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Hydrodynamic coefficients of compound circular cylinders in heave motion Venugopal, Madan 1984

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HYDRODYNAMIC COEFFICIENTS OF COMPOUND CIRCULAR CYLINDERS IN HEAVE MOTION by MADAN VENUGOPAL B.Tech(NA&SB), University of Cochin, 1 9 8 0 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF APPLIED SCIENCE i n FACULTY OF GRADUATE STUDIES Department of Mechanical Engineering We accept this thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA 27th July, 1984 © Madan Venugopal, 1984 . «6 In presenting this thesis in p a r t i a l fulfilment of the requirements for an advanced degree at the THE UNIVERSITY OF BRITISH COLUMBIA, I agree that the Library s h a l l make i t freely available for reference and study. I further agree that permission for extensive copying of t h i s thesis for scholarly purposes may be granted by the Head of my Department or by his or her representatives. It i s understood that copying or publication of t h i s thesis for f i n a n c i a l gain s h a l l not be allowed without my written permission. Department of Mechanical Engineering THE UNIVERSITY OF BRITISH COLUMBIA 2075 Wesbrook Place Vancouver, Canada V6T 1W5 Date: 27th July, 1984 ABSTRACT The added mass and damping c o e f f i c i e n t s of a compound c i r c u l a r cylinder in heave motion are computed t h e o r e t i c a l l y using a semi a n a l y t i c a l potential flow method. The method uses continuity of pressures and v e l o c i t i e s between adjacent regions of the flow f i e l d . The heave exciting forces on the compound cylinder are calculated from the heave damping c o e f f i c i e n t . The hydrodynamic c o e f f i c i e n t s and the heave exciting forces are compared to theoretical results obtained from a boundary element method. The hydrodynamic c o e f f i c i e n t s of the compound c i r c u l a r cylinder are determined experimentally by forced harmonic o s c i l l a t i o n of the cylinder model. The wave height at a point in the flow f i e l d was also measured during the experiment. The effects of variation of amplitude and frequency of o s c i l l a t i o n and draft are also studied experimentally. The results are compared to the theoret i c a l predictions. The heave exciting forces on a compound cylinder model due to small amplitude, sinusoidal waves are measured experimentally in a towing tank. The results are compared to predictions by the theoretical method presented in th i s thesis and by a boundary element method. The heave exciting forces on single and double cylinder models are also determined experimentally. These results are compared to theo r e t i c a l predictions by a boundary element method. i i The comparisons between theory and experiment show the a p p l i c a b i l i t y of linear potential flow theory in the determination of the hydrodynamic c o e f f i c i e n t s of the compound cylinder model. i i i Table of Contents INTRODUCTION 1 THEORETICAL MODEL AND SOLUTION 8 2.1 General formulation of the boundary value problem 9 2.2 Loads and motions 12 2.3 Heave motion of compound cylinder 14 2.3.1 D e f i n i t i o n of flow f i e l d 15 2.3.2 Governing equations and boundary conditions . 15 2.3.3 De f i n i t i o n of potentials 16 2.3.3.1 Region 1 16 2.3.3.2 Region 2 17 2.3.3.3 Region 3 19 2.3.3.4 Exterior region 21 2.4 Solution for unknown c o e f f i c i e n t s for potentials 22 2.5 Calculation of the added mass and damping c o e f f i c i e n t s 23 2.6 Calculation of heave exciting force 25 2.7 Calculation of wave amplitude 25 2.8 Computer program 26 EXPERIMENTAL WORK 27 3.1 Purpose of experiments 27 3.2 Determination of hydrodynamic c o e f f i c i e n t s 28 3.2.1 Data analysis 30 3.3 Determination of heave exciting force 31 3.4 Flow v i s u a l i z a t i o n 32 RESULTS AND DISCUSSION 33 4.1 Presentation of data 34 iv 4.2 Discussion of t h e o r e t i c a l results 35 4.3 Discussion of experimental results 37 4.3.1 Hydrodynamic c o e f f i c i e n t s 37 4.3.2 Flow v i s u a l i s a t i o n tests 42 4.3.3 Heave exciting forces 43 CONCLUSIONS 48 RECOMMENDATIONS 53 NOMENCLATURE FOR PLOTS 54 LIST OF SYMBOLS 55 BIBLIOGRAPHY 56 5. APPENDIX 1 - EVALUATION OF POTENTIAL FUNCTIONS 58 5.1 Potentials 60 5.1.1 Region 1 .60 5.1.2 Region 2 60 5.1.3 Region 3 61 5.1.4 Exterior region 62 5.2 Generation of systems of equations for solution .63 5.2.1 Continuity of pressure between regions 1 and 2 63 5.2.2 Continuity of v e l o c i t y between regions 1 and 2 65 5.2.3 Continuity of pressure between region 2 and the exterior region .66 5.2.4 Continuity of pressure between region 3 and the exterior region 68 5.2.5 Continuity of v e l o c i t i e s between regions 2, 3 and the exterior region 70 5.2.6 Solution for c o e f f i c i e n t s of series 73 5.3 Integrals for evaluation of hydrdoynamic c o e f f i c i e n t s 74 v 6. APPENDIX 2 - EXPERIMENTAL SET-UP 76 6.1 Experimental f a c i l i t i e s 76 6.1.1 The towing tank 76 6.1.2 Wavemaker 76 6.1.2.1 The Wave Signal Generator 76 6.1.2.2 Wave Synthesizer 77 6.1.2.3 Wave Paddle 77 6.2 Motion generator 78 6.3 Data c o l l e c t i o n system 80 6.3.1 Amplifiers and signal conditioners 80 6.3.2 MINC-11 computer 80 6.3.2.1 Main Console 80 6.3.2.2 Dual Floppy Disk Drive System 81 6.3.2.3 VT105 Video Terminal 81 6.3.2.4 Line Printer 81 6.3.2.5 Tektronix Screen Dump Printer 81 6.4 Models 82 6.5 Equipment used 82 6.5.1 Strain indicators 83 6.5.2 Sonar l e v e l monitor 83 6.5.3 Wave probe 84 6.5.4 Force recording equipment 84 6.5.4.1 80 l b . Dynamometer 84 6.5.4.2 Universal Shear Beam 85 6.5.5 Calibration 86 6.6 Software used 86 6.6.1 Data c o l l e c t i o n software 87 v i 6.6.1.1 ADCAL 87 6.6.1.2 ADMAIN 87 6.6.1.3 ADMUX 88 6.6.1.4 GRAPH 88 6.6.2 Data analysis software 88 6.6.2.1 DEMUX 88 6.6.2.2 AMP 89 6.6.2.3 PHAMP 89 6.6.2.4 DELAY 89 6.6.2.5 SPECADD 90 6.6.2.6 ZERO 90 6.6.2.7 TABLE 90 6.6.2.8 FINAL 90 6.6.2.9 STAT 91 vi i LIST OF FIGURES Fig.1 D e f i n i t i o n of motions ... Fig.2 Compound cylinder geometry ... Fig.3 Subdivision of flow f i e l d ... Fig.4 Compound cylinder model ... Fig.5 Double cylinder model ... Fig.6 Single cylinder model ... Fig.7 Towing tank at B.C.Research ... Fig.8 Motion Generator ... Fig.8a Load c e l l positioning ... Fig.9 Data c o l l e c t i o n equipment ... Fi g . 10 Wave paddle ... Fig.11 Flow v i s u a l i z a t i o n ... Fi g . 12 Displacement record ... Fig.12a Displacement spectrum ... Figs.13 & 14 Hydrodynamic c o e f f i c i e n t s of compound cylinder - theore t i c a l comparison ... Figs.15 to 26 Hydrodynamic c o e f f i c i e n t s and wave amplitudes for compound cylinder ... Figs.27 to 30 Heave exciting force on single cylinder ... Figs. 31 to 34 Heave exciting force on double cyli n d e r . . . Figs. 35 to 42 Heave exciting force on compound cylinder ... v.i i i Fig. 4 3 Hydrodynamic c o e f f i c i e n t s for 1 3 6 single cylinder (McCormick)... ix ACKNOWLEDGEMENTS I am deeply indebted to Dr.S.M.Calisal for his patient guidance throughout the course of t h i s research.I am also indebted to NSERC for the f i n a n c i a l support for t h i s project. My sincere gratitude to Michael Desandoli for his assistance with the experiments and data analysis. Thanks are also due to Johnson Chan for the use of his boundary element program for computation of heave exciting forces on axisymmetric f l o a t i n g objects. I would also l i k e to thank Fraser Elhorn for redesigning the motion generator and John Hoar, and his technicians at the Machine Shop at the Mechanical Engineering Department for fabrication of the motion generator and the models. Special thanks are due to Bruce Hanson for doing an excellent job in reconstructing the motion generator. I would l i k e to thank Gerry Stensgaard for permission to use the towing tank at B.C.Research and George Roddan for his expert advice on the instrumentation. I would l i k e to thank Marcel Lefrangois for his assistance with the instrumentation for data c o l l e c t i o n and Steve Thomson for assistance with the equipment handling. F i n a l l y , I wish to thank Angela Runnals and Jon Nightingale for their assistance with the text processor TEXTFORM on which th i s thesis was prepared. x DEDICATION This thesis is dedicated to my parents for their love and a f f e c t i o n , and to my wife for her patient forebearance during the course of my studies. xi 1. INTRODUCTION In the l a s t few decades there has been an accelerated growth in the offshore industry. A wide variety of offshore structures have been designed and constructed, primarily for o i l exploration, d r i l l i n g , production and storage. The main features in the continuing growth of offshore structures are their size and the depth at which they are capable of operating. Since there i s considerable investment going into a large offshore structure and i t i s expected to have a f a i r l y long design l i f e (upto 100 years in some cases), considerable research has been and i s being devoted to the problems associated with the design of these structures. Of primary importance are the loads which the structure experiences and the motions i t undergoes. The loading on the structure i s primarily hydrodynamic, i . e . due. to the action of the ocean waves on the structure. Even i f the extreme sea states were known accurately, i t would s t i l l be d i f f i c u l t to estimate exactly the nature of the loading on the structure. To mathematically model the actual flow around a f u l l scale offshore structure poses considerable d i f f i c u l t i e s and an exact picture- of the problem i s beyond the reach of the present state of the a n a l y t i c a l techniques. Several s i m p l i f i c a t i o n s have to be made in the mathematical modelling of the flow. These are discussed in relevance to the topic of t h i s thesis in Chapter 2. 1 2 Most of the offshore structures in current use have component members of c i r c u l a r section. So, the study of flows around c i r c u l a r cylinders are of considerable importance and much research has been devoted to t h i s topic. To study the hydrodynamic loads and r e s u l t i n g motions of any structure f l o a t i n g in a f l u i d medium i t i s neccessary to determine two quantities (known as the hydrodynamic c o e f f i c i e n t s ) : the added mass and damping c o e f f i c i e n t s . In addition, i t i s also neccessary to know the exciting forces on the structure. A body accelerating in a f l u i d medium experiences an additional force due to the acceleration of a part of the f l u i d also.This additional hydrodynamic force can be expressed as an added mass to the body's mass, being accelerated at the same rate. This fact was f i r s t recognized by Chevalier du Buat about 200 years ago. Since then several researchers including Bessel, Green, Plana, Stokes, Lamb, and even S i r Charles Darwin have worked on the problem /8/. Sir Charles Darwin showed that a cylinder moving through a f l u i d medium displaces f l u i d p a r t i c l e s in the d i r e c t i o n of i t s motion. Further, he showed that t h i s permanently displaced mass of the f l u i d enclosed between the i n i t i a l and f i n a l positions of the f l u i d p a r t i c l e s i s the added mass i t s e l f . Added mass can be expressed mathematically in various ways. One of the d e f i n i t i o n s is given l a t e r . If we consider the Cartesian co-ordinate system shown in F i g . 1, translatory o s c i l l a t i o n s in the x, y, and z 3 directions are known as surge, sway and heave, respectively.Rotational motions about the same axes are known as r o l l , p i t c h and yaw, respectively. For each direction of motion there exist six added mass c o e f f i c i e n t s corresponding to the displacement of the f l u i d in the six degrees of freedom. Hence added mass may be expressed as a tensor a^j where the f i r s t subscript refers to the di r e c t i o n of the body motion and the second to the d i r e c t i o n of the hydrodynamic force. Of the 36 added mass c o e f f i c i e n t s , i t can be shown that a^j = a ^ and hence there are only 21 independent c o e f f i c i e n t s . These may be further reduced for a body symmetric about one or more axes /5/. When a body moves p e r i o d i c a l l y in a f l u i d medium near the free surface, the hydrodynamic force develops frequency dependent components in-phase and out-of-phase with the acceleration. The component in-phase with the acceleration contributes to the added mass and the component in-phase with the v e l o c i t y contributes to the damping c o e f f i c i e n t . Further, for a body with separated flow about i t , the wake or cavity induces an added mass. This cavity induced mass varies with the instantaneous shape and volume of the cavity and i t s rate of change. The instantaneous value of the added mass depends on the time history of the motion.Also, for motion in a viscous f l u i d there exists some viscous damping too. The added mass c o e f f i c i e n t s depend, in general, on the parameters characterizing the motion, time, a suitably 4 defined Reynold's number, etc. The determination of an expression for the the time dependent force on a body undergoing an arb i t r a r y motion i s a very complex problem. This problem i s si m p l i f i e d by considering simpler time dependent motions l i k e harmonic o s c i l l a t i o n s . A further s i m p l i f i c a t i o n i s achieved by using the potential flow theory to describe the flow. Several t h e o r e t i c a l and experimental techniques have been devised to determine the hydrodynamic c o e f f i c i e n t s of a wide range of bodies. The common approaches to solution of boundary value problems in potential flow theory are a n a l y t i c a l methods l i k e conformal mapping and numerical procedures l i k e s i n g u l a r i t y techniques. In recent years the f i n i t e element methods have also been successfully used. Conformal mapping can be used only for two dimensional problems. The si n g u l a r i t y methods do not suffer from this l i m i t a t i o n . They have been used extensively in hydrodynamic problems in recent years /1/. Havelock(1955) determined the hydrodynamic c o e f f i c i e n t s of a sphere /12/. Kim (1974) studied the hydrodynamic c o e f f i c i e n t s for e l l i p s o i d a l bodies o s c i l l a t i n g near the free surface /18/. Wang and Shen (1966) calculated the added mass and damping c o e f f i c i e n t s of a sphere in water of f i n i t e depth /21/. Garrison (1975) used di s t r i b u t e d s i n g u l a r i t i e s to calculate the hydrodynamic c o e f f i c i e n t s of v e r t i c a l c i r c u l a r 5 cylinders in water of f i n i t e depth /15/. He formulated the general problem for arb i t r a r y forms too. Bai and Yeung (1974) calculated added mass and damping c o e f f i c i e n t s for horizontal and v e r t i