o r ka=0.1. The r e f l e c t i o n c o e f f i c i e n t decreases s l i g h t l y with i n c r e a s i n g angle of i n c i d e n c e before i n c r e a s i n g to one at 0=90°. The e x c i t i n g f o r c e and hydrodynamic c o e f f i c i e n t s of a r e c t a n g u l a r c y l i n d e r with a d r a f t of 0.265a i n water of i n f i n i t e depth were a l s o computed and compared with the r e s u l t s of G a r r i s o n (1984) i n F i g s . 13-21. G a r r i s o n (1984) used a Green's f u n c t i o n which s a t i s f i e s the f r e e s u r f a c e and r a d i a t i o n boundary c o n d i t i o n s and thus r e q u i r e s the d i s c r e t i z a t i o n of the c y l i n d e r s u r f a c e o n l y . The Green's f u n c t i o n used i n the present procedure i s r e l a t i v e l y simple 50 w h i l e the Green's f u n c t i o n used by G a r r i s o n (1984) i s q u i t e complex and i s o n l y v a l i d f o r water of i n f i n i t e depth. A water depth d=7r/k+b, where k i s the wavenumber i s used i n the present procedure to simulate i n f i n i t e water depth. The d i s c r e t i z e d s u r f a c e had 40 node p o i n t s on the fr e e s u r f a c e , 40 node p o i n t s on the r a d i a t i o n surface and 16 node p o i n t s on the body s u r f a c e . The c o e f f i c i e n t s are p l o t t e d as a f u n c t i o n of the frequency parameter ka f o r angles of i n c i d e n c e 0=0°, 30° and 60°. The computed e x c i t i n g f o r c e c o e f f i c i e n t s agree q u i t e w e l l with G a r r i s i o n ' s (1984) r e s u l t s . The added mass and damping c o e f f i c i e n t s g e n e r a l l y show good agreement with G a r r i s o n ' s r e s u l t s . The sway added mass c o e f f i c i e n t at 60° d e v i a t e d by as much as 15% while the r o l l damping c o e f f i c i e n t s d e v i a t e d s u b s t a n t i a l l y from G a r r i s o n ' s r e s u l t s with d i f f e r e n c e s of up to 25%. G a r r i s o n ' s r e s u l t s however agreed much b e t t e r with the Haskind r e l a t i o n s . The r e s u l t s were s l i g h t l y s e n s i t i v e to the l o c a t i o n of the r a d i a t i o n d i s t a n c e which was estimated e m p i r i c a l l y . The use of elements with higher order v a r i a t i o n s of the p o t e n t i a l should improve the accuracy of the present method. The e x c i t i n g f o r c e c o e f f i c i e n t s show the expected t e n d e n c i e s , d e c r e a s i n g with i n c r e a s i n g angle of i n c i d e n c e . The maximum r o l l moment occurs at about ka = ?r/4. This r e s u l t agrees with i n t u i t i o n s i n c e one would expect the maximum moment to occur when the trough of a wave i s at the o r i g i n and the c r e s t at the s i d e s of the c y l i n d e r . 51 The added mass c o e f f i c i e n t s tended to i n c r e a s e , while the damping c o e f f i c i e n t decreased with i n c r e a s i n g angle of inci d e n c e f o r most of the frequency range s t u d i e d . The damping c o e f f i c i e n t s should v a n i s h at 0=90° s i n c e the wave c r e s t s are normal to the a x i s of the c y l i n d e r and hence no energy i s propagated away from the c y l i n d e r i n the ±x d i r e c t i o n s . The e x c i t i n g f o r c e c o e f f i c i e n t s , hydrodynamic c o e f f i c i e n t s and wave amplitude r a t i o s of a semi-immersed c i r c u l a r c y l i n d e r i n water of i n f i n i t e depth were computed and are compared with the r e s u l t s of B o l t o n and U r s e l l (1973), and G a r r i s o n (1984) i n Tables 1-4. G a r r i s o n ' s r e s u l t s were estimated from the f i g u r e s presented i n h i s paper. The r e s u l t s are shown for ka=0.25, 0.75 and 1.25 with angles of i n c i d e n c e 0=0°, 35° and 55°. The boundary was modelled with 40 node p o i n t s on the f r e e s u r f a c e , 40 node p o i n t s on the r a d i a t i o n s u r f a c e and 16 s t r a i g h t l i n e segments on the surface of the c y l i n d e r . Agreement between the d i f f e r e n t methods i s g e n e r a l l y good with d i f f e r e n c e s of l e s s than 15%. I t i s i n t e r e s t i n g to note that the wave amplitude r a t i o s i n c r e a s e with angle of i n c i d e n c e . T h i s i n d i c a t e s t h a t as the wavelength along the c y l i n d e r decreases, the waves generated by the motions of the c y l i n d e r become more a m p l i f i e d . 52 4.2 MOTIONS OF AN UNRESTRAINED BODY The equations of motion were s o l v e d to g i v e the amplitudes of motion of a long f l o a t i n g box (a=7.5m, b=3m, l=75m) i n water of depth d=12m. The box i s assumed to be r i g i d and hence the added mass and damping c o e f f i c i e n t s f o r beam seas (/3=0°) are used. The mass of the box i s pV where V i s the d i s p l a c e d volume. The centre of g r a v i t y i s assumed to be at the s t i l l water l e v e l and the r o l l r a d i u s of g y r a t i o n i s given as 19.5m. F i g s . 