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Numerical simulation of nonlinear waves and ship motions Fitz-Clarke, John R. 1986

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NUMERICAL SIMULATION OF NONLINEAR WAVES AND SHIP MOTIONS by JOHN R. FITZ-CLARKE B . A . S c , U n i v e r s i t y of B r i t i s h Columbia (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 M e c h a n i c a l E n g i n e e r i n g We ac c e p t t h i s t h e s i s as c o n f o r m i n g t o the r e q u i r e d s t a n d a r d UNIVERSITY OF BRITISH COLUMBIA Oc t o b e r , 1986 © JOHN R. FITZ-CLARKE, 1986 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 r e q u i r e m e n t s f o r an advanced degree a t the UNIVERSITY OF BRITISH COLUMBIA, I agree t h a t t h e L i b r a r y s h a l l make i t f r e e l y a v a i l a b l e f o r r e f e r e n c e and s t u d y . I f u r t h e r agree t h a t p e r m i s s i o n f o r e x t e n s i v e c o p y i n g 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 g r a n t e d by t h e 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 un d e r s t o o d t h a t c o p y i n g 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 g a i n s h a l l not be a l l o w e d w i t h o u t my w r i t t e n p e r m i s s i o n . Department of M e c h a n i c a l E n g i n e e r i n g UNIVERSITY OF BRITISH COLUMBIA 2075 Wesbrook P l a c e Vancouver, Canada V6T 1W5 Date: O c t o b e r , 1986 ABSTRACT A n u m e r i c a l method i s p r e s e n t e d f o r s i m u l a t i n g the be h a v i o u r of l a r g e a m p l i t u d e n o n l i n e a r f r e e s u r f a c e waves i n c l u d i n g wave b r e a k i n g . V a r i o u s i n i t i a l c o n d i t i o n s are g i v e n and the subsequent s u r f a c e p r o f i l e s a r e c a l c u l a t e d by a time s t e p p i n g s i m u l a t i o n . The f l o w f i e l d i s s o l v e d as a boundary v a l u e problem f o r the v e l o c i t y p o t e n t i a l u s i n g a complex v a r i a b l e method based on the Cauchy i n t e g r a l theorem. Waves of v a r y i n g shape, h e i g h t , and l e n g t h a re i n v e s t i g a t e d t o det e r m i n e the parameters n e c e s s a r y f o r wave b r e a k i n g and the r e s u l t i n g f l u i d v e l o c i t i e s . The t e c h n i q u e has proven t o be v e r y a c c u r a t e and s t a b l e . The method i s extended t o p r e d i c t the motions of a two d i m e n s i o n a l f l o a t i n g body i n l a r g e a m p l i t u d e seas a c c o u n t i n g f o r n o n - l i n e a r e f f e c t s and f l u i d - b o d y i n t e r a c t i o n . The presence of s i n g u l a r i t i e s a t the f r e e s u r f a c e i n t e r s e c t i o n p o i n t s was found t o s e v e r e l y l i m i t a c c u r a c y of the s o l u t i o n and a t t e m p t s t o overcome t h i s problem a r e d i s c u s s e d . An e x t e n s i o n t o handle t h r e e d i m e n s i o n a l s h i p s i s a l s o d e s c r i b e d . T a b l e of C o n t e n t s ABSTRACT i i LIST OF FIGURES i v NOMENCLATURE v i i ACKNOWLEDGEMENT x 1. INTRODUCTION 1 1.1 I n t r o d u c t i o n 1 1.2 L i t e r a t u r e Survey 3 1.2.1 Wave S o l u t i o n s 3 1.2.2 Body M o t i o n s 4 1 . 3 O b j e c t i v e s 6 2. POTENTIAL FLOW SOLUTION 9 2.1 I n t r o d u c t i o n 9 2.2 Cauchy I n t e g r a l Method 10 2.2.1 F o r m u l a t i o n 10 2.2.2 Boundary C o n d i t i o n s 13 2.2.3 S o l u t i o n 14 2.2.4 V e l o c i t i e s 15 2.2.5 I n t e r i o r 16 2.3 Test Case 17 3. WAVE SIMULATION 23 3.1 I n t r o d u c t i o n 23 3.1.1 The Problem 23 3.1.2 L i n e a r Wave Theory 24 3.1.3 S t o k e s Wave Theory 25 3.1.4 Waves i n Nature 26 3.2 N u m e r i c a l S o l u t i o n 27 3.2.1 F o r m u l a t i o n 27 i i i 3.2.2 C o n s t r u c t i o n of M a t r i x 29 3.2.3 C h o i c e of I n i t i a l C o n d i t i o n s 29 3.2.4 Time S t e p p i n g Procedure 30 3.2.5 Segment S i z e and Time Step 32 3.2.6 N u m e r i c a l Adjustments 34 3.2.7 Energy 35 3.2.8 Computer S o l u t i o n 37 3.3 R e s u l t s 37 4. FLOATING BODY MOTION 59 4.1 I n t r o d u c t i o n 59 4.2 N u m e r i c a l S o l u t i o n 59 4.2.1 F o r m u l a t i o n 59 4.2.2 The C l o s u r e Problem 62 4.2.3 Body P o s i t i o n 65 4.2.4 I n i t i a l C o n d i t i o n s 66 4.2.5 I n t e r s e c t i o n S i n g u l a r i t i e s 67 4.2.6 I n t e r s e c t i o n S o l u t i o n 68 4.2.7 A N u m e r i c a l P e r t u r b a t i o n C o r r e c t i o n .......70 4.3 R e s u l t s 72 4.4 E x t e n s i o n t o Three Dimensions 76 5. CONCLUSIONS AND RECOMMENDATIONS 99 REFERENCES 102 APPENDIX I 106 APPENDIX I I 1 08 APPENDIX I I I 113 APPENDIX IV 115 APPENDIX V 117 i v APPENDIX VI APPENDIX V I I APPENDIX V I I I APPENDIX IX APPENDIX X LIST OF FIGURES 1. Tour s h i p b e i n g b u f f e t t e d by " f r e a k waves" 8 2. G e n e r a l f l u i d domain i n complex p l a n e 19 3. D i s t r i b u t i o n of complex p o t e n t i a l on elements 19 4. F i n a l c o n s t r u c t i o n of complex p o t e n t i a l s o l u t i o n on c o n t o u r 19 5. Test case of c i r c u l a r c y l i n d e r i n u n i f o r m f l o w 20 6. Geometry and boundary c o n d i t i o n s f o r t e s t case 20 7. V e l o c i t y p o t e n t i a l and stream f u n c t i o n c a l c u l a t e d i n t e s t case 21 8. Comparison of n u m e r i c a l and t h e o r e t i c a l s o l u t i o n s f o r v e l o c i t y on c y l i n d e r s u r f a c e 22 9. D e f i n i t i o n of wave v a r i a b l e s 42 10. Regions of v a l i d i t y of wave t h e o r i e s 42 11. C l a s s i f i c a t i o n of b r e a k i n g waves 43 12. Flow c h a r t of wave s i m u l a t i o n a l g o r i t h m 44 13. D e f i n i t i o n of wave c o n t r o l volume 45 14. Two t y p e s of c o n t r o l volume d i s t o r t i o n r e q u i r i n g n o d a l p o i n t a djustment 45 15. CPU time per s t e p v e r s u s number of elements f o r wave s i m u l a t i o n 46 16. T r a n s l a t i n g f i f t h o r d e r S t o k e s wave w i t h H/L = 0.06 s i m u l a t e d f o r one p e r i o d 47 17. T r a j e c t o r y of marked p a r t i c l e i n S t o k e s f i f t h o r d e r wave w i t h H/L = 0.06 48 18. Deep water s p i l l i n g b r e a k e r s i m u l a t e d from c o s i n e i n i t i a l c o n d i t i o n H/L = 0.10 49 i v 19. Deep water p l u n g i n g b r e a k e r s i m u l a t e d from c o s i n e i n i t i a l c o n d i t i o n H/L = 0.13 50 20. S h a l l o w water s p i l l i n g b r e a k e r s i m u l a t e d * from c o s i n e i n i t i a l c o n d i t i o n H/L = 0.10 51 21. S h a l l o w water p l u n g i n g b r e a k e r s i m u l a t e d from c o s i n e i n i t i a l c o n d i t i o n H/L = 0.13 52 22. L a s t s i m u l a t e d s t e p f o r deep water waves of v a r y i n g h e i g h t r a t i o s 53 23. Maximum f l u i d p a r t i c l e v e l o c i t i e s v e r s u s time f o r deep water waves from c o s i n e i n i t i a l c o n d i t i o n s 54 24. Maximum f l u i d p a r t i c l e v e l o c i t i e s v e r s u s time f o r s h a l l o w water waves from c o s i n e i n i t i a l c o n d i t i o n s 55 25. F l u i d v e l o c i t i e s around p l u n g i n g j e t 56 26. C o n t r o l volume energy v e r s u s time f o r deep water b r e a k i n g wave from c o s i n e i n i t i a l c o n d i t i o n H/L = 0.13 57 27. C o n t r o l volume energy v e r s u s time f o r s h a l l o w water b r e a k i n g wave from c o s i n e i n i t i a l c o n d i t i o n H/L = 0.13 58 28. C o n t r o l volume f o r body motion s i m u l a t i o n 78 29. D e f i n i t i o n of c o o r d i n a t e systems 72 30. D e c o m p o s i t i o n i n t o f o u r independent problems 79 31. Flow c h a r t f o r body motion s i m u l a t i o n 80 32. Boundary c o n d i t i o n s t e s t e d 81 33. S i m u l a t i o n of r o l l motion i n calm water 82 v 34. P l o t of r o l l h i s t o r y i n calm water 83 35. S i m u l a t i o n of motion on wave of H/L = 0.04 84 36. S i m u l a t i o n of motion on wave of H/L = 0.04 85 37. P r e s s u r e d i s t r i b u t i o n on h u l l H/L = 0.04 87 38. P l o t of motions f o r H/L = 0.04 89 39. S i m u l a t i o n of motion on wave of H/L = 0.08 ....90 40. S i m u l a t i o n of motion on wave of H/L = 0.08 91 41. P l o t of motions f o r H/L = 0.08 93 42. S i m u l a t i o n of motion on wave of H/L = 0.12 94 43. S i m u l a t i o n of motion on wave of H/L = 0.12 95 44. P l o t of motions f o r H/L = 0.12 96 45. R e s u l t of z e r o i n i t i a l a n g u l a r v e l o c i t y 97 46. E q u i v a l e n t p r i s m a t i c r e p r e s e n t a t i o n of s h i p 98 v i NOMENCLATURE a^,By = body a c c e l e r a t i o n a ^ j = r e a l p a r t of m a t r i x element b ^ j = i m a g i n a r y p a r t of m a t r i x element c = phase v e l o c i t y of wave C = c o n t o u r of i n t e g r a t i o n d = water depth E^ = k i n e t i c energy Ep = p o t e n t i a l energy E f c = t o t a l energy F x , F = body f o r c e s g = g r a v i t a t i o n c o n s t a n t G = c e n t r e of g r a v i t y of body H = wave h e i g h t h ^ j = r e a l m a t r i x I = mass moment of i n e r t i a of body k = wave number = 2-n/L L = wave l e n g t h m = body mass M = body moment about G n = normal u n i t v e c t o r N = t o t a l number of elements NB = number of bottom elements NH = number of h u l l elements NS = number of f r e e s u r f a c e elements NV = number of v e r t i c a l s i d e elements vi 1 P = f l u i d p ressure Q = volume flow r a t e per u n i t width R = r a d i u s from G to body s u r f a c e s = arc len g t h running v a r i a b l e t = time At = time s t e p i n t e r v a l T = wave p e r i o d t„ = time to wave breaking u = h o r i z o n t a l f l u i d v e l o c i t y u Q = h o r i z o n t a l body v e l o c i t y v = v e r t i c a l f l u i d v e l o c i t y v G = v e r t i c a l body v e l o c i t y V n = normal component of body v e l o c i t y V = volume w = complex f l u i d v e l o c i t y = u - i v * w = complex conjugate of w x = h o r i z o n t a l d i s t a n c e Ax = element s i z e y = v e r t i c a l d i s t a n c e z = complex s p a t i a l c o o r d i n a t e = x+iy a = i n c l u d e d angle at node 0 = complex p o t e n t i a l = <p+i\p <j> . = v e l o c i t y p o t e n t i a l \jj = stream f u n c t i o n * " i j = c o m P l e x i n f l u e n c e c o e f f i c i e n t p = f l u i d d e n s i t y v i i i e = r o l l a n g l e V = wave e l e v a t i o n above s t i l l water A. . 1 3 = i n f l u e n c e c o e f f i c i e n t n. . 1 D = i n f l u e n c e c o e f f i c i e n t = a n g u l a r f r e q u e n c y = 2TI/T = f l u i d domain i x ACKNOWLEDGEMENT The a u t h o r would l i k e t o ext e n d s i n c e r e thanks t o h i s s u p e r v i s o r Dr. Sander M. C a l i s a l f o r p a t i e n t guidance throughout the co u r s e of t h i s work. The au t h o r would a l s o l i k e t o e x p r e s s h i s g r a t i t u d e t o the B r i t i s h Columbia S c i e n c e C o u n c i l and the Defence Research E s t a b l i s h m e n t A t l a n t i c , H a l i f a x f o r f i n a n c i a l s u p port which made t h i s work p o s s i b l e . x 1. INTRODUCTION 1.1 INTRODUCTION D e s p i t e advances i n s h i p d e s i g n , s m a l l v e s s e l s are s t i l l no match f o r rough seas and c a p s i z i n g s c o n t i n u e t o o c c u r each year w i t h the l o s s of l i v e s and p r o p e r t y . Fundamental t o r e d u c i n g the r i s k of such t r a d g e d i e s i s a b e t t e r u n d e r s t a n d i n g of the k i n e m a t i c s and dynamics of extreme waves and the r e s u l t i n g f o r c e s and response of a v e s s e l t o them. C a p s i z i n g of a v e s s e l may o c c u r due t o any of s e v e r a l phenomena i n c l u d i n g extreme r o l l i n g i n beam seas, r o l l e x c i t e d by a n e a r - r e s o n a n t e n c o u n t e r f r e q u e n c y w h i l e underway i n f o l l o w i n g or q u a r t e r i n g seas ( t h e M a t h i e u e f f e c t ) , or b r o a c h i n g or p i t c h - p o l i n g i n v o l v i n g s t e e p o v e r t a k i n g s t e r n waves and l o s s of d i r e c t i o n a l s t a b i l i t y . These phenomena f r e q u e n t l y r e s u l t i n water on deck and subsequent d o w n f l o o d i n g of deck o p e n i n g s . Model e x p e r i m e n t s of f i s h i n g b o a t s c a r r i e d out a t the B.C. R e s e a r c h Ocean E n g i n e e r i n g C e n t r e have demonstrated t h a t extreme r o l l c a p s i z i n g i n beam seas i s v e r y u n l i k e l y f o r a v e s s e l l o a d e d w i t h i n the recommended l i m i t s . No such c a p s i z i n g s have been ob s e r v e d i n the b a s i n under e x t e n s i v e t e s t s i n v o l v i n g l a r g e a m p l i t u d e r e g u l a r wave c o n d i t i o n s [ 2 6 ] . The presence of b r e a k i n g waves i n the v i c i n i t y of the v e s s e l , however, can l e a d t o v e r y l a r g e a d d i t i o n a l f o r c e s from the p l u n g i n g j e t i m p a c t i n g on the s i d e , p o s s i b l y s u f f i c i e n t t o cause 1 2 c a p s i z i n g . Fishermen who have s u r v i v e d a c c i d e n t s have o c c a s i o n a l l y r e p o r t e d b e i n g h i t by " f r e a k waves", t h a t i s one or more waves u n u s u a l l y l a r g e r than those i n the normal p r e c e d i n g sea s t a t e . Such anomalous waves have indeed been documented ( f i g u r e 1) and may have c o n t r i b u t e d as w e l l t o u n w i t n e s s e d a c c i d e n t s . The p r e s e n t work c o n c e n t r a t e s on ex a m i n i n g the motions of a v e s s e l i n extreme beam seas. C u r r e n t s t a b i l i t y c r i t e r i a a r e u s u a l l y based on q u a s i - s t a t i c d e f i n i t i o n s i n v o l v i n g the m e t a c e n t r i c h e i g h t and c r i t i c a l r o l l a n g l e s i n calm water. These r u l e s a r e l a r g e l y e m p i r i c a l and become somewhat m e a n i n g l e s s i n dynamic c o n d i t i o n s e n c o u n t e r e d i n the r e a l sea, e s p e c i a l l y where l a r g e s t e e p waves a r e i n v o l v e d . More r e c e n t l y , e f f o r t s have been made t o d e f i n e s t a b i l i t y i n terms of dynamic parameters but much work remains t o be done. O c c a s i o n a l l y model t e s t s are performed when e x p e r i e n c e w i t h a p a r t i c u l a r d e s i g n i s l a c k i n g , however, such t e s t s a r e e x p e n s i v e and time consuming, and f r e q u e n t l y l i m i t e d by f a c i l i t y s i z e and equipment. The t h e o r e t i c a l a n a l y s i s of wave and body motions p r o v i d e s i m p o r t a n t i n s i g h t i n t o the fundamental p r o c e s s e s i n v o l v e d and p e r m i t s e s t i m a t e s of dynamic b e h a v i o u r i n c o n d i t i o n s t h a t cannot r e a d i l y be t e s t e d . I n a d d i t i o n , the r e l a t i v e e f f e c t s of t h e d i f f e r e n t g o v e r n i n g parameters can be seen and a s s e s s e d i n t h e o r e t i c a l models whereas i n a c t u a l model t e s t s the i n f l u e n c e of c o n t r i b u t i n g e f f e c t s u s u a l l y cannot r e a d i l y be decomposed. 