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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), U n i v e r s i t y of Cochin,  1980  A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF APPLIED SCIENCE  in  FACULTY OF GRADUATE STUDIES Department of Mechanical E n g i n e e r i n g  We accept t h i s t h e s i s as conforming to the r e q u i r e d  standard  THE UNIVERSITY OF BRITISH COLUMBIA 27th J u l y , 1984 ©  Madan Venugopal, 1984  .  «6  In p r e s e n t i n g  this  thesis  in  partial  fulfilment  of the  requirements f o r an advanced degree a t the THE UNIVERSITY OF BRITISH COLUMBIA, I agree that the freely  available  f o r reference  Library  shall  and study. I f u r t h e r agree  that p e r m i s s i o n f o r e x t e n s i v e copying  of  this  scholarly  by  the  purposes  Department  or  understood  that  f i n a n c i a l gain  by  may  be  h i s or  copying  shall  not  granted her  thesis Head  representatives.  of It  for my is  or p u b l i c a t i o n of t h i s t h e s i s f o r be  allowed  permission.  Department of Mechanical E n g i n e e r i n g THE UNIVERSITY OF BRITISH COLUMBIA 2075 Wesbrook Place Vancouver, Canada V6T 1W5  Date: 27th J u l y , 1984  make i t  without  my  written  ABSTRACT The added mass and damping c o e f f i c i e n t s of a compound circular cylinder  i n heave motion are computed  using a semi a n a l y t i c a l p o t e n t i a l uses c o n t i n u i t y  theoretically  flow method. The method  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 e x c i t i n g compound c y l i n d e r  are c a l c u l a t e d  f o r c e s on the  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  f o r c e s are compared to t h e o r e t i c a l  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 e x p e r i m e n t a l l y by forced harmonic  o s c i l l a t i o n of the c y l i n d e r point  model. The wave height at a  i n the flow f i e l d was a l s o measured d u r i n g the  experiment. The e f f e c t s of v a r i a t i o n  of amplitude and  frequency of o s c i l l a t i o n and d r a f t are a l s o  studied  e x p e r i m e n t a l l y . The r e s u l t s are compared to the t h e o r e t i c a l predictions. The heave e x c i t i n g  f o r c e s on a compound c y l i n d e r  due to small amplitude, s i n u s o i d a l experimentally predictions  model  waves are measured  i n a towing tank. The r e s u l t s are compared to  by the t h e o r e t i c a l  method presented i n t h i s  t h e s i s and by a boundary element method. The heave f o r c e s on s i n g l e  and double c y l i n d e r  exciting  models are a l s o  determined e x p e r i m e n t a l l y . These r e s u l t s are compared to theoretical  predictions  by a boundary element method.  ii  The comparisons between theory and experiment show the a p p l i c a b i l i t y of l i n e a r p o t e n t i a l  flow theory i n the  determination of the hydrodynamic  c o e f f i c i e n t s of the  compound c y l i n d e r  model.  i ii  Table of Contents INTRODUCTION  1  THEORETICAL MODEL AND SOLUTION  8  2.1 General f o r m u l a t i o n of the boundary value problem 9 2.2 Loads and motions  12  2.3 Heave motion of compound c y l i n d e r  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 c o n d i t i o n s . 15 2.3.3 D e f i n i t i o n of p o t e n t i a l s  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 E x t e r i o r region  21  2.4 S o l u t i o n  f o r unknown c o e f f i c i e n t s f o r p o t e n t i a l s 22  2.5 C a l c u l a t i o n of the added mass and damping coefficients  23  2.6 C a l c u l a t i o n of heave e x c i t i n g f o r c e  25  2.7 C a l c u l a t i o n 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 a n a l y s i s  30  3.3 Determination of heave e x c i t i n g f o r c e  31  3.4 Flow v i s u a l i z a t i o n  32  RESULTS AND DISCUSSION  33  4.1 P r e s e n t a t i o n of data  34 iv  4.2 D i s c u s s i o n of t h e o r e t i c a l r e s u l t s  35  4.3 D i s c u s s i o n  37  of experimental  results  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  42  tests  4.3.3 Heave e x c i t i n g f o r c e s  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 P o t e n t i a l s  60  5.1.1 Region 1  .60  5.1.2 Region 2  60  5.1.3 Region 3  61  5.1.4 E x t e r i o r region  62  5.2 Generation  of systems of equations f o r s o l u t i o n .63  5.2.1 C o n t i n u i t y of pressure and 2  between regions 1 63  5.2.2 C o n t i n u i t y of v e l o c i t y between regions 1 and 2  65  5.2.3 C o n t i n u i t y of pressure the e x t e r i o r region  between region 2 and .66  5.2.4 C o n t i n u i t y of pressure the e x t e r i o r r e g i o n  between region  3 and  68  5.2.5 C o n t i n u i t y of v e l o c i t i e s between regions 2, 3 and the e x t e r i o r region 70 5.2.6 S o l u t i o n f o r c o e f f i c i e n t s of s e r i e s 5.3 I n t e g r a l s f o r e v a l u a t i o n of hydrdoynamic coefficients v  73 74  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 S i g n a l Generator  76  6.1.2.2 Wave S y n t h e s i z e r  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 A m p l i f i e r s and s i g n a l c o n d i t i o n e r s  80  6.3.2 MINC-11 computer  80  6.3.2.1 Main Console  80  6.3.2.2 Dual Floppy Disk D r i v e System  81  6.3.2.3 VT105 Video Terminal  81  6.3.2.4 L i n e P r i n t e r  81  6.3.2.5 T e k t r o n i x Screen Dump P r i n t e r  81  6.4 Models  82  6.5 Equipment  used  6.5.1 S t r a i n  82  indicators  83  6.5.2 Sonar l e v e l monitor  83  6.5.3 Wave probe  84  6.5.4 Force r e c o r d i n g equipment  84  6.5.4.1 80 l b . Dynamometer  84  6.5.4.2 U n i v e r s a l Shear Beam  85  6.5.5 C a l i b r a t i o n  86  6.6 Software used  86  6.6.1 Data c o l l e c t i o n software vi  87  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 a n a l y s i s 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 c y l i n d e r geometry  Fig.3  Subdivision  Fig.4  Compound c y l i n d e r model  Fig.5  Double c y l i n d e r model  ...  Fig.6  Single  ...  Fig.7  Towing tank at B.C.Research  Fig.8  Motion Generator  of flow f i e l d  c y l i n d e r model  ...  ...  ...  Data c o l l e c t i o n equipment  F i g . 10 Wave paddle  ...  ...  Fig.8a Load c e l l p o s i t i o n i n g Fig.9  ...  ...  ...  Fig.11 Flow v i s u a l i z a t i o n ... F i 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 c y l i n d e r - t h e o r e 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 f o r compound c y l i n d e r Figs.27 to 30 Heave e x c i t i n g f o r c e single cylinder  ...  on  ...  F i g s . 31 to 34 Heave e x c i t i n g f o r c e on double cylinder... F i g s . 35 to 42 Heave e x c i t i n g f o r c e compound c y l i n d e r  ...  v.i i i  on  F i g . 4 3 Hydrodynamic c o e f f i c i e n t s f o r single cylinder  (McCormick)...  ix  136  ACKNOWLEDGEMENTS I am deeply guidance  indebted t o D r . S . M . C a l i s a l f o r h i s p a t i e n t  throughout  the course of t h i s research.I am a l s o  indebted t o NSERC f o r the f i n a n c i a l  support  for t h i s  project. My s i n c e r e g r a t i t u d e t o Michael Desandoli a s s i s t a n c e with the experiments  for h i s  and data a n a l y s i s . Thanks  are a l s o due t o Johnson Chan f o r the use of h i s boundary element program f o r computation axisymmetric  of heave e x c i t i n g f o r c e s on  floating objects.  I would a l s o l i k e t o thank F r a s e r E l h o r n f o r r e d e s i g n i n g the motion generator and John Hoar, and h i s t e c h n i c i a n s at the Machine Shop a t the Mechanical E n g i n e e r i n g Department f o r f a b r i c a t i o n of the motion generator and the models. S p e c i a l thanks Hanson f o r doing an e x c e l l e n t motion  are due t o Bruce  j o b i n r e c o n s t r u c t i n g the  generator.  I would l i k e t o thank Gerry Stensgaard to use the towing  f o r permission  tank a t B.C.Research and George Roddan f o r  h i s expert advice on the i n s t r u m e n t a t i o n . I would l i k e t o thank Marcel L e f r a n g o i s f o r h i s a s s i s t a n c e with the i n s t r u m e n t a t i o n f o r data c o l l e c t i o n and Steve Thomson f o r a s s i s t a n c e with the equipment h a n d l i n g . F i n a l l y , I wish t o thank Angela Runnals and Jon N i g h t i n g a l e f o r t h e i r a s s i s t a n c e with the text p r o c e s s o r TEXTFORM on which t h i s t h e s i s was prepared.  x  DEDICATION T h i s t h e s i s i s d e d i c a t e d to my and a f f e c t i o n , and to my d u r i n g the course of my  parents f o r t h e i r love  wife f o r her p a t i e n t studies.  