WAVE LOADS AND MOTIONS OF LONG STRUCTURES IN DIRECTIONAL SEAS by OKEY U. NWOGU B.A.Sc, University o f Ottawa, 1983 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF APPLIED SCIENCE in FACULTY OF GRADUATE Department We a c c e p t to of C i v i l this Engineering thesis the r e q u i r e d STUDIES as c o n f o r m i n g standard THE UNIVERSITY OF BRITISH COLUMBIA J u l y 1985 © OKEY U. NWOGU, 1985 In presenting requirements this that I agree that available permission scholarly for partial purposes or understood that gain by may his be or copying shall the reference f o r extensive Department financial in not of C i v i l 1985 shall and s t u d y . of the this granted by the her be allowed Engineering Head i t agree thesis representatives. or p u b l i c a t i o n make I further of THE UNIVERSITY OF BRITISH COLUMBIA 2075 Wesbrook P l a c e V a n c o u v e r , Canada V6T 1W5 Date: J u l y Library copying permission. Department fulfilment f o r an a d v a n c e d d e g r e e a t t h e THE UNIVERSITY OF BRITISH COLUMBIA, freely thesis of It for my is of t h i s t h e s i s f o r without my written ABSTRACT The effects of wave directionality on t h e l o a d s a n d motions of long s t r u c t u r e s i s i n v e s t i g a t e d i n t h i s A numerical method based on d e v e l o p e d t o compute t h e e x c i t i n g coefficients due to the previous The of diffraction of finite obtained problem and long, shape. results obtained results functions arbitrary forces using other are used hence to the i s expressed as dependent r e d u c t i o n ratio for techniques. determine operators loads a the cosine power t y p e r e s u l t s of and amplitudes of seas. short-crested seas results for long-crested seas. directionally averaged, frequency presented f a c t o r and response r a t i o o f a l o n g t o a d i r e c t i o n a l wave s p e c t r u m energy spreading the transfer f a c t o r f o r t h e wave l o a d s a n d a r e s p o n s e the force reduction box s u b j e c t wave f o ra structure f o r t h e body m o t i o n s . N u m e r i c a l r e s u l t s a r e floating the floating a r e made w i t h solution wave l o a d s a n d body m o t i o n s i n compared t o c o r r e s p o n d i n g This oblique from t h e s o l u t i o n o f t h e o b l i q u e are is hydrodynamic semi-immersed motion of t h e s t r u c t u r e i n s h o r t - c r e s t e d The and Comparisons response amplitude l e n g t h and theorem i n t e r a c t i o n of a r e g u l a r wave t r a i n w i t h a n i n f i n i t e l y cylinder Green's thesis. present i i and a f u n c t i o n . A p p l i c a t i o n s of procedure s t r u c t u r e s as f l o a t i n g b r i d g e s with include breakwaters. such long Table of Contents ABSTRACT . i i L I S T OF TABLES v L I S T OF FIGURES . . vi NOMENCLATURE viii ACKNOWLEDGEMENTS 1. x i i INTRODUCTION 1 1 . 1 GENERAL 1 1.2 LITERATURE SURVEY 1.2.1 3 DIFFRACTION THEORY 3 1.2.2 EFFECTS OF DIRECTIONAL WAVES 5 1.3 DESCRIPTION OF METHOD 2. 8 DIFFRACTION THEORY 11 2.1 11 INTRODUCTION 2.2 THEORETICAL FORMULATION 2.2.1 13 WAVE DIFFRACTION PROBLEM 13 2.2.2 FORCED MOTION PROBLEM 2.3 GREEN'S FUNCTION 17 SOLUTION 2.4 EXCITING FORCES, COEFFICIENTS ADDED 19 MASSES AND DAMPING 2.5 EQUATIONS OF MOTION 25 2.6 REFLECTION AND TRANSMISSION 3. COEFFICIENTS 28 2.7 NUMERICAL PROCEDURE 30 2.8 EFFECT OF F I N I T E STRUCTURE LENGTH 35 EFFECTS OF DIRECTIONAL WAVES 39 3.1 4. 21 REPRESENTATION OF DIRECTIONAL SEAS 39 3.2 RESPONSE TO DIRECTIONAL WAVES 44 RESULTS AND DISCUSSION 48 iii 4.1 EXCITING FORCES, COEFFICIENTS 5. ADDED MASS AND DAMPING 48 4.2 MOTIONS OF AN UNRESTRAINED BODY 52 4.3 EFFECTS OF DIRECTIONAL WAVES 53 CONCLUSIONS AND RECOMMENDATIONS 57 5.1 CONCLUSIONS 57 5.2 RECOMMENDATIONS FOR FURTHER STUDY 59 BIBLIOGRAPHY 61 APPENDIX I .65 i v L I S T OF TABLES Table 1. 2. 3. 4. page C o m p a r i s o n o f t h e sway a d d e d mass a n d damping c o e f f i c i e n t s o f a s e m i - c i r c u l a r c y l i n d e r (d/a=») obtained i n the present study with the r e s u l t s of GAR ( G a r r i s o n , 1 984) 68 C o m p a r i s o n o f t h e h e a v e a d d e d mass a n d damping c o e f f i c i e n t s o f a s e m i - c i r c u l a r c y l i n d e r (d/a=°°) o b t a i n e d i n the present study with the r e s u l t s of B&U ( B o l t o n and U r s e l l , 1 973) 69 C o m p a r i s o n o f t h e sway e x c i t i n g f o r c e c o e f f i c i e n t and wave a m p l i t u d e r a t i o o f a s e m i - c i r c u l a r c y l i n d e r (d/a=°°) o b t a i n e d i n t h e p r e s e n t s t u d y w i t h t h e r e s u l t s o f GAR ( G a r r i s o n , 1 984) Comparison of the heave e x c i t i n g f o r c e c o e f f i c i e n t and wave a m p l i t u d e r a t i o o f a s e m i - c i r c u l a r c y l i n d e r (d/a=») o b t a i n e d i n t h e p r e s e n t s t u d y w i t h t h e r e s u l t s o f B&U ( B o l t o n a n d U r s e l l , 1973) v 70 71 L I S T OF FIGURES Figure page 1. Definition sketch for a rectangular 2. Definition sketch for floating component cylinder cylinder motions showing . 73 3. Sketch of c l o s e d 4. Sketch showing 5. A t y p i c a l b o u n d a r y e l e m e n t mesh f o r a r e c t a n g u l a r cylinder (b/a=1 ,d/a=2) Square of r e d u c t i o n f a c t o r r f o r d i f f e r e n t v a l u e s 6. of 72 surface 73 relationship between x, £, and 74 74 0 75 7. Sketch of a d i r e c t i o n a l 8. D i r e c t i o n a l spreading function f o r d i f f e r e n t values of the parameter s.. Sway e x c i t i n g f o r c e c o e f f i c i e n t f o r a r e c t a n g u l a r cylinder (b/a=1,d/a=2) 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. wave s p e c t r u m Heave e x c i t i n g f o r c e c o e f f i c i e n t cylinder (b/a=1 ,d/a=2) for a R o l l e x c i t i n g moment c o e f f i c i e n t cylinder (b/a=1,d/a=2) for a Reflection coefficient (b/a=1 ,d/a=2) 75 76 76 rectangular 77 rectangular 77 for a rectangular cylinder ....78 Sway e x c i t i n g f o r c e c o e f f i c i e n t cylinder (b/a=0.265,d/a=») for a 78 Heave e x c i t i n g force, c o e f f i c i e n t cylinder (b/a=0 . 265 ,d/a=») for a R o l l e x c i t i n g moment c o e f f i c i e n t cylinder (b/a=0 .265,d/a==>) for a rectangular 79 rectangular 80 rectangular Heave a d d e d mass c o e f f i c i e n t f o r a cylinder (b/a=0.265,d/a=») vi rectangular 79 Sway a d d e d mass c o e f f i c i e n t f o r a cylinder (b/a=0.265,d/a=») Sway damping c o e f f i c i e n t f o r a cylinder (b/a=0.265,d/a=») rectangular 80 rectangular 81 19. 20. 21. 22. 23. 24. 25. 26. 27. Heave damping c o e f f i c i e n t f o r a cylinder (b/a=0.265,d/a=») rectangular 81 R o l l a d d e d mass c o e f f i c i e n t f o r a cylinder (b/a=0.265,d/a=«) R o l l damping c o e f f i c i e n t f o r a cylinder (b/a=0.265,d/a=») rectangular 82 rectangular Sway r e s p o n s e a m p l i t u d e o p e r a t o r f o r a l o n g f l o a t i n g box (a=7.5m,b=3m,l=75m,d=12m) Heave r e s p o n s e a m p l i t u d e o p e r a t o r f o r a l o n g f l o a t i n g box (a=7 . 5m, b=3m, l = 75m,d= 1 2m) R o l l response amplitude operator f o r a long f l o a t i n g box (a=7.5m,b=3m,l=75m,d=l2m) F o r c e and moment r e d u c t i o n f a c t o r s f o r a l o n g f l o a t i n g box (a=7 . 5m, b=3m, l=75m,d= 1 2m) 82 83 83 84 84 F o r c e r e d u c t i o n f a c t o r s f o r a l o n g f l o a t i n g box (a=7.5m,b=3m,l=75m,d=l2m) i n n o r m a l a n d o b l i q u e mean s e a s 86 R e s p o n s e r a t i o s f o r a l o n g f l o a t i n g box (a=7.5m,b=3m,l = 75m,d=1 2m) 87 vi i NOMENCLATURE a = half a. . = matrix A = displaced A beam o f cylinder coefficient volume p e r u n i t = complex a m p l i t u d e 0 A^j = complex wave b = draft of of v e l o c i t y potential amplitude cylinder b. . 3 = matrix B = beam o f c^j = hydrostatic Cj = exciting C(s),C'(s) = normalizing coefficients spreading functions d = water f = circular 1 length coefficient cylinder s t i f f n e s s matrix force coefficient coefficient for directional depth frequency f^ * = coefficient Fj = exciting F.. ^ = f o r c e i n t h e i t h d i r e c t i o n due t o t h e j t h mode o f m o t i o n o f c y l i n d e r g = gravitational G(fa>,0) = directional G(x;£) = Green's function H = incident wave Hj = system i = v/(-D k ; I 0 defined i n e q n . (2.92) force acceleration spreading function \ height response function = p o l a r mass moment per u n i t l e n g t h k = incident K = Keulegan-Carpenter of i n e r t i a wavenumber viii number about the y axis K ,K_ = D H reflection and transmission K ,K, = modified one 1* = length L = incident m = mass p e r m^j = mass m a t r i x 0 of Bessel f u n c t i o n s of unit coefficient = u n i t normal n.n = direction p = pressure q(kl,/3) = factor defined r = distance Ty = r a d i u s of g y r a t i o n of r(kl,/3) = factor r' = distance Rp = Rj^ = response s = cosine S(CJ) = s p e c t r a l energy S(co,/3) = directional S D = immersed S D = seabed S F = free segments on vector cosines S +Sp+S B R d i r e c t e d out of of fluid region n i n eqn. (2.103) between x and J[ between x force reduction c y l i n d e r about the y axis and factor ratio power o f spreading wave body function density spectrum surface surface S_ = radiation surface S = w a t e r p l a n e a r e a moment of a x i s per u n i t l e n g t h = and l e n g t h of c y l i n d e r n t zero wavelength number o f reduction orders structure N N coefficients 1 time ix inertia about the x T = wave p e r i o d u = fluid U = wind velocity vector speed U" = maximum p a r t i c l e V = d i s p l a c e d volume o f c y l i n d e r V = normal v e l o c i t y x = horizontal coordinate x = vector x^ = c e n t r o i d of the w a t e r p l a n e the c e n t r e of g r a v i t y X = x c o o r d i n a t e of t h e r a d i a t i o n m R velocity o f body normal t o c y l i n d e r axis of p o i n t ( x , z ) line measured from surface y = horizontal coordinate p a r a l l e l z = v e r t i c a l c o o r d i n a t e measured upwards f r o m t h e s t i l l water l e v e l z axis = z c o o r d i n a t e of t h e c e n t r e of buoyancy B ZQ = Z c o o r d i n a t e of t h e c e n t r e of g r a v i t y = response amplitude (5 = angle axis P = principal 77 = water water 77^ = asymptotic 0 7 j to cylinder R , 7 j T operator of i n c i d e n c e measured direction o f wave from t h e p o s i t i v e x propagation s u r f a c e e l e v a t i o n measured level = reflected wave from t h e s t i l l amplitude and t r a n s m i t t e d wave amplitudes 5.. = Kronecker d e l t a A = phase angle 7 = angle between x-jj_ a n d n; a l s o E u l e r ' s 7' = angle between x - £ ' a n d n' X^j = damping 3 1 function coefficient x constant u = nondimensional 2.14) frequency parameter u- • D = a d d e d mass c o e f f i c i e n t v = nondimensional 2.14) 6 = a n g l e o f i n c i d e n c e measured direction p = d e n s i t y of 4> = velocity 0^ = complex v e l o c i t y cj = wave a n g u l a r £ = v e c t o r of p o i n t (£,$) on f l u i d £' = v e c t o r of p o i n t U,-($+2d)] 1 frequency parameter from (see eqn. (see eqn. principal fluid potential potentials frequency = nondimensional boundary amplitude o f body or r o t a t i o n o f body Hj = displacement 5^ = complex wave a m p l i t u d e xi ratio motion wave ACKNOWLEDGEMENTS The Dr. author Michael throughout wishes t o express de S t . Q. I s a a c s o n the preparation Financial assistantship support from the Natural to thesis. the form of Sciences and i s gratefully xii gratitude f o r h i s g u i d a n c e and a d v i c e of t h i s in R e s e a r c h C o u n c i l of Canada h i s immense a research Engineering acknowledged. 1. 