3 1.2 LITERATURE SURVEY 1.2.1 WAVE SOLUTIONS A n a l y t i c a l s o l u t i o n s t o f r e e s u r f a c e waves have been de v e l o p e d t o a f a i r l y h i g h degree b e g i n n i n g w i t h the c l a s s i c a l l i n e a r i z e d t h e o r y of A i r y . S t o k es (1847) d e v e l o p e d a p e r t u r b a t i o n e x p a n s i o n t o e x t e n d the s o l u t i o n t o f i n i t e a m p l i t u d e s t e a d y waves and c a l c u l a t e d the t h i r d o r d e r c o r r e c t i o n . To date many h i g h e r o r d e r s o l u t i o n s have been c a l c u l a t e d , more r e c e n t l y u s i n g a computer t o p e r f o r m the c o e f f i c i e n t a r i t h m e t i c as by Schwartz (1974) i n an attempt t o f i n d the e l u s i v e h i g h e s t p o s s i b l e s t e a d y p r o g r e s s i v e wave. Other n o n l i n e a r t h e o r i e s have been d e v e l o p e d i n c l u d i n g Dean's stream f u n c t i o n t h e o r y f o r i n t e r m e d i a t e and deep water and c n o i d a l t h e o r y f o r s h a l l o w w a t e r . Each method has i t s l i m i t a t i o n s and the most a p p r o p r i a t e depends on the i n t e n d e d a p p l i c a t i o n . Sarpkaya and I s a a c s o n (1981) p r o v i d e an o v e r v i e w of t h e s e and o t h e r wave t h e o r i e s . P r o g r e s s i n t r a n s i e n t wave b e h a v i o u r has so f a r r e q u i r e d n u m e r i c a l s i m u l a t i o n on a computer. The f i r s t a ttempt was a marker and c e l l s o l u t i o n t o the i n c o m p r e s s i b l e N a v i e r S t o k e s e q u a t i o n s c a r r i e d out by Harlow i n 1965 a t the Los Alamos L a b o r a t o r i e s . The method r e q u i r e d u n r e a l i s t i c a l l y h i g h v i s c o s i t y t o compensate f o r n u m e r i c a l i n s t a b i l i t y and a c c u r a c y was poor. Chan and S t r e e t (1970) improved the g r i d a d j u s t m e n t s and a c h i e v e d a r e a s o n a b l e s i m u l a t i o n of a s o l i t a r y wave on a beach up t o the p o i n t where the f r e e 4 s u r f a c e became v e r t i c a l . On a s e p a r a t e f r o n t , L o n g u e t - H i g g i n s and C o k e l e t (1976) c o n c e n t r a t e d on the i n v i s c i d s o l u t i o n u s i n g a p o t e n t i a l f l o w boundary i n t e g r a l method based on Green's theorem t o s i m u l a t e one wavelength of a p e r i o d i c deep water b r e a k i n g wave, a g a i n up t o a v e r t i c a l f a c e . F u r t h e r a p p l i c a t i o n s of t h e i r work may be found i n L o n g u e t - H i g g i n s (1977). V i n j e and B r e v i g (1980) extended the s o l u t i o n by d e v e l o p i n g a complex v a r i a b l e boundary i n t e g r a l method based on the Cauchy theorem t o s i m u l a t e the complete b r e a k i n g wave i n f i n i t e water depth i n c l u d i n g o v e r t u r n i n g of the c r e s t . V i n j e and B r e v i g (1981a) d e s c r i b e how the method might be used t o n u m e r i c a l l y e s t i m a t e b r e a k i n g wave f o r c e s on a f i x e d o b j e c t and some ex p e r i m e n t s t o measure such f o r c e s a r e p r e s e n t e d i n K j e l d s e n and Myrhaug (1979), and K j e l d s e n (1981). 1.2.2 BODY MOTIONS C a l c u l a t i o n of f l o a t i n g body motions poses a v e r y d i f f i c u l t m a t h e m a t i c a l problem owing t o the c o m p l e x i t y and n o n l i n e a r i t y of the g o v e r n i n g e q u a t i o n s . K o r v i n - K r o u k o v s k y (1955) p r e s e n t e d the c o u p l e d e q u a t i o n s of body motion f o r the s i x degrees of freedom, however, a n a l y t i c a l d e t e r m i n a t i o n of f o r c e s , added masses, and damping c o e f f i c i e n t s were o n l y c r u d e l y p o s s i b l e i n i d e a l i z e d c a s e s . A n a l y t i c a l s o l u t i o n s d e v e l o p e d t o date have i n v a r i a b l y r e q u i r e d some form of l i n e a r i z a t i o n , where body and f r e e s u r f a c e boundary c o n d i t i o n s a r e a p p l i e d on f i x e d s u r f a c e s 5 and body ge o m e t r i e s a r e s i m p l e shapes such as c i r c u l a r c y l i n d e r s or s p h e r e s . Examples i n c l u d e MacCamy (1964), U r s e l l (1964), Lee (1969), and M a s k e l l and U r s e l l (1970). Wehausen (1971) g i v e s an o v e r v i e w of t h e s e and o t h e r f o r m u l a t i o n s , and a d d i t i o n a l r e f e r e n c e s . A p p l i c a t i o n s of t h e s e methods a r e , of c o u r s e , l i m i t e d t o p e r i o d i c s m a l l a m p l i t u d e waves and m o t i o n s . N u m e r i c a l p a n e l methods based on Green's theorem have been used by Kim (1966), and G a r r i s o n (1975) t o handle the l i n e a r i z e d , s t e a d y harmonic motions of a r b i t r a r i l y shaped b o d i e s . The methods d e s c r i b e d so f a r are frequency-domain t e c h n i q u e s which p r e d i c t the response i n r e g u l a r monochromatic waves. For the more g e n e r a l wave c o n d i t i o n s a t r a n s f e r f u n c t i o n can be c o n s t r u c t e d and used t o o b t a i n a f r e q u e n c y spectrum of motions g i v e n an i n p u t spectrum of the sea s t a t e . S t a t i s t i c a l q u a n t i t i e s can then be e s t i m a t e d . These t e c h n i q u e s a r e n e c e s s a r i l y l i n e a r , y e t can p r o v i d e r e a s o n a b l e r e s u l t s i n waves t h a t a r e not too l a r g e , and are w i d e l y used i n p r a c t i c e . D e t a i l s may be found i n B h a t t a c h a r y y a (1979) and Newman (1980). The c a l c u l a t i o n of t r a n s i e n t n o n l i n e a r motions a g a i n r e q u i r e s a p p l i c a t i o n of a n u m e r i c a l t i m e - s t e p p i n g s i m u l a t i o n from g i v e n i n i t i a l c o n d i t i o n s . Examples i n c l u d e F a l t i n s e n ( 1 9 77), Chapman (1979), and I s a a c s o n (1982). V i n j e and B r e v i g (1981b,c) extended t h e i r n o n l i n e a r wave s i m u l a t i o n t e c h n i q u e t o i n c l u d e the presence of an a r b i t r a r y two d i m e n s i o n a l f l o a t i n g body and d e s c r i b e i t s a p p l i c a t i o n t o a 6 h e a v i n g c y l i n d e r . Greenhow, B r e v i g , and T a y l o r (1982) a p p l i e d the method t o s t u d y the extreme motions of a r o t a t i n g wave energy d e v i c e , however, f r e e s u r f a c e b e h a v i o u r near the body had t o be s p e c i f i e d e m p i r i c a l l y . V e ry few q u a n t i t a t i v e r e s u l t s of body motion s i m u l a t i o n s have been p r e s e n t e d i n the l i t e r a t u r e making comparisons w i t h p r e v i o u s work d i f f i c u l t . S e v e r a l problems have y e t t o be overcome i n v o l v i n g the c h o i c e of i n i t i a l c o n d i t i o n s and the p r e s ence of s i n g u l a r p o i n t s where the body i n t e r s e c t s the f r e e s u r f a c e . T h i s t h e s i s a d d r e s s e s t h e s e problems, and v a r i o u s a t t e m p t s t o handle them a r e d i s c u s s e d . 1.3 OBJECTIVES A t h e o r e t i c a l model f o r p r e d i c t i n g s h i p motions must s o l v e f o r the c o u p l e d body motions and f l u i d b e h a v i o u r . An e x a c t s o l u t i o n i n c l u d i n g s m a l l s c a l e f l u i d m otions and v i s c o u s e f f e c t s i s beyond the s t a t e of the a r t and v a r i o u s a p p r o x i m a t i o n s must be made t o keep th e mathematics t r a c t i b l e and w i t h i n the c a p a c i t y of a v a i l a b l e computing r e s o u r c e s . The u s u a l a ssumptions are t h a t the f l u i d i s i n c o m p r e s s i b l e and i n v i s c i d and thus s a t i s f i e s L a p l a c e ' s e q u a t i o n f o r a v e l o c i t y p o t e n t i a l . The f i r s t t a s k i n the p r e s e n t work i s t o d e v e l o p a n u m e r i c a l t e c h n i q u e f o r s o l v i n g L a p l a c e ' s e q u a t i o n and c a l c u l a t i n g the v e l o c i t y p o t e n t i a l i n a g e n e r a l two d i m e n s i o n a l f l u i d domain under a r b i t r a r y boundary c o n d i t i o n s u s i n g a complex v a r i a b l e boundary i n t e g r a l method. T h i s 7 L a p l a c e s o l v i n g r o u t i n e i s then used t o c a l c u l a t e the f l u i d v e l o c i t i e s of a f r e e s u r f a c e wave a t d i s c r e t e time i n t e r v a l s and a l l o w a t i m e - s t e p p i n g s i m u l a t i o n of the b e h a v i o u r of an a r b i t r a r y wave up t o and i n c l u d i n g b r e a k i n g . The e f f e c t of d i f f e r e n t i n i t i a l wave c o n d i t i o n s can then be a s s e s s e d t o e s t a b l i s h a domain of dependence of b r e a k i n g waves and the d e t a i l e d k i n e m a t i c s of t h e f l o w development. F i n a l l y , the wave s i m u l a t i o n t e c h n i q u e i s extended t o p e r m i t the i n c l u s i o n of an a r b i t r a r y two d i m e n s i o n a l body on the f r e e s u r f a c e . F l u i d v e l o c i t i e s and p r e s s u r e s a r e c a l c u l a t e d around the body a t each time s t e p t o det e r m i n e the r e s u l t i n g hydrodynamic f o r c e s . The e q u a t i o n s of r i g i d body motion a r e then i n t e g r a t e d a t each time s t e p t o c a l c u l a t e a time h i s t o r y of the n o n l i n e a r body response i n the wave f i e l d . 8 Figure 1. Tour ship being buffetted by "freak waves". 2. POTENTIAL FLOW SOLUTION 2. 1 INTRODUCTION The e x a c t c a l c u l a t i o n of an a r b i t r a r y f l o w f i e l d i n c l u d i n g v i s c o u s and t u r b u l e n t e f f e c t s i s an e x t r e m e l y d i f f i c u l t t a s k due t o the c o m p l e x i t y of the g o v e r n i n g N a v i e r - S t o k e s e q u a t i o n s and can o n l y be done i n v e r y l i m i t e d c a s e s . In g e n e r a l , one must make assumptions t h a t s i m p l i f y the problem and p e r m i t a t r a c t i b l e m a t h e m a t i c a l s o l u t i o n . The u s u a l assumptions a r e t h a t the f l u i d i s i n c o m p r e s s i b l e and i r r o t a t i o n a l , the l a t t e r a consequence of n e g l e c t i n g the e f f e c t s of v i s c o s i t y . These two c o n d i t i o n s d i c t a t e t h a t the f l o w f i e l d s a t i s f i e s L a p l a c e ' s e q u a t i o n V** - 0 { 2 ' 1 ) or i n C a r t e s i a n c o o r d i n a t e s +iii - o 3x2 3 y 2 f o r a s c a l a r v e l o c i t y p o t e n t i a l 0. V e l o c i t i e s a r e then found as the g r a d i e n t of <p. The problem i s completed by s p e c i f y i n g the a p p r o p r i a t e boundary c o n d i t i o n s and s o l v i n g the r e s u l t i n g boundary v a l u e problem. A n a l y t i c a l s o l u t i o n s t o (2.1) a r e p o s s i b l e i n many ca s e s u s i n g c l a s s i c a l methods such as s e p a r a t i o n of v a r i a b l e s , l i n e a r s u p e r p o s i t i o n of s o u r c e s and v o r t i c e s , p e r t u r b a t i o n methods, and c o n f o r m a l mappings, however, f o r t r a n s i e n t problems i n v o l v i n g c o m p l i c a t e d g e o m e t r i e s and boundary c o n d i t i o n s one must r e s o r t t o a n u m e r i c a l 9 10 p r o c e d u r e . S e v e r a l n u m e r i c a l methods have been deve l o p e d over the y e a r s t o s o l v e L a p l a c e ' s e q u a t i o n i n c l u d i n g the t r a d i t i o n a l f i n i t e d i f f e r e n c e and f i n i t e element methods which s o l v e f o r the p o t e n t i a l a l o n g a g r i d of d i s c r e t e i n t e r i o r p o i n t s . More r e c e n t l y , a t t e n t i o n has t u r n e d t o boundary i n t e g r a l methods which have the p o w e r f u l advantage t h a t the v a l u e s of <p and an o r t h o g o n a l f u n c t i o n a r e c a l c u l a t e d o n l y on the domain boundary. The problem i s t h e r e b y reduced by one di m e n s i o n a l l o w i n g much g r e a t e r r e s o l u t i o n f o r a g i v e n c o m p u t a t i o n a l e f f o r t . Element g e n e r a t i o n i s a l s o g r e a t l y s i m p l i f i e d . F u r t h e r m o r e , i n many problems of f l u i d dynamics one i s i n t e r e s t e d o n l y i n the f l u i d v e l o c i t i e s and p r e s s u r e s a t the boundary. The i n t e r i o r f l o w f i e l d can be found i f d e s i r e d knowing o n l y the boundary v a l u e s , however, t h i s i s u s u a l l y of l i t t l e p r a c t i c a l i n t e r e s t . The method used i n t h i s work i s a complex v a r i a b l e boundary i n t e g r a l method based on the Cauchy i n t e g r a l theorem. 2.2 CAUCHY INTEGRAL METHOD 2.2.1 FORMULATION The o b j e c t i v e of the n u m e r i c a l p r o c e d u r e i s t o o b t a i n <j> v a l u e s around the c o n t o u r C, however, t h i s a l o n e i s i n s u f f i c i e n t t o dete r m i n e v e l o c i t i e s a t the boundary s i n c e d i f f e r e n t i a t o n a l o n g C p r o v i d e s o n l y t h e t a n g e n t i a l 11 v e l o c i t y . The normal component must a l s o be d e t e r m i n e d , c o n s e q u e n t l y one must e v a l u a t e an o r t h o g o n a l f u n c t i o n a l o n g the boundary as w e l l . Green's f u n c t i o n approaches c a r r y out t h e c a l c u l a t i o n s f o r <p and 90/9n, w h i l e t h e c u r r e n t approach s o l v e s f o r the v e l o c i t y p o t e n t i a l <j> and the stream f u n c t i o n The normal v e l o c i t y a t the boundary can then be found by d i f f e r e n t i a t i n g \p a l o n g the c o n t o u r s i n c e , by the Cauchy-Riemann p r o p e r t y , The stream f u n c t i o n can a l s o be shown t o s a t i s f y L a p a l a c e ' s e q u a t i o n hence b o t h <f> and \p are harmonic f u n c t i o n s . By d e f i n i n g a complex p o t e n t i a l the problem reduces t o one of f i n d i n g the f u n c t i o n /3 t h a t i s a n a l y t i c i n the domain J2 of the complex p l a n e z=x+iy, and s a t i s f i e s the boundary c o n d i t i o n s of <t> or ^ g i v e n . S i n c e 0 i s r e q u i r e d t o be a n a l y t i c , i t must s a t i s f y the well-known Cauchy i n t e g r a l theorem [9] 3£ as 3£ V 2 * 0 0 - <|> + ! • dz - 0 c 0 (2.2) p r o v i d e d z 0 i s o u t s i d e the c o n t o u r C. I f z 0 i s a l l o w e d t o 1 2 approach C i n the l i m i t , t h i s e q u a t i o n becomes { - § — dz - 1 a B(z0) Jc z~ zo w i t h z 0 on C. D e r i v a t i o n of t h i s e x p r e s s i o n can be found i n Appendix I . /3 must be found t h a t s a t i s f i e s t h i s i n t e g r a l equat i o n . To s o l v e (2.2) n u m e r i c a l l y , the c o n t o u r C i s d i s c r e t i z e d i n t o N l i n e a r elements bounded by no d a l p o i n t s a t each end. The i n t e g r a l can then be r e p r e s e n t e d as the sum of the i n t e g r a l s over each element 1 \l I=zTdz> " 1 a B ( z o ) (2.3) j = l Z j 0 A l i n e a r v a r i a t i o n of /3 i s assumed over each element (a h i g h e r o r d e r p o l y n o m i a l d i s t r i b u t i o n c o u l d be used, however t h i s has not been found n e c e s s a r y ) . D e f i n i n g upper and lower n o d a l v a l u e s as z., z.^., p., and as i n f i g u r e 3, the 3 3 + 1 ] 3 + 1 d i s t r i b u t i o n of 0 on the element can be e x p r e s s e d as j + l J ^ J+l J 3 D T h i s e x p r e s s i o n i s then s u b s t i t u t e d i n t o the l e f t h a n d s i d e of ( 2 . 3 ) . |3j a r e c o n s t a n t s so they can be removed from the i n t e g r a l s and the r e m a i n i n g k e r n e l f u n c t i o n s e v a l u a t e d . A f t e r a l g e b r a i c m a n i p u l a t i o n the i n t e g r a l e q u a t i o n (2.2) reduces t o the l i n e a r e q u a t i o n N I r. B. - 0 j-1 J J 1 3 By l e t t i n g each node t a k e on the v a l u e z 0 i n t u r n , o b t a i n N complex e q u a t i o n s f o r N unknown /3 N j=l 1 J J where the i n f l u e n c e c o e f f i c i e n t s a r e D e t a i l s can be found i n Appendix I I . Care must be e x e r c i s e d i n e v a l u a t i n g the term t o a v o i d problems w i t h the m u l t i p l e - v a l u e d complex l o g a r i t h m f u n c t i o n as d e s c r i b e d i n Appendix I I I . 2.2.2 BOUNDARY CONDITIONS S i n c e L a p l a c e ' s e q u a t i o n i s e l l i p t i c i n n a t u r e , each p o i n t a f f e c t s every o t h e r p o i n t and boundary c o n d i t i o n s must be s p e c i f i e d on a l l b o u n d a r i e s . Due t o l i n e a r i t y , problems can a l s o be hand l e d where the boundary c o n d i t i o n s on segments a r e l i n e a r l y r e l a t e d such as b e i n g p r o p o r t i o n a l or one can (2.4) 14 e q u a l by lumping t o g e t h e r the unknown q u a n t i t i e s a p p r o p r i a t e l y i n the f i n a l l i n e a r e q u a t i o n s . The s o l u t i o n of L a p l a c e ' s e q u a t i o n i s unique o n l y t o w i t h i n an a r b i t r a r y c o n s t a n t . For example i f 0{z) i s a s o l u t i o n then so i s 0(z) + /30 where /30 i s any c o n s t a n t . The n u m e r i c a l method does not g e n e r a t e t h i s c o n s t a n t and t h e r e b y f i x the unique s o l u t i o n , c o n s e q u e n t l y the boundary c o n d i t i o n s g i v e n must i n c l u d e c o n t r i b u t i o n s from both tf> and t o e l i m i n a t e any a m b i g u i t y . 