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Non-linear wave forces on floating breakwaters Niwinski, Chris T. 1982

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NON-LINEAR WAVE FORCES ON FLOATING BREAKWATERS by CHRIS T. NIWINSKI B.Ap.Sc, U n i v e r s i t y Of B r i t i s h Columbia, 1980 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF APPLIED SCIENCE i n THE FACULTY OF GRADUATE STUDIES Department Of C i v i l E n g i n e e r i n g We accept t h i s t h e s i s as conforming to the r e q u i r e d standard THE UNIVERSITY OF BRITISH COLUMBIA A p r i l 1982 © C h r i s T. N i w i n s k i , 1982 In presenting t h i s thesis i n p a r t i a l f u l f i l m e n t of the requirements for an advanced degree at the University of B r i t i s h Columbia, I agree that the Library s h a l l make i t f r e e l y available for reference and study. I further agree that permission f o r extensive copying of t h i s thesis f o r s c h o l a r l y purposes may be granted by the head of my department or by h i s or her representatives. I t i s understood that copying or publication of t h i s thesis for f i n a n c i a l gain s h a l l not be allowed without my written permission. Department of C/vt^ ^rjineenr^ The University of B r i t i s h Columbia 1956 Main Mall Vancouver, Canada V6T 1Y3 Date M«V 3T, DE-6 (3/81) A b s t r a c t A n o n - l i n e a r numerical method of c a l c u l a t i n g wave f o r c e s on f l o a t i n g bodies i s i n v e s t i g a t e d . A computer program has been developed that w i l l time s t e p a given wave past a f i x e d two-dimensional r e c t a n g u l a r breakwater. The e f f e c t of v a r i o u s input parameters on the accuracy of r e s u l t s i s i n v e s t i g a t e d , and optimal v a l u e s of the parameters are determined. The r e q u i r e d i n i t i a l c o n d i t i o n f o r the time stepping procedure has a s p e c i f i e d i n c i d e n t wave atte n u a t e d to zero flow i n the v i c i n i t y of the body. Incident waves p r e d i c t e d by both l i n e a r wave theory and Stokes' f i f t h order wave theory are t e s t e d . The decay l e n g t h and group v e l o c i t y of the i n c i d e n t wave t r a i n are v a r i e d . I t i s found that a decay l e n g t h of approximately one wavelength i s o p t i m a l , and that the group v e l o c i t y of the developed flow i s r e l a t i v e l y independent of the i n i t i a l l y s p e c i f i e d group v e l o c i t y . D i f f e r e n t time step s i z e s , segment le n g t h s , and time stepping equations are i n v e s t i g a t e d . The t e s t s conducted i n d i c a t e t h at t w e n t y - f i v e time steps and ten segments per wave p e r i o d are s u f f i c i e n t f o r ac c u r a t e r e s u l t s . The Adams-Bashforth three step method i s found to be the p r e f e r r e d time stepping technique. Forces on the f i x e d body and t r a n s m i t t e d wave h e i g h t s are obtained from the program. The r e s u l t s compare w e l l with p r e v i o u s l y p u b l i s h e d r e s u l t s and c l e a r l y demonstrate the n o n - l i n e a r i t y of the method, with d i f f e r e n t f o r c e curves r e s u l t i n g from v a r y i n g wave h e i g h t s . i i \ TABLE OF CONTENTS page ABSTRACT i i TABLE OF CONTENTS . i i i LIST OF FIGURES v NOMENCLATURE v i ACKNOWLEDGEMENT ix CHAPTER 1 INTRODUCTION 1 1.1 I n t r o d u c t i o n 1 1.2 Survey of L i t e r a t u r e 4 1.3 General D e s c r i p t i o n of Method 5 1.4 Areas of I n v e s t i g a t i o n 7 CHAPTER 2 THEORETICAL DEVELOPMENT 10 2.1 Governing Equations 10 2.2 Green's F u n c t i o n R e p r e s e n t a t i o n 13 2.3 C o n t r o l Surface C o n d i t i o n s 15 2.4 Forces on a F i x e d Body 16 2.5 Summary of Time Stepping Procedure 17 CHAPTER 3 NUMERICAL PROCEDURE 19 3.1 Numerical Time Stepping Procedure 19 3.2 Numerical Procedure - Computer S o l u t i o n .. 24 3.2.1 Segment G r i d 24 3.2.2 In c i d e n t Wave S p e c i f i c a t i o n 27 3.2.3 Time Stepping Equation 28 3.2.4 Matrix S o l u t i o n 29 CHAPTER 4 INPUT PARAMETER CONSIDERATIONS 31 4.1 I n t r o d u c t i o n 31 i i i Page 4.2 Group V e l o c i t y 32 4.3 Incident Wave Decay Length 35 4.4 Time Step Size 37 4.5 Segment Length 39 4.5.1 Global Segment Length 39 4.5.2 Body Segments 40 CHAPTER 5 OTHER CONSIDERATIONS 42 5.1 Time Stepping Equation 42 5.2 Incident Wave Type 44 CHAPTER 6 RESULTS 46 6.1 E x c i t i n g Forces 46 6.2 Transmission C o e f f i c i e n t 49 CHAPTER 7 CONCLUSIONS AND FURTHER STUDIES 50 7.1 Conclusions 50 7.2 Recommendations f o r Further Study 54 BIBLIOGRAPHY 55 APPENDIX I 58 APPENDIX II 66 i v LIST OF FIGURES page F i g . 1 RUBBLE MOUND AND FLOATING BREAKWATERS 68 F i g . 2 FLOW CHART FOR DYNAMIC ANALYSIS 69 F i g . 3 CLOSED SURFACE FOR GREEN'S IDENTITY 70 F i g . 4 TYPICAL INCIDENT WAVE PROFILE 71 F i g . 5 DEFINITION SKETCH 72 F i g . 6 GEOMETRICAL RELATIONSHIP BETWEEN x, and £' 73 F i g . 7 SEGMENT GRID 74 F i g . 8 FLOW CHART FOR COMPUTER PROGRAM 75 F i g . 9 CPU SECONDS PER TIME STEP 76 F i g . 10 EFFECT OF VARYING DECAY LENGTH 77 F i g . 11 EFFECT OF VARYING TIME STEP SIZE 78 F i g . 12 EFFECT OF VARYING GLOBAL SEGMENT LENGTH 79 F i g . 13 EFFECT OF VARYING BODY SEGMENT LENGTH 80 F i g . 14 EFFECT OF VARYING TIME STEPPING METHODS 81 F i g . 15 WAVE THEORY RANGES OF VALIDITY 82 F i g . 16 LINEAR AND STOKES' FIFTH ORDER WAVES 83 F i g . 17 FLOW DEVELOPMENT WITHOUT BODY 84 F i g . 18 TYPICAL HORIZONTAL FORCE DEVELOPMENT 85 F i g . 19 TYPICAL VERTICAL FORCE AND MOMENT DEVELOPMENT ... 86 F i g . 20 FORCE RESULTS FOR BEAM/DRAUGHT = 2.0 87 F i g . 21 FORCE RESULTS FOR BEAM/DRAUGHT = 4.0 88 F i g . 22 FORCE RESULTS FOR BEAM/DRAUGHT = 8.0 89 F i g . 23 TRANSMISSION COEFFICIENT RESULTS 90 v NOMENCLATURE a ,b = matrix elements A = immersed c r o s s - s e c t i o n a l area of body A = matrix element [A] = c o e f f i c i e n t matrix B = beam of breakwater c. = value of Green's i d e n t i t y n u m e r i c a l l y i n t e g r a t e d 1 on c o n t r o l s u r f a c e d = s t i l l water depth D = draught of breakwater F = f o r c e v e c t o r a c t i n g on body F x = x-component of F F z = z-component of F g = a c c e l e r a t i o n due to g r a v i t y G = Green's f u n c t i o n H = i n c i d e n t wave height Hfc = t r a n s m i t t e d wave height i , j = segment i n d i c e s k = 2 i r / L = i n c i d e n t wave number K t = t r a n s m i s s i o n c o e f f i c i e n t L b = i n c i d e n t wave l e n g t h L d = body dimension L = decay l e n g t h of i n c i d e n t wave M = moment about x-z o r i g i n a c t i n g on body n = d i r e c t i o n normal to s u r f a c e n ,n = d i r e c t i o n c o s i n e s of n X z — n = u n i t normal v e c t o r d i r e c t e d outward from f l u i d r e g i o n v i u n i t n o r m a l v e c t o r d i r e c t e d o u t w a r d f r o m t h e f l u i d r e g i o n r e f l e c t i o n i n t h e s e a b e d number o f s e g m e n t s on S. + S, f b number o f s e g m e n t s on S, + S + S X. b c number o f s e g m e n t s on S, b number o f s e g m e n t s on S c number o f s e g m e n t s on p r e s s u r e d i s t a n c e b e t w e e n x a n d e d i s t a n c e b e t w e e n x a n d £• a c l o s e d s u r f a c e i m m e r s e d body s u r f a c e c o n t r o l s u r f a c e f r e e s u r f a c e t i m e wave p e r i o d h o r i z o n t a l c o o r d i n a t e m e a s u r e d i n d i r e c t i o n o f wave p r o p a g a t i o n f r o m m i d p o i n t o f body beam v e c t o r o f p o i n t ( x , z ) v e r t i c a l c o o r d i n a t e m e a s u r e d u p w a r d s f r o m s t i l l w a t e r l e v e l segment l e n g t h segment l e n g t h on S b t i m e s t e p v e l o c i t y 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 o f i n c i d e n t wave f r e e s u r f a c e e l e v a t i o n a b o v e , s t i l l w a t e r l e v e l i n c i d e n t wave a n g u l a r f r e q u e n c y a n g l e b e t w e e n x-£_ a n d n v i i angle between x-£' and n' x-component of v e c t o r of p o i n t v e c t o r of p o i n t U , - [ c+2d]) z-component of £ s o l u t i o n v e c t o r element s o l u t i o n v e c t o r c o e f f i c i e n t v e c t o r element c o e f f i c i e n t v e c t o r v i i i ACKNOWLEDGEMENT The author wishes to thank Dr. M. de S t . Q. Isaacson and Dr. M. D. Olson f o r t h e i r a s s i s t a n c e and advice i n the p r e p a r a t i o n of t h i s t h e s i s . i x 1 CHAPTER I; INTRODUCTION 1 .1 I n t r o d u c t i o n Wave-structure i n t e r a c t i o n i s a t o p i c of major importance i n the design of marine s t r u c t u r e s . The f o r c e s generated by s u r f a c e g r a v i t y waves on both f i x e d and f l o a t i n g s t r u c t u r e s are needed to p r e d i c t the motions, s t r e s s e s , and s t r a i n s on a given s t r u c t u r e . The purpose of t h i s t h e s i s i s to examine one method of p r e d i c t i n g the f o r c e s generated by f i n i t e amplitude water waves on a f i x e d breakwater. Breakwaters are used to p r o t e c t harbours and other c o a s t a l f a c i l i t i e s . By i n t e r f e r i n g with an i n c i d e n t wave t r a i n , a breakwater reduces the wave h e i g h t s w i t h i n the p r o t e c t e d a r e a . One can c o n s t r u c t a f i x e d s t r u c t u r e such as a rubble mound breakwater, or a l t e r n a t e l y , p l a c e a f l o a t i n g breakwater and moor i t with c a b l e s . The two a l t e r n a t i v e s are shown s c h e m a t i c a l l y i n F i g u r e 1. In c e r t a i n a p p l i c a t i o n s , the c h o i c e of a f l o a t i n g type of breakwater o f f e r s some advantages over the rubble mound type of breakwater. I f one wants to e f f e c t i v e l y p r o t e c t a s p e c i f i e d marine area from a f u l l range of long and short p e r i o d waves, then a f i x e d rubble mound type of breakwater i s u s u a l l y necessary. However, i f one i s mainly concerned with waves of s h o r t e r p e r i o d s , then a f l o a t i n g 2 breakwater i s u s u a l l y a v i a b l e a l t e r n a t i v e which i s often cheaper, p a r t i c u l a r l y i n deeper water where a f i x e d rubble mound type of breakwater r e q u i r e s l a r g e amounts of m a t e r i a l . In a d d i t i o n to the economic advantage, a f l o a t i n g breakwater i n t e r f e r e s much l e s s with sediment tra n s p o r t and marine l i f e m i gration than does a rubble mound breakwater. Considerations i n designing a f l o a t i n g breakwater include the s i z e , shape, c o n s t r u c t i o n , c o n f i g u r a t i o n , and connections between i t s u n i t s . The behaviour of the f l o a t i n g breakwater under the a c t i o n of waves must be p r e d i c t e d . The forces generated by the waves on the breakwater, or e x c i t i n g f o r c e s , can be p r e d i c t e d using experimental or a n a l y t i c a l f o r c e r e s u l t s . Using the e x c i t i n g f o r c e s with the system's dynamic p r o p e r t i e s (added mass and damping c o e f f i c i e n t s ) and the mooring system p r o p e r t i e s , one can obtain p r e d i c t i o n s for the motions of the breakwater, the s t r e s s e s and s t r a i n s on the s t r u c t u r e and mooring system, and the t r a n s m i t t e d wave heights expressed as transmission c o e f f i c i e n t s . A flow chart of t h i s a n a l y t i c a l procedure i s shown i n Figure 2. The general approach described above f o r s o l v i n g the f l o a t i n g breakwater problem i s not unique. Other s o l u t i o n s using d i f f e r e n t approaches e x i s t and are b r i e f l y r e f e r r e d to i n Section 1 .2. The method used i n t h i s t h e s i s to p r e d i c t the e x c i t i n g forces i s set apart from other c u r r e n t l y used techniques by the f a c t that i t i s a non-linear