xi  forebearance  1. INTRODUCTION In the l a s t  few decades there has been an a c c e l e r a t e d  growth i n the o f f s h o r e i n d u s t r y . A wide v a r i e t y of o f f s h o r e s t r u c t u r e s have been designed o i l exploration, d r i l l i n g ,  and c o n s t r u c t e d , p r i m a r i l y f o r  p r o d u c t i o n and s t o r a g e . The main  f e a t u r e s i n the c o n t i n u i n g growth of o f f s h o r e s t r u c t u r e s are t h e i r s i z e and the depth at which they are capable of operating. Since there i s c o n s i d e r a b l e investment l a r g e o f f s h o r e s t r u c t u r e and i t i s expected long design  life  (upto  100 years  going  into a  t o have a f a i r l y  i n some c a s e s ) ,  c o n s i d e r a b l e r e s e a r c h has been and i s being devoted to the problems a s s o c i a t e d with the design of these s t r u c t u r e s . Of primary experiences  importance are the loads which the s t r u c t u r e  and the motions i t undergoes. The l o a d i n g on the  s t r u c t u r e i s p r i m a r i l y hydrodynamic, i . e . due. to the a c t i o n of  the ocean waves on the s t r u c t u r e . Even i f the extreme sea  s t a t e s were known a c c u r a t e l y , i t would s t i l l  be d i f f i c u l t t o  estimate e x a c t l y the nature of the l o a d i n g on the s t r u c t u r e . To mathematically  model the a c t u a l flow around a f u l l  s c a l e o f f s h o r e s t r u c t u r e poses c o n s i d e r a b l e d i f f i c u l t i e s and an exact p i c t u r e - of the problem i s beyond the reach of the present  s t a t e 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 i n the mathematical modelling of the flow. These a r e d i s c u s s e d i n relevance t o the t o p i c of t h i s t h e s i s i n Chapter 2.  1  2 Most of the o f f s h o r e  structures  in c u r r e n t  component members of c i r c u l a r s e c t i o n . So, flows around c i r c u l a r c y l i n d e r s are of importance and To of any  much research  has  been devoted to t h i s t o p i c .  quantities  r e s u l t i n g motions  (known as the  c o e f f i c i e n t s ) : the added mass and  on the  study of  f l o a t i n g in a f l u i d medium i t i s neccessary  to determine two  addition,  have  considerable  study the hydrodynamic loads and structure  the  use  hydrodynamic  damping c o e f f i c i e n t s . In  i t i s a l s o neccessary to know the  exciting  forces  structure.  A body a c c e l e r a t i n g a d d i t i o n a l f o r c e due f l u i d also.This  in a f l u i d medium experiences  to the a c c e l e r a t i o n  a d d i t i o n a l hydrodynamic f o r c e can  expressed as an added mass to the accelerated  at the  by C h e v a l i e r  of a p a r t  same r a t e . T h i s  du Buat about 200  researchers including Bessel,  of  an the  be  body's mass, being f a c t was  years ago.  first  recognized  Since then  several  Green, Plana, Stokes, Lamb,  and  even S i r Charles Darwin have worked on  Sir  C h a r l e s Darwin showed that a c y l i n d e r moving through a  f l u i d medium d i s p l a c e s i t s motion. Further, displaced  /8/.  f l u i d p a r t i c l e s i n the d i r e c t i o n of  he  mass of the  the problem  showed that t h i s permanently  fluid  enclosed between the  initial  f i n a l p o s i t i o n s 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 i n  various  ways. One  If we in F i g . 1,  of the d e f i n i t i o n s i s given  consider  the C a r t e s i a n  and  later.  c o - o r d i n a t e system shown  t r a n s l a t o r y o s c i l l a t i o n s in the  x, y, and  z  3 d i r e c t i o n s are known as surge, sway and heave, r e s p e c t i v e l y . R o t a t i o n a l motions about the same axes are known as r o l l ,  p i t c h and yaw, r e s p e c t i v e l y . For each  d i r e c t i o n of motion there e x i s t  s i x 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  i n the s i x  degrees of freedom. Hence added mass may be expressed as a tensor a ^ j where the f i r s t of  s u b s c r i p t r e f e r s to the d i r e c t i o n  the body motion and the second to the d i r e c t i o n of the  hydrodynamic f o r c e . 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 f u r t h e r reduced f o r 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 the  i n a f l u i d medium near  f r e e s u r f a c e , the hydrodynamic f o r c e develops frequency  dependent components  in-phase and out-of-phase with the  a c c e l e r a t i o n . The component  in-phase with the a c c e l e r a t i o n  c o n t r i b u t e s t o the added mass and the component with the v e l o c i t y c o n t r i b u t e s t o the damping  in-phase  coefficient.  F u r t h e r , f o r a body with separated flow about i t , the wake or c a v i t y  induces an added mass. T h i s c a v i t y  induced  mass v a r i e s with the instantaneous shape and volume of the c a v i t y and i t s r a t e of change. The instantaneous value of the  added mass depends on the time h i s t o r y of the  motion.Also, f o r motion i n a v i s c o u s f l u i d  there e x i s t s some  v i s c o u s damping t o o . The added mass c o e f f i c i e n t s depend, i n g e n e r a l , on the parameters c h a r a c t e r i z i n g the motion, time, a s u i t a b l y  4 d e f i n e d Reynold's number, e t c . The d e t e r m i n a t i o n of an e x p r e s s i o n f o r the the time dependent undergoing an a r b i t r a r y  f o r c e on a body  motion i s a very complex  T h i s problem i s s i m p l i f i e d  by c o n s i d e r i n g simpler time  dependent motions l i k e harmonic o s c i l l a t i o n s . A simplification  problem.  further  i s achieved by using the p o t e n t i a l  flow  theory to d e s c r i b e the flow. Several t h e o r e t i c a l  and experimental techniques have  been d e v i s e d t o determine the hydrodynamic c o e f f i c i e n t s  of a  wide range of b o d i e s . The common approaches to s o l u t i o n problems i n p o t e n t i a l l i k e conformal mapping singularity  of boundary value  flow theory are a n a l y t i c a l methods and numerical procedures l i k e  techniques. In recent years the f i n i t e element  methods have a l s o  been s u c c e s s f u l l y  Conformal mapping  used.  can be used only f o r two dimensional  problems. The s i n g u l a r i t y methods do not s u f f e r l i m i t a t i o n . They have been used e x t e n s i v e l y  from t h i s  i n hydrodynamic  problems i n recent years /1/. Havelock(1955) determined the hydrodynamic  coefficients  of a sphere /12/. Kim (1974) s t u d i e d the hydrodynamic coefficients free  f o r 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  s u r f a c e /18/. Wang and Shen  mass and damping  coefficients  (1966) c a l c u l a t e d  the added  of a sphere i n water of f i n i t e  depth /21/. Garrison calculate  (1975) used d i s t r i b u t e d  s i n g u l a r i t i e s to  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  i n water of f i n i t e depth /15/.  general problem for a r b i t r a r y (1974) c a l c u l a t e d  He  forms too. Bai and  added mass and  and v e r t i c a l c i r c u l a r c y l i n d e r s  calculated  the hydrodynamic c o e f f i c i e n t s /13/.  Kritis  method of Yeung (1975) /22/ hydrodynamic c o e f f i c i e n t s  /14/.  (1979) used the h y b r i d  integral  to n u m e r i c a l l y c a l c u l a t e  the  /19/.  methods have s e v e r a l s i g n i f i c a n t  they a l s o  have some  of  disadvantages.  There i s c o n s i d e r a b l e computation i n v o l v e d and the d i s c r e t i z a t i o n  (1976)  of axisymmetric  advantages and are e x t e n s i v e l y used i n the design offshore structures,  for  Bai  for a c i r c u l a r cylinder  While the s i n g u l a r i t y  the  Yeung  damping c o e f f i c i e n t s  horizontal  ocean p l a t f o r m s  formulated  the c h o i c e of  of the body s u r f a c e must be made  c a r e f u l l y . F u r t h e r , these methods sometimes g i v e  numerical  problems at c e r t a i n  "irregular  f r e q u e n c i e s which are c a l l e d  f r e q u e n c i e s " f o r the method. The not have t h i s disadvantage, f i e l d to be d i s c r e t i z e d . unless  but  f i n i t e element methods do they  r e q u i r e the e n t i r e  This i s considerably  flow  difficult  i t i s handled by a computer.  Analytic  methods can handle only a small number of w e l l  d e f i n e d geometric shapes. Even f o r these, sometimes c o n s i d e r a b l e mathematical This thesis  there i s ,  difficulty.  d i s c u s s e s the determination  mass and  damping c o e f f i c i e n t s  cylinder  undergoing simple  of the added  f o r a compound c i r c u l a r  harmonic heave motion at the  free  s u r f a c e i n 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 t h e o r e t i c a l  method as w e l l  6 as experiments. Experiments were a l s o conducted to determine the heave e x c i t i n g force due to harmonic r i g i d model. These  waves i n c i d e n t  on a  r e s u l t s are a l s o compared to t h e o r e t i c a l  predict ions. The t h e o r e t i c a l method d i s c u s s e d here i s a p p l i c a b l e  to  a s p e c i a l c l a s s of axisymmetric b o d i e s . I t can be termed  an  a n a l y t i c method as well as a macro-element technique, i n the sense that  the flow f i e l d  The value of the p o t e n t i a l  i s divided  i n t o a few  is explicitly  i n t e r i o r domain as a f u n c t i o n  elements.  