22-24 show the amplitudes of motion f o r the sway, heave and r o l l modes r e s p e c t i v e l y . The amplitudes are p l o t t e d as a f u n c t i o n of ka for 0=0°, 30° and 60°. At low f r e q u e n c i e s (ka:$0.1), the sway and heave motions have the same amplitudes as the h o r i z o n t a l and v e r t i c a l motions of a p a r t i c l e at the f r e e s u r f a c e . The sway amplitude i s maximum as ka—>0 and decreases as ka i n c r e a s e s . The heave amplitude for beam seas i n c r e a s e s with ka up t o maximum before d e c r e a s i n g , while the response amplitudes f o r 0=30° and 60° decrease with i n c r e a s i n g ka. There are l o c a l zeros of the response f o r o b l i q u e waves corresponding to the zeros of the f a c t o r r ( k l , j 3 ) . The r o l l amplitude a t resonance i s e x c e s s i v e l y h i g h . T h i s i s because v i s c o u s damping which i s present i n p r a c t i c a l s i t u a t i o n s was n e g l e c t e d i n the computations. In s o l v i n g the equations of motion, i t was observed that the heave response i s uncoupled from the sway and r o l l responses while c o u p l i n g between the sway and r o l l modes was weak except c l o s e to the r o l l resonance frequency 53 where t h e r e i s a s u d d e n d r o p i n t h e sway a m p l i t u d e . 4.3 E F F E C T S O F D I R E C T I O N A L W A V E S T h e r e a r e two f a c t o r s t h a t c o n t r i b u t e t o t h e r e d u c t i o n of wave l o a d s e x p e r i e n c e d by l o n g s t r u c t u r e s i n s h o r t - c r e s t e d s e a s compared t o l o n g - c r e s t e d s e a s : ( 1 ) t h e s i n u s o i d a l v a r i a t i o n o f t h e wave f o r c e s a l o n g t h e l e n g t h o f t h e s t r u c t u r e , and ( 2 ) t h e v a r i a t i o n of t h e t w o - d i m e n s i o n a l f o r c e s w i t h a n g l e o f i n c i d e n c e f o r a g i v e n c r o s s - s e c t i o n . The i n t e g r a t i o n o f t h e t w o - d i m e n s i o n a l f o r c e a l o n g t h e l e n g t h o f the s t r u c t u r e r e s u l t s i n a r e d u c t i o n f a c t o r r ( k l , / 3 ) . The s q u a r e o f t h e r e d u c t i o n f a c t o r r ( k l , / 3 ) i s p l o t t e d as a f u n c t i o n of k l f o r /3=0°, 1 5 ° , 30° and 60° i n F i g . 6. F o r a g i v e n s t r u c t u r e of f i n i t e l e n g t h , t h e f a c t o r r ( k l , / 3 ) r e s u l t s i n t h e r e d u c t i o n o f t h e wave l o a d s p e r u n i t l e n g t h f o r o b l i q u e waves even i f t h e r e i s no v a r i a t i o n o f t h e s e c t i o n a l f o r c e w i t h a n g l e o f i n c i d e n c e . I t a l s o r e s u l t s i n t h e d e c r e a s e o f t h e wave l o a d s p e r u n i t l e n g t h as ka i n c r e a s e s i f we i g n o r e t h e v a r i a t i o n of t h e s e c t i o n a l f o r c e s w i t h k a . The v a r i a t i o n o f t h e s e c t i o n a l f o r c e s w i t h t h e f r e q u e n c y p a r a m e t e r ka and a n g l e of i n c i d e n c e /3 has been d i s c u s s e d p r e v i o u s l y i n s e c t i o n 4 . 1 . The c o m b i n a t i o n of t h e f a c t o r r ( k l , 0 ) w i t h t h e s e c t i o n a l f o r c e v a r i a t i o n w i t h a n g l e o f i n c i d e n c e r e s u l t s i n t h e t o t a l f o r c e r e d u c t i o n f a c t o r R_. F The f r e q u e n c y d e p e n d e n t f o r c e r e d u c t i o n f a c t o r Rp, has been computed f o r t h e l o n g f l o a t i n g box d e s c r i b e d i n s e c t i o n 4 . 2 . The computed R „ v a l u e s f o r t h e c o s i n e power t y p e e n e r g y 54 s p r e a d i n g f u n c t i o n i s p l o t t e d a s a f u n c t i o n o f ka i n F i g s . 2 5 ( a ) - ( c ) f o r t h e sway, heave and r o l l f o r c e s ( o r moment) r e s p e c t i v e l y . The r e s u l t s a r e shown f o r s=1,3,6 i n o r d e r t o a s s e s s t h e i n f l u e n c e o f t h e d e g r e e of wave s h o r t - c r e s t e d n e s s . A p r i n c i p a l d i r e c t i o n /3o = 0° was u s e d i n t h e c o m p u t a t i o n s . Simpson's r u l e was u s e d t o c a r r y o u t t h e n u m e r i c a l i n t e g r a t i o n i n eqn. (3.25) w i t h an i n t e r v a l of 1 0 ° . At low f r e q u e n c i e s , t h e heave f o r c e r e d u c t i o n f a c t o r a p p r o a c h e s a l i m i t i n g v a l u e o f one. T h i s c o n f i r m s t h e f a c t t h a t t h e heave e x c i t i n g f o r c e i s i n d e p e n d e n t o f d i r e c t i o n f o r low v a l u e s of ka. As ka ( o r k l ) i n c r e a s e s , t h e r e i s a s i g n i f i c a n t r e d u c t i o n o f t h e heave f o r c e m o s t l y due t o t h e f a c t o r r ( k l , / 3 ) . The s i n u s o i d a l v a r i a t i o n a l o n g t h e l e n g t h t h u s makes i t i m p o r t a n t t o a c c o u n t f o r d i r e c t i o n a l s p r e a d i n g p a r t i c u l a r l y f o r l o n g s t r u c t u r e s . I t c a n a l s o be seen from F i g s . 