method. In a d d i t i o n , i t must be s t r e s s e d that the s o l u t i o n discussed 3 here i s f o r a monochromatic wave approaching normal to the le n g t h of the breakwater. The relevance of t h i s approach to a c t u a l f i e l d c o n d i t i o n s must be c o n s i d e r e d when a p p l y i n g the r e s u l t s o b t a i n e d here to a given d e s i g n . For purposes of t h i s t h e s i s , we r e s t r i c t o u r s e l v e s to a breakwater having a two-dimensional c r o s s - s e c t i o n and i n v e s t i g a t e the e x c i t i n g f o r c e s generated by a beam wave t r a i n . We use a time stepping procedure developed by Isaacson* 1 } which uses a Green's boundary i n t e g r a l i d e n t i t y combined with the usual governing equations and boundary c o n d i t i o n s t o time s t e p an i n c i d e n t wave t r a i n past a f i x e d r e c t a n g u l a r c r o s s - s e c t i o n . More s p e c i f i c a l l y , we i n v e s t i g a t e the l i m i t a t i o n s of the method, v a r y i n g parameters such as the time step s i z e , segment l e n g t h , i n c i d e n t wave form, and time s t e p p i n g e q u a t i o n . Force r e s u l t s on a f i x e d breakwater are obtained using the method and are compared to e x i s t i n g experimental r e s u l t s . The f o r c e s are obtained from an a n a l y s i s that assumes a m o t i o n l e s s body. The body motions and a c t u a l t r a n s m i s s i o n c o e f f i c i e n t s are not i n v e s t i g a t e d . However, the method does pro v i d e a lower l i m i t f o r the t r a n s m i s s i o n c o e f f i c i e n t , and hence an i n d i c a t i o n of the r e l a t i v e v a l u e s of the t r a n s m i s s i o n c o e f f i c i e n t . The e f f e c t i v e n e s s of reducing wave h e i g h t s c o u l d a l s o be i n v e s t i g a t e d f o r v a r i o u s breakwater c r o s s - s e c t i o n s , but t h i s i s not done here. 4 1.2 Survey of Literature Much has been written on the topic of f l o a t i n g breakwaters. Topics discussed have included the advantages and disadvantages ( J o l y ( 2 > and Adee ( 3 > ), case studies and operational experience (Adee < 3 ), and M i l l e r ' 4 > ), experimental evaluation of performance (Brebner & Ofuya' 5', Hom-ma, Horikawa, & M o c h i z u k i ( 6 } , S k e y < 7 ) , Sutko' a ), and Harris & Webber ( 9 > ), and the t h e o r e t i c a l prediction of performance (Mei & B l a c k ' 1 0 ' , Longuet-Higgins' 1 1', Nichols & H i r t < 1 2 ) , and Ijima, Chou, & Yoshida' 1 3>). Recently, with the advent of the computer as an aid to methods involving lengthy computations, numerical methods have evolved that were previously impractical ( i f not impossible) due to the large volume of c a l c u l a t i o n s . Numerical methods have been used by Mei & Black' 1 0>, Nichols & H i r t ' 1 2 > , Garrison' 1 4>, and F r a s e r ' 1 5 ' , to obtain forces and motions of f l o a t i n g breakwaters. Boundary integral methods based on Green's identity have been used by Ijima & Chou' 1 6', and Ijima, Yoshida, & Yamamoto' 1 7 }. A l l of the methods referred to thus far are l i n e a r . The non-linear boundary integral method used in t h i s thesis was developed by Isaacson' 1 >. The results for the forces on a fixed breakwater obtained here are compared to those of F r a s e r ' 1 5 ' , and V u g t s ' 1 9 } . 5 1.3 General D e s c r i p t i o n of Method The method i s based on the usual governing equations. L a p l a c e ' s equation, the kinematic and dynamic f r e e s u r f a c e boundary c o n d i t i o n s , and the seabed and body s u r f a c e boundary c o n d i t i o n s must a l l be s a t i s f i e d . These equations are given i n Chapter I I . In order to c o n s i d e r a c l o s e d boundary over which the Green's i d e n t i t y used can be i n t e g r a t e d , we choose v e r t i c a l c o n t r o l s u r f a c e s some d i s t a n c e away from the body, and r e f l e c t about the seabed the bounding s u r f a c e d e f i n e d by the c o n t r o l s u r f a c e s , f r e e s u r f a c e , and body s u r f a c e . The r e s u l t i n g c l o s e d s u r f a c e i s shown i n F i g u r e 3. The problem i s s o l v e d by f i r s t s p e c i f y i n g an i n c i d e n t wave us i n g a chosen wave theory i n a manner that p r e s c r i b e s zero flow i n the v i c i n i t y of the body. Using an a p p r o p r i a t e wave theory, we s p e c i f y the i n c i d e n t flow i n terms of a wave height and p e r i o d which d e f i n e the i n i t i a l flow p o t e n t i a l and outward normal d e r i v a t i v e of the p o t e n t i a l along the e n t i r e c l o s e d boundary. The chosen wave theory a l s o d e f i n e s the f r e e s u r f a c e p r o f i l e along the f r e e s u r f a c e on the i n c i d e n t s i d e of the body. The c o n t r o l s u r f a c e s are p l a c e d s u f f i c i e n t l y d i s t a n t so that the s c a t t e r e d wave p o t e n t i a l due to the i n t e r a c t i o n of the p r e s c r i b e d i n c i d e n t wave with the body does not have enough time to propagate to the c o n t r o l s u r f a c e s d u r i n g the t o t a l flow development time chosen. Hence, we can p r e s c r i b e v a l u e s of the p o t e n t i a l and outward normal d e r i v a t i v e of the p o t e n t i a l along the c o n t r o l 6 s u r f a c e s t h r o u g h o u t t h e t i m e s t e p p i n g p r o c e d u r e , a c c o r d i n g t o o u r c h o s e n wave t h e o r y . The f r e e s u r f a c e , body s u r f a c e , and c o n t r o l s u r f a c e s a r e d i v i d e d i n t o an a p p r o p r i a t e number o f s e g m e n t s . Segment m i d p o i n t v a l u e s o f t h e p o t e n t i a l a n d i t s o u t w a r d n o r m a l d e r i v a t i v e a r e assumed t o be c o n s t a n t o v e r e a c h segment l e n g t h . The k i n e m a t i c a n d d y n a m i c f r e e s u r f a c e b o u n d a r y c o n d i t i o n s , t o g e t h e r w i t h t h e G r e e n ' s i d e n t i t y a p p l i e d a t e a c h m i d p o i n t , a r e u s e d t o o b t a i n new v a l u e s f o r t h e p o t e n t i a l a n d i t s n o r m a l d e r i v a t i v e a s w e l l a s t h e f r e e s u r f a c e e l e v a t i o n a t t h e end o f e a c h t i m e s t e p . From t h e v a r i a t i o n i n p o t e n t i a l v a l u e s b e t w e e n a d j a c e n t s e g m e n t s , t h e f l u i d v e l o c i t i e s c a n be a p p r o x i m a t e d . U s i n g B e r n o u l l i ' s e q u a t i o n , t h e f l u i d v e l o c i t i e s a l o n g t h e body s u r f a c e g i v e t h e p r e s s u r e s a c t i n g on e a c h s e g m e n t , and t h e p r e s s u r e s a r e n u m e r i c a l l y i n t e g r a t e d t o g i v e t h e t o t a l f o r c e s a c t i n g on t h e b o d y . We t h u s o b t a i n t h e h y d r o d y n a m i c f o r c e s a c t i n g on t h e body a t e a c h t i m e s t e p . The t i m e s t e p p i n g p r o c e d u r e b e g i n s by s p e c i f y i n g an i n c i d e n t wave f l o w f o r t h e f i r s t two o r more t i m e s t e p s , a n d c o n t i n u e s u n t i l s u f f i c i e n t t i m e h a s e l a p s e d f o r a t l e a s t one c o m p l e t e wave t o i n t e r a c t w i t h t h e b o d y . A f u l l d e v e l o p m e n t o f t h e t h e o r y a n d method i s g i v e n i n C h a p t e r s I I a n d I I I . 7 1.4 Areas of Investigation In t h i s thesis, we examine various aspects of the time stepping procedure used. The effects of time step size and segment length on the accuracy of results is investigated. The time step size is treated in the dimensionless time step to wave period r a t i o , At/T, while the segment length is s i m i l a r l y treated as the segment length to wave length r a t i o , AS/L. By varying the two parameters over an appropriate range for a given test case, we i d e n t i f y the desirable values of both At/T and AS/L that w i l l ensure good accuracy within the constraints of the method i t s e l f , and we v e r i f y the condition on the method stated by Isaacson, that At/T « AS/L. The global segment length parameter AS/L i s limited both by the condition that At/T « AS/L and the need to have enough segments on the free surface to approximate the free surface p r o f i l e . In addition, there must be enough segments on the body surface to allow reasonably accurate numerical integration of the t o t a l forces on the body from the pressure d i s t r i b u t i o n defined by the v e l o c i t i e s at each segment on the body surface. This minimum segment number requirement on the body surface, expressed as a body segment length to body dimension r a t i o , AS B/L B, is determined. The kinematic and dynamic free surface boundary conditions are used to obtain time stepping equations for the free surface elevation and the potential respectively. There are a number of techniques available that w i l l 8 e x p l i c i t l y e x t r a p o l a t e v a l u e s at t + A t from va l u e s at pr e v i o u s times given a d e f i n i n g equation f o r the time d e r i v a t i v e of a given parameter. These numerical methods are d e s c r i b e d i n Burden, F a i r e s , & R e y n o l d s ' 2 0 ' . We t e s t methods that r e q u i r e anywhere from two to f i v e p r e v i o u s p o i n t s to e x t r a p o l a t e a forward value, and examine and compare the r e s u l t s from each method. D i f f e r e n t wave t h e o r i e s can be used to d e s c r i b e the i n c i d e n t wave flow. In p a r t i c u l a r , , a higher order theory such as Stokes' f i f t h order theory i s d e s i r a b l e f o r steeper waves. The r e s u l t s o b tained u s i n g l i n e a r and Stokes' f i f t h order t h e o r i e s f o r d i f f e r e n t wave steepnesses are compared, and the range of v a l i d i t y of each i s v e r i f i e d . When s p e c i f y i n g an i n c i d e n t wave t r a i n , the n e c e s s i t y fo r z ero flow i n the v i c i n i t y of the body r e q u i r e s that a c o n v e n t i o n a l harmonic wave t r a i n be decayed to zero flow near the body when s p e c i f y i n g the p r e c i s e i n c i d e n t wave form. A t y p i c a l i n c i d e n t wave form to be used i s shown i n F i g u r e 4. The l e n g t h over which the decay of the i n c i d e n t wave occurs i s examined. The decay l e n g t h i s t r e a t e d as a m u l t i p l e of the wavelength, and i s desig n a t e d L d as shown i n F i g u r e 4. The e f f e c t of v a r y i n g the d i f f e r e n t parameters mentioned above i s i n v e s t i g a t e d and d i s c u s s e d i n Chapter IV. A f t e r determining the c o n s t r a i n t s on the v a r i o u s parameters mentioned above, we o b t a i n the f o r c e s on a f i x e d body and p l o t them as a f u n c t i o n of frequency, a l l q u a n t i t i e s being expressed i n dime n s i o n l e s s form. The f o r c e 9 r e s u l t s are compared to p r e v i o u s l y p u b l i s h e d experimental r e s u l t s . S i n c e the method i s n o n - l i n e a r , f o r c e r e s u l t curves are p l o t t e d f o r d i f f e r e n t values of H/d. C a l c u l a t i n g the e x c i t i n g f o r c e s on the f l o a t i n g body by t r e a t i n g i t as f i x e d a l s o p r o v i d e s a lower l i m i t on the t r a n s m i s s i o n c o e f f i c i e n t . By r e c o r d i n g the wave h e i g h t s on the s i d e of the body o p p o s i t e t o the i n c i d e n t wave t r a i n , a t r a n s m i s s i o n c o e f f i c i e n t i s obtained that p r o v i d e s a r e l i a b l e lower l i m i t f o r the a c t u a l c o e f f i c i e n t value (which depends on the motions of the body). T h i s t r a n s m i s s i o n c o e f f i c i e n t lower l i m i t i s compared to experimental r e s u l t s p u b l i s h e d by Nece & Richey^ 2 1^ . Although a lower l i m i t to the t r a n s m i s s i o n c o e f f i c i e n t i s of l i t t l e p r a c t i c a l v a l u e , i t does p r o v i d e a r e l a t i v e i n d i c a t i o n of the e f f e c t i v e n e s s of v a r i o u s s e c t i o n s i n reducing wave h e i g h t s . The f o r c e and t r a n s m i s s i o n c o e f f i c i e n t r e s u l t s are presented and d i s c u s s e d i n Chapter V I . 