@prefix vivo: . @prefix edm: . @prefix ns0: . @prefix dcterms: . @prefix skos: . vivo:departmentOrSchool "Applied Science, Faculty of"@en, "Mechanical Engineering, Department of"@en ; edm:dataProvider "DSpace"@en ; ns0:degreeCampus "UBCV"@en ; dcterms:creator "Fitz-Clarke, John R."@en ; dcterms:issued "2010-07-10T16:54:46Z"@en, "1986"@en ; vivo:relatedDegree "Master of Applied Science - MASc"@en ; ns0:degreeGrantor "University of British Columbia"@en ; dcterms:description """A numerical method is presented for simulating the behaviour of large amplitude nonlinear free surface waves including wave breaking. Various initial conditions are given and the subsequent surface profiles are calculated by a time stepping simulation. The flow field is solved as a boundary value problem for the velocity potential using a complex variable method based on the Cauchy integral theorem. Waves of varying shape, height, and length are investigated to determine the parameters necessary for wave breaking and the resulting fluid velocities. The technique has proven to be very accurate and stable. The method is extended to predict the motions of a two dimensional floating body in large amplitude seas accounting for non-linear effects and fluid-body interaction. The presence of singularities at the free surface intersection points was found to severely limit accuracy of the solution and attempts to overcome this problem are discussed. An extension to handle three dimensional ships is also described."""@en ; edm:aggregatedCHO "https://circle.library.ubc.ca/rest/handle/2429/26286?expand=metadata"@en ; skos:note "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 = . = 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 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

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 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 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 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 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 = 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 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 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 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 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,

(x) as t h e r e i s a one t o one correspondence 30 between and w on the s u r f a c e . S e l e c t i o n of 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 . 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 - 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 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 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/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 and ^ a r e harmonic f u n c t i o n s , then so must be d/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/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 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 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 (,\\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/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 . Dep't 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 Columbia, Vancouver, Canada. 4. Chan, R.K.C. & S t r e e t , R.L. (1970) \"A Computer Study of F i n i t e A m p l i t u d e Water Waves\", /. Comp. Phys., V o l . 6, pp. 68-79. 5. Chapman, R.B. (1979) \"Large A m p l i t u d e T r a n s i e n t M o t i o n of Two D i m e n s i o n a l F l o a t i n g B o d i e s \" , /. Ship Res., V o l . 23, No. 1, pp. 20-31. 6. C o k e l e t , E.D. (1977) \" B r e a k i n g Waves,\" Nature, V o l . 267, pp. 769-774. 7. Comstock, J.P. (1967) P r i n c i p l e s of Naval Architecture, Soc. of N a v a l A r c h i t e c t s and Marine E n g i n e e r s . , New York. 8. F a l t i n s e n , O.M. (1977) \" N u m e r i c a l S o l u t i o n s of T r a n s i e n t N o n l i n e a r Free S u r f a c e M o t i o n O u t s i d e or I n s i d e Moving B o d i e s \" , 2nd Int. Conf. Numerical Ship Hydrodynamics, U n i v . of C a l i f o r n i a , B e r k e l e y . 