2 5 ( a ) - ( c ) t h a t a s s i n c r e a s e s , t h e f o r c e s a p p r o a c h t h e r e s u l t s f o r l o n g - c r e s t e d s e a s . B a t t j e s (1982) d e r i v e d an e x p r e s s i o n f o r t h e a s y m p t o t i c f o r m o f R p a t h i g h f r e q u e n c i e s . T h i s i s g i v e n a s R| = 2 7 r C ( s ) c o s 2 s / 3 0 A l a s k l ^ = ° (4.1) A t h i g h e r f r e q u e n c i e s (ka>1), t h e sway, h e a v e a n d r o l l f o r c e ( o r moment) r e d u c t i o n f a c t o r s a l l c o n v e r g e t o a v a l u e w hich i s s l i g h t l y l e s s t h a n t h e a s y m p t o t i c v a l u e . The sway and r o l l f o r c e ( o r moment) r e d u c t i o n f a c t o r s a p p r o a c h a v a l u e of 0.866 as k a —> 0 . T h i s r e s u l t was e x p e c t e d s i n c e t h e s e c t i o n a l 55 sway and r o l l e x c i t i n g f o r c e (or moment) i s p r o p o r t i o n a l to cos/3 at low f r e q u e n c i e s (ka<0.1). The f o r c e r e d u c t i o n f a c t o r s f o r a l l three modes decrease with i n c r e a s i n g ka up to a value of 0.4 at ka=2. The sway and heave for c e r e d u c t i o n f a c t o r s were a l s o computed f o r one case of o b l i q u e mean i n c i d e n c e (/3o = 30 o) and the r e s u l t s are shown in F i g s . 2 6 ( a ) - ( b ) . The r e d u c t i o n f a c t o r s f o r normal mean inc i d e n c e are i n c l u d e d f o r comparison. At low f r e q u e n c i e s , the sway f o r c e r e d u c t i o n f a c t o r has a value of 0.79 f o r /3o = 30° compared to 0.866 f o r normal mean i n c i d e n c e . The heave f o r c e r e d u c t i o n f a c t o r at ka=0 was 0.985 f o r /3o = 30° compared to 1.0 f o r /3o = 0°. The s l i g h t r e d u c t i o n of the heave f o r c e a r i s e s from the f a c t that the spreading f u n c t i o n was cut o f f to ensure that the waves approach the s t r u c t u r e from one s i d e o n l y . As ka i n c r e a s e s , the d i f f e r e n c e between the heave f o r c e r e d u c t i o n f a c t o r f o r o b l i q u e mean waves and normal mean waves increases, up to an asymptotic r a t i o of cos/3 0. The response r a t i o f o r the body motions has been computed f o r the case of the f l o a t i n g box s u b j e c t to a B r e t s c h n e i d e r spectrum with c o s i n e power energy spreading. The i n c i d e n t u n i d i r e c t i o n a l wave spectrum i s given as S ( w ) = T ^ f , V _ exp [ - | (4 ) " 4 ] (4.2) 1 6 f ° ( f / f 0 ) 5 4 f ° where H s i s the s i g n i f i c a n t wave height and f 0 i s the peak frequency. The r e s u l t s are p l o t t e d as a f u n c t i o n of s i n F i g s . 2 7 ( a ) - ( c ) f o r the sway, heave and r o l l responses 56 r e s p e c t i v e l y assuming normal mean i n c i d e n c e . A s i g n i f i c a n t wave height H =2m and a peak frequency f o=0.2Hz were used i n the computations. In the numerical i n t e g r a t i o n , f i v e f r e q u e n c i e s between 0.14Hz and 0.26Hz and an angle i n t e r v a l of 10° were used. The rms amplitudes i n l o n g - c r e s t e d seas are 0.22m, 0.32m and 0.60rad f o r the sway, heave and r o l l responses r e s p e c t i v e l y . F i g s . 27(a)-(c) show re d u c t i o n s of 43%, 42.5% and 41.5% i n the rms value of the sway, heave and r o l l responses r e s p e c t i v e l y i n s h o r t - c r e s t e d seas with s=1 compared to l o n g - c r e s t e d seas. As s i n c r e a s e s , the response r a t i o s approach a l i m i t i n g value of one i n d i c a t i n g that the amplitudes of motion of the s t r u c t u r e i n s h o r t - c r e s t e d seas approach the l o n g - c r e s t e d r e s u l t s as s—>-<=°. 5. CONCLUSIONS AND RECOMMENDATIONS 5.1 CONCLUSIONS The e f f e c t s of wave d i r e c t i o n a l i t y on the loads and motions of long s t r u c t u r e s has been s t u d i e d . A numerical method based on Green's theorem has been developed to compute the e x c i t i n g f o r c e s and hydrodynamic c o e f f i c i e n t s a s s o c i a t e d w i t h the i n t e r a c t i o n of a r e g u l a r o b l i q u e wave t r a i n with an i n f i n i t e l y long, f l o a t i n g semi-immersed c y l i n d e r of a r b i t r a r y shape. The method i s q u i t e general and can be a p p l i e d to cases of v a r i a b l e water depth. Numerical r e s u l t s o b t a i n e d from the present method have been compared with those o b t a i n e d by Bai (1975) u s i n g