1.1 GENERAL With the resources, design in of of both of these complex The ocean have structures unidirectional random and different structure forces, could could also accepted economic the acceptable to the determination i n t e r a c t i o n with The a structure the the loads i t s from motions limits. the design incident wave offshore field t o be have and frequencies different of leads to affect or rejected waves approach property a can a i s also spectra reduction case the i n wave force the i n construction as in feasibility 1 in by of long-crested savings decisions significantly experienced directional the to significant the motions to may short-crestedness. of and use but a l s o d i r e c t i o n s . This t o a s wave loads often seas of m u l t i - d i r e c t i o n a l , i . e . t h e waves n o t o n l y The Real aspect both compared lead and a r e , however, the calculations offshore but i n addition approach directionality structure. t o withstand assumes referred influence d u e t o wave or long-crested. from The involves of important of the s t r u c t u r e . t o be w i t h i n amplitudes sometimes forces environment, often f o r the safe structures not only traditional development s t r u c t u r e s . An and the response be d e s i g n e d the a need offshore the e x c i t i n g generally in has been various body should growth there the design fixed the INTRODUCTION to whether studies. With computed waves. costs. designs the This It are recent 2 developments spectra in methods (Borgman of part of spectra the When semi-immersed motion), axis). with structures, the et forces. If incident wave is waves, design the structure (vertical motion only waves the and the i s often structure the usually about the (beamwise longitudinal due to slender significantly Morison's to estimate the forces s t r u c t u r e . For body d o e s n o t used three hydrodynamic the long in sway forces separation problem linearized established the equation exciting i s l a r g e enough t o d i f f r a c t flow ( K e l l o g g , 1 9 2 9 ) . The of infinitely motion), also r e s p o n s e of field, basins use responds exciting but p r e s e n c e of al ,1950) the and process. (angular not wave i s soon b e c o m i n g an i n c i d e n t wave k i n e m a t i c s and (Morison theory directional : heave are associated neglected laboratory the the the b u i l d i n g of (1975), Leblanc structure, There affect al i s i n c i d e n t upon an roll of et d i r e c t i o n a l wave wave t r a i n freedom and presence the models offshore a d e g r e e s of and generating directional determining (1969), M i t s u y a s u Middleton(1982)) capable of i s solved effects using complete problem by assuming a are the often potential flow i s nonlinear small amplitude and wave train. A this n u m e r i c a l method b a s e d on thesis to hydrodynamic cylinder first of solve for coefficients a r b i t r a r y shape extended to Green's theorem the of an exciting infinite in oblique structures of forces in and semi-immersed s e a s . The finite i s used length results and then are to 3 directional seas u s i n g the t r a n s f e r f u n c t i o n approach. wave l o a d s and motions of the s t r u c t u r e i n d i r e c t i o n a l The seas a r e compared with those of l o n g - c r e s t e d waves. The a p p l i c a t i o n s of the r e s u l t s of t h i s t h e s i s such long structures as floating include breakwaters, floating b r i d g e s and p i p e l i n e s . I t c o u l d a l s o be used i n the study of ship motions where Korvin-Kroukovsky's (1955) s t r i p theory i s o f t e n used to reduce the t h r e e - d i m e n s i o n a l problem two-dimensional 1.2 LITERATURE 1.2.1 a one. SURVEY DIFFRACTION THEORY A number of authors ( U r s e l l Kim to (1949), MacCamy (1964), (1965), Bai (1972), I j i m a et al (1976)) have t r e a t e d the two-dimensional wave-structure i n t e r a c t i o n problem. Much l e s s work has however been r e p o r t e d f o r the case of obliquely i n c i d e n t waves. Previous interaction (1970), studies of i n c l u d e those conducted Bai (1975), Leonard water depth, and by G a r r i s o n (1973), and G a r r i s o n Garrison to compute coefficients, coefficients oblique et for Mei and Ursell depth. function procedure f o r c e s , added mass and damping reflection a and al (1983) f o r f i n i t e (1984) f o r i n f i n i t e the e x c i t i n g Black (1969), B o l t o n (1969) used a Green's and by wave-structure and shallow d r a f t c y l i n d e r transmission f l o a t i n g at 4 the free surface. potential The a t any p o i n t continuous method i n the f l u i d distribution of s u r f a c e . The G r e e n ' s f u n c t i o n of unit strength. involves The expressing region sources equation solved numerically the source obtain hence the v e l o c i t y p o t e n t i a l . G a r r i s o n approach Bolton solve problem oscillating varying The based with field then a technique to solve and t r a n s m i s s i o n depth. The method amplitude functional. The f l u i d domain and of variables used to method t o cylinder of motion relate this problem. variational principle technique t o obtain the i s used to define variables f o r the e x c i t i n g coefficients forces i n water of a s t h e minimum into functions the of some subregions with nodal the v e l o c i t y p o t e n t i a l over the f u n c t i o n a l yields used expressing i s divided interpolation domain. M i n i m i s i n g nodal and of the c y l i n d e r . involves equation set be extended s o l u t i o n o f t h e p r o b l e m . B a i (1975) a l s o governing d i f f e r e n t i a l the strengths a circular t o t h e wave d i f f r a c t i o n a n d M e i (1970) u s e d reflection a can (1984) the length on S c h w i n g e r ' s v a r i a t i o n a l finite the with the along r e l a t i o n s were a variational and heave problem Black which (1973) u s e d a m u l t i p o l e associated sinusoidally radiation far in Haskind source t o c y l i n d e r s of a r b i t r a r y shape. and U r s e l l the body b o u n d a r y c o n d i t i o n on t h e body r e s u l t s i n an i n t e g r a l this the a point surface to i n terms of a along represents the with respect a set of l i n e a r to equations 5 which can be variational larger using that i s however efficient approach case approach than matrix of A boundary diffraction used by (1979) that Green's in et results The al the (1975) Ursell of field. The equations much e q u a t i o n method. banded Bai and et al (1975) can be (1983) in The solved used studying approach wave the has in solve an the Finnigan problems to water procedure second the wave previously and and been Yammamoto by Isaacson interaction. complexity has for finite (1973) and and Green's to wave-structure which present involving thesis (1976) avoids conditions of and this nonlinear function boundary system method for two-dimensional method potential integral of integral Ijima present the cylinders. problem. for and the symmetric used (1981) Bai give leads to a of to multiple is to techniques. Leonard similar identity of solved of satisfy deriving the of finite are compared a various depth. well The with water depth, as as Garrison (1984) for infinite The those Bolton water depth. 1.2.2 EFFECTS OF Previous structures scattered There determine studies in of the directional i n the have the DIRECTIONAL WAVES loading seas are and few response and of widely literature. been response two of general structures approaches in used to short-crested 6 s e a s . The more common a p p r o a c h approach where transfer functions spectra linear theories which t o the response n a t u r e . Time involve Fast either Fourier analysis is seas. the on a transfer found Thompson vertical of of a when the nonlinear directional seas of white noise The t i m e or domain more e x p e n s i v e than the (1976) computed t h e wave cylinder diffraction functions. to is filtering generally and large Linear wave approach. Huntington loads used (FFT) techniques. however to determine incident often process the d i g i t a l f r e q u e n c y domain are description Transform the domain spectra. interaction domain frequency a r e used relate Time domain s i m u l a t i o n s wave-structure i s the be in theory The in was short-crested used t o determine theoretical results were good agreement with experimental proposed a hybrid method of computing which incorporates measurements. Dean the (1977) wave l o a d s on o f f s h o r e structures both the n o n l i n e a r i t y and d i r e c t i o n a l i t y linearized form of Morison's determine . the effect of f a c t o r s were p r e s e n t e d spreading function. waves on (1982) s t u d i e d the loads f a c t o r s were p r e s e n t e d on equation for the the e f f e c t s a was directional reduction Battjes o f t h e waves. A long of wall to waves. Force cosine power directional structure. for a vertical used Reduction occupying the 7 entire water d e p t h and type d i r e c t i o n a l Dallinga directional barge used et al (1984) i n v e s t i g a t e d spreading for the transfer functions. in equation was Hackley response water by the reduction compared in to significant the determine inline long-crested Georgiadis the used to response seas. al of (1979) used a loading and s e a s . The Fast determine the f o r use i n found in short-crested There the scheme. (1979) (1984) u s e d a Monte C a r l o s e a s . The determined accelerations et to differential difference et al a was a seas also a simulation to response. the a p p r o p r i a t e short-crested and Shinozuka transverse directions in short-crested T r a n s f o r m t e c h n i q u e was Morison's equation. (1982) used the simulate velocities horizontal response solving the (1982) b o t h flexible and of a obtain Lambrakos Shinozuka to structures to of platform. wave l o a d s were using a f i n i t e and motions jackup used slender and obtained particle a frequencies approach of power effects and s e a s . Lambrakos equation (1979) domain Fourier wave of motion of (1984) a n d directional M o r i son's cosine the loads was t h e s e a s u r f a c e . The structure time theory and G r e a t e d number o f describe from the the r e s p o n s e of l o n g cylinders finite on transport diffraction Bryden f o r the spreading function. Linear studied a pipeline nodal f o r c e s on r e s p o n s e of the structures in structure was 8 then evaluated using a d e t e r m i n i s t i c analysis. 1.3 DESCRIPTION OF METHOD The analysis o f t h e dynamic response of long i n d i r e c t i o n a l s e a s c a n be d i v i d e d The first diffraction of semi-immersed Green's forces part involves a regular identity and hydrodynamic solving the problem of the o b l i q u e wave t r a i n b y a n infinite fluid motion diffracted i s used i sdescribed wave, their the values normal function and of forced The finite incident waves f o r e a c h mode o f second i d e n t i t y i s used Green's on a boundary free to function surface only has the of and to Green's consists radiation s a t i s f y the equation which i s t h e two-dimensional H e l m h o l t z e q u a t i o n . The b o u n d a r y number to t h e unknown v e l o c i t y p o t e n t i a l s a n d body s u r f a c e , governing d i f f e r e n t i a l modified the i t s n o r m a l d e r i v a t i v e s . The b o u n d a r y t h e immersed surface. of derivatives and i n terms of a v e l o c i t y o f components due t o motion of t h e c y l i n d e r . Green's relate t o compute t h e e x c i t i n g coefficients. p o t e n t i a l which c o n s i s t s wave, i n t o two p a r t s . c y l i n d e r . An i n t e g r a l e q u a t i o n m e t h o d b a s e d on second The structures segments. Application i sdivided of the into a various boundary conditions on t h e v a r i o u s s u r f a c e s y i e l d s a s e t o f algebraic equations which velocity potentials. Bernoulli's pressures and equation hence can be i s then solved used the e x c i t i n g forces to to obtain the compute the and hydrodynamic 9 forces forces due t o t h e m o t i o n s o f t h e c y l i n d e r . The h y d r o d y n a m i c c a n be e x p r e s s e d i n t e r m s o f components i n phase with the body a c c e l e r a t i o n and v e l o c i t y . T h e s e a r e r e f e r r e d t o a s the a d d e d mass and damping c o e f f i c i e n t s r e s p e c t i v e l y . The reflection determined the with surface the pressure combined added with elevation and equations of motion a r e then coefficients of i n e r t i a motion solved response amplitude two-dimensional obtain the t o t a l The results for a multi-directional approach. The three body. The r e f e r r e d t o as oblique seas using the short-crested frequency length, the t h e body a x i s t o on t h e s t r u c t u r e . of the a n a l y s i s regular along involves wave sea spectrum function. A cosine extending train linear to transfer surface s p e c t r u m c a n be e x p r e s s e d a s t h e p r o d u c t spreading then the amplitudes finite t e r m s o f a d i r e c t i o n a l wave s p e c t r u m . The one-dimensional obtain the often of are integrated wave l o a d s second part are operator. structure forces to to obtain the rigid used t o o f t h e body and for body m o t i o n p e r u n i t wave a m p l i t u d e a is at surface. coefficients of amplitudes equation of For wave are t o the v e l o c i t y p o t e n t i a l damping stiffness coefficients a t the free t h e mass o r moment linear equations Bernoulli's set to zero mass hydrostatic coupled transmission the asymptotic surface. t h e water The the by e v a l u a t i n g radiation relate and the random function i s described in directional of the and power s p r e a d i n g a wave conventional directional function which 1 0 is independent The of frequency i s used exciting obtained force by m u l t i p l y i n g appropriate transfer and i n t h i s study. body the i n c i d e n t function or response wave s p e c t r u m response o p e r a t o r . The e f f e c t s o f wave d i r e c t i o n a l i t y a directionally factor The t o be a p p l i e d mean are seas. averaged, also frequency compared are with the amplitude i s e x p r e s s e d as dependent t o the one-dimensional square v a l u e s of t h e r e s p o n s e spectra force reduction spectrum. in short-crested seas to corresponding results for long-crested 2. 