c a l c i r c u l a r cylinders /14/. Bai (1976) calculated the hydrodynamic c o e f f i c i e n t s of axisymmetric ocean platforms /13/. K r i t i s (1979) used the hybrid integral method of Yeung (1975) /22/ to numerically calculate the hydrodynamic c o e f f i c i e n t s for a c i r c u l a r cylinder /19/. While the si n g u l a r i t y methods have several s i g n i f i c a n t advantages and are extensively used in the design of offshore structures, they also have some disadvantages. There is considerable computation involved and the choice of the d i s c r e t i z a t i o n of the body surface must be made c a r e f u l l y . Further, these methods sometimes give numerical problems at certain frequencies which are c a l l e d "irregular frequencies" for the method. The f i n i t e element methods do not have t h i s disadvantage, but they require the entire flow f i e l d to be d i s c r e t i z e d . This i s considerably d i f f i c u l t unless i t i s handled by a computer. Analytic methods can handle only a small number of well defined geometric shapes. Even for these, there i s , sometimes considerable mathematical d i f f i c u l t y . This thesis discusses the determination of the added mass and damping c o e f f i c i e n t s for a compound c i r c u l a r cylinder undergoing simple harmonic heave motion at the free surface in water of f i n i t e depth.The hydrodynamic c o e f f i c i e n t s are obtained using a theoretical method as well 6 as experiments. Experiments were also conducted to determine the heave exc i t i n g force due to harmonic waves incident on a r i g i d model. These results are also compared to theoret i c a l predict ions. The t h e o r e t i c a l method discussed here i s applicable to a special class of axisymmetric bodies. It can be termed an analytic method as well as a macro-element technique, in the sense that the flow f i e l d i s divided into a few elements. The value of the potential is e x p l i c i t l y known in the i n t e r i o r domain as a function of the v e r t i c a l and r a d i a l coordinates. The basis of the method i s suggested by Garrett in his paper on wave loads on a c i r c u l a r dock /4/. This method has been applied to the case of a single c i r c u l a r cylinder by Sabuncu and C a l i s a l with good results /./. The method has also been applied by K.Kokkinowrachos et. a l /3/ in the predicion of hydrodynamic c o e f f i c i e n t s of bodies of several a r b i t r a r y axisymmetric shapes. The experimental results presented in t h i s thesis correspond to the forced harmonic o s c i l l a t i o n of a compound cylinder model in a towing tank. This gave the hydrodynamic c o e f f i c i e n t s of the cylinder model. A q u a l i t a t i v e study of the vortex shedding during the cylinder motion was also made with the help of an underwater window at the towing tank. The heave exc i t i n g force due to small amplitude waves on three cylinder models of d i f f e r e n t shapes was determined at varying drafts for the models and d i f f e r e n t amplitudes of 7 the waves. These results were compared to a theoret i c a l prediction using a boundary element technique applied by Chan /11/. The heave exciting forces for the compound c i r c u l a r cylinder model were also compared to predictions using the present theory. 2. THEORETICAL MODEL AND SOLUTION Using dimensional analysis r e l a t i n g to the wave force on a fixed body, one can conveniently compare the r e l a t i v e importance of flow separation and d i f f r a c t i o n e f f e c t s /8/. A time invariant force on a fixed structure due to an incident wave can be expressed as, F/pgHD2 = f(d/L, H/L, D/L, Rn), ...(2.1) where, p = Density of water; g = Acceleration due to gravity; d = Depth of water; L = Wavelength; H = Wave height; D = Representative diameter of body; Rn = Reynolds number. The body size to wavelength r a t i o , D/L i s termed the d i f f r a c t i o n parameter. As D/L becomes large, d i f f r a c t i o n e f f e c ts become important. When d i f f r a c t i o n e f f e c t s are important, flow separation i s r e l a t i v e l y unimportant and the problem can be studied by potential flow methods. Further, i t i s assumed that the wave steepness, H/L i s small corresponding to linear wave theory /8/,/3/. A l i n e a r i s e d boundary value problem i s thus formulated for the v e l o c i t y potential in the flow f i e l d . 8 9 2.1 GENERAL FORMULATION OF THE BOUNDARY VALUE PROBLEM Consider the compound c i r c u l a r cylinder shown in Fig.2 with a sinusoidal wave incident on i t . The f l u i d i s assumed to be homogeneous, i n v i s c i d and incompressible. An earth fixed c y l i n d r i c a l coordinate system with o r i g i n on the seabed i s defined as shown in Fig.2 with the z-axis coinciding with the v e r t i c a l axis of the cylinder. It i s further assumed that the amplitude of the cylinder motion is small compared to the wavelength and that the amplitude of the incident wave i s also small compared to the wavelength. The v e l o c i t y potential for the flow f i e l d in the presence of the cylinder can, in general, be expressed in the form: $(x,y,z,t) =* 0(x,y,z,t) + (x,y,z,t) ...(2.2) In the above $ 0 i s the potential of the undisturbed waveform and i s the disturbance p o t e n t i a l . As a consequence of l i n e a r i s a t i o n , the potential can be further expressed as, 6 (x,y,z,t) = $„(x,y,z,t) + Z s. <j>. (x,y,z,t) (2.3) a v j = 1 J 0 J where, $ F i s the potential due to a stationary body in the wavefield and <t>^ are the potentials due to the motions of the body in the six component directions with unit v e l o c i t y , 10 in undisturbed water, s. i s the magnitude of the vel o c i t y 3 0 th of the body motion in the j d i r e c t i o n . Hence, *(x,y,z,t) =*o(x,y,z,t) + $ F(x,y,z,t) 6 + Z s . <t>. (x,y,z,t) ... (2.4) j=1 3 0 D The potentials $o, , c are harmonic, with r 1 . . b the frequency of the incident wave and hence, - i c j t <f>(x,y,z,t) = 4>(x,y,z) e ...(2.5) With the assumption that the cylinder motion is harmonic, - 10>t s . = s . e D Do The problem can now be formulated in two parts. One due to the wave incident on a fixed structure ( d i f f r a c t i o n problem), and the second due to the motion of the body in undisturbed water. The assumption of l i n e a r i s a t i o n enables us to consider the individual degrees of freedom independent of each other . The cross-coupling terms have to be considered i f they are l i n e a r . The potentials <t>^ must s a t i s f y the governing Laplace's equation, V 2 <S>^ (x,y,z) = 0 ...(2.6) and the following boundary conditions, (i) Linearised free surface kinematic and dynamic boundary conditions. d<f> • 3TJ for z=d 9z 9t d<t> • 9t = -qrj for z=d 1 1 . . .(2.7) ...(2.8) The above can be written as, 92S> . 1 9t ; 9$. 9z = 0 or, d2<fi. -cj2<j>. + g 1 = 0 for z=d 3 9z 2 ...(2.9) ( i i ) Ocean bottom boundary condition. 90 — = 0 for z = 0 . . . (2. 10) 9z ( i i i ) Radiation condition. 90 . lim / r ( — 1 - i(co 2/g) 4> • ) = 0 . . . ( 2 . 1 1 ) r <» 9r ^ The following boundary conditions must also be s a t i s f i e d on the body surface. 1 2 (iv) For a wave incident on a fixed structure, on the body surface S, - . ( 2 . 1 2 ) 3 n 9n (v) For a f l o a t i n g body free to move in calm water, 9*. _. —L_ = n. e l w t |_ ...(2.13) 9n 11 b or , 90 . = n j Is 9n where, n. is the component of the outward normal on the body 3 th surface S, in the j d i r e c t i o n as shown in Fig.1. 2.2 LOADS AND MOTIONS The instantaeneous l i n e a r i s e d pressure on the body is given by, Pinst " ~P 9t ...(2.14) Using (2.4), we can rewrite t h i s as, 6 _. i n S t F j - l j o j It i s expedient to divide the force into that due to the d i f f r a c t i o n and that due to the motion. The d i f f r a c t i o n problem gives the exciting force due to the incident wave, 13 F E k ( t ) = " " s Pinst. nk d S = - i c o p e ~ l w t ;/ s (0 O + 0 p ) n k dS ...(2.15) where, k=1,2 , . . 6 . The hydrodynamic forces due to the motion of the body in undisturbed water can be calculated using the potentials F H = - i c o p e 1U>Z JSS 2 s 0 n R dS ...(2.16) k j=1 jo j Now, we define the added masses, a^^ and damping c o e f f i c i e n t s , b^j as in /3/. - p / ; s 0^  n k dS = akj + (i/co) b k j ...(2.17) where, a k j = - p Re [ J 7 S 0j n k dS] ...(2.17a) b k j = - p Im [ J 7 S 0^  n k dS] ...(2.17b) Now, we can write (2.16) in the form, 6 F u (t) = L (a. . s. + b, . s. ) ...(2.18) H k j = 1 k ] 3 D a, • represent added mass in the case of tra n s l a t i o n and K 3 added mass moments of i n e r t i a in the case of rotational motions, b, . are the damping c o e f f i c i e n t s , k indicates the 1 4 direction of the force and j indicates the di r e c t i o n of the motion. Using the above we can, in general, write a system of six coupled d i f f e r e n t i a l equations for the motion of a flo a t i n g body due to an incident wave as, * ( m k j + a k j } S j + b k j S j + C k j S j = FE„ .-.(2.19) where, c ^ are the restoring force c o e f f i c i e n t s and m^ are the masses or the mass moments of i n e r t i a of the body. In the special case of a body restrained to undergo heave motion only, the above system reduces to an ordinary d i f f e r e n t i a l equation, (m + a 2 2 ) q + b 2 2 q + c 2 2 q = F^ , (t) ...(2.20) where q denotes heave motion. 2.3 HEAVE MOTION OF COMPOUND CYLINDER Consider the compound cylinder with geometry as shown in Fig.2. The problem i s to define the potential during the heave motion of the cylinder. The flow f i e l d i s subdivided as follows. 15 2.3.1 DEFINITION OF FLOW FIELD The subdivision i s shown in Fig.3. Though only a section i s shown because of the a x i a l symmetry, the elements are actually c y l i n d r i c a l . Region 1 is defined as the volume within the surface r=a,, 0<z^d,. Region 2 i s defined as the volume within the surfaces r=a 1 r 0^z^d 2 and r=a 2, 0^z<d2. Region 3 is defined as the volume within the surfaces r=a 3, d 3^z<d and r=a 2, d3<z<d. The exterior region is defined as the volume exterior to the surface r=a 2, 0<z<d. Using Garrett's method potentials are defined in the four regions as , $ 2, $ 3 , $ E , respectively. 2.3.2 GOVERNING EQUATIONS AND BOUNDARY CONDITIONS The ve l o c i t y potentials <i> must s a t i s f y Laplace's equation from the i r r o t a t i o n a l assumption made e a r l i e r /9/. V • • O/r) ! £ • <1/r»> fjt ...<2.6) The equation i s defined in c y l i n d r i c a l coordinates defined as in Fig.2. <j> can be written as, 0(r,0,z) = R(r) T(6) Z(z) ...(2.21) i f the variables are assumed to be independent of each other. Using separation of variables, (2.6) can be written 16 as three ordinary linear d i f f e r e n t i a l equations as, + A 2T = 0 ...(2.22) ^ r f 2 - B 2Z = 0 ...(2.23) ^ f 2 + (1/r) + (B 2-A 2/r 2)R = 0 ...(2.24) Since the flow i s axisymmetric the a x i a l function has to be an even function. Hence, T(6) = cos(mfl) ... (2.25) We can write the potential $ as *(r,0,z,t) = 4>(r,z) e ~ l u t ...(2.26) The c o e f f i c i e n t m i s taken as zero on the assumption that the most s i g n i f i c a n t contribution comes from the f i r s t term of the cosine series for 6. 2.3.3 DEFINITION OF POTENTIALS The potentials , 4>2, * 3 , * E are defined such that they s a t i s f y the governing equations and boundary conditions as below. 2.3.3.1 Region 1 The potential *, = ^, V d e l c j t has to s a t i s f y the H following boundary conditions. V"H i s the v e l o c i t y in the heave d i r e c t i o n . I at z-d, ...(2.27) = 0 at z=0 ....(2.28) The potential s a t i s f i e s the governing equation (2.22) and (2.23) and the boundary conditions (2.27) and (2.28). The potential 0, comprises of the p a r t i c u l a r solution, 17 0 = [(z 2 - r 2 / 2 ) / 2 d d 1 ] ...(2.29a) and the homogeneous solution, 0 1 h = A0/2 + g = 1 A n ( I 0 (nTrr/d, ) / I 0 (n7r a T/d, ) )cos(n7rz/d 1 ) ] ...(2.29b) Hence, = V d e~ i c J t [ ( z 2-r 2/2)/2dd 1 + A0/2 H oo + £ = 1 A n (I 0(n7rr/d , )/I 0 (nrca ,/d, ) ) cosfnirz/d, ) ] ...(2.29) The c o e f f i c i e n t s A 0 and A n are unknown and w i l l be fixed l a t e r in the solution. I 0 i s the modified Bessel function of the f i r s t kind. 2.3.3.2 Region 2 The potential $ 2 = <f>2 V d e~ l c J t has to s a t i s f y the H following boundary conditions. = 1 at z=d2 ...(2.30) = 0 at z=0 ...(2.31) The potential 4>2 s a t i s f i e s the governing equation (2.22) and (2.23) and the boundary conditions (2.30) and (2.31). The potential 4>2 comprises of the p a r t i c u l a r solut ion 0 = (z 2 - r 2 / 2 ) / 2 d d 2 ...(2.32a) 18 0 2 h = B0+ Z = 1 {B n V n (r)+C p Wn (r)} c o s ( n n z / d 2 ) ...(2.32b) and adding the above, * 2 = V d e" i a J t [(z 2-r 2/2)/2dd, H 00 + B0+ 2 = 1 (B n V n (r)+C n Wn (r)} cos (n7rz/d2) ] ...(2.32) where V n (r) -I 0(n7rr/d 2) +( I i(nita,/d2) / (nira ,/d2) ) K 0(n7rr/d 2) I 0 ( n 7 r a 2 / d 2 ) +( I^nn-a,/^) / K ^ n n c ^ / d ; , ) ) K 0 ( n 7 r a 2/d 2) ...(2.33) Wn (r) . I 0(n7rr/d 2) + ( I 0(n7ra 2/d 2) / K 0(n7ra 2/d 2) ) K 0(n7rr/d 2) I , (nwa iVd 2) +( I 0(n7ra 2/d 2) / K 0(n7ra 2/d 2) ) K^nn-a,/^) ...(2.34) B 0 and B n are unknown c o e f f i c i e n t s which are determined by the r a d i a l boundary conditions. <i>2 as defined in (2.32) s a t i s f i e s (2.22), (2.23), (2.30) and (2.31). 19 2.3.3.3 Region 3 The potential in thi s region $ 3 = <j>3 V d e l w t H s a t i s f i e s the governing equation and the boundary conditions = 1 at z=d3 ... (2.35) and -co2 03 + g = 0 at z=d ...(2.36) The vel o c i t y potential in thi s region i s defined by a par t i c u l a r solution, 0 3 p = (z/d+g/(co2d)-1 ) ...(2.37a) and a homogeneous solution, 00 0 3 h = DoX0Yo+ I = 1 D n X n Y n ...(2.37b) Combining the above, *>3 = V d e" i t J t [ (z/d+g/(co2d)-1 ) H + D 0X 0Yo + 2 = 1D n X n Y n ] ...(2.37) where, J 0(m 0r) - ( J,(moa3) / H,(m0a3) ) H 0(m 0r) X Q = . . . (2.38) J 0(m 0a 2) - ( J,(m 0a 3) / H,(moa3) ) H 0(m 0a 2) 20 I 0 ( % r) + ( I,(mn a 3) / K,(mn a 3) ) K 0(m n r) X = n I 0(m n a 2) - ( I 1(m n a 3) / K i ( m n a 3) ) K 0(m n a 2) ...(2.39) In t h i s formulation, Y 0 = M 0 " 1 / 2 cosh[m 0(z-d 3)] ...(2.40) M 0 = [1+sinh{2m 0(d-d 3)}/2m 0(d-d 3)]/2 ...(2.40a) Y n = M N 1 / 2 cos[m n (z-d 3)] ...(2.41) M N = [1+sin{2m 0(d-d 3)}/2m 0(d-d 3)]/2 ...(2.41a) where m0 and mn are the roots of, a)2 - g m0 tanh[m 0 (d-d 3) ] = 0 ...(2.42) co2 + g mn tan[m n (d-d 3)] = 0 ...(2.42a) respectively. 21 2.3.3.4 Exterior region The potential in the exterior region S> = <t>„ V d e 1 £ J t E E H has to s a t i s f y the governing equation (2.6) and the following boundary conditions, - s - = 0 at z = 0 . . . (2.43) 3z 90 p - C J 2 0 „ + g — - = 0 at z=d ...(2.44) E 3z The potential <l> describing the free wave and the waves refle c t e d off the r i g i d cylinder i s given by, = V d e l c J t [ - E 0 ( H 0 ( k 0 r ) / H 1 ( k 0 a 2 ) ) Z 0 ( z ) E H £ = 1 E n ( K 0( k n r)/K, (k p a 2 ) ) Z p (z)] ...