a f i n i t e element method f o r a r e c t a n g u l a r s e c t i o n c y l i n d e r i n water of f i n i t e depth. The present r e s u l t s have a l s o been compared to those o b t a i n e d f o r i n f i n i t e water depth by Bolton and U r s e l l (1973) u s i n g a m u l t i p o l e method f o r a semi-immersed c i r c u l a r c y l i n d e r as w e l l as G a r r i s o n (1984) using a Green's f u n c t i o n procedure f o r a r e c t a n g u l a r c y l i n d e r and a semi-immersed c i r c u l a r c y l i n d e r . The present method i s q u i t e e f f i c i e n t and g i v e s r e s u l t s which compare f a v o r a b l y w i t h a l l the p r e v i o u s r e s u l t s over a wide range of f r e q u e n c i e s c o v e r i n g the usual range of design c o n d i t i o n s . The present procedure i s not as e f f i c i e n t f o r very high f r e q u e n c i e s due to the l a r g e number of node p o i n t s r e q u i r e d to give accurate r e s u l t s . The p r e s e n t procedure i s 57 58 however not v a l i d f o r head seas s i n c e the wavelength along the body a x i s becomes of the same order of magnitude as a t y p i c a l c r o s s - s e c t i o n a l dimension. The two-dimensional r e s u l t s have been i n t e g r a t e d along the body a x i s to o b t a i n the wave loads on s t r u c t u r e s of f i n i t e l e n g t h . The wave loads and motions of a r i g i d s t r u c t u r e i n s h o r t - c r e s t e d seas have been obtained using the l i n e a r t r a n s f e r f u n c t i o n approach. The e f f e c t s of wave d i r e c t i o n a l i t y i s expressed as a frequency dependent, d i r e c t i o n a l l y averaged r e d u c t i o n f a c t o r f o r the wave loads and a response r a t i o f o r the body motions. The re d u c t i o n f a c t o r s have been eva l u a t e d n u m e r i c a l l y f o r the cosine-power type d i r e c t i o n a l spreading f u n c t i o n . Response r a t i o s were a l s o computed f o r a B r e t s c h n e i d e r i n c i d e n t wave spectrum with c o s i n e power spre a d i n g . For the given s t r u c t u r e , the sway and r o l l f o r c e r e d u c t i o n f a c t o r s v a r i e d from 0.87 at ka=0 to 0.41 at ka=2 for a co s i n e - s q u a r e d d i s t r i b u t i o n with normal mean i n c i d e n c e . The heave r e d u c t i o n f a c t o r v a r i e d from 1.0 at ka=0 to 0.40 at ka=2. The r a t i o of the amplitudes of motion of the s t r u c t u r e f o r the s p e c i f i e d s h o r t - c r e s t e d sea s t a t e with a c o s i n e - s q u a r e d d i s t r i b u t i o n were 57%, 57.5% and 58.5% of the response i n l o n g - c r e s t e d seas, f o r the sway, heave and r o l l modes r e s p e c t i v e l y . A f u r t h e r r e d u c t i o n of the f o r c e s and amplitudes of motions i s obtained f o r o b l i q u e mean waves { ( $ 0 * 0 ° ) . These r e d u c t i o n s are q u i t e s i g n i f i c a n t p a r t i c u l a r l y f o r long r e l a t i v e s t r u c t u r e lengths and need to be 59 considered i n the design p r o c e s s . As the parameter s which d e s c r i b e s the degree of s h o r t - c r e s t e d n e s s i n c r e a s e s , the loads and motions i n s h o r t - c r e s t e d seas approach the r e s u l t s f o r l o n g - c r e s t e d seas. 5 . 2 R E C O M M E N D A T I O N S F O R F U R T H E R S T U D Y There are s e v e r a l areas i n which f u r t h e r s t u d i e s c o u l d be made to improve the present method. The accuracy of the numerical scheme used i n the s o l u t i o n of the o b l i q u e wave d i f f r a c t i o n problem c o u l d be improved by u s i n g higher order elements. T h i s however r e q u i r e s an i n c r e a s e d computing e f f o r t . The present study c o n s i d e r e d the e f f e c t s of wave d i r e c t i o n a l i t y on the loads and motions of a r i g i d body even though hydrodynamic c o e f f i c i e n t s have been presented f o r s t r u c t u r e s with s i n u s o i d a l mode shapes. A numerical procedure c o u l d be developed to determine the dynamic response of a f l e x i b l e s t r u c t u r e such as a f l o a t i n g bridge i n s h o r t - c r e s t e d seas u s i n g the e x c i t i n g f o r c e s and hydrodynamic c o e f f i c i e n t s given by the p r e s e n t method. A d d i t i o n a l f o r c e s due to moorings and v i s c o u s damping could be i n c l u d e d i n the a n a l y s i s . The present method assumes a small amplitude wave t r a i n . For steep waves, n o n l i n e a r e f f e c t s have to be c o n s i d e r e d . Developing a theory that i n c o r p o r a t e s both the n o n l i n e a r i t y and d i r e c t i o n a l i t y of the waves i s however 60 q u i t e d i f f i c u l t . The present l i n e a r d i f f