10 CHAPTER 1 1 : THEORETICAL DEVELOPMENT 2.1 Governing Equations For our two-dimensional example, we have a f i x e d body of r e c t a n g u l a r c r o s s - s e c t i o n with beam B, draught D, and f l o a t i n g i n water of uniform s t i l l water depth d. We a t t a c h an x-z c o o r d i n a t e system to the body, with x measured h o r i z o n t a l l y in the d i r e c t i o n of wave propagation and z measured upward v e r t i c a l l y from the s t i l l water l e v e l . The o r i g i n i s l o c a t e d at s t i l l water l e v e l midway along the beam of the body. We l e t n denote the f r e e s u r f a c e e l e v a t i o n above s t i l l water l e v e l . The d e f i n i t i o n sketch corresponding to the above i s shown i n F i g u r e 5 . The f l u i d i s assumed to be non-viscous and inc o m p r e s s i b l e , with the flow assumed to be i r r o t a t i o n a l . The f l u i d motion may thus be represented by the v e l o c i t y p o t e n t i a l 4> . In the usual manner, i t can be shown that <(> s a t i s f i e s the Laplace equation w i t h i n the boundaries of the f l u i d , V2<(> = 0 (1 ) o r , f o r our two-dimensional case, = 0 (1a) 11 The seabed i s assumed to be h o r i z o n t a l and impermeable. Hence, no flow can occur normal to the seabed. T h i s c o n d i t i o n i s expressed as | i = 0 at z = -d (2) dZ The body s u r f a c e i s a l s o assumed to be impermeable, g i v i n g | i = 0 on S, . (3) d n D Here, n denotes the d i r e c t i o n normal to the s u r f a c e . The f r e e s u r f a c e of the f l u i d i s governed by kinematic and dynamic f r e e s u r f a c e boundary c o n d i t i o n s . The kinematic f r e e s u r f a c e boundary c o n d i t i o n r e q u i r e s that the normal v e l o c i t y of the f r e e s u r f a c e e l e v a t i o n correspond to the normal flow v e l o c i t y , that i s , that p a r t i c l e s at the f r e e s u r f a c e remain a t t a c h e d to the f r e e s u r f a c e . T h i s c o n d i t i o n i s u s u a l l y expressed i n two dimensions as The form of (4) we use f o r our purposes i s one used by (22) Isaacson v 1,-given by (4a) 12 Here, t denotes time and n i s the d i r e c t i o n c o s i n e i n the z z d i r e c t o n of the v e c t o r n , which we d e f i n e as the u n i t outward normal to the s u r f a c e . S i m i l a r l y , n x w i l l denote the d i r e c t i o n c o s i n e of n i n the x d i r e c t i o n . The dynamic f r e e s u r f a c e boundary c o n d i t i o n r e q u i r e s that the pressure at the s u r f a c e be uniform (atmospheric), | i + gn + (V<j>)2 = constant on S, (5) Here, g i s the a c c e l e r a t i o n due to g r a v i t y , and the constant i s the B e r n o u l l i constant which we set equal to zero f o r convenience. 13 2.2 Green's Function R e p r e s e n t a t i o n The v e l o c i t y p o t e n t i a l <|> at any p o i n t along a c l o s e d two-dimensional f l u i d boundary can be expressed as f o l l o w s using a r e d u c t i o n of the second form of Green's theorem, •(x) = - A /(GCx,i)-f£«) - KD-f(x ,D)ds (6) s Here x i s the p o i n t (x,z) being c o n s i d e r e d , i i s a p o i n t (E,C) on the c l o s e d curve i n the x-z plane over which the i n t e g r a t i o n i s being performed, dr i s measured along the c l o s e d curve, and G i s an a p p r o p r i a t e Green's f u n c t i o n . The o r i g i n of the (E,e) c o o r d i n a t e s i s the same as f o r the (x,z) c o o r d i n a t e s . The seabed boundary c o n d i t i o n given by (2) allows us to r e f l e c t the p o i n t s on the boundary d e f i n e d by S ,S , and £ b S about the seabed and o b t a i n a c l o s e d curve over which the c Green's i d e n t i t y given by (6) can be i n t e g r a t e d . T h i s c l o s e d curve i s shown i n F i g u r e 3. The Green's f u n c t i o n which accounts f o r the symmetry about the seabed i s given by G(x, O = l n ( r ) + l n ( r ' ) (7) Here, r i s the d i s t a n c e between x and |_ given by r = |x-gj = A X - E ) ' + l z - 5 ) ' ( 8 ) 1 4 and r' i s the d i s t a n c e between x and V = (£,-[c+2d]) (the r e f l e c t i o n of £ i n the seabed) expressed as r' = |x-£| = /(x-£) 2+(z+c+2d) 2' . (9) A d e f i n i t i o n sketch of the v a r i o u s c o o r d i n a t e s and d i s t a n c e s d e s c r i b e d here i s shown i n F i g u r e 6. R e f e r r i n g to F i g u r e 6, we obt a i n ^ - ( X , E ) = C O S Y + C O S Y ' ( 1 0 ) 9n — — x r ' Here, Y i s the angle between a v e c t o r from x to i and the ve c t o r n , given by n•(E-x) /, , \ C O S Y = - - - ( 1 1 ) and Y' i s the angle between a v e c t o r from x to £' and the ve c t o r n', given by C O S y = = ", ( 1 2 ) 15 2.3 C o n t r o l Surface C o n d i t i o n s As mentioned i n S e c t i o n 1.3, the v e r t i c a l c o n t r o l s u r f a c e s to the r i g h t and l e f t s i d e s of the body c r o s s -s e c t i o n are p l a c e d s u f f i c i e n t l y d i s t a n t t o ensure t h a t the s c a t t e r e d waves from the body do not reach the c o n t r o l s u r f a c e s d u r i n g the e n t i r e time s t e p p i n g procedure. I f , as we s h a l l assume, the i n c i d e n t wave t r a i n propagates from l e f t to r i g h t , and the i n i t i a l c o n d i t i o n s s p e c i f y zero flow in the v i c i n i t y of the body, then the r i g h t s i d e c o n t r o l s u r f a c e w i l l have zero flow a c r o s s i t throughout the e n t i r e time stepping procedure. The l e f t c o n t r o l s u r f a c e w i l l have flow c o n d i t i o n s s p e c i f i e d by the chosen i n c i d e n t wave theory, ( W ) 3n 3n on S f o r a l l t (13) c Here, the s u p e r s c r i p t < w ) denotes a value s p e c i f i e d by a chosen wave theory at a given time, t . Apply i n g the c o n d i t i o n s from (8) to the Green's i d e n t i t y given by ( 6 ) , we o b t a i n • <£> - - 7 / ( G C X . D ^ I ) - • f wh)fOc.i)}ds s c - 7 /(G(x,i)f£<I) - <0(I)f<x,l))dS (14) S f + S b 16 2.4 Forces on a F i x e d Body We wish to o b t a i n the h o r i z o n t a l and v e r t i c a l f o r c e s as w e l l as the moment on a f i x e d body. From the p o t e n t i a l d i s t r i b u t i o n as a f u n c t i o n of time and l o c a t i o n i n the f l u i d r e g i o n , we can o b t a i n the pr e s s u r e d i s t r i b u t i o n using the unsteady B e r n o u l l i equation, P - " P t ^ + ^ V * ) 2 3 (15) 5 I n t e g r a t i n g the pr e s s u r e , as given by (15), over the body su r f a c e y i e l d s the f o r c e s on the body, F = / pn dS (16) S b M = j p(rxn) dS (17) S b The h y d r o s t a t i c component of the pressure on the body i s taken to be constant and i s excluded from the above. 17 2 . 5 Summary o f T ime S t e p p i n g P r o c e d u r e To o b t a i n t h e f l o w d e v e l o p m e n t , t h e t i m e s t e p p i n g p r o c e d u r e i s i m p l e m e n t e d a s f o l l o w s . F i r s t , t h e c l o s e d b o u n d a r y shown i n F i g u r e 3 i s d i v i d e d i n t o s e g m e n t s i n a manner t h a t i s s y m m e t r i c a b o u t t h e s e a b e d . On e a c h s i d e o f t h e s e a b e d we h a v e N' s e g m e n t s , w i t h N s e g m e n t s on t h e b o d y b s u r f a c e , N f s e g m e n t s on t h e f r e e s u r f a c e , a n d N c s e g m e n t s on t h e c o n t r o l s u r f a c e s . A t y p i c a l segment g r i d i s shown i n F i g u r e 7 . T h e i n c i d e n t wave f o r m i s s p e c i f i e d a t t i m e s t a n d t - A t by a s s i g n i n g v a l u e s o f 9 $ / 9 n , and n ( a c c o r d i n g t o o u r c h o s e n wave t h e o r y ) a t t h e m i d p o i n t o f e a c h s e g m e n t . The b o d y s u r f a c e b o u n d a r y c o n d i t i o n g i v e n by (3) t h e n y i e l d s 9<}>/9n = 0 on S L . b T h e k i n e m a t i c f r e e s u r f a c e b o u n d a r y c o n d i t i o n g i v e n by (4a) i s u s e d t o o b t a i n v a l u e s o f n a t t + A t a t t h e f r e e s u r f a c e segment m i d p o i n t s . The e x t r a p o l a t e d v a l u e s c a n be o b t a i n e d u s i n g a c e n t r a l d i f f e r e n c e a p p r o x i m a t i o n o r a n o t h e r s u i t a b l e m e t h o d . V a l u e s o f <f> a t t + At a t t h e f r e e s u r f a c e segment m i d p o i n t s a r e o b t a i n e d i n a s i m i l a r manner u s i n g t h e d y n a m i c f r e e s u r f a c e b o u n d a r y c o n d i t i o n g i v e n by ( 5 ) . The G r e e n ' s i d e n t i t y g i v e n by (9) i s t h e n a p p l i e d a t t h e f r e e s u r f a c e a n d body s u r f a c e segment m i d p o i n t s t o y i e l d N e q u a t i o n s (where N = N +N ) i n N u n k n o w n s , t h a t i s , f b i v a l u e s o f 9<J>/9n a t t+At a n d N b v a l u e s o f <t> a t t + A t . The v a l u e s o f $ a n d 9<t>/ 9n a t t+At on S c a r e known f r o m t h e p r e s c r i b e d i n c i d e n t wave f l o w , a s p r e v i o u s l y e x p l a i n e d i n S e c t i o n 2 . 3 . The N e q u a t i o n s a r e s o l v e d on a c o m p u t e r a n d 18 hence we have a l l segment midpoint values f o r <J> and 9<)>/9n at t+At. With values of <j> and 9<j>/3n at times t - A t , t , and t+At we o b t a i n the pressure d i s t r i b u t i o n at time t using a c e n t r a l d i f f e r e n c e approximation of (15). From the pressure d i s t r i b u t i o n , we o b t a i n the f o r c e s a c t i n g on the body at time t using a numerical i n t e g r a t i o n of equations (16) and (17). The values of <$>, 9<j>/an, and n at time t become those at t - A t , while those at t+At become those at t , thus advancing the flow c o n d i t i o n s one time s t e p . The process f o r o b t a i n i n g the r e q u i r e d values at t+At i s repeated and i n t h i s manner we can develop the flow f o r as long as d e s i r e d or c o n s t r a i n e d by accuracy or computer c o s t . The d e t a i l s of the complete procedure and i t s computer s o l u t i o n are presented and d i s c u s s e d i n Chapter I I I . A user guide to the computer program developed f o r t h i s t h e s i s i s given i n Appendix I. 0 19 CHAPTER I I I : NUMERICAL PROCEDURE 3.1 Numerical Time Stepping Procedure We begin with the c l o s e d boundary d i v i d e d i n t o segments as i n F i g u r e 7, with Nfe body s u r f a c e segments, N f f r e e surface.segments, and N c o n t r o l s u r f a c e segments. c R e c a l l t h a t N = N+N , N' = N+N , and that a l l v a l u e s of 4 % D C and 3<j>/3n at segment midpoints are assumed to be constant over the segment le n g t h S. I n i t i a l v alues of and 3<j>/3n at times t-£t and t are a s s i g n e d t o a l l segments as d e f i n e d by a chosen wave theory. There must be zero flow i n the v i c i n i t y of the body and the wave propagation here i s assumed to flow from l e f t to r i g h t . A t y p i c a l i n i t i a l wave p r o f i l e i s shown i n F i g u r e 4. As d e s c r i b e d i n S e c t i o n 2.5, we need to develop e x p r e s s i o n s from the kinematic and dynamic f r e e s u r f a c e boundary c o n d i t i o n s which w i l l g i ve us e x t r a p o l a t e d values of n and <J> at t+At. We can r e w r i t e (4a) and (5) using a c e n t r a l d i f f e r e n c e approximation to o b t a i n W = nt-At + 2 A t and 1 3<J> n 3n (18) •t+At " • t - A t - 2 A t [ g n + i W ] t (19) 20 We note that to r e t a i n a non-zero B e r n o u l l i constant i n (5) would not a f f e