2.1 long t r e a t i n g the problem structures the with infinite an arbitrary The interaction cylinder the assumed a semi-immersed surface. The and a regular shall oblique horizontal the the Keulegan-Carpenter amplitude of large field. the thin flow indication is flow and to fluid irrotational effects considered negligible confined An i s incident assumed body, of we response of first wave train cylinder of shape. diffract theory. of the dynamic i n m u l t i - d i r e c t i o n a l seas, consider can enough Flow of layer thus considered be using of the importance of Keulegan-Carpenter fluid i s defined motion to a typical to potential flow the are on t h e body be flow separation number, as to effects viscosity boundary solved as separation effects problem number so R. ratio The of dimension the of the that i s K where U m period = U T/B (2.1) m i s t h e maximum and B i s range of frequencies less than Sarpkaya For this THEORY I N T R O D U C T I O N Before are D I F F R A C T I O N two and a typical used and velocity, dimension in this flow is the of t h e body. study, separation T K will should wave For the usually not occur be (see Isaacson,1981). rectangular study, particle vortices section are cylinders usually 11 which formed at a r e used i n the sharp 12 corners. Various authors Jamieson ( 1 9 7 6 ) ) have however potential flow cylinders For when coefficient It exciting and still wave s m a l l so t h a t with a f l o a t i n g superposition cylinder oscillation of two-dimensional obtained i s assumed periodic three-dimensional along problem one. Even f o r an i n f i n i t e structures ships, by a f i x e d added with the (1) cylinder, an infinite i n an otherwise mass and damping motions of the the complete cylinder motion. two p a r t s : The wave h e i g h t and o s c i l l a t o r y a r e assumed of the wave-structure sway and r o l l i n terms of near v i s c o u s damping into associated i n heave, of motion i s particularly an e m p i r i c a l bodies vortices. problem of c a n be r e p r e s e n t e d diffraction and forced problems. The axis, f o r such to separate forces expressed interaction motion and for floating oscillating a linear results amplitude by v i s c o u s damping convenient problem coefficients. cylinder roll f o r c e s due t o wave d i f f r a c t i o n water between s h o u l d be i n c l u d e d i n t h e e q u a t i o n s (2) h y d r o d y n a m i c cylinder agreement experimental the frequency i s usually interaction good Mogridge and d e s p i t e the formation of the affected resonance et al ( 1 9 7 9 ) , found and cylinders, significantly by theory fixed floating the (Bearman of ignoring finite flexible with i t s amplitude the a x i s of the c y l i n d e r , can be reduced though the n u m e r i c a l cylinder, they are l e n g t h by i n t e g r a t i n g end e f f e c t s . Korvin-Kroukovsky's For non-uniform (1955) s t r i p of so t h e to a results are extended along bodies to t h e body such t h e o r y c a n be as used 13 w i t h t h e t w o - d i m e n s i o n a l r e s u l t s , F o r head normal t o the length of magnitude cylinder the axis), cylinder the becomes as a t y p i c a l c r o s s the sectional wave direction crests along same the order of and the dimension A t h r e e - d i m e n s i o n a l model w h i c h c o n s i d e r s end e f f e c t s w o u l d h a v e t o incident (wave wavelength of p r o c e d u r e i s no l o n g e r a p p l i c a b l e . seas be used as the moves s u b s t a n t i a l l y away f r o m t h e beam d i r e c t i o n . 2.2 THEORETICAL FORMULATION 2.2.1 WAVE DIFFRACTION PROBLEM A regular small a m p l i t u d e wave t r a i n o f h e i g h t H a n d a n g u l a r f r e q u e n c y co i s o b l i q u e l y . i n c i d e n t infinitely propagate long fixed horizontal upon c y l i n d e r . The i n water of depth d i n a d i r e c t i o n waves making angle 0 system i s r i g h t handed w i t h z measured upwards from still water with the x axis level i s p a r a l l e l to the a x i s The origin still of ( s e e F i g . 1 ) . The c o o r d i n a t e The fluid is the flow i n c o m p r e s s i b l e and m o t i o n may potential of the infinite the above or below i s at the the centre a s s u m e d t o be i n v i s c i d a n d irrotational. t h e r e f o r e be d e s c r i b e d defined y cylinder. the ( x , y , z ) c o o r d i n a t e system water l e v e l v e r t i c a l l y gravity. an and t h e x-y p l a n e h o r i z o n t a l . The axis of an The fluid i n terms of a v e l o c i t y by u = V#(x,y,z,t) (2.2) 1 4 where the u i s the f l u i d Laplace and # must = 0 2 the f l u i d sufficiently (2.3) region. small The so wave that height linear boundary g constant The that fluid free be equal to given to zero surface f o r convenience. boundary a t the f r e e + surface. Hi2 the velocity + square eqn. the s t i l l i n s t a n t a n e o u s water can surface then boundary free surface This fluid c a n be e x p r e s s e d a s !f!? < 2 neglecting -5) the t e r m i n e q n . (2.4) and t h e wave conditions at condition the normal v e l o c i t y of a in free is (2.4) (2.4) and (2.5) a r e l i n e a r i z e d by equations usual = R 2 s t e e p n e s s terms the the pressure the normal v e l o c i t y of H =& Eqns. t h e dynamic s e t equal kinematic elevation particle is i s the g r a v i t a t i o n a l a c c e l e r a t i o n and R i s the Bernoulli requires theory equation | | + grj + ^ ( V # ) where to assumed conditions. On t h e f r e e s u r f a c e , by t h e B e r n o u l l i is wave a p p l i c a b l e and c o n s e q u e n t l y $ i s s u b j e c t linearized satisfy equation V <I>(x,y,z,t) within v e l o c i t y vector (2.5), water surface and level applying z=0 r a t h e r elevation be combined condition by to give z=7j. the than at The two the l i n e a r i z e d 15 || for simple The and = 0 hence S (2.6) harmonic motion. immersed surface, at z = 0 body surface is the normal v e l o c i t y D must equal assumed impermeable of the f l u i d on t h e body zero a ||=0 where n is directed and on S a direction normal i n t o t h e body. The impermeable addition satisfy surface horizontal a t z=-d (2.8) t o t h e above b o u n d a r y c o n d t i o n s , condition $ has to at the f a r f i e l d t o ensure t o assume t h e v e l o c i t y potential solution. It to body giving a radiation a unique the s e a b e d i s assumed || = 0 In to (2.7) B is convenient be o f t h e form * = # where 0 + *a 4> and 4>j, a r e 0 incident (2.9) the velocity potentials and d i f f r a c t e d waves r e s p e c t i v e l y . wave p o t e n t i a l i s given •„(,.y.«,t) r The by l i n e a r wave t h e o r y - Ret^gH c for angular k is the frequency wave number incident as ° ^ d f " x e x p { i (kxcos0+kysin/3-o>t)} ] where the which i s r e l a t e d u by t h e d i s p e r s i o n relation (2.10) t o the 16 k tanh(kd) The radiation diffracted is given a (2.11) condition + ikcos/3 away from = 0 that the at a f i n i t e distance, modes due analysis to radiation condition The find fluid *(x,y,z,t) (2.12) boundary from t h e o r i g i n An a t which R is considered periodic t h u s be d e f i n e d by x' = x/a, d' = d/a, For now and beam o f the c y l i n d e r , z' = z/a, u = ^g -, 3 y' as nondimensional „ variables a. = y/a, k' = ka (2.14) v = kasin/3 the primes i t i s understood nondimensional. h e n c e f o r t h be I. = R e [ ^ i r g ^ > ( x , z ) e x p { i (kysin/3 - cot)}] (2.13) convenience, variables the i n time i s a l s o convenient to n o n d i m e n s i o n a l i z e the the h a l f body approximate i s given i n appendix of t h e c y l i n d e r . A is where of t h e t h e sufficiently. i s applied motion <t> c a n X infinite t h e optimum d i s t a n c e X as a l o n g t h e a x i s potential, the to the presence a r e assumed t o have d e c a y e d using the cylinder a t x = ±» numerical approximation, the evanescent It ensures by truncated well which waves a r e t r a v e l l i n g j-— In = £p have been d r o p p e d that Dimensional the from variables variables b a r r e d where n e c e s s a r y f o r c l a r i t y . the are will 17 The boundary potential can V 0« now - 2 value be problem 2 the diffracted s t a t e d i n nondimensional = 0 v 4>k for i n the form fluid as (2.15a) -g-pj— = u<t>* a t z=0 (2. 1 5b) •g-^- = 0 a t z=-d (2.15c) •g^- = ikcos/3tf> at x = ± X a d(f>n (2.15d) R d<t> 0 1TT -cTfr = o n (2.l5e) S B where *° • ° c S s h U d f C The three-dimensional been r e d u c e d equation 2.2.2 to the Laplace (2.16) equation two-dimensional (2.3) modified has now Helmholtz FORCED MOTION PROBLEM an h e a v e , sway and motion infinitely roll is periodic the c y l i n d e r . The mode i s g i v e n by E (y,t) = £^ cylinder roll * as long c y l i n d e r shown i n F i g . 2. i n time displacement or r o t a t i o n k s with the complex a m p l i t u d e k=1,2,3 c o r r e s p o n d i n g modes oscillating in Each of as w e l l as a l o n g ReU exp{iUy-cot)}] k and exp(ikxcos^) (2.15a). Consider where ) ] {J mode t h e a x i s of in the I \' } 2 of o s c i l l a t i o n to the r e s p e c t i v e l y . Throughout sway, kth (2.17) of the heave the f o l l o w i n g 18 development, with the upper t e r m s i n the c u r l y e a c h o t h e r , and s e p a r a t e l y t h e with each other. by a, w h i l e i- 3 (•, and corresponds The v e l o c i t y is given V lower have been 2 t o the r o l l o f t h e body apply terms apply nondimensionalized angle in radians. surface i n the d i r e c t i o n n by 3 _, = Z |fn k=1 n £ brackets = R e [ 1 k 3 -i"a£ n exp{i(vy-ut)}] k=1 k (2.18) k where n 1 = x ) n n~ = n„ n and n 3 , n vector } = (z-e)n - xn x z ) are the d i r e c t i o n n on t h e immersed (2.19) c o s i n e s of t h e u n i t body the p o i n t about which the r o l l normal velocity surface must e q u a l of the the s u r f a c e and normal (0,e) d e n o t e s m o t i o n i s p r e s c r i b e d . The fluid normal on the velocity immersed body of body the yielding ^ 3n This = V on S. n boundary c o n d i t i o n i s s a t i s f i e d position o f t h e body position o f t h e body. From potential equations (2.20) B r a t h e r than (2.18) and at (2.20), at the the equilibrium instantaneous the f o r c e d motion f o r t h e k t h mode o f m o t i o n c a n be e x p r e s s e d = Re[-icja ^.^. 2 exp(iUy-cot)} ] as (2.21) 19 The linearized boundary c o n d i t i o n can t h u s be e x p r e s s e d a s 30v The potentials value is in and for hence order method f o r the <j>^ (k= 1 ,2, 3 , 4 ) . the f l u i d </>(Jj.) surface (2.22) the forced governed motion by eqns. (2.22). integral may be a p p l i e d o v e r in body FUNCTION SOLUTION used as t h e b a s i s region is c boundary potentials problem 0j (k=1,2,3) ( 2 . l 5 a - d ) and e q n . A the k=1,2,3 boundary 2.3 GREEN'S on i n v o l v i n g Green's numerical The s e c o n d a closed surface t o the boundary i t s normal evaluation S containing the theorem the fluid o f t h e p o t e n t i a l <f>(x) values of the 3t/>(JL)/9n. derivative of form o f G r e e n ' s t o r e l a t e the values region identity potential This can be expressed as *<i> = ^ where G ( x ; £ ) the point (£/$) over surface free as fU(i)f§(x;i) i s an a p p r o p r i a t e (x,z) being i$(JL)G(x;I)]dS " Green's considered which the i n t e g r a t i o n i s S surface comprises t h e immersed function, (2.23) x denotes and £ d e n o t e s t h e p o i n t performed. body S„, the r a d i a t i o n surface F surface The closed S , t h e mean fi S_., a n d t h e s e a b e d S_. R D shown i n F i g . 3. When t h e i n t e r i o r within, eqn. (2.23) point x approaches reduces to the the boundary following from integral 20 equation • (£> - i SSi^V^liV The Green's equation the function (2.15a) point " U i *''i { which s a t i s f i e s i n an unbounded )G{ ' )]dS (2 24) the modified Helmholtz fluid and is singular x=i. i s g i v e n by G ( x ; £ ) = -K (vr) (2.25) 0 where K is 0 i s the modified the distance r The = function between |1 - x| = K (x) Bessel function the p o i n t s [ U - x ) + 2 of order zero and r x and £ ( S - Z ) ] 2 1 / (2.26) 2 - I n x a s x —s» 0. The 0 which s a t i s f i e s Green's the two-dimensional Laplace function equation G ( x ; l ) = In r is at thus obtained Since the computationally (2.27) a s 0 —> 0 ° . seabed is assumed more e f f i c i e n t and an a l t e r n a t i v e Green's horizontal, t o exclude function i t is t h e seabed from S which takes into account symmetry a b o u t t h e s e a b e d c a n be d e f i n e d G(x;£) = -[K (;/r) + K ( * » r ' ) ] 0 where r' is the (£/~($ 2d)) which + r« If distance eqn. to = |£' - x| = be i n c l u d e d (2.25) u s e d between i s the r e f l e c t i o n the depth v a r i a t i o n s have (2.28) 0 [ U - x ) 2 the points of £ about t h e seabed: + ($ + 2 d + z ) ] are significant, i n S and t h e Green's instead. x and £' = 2 the (2.29) l / 2 seabed function would g i v e n by 21 The i n t e g r a l numerically fluid to equation give the and h e n c e p r o v i d e (2.24) can potential the solution now be evaluated </> a t any p o i n t t o the i n the boundary value problem. 2.4 EXCITING FORCES, ADDED MASSES AND Once the velocity DAMPING potential h y d r o d y n a m i c p r e s s u r e c a n be computed Bernoulli the the linearized = - p | | = iw/o$ forces and integrating surface from obtained, equation p The is COEFFICIENTS (2.30) moments per u n i t length a r e d e t e r m i n e d by the hydrodynamic p r e s s u r e over t h e immersed body S„. a The e x c i t i n g incident height force and s c a t t e r e d i s given F per unit length waves and i s p r o p o r t i o n a l by j HMf * j = n Fj(j=1,2) respectively of while F equations Fj(y,t) = denotes 3 (2.9) and the the t o t h e wave {i - 3 I - d s ,2 B where w h i c h i s due t o sway and (2 heave force denotes the r o l l moment. Substitution (2.13) i n t o e q n . (2.31) y i e l d s pg§{f[2}Re[;(0 + >„)n .exp{i(vy-cjt)}dS]|1:3' |(2.32) 2 o S t :] B The d i m e n s i o n l e s s e x c i t i n g force amplitude i s given by 31) 22 F.