2.2.3 SOLUTION D e f i n i n g r \ j = a ^ j + i ^ , e q u a t i o n (2.4) can be w r i t t e n as N w i t h the r e a l p a r t and the i m a g i n a r y p a r t I t i s c l e a r now t h a t w h i l e t h e r e a r e N unknown q u a n t i t i e s ( e i t h e r #j or a t each node) t h e r e a r e a c t u a l l y 2N r e a l e q u a t i o n s a v a i l a b l e . The problem i s o v e r s p e c i f i e d and one must make a c h o i c e as t o which N e q u a t i o n s t o s a t i s f y . i 5 S e l e c t i o n of o n l y t h e r e a l p a r t s p r o v i d e s a s a t i s f a c t o r y s o l u t i o n , as does s e l e c t i o n of o n l y t h e i m a g i n a r y p a r t s , a l b e i t s l i g h t l y d i f f e r e n t , however an improved s o l u t i o n i s p o s s i b l e i f one s e l e c t s the r e a l p a r t f o r e q u a t i o n i when <p^ i s the unknown q u a n t i t y and the i m a g i n a r y p a r t when i s unknown. Such s e l e c t i o n w i l l ensure t h a t o n l y the inhomogeneous i n t e g r a l e q u a t i o n s a r e b e i n g chosen i n each case and one would expect a more s t a b l e s o l u t i o n as a r e s u l t . R e c e n t l y , S c h u l t z e t a l (1986) f o r m u l a t e d a s o l u t i o n u t i l i z i n g a l l 2N e q u a t i o n s which a r e s o l v e d i n a l e a s t s quares sense, r e s u l t i n g i n a f u r t h e r improvement. The N s e l e c t e d e q u a t i o n s c o n t a i n the unknown v a l u e s X j as w e l l as terms i n v o l v i n g the known boundary c o n d i t i o n s . I f thes e l a t t e r terms are t r a n s p o s e d t o the r i g h t h a n d s i d e and summed one o b t a i n s a s e t of N l i n e a r e q u a t i o n s f o r N unknowns which can be s o l v e d i n m a t r i x form as N I h X - g j-1 3 3 2 The complete s o l u t i o n /3j i s c o n s t r u c t e d by combining the c a l c u l a t e d X j w i t h the known boundary c o n d i t i o n s as shown i n f i g u r e 5. 2.2.4 VELOCITIES Once /3 has been c a l c u l a t e d on the boundary, the v e l o c i t i t e s can be d e t e r m i n e d as 16 w (2.5) The d e r i v a t i v e of a complex f u n c t i o n i s independent of d i r e c t i o n so t h i s d i f f e r e n t i a t i o n can be c a r r i e d out a l o n g the c o n t o u r C. S i n c e 0 has been assumed p i e c e w i s e l i n e a r , 30/3Z i s d i s c o n t i n u o u s a t the n o d a l p o i n t s . A c e n t r a l d i f f e r e n c e scheme i s used based on a T a y l o r s e r i e s e x p a n s i o n of /3 about the p o i n t of e v a l u a t i o n . D e t a i l s may be found i n Appendix IV. 2.2.5 INTERIOR The method p r e s e n t e d a l l o w s c a l c u l a t i o n of /3 o n l y on the boundary C. For p r a c t i c a l purposes t h i s i s u s u a l l y s u f f i c i e n t s i n c e o f t e n one i s o n l y i n t e t r e s t e d i n the v e l o c i t i e s and p r e s s u r e s on the bounding s u r f a c e s . The i n t e r i o r f l o w f i e l d can be found however, i f d e s i r e d , by r e a p p l y i n g Cauchy's theorem as B(z k) B dz z-z, k C and The i n t e g r a l s here can a g a i n be e v a l u a t e d n u m e r i c a l l y as l i n e a r sums i n v o l v i n g the known n o d a l v a l u e s /3j i n a s i m i l a r f a s h i o n as b e f o r e r e s u l t i n g i n e q u a t i o n s of the form 17 N j=l 2 3 and N W ( z ) - i n B. where the i n f l u e n c e c o e f f i c i e n t s a r e f u n c t i o n s of the c o n t o u r geometry z... I n t e r i o r v a l u e s c a l c u l a t e d as such a r e not used i n the p r e s e n t work. 2.3 TEST CASE The Cauchy i n t e g r a l method p r o v i d e s a p o w e r f u l t e c h n i q u e f o r s o l v i n g an i n t e r i o r f l o w f i e l d t h a t can be c a s t i n t o t h e form of a mixed boundary v a l u e problem i n v o l v i n g <j> and i//. To t e s t the method an example i s chosen of u n i f o r m f l o w p a s t a c i r c u l a r c y l i n d e r as i n f i g u r e 5, the a n a l y t i c a l s o l u t i o n of which i s w e l l known. The v e l o c i t y p o t e n t i a l can be shown t o be B(z) - U(z + |^) w i t h the o r i g i n a t the c e n t r e of the c y l i n d e r , and the v e l o c i t y R 2 w(z) - U(l - — ) *2 For the purpose of n u m e r i c a l s o l u t i o n , b o u n d a r i e s are p l a c e d i n the f l u i d and assumed f a r enough away from the c y l i n d e r 18 t h a t t h e i r e f f e c t i s s m a l l . By symmetry, the n u m e r i c a l s o l u t i o n can be s e t up c o n s i d e r i n g o n l y the upper l e f t q uadrant as shown i n f i g u r e 6. The upper boundary i s assumed a s t r e a m l i n e as i s the m i d l i n e a x i s w i t h i//, and \p2 chosen such t h a t t h e i r d i f f e r e n c e e q u a l s the f l o w between the s t r e a m l i n e s Q = ^\~^2- The r i g h t h a n d s i d e i s an e q u i p o t e n t i a l l i n e by symmetry, w i t h <j> chosen as an a r b i t r a r y c o n s t a n t . The l e f t h a n d s i d e i s assumed f a r enough upstream t h a t u n i f o r m f l o w c o n d i t i o n s p r e v a i l and */> can be c o n s i d e r e d t o v a r y l i n e a r l y . A t e s t case has been run w i t h R=1, L=5, H=5, and U=1. Elements a r e p l a c e d on the c y l i n d e r a t 5° i n t e r v a l s w i t h a t o t a l of 80 elements on the c o n t o u r . The c a l c u l a t e d v e l o c i t y p o t e n t i a l and stream f u n c t i o n a r e shown i n f i g u r e 7 and agree w e l l w i t h the a n a l y t i c a l s o l u t i o n . V e l o c i t i e s a l o n g the c y l i n d e r s u r f a c e , c a l c u l a t e d u s i n g the n u m e r i c a l d i f f e r e n t i a t i o n t e c h n i q u e d e s c r i b e d above a r e p l o t t e d i n f i g u r e 8 a l o n g w i t h the t h e o r e t i c a l v a l u e s . Agreement i s g e n e r a l l y good, w i t h the d i s c r e p e n c i e s due p r i m a r i l y t o the f i n i t e f a r f i e l d b o u n d a r i e s imposed i n the n u m e r i c a l s o l u t i o n . 19 C Figure 2. General f l u i d domain in complex plane. Figure 3. D i s t r ibut ion of complex potential on elements. Figure 4. Final construction of complex potential solut ion on contour. Figure 5. Test case of circular cylinder in uniform flow. 0 = 1 0 0 linear variation <t> = 0 0 = 0 Figure 6 . Geometry and boundary conditions for test case. Figure 7. Velocity potential and stream function calculated in test case. Figure 8. Comparison of numerical and theoret ical solutions for ve loc i t y on cyl inder surface. Angle i s measured from forward stagnation point. 3. WAVE SIMULATION 3.1 INTRODUCTION 3.1.1 THE PROBLEM A wave i s c o n s i d e r e d of wavelength L, water depth d, and s u r f a c e e l e v a t i o n 7 j ( x , t ) measured from the s t i l l water l e v e l . The wave i s p e r i o d i c of p e r i o d T, and t r a n s l a t e s w i t h a' phase speed c as i n f i g u r e 9. The f l u i d i s c o n s i d e r e d i n c o m p r e s s i b l e and i r r o t a t i o n a l and a g a i n s a t i s f i e s L a p l a c e ' s e q u a t i o n i l i + i i i = o s u b j e c t t o the f o l l o w i n g boundary c o n d i t i o n s . The seabed i s impermeable and t h e r e f o r e can have no normal v e l o c i t y | | = 0 y = - d (3.1) The f r e e s u r f a c e must s a t i s f y a dynamic c o n d i t i o n t h a t B e r n o u l l i ' s e q u a t i o n i s obeyed ! F + T [<|£>2 + ( f ^ ) 2 ] + s n - f ( t ) y = V (3.2) as w e l l as a k i n e m a t i c c o n d i t i o n which s t a t e s t h a t s u r f a c e f l u i d p a r t i c l e s have v e l o c i t i e s i d e n t i c a l t o the wave p r o f i l e v e l o c i t i e s I f t he wave i s c o n s i d e r e d p e r i o d i c i n space then e x p l i c i t 23 24 boundary c o n d i t i o n s on the v e r t i c a l c o n t r o l volume segments ar e u n n e c e s s a r y . 3.1.2 LINEAR WAVE THEORY The problem d e f i n e d above i s v e r y d i f f i c u l t t o s o l v e because the two f r e e s u r f a c e boundary c o n d i t i o n s a r e n o n l i n e a r i n <j> and the s u r f a c e e l e v a t i o n 17 i s unknown a p r i o r i . The c l a s s i c a l s o l u t i o n of A i r y assumes the wave a m p l i t u d e i s s m a l l so the problem can be l i n e a r i z e d . A s i n u s o i d a l s u r f a c e i s assumed j] » -j cos(kx-o)t) and the s l o p e i s c o n s i d e r e d s m a l l so the n o n l i n e a r terms a r e n e g l i g i b l e . The boundary c o n d i t i o n s then reduce t o |£ + gn - 0 y = 0 and | i - | J L = 0 y = 0 3y 3t S o l u t i o n of the boundary v a l u e problem by s e p a r a t i o n of v a r i a b l e s y i e l d s : •nH cosh k(y+d) . k X * = KT slnh kd • inC** - " * ) (3.4) C 2 - tanh(kd) 25 3.1.3 STOKES WAVE THEORY E a r l y work by S t o k e s (1847) extended the a n a l y t i c a l s o l u t i o n t o i n c l u d e n o n l i n e a r e f f e c t s by expanding v a r i a b l e s i n a p e r t u r b a t i o n s e r i e s . V a r i a b l e s are e x p r e s s e d as $ = e o>1 + E 2 4>2 + .. • n = e rij + e 2 ri 2 + • • • where e i s a s m a l l parameter of the o r d e r H/L. These e x p r e s s i o n s are s u b s t i t u t e d i n t o e q u a t i o n s (3.2) and ( 3 . 3 ) , and terms of l i k e o r d e r of magnitude are g a t h e r e d y i e l d i n g s u c c e s s i v e l y h i g h e r o r d e r s o l u t i o n s , l i n e a r t h e o r y p r o v i d i n g the f i r s t a p p r o x i m a t i o n . S t o k e s o r i g i n a l l y c a l c u l a t e d a t h i r d o r d e r s o l u t i o n , as the a l g e b r a q u i c k l y becomes i n v o l v e d . S k j e l b r e i a and H e n d r i c k s o n (1960) p r e s e n t e d a S t o k e s f i f t h o r d e r s o l u t i o n of the form: 5 n = I n cos(nKx) n=l n 5 4> = I <J> sin(nKx) n=l n The e f f e c t of h i g h e r o r d e r terms i s t o sharpen the c r e s t s and f l a t t e n the t r o u g h s . The r e g i o n s of v a l i d i t y of l i n e a r and h i g h e r o r d e r s o l u t i o n s can be seen i n f i g u r e 10 from Sarpkaya and I s a a c s o n (1981). 26 3.1.4 WAVES IN NATURE Most ocean waves are g e n e r a t e d by winds e x e r t i n g shear on the sea s u r f a c e forming s m a l l d i s t u r b a n c e s which then grow as a r e s u l t of work done by aerodynamic f o r c e s . Steady p e r i o d i c p r o g r e s s i v e waves have been w e l l s t u d i e d i n t h e o r y , but p r o b a b l y e x i s t o n l y under i d e a l c o n d i t i o n s . The r e a l s e a , by c o n t r a s t , i s ve r y unsteady. More g e n e r a l l y , r e a l wave t r a i n s a r e i r r e g u l a r and undergo d i s t o r t i o n over time due t o a m p l i t u d e and frequ e n c y d i s p e r s i o n e f f e c t s . In a d d i t i o n , the r e a l ocean i s c h a r a c t e r i z e d by many i n t e r s e c t i n g wave t r a i n s from storms i n d i f f e r e n t l o c a t i o n s and from changes i n wind speed and d i r e c t i o n . Wave b r e a k i n g i s a form of i n s t a b i l i t y t h a t can d e v e l o p whenever the l o c a l energy d e n s i t y of the wave f i e l d exceeds some c r i t i c a l l i m i t . Examples of when t h i s may occur a r e when deep water waves o v e r t a k e or c o l l i d e r a i s i n g t he s u r f a c e t o u n s t a b l e h e i g h t s , or due t o s h o a l i n g i n s h a l l o w water. Winds may a l s o induce shear f o r c e s a t the wave c r e s t s . B r e a k i n g waves a r e commonly c l a s s i f i e d as s u r g i n g , s p i l l i n g , or p l u n g i n g as shown i n f i g u r e 11. S u r g i n g b r e a k e r s u s u a l l y o c c u r o n l y on s t e e p l y s l o p e d beaches and a r e not c o n s i d e r e d i n the p r e s e n t work. S p i l l i n g and p l u n g i n g b r e a k e r s may occur i n deep or s h a l l o w water, the r e s u l t i n g t y pe depending on the a v a i l a b l e energy. S p i l l i n g b r e a k e r s a r e c h a r a c t e r i z e d by a s h a r p e n i n g of the wave c r e s t u n t i l the f o r w a r d f a c e b e g i n s t o c u r l o v e r . 27 The e j e c t e d j e t i s weak and im m e d i a t e l y succumbs t o g r a v i t y , s i m p l y f l o w i n g down the f o r w a r d f a c e d i s s i p a t i n g energy i n v i s c o u s t u r b u l e n c e . T h i s i s the c l a s s i c w h i t e cap. More d r a m a t i c a r e p l u n g i n g b r e a k e r s which c o n t a i n much more energy and are a b l e t o e j e c t a w e l l d e f i n e d j e t ahead of the fo r w a r d f a c e r e s u l t i n g i n the s u r f a c e c o m p l e t e l y o v e r t u r n i n g on i t s e l f . The momentum of the p l u n g i n g j e t can be ve r y l a r g e . 3.2 NUMERICAL SOLUTION 3.2.1 FORMULATION The a n a l y t i c a l t h e o r i e s d e s c r i b e d above s u f f e r the se v e r e r e s t r i c t i o n t h a t they can handle o n l y s t e a d y s t a t e symmetric waves. To overcome t h i s l i m i t a t i o n one must r e s o r t t o a n u m e r i c a l t i m e - s t e p p i n g s i m u l a t i o n of the wave from a g i v e n i n i t i a l c o n d i t i o n . The problem i s s o l v e d as an i n i t i a l v a l u e problem where e q u a t i o n s (3.2) and (3.3) a r e i n t e g r a t e d n u m e r i c a l l y i n time and a boundary v a l u e problem f o r the p o t e n t i a l f i e l d i s s o l v e d a t each time s t e p t o p r o v i d e the r i g h t - h a n d s i d e p a r a m e t e r s . A c o n t r o l volume i s c o n s i d e r e d c o n s i s t i n g of a segment of the sea s u r f a c e , the seabed, and v e r t i c a l b o u n d a r i e s t h r o u g h the water column. T h i s r e g i o n i s c o n s i d e r e d t o be p e r i o d i c i n space f u r n i s h i n g the r e m a i n i n g n e c e s s a r y c o n d i t i o n . The seabed i m p e r m e a b i l i t y c o n d i t i o n , e q u a t i o n ( 3 . 1 ) , can be r e w r i t t e n as t h e seabed b e i n g a s t r e a m l i n e of 28 a r b i t r a r y c o n s t a n t v a l u e The f r e e s u r f a c e k i n e m a t i c c o n d i t i o n , e x p r e s s e d i n terms of the m a t e r i a l d e r i v a t i v e , becomes £ | = w* (3.5) s t a t i n g t h a t the f r e e s u r f a c e p a r t i c l e s move a c c o r d i n g t o t h e i r f l u i d v e l o c i t i e s , and the dynamic c o n d i t i o n , e q u a t i o n ( 3 . 2 ) , can be r e w r i t t e n as 3<fr ww* 3t 2 8 7 o r , as seen by the f l u i d p a r t i c l e s , Dft ww* _ D t 2 8 y (3.6) E q u a t i o n s (3.5) and (3.6) a r e e v a l u a t e d a t each time s t e p t o d e t e r m i n e the new p o s i t i o n s and p o t e n t i a l s of the f r e e s u r f a c e nodes. V e l o c i t i e s a r e then d e t e r m i n e d as 3B_ 3z f o l l o w i n g the procedure d e s c r i b e d i n Chapter 2. 29 A f l o w c h a r t f o r the time s t e p p i n g a l g o r i t h m i s shown i n f i g u r e 12. 3.2.2 CONSTRUCTION OF MATRIX The c o n t r o l volume c o n s i s t s of one w a v e l e n g t h , w i t h n o d a l p o i n t s numbered as shown i n f i g u r e 13. The f r e e s u r f a c e nodes, 1 t o N2, have known v a l u e s of <t> and the r e a l p a r t s of e q u a t i o n (2.4) are s e l e c t e d h e r e , w h i l e the seabed nodes, N3 t o N4, have known v a l u e s of \p and the i m a g i n a r y p a r t s a r e chosen. The v e r t i c a l b o u n d a r i e s , however, have both <t> and 0 as unknowns, hence both the r e a l and i m a g i n a r y e q u a t i o n s must be used f o r t h e s e p o i n t s . When the c o l l o c a t i o n p o i n t z 0 i s on the l e f t h a n d boundary, <p i s c o n s i d e r e d t o be the unknown q u a n t i t y and the r e a l e q u a t i o n i s t a ken h e r e . S i m i l a r l y , on the r i g h t h a n d s i d e 0 i s c o n s i d e r e d as the unknown q u a n t i t y and the i m a g i n a r y e q u a t i o n s a r e s e l e c t e d h e r e . A s e t of N e q u a t i o n s r e s u l t s f o r the N unknowns. An improvement can be made however by r e c o g n i z i n g t h a t 1 and N2 a r e the same p o i n t as a r e N3 and N4 a l l o w i n g e l i m i n a t i o n of two unknowns. The a c t u a l e q u a t i o n s used a r e g i v e n i n Appendix V. 3.2.3 CHOICE OF INITIAL CONDITIONS I n i t i a l i z a t i o n of the s i m u l a t i o n r e q u i r e s s t a r t i n g v a l u e s f o r the s u r f a c e p o s i t i o n T J ( X ) and s u r f a c e v e l o c i t y . The l a t t e r i s a c h i e v e d i n d i r e c t l y by s p e c i f y i n g the v e l o c i t y p o t e n t i a l 4>(x) as t h e r e i s a one t o one correspondence 30 between <j> and w on the s u r f a c e . S e l e c t i o n of <t> t o match a g i v e n v e l o c i t y d i s t r i b u t i o n , however, would be an i n v e r s e problem r e q u i r i n g e i t h e r i n t e g r a t i o n of the t a n g e n t i a l v e l o c i t y or t r i a l and e r r o r . 