known i n the  of the v e r t i c a l and r a d i a l  coordinates. The b a s i s  of the method i s suggested by G a r r e t t  in h i s  paper on wave loads on a c i r c u l a r dock /4/. T h i s method has been a p p l i e d  to the case of a s i n g l e c i r c u l a r c y l i n d e r  by  Sabuncu and C a l i s a l with good r e s u l t s /./. The method has a l s o been a p p l i e d predicion  by K.Kokkinowrachos e t . a l /3/ i n the  of hydrodynamic  a r b i t r a r y axisymmetric  c o e f f i c i e n t s of bodies of  several  shapes.  The experimental r e s u l t s p r e s e n t e d i n t h i s t h e s i s correspond t o the forced harmonic  o s c i l l a t i o n of a compound  c y l i n d e r model i n a towing tank. T h i s  gave the  hydrodynamic  c o e f f i c i e n t s of the c y l i n d e r model. A q u a l i t a t i v e study of the vortex shedding d u r i n g the c y l i n d e r motion was with the h e l p of an underwater  window at the towing  a l s o made tank.  The heave e x c i t i n g force due to small amplitude waves on three c y l i n d e r models of d i f f e r e n t shapes was varying  drafts  determined at  f o r the models and d i f f e r e n t amplitudes of  7 the waves. These r e s u l t s were compared to a t h e o r e t i c a l prediction  u s i n g a boundary element technique a p p l i e d  Chan /11/. The heave e x c i t i n g f o r c e s  by  f o r the compound  c i r c u l a r c y l i n d e r model were a l s o compared to p r e d i c t i o n s using the present  theory.  2. THEORETICAL MODEL AND  SOLUTION  Using dimensional a n a l y s i s r e l a t i n g to the wave f o r c e on a f i x e d body, one can c o n v e n i e n t l y compare the importance time  of flow s e p a r a t i o n and d i f f r a c t i o n e f f e c t s /8/. A  invariant  f o r c e on a f i x e d s t r u c t u r e due  wave can be expressed F/pgHD  2  relative  to an  incident  as,  = f ( d / L , H/L,  D/L,  Rn), ...(2.1)  where, p  = Density of water;  g  = A c c e l e r a t i o n due  d  = Depth of water;  L  = Wavelength;  H  = Wave height;  D  = Representative diameter  Rn  = Reynolds number.  to g r a v i t y ;  of body;  The body s i z e to wavelength r a t i o , D/L d i f f r a c t i o n parameter. As D/L  i s termed the  becomes l a r g e ,  diffraction  e f f e c t s become important. When d i f f r a c t i o n e f f e c t s are important,  flow s e p a r a t i o n i s r e l a t i v e l y unimportant  problem can be s t u d i e d by p o t e n t i a l it  and  the  flow methods. F u r t h e r ,  i s assumed that the wave steepness, H/L  i s small  corresponding to l i n e a r wave theory /8/,/3/. A l i n e a r i s e d boundary value problem i s thus f o r the v e l o c i t y p o t e n t i a l  i n the flow  8  field.  formulated  9 2.1 GENERAL FORMULATION OF THE BOUNDARY VALUE PROBLEM Consider the compound c i r c u l a r c y l i n d e r a s i n u s o i d a l wave i n c i d e n t The f l u i d  on i t .  i s assumed to be homogeneous, i n v i s c i d and  i n c o m p r e s s i b l e . An e a r t h  f i x e d c y l i n d r i c a l c o o r d i n a t e system  with o r i g i n on the seabed i s d e f i n e d the  shown i n Fig.2 with  z-axis c o i n c i d i n g  as shown i n Fig.2 with  with the v e r t i c a l a x i s of the  cylinder. It  i s f u r t h e r assumed that  the amplitude of the  c y l i n d e r motion i s small compared the amplitude of the i n c i d e n t the  to the wavelength and  that  wave i s a l s o small compared  to  wavelength. The v e l o c i t y p o t e n t i a l  f o r the flow f i e l d  i n the  presence of the c y l i n d e r can, i n g e n e r a l , be expressed i n the  form:  $(x,y,z,t) = * ( x , y , z , t ) + 0  In the above $ and  ...(2.2)  i s the p o t e n t i a l of the undisturbed waveform  0  i s the d i s t u r b a n c e 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 p o t e n t i a l  a  (x,y,z,t)  can be f u r t h e r expressed as,  6 (x,y,z,t) = $„(x,y,z,t) + Z s. j=1 J  where, $  v  F  0  <j>. (x,y,z,t) J  i s the p o t e n t i a l due to a s t a t i o n a r y  body  (2.3) i n the  w a v e f i e l d and <t>^ are the p o t e n t i a l s due to the motions of the body  i n the s i x component d i r e c t i o n s with u n i t v e l o c i t y ,  10 in u n d i s t u r b e d water, s. 3 0  i s the magnitude th  of the body motion i n the j  of the v e l o c i t y  d i r e c t i o n . Hence,  *(x,y,z,t) =*o(x,y,z,t) + $ ( x , y , z , t ) F  6  + Z s. j=1 The p o t e n t i a l s 3 0  <t>. (x,y,z,t)  ... (2.4)  D  $o,  r  ,  c  1. . b  are harmonic, with  the frequency of the i n c i d e n t wave and hence, - icjt  <f>(x,y,z,t) = 4>(x,y,z) e  ...(2.5)  With the assumption that the c y l i n d e r motion  i s harmonic,  - 10>t s . = s. e D  Do  The problem can now be formulated i n two p a r t s . One due to the wave i n c i d e n t on a f i x e d s t r u c t u r e  (diffraction  problem), and the second due t o the motion of the body i n undisturbed water. The assumption of l i n e a r i s a t i o n enables us t o c o n s i d e r the i n d i v i d u a l degrees of freedom  independent  of each other . The c r o s s - c o u p l i n g terms have to be c o n s i d e r e d i f they are l i n e a r . The p o t e n t i a l s <t>^ must s a t i s f y the governing L a p l a c e ' s equation, V  2  <S>^  (x,y,z) = 0 ...(2.6)  and the f o l l o w i n g boundary c o n d i t i o n s ,  (i) Linearised  surface kinematic and dynamic  conditions.  boundary  free  1 1 d<f> •  3TJ  9z  d<t> •  . . .(2.7)  for z=d  9t  = -qrj  f o r z=d  9t  ...(2.8)  The above can be w r i t t e n as, 9S> . 1 9t  9$.  2  = 0  9z  ;  or, d <fi. 2  -cj <j>.  +  2  g  1  9z  3  =  0  for  ( i i ) Ocean bottom boundary 90 — = 9z  z=d  condition.  0 for z = 0  ( i i i ) Radiation 90 . lim / r ( — r <» 9r 1  ...(2.9)  2  . . . (2. 10)  condition.  - i(co /g) 4> • ) = 0 ^ 2  ...(2.11)  The f o l l o w i n g boundary c o n d i t i o n s must a l s o be s a t i s f i e d on the body  surface.  12 ( i v ) For a wave i n c i d e n t on a f i x e d s t r u c t u r e , on the body surface S,  -.(2.12)  9n  3 n  (v) For a f l o a t i n g body f r e e to move i n calm water, 9*. —L_ 9n  _. = n. e l  w  t  11  |_  ...(2.13)  b  or , 90 . =  n  j  Is  9n where, n. i s the component of the outward normal on the body th surface S, i n the j d i r e c t i o n as shown i n F i g . 1 . 3  2.2 LOADS AND MOTIONS The  instantaeneous l i n e a r i s e d p r e s s u r e on the body i s given  by, P i n s t " ~P 9t Using  (2.4),  ...(2.14)  we can r e w r i t e  t h i s as,  6 i  n  S  t  F  It  the  jo  j  i s expedient to d i v i d e the f o r c e  the d i f f r a c t i o n The  j - l  _.  i n t o that due t o  and that due to the motion.  d i f f r a c t i o n problem g i v e s  i n c i d e n t wave,  the e x c i t i n g f o r c e due to  13  F  E  (  t  = ""s  )  k  Pinst. k n  = -icop e ~  ;/  l w t  d  s  S  (0  O  + 0  p  ) n  k  dS  ...(2.15)  where, k=1,2 , . . 6 . The hydrodynamic f o r c e s due to the motion of the body in undisturbed water can be c a l c u l a t e d using the p o t e n t i a l s  F  = -icop e  H  JS  1U>Z  S  k  2 s 0 n j=1 jo j  R  dS  ...(2.16)  Now, we d e f i n e the added masses, a^^ and damping c o e f f i c i e n t s , b ^ j as i n /3/. -p/;  0^ n  s  dS = akj  k  + (i/co) b  k j  ...(2.17)  where, a j k  = - p Re [ J 7 0j n  k  dS]  ...(2.17a)  b j  = - p Im [ J 7 0^ n  k  dS]  ...(2.17b)  k  S  S  Now, we can w r i t e (2.16) i n the form,  F  6 (t) = L  u  H  k  j  =  1  (a. . s. + b, . s. ) k] 3 D  ...(2.18)  a, • represent added mass i n the case of t r a n s l a t i o n and 3 K  added mass moments of i n e r t i a  i n the case of r o t a t i o n a l  motions, b, . are the damping c o e f f i c i e n t s , k i n d i c a t e s the  14 d i r e c t i o n of the f o r c e and j i n d i c a t e s the d i r e c t i o n of the motion. Using the above we can, i n g e n e r a l , w r i t e a system of s i x coupled d i f f e r e n t i a l  equations f o r the motion of a  f l o a t i n g body due to an i n c i d e n t wave as,  *  (  m  k j  +  a  k j  }  S  j  +  b  k j Sj  +  C  k j  where, c ^ are the r e s t o r i n g the  S  j  =  F  E„  f o r c e c o e f f i c i e n t s and m^ are  masses or the mass moments of i n e r t i a In  .-.(2.19)  of the body.  the s p e c i a l case of a body r e s t r a i n e d to undergo  heave motion only, the above system reduces to an o r d i n a r y 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 c y l i n d e r with geometry  as shown i n  F i g . 2 . The problem i s to d e f i n e the p o t e n t i a l d u r i n g the heave motion of the c y l i n d e r . The flow f i e l d as f o l l o w s .  i s subdivided  15 2.3.1  DEFINITION OF FLOW FIELD The s u b d i v i s i o n i s shown i n F i g . 3 . Though only a  section  i s shown because of the a x i a l  symmetry, the elements  are a c t u a l l y c y l i n d r i c a l . Region 1 i s d e f i n e d as the volume w i t h i n the surface 0<z^d,.  r=a,,  Region 2 i s d e f i n e d as the volume w i t h i n the s u r f a c e s r=a  1 r  0^z^d  2  and r = a , 0^z<d . 2  2  Region 3 i s d e f i n e d as the volume w i t h i n the s u r f a c e s r = a , d ^z<d and r = a , d <z<d. 3  3  2  3  The e x t e r i o r region  i s d e f i n e d as the volume e x t e r i o r  0<z<d.  to the s u r f a c e r = a , 2  Using G a r r e t t ' s method p o t e n t i a l s are d e f i n e d four regions as  , $ , $3,$ 2  E  i n the  , respectively.  