(y,t) Cj = — ^•pgHa The e x c i t i n g force Ull' } (- > 2 = J ( * o + * « > n j dS B 2 33 s could a l t e r n a t i v e l y be d e f i n e d by F • (y,t) r~7"{i} lCjl = {1l 3 ' 2 } cos(,y- t A.) W + (2.34) 2PgHa where t h e p h a s e a n g l e A j i s d e f i n e d Aj There =tan" are also motions of the amplitude due Im(C..) 1 hydrodynamic cylinder forces which are associated * j i n ji=3' j 2 d s 1 2 S ; one component in can a l s o phase w i t h F where i j p^j ( 2 - 3 6 ) forced motion i B 2 force 3 « .n exp{i(vy-wt)}dS] )i=3' | This 2 ' B pcj ^|^a}Re[/ i the of the f o r c e 3- ' ' Substitution of t h e e q u a t i o n for the p o t e n t i a l s (2.21) i n t o e q n . (2.36) y i e l d s F y" to o f m o t i o n c a n be e x p r e s s e d a s = S with the proportional m o t i o n . The i t h component t o t h e j t h component i j (2.35) Re(Cj) of c y l i n d e r F by be e x p r e s s e d i n phase w i t h 3=1,2,3 (2.37) i n t e r m s o f two the a c c e l e r a t i o n components; and the o t h e r the v e l o c i t y *--OijSj and X^j - X i j j are 3" ' ' E the 1 added 2 mass 3 ( and 2 ' 3 8 ) damping 23 coefficients eqn. (2.38) F respectively. Substitution o f e q n . (2.17) into gives i j {*} = R e [ { c j 2 M ij*j + i a , i j^j X ) jj='j e x P {1 ( f'Y-^t)} ] 2 i=1,2,3 3 Comparing eqn. (2.37) with eqn (2.39) n o n d i m e n s i o n a l added mass a n d damping M i j (2.39) gives c o e f f i c i e n t s as = R e [ / 4>.n. dS] S 3 m pa the (2.40) B = Im[J 4>.n. dS] — pwa m s where t h e c o n s t a n t (2.41) B m i s given as 2 f o r ( i , j ) = (1,1) a n d (2,2) m = {3 f o r ( i , j ) = ( 1 , 3 ) a n d (3,1) (2.42) 4 f o r ( i , j ) = (3,3) The H a s k i n d an (1953) r e l a t i o n s alternate Applying way of s e e Newman,1962) calculating the Green's theorem t o t h e d i f f r a c t i o n j - 5 r r " *«-9n ( 0 Substituting 1 + Applying eliminates ) d S the boundary force provide exciting forces. potential 3 = 1/2,3 0 90K *j7Jn- into potential gives (2.43) e q n . (2.32) gives ) d s condition the d i f f r a c t i o n exciting = t h e above e x p r e s s i o n 9#/ (0O-3W the (also { given by eqn. 2 ' 4 4 ) (2.l5e) from t h e e x p r e s s i o n f o r 24 d<j> • / S There is coefficient at " 0 30o j ? T a direct (2.45) relation and t h e amplitude cylinder to between about as the r a t i o the amplitude the damping o f t h e waves g e n e r a t e d symmetrical |$^| c a n be d e f i n e d |x|=°° r)dS B also oscillating ratio U O - S T P by an x=0. An a m p l i t u d e o f t h e wave of o s c i l l a t i o n amplitude of the c y l i n d e r , that i s where for to |7j^| i s the amplitude of t h e r a d i a t e d t h e i t h mode o f o s c i l l a t i o n By equating the energy infinity, the oblique to the the does negative not vanish does not s a t i s f y motion energy x=-X R in oscillating radiated ratio potential across the a control | x | =°° cylinder surface at ( s e e Newman,1977) by e v a l u a t i n g radiation since can a l s o surface the i n c i d e n t the radiation condition. i s proportional the e x c i t i n g force be r e l a t e d the i n t e g r a l i n eqn. (x=-X ). R wave Since t o t h e square f l u x , e q n . (2.45) c a n be i n t e g r a t e d to give at of the c y l i n d e r . c a s e . The e x c i t i n g f o r c e amplitude at integral flux i t c a n be shown t h a t for (2.45) t h e work done waves over c o e f f i c i e n t as The potential the forced root of the the depth a t 25 Equations direct the (2.47) and relation damping (2.48) c a n between t h e be combined exciting to provide force c o e f f i c i e n t s a and coefficients X. . I il = c 1 Equations results EQUATIONS OF The (m. 2 J 1 now .+fi..) - and c^j 1 J matrix coefficients or present. k a be ico\.. of obtained l are be the t h e mass and noted frequency assumed the due to the the equations form i = 1,2,3 (2.50) stiffness Additional forces be that and the hydrostatic due i n c l u d e d i n eqn. for the case i s important would to (2.50) of roll particularly have t o that in deriving the the be cylinder a d d e d mass was of motion v a r y i n g s i n u s o i d a l l y cylinder thought on included applications. coefficients amplitude of A resonance was are 1 respectively. It should solving c. . ] S . = F . ( y , t ) J J + •J near It check cylinder by of m o t i o n v i s c o u s damping may practical useful the n o n l i n e a r v i s c o u s damping in (2.49) 1 / 2 . motion, the cos/3] d provide equations m^j if 2 response where moorings )tanhkd sinh2kd + MOTION waves c a n of m o t i o n . The [-co 1 obtained. dynamic exciting 3 2 j_1 < (2.47)-(2.49) numerical 2.5 f — m pcua of as as well the ratio as of in the time. incident and damping flexible with along length of sin/3 c a n be The the term wave l e n g t h to its the 26 the wave l e n g t h cylinder has cylinder and along an the axis of the cylinder. i n f i n i t e wavelength along hence c o r r e s p o n d s to a flexible the A rigid axis of cylinder the with 0=0°. The components of m m. 1 D -mz, 'G where m i s the coordinate moment o f be of I where axis. 0 r^ the m 0 are given as (2.51) 0 ~ unit of the length of the g r a v i t y and y axis per I body, z^ i s the 0 unit i s the polar length. I z mass may 0 as = m(r calculating stiffness -mz. about 2 y + z*) G the • radius hydrostatic equilibrium 0 centre i s the The mass m a t r i x mass p e r inertia expressed the of gyration stiffness forces small m a t r i x components a r e c. . = D 1 0 0 0 C 2 2 0 C 2 3 matrix required position for of to is restore amplitude given the (2.52) body a b o u t the determined the body y by to i t s displacements. The as 0 C (2.53) 23 C 3 3 where ' c 2 2 = pgB c 2 3 = pgBx (2.54a) f (2.54b) 27 = pgA[(S,,/A) + z Q where B i s t h e beam of t h e c y l i n d e r , the waterplane symmetrical of line about x=0, z f i is the waterplane a x i s per u n i t 2 stability positive. metacentre centre bodies i s t h e z c o o r d i n a t e of t h e centre of inertia l e n g t h , and about the x (S^/A) (2.55) 3 + requires (2.54c), z f i i t z^ f o r t h e f l o a t i n g of motion t h e complex a m p l i t u d e s given wave frequency terms of is evident the body t o be oscillation, and d i r e c t i o n of the c o e f f i c i e n t that c 3 3 the stable. (2.50) c a n now be s o l v e d t o t e c h n i q u e . The a m p l i t u d e in that has t o be l o c a t e d h i g h e r t h a n t h e obtain described volume p e r u n i t = B /12 in roll equations inversion zero of for a r e a moment From e q n . of g r a v i t y The to centroid length, that i s = / x dx B Static x^ i s t h e equal buoyancy, A i s the d i s p l a c e d is be and (2.54c) - z ] f i £j for u s i n g a complex o f body m o t i o n response amplitude is any matrix often operator d e f i n e d as Z.(u,/3) = The body (2.56) response, a m p l i t u d e motion travelling due to operator represents the amplitude a unit i n d i r e c t i o n /3. amplitude of wave o f f r e q u e n c y co, 28 2.6 REFLECTION AND TRANSMISSION COEFFICIENTS Another for such two q u a n t i t i e s structures reflection and as of p h y s i c a l floating transmission a r e o b t a i n e d by e v a l u a t i n g coefficients. this a s y m p t o t i c wave a m p l i t u d e f r o m : the cylinder related for is wave by a f i x e d equation, the body. wave amplitude i n t o e q n . (2.57) is p o t e n t i a l by < - > 2 the equation f o r the forced yields motion 57 potentials t h e a s y m p t o t i c wave a m p l i t u d e e a c h mode o f m o t i o n exp{iUy-ot)}] = Ret ^-a ^ t> (x. 0) 2 i< i wave a m p l i t u d e r a t i o r previously defined (2.58) by eqn. (2.46) now g i v e n as l*il evaluated = TaTtl at x=±X . R = ^i^i ( x and t h e v e l o c i t y 0(x,z) = A o I '°> At the r a d i a t i o n modes a r e assumed t o have d e c a y e d I) (1) t h e o s c i l l a t i o n s o f "g- f f ^ ' Y ' O ' t ) TJ. The at i n i t s t h r e e modes, a n d (2) t h e r e f l e c t i o n and t o the v e l o c i t y Substituting (2.21) The c o e f f i c i e n t s R of t h e i n c i d e n t = the ( x = ± X ) . There are c o n t r i b u t i o n s t o Bernoulli's * are t h e component wave a m p l i t u d e s radiation From especially breakwaters the transmission surfaces interest { 2 surface, - 5 9 ) the evanescent sufficiently (see appendix p o t e n t i a l s a r e of t h e form cosh[k(z+d)] cosh(kd) e x p ( ± i k x c o s g ) a t x = ±X (2.60) R K 29 where A Given i s t h e complex 0 a m p l i t u d e of the p o t e n t i a l = s i n h ( 2 k d ) + 2kd 2 the coefficient orthogonality A i s given °i = A 4 can 0 condition be obtained of the sinhtSkdi^Zkd applying cosine exp(±ikxcos/5) obtained be the to in a similar obtained waves (x=-X ). Substitution potential given R TJ The r reflected is The the ) the /«.cosh[ k (z+d) ]dz J x=+X (2.62) R reflection presence of and a at the asymptotic the of the by e q n . (2.13) body a r e c o e f f i c i e n t can wave a m p l i t u d e o f radiation surface form o f t h e d i f f r a c t e d into e q n . (2.57) i s d e f i n e d as t h e to the incident wave yields ,0)_ e x p { i U y - u t ) } ] coefficient wave a m p l i t u d e negative transmission fixed manner. The r e f l e c t i o n = Re[.§0,(-X reflection l (2.63) by e v a l u a t i n g scattered g r a t i o c a n t h u s be e v a l u a t e d a s e a c h mode o f m o t i o n . due # function ISjl = ^ I A O J I coefficients 2 as at The wave a m p l i t u d e by hyperbolic f o r the j t hp o t e n t i a l ( K -d for z=0. that Jcosh [k(z+d)]dz -d and at (2.64) ratio of the wave a m p l i t u d e a n d t h u s g i v e n by K IV I R = -i-^- H/ 2 = |0„(-X O) | R f (2.65) 30 The transmission amplitude positive of the = T 0 wave K After R as reflection boundary the freely floating = 3 |0,(-X ,O) + Z $.Z.(u,/3)| j=1 3 D square of transmission K T = |0 (X ,O) and body R the transmission are determined (2.68a) O R 3 + 0,(X ,O) + Z S j Z j ( w , 0 ) | (2.68b) R PROCEDURE to evaluate i s divided into considered constant value at the midpoint replaced and reflection D order remembering of body m o t i o n by s o l v i n g K NUMERICAL (2.66) (2.67) a O by by motion, respectively In principles, = 1 of for 90/3n the are related coefficients given (2.65) and i n a wave i s p r o p o r t i o n a l t o t h e 2 the (2.66) s i d e of eqns. o b t a i n i n g the amplitudes equations at 0)| c o n s e r v a t i o n of e n e r g y + K wave (2.62). amplitude, coefficients R f waves is similarly R on t h e r i g h t the energy the scattered ( x = X ) and R e v a l u a t e d u s i n g eqn. Applying 2.7 surface and |# (X ,0) + * , ( X The e x p r e s s i o n s that i s due t o t h e a s y m p t o t i c incident radiation K are coefficient the i n t e g r a l N segments w i t h over equation (2.24), the v a l u e of e a c h segment o f t h e segment. by t h e summation equation Eqn. and e q u a l (2.24) the <t> o r t o the can be 31 i 1 0 (x.) = k ar N 90i Ig^i^^dS 2 {*k(£jU - ^ / G(x.;x.)dS} j j k=1,2,3,4 where t h e summation (2.69) i n e q n . (2.69) i s p e r f o r m e d i n a c o u n t e r c l o c k w i s e manner a r o u n d t h e b o u n d a r y . Eqn. (2.69) can be r e w r i t t e n as N • 5 where b<j>[ m { ( a 1 i j + i j 6 ) 0 j ijJn + h k) } 3 S^j i s the K r o n e c k e r d e l t a 6^ (1 = { = k=1,2,3,4 0 function (2.70) g i v e n by i - j (2.71) The c o e f f i c i e n t s a ^ j a n d b ^ j a r e d e f i n e d a s i j a ij InlKot^r.j) - 4 b. • = + KoUrl^JdS •) + . K o U r ! -)]dS S^oivr. (2.72) (2.73) r . . and r ! . a r e g i v e n a s 3 3 1 1 y r-j = [(Xj-x.) rlj = [(Xj-x^ + ( 2 -z.) ] 2 2 j l / + (z +2d+z.) ] 2 2 angles (2.75) o f e a c h segment. The 9G/9n may be e x p r e s s e d a s !§<£i?£j> where l / 2 j x^ and X j a r e e v a l u a t e d a t t h e m i d p o i n t gradient (2.74) 2 - H c o s 7 + If' COST' 7 a n d 7' a r e a s shown i n F i g . 4 a n d c o r r e s p o n d between respectively, (2.76) t o the n . a n d r=x.-x., and between n'. and r'=x'.-x. -3 3 - i ' -D -3 - 1 that i s 32 COST n . • ( x .-x. ) ^ — ~ 3 ~* = COST' = =3—7 1 - (2.77) (2.78) 1 where n. = n i + n k ~1 ~ ~ n'. = n i - n k —3 x— z— X X j i s the point and is (2.79) 2 (xj(zj+2d)). The u n i t normal v e c t o r n g i v e n by n = || i - || k (2.80a) The above e x p r e s s i o n c a n be a p p r o x i m a t e d a s Az • _ 2S i The d e r i v a t i v e Ax , 715" - (2.80b) o f t h e G r e e n ' s f u n c t i o n i s g i v e n by IfKodr) = - J » K , (vr) (2.81 ) where K , i s t h e m o d i f i e d B e s s e l i*j, the integrals in a p p r o x i m a t e d by e v a l u a t i n g normal derivative coefficient at K,Ur..) 1 K, v the (2.72) Green's the midpoint of and When (2.73) are function and i t s e a c h s e g m e n t . The (x .-x. )n„ + ( z . - z . ) n l A S . 3 i x ] i ' z' 3 Ur! .) Trr!. i] Substituting given eqns. one. a ^ j i s thus given as a. 3 . = - v- Trr.j - f u n c t i o n of order [ (x .-x. )n + ( z . + 2d+z.)n ]AS . D i x 3 1 2 3 the approximation i n eqn. (2.80b), a ^ j becomes i * j (2.82a) f o r the d i r e c t i o n cosines 33 = -v a.. *D where v (T (x .-x. ) A z . D D Az. = D z ... D 1 z. D A = x x. = [(A X j + b i : j i=j, singular. - 1 Z 0 + 2 b.^ the 7 given as j ) 2 ] i s given 0=0°, l / as before » as i n egns. (2.72) -{lnUr/2) ASi>AS. = - y l l l n - ^ - i about coefficients + components 0 f o r 0=0° and formula f o r as the diagonal vr^O + 7 ~ 1 - K {2i»(z.+d)}] (2.86) 0 function are given are (2.85) to the t y p i c a l seabed (2.84) coefficients 1 reduces Green's the become K ('i>r') the asymptotic 7} 1 the problem the (2.73) K,[2v(z.+d)]Ax. b Using and (2.83) and Stegun,l964) = £ H i * j 0 and using a.. 7T (2.82b) 2 + K (»rJj)]ASj i s Euler's constant, 1 1 symmetry X integrals 0 where ( A = "^[Kod^r.j) K (*>r) one. ) (see Abramowitz Q For j i#j 1 Evaluating the nonsingular 9K (j>r' )/3n K {vr) 1 ( z - + 2 d + z . ) A x .] D D 1 + j (z:-z.)Ax.] D D I • ) coefficient When [ ( x .-x . ) A z . D I D 7rr!^j ASj The .) ,rr. . ID 1 K - K,(vr. given taken by two i n eqn. into dimensional (2.27) account, with the 34 1 i j [ (XyX.)Az. b j - (2j+2d+z )AXj] i = l ( l n r . j + In r l ^ A S j t i * j (2.87) i * j (2.88) For i = j Ax. a ii b.. With the provides AS. AS. = — i t l n - T i - 1 + In 2(z.