3.2.4 TIME STEPPING PROCEDURE S e v e r a l s t a n d a r d n u m e r i c a l p r o c e d u r e s were compared f o r i n t e g r a t i n g e q u a t i o n s (3.5) and (3.6) w i t h r e s p e c t t o t i m e . The p r e l i m i n a r y v e r s i o n of the s i m u l a t i o n program used the s i n g l e s t e p E u l e r method which uses time d e r i v a t i v e v a l u e s a t the p r e s e n t s t e p t o p r e d i c t the new f u n c t i o n v a l u e s . T h i s s i m p l e scheme y i e l d e d s a t i s f a c t o r y r e s u l t s . H i g h e r o r d e r methods were then t e s t e d f o r comparison b e g i n n i n g w i t h a second o r d e r Heun method [ 2 ] . T h i s method c a l c u l a t e s d e r i v a t i v e s a t the c u r r e n t time and uses them t o make an e s t i m a t e of the v a l u e s of z. and <t>. a f t e r At where the I ^ i d e r i v a t i v e s a re a g a i n c a l c u l a t e d . An average of these d e r i v a t i v e v a l u e s b e f o r e and a f t e r At i s used t o take the a c t u a l s t e p f o r w a r d i n t i m e . S p e c i f i c a l l y , * ' = A + * At Z ' = 2 + \» At n+1 n n <t> - 4> + [*' + I 1 rn+l *n l*n+l *nJ 2 * * At 31 where s u b s c r i p t n i n d i c a t e s c u r r e n t v a l u e s and s u b s c r i p t n+1 r e f e r s t o the new v a l u e s of s u r f a c e n o d a l p o i n t s . T h i s scheme p r o v i d e s a s l i g h t improvement over the s i n g l e s t e p E u l e r method and r e s u l t s i n a v e r y s t a b l e s o l u t i o n i n terms of both smoothness and s t a t i o n a r i t y of energy as d i s c u s s e d i n a l a t e r s e c t i o n . To t e s t the adequacy of the two s t e p scheme t e s t runs were then c a r r i e d out u s i n g a f o u r t h o r d e r Adams-Moulton p r e d i c t o r - c o r r e c t o r method [ 2 ] , V a l u e s from t h r e e p r e v i o u s s t e p s a r e r e q u i r e d and new v a l u e s are p r e d i c t e d as a f i r s t a p p r o x i m a t i o n . New d e r i v a t i v e s a r e then c a l c u l a t e d and a c o r r e c t i o n made t o o b t a i n the a c t u a l time d e r i v a t i v e s used t o s t e p f o r w a r d . S p e c i f i c a l l y , *n+l = 1 4 <55*n - 5 9 V l + 37*n-2 " 9*n-3> (55w - 59w . + 37w , - 9w -) 24 n n-1 n-^  n J n+1 *n+l = *n+*n+l A t z' z + w' At n+1 n n+1 *n+l - 1 4 <9*n+l + 19*n ' 5 V l + • « - 2 )  wn+l = h ^9wn+l + 1 9 wn ' 5wn-l + n-2> o>' . = ()> + i|> At Tn+1 Tn Yn z'., = z +w At n+1 n n The r e s u l t s of the second o r d e r method have been found t o be 32 v i r t u a l l y i n d i s t i n g u i s h i b l e from those of the f o u r t h o r d e r method. The l a t t e r r e q u i r e s more programming e f f o r t w i t h the o n l y p o s s i b l e advantage of i n c r e a s i n g the p e r m i s s i b l e time s t e p i n t e r v a l A t , t h e r e b y r e d u c i n g c o m p u t a t i o n a l t i m e . As w i l l be shown i n the next s e c t i o n , however, At i s d i c t a t e d by a n u m e r i c a l s t a b i l i t y c r i t e r i o n . I t i s c o n c l u d e d t h a t the two s t e p i n t e g r a t i o n scheme i s s u f f i c i e n t and h i g h e r o r d e r t e c h n i q u e s a r e not n e c e s s a r y . Each time s t e p i n the s i m u l a t i o n as such r e q u i r e s two m a t r i x s o l u t i o n s . 3.2.5 SEGMENT SIZE AND TIME STEP Program perfomance depends on the segment s i z e s e l e c t e d f o r the e l e m e n t s . Too few elements r e s u l t i n poor r e s o l u t i o n and l a r g e e r r o r s w h i l e t o o many elements r e s u l t i n e x c e s s i v e c o m p u t a t i o n a l time and the r i s k of n u m e r i c a l i n s t a b i l i t y . The optimum number of elements s e l e c t e d depends somewhat on the s i t u a t i o n . A n o n - b r e a k i n g wave w i t h r e l a t i v e l y low s u r f a c e c u r v a t u r e can be s i m u l a t e d r e a s o n a b l y w e l l w i t h as few as 30 s u r f a c e p o i n t s f o r one w a v e l e n g t h . Fewer than t h i s r e s u l t s i n u n d e s i r a b l e cusps i n the r e g i o n s of h i g h c u r v a t u r e , most n o t a b l y a t the wave c r e s t . E x c e s s i v e e r r o r s i n the c a l c u l a t e d v e l o c i t i e s f o l l o w w i t h subsequent i n s t a b i l i t y and breakdown. B r e a k i n g waves have r e g i o n s of h i g h c u r v a t u r e and t h e r e f o r e r e q u i r e more e l e m e n t s . F o r t u n a t e l y , n o d a l p o i n t s t e n d t o m i g r a t e i n t o the c r e s t r e g i o n as the s i m u l a t i o n p r o c e e d s , a u t o m a t i c a l l y p r o v i d i n g b e t t e r r e s o l u t i o n here 33 where i t i s needed. Most b r e a k i n g waves r e q u i r e about 60 s u r f a c e e l e m e n t s . O c c a s i o n a l l y more may be needed, e s p e c i a l l y i n cases where the p l u n g i n g j e t i s t h i n and the e l l i p t i c a l s u r f a c e under the wave c r e s t becomes p o o r l y r e s o l v e d due t o n o d a l p o i n t m i g r a t i o n away from t h i s a r e a . Such s i t u a t i o n s may r e q u i r e up t o 100 s u r f a c e elements. S i m p l y i n c r e a s i n g the number of s u r f a c e e l e m e n t s , however, can l e a d t o a d d i t i o n a l problems as the i n c r e a s i n g element d e n s i t y i n the c r e s t r e g i o n may r e s u l t i n nodes c r o s s i n g over c r e a t i n g a m u l t i p l y - c o n n e c t e d f l u i d domain and immediate breakdown of the s i m u l a t i o n . A s o l u t i o n t o t h i s problem would be t o remove elements from the h i g h d e n s i t y c r e s t r e g i o n t o the more s p a r s e t r o u g h r e g i o n . T h i s p r o c e d u r e would be u s e f u l i f the d e t a i l e d s t r u c t u r e of the j e t t i p was b e i n g examined. The number of elements recommended f o r the v e r t i c a l boundary depends on the water d e p t h . For deep water waves at l e a s t 15 s h o u l d be used i f elements a r e u n i f o r m l y spaced, or 20 f o r h i g h e r waves. T h i s number c o u l d be reduced s l i g h t l y by u s i n g p r o g r e s s i v e l y l a r g e r elements as the depth i n c r e a s e s . On the seabed 20 elements u s u a l l y proves s u f f i c i e n t f o r deep water waves. In s h a l l o w e r water, however, where seabed v e l o c i t i e s become s i g n i f i c a n t , more elements a r e r e q u i r e d w i t h 30 b e i n g used t y p i c a l l y . The g r e a t e s t number of e l e m e n t s , t h e r e f o r e , a r e r e q u i r e d f o r l a r g e a m p l i t u d e s h a l l o w water b r e a k i n g waves where N may be up t o 180. 34 Once element s i z e s have been d e t e r m i n e d the time s t e p i n t e r v a l At i s s e l e c t e d a c c o r d i n g t o the Courant s t a b i l i t y c r i t e r i o n which s t a t e s t h a t a p a r t i c l e s h o u l d not be p e r m i t t e d t o move a d i s t a n c e g r e a t e r than a p p r o x i m a t e l y the element s i z e . T h i s c o n d i t i o n can be e x p r e s s e d r o u g h l y as if < c or At . Ax where Ax i s the element s i z e and c i s the phase v e l o c i t y of the wave. A c o n v e n i e n t time s t e p i n t e r v a l i s s e l e c t e d f o l l o w i n g t h i s c r i t e r i o n based on the i n i t i a l element s i z e . One f u r t h e r d i s a d v a n t a g e of t o o many elements on the f r e e s u r f a c e apparent now i s the s m a l l e r time s t e p r e q u i r e d and the r e s u l t i n g i n c r e a s e i n c o m p u t a t i o n t i m e . 3.2.6 NUMERICAL ADJUSTMENTS S e v e r a l checks and a d j u s t m e n t s of elements must be made a t each time s t e p t o ensure smooth e x e c u t i o n of the s i m u l a t i o n . The v e r t i c a l nodes a r e f i x e d p o i n t s , however, the s u r f a c e c o r n e r nodes 1 and N2 a r e f r e e t o move and w i l l t e n d t o s t r e t c h and compress t h e uppermost v e r t i c a l element r e s u l t i n g i n poor a c c u r a c y i n t h i s r e g i o n . To overcome t h i s problem the elements on the v e r t i c a l boundary a r e e v e n l y r e d i s t r i b u t e d at each time s t e p by d i v i d i n g the v e r t i c a l boundary l e n g t h by the number of elements on the s i d e . 35 Element s i z e here then becomes (d+rj)/NV. F a i l u r e t o do so w i l l u s u a l l y r e s u l t i n "sawtooth" i n s t a b i l i t i e s d e v e l o p i n g on the s u r f a c e near the edges. A s i m i l a r problem r e s u l t s from e x c e s s i v e h o r i z o n t a l e x c u r s i o n of the s u r f a c e c o r n e r p o i n t s . The s i t u a t i o n i s compounded by n o n l i n e a r e f f e c t s c a u s i n g a g r a d u a l downstream m i g r a t i o n of s u r f a c e p a r t i c l e s . I f l e f t unchecked the c o n t r o l volume w i l l d i s t o r t as shown i n f i g u r e 14. A check of the r e g i o n i s made a t each s t e p and the c o r n e r p o i n t s r e a d j u s t e d i f n e c e s s a r y t o ensure t h a t they a re always those c l o s e s t t o the v e r t i c a l boundary. S u r f a c e n o d a l p o i n t i n d i c e s a r e incremented i f such a s h i f t i s r e q u i r e d . The s i m u l a t i o n i s v e r y s t a b l e and smoothing of the s u r f a c e has not been found n e c e s s a r y i n most c a s e s . O c c a s i o n a l l y , a l t e r i n g i n i t i a l element s i z e s l i g h t l y w i l l c o r r e c t r a r e u n s t a b l e s i t u a t i o n s . 3.2.7 ENERGY A wave c o n t a i n s k i n e t i c energy due t o f l u i d motion and p o t e n t i a l energy due t o d i s p l a c e m e n t of the f r e e s u r f a c e . Under the assumption of z e r o v i s c o s i t y i n p o t e n t i a l f l o w t h e r e i s no mechanism f o r energy d i s s i p a t i o n , and the t o t a l c o n t r o l volume energy E t " EK + Ep must i n t h e o r y remain c o n s t a n t . Due to n u m e r i c a l a p p r o x i m a t i o n s and computer r o u n d o f f e r r o r s , however, one 36 would expect a s l i g h t " n u m e r i c a l v i s c o s i t y " t o cause a r t i f i c i a l changes i n energy. A p e r f e c t s o l u t i o n s h o u l d e x h i b i t no such change, t h e r e f o r e , s t a t i o n a r i t y of the t o t a l energy p r o v i d e s an e x c e l l e n t assessment of s i m u l a t i o n a c c u r a c y . K i n e t i c energy can be e x p r e s s e d as » « £. / / V2 dV K 1 a w h i c h , by i n v o k i n g Green's theorem, can be e v a l u a t e d n u m e r i c a l l y as N2 1=1 P o t e n t i a l energy i s g i v e n by (3.8) or n u m e r i c a l l y as N2 h - • x i + i ) ( y i + i y i + y i + i 2 + y i 2 ) D e r i v a t i o n of t h e s e e q u a t i o n s may be found i n Appendix V I . L i n e a r t h e o r y p r e d i c t s k i n e t i c and p o t e n t i a l e n e r g i e s a r e e x a c t l y e q u a l , w h i l e n o n l i n e a r t h e o r i e s p r e d i c t t h a t the k i n e t i c component i s s l i g h t l y l a r g e r . B r e a k i n g waves e x h i b i t a s h i f t from p o t e n t i a l t o k i n e t i c energy as time p r o c e e d s . 37 3.2.8 COMPUTER SOLUTION The wave s i m u l a t i o n l i b r a r y d e v e l o p e d c o n s i s t s of p r e p r o c e s s i n g programs f o r g e n e r a t i n g the v a r i o u s t y p e s of wave i n i t i a l c o n d i t i o n s which a r e then passed t o the main s i m u l a t i o n program. The p r i m a r y output f i l e p r o v i d e s a l l n u m e r i c a l v a l u e s f o r each time s t e p i n f o r m a t t e d t a b l e s s u i t a b l e f o r a n a l y s i s . A secondary output f i l e can be used f o r hard-copy p l o t t i n g of the waves or animated d i s p l a y on a g r a p h i c s t e r m i n a l . A l l programs were w r i t t e n i n FORTRAN, and c o m p i l e d and run on a VAX 11/750. M a t r i x s o l u t i o n s were o b t a i n e d u s i n g s t a n d a r d G a u s s i a n e l i m i n a t i o n w i t h double p r e c i s i o n v a r i a b l e s used t h r o u g h o u t . R e q u i r e d CPU ti m e s f o r d i f f e r e n t numbers of elements a r e shown i n f i g u r e 15. CPU time f o r a s i n g l e s t e p i n c r e a s e s r o u g h l y as N 2 , however, as N i n c r e a s e s a s m a l l e r At i s r e q u i r e d so CPU time a c t u a l l y i n c r e a s e s by a power a p p r o a c h i n g N 3. S i m u l a t i o n of a t y p i c a l b r e a k i n g wave w i t h N=120 r e q u i r e s about f o u r h o u r s . 3.3 RESULTS The s i m u l a t i o n p r o c e d u r e d e s c r i b e d p r o v i d e s a p o w e r f u l t o o l f o r a n a l y s i n g the b e h a v i o u r of a r b i t r a r y n o n l i n e a r waves under the assumptions d e s c r i b e d p r e v i o u s l y . To t e s t the a c c u r a c y of the s i m u l a t i o n a S t o k e s f i f t h o r d e r wave was chosen. L i n e a r wave t h e o r y p r e d i c t s t h a t f l u i d m otions s h o u l d be n e g l i g i b l e below a depth of about d/L =0.5 which i s c o n s i d e r e d the t r a n s i t i o n between s h a l l o w 38 and deep water. S i n c e Stokes t h e o r y i s v a l i d o n l y f o r deep water, the sea bed was p l a c e d a t a depth of d/L = 0.6 and n e g l i g i b l e f l u i d m otions were c o n f i r m e d . The h e i g h t r a t i o was s e l e c t e d t o be H/L = 0.06 which can be seen from f i g u r e 10 t o l i e w e l l w i t h i n the r e g i o n of v a l i d i t y of Stokes t h e o r y . As such, one would expect the wave t o t r a n s l a t e s t e a d i l y w i t h no d e f o r m a t i o n over t i m e . F i g u r e 16 shows the r e s u l t s of a s i m u l a t i o n c a r r i e d out u s i n g NS=60, NV=15, NB=20, L=100 f e e t , and At = 0.05 s e c . The r e s u l t i n g s u r f a c e p r o f i l e a f t e r one wave p e r i o d i s shown superimposed on the i n i t i a l wave. The s i m u l a t i o n i n t h i s case i s remarkably a c c u r a t e showing l i t t l e d i s t i n g u i s h i b l e d i f f e r e n c e a f t e r one p e r i o d . F o l l o w i n g t h r e e complete wave p e r i o d s the wave s t i l l showed n e g l i g i b l e d i s t o r t i o n . T o t a l c o n t r o l volume energy e x h i b i t e d f l u c t u a t i o n s of l e s s than 0.1%. F i g u r e 17 shows the t r a j e c t o r y over time of a t y p i c a l n o d a l p o i n t r e p r e s e n t i n g a marked f l u i d p a r t i c l e . The n o n l i n e a r Stokes d r i f t r e s u l t s i n a net m i g r a t i o n of f l u i d i n the downwave d i r e c t i o n . Net e x c u r s i o n a t the s u r f a c e i n t h i s case was x/L = 0.036 and i n c r e a s e d w i t h wave h e i g h t . G e n e r a t i o n of a b r e a k i n g wave r e q u i r e s an i n i t i a l c o n d i t i o n t h a t i s u n s t a b l e . As d i s c u s s e d e a r l i e r t h i s s e l e c t i o n i s somewhat a r b i t r a r y , and many waves would s a t i s f y t h i s c r i t e r i o n . To be s p e c i f i c , however, a p a r t i c u l a r c l a s s has been chosen of a c o s i n e s u r f a c e p r o f i l e w i t h <j> from l i n e a r t h e o r y a p p l i e d a t the ex a c t f r e e s u r f a c e . Such waves cannot remain s t e a d y i n form and would be 39 e x p e c t e d t o break i f g i v e n s u f f i c i e n t i n i t i a l h e i g h t . A deep water c o s i n e wave was run w i t h H/L = 0.06. In t h i s case n o n l i n e a r e f f e c t s were q u i t e s m a l l and a f t e r two wave p e r i o d s t h e o n l y d i s c e r n i b l e change was a s l i g h t i n c r e a s e i n t h e s l o p e of the f o r w a r d s u r f a c e . I n c r e a s i n g the i n i t i a l wave h e i g h t t o H/L = 0.10 produced a s p i l l i n g b r e a k e r as shown i n F i g u r e 18. F l u i d p a r t i c l e v e l o c i t i e s a t the wave c r e s t reached the phase speed around t h i s t i m e . The h o r i z o n t a l a c c e l e r a t i o n of the f l u i d here was about 0.04 g and the v e r t i c a l a c c e l e r a t i o n -0.22 g, h a v i n g changed l i t t l e from the i n i t i a l c o n d i t i o n . A c c e l e r a t i o n i n the f o r w a r d d i r e c t i o n was v e r y s m a l l and f l u i d i n the i n c i p i e n t j e t would s i m p l y f l o w down the f o r w a r d f a c e of the wave under the i n f l u e n c e of g r a v i t y . F u r t h e r c o m p u t a t i o n beyond t h i s p o i n t i s not p o s s i b l e as the n o d a l p o i n t s a t the c r e s t f a l l i n s i d e the c o n t r o l volume p r o d u c i n g a m u l t i p l y c o n n e c t e d r e g i o n . The r e s u l t s of a deep water c o s i n e wave w i t h H/L = 0.13 a r e shown i n F i g u r e 19. The i n i t i a l energy i n t h i s case i s much h i g h e r and a w e l l d e f i n e d p l u n g i n g b r e a k e r r e s u l t s . The s i m u l a t i o n i n t h i s case l o o k e d q u a l i t a t i v e l y i d e n t i c a l t o t h a t p r e s e n t e d i n V i n j e and B r e v i g (1980) f o r the same i n i t i a l c o n d i t i o n , however, no n u m e r i c a l r e s u l t s were g i v e n i n t h e i r work. At the time of the f o r w a r d f a c e becoming v e r t i c a l t he h o r i z o n t a l f l u i d a c c e l e r a t i o n near the c r e s t was 0.58 g w h i l e the v e r t i c a l a c c e l e r a t i o n was -0.76 g. As the j e t became w e l l d e v e l o p e d , the h o r i z o n t a l a c c e l e r a t i o n 40 at the t i p dropped t o near z e r o w h i l e the v e r t i c a l a c c e l e r a t i o n approached -0.98 g c h a r a c t e r i s t i c of a pure g r a v i t y j e t . For compa r i s o n , the same i n i t i a l c o n d i t i o n s were run i n s h a l l o w water u s i n g d/L = 0.25. A s p i l l i n g b r e a k e r of H/L = 0.10 i s shown i n f i g u r e 20 and i s rem a r k a b l y s i m i l a r t o the deep water case d e m o n s t r a t i n g t h a t s p i l l i n g i s a l o c a l phenomenon and not v e r y s e n s i t i v e t o water d e p t h . As can be seen i n f i g u r e 21 f o r the s h a l l o w water p l u n g i n g b r e a k e r , the j e t r e s u l t e d i n a g r e a t e r f l u i d volume e j e c t e d a t a s l i g h t l y h i g h e r v e l o c i t y . Numerous wave s i m u l a t i o n s have been run u s i n g the deep .water c o s i n e i n i t i a l c o n d i t i o n s . Waves of s m a l l a m p l i t u d e e x h i b i t o n l y a g r a d u a l s h a r p e n i n g of the c r e s t over s e v e r a l wave p e r i o d s . As the i n i t i a l h e i g h t r a t i o i s i n c r e a s e d , however, a t r a n s i t i o n from s p i l l i n g t o p l u n g i n g b e h a v i o u r o c c u r s as shown i n f i g u r e 22. S i m u l a t i o n s t e r m i n a t e when v e l o c i t i e s cannot be r e s o l v e d i n t h e j e t t i p r e g i o n due t o no d a l p o i n t c r o s s o v e r or when the p l u n g i n g j e t touches the f o r w a r d f a c e . F i g u r e 23 shows the time c o u r s e of maximum f l u i d v e l o c i t i e s on the s u r f a c e f o r deep water waves of i n c r e a s i n g i n i t i a l h e i g h t r a t i o s . F i g u r e 24 shows the same t h i n g f o r s h a l l o w water waves of depth r a t i o d/L = 0.25. F l u i d v e l o c i t i e s i n the s h a l l o w water waves a r e g r e a t e r f o r the same i n i t i a l h e i g h t r a t i o s . B r e a k i n g wave j e t v e l o c i t i e s can approach t w i c e the phase speed of the c o r r e s p o n d i n g l i n e a r wave. For the p l u n g i n g b r e a k e r s t h e s e maximum v e l o c i t i e s 41 t e n d t o o c c u r on the a d v a n c i n g t o p s u r f a c e j u s t above the j e t t i p and a r e d i r e c t e d almost h o r i z o n t a l l y as seen i n f i g u r e 25. For the c l a s s of waves s t u d i e d , b r e a k i n g u s u a l l y o c c u r s i n l e s s than one wave p e r i o d . The b r e a k i n g l i m i t s i n d i c a t e d i n f i g u r e s 23 and 24 a r e not r e a l l y w e l l d e f i n e d but i n d i c a t e r o u g h l y a t r a n s i t i o n between the p l u n g i n g j e t t o u c h i n g t h e f o r w a r d f a c e and s p i l l i n g b r e a k e r s r e a c h i n g the c r i t i c a l p o i n t where f u r t h e r c o m p u t a t i o n i s not p o s s i b l e due t o problems i n s p a t i a l or t e m p o r a l r e s o l u t i o n . The energy h i s t o r i e s f o r p l u n g i n g b r e a k e r s of H/L = 0.13 i n deep and s h a l l o w water a r e shown i n f i g u r e s 26 and 27. At t = 0 the c o s i n e i n i t i a l c o n d i t i o n has a k i n e t i c component o n l y s l i g h t l y l a r g e r than the p o t e n t i a l component. As the wave b r e a k s , however, t h e r e i s a t r a n s f e r from p o t e n t i a l t o k i n e t i c energy which becomes i n c r e a s i n g l y r a p i d as the p l u n g i n g j e t forms. The t o t a l energy remains n e a r l y c o n s t a n t t h r o u g h o u t most of the s i m u l a t i o n showing a t y p i c a l s l i g h t i n c r e a s e near t e r m i n a t i o n due t o i m p e r f e c t r e s o l u t i o n i n the j e t r e g i o n . Figure 9 . Definition of wave variables. Figure 10. Regions of v a l i d i t y of wave theories (27). SURGING Figure 11. C l a s s i f i c a t i o n of breaking waves. 44 GENERATE ELEMENTS SET UP BOUNDARY CONDITIONS INITIAL CONDITION SOLVE MATRIX CALCULATE VELOCITIES *•¥• it TIME STEP t • t + A t CALCULATE Dt Dt CALCULATE NEW . SURFACE z * STOP Figure 12. Flow chart of wave simulation algorithm. 45 $ known N3 N4 i|) » o Figure 13. Definition of wave control volume. ( a ) (b) Figure 14. Two types of control volume distortion requiring nodal point adjustment: (a) horizontal drift, and (b) vertical stretch. 200 Ld ±1 100 0 0 50 100 150 200 NUMBER OF ELEMENTS Figure 15. CPU time per step versus number of elements for wave s imulat ion. F i g u r e 16. T r a n s l a t i n g Stokes f i f t h o r d e r wave o f H/L = 0.06 s i m u l a t e d f o r one p e r i o d and superposed on I n i t i a l c o n d i t i o n . V e r t i c a l s c a l e has been doub l e d f o r c l a r i t y . 48 0.10 <J1 o F i g u r e 20. S h a l l o w water s p i l l i n g b r e a k e r s i m u l a t e d from c o s i n e i n i t i a l c o n d i t i o n H/L = 0.10. Depth r a t i o d/L = 0 . 2 5 . 0.10 r 0.00 -0.10 0.5 X/L F i g u r e 21. S h a l l o w water p l u n g i n g b r e a k e r s i m u l a t e d from c o s i n e i n i t i a l c o n d i t i o n H/L = 0.13. Depth r a t i o d/L = 0.25. Figure 22. Last simulated step for deep water waves of varying height ratios showing t r a n s i t i o n from s p i l l i n g to plunging breakers. 2.5 F i g u r e 23. 0.4 0.5 o.6 TIME t/T 0.7 0.8 0.9 Maximum f l u i d p a r t i c l e v e l o c i t i e s v e r s u s time f o r deep water waves from c o s i n e i n i t i a l c o n d i t i o n s . V a r i a b l e s a r e n o r m a l i z e d w i t h v a l u e s from l i n e a r t h e o r y . -pi 2.5 2 . TIME t/T t n t n F i g u r e 24. Maximum f l u i d p a r t i c l e v e l o c i t i e s v e r s u s time f o r s h a l l o w water waves from c o s i n e i n i t i a l c o n d i t i o n s . V a r i a b l e s a r e n o r m a l i z e d w i t h deep w a t e r l i n e a r t h e o r y . F i g u r e 25. F l u i d v e l o c i t i e s around deep water p l u n g i n g j e t f o r c a s e H/L = 0.13. CTi 57 1.2 P 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 TIME t/T F i g u r e 26. C o n t r o l volume energy v e r s u s time f o r deep water b r e a k i n g wave from c o s i n e i n i t i a l c o n d i t i o n H/L = 0.13. 58 1.2 r TIME t/T F i g u r e 27. C o n t r o l volume e n e r g y v e r s u s t i m e f o r s h a l l o w water b r e a k i n g wave from c o s i n e i n i t i a l c o n d i t i o n H/L = 0.13. Depth r a t i o d/L = 0.25. 4. FLOATING BODY MOTION 4.1 INTRODUCTION The f o r m u l a t i o n of a g e n e r a l two d i m e n s i o n a l time s t e p p i n g body motion s i m u l a t i o n i s p r e s e n t e d . Development of such a s i m u l a t i o n would p e r m i t extreme s h i p motions t o be a s s e s s e d i n a r b i t r a r y s t e e p n o n l i n e a r waves. Water on deck and c a p s i z i n g c o u l d be m o d e l l e d , and f u r t h e r e x t e n s i o n s would a l l o w b r e a k i n g wave f o r c e s a g a i n s t the h u l l t o be i n c l u d e d . 4.2 NUMERICAL SOLUTION 4.2.1 FORMULATION The c o n t r o l volume used f o r the wave s i m u l a t i o n i s m o d i f i e d t o i n c l u d e an a r b i t r a r y t w o - d i m e n s i o n a l s u r f a c e p i e r c i n g body i n the c o n t o u r of i n t e g r a t i o n as shown i n f i g u r e 28. The domain i s a g a i n c o n s i d e r e d p e r i o d i c w i t h the f r e e s u r f a c e s a t i s f y i n g the k i n e m a t i c and dynamic c o n d i t i o n s d i s c u s s e d p r e v i o u s l y The stream f u n c t i o n d i s t r i b u t i o n on the body can be found by i n t e g r a t i n g the normal v e l o c i t i e s a l o n g the s u r f a c e g i v i n g 3 t * < 3 x ' C U ' ~ 3 y " (4.1) + f [ ( | i ) 2 + ( | ± ) 2 ] + g n = f ( t ) (4.2) 59 6 0 t = - ~2~+ u G ( y - y G ) - v i x - x G ) + * o (4.3) D e t a i l s a r e d e s c r i b e d i n Appendix V I I . The c o n s t a n t of i n t e g r a t i o n \jj0 can t a k e on any v a l u e and i s a r b i t r a r i l y s e t t o z e r o . The seabed i s a g a i n a s t r e a m l i n e , however, i n t h i s case the v a l u e cannot be chosen a r b i t r a r i l y s i n c e was f i x e d above and \p here must be d e t e r m i n e d as p a r t of the s o l u t i o n . The m a t r i x f o r s o l v i n g the complex p o t e n t i a l i s c o n s t r u c t e d such t h a t terms i n v o l v i n g \p on the seabed a r e lumped t o g e t h e r a p p r o p r i a t e l y and 4> here i s c o n s i d e r e d a s i n g l e unknown. S o l u t i o n of the r e s u l t i n g boundary v a l u e problem u s i n g the Cauchy method p r o v i d e s v e l o c i t i e s a l o n g the c o n t o u r which a r e used t o s t e p the f r e e s u r f a c e n o d a l p o i n t s as i n the wave s i m u l a t i o n The p r e s s u r e d i s t r i b u t i o n on the h u l l i s found by a p p l y i n g B e r n o u l l i ' s e q u a t i o n D<ft _ ww* Dt ~ 2 gy p = + gy) (4.4) where b<t>/dt remains t o be d e t e r m i n e d . The o b v i o u s c h o i c e i s t o use a backwards f i n i t e d i f f e r e n c e a p p r o x i m a t i o n , however, 61 V i n j e and B r e v i g (1981b) c l a i m such a scheme i s u n s t a b l e and ano t h e r t e c h n i q u e must be used. An a l t e r n a t i v e method can be o b t a i n e d by r e c o g n i z i n g t h a t s i n c e <j> and ^ a r e harmonic f u n c t i o n s , then so must be d<f>/dt and b\p/dt, c o n s e q u e n t l y , 3/3/3t i s a l s o an a n a l y t i c f u n c t i o n i n the f l u i d domain. The Cauchy i n t e g r a l method i s a g a i n a p p l i e d i n an i d e n t i c a l f a s h i o n as f o r /3 u s i n g the time d e r i v a t i v e s i n t h i s c a s e . On the f r e e s u r f a c e d<j>/bt i s found from 3$ WW* 3t 2 8 7 and on the body 3\/>/3t may be found by d i f f e r e n t i a t i n g (4.3) g i v i n g It = ( y" yG ) ax - ( x" xG ) ay " + u Gv - v Gu + [(x-x G)(u G-u) + (y-y G)(v G-v)]e (4.5) The seabed has c o n s t a n t but unknown 3i///3t which comes out from the s o l u t i o n . The r e s u l t y i e l d s 3#/3t on the body from which the p r e s s u r e d i s t r i b u t i o n can be found u s i n g (4.4) Body f o r c e s and moment about G a r e found as F = - / P n ds 8 M = - J P rxn ds s 62 which are evaluated numerically as N6-1 P -P F - - Z 1( , 1 ) ( z , + 1 - 0 i=N5 i = N j Derivations are found in Appendix VIII. Accelerations are then determined from the equations of motion F _ _x x m F a = y m " M e = T which can then be integrated twice in time to fin d the heave, surge and r o l l motions. 4.2.2 THE CLOSURE PROBLEM The technique outlined above suffers a closure problem in that the accelerations a x , a^ and 69 in (4.5) are unknown a p r i o r i and must come out as part of the solution. Fortunately Cauchy's theorem i s linear and 3/3/9t can be considered as being composed of four p h y s i c a l l y unreal components as 63 The problem i s then decomposed i n t o f o u r independent s o l u t i o n s f o r the f o u r c o n t r i b u t i n g terms as shown s c h e m a t i c a l l y i n f i g u r e 30. The boundary c o n d i t i o n s on the s u r f a c e a r e • (!£>2 - 0 w h i l e those on the body a r e (!£>! - U G V ~ VGU + [(^- G^)(vu) + (y-yG)(vG-v)]e (|t)2 - (y-yG) ,3K _ 9R 2 vat;,» 2 The body p r e s s u r e d i s t r i b u t i o n (4.4) can be w r i t t e n as P - p i + P 2 a x + P 3 a y + V where 64 F o r c e s and moments a r e then c a l c u l a t e d f o r each i n d i v i d u a l problem u s i n g e q u a t i o n s ( 4 . 5 ) , and s u b s t i t u t e d i n t o the e q u a t i o n s of motion (4.7) which t a k e on the form F + F a + F a + F 6 = m a x i x 2 x x3 y x<* x F + F a F a + F 6 - W = ma yi y 2 x y 3 y y* y M + M„a + M3a + M 6 = I e 1 2 x 3 y »• These t h r e e e q u a t i o n s can be s o l v e d f o r the t h r e e unknown a c c e l e r a t i o n s a , a and 6, t h e r e b y c l o s i n g the problem. x y Giv e n a c c e l e r a t i o n s a t each time s t e p , v e l o c i t i e s and motions a r e determined by s u c c e s s i v e i n t e g r a t i o n s . A s i m p l e f i n i t e d i f f e r e n c e E u l e r scheme i s used f o r th e s e i n t e g r a t i o n s Un+1 = U n + a x A t - v_ + a_ At n y e - e + e At n+1 n a x = x + u At + [At] 2 n+1 n n 2 ^ ; yn+l = y n + V n A t + en+l = e n + ° n A t + f ( A t ) 2 65 A f l o w c h a r t f o r t h e body motion s i m u l a t i o n i s p r e s e n t e d i n f i g u r e 31. 4.2.3 BODY POSITION The body geometry i s s t o r e d as a s e t of p o i n t s d e f i n e d i n a s h i p c o o r d i n a t e system (x , y ) a t t a c h e d t o the body w i t h the o r i g i n a t G as shown i n f i g u r e 29. Body p o s i t i o n i s s p e c i f i e d by the l o c a t i o n of G i n the g l o b a l frame ( x , y) and the r o l l a n g l e 8 measured c o u n t e r c l o c k w i s e from the v e r t i c a l . These t h r e e v a r i a b l e s a r e known a t each time s t e p a l l o w i n g the l o c a t i o n of s h i p n o d a l p o i n t s t o be c a l c u l a t e d i n the g l o b a l frame u s i n g the f o l l o w i n g t r a n s f o r m a t i o n x = x cosB - y sin6 + x„ s s G y = x sin6 + y cos6 + y_ s s G The i n t e r s e c t i o n p o i n t s between the f r e e s u r f a c e and the body a r e d i f f i c u l t t o determine due t o the p o s s i b l e presence of s i n g u l a r i t i e s here and f o r p r a c t i c a l purposes approximate methods must be used as d i s c u s s e d i n a l a t e r s e c t i o n . Once the i n t e r s e c t i o n p o i n t s are l o c a t e d , nodes a r e p l a c e d t h e r e on each s i d e and o t h e r s added or removed a t each time s t e p t o keep the same number of elements NH on the h u l l as the w e t t e d s u r f a c e changes throughout the s i m u l a t i o n . Depending on the d i r e c t i o n of f l u i d motion near the h u l l , a f r e e s u r f a c e n o d a l p o i n t may e n t e r the body. T h i s c o n d i t i o n must 6 6 be checked a t each time s t e p and i f n e c e s s a r y the o f f e n d i n g p o i n t r e p l a c e d o u t s i d e the body. C o n v e r s e l y , i f a n o d a l p o i n t m i g r a t e s e x c e s s i v e l y f a r from the body, the r e s u l t i n g l a r g e f r e e s u r f a c e element s h o u l d be s u b d i v i d e d a p p r o p r i a t e l y . 4.2.4 INITIAL CONDITIONS I n i t i a l i z a t i o n of the s i m u l a t i o n r e q u i r e s a f r e e s u r f a c e p r o f i l e T?(X) and p o t e n t i a l 0 ( x ) , as w e l l as body p o s i t i o n ( x Q , y Q , e) and v e l o c i t i e s ( u G , v Q , 8). There a r e c o n s e q u e n t l y s i x f r e e parameters t o s p e c i f y when p l a c i n g the body on the wave. U n l i k e the wave s i m u l a t i o n , however, these i n i t i a l c o n d i t i o n s cannot be chosen a r b i t r a r i l y . The f r e e s u r f a c e v e l o c i t y i s s p e c i f i e d by <p(s) and must match the body v e l o c i t y a t the i n t e r s e c t i o n p o i n t s t o a v o i d e r r o n e o u s l y f o r c i n g f l u i d i n t o or away from the body. T h i s c o n d i t i o n i s ensured by a d j u s t i n g the p o t e n t i a l d i s t r i b u t i o n i n the neighbourhood of the body t o s a t i s f y the r e l a t i o n | t - v • % 3s n a t the i n t e r s e c t i o n p o i n t P, where % i s the u n i t v e c t o r a l o n g the f r e e s u r f a c e . The <t> d i s t r i b u t i o n i s a d j u s t e d by a q u a d r a t i c p o l y n o m i a l e x t r a p o l a t i o n over a r e g i o n of l e n g t h Lc on e i t h e r s i d e of the body t o match the d e r i v a t i v e s at the i n t e r s e c t i o n p o i n t s . 