2.3.2 GOVERNING EQUATIONS AND BOUNDARY CONDITIONS The v e l o c i t y p o t e n t i a l s <i> must s a t i s f y equation from the i r r o t a t i o n a l  V  •  in c y l i n d r i c a l coordinates  ...<2.6)  defined  i n F i g . 2 . <j> can be w r i t t e n as,  0(r,0,z) = R(r) T(6) if  assumption made e a r l i e r /9/.  • O / r ) ! £ • <1/r»> fjt  The equation i s d e f i n e d as  Laplace's  Z(z)  ...(2.21)  the v a r i a b l e s are assumed to be independent of each  o t h e r . Using s e p a r a t i o n of v a r i a b l e s , (2.6) can be w r i t t e n  16 as three o r d i n a r y l i n e a r d i f f e r e n t i a l equations as, + A T  = 0  ...(2.22)  - BZ  = 0  ...(2.23)  2  ^rf ^ f  2  2  2  + (1/r)  + (B -A /r )R 2  2  2  = 0  Since the flow i s axisymmetric  ...(2.24)  the a x i a l f u n c t i o n has to be  an even f u n c t i o n . Hence,  T(6)  = cos(mfl)  ... (2.25)  We can w r i t e the p o t e n t i a l $ as *(r,0,z,t)  = 4>(r,z)  e~ lut  ...(2.26)  The c o e f f i c i e n t m i s taken as  zero  on the assumption  the most s i g n i f i c a n t c o n t r i b u t i o n comes from the f i r s t  that term  of the c o s i n e s e r i e s f o r 6.  2.3.3 DEFINITION  OF POTENTIALS  The p o t e n t i a l s  , 4> , * 3 , 2  *  E  are d e f i n e d such that  they s a t i s f y the governing equations and boundary c o n d i t i o n s as below. 2.3.3.1 Region 1 The p o t e n t i a l *, = ^, V  d e  l c j t  has to s a t i s f y the  H f o l l o w i n g boundary c o n d i t i o n s . V"  H  i s the v e l o c i t y  i n the  heave d i r e c t i o n . I at z-d, = 0 at z=0 The p o t e n t i a l  ...(2.27) ....(2.28) s a t i s f i e s the governing equation  (2.22) and (2.23) and the boundary c o n d i t i o n s (2.27) and (2.28). The p o t e n t i a l 0, comprises of the p a r t i c u l a r solution,  17  0  = [(z -r /2)/2dd ] 2  ...(2.29a)  2  1  and the homogeneous s o l u t i o n , 0  = A /2  1 h  0  + g  A  = 1  n  ( I (nTrr/d, ) / I (n7r a T/d, ) )cos(n7rz/d ) ] 0  0  1  ...(2.29b) Hence, = V  d e~  H  [(z -r /2)/2dd  i c J t  2  2  1  + A /2 0  oo  + £  = 1  A  n  (I (n7rr/d 0  , )/I (nrca ,/d, ) ) cosfnirz/d, ) ] 0  ...(2.29) The c o e f f i c i e n t s A fixed later function  0  and A  i n the s o l u t i o n . I  of the f i r s t  are unknown and w i l l be  n  0  i s the m o d i f i e d B e s s e l  kind.  2.3.3.2 Region 2 The p o t e n t i a l $  = <f>  V H boundary c o n d i t i o n s .  following  = 1 at z=d  2  2  d e~  l c J t  has to s a t i s f y the  ...(2.30)  2  = 0 a t z=0  ...(2.31)  The p o t e n t i a l 4> s a t i s f i e s the governing 2  equation  (2.22) and (2.23) and the boundary c o n d i t i o n s (2.30) and (2.31). The p o t e n t i a l  4>2 comprises of the p a r t i c u l a r  s o l u t ion 0  = (z -r /2)/2dd 2  2  2  ...(2.32a)  18 0  2 h  and *  = B+  Z  0  {B  = 1  V  n  (r)+C  n  p  (r)} c o s ( n n z / d 2 )  W  n  ...(2.32b)  adding the above, = V d e" H  2  [(z -r /2)/2dd,  i a J t  2  2  00  + B+  2  0  = 1  (B  n  V  (r)+C  n  n  W  (r)} cos (n7rz/d ) ]  n  2  ...(2.32) where V  n  (r) -  I ( n 7 r r / d ) +( I i(nita,/d2) 0  I0(n7ra /d ) 2  W  /  2  2  +( I^nn-a,/^)  (nira ,/d2)  )  / K^nnc^/d;,)  K (n7rr/d ) 0  2  ) K (n7ra /d ) ...(2.33) 0  2  2  (r) .  n  I (n7rr/d ) + ( I (n7ra /d ) / K (n7ra /d ) ) K (n7rr/d ) 0  2  0  2  2  0  2  2  0  2  I , (nwa i V d ) +( I ( n 7 r a / d ) / K ( n 7 r a / d ) ) K^nn-a,/^) ...(2.34) 2  B  0  the  and B  n  0  2  2  0  are unknown c o e f f i c i e n t s  2  2  which are determined by  r a d i a l boundary c o n d i t i o n s . <i> as d e f i n e d 2  (2.30) and (2.31).  i n (2.32) s a t i s f i e s (2.22),  (2.23),  19 2.3.3.3 Region 3 The p o t e n t i a l  i n t h i s region $  s a t i s f i e s the governing equation = 1 at z=d  = <j> V d e H 3  3  l  w  t  and the boundary c o n d i t i o n s ... (2.35)  3  and -co  2  03 + g  = 0 at z=d  ...(2.36)  The v e l o c i t y p o t e n t i a l particular 0  i n t h i s region  i s d e f i n e d by a  solution,  = (z/d+g/(co d)-1 )  ...(2.37a)  2  3 p  and  a homogeneous  solution,  00  0  3 h  = DoX Yo+ I 0  =  1  D  n  X  Y  n  ...(2.37b)  n  Combining the above, *>  = V d e" H  3  + D X Yo + 2 0  0  [ (z/d+g/(co d)-1 )  i t J t  2  = 1  D  n  X  n  Y  ]  n  ...(2.37)  where,  XQ =  J ( m r ) - ( J,(moa ) / H,(m a ) 0  0  3  0  ) H (m r)  3  0  0  . . . (2.38)  J ( m a ) - ( J , ( m a ) / H,(moa ) ) H ( m a ) 0  0  2  0  3  3  0  0  2  20 I ( %  r ) + ( I,(m  0  X  n  n  a ) / K,(m 3  a ) ) K (m  n  3  0  n  r)  =  I (m 0  a ) - ( I 1(m a ) / i ( K  n  2  n  m  3  n  a ) ) K (m 3  0  n  a ) 2  ...(2.39) In t h i s  formulation,  Y  0  = M "  M  0  = [1+sinh{2m (d-d )}/2m (d-d )]/2  Y  n  = M  M  N  = [1+sin{2m (d-d )}/2m (d-d )]/2  cosh[m (z-d )]  1 / 2  0  0  0  1 N  co  2  -  3  cos[m  and m  0  2  2  0  where m  a)  /  n  0  0  n  respectively.  3  ...(2.41a)  a r e the roots o f ,  0  tan[m  ...(2.40a)  ...(2.41)  3  g m tanh[m (d-d ) ] = 0  + g m  3  (z-d )]  3  n  0  ...(2.40)  3  n  3  (d-d )] = 0 3  ...(2.42)  ...(2.42a)  21 2.3.3.4 E x t e r i o r  region  The p o t e n t i a l  i n the e x t e r i o r  region  S> = <t>„ V E E  d e  1 £ J t  H  has to s a t i s f y the governing equation following -s3z  boundary  (2.6) and the  conditions,  = 0 at z = 0  . . . (2.43)  90 + g — 3z  ...(2.44)  p  -CJ 0„ 2  E  = 0 at z=d  The p o t e n t i a l <l> d e s c r i b i n g  the free wave and the waves  r e f l e c t e d o f f the r i g i d c y l i n d e r  = V d e H  E  £  = 1  l c J t  [-E (H (k r)/H 0  E  0  (K (k  n  0  0  n  1  i s given by,  (k a ))Z (z) 0  2  0  r)/K, ( k a ) ) Z p  2  p  (z)]  ...(2.45)  where,  Z  0  = N ~  N  0  = [1+sinh{2k d}/2k d]/2  0  1 / / 2  cosh[k z]  ...(2.46a)  0  0  0  ...(2.46b)  22 Z  n  = N  ~  N  n  = [1+sin{2k d}/2k d]/2  E  0  and E  n  1  /  cos[k  2  n  0  n  2  0  0  tan[k  n  ....(2.47b)  0  and k are the r o o t s of n  0  cj -gk tanh[ k d]  2  ...(2.47a)  are unknown c o e f f i c i e n t s determined as d e s c r i b e d  l a t e r and k  w +gk  z]  n  = 0  d] = 0  respectively. $  E  as d e f i n e d i n (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 m o d i f i e d first  kind, K i s the m o d i f i e d  Bessel f u n c t i o n of the  B e s s e l f u n c t i o n of the second  k i n d , J i s the B e s s e l f u n c t i o n of the f i r s t the Hankel f u n c t i o n of the f i r s t  kind and H i s  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 s e r i e s f o r the p o t e n t i a l s  A_ , B_ , C , D and E„ are determined by matching the n n n n n 1  pressures The  and v e l o c i t i e s between adjacent procedure i s d e t a i l e d i n Appendix  of the p r e s s u r e s  3  regions. 1. The matching  and v e l o c i t i e s between adjacent  regions  23 r e s u l t s i n f i v e systems of l i n e a r simultaeneous e q u a t i o n s . The number of equations i n each system i s equal to the number of terms taken f o r the s e r i e s and i s a l s o 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 elimination  routine  p r e c i s i o n complex  u s i n g a Gaussian  CDSOLN which permits the use of double  v a r i a b l e s . The r o u t i n e  i s a v a i l a b l e on the  Michigan T e r m i n a l System at the U n i v e r s i t y  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 , f o r 20 terms i n the s e r i e s , the procedure i n v o l v e s  s o l v i n g a 100 X 100 matrix  and the Amdahl 470 V/8 computer  accomplishes t h i s i n 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 follows a  22  /  p  V  i n heave motion are r e l a t e d to the p o t e n t i a l as from (2.15).  +  i b  22 /  p  V  w  =  1  /  V  /J*s  0  P  D e n s i t y of the medium.  V  Volume of the c y l i n d e r  (  r  '  6  '  Z  )  n  2  d  S  Frequency of the motion. a  22  b '22 n.  Added mass i n heave  motion.  Damping c o e f f i c i e n t i n heave Unit  normal  motion.  i n the z d i r e c t i o n .  ...(2.49)  24 S denotes i n t e g r a l over the s u r f a c e of the c y l i n d e r . The added mass and damping c o e f f i c i e n t are non-dimensionalized  as shown.  The p o t e n t i a l 0 i s d e f i n e d as d i f f e r e n t p o t e n t i a l s i n different  r e g i o n s . Hence the i n t e g r a l  i n (2.49) can be  w r i t t e n as, 2ir j;  s  0(r,  9,  z)  n  dS  z  =  a! ioS  J  0  S  + a  a  a * i  S  + a  2  <t>2  2  * ( - l ) ] r dr dfl 3  3  3  ...(2.50) Using  the a x i a l symmetry  of the p o t e n t i a l f u n c t i o n s  (50) can be w r i t t e n as, SSe s  4>(r,  0,  „,  z)  n^  dS  =  .. _ z  a,  2TT [ J  - ...  0  l 0  ;  *I  +  0, +  a2  <t>2  J  a  a i  a z  +  03(-1)1 r dr ae  J 3  a  ...(2.51) E v a l u a t i n g the three we can r e w r i t e  a  22  /  p V  +  i b  22 /  independently  (2.49) as below,  p  V  u  where IN,, I N , IN 2  i n t e g r a l s i n (2.51)  =  3  (  2  7  f  /  V  )  ^  I  N  1  +  I  N  are as e v a l u a t e d  2  +  I  N  3 ]  ...(2.52)  i n Appendix 1.  25 2.6 CALCULATION OF HEAVE EXCITING FORCE The amplitude of the heave e x c i t i n g force can be c a l c u l a t e d from the damping c o e f f i c i e n t b  2 2  using  a r e l a t i o n s h i p given  by Newman /8/ as f o l l o w s . F  = (pgH b u>/2k ) [1 + 2kd/sinh( 2kd) ]}~ 2  2 2  2  1 / / 2  22  ...