(2.45) where, Z 0 = N 0~ 1 / / 2 cosh[k 0z] ...(2.46a) N 0 = [1+sinh{2k 0d}/2k 0d]/2 ...(2.46b) 22 Z n = N n ~ 1 / 2 cos[k n z] ...(2.47a) N n = [1+sin{2k 0d}/2k 0d]/2 ....(2.47b) E 0 and E n are unknown c o e f f i c i e n t s determined as described later and k 0 and k are the roots of n c j 2-gk 0tanh[ k 0d] = 0 w2+gkn tan[k n d] = 0 respectively. $ E as defined in (2.43) s a t i s f i e s (2.22), (2.23), (2.43) and (2.44). In the above, I i s the modified Bessel function of the f i r s t kind, K i s the modified Bessel function of the second kind, J i s the Bessel function of the f i r s t kind and H i s the Hankel function of the f i r s t kind. 2.4 SOLUTION FOR UNKNOWN COEFFICIENTS FOR POTENTIALS The unknown c o e f f i c i e n t s of the series for the potentials A_ , B_ , C , D and E„ are determined by matching the n n n n n 1 3 pressures and v e l o c i t i e s between adjacent regions. The procedure i s detailed in Appendix 1. The matching of the pressures and v e l o c i t i e s between adjacent regions 23 results in five systems of linear simultaeneous equations. The number of equations in each system i s equal to the number of terms taken for the series and i s also equal to the number of unknown c o e f f i c i e n t s . The system of equations i s solved using a Gaussian elimination routine CDSOLN which permits the use of double precision complex variables. The routine i s available on the Michigan Terminal System at the University of B r i t i s h Columbia and i s e f f i c i e n t . T y p i c a l l y , for 20 terms in the series, the procedure involves solving a 100 X 100 matrix and the Amdahl 470 V/8 computer accomplishes t h i s in 2.6 seconds. 2.5 CALCULATION OF THE ADDED MASS AND DAMPING COEFFICIENTS The added mass and damping c o e f f i c i e n t s of the compound cylinder in heave motion are related to the potential as follows from (2.15). a22 / p V + i b22 / p V w = 1 / V /J*s 0 ( r ' 6 ' Z ) n2 d S ...(2.49) Density of the medium. Volume of the cylinder Frequency of the motion. Added mass in heave motion. Damping c o e f f i c i e n t in heave motion. Unit normal in the z d i r e c t i o n . P V a b n. 22 '22 24 S denotes integral over the surface of the cylinder. The added mass and damping c o e f f i c i e n t are non-dimensionalized as shown. The potential 0 i s defined as di f f e r e n t potentials in dif f e r e n t regions. Hence the integral in (2.49) can be written as, 2ir a! a 2 j ; s 0(r, 9, z) n z dS = 0 J ioS * i + a S <t>2 a 2 + a S * 3 ( - l ) ] r dr dfl 3 3 ...(2.50) Using the a x i a l symmetry of the potential functions (50) can be written as, a, a 2 s „, . . z _ - ... l 0 ; 0, + a i SSe 4>(r, 0, z) n^ dS = 2TT [ 0 J *I + a J <t>2 a z + J 03(-1)1 r dr ae a 3 ...(2.51) Evaluating the three integrals in (2.51) independently we can rewrite (2.49) as below, a22 / p V + i b22 / p V u = ( 2 7 f / V ) ^ I N 1 + I N 2 + I N 3 ] ...(2.52) where IN,, IN 2, IN 3 are as evaluated in Appendix 1. 25 2.6 CALCULATION OF HEAVE EXCITING FORCE The amplitude of the heave exc i t i n g force can be calculated from the damping c o e f f i c i e n t b 2 2 using a relationship given by Newman /8/ as follows. F 2 2 = (pgH2b22u>/2k2) [1 + 2kd/sinh( 2kd) ]}~ 1 / / 2 ...(2.53) The heave exciting force computed as above i s compared to the experimental results and a prediction by a boundary element program, l a t e r . In (2.53), F 2 2 = Heave exciting force b 2 2 = Heave damping c o e f f i c i e n t p = density of medium g = acceleration due to gravity H = incident wave height u> = wave c i r c u l a r frequency d = water depth k = Wave number 2.7 CALCULATION OF WAVE AMPLITUDE The wave elevation at any point on the free surface can be computed from the potential for the flow f i e l d using the following equation from linear theory /9/. .(2.54) 26 where, 7? i s the wave elevation and g is the acceleration due to gravity. $ i s the potential function for the region in the flow f i e l d where the wave elvation i s desired. 2.8 COMPUTER PROGRAM A computer program, CYLINDER was written to set up and solve the system of equations to determine the unknown c o e f f i c i e n t s for the potentials as described e a r l i e r in t h i s chapter. The program also computes the potentials and their normal derivatives at any point in the flow f i e l d . F i n a l l y , the program computes the added mass and damping c o e f f i c i e n t s for the compound c i r c u l a r cylinder in heave motion. The program requires as input the dimensions a n , a 2 , a 3 , d,, d 2 and d 3 of the compound c i r c u l a r cylinder as shown in F i g . 2. It also requires the frequency of the motion, the depth of water,d and the number of terms to be taken in the series for the po t e n t i a l . The program takes 2.8 seconds to compute the added mass and damping c o e f f i c i e n t for the compound cylinder at one frequency. 3. EXPERIMENTAL WORK Experiments were conducted in the towing tank at the Ocean Engineering Centre (O.E.C.) at B.C.Research, Vancouver to v e r i f y the theoretical r e s u l t s . 3.1 PURPOSE OF EXPERIMENTS Two d i f f e r e n t sets of experiments were conducted. The f i r s t set of experiments were performed to evaluate the hydrodynamic c o e f f i c i e n t s in heave motion of a compound c i r c u l a r cylinder model. The experiments were conducted to v e r i f y the theoretical predictions of the hydrodynamic c o e f f i c i e n t s . The compound cylinder model was tested at four d i f f e r e n t drafts to estimate the effect of the depth of Region 3 ( f i g . 3) on the added mass and damping c o e f f i c i e n t . Wave heights at a fixed distance from the cylinder centreline were measured to compare them with the t h e o r e t i c a l l y predicted wave heights using Egn. 2.54. The second set of experiments were conducted to evaluate the heave exciting force on three models - a compound cylinder, a double cylinder and a single cylinder. Dimensions of the three cylinders are shown on Figs. 4, 5, and 6. Descriptions of the models are given in Appendix 2. The exciting forces on the compound cylinder model were used to compare with the values t h e o r e t i c a l l y computed from the heave damping c o e f f i c i e n t calculated using the matching technique. This provides a method for v e r i f y i n g the heave damping c o e f f i c i e n t computed using the matching technique, 27 28 as well as a v e r i f i c a t i o n of the relationship between the heave damping c o e f f i c i e n t and the heave exciting force (Eqn. 2.53). The experiments with the single and double cylinder models were used to v e r i f y a prediction using the boundary element method. The experimental f a c i l i t i e s used are described in Appendix 2. Besides the experiments mentioned above, a qu a l i t a t i v e observation of the vortex shedding during the compound cylinder motion was also made. 3.2 DETERMINATION OF HYDRODYNAMIC COEFFICIENTS Forced harmonic o s c i l l a t i o n s of the compound cylinder model were used to determine i t s hydrodynamic c o e f f i c i e n t s in heave motion. Fig.7 shows a photograph of the towing tank. Its dimensions and other p a r t i c u l a r s are given in Appendix 2. Fig . 8 shows a photograph of the motion generator used along with the associated pump and control gear. A brief description i s given in Appendix 2. The cylinder model was mounted as shown , ballasted suitably so that i t was nearly neutrally buoyant. Fig.9 shows the equipment on board the O.E.C. towing carriage which was used for the data c o l l e c t i o n . These are also described in Appendix 2. The motion generator gives a small amplitude sinusoidal motion to the cylinder model at a fixed frequency. The amplitude of o s c i l l a t i o n can be varied. The force on the cylinder i s measured continuously with the help of 29 dynamometers. Two types of dynamometers were used. One i s a 3-component measuring device which i s capable of reading a v e r t i c a l force, a horizontal force and moment simultaneously. Interaction effects of the three quantities are minimized by a suitable c i r c u i t design. The second type of dynamometer used were Universal Shear Beams manufactured by HBM Inc. of Framingham, MA. These were capable of measuring only v e r t i c a l forces. Two of these used in conjunction. Moments and horizontal forces were excluded by design. The dynamometers are described in Appendix 2. The cylinder motion was measured by a sonar displacement measurement device positioned over the cylinder model. Tests were performed at two d i f f e r e n t amplitudes of motion of 10 mm. and 15 mm. to check the dependence of the force measurements and hence the added mass and damping c o e f f i c i e n t on the amplitude of motion. Thus, continuous simultaneous records of the v e r t i c a l forces on the cylinder and the displacement of the cylinder were obtained. The signals were suitably amplified using amplifiers on the O.E.C. towing carriage and stored on the O.E.C. MINC-11 computer. The data c o l l e c t i o n software i s described in Appendix 2. The data was multiplexed and stored on disk. 30 3.2.1 DATA ANALYSIS The data was demultiplexed and a spectral analysis was performed on the force and displacement records, using Fast Fourier Transforms. This was done to eliminate noise and extraneous measurements other than at the driving frequency. The spectral analysis yielded the force and displacement amplitudes and phases at the driven frequency. The phases were corrected for the phase s h i f t s induced by the amplifiers. Knowing these the forces in phase with the acceleration, F A and in phase with the v e l o c i t y , F^ were computed. Then, F A = (m+a22 )x = -(m+a22 ) c o 2 x 0 e~ lut ...(3.1) and, F v = b 2 2 x = -ia>(b 2 2 x 0) e 1 C J t ...(3.2) where, x 0 = Amplitude of the displacement. m = mass of cylinder to = Driven frequency a 2 2 = Added mass c o e f f i c i e n t b 2 2 = Damping c o e f f i c i e n t From the above a 2 2 a n <3 b 2 2 ^ o r t* i e driven frequency are calculated as, a 2 2 = _ ( F A /^ 2x 0) " m t>22 = F y /cox0 Experiments were performed at about ten frequencies for each d r a f t . The experiments were repeated for d i f f e r e n t d r a f t s . The results are plotted in non-dimensional form as 31 described l a t e r . The software used for the data c o l l e c t i o n and analysis are described in Appendix 2. 3.3 DETERMINATION OF HEAVE EXCITING FORCE For these experiments the cylinder was held stationary at a specified d r a f t . Small amplitude sinusoidal waves were generated using the paddle type wavemaker at the O.E.C. Fig.10 shows a photograph of the wavemaker. A brief description of i t s operation and components i s given in Appendix 2. A resistance wave probe was used to measure the wave height. A description i s given in Appendix 2. The v e r t i c a l exciting force on the model was recorded using the 3-component dynamometer. The MINC-11 computer and the O.E.C. towing carriage were used for data c o l l e c t i o n as before. A spectral analysis on the wave record using Fast Fourier Transforms gave the wave amplitude and phase at a pa r t i c u l a r frequency. The same analysis on the force record gave the force amplitude and phase. The wave phase was adjusted for the position for the wave probe. Thus, the v e r t i c a l e xciting force due to a wave at a specified amplitude and frequency was determined. The experiments were repeated for d i f f e r e n t frequencies for each draft for three cylinder models. The configurations were as shown in Figs. 4,5, and 6. The non-dimensional exciting force per foot of wave height was plotted versus frequency. The results are 32 compared with a theoretical prediction by a boundary element method by Chan /11/ . The results are discussed l a t e r . The software used i s described in Appendix 2. 3.4 FLOW VISUALIZATION Flow v i s u a l i z a t i o n tests were conducted to observe q u a l i t a t i v e l y the vortex shedding process on a single cylinder model and a compound cylinder model. Dye was injected through points on the cylinder's surface by means of c a p i l l a r y tubing. The cylinder was o s c i l l a t e d and the vortex shedding was observed through an illuminated underwater window at the towing tank. A record was made on video tape. F i g . 11 shows a photograph i l l u s t r a t i n g the vortex shedding. 4. RESULTS AND DISCUSSION The added mass and damping c o e f f i c i e n t computed numerically using the matching technique are compared to the results from a boundary element method by Chan /11/ . The heave added mass and damping c o e f f i c i e n t are determined experimentally for the compound cylinder model for four drafts of 35.5", 39.5", 43.5", 47.5". These correspond to a step size, D' of 6", 10", 14" and 18" respectively. D' i s as defined in Fig.2. The heave added mass and damping c o e f f i c i e n t are determined for two dif f e r e n t amplitudes of o s c i l l a t i o n of 10mm and 15mm for the 35.5" d r a f t . This is done to check the effect of the amplitude of o s c i l l a t i o n on the hydrodynamic c o e f f i c i e n t s . For each draft the tests are conducted for over 10 frequencies in the range 0.2 to 2.5 Hz. Wave amplitude at a distance of 33" from the centreline of the o s c i l l a t i n g cylinder i s measured during the tests. Measurement of a l l quantities were done for at least 10 frequencies at each dr a f t . The hydrodynamic c o e f f i c i e n t s determined experimentally and the wave height measured during the experiments are compared to theoretical predictions using the matching technique. Flow v i s u a l i s a t i o n tests were conducted during the o s c i l l a t i o n of the compound cylinder model. They were conducted to observe the vortex shedding process. Heave exciting forces due to small amplitude sinusoidal waves on a single cylinder, a double cylinder and a compound 33 34 cylinder were measured. The single cylinder was tested at two drafts of 7" and 10.5". The double cylinder was tested at two drafts of 23.5" and 27.5". The compound cylinder was tested at four drafts of 35.5", 38.375", 42.625" and 49.5". The configurations for the three cylinder models were as shown in Figs. 6,5 and 4 respectively. For each draft, tests were performed for at least two d i f f e r e n t amplitudes to check the dependence of the exciting forces on the amplitudes of the waves. For each draft, at each amplitude setting, the tests were performed for at least seven wave frequencies over the range of 0.25 to 2.5 Hz. The experimental results for the compound cylinder are compared to theoretical predictions by the matching technique and a boundary element method. The experimental results for the single and double cylinder are compared only to the the o r e t i c a l predictions by the boundary element method. The theoret i c a l method using the matching technique i s as described in Chapter 2 and Appendix 1. The experimental technique, equipment and software are described in Chapter 3 and Appendix 2. 