9. F l a n i g a n , F . J . (1972) Complex Variables: Harmonic and A n a l y t i c F u n c t i o n s . Dover P u b l i c a t i o n s I n c . , New York. 10. G a r r i s o n , C.J. (1975) \"Hydrodynamics of Large O b j e c t s i n the Sea. P a r t I I : M o t i o n of F r e e - F l o a t i n g B o d i e s \" , /. Hydr onaut i cs, V o l . 9, pp.58~63. 11. Greenhow, M., B r e v i g , P. & T a y l o r . J . (1982) \"A T h e o r e t i c a l and E x p e r i m e n t a l Study of the C a p s i z e of S a l t e r ' s Duck i n Extreme Waves\", /. F l u i d Mech. V o l 118, pp. 221-239. 12. Greenhow, M. & Woei-Min, L. (1985) \" N u m e r i c a l S i m u l a t i o n of N o n l i n e a r F r e e S u r f a c e Flows G e n e r a t e d by Wedge E n t r y 102 1 03 and Wavemaker M o t i o n s \" , P r e p r i n t , 4th Int. Conf. Numerical Ship Hydrodynamics, Washington, D.C. 13. I s a a c s o n , M. de S. Q. (1982) \" N o n l i n e a r Wave E f f e c t s on F i x e d and F l o a t i n g B o d i e s \" , /. F l u i d Mech. V o l . 120, pp. 267-281. 14. Kim, W.D. (1966) \"On a Free F l o a t i n g S h i p i n Waves\", /. Ship Res. V o l . 10, pp.182-191. 15. K j e l d s e n , P. (1981) \"Shock P r e s s u r e s from Deep Water B r e a k i n g Waves\", Int. Symp. Hydrodynamics in Ocean Eng., Norwegian I n s t . Tech., pp. 567-584. 16. K j e l d s e n , S.P. & Myrhaug, D. (1979) \" B r e a k i n g Waves i n Deep Water and R e s u l t i n g Wave F o r c e s \" , OTC 3646, 11th O f f s h o r e Tech. Conf., Houston, Texas, pp. 2515-2522. 17. K o r v i n - K r o u k o v s k y , B.V. (1955) \" I n v e s t i g a t i o n s of S h i p M o t i o n s i n Re g u l a r Waves\", Trans. SHAME, V o l . 63, pp. 386-435. 18. Lee, C M . (1969) \"The Second Order Theory of Heaving C y l i n d e r s i n a Free S u r f a c e \" , J. Ship Res. V o l . 12, pp. 313-327. 19. L i n , W.M., Newman, J.N., & Yue, D.K. (1984) \" N o n l i n e a r F o r c e d M o t i o n s of F l o a t i n g B o d i e s \" , 15th Symp. on Naval Hydrodynamics, Hamburg, Germany, pp. 33-49. 20. L o n g u e t - H i g g i n s , M.S. (1977) \"Advances i n the C a l c u l a t i o n of Steep S u r f a c e Waves and P l u n g i n g B r e a k e r s \" , 2nd Int. Conf. Numerical Ship Hydrodynamics, U n i v . of C a l i f o r n i a , B e r k e l e y . 21. L o n g u e t - H i g g i n s , M.S. & C o k e l e t , E.D. (1976) \"The D e f o r m a t i o n of Steep S u r f a c e Waves on Water, I . A N u m e r i c a l Method of Computation\", Proc. R. Soc. Lond. Series A. V o l 350, pp. 1-26. 22. MacCamy, R.C. (1964) \"The M o t i o n of C y l i n d e r s of Sh a l l o w D r a f t \" , /. Ship Res. V o l . 7, No. 3, pp. 1-11. 23. M a s k e l l , S.J. & U r s e l l , F. (1970) \"The T r a n s i e n t 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 . 44, pp. 303-313. 24. Newman, J.N. (1980) Marine Hydrodynamics. 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 ) 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 ^ 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 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>*} 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 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 ] } "@en ; edm:hasType "Thesis/Dissertation"@en ; edm:isShownAt "10.14288/1.0096909"@en ; dcterms:language "eng"@en ; ns0:degreeDiscipline "Mechanical Engineering"@en ; edm:provider "Vancouver : University of British Columbia Library"@en ; dcterms:publisher "University of British Columbia"@en ; dcterms:rights "For non-commercial purposes only, such as research, private study and education. Additional conditions apply, see Terms of Use https://open.library.ubc.ca/terms_of_use."@en ; ns0:scholarLevel "Graduate"@en ; dcterms:title "Numerical simulation of nonlinear waves and ship motions"@en ; dcterms:type "Text"@en ; ns0:identifierURI "http://hdl.handle.net/2429/26286"@en .