r a c t i o n theory f o r obl i q u e waves c o u l d be extended t o n o n l i n e a r waves and a h y b r i d method such as t h a t proposed by Dean (1977) can be used to i n c l u d e the e f f e c t s of wave d i r e c t i o n a l i t y . F i n a l l y , experimental i n v e s t i g a t i o n s c o u l d be c a r r i e d out to measure the loads and response of long s t r u c t u r e s i n s h o r t - c r e s t e d seas to h e l p v e r i f y the present t h e o r e t i c a l r e s u l t s . BIBLIOGRAPHY 1. Abramowitz, M. and Stegun, I.A. 1964. Handbook of Mathematical Functions. Dover P u b l i c a t i o n s , New York. 2. B a i , K.J. 1972. A v a r i a t i o n a l method i n p o t e n t i a l flows with a f r e e s u r f a c e . Report No. NA72-2, Co l l e g e of E n g i n e e r i n g , U n i v e r s i t y of C a l i f o r n i a , B e r k e l e y . 3. B a i , K.J. 1975. D i f f r a c t i o n of o b l i q u e waves by an i n f i n i t e c y l i n d e r . /. Fluid Mech. 6 8 , pp. 513-535. 4. B a t t j e s , J.A. 1982. E f f e c t s of s h o r t - c r e s t e d n e s s on wave loads on long s t r u c t u r e s . Applied Ocean Res earch. 4(3), pp. 165-172. 5. Bearman, P.W., Graham, J.M.R., and Singh, S. 1979. Forces on c y l i n d e r s i n h a r m o n i c a l l y o s c i l l a t i n g flow. In Mechanics of Wave Induced Forces on Cylinders, ed. T.L. Shaw, Pitman, London, pp. 437-449. 6. Bhattacharyya, R. 1978. Dynamics of Marine Vehicles. John Wiley and Sons, New York. 7. Black, J.L. and Mei, C C . 1970. S c a t t e r i n g and r a d i a t i o n of water waves. Rep. No. 121, Water Resources and Hydrodynamics Laboratory, Dept. of C i v i l E n g i n e e r i n g , Massachusetts I n s t i t u t e of Technology. 8. Bolton, W.E. and U r s e l l , F. 1973. The wave f o r c e on an i n f i n i t e l y long c y l i n d e r i n an o b l i q u e sea. /. Fluid Mech. 57, pp. 241-256. 9. Borgman, L.E. 1969. D i r e c t i o n a l s p e c t r a models f o r design use. Proc. Offshore Tech. Conf., Houston, Paper No. OTC1069, pp. 721-746. 10. Bryden, I.G. and Greated, C A . 1984. Hydrodynamic response of long s t r u c t u r e s to random seas. Proc. Symp. on Description and Modelling of Directional Seas, Copenhagen. 11. Cartwright, D.E. and Smith, N.D. 1964. Buoy techniques f o r o b t a i n i n g d i r e c t i o n a l wave s p e c t r a . Buoy Technol. , Mar. Technol. Soc, pp. 173-182. 12. Cote, L . J . et al. 1960. The d i r e c t i o n a l spectrum of a wind generated sea as determined from d a t a obtained by the Stereo Wave Observation P r o j e c t . Meteorological Paper, 2(6), C o l l e g e of E n g i n e e r i n g , New York U n i v e r s i t y . 13. D a l l i n g a , R.P., A a l b e r s , A.B., and van der Vegt, J.W.W. 61 62 1984. D e s i g n a s p e c t s f o r t r a n s p o r t o f j a c k - u p p l a t f o r m s on a b a r g e . Proc. Offshore Tech. Conf., H o u s t o n , P a p e r No. OTC4733, pp. 19 5 - 2 0 2 . 14. D a v i s , R.E. and R e g i e r , L.A. 1977. M e t h o d s f o r e s t i m a t i n g d i r e c t i o n a l wave s p e c t r a f r o m m u l t i - e l e m e n t a r r a y s . /. Marine Research. 3 5 ( 3 ) , pp. 453 - 4 7 7 . 15. Dean, R.G. 1977. H y b r i d m e thod o f c o m p u t i n g wave l o a d i n g . Proc. Offshore Tech. Conf., H o u s t o n , P a p e r No. OTC3029, pp. 4 8 3 - 4 9 2 . 16. F i n n i g a n , T.D. a n d Yamamoto, T. 1979. A n a l y s i s o f s e m i - s u b m e r g e d p o r o u s b r e a k w a t e r s . Proc. Civil Engineering in the Oceans IV, ASCE, San F r a n s i s c o , pp. 380-397. 17. F o r r i s t a l l , G.Z., Ward, E.G., C a r d o n e , V . J . , and Borgman, L.E. 1978. The d i r e c t i o n a l s p e c t r a and k i n e m a t i c s o f s u r f a c e g r a v i t y waves i n t r o p i c a l s t o r m D e l i a . /. Phys. Oceanography. 8, p p . 8 8 8 - 9 0 9 . 18. G a r r i s o n , C . J . 1969. On t h e i n t e r a c t i o n o f an i n f i n i t e s h a l l o w - d r a f t c y l i n d e r o s c i l l a t i n g a t t h e f r e e s u r f a c e w i t h a t r a i n o f o b l i q u e w a v es. /. Fluid Mech. 39, pp. 227- 2 5 5 . 19. G a r r i s o n , C . J . 1984. I n t e r a c t i o n o f o b l i q u e waves w i t h an i n f i n i t e c y l i n d e r . Applied Ocean Research. 6 ( 1 ) , pp. 4-15. 20. G e o r g i a d i s , C. 1984. Time a n d f r e q u e n c y d o m a i n a n a l y s i s o f m a r i n e s t r u c t u r e s i n s h o r t - c r e s t e d s e a by s i m u l a t i n g a p p r o p r i a t e n o d a l l o a d s . Proc. 3rd Int. Symp. on Offshore Mechanics and Arctic Engineering, New O r l e a n s , pp. 177-183. 2 1 . H a c k l e y , M.B. 1979. Wave f o r c e s i m u l a t i o n s i n random d i r e c t i o n a l s e a s . Proc. 2nd Int. Conf. on the Behaviour of Offshore Structures, BOSS' 79, L o n d o n , p p . 