4.1 PRESENTATION OF DATA The data i s presented in the form of non-dimensional p l o t s . Exceptions are the plots of wave height and heave exciting forces. 35 The heave added mass is plotted as a 2 2/pV versus w2a/g. a 2 2 i s the heave added mass, p i s the density of fresh water, and V i s the volume of buoyancy for the cylinder at the respective d r a f t , CJ is the c i r c u l a r frequency of o s c i l l a t i o n of the cylinder in radians/second, a is the maximum radius of the cylinder and g i s the acceleration due to gravity. The heave damping c o e f f i c i e n t i s non-dimensionalised as b 2 2 / pVco, where, b 2 2 i s the heave damping c o e f f i c i e n t . The damping c o e f f i c i e n t i s plotted against the same non-dimensional frequency as the added mass. The wave amplitude in inches i s plotted against the same non-dimensional frequency. The heave exciting force, F is plotted as F/pVgA versus co2a/g. Here u> i s the wave c i r c u l a r frequency and A i s the wave amplitude in feet. A l l other quantities are as above . 4.2 DISCUSSION OF THEORETICAL RESULTS The computer program CYLINDER computes the added mass and damping c o e f f i c i e n t in heave motion for the compound c i r c u l a r cylinder using the matching technique. For a l l results reported here, 20 terms were taken for each of the Fourier series for the potentials. The program was run for 5, 10, 15, 20, and 30 terms for the series and i t was seen that s a t i s f a c t o r y convergence of the hydrodynamic c o e f f i c i e n t s were acheived with 20 terms. The program takes 2.8 seconds of CPU time on the Amdahl 470 V/8 computer to calculate the hydrodynamic c o e f f i c i e n t s for the compound 36 cylinder at one frequency of o s c i l l a t i o n when 20 terms are taken for the series. The matching technique s a t i s f i e s the continuity of pressure and ve l o c i t y between adjacent regions in which potentials are defined (Chapter 2). But, t h i s continuity i s s a t i s f i e d as an integral over the depth and not at every point on the common boundary between adjacent regions. Hence, when pressures and v e l o c i t i e s are computed using the solved potentials, they are not continuous r a d i a l l y at points along the boundary between adjacent regions defined in Chapter 2. This discontinuity i s more apparent in the ra d i a l derivatives of the potentials, than in the potentials themselves. Since the potentials are not affected s i g n i f i c a n t l y by this fact, the hydrodynamic c o e f f i c i e n t s which are computed using the potentials are not affected. The computer program CYLINDER was run for the compound cylinder configuration shown in F i g . 4 at d i f f e r e n t d r a f t s . Numerical d i f f i c u l t i e s were observed with step sizes, D', less than 6". These are due to the d i f f i c u l t y solving the dispersion r e l a t i o n for waves generated by a very shallow step size, D'. The comparison of the results using the matching technique with predictions by the boundary element method show very good agreement. Fig.13 shows added mass values for the compound cylinder at a draft of 42.625" (D'=13.125"). The matching technique overpredicts the added mass in comparison to the boundary element method by approximately 37 4% at the higher frequencies. The agreement improves as the frequency decreases. Fig.14 shows a comparison of the damping c o e f f i c i e n t s computed by the matching technique and the boundary element methods. Here, the matching technique underpredicts the damping c o e f f i c i e n t in comparison to the boundary element method. But, the difference i s within 1% for most frequencies. 4.3 DISCUSSION OF EXPERIMENTAL RESULTS 4.3.1 HYDRODYNAMIC COEFFICIENTS The experiments to determine the hydrodynamic c o e f f i c i e n t s in heave motion for the compound cylinder were f i r s t conducted in the Summer of 1983. However, the motion generator used for the experiments did not perform s a t i s f a c t o r i l y . The hydraulic pump and motor used for running the motion generator could not handle the load adequately. The scotch-yoke mechanism used in the motion generator induced considerable extraneous loads on the dynamometers due to vibrations. Further, i t f a i l e d to produce a smooth sinusoidal motion. For the above reasons the motion generator was redesigned and reconstructed and the tests repeated in June, 1984. The redesigned motion generator gave a smooth sinusoidal motion over a frequency range of 0.2 to 2.5 Hz. It was also capable of handling the imposed loads well. Fig.12 shows a time record of the motion and Fig.12a shows 38 an amplitude spectrum of the time record. It can be seen that the spectrum does not show any prominent peaks except at the driving frequency. Descriptions of the motion generator are given in Appendix 2. Two 500 lb load c e l l s were added to the system to v e r i f y the force measurement by the 801b force block. They corroborated the readings of the 80 l b . force block which was used primarily for the data analysis. Details of the mounting of the 500 l b . force blocks are v i s i b l e in Fig.8a. Fig.15 shows a plot of the heave added mass of the compound cylinder model at a draft of 35.5" (D'=6"). The theoretical prediction i s by the matching technique. The experimental plots show results for two amplitudes of o s c i l l a t i o n of 0.39" and 0.5". The experimental results show an apparent increase of added mass with frequency. Reliable experimental results could not be obtained at frequencies below 1 Hz, because the magnitude of the forces measured were too small and the s e n s i t i v i t y of the dynamometers was not s u f f i c i e n t enough to measure values less than 0.5% of the f u l l scale range accurately . The deviation in the added mass results with change in amplitude setting i s not s u f f i c i e n t enough within the l i m i t s of the experimental accuracy to warrant any conclusions regarding non-linearity. The experiments at the remaining three drafts were conducted at the 0.5" amplitude setting, so that the forces would be large enough to measure at the lower frequencies. The theoretical curve shows a well defined peak at a frequency 39 of about 0.75 Hz. The added mass c o e f f i c i e n t then drops sharply t i l l a frequency of about 1.5 Hz and remains f a i r l y constant t i l l about 2.5 Hz, the upper end of the frequency range. The best agreement between the th e o r e t i c a l and experimental results is at frequencies close to 1.25 Hz. In general the experimental values are higher than the the o r e t i c a l r e s u l t s . Fig.16 shows a plot of the heave damping c o e f f i c i e n t for the compound cylinder model at a draft of 35.5". The experimental results are compared to the th e o r e t i c a l prediction by the matching technique. The experimental results follow the same trend as the th e o r e t i c a l curve, but the experimental values are considerably higher than the the o r e t i c a l values. Though the peak value appears at the same frequency for both experiment and theory, the magnitude of the damping c o e f f i c i e n t as shown by experiment i s higher by about 80% when compared to the th e o r e t i c a l r e s u l t . Fig.17 shows plots of the wave amplitude measured at a distance of 33" from the o s c i l l a t i n g cylinder during the experiments. The theore t i c a l curve i s a prediction by the matching technique. The t h e o r e t i c a l prediction i s considerably higher than the experimentally measured values. They d i f f e r by about 50%. The difference may possibly be due to the interference from the tank walls. Figs. 18,21 and 24 are plots of the added mass for the compound cylinder at drafts of 39.5", 43.5" and 47.5", respectively. The amplitude of o s c i l l a t i o n i s 0.5" in a l l 40 cases. The comparison between experiment and theory i s much the same as for the 35.5" draft. However, the absolute value of the non-dimensional added mass decreases with increasing draft, according to the theoreti c a l prediction. The experimental re s u l t s , however, show the same range of variation for a l l the dra f t s . Figs. 19, 22 and 25 are plots of the heave damping c o e f f i c i e n t for cylinder drafts of 39.5", 43.5" and 47.5" respectively. The trends in the comparison between the theory and experiment are similar to those for the 35.5" draf t . The value of the non-dimensional damping c o e f f i c i e n t decreases with increasing draft. This i s shown by both theory and experiment. This can be explained by the increase in step si z e , D' with increasing d r a f t . This leads to a decreasing wavemaking action and hence a smaller damping c o e f f i c i e n t at the deeper dr a f t s . Figs. 20, 23 and 26 are plots of the wave amplitude measured at a distance of 33" from the compound cylinder centreline for drafts of 39.5", 43.5" and 47.5". These plots show the same trend as Fig.17 for the 35.5" dra f t . The theo r e t i c a l prediction i s higher than the experimentally measured wave amplitudes. But, the experimental values show the same trend as the theoreti c a l curve. The the o r e t i c a l maximum wave amplitudes steadily decrease from about 0.3" at the 35.5" draft to about 0.1" at the maximum draft of 47.5". This can be attributed to the decreasing wavemaking action with increasing draft as mentioned in the previous 41 paragraph. The experimental results also show the same trend, decreasing from a maximum wave amplitude of about 0.15" at a 35.5" draft to a maximum of about 0.04" at the maximum draft of 47.5". The added masses and damping c o e f f i c i e n t s determined experimentally do not show very close agreement with the theoretical r e s u l t s . This can be due to two reasons. The theory does not take into account the v i s c o s i t y of the f l u i d medium, which exists in actual fac t . The v i s c o s i t y induces vortex shedding during the cylinder motion. This was observed during the flow v i s u a l i s a t i o n tests conducted on the compound cylinder model (Chapter 3 and Fig.11). Vortex shedding takes place at the corners of the cylinder during the cylinder motion. This introduces viscous damping in addition to the potential damping due to the wavemaking action. This may explain why the experimentally determined damping c o e f f i c i e n t i s considerably higher than the th e o r e t i c a l l y predicted value. The extent of the viscous damping can be determined by modelling the vortex shedding at the corners t h e o r e t i c a l l y . The second cause for the difference between experimental and theoreti c a l values may be due to the effect of the walls of the towing tank on the flow f i e l d during the cylinder motion. The walls are at a distance of 62" from the cylinder centreline on one side and 82" on the other side. The effect of the presence of the walls can only be isolated by performing the same tests in a very large basin and 42 comparing the results. A further cause for the difference between the experimental and theoretical results may be inaccuracies in the measurement of the force, displacement and the phase s h i f t s of the amplifiers. Care was taken in the c a l i b r a t i o n process to eliminate these as much as possible. However, a very small change of phase introduced e l e c t r o n i c a l l y in the data c o l l e c t i o n process may a f f e c t the force in phase with the acceleration more than the force in phase with the v e l o c i t y , since the former i s proportional to the cosine of the phase and the l a t t e r i s proportional to the sine of the phase. Thus, i f a small error i s introduced in the phase this w i l l a f f e c t the added mass more than the damping c o e f f i c i e n t and the error w i l l increase as the forces measured increase. This may explain the reason for the increase in the added mass with frequency. 4.3.2 FLOW VISUALISATION TESTS These tests were conducted in June,1983. A description of the tests i s given in Chapter 3. They show the presence of vortex shedding at the corners of the compound c i r c u l a r cylinder. They suggest that the effects of v i s c o s i t y may be important. The tests were recorded on video tape and photographs were also taken during the t e s t s . Fig.11 shows one such photograph. 43 4.3.3 HEAVE EXCITING FORCES Heave exciting forces due to small amplitude sinusoidal waves were measured on three d i f f e r e n t cylinder models. These experiments were performed in July, 1983 at the towing tank at B.C.Research. The purpose of these experiments was to v e r i f y the relationship between the heave damping c o e f f i c i e n t and the heave exciting force for the compound cylinder model. These experiments were also intended to be a further v e r i f i c a t i o n of the heave damping c o e f f i c i e n t for the compound cylinder model t h e o r e t i c a l l y calculated using the matching technique. The experimental results for the other cylinder models were intended as a v e r i f i c a t i o n of the heave exciting forces computed by a boundary element method. The boundary element method applied by Chan /11/ used linear d i f f r a c t i o n theory and hence can be considered a more dire c t computation of the exciting force. The experimental procedure i s described in Chapter 3 and a description of the equipment i s given in Appendix 2. Fig.27 i s a plot of the heave ex c i t i n g force on single cylinder model at a draft of 7". The range of wave frequency is from 0 to 2.5 Hz. The experimental results are compared to a th e o r e t i c a l prediction by a boundary element method. The experimental measurements were made for waves of three di f f e r e n t amplitude settings of 0.12", 0.2" and 0.4". The amplitudes are approximate since the wave amplitudes cannot be duplicated exactly. The waveform obtained was also s l i g h t l y i r r e g u l a r . The causes for t h i s are discussed in 44 Appendix 2. The results show quite good agreement with the theoretic a l prediction by the boundary element method. The experimentally measured peak exciting forces are at about 0.5 Hz and are higher than the t h e o r e t i c a l l y predicted peak value which occurs at the same frequency. The experiments results show a scatter of about 15% at a frequency of 1.25 Hz. These can be due to the inaccuracies in the measurement of the forces and the wave heights caused by the i r r e g u l a r i t y of the generated waveform. Fig.28 i s a plot of the phase difference between the exciting force and the wave. The experimentally measured values show considerable scatter and deviation from the theo r e t i c a l prediction by the boundary element method. This scatter may again be due to inaccuracies in the measurement of the waves because