c t our f o r c e r e s u l t s , and thus set i t equal to zero f o r convenience. We now need to have the Green's boundary i n t e g r a l i d e n t i t y as expressed by (11) i n a form that can be n u m e r i c a l l y e v a l u a t e d . S e p a r a t i n g the boundary i n t o segments, we get from (14) N / G(x.,0 |J<I)dS - / <Kl)^(x.»I)dS AS 1 9 n AS. d n 1 3 N I L(w) AS. J j=N+l f o r i = 1,2, N (20) We assume the integrands to be constant over each segment, excepting when i = j , and represent the N equations generated by (20) i n the matrix form, where [A] U ) = (X) (21 ) <j> f o r j = 1 ,2 , ... I ( 3<j>/3n)j for j=Nfe + 1 ,Nb + 2, . . .N (22) 21 A i j " < a f o r j=1,2,...N i j b b±i f o r j=Nb+1 ,Nb + 2,.. .N (23) N J b (24) (23) and (24), 3 i j 3n i j AS^cosy* 2Tr(z±+d) (25) (26) i j AS. TT : : G i j (27) b ± i = ( lnCAS±) + lnCz±+d) - 1 } (28) N1 AS. I —^  j-N+1 (w) (29) G and (3G/3n) as given by equations (7) and 10) are n u m e r i c a l l y represented by 22 In (20), the <\>± from the l e f t - h a n d s i d e of (14) shows up i n a., f o r i = 1 , 2, . . . R , and b f o r i=N +1,...N. The terms a i i b i i b ±i and b i : L are obtained by e x p l i c i t l y i n t e g r a t i n g the a p p r o p r i a t e q u a n t i t i e s i n (20) over a segment l e n g t h . The matrix s o l u t i o n represented by (20) through (31), • i s a p p l i e d at t+At. At t + At, the terms ( 3^/3n) (w' and <j> tw) ( f o r j=N+1,...N') i n (29) are obtained from the chosen i n c i d e n t wave theory and the <j> • i n (24) are obtained from i (19). The matrix equation represented by (20) i s sol v e d and y i e l d s ( 3^/Sn) at t+At for i = 1,...Rb and <j> at t + At f o r i=N b+1,...N. With * | A and (9d>/3n) known at a l l p o i n t s on the t+At t+At body, the f o r c e s on the body can be c a l c u l a t e d at time t using (15), (16), and (17), thus F = J p.AS. (n ). (32) x r i 1 x i N h F = F p.AS. (n ).. (33) z . L * i 1 z i i=l \ M = r p.AS. [x. (n ). -z. (n ). ] (34) . , 1 1 1 z 1 1 X I i=l 23 where p = -P 2At (8> i-1 i + l ~ 2 A S > (35) The time i s now advanced one s t e p , At. The t+At terms become t terms, t becomes t - A t , and the p r o c e s s i s r e p e a t e d as many ti m e s as needed f o r the d e s i r e d f l o w development. 24 3.2 Numerical Procedure - Computer S o l u t i o n The time st e p p i n g procedure f o r flow development, given i n S e c t i o n 3.1, i s r e a d i l y programmed f o r a computer s o l u t i o n . The computer program developed f o r t h i s t h e s i s i s w r i t t e n i n FORTRAN IV. The program i s approximately 1000 l i n e s long and i s compiled on the FORTRAN H compiler a t t a c h e d to an Amdahl 470 V/6 Model II computer. The compiled program r e q u i r e s approximately 10 6 bytes of storage space. The flow c h a r t shown i n F i g u r e 8 re p r e s e n t s the b a s i c s t r u c t u r e of the program, which i s d r i v e n by a main r o u t i n e c a l l i n g the r e l e v a n t s u b r o u t i n e s . For a t y p i c a l example i n v o l v i n g a t o t a l of 100 segments, a matrix of dimension N=80, and 100 time st e p s , the s o l u t i o n on the Amdahl 470 takes approximately 50 seconds CPU time, or about 0.5 seconds per time ste p . C o n s i d e r a t i o n s f o r program s t r u c t u r e and input parameters are d i s c u s s e d i n t h i s s e c t i o n . 3.2.1 Segment G r i d When s e t t i n g up the segment g r i d , the f i r s t c o n s i d e r a t i o n i s choosing an a p p r o p r i a t e segment l e n g t h . I t i s c l e a r that the segment l e n g t h to wavelength r a t i o A S / L i s the r e l e v a n t parameter t h a t d e f i n e s the number of segments that w i l l be r e q u i r e d on the f r e e s u r f a c e . I f one c o n s i d e r s the c y c l i c a l motion of the f l u i d p a r t i c l e s over a wave p e r i o d , i t i s r e a d i l y apparent that the time st e p At must be s u f f i c i e n t l y s m a l l to ensure that the motion of the 25 p a r t i c l e s i s small compared to segment l e n g t h , and hence th a t At/T should be l e s s than AS / L f o r good accuracy i n the flow development. Thus, f o r a given At/T, AS / L must be small enough to y i e l d good accuracy while being l a r g e enough to remain g r e a t e r than the given At/T. The r e q u i r e d ranges of At/T and A S / L are i n v e s t i g a t e d and d i s c u s s e d i n S e c t i o n s 4.4 and 4.5. Our p a r t i c u l a r example i s a r e c t a n g u l a r c r o s s -s e c t i o n . For t h i s case, the segment le n g t h s along the c o n t r o l s u r f a c e and s i d e s of the body w i l l vary as the s u r f a c e e l e v a t i o n at those l o c a t i o n s changes. I t i s a s t r a i g h t f o r w a r d procedure t o s c a l e the segment l e n g t h s along the c o n t r o l s u r f a c e and body s i d e s at each time s t e p i n order to account f o r the f r e e s u r f a c e e l e v a t i o n change. The s c a l i n g of the segments r e s u l t s i n a small change i n the l o c a t i o n of the segment midpoints along the body s i d e s and c o n t r o l s u r f a c e . A refinement t o the program c o u l d be i n c o r p o r a t e d that would i n t e r p o l a t e v a l u e s of <(> and 94>/9n between segment midpoints, so that i n t e r p o l a t e d v a l u e s c o u l d be a s s i g n e d t o new segment midpoint l o c a t i o n s . R e c a l l i n g that 8<j>/3n = 0 on S, , that we use the matrix s o l u t i o n f o r b new va l u e s of <j> on S . and that the c o n t r o l s u r f a c e b c o n d i t i o n s are s p e c i f i e d throughout the flow development u s i n g our chosen wave theory, we have no need of the refinement f o r our present s i t u a t i o n . The f r e e s u r f a c e segment lengths must a l s o be a d j u s t e d at each time step, as the slopes of the segments 26 change with changing f r e e s u r f a c e e l e v a t i o n . For the r e c t a n g u l a r body c r o s s - s e c t i o n i n t h i s study, the x c o o r d i n a t e s of the f r e e s u r f a c e remain the same throughout the time s t e p p i n g procedure. However, i f the program were w r i t t e n f o r a procedure that accounted f o r body motions, or i f the geometry of the body i n our procedure were a l t e r e d so th a t the s i d e s were not v e r t i c a l at the w a t e r l i n e , then we would have to account at each time s t e p f o r the change i n h o r i z o n t a l f r e e s u r f a c e l e n g t h t o each s i d e of the body. „The change i n h o r i z o n t a l f r e e s u r f a c e l e n g t h would r e l o c a t e the x c o o r d i n a t e s of the f r e e s u r f a c e segment midpoints, which c o u l d be accounted f o r by u s i n g a refinement to the program as d i s c u s s e d i n the pr e c e d i n g paragraph. As the body dimensions are s i g n i f i c a n t l y s m a l l e r than the wavelength f o r waves of longer p e r i o d , the segment le n g t h parameter A S / L found to be a p p r o p r i a t e on the f r e e s u r f a c e may not be very u s e f u l on the body s u r f a c e i f only 1 or 2 segments are needed to meet the s p e c i f i e d g l o b a l A S / L requirement. R e c a l l i n g that <j> and 94>/9n (and hence p) are constant over each segment, a minimum number of segments are needed on the body s u r f a c e to adequately d e s c r i b e the pre s s u r e d i s t r i b u t i o n over the body s u r f a c e . The moment a c t i n g on the body i s most l i k e l y t o be s e n s i t i v e to t h i s c o n s i d e r a t i o n . " - For purposes of the body segments we d e f i n e a parameter A S B / L B , where A S B i s the body s u r f a c e segment l e n g t h , and L B i s the dimension of the body along which the segment l i e s ( f o r our example, B or D). The r e q u i r e d range 27 for AS./L, i s i n v e s t i g a t e d and d i s c u s s e d i n S e c t i o n 4.5.2. p. b 3.2.2 I n c i d e n t Wave S p e c i f i c a t i o n When s p e c i f y i n g the i n i t i a l c o n d i t i o n s w i t h i n the f l u i d r e g i o n , and hence the i n i t i a l i n c i d e n t wave p r o f i l e , we have a c h o i c e of s e v e r a l wave t h e o r i e s . For purposes of t h i s t h e s i s , we i n v e s t i g a t e the r e s u l t s from harmonic i n c i d e n t wave t r a i n s , such as those p r e d i c t e d by l i n e a r or higher order wave t h e o r i e s , which are m o d i f i e d to s a t i s f y the s t i l l water c o n d i t i o n by a t t e n u a t i n g the flow p r e d i c t e d by the theory with a c o s i n e modulation envelope, as shown i n Fi g u r e 4. We can vary the le n g t h of the modulation envelope so that the number of wavelengths to f u l l wave height i s v a r i e d . The time s t e p p i n g procedure can be run f o r a s u f f i c i e n t l e n g t h of time to allow one or more waves to pass the body. The i n c r e a s e i n f o r c e s from the f i r s t to the second and subsequent waves can thus be observed i f i t occ u r s . The major drawback of t h i s approach i s that the number of time steps r e q u i r e d i n c r e a s e s with the number of wavelengths that pass the body, thus s i g n i f i c a n t l y i n c r e a s i n g the co s t of the computer s o l u t i o n . The inc r e a s e i n c o s t i s compounded by the f a c t that to be able to run more wavelengths past the body, a longer f r e e s u r f a c e and hence more segments are r e q u i r e d t o s a t i s f y the c o n d i t i o n t h a t r e f l e c t e d waves not reach the c o n t r o l s u r f a c e . T h i s i n c r e a s e s the s i z e of the matrix to be so l v e d at each time step, thus adding to the computer c o s t . 28 If the modulation envelope i s t r e a t e d as t r a v e l l i n g with the group v e l o c i t y c , then the number of wavelengths r e q u i r e d f o r a f u l l wave height t o reach the body w i l l be about twice the decay l e n g t h , s i n c e the group v e l o c i t y i s about h a l f the wave speed c. As a p o s s i b l e means of reducing the c o s t of the method, we i n v e s t i g a t e the e f f e c t of s p e c i f y i n g a speed f a s t e r than c as the speed of the modulation envelope. Since there i s no r i g o r o u s l y d e f i n e d group v e l o c i t y f o r Stokes' f i f t h order wave theory, we must s p e c i f y a value f o r the group v e l o c i t y when we use Stokes' f i f t h order wave theory f o r our i n c i d e n t wave. The e f f e c t of s p e c i f y i n g a d i f f e r e n t v alue f o r c other than that p r e d i c t e d by l i n e a r wave theory and of v a r y i n g the decay l e n g t h i s i n v e s t i g a t e d and d i s c u s s e d i n S e c t i o n s 4.2 and 4.3 r e s p e c t i v e l y . 