+d)] N equations The a^j and b ^ j now relating various the remaining the values e q n . (2.15) into N equations e q n . (2.70) 2 N 1 Z ( . . + 6. •+^r -b. . ) * • + a a N 3 Z ( a j=N2+1 1 of < $ > and 9</>/9n o v e r needed 2« to solve f o r 4> and boundary condition's given yields N2 ,,x S ( a - . + S.-U- + N 4 K i i 3 (a. j=N4+1 known, e q n . (2.70) (k) (k) +6..+2gab .)^ ' + I ( a . . + 6 ,mcos0b.. >^ j=N3+1 W J , u i 3 N I (2.90) b o u n d a r y c o n d i t i o n s a r o u n d S +S +S_ d<j>/dn. S u b s t i t u t i o n o f t h e v a r i o u s in .+.6. J 3 i 3 3 .+ikcos0b. ,)0-(k)= ; J i s d e f i n e d as i 1 N J 3 (k) b..t\ 2 - Z j=N1+T J for (k) where f . '89) ( 2 coefficients S +S+S_. provide 2*U*+d) = K> J J i=1,....,N; k=1,2,3,4 (2.91) 35 k=1,2,3 f (k) (2.92) k=4 The expressions are given on the right-hand s i d e of the above equation as a/4) TTn^ = ik c o s h [ k ( z .+d) ] osh(kd) exp( ikcos/3x .) cos/3 sinh[k(z .+d)] Ax. exp(ikcos/3x . ) — . 3 AS coshUd) k Az. C (2.93) and AZj/ASj k=1 -AXj/ASj k=2 Az (z . - e ) — : 3 AS Fig. 5 shows constants a N1, typical N2, N3 Ax . x.—1 3 AS . D + k=3 discretized and N4 (2.94) boundary shown. Eqn. with the (2.91) y i e l d s N (k) equations solved f o r N unknown <f>\ using a matrix unknown v e l o c i t y forces, (k=1,2,3,4) v a l u e s inversion p o t e n t i a l s on added mass and damping and transmission coefficients the expressions 2.8 E F F E C T OF Let structure is given i n the boundary. coefficients, can now preceding be be to obtain the The and exciting reflection determined using sections. F I N I T E STRUCTURE LENGTH us now of the technique w h i c h can consider finite assumed t o be the l e n g t h , 1. much g r e a t e r f o r c e s and The than response length of the the incident of a rigid structure wavelength. 36 The force per u n i t 2 by e q n . (2.32) a s ^]ll' \ 2 1 j = 1,2,3 exciting corresponds forces structure along i s given = pq%{l }Re[C.(to,ti)exp{i( ,y-tot)}] Fj(y,t) where length is (or to the sway, heave and moment). The total force on the by i n t e g r a t i n g t w o - d i m e n s i o n a l force obtained i t s length, (2.95) roll i g n o r i n g end e f f e c t s 1/2 F.(t) = / F.(y,t)dy -1/2 3 Substitution of eqn. (2.96a) 3 (2.95) i n t o e q n . (2.96a) y i e l d s , , 2sin(- -isin^) F.(t) = pg§l\l C?e x p ( - i u t ) \ ]Z' J ^ -1 klsin/J 13--* ?\ \ J k p a 2 a for p*0°. product The above e x p r e s s i o n of structure, the force per can be unit length, and a f a c t o r r ( k l , / 3 ) d e f i n e d 2sin(^isin/3) ^ the load structure unit f o r a given obliqueness angle of r The influences zeros at given of against l e n g t h c a n be s e e n . The behavior large by as a reduction l e n g t h due t o t h e f i n i t e 6 shows a p l o t separate the l e n g t h of the t o be approach, of t h e waves f o r a g i v e n 2 the 0=0° r ( k l , 0 ) can be c o n s i d e r e d per as (2.97) 1 factor of (1*0° klsin/3 r(kl,/3) = The thought (2.96b) due to the 1 5 ° , 30° and 60°. k l and /3 on t h e l o a d p e r u n i t f a c t o r r(kl,/3) values l e n g t h of the structure length. F i g . k l f o r /3=0°, of or of of k l w i t h has an an oscillatory infinite number o f 37 ^sinj3 For an length = n it infinite n=1,2,... span structure, (2.98) the t o t a l load t e n d s t o z e r o . The maximum t o t a l load per unit occurs on a span of l e n g t h 1 = L/2sin/3 where L=27r/k Fj(t) The (2.99) i s t h e w a v e l e n g t h . T h i s maximum f o r c e i s = P g j ^ C j U ^ T f g motions of a r i g i d s e a s c a n be d e s c r i b e d addition seas, to the cylinder can also to and motions about rotational section added with motion along damping sway, structure heave are coefficients forces motions the (2.100) i n oblique and yaw damping variation of a r i g i d obtained roll by respectively. of of multiplying c a n be o b t a i n e d coefficients from a flexible a finite the The strip and to the coefficients f o r the p i t c h the sectional heave and sway m o t i o n s u s i n g a correspond with the length of the s t r u c t u r e . and h y d r o d y n a m i c derived in The added mass cylinder motions axis of the amplitude of ( 0 = 0 ° ) . The h y d r o d y n a m i c and pitch the y coefficient to the o s c i l l a t i o n s i n beam and motion along the l e n g t h of t h e c y l i n d e r . seas length t h e z and x axes sinusoidal coefficients c a s e o f beam the mass a 2 modes p r e s e n t surge, the t r a n s l a t i o n a l 2.4 c o r r e s p o n d s cylinder of f i n i t e t h e sway, heave a n d r o l l cylinder {]ll' } exp(-icot) i n t e r m s o f s i x d e g r e e s o f f r e e d o m . In corresponding The ^ 2 for length sectional exciting and yaw coefficients for theory approach 38 described exciting in Bhattacharyya exciting moment (1978). coefficients The yaw and pitch are given as 1/2 F where j + 3 (t) j=4,5 = J corresponds respectively. Substitution two-dimensional forces Fj ( t ) + 3 where q ( k l , 0 ) (2.101) to the of (2.95) i n t o yaw and expression e q n . (2.101) pg5al Cj-q(kl,/3)exp(-icjt) 2 pitch modes for the yields j = 1,2 (2.102) i s d e f i n e d as 2i (klsin/3) = j=1,2 . yF.(y,t)dy -1/2 2 [^sinj3cos(^sin/3) sin(^sin/3)] 0*0' q = (2.103) 0=0' 3. EFFECTS OF DIRECTIONAL 3.1 REPRESENTATION Before proceeding structures and to in directional mathematical The OF DIRECTIONAL seas, representation preceding response chapter of a WAVES SEAS determine the we first shall of d i r e c t i o n a l dealt with structure response the exciting subject pattern i s h i g h l y complex and i r r e g u l a r . surface i s often modelled long-crested point waves o f a l l p o s s i b l e from a l l d i r e c t i o n s . assumed to Gaussian process. The implies symmetry about realistic A relative however by a l i n e a r be a f o r small zero The mean, the s t i l l amplitude long-crested wave surface of water a T h i s complex a elevation ergodic Gaussian level train travelling relation A random sum of l i n e a r random process which at be r e p r e s e n t e d where A i s t h e complex wave a m p l i t u d e w i t h dispersion is i s only waves. t o t h e p o s i t i v e x a x i s may t h e wave number of approaching a angle 0 by r ? ( x , y , t ) = Re [A exp{ i (kxcos0+kysin/3-cot)} ] is wave superposition stationary, assumption regular exhibit frequencies sea a forces to waves. Ocean waves sea present seas. unidirectional which of (3.1) a random p h a s e , k r e l a t e d t o t h e f r e q u e n c y co by t h e l i n e a r (eqn. 2.11). sea surface c a n be c o n s i d e r e d waves o f d i f f e r e n t frequencies t o be a d i s c r e t e and d i r e c t i o n s 77 = R e [ Z Z A ^ e x p { i ( k ^ c o s / 3 j + k ^ s i n / 3 ^ - c o ^ ) } ] i 39 (3.2) 40 where k^ d e n o t e s travelling in amplitude. infinity be direction while frequencies number of h a r m o n i c s the d i f f e r e n c e between a d j a c e n t t e n d s t o z e r o , t h e summation replaced component /3j, co^ i t s f r e q u e n c y a n d A ^ j I f we l e t t h e t o t a l and d i r e c t i o n s can t h e wave number o f t h e i - t h wave by an i n t e g r a l in tend to frequencies eqn. (3.2) o v e r a c o n t i n u o u s r a n g e of and d i r e c t i o n s r?(x,y,t) = R e [ J/exp{ i (kxcos/3+kysin/3-a>t) }dA(co, 0) ] where dA r e p r e s e n t s t h e d i f f e r e n t i a l two-dimensional (/3,/3+d|3). (co,/3) The elevation its mean space square wave a m p l i t u d e bounded value of in ico,co+dco) by the (3.3) water the and surface i s g i v e n by T7 7 = i//dA(w,/3)dA*(u,/5) = U S(co,/3)dcod/J (3.4) -TTO where dA*(co,/3) is the S(co,/3) i s a d i r e c t i o n a l energy d e n s i t y complex wave conjugate spectrum. o f dA(to,/3) and Since i n t h e waves i s p r o p o r t i o n a l the average t o t h e s q u a r e of t h e wave a m p l i t u d e , t h e p r o d u c t S ( C J , j3)dcod/3 c a n be c o n s i d e r e d to be t h e c o n t r i b u t i o n to the t o t a l mean e n e r g y t o waves w i t h f r e q u e n c i e s between co and co+dco, directions between directional wave s p e c t r u m The and one-dimensional integrating directions /3 the /3 dj3. + A d e n s i t y due travelling sketch of a in typical i s shown i n F i g . 7. spectrum, directional wave S(co) c a n be o b t a i n e d by spectrum over a l l 41 7T S(w) The = / S(cj,/3)d0 one-dimensional wave s p e c t r u m measurements o f t h e in space; buoy. free In order complicated evaluating to techniques. directional analysis of the analysis of elevation, floating elevation at a s i n g l e point water slope Smith Sand of curvature measurements Holthujsen (1981)). is convenient spectrum often i n terms o f an e n e r g y the o n e - d i m e n s i o n a l S(u,0) = to methods for the from of Regier and the (1980)). (1964), M i t s u y a s u stereophotographs more observation point water surface the motions Longuet-Higgins f r o m an a r r a y o f guages of the to elevation a t an elevation means resort common surface ( 1 9 7 1 ) , D a v i s and about include of by the to most measurements (e.g. and has velocities and of a h e a v i n g information one The the motions analysis Panicker It from e t a l (1978), the buoy Cartwright 4. determined wave s p e c t r a orbital (e.g. F o r r i s t a l l 3. be obtain o f t h e waves, horizontal 2. surface can f o r i n s t a n c e by r e c o r d i n g directionality 1. (3.5) et et the al al of a (1961), (1975)). water surface ( e . g . Borgman (1969), (1977)). (e.g. Cot6 express et al (1960), the d i r e c t i o n a l spreading function wave applied to spectrum S(u)G(u,0) where G(co,|3) i s a d i r e c t i o n a l (3.6) spreading function. 42 It follows from eqn. (3.5) that G(w,0) must satisfy 7T / G(u,0)d0 = 1 (3.7) -it Various one-dimensional frequency describe ocean most commonly the waves. The Bretschneider, These spectra Isaacson also been described and several various authors. outlined below 1. C o s i n e - s q u a r e d St. D e n i s and which 2. spectrum A C(e) that eqn. used and detail used ones include JONSWAP spectra. and a r e not g i v e n h e r e . T h e r e have for the in G(CJ,0) commonly proposed by used ones a r e spreading function formulation (1953) p r o p o s e d a of f r e q u e n c y f o r I 0 I < TT/2 ( 0 i s centred (3.81 otherwise about 0=0°. formulation et al (1961) proposed the following spreading function = C(s) c o s where 8 i s m e a s u r e d propagation. have been Sarpkaya of ( | cos 2 0 =I * Longuet-Higgins directional hence few Pierson Cosine-power in formulations i s independent G(0) The Pierson-Moskowitz are (1981) spectra C(s) (3.7) 2 s (0) from t h e (3.9) principal direction i s a normalizing coefficient i s s a t i s f i e d and i s g i v e n by that of wave ensures 43 C(s) r is the spreading that s = ] 2/TT r ( s + ^ ) r gamma On the + (3.10) ) Fig. 8 fordifferent describes with s function. function direction ( the degree basis of of s p r e a d about their measurements et al (1975) f o u n d depend on t h e d i m e n s i o n l e s s the p r i n c i p a l 2 5 / s to (3.11) m v , 5 b f o r T-<I frequency = d i m e n s i o n l e s s modal U = wind speed the parameter f o r I>I 5 b where T = d i m e n s i o n l e s s f o r wind d r i v e n frequency ( 0.116(T)~ * s = { _ _ ( 0.116(1) ( T J m m v a l u e s o f s . I t c a n be seen s — r e p r e s e n t i n g l o n g - c r e s t e d waves. o c e a n waves, M i t s u y a s u T shows t h e d i r e c t i o n a l m m = Uf/g frequency a t 19.5m above = Uf^/g sea l e v e l Hasselmann et al (1980) on t h e b a s i s o f the from Joint (JONSWAP) f o u n d t h e the parameter proposed a different (1969) formula used on f / f m rather than obtained I and f o r s. an alternative cosine power g i v e n as G(6) The Sea Wave P r o j e c t s t o depend m a i n l y Borgman function North data •ir (s) cos 2 s (0) f o r |0|<7r/2 ={ (3.12) otherwise normalizing coefficient C(s) = -1 H r(s+ l> r(s+£) C'(s) i s given as (3.13) 44 3. SWOP f o r m u l a t i o n Cote et al (1960) p r o p o s e d which i s dependent data obtained a directional on b o t h from frequency the Stereo spreading and d i r e c t i o n Wave function based Observation on Project (SWOP). 1 + acos20 + bcos40] for|0|<7r/2 (3.14) 0 otherwise where a = 0.50 + 0.82exp(-^u*) b = 0.32exp(-2^ ) a co = n o n d i m e n s i o n a l frequency = Uco/g 3.2 RESPONSE TO DIRECTIONAL WAVES The length and exciting due to direction force on a regular a rigid o b l i q u e wave t r a i n finite o f f r e q u e n c y co = Hj(a>,0)7?(t) (3.15) where Hj(co,j3) i s a c o m p l e x - v a l u e d system response function by e q n . (2.96b) a s Hj(co,/3) Since the linear, = frequency to spectrum interaction t h e v a l u e o f any f o r c e be due to but p r o p a g a t i n g spectrum {ill' } wave S (co,0) by process at components from a l l p o s s i b l e S„ (co) i s t h u s related * 2 pql{l^Cj(u,fi)r(kl,p) wave-structure we e x p e c t frequency force of 0 c a n be e x p r e s s e d a s Fj(t) given structure a at (3 16) i s assumed given wave that same directions. The to the incident wave 45 S where the 2 jH^ (co, j3) | subscript noted that i s the t r a n s f e r 2 j will water Gaussian process, function. For convenience, h e n c e f o r t h be d r o p p e d and i t s h o u l d a l l following S i n c e the (3.17) (u,0)d0 (co) = / |H.