of their i r r e g u l a r i t y . Fig.29 shows the heave exciting force results for the single cylinder model at a draft of 10.5". The scatter in the experimental results i s less than for the 7" dra f t . The experimental results show quite good agreement with the the o r e t i c a l prediction by the boundary element method. Fig.30 shows a plot of the phase difference between the heave exciting force and the wave for the 10.5" dra f t . The phase values do not show good agreement with the theoreti c a l predict ion. Figs. 31 and 33 show the plots of heave exciting forces for the double cylinder model (Fig.5) at drafts of 23.5" and 27.5" respectively. Here, the agreement with th e o r e t i c a l 45 prediction by the boundary element method i s not very good. There i s considerable scatter in the data. In general, the experimental results are higher than the theore t i c a l prediction. Figs. 32 and 34 show plots of the phase difference between the exciting force and the wave. The experimental values show considerable scatter and do not show good agreement with the theory. Fig.35 i s a plot of the heave exciting force for the compound cylinder model (Fig.4) at a draft of 35.5". Experimental results are compared with theore t i c a l predictions by the matching technique and the boundary element method. Both th e o r e t i c a l methods overpredict the exciting force at frequencies above 1.25 Hz. The boundary element method gives a lower value of the exciting force except at the lower frequencies. Both theoretical methods show good agreement. The agreement i s best at the lowest and highest frequencies. At frequencies lower than 0.75 Hz, the experimentally measured exciting forces are higher than the theoret i c a l prediction. Fig.36 shows a plot of the force-wave phase difference for the compound cylinder at a draft of 35.5". The phase plot shows considerable scatter in the experimental values and the agreement between theory and experiment i s not good. Fig.37 shows heave exciting force for the compound cylinder at a draft of 38.375". Here the theore t i c a l results by the boundary element method underpredicts the exciting force in comparison to the matching technique at most 46 frequencies. The boundary element method results show good agreement with the results from the matching technique. In general, the experimental results show good agreement with the t h e o r e t i c a l predictions. The scatter in the experimental data i s low except at around 0.25 Hz. Fig.38 shows the force-wave phase difference for the compound cylinder at 35.5" d r a f t . The experimental results show scatter and do not agree well with the th e o r e t i c a l prediction. Fig.39 shows the heave exciting force for the compound cylinder at a draft of 42.625". Here, the prediction by the boundary element method i s close to that by the matching technique at a l l frequencies. The agreement i s best at the highest and lowest frequencies. The experimental results are mostly lower than the th e o r e t i c a l predictions. They show good agreement with the theory at two frequencies. Fig.40 shows the force-wave phase difference for the same draft. Again, the agreement with theory i s not good. Fig.41 shows the heave exciting force for the compound cylinder at a draft of 49.5". The theoreti c a l prediction by the boundary element method i s close to that by the matching technique. The agreement i s best at the higher frequencies. In general, the experimental results show good agreement with the the o r e t i c a l predictions. However, there are a few scattered points which show considerably higher exciting forces than the theory. Fig.42 shows a plot of the force-wave phase difference. Again, the experimental values show scatter and do not agree well with theory. 47 In general, the exciting force measurements show good agreement with theoretical predictions. The exciting forces computed from the damping c o e f f i c i e n t s calculated by the matching technique are close to the values computed d i r e c t l y by the boundary element method. Hence, i t can be inferred that the relationship suggested by Newman i s good. It can be inferred that the the o r e t i c a l results computed using linear d i f f r a c t i o n theory can be corroborated by experimental r e s u l t s . Though there i s some scatter in the experimental forces measured from d i f f e r e n t amplitude waves, there i s enough evidence to suggest v a l i d i t y of the lin e a r theory for waves of these amplitudes. The measurement of phases is not s u f f i c i e n t l y r e l i a b l e to enable any d e f i n i t e conclusions regarding their trends. CONCLUSIONS 1 . The t h e o r e t i c a l method presented here, using the continuity of pressure and ve l o c i t y between regions of the flow f i e l d and a Fourier series representation for the potentials, provides a good method for determining the hydrodynamic c o e f f i c i e n t s in heave motion for the compound c i r c u l a r cylinder t h e o r e t i c a l l y . The comparison with the boundary element shows good agreement. The input data required for the present method i s minimal. The method is capable of computing the pressures and v e l o c i t i e s at any point in the flow f i e l d , and hence the wave height at any point on the free surface also. 2. The computer program CYLINDER, used to compute the added mass and damping c o e f f i c i e n t for the compound c i r c u l a r cylinder in heave motion i s e f f i c i e n t . It takes 2.8 seconds of CPU time on the Amdahl V-8/470 computer to compute the hydrodynamic c o e f f i c i e n t s for one frequency of o s c i l l a t i o n . 3. The t h e o r e t i c a l method being a potential flow method i s unable to account for the v i s c o s i t y of the f l u i d medium and hence the vortex shedding during the o s c i l l a t i o n fo the cylinder. The method also gives numerical problems when the step si z e , D' i s made very small. Further, in the computation of r a d i a l v e l o c i t i e s in the flow f i e l d , exact continuity of v e l o c i t y i s not acheived at every point on the 48 49 common boundary between two adjacent regions in the flow f i e l d . 4. The experimental results for the added mass show larger values than the theoreti c a l prediction by the matching technique. The experimental results also show an increase with frequency. This can be due to an error in measurement of the phase s h i f t . Experimental values could not be determined at frequencies lower than 1 Hz, due to lack of s u f f i c i e n t s e n s i t i v i t y in the force recording equipment. 5. The experimental heave damping c o e f f i c i e n t values show the same trends as the theoreti c a l values, but are considerably higher than the theoreti c a l values. This can partly be due to viscous e f f e c t s during the cylinder motion and partly due to the ef f e c t s of the walls of the towing tank on the flow f i e l d . 6 . The flow v i s u a l i s a t i o n tests show the vortex shedding occuring at the corners of the cylinder. This suggests the presence of viscous damping, which i s not accounted for theoret i c a l l y . 7. The added masses and damping c o e f f i c i e n t s obtained using d i f f e r e n t amplitudes of o s c i l l a t i o n do not show s i g n i f i c a n t deviation. 8. 50 The wave amplitudes measured during o s c i l l a t i o n of the compound cylinder model do not show good agreement with the theoretical prediction by the matching technique. During the tests interference with waves re f l e c t e d off the side walls of the towing tank was observed. This may be one of the reasons for the large difference between the o r e t i c a l and experimental r e s u l t s . Further, the actual magnitude of the wave heights measured i s small. Hence, the s e n s i t i v i t y of the wave probe may have some eff e c t on the recorded data. 9. The heave damping c o e f f i c i e n t s and the wave heights for the compound cylinder model decrease s i g n i f i c a n t l y as the draft increases due to decreasing wavemaking action. Both theoretical and experimental results show t h i s trend. 10. The heave exciting forces measured experimentally for the compound cylinder model show good agreement with theoretical predictions by the matching technique and the boundary element method. This suggests that linear d i f f r a c t i o n theory provides results which can be corroborated by experiment. The agreement with the results from the matching technique show the v a l i d i t y of the rel a t i o n s h i p between the heave damping c o e f f i c i e n t and the heave exciting force. 11. The heave exciting forces measured on the single cylinder model show very good agreement with theore t i c a l predictions by the boundary element method. The agreement between theory 51 and experiment i s not very good for the tests on the double cylinder model. 12. The phases measured during the heave exciting force tests on a l l three cylinder models do not show good agreement with theoret i c a l r e s u l t s . They also show considerable scatter. No r e l i a b l e inference can be made from these r e s u l t s . The scatter can be attributed to inaccuracy in measurement of the wave elevation due to i r r e g u l a r i t y of the waveform generated. 13. Though the heave exciting forces measured for the d i f f e r e n t amplitude settings show some scatter there i s enough evidence to j u s t i f y the assumption of l i n e a r i t y for small amplitude waves. 14. The tests show that the e f f e c t s of v i s c o s i t y of the f l u i d medium are more important for the measurement of forces on the o s c i l l a t i n g cylinder than for the measurement of exciting forces due to small amplitude waves. 15. F i g . 43 shows hydrodynamic c o e f f i c i e n t s for a single cylinder measured by McCormick /1/, /20/. These tests were conducted in a towing tank at the U.S. Naval Academy. The frequency of o s c i l l a t i o n i s 3 radians/sec. The experimental results show very much higher values than the theoretical r e s u l t s . These results are comparable to present comparison 52 between theory and experiment for the compound cylinder model. ANU/MH in Fig.43 i s the added mass non-dimensionalized by mass of the cylinder of height equal to the depth of water. RECOMMENDATIONS The experimental results for the heave added mass and damping c o e f f i c i e n t s do not show very good agreement with the theoretical predictions by the matching technique. The heave exciting forces computed t h e o r e t i c a l l y from the damping c o e f f i c i e n t show better agreement with the experimental r e s u l t s . The flow v i s u a l i s a t i o n tests show the presence of vortex shedding during the cylinder o s c i l l a t i o n . These suggest that the eff e c t s of v i s c o s i t y may be r e l a t i v e l y important. Also, the ef f e c t s of the interference due to the presence of the side walls of the towing tank are not assessed at present. To estimate the effects of the above i t would be desirable to do the following. 1 . Theoretically model the vortex shedding using ring vortices at the corners of the cylinder. 2. Theoretically model the ef f e c t s of the tank walls by a potential flow method such as the method of images. 3. Repeat the same tests for the o s c i l l a t i n g cylinder in an open basin where the restraining efects of the tank walls would not be f e l t . 53 NOMENCLATURE FOR PLOTS HEAVE EXCITING FORCE PLOTS F =Heave exciting force RO =Density of medium V =Buoyancy in cu. f t . of cylinder model g =Acceleration due to gravity AMP =Wave amplitude in f t . W =Wave frequency A =Maximum radius of cylinder model HYDRODYNAMIC COEFFICIENTS AND WAVE AMPLITUDE PLOTS a22 =Heave added mass RHO =Density of fresh water V =Buoyancy in cu. f t . of cylinder model g =Acceleration due to gravity b22 =Heave damping c o e f f i c i e n t W =Cylinder o s c i l l a t i o n frequency A =Maximum radius of cylinder model 54 LIST OF SYMBOLS p = Density of water. u> = Circular frequency of cylinder or wave motion. k 0, k n = Wave number. m0, mn = Wave number. <i>, <t> = Velocity potential J 0 , J i = Bessel function of the f i r s t kind. I 0 , I, = Modified Bessel function of the f i r s t kind K 0, K, = Modified Bessel function of the second kind H 0, H, = Hankel function of the f i r s t kind, d = Depth of water a,, a 2 , a 3 = Cylinder r a d i i . d 1 , d 2, d 3 = Depth from cylinder, g = Acceleration due to gravity NOTE This l i s t of symbols i s not complete. The various symbols are defined at the points in the text where they are used. 55 BIBLIOGRAPHY 1. SABUNCU,T. and CALISAL, S.M., "Hydrodynamic c o e f f i c i e n t s for v e r t i c a l c i r c u l a r cylinders at f i n i t e depth," Ocean  Engineering vol.8, 1981. 2. KOKKINOWRACHOS,K. et a l . "Belastungen und bewegungen gro|3volumiger seebauwerke durch wellen," Nr. 2905/ Fachgruppe  Maschinenbau / Verfahrenstechnik Forschungsbericht des  Landes Nordrhein-Westfalen, 1980 3. KOKKINOWRACHOS,K., HOEFELD, J. , "Theoretische und experimentelle untersuchungen des bewegungsverhaltens von Halbtauchern," Nr.2915/ Fachgruppe Umwelt / Verkehr  Forschungsbericht des Landes Nordrhein-Westfalen, 1980 4. GARRETT,C.J.R., "Wave forces on a c i r c u l a r dock," Journal  of F l u i d Mechanics, vol.46, part 1, 1971 5. WEHAUSEN, J.V., "The motion of f l o a t i n g bodies," Annual  Review of F l u i d Mechanics, vol.3, 1971. 6. KRITIS, I r . B., "Heaving motion of axisymmetric bodies," Journal of Ship Research, vol.5, No.3, 1979. 7. RAMBERG, S.E., and NIEDZWECKI, J.M., Ocean Engineering vol.9, No.1,1982 8. ISAACSON, M., and SARPKAYA,T., "Mechanics of Wave Forces  on Structures," Van Nostrand Publications, 1980 9. NEWMAN, J.N., "Marine Hydrodynamics," The MIT Press, 1980. 10. MacCAMY, R.C., FUCHS,R.A., "Wave Forces on P i l e s : A D i f f r a c t i o n Theory," Tech. Memorandum No.69 Beach Erosion  Board 1954 11. CHAN, J.L.K.," Hydrodynamic c o e f f i c i e n t s for axisymmetric bodies,"M.A.Sc. Thesis, University of B r i t i s h  Columbia, 1984 12. HAVELOCK, T.H., "Collected papers," ONR/ACE-103, U.S.  Government Printing O f f i c e , Washington D.C., 1963 13. BAI, J . , "The added mass and damping c o e f f i c i e n t s of and the exciting forces on four axisymmetric ocean platforms," Naval Ship Research and Development Center, Report No.  SPD-670-01, 1974 14. BAI, K.J. and YEUNG, R.W., "Numerical solutions to free-surface flow problems," Tenth Symposium Naval  Hydrodynamics, Cambridge, Mass. 1974 56 57 15. GARRISON, C.J., "Hydrodynamics of large objects in the sea, Part I I . Motion of fr e e - f l o a t i n g bodies," J .  