(2.53)  The heave e x c i t i n g f o r c e computed as above i s compared to the experimental r e s u l t s and a p r e d i c t i o n by a boundary element program, l a t e r . In (2.53), F b  2 2  2 2  = Heave e x c i t i n g  force  = Heave damping c o e f f i c i e n t  p  = density  of medium  g  = a c c e l e r a t i o n due t o g r a v i t y  H  = i n c i d e n t 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 e l e v a t i o n a t any p o i n t  on the free s u r f a c e  can be  computed from the p o t e n t i a l f o r the flow f i e l d using the following  equation from l i n e a r theory /9/.  .(2.54)  26 where, 7? i s the wave e l e v a t i o n and g i s the a c c e l e r a t i o n due to g r a v i t y . $ i s the p o t e n t i a l f u n c t i o n the  f o r the region i n  flow f i e l d where the wave e l v a t i o n i s d e s i r e d .  2.8 COMPUTER PROGRAM A computer program, CYLINDER was w r i t t e n the  t o s e t up and solve  system of equations to determine the unknown  c o e f f i c i e n t s f o r the p o t e n t i a l s as d e s c r i b e d  earlier  in this  chapter. The program a l s o computes the p o t e n t i a l s and t h e i r normal d e r i v a t i v e s at any p o i n t  i n 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 c y l i n d e r The program r e q u i r e s  a , 3  d,, d  2  and d  3  as input  i n heave motion. the dimensions a , a , n  2  of the compound c i r c u l a r c y l i n d e r as shown  in F i g . 2. I t a l s o r e q u i r e s  the frequency of the motion, the  depth of water,d and the number of terms t o be taken i n the s e r i e s f o r the p o t e n t i a l . The program takes 2.8 seconds t o compute the added mass and  damping c o e f f i c i e n t f o r the compound c y l i n d e r a t one  frequency.  3. EXPERIMENTAL WORK Experiments were conducted i n the towing tank at the Ocean E n g i n e e r i n g Centre (O.E.C.) at B.C.Research, to v e r i f y  the t h e o r e t i c a l  3.1  PURPOSE OF EXPERIMENTS  Two  different  Vancouver  results.  sets of experiments were conducted. The  first  set of experiments were performed to e v a l u a t e the hydrodynamic  c o e f f i c i e n t s i n heave motion of a compound  c i r c u l a r c y l i n d e r model. The experiments were conducted to v e r i f y the t h e o r e t i c a l p r e d i c t i o n s of the  hydrodynamic  c o e f f i c i e n t s . The compound c y l i n d e r model was  t e s t e d at four  d i f f e r e n t d r a f t s to estimate the e f f e c t 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 h e i g h t s at a f i x e d d i s t a n c e from the c y l i n d e r c e n t r e l i n e were measured to compare them with the t h e o r e t i c a l l y p r e d i c t e d wave h e i g h t s using Egn.  2.54.  The second set of experiments were conducted to evaluate the heave e x c i t i n g f o r c e on three models - a compound c y l i n d e r , a double c y l i n d e r and a s i n g l e Dimensions  cylinder.  of the three c y l i n d e r s are shown on F i g s . 4, 5,  and 6. D e s c r i p t i o n s of the models are given i n Appendix The e x c i t i n g  2.  f o r c e s on the compound c y l i n d e r model were used  to compare with the v a l u e s 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 c a l c u l a t e d using the matching technique. T h i s p r o v i d e s a method f o r 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 w e l l as a v e r i f i c a t i o n  of the r e l 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 e x c i t i n g  f o r c e (Eqn.  2.53). The experiments with the s i n g l e and double c y l i n d e r models were used to v e r i f y a p r e d i c t i o n using the boundary element method. The experimental f a c i l i t i e s used are d e s c r i b e d i n Appendix  2. Besides the experiments mentioned above, a  q u a l i t a t i v e o b s e r v a t i o n of the vortex shedding during the compound c y l i n d e r motion was a l s o made.  3.2 DETERMINATION OF HYDRODYNAMIC COEFFICIENTS Forced harmonic  o s c i l l a t i o n s of the compound c y l i n d e r model  were used t o determine i t s hydrodynamic heave  coefficients in  motion. F i g . 7 shows a photograph of the towing tank. I t s  dimensions and other p a r t i c u l a r s are given i n Appendix 2. F i g . 8 shows a photograph of the motion generator used along with the a s s o c i a t e d pump and c o n t r o l gear. A b r i e f description  i s given i n Appendix  2. The c y l i n d e r model was  mounted as shown , b a l l a s t e d s u i t a b l y so that i t was n e a r l y n e u t r a l l y buoyant. F i g . 9 shows the equipment  on board the  O.E.C. towing c a r r i a g e which was used f o r the data c o l l e c t i o n . These are a l s o d e s c r i b e d  i n Appendix 2.  The motion generator g i v e s a small amplitude s i n u s o i d a l motion t o the c y l i n d e r model a t a f i x e d frequency. The amplitude of o s c i l l a t i o n can be v a r i e d . The f o r c e on the cylinder  i s measured c o n t i n u o u s l y with the h e l p of  29 dynamometers. Two types of dynamometers were used. One i s a 3-component measuring vertical  d e v i c e which i s capable of reading a  f o r c e , a h o r i z o n t a l f o r c e and moment  s i m u l t a n e o u s l y . I n t e r a c t i o n e f f e c t s of the three q u a n t i t i e s are minimized by a s u i t a b l e c i r c u i t  d e s i g n . The second  of dynamometer used were U n i v e r s a l Shear Beams  type  manufactured  by HBM Inc. of Framingham, MA. These were capable of measuring  only v e r t i c a l  f o r c e s . Two of these used i n  c o n j u n c t i o n . Moments and h o r i z o n t a l f o r c e s were excluded by d e s i g n . The dynamometers are d e s c r i b e d i n Appendix 2. The c y l i n d e r motion  was measured by a sonar  displacement measurement device p o s i t i o n e d over the c y l i n d e r model. T e s t s were performed motion  of 10 mm.  at two d i f f e r e n t amplitudes of  and 15 mm.  t o check  the dependence of the  f o r c e 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  f o r c e s on the c y l i n d e r and the displacement of the c y l i n d e r were o b t a i n e d . The s i g n a l s were s u i t a b l y a m p l i f i e d u s i n g a m p l i f i e r s on the O.E.C. towing c a r r i a g e and s t o r e d on the O.E.C. MINC-11 computer. The data c o l l e c t i o n software i s d e s c r i b e d i n Appendix 2. The data was m u l t i p l e x e d and s t o r e d on d i s k .  30 3.2.1  DATA ANALYSIS The data was  performed  demultiplexed and a s p e c t r a l a n a l y s i s  on the f o r c e and displacement  F o u r i e r Transforms.  T h i s was  r e c o r d s , using Fast  done to e l i m i n a t e noise and  extraneous measurements other than at the d r i v i n g The  was  s p e c t r a l a n a l y s i s y i e l d e d the f o r c e  frequency.  and  displacement amplitudes and phases at the d r i v e n frequency. The phases were c o r r e c t e d f o r the phase s h i f t s induced by the a m p l i f i e r s . Knowing these the f o r c e s i n phase with the acceleration, F  and  A  i n phase with the v e l o c i t y , F^ were  computed. Then, F  = (m+a  A  22  )x = -(m+a  )co x 2  22  0  e~ lut  ...(3.1)  and, F  = b  v  2 2  x = -ia>(b  22  x ) 0  e  ...(3.2)  1 C J t  where, x  = Amplitude  0  of the  m  = mass of c y l i n d e r  to  = Driven  a 2  frequency  =  Added mass c o e f f i c i e n t  =  Damping c o e f f i c i e n t  2  b 2  displacement.  2  From the above a 2 2  a n <  3 b  2 2  ^  o r  t  *  i e  d r i v e n frequency are  c a l c u l a t e d as, a 2  =  t>  = F  2  22  (F  _  y  /^ x ) " m 2  A  0  /cox  0  Experiments  were performed  each d r a f t . The experiments d r a f t s . The  at about  ten f r e q u e n c i e s f o r  were repeated f o r d i f f e r e n t  r e s u l t s are p l o t t e d i n non-dimensional  form as  31 d e s c r i b e d l a t e r . The software used f o r the data c o l l e c t i o n and a n a l y s i s are d e s c r i b e d i n Appendix  2.  3.3 DETERMINATION OF HEAVE EXCITING FORCE For these experiments  the c y l i n d e r was  h e l d s t a t i o n a r y at a  s p e c i f i e d d r a f t . Small amplitude s i n u s o i d a l waves were generated u s i n g the paddle type wavemaker at the O.E.C. Fig.10 shows a photograph  of the wavemaker. A b r i e f  d e s c r i p t i o n of i t s o p e r a t i o n and components i s given i n Appendix  2.  A r e s i s t a n c e wave probe was height. A description exciting  used to measure the wave  i s given i n Appendix 2. The  f o r c e on the model was  vertical  recorded using the  3-component dynamometer. The MINC-11 computer and the O.E.C. towing c a r r i a g e were used f o r data c o l l e c t i o n as b e f o r e . A s p e c t r a l a n a l y s i s on the wave r e c o r d u s i n g Fast F o u r i e r Transforms particular  gave the wave amplitude and phase at a  frequency. The  same a n a l y s i s on the f o r c e r e c o r d  gave the f o r c e amplitude and phase. The wave phase a d j u s t e d f o r the p o s i t i o n  was  f o r the wave probe. Thus, the  v e r t i c a l e x c i t i n g f o r c e due to a wave at a s p e c i f i e d amplitude and  frequency was  repeated f o r d i f f e r e n t  determined. The experiments  f r e q u e n c i e s f o r each d r a f t  were  f o r three  c y l i n d e r models. The c o n f i g u r a t i o n s were as shown i n F i g s . 