(co,/3) | S p expressions are v a l i d surface elevation the forces is will be f o r j = 1,2,3. assumed possess to a be a Gaussian probability distribution. Using in the eqn. (3.6), form o f t h e d i r e c t i o n a l wave s p e c t r u m given e q n . (3.17) r e d u c e s t o (3.18) The factor i n the brackets directionally the = 0- 0 root the usually The can of force the of d i s measured from and i s t h u s r e l a t e d t o 0 by force spectrum (rms) from of can be obtained by over t h e f r e q u e n c y co. The the force represents w h i c h extreme v a l u e a predictions made. effects of wave d i r e c t i o n a l i t y be e x p r e s s e d a s a f o r c e ratio function, dependent, (3.19) force mean s q u a r e v a l u e characteristic O a frequency O mean s q u a r e v a l u e integrating are transfer p r i n c i p a l wave d i r e c t i o n 0 6 The averaged represents the frequency transfer function in function for long-crested, reduction on t h e wave factor defined as loads the dependent, d i r e c t i o n a l l y averaged short-crested normally seas to the i n c i d e n t waves transfer 46 7T J R = — 2 the rms G(co, 0)d0 2 (3.20) 2 r a t i o R^^ c a n value corresponding | |H(co,0)| F A body r e s p o n s e 0) |H(CJ, of the results a l s o d e f i n e d as response the ratio of i n s h o r t - c r e s t e d seas to for long-crested seas, that i s / /|Z(co,0) | G ( w , e ) S ( c o ) d 0 d c o 2 r? R^ = °~* (3.21) 7| Z(w, 0) | S 0 (u)dw 2 where Z ( t o , 0 ) previously The is i n eqn. first at any 0 angle function Substitution R is = 2 in = C'(s) the case distribution approach |H(o>,0)| transfer cos 0 (3.22) 2 2 (3.12) is for directional used the in / 2 / cos 0 cos 2 spreading this transfer f u n c t i o n (3.12) i n t o eqn. * The force as cosine-power eqn. an sinusoidal horizontal cos0. to on study. function (3.20) y i e l d s 2s (0)d0 (3.23) s -TT/2 F For defined force s t r u c t u r e w i t h the of the e x p r e s s i o n s spreading 2 a i s t h e wave proportional independent given (3.22) and of t h u s be e x p r e s s e d frequency operator t h e l e n g t h n e g l e c t e d . The |H(w,0)| The amplitude (2.56). segment along f u n c t i o n can response example c o n s i d e r e d infinitesimal variation the n the of will oblique be cut mean i n c i d e n c e , t h e off s t r u c t u r e from one to ensure directional that s i d e o n l y . I f the the waves principal 47 direction of wave integrated to give propagation i s zero, e q n . (3.23) c a n be C (s) (3.24) C(s+1 ) For any length, given the (3.20) w i l l force the structure dependence arbitrary shape on 0 i s no l o n g e r reduction factor. transfer function / to finite and e q n . give the S u b s t i t u t i o n of t h e e x p r e s s i o n f o r (3.16) i n t o e q n . (3.20) 2 J and explicit have t o be i n t e g r a t e d n u m e r i c a l l y * yields 2s |C .(w,0) 1 r ( k l , / 3 ) c o s ( 0 ) d 0 -TT/2 2 the cosine-power 2 z s 3 C (s) for of (3.25) ICj^O)!2 type spreading function. 4. RESULTS AND DISCUSSION 4 . 1 EXCITING FORCES, ADDED MASS AND DAMPING COEFFICIENTS A computer the p r e c e d i n g forces, compare the other The cylinder water of coefficients accuracy solution first the element case 0.4. results of Bai's and e f f i c i e n c y surface, The angle f o r c e and by a rectangular section reflection 1, (1975) u s i n g a are finite plotted as results solid a under t h e M i c h i g a n for the exciting and angle a force coefficients matrix System of dimension an 88 mesh w i t h a CPU t i m e procedure 48 i s thus central (MTS) t o f o r a given of i n c i d e n c e . B a i (1975) u s e d free s u r f a c e and 16 node 3.0s on t h e Amdahl V 8 - I I Terminal and a r e shown a s p o i n t s . s u r f a c e h a d 40 node p o i n t s on t h e computer. The p r e s e n t in coefficients a r e r e p r e s e n t e d by t h e approximately element of o f i n c i d e n c e 0 f o r ka=0.1, 0.2 and processor 370 method 20 node p o i n t s on t h e r a d i a t i o n finite i n order to 9-12 show a c o m p a r i s o n Bai p o i n t s on t h e body s u r f a c e y i e l d i n g node cases and of t h e p r e s e n t coefficients the present discretized N=76. I t took exciting reflection (b/a) r a t i o (d/a=2). F i g s . obtained while is t o half-beam (1975) r e s u l t s dashed curves The the the and for several test considered depth technique. function determine described in techniques. with a d r a f t of f i n i t e to coefficients, t h e computed e x c i t i n g with on t h e p r o c e d u r e s e c t i o n s was u s e d hydrodynamic transmission with program based solve wavenumber element, 325 o f 12s on an IBM relatively quite 49 efficient. The and computed the Bai's was sway a n d heave e x c i t i n g reflection coefficient (1975) r e s u l t s . consistently The r o l l greater w i t h a maximum d i f f e r e n c e larger set present results. the From F i g s . coefficients vanishing 0=0°. The d i f f e r e n c e constant 0=90°, while coefficients cos/3 with exciting up t o c e r t a i n for slightly angle ka=0.1. The presented coefficient by B a i (1975) a much that with the e x c i t i n g increasing force force or moment occurs was before decreasing exciting to reflection be to force of at fairly zero at ( o r moment) proportional coefficient angle force a n g l e of i n c i d e n c e coefficient 0 seemed increasing to diminish to decreases incidence before t o one a t 0 = 9 0 ° . exciting rectangular f o r c e and h y d r o d y n a m i c cylinder with infinite d e p t h were a l s o results of Garrison used a Green's radiation moment with computations. sway a n d r o l l at, any a n g l e with increasing The the closely 7.5%. The u s e o f 9-11 i t c a n be s e e n heave quite i s expected mesh i n B a i ' s decrease coefficients d i d n o t s i g n i f i c a n t l y change t h e a t 0 = 9 0 ° . The maximum The that of about o f node p o i n t s use of a f i n e r agree exciting than force computed o f 0.265a and conditions of the c y l i n d e r function i n the present compared and surface procedure the free thus only. of a i n water of with 13-21. G a r r i s o n which s a t i s f i e s discretization used draft (1984) i n F i g s . function boundary a coefficients the (1984) s u r f a c e and requires The i s relatively the Green's simple 50 while the G r e e n ' s c o m p l e x and water the i s only v a l i d depth present d=7r/k+b, procedure discretized 40 the body function of incidence results. A where k i s t h e wavenumber i s used in radiation 30° The and infinite the s u r f a c e and coefficients parameter 6 0 ° . The quite water depth. node p o i n t s on frequency agree The 40 of well a d d e d mass and Garrison's results results however relations. location The of empirically. variations the present The slightly distance of potential with the c r e s t force decreasing should The of force (1984) sway 15% t o 25%. with added while Garrison's the was with improve Haskind to the estimated higher the the from sensitive which elements at coefficients with increasing moment o c c u r s intuition moment t o o c c u r and were exciting maximum r o l l agrees results the a order accuracy of method. tendencies, The of up better use angles substantially much radiation as exciting as much a s deviated with d i f f e r e n c e s The of d e v i a t e d by agreed the plotted damping c o e f f i c i e n t s g e n e r a l l y 60° coefficients points Garrision's mass c o e f f i c i e n t damping are for with G a r r i s o n ' s r e s u l t s . roll free surface, node computed with The 16 ka show good a g r e e m e n t at (1984) i s q u i t e depth. surface. 0=0°, coefficients f o r water to simulate the the by G a r r i s o n infinite s u r f a c e had node p o i n t s on on f u n c t i o n used when t h e the a t about show the angle of i n c i d e n c e . ka = ?r/4. T h i s since one trough of a wave i s a t s i d e s of the would e x p e c t cylinder. expected result the maximum the origin 51 The added mass c o e f f i c i e n t s t h e damping c o e f f i c i e n t incidence for most of damping c o e f f i c i e n t s crests energy are decreased the tended with normal t o the i s propagated increase, while increasing frequency should vanish at to range from 0=90° s i n c e the of studied. a x i s of t h e c y l i n d e r away angle the and cylinder The wave hence no the ±x in directions. The exciting coefficients circular and and and results were paper. The angles of modelled with Garrison are incidence with on segments on 40 ratios of infinite the results (1984) estimated results coefficients, amplitude i n water compared (1973), points wave cylinder are force from hydrodynamic of a depth semi-immersed were of B o l t o n and 1-4. Tables the figures presented 35° node p o i n t s on the radiation the s u r f a c e of Ursell in shown f o r ka=0.25, 0.75 0=0°, computed and and 5 5 ° . The the free surface and the c y l i n d e r . Garrison's 1.25 with boundary surface, 16 in his was 40 node straight line Agreement between the different methods i s g e n e r a l l y good w i t h d i f f e r e n c e s less than It is 15%. amplitude ratios indicates that decreases, cylinder the interesting increase as the waves with to angle wavelength generated become more a m p l i f i e d . note by of that the of wave incidence. This along the the motions cylinder of the 52 4.2 MOTIONS OF The AN UNRESTRAINED BODY equations of a m p l i t u d e s of motion l=75m) rigid in and water the d i s p l a c e d be at the s t i l l g i v e n as Figs. depth a i s assumed t o be for i s pV where V volume. The c e n t r e of g r a v i t y i s assumed t o water and level the r o l l modes respectively. function r a d i u s of at the ka—>0 and for beam free decreasing, surface. d e c r e a s e s as seas increases ka The gyration 30° and heave and with f o r t h e sway, amplitudes vertical up to amplitudes ka. There low have the motions of a sway a m p l i t u d e ka are 6 0 ° . At motions i n c r e a s e s . The w h i l e the r e s p o n s e decrease with i n c r e a s i n g The of ka f o r 0 = 0 ° , (ka:$0.1), t h e sway and as factor b=3m, damping c o e f f i c i e n t s a m p l i t u d e s as t h e h o r i z o n t a l response box the mass of t h e box roll as particle is heave maximum amplitude maximum before f o r 0=30° and are l o c a l zeros 60° of the f o r o b l i q u e waves c o r r e s p o n d i n g t o t h e z e r o s of r(kl,j3). The excessively high. present practical in computations. observed and d=12m. The give (a=7.5m, 22-24 show t h e a m p l i t u d e s o f m o t i o n frequencies same of box to 19.5m. and plotted solved of a l o n g f l o a t i n g (/3=0°) a r e u s e d . The is heave were hence t h e added mass and beam s e a s is motion roll modes was that In roll This amplitude i s because situations solving the t h e heave r e s p o n s e resonance is v i s c o u s damping w h i c h was neglected in e q u a t i o n s of motion, i s uncoupled responses while coupling weak e x c e p t c l o s e at the between t o the r o l l the i t was from t h e t h e sway and resonance is sway roll frequency 53 where there is a EFFECTS 4.3 OF There of are wave sinusoidal The that factors compared (2) and angle of of the The square plotted as a 6. For r(kl,/3) length the in a the with ka. i f we The of the ignore parameter discussed previously of r(kl,0) incidence with ka in the results in the given in the reduction kl for length of finite of i f there of and is per section 4.1. of The the It in factor per also unit of results length as sectional ka forces forces with the incidence /3 h a s been combination of sectional force v a r i a t i o n with the force reduction total is 60° variation unit the the factor and loads no sectional angle 30° wave along r(kl,/3) length, incidence. loads the 15°, the of cross-section. factor /3=0°, the two-dimensional a v a r i a t i o n of v a r i a t i o n of frequency factor the along in (1) seas: reduction angle wave structures results even with a reduction force reduction waves force decrease increases the forces for the two-dimensional the of to long v a r i a t i o n of the amplitude. long-crested wave structure in oblique sectional of of given by incidence function results for the structure r(kl,/3). Fig. to the sway contribute experienced integration length the two v a r i a t i o n of with in WAVES seas structure, forces drop DIRECTIONAL loads short-crested the sudden the angle factor R_. F The frequency been computed 4.2. The for computed the R„ dependent long values force floating for the reduction box factor described cosine power in type Rp, has section energy 54 spreading function 25(a)-(c) i s plotted f o r t h e sway, heave respectively. The assess influence the computations. numerical and A of integration in was eqn. ka used (3.25) = o moment) i n order degree /3 in Figs. (or f o r s=1,3,6 direction rule of forces the principal Simpson's function roll r e s u l t s a r e shown short-crestedness. the as a of 0° with wave was to carry an to used i n out the interval of 10°. At low frequencies, approaches a that heave for the limiting low v a l u e s significant factor thus makes The that for frequencies. This R| At higher (or is = slightly 0.866 force as s the than This along due the be the forces Battjes of is a t o the length spreading seen from approach the derived an (1982) form /3 Al as 0 t h e sway, factors R p at reduction result was kl^=° heave a l l converge the asymptotic ( o r moment) ka—>0. 