187-219. 22. H a s k i n d , M.D. 1953. Oscillation of a ship i n a calm sea. E n g l i s h t r a n s l a t i o n , S o c . o f N a v a l A r c h i t e c t s a n d M a r i n e E n g i n e e r s , T&R B u l l e t i n 1-12. 23. H a s s e l m a n n , K., D u n c k e l , M., a n d E w i n g , J.A. 1980. D i r e c t i o n a l wave s p e c t r a o b s e r v e d d u r i n g JONSWAP. /. Phys. Oceanography. 1 0 , pp. 1264-1280. 24. H o l t h u j s e n , L.H. 1981. The d i r e c t i o n a l e n e r g y d i s t r i b u t i o n o f w i n d g e n e r a t e d w aves a s i n f e r r e d f r o m s t e r e o p h o t o g r a p h i c o b s e r v a t i o n s o f t h e s e a s u r f a c e . Rep. No. 8 1 - 2 , D e p t . o f C i v i l E n g i n e e r i n g , D e l f t U n i v . o f T e c h n o l o g y . 6 3 25. Huntington, S.W. and Thompson, D.M. 1976. Forces on a l a r g e v e r t i c a l c y l i n d e r i n m u l t i - d i r e c t i o n a l random waves. Proc. Offshore Tech. Conf., Houston, Paper No. OTC2539, pp. 169-183. 26. I j i m a , T., Chou, C.R., and Yoshida, A. 1976. Method of a n a l y s i s f o r two-dimensional water wave problems. Proc. 15th Coastal Engineering Conference, Honolulu, pp. 2717-2736. 27. Isaacson, M. de St. Q. 1981. Nonlinear 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 . Coastal/Ocean E n g i n e e r i n g r e p o r t , Dept. of 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 Columbia. 28. K e l l o g g , O.D. 1929. Foundat i ons of Potential Theory. S p r i n g e r , B e r l i n . 29. Kim, W.D. 1965. On the harmonic o s c i l l a t i o n of a r i g i d body on the f r e e s u r f a c e . J. Fluid Mech. 2 1 , pp. 427-451. 30. Korvin-Kroukovsky, B.V. 1955. I n v e s t i g a t i o n of s h i p motions i n r e g u l a r waves. Trews. SNAME. 63, pp. 386-435. 31. Lambrakos, K.F. 1982. Marine p i p e l i n e dynamic response to waves from d i r e c t i o n a l wave s p e c t r a . Ocean Engineering. 9(4), pp. 385-405. 32. Leblanc, L.R. and Middleton, F.H. 1982. P i t c h - r o l l buoy wave d i r e c t i o n a l s p e ctra a n a l y s i s . In Measuring Ocean Waves, N a t l . Acad. Press, Washington, D.C., pp. 181-193. 33. Leonard, J.W., Huang, M.-C, and Hudspeth, R.T. 1983. Hydrodynamic i n t e r f e r e n c e between f l o a t i n g c y l i n d e r s in o b l i q u e seas. Applied Ocean Research. 5(3), pp. 158-167. 34. Longuet-Higgins, M.S., C a r t w r i g h t , D.E., and Smith, N.D. 1961. Observations of the d i r e c t i o n a l spectrum of sea waves u s i n g the motions of a f l o a t i n g buoy. In Ocean Wave Spectra, P r e n t i c e - H a l l , Englewood C l i f f s , New J e r s e y , pp. 111 -132. 35. MacCamy, R.C. 1964. The motions of c y l i n d e r s of shallow d r a f t . J. Ship Research. 7(3), pp. 1-11. 36. Mitsuyasu, H. et al. 1975. Observations of the d i r e c t i o n a l spectrum of ocean waves using a c l o v e r l e a f buoy. /. Phys. Oceanography. 5, pp. 750-760. 37. Mogridge, G.R. and Jamieson, W.W. 1976. Wave f o r c e s on square c a i s s o n s . Proc. 15th Coastal Engineering Conference, Honolulu, pp. 2271-2289. 64 38. Morison, J.R., O'Brien, M.P., Johnson, J.W., and Schaaf, S.A. 1950. The f o r c e s e x e r t e d by sur f a c e waves on p i l e s . Petrol eum Trans., AIME, 1 8 9 , pp. 149-157. 39. Newman, J.N. 1962. The e x c i t i n g f o r c e s on f i x e d bodies i n waves. /. Ship Research. 6 ( 3 ) , pp. 10-17. 40. Newman, J.N. 1977. Marine Hydrodynamics. MIT Press, Cambridge, Massachusetts. 41. Panicker, N.N. 1971. Determination of d i r e c t i o n a l s p e ctra of ocean waves from ocean a r r a y s . Rep. HEL1-18, Hydrogr. Eng. Lab., Univ. of C a l i f o r n i a , B e rkeley. 42. St. Denis, M. and P i e r s o n , W.J. 1953. On the motions of ships i n confused seas. Trans. SNAME. 6 1 , pp. 280-357. 43. Sand, S.E. 1980. Three-dimensional d e t e r m i n i s t i c s t r u c t u r e of ocean waves. S e r i e s paper 24, I n s t . Hydrody. and H y d r a u l i c Eng., Tech. Univ. of Denmark. 44. Sarpkaya, T. and Isaacson, M. 1981. Mechanics of Wave Forces on Offshore Structures. Van Nostrand Reinhold, New York. 45. Shinozuka, M., Fang, S.-L.S., and N i s h i t a n i , A. 1979. Time-domain s t r u c t u r a l response s i m u l a t i o n i n a s h o r t - c r e s t e d sea. /. Energy Res. Tech., Trans. ASME, 1 0 1 , pp. 270-275. 