3.2.3 Time Stepping Equation As d i c u s s e d b r i e f y i n S e c t i o n 3.1, the time stepping equation used i n e x t r a p o l a t i n g new values of n and <j> on the f r e e s u r f a c e from the kinematic and dynamic f r e e s u r f a c e boundary c o n d i t i o n s can be a c e n t r a l d i f f e r e n c e method which y i e l d s equations (18) and (19) or other methods of the same or h i g h e r order. For our purposes, the Adams-Bashforth m u l t i s t e p methods d e s c r i b e d i n Burden, F a i r e s , & Reynolds are u s e f u l , and we examine the e f f e c t of u s i n g these higher order time stepping methods on our r e s u l t s . The d e r i v a t i o n of the r e l e v a n t time s t e p p i n g equations i s d e s c r i b e d i n 29 Appendix II and r e s u l t s using higher order time stepping equations are d i s c u s s e d i n S e c t i o n 5.1. o 3.2.4 Matrix S o l u t i o n At each s t e p of the t i m e - s t e p p i n g procedure, the NxN matrix generated by the Green's i n t e g r a l i d e n t i t y must be s o l v e d . V a r i o u s packaged programs e x i s t f o r matrix s o l u t i o n , each with i t s own advantages and disadvantages. In our case, the matrix i s q u i t e f u l l and without any symmetry that can be e x p l o i t e d . The elements of the matrix can vary by s e v e r a l orders of magnitude, p a r t i c u l a r l y a c r o s s the columns. I t i s advantageous to use a matrix s o l u t i o n procedure that s c a l e s the columns to a c l o s e r order of magnitude before s o l v i n g and then s c a l e s the s o l u t i o n a c c o r d i n g l y . The matrix s o l u t i o n i s by f a r the most expensive component of the computer program i n terms of computing time. I t i s t h e r e f o r e necessary "that every e f f o r t be made to reduce the computing e f f o r t i n s o l v i n g the matrix. In our program, a d i r e c t matrix s o l u t i o n which f i r s t s c a l e s and decomposes the matrix i s employed as the p r e f e r r e d s o l u t i o n technique. Rounding e r r o r s i n the technique may r e s u l t i n a s i n g u l a r matrix being undetected, and hence the c o n d i t i o n of the matrix i s checked at each time step. I f the p r e f e r r e d technique i s unable to s o l v e the matrix a c c u r a t e l y , an i t e r a t i v e s o l u t i o n technique i s c a l l e d . For our p a r t i c u l a r method and computer s o l u t i o n , the computer time per step i s 30 p l o t t e d as a f u n c t i o n of N i n F i g u r e 9 , which thus g i v e s a r e l a t i v e i n d i c a t i o n of the i n c r e a s e i n cost with matrix s i z e . 31 CHAPTER IV; INPUT PARAMETER CONSIDERATIONS 4.1 I n t r o d u c t i o n In t h i s Chapter we i n v e s t i g a t e v a r i o u s input parameters used i n the computer program developed from the method d e s c r i b e d i n Chapter I I I . The e f f e c t of v a r y i n g the group v e l o c i t y , i n c i d e n t wave decay l e n g t h , time s t e p s i z e , and segment l e n g t h on both the f r e e s u r f a c e and the body s u r f a c e i s i n v e s t i g a t e d . In each case we determine an optimal value f o r the parameter that w i l l y i e l d a c c u r a t e r e s u l t s at minumum c o s t . Unless otherwise s t a t e d i n a given S e c t i o n of t h i s and the next Chapter, t e s t runs r e f e r r e d to are f o r an i n c i d e n t wave p r e s c r i b e d by l i n e a r wave theory, of height H = 1.0 m , angular frequency u> = 1.5 rad/sec , and decay l e n g t h L, = 1.0 wavelength , with one f u l l y developed wave d i n t e r a c t i n g with the body. The body i s r e c t a n g u l a r , with beam B = 10.0 m and draught D = 2.5 m , i n water of depth d = 20.0 m . The time s t e p r a t i o At/T i s 0.04 , the segment l e n g t h r a t i o AS/L i s 0.1 , and the body segment l e n g t h r a t i o AS /L i s 0.2 . The time s t e p p i n g equation used i s the b c e n t r a l d i f f e r e n c e method given i n (18) and (19). The user guide f o r the program, which d e s c r i b e s a l l the necessary input parameters and o p t i o n s , i s given i n Appendix I. 32 4.2 Group V e l o c i t y The i n c i d e n t wave t r a i n i s s p e c i f i e d a c c o r d i n g to a chosen wave theory. Throughout our study, we use e i t h e r l i n e a r wave theory or Stokes' f i f t h order wave theory to a s s i g n i n i t i a l v alues of <j>,- 8<j>/8n, and n at the segment midpoints. We attenuate the i n c i d e n t wave as we go from the l e f t c o n t r o l s u r f a c e towards the body. The i n i t i a l c o n d i t i o n s are s p e c i f i e d at two or more i n i t i a l time steps, depending on the order of the time s t e p p i n g procedure (see Appendix I I ) . When using l i n e a r wave theory, we can apply the group v e l o c i t y c as p r e d i c t e d by the theory to the i n i t i a l motion of the modulation envelope. However, when using Stokes' f i f t h order wave theory, an e x p l i c i t e x p r e s s i o n f o r the group v e l o c i t y i s not a v a i l a b l e . Hence, fo r Stokes' f i f t h order theory, we must assume a value f o r c that seems reasonable. g When us i n g l i n e a r wave theory, one might expect that s p e c i f y i n g the wave v e l o c i t y c as the group v e l o c i t y f o r the modulation envelope would cause a f u l l wave height to reach the body sooner than i f the normal group v e l o c i t y c - c/2 were used. To i n v e s t i g a t e t h i s p o s s i b i l i t y , the program developed f o r t h i s t h e s i s was run twice with the input v a l u e s as d e f i n e d at the beginning of t h i s Chapter, changing o n l y the i n i t i a l group v e l o c i t y of the modulation envelope between runs. In the f i r s t case, the group v e l o c i t y given by 33 c C S = 2 1 + k d s i n h ( k d ) was used as the i n i t i a l group v e l o c i t y of the modulation envelope. The f o r c e s on the body a t each time s t e p as w e l l as the wave runup on each s i d e of the body were recorded. The second t e s t case used the wave v e l o c i t y c as the i n i t i a l group v e l o c i t y of the modulation envelope, and corresponding r e s u l t s were recorded. The. two s e t s of r e s u l t s were compared and found to be almost i d e n t i c a l . The c o n c l u s i o n we draw i s that the method i t s e l f with i t s p h y s i c a l c o n s t r a i n t s expressed by the boundary c o n d i t i o n s and the Green's i d e n t i t y very s t r o n g l y d e f i n e s the flow development, and o v e r r i d e s the attempt (by s e t t i n g the i n i t i a l group v e l o c i t y equal to the wave speed) to f o r c e a f a s t e r i n c r e a s e to f u l l flow development i n the v i c i n i t y of the body. The c o n c l u s i o n s t a t e d above was f u r t h e r v e r i f i e d by using a f i v e s t e p time s t e p p i n g procedure ( d e s c r i b e d i n Appendix I I ) , and thus d e f i n i n g v a l u e s of <j> , 9<f>/an, and n at the f i r s t f i v e time steps based on a group v e l o c i t y c = c. A l l other input v a l u e s were kept the same. The r e s u l t s f o r the f i r s t ten or so time steps d i f f e r e d s i g n i f i c a n t l y from p r e v i o u s r e s u l t s , but subsequent flow development y i e l d e d f o r c e s and wave runup v a l u e s w i t h i n 1% of the previous r e s u l t s at any given time st e p . The r e s u l t s of t h i s S e c t i o n show that the flow development i s r e l a t i v e l y independent of the modulation envelope i n i t i a l group v e l o c i t y . For purposes of our 34 program, we set the group v e l o c i t y equal to the value p r e d i c t e d by l i n e a r wave theory when using Stokes' f i f t h order wave theory. 35 4.3 I n c i d e n t Wave Decay Length When using l i n e a r or Stokes' f i f t h order wave theory to d e f i n e our i n c i d e n t wave p r o f i l e s , the decay l e n g t h L of d the i n i t i a l wave p r o f i l e (as shown i n F i g u r e 4) w i l l determine the time to f u l l flow development. By reducing the decay l e n g t h we reduce the number of segments needed to d e f i n e the f r e e s u r f a c e , and a l s o shorten the time to f u l l flow development, thus reducing the c o s t of running the program. Too short a decay l e n g t h c o u l d e x c e s s i v e l y a f f e c t the accuracy of our r e s u l t s , while too long a decay l e n g t h would u n n e c e s s a r i l y make the c o s t of f u l l flow development too expensive. Using the input values given i n S e c t i o n 4.1, we now vary the decay l e n g t h L d from L d = 0.25 up to L d = 1.75 . The maximum f o r c e s generated by the f i r s t f u l l y developed wave c r e s t to i n t e r a c t with the body are p l o t t e d as a f u n c t i o n of decay l e n g t h i n F i g u r e 10 . In F i g u r e 10, the f o r c e s a re expressed as a percent v a r i a t i o n from the value to which a given f o r c e converges. The r e s u l t s of F i g u r e 10 i n d i c a t e that a decay l e n g t h of about 0.75 and l e s s y i e l d s somewhat u n r e l i a b l e r e s u l t s , while a decay l e n g t h of 1.25 and g r e a t e r does not a f f e c t r e s u l t s s i g n i f i c a n t l y . We w i l l use L d = 1.0 f o r our f u r t h e r i n v e s t i g a t i o n s r e g a r d i n g numerical parameters, and w i l l use L d = 1.-25 to o b t a i n a c c u r a t e r e s u l t s f o r p l o t t i n g . I t i s noted that f o r L d = 1.0 , i t w i l l take approximately two wave p e r i o d s f o r a f u l l wave height to 36 reach the body. T h i s i s due to the a c t u a l group v e l o c i t y of the wave t r a i n being approximately c/2, as was seen i n S e c t i o n 4.2. As a r e s u l t , f o r L. = 1.0 , the second wave d c r e s t to reach the body w i l l be the f i r s t f u l l y developed wave c r e s t i n t e r a c t i n g with the body. 37 4.4 Time Step S i z e The accuracy of our numerical s o l u t i o n i s dependent on, among other f a c t o r s , the s i z e of the time step, which we express as a f r a c t i o n of the wave p e r i o d , by At/T . - I t i s reasonable to expect t h a t as the time step r a t i o At/T i s reduced ( h o l d i n g a l l other input parameters e q u a l ) , the f o r c e r e s u l t s should converge t o r e l a t i v e l y uniform v a l u e s . We t e s t the e f f e c t of v a r y i n g At/T by us i n g the input parameters given i n S e c t i o n 4.1, and run the program changing only the value of At/T . R e c a l l i n g from S e c t i o n 3.2.1 that we expect that At/T should be smal l e r than AS/L for a c c u r a t e flow development, the l a r g e s t value of At/T that we t e s t i s At/T = AS/L = 0.1 . In a d d i t i o n , we t e s t s u c c e s s i v e l y s m a l l e r values of At/T r e c o r d i n g the f o r c e and runup r e s u l t s at each time s t e p . As f o r F i g u r e 10, the maximum f o r c e s generated by the f i r s t f u l l y developed wave to i n t e r a c t with the body are p l o t t e d i n F i g u r e 11 as a f u n c t i o n of At/T. The r e s u l t s of F i g u r e 11 co n f i r m our e a r l i e r d i s c u s s i o n . For At/T = A S / L =0.1 , the r e s u l t s d i v e r g e d to the p o i n t where the program would not continue the flow development. T h i s was expected f o r At/T being i n the same order of magnitude as A S / L, s i n c e the p a r t i c l e motions f o r the flow development would be of the same order of magnitude as the segment lengths themselves. As we decrease A t / T , the r e s u l t s q u i c k l y converge to uniform v a l u e s . L i t t l e accuracy i s gained by using At/T < 0.05 , and f o r At/T = 0.04 , the 38 r e s u l t s o b t a i n e d are w i t h i n 3% of the most a c c u r a t e r e s u l t s . We conclude that i f we use At/T < 0.04 f o r our subsequent i n v e s t i g a t i o n s , we w i l l have s u f f i c i e n t accuracy. 39 4.5 Segment Length The importance of reducing the number of segments used i s r e a d i l y apparent when c o n s i d e r i n g the c o s t of running the program. As d i s c u s s e d i n S e c t i o n 3.2.1, the segment l e n g t h parameter AS/L i s g l o b a l l y c o n s t r a i n e d by an upper and lower l i m i t . In a d d i t i o n , the body segments must be small enough to have a s u f f i c i e n t number to a c c u r a t e l y approximate the pressure d i s t r i b u t i o n on the body s u r f a c e at each time s t e p . Both these c o n s i d e r a t i o n s are i n v e s t i g a t e d i n t h i s S e c t i o n . 