Hydronautics 9 (2), 1975 16. ISSHIKI, H. and HWANG, J.H., "An axi-symmetric dock in waves," Seoul National University, Korea, College of  Engineering, Dept. of Naval Architecture, Report No. 73-1, 1973 17. NEWMAN, J.N., "The interaction of stationary vessels with regular waves," Eleventh Symposium Naval Hydrodynamics,  London, 1976 18. KIM, W.J., "On the harmonic o s c i l l a t i o n s of a r i g i d body on a free surface," J. F l u i d Mech. 21, 1974 19. ABRAMOWITZ, M. and STEGUN, I.A., "Handbook of mathematical functions," National Bureau of Standards,  Washington, D.C. 1964 20. MCCORMICK, M.E., COFFEY, J.P. and RICHARDSON, J.B., "An experimental study of wave power conversion by a heaving v e r t i c a l c i r c u l a r cylinder in r e s t r i c t e d waters," U.S. Naval  Academy, Engineering Report EW 10-80 1980 21. WANG and SHEN, "The hydrodynamic forces and pressure d i s t r i b u t i o n s for an o s c i l l a t i n g sphere in a f l u i d of f i n i t e depth,"M.I.T. Department of Naval Architecture and Marine  Engineering, Doctoral Thesis 1966 22. YEUNG, R.W., "A hybrid integral-equation method for time harmonic free surface flow,"1st Int. Conf. Numer. Ship  Hydrodynamics, Gaithersburg, Maryland 1975 5. APPENDIX 1 - EVALUATION OF POTENTIAL FUNCTIONS The c o e f f i c i e n t s of the potential functions described in chapter 2 are obtained by solving a system of linear simultaeneous equations. Five systems of equations are written and solved to give the c o e f f i c i e n t s of the fi v e series needed to describe the potentials for the flow model. These are obtained by equating the pressures (potentials) and v e l o c i t i e s (normal derivatives of potentials) along the boundaries of the regions defined in chapter 2. For example, l e t $ A and ^ be the potentials in two adjoining regions given by, T T, j -icot , *A = VH d e *A and , TT j -icot , *B = VH d e *B Along a common boundary for continuity of pressure, P A ° ~ p *A,t = 1 C J P VH d e " 1 C J t *A = PB = ~ p *B ft = 1 W P VH d e +B Or, 0 A = 0 B ...(1.1) And for continuity of r a d i a l v e l o c i t y i ,T j -icot , *A,r = VH d e ^A,r = , T T , -icot , *B,r " VH d e ^B,r where the s u f f i x 'r' denotes derivative with respect to 58 59 r.Or, A,r = 0 B,r (1 .2 ) However,the above denotes continuity of pressure and vel o c i t y at a point along a common boundary. Actually,in order to separate the c o e f f i c i e n t s of the series, the orthonormal properties of the potential functions are made use of. This i s done by equating the integral of the function over the depth as shown below. Using a continuity of pressure and integrating as below, Here, Z(z) i s an orthonormal function for d-^  <z<du . By substituting the normal derivative of the potential for the potential we have a similar r e l a t i o n for continuity of v e l o c i t y . Because the problem i s formulated in t h i s manner, the solution does not s a t i s f y continuity of pressure and ve l o c i t y at a l l points along the boundary exactly. But, the integral of the velocity and pressure along the depth for one region w i l l be equal to the respective integral in the adjoining region. An alternative formulation would be to equate the pressures and v e l o c i t i e s in adjoining regions at a number of points along the boundary. The exact number of points would be equal to the number of unknown c o e f f i c i e n t s d u <j>u Z(z)dz one obtains a set of equations. 60 in the series for the pot e n t i a l . However,the method of solution used here involves no d i s c r e t i z a t i o n of the boundary of adjoining regions. Also, i t enables one to use the orthonormal properties of the potential functions. 5.1 POTENTIALS The potentials in the various regions are as given below. 5.1.1 REGION 1 *, = V d e ~ i w t [(z 2-r 2/2)/2dd, + A 0/2 H 00 + £ A n ( I 0 (nwr/di ) / I 0 (n7T a,/d, ) )cos(niz/d, ) ] ... (I.3) 5.1.2 REGION 2 $ 2 = V d e " i w t [(z 2-r 2/2)/2dd 2+ B 0 H OS + £ = 1 {B n V n (r) + C n Wn (r)} cos (nirz/d 2) ] ... (1.4) where, v n (r) = I 0(n7rr/d 2) +( I 1(n7ra 1/d 2) / K^nTra^dz) ) K 0(n7rr/d 2) I 0(n7ra 2/d 2) +( I,(unai/d2) / K^ n j r a ^ d j ) ) K 0(n7ra 2/d 2) Wn (r) -I 0(n7rr/d 2) +( I 0(n7ra 2/d 2) / K 0(n7ra 2/d 2) ) K 0(n7rr/d 2) I 1(nira 1/d 2) +( I 0(n7ra 2/d 2) / K 0 (n7ra 2/d 2) ) K 1(n7ra,/d 2) 61 It i s e a s i l y seen that V (r) and W (r) have the •* n n following properties. At r=a, V ^ ( a , ) = 0, W j ( a , ) ='mr/d2 where the primes indicate derivative with respect to r. At r=a 2, Wn (a 2) = 0 V n ( a 2 ) = 1 5.1.3 REGION 3 The v e l o c i t y potential in t h i s region i s defined as: $ 3 = V d e ~ i w t [(z/d+g/(w 2d)-1) H 00 + D 0X 0Yo + E = 1 D n X n Y n ] ...(1.5 where J 0(m 0r) -( J , ( m 0 a 3 ) / H,(m 0a 3) ) H 0(m 0r) X 0 = J 0(m 0a 2) -( J,(moa 3) / H^moas) ) H 0(m 0a 2) I 0(m n r) +( I i ( m n a 3) / K 1 ( m n a 3) ) K 0(m n r) X = n I 0(m n a 2) -( I , ( m n a 3) / K i ( m n a 3) ) K 0(m n a 2) In t h i s formulation, Yo = M 0~ 1 / 2 cosh[m 0(z-d 3)] 62 M0 = [l+sinh{2m 0(d-d 3)}/2m 0(d-d3)]/2 Y n = Mn ~ 1 / 2 cos[m n (z-d 3)] Mn = [1+sin{2mo(d-d 3)}/2m 0(d-d 3)3/2 where m0 and mn are the roots of CL)2-gm0tanh[m0 (d-d 3 ) ] = 0 w2+gmn tan[m n (d-d 3)j = 0 respectively. It i s e a s i l y seen that X 0 and X n have the following properties. At r=a 3 X 0'(a 3) = 0, X p '(a 3) = 0, and at r=a 2 X 0 ( a 2 ) = 1, X n (a 2) = 1 . The primes denote derivative with respect to r. 5.1.4 EXTERIOR REGION The free wave and standing waves are defined with the potential * E as: *_ = V d e 1 U ) t [-E 0(H 0(ko r)/H, ( k 0 a 2 ) ) Z 0 ( z ) E H. - Z = 1 E n ( K 0 ( k n r)/K,(k n a 2 ) ) Z n (z)] ...(1.6) 63 where, Z 0 = N 0~ 1 / / 2 cosh[k 0z] N 0 = [1+sinh{2k 0d}/2k 0d]/2 Zn = N n " 1 / 2 c o s [ k n z ] N n = [1+sin{2k 0d}/2k 0d]/2 where k 0 and kfi are the roots of co 2-gk 0tanh[ k 0d] = 0 co2+qk tan[k d] = 0 3 n n respectively. 5.2 GENERATION OF SYSTEMS OF EQUATIONS FOR SOLUTION The five systems of equations neccessary to solve for the c o e f f i c i e n t s of the five series for the potentials are obtained by matching the pressures and v e l o c i t i e s at common boundaries between the regions defined . 5.2.1 CONTINUITY OF PRESSURE BETWEEN REGIONS 1 AND 2 Expressing the vel o c i t y potential $ as $ = V d e l a ) t <t>, for continuity of pressure between regions H 1 and 2, we can express 64 Pressure,p = -p4>1 fc = ~p^>2 t using l i n e a r i s e d Bernoulli's equation. As proved in equation (1.1) we can write </>n = <S>2 , for continuity of pressure at points on the interface r=a,, 0<z<di. Integrating the above equation between the l i m i t s 0 and d, after multiplying by the orthogonal function 2cos ( kirz/d n ) /d} we have, a, a, ( 2 / d 1 ) 0 / c61cos(k7rz/d1 )dz = ( 2 / d 1 ) 0 / c62cos(kwz/d, )dz Using the orthonormal properties of the cosine function we can rewrite the above equation as , Ak " fi-0 akn Bn ^kn C n = ak ...(1.7) where, a00 = 1 ' ak0 = 0 a Q n = 2d 2[V (a, ) ]sin(n7rd, /d2)/7rd,n n=1,2,3,.. j3 Q n = 2d 2[W n (a, ) Isindurd, /d2)/ird^n n=1,2,3,.. a k n = 2d 1d 2n[V n (a,)] sinfnTrd, / d 2 ) ( - D /via, 2 n 2 - d 2 2 k 2 ) 0 k n = 2d!d2n[Wn (a,)] sin(n7rd 1 / d 2 ) ( - l ) /ir(d, 2 n 2 - d 2 2 k 2 ) 65 a 0 = -d,(1-(d,/d 2)[1/3 - (a,/d,) 2]/d k a k = -2d l(1-d 1/d 2)(-1 ) /d U T T ) 2 (1.7) gives the f i r s t system of equations. 5.2.2 CONTINUITY OF VELOCITY BETWEEN REGIONS 1 AND 2 For continuity of v e l o c i t i e s between regions 1 and 2 along the boundary r=a, 0^z<d, , we have, following equation (1.2) V r = *2,r Multiplying by cos (k7rz/d 2) and integrating as follows, d, d 2 of <t> « cos(k7rz/d 2)dz = oJ" <t> o cos(k7rz/d 2 )dz at r=a, i , r z , r we have the second system of equations. The second integral i s from 0 to d 2 . Hence the normal v e l o c i t y on the surface r=a, , di^z<d 2 i s i m p l i c i t l y set to zero. Using the orthonormal properties of cos(k7rz/d 2) we arrive at the second system of equations as 00 Ck " n=0 7kn = bk where .(1.8) 66 700 = 0 ; 7 On = 0 for n=1,2 7k0 = 0 for k=1,2 n 7 t n = 2d,d 2 n(-l) sin(k7rd,/d 2) I 1 (nira ,/d, )/7r I o l n ^ / d , ) b 0 = 0 b^ = -a,d 2 sin (mrd, /d 2) /d, d (n7r) 2 5.2.3 CONTINUITY OF PRESSURE BETWEEN REGION 2 AND THE  EXTERIOR REGION Using a procedure similar to the one used for equation (1.7) we equate the pressure on the interface r=a 2, 0<z<d2 between region 2 and the exterior region. Multiplying by the function cos(k7rz/d 2) and integrating from 0 to d 2 one obtains: [ d ^ k 2 - d 2 2 n 2 ] for k=n=1,2 ( 2 / d 2 ) 0 / </» 2cos (kffz/d 2 )dz = (2/d 2) 0J" <S> E cos (k7rz/d2) dz for r=a 2 Using the orthonormal properties of the cosine function we can rewrite the above equation in the form, 6 7 B, - £ , e, E = c, ...(1.9) k n=l kn n k (1.9) i s the t h i r d system of equations, where, ek0 = - ( H o ( k o a 2 ) / H i ( k 0 a 2 ) ) G k Q for k=0,1,2,.. ; n=0 ' k n = -(K 0 (k n a 2)/H,(k n a 2 ) ) G k n for k=0,1,2,.. ; n=1,2, d 2 G Q 0 = (2/d 2) o / Z 0 dz = 2N0 1 / 2 s i n h ( k o d 2 ) / k 0 d 2 d 2 G k Q = (2/d 2) o / Zo cos ( k 7 r z/d 2) dz = 2N0 1 / 2 s i n h ( k 0 d 2 ) k o d 2 ( - 1 ) / [ ( k 0 d 2 ) 2 + (kvr)2 ] d 2 G Q n = (2/d 2) 0 ; Z n dz = 2N n 1 / 2 s i n ( k n d 2 ) / k n d 2 d 2 Gkn = < 2 / d a ) o/ Z n cos(k7rz/d 2) dz n = 2N 1 / 2 sin(k d 2)k d 2(-1) / [(k d 2 ) 2 - (kjr) 2 ] •n J 2 ' * n U 2 n 68 c 0 = - ( d 2 / d ) [ l / 3 - ( a 2 / d 2 ) 2 / 2 ] k c k = -2d 2 (-1 ) /d(k)r) 2 5.2.4 CONTINUITY OF PRESSURE BETWEEN REGION 3 AND THE  EXTERIOR REGION We have the fourth system of equations by equating the pressures on the boundary r=a 2 , d 3^z<d , between region 3 and the exterior region. Following eqn. (1.1) we have, 03 = <t>E Multiplying by the function Y(z-d 3) and integrating between d 3 and d , we have, d (1/(d-d 3)) / 0 3 Y(z-d 3)dz d 3 d = d/(d-d 3)) / 0 P Y(z-d 3)dz d 3 E Using the orthonormal properties of the function Y(z-d 3) we can rewrite the above equation to give the fourth system of equations as, Dk " 5kn E n = dk ....(1.10) where, 5kO = " < H o ( k o a 2 ) / H i ( k o a 2 ) ) L k Q for k=0,1,2,.. ; n=0 5 k n = -(K 0 (k n a 2 ) / K 1 ( k n a 2 ) ) L R n for k=0,1,2,.. ; n=1 L k n i s a special function defined as below. L Q 0 = ( l / ( d - d 3 ) ) / Z o ( z ) Y 0 (z-d 3)dz d 3 = -(M Q N Q ) 1 / 2 k 0 s i n h ( k 0 d 3 ) / [ ( d - d 3 ) ( k 0 2 - m 0 2)] L k Q = d/(d-d 3)) / Z 0 ( z ) Y k (z-d 3)dz d 3 = -(M k N Q ) " 1 / 2 k 0 s i n h ( k 0 d 3 ) / [ ( d - d 3 ) ( k 0 2 + mR 2 ) ] L0n = ( 1/(d-d 3)) / Z n (z)Y Q (z-d 3)dz d 3 = -(M Q N n ) ~ 1 / 2 k n s i n ( k n d 3)/[(d-d 3) (k„ 2 + m 0 2)] L k n = d/(d-d 3)) / Z n (z)Y k (z-d 3)dz d 3 = -(M k N n ) " 1 / 2 k n s i n ( k n d 3)/[(d-d 3) (k n 2 - mR 2 ) ] d 0 = -M0 1 / 2 / d(d-d 3)(m 0) 2 d k = -Mk " 1 / 2 / d(d-d 3)(m k ) 2 70 5.2.5 CONTINUITY OF VELOCITIES BETWEEN REGIONS 2, 3 AND THE  EXTERIOR REGION We have the f i f t h and f i n a l system of equations by equating the r a d i a l v e l o c i t i e s from regions 2 and 3 to the rad i a l v e l o c i t y from the exterior region on the boundary r=a 2, 0<z<d2, d3<z<d. The normal v e l o c i t y on the surface r=a 2, d 2^z<d 3 i s i m p l i c i t l y assigned to be zero. Using equation (2) we have, <t>2 r = 4>E r > for r=a 2 0<z<d2 ...(1.11a) c6_ ^ = ^ , for r=a ? d3<z<d ...(1.11b) o , r Ei, r Adding (1.11a) and (I.11b) and multiplying throughout by the function (z) and integrating between the l i m i t s as shown we have the f i f t h system of equations. d 2 d (l / d k k ) o/ 0 2,r Zk ( z ) d z + ( l / d k k )d3 S 03,r Zk ( z ) d z d = d/dk k ) 0 ; <fiKir Zk ( z ) d z Integrating using the orthonormal properties of the function Z k (z), and rewriting we have the f i f t h system of equations as (1 .11). -K, B - 0. C - \p. D + E. =e. ...(1.11 kn n kn n rkn n k k The terms in (1.11) are defined as below. Kkn = ( n 7 r / 2 d k k ) v n ( a 2 ) Gkn f o r k =°. 1' 2.--? n=1,2,..; 6kn = < n 7 r/ 2 d k k >Wn(a2) G k n for k=0,1,2,..; n=1,2,..; *kn " r<d-d 3)m n /dk, ) X n '(a 2> F k n e 0 = -a 2N 0 1 / 2 s i n h ( k 0 d 2 ) / 2 ( k 0 d ) 2 d 2 e k = -a 2N R 1 / 2 s i n ( k R d 2)/2(k k d ) 2 d 2 v;(a 2) -I ^ n T r a z / d ; ) - ( I^nn-ai / d z ) / I , (una,/d2) ) K,(n7ra 2 / d 2 ) I 0(n7ra 2 / d 2 ) + ( I i (n7ra, / d 2 ) / Ki(n7rai / d 2 ) ) K 0(n7ra 2 / d 2 ) w;(a 2) = I 1 ( n 7 r a 2 / d 2 ) + ( I 0(n7ra 2 / d 2 ) / I 0 ( n 7 r a 2 / d 2 ) ) K ^ n T r a z / d ; ) I,(n7rai / d 2 ) + ( I 0 ( n 7 r a 2 / d 2 ) / K 0(n7ra 2 / d 2 ) ) K ^ n T r a i / d j ) Ji(m 0a 2) + ( J,(m 0a 3) / H,(moa3) ) H, (m 0a 2) J 0(m 0a 2) + ( J 1(m 0a 3) / H%{viy0a3) ) H 0(m 0a 2) Xn'(a 2) = 72 I , ( in a 2 ) + ( I , ( m a 3 ) / K ^ n t a 3 ) ) K ^ n i a 2 ) X n ' ( a 2 ) n " * n I 0 ( m n a 2 ) + ( I , ( m n a 3 ) / K t d n a 3 ) ) K 0 ( m n a 2 ) F Q 0 = [ l / ( d - d 3 ) ] d / Y Q ( z - d 3 ) Z Q (z)dz = (MQ N Q ) 1 / 2 k Q sinh(k 0 d 3 ) / [ (d-d 3)(k Q 2 - m0 2 ) ] F k Q = [ 1 / ( d - d 3 ) ] d J * 0 ( z - d 3 ) Z k (z)dz = (MQ N k ) 1 / 2 k k s i n ( k k d 3 ) / [ ( d - d 3 ) ( k k 2 + m 0 2)] d F n„ = [ l / ( d - d 3 ) ] , i / Y n ( z - d 3 ) Z n (z)dz = (Mn N Q ) 1 / 2 k Q sinh(k 0 d 3 ) / [ (d-d 3)(k Q 2 + mn 2 ) ] Fkn = [ l / ( d _ d 3 ) ] d 3 1 ? n ( Z _ D 3 ) Z k (z)dz = (M„ N. ) 1 / 2 k, sin(k. d 3 ) / [ (d-d 3)(k, 2 - iri 2 ) ] d 2 G Q n = (2/d 2) 0 ; Z 0(z) cos(n7rz/d 2) dz = 2N0 1 / 2 s i n h ( k 0 d 2 ) k 0 d 2 ( - 1 ) n / [ ( k 0 d 2 ) 2 + (nvr) 2] 73 d 2 G k n = (2/d 2) 0 / Z R (z) cos(n7rz/d 2) dz = 2N, ~ 1 / 2 sin(k, d 2)k. d 2 ( - D * / [ ( k . d 2 ) 2 - (nTr)2] 5.2.6 SOLUTION FOR COEFFICIENTS OF SERIES Equations (1.7) to (1.11) form the five systems of linear equations which are solved simultaneously to obtain the c o e f f i c i e n t s A , B , C , D and E of the series for n n n n n the potential functions. The above system of equations are equivalent to [M]|x| = |v| ...(1.12) where [M] i s a complex square matrix of order (5m-l), |x| i s a complex vector comprising of the unknown c o e f f i c i e n t s of the series and |v| i s a complex vector comprising of the right hand side of the system of equations (1.7) to (1.11). The system of equations (1.12) i s solved by a complex, double precision routine CDSOLN available on the UBC Michigan Terminal System. The routine solves the system by Gaussian elimination. CPU time required for solving a system of 100 equations, corresponding to 20 terms for each series for the potentials, i s t y p i c a l l y 2.6 seconds on the Amdahl 470 V/8 computer. Details of the formulation of the system 74 of equations (1.12) are given in the write-up on the computer program. 5.3 INTEGRALS FOR EVALUATION OF HYDRDOYNAMIC COEFFICIENTS The integrals I N 1 f IN 2 and IN 3 in equation (2.51) of chapter 2 are as given below. !Ni = o/ 01 r dr at z=d. = [a , 2 d,/4d - a^/iedd, + A 0a, 2/4 co n + ^ = 1 A n (-1) ( a ^ / n i r ) I , (nira , /d, ) /I 0 (n/ra , /d, ) ] ...(1.13) a 2 IN 2 = / 0 2 r dr at z=d 2 a 1 = [ ( a 2 2 - a! 2)d 2/4d - ( a 2 " - a )/-\ 6 a a : + B 0 2 ( a 2 2 - a, 2)/4 + 2 = 1 B n (-1 ) (d 2/n7r) V n (a 2) + ^ = 1 C n (-1) (d2/n7r) Wn ( a 2 ) ] ...