67 4.2.5 INTERSECTION SINGULARITIES The i n t e r s e c t i o n p o i n t s between the body and the f r e e s u r f a c e p r e s e n t s p e c i a l problems because the f l u i d v e l o c i t y w(z) may be s i n g u l a r h e r e . Yim (1985) d e s c r i b e s the a n a l y t i c a l s o l u t i o n t o a s e m i - i n f i n i t e f l a t e p l a t e moving downward on the f r e e s u r f a c e . The complex v e l o c i t y has a square r o o t s i n g u l a r i t y a t the p l a t e edge where the v e l o c i t y becomes i n f i n i t e . Greenhow and L i n (1985) d e s c r i b e a s e r i e s s o l u t i o n t o an i m p u l s i v e l y moved v e r t i c a l boundary t h a t p r e d i c t s a l o g a r i t h m i c a l l y i n f i n i t e s u r f a c e e l e v a t i o n next t o the w a l l . These two i d e a l i z e d c a s e s are a n a l o g o u s t o the heave and surge of a f l o a t i n g body, which f o r g e n e r a l motions i s o b v i o u s l y a much more c o m p l i c a t e d problem. S i n g u l a r i t y b e h a v i o u r a t the f r e e s u r f a c e i n the more g e n e r a l case i s l e s s c l e a r . L i n e t a l (1984) d e s c r i b e e x p e r i m e n t s which c l o s e l y examined the f l o w b e h a v i o u r next t o an i m p u l s i v e l y moved wavemaker, s u g g e s t i n g t h a t the s i n g u l a r i t y may a c t u a l l y have a p h y s i c a l i n t e r p r e t a t i o n . P hotographs show a v e r y s m a l l f l u i d j e t b e i n g e j e c t e d p e r p e n d i c u l a r l y from the p a d d l e a t the i n t e r s e c t i o n . T h i s p e c u l i a r phenomenon i s an example of how p o t e n t i a l f l o w m o d e l l i n g can break down i n c e r t a i n r e g i o n s , where e f f e c t s such as v i s c o s i t y and s u r f a c e t e n s i o n keep the r e a l f l u i d b e h a v i o u r f i n i t e . One f u r t h e r problem a t the i n t e r s e c t i o n p o i n t s i s t h a t even i n the absence of s i n g u l a r i t i e s , the v a r i a t i o n of 4> or i> may be r a p i d c l o s e t o the body. M a s k e l l and U r s e l l ( 1970) 68 g i v e a second o r d e r s o l u t i o n f o r a h e a v i n g c y l i n d e r which shows a s t e e p g r a d i e n t of 0 and h i g h v e l o c i t i e s near the body. High element d e n s i t y would be r e q u i r e d here t o a c h i e v e good r e s o l u t i o n , though i t i s not c l e a r i f t h i s i s r e a l l y n e c e s s a r y f o r a s t a b l e s o l u t i o n . F u r t h e r m o r e , e x c e s s i v e element d e n s i t y can cause a d d i t i o n a l n u m e r i c a l problems b o t h i n c o m p u t a t i o n a l s t a b i l i t y and f u r t h e r c o m p l i c a t i o n of element r e d i s t r i b u t i o n near the body. I t has been found t h a t r e d u c i n g the number of f r e e s u r f a c e elements o f t e n improves n u m e r i c a l s t a b i l i t y , though r e s o l u t i o n i s o b v i o u s l y the c o s t . One may have t o be s a c r i f i c e d f o r the o t h e r . 4.2.6 INTERSECTION SOLUTION S e v e r a l t e c h n i q u e s were t r i e d t o h a n d l e the n u m e r i c a l s o l u t i o n i n the i n t e r s e c t i o n r e g i o n , each r e q u i r i n g a s e p a r a t e s i m u l a t i o n program. Boundary c o n d i t i o n s f o r t h e t h r e e main v e r s i o n s a r e shown i n f i g u r e 32. V e r s i o n I was the same as t h a t used by V i n j e and B r e v i g (1981c) where a node i s p l a c e d a t the i n t e r s e c t i o n p o i n t and the stream f u n c t i o n s p e c i f i e d as a boundary c o n d i t i o n . The v e l o c i t y p o t e n t i a l here comes out from the s o l u t i o n and as such does not n e c e s s a r i l y s a t i s f y the f r e e s u r f a c e boundary c o n d i t i o n . E x p e r i e n c e w i t h t h i s f o r m u l a t i o n showed the complex p o t e n t i a l t o be p o o r l y r e s o l v e d near the body, and the v e l o c i t y p o t e n t i a l i n d e e d d i d not s a t i s f y the f r e e s u r f a c e boundary c o n d i t i o n a t the i n t e r s e c t i o n . The d i s c r e p e n c y was o f t e n l a r g e . E x a m i n a t i o n of the n u m e r i c a l s o l u t i o n a t each 69 time s t e p r e v e a l e d t h a t the f l u i d v e l o c i t i e s were p o o r l y c a l c u l a t e d i n the i n t e r s e c t i o n r e g i o n s , o f t e n y i e l d i n g u n r e a l i s t i c a l l y l a r g e v a l u e s t h a t were not even of the c o r r e c t s i g n . V e r s i o n I I a t t e m p t e d t o reduce the s i n g u l a r i t y problem by d i s p l a c i n g nodes a s m a l l d i s t a n c e on e i t h e r s i d e of the t r u e i n t e r s e c t i o n p o i n t s and t h e r e b y a v o i d i n t e g r a t i n g t h r o u g h the s i n g u l a r i t y . T h i s method d i d not appear t o o f f e r any improvement. S i n c e the s m a l l d i a g o n a l c o n n e c t i n g elements are i n c l u d e d i n the c o n t o u r i n t e g r a t i o n , t h i s f o r m u l a t i o n i s r e a l l y e q u i v a l e n t t o v e r s i o n I w i t h the nodes s h i f t e d a p p r o p r i a t e l y . In the u s u a l f o r m u l a t i o n each node i s c o n s i d e r e d t o have c o n t r i b u t i o n s from each of t h e a d j a c e n t e l e m e n t s . To a v o i d i n t e g r a t i n g t h r o u g h t h e s i n g l a r i t i e s , however, th e s e s m a l l elements must be l e f t out of the c o n t o u r . As such, the m a t r i x c o e f f i c i e n t s f o r t h e s e s p e c i a l p o i n t s must be r e d e r i v e d c o n s i d e r i n g the c o n t r i b u t i o n t o 0 from one element o n l y . V e r s i o n I I I a t t e m p t s t o improve the s o l u t i o n by p l a c i n g a node d i r e c t l y a t each i n t e r s e c t i o n and s p e c i f y i n g b oth <j> and ii h e r e . L i n e t a l ( 1984) used t h i s f o r m u l a t i o n f o r m o d e l l i n g a wave maker i n a b a s i n and c l a i m e d g r e a t l y improved r e s u l t s . The number of unknowns i n t h i s case i s two l e s s and the m a t r i x must be r e c o n s t r u c t e d a c c o r d i n g l y . The a c t u a l e q u a t i o n s used are g i v e n i n Appendix IX. T h i s f o r m u l a t i o n r e s u l t e d i n a more s t a b l e s o l u t i o n , a l t h o u g h the f l u i d v e l o c i t i e s near the i n t e r s e c t i o n s were s t i l l q u e s t i o n a b l e . In a l l the above f o r m u l a t i o n s the v e l o c i t i e s a t the i n t e r s e c t i o n p o i n t s c o u l d not be de t e r m i n e d d i r e c t l y by the n u m e r i c a l p r o c e d u r e and the v a l u e s c a l c u l a t e d a t these two p o i n t s were d i s c a r d e d . Without th e s e v e l o c i t e s , however, the f r e e s u r f a c e p o s i t i o n a t the next time s t e p cannot be d e t e r m i n e d . T h i s i s c e r t a i n l y the most c h a l l e n g i n g problem i n a c h i e v i n g a good s i m u l a t i o n . For p r a c t i c a l p u r p o s e s , the s e v e l o c i t i e s must be e s t i m a t e d by o t h e r means u s i n g some s o r t of a p p r o x i m a t i o n . The most o b v i o u s c h o i c e i s t o use a p o l y n o m i a l e x t r a p o l a t i o n i n t o the body, however, s i n c e the i m m e d i a t e l y a d j a c e n t p o i n t s a r e a l s o under the i n f l u e n c e of the s i n g u l a r i t i e s t h e i r p o s i t i o n s are q u e s t i o n a b l e as w e l l . F u r t h e r m o r e , s u r f a c e i r r e g u a r i t i e s can produce l a r g e e r r o r s , e s p e c i a l l y i f the element n e a r e s t the body r e q u i r e s a l o n g e x t r a p o l a t i o n . A more t e n a b l e s o l u t i o n i s t o s i m p l y e x t e n d the a d j a c e n t nodes h o r i z o n t a l l y i n t o the body. T h i s a l t e r n a t i v e works w e l l p r o v i d e d the s u r f a c e i s not e x c e s s i v e l y s t e e p and a l l o w s the s i m u l a t i o n t o p r o c e e d . For s t e e p e r waves t h r e e p o i n t smoothing must be used i n the r e g i o n next t o the body. R e s u l t s are p r e s e n t e d i n a l a t e r s e c t i o n . 4.2.7 A NUMERICAL PERTURBATION CORRECTION Throughout t h e s i m u l a t i o n s both the f l u i d volume and t o t a l system energy were m o n i t o r e d as a measure of n u m e r i c a l s t a b i l i t y . The volume was u s u a l l y q u i t e s t a b l e a l t h o u g h the 71 t o t a l energy o f t e n grew s i g n i f i c a n t l y . These e r r o r s were no doubt due l a r g e l y t o the problems of a c c u r a c y i n the i n t e r s e c t i o n r e g i o n s and the need t o impose a r t i f i c i a l v a l u e s h e r e . W h i l e t h e s e e r r o r s a r e o b v i o u s l y u n d e s i r a b l e they do, however, p r o v i d e a p o s s i b l e means f o r c o r r e c t i n g the p o s i t i o n s of t h e i n t e r s e c t i o n p o i n t s . C o n s e r v a t i o n of mass and energy can be c o n s i d e r e d as two e q u a t i o n s f o r the two unknown i n t e r s e c t i o n l o c a t i o n s . R e f e r r i n g t o e q u a t i o n s ( 3 . 7 ) and ( 3 . 8 ) i t can be seen t h a t the energy of the f l u i d i s of the form E = E (x,y) + P fc U n f o r t u n a t e l y , t h i s e x p r e s s i o n cannot be c a s t i n t o a c l o s e d form s o l u t i o n f o r the unknown i n t e r s e c t i o n p o s i t i o n s s i n c e the c o n t o u r geometry (x,y) and the complex p o t e n t i a l (<t>,\p) a r e l i n k e d n u m e r i c a l l y . An i t e r a t i v e t r i a l and e r r o r s o l u t i o n t o s a t i s f y t h e two c o n s t r a i n t s might be p o s s i b l e , however, c o m p u t a t i o n a l time would become p r o h i b i t i v e w i t h no g uarantee of convergence. An a l t e r n a t i v e i s t o c o n s i d e r a " n u m e r i c a l p e r t u r b a t i o n " method where s m a l l changes i n system energy and volume can be d e r i v e d as f u n c t i o n s of s m a l l changes i n the i n t e r s e c t i o n p o s i t i o n s . That i s , AV = f ( A y L , Ay R, J^, J^) AE = f ( A y L , Ay R, * L , i R ) 7 2 where A y L and A y R r e p r e s e n t upward v e r t i c a l d i s p l a c e m e n t s of the l e f t and r i g h t i n t e r s e c t i o n p o i n t s and l L a n d 1 R are the l e n g t h s of the elements next t o the body. These e x p r e s s i o n s t a k e on the form AV = A(Ay L) + B(Ay R) AE = C(Ay L) + D(Ay R) + E(Ay^) 2 + F ( A y R ) 2 where the c o e f f i c i e n t s a r e f u n c t i o n s of known q u a n t i t i e s . These e q u a t i o n s can then be s o l v e d f o r Ay r and Ay_. D e t a i l e d Li K d e r i v a t i o n s may be found i n Appendix X. To implement the p r o c e d u r e , a p r e l i m i n a r y s o l u t i o n i s o b t a i n e d by the p r e v i o u s l y d e s c r i b e d method and the changes i n volume and energy n o t e d . These a r e then used t o c a l c u l a t e c o r r e c t i o n s t o the i n t e r s e c t i o n p o s i t i o n s . As o n l y two c o n s t r a i n t s a r e a v a i l a b l e , o n l y the two elements a d j a c e n t t o the body a r e i n v o l v e d . A more e l a b o r a t e c o r r e c t i o n scheme i n v o l v i n g a d j a c e n t p o i n t s as w e l l would be p o s s i b l e u s i n g p o l y n o m i a l segments, however, the d e r i v a t i o n would be much more c o m p l i c a t e d . Only the f i r s t o r d e r a p p r o x i m a t i o n d e s c r i b e d above i s c o n s i d e r e d i n the p r e s e n t work. T h i s s e l f - c o r r e c t i n g s i m u l a t i o n i s r e f e r r e d t o as v e r s i o n IV. 73 4.3 RESULTS For the purpose of t e s t i n g the s i m u l a t i o n a s i m p l e case i s examined f i r s t of a r e c t a n g u l a r body h e e l e d over i n calm w a t e r . Upon r e l e a s e the body s h o u l d undergo r o l l o s c i l l a t i o n s w h i l e d i s t u r b a n c e waves r a d i a t e outward removing energy from the body and damping the m o t i o n . The r o l l p e r i o d i n t h i s case can be c a l c u l a t e d t h e o r e t i c a l l y from the s t a n d a r d f o r m u l a f o r s m a l l a n g l e s found i n any t e x t such as Comstock (1967). For the p r e s e n t case the f o l l o w i n g c o n d i t i o n s a p p l i e d : beam = 10 f e e t , d r a f t = 3 f e e t , weight = 1900 l b / f t , r a d i u s of g y r a t i o n = 2.5 f e e t , and i n i t i a l a n g l e = 30°. The t h e o r e t i c a l r o l l p e r i o d f o r t h i s body i s 2.5 s e c . For p r e l i m i n a r y t e s t s t h e s h a r p c o r n e r s were rounded t o keep v e l o c i t i e s w e l l - b e h a v e d and a v o i d c r e a t i n g unnecessary a d d i t i o n a l c o m p l i c a t i o n s . F i g u r e 33 shows the r e s u l t s of a s i m u l a t i o n c a r r i e d out u s i n g v e r s i o n I I I w i t h L=100 f e e t , NS=25, NH=15, N=100, and At = 0.05 s e c . C l o s e i n s p e c t i o n of the f i g u r e r e v e a l s v e r y low d i s t u r b a n c e s r a d i a t i n g outward as the body o s c i l l a t e s . As e x p e c t e d , f l u i d v e l o c i t i e s were p o o r l y r e s o l v e d i n the i n t e r s e c t i o n r e g i o n , however, as the f o r c e s i n t h i s s i m p l e case were p r i m a r i l y h y d r o s t a t i c the b e h a v i o u r was good. The a c t u a l r o l l p e r i o d was about 2.6 seconds d u r i n g which t h e system g a i n e d energy c a u s i n g an i n c r e a s e i n r o l l a m p l i t u d e of about f i v e degrees over one c y c l e . The c a l c u l a t e d motions a r e p l o t t e d i n f i g u r e 34. The second t e s t c a s e examines the motion of the same body on a low a m p l i t u d e wave. F i g u r e s 35 and 36 show the 74 r e s u l t s of a motion s i m u l a t i o n on a c o s i n e wave of l e n g t h L = 100 f e e t and h e i g h t r a t i o H/L = 0.04 u s i n g v e r s i o n I I I . The i n i t i a l c o n d i t i o n s i n t h i s case were chosen c a r e f u l l y by p l a c i n g the body a t the m i d p o i n t of the t r o u g h where the v e l o c i t y p o t e n t i a l g r a d i e n t s on the f r e e s u r f a c e a re e q u a l on both s i d e s of the body and match the i n i t i a l h o r i z o n t a l body v e l o c i t y . The i n i t i a l r o l l a n g l e was chosen a r b i t r a r i l y t o be 15°. As can be seen i n the f i g u r e the n o n l i n e a r e f f e c t s were l a r g e and the wave underwent c o n s i d e r a b l e m o d i f i c a t i o n as the c r e s t passed the body. Energy was impar t e d t o the body r e s u l t i n g i n an i n c r e a s e d r o l l a n g l e and water on deck a f t e r one c y c l e . H u l l p r e s s u r e d i s t r i b u t i o n s a r e shown i n f i g u r e 37 f o r v a r i o u s p o s i t i o n s on the wave and the motions f o r sway, heave, and r o l l a r e p l o t t e d i n f i g u r e 38. Of p a r t i c u l a r i n t e r e s t a r e the r a t h e r l a r g e h o r i z o n t a l d i s p l a c e m e n t s . A c c e l e r a t i o n s i n t h i s case were as much as 0.5 g. These c h a r a c t e r i s t i c s have a l s o been noted i n model e x p e r i m e n t s [ 2 6 ] . The t h i r d t e s t case i n v o l v e d i n c r e a s i n g the wave h e i g h t f u r t h e r t o H/L = 0.08 t o in v o k e l a r g e n o n l i n e a r e f f e c t s and examine the l i m i t a t i o n s of t h e model. P l a c i n g the body i n the t r o u g h w i t h an i n i t i a l h o r i z o n t a l v e l o c i t y t o match the v e l o c i t y p o t e n t i a l g r a d i e n t r e s u l t e d i n almost immediate water on deck. By g i v i n g the body an a d d i t i o n a l n e g a t i v e a n g u l a r v e l o c i t y of -0.15 r a d / s e c deck w e t t i n g was d e l a y e d and e v e n t u a l l y came about from the downwave s i d e as the body r o l l e d t o o f a r back. F i g u r e s 39 and 40 show the a c t u a l 75 s i m u l a t i o n . The r e s u l t i n g motions a r e p l o t t e d i n f i g u r e 41. One f i n a l t e s t case i s t h a t of the body i n a v e r y s t e e p wave of