4.5.1 G l o b a l Segment Length The segment l e n g t h parameter AS/L i s c o n s t r a i n e d i n two ways. The f i r s t c o n s t r a i n t i s s i m i l a r t o that on At/T, that i s , AS/L should be small enough to a c c u r a t e l y approximate the f r e e s u r f a c e p r o f i l e , and hence the flow development. The second c o n s t r a i n t i s the r e l a t i o n s h i p of A S / L to At/T ( d i s c u s s e d i n S e c t i o n 3.2.1) that r e q u i r e s that At/T << As/L. Thus, there i s an upper and a lower l i m i t expected on AS/L. We t e s t the e f f e c t of v a r y i n g A S / L by again using the input v a l u e s given i n S e c t i o n 4.1, with At/T = 0.03, and l e t t i n g AS/L take on valu e s ranging from 0.05 to 0.25 . The maximum value s f o r the f o r c e s generated by the f i r s t f u l l y developed wave to i n t e r a c t with the body, the f o r c e s 40 expressed i n dime n s i o n l e s s form, are p l o t t e d as a f u n c t i o n of A S / L i n F i g u r e 12. Our e a r l i e r d i s c u s s i o n concerning upper and lower l i m i t s f o r A S / L i s confirmed as the r e s u l t s remain r e l a t i v e l y c o n s i s t e n t when A S / L i s i n the v i c i n i t y of 0.1, and d i v e r g e as one i n c r e a s e s or decreases A S / L beyond a l i m i t e d r e g i o n . As b e f o r e , when A S / L i s set to a value that i s l e s s than approximately twice the value of At/T, the r e s u l t s d i v e r g e t o the p o i n t where the computer s o l u t i o n i s not a b l e t o c o n t i n u e . In t h i s s et of t e s t s , t h i s was the case f o r A S / L = 0.05 . We conclude from F i g u r e 12 that a value of 0.1 f o r A S / L w i l l y i e l d good r e s u l t s , and that A S / L should a l s o be at l e a s t 2 to 3 times g r e a t e r than A t/T. 4.5.2 Body Segments As d i s c u s s e d i n S e c t i o n 3.2.1, a g l o b a l l i m i t on A S / L may not p r o v i d e enough body segments to a c c u r a t e l y c a l c u l a t e the f o r c e r e s u l t s , p a r t i c u l a r l y the moment M. Using the input v a l u e s given i n S e c t i o n 4.1, with At/T = 0.04 and A S / L =0.1, we t e s t d i f f e r e n t v a l u e s of A S ^ / L , ranging from b b 0.1 to 0.25 . We should a l s o check that the segment l e n g t h s on the body s u r f a c e d e f i n e d by A S B / L B f a l l w i t h i n the g l o b a l c o n s t r a i n t s on A S / L determined i n S e c t i o n 4.5.1 . Along the h o r i z o n t a l segments d e f i n i n g the base of the body s e c t i o n , 41 we are concerned with h o r i z o n t a l p a r t i c l e motions, and thus the g l o b a l l i m i t should apply to the h o r i z o n t a l segments. For the v e r t i c a l body segments, we are concerned with v e r t i c a l p a r t i c l e motions, which can be c o n s i d e r e d t o be sma l l e r than the h o r i z o n t a l p a r t i c l e motions f o r a given time s t e p by a f a c t o r of approximately 2H/L ( a l l o w i n g f o r a wave runup at the v e r t i c a l body s u r f a c e of as much as twice the wave h e i g h t ) . The program whose user guide i s given i n Appendix I conducts t h i s check and i f necessary, r e s e t s the value of A S ^ I ^ to f a l l w i t h i n the g l o b a l c o n s t r a i n t s . Force r e s u l t s are p l o t t e d as f o r F i g u r e s 10 and 11, t h i s time as a f u n c t i o n of hS^/L^, i n F i g u r e 13. We observe from F i g u r e 13 that the moment i s the r e s u l t most a f f e c t e d by v a r y i n g A S b / L b . The sharp v a r i a t i o n i n r e s u l t s as one moves towards lower v a l u e s of A S b / L b can be accounted f o r by the f a c t that f o r A S b / L b < 0.2 , A S b / L < 2At/T, which v i o l a t e s our p r e v i o u s l y v e r i f i e d c o n d i t i o n that AS/L should be at l e a s t twice the value of At/T. We conclude that we should use as many segments as our g l o b a l c o n s t r a i n t on AS/L a l l o w s , and set AS f e/L b = 0.2 as a maximum value, g i v i n g a minimum of f i v e segments along each body dimension. 42 CHAPTER V: OTHER CONSIDERATIONS 5.1 Time Stepping Equation As d i s c u s s e d i n S e c t i o n 2.5, we use the kinematic and dynamic f r e e s u r f a c e boundary c o n d i t i o n s to time step the values of n and <f> on the f r e e s u r f a c e . The time stepping technique used to o b t a i n (18) and (19) i s a c e n t r a l d i f f e r e n c e method. For our purposes, we f i n d t h a t the Adams-Bashforth m u l t i s t e p methods d e s c r i b e d i n Appendix II are a l s o u s e f u l . We now i n v e s t i g a t e the e f f e c t on our r e s u l t s of us i n g each of the methods l i s t e d . Using the input v a l u e s given i n S e c t i o n 4.1 with At/T ~ 0.04 , and AS/L = 0.2, we run our program using each of the time st e p p i n g methods g i v e n , that i s , the c e n t r a l d i f f e r e n c e method, and the Adams-Bashforth two, thr e e , f o u r , and f i v e s t e p methods. We p l o t the r e s u l t s f o r the f o r c e s as a f u n c t i o n of time f o r each method. The r e s u l t i n g p l o t s are shown i n F i g u r e 14 . From our r e s u l t s of F i g u r e 14, we see that the c e n t r a l d i f f e r e n c e method produces a s l i g h t l y uneven p l o t , p a r t i c u l a r l y f o r the f i r s t s e v e r a l time s t e p s . The two, th r e e , and four step methods g i v e smoother r e s u l t s . The f i v e s t e p method produces h i g h l y e r r a t i c r e s u l t s which sawtooth about values c o i n c i d e n t with the other r e s u l t s . The sawtooth r e s u l t s of the f i v e s t e p method are probably 43 caused by small p e r t u r b a t i o n s from smooth r e s u l t s being magnified by the f i t t i n g of a f i f t h order polynomial to pr e v i o u s p o i n t s when p r o j e c t i n g forward at each time s t e p f o r v a l u e s at t+A.t. A n t i c i p a t i n g t h a t the same e f f e c t c o u l d occur f o r any of the higher order methods, we w i l l use the three s t e p method as the p r e f e r r e d method f o r subsequent work, and d e f a u l t to the two step method i f the three step method leads to a d i v e r g e n t i n s t a b i l i t y s i m i l a r to the one observed here f o r the f i v e s t e p method. 44 5.2 I n c i d e n t Wave Type The wave theory chosen to d e s c r i b e the i n c i d e n t wave t r a i n depends on the wave steepness and H/d, as shown i n F i g u r e 15. The two t h e o r i e s employed as o p t i o n s i n our program are l i n e a r wave theory and Stokes' f i f t h order wave theo r y . We seek t o v e r i f y the range of v a l i d i t y f o r these t h e o r i e s by f i r s t running the program f o r a given wave c o n d i t i o n f a l l i n g w i t h i n the normal range of l i n e a r wave, theory, and comparing f o r c e r e s u l t s obtained u s i n g both l i n e a r and Stokes' f i f t h order t h e o r i e s f o r the i n c i d e n t waves. We then do the same f o r a wave w i t h i n the normal range of Stokes' f i f t h order wave theory. A wave with co = 0.5 rad/sec, H = 1.0 m , i n water of depth d = 20.0 m , f a l l s w i t h i n the normal range of l i n e a r wave theory. A wave with co = 2.0 rad/sec, H = 1.0 m , i n water of depth d = 5.0 m , f a l l s w i t h i n the normal range of Stokes' f i f t h order wave theory. We t e s t both waves using l i n e a r and Stokes' f i f t h order wave theory. The v e r t i c a l f o r c e r e s u l t s u s i n g each theory are compared f o r both waves. These r e s u l t s are shown i n F i g u r e 16. The wave i n • the l i n e a r theory range produces v i r t u a l l y no d i f f e r e n c e between r e s u l t s u s i n g l i n e a r wave theory and r e s u l t s using Stokes' f i f t h order wave theory, while the wave i n the Stokes' f i f t h order range produces a n o t i c a b l e d i f f e r e n c e i n the magnitude and phase of the r e s u l t s . We conclude that i n order t o o b t a i n a c c u r a t e r e s u l t s , i t i s important to use the a p p r o p r i a t e wave theory f o r a given i n c i d e n t wave. 46 CHAPTER VI; RESULTS 6.1 E x c i t i n g Forces The primary purpose of t h i s t h e s i s was to ex p l o r e the v i a b i l i t y of the method d e s c r i b e d to p r e d i c t the e x c i t i n g wave f o r c e s on a f i x e d s t r u c t u r e . Having i n v e s t i g a t e d the range of the v a r i o u s numerical parameters and other c o n s i d e r a t i o n s needed to o b t a i n a c c u r a t e r e s u l t s w i t h i n the l i m i t a t i o n s of the method i t s e l f , we now proceed t o generate f o r c e r e s u l t s over a range of co. For a l l our r e s u l t s i n t h i s Chapter, we run the program d e s c r i b e d i n Appendix I with L = 1.25, At/T = 0.04 , AS/L = 0.1 , and d AS /L = 0.2 . We use the Adams-Bashforth t h r e e - s t e p method b b as our p r e f e r r e d time s t e p p i n g technique, and use e i t h e r l i n e a r or Stokes' f i f t h order wave theory, depending on the wave steepness, to d e s c r i b e our i n c i d e n t wave. In a d d i t i o n , care was taken to ensure that the magnitude of the the wave runup d i d not exceed the draught of the body, as t h i s c o n d i t i o n would render the flow development u n s o l v a b l e . We begin by using the program i n a form m o d i f i e d to g i v e the flow development without a body. A t y p i c a l flow development f o r an i n c i d e n t wave p r e s c r i b e d by l i n e a r wave theory i s shown i n F i g u r e 17. The i n i t i a l wave form i s shown at the top of F i g u r e 17, with the f r e e s u r f a c e p r o f i l e a f t e r 10, 20, 30, and 40 time steps shown below i t . 47 Next, we compare r e s u l t s of f o r c e development as a f u n c t i o n of time with l i n e a r r e s u l t s o b tained from a f i n i t e element method used by F r a s e r . For an i n c i d e n t wave of h e i g h t H = 1.0 m and <»> = 1.5 rad/sec on a body with B = 10.0 m and D = 2.5 m i n water of depth d = 20.0 m , we produce the r e s u l t s shown i n F i g u r e s 18 and 19. We see that the r e s u l t s o b t a i n e d from our program, once f u l l y developed, compare f a v o r a b l y with those obtained by F r a s e r both i n magnitude and i n phase. The notable d i f f e r e n c e between the r e s u l t s i s that the r e s u l t s from our method are c l e a r l y non-l i n e a r , p a r t i c u l a r l y f o r F z and M. We now use our program to generate f o r c e r e s u l t s over a range of f r e q u e n c i e s . For each frequency t e s t e d , the maximum f o r c e s from the second f u l l y developed wave to i n t e r a c t with the body are recorded. We compare our f o r c e r e s u l t s with F r a s e r ' s l i n e a r r e s u l t s and Vugts' experimental r e s u l t s . F r a s e r obtained h i s f o r c e r e s u l t s u s ing a l i n e a r f i n i t e element method, and hence h i s r e s u l t s are not depth dependent. Vugts' r e s u l t s were obtained f o r deep water c o n d i t i o n s . Since our' method i s n o n - l i n e a r , we o b t a i n d i f f e r e n t f o r c e curves f o r d i f f e r e n t v a l u e s of H/d. For a beam to draught r a t i o B/D = 2.0, f o r c e r e s u l t s generated by our program f o r H/d = 0.05, 0.1, and 0.2 are p l o t t e d i n F i g u r e 20, with both F r a s e r ' s and Vugts' r e s u l t s i n c l u d e d f o r comparison. S i m i l a r l y , f o r c e r e s u l t s f o r B/D = 4.0 and 8.0 are shown i n F i g u r e s 21 and 22. The r e s u l t s of F i g u r e s 20, 21, and 22 show that the 48 method used here has a f a i r l y f u l l range of a p p l i c a b i l i t y , and t h a t the magnitude of most of the f o r c e r e s u l t s compares w e l l to p r e v i o u s experimental and t h e o r e t i c a l r e s u l t s . While the r e s u l t s f o r the h o r i z o n t a l f o r c e c o i n c i d e c l o s e l y with F r a s e r ' s l i n e a r r e s u l t s , our n o n - l i n e a r method produces curves f o r the v e r t i c a l f o r c e and the moment th a t d i f f e r s i g n i f i c a n t l y i n slope from the l i n e a r p r e d i c t i o n s . 