2 s there high as (ka>1), reduction less seas. fact of d i r e c t i o n mostly I t can a l s o factor the for directional asymptotic 27rC(s)cos force variation increases, i s given confirms (or k l ) increases, to account as reduction i s independent structures. frequencies moment) roll ka for long-crested expression force sinusoidal f o r long force of one. T h i s of t h e heave i t important 25(a)-(c) results exciting reduction particularly Figs. value of ka. As r(kl,/3). the heave roll force to a value which value. factors expected (4.1) and The approach since sway a value and of the sectional 55 sway and cos/3 roll at factors low The of 0.4 at sway and computed shown for factor has force low in Figs. mean of 0.79 ka=0 /3 = 30° slight that for reduction the of waves a p p r o a c h t h e factor the for The for Bretschneider force was to mean waves ratio for case of 1.0 the = T^f 1 6 f where H s i s the f r e q u e n c y . The Figs. ° , V 5 from the that the As ka only. force fact reduction mean power 4 0 the f box energy wave h e i g h t sway, a heave been subject to a spreading. i s given as (4.2) ° p l o t t e d as waves has _ exp[-|(4 )" ] significant for The o 4 (f/f ) r e s u l t s are 27(a)-(c) f o r /3 = 0 ° . motions i n c i d e n t u n i d i r e c t i o n a l wave s p e c t r u m S ( w ) at 0 floating spectrum with c o s i n e 0.866 f o r cos/3 . body the reduction to normal of for factor side and ratio included force heave and reduction o f f to ensure one also o The arises cut were reduction compared from asymptotic the up o compared d i f f e r e n c e between the response computed The structure t o an ka (/3 = 3 0 ) are sway heave f o r c e function oblique i n c r e a s e s , up the heave o the spreading increases, incidence o The 0.985 26(a)-(b). f o r /3 = 30° n o r m a l mean i n c i d e n c e . was reduction factors mean i n c i d e n c e frequencies, a value force increasing reduction oblique normal c o m p a r i s o n . At The to ka=2. c a s e of r e s u l t s are (ka<0.1). modes d e c r e a s e w i t h heave f o r one factors (or moment) i s p r o p o r t i o n a l frequencies for a l l three to a value the e x c i t i n g force and f 0 i s the function and roll of peak s in responses 56 respectively assuming wave h e i g h t H =2m the and computations. f r e q u e n c i e s between of 10° were are 0.22m, 0.32m and normal a peak In u s e d . The 43%, and roll responses compared ratios numerical integration, 0.26Hz and an interval in long-crested heave and v a l u e of t h e sway, heave in short-crested s increases, v a l u e of one of the s t r u c t u r e the l o n g - c r e s t e d angle five seas roll 2 7 ( a ) - ( c ) show r e d u c t i o n s of s e a s . As a limiting o f o r t h e sway, Figs. significant f = 0 . 2 H z were used i n amplitudes respectively a m p l i t u d e s of motion approach rms i n t h e rms to long-crested approach the 0.60rad respectively. 41.5% frequency 0.14Hz and responses 42.5% mean i n c i d e n c e . A results seas with the indicating s=1 response that in short-crested as s—>-<=°. and the seas 5. CONCLUSIONS AND RECOMMENDATIONS 5.1 CONCLUSIONS The effects m o t i o n s of long of wave d i r e c t i o n a l i t y s t r u c t u r e s has A numerical developed to coefficients oblique been method b a s e d compute the train with Green's f o r c e s and the interaction an infinitely arbitrary quite a p p l i e d to cases and can be loads theorem semi-immersed c y l i n d e r of general the and studied. exciting associated with wave on on has hydrodynamic of a long, shape. regular floating The of been method is v a r i a b l e water depth. Numerical results been compared w i t h finite finite compared to and a cylinder The those and present wide r a n g e of frequencies required have using a also been water d e p t h method w e l l as Garrison for method i s q u i t e e f f i c i e n t favorably with high have section cylinder in multipole procedure w h i c h compare very a method (1975) infinite c y l i n d e r as function Bai results for (1973) u s i n g present a by for a (1984) rectangular a semi-immersed c i r c u l a r c y l i n d e r . present conditions. by rectangular obtained circular Green's from t h e obtained d e p t h . The Ursell semi-immersed using those e l e m e n t method f o r a w a t e r of Bolton obtained The present frequencies due to give accurate a l l the covering t o the usual i s not gives results r a n g e of as The present results over node a design efficient l a r g e number of results. 57 previous the procedure and for points procedure is 58 however n o t v a l i d the body typical axis finite to obtain length. structure linear The transfer the wave function is averaged and ratio a response wave For power the given cosine-squared of The heave The structures of factor of rigid using the of wave dependent, f o r t h e wave motions. function. a effects frequency numerically The loads reduction f o r the cosine-power Response ratios i n c i d e n t wave the sway and were spectrum roll from 0.87 a t ka=0 t o 0.41 distribution reduction factor t o 0.40 a t ka=2. The r a t i o the s t r u c t u r e a body structure, for ka=0 on along spreading. factors varied incidence. as Bretschneider reduction a integrated motions approach. f o r the a and reduction type d i r e c t i o n a l spreading for along o f magnitude as a s e a s have been o b t a i n e d f a c t o r s have been e v a l u a t e d computed loads loads expressed directionally with cosine wavelength r e s u l t s have been in short-crested directionality also the dimension. two-dimensional body a x i s seas s i n c e becomes o f t h e same o r d e r cross-sectional The the f o r head with varied force at ka=2 normal mean from of the amplitudes f o r the s p e c i f i e d s h o r t - c r e s t e d 1.0 at of motion sea state w i t h a c o s i n e - s q u a r e d d i s t r i b u t i o n were 57%, 57.5% and 58.5% of the response roll and modes for long seas, f o r t h e sway, heave and respectively. A further amplitudes {($0*0°). in long-crested of motions These r e d u c t i o n s relative reduction i s obtained are quite structure of the forces for oblique significant lengths and mean waves particularly need to be 59 considered As i n the the design process. parameter short-crestedness short-crested s which describes increases, the loads approach the results seas the and degree motions for of in long-crested seas. 5.2 RECOMMENDATIONS There are be made to numerical FOR several areas improve the scheme u s e d diffraction elements. FURTHER i n which present i n the problem c o u l d This STUDY however further studies method. The s o l u t i o n of be improved requires an accuracy the by could using of the oblique wave higher order increased computing effort. The present directionality on study the considered loads and though h y d r o d y n a m i c coefficients structures sinusoidal with procedure could response in of Additional be train. considered. nonlinearity mode to using been shapes. a t o m o o r i n g s and the body even A for numerical the dynamic floating exciting by wave presented determine the given rigid of bridge forces present viscous and method. damping could i n the a n a l y s i s . present For have s t r u c t u r e such as coefficients f o r c e s due included The seas effects m o t i o n s of a developed a flexible short-crested hydrodynamic be the steep method waves, Developing and assumes a nonlinear a theory that directionality of small amplitude effects have incorporates the waves wave to both be the i s however 60 quite difficult. oblique waves c o u l d hybrid method used t o i n c l u d e Finally, out The p r e s e n t such be e x t e n d e d as t h a t the e f f e c t s results. seas to to d i f f r a c t i o n theory f o r nonlinear proposed o f wave experimental t o measure t h e l o a d s short-crested linear by Dean a (1977) c a n be investigations could verify and directionality. a n d r e s p o n s e of l o n g help waves be carried structures the p r e s e n t in theoretical BIBLIOGRAPHY 1. Abramowitz, Mathematical M. and Functions. Stegun, I.A. 1964. Handbook of Dover P u b l i c a t i o n s , New York. 2. B a i , K . J . 1972. A v a r i a t i o n a l method i n p o t e n t i a l flows with a free surface. Report No. NA72-2, C o l l e g e of E n g i n e e r i n g , U n i v e r s i t y of C a l i f o r n i a , B e r k e l e y . 3. B a i , K . J . 1975. Diffraction of i n f i n i t e c y l i n d e r . /. Fluid Mech. 4. B a t t j e s , J.A. l o a d s on l o n g pp. 165-172. 5. Bearman, P.W., Graham, J.M.R., and Singh, F o r c e s on c y l i n d e r s i n h a r m o n i c a l l y o s c i l l a t i n g Mechanics of Wave Induced Forces on Cylinders, Shaw, P i t m a n , L o n d o n , pp. 437-449. 6. Bhattacharyya, John W i l e y and 7. B l a c k , J . L . and M e i , C C . 1970. S c a t t e r i n g and r a d i a t i o n of water waves. Rep. No. 121, Water Resources and Hydrodynamics Laboratory, Dept. of C i v i l E n g i n e e r i n g , M a s s a c h u s e t t s I n s t i t u t e of T e c h n o l o g y . 8. B o l t o n , W.E. and U r s e l l , F. infinitely long cylinder Mech. 57, pp. 241-256. 9. Borgman, L.E. 1969. Directional spectra models for design use. Proc. Offshore Tech. Conf., H o u s t o n , Paper No. OTC1069, pp. 721-746. 10. Bryden, I.G. and Greated, CA. 1984. Hydrodynamic response of l o n g s t r u c t u r e s t o random s e a s . Proc. Symp. on Description and Modelling of Directional Seas, Copenhagen. 11. Cartwright, D.E. and S m i t h , N.D. 1964. f o r o b t a i n i n g d i r e c t i o n a l wave s p e c t r a . Mar. Technol. Soc, pp. 173-182. 12. Cote, L.J. et al. 1960. The d i r e c t i o n a l s p e c t r u m of a wind g e n e r a t e d sea a s d e t e r m i n e d from d a t a obtained by the Stereo Wave Observation Project. Meteorological Paper, 2(6), College of Engineering, New York University. 13. Dallinga, oblique 6 8 , pp. waves by 513-535. 1982. E f f e c t s of s h o r t - c r e s t e d n e s s s t r u c t u r e s . Applied Ocean Res earch. R. 1978. Sons, New R.P., Dynamics York. Marine wave 4(3), S. 1979. f l o w . In ed. T.L. Vehicles. 1973. The wave f o r c e on an in an o b l i q u e s e a . /. Fluid A a l b e r s , A.B., 61 of on an and van Buoy Buoy der techniques Technol. , V e g t , J.W.W. 62 1984. D e s i g n a s p e c t s f o r t r a n s p o r t on a b a r g e . Proc. Offshore Tech. No. O T C 4 7 3 3 , p p . 1 9 5 - 2 0 2 . of jack-up platforms Conf., H o u s t o n , P a p e r 14. Davis, R.E. a n d R e g i e r , L.A. 1977. Methods for estimating directional wave s p e c t r a f r o m multi-element a r r a y s . /. Marine Research. 3 5 ( 3 ) , pp. 453-477. 15. Dean, R.G. 1 9 7 7 . H y b r i d method of computing wave loading. Proc. Offshore Tech. Conf., H o u s t o n , P a p e r N o . OTC3029, p p . 483-492. 16. Finnigan, T.D. a n d Y a m a m o t o , T. 1979. A n a l y s i s of semi-submerged porous breakwaters. Proc. Civil Engineering in the Oceans IV, A S C E , S a n F r a n s i s c o , p p . 380-397. 17. Forristall, G.Z., Ward, E.G., C a r d o n e , V.J., and Borgman, L.E. 1978. The directional spectra and kinematics of surface g r a v i t y waves i n t r o p i c a l storm D e l i a . /. Phys. Oceanography. 8, p p . 8 8 8 - 9 0 9 . 18. G a r r i s o n , C . J . 1 9 6 9 . On t h e i n t e r a c t i o n o f a n infinite shallow-draft cylinder o s c i l l a t i n g a t the free surface w i t h a t r a i n o f o b l i q u e w a v e s . /. Fluid Mech. 39, pp. 227-255. 19. Garrison, C . J . 1984. Interaction of oblique a n i n f i n i t e c y l i n d e r . Applied Ocean Research. 4-15. 20. Georgiadis, C. 1 9 8 4 . T i m e a n d f r e q u e n c y d o m a i n a n a l y s i s of m a r i n e s t r u c t u r e s i n s h o r t - c r e s t e d s e a by simulating appropriate nodal loads. Proc. 3rd Int. Symp. on Offshore Mechanics and Arctic Engineering, New Orleans, pp. 177-183. 21. Hackley, M.B. 1 9 7 9 . Wave f o r c e s i m u l a t i o n s i n random d i r e c t i o n a l s e a s . Proc. 2nd Int. Conf. on the Behaviour of Offshore Structures, BOSS' 79, L o n d o n , p p . 1 8 7 - 2 1 9 . 22. H a s k i n d , M.D. 1 9 5 3 . Oscillation of a ship i n a calm sea. E n g l i s h t r a n s l a t i o n , Soc. o f N a v a l A r c h i t e c t s a n d Marine E n g i n e e r s , T&R B u l l e t i n 1 - 1 2 . 23. Hasselmann, K., D u n c k e l , M., and Ewing, J.A. 1980. D i r e c t i o n a l wave s p e c t r a observed during JONSWAP. /. Phys. Oceanography. 1 0 , pp. 1264-1280. 24. Holthujsen, L.H. 1981. The directional energy d i s t r i b u t i o n o f wind g e n e r a t e d waves as inferred from stereophotographic o b s e r v a t i o n s o f t h e s e a s u r f a c e . Rep. No. 8 1 - 2 , D e p t . o f C i v i l Engineering, Delft Univ. of Technology. waves 6(1), with pp. 63 25. H u n t i n g t o n , S.W. and Thompson, D.M. 1976. Forces on a large vertical cylinder in m u l t i - d i r e c t i o n a l random waves. Proc. Offshore Tech. Conf., Houston, Paper No. OTC2539, pp. 169-183. 26. Ijima, T., Chou, C.R., and a n a l y s i s f o r two-dimensional 15th Coastal Engineering 2717-2736. 27. I s a a c s o n , M. de S t . Q. 1981. N o n l i n e a r wave forces on large offshore structures. Coastal/Ocean Engineering report, Dept. of Civil Engineering, University of British Columbia. 28. Kellogg, O.D. 1929. Springer, B e r l i n . 29. Kim, W.D. body on 427-451. 30. Korvin-Kroukovsky, B.V. 1955. m o t i o n s i n r e g u l a r waves. T r e w s . 31. Lambrakos, to waves Engineering. 32. Leblanc, L.R. and M i d d l e t o n , F.H. 1982. P i t c h - r o l l buoy wave d i r e c t i o n a l s p e c t r a a n a l y s i s . In Measuring Ocean Waves, N a t l . A c a d . P r e s s , W a s h i n g t o n , D.C., pp. 181-193. 33. Leonard, J.W., Huang, M.