46. U r s e l l , F. 1949. On the heaving motion of a c i r c u l a r c y l i n d e r on the s u r f a c e of a f l u i d . Quart. J. Mech. Appl. Math. 2, pp. 218-231. APPENDIX I ANALYSIS TO DETERMINE OPTIMUM RADIATION DISTANCE C o n s i d e r t h e o b l i q u e waves g e n e r a t e d by t h e o s c i l l a t i o n o f an i n f i n i t e l y l o n g c y l i n d e r i n any one o f i t s t h r e e modes w i t h e a c h mode o f m o t i o n p e r i o d i c i n t i m e a s w e l l a s a l o n g t h e a x i s o f t h e c y l i n d e r . The p o t e n t i a l a s s o c i a t e d w i t h t h e f o r c e d m o t i o n s c a n be e x p r e s s e d as * ( x , y , z , t ) = R e [ # ( x , z ) exp{ i ( k y s i n / 3 - t o t ) } ] ( 1 1 ) where k i s t h e wavenumber w h i c h i s r e l a t e d t o t h e a n g u l a r f r e q u e n c y a> by t h e d i s p e r s i o n r e l a t i o n ( e q n . 2 . 1 1 ) . The t w o - d i m e n s i o n a l p o t e n t i a l 0 ( x , z ) c a n be e x p r e s s e d i n t e r m s o f an e i g e n f u n c t i o n e x p a n s i o n a s *(x'z> " A°C°cSsh(kd))] e x p ( i k x c o s / 3 ) + a c o s [ k ( z + d ) ] Vm cosTk d) e x p ( - k * x ) x>0 (12) m= 1 m where k a n d k* a r e wavenumbers d e f i n e d by m m -k t a n ( k d ) = (13) m m g a n d k* = [ k 2 + ( k s i n / 3 ) 2 ] l / 2 ( 1 4 ) m m A 0 i s t h e c o m p l e x a m p l i t u d e o f t h e p o t e n t i a l a t t h e f a r f i e l d a n d t h e c o e f f i c i e n t s A m a r e i n c l u d e d t o a c c o u n t f o r m t h e e v a n e s c e n t modes of wave m o t i o n n e a r t h e c y l i n d e r . 65 66 S i n c e t h e l o w e s t e i g e n v a l u e k* g i v e s t h e s l o w e s t d e c a y amongst a l l t h e e v a n e s c e n t modes, a de c a y f a c t o r c a n be d e f i n e d a s d ( x ) = e x p ( - k * x ) (15) where J < k*d < Tr (16) In o r d e r t o a c h i e v e a de c a y r a t e o f exp(-27r) o r 0.01 t i m e s t h e v a l u e a t x=0, t h e i n f i n i t e b o u n d a r y i s t r u n c a t e d a t a d i s t a n c e .X R g i v e n by X = ^ — (17) [ ( k s i n / 3 ) 2 + ( k * ) 2 ] 1 / 2 A maximum d i s t a n c e of four, t i m e s t h e d e p t h i s o b t a i n e d when k*d = 7r/2 and 0=0°. The above a p p r o x i m a t i o n was f o u n d t o g i v e good r e s u l t s i n water o f f i n i t e d e p t h . I n deep w a t e r , eqn. (17) g i v e s a d i s t a n c e w h i c h i s t o o l a r g e . B a i (1975) n o t e d t h a t an e i g e n f u n c t i o n e x p a n s i o n c a n n o t be u s e d i n water o f i n f i n i t e d e p t h f o r 0=0°. A p u l s a t i n g s o u r c e s h o u l d r a t h e r be u s e d t o o b t a i n u s e f u l i n f o r m a t i o n a b o u t t h e optimum d i s t a n c e f o r t r u n c a t i o n o f t h e i n f i n i t e b o u n d a r y . The f o l l o w i n g e m p i r i c a l e x p r e s s i o n f o r t h e r a d i a t i o n d i s t a n c e i s u s e d i n t h i s s t u d y f o r deep w a t e r c o n d i t i o n s X R = H (18) [(ksin0) 2 + ( 7 r / m a ) 2 ] 1 / 2 where a i s t h e h a l f - b e a m of t h e c y l i n d e r a n d m i s g i v e n a s 67 ka<0.5 0.5£ka<1.5 (19) ka>1.5 68 M.J pa2 \../pcoa2 0° present present ka r e s u l t s GAR r e s u l t s GAR 0.25 5 1 .97 2.10 0.57 0.60 35 2.04 2.16 0.46 0.53 55 2.08 2.21 0.30 0.38 0.75 5 1 .00 0.93 1.31 1 .39 35 1.14 1.19 1 .40 1 .51 55 1 .84 1 .74 1 .44 1 .56 1 .25 5 0.45 0.43 0.93 0.99 35 0.61 0.59 1.01 1.14 55 1 .09 0.93 1 .34 1 .40 T a b l e ! . Comparison of the sway added mass and damping c o e f f i c i e n t s of a s e m i - c i r c u l a r c y l i n d e r (d/a=°°) obtained i n the present study with the r e s u l t s of GAR (Garrison,1984) 6 9 M 2 2 / p a 2 X 2 2/pcoa 2 0° p r e s e n t p r e s e n t ka r e s u l t s B&U r e s u l t s B&U 0.25 5 1 .38 1 .38 1 .99 1 .96 35 1.61 1 .60 2.51 2.38 55 2.64 2.32 3.23 3.06 0.75 5 0.97 0.94 0.94 0.88 35 1 .04 1 .06 0.93 0.92 55 1 .43 1 .32 1.10 1 .02 1 .25 5 1 .01 0.98 0.49 0.44 35 0.92 0.90 0.39 0.40 55 0.98 0.90 0.46 0.42 T a b l e 2. C o m p a r i s o n o f t h e heave a d d e d mass a n d damping c o e f f i c i e n t s o f a s e m i - c i r c u l a r c y l i n d e r (d/a=°°) o b t a i n e d i n t h e p r e s e n t s t u d y w i t h t h e r e s u l t s o f B&U ( B o l t o n a n d U r s e l l , 1 9 7 3 ) 70 1^1 present present ka r e s u l t s GAR r e s u l t s GAR 0.25 5 0.75 0.77 0.18 0.19 35 0.63 0.65 0.18 0.19 55 0.44 0.46 0.18 0.19 0.75 5 1.17 1.18 0.85 0.89 35 1 .07 1.11 0.97 1 .02 55 0.94 0.94 1.17 1 .26 1 .25 5 0.99 0.99 1.19 1 .26 35 0.95 0.95 1 .37 1 .56 55 0.91 0.90 1 .89 1 .96 Table 3. Comparison of the sway e x c i t i n g f o r c e c o e f f i c i e n t and wave amplitude r a t i o of a s e m i - c i r c u l a r c y l i n d e r (d/a=°°) obtained i n the present study with the r e s u l t s of GAR (Garrison,1984) 71 |c2l U2| 0° present present ka r e s u l t s B&U r e s u l t s B&U 0.25 5 1 .40 1 .40 0.34 0.35 35 1.41 1 .40 0.42 0.43 55 1 .29 1 .32 0.58 0.58 0.75 5 0.95 0.94 0.71 0.70 35 0.87 0.87 0.78 0.80 55 0.77 0.76 1 .02 1 .00 1 .25 5 0.68 0.67 - 0.85 0.84 35 0.54 0.57 0.85 0.87 55 0.49 0.49 1.10 1 .07 Table 4. Comparison of the heave e x c i t i n g f o r c e c o e f f i c i e n t