49 6.2 Transmission C o e f f i c i e n t An i n t e r e s t i n g byproduct of the method i s the c a p a b i l i t y to monitor the t r a n s m i s s i o n c o e f f i c i e n t f o r the f i x e d case. Since the f i x e d case i s used here o n l y as an i d e a l i z a t i o n t o c a l c u l a t e e x c i t i n g f o r c e s that can be used f o r a subsequent a n a l y s i s of the a c t u a l response of a moored breakwater, the observed t r a n s m i s s i o n c o e f f i c i e n t s are of l i t t l e p r a c t i c a l v a l u e , but r a t h e r , p r o v i d e a lower l i m i t f o r p r e d i c t e d t r a n s m i s s i o n c o e f f i c i e n t s i n an a c t u a l moored example. In our case of a f i x e d r e c t a n g u l a r breakwater, we can a l s o o b t a i n from the t r a n s m i s s i o n c o e f f i c i e n t r e s u l t s an i n d i c a t i o n of p r e f e r r e d v a l u e s of B/d f o r v a r i o u s to. The t r a n s m i s s i o n c o e f f i c i e n t , K t, obtained from our r e s u l t s i s p l o t t e d as a f u n c t i o n of <*> f o r B/D = 2.0 , 4.0 , and 8.0 i n F i g u r e 23, with K t=H t/H, where H t i s the t r a n s m i t t e d wave h e i g h t . A l s o p l o t t e d on F i g u r e 23 are r e s u l t s from Nece and Richey f o r B/D =5.0 . The experimental r e s u l t s shown are f o r an i n c i d e n t wave of the same steepness f o r a given to as f o r our r e s u l t s . The r e s u l t s o b tained f o r the t r a n s m i s s i o n c o e f f i c i e n t were found to be r e l a t i v e l y independent of depth over the range t e s t e d . As expected, our r e s u l t s f o r a f i x e d body g i v e lower v a l u e s f o r K t than Nece and Richey's experimental r e s u l t s f o r a responding'cable moored body, and thus appear t o pr o v i d e a r e l i a b l e lower l i m i t f o r K t. 50 CHAPTER V I I ; CONCLUSIONS AND FURTHER STUDIES 7.1 C o n c l u s i o n s We have examined the Green's i d e n t i t y flow development method developed by Isaacson, a p p l y i n g the method t o a two-dimensional f i x e d r e c t a n g u l a r c r o s s - s e c t i o n . The most s i g n i f i c a n t l i m i t a t i o n of the method i s the very l a r g e computing e f f o r t r e q u i r e d t o develop the flow f o r a few wave p e r i o d s . We have determined the d e s i r a b l e range of v a r i o u s parameters t o ensure accuracy of r e s u l t s . The e f f e c t of s p e c i f y i n g the group v e l o c i t y as a v a l u e other than that p r e d i c t e d by l i n e a r wave theory was i n v e s t i g a t e d . The only stage i n the method where the group v e l o c i t y i s used i s to s p e c i f y the i n i t i a l group v e l o c i t y of the modulation envelope. I t was d i s c o v e r e d that the method i t s e l f as d e f i n e d by the p h y s i c a l c o n d i t i o n s and boundary i n t e g r a l s t r o n g l y d e f i n e d the group v e l o c i t y of the flow development, and that s p e c i f y i n g a d i f f e r e n t v alue f o r the group v e l o c i t y a f f e c t e d the r e s u l t s of only the f i r s t few time s t e p s . For the decay l e n g t h , a value of L d = 1.0 was found to be adequate, with L = 1.25 p r e f e r r e d . Longer decay l e n g t h s r e s u l t e d i n e x c e s s i v e computing e f f o r t due to the i n c r e a s e d number 'of segments and time steps necessary, while s h o r t e r decay lengths produced l e s s a c c u r a t e r e s u l t s . 51 The time step parameter At/T of 0.04 or l e s s was found t o g i v e good r e s u l t s . The g l o b a l segment l e n g t h parameter A S / L was c o n s t r a i n e d by both an upper and a lower bound. Higher v a l u e s of A S / L d i d not p r o v i d e enough segments t o a c c u r a t e l y approximate the f r e e s u r f a c e wave p r o f i l e , while lower v a l u e s of A S / L c o n f l i c t e d with the requirement that A S / L « At/T. For accurate r e s u l t s , A S / L should be at l e a s t two or three times g r e a t e r than At/T, and should not be g r e a t e r than 0.1 . The g l o b a l v a l u e f o r A S / L was found not to be s u f f i c i e n t t o d e s c r i b e the p r e s s u r e d i s t r i b u t i o n on the body s u r f a c e . For the body s u r f a c e , the parameter A S B / L F E = 0.2 was found to g i v e enough segments on the body s u r f a c e to allow f a i r l y a c c u r a t e c a l c u l a t i o n of the f o r c e s on the f i x e d body. D i f f e r e n t time s t e p p i n g techniques were i n v e s t i g a t e d , i n c l u d i n g a c e n t r a l d i f f e r e n c e method and the Adams-Ba s h f o r t h m u l t i s t e p methods given i n Appendix I I . The higher order Adams-Bashforth methods seemed to f r e q u e n t l y generate i n s t a b i l i t i e s a p p a r e n t l y due to the f i t t i n g of a higher order polynomial to small p e r t u r b a t i o n s i n pre v i o u s time s t e p s . Methods employing only two or th r e e p r e v i o u s v a l u e s were found to be s a t i s f a c t o r y . The importance of using a wave theory a p p r o p r i a t e to the regime of a given wave was v e r i f i e d . In p a r t i c u l a r , l i n e a r wave theory and Stokes' f i f t h order wave theory were used, and the v a l i d i t y of each i n t h e i r r e s p e c t i v e range of 52 a p p l i c a b i l i t y was confirmed. A t y p i c a l f o r c e - t i m e p l o t generated by our method compared f a v o r a b l y both i n magnitude and i n phase to l i n e a r r e s u l t s o b t a i n e d by F r a s e r . The flow development i n the v i c i n i t y of the body as c a l c u l a t e d by the present method was c l e a r l y n o n - l i n e a r , even when the i n i t i a l i n c i d e n t wave flow parameters and the c o n t r o l s u r f a c e parameters throughout the flow development were p r e s c r i b e d by l i n e a r wave the o r y . The maximum dime n s i o n l e s s f o r c e s generated by i n c i d e n t waves of v a r i o u s f r e q u e n c i e s as determined by our method compared w e l l i n magnitude t o pre v i o u s numerical and experimental r e s u l t s . We o b t a i n e d d i f f e r e n t f o r c e curves f o r d i f f e r e n t v a l u e s of H/d, and the slopes of many of our fo r c e curves d i f f e r e d s i g n i f i c a n t l y from p r e v i o u s t h e o r e t i c a l l i n e a r and deep water experimental r e s u l t s , p a r t i c u l a r l y f o r the v e r t i c a l f o r c e and the moment. We conclude that the n o n - l i n e a r method used here has a s i g n i f i c a n t p o t e n t i a l f o r improved accuracy i n the p r e d i c t i o n of f o r c e s on f l o a t i n g b o d i e s . The method as presented here allows only f o r a r e l a t i v e comparison of the t r a n s m i s s i o n c o e f f i c i e n t Kfc f o r d i f f e r e n t v a l u e s of beam to draught r a t i o and frequency. The t r a n s m i s s i o n c o e f f i c i e n t f o r our f i x e d breakwater appeared to p r o v i d e a v a l i d lower l i m i t f o r K t. In order order t o use the method to a c c u r a t e l y p r e d i c t t r a n s m i s s i o n c o e f f i c i e n t s , the motions of the body must be determined, e i t h e r by using the e x c i t i n g f o r c e s obtained here together 53 with the hydrodynamic p r o p e r t i e s of the body to c a l c u l a t e the motions, as done by F r a s e r , or by r e f i n i n g the method to take motions i n t o account at each time ste p . .54 7.2 Recommendations f o r Fu r t h e r Study In t h i s work we have i n v e s t i g a t e d the a p p l i c a t i o n of the Green's i d e n t i t y flow development method to a f i x e d r e c t a n g u l a r c r o s s - s e c t i o n . The method d e s c r i b e d here c o u l d be extended i n s e v e r a l ways. Adding d i f f e r e n t wave t h e o r i e s such as s o l i t a r y wave theory or stream f u n c t i o n wave theory to the o p t i o n s f o r the i n c i d e n t wave t r a i n would extend the range of a p p l i c a b i l i t y of the method. The method c o u l d a l s o be extended to a r b i t r a r y c r o s s - s e c t i o n s . An e x t e n s i o n of the method, i n c o r p o r a t i n g body motions i n t o the time st e p p i n g procedure, c o u l d be developed and a p p l i e d to o b t a i n body motions and s t r e s s e s d i r e c t l y as w e l l as accurate p r e d i c t i o n s of the t r a n s m i s s i o n c o e f f i c i e n t . By i n c o r p o r a t i n g a s l i t from the body s u r f a c e to a submerged body, a c l o s e d s u r f a c e c o u l d be d e f i n e d that would allow the method t o be a p p l i e d to submerged bodies. The present method has a l l r e a d y been a p p l i e d t o f l o a t i n g t h r e e - d i m e n s i o n a l bodies by I s a a c s o n ( 2 2 ' , though a c o r r e s p o n d i n g l y g r e a t e r computer e f f o r t i s r e q u i r e d to maintain accuracy. With e x t e n s i o n s and improvments to the method, a complete numerical a n a l y s i s of v a r i o u s f l o a t i n g and submerged s t r u c t u r e s c o u l d be developed. 55 BIBLIOGRAPHY 1 Isaacson, Michael de S t . Q. "Steep Wave E f f e c t s on Large O f f s h o r e S t r u c t u r e s " , Proc. T h i r t e e n t h Annual O f f s h o r e Technology Conference, V o l . I, 1981, pp. 21-29. 2 J o l y , J . "On F l o a t i n g Breakwaters", Royal D u b l i n S o c i e t y S c i e n t i f i c Proceedings, V o l . 10, 1905, pp.378-383. 3 Adee, Bruce H. " O p e r a t i o n a l Experience with F l o a t i n g Breakwaters", Marine Technology, V o l . 14, No. 4, Oct. 1977, pp. 379-386 4 M i l l e r , D. S. " F l o a t i n g Breakwaters", C i v i l E n g i n e e r i n g , V o l . 44, No. 3, March 1974, pp.77-79. 5 Brebner, A. and Ofuya, A. 0. " F l o a t i n g Breakwaters", Proc. E l e v e n t h Conference on C o a s t a l E n g i n e e r i n g , ASCE, V o l . I I , 1968, pp.1054-1094. 6 Hom-ma, M., Horikawa, K. and Mochizuki, H. "An Experimental Study on F l o a t i n g Breakwaters", C o a s t a l E n g i n e e r i n g i n Japan, V o l . 7, 1964, pp. 85-94. 7 Skey, S. " C y l i n d r i c a l F l o a t i n g Breakwater", Dock and Harbour A u t h o r i t y , V o l . 56, No. 658, August 1975, pp. 138-139. 8 Sutko, A. A. " F l o a t i n g Breakwaters - A Wave Tank Study", J o u r n a l of Petroleum Technology, V o l . 27, March 1975, pp. 269-273. 9 H a r r i s , A. J . and Webber, N. B. "A F l o a t i n g Breakwater", Proc. E l e v e n t h Conference on C o a s t a l E n g i n e e r i n g , ASCE, V o l . I I , 1968, pp. 1049-1054. 10 Mei, Chiang C. and Black, J a r e d L. " S c a t t e r i n g of Surface Waves by Rectangular O b s t a c l e s i n Waters of F i n i t e Depth", J o u r n a l of F l u i d Mechanics, V o l . 38, 1969, pp.499-511. 11 Longuet-Higgins, M. "The Mean Forces Exerted by Waves on F l o a t i n g or 5.6 Submerged Bodies with A p p l i c a t i o n s to Sand Bars and Wave Power Machines", Proc. Royal Soc. London A, V o l . 352, 1977, pp. 463-480. 12 N i c h o l s , B. D. and H i r t , C. W. "Numerical C a l c u l a t i o n of Wave Forces on S t r u c t u r e s " , Proc. F i f t e e n t h C o a s t a l E n g i n e e r i n g Conference, V o l . I l l , 1975, pp. 2254-2270. 13 I j i m a , T., Chou, C R . and Yoshida, A. "Method of A n a l y s i s f o r Two-Dimensional Water Wave Problems", Proc. F i f t e e n t h C o a s t a l E n g i n e e r i n g Conference, V o l . I l l , 1975, pp. 2717-2736. 14 G a r r i s o n , C. J . "Dynamic Response of F l o a t i n g Bodies", Proc. S i x t h O f f s h o r e Technology Conference, 1974. 