4,5,  and  6.  The non-dimensional height was  exciting  f o r c e per foot of wave  p l o t t e d versus frequency. The  r e s u l t s are  32 compared with a t h e o r e t i c a l p r e d i c t i o n  by a boundary  element  method by Chan /11/ . The r e s u l t s are d i s c u s s e d l a t e r . The software used i s d e s c r i b e d i n Appendix  2.  3.4 FLOW VISUALIZATION Flow v i s u a l i z a t i o n t e s t s were conducted to observe q u a l i t a t i v e l y the vortex shedding process on a s i n g l e c y l i n d e r model and a compound c y l i n d e r model. Dye was i n j e c t e d through p o i n t s on the c y l i n d e r ' s s u r f a c e by means of c a p i l l a r y t u b i n g . The c y l i n d e r 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 v i d e o tape. F i g . 11 shows a photograph illustrating  the vortex shedding.  4. RESULTS AND  DISCUSSION  The added mass and damping c o e f f i c i e n t  computed  n u m e r i c a l l y 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 e x p e r i m e n t a l l y f o r the compound for four d r a f t s  model  of 35.5", 39.5", 43.5", 47.5". These  correspond t o a step s i z e , D' of 6", respectively.  cylinder  10", 14" and 18"  D' i s as d e f i n e d i n F i g . 2 . The heave added  mass and damping c o e f f i c i e n t are determined f o r two different  amplitudes of o s c i l l a t i o n of 10mm  and 15mm  f o r the  35.5" d r a f t . T h i s i s done to check the e f f e c t 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 d r a f t  the t e s t s are conducted f o r over 10  f r e q u e n c i e s i n the range 0.2 to 2.5 Hz. Wave amplitude at a d i s t a n c e of 33" from the c e n t r e l i n e cylinder quantities  of the o s c i l l a t i n g  i s measured d u r i n g the t e s t s . Measurement of a l l were done f o r at l e a s t  10 f r e q u e n c i e s at each  d r a f t . The hydrodynamic c o e f f i c i e n t s  determined  e x p e r i m e n t a l l y and the wave height measured d u r i n g the experiments are compared t o t h e o r e t i c a l the  p r e d i c t i o n s using  matching technique. Flow v i s u a l i s a t i o n  t e s t s were conducted d u r i n g 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 p r o c e s s . Heave e x c i t i n g waves on a s i n g l e  f o r c e s due to small amplitude  cylinder,  a double c y l i n d e r  33  sinusoidal  and a compound  34 c y l i n d e r were measured. The s i n g l e c y l i n d e r was t e s t e d at two d r a f t s of 7" and 10.5". The double c y l i n d e r was  tested  at two d r a f t s of 23.5" and 27.5". The compound c y l i n d e r tested  was  at four d r a f t s of 35.5", 38.375", 42.625" and 49.5".  The c o n f i g u r a t i o n s  f o r the three c y l i n d e r models were as  shown i n F i g s . 6,5 and 4 r e s p e c t i v e l y . For each d r a f t ,  tests  were performed f o r at l e a s t two d i f f e r e n t amplitudes to check the dependence of the e x c i t i n g f o r c e s  on the  amplitudes of the waves. For each d r a f t , at each amplitude s e t t i n g , the t e s t s were performed f o r at l e a s t seven wave f r e q u e n c i e s over the range of 0.25 to 2.5 Hz. The experimental r e s u l t s f o r the compound c y l i n d e r are compared to t h e o r e t i c a l p r e d i c t i o n s  by the matching  technique and a boundary element method. The experimental r e s u l t s f o r the s i n g l e and double c y l i n d e r are compared only to the t h e o r e t i c a l p r e d i c t i o n s  by the boundary  element  method. The t h e o r e t i c a l method using the matching technique i s as d e s c r i b e d  i n Chapter 2 and Appendix  technique, equipment and Appendix  1. The experimental  and software are d e s c r i b e d  i n Chapter 3  2.  4.1 PRESENTATION OF DATA The data i s presented i n the form of non-dimensional p l o t s . Exceptions are the p l o t s of wave height and heave e x c i t i n g forces.  35 The heave added mass i s p l o t t e d as a / p V versus w a/g. 2  2 2  a 2 2  i s the heave added mass, p i s the d e n s i t y of f r e s h  water, and V i s the volume of buoyancy f o r the c y l i n d e r at the  r e s p e c t i v e d r a f t , CJ i s the c i r c u l a r frequency of  o s c i l l a t i o n of the c y l i n d e r  i n radians/second, a i s the  maximum r a d i u s of the c y l i n d e r and g i s the a c c e l e r a t i o n  due  to g r a v i t y . 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 p l o t t e d a g a i n s t the same non-dimensional frequency as the added mass. The wave amplitude i n inches i s p l o t t e d a g a i n s t the same non-dimensional frequency. The heave e x c i t i n g f o r c e , F i s p l o t t e d as F/pVgA versus co a/g. Here u> i s the wave c i r c u l a r frequency and A i s the 2  wave amplitude i n f e e t . A l l other q u a n t i t i e s  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 i n heave motion f o r the compound circular cylinder  using the matching technique. For a l l  r e s u l t s r e p o r t e d here, 20 terms were taken f o r each of the Fourier  s e r i e s f o r the p o t e n t i a l s . The program was run f o r  5, 10, 15, 20, and 30 terms f o r the s e r i e s 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 c a l c u l a t e the hydrodynamic c o e f f i c i e n t s f o r the compound  36 c y l i n d e r at one taken f o r the  frequency of o s c i l l a t i o n when 20 terms are  series.  The matching  technique s a t i s f i e s the c o n t i n u i t y of  pressure and v e l o c i t y between adjacent r e g i o n s i n which p o t e n t i a l s are d e f i n e d (Chapter 2 ) . But, t h i s c o n t i n u i t y i s s a t i s f i e d as an i n t e g r a l over the depth and not at every p o i n t on the common boundary between adjacent r e g i o n s . Hence, when p r e s s u r e s and v e l o c i t i e s are computed u s i n g the solved p o t e n t i a l s , they are not continuous r a d i a l l y at p o i n t s along the boundary between adjacent r e g i o n s d e f i n e d in Chapter  2. T h i s d i s c o n t i n u i t y  i s more apparent  i n the  r a d i a l d e r i v a t i v e s of the p o t e n t i a l s , than i n the p o t e n t i a l s themselves. Since the p o t e n t i a l s are not a f f e c t e d significantly  by t h i s f a c t , the hydrodynamic  coefficients  which are computed using the p o t e n t i a l s are not The computer program CYLINDER was  affected.  run f o r the compound  c y l i n d e r c o n f i g u r a t i o n shown i n F i g . 4 at d i f f e r e n t  drafts.  Numerical d i f f i c u l t i e s were observed with step s i z e s ,  D',  l e s s than 6".  the  These are due  dispersion  relation  step s i z e ,  D'.  The comparison  to the d i f f i c u l t y  solving  f o r waves generated by a very shallow  of the r e s u l t s using the  matching  technique with p r e d i c t i o n s by the boundary element  method  show very good agreement. Fig.13 shows added mass v a l u e s f o r the compound c y l i n d e r at a d r a f t of 42.625" (D'=13.125"). The matching comparison  technique o v e r p r e d i c t s the added mass i n  to the boundary element  method by approximately  37 4% at the higher f r e q u e n c i e s . The agreement  improves as the  frequency d e c r e a s e s . 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  u n d e r p r e d i c t s the damping c o e f f i c i e n t  i n comparison to the  boundary element method. But, the d i f f e r e n c e i s w i t h i n 1% for most f r e q u e n c i e s .  4.3 DISCUSSION OF EXPERIMENTAL RESULTS  4.3.1  HYDRODYNAMIC COEFFICIENTS The experiments t o determine the hydrodynamic  c o e f f i c i e n t s i n heave motion f o r the compound c y l i n d e r were f i r s t conducted i n the Summer of 1983. However, the motion generator used f o r the experiments d i d not perform s a t i s f a c t o r i l y . The h y d r a u l i c pump and motor used f o r running the motion generator c o u l d not handle the load adequately. The scotch-yoke mechanism used i n the motion generator induced c o n s i d e r a b l e extraneous loads on the dynamometers due t o v i b r a t i o n s . F u r t h e r , i t f a i l e d to produce a smooth s i n u s o i d a l motion. For  the above reasons the motion generator was  redesigned and r e c o n s t r u c t e d and the t e s t s repeated i n June, 1984. The redesigned motion generator gave a smooth s i n u s o i d a l motion over a frequency range of 0.2 t o 2.5 Hz. It was a l s o capable of handling the imposed  loads w e l l .  Fig.12 shows a time record of the motion and Fig.12a shows  38 an amplitude  spectrum  that the spectrum at  the d r i v i n g  of the time r e c o r d . I t can be  does not show any prominent peaks except  frequency. D e s c r i p t i o n s of the motion  generator are given i n Appendix Two  500  seen  2.  