H/L = 0.12. U n d i s t u r b e d , t h i s i n i t i a l c o n d i t i o n would r e s u l t i n a d i s t i n c t p l u n g i n g b r e a k e r as shown e a r l i e r i n c h a p t e r 3, however, as can be seen i n f i g u r e s 42 and 43 the presence of the body m o d i f i e d the wave c o n s i d e r a b l y and b r e a k i n g d i d not oc c u r b e f o r e water on deck t e r m i n a t e d the s i m u l a t i o n . S e v e r a l i n i t i a l c o n d i t i o n s were t r i e d f o r t h i s wave and a l l r e s u l t e d i n water on deck i n l e s s than one second. The motions f o r t h i s case a r e p l o t t e d i n f i g u r e 44. For s t e e p waves one would expect the body t o a c t somewhat as a wave damping d e v i c e removing energy much l i k e a f l o a t i n g b r e a k w a t e r . T h i s wave energy, of c o u r s e , i s c o n v e r t e d t o body motion and, as a r e s u l t , would tend t o keep the wave from b r e a k i n g . Achievement of a b r e a k i n g wave next t o the body w i l l r e q u i r e some o t h e r type of i n i t i a l wave c o n f i g u r a t i o n . V e r s i o n IV of the s i m u l a t i o n employing the s e l f - c o r r e c t i o n scheme met w i t h l i m i t e d s u c c e s s and was u s e f u l o n l y f o r n e a r l y calm w a t e r . I t became apparent t h a t c o r r e c t i n g the p o s i t i o n s and boundary c o n d i t i o n s of o n l y the i n t e r s e c t i o n n o d a l p o i n t s was i n s u f f i c i e n t , and r e a l l y the method s h o u l d be used i n c o n j u n c t i o n w i t h p o l y n o m i a l segments t o a l l o w the a d j a c e n t f r e e s u r f a c e p o i n t s t o be c o r r e c t e d as w e l l , b o t h i n p o s i t i o n and v e l o c i t y p o t e n t i a l . Having demonstrated the a b i l i t y t o s i m u l a t e extreme body motions i n s t e e p n o n l i n e a r waves, the next s t e p was t o 76 use the s i m u l a t i o n program t o c a r r y out n u m e r i c a l e x p e r i m e n t s under v a r y i n g i n i t i a l c o n d i t i o n s . The c u r r e n t v e r s i o n s have been found t o accumulate energy due t o n u m e r i c a l problems a t the i n t e r s e c t i o n s i n g u l a r i t i e s and a r e t h e r e f o r e r e a l l y o n l y u s e f u l f o r s h o r t time p e r i o d s . F u r t h e r m o r e , p r e v i o u s e x p e r i e n c e w i t h a s i m p l e r l i n e a r i z e d s i m u l a t i o n [3] has shown t h a t r o u g h l y two wave p e r i o d s a r e r e q u i r e d f o r t r a n s i e n t e f f e c t s t o decay and a l l o w the body t o a c h e i v e steady s t a t e r o l l i n g i n a wave t r a i n . T h e r e f o r e , i f c a p s i z i n g i s t o be m o d e l l e d w i t h the p r e s e n t v e r s i o n of the s i m u l a t i o n i t must be i n duced i n a s h o r t p e r i o d of t i m e . •Bearing t h i s i n mind, the s e l e c t i o n of i n i t i a l c o n d i t i o n s becomes v e r y i m p o r t a n t . I n i t i a l p o s i t i o n placement can be a r b i t r a r y , however, the c h o i c e of the t h r e e c o r r e s p o n d i n g i n i t i a l v e l o c i t i e s and t h e i r r e l a t i v e phases w i l l have a g r e a t i n f l u e n c e on the subsequent m o t i o n s . For example, f i g u r e 45 shows the e f f e c t of z e r o i n i t i a l a n g u l a r v e l o c i t y . Water on deck i s almost immediate due t o r o t a t i o n a l i n e r t i a . More l i k e l y , a s h i p a t t h i s p o i n t would have a n e g a t i v e a n g u l a r v e l o c i t y , a l t h o u g h t h i s would r e a l l y depend on the p r i o r unsteady response h i s t o r y . I t i s c l e a r now t h a t c o n d i t i o n s l e a d i n g t o water on deck or c a p s i z i n g can indeed be s i m u l a t e d and one must now ask whether or not the chosen i n i t i a l c o n d i t i o n s a r e l i k e l y t o o c c u r i n n a t u r e . 77 4.4 EXTENSION TO THREE DIMENSIONS Comparisons made w i t h a c o n s t a n t element Green's f u n c t i o n method used by o t h e r i n v e s t i g a t o r s [ 3 ] , t o s i m u l a t e a b r e a k i n g wave produced somewhat d i f f e r i n g r e s u l t s . The Cauchy i n t e g r a l method remained r e m a r k a b l y c o n s t a n t i n both energy and volume and t h e r e f o r e f o r two d i m e n s i o n a l f r e e s u r f a c e problems appears t o be the p r e f e r r e d t e c h n i q u e . U n f o r t u n a t e l y , because the method uses complex v a r i a b l e s i t cannot be extended t o t h r e e d i m e n s i o n s . For some c a s e s , however, one may be a b l e t o make an a p p r o x i m a t i o n . S i n c e s h i p motions and s t a b i l i t y a re dominated by t h e midbody, end . e f f e c t s would be e x p e c t e d t o be of minor s i g n i f i c a n c e , e s p e c i a l l y f o r s h o r t time p e r i o d s i n v o l v e d i n s h i p c a p s i z i n g . I f i n a d d i t i o n the beam/length r a t i o i s r e a s o n a b l y low one can use a " s l e n d e r s h i p " a p p r o x i m a t i o n where the c r o s s f l o w term d2<j>/b2z i n L a p l a c e ' s e q u a t i o n i s assumed s m a l l enough t o n e g l e c t and the f l o w i s l o c a l l y two d i m e n s i o n a l . S t r i p t h e o r y can then be used where the s h i p i s c o n s i d e r e d e q u i v a l e n t t o a composite of r e p r e s e n t a t i v e p r i s m a t i c s e c t i o n s as shown i n f i g u r e 46 and independent s i m u l a t i o n s a r e c a r r i e d out f o r each. The f o r c e s and moments can then summed over each s e c t i o n t o o b t a i n the t o t a l body a c c e l e r a t i o n s and r e s u l t i n g m o t i o n s . 78 • known $ - c o n s t a n t F i g u r e 28. C o n t r o l volume f o r body motion s i m u l a t i o n . 79 Figure 30. Decomposition of time derivat ives into four independent problems. 80 GENERATE ELEMENTS SET UP BOUNDARY CONDITIONS INITIAL CONDITION • V UG VG e CALCULATE VELOCITIES SOLVE MATRIX CALCULATE Dz D£ Dt Dt CALCULATE it St CALCULATE NEW SURFACE CALCULATE TIKE STEP t = t + At CALCULATE ' l F2 F3 '» M l M2 », M , SOLVE FOR sx ay e DETERMINE NEW BODY POSITION STOP gure 31. Flow c h a r t f o r body motion s i m u l a t i o n a l g o r i t h m . F i g u r e 32. Boundary c o n d i t i o n s t e s t e d . 0.0 sec 0.6 sec 1.2 sec 1.8 sec 2.4 sec Figure 33. Simulation of r o l l motion in calm water. -40 T I M E ( s e c ) Figure 34. Roll motion for calm water case. F i g u r e 35. S i m u l a t i o n o f motion i n wave H/L = 0.04. CD F i g u r e 36. S i m u l a t i o n o f m o t i o n i n wave H/L = 0.04. 86 F i g u r e 37. H u l l p r e s s u r e d i s t r i b u t i o n s f o r H/L = 0.04. 88 T I M E ( s e c ) F i g u r e -38. M o t i o n s f o r c a s e H/L = 0.04. F i g u r e 39. S i m u l a t i o n o f m o t i o n i n wave H/L = 0.08. F i g u r e 40. S i m u l a t i o n o f m o t i o n i n wave H/L = 0.08. 92 40 -5 • -10. 101 -5' -10. 0 0.5 1 1.5 2 2.5 3 3.5 T I M E ( s e c ) F i g u r e 41. M o t i o n s f o r c a s e H/L = 0.08. F i g u r e 42. S i m u l a t i o n o f motion i n wave H/L = 0.12. 95 F i g u r e 43. S i m u l a t i o n o f m o t i o n i n wave H/L = 0.12. 4 0 F i g u r e 44. M o t i o n s f o r c a s e H/L = 0.12. Figure 45. Result of zero i n i t i a l angular velocity. F i g u r e 46. E q u i v a l e n t p r i s m a t i c r e p r e s e n t a t i o n o f s h i p . 5. CONCLUSIONS AND RECOMMENDATIONS A complex v a r i a b l e boundary i n t e g r a l method has been used t o n u m e r i c a l l y s i m u l a t e the b e h a v i o u r of n o n l i n e a r f r e e s u r f a c e waves. B r e a k i n g waves i n deep and s h a l l o w water have been s i m u l a t e d and p r o f i l e v e l o c i t i e s d e t e r m i n e d . The method has proven t o be p o w e r f u l and r o b u s t . V i r t u a l l y any c o n t i n u o u s smooth wave can be s i m u l a t e d p r o v i d e d i n i t i a l c o n d i t i o n s can be a s s i g n e d . The method was extended t o i n c l u d e the n o n l i n e a r m otions of a body on the f r e e s u r f a c e , and s i m u l a t i o n s were c a r r i e d out f o r s e v e r a l t e s t c a s e s . The presence of s i n g u l a r i t i e s a t the f r e e s u r f a c e i n t e r s e c t i o n p o i n t s p r e v e n t e d the d i r e c t d e t e r m i n a t i o n of v e l o c i t i e s i n t h e s u r r o u n d i n g r e g i o n s , l i m i t i n g a c c u r a c y of the s i m u l a t i o n . No a c c u r a t e s o l u t i o n s t o t h i s i n t e r s e c t i o n problem have been d e v e l o p e d t o d a t e . A two s t e p p e r t u r b a t i o n c o r r e c t i o n p r o c e d u r e was i n t r o d u c e d t o f o r c e the a d d i t i o n a l c o n s t r a i n t s of mass and energy c o n s e r v a t i o n . T h i s f i r s t o r d e r c o r r e c t i o n was used f o r l o c a t i n g the two i n t e r s e c t i o n p o i n t s . The a c c u r a c y of the i m m e d i a t e l y a d j a c e n t p o i n t s , however, was a l s o q u e s t i o n a b l e and work i s needed t o make the c o r r e c t i o n p r o c e d u r e more r o b u s t t o a l l o w t h e s e a d d i t i o n a l p o i n t s t o be i n c l u d e d . L o c a l smoothing of f u n c t i o n d i s t r i b u t i o n s may be r e q u i r e d . F r e e s u r f a c e smoothing may be h e l p f u l as w e l l , however, c a r e must be taken t h a t i m p o r t a n t d e t a i l s of the s u r f a c e b e h a v i o u r a r e not l o s t i n the p r o c e s s . E m p i r i c a l 99 100 i n f o r m a t i o n would be u s e f u l here as a g u i d e . There i s wide scope f o r f u r t h e r work on the body motion problem. One p o s s i b l e a l t e r n a t i v e s o l u t i o n method t h a t has so f a r not been attempted i s t o r e f o r m u l a t e the problem as an i n n e r and o u t e r s o l u t i o n . The o u t e r s o l u t i o n would i n v o l v e the u s u a l Cauchy i n t e g r a l around a p a t h t h a t i s c l e a r of the s i n g u l a r p o i n t s and w e l l behaved, w h i l e i n n e r s o l u t i o n s on each s i d e of the body c o u l d u t i l i z e a d d i t i o n a l i n f o r m a t i o n such as c o n s e r v a t i o n of mass, momentum f l u x , and energy w h i l e e n s u r i n g matched v a l u e s of complex p o t e n t i a l and v e l o c i t i e s a l o n g the common b o u n d a r i e s . P o l y n o m i a l d i s t r i b u t i o n s of f u n c t i o n s c o u l d be assumed c l o s e t o the body p r o v i d e d t h e i r c o e f f i c i e n t s c o u l d be d e t e r m i n e d . The f o r c e s due t o a wave b r e a k i n g on t h e s i d e of a v e s s e l a r e of g r e a t i n t e r e s t i n s t u d y i n g the s a f e t y of s h i p s a t s e a . These f o r c e s c o u l d be e s t i m a t e d by a s i m u l a t i o n e i t h e r by means of a p p l y i n g an e x p l i c i t boundary c o n d i t i o n on the r e c i p i e n t s u r f a c e , or more s i m p l y by c o n s i d e r i n g the b r e a k i n g wave j e t t o be a p p r o x i m a t e l y e q u i v a l e n t t o an i d e a l i z e d j e t whose f o r c e on a f l a t p l a t e can be c a l c u l a t e d from momentum c o n s i d e r a t i o n s knowing the f l u i d v e l o c i t y and e f f e c t i v e f l o w r a t e as a f u n c t i o n of t i m e . E m p i r i c a l i n p u t i s s t i l l r e q u i r e d t o d e f i n e the f r e e s u r f a c e b e h a v i o u r near the i n t e r s e c t i o n s , but i s l a c k i n g . I t i s recommended t h a t e x p e r i m e n t s be c a r r i e d out on two d i m e n s i o n a l b o d i e s i n a wave b a s i n under both s m a l l and l a r g e a m p l i t u d e waves so t h a t d e t a i l e d e m p i r i c a l r e s u l t s may 101 be o b t a i n e d as a b a s e l i n e f o r n u m e r i c a l e x p e r i m e n t s . S t r o b e photographs and f l o a t i n g marker p a r t i c l e s a r e needed t o a c c u r a t e l y o b t a i n the f r e e s u r f a c e p r o f i l e s and f l u i d v e l o c i t i e s . T h i s i n f o r m a t i o n would be u s e f u l f o r e s t a b l i s h i n g r e a l i s t i c i n i t i a l c o n d i t i o n s and a s s e s s i n g the p r o g r e s s of n u m e r i c a l s i m u l a t i o n s as v a r i o u s f o r m u l a t i o n s a r e t r i e d . The importance of v i s c o u s e f f e c t s such as boundary l a y e r development and v o r t e x shedding remain t o be d e t e r m i n e d and may be q u i t e s i g n i f i c a n t , e s p e c i a l l y i n c ases where the h u l l geometry i s not smooth. Flow v i s u a l i z a t i o n s t u d i e s s h o u l d be c a r r i e d out u s i n g dye i n j e c t i o n t o e l u c i d a t e the f l o w s t r u c t u r e . Such e x p e r i m e n t s would a l s o p r o v i d e a u s e f u l assessment of t h e v a l i d i t y of s t r i p t h e o r y by a l l o w i n g o b s e r v a t i o n of the l o n g i t u d i n a l c r o s s f l o w component. REFERENCES 1. B h a t t a c h a r y y a , R. (1979) Dynamics of Marine Vehicles. John W i l e y & Sons, New York. 2. Carnahan, B. (1969) A p p l i e d Numerical Methods. John W i l e y & Sons, New York. 3. Chan, J.L.K. (1986) U n p u b l i s h e d r e s u l t s . 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MIT P r e s s , Cambridge, Mass. 1 04 25. Rawson, K . J . & Tupper, E.C. (1968) B a s i c Ship Theory. Longmans, London. 26. R o h l i n g , G.F. (1986) " E x p e r i m e n t a l I n v e s t i g a t i o n of F i s h i n g V e s s e l S t a b i l i t y i n a T r a n s v e r s e Seaway", M.A.Sc. T h e s i s , Dept. of M e c h a n i c a l E n g i n e e r i n g , U n i v e r s i t y of B r i t i s h C olumbia, Vancouver, Canada. 27. S a r p k a y a , T. & I s a a c s o n , M. (1981) Mechanics of Wave Forces on Offshore Structures. Van N o s t r a n d R e i n h o l d Co., New York. 28. S c h u l t z , W.W., Ramberg, S.E. & G r i f f i n , O.M. (1986) "Steep and B r e a k i n g Deep Water Waves", P r e p r i n t 16i h Symp. Naval Hydrodynamics, B e r k e l e y , C a l i f o r n i a . 29. S c h w a r t z , L.W. (1974) "Computer E x t e n s i o n and A n a l y t i c C o n t i n u a t i o n of S t o k e ' s E x p a n s i o n f o r G r a v i t y Waves", J. Fluid Mech. V o l . 62, pp. 553-578. 30. S k j e l b r e i a , L. & H e n d r i c k s o n , J . (1960) " F i f t h Order G r a v i t y Wave Theory", Proc. 7th Coastal Eng. Conf., The Hague, pp.184-196. 31. S t o k e s , G.G. (1847) "On the Theory of O s c i l l a t o r y Waves", Trans. Camb. P h i l . Soc. V o l . 8, pp. 441-455. 32. U r s e l l , (1964) "The Decay of the Free M o t i o n of a F l o a t i n g Body", /. F l u i d Mech. V o l . 19, pp. 305-319. 33. V i n j e , T. & B r e v i g , P. (1980) " B r e a k i n g Waves on F i n i t e Water Depth - A N u m e r i c a l Study", SIS Report, Norwegian Hydrodynamic L a b o r a t o r i e s . 34. V i n j e , T. & B r e v i g , P. (1981a) "Numerical C a l c u l a t i o n of F o r c e s from B r e a k i n g Waves", Int. Symp. Hydrodynamics in Ocean Eng., Norwegian I n s t . Tech., pp. 547-565. 35. V i n j e , T. & B r e v i g , P. (1981b) " N o n l i n e a r S h i p M o t i o n s " , 3rd Int. Conf. Numerical Ship Hydrodynamics, P a r i s , pp. 257-266. 36. V i n j e , T. & B r e v i g , P. (1981c) " N o n l i n e a r Two D i m e n s i o n a l S h i p M o t i o n s " , Study", SIS Report, Norwegian Hydrodynamic L a b o r a t o r i e s . 37. Wehausen, J.V. (1971) "The M o t i o n of F l o a t i n g B o d i e s " , Ann. Rev. Fluid Mech. V o l . 3, pp.237-268. 38. W i e g e l , R.L. (1964) Oceanographical E n g i n e e r i n g . 