(1.14) where, I ,(nira2/d2) - ( I , (n.7ra j/d 2) / I 1 (n7ra ,/d2) ) K,(n7ra 2/d 2) a 2 I o {n.7ra2/d2) + ( I 1(n7ra 1/d 2) / K ^ n ^ A ^ ) ) K 0(ri7ra 2/d 2) W n ( a 2 ) = Ii(n7ra 2/d 2) + ( I 0(nira 2/d 2) / I 0(ri7ra 2/d 2) ) K, (n.7ra2/d2) a 2 Ii(n7rai/d 2) + ( I 0(nira 2/d 2) / K 0(n7ra 2/d 2) ) Ki(n7ra,/d 2) - a a 2 IN 3 = / </»3 r dr at z=d 3 a 3 = [d 3/d + q/co2d - 1 ] ( a 2 2 - a 3 2 ) + ( D 0 M 0 " 1 / 2 a 2 / m 0 ) J i ( m 0 a ; ) + ( J 1 ( m 0 a 3 ) / H , ( m 0 a 3 ) ) H ^ m p a ; ) J 0 ( m o a 2 ) + ( J, ( m o a j ) / H,(moa3) ) H 0 ( m o a 2 ) -1/2 n=1 "n "n + ( L_, D „ M „ a 2 / m n ) I 1 ( m a 2 ) + ( I 1 ( in a 3 ) / K ^ m a 3 ) ) K , ( m a 2 ) n n n n ...(1.15) I 0 ( m n a 2 ) + ( I , ( m n a 3) / K 1 ( m n a 3) ) K 0 ( m n a 2) 6. APPENDIX 2 - EXPERIMENTAL SET-UP 6.1 EXPERIMENTAL FACILITIES The f a c i l i t i e s of the Ocean Engineering Centre at B.C. Research, Vancouver, were u t i l i s e d for the experiments. The f a c i l i t i e s used are described below. 6.1.1 THE TOWING TANK This i s a 220'X12 ,X10' tank primarily used for ship model resistance tests. It i s equipped with a towing carriage f i t t e d with data c o l l e c t i o n equipment, which traverses the length of the tank on r a i l s . A photograph of the tank i s shown in Fig.7. The tank i s equipped with a hinged paddle type wavemaker at one end and a wave damping beach at the opposite end. The tank i s also equipped with underwater windows for flow v i s u a l i z a t i o n experiments and an overhead l i f t i n g hook fixed at a position halfway along the length of the tank but capable of moving transversely. The l a t t e r was used for equipment handling during the tests. 6.1.2 WAVEMAKER The wavemaker consists b a s i c a l l y of three units. 6.1.2.1 The Wave Signal Generator This device generates a time varying voltage signal to represent the wave. The device has the c a p a b i l i t y to generate quasi-random waves or regular repeated waveforms. The operation of this device i s complicated. Instead, a 76 77 sinusoidal wave generator was used. This enables easy variation of amplitudes and frequencies for sinusoidal waveforms and is much simpler to operate. 6.1.2.2 Wave Synthesizer The wave synthesizer serves the dual purpose of boosting the input signal and correcting for any i r r e g u l a r i t i e s in the motion. The input signal is boosted to voltage and current levels appropriate to the hydraulic actuator. A displacement transducer on the wave paddle provides the synthesizer with the actual position of the paddle at any instant. The wave synthesizer sends a corrected signal to the hydraulic actuator after comparing the actual position of the paddle to the desired paddle location. 6.1.2.3 Wave Paddle A photograph of the wave paddle i s shown in Fig.10. This i s an aluminium paddle which spans the width of the towing tank. The paddle i s hinged approximately four feet below the water surface. The paddle i s o s c i l l a t e d by a hydraulic piston controlled by the hydraulic actuator which in turn i s controlled by the wave synthesizer. Any i r r e g u l a r i t i e s in the waves generated and propagating along the tank may•be due to one or more of the following reasons. The wave paddle has a natural, undriven o s c i l l a t i o n . The amplitude of thi s o s c i l l a t i o n varies with time and and 78 becomes more pronounced as the actuator heats up. The o s c i l l a t i o n i s approximately 0.4" at a frequency of 2.3Hz. Except for experiments near t h i s frequency, the effects can be isolated by spectral analysis. This o s c i l l a t i o n decreased considerably on i n s t a l l a t i o n of a new hydraulic actuator valve. So, only a few experiments were affected by thi s o s c i l l a t i o n . The above o s c i l l a t i o n causes an irregular waveform. At certain frequencies, beats were observed in the waveform. This can be d i r e c t l y attributed to the presence of a large seakeeping basin adjacent to the towing tank. The two are separated only by an aluminium h a l f - w a l l . Beats are generated in the waveforms at certain frequencies, normally higher than 1Hz. The frequency of the in t e r f e r i n g waveforms d i f f e r s by about 1Hz. from the driven frequency. The beats in the waveforms may possibly have been generated by a cross-flow between the tanks. Other i r e g u l a r i t i e s in the waveform may be induced by re f l e c t i o n s from the tank walls or disturbances due to a i r currents or gusts disturbing the water surface. 6.2 MOTION GENERATOR The motion generator i s a hyd r a u l i c a l l y driven scotch yoke mechanism. F i g . 8 shows a photograph of the unit.' The unit was redesigned since i t did not provide a smooth sinusoidal o s c i l l a t i o n in i t s o r i g i n a l form. 79 At present i t consists of a horizontal fixed frame 12'x4' made up of two I-beams. A v e r t i c a l frame i s mounted on this frame and holds the moving frame which provides the v e r t i c a l harmonic o s c i l l a t i o n . The moving frame i s connected to the scotch yoke mechanism which drives i t . The moving frame i s also connected to the cylinder model through a connecting block. The scotch yoke i t s e l f i s driven by a r a d i a l piston hydraulic motor having a displacement of 12.7 cu. i n . / revolution. The motor i s capable of an output torque of 500 f t . - l b s . A closed loop variable displacement hydrostatic transmission c i r c u i t i s used to ensure tight control of angular v e l o c i t y over the speed range of 3 to 150 rpm. Owing to l i m i t a t i o n s in the available e l e c t r i c a l power supply the hydraulic motor can only deliver rated torque upto a frequency of 0.83 Hz, with torque capacity diminishing to 167 f t . - l b s at a frequency of 2.5 Hz. The hydraulic power unit consists of a variable displacement a x i a l piston pump having a maximum displacement of 2.5 cu. i n . / revolution, driven by a 1750 rpm 5 H.P. e l e c t r i c motor. The pump displacement i s limited by external mechanical stops to a maximum of 1.25 cu. i n . / revolution to prevent overloading of the motor. A cross-port r e l e i f valve set at 3000 psi i s used to prevent excessive c i r c u i t pressures and to l i m i t motor torque to 500 f t - l b s . The scotch yoke mechanism and the hydraulic system were redesigned by Fraser Elhorn, a graduate of the Mechanical 80 Engineering Department at UBC. The moving frame was constructed at the Machine Shop at the Mechanical Engineering Department. The hydraulic motor and power unit were supplied by Fleck Hydraulics Inc., Vancouver B.C. 6.3 DATA COLLECTION SYSTEM The O.E.C. data c o l l e c t i o n system was used. This consisted of a MINC-11 computer and amplifiers and amplifiers and signal conditioners mounted on the towing carriage. 6.3.1 AMPLIFIERS AND SIGNAL CONDITIONERS The towing carriage c a r r i e s ten signal conditioners. Internal registers can be set to high,low or band-pass. Externally, d i a l s allow amplification of the input voltage signal by a factor of 10. A few of the amplifiers are d i f f e r e n t i a l amplifiers and permit amplifications upto 1000. For these experiments, low pass f i l t e r i n g was chosen with varying amplification. The amplifiers induce a phase s h i f t in the signal which was analyzed and corrected for. 6.3.2 MINC-11 COMPUTER The hardware for the system comprises of the following. 6.3.2.1 Main Console This incorporates analog and d i g i t a l input ports with clocking c a p a b i l i t i e s . A maximum of 16 analog ports can be used. Each port w i l l accept as input ±5.12 vol t s with a resolution of 2.5mV. Voltages in excess of the l i m i t s are 81 removed by the MINC-11. Analog voltages are converted to integer values for internal use, the voltage range being translated from 0 to 4096. Time difference between the sampling of two channels i s of the order of microseconds and so was not of concern for these experiments. 6.3.2.2 Dual Floppy Disk Drive System System and often used programs are stored on one disk and data and development programs are stored on the other. 6.3.2.3 VT105 Video Terminal The terminal has graphic display c a p a b i l i t i e s . It can display upto a maximum of two single valued functions. This enabled v i s u a l observation of the recorded signal and comparison of two d i f f e r e n t channels. 6.3.2.4 Line Printer A high speed variable character size l i n e printer i s used to obtain hard copies of data and program l i s t i n g s . 6.3.2.5 Tektronix Screen Dump Printer This printer generates a duplicate of the video screen contents on heat sensitive paper. This i s useful for obtaining plots of the recorded data. Software used for the experiments were partly existing on the system and partly written or modified. These are described l a t e r . 82 6.4 MODELS Three types of models were used for the experiments. Their dimensions and shapes are as shown in Figs. 4, 5, and 6.The models were constructed by the machine shop at the Department of Mechanical Engineering. The single cylinder model was made of PVC tubing 15" O.D. The ends were sealed with aluminium p l a t i n g . The cylinder was connected to the block on the motion generator by means of four threaded rods fixed on the bottom plate. These rods also served as a mounting for lead weights used to ballast the cylinder to achieve neutral buoyancy. The draft T, was varied during the experiments. The double cylinder model was similar in construction except for the fact that the top plate was replaced by an aluminium unit comprising of an 8.625" O.D. aluminium cylinder mounted on a plate. The dimensions of the model are shown in F i g . 5. The compound cylinder model had, in addition, another unit mounted on the bottom, similar to the aluminium unit mounted on the top for the double cyli n d e r . The dimensions are as shown in F i g . 6. 6.5 EQUIPMENT USED Electronic equipment used in association with the data c o l l e c t i o n system are described below. 83 6.5.1 STRAIN INDICATORS Strain indicators were use in association with the wire resistance wave probe and the dynamometers used to measure forces. These were model P-350A st r a i n indicators, manufactured by Vishay Instruments Inc. The output ports of the indicator excites the connected bridge c i r c u i t s with a 1.5 volts R.M.S., 1000Hz square wave. The output is a maximum of ± 250 mV DC. This i s too low to be used in association with the MINC-11, and hence had to be amplified. The s p e c i f i c a t i o n s are as follows. Accuracy : ±0.5% of reading or 5ye (whichever i s greater.) S e n s i t i v i t y : 0.2 to 20Me/mV Output : Linear range ±250 mV DC Noise & Ripple : 3/xe, 1mV. 6.5.2 SONAR LEVEL MONITOR This i s a s o l i d state device used for measurement of displacement. The device was manufactured by Wesmar Marine Electronics Inc.,Seattle,USA. The device measures the time required for a sonar pulse to travel to and from an object. Various range settings are available with a manual mu l t i p l i e r d i a l which gives a continuous control of the s e n s i t i v i t y . The sp e c i f i c a t i o n s are as follows. Output ripple : <0.1% of f u l l s c a l e Output D r i f t s t a b i l i t y 84 : Better than 0.25% of f u l l scale/hour after warm-up. Resolution Better than 0.5% of measured range. Lin e a r i t y Better than 0.5% of f u l l scale. Operating temperature 45° to 140°F Beam emitted 3° in span. Pulse re p e t i t i o n rate 160 Hz at 7 to 30 range. 6.5.3 WAVE PROBE This i s a simple device used to measure the water surface elevation and hence the wave p r o f i l e . The device b a s i c a l l y consists of two separated wires in the water, which act as the fourth arm of a Wheatston bridge. The changing water l e v e l provides a recordable change in c i r c u i t resistance. A Vishay indicator i s used to input a 1.5 Volts R.M.S. 1000 Hz square wave. 6.5.4 FORCE RECORDING EQUIPMENT These consisted primarily of an 80 l b . dynamometer and two 500 l b . Universal Shear beams used to measure v e r t i c a l forces. 6.5.4.1 80 l b . Dynamometer The dynamometer i s capable of measuring a v e r t i c a l force, a horizontal force and a moment. 80 lbs . i s the maximum range for the v e r t i c a l force. The effects of any one 85 of the forces on the others are excluded by a suitably designed c i r c u i t . The measuring devices are f o i l bonded strai n gauges Type CEA-06- 125-OT-350. A f u l l Wheatstone bridge arrangement i s used for each parameter the bridge c i r c u i t s were excited by voltages from Vishay s t r a i n indicators. The dynamometer was designed and constructed by the Ocean Engineering Centre at B.C.Research. A l l the bridges have self compensating gauges to provide self temperature correction. This dynamometer was used for the wave force tests on s t a t i c cylinders. 6.5.4.2 Universal Shear Beam Two Universal Shear Beams (USB) were used to measure the v e r t i c a l forces on the o s c i l l a t i n g cylinder model. The USBs were manufactured by Hottinger Baldwin Measurements Inc. of Framingham, MA. The s p e c i f i c a t i o n s as supplied by the manufacturer are as follows. Rated Capacity Rated Output Non-linearity Hysteresis Non repeatability Excitation voltage Side Load rejection Max. Load, Safe 5001bs 1.9925 mV/V -0.02% of rated ouptut. 0.01% of rated output. 0.01% of rated output 18V DC or AC RMS max. 500:1 150% The USBs were mounted below the two v e r t i c a l shafts of the moving frame on the motion generator as shown in F i g . 8a. They were pin-jointed to exclude moments and measure only 86 v e r t i c a l forces. 6.5.5 CALIBRATION The sonar l e v e l recorder and the wave probe were ca l i b r a t e d before each set of experiments by moving them known distances and recording the output voltages. A c a l i b r a t i o n program existing on the O.E.C. data c o l l e c t i o n system was used to obtain the slope and intercept of the c a l i b r a t i o n data. The 80 l b . dynamometer and the USBs were calibr a t e d s t a t i c a l l y in compression on the Universal Testing Machine at the Mechanical Engineering Department, and also dynamically using the motion generator. They were also c a l i b r a t e d in tension by suspending known weights from them when the dynamometers were mounted on the motion generator. 6.6 SOFTWARE USED Two d i f f e r e n t sets were used for the experiments. One set was used for data c o l l e c t i o n on the O.E.C. MINC-11 system. The other set was used for data analysis on the PDP-11 system at the Department of Mechanical Engineering at UBC, which was similar to the MINC-11. Part of the software neccessary was already existing at the O.E.C. Some of i t had to be adapted for these experiments. However, some of the software was developed s p e c i f i c a l l y for these experiments. 