l b load c e l l s were added to the system to  v e r i f y the f o r c e measurement by the 801b  f o r c e b l o c k . They  c o r r o b o r a t e d the readings of the 80 l b . f o r c e block which was  used p r i m a r i l y f o r the data a n a l y s i s . D e t a i l s of the  mounting of the 500  l b . f o r c e b l o c k s are v i s i b l e  in Fig.8a.  Fig.15 shows a p l o t of the heave added mass of the compound c y l i n d e r model at a d r a f t of 35.5" theoretical prediction  (D'=6"). The  i s by the matching technique.  experimental p l o t s show r e s u l t s f o r two amplitudes of 0.39"  an apparent  i n c r e a s e of added mass with frequency.  below 1 Hz,  0.5".  The experimental  of  oscillation  experimental  and  of the dynamometers  enough to measure v a l u e s l e s s than 0.5%  was of  s c a l e range a c c u r a t e l y . The d e v i a t i o n i n the added  mass r e s u l t s with change i n amplitude sufficient  enough w i t h i n the l i m i t s  accuracy to warrant  at  Reliable  because the magnitude of the f o r c e s measured  not s u f f i c i e n t  The  r e s u l t s show  r e s u l t s c o u l d not be obtained at f r e q u e n c i e s  were too small and the s e n s i t i v i t y  the f u l l  The  experiments the 0.5"  setting  of the  i s not  experimental  any c o n c l u s i o n s r e g a r d i n g n o n - l i n e a r i t y .  at the remaining  amplitude  three d r a f t s were  conducted  s e t t i n g , so that the f o r c e s would be  l a r g e enough to measure at the lower  f r e q u e n c i e s . The  t h e o r e t i c a l curve shows a w e l l d e f i n e d 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  sharply t i l l  a frequency of about  constant t i l l range. The  about 2.5 Hz,  1.5 Hz and  drops  remains  the upper end of the  frequency  best agreement between the t h e o r e t i c a l  experimental  results  i s at f r e q u e n c i e s c l o s e  theoretical  and  to 1.25  general the experimental v a l u e s are higher than  fairly  Hz.  In  the  results.  Fig.16 shows a p l o t of the heave damping c o e f f i c i e n t for the compound c y l i n d e r experimental prediction results  model at a d r a f t of 35.5". The  r e s u l t s are compared to the  theoretical  by the matching technique. The  follow  experimental  the same t r e n d as the t h e o r e t i c a l  curve, but  the experimental values are c o n s i d e r a b l y higher than theoretical  v a l u e s . Though the peak value appears  same frequency  f o r both experiment  by about 80% when compared to the t h e o r e t i c a l  experiments.  measured at a  from the o s c i l l a t i n g c y l i n d e r  The  theoretical  curve  i s higher  result.  shows p l o t s of the wave amplitude  d i s t a n c e of 33"  at the  and theory, the magnitude  of the damping c o e f f i c i e n t as shown by experiment  Fig.17  the  d u r i n g the  i s a prediction  matching technique. The t h e o r e t i c a l  prediction  by the  is  c o n s i d e r a b l y higher than the e x p e r i m e n t a l l y measured v a l u e s . They d i f f e r by about 50%. The d i f f e r e n c e to the i n t e r f e r e n c e Figs.  18,21  compound c y l i n d e r respectively.  from the tank  and 24 are p l o t s at d r a f t s  The amplitude  may  possibly  be  due  walls. of the added mass f o r the  of 39.5", 43.5" of o s c i l l a t i o n  and  47.5",  i s 0.5"  in a l l  40 cases. The comparison between experiment and theory i s much the same as f o r the 35.5" d r a f t . However, the a b s o l u t e value of the non-dimensional  added mass decreases  d r a f t , a c c o r d i n g to the t h e o r e t i c a l experimental variation  increasing  p r e d i c t i o n . The  r e s u l t s , however, show the same range of  f o r a l l the d r a f t s .  Figs.  19, 22 and 25 are p l o t s  coefficient  for cylinder  respectively.  drafts  of the heave damping  of 39.5", 43.5" and 47.5"  The trends i n the comparison between the  theory and experiment a r e s i m i l a r  to those f o r the 35.5"  d r a f t . The v a l u e of the non-dimensional decreases  with  with  increasing  damping c o e f f i c i e n t  d r a f t . T h i s i s shown by both  theory and experiment. T h i s can be e x p l a i n e d by the i n c r e a s e in step s i z e , D' with i n c r e a s i n g d e c r e a s i n g wavemaking a c t i o n c o e f f i c i e n t at the deeper Figs.  d r a f t . T h i s leads to a  and hence a smaller damping  drafts.  20, 23 and 26 are p l o t s  of the wave  measured at a d i s t a n c e of 33" from the compound centreline  for drafts  prediction  f o r the 35.5" d r a f t . The  i s higher than the e x p e r i m e n t a l l y  measured wave amplitudes.  But, the experimental  the same t r e n d as the t h e o r e t i c a l maximum wave amplitudes the 35.5" d r a f t  values show  curve. The t h e o r e t i c a l  s t e a d i l y decrease  from about 0.3" at  t o about 0.1" at the maximum d r a f t  T h i s can be a t t r i b u t e d with i n c r e a s i n g  cylinder  of 39.5", 43.5" and 47.5". These p l o t s  show the same t r e n d as Fig.17 theoretical  amplitude  t o the d e c r e a s i n g wavemaking  d r a f t as mentioned i n the p r e v i o u s  of 47.5". action  41 paragraph.  The experimental  r e s u l t s a l s o show the same  trend, d e c r e a s i n g from a maximum wave amplitude 0.15"  of about  at a 35.5" d r a f t t o a maximum of about 0.04" at the  maximum d r a f t of 47.5". The  added masses and damping c o e f f i c i e n t s  determined  e x p e r i m e n t a l l y do not show very c l o s e agreement with the t h e o r e t i c a l r e s u l t s . T h i s can be due t o two reasons. The theory does not take  i n t o account  the v i s c o s i t y of the f l u i d  medium, which e x i s t s i n a c t u a l f a c t . The v i s c o s i t y vortex shedding observed  during the c y l i n d e r motion. T h i s was  d u r i n g the flow v i s u a l i s a t i o n t e s t s conducted  the compound c y l i n d e r model (Chapter shedding  induces  on  3 and F i g . 1 1 ) . Vortex  takes place at the c o r n e r s of the c y l i n d e r during  the c y l i n d e r motion. T h i s i n t r o d u c e s v i s c o u s damping i n a d d i t i o n to the p o t e n t i a l damping due to the wavemaking a c t i o n . T h i s may e x p l a i n why the e x p e r i m e n t a l l y damping c o e f f i c i e n t  determined  i s c o n s i d e r a b l y higher than the  t h e o r e t i c a l l y p r e d i c t e d v a l u e . The extent of the v i s c o u s damping can be determined at  the c o r n e r s The  shedding  theoretically.  second cause f o r the d i f f e r e n c e between  experimental of  by modelling the vortex  and t h e o r e t i c a l values may be due to the e f f e c t  the w a l l s of the towing  tank on the flow f i e l d  during the  c y l i n d e r motion. The w a l l s a r e at a d i s t a n c e of 62" from the c y l i n d e r c e n t r e l i n e on one s i d e and 82" on the other The  e f f e c t of the presence  by performing  side.  of the w a l l s can only be i s o l a t e d  the same t e s t s i n a very l a r g e b a s i n and  42 comparing the r e s u l t s . A f u r t h e r cause f o r the d i f f e r e n c e between experimental and  t h e o r e t i c a l r e s u l t s may  the measurement of the  be  i n a c c u r a c i e s in  f o r c e , displacement and  s h i f t s of the a m p l i f i e r s . Care was  the  the phase  taken in the c a l i b r a t i o n  process to e l i m i n a t e these as much as p o s s i b l e . However, a very  small change of phase introduced  data c o l l e c t i o n process may  s i n c e the  the phase and  in the  a f f e c t the f o r c e i n phase with  the a c c e l e r a t i o n more than the velocity,  electronically  f o r c e in phase with  the  former i s p r o p o r t i o n a l to the cosine  the l a t t e r  i s p r o p o r t i o n a l to the  phase. Thus, i f a small e r r o r i s introduced  sine of  of the  i n the phase  t h i s 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 e r r o r w i l l  measured i n c r e a s e . T h i s may increase  4.3.2  i n c r e a s e as the  forces  e x p l a i n the reason f o r the  i n the added mass with  frequency.  FLOW VISUALISATION TESTS These t e s t s were conducted i n June,1983. A d e s c r i p t i o n  of the t e s t s i s given of vortex  i n Chapter 3. They show the  shedding at the corners  presence  of the compound c i r c u l a r  c y l i n d e r . They suggest that the e f f e c t s of v i s c o s i t y may important. The  t e s t s were recorded  photographs were a l s o taken during one  such photograph.  on video  tape  be  and  the t e s t s . Fig.11 shows  43 4.3.3  HEAVE EXCITING FORCES Heave e x c i t i n g f o r c e s due to small amplitude s i n u s o i d a l  waves were measured on three d i f f e r e n t c y l i n d e r These experiments were performed i n J u l y ,  models.  1983 at the towing  tank at B.C.Research. The purpose of these experiments  was  to v e r i f y the r e l a t i o n s h i p between the heave damping coefficient  and the heave e x c i t i n g f o r c e  f o r the compound  c y l i n d e r model. These experiments were a l s o  intended to be a  f u r t h e r 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 f o r the compound c y l i n d e r model t h e o r e t i c a l l y c a l c u l a t e d  using  the matching technique. The experimental r e s u l t s f o r the other c y l i n d e r models were intended as a v e r i f i c a t i o n of the heave e x c i t i n g f o r c e s computed by a boundary The boundary element method a p p l i e d diffraction  element method.  