105 P r e n t i c e - H a l l , Englewood C l i f f s , N.J. Yim, B. (1985) "Numerical S o l u t i o n f o r Two D i m e n s i o n a l Wedge Slamming W i t h a N o n l i n e a r F r e e S u r f a c e C o n d i t i o n " , P r e p r i n t , 4th Int. Conf. Numerical Ship Hydrodynamics, Washington, D.C. APPENDIX I THE CAUCHY INTEGRAL THEOREM A c c o r d i n g t o the Cauchy Theorem the pa t h i n t e g r a l of an a n a l y t i c f u n c t i o n around a c l o s e d c o n t o u r i s z e r o so l o n g as z 0 i s o u t s i d e the e n c l o s e d r e g i o n For t h e purpose of n u m e r i c a l s o l u t i o n z 0 must take on the v a l u e s of the nod a l p o i n t s on the co n t o u r and hence must be a l l o w e d t o approach C. The c o n t o u r C can be c o n s i d e r e d as composed of C, and C 2 where C 2 subtends z 0 w i t h a c i r c u l a r a r c of r a d i u s e. f(z) dz t h e r e f o r e - 0 (A1.1) In t h i s c a s e Cauchy's Theorem can be w r i t t e n as 106 107 ' z-z d z + ' ^ " d z " ° ( A 1' 2> C1 2 Z 0 c 2 Z Z 0 On C 2 1 6 z 0 + e e 16 ,„ dz = i e e do so e q u a t i o n (A1.2) reduces t o f -2— dz + f" —JLy i e e 1 6 de - o C, Z " Z 0 o c e 1 6 where a i s the i n t e r i o r a n g l e a t z 0 e q u a l t o n when C i s smooth and ir/2 at a c o r n e r . E v a l u a t i n g i n the l i m i t as e —**• 0, becomes C and 8 |^z7 d z = 1 a ^ z o > APPENDIX I I FORMULATION OF INTEGRAL EQUATION The c o n t o u r i s d i s c r e t i z e d i n t o N elements and the v a l u e s of 0(z) a r e c a l c u l a t e d a t the nodes j o i n i n g each. The i n t e g r a l e q u a t i o n (2.2) can then be reduced t o a l i n e a r e q u a t i o n f o r n u m e r i c a l s o l u t i o n , as d e s c r i b e d below. As o u t l i n e d i n Chapter 2, Cauchy's theorem can be w r i t t e n <£ -2— dz = 1 a B(z n) (A2. 1 ) <J z-z n u c o N Zj+1 1 f / d b - d z * " 1 ° B ( z o ) ( A 2 - 2 > N E y 8 = i « B 4 j=l J J To e v a l u a t e t h e i n t e g r a l i n (A2.2), and hence c a l c u l a t e Y j y a l i n e a r d i s t r i b u t i o n of 0 i s assumed over each element. /3(z) can then be e x p r e s s e d by the l i n e a r i n t e r p o l a t i o n f o r m u l a Z-Z. Z 4 + 1 ~ Z j+1 j ^ J+1 J 3 3 + 1 which i s then s u b s t i t u t e d i n t o (A2.2). I n t e g r a t i o n i s then c a r r i e d out as . 108 109 B . Zj+1 Z...-Z B,., j+1 z-z _ — 1 — _ / r-Ji^-i dz + r J + 1 > / (—i) dz B i B I, + where i = / j + 1 l i t l d z - f 1 J z-z n J z-z n Z j 0 Z j 0 dz Z "Z 2 j o 2 « Z q («JH-«j> - ( z j - z o ) i n V z o S u b s t i t u t i n g I , and I 2 back i n t o (A2.4) g i v e s / Z j + 1 d Z = B rrV 1" 2 0) l n rV 1" 0] - n > 1 z j z 0 J ^ X Zj+1 Z 0 z j z 0 z ., ,-z r R e t u r n i n g t o (A2.2) and u s i n g the above r e s u l t one o b t a i n s 1 a B(z Q) = j z - Z q J -z-N 1 a B = I v e j=l where 1 10 i , J J J-1 J - l i J+l J z j + r z i  Z J _ Z I (A2.3) T h i s e x p r e s s i o n r e p r e s e n t s the i n f l u e n c e c o e f f i c i e n t f o r /3j a t any Z j away from the c o n t r o l p o i n t z^. I t can be seen however t h a t problems a r i s e when the c o n t r o l p o i n t i s i n the neighbourhood of Z j and t h e s e s p e c i a l c a s e s must be examined s e p a r a t e l y . i = i ~ 1 In t h i s c a s e , ^j-i j e q u a l s the e x p r e s s i o n (A2. ) i n the l i m i t as z . ^ z ^ _ 1 . The f i r s t term becomes z-z Z l * Z j - l ZJ" ZJ-1 zj+rz± which i s of the form l i m [z l n ( 1 / z ) ] = 0, and hence v a n i s h e s z-~o l e a v i n g o n l y the second term , -z, V l . J - f 1 ^ ^ ) r , J Zj+1 Z J Z j 2 j - 1 i = " The q u a n t i t i e s p r e m u l t i p l y i n g the In terms i n (A2. ) reduce t o 1 i n the l i m i t i j l e a v i n g . 2 , 3 i+j Z j - 1 Z i Z j Z i 111 J - l Z i j - l z j i = j + 1 Here T j e q u a l s (A2.3) i n the l i m i t as i _ * . j + 1. The second term becomes Z j , . — z , • L n v z - z ; J  z i * z j + l J 4" 1 J J 1 which i s of the form l i m [z l n ( z ) ] = 0, and hence v a n i s h e s 2 - 0 l e a v i n g o n l y the f i r s t term V l J = ^ ^ ^ ^ 1» z j z j - l z j - l 2j+1 The l e f t h a n d s i d e of (A2.3) has a c t u a l l y been t a k e n c a r e of when the l i m i t was t a k e n of i j and i s s e t t o z e r o t o p r e v e n t d u p l i c a t i o n , r e s u l t i n g i n the f i n a l form f o r s o l u t i o n . N j=l 1 J J where 1 12 z - z zr zi J+1 2 j Z j Z j - 1 i = j - 1 i n [IXLll) Z J - 1 " Z J 1 = j j j - l Z j - 1 Zj+1 i = j + 1 APPENDIX I I I CALCULATION OF r.. TERM C a l c u l a t i o n of the T^j terms i s s t r a i g h t f o r w a r d u s i n g the Cauchy p r i n c i p l e v a l u e s f o r the complex l o g a r i t h m f u n c t i o n . C a l c u l a t i n g however, r e q u i r e s s p e c i a l a t t e n t i o n s i n c e i t c o n t a i n s the term - i a from e q u a t i o n (2.3) as i t s i m a g i n a r y p a r t . To c l a r i f y the problem can be w r i t t e n i n p o l a r form as The f i g u r e on the l e f t above shows the a n g l e t h a t r e s u l t s when the branch c u t a t -it i s i n s i d e a. Here (# A e?B) e r r o n e o u s l y y i e l d s the e x t e r i o r a n g l e . I f the branch c u t i s o u t s i d e a, as i n the f i g u r e on the r i g h t , then t h e r e i s no problem. In g e n e r a l , t h i s e r r o r o c c u r s whenever the where (0. - d O s h o u l d e q u a l -a. 1 13 1 1 4 bra n c h c u t of the l o g f u n c t i o n l i e s i n s i d e the i n c l u d e d a n g l e a. In the s e c a s e s 2 i r i must be s u b t r a c t e d from the i m a g i n a r y p a r t of . A l t e r n a t i v e l y , s i n c e a i s by d e f i n i t i o n a p o s i t i v e number, an e q u i v a l e n t r u l e i s t h a t i f Im ( r^) < 0 then s u b t r a c t 2ni t o ensure t h a t i t i s e q u a l t o - i a . APPENDIX IV CALCULATION OF VELOCITIES The v e l o c i t y of each p o i n t on the co n t o u r i s found as 36 w — —— 3z The complex p o t e n t i a l /J has been assumed p i e c e w i s e l i n e a r so 3/3/3z i s d i s c o n t i n u o u s a t each node. P r o v i d e d elements a r e s m a l l , 3/3/3z can be approximated by (f }j ~ 3 j-l 6 J-l + V j + 'jflVl (A4.1) where the c o e f f i c i e n t s a re t o be d e t e r m i n e d . + 1 a n < 3 ^ j - 1 i e s (n) can be w r i t t e n as T a y l o r s e r e x p a n s i o n s away from /3j B j - l " n Z = 0 '-it- ( z j - l " z f j n Bj+1 = n= 0 n! ( z j + l " Zj> which are then s u b s t i t u t e d i n t o (A4.1) g i v i n g + l a M ( z M - z j ) + V i ( v r z j ) ] B j + i [ a j - i ( z j - i - z j ) 2 + V i ( z ^ i " zj)2]e5 + 115 1 16 C o e f f i c i e n t s can be equated g i v i n g t h r e e l i n e a r e q u a t i o n s f o r the t h r e e unknown q u a n t i t i e s a j - l + a j + a j + l = ° a j - l ( z j - l • zj> + Vl ( zj+1 ~ z j } = 1 a j - i ( z j - l - z j ) 2 + a j f l ( z j + l - z j ) 2 = ° S o l u t i o n y i e l d s Z,i+1 - 2 i " l zj-l ~ ZjJ ^Zj+1 " Zj-J 21-l - 2 i '*1 = I ^  " 2jJ l ^ i " VlJ a j = " a j - i " V i APPENDIX V EQUATIONS FOR WAVE SIMULATION The s e t of l i n e a r e q u a t i o n s t o be s o l v e d t o o b t a i n the unknown q u a n t i t i e s X j which a r e e i t h e r <f>^ or 0^, can be w r i t t e n as There a r e N no d a l p o i n t s i n the c o n t o u r , however, t h e r e a re a c t u a l l y N-2 unknowns s i n c e p o i n t s 1 and N2 a r e i d e n t i c a l as are N3 and N4. The number of l i n e a r e q u a t i o n s i s t h e r e f o r e a l s o N-2. The a c t u a l e q u a t i o n s a r e : 1 < i < N2 N4 < i < N N2-1 N3-1 N4-1 j-1 i j j j=N2+l i J l k 2 j=N3 1 J 2 N N2 + E ^ i k ^ i i ^ i = " 1 V j j=N4+l l k 1 J 2 j-1 J J 117 118 N2 < i < N4 N2~l N3-1 m _ ± J-1 J J j=N2+l l k J j = N 3 N N2 j=N4+l " * V » J " " £ V j where k r e p r e s e n t s the n o d a l p o i n t on the v e r t i c a l boundary e x a c t l y o p p o s i t e j and i s e q u a l t o k=N+NS+2-j By p e r i o d i c i t y the f o l l o w i n g r e l a t i o n s a l s o h o l d *k = •j *N2 = +1 *k = *j <t>N4 = *N3 APPENDIX VI CALCULATION OF ENERGY The k i n e t i c energy f o r a f l u i d volume i s \ - J ! J v 2 dV a In o r d e r t o e v a l u a t e t h i s e x p r e s s i o n the volume i n t e g r a l must be r e w r i t t e n as a c o n t o u r i n t e g r a l i n v o l v i n g f u n c t i o n v a l u e s on the boundary. The f l u i d v e l o c i t y a t any p o i n t can be e x p r e s s e d as V2 m Vd) • Vd> which can be s u b s t i t u t e d i n t o Green's f i r s t i d e n t i t y J / [4>V2<|> + vi|> • V $ ] d v = / [d>Vo>] • n d s v c y e i l d i n g - f / • (|t) as C c - f / • d * c 119 120 S i n c e the normal v e l o c i t y component i s z e r o on the seabed and c a n c e l s on the v e r t i c a l b o u n d a r i e s , t h i s i n t e g r a l need o n l y be e v a l u a t e d on the f r e e s u r f a c e . C o n s i d e r i n g the s u r f a c e t o be p i e c e w i s e l i n e a r the i n t e g r a l becomes N2 = £ . J 4. A* i=l N2 * + *. " 2 1 ( 2 ) ( * i + l - V 1=1 which by p e r i o d i c i t y reduces t o N2 The p o t e n t i a l energy f o r the f l u i d volume i s L By c o n s i d e r i n g the volume t o be made up of t r a p e z o i d a l s e c t i o n s under each element t h i s e x p r e s s i o n becomes N2 x i + l |* I / y2 dx i = l Ti ££. T r l l i i + 1 £x 2 ^ l3 J x ± Ay 121 y i e l d i n g f i n a l l y EP = 6* Z ( X1 " X l + l ) ( y i + l y i + W + y i 2 ) A g a i n o n l y f r e e s u r f a c e v a l u e s a r e r e q u i r e d . The f l o a t i n g body k i n e t i c energy i n c l u d e s both t r a n s l a t i o n a l and r o t a t i o n a l c o n t r i b u t i o n s . P o t e n t i a l energy can be measured from any a r b i t r a r y datum w i t h the s t i l l water l e v e l b e i n g chosen f o r c o n v e n i e n c e . The t o t a l body energy i s t h e r e f o r e - f < UG 2 + V G 2 ) + 1 1 1 + m g yG APPENDIX V I I CALCULATION OF BODY BOUNDARY CONDITIONS The p o s i t i o n and v e l o c i t i e s of the body a r e assumed known a t each time s t e p hence the v e l o c i t y component normal t o the body can be w r i t t e n as n 3n 3s T h i s e x p r e s s i o n i s then i n t e g r a t e d n u m e r i c a l l y t o determine 0 as ij> = / v ds + il> J n o 6 The normal v e l o c i t y component V n i s found as f o l l o w s . The v e l o c i t y of a p o i n t P on the h u l l i s 122 123 where *G = V + V v , = 6 k x R P/G R = (x-x G)i + (y-y G)J The u n i t normal v e c t o r n = k x swhere a _ dx 1 + dy j  s ds The v e l o c i t y a t P i s t h e r e f o r e \ = {UG 1 + VG J l + * [-(y-yG)1 + (X-VN (A?. 1) and the normal component i s 3* -T r = v • n 9 s p The stream f u n c t i o n i s found by i n t e g r a t i n g d+ - u G dy - v G dx - 6[(y-y G)dy + (x-xG>dx] t o f i n a l l y o b t a i n 6 R 2 + = " — + U G I Y ~ Y G ) " V(-X~XG> + * 0 ( A 7 . 2 ) 124 where R2 = ( X - x G ) 2 + ( y - y G ) 2 ; For time s t e p p i n g purposes the time d e r i v a t i v e 3<^/8t i s needed. E q u a t i o n (A7.2) r e p r e s e n t s \p on the moving f l u i d boundary so the m a t e r i a l d e r i v a t i v e must be t a k e n f o l l o w i n g the f l u i d p a r t i c l e s . 6R2 where = ( y _ y G ) a x ' ( x " X G ) a y " — + UG h ( ^ G ) " VG 57 (X"XG} v_ , s Dx Dt ( X~V " DF " UG = - e < y - y G > ^ ( y - y ^ = | f - v G » 6 ( x - x G ) The l a s t s t e p was o b t a i n e d by r e f e r r i n g t o e q u a t i o n (A7.1) s i n c e Dx/Dt and Dy/Dt a re j u s t the v e l o c i t y components of p o i n t s on the boundary V p . T h e r e f o r e , 2£ - ( y - y G ) a x - ( x - x G ) a y - - r + 6 [ u G ( x - x G ) + v G ( y - y G ) ] For a p p l y i n g a boundary c o n d i t i o n i n the n u m e r i c a l s o l u t i o n , 125 however, the d e r i v a t i v e i s needed as seen by the i n s t a n t a n e o u s boundary. That i s , 8iJ> Dili 97 - DF " V ' V * where the a d v e c t i v e term i n t h i s c a s e i s due t o motion of the r i g i d body and i t s v e l o c i t y must be used f o r v g i v i n g = + <|*)j = -v i + u j v - [u G-6(y-y G)]i + [v G+9(x-x G)]j C o m p l e t i n g the a l g e b r a y i e l d s f o r the f i n a l boundary c o n d i t i o n If - (y-y G)a x - ( „ 6 ) . y - I f i + u Gv - v Gu + [(x-x G)(u G-u) + (y-y G)(v G-v)]6 APPENDIX V I I I CALCULATION OF BODY FORCES AND MOMENTS The body f o r c e v e c t o r i s de t e r m i n e d by i n t e g r a t i n g the p r e s s u r e s over the wetted h u l l as F = - / P n-ds o r , f o r n u m e r i c a l e v a l u a t i o n , as the sum of the c o n t r i b u t i o n s over each element. The p r e s s u r e i s taken as the average of the v a l u e s a t the bounding n o d a l p o i n t s and a c t s i n the d i r e c t o n normal t o the s F = - I P n As s h u l l . The u n i t normal v e c t o r i s found n = i s where s Az As 126 127 The t o t a l f o r c e F + i F can t h e r e f o r e be computed as x y N6-1 Tf -If F . _ E i(^±L-i) ( Z ) i=N5 The moment a c t i n g on the h u l l about G i s e q u a l t o M = - / P rxri ds s where the p r e s s u r e over each element i s a g a i n t a k e n as the average of the bounding n o d a l v a l u e s and c o n s i d e r e d t o a c t at the c e n t r o i d of t h e p r e s s u r e d i s t r i b u t i o n . R" i s the r a d i u s v e c t o r from G t o the body s u r f a c e p o i n t where the p r e s s u r e i s a c t i n g M = - Z P(r x fi)As s The c r o s s p r o d u c t i s e v a l u a t e d by c o n s i d e r i n g R" and n t o be complex numbers and u s i n g the i d e n t i t y Zft X Z f i = I m ^ Z A ZB^* The moment i s t h e r e f o r e computed as N 6 - 1 p1 + i _ p 1 z i + r z i M = " i= N 5 ( - ^ } X J 1 ( J i H " zG><zi-rzi>*} APPENDIX IX EQUATIONS FOR BODY MOTION SIMULATION The s e t of l i n e a r e q u a t i o n s used i n v e r s i o n I I I of the body motion s i m u l a t i o n a r e g i v e n below. In t h i s v e r s i o n b oth <t> and $ a r e s p e c i f i e d a t the i n t e r s e c t i o n p o i n t s N5 and N6 hence t h e r e a re two l e s s unknowns than i n the wave s i m u l a t i o n . The stream f u n c t i o n on the seabed i s an unknown c o n s t a n t e v a l u a t e d a t N3. There a r e c o n s e q u e n t l y N-3 equat i o n s . The a c t u a l e q u a t i o n s a r e : 1 ^ i < N5  N5 < i <: N2  N4 < i ^  N  i = N3 N5-1 N6-1 N2-1 j-1 Z j=N5+l 2 j=N6+l N3-1 I j=N2+l N4-1 N I j=N4+l N4-1 + ( W * j + j=N3 (a +a )il> + I vaik ij^j ,_, j=N3 N 5 N6 N2 j=N5 - I j=N6 128 129 N5 < i < N6 N2 < i < N3 N3 i < N4 N5-1 N6-1 N2-1 j - l i j j j=N5+l ± J 3 j=N6+l 1 3 J N3-1 N4-1 N N4-1 + I (a. .+a..H,+ I a A - I (b +b.)^, " I b * j=N2+l 1 : 1 i k 3 j=N3 i j j j=N4+l i k i J 3 j=N3 1 3 N5 N6 N2 = " £ a A . + Z b i|> - E ad> j=l ^ 3 j=N5 1 3 j j=N6 1 J 3 where k r e p r e s e n t s the nod a l p o i n t on the v e r t i c a l boundary e x a c t l y o p p o s i t e j and i n t h i s case i s eq u a l t o k=N+NS+NH+2-j By p e r i o d i c i t y the f o l l o w i n g r e l a t i o n s a l s o h o l d *k " *j *H2 = *1 *k = *j *m = *N3 1 30 The stream f u n c t i o n on the seabed i s a c o n s t a n t e v a l u a t e d a t N3 so * j = * N 3 j = N 3 , N 4 APPENDIX X NUMERICAL PERTURBATION CORRECTION The c o n t r o l volume energy f o r the f i r s t a pproximate s o l u t i o n i s E and the volume i s V. The two i n t e r s e c t i o n p o i n t s a r e then d i s p l a c e d upward on the body by A y L and A y R r e s u l t i n g i n new v a l u e s f o r the energy and volume of E' and V . The t o t a l change i n energy i n c l u d i n g t h e known change i n body energy i s AE = AE B + AE R + AEp w h i l e the change i n volume i s AV = V - V The problem i s t o f i n d the p o s i t i o n c o r r e c t i o n s A y L and A y R r e q u i r e d t o o f f s e t the e r r o r s i n energy and volume r e s u l t i n g from the n u m e r i c a l s o l u t i o n a t each time s t e p . The two elements a d j a c e n t t o the body a r e a d j u s t e d a c c o r d i n g l y . 131 1 32 R e f e r r i n g t o the above f i g u r e the change i n volume i s the a r e a of the two t r i a n g l e s AV = \ U R Ay R + £ L Ay L) where 1 L and 1 R a r e the l e f t and r i g h t element l e n g t h s . T h i s i s the f i r s t c o n s t r a i n t . U s i n g e q u a t i o n (3.8) •p " - W ( W i + W + y i 2 ) the p o t e n t i a l energy f o r the p r e l i m i n a r y c o n f i g u r a t i o n can be expanded and s u b t r a c t e d from the e x p a n s i o n f o r the d i s p l a c e d c o n f i g u r a t i o n . The d i f f e r e n c e i s then AE p = f {(3 £ R y N 5 ) ( A y R ) + * R ( A y R ) 2 S i m i l a r l y , u s i n g e q u a t i o n (3.7) N2 \ " i ± l ± <*i*i+l ' • i + l V t h e k i n e t i c energy b e f o r e d i s p l a c e m e n t i s s u b t r a c t e d from the e x p r e s s i o n r e s u l t i n g a f t e r d i s p l a c e m e n t . By u s i n g the f o l l o w i n g Cauchy Riemann a p p r o x i m a t i o n s *a ~ *N5 *N5 ~ *N5-1 *N5-1 " *N5 A yR *R •b " *N6 *N6+1 *N6 A y L £L +b " *N6 *N6+1 " A y L K and c a r r y i n g out the a l g e b r a , t he d i f f e r e n c e works out t o R + I T ^ N 6 - * N 6 + l + ( W l - * N 6 ) 2 ] } 

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