-C, and H u d s p e t h , R.T. 1983. H y d r o d y n a m i c i n t e r f e r e n c e between f l o a t i n g c y l i n d e r s in o b l i q u e s e a s . Applied Ocean Research. 5 ( 3 ) , pp. 158-167. 34. L o n g u e t - H i g g i n s , M.S., C a r t w r i g h t , D.E., and S m i t h , N.D. 1961. O b s e r v a t i o n s of t h e d i r e c t i o n a l spectrum of sea waves using the motions of a f l o a t i n g buoy. In Ocean Wave Spectra, Prentice-Hall, Englewood Cliffs, New J e r s e y , pp. 111 -132. 35. MacCamy, R.C. d r a f t . J. Ship 36. Mitsuyasu, H. et al. 1975. Observations directional spectrum of ocean waves u s i n g a buoy. /. Phys. Oceanography. 5, pp. 750-760. of the cloverleaf 37. M o g r i d g e , G.R. and J a m i e s o n , W.W. 1976. Wave square caissons. Proc. 15th Coastal Conference, H o n o l u l u , pp. 2271-2289. forces on Engineering Y o s h i d a , A. 1976. Method of w a t e r wave p r o b l e m s . Proc. Conference, Honolulu, pp. Foundat i ons of Potential Theory. 1965. On t h e h a r m o n i c o s c i l l a t i o n of the free s u r f a c e . J. Fluid Mech. Investigation SNAME. 63, pp. K.F. 1982. M a r i n e p i p e l i n e from directional wave 9 ( 4 ) , pp. 385-405. 1964. The Research. a rigid 2 1 , pp. of ship 386-435. dynamic r e s p o n s e spectra. Ocean motions of c y l i n d e r s 7 ( 3 ) , pp. 1-11. of shallow 64 38. M o r i s o n , J.R., O ' B r i e n , M.P., J o h n s o n , J.W., and S c h a a f , S.A. 1950. The f o r c e s e x e r t e d by s u r f a c e waves on p i l e s . Petrol eum Trans., AIME, 1 8 9 , p p . 149-157. 39. Newman, J.N. 1962. The e x c i t i n g f o r c e s on f i x e d i n waves. /. Ship Research. 6 ( 3 ) , p p . 10-17. bodies 40. Newman, J.N. 1977. Marine Cambridge, M a s s a c h u s e t t s . Press, Hydrodynamics. MIT 41. P a n i c k e r , N.N. 1971. D e t e r m i n a t i o n of directional s p e c t r a o f ocean waves from o c e a n a r r a y s . Rep. HEL1-18, Hydrogr. Eng. Lab., U n i v . of C a l i f o r n i a , B e r k e l e y . 42. S t . D e n i s , M. a n d P i e r s o n , W.J. 1953. On t h e m o t i o n s o f s h i p s i n c o n f u s e d s e a s . Trans. SNAME. 6 1 , p p . 280-357. 43. Sand, S.E. 1980. structure of ocean Hydrody. and H y d r a u l i c Three-dimensional deterministic waves. Series paper 24, I n s t . Eng., T e c h . U n i v . o f Denmark. 44. S a r p k a y a , T. a n d I s a a c s o n , M. 1981. Mechanics of Wave Forces on Offshore Structures. Van N o s t r a n d R e i n h o l d , New Y o r k . 45. S h i n o z u k a , M., F a n g , S.-L.S., a n d N i s h i t a n i , A. Time-domain structural response simulation s h o r t - c r e s t e d s e a . /. Energy Res. Tech., Trans. 1 0 1 , pp. 270-275. 1979. in a ASME, 46. U r s e l l , F. 1949. On the heaving motion of a c i r c u l a r c y l i n d e r on t h e s u r f a c e o f a fluid. Quart. J. Mech. Appl. Math. 2, p p . 218-231. APPENDIX ANALYSIS TO D E T E R M I N E OPTIMUM R A D I A T I O N Consider of an t h e o b l i q u e waves infinitely long c y l i n d e r with each mode o f m o t i o n the axis of the c y l i n d e r . forced motions a> by two-dimensional of xz A the m i t st h r e e ° °cSsh(kd) C tan(kd) m m as )] modes along associated with the exp{ i (kysin/3-tot)} ] (11) relation to the angular (eqn. 2.11). 0 ( x , z ) c a n be e x p r e s s e d in The terms as exp(ikxcos/3) + cos[k (z+d)] cosTkm d ) = as w e l l i s related a n d k* a r e w a v e n u m b e r s m -k i n time oscillation as which potential Vm m= 1 k of dispersion a where i n any one potential an e i g e n f u n c t i o n e x p a n s i o n *( '> " the = Re[#(x,z) k i s t h e wavenumber frequency by periodic The DISTANCE generated c a n be e x p r e s s e d *(x,y,z,t) where I exp(-k*x) x>0 (12) d e f i n e d by (13) g and k* m A 0 [k 2 m i s the complex field the = and the evanescent + (ksin/3) ] 2 amplitude of coefficients (14) l / 2 the A potential at the far are included to account f o r m modes o f w a v e m o t i o n n e a r t h e c y l i n d e r . 65 m 66 Since the lowest eigenvalue amongst a l l the evanescent defined as d(x) k* g i v e s modes, a the slowest decay factor decay can = exp(-k*x) be (15) where J In order the value < k*d < to achieve a t x=0, d i s t a n c e .X R X given (16) Tr a decay the i n f i n i t e = maximum k*d give distance eqn. good 0=0°. results (17) g i v e s noted water that of infinite be used optimum distance following distance i s used X R a + 2 ( k * ) ] o f four, t i m e s The above i n water times truncated depth to at a i s obtained was depth. of Bai cannot expression f o r deep water, (1975) should about the infinite boundary. for radiation water the conditions H 2 i s the half-beam + to be u s e d i n source information the when found In deep large. A pulsating useful study = the depth expansion for truncation in this (17) 2 i s too 0=0°. obtain empirical / of f i n i t e which for 1 approximation eigenfunction [(ksin0) where i s o r 0.01 — 2 a distance an rather The boundary ^ = 7r/2 and o f exp(-27r) by [(ksin/3) A rate (18) (7r/ma) ] 2 1 / 2 of t h e c y l i n d e r and m i s given as 67 ka<0.5 0.5£ka<1.5 ka>1.5 (19) 68 M.J pa ka 0.25 0.75 1 .25 0° present results \../pcoa 2 2 GAR present results GAR 5 1 .97 2.10 0.57 0.60 35 2.04 2.16 0.46 0.53 55 2.08 2.21 0.30 0.38 5 1 .00 0.93 1.31 1 .39 35 1.14 1.19 1 .40 1 .51 55 1 .84 1 .74 1 .44 1 .56 5 0.45 0.43 0.93 0.99 35 0.61 0.59 1.01 1.14 55 1 .09 0.93 1 .34 1 .40 Table!. C o m p a r i s o n o f t h e sway added mass and damping c o e f f i c i e n t s of a s e m i - c i r c u l a r cylinder (d/a=°°) o b t a i n e d i n t h e p r e s e n t s t u d y w i t h t h e r e s u l t s o f GAR (Garrison,1984) 69 M ka 0.25 0.75 1 .25 2 2 0° present results 5 /pa X /pcoa 2 2 2 2 B&U present results B&U 1 .38 1 .38 1 .99 1 .96 35 1.61 1 .60 2.51 2.38 55 2.64 2.32 3.23 3.06 5 0.97 0.94 0.94 0.88 35 1 .04 1 .06 0.93 0.92 55 1 .43 1 .32 1.10 1 .02 5 1 .01 0.98 0.49 0.44 35 0.92 0.90 0.39 0.40 55 0.98 0.90 0.46 0.42 T a b l e 2. Comparison of t h e heave added mass and damping c o e f f i c i e n t s of a semi-circular c y l i n d e r (d/a=°°) o b t a i n e d i n the present study with the results of B&U (Bolton and Ursell,1973) 70 1^1 present results ka 0.25 0.75 1 .25 GAR present results GAR 5 0.75 0.77 0.18 0.19 35 0.63 0.65 0.18 0.19 55 0.44 0.46 0.18 0.19 5 1.17 1.18 0.85 0.89 35 1 .07 1.11 0.97 1 .02 55 0.94 0.94 1.17 1 .26 5 0.99 0.99 1.19 1 .26 35 0.95 0.95 1 .37 1 .56 55 0.91 0.90 1 .89 1 .96 Table 3. C o m p a r i s o n of t h e sway e x c i t i n g f o r c e c o e f f i c i e n t and wave a m p l i t u d e r a t i o o f a s e m i - c i r c u l a r cylinder (d/a=°°) obtained i n the present study with the r e s u l t s o f GAR (Garrison,1984) 71 U| |c l 2 2 ka 0.25 0.75 1 .25 0° 5 present results present results B&U B&U 1 .40 1 .40 0.34 0.35 35 1.41 1 .40 0.42 0.43 55 1 .29 1 .32 0.58 0.58 5 0.95 0.94 0.71 0.70 35 0.87 0.87 0.78 0.80 55 0.77 0.76 1 .02 1 .00 5 0.68 0.67 0.85 0.84 35 0.54 0.57 0.85 0.87 55 0.49 0.49 1.10 1 .07 - T a b l e 4. C o m p a r i s o n of t h e heave e x c i t i n g f o r c e c o e f f i c i e n t and wave a m p l i t u d e r a t i o of a s e m i - c i r c u l a r cylinder (d/a=°°) obtained i n the present study with the results of B&U ( B o l t o n and U r s e l l , 1 9 7 3 ) Figure 1. Definition sketch for a rectangular cylinder 73 incident wave reflected wave f£2(heave) transmitted wave _* (roll) ' 3 / ^(sway)/ ///////////// Figure 2. Definition component m o t i o n s sketch for floating cylinder S showing F SR s Figure 3. Sketch of c l o s e d D surface 74 Figure 4. Sketch showing relationship between x, £ , a n d £' j=N1 j=N3 i = N 2 j»1 l<l»l'l'l'l-l'l'l'l'l'l'l'l'l'l'l'l'l'l- J-N j=N4l Figure 5 . cylinder A t y p i c a l boundary (b/a=1,d/a=2) e l e m e n t mesh f o r a rectangular Figure 7. S k e t c h of a d i r e c t i o n a l wave spectrum 76 0(degrees) Figure values 8. Directional of the parameter spreading function for s different 1 2-1 ka=0.1 BAI(1975) Angle of incidence, /S (degrees) F i g u r e 9. Sway e x c i t i n g cylinder (b/a=1,d/a=2) force coefficient for a rectangular 0 15 —r30 30 45 - T — 75 75 60 60 90 Angle of incidence, /? (degrees) Figure 10. Heave exciting force r e c t a n g u l a r c y l i n d e r (b/a=1,d/a=2) coefficient for 0.5- 0.4 — ko=0.1 BAI(1975) — ka=0.2 BAI(1975) — ka=0.4 BAI(1975) El PRESENT RESULTS 0.3o 0.2- 0.1- 0.0-+ 15 30 45 60 /5 Angle of incidence, (1 (degrees) Figure 11. R o l l exciting moment coefficient r e c t a n g u l a r c y l i n d e r (b/a=1,d/a=2) 90 for 78 1.2 ka=0.1 BAI(1975) k a = 0 . 2 BAI(1975) k a = 0 . 4 BAI(1975) PRESENT RESULTS 0.8 a: o.6- 0.4 —r15 Figure cylinder -~T~ 30 —r- 60 45 Angle of incidence, /S (degrees) 12. R e f l e c t i o n (b/a=1,d/a=2) coefficient i 75 for 90 a rectangular 0.6 A 0.5- P R E S E N T RESULTS G A R R I S O N (1984) /? = 0 ° 0.4 c_f 0.3 0.2 0.0- Figure 13. Sway exciting force r e c t a n g u l a r c y l i n d e r (b/a=0.265,d/a=») coefficient for a 79 1.5 A PRESENT RESULTS GARRISON (1984) o A " " \ ^ ^ 0.5- 1 1 1 r r — -i r• i 0.5 i i i i -i r- 1 j • 1.5 ka Figure 14. Heave exciting force coefficient r e c t a n g u l a r c y l i n d e r (b/a=0.265,d/a=°°) for 0.35- A PRESENT RESULTS 0.30 GARRISON (1984) 0.25 a--l / ^ = 0 ° -TT-lo^^- 0.20o 0.15- 0.10- 0.05 0.00 1.5 0.5 Figure 15. R o l l exciting moment coefficient r e c t a n g u l a r c y l i n d e r (b/a=0.265,d/a=° ) o for a 80 0.5 A PRESENT RESULTS GARRISON (1984) 0.4 0.3 H O >9. 0.2 H o.H 0.0 0.5 —I— 1 ka —r~ 1.5 Figure 16. Sway added mass c o e f f i c i e n t c y l i n d e r (b/a=0.265,d/a=») f o r a rectangular 0.30 0.25 H A PRESENT RESULTS GARRISON (1984) 0.20 H o ^ 0.15- 0.10- 0.05 Figure 17. Sway damping c y l i n d e r (b/a=0.265,d/a=») coefficient for a rectangular 81 A PRESENT RESULTS - GARRISON (1984) \fl=60° 3- \ A D —g. ^-^^^ ^ ^ ^ ^ ^ 1- 1 , , 1 1 , , 0.5 , 1- i . . . . . . i i i 1.5 1 ka F i g u r e 18. Heave added mass c o e f f i c i e n t c y l i n d e r (b/a=0.265,d/a=») for a rectangular A PRESENT RESULTS 2.5 GARRISON (1984) 2o 3 v9. 1.5 AVV 0.5- o- 0.5 1 1.5 ka F i g u r e 19. Heave damping c y l i n d e r (b/a=0.265,d/a=») coefficient for a rectangular 82 0.4-1 A PRESENT RESULTS GARRISON (1984) Figure 21. R o l l damping cylinder (b/a=0.265,d/a=») ka coefficient for a rectangular 83 for a long 84 Figure 24. R o l l response amplitude f l o a t i n g box (a=7.5m,b=3m,l=75m,d=12m) operator f o r a long 0.5- Figure 25. F o r c e a n d moment r e d u c t i o n f l o a t i n g box (a=7.5m,b=3m,l=75m,d=12m) factors f o r a long 85 ka F i g u r e 25.(cont.) l o n g f l o a t i n g box F o r c e a n d moment r e d u c t i o n (a=7.5m,b=3m,l=75m,d=12m) factors for a 86 (a) SWAY Figure 26. F o r c e r e d u c t i o n f a c t o r s f o r a l o n g f l o a t i n g box (a=7.5m,b=3m,l=75m,d=l2m) i n n o r m a l a n d o b l i q u e mean s e a s 87 Figure 27. R e s p o n s e ratios (a=7.5m,b=3m,l=75m,d=l2m) for a long floating 88 Figure 27.(cont.) Response (a=7.5m,b=3m,l=75m,d=l2m) ratios f o r a long f l o a t i n g box
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Wave loads and motions of long structures in directional seas Nwogu, Okey U. 1985
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Title | Wave loads and motions of long structures in directional seas |
Creator |
Nwogu, Okey U. |
Publisher | University of British Columbia |
Date | 1985 |
Date Issued | 2010-05-28 |
Description | The effects of wave directionality on the loads and motions of long structures is investigated in this thesis. A numerical method based on Green's theorem is developed to compute the exciting forces and hydrodynamic coefficients due to the interaction of a regular oblique wave train with an infinitely long, semi-immersed floating cylinder of arbitrary shape. Comparisons are made with previous results obtained using other solution techniques. The results obtained from the solution of the oblique wave diffraction problem are used to determine the transfer functions and response amplitude operators for a structure of finite length and hence the loads and amplitudes of motion of the structure in short-crested seas. The wave loads and body motions in short-crested seas are compared to corresponding results for long-crested seas. This is expressed as a directionally averaged, frequency dependent reduction factor for the wave loads and a response ratio for the body motions. Numerical results are presented for the force reduction factor and response ratio of a long floating box subject to a directional wave spectrum with a cosine power type energy spreading function. Applications of the results of the present procedure include such long structures as floating bridges and breakwaters. |
Subject |
Ocean Waves Offshore Structures - Hydrodynamics |
Genre |
Thesis/Dissertation |
Type |
Text |
Language | Eng |
Collection |
Retrospective Theses and Dissertations, 1919-2007 |
Series | UBC Retrospective Theses Digitization Project |
Date Available | 2010-05-28 |
Provider | Vancouver : University of British Columbia Library |
Rights | For non-commercial purposes only, such as research, private study and education. Additional conditions apply, see Terms of Use https://open.library.ubc.ca/terms_of_use. |
DOI | 10.14288/1.0062835 |
Degree |
Master of Applied Science - MASc |
Program |
Civil Engineering |
Affiliation |
Applied Science, Faculty of Civil Engineering, Department of |
Degree Grantor | University of British Columbia |
Campus |
UBCV |
Scholarly Level | Graduate |
URI | http://hdl.handle.net/2429/25129 |
Aggregated Source Repository | DSpace |
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