15 F r a s e r , Glen A. "Dynamic Response of Moored F l o a t i n g Breakwaters", M. Ap. Sc. T h e s i s , U n i v e r s i t y of B r i t i s h Columbia, 1979. 16 I j i m a , T. and Chou, C R . "Analyses of Two-Dimensional Wave Problems by Means of Green's I d e n t i t y Formula", Proc. JSCE, No. 252, Aug. 1976, pp. 57-71. 17 I j i m a , T., Yoshida, A. and Yamamoto, T. "Two-Dimensional Motions of Moored and F l o a t i n g Bodies with A r b i t r a r y C r o s s - S e c t i o n " , Memoirs of the F a c u l t y of E n g i n e e r i n g , Kyushu U n i v e r s i t y , V o l . 37, No. 3, Fukuoka, Japan, 1977, pp. 107-132. 18 Newman, J . N. "The E x c i t i n g Forces on F i x e d Bodies i n Waves", J o u r n a l of Ship Research, V o l . 6, No. 3, Dec. 1962, pp. 10-17. 19 Vugts, J . H. "The Hydrodynamic C o e f f i c i e n t s f o r Swaying, Heaving, and R o l l i n g C y l i n d e r s i n a Free S u r f a c e " , I n t e r n a t i o n a l S h i p b u i l d i n g Progress, V o l . 15, No. 167, J u l y 1968, pp. 251-276. 20 Burden, R. L., F a i r e s , J . D., and Reynolds, A. C , Numerical A n a l y s i s , Boston, P r i n d l e Weber & Schmidt Inc., 1978. 21 Nece, R. E. and Richey, E. P. "Wave Transmission T e s t s of F l o a t i n g Breakwaters f o r Oak Harbour", C W. H a r r i s H y d r a u l i c Laboratory T e c h n i c a l Report 32, U n i v e r s i t y of 57 Washington, S e a t l e , Washington, 1972. Isaacson, M. de S t . Q. "Nonlinear Wave Forces on Large O f f s h o r e S t r u c t u r e s " , Coastal/Ocean E n g i n e e r i n g Report, Department of C i v i l E n g i n e e r i n g , U n i v e r s i t y of B r i t i s h Columbia, 1981. •58 APPENDIX I USER GUIDE FOR THE COMPUTER PROGRAM BWFIX I.1 I n t r o d u c t i o n T h i s program was developed to time step a given i n c i d e n t wave past a f i x e d r e c t a n g u l a r breakwater. The numerical method using Green's i d e n t i t y i s d e s c r i b e d i n Chapter 3 of t h i s t h e s i s . The r e q u i r e d content and format f o r the input as w e l l as the o p t i o n s and content f o r the output are given i n d e t a i l below. The program i s w r i t t e n i n F o r t r a n IV, and i s s t o r e d i n the f i l e BWFIX. The compiled v e r s i o n of BWFIX i s s t o r e d i n BWFIX.C . Information concerning access to these f i l e s may be obtained by con t a c t i n g : Michael de S t . Q. Isaacson Department of C i v i l E n g i n e e r i n g U n i v e r s i t y of B r i t i s h Columbia Vancouver, B.C., Canada V6T 1W5 59 1.2 Running the Program 1 . 2 . 1 RUN COMMAND To run the program the f o l l o w i n g command may be used: $RUN BWFIX.C 5 = d a t a f i l e 6=-6 8=-8* 9=-9* 1.2.2 INPUT/OUTPUT DEVICES The I/O d e v i c e s are as s i g n e d as f o l l o w s : UNIT 5 - Assigned to the d a t a f i l e UNIT 6 - Assigned to the primary output f i l e UNIT 8* - Assigned to the secondary output f i l e UNIT 9* - Assigned to the p l o t f i l e * o p t i o n a l - see LINE 1 under Required Input 60 1.3 Required Input The input d a t a f i l e i s read on UNIT 5 and i s s p e c i f i e d i n f r e e format as f o l l o w s : LINE 1: NINIT,NDUMP,NPLOT, NINIT = NINIT = 0 f o r no i n i t i a l segment g r i d output 1 f o r output of i n i t i a l segment g r i d on UNIT 8 NDUMP NDUMP 0 f o r no int e r m e d i a t e value output 1 f o r output of a l l in t e r m e d i a t e v a l u e s of segment data (n, <t>, e t c . ) on UNIT 8 NPLOT = 0 f o r no p l o t data output NPLOT = 1 f o r output of p l o t data on UNIT 9 Note: I f NINIT and NDUMP = 0, UNIT 8 need not be assigne d i n the $RUN command. I f NPLOT = 0, UNIT 9 need not be ass i g n e d i n the $RUN command. LINE 2: NWVTP,NSTEP,NWAV, NWVTP = 0 f o r i n c i d e n t wave given by l i n e a r 61 NWVTP = NSTEP = NSTEP = NSTEP = NSTEP = NSTEP = NWAV = LINE 3 : OMEGA,H, OMEGA = H LINE 4 : D,B,BD,G,RHO, D B BD = wave theory 1 f o r i n c i d e n t wave given by Stokes' f i f t h order wave theory 1 f o r c e n t r a l d i f f e r e n c e time s t e p p i n g procedure 2 f o r Adams-Bashforth two step time s t e p p i n g procedure 3 f o r Adams-Bashforth three step time s t e p p i n g procedure 4 f o r Adams-Bashforth four step time stepping procedure 5 f o r Adams-Bashforth f i v e step time s t e p p i n g procedure number of f u l l y developed waves to i n t e r a c t with body angular frequency of i n c i d e n t wave t r a i n h e i g h t of i n c i d e n t wave s t i l l water depth beam of breakwater draught of breakwater 62 G = a c c e l e r a t i o n due to g r a v i t y RHO = d e n s i t y of water LINE 5 : DSL,DTT,DL,DSBLB, DSL = g l o b a l segment l e n g t h t o wavelength r a t i o DTT = time step t o wave p e r i o d r a t i o DL = decay l e n g t h of i n c i d e n t wave t r a i n DSBLB = body segment l e n g t h to body dimension r a t i o 63 I . 4 Output The output from t h i s program i s as f o l l o w s : UNIT 6 : Echos a l l the input v a l u e s p l u s the f o l l o w i n g c a l c u l a t e d v a l u e s : AK = wave number C = wave speed CG = i n i t i a l group v e l o c i t y of modulation envelope DT = time s t e p s i z e FSL = f r e e s u r f a c e l e n g t h on each si d e of body IT = t o t a l number of time steps NBB = number of segments along body base NBS = number of segments along each body s i d e NCS = number of segments along each c o n t r o l s u r f a c e NFS = number of segments along f r e e s u r f a c e at each si d e of body NP = t o t a l number of segments T = wave p e r i o d TTOT = t o t a l flow development time In a d d i t i o n , the f o l l o w i n g v a l u e s are p r i n t e d at each time step: 64 FHDIM = FVDIM = FMDIM = ETDIM = RNLDIM = RNRDIM = F /(.5pgHAk) - dimensionless h o r i z o n t a l f o r c e on breakwater F /(.5pgHB) - dimensionless v e r t i c a l f o r c e on breakwater M/(.5pgHkB 3/12) - dimensionless moment about c o o r d i n a t e o r i g i n of breakwater dim e n s i o n l e s s f r e e s u r f a c e e l e v a t i o n at c o o r d i n a t e o r i g i n i n absence of body di m e n s i o n l e s s wave runup on l e f t s i d e of body di m e n s i o n l e s s wave runup on r i g h t s i d e of body UNIT 8: I f NINIT = 1, UNIT 8 s t o r e s the i n i t i a l segment g r i d i n f o r m a t i o n before the time stepping procedure begins. The data s t o r e d f o r each segment i s as f o l l o w s : I = segment number X(I) = x - c o o r d i n a t e at midpoint of segment I Z(I) = z - c o o r d i n a t e at midpoint of segment I XN(I) = x - d i r e c t i o n c o s i n e of n at midpoint 65 of segment I ZN(I) = z-direction cosine of n at midpoint of segment I SLEN(I) = length of segment I If NDUMP = 1, the information l i s t e d above (for NINIT=1) together with the following i s stored at each time step: PH(I) = v e l o c i t y potential at midpoint of segment I PHN(I) = normal derivative of v e l o c i t y p o t e n t i a l at midpoint of segment I ET(I) = free surface elevation of free surface segment midpoint At each time step, UNIT 9 stores the forces, the theoretic a l free surface elevation at the o r i g i n without the body, and the wave runup at each side of the body. The res u l t s are stored in a format suitable for a p l o t t i n g routine. 66 APPENDIX II TIME STEPPING EQUATIONS Given a f i r s t order d i f f e r e n t i a l equation of the form, y' = f ( x , t ) the f o l l o w i n g methods e x i s t f o r approximate s o l u t i o n s : Adams-Bashforth two step method: Adams-Bashforth three step method: y n + i • * n + H [ 2 3yn- i 6y;-i + 5y;-2 ] Adams-Bashforth four step method: y n + i • y n + § [ 5 5 y n - 5 9 y n - i + 3 7 y n - 2 - 9 y n - 3 ] Adams-Bashforth f i v e step method: -1274y i;_3 +25ly i;_ 4] For our given problem, the kinematic free surface boundary c o n d i t i o n given by Equation (4a) i n Chapter 2 gives us an expession f o r 9 n / 3 t , 67 With known value s of n z and 9<l>/9n on the f r e e s u r f a c e at p r e v i o u s time s t e p s , we o b t a i n from the Adams-Bashforth two step method, S i m i l a r l y , we can o b t a i n corresponding time stepping equations f o r n. t +^. t using the Adams-Bashf o r t h t h r e e , f o u r , and f i v e s t e p methods. The dynamic f r e e s u r f a c e boundary c o n d i t i o n given by Equation (5) i n Chapter 2 g i v e s us an e x p r e s s i o n f o r a^/at. t + At = n t on S (5) For known va l u e s of n. and v<f> at p r e v i o u s time s t e p s , we e a s i l y o b t a i n the time s t e p p i n g formaulae f o r <t>t+At as we d i d f o r n. ... above. 68 l ( a ) : R u b b l e Mound B r e a k w a t e r 1 ( b ) : F l o a t i n g B r e a k w a t e r FIGURE 1 ; RUBBLE MOUND AND FLOATING BREAKWATERS 69 I N C I D E N T WAVE MOORING SYSTEM P R O P E R T I E S E X C I T I N G FORCES 1 ^ DYNAMIC A N A L Y S I S OF S Y S T E M BODY GEOMETRY ADDED MASS AND DAMPING C O E F F I C I E N T S _JL BODY MOTIONS T T R A N S M I S S I O N C O E F F I C I E N T F I G U R E 2: FLOW CHART FOR DYNAMIC A N A L Y S I S 70 BODY SURFACE, S, FREE SURFACE, S CONTROL SURFACES, S W/ // // // // // // // ///////// FIGURE 3: CLOSED SURFACE FOR GREEN'S IDENTITY 71 FIGURE 4 : TYPICAL INCIDENT WAVE PROFILE 72 FIGURE 5: DEFINITION SKETCH 73 74 75 ASSIGN I N I T I A L VALUES FOR <j>,3<t>/8n,&ri OBTAIN <0t + A t , 3<t»/9n t + A t ON CONTROL SURFACE SET UP SEGMENT GRID CALCULATE n T t + At ON FREE t + At SURFACE! READ INPUT PARAMETERS ADJUST SEGMENT LENGTHS TIME STEP BEGIN BUILD AND MATRIX SOLVE] FOR V t + At a$/an ON S b' t+At ON S\ CALCULATE FORCES ON BODY AT TIME t STOP FIGURE 8: FLOW CHART FOR COMPUTER PROGRAM FIGURE 9: CPU SECONDS PER TIME STEP O o X '—s in • r—1 m II • • J r H s ll X fe <— 1 y. fe X fe 0 . 0 © 0 0 . 5 © 1.0 © — © 1.5 2.0 O O m • 00 • /—s rH m II • . rH i — i . H >-/ ll N • fe 1 H J N fe • N o 0 . 0 0 . 5 © 1.0 "TET -©—©-- i r 1.5 2.0 o o x in 1 m T3 CM CM rH I CM i 0.0 © o 0.5 O 1.0 o - © — e -1.S 2.0 FIGURE 1 0 : EFFECT OF VARYING DECAY LENGTH O O X u-i O o s—S II o •UlH o < l • V ' II X 4J|H fa 1 < l X fa X fa CN oo CN CD CD 0 .0 0 . 0 2 CD CD CD 0.Q4 0 . 0 6 CD 1 ) At/T fl.08 O O X! 00 /—. <r in o o • II in o m J - i | H o < l • • *w il CN N 4J|H fa < l CN • 1 N i—( fa N fa t CD a.o 0 . 0 2 CD CD 0 . 0 4 CD O . 0 6 CD 1 1 j At / T 0 . 0 8 O o m o o II < l S I s m o o < CN 00 CD CD CD CD 0 . 0 0 . 0 2 CD At/T 0 . Q A 0 . 0 6 0 . 0 8 FIGURE 11; EFFECT O F VARYING TIME S T E P SIZE 79 x •jpgHAk 10 rv ro fl.O CD© o © 1 1 — ' — 1 — 1 — i A s / L 0.08 0.16 0.24 0.32 jPgHB ro to C3 0.0 o o o © 0 © ^ i r— 0.08 D.16 O ~ l 1 J AS/L 0.24 fl.32 M y P g H k — to d PM O CJ © 0 © o o o o °'° 0.16 0.24 0,32 FIGURE 12: EFFECT OF VARYING GLOBAL SEGMENT LENGTH AS/L 80 O O CO < fa I 00 I u-i-. CD CD —i—©- i 1 r— 05 0.11 0.17 0 0.23 1 A S b / L b 0.3 O o X o CN o -1 i—t CO • < 1 • IT) N CO fa < • 1 N o O.OS -©-JZL 0.11 CD 0.17 CD 0.23 0.3 1 A S b / L b m — o cn o i—i • oo — X CN .—I CJ • /•— i .—1 • .O CO • J 1 .—1 < CO i J • < — ' 1 0.05 CD -©-CD CD 0.11 0.17 CD 0.23 0.3 A S b / L b FIGURE 13: EFFECT OF VARYING BODY SEGMENT LENGTH FIGURE 14: TIME EFFECT OF VARYING TIME STEPPING METHODS 82 d g T 2 FIGURE 15: WAVE THEORY RANGES OF VALIDITY 83 t ime FIGURE 16: LINEAR AND STOKES' F I F T H ORDER WAVES FIGURE 17 ; FLOW DEVELOPMENT WITHOUT BODY FIGURE 18: TYPICAL HORIZONTAL FORCE DEVELOPMENT 86 FIGURE 19: TYPICAL VERTICAL FORCE AND MOMENT DEVELOPMENT 87 -] 1 1 , 1 1 1 1 1 1 1 a)/B/2g" 0.0 0,32 0,64 0.96 1.28 1.6 - | , 1 , , , , 1 1 1 , ^/B / 2 g 0.0 0.32 0.64 0.96 1.28 1.6 FIGURE 20 : FORCE RESULTS FOR BEAM/DRAUGHT = 2 . 0 88 FIGURE 21: FORCE RESULTS FOR BEAM/DRAUGHT = 4.0 89 0,0 0.48 0.96 144 192 2.4 to n 1 1 1 1 1 1 1 1 1 1 w/B/2g" 0.0 0.48 0.96 1.44 1.92 2.4 FIGURE 22: FORCE RESULTS FOR B.E.AM./DR.AUGHT = 8.0 90 FIGURE 23: TRANSMISSION COEFFICIENT RESULTS 

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