and wave amplitude r a t i o of a s e m i - c i r c u l a r c y l i n d e r (d/a=°°) obtained i n the present study with the r e s u l t s of B&U (Bolton and Urse l l , 1 9 7 3 ) F i g u r e 1. D e f i n i t i o n sketch f o r a r e c t a n g u l a r c y l i n d e r 73 i n c i d e n t wave r e f l e c t e d wave / ^ ( s w a y ) / t r a n s m i t t e d f£2(heave) wave _ * 3 ( r o l l ) ' ///////////// F i g u r e 2. D e f i n i t i o n component motions s k e t c h f o r f l o a t i n g c y l i n d e r showing S F S R s D F i g u r e 3 . Sketch of c l o s e d s u r f a c e 74 Figure 4. Sketch showing r e l a t i o n s h i p between x, £, and £' j=N3 j=N1 j»1 j=N4l i = N 2 l < l » l ' l ' l ' l - l ' l ' l ' l ' l ' l ' l ' l ' l ' l ' l ' l ' l ' l - J-N Figure 5 . A t y p i c a l boundary element mesh f o r a r e c t a n g u l a r c y l i n d e r ( b / a = 1 , d / a = 2 ) F i g u r e 7 . Sketch of a d i r e c t i o n a l wave spectrum 76 0(degrees) F i g u r e 8. D i r e c t i o n a l s p r e a d i n g f u n c t i o n f o r d i f f e r e n t v a l u e s o f t h e p a r a m e t e r s 2-1 1 ka=0.1 BAI(1975) Angle of incidence, /S (degrees) F i g u r e 9. Sway e x c i t i n g f o r c e c o e f f i c i e n t f o r a r e c t a n g u l a r c y l i n d e r (b/a=1,d/a=2) 0 —r- 30 15 30 45 60 75 90 Angle of incidence, /? (degrees) F i g u r e 10. Heave e x c i t i n g f o r c e c o e f f i c i e n t f o r re c t a n g u l a r c y l i n d e r (b/a=1,d/a=2) 0.5- - T —  0.4 0.3- o 0.2- 0 . 1 - 0.0-+ — ko=0.1 BAI(1975) — ka=0.2 BAI(1975) — ka=0.4 BAI(1975) El PRESENT RESULTS 15 30 45 60 /5 90 Angle of incidence, (1 (degrees) F i g u r e 11. R o l l e x c i t i n g moment c o e f f i c i e n t r e c t a n g u l a r c y l i n d e r (b/a=1,d/a=2) f o r 78 1.2 0.8 a: o.6- 0.4 ka=0.1 BAI(1975) ka=0.2 BAI(1975) ka=0.4 BAI(1975) PRESENT RESULTS —r- 15 -~T~ 30 —r- 45 60 i 75 90 Angle of incidence, /S (degrees) Fi g u r e 12. R e f l e c t i o n c o e f f i c i e n t f o r a r e c t a n g u l a r c y l i n d e r (b/a=1,d/a=2) 0.6 0 . 5 - 0.4 c_f 0.3 0.2 0 . 0 - /? = 0 ° A PRESENT RESULTS GARRISON (1984) F i g u r e 13. Sway e x c i t i n g f o r c e c o e f f i c i e n t f o r a rec t a n g u l a r c y l i n d e r (b/a=0.265,d/a=») 79 1.5 o 0.5- A PRESENT RESULTS GARRISON (1984) 1 1 1 r r— -i r • i i i i i A " " \ ^ ^ - i r - j • 0.5 1 k a 1.5 F i g u r e 14. Heave e x c i t i n g f o r c e c o e f f i c i e n t f o r r e c t a n g u l a r c y l i n d e r (b/a=0.265,d/a=°°) 0.35- 0.30 0.25 0.20- o 0.15- 0.10- 0.05 0.00 A PRESENT RESULTS a - - l = 0 ° GARRISON (1984) / ^ -TT-lo^^- 0.5 1.5 F i g u r e 15. R o l l e x c i t i n g moment c o e f f i c i e n t f o r a r e c t a n g u l a r c y l i n d e r (b/a=0.265,d/a=° o) 80 0.5 0.4 O >9. 0.3 H 0.2 H o.H 0.0 A PRESENT RESULTS GARRISON (1984) 0.5 — I — 1 k a — r ~ 1.5 F i g u r e 16. Sway added mass c o e f f i c i e n t f o r a re c t a n g u l a r c y l i n d e r (b/a=0.265,d/a=») 0.30 0.25 H 0.20 H o ^ 0.15- 0.10- 0.05 A PRESENT RESULTS GARRISON (1984) F i g u r e 17. Sway damping c o e f f i c i e n t f o r a r e c t a n g u l a r c y l i n d e r (b/a=0.265,d/a=») 81 3- D 1- A PRESENT RESULTS - GARRISON (1984) \fl=60° \ A 1 , , 1 1 , , , 1 - —g. ^ - ^ ^ ^ ^ ^ ^ ^ ^ i . . . . . . i i i 0.5 1 k a 1.5 Figure 18. Heave added mass c o e f f i c i e n t f o r a rectangular c y l i n d e r (b/a=0.265,d/a=») 2.5 2- o 3 v9. 1.5 0.5- o- A PRESENT RESULTS GARRISON (1984) A V V 0.5 1 k a 1.5 Figure 19. Heave damping c o e f f i c i e n t f o r a rectangular c y l i n d e r (b/a=0.265,d/a=») 8 2 0.4-1 A PRESENT RESULTS GARRISON (1984) ka F i g u r e 21. R o l l damping c o e f f i c i e n t f o r a r e c t a n g u l a r c y l i n d e r (b/a=0.265,d/a=») 83 f o r a long 8 4 F i g u r e 24. R o l l response amplitude operator f o r a long f l o a t i n g box (a=7.5m,b=3m,l=75m,d=12m) 0.5- F i g u r e 25. Force and moment r e d u c t i o n f a c t o r s f o r a long f l o a t i n g box (a=7.5m,b=3m,l=75m,d=12m) 85 ka F i g u r e 25.(cont.) Force and moment r e d u c t i o n f a c t o r s f o r a long f l o a t i n g box (a=7.5m,b=3m,l=75m,d=12m) 86 (a) SWAY F i g u r e 26. F o r c e r e d u c t i o n f a c t o r s f o r a l o n g f l o a t i n g box (a=7.5m,b=3m,l=75m,d=l2m) i n normal and o b l i q u e mean seas 87 F i g u r e 27. Response r a t i o s f o r a long f l o a t i n g (a=7.5m,b=3m,l=75m,d=l2m) 88 F i g u r e 27.(cont.) Response r a t i o s f o r a long f l o a t i n g box (a=7.5m,b=3m,l=75m,d=l2m)

Cite

Citation Scheme:

    

Usage Statistics

Country Views Downloads
Japan 4 0
United States 3 0
China 1 17
City Views Downloads
Tokyo 4 0
Beijing 1 0
Mountain View 1 0
Redmond 1 0
Ashburn 1 0

{[{ mDataHeader[type] }]} {[{ month[type] }]} {[{ tData[type] }]}

Share

Share to:

Comment

Related Items