87 6.6.1 DATA COLLECTION SOFTWARE The data c o l l e c t i o n software was used on the MINC-11 system at the O.E.C.. The data f i l e s were stored on disk and transferred to the PDP-11 at the Mechanical Engineering Department at UBC. 6.6.1.1 ADCAL This i s a c a l i b r a t i o n program that i s run before data c o l l e c t i o n . This program reads input voltage on a spec i f i e d channel which can be indexed to any desired value. By varying the range of measurements for a device with the corresponding input of such variations a series of c a l i b r a t i o n points i s established. At least f i v e points are required. The program then establishes the best f i t straight l i n e through these points by least squares minimization. The slope, intercept, variance and delta of the data l i n e i s output. This information i s also stored in a matrix on a user s p e c i f i e d f i l e . The f i l e contains c a l i b r a t i o n data for several channels. If c a l i b r a t i o n cannnot be ca r r i e d out before the experiments a dummy c a l i b r a t i o n f i l e has to be created to run the demultiplexing program. 6.6.1.2 ADMAIN This i s the p r i n c i p a l program for data c o l l e c t i o n . The program c o l l e c t s data on several channels, upto a maximum of 16 simultaneously. The sampled data i s stored in a multiplexed group for memory minimization. The program also 88 allows real time viewing of any channel prior to sampling. This is useful as a check to ensure that recording is occuring. 6.6.1.3 ADMUX This program i s used to separate each channel's unique signal and store the data points as numeric values on user specified f i l e s . The program demultiplexes the stored data. This program i s available on the MINC-11 system. 6.6.1.4 GRAPH This program displays upto two graphs on the video terminal. The program has the a b i l i t y to display either an x-y graph or real time plot of a channel. It can also shade from various portions of the graph f i e l d to the data points. Further, the program can calculate and display a cubic moving spline f i t to the data. Comments, t i t l e and axes labels can be input. The program can also display various portions of a curve, i e , scaling can be changed to magnify some portion of a curve. 6.6.2 DATA ANALYSIS SOFTWARE The data analysis software was used on the PDP-11 at the Mechanical Engineering Department. They were mostly written for the s p e c i f i c purpose of these experiments. 6.6.2.1 DEMUX This program demultiplexes one or more signals of the collected data, converts the d i g i t i z e d voltage lev e l s to 89 user values and stores the time record on a user sp e c i f i e d f i l e . If the c a l i b r a t i o n was not done before the experiments, but l a t e r , these c a l i b r a t i o n values can be input before the demultiplexing. 6.6.2.2 AMP This program takes as input a time record of some measured data and calculates the Fourier spectrum of the record using the Fast Fourier Transform algorithm (FFT). The program outputs the frequency vs amplitude and the frequency vs phase (in radians) to two separate data f i l e s . The phase depends on the starting time of the experiment. It can be used to judge the the r e l a t i v e phases of two simultaneously sampled channels. 6.6.2.3 PHAMP This program removes phase s h i f t s caused by the O.E.C. signal conditioners. The amplifiers were calibr a t e d prior to the experiments to determine their phase s h i f t s . In general, a l l amplifiers were set up as low pass f i l t e r s to remove high frequency noise and interference. The program takes as input a phase f i l e and the slope and intercept of the phase s h i f t caused by the amplifier used. The output f i l e contains the corrected phase values. 6.6.2.4 DELAY This program adjusts the phase of the wave probe record to account for the distance between the centreline of the cylinder and the wave probe. It takes as input the phase 90 record of the wave and the distance from the wave probe to the cylinder centreline. The output i s the adjusted phase of the wave. 6.6.2.5 SPECADD This program adds or subtracts spectra. For the purpose of these experiments i t was used to remove a reference phase from one or more phase spectra. Thus the phase of one record can be expressed r e l a t i v e to another. 6.6.2.6 ZERO This i s a program used to zero the spectrum of some record. Here, i t was used to zero the reference phase spectrum. 6.6.2.7 TABLE This program scans an amplitude spectrum for l o c a l maxima. A cut off amplitude value can be input and the user can s e l e c t i v e l y chose the frequency/amplitude point to output.This way driving frequency harmonics can be ignored. The harmonics are a result of a f i n i t e time record sampled at d e f i n i t e i n t e r v a l s . The program outputs the amplitude and phase of the record for the chosen frequency. The 90° and 180° components of the vector can also be output i f required. 6.6.2.8 FINAL This program calculates the added mass and damping coefficents from the 90° and 180° components of the v e r t i c a l 91 force. The cylinder mass, the cross sectional area of the cylinder, and the buoyancy are input for non-dimensionalisation. 6.6.2.9 STAT This program performs the same function as FINAL, but for the ca l c u l a t i o n of the non-dimensional exciting force from the v e r t i c a l exciting force and the wave amplitude records. 92 F i g . l D e f i n i t i o n of motions Fig.2 Compound c y l i n d e r geometry E X T E R I O R R E G I O N ® E X T E R I O R R E G I O N i R E G I O N Fig.3 S u b d i v i s i o n of flow f i e l d 95 F i g . 4 C o m p o u n d c y l i n d e r m o d e l F i g . 5 Double c y l i n d e r model Fig.6 S i n g l e c y l i n d e r model F i g . 7 T o w i n g t a n k a t B . C . R e s e a r c h F i g . 8 M o t i o n g e n e r a t o r ID to 100 F i g . 8 a P o s i t i o n i n g o f l o a d c e l l s 1 0 1 P i g . 9 D a t a c o l l e c t i o n e q u i p m e n t F i g . 1 0 W a v e p a d d l e o F i g . 1 1 F l o w v i s u a l i z a t i o n o w F i g . 1 2 Displacement record Legend - Displacement 1 1 1 l I 1 1 1 J L i I I i 1 1 I 1.75 2.33 2.92 3.50 Time, sees. Compound c y l i n d e r T=35.5" f=2.5Hz 0.70 _ 0.61 0 .52£-0.44 0.35 0.26 0.17 0.09 0.00 Legend Disp. spectrum T I r • • i i i i i I i i t i '-4- J L J 16.67 20.00 0.00 3.33 6.67 10.00 13.33 Freq. , Hz Fig.12a Disp„ amplitude spectrum. Compound c y l i n d e r T=35.5" f=2 05Hz Legend ° - 3 0 rr - Theory-B.E.M. 0 26 H " Present theory 0.22 n t K 0.19 -Jl 0.15 t. f-6,1 i j i 0.08 fl L U 0.04 L O . O O Q L j |_ j ..I. I . J . - l . ..I I J . L .. I l I t I I. I.I I—I —J—I — I 1 1 1 1 1 0.00 1.17 2.33 3.50 4.67 5.83 7.00 WxWxA/G Fig.13 Heave added mass. Compound c y l i n d e r T=42 0625" D'=13.125" > X O I GC x CM CM CO 0.20 0.17 0.15 £ r-U 0.12 t-E 0.10 " 0.08 0.05 0.03 Legend - Theory-B.E.M. - Present theory \\ *.\ V 0.00 0.00 j i I .i.'S.' t * T T r . ? i u ; . n a . .Lziz-i. . u ^ - L - . 1 . . - -1.17 2.33 3.50 4.67 WxWxA/G ... J. .-I rtr-.L- I r- J 5.83 7.00 Fig.14 Damping c o e f f i c i e n t . Compound c y l i n d e r T=42.625" D'=13.125" o -j 0.80 0.70 r F 0.60 ^-r 0.5oE-0.40 n-t 0.30 E-0.201-0 . 1 0 -Legend - Present theory A Expt. amp. - 0.5in. -I- Expt. amp. - 0.39in. o.oo r. i i i i J 1 . 1 i . i . J i. i i i i i i i . i .! i _ _ L - ^ _ i _ J 0.00 1.17 2.33 3.50 4.67 5.83 7.00 WxWxA/G Fig.15 Heave added mass. Compound c y l i n d e r T=35.5" D*=6" Legend - Present theory ^ Expt. amp. -0 .5 in . + Expt. amp. -0 .39in . i i i . -I L L-5.83 7.00 WxWxA/G Fig.16 Damping c o e f f i c i e n t . Compound c y l i n d e r T=35 05" D'=6" 0.05 f-Legend - Present theory A Expt. amp. - 0.5in. + Expt. amp. - 0.39in. °"0« nn " 1 1 1 ' ' ' 1 ' ' l ' ' A - ' 1 - - 1 ^ - 1 ^ °-00 1.17 2.33 3.50 4.67 5.83 7.00 WxWxA/G Fig.17 Wave amplitude. Compound c y l i n d e r T=35.5" D'=6" > X O I DC CM CM < 0.80 0.70 r b r 0.60 r 0.50 r-t 0.40 r r-c 0.30 |r 0.20 fT 0.10 r w i o.oo E-1 0.00 Legend _ Present theory A Expt. amp. -0 .5 in . .1.... I. ... 1 1 . 1 -I .1 I—L . J — J . I I. I- J I .J 1—1 1 A—I 1 L 1.17 2.33 3.50 WxWxA/G 4.67 5.83 7.00 Fig.18 Heave added mass. Compound c y l i n d e r T=39„5" D'=10" 0.10 0.09 F Legend - Present theory * Expt. amp. - 0.5in. 0.08 C. 0.06 P-0.05 0.04 i-0.03 r_ 0.01 0.00 J = 4 i i i I i i i , i , ° - 0 0 1-17 2.33 3 - 5 (> 4.67 5.83 WxWxA/G 7.00 Fig.19 Damping c o e f f i c i e n t . Compound c y l i n d e r T » 39.5" D'=10" 0.20 Legend - Present theory A Expt. amp. -0 .5 in . 0.00 1.17 2.33 3.50 4.67 5.83 7.00 WxWxA/G Fig.20 Wave amplitude. Compound c y l i n d e r T=39.5" D'=10" Legend - Present theory A Expt. amp. - 0.5in. 0.80 0.70 0.60 0.50 0.40 0.30 0.20 0.10 o.oo h . . . . » • • • • i t i i t i i i i i i i i » i i i ' i 0.00 1.17 2.33 3.50 4.67 5.83 7.00 WxWxA/G Fig.21 Heave added mass. Compound c y l i n d e r T=43.5" D'=14" X > X O I X CM CM CD 0.50 _ 0.44 0.37 0.31 p.25 0.19 0.12 0.06 0.00 Legend - Present theory A Expt. amp. - 0.5in. J L J I I I 1 1 L J I I I 1 L J 0.00 1.17 4.67 5.83 2.33 3.50 WxWxA/G Fig.22 Damping c o e f f i c i e n t . Compound c y l i n d e r T=43.5" D'=14 7.00 Legend - Present theory A Expt. amp. - 0.5in. 0.17 r 0.15 1 0.12 L 7.00 > X O x CM CM < 0.80 0.70 0.60 0.50 0.40 0.30 0.20 0.10 (l 0.00 Legend - Present theory * Expt. amp. - 6.5in. J I I I 1 I 1 I I I I 1 I I I I I I I I L 0.00 1.17 2.33 3.50 WxWxA/G 4.67 J I I I I L 5.83 7.00 Fig.24 Heave added mass. Compound c y l i n d e r T=47.5" D ^ I B " 0.70 0.61 0.52 0.44 0.35 0.26 Legend - Present theory A Expt. amp. -0 .5 in . 0.17 0.09 0.00 \—A i 1 1 1 1 1 1 ~ 0.00 1.17 2.33 j i i l 1 1 '•-3.50 4.67 i — i — 1 — ' — ' — ' ' ^ 7.00 5.83 Fig.25 Damping WxWxA/G c o e f f i c i e n t . Compound c y l i n d e r T=47.5» D'=18» Legend - Present theory A Expt. amp. - 0.5in. . 1 • . • > ' • • • • i i i 1 i i i ' 1 0.00 1.17 2.33 3.50 WxWxA/G 4.67 5.83 7.00 Fig.26 Wave amplitude. Compound c y l i n d e r T-47.5" D'=18" a E < x O x > X O DC u. 2.00 0.25 0.00 LJ_ 0.00 Legend © Expt. amp., - ° - 1 2 i n . A Expt. amp. - 0.2in. + Expt. amp. ^0.4in. - Theory-B.E.M. j i_a IT83 J I L 1.67 2.50 WxWxA/G 3.33 i I i i i i—I 4.17 5.00 Fig»27 Heave e x c i t i n g f o r c e on s i n g l e c y l i n d e r . T-7" M o - 4 . 5 0 -Legend © Expt. amp. - 0.12in. Expt. amp. - 0.2in. + Expt. amp. - 0.4in. Theory-B.E.M. - 6 . 0 0 - j i I i—i—i—i—I 0.00 0.83 1.67 2.50 3.33 4.17 5.00 i i i i i i i i—i—i—i—i—i—i WxWxA/G Fig.28 Heave e x c i t i n g force phase. Si n g l e c y l i n d e r T=7" Q. E < x O x > X O GC 2.00 • 1.75 1.50 1.25 1.00 0.75 0.50 0.25 0.00 0 Li \ • CD J I 1 L I I I I L .00 0.83 1.67 2.50 WxWxA/G Legend © Expt. amp. - 0.35in. & Expt. amp. - 0.5in. — Theory-B.E.M. I | • • i — J I I 1 1 L _ J . 3.33 4.17 5.00 Fig„29 Heave e x c i t i n g f o r c e on s i n g l e c y l i n d e r . T-10.5" rO 6.00 _ 4.50 <D 3.00 1.50 0.00 L - 1 . 5 0 L - 3 . 0 0 - 4 . 5 0 - 6 . 0 0 CD Legend o Expt. amp. -o.35in. A Expt. amp. - 0.5in. - Theory-B.E.M. ! i i t i L J A l i i i * J L 0.00 0.83 1 i i • • 1 • i i ' 1 4.17 5.00 Fig.30 Heave e x c i t i n g f o r c e phase. 1.67 2.50 3.33 WxWxA/G S i n g l e c y l i n d e r T=10.5" 0.40 _ Legend o Expt. amp. -0 .2 in . A Expt. amp. ~0.4in. + Expt. amp. -0 .6 in . x Expt. amp. - 1.0in. - Theory-B.E.M. 0.00 j i i i I i i i—i—I—i—i—i—i I i t i I I I I I 1 1 1 1 1 L 0.00 0.83 1.67 2.50 WxWxA/G 3.33 4.17 5.00 Fig.31 Heave e x c i t i n g f o r c e on double c y l i n d e r . T-23.5" CO CO CD > CO I <D O •o -<D CO CO SI Q. 5.00 3.75 2.50 1.25 0.00 1.25 2.50 3.75 V S Legend © A 4 -X CD -5.0.0 r t i i a> I i i i i I Expt. amp. - 0.2in. Expt. amp. - 0.4in. Expt. amp. - 0.6in. Expt. amp. - 0.7in. Theory-B.E.M. 0.00 0.83 J I I L J I I L 1.67 2.50 WxWxA/G J 1 I L 1 J 1 1 L 3.33 4.17 5.00 Fig.32 Heave e x c i t i n g f o r c e phase. Double c y l i n d e r . T=23.5" H 1 in 0.50 0.44 0.37 0.31 0.25 _ X 0.00 0.00 o Legend © Expt. amp. - O.zin. A Expt. amp. - o.4in. + Expt. amp. - 0.6in. x Expt. amp. - 0.7in, - Theory-B.E.M. 0.83 1.67 2.50 WxWxA/G 3.33 4.17 5.00 Fig.33 Heave e x c i t i n g f o r c e on double c y l i n d e r . T=27.5" CD J I I L 1 I I I Legend © * Expt. amp. - 0.2in. ^ Expt. amp. - 0.4in + Expt. amp. - 0.6in. X Expt. amp. - 0.7'm. - Theory-B.E.M. J I I L 1 0.00 0.83 1.67 2.50 3.33 J L J I 1 L 4.17 5.00 WxWxA/G Fig.34 Heave e x c i t i n g f o r c e phase. Double c y l i n d e r . T-27.5" a . E < x O x > X O DC Legend - Theory-B.E.M. — Present theory + Expt. amp. - 0.12in. xExpt . amp. "0.30in. 4>Expt. amp. - 0.5in.l 4 Expt. amp. - 0 . 7 i n . XExpt. amp. - ^ i n . 0.00 0.83 1.67 2.50 WxWxA/G 3.33 4.17 5.00 Fig.35 Heave e x c i t i n g f o r c e on compound c y l i n d e r . T=35.5" to 00 5.00 Legend © Expt. amp. - 0.12in. A Expt. amp. -0.30in. •+• Expt. amp. - 0 . 5 m . x Expt. amp. -0.7in. «> Expt. amp. "LQ in . - Theory-B.E.M. •5.00 t_i—i—i—i—I—i—i—i—i—I—i i i i I i i i i l i i i i I i i i i l 0.00 0.83 1.67 2.50 3.33 4.17 5.00 WxWxA/G Fig.36 Heave e x c i t i n g f o r c e phase. Compound c y l i n d e r . T-35.5" 0.50 0.44 0.37 0.31 fj.25 0.19 Legend - Theory-B.E.M. - - Present theory + Expt. amp. - 0 . 2 i n . ; x Expt. amp. - 0.5'mJ * Expt. amp. -0.7Sin. * Expt. amp. - I.Oin. 0.00 J I I—I—l—J—I—I—I 0.83 I , , i i I* i i f f L L t U —1 3.33 4.17 Fig.37 Heave e x c i t i n g force on co 1.67 2.50 WxWxA/G mpound c y l i n d e r . T=38.375" 5.00 CO CO CD > CO I CD O CD CO CO J C Q. 5.00 o 4-X Legend Expt. amp. -0.2in. Expt. amp. -0 .5 in . Expt. amp. - 0 7 5 j n Expt. amp. -join. Theory-B.E.M. ' ^ • ^ J j i i t 1 i i i i I i i i i 1 i » i i I i • i i I i i . , 0.00 0.83 1.67 2.50 3.33 4.17 5.00 WxWxA/G Fig.38 Heave e x c i t i n g force phase. Compound c y l i n d e r . T=38.375" 0.20 _ 0.00 0.83 1.67 2.50 WxWxA/G o Legend Expt. amp. - 0.5in. Expt. amp. - I.Oin. Theory-B.E.M. Present theory 3.33 4.17 5.00 Fig.39 Heave e x c i t i n g force on compound c y l i n d e r T=42.625" 5.00 , 3.75 t b Legend © Expt. amp. -0.5in. A Expt. amp. -1.0in. - Theory-B.E.M. 2.50 1.25 0.00 - 1 . 2 5 - 2 . 5 0 - 3 . 7 5 '5.00 t_j—i—i—i—I—i—i—i—i—I—i—i i i | i i i i | i i i i I i i i i l 0.00 0.83 1.67 2:50 3.33 4.17 5.00 WxWxA/G Fig.40 Heave e x c i t i n g f o r c e phase. Compound c y l i n d e r T=42.625" 0.40 0.35 ~ 0.30 Q E < x O x > X O 0.25 0.20 - X 0.15 0.10 0.05 0.00 t _ J — i — i I i i i i 0.00 0.83 1.67 Legend - Theory-B.E.M. -- Present theory + Expt. amp. -0.12in. x Expt. amp. -0.5in. <> Expt. amp. -0.7inJ 4 Expt. amp. -0.90in. 1 ' . I i i—i—i I i • • • t • i i i i 2.50 3.33 4.17 5.00 WxWxA/G F i g 0 4 1 Heave e x c i t i n g force on compound c y l i n d e r . T=49.5« W A Expt. amp. Legend © Expt. amp. - 0.12in. _ 0.5in. _0.7in. + Expt. amp. x Expt. amp. ~o.90in - Theory-B.E.M. • f i . O O J - . • i i I i i i i _ J I I I I I L _ l 1 1 1 1 1 — I 1 1 L _ 0.00 0.83 1.67 2.50 3.33 4.17 WxWxA/G J L 5.00 Fig.42 Heave e x c i t i n g force phase. Compound c y l i n d e r . T=49.5" 1 3 6 Adctoa mms ratio for dfott» T = J . f T , T = J 5 F T 5 • E i p « n m t n t o l 0 (McCormick) O Triii tritory Omtgo = 30 rod /tec J ! _ 1 L J L »» «* D t p r h / r o d u s Fig.43 Hydrodynamic c o e f f i c i e n t s f o r s i n g l e c y l i n d e r . 

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