by Chan /11/ used l i n e a r  theory and hence can be c o n s i d e r e d a more d i r e c t  computation of the e x c i t i n g f o r c e . The experimental procedure i s d e s c r i b e d i n Chapter 3 and a d e s c r i p t i o n of the equipment  i s given i n Appendix  2.  Fig.27 i s a p l o t of the 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 model at a d r a f t of 7". The range of wave frequency i s from 0 to 2.5 Hz. The experimental r e s u l t s are compared to a t h e o r e t i c a l p r e d i c t i o n by a boundary  element method.  The experimental measurements were made f o r waves of three d i f f e r e n t amplitude s e t t i n g s of 0.12", 0.2"  and 0.4".  The  amplitudes are approximate s i n c e the wave amplitudes cannot be d u p l i c a t e d slightly  e x a c t l y . The waveform o b t a i n e d was  also  i r r e g u l a r . The causes f o r t h i s are d i s c u s s e d i n  44 Appendix 2. The r e s u l t s show q u i t e theoretical prediction  good agreement with the  by the boundary element method. The  e x p e r i m e n t a l l y measured peak e x c i t i n g  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 p r e d i c t e d  peak  value which occurs at the same frequency. The experiments r e s u l t s show a s c a t t e r Hz.  of about 15% at a frequency of 1.25  These can be due to the i n a c c u r a c i e s  of the f o r c e s  i n the measurement  and the wave h e i g h t s caused by the  i r r e g u l a r i t y of the generated waveform. Fig.28 i s a p l o t of the phase d i f f e r e n c e exciting  f o r c e and the wave. The e x p e r i m e n t a l l y measured  values show c o n s i d e r a b l e s c a t t e r theoretical prediction scatter  and d e v i a t i o n  may again be due to i n a c c u r a c i e s  i n the measurement  irregularity.  Fig.29 shows the heave e x c i t i n g single cylinder  from the  by the boundary element method. T h i s  of the waves because of t h e i r  the  between the  force  r e s u l t s f o r the  model at a d r a f t of 10.5". The s c a t t e r i n  experimental r e s u l t s i s l e s s than f o r the 7" d r a f t . The  experimental r e s u l t s show q u i t e theoretical prediction  good agreement with the  by the boundary element method.  Fig.30 shows a p l o t of the phase d i f f e r e n c e heave e x c i t i n g  between the  force and the wave f o r the 10.5" d r a f t . The  phase v a l u e s do not show good agreement with the t h e o r e t i c a l predict ion. Figs.  31 and 33 show the p l o t s of heave e x c i t i n g  for the double c y l i n d e r 27.5"  respectively.  forces  model (Fig.5) at d r a f t s of 23.5" and  Here, the agreement with  theoretical  45 prediction  by the boundary element method i s not very good.  There i s c o n s i d e r a b l e s c a t t e r experimental prediction. difference  i n the data. In g e n e r a l , the  r e s u l t s are higher than the Figs.  32 and  theoretical  34 show p l o t s of the phase  between the e x c i t i n g  force  and the wave. The  experimental values show c o n s i d e r a b l e s c a t t e r  and do not  show good agreement with the theory. Fig.35  i s a p l o t of the heave e x c i t i n g  compound c y l i n d e r Experimental predictions  force  f o r the  model (Fig.4) at a d r a f t of 35.5".  r e s u l t s are compared with  theoretical  by the matching technique and the boundary  element method. Both t h e o r e t i c a l methods o v e r p r e d i c t exciting  f o r c e at f r e q u e n c i e s above 1.25  Hz. The  boundary  element method g i v e s a lower value of the e x c i t i n g except at the lower  the  force  f r e q u e n c i e s . Both t h e o r e t i c a l methods  show good agreement. The agreement i s best at the lowest highest f r e q u e n c i e s . At f r e q u e n c i e s lower than 0.75 e x p e r i m e n t a l l y measured e x c i t i n g theoretical prediction.  forces  and  Hz,  the  are higher than  the  Fig.36 shows a p l o t of the  force-wave phase d i f f e r e n c e  f o r the compound c y l i n d e r  at a  d r a f t of 35.5". The phase p l o t shows c o n s i d e r a b l e s c a t t e r i n the experimental values and the agreement between theory experiment  i s not good.  Fig.37 shows heave e x c i t i n g cylinder  force  f o r the compound  at a d r a f t of 38.375". Here the t h e o r e t i c a l  by the boundary element method u n d e r p r e d i c t s the force  and  results  exciting  i n comparison to the matching technique at most  46 f r e q u e n c i e s . The boundary element method r e s u l t s show good agreement with the r e s u l t s from the matching technique. In g e n e r a l , the experimental  r e s u l t s show good agreement with  the t h e o r e t i c a l p r e d i c t i o n s .  The s c a t t e r  i n the experimental  data i s low except at around 0.25 Hz. Fig.38 force-wave phase d i f f e r e n c e 35.5"  f o r the compound c y l i n d e r at  d r a f t . The experimental  not agree w e l l  r e s u l t s show s c a t t e r  with the t h e o r e t i c a l  and do  prediction.  Fig.39 shows the heave e x c i t i n g cylinder  shows the  force  f o r the compound  at a d r a f t of 42.625". Here, the p r e d i c t i o n  boundary element method i s c l o s e  to that  by the  by the matching  technique a t a l l f r e q u e n c i e s . The agreement i s best at the highest and lowest mostly  lower  f r e q u e n c i e s . The experimental  than the t h e o r e t i c a l p r e d i c t i o n s .  r e s u l t s are  They show  good agreement with the theory a t two f r e q u e n c i e s . Fig.40 shows the force-wave phase d i f f e r e n c e  f o r the same d r a f t .  Again, the agreement with theory i s not good. Fig.41 cylinder  shows the heave e x c i t i n g  force  f o r the compound  a t a d r a f t of 49.5". The t h e o r e t i c a l p r e d i c t i o n by  the boundary element method i s c l o s e  to that  by the matching  technique. The agreement i s best at the higher f r e q u e n c i e s . In g e n e r a l , the experimental  r e s u l t s show good agreement  with the t h e o r e t i c a l p r e d i c t i o n s . scattered forces  However, there are a few  p o i n t s which show c o n s i d e r a b l y higher  than the theory. Fig.42  force-wave phase d i f f e r e n c e . show s c a t t e r  exciting  shows a p l o t of the  Again, the experimental  and do not agree w e l l  with theory.  values  47 In g e n e r a l , the e x c i t i n g  force measurements show good  agreement with t h e o r e t i c a l p r e d i c t i o n s .  The e x c i t i n g  computed from the damping c o e f f i c i e n t s c a l c u l a t e d matching technique are c l o s e  forces  by the  to the v a l u e s computed d i r e c t l y  by the boundary element method. Hence, i t can be i n f e r r e d that  the r e l a t i o n s h i p  i n f e r r e d that diffraction  suggested  by Newman i s good. I t can be  the t h e o r e t i c a l r e s u l t s computed using l i n e a r  theory can be c o r r o b o r a t e d by experimental  r e s u l t s . Though there i s some s c a t t e r forces  i n the experimental  measured from d i f f e r e n t amplitude  enough evidence to suggest  waves, there i s  v a l i d i t y of the l i n e a r theory f o r  waves of these amplitudes. The measurement sufficiently  of phases i s not  r e l i a b l e to enable any d e f i n i t e c o n c l u s i o n s  regarding t h e i r t r e n d s .  CONCLUSIONS 1 . The of  t h e o r e t i c a l method presented here, using the c o n t i n u i t y pressure and v e l o c i t y between regions of the flow  field  and a F o u r i e r s e r i e s r e p r e s e n t a t i o n f o r the p o t e n t i a l s , p r o v i d e s a good method f o r determining the hydrodynamic coefficients  i n heave motion f o r the compound c i r c u l a r  c y l i n d e r 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 r e q u i r e d f o r  the present method i s minimal. The method i s capable of computing the pressures and v e l o c i t i e s at any p o i n t i n the flow f i e l d ,  and hence the wave height at any p o i n t on  the  free s u r f a c e a l s o . 2. The computer program CYLINDER, used to compute the added mass and damping c o e f f i c i e n t cylinder of  CPU  f o r the compound c i r c u l a r  i n heave motion i s e f f i c i e n t . I t takes 2.8  seconds  time on the Amdahl V-8/470 computer to compute the  hydrodynamic c o e f f i c i e n t s f o r one  frequency of  oscillation.  3. The  t h e o r e t i c a l method being a p o t e n t i a l flow method i s  unable  to account  f o r the v i s c o s i t y of the f l u i d medium and  hence the vortex shedding d u r i n g the o s c i l l a t i o n  fo the  c y l i n d e r . The method a l s o g i v e s numerical problems when the step s i z e , D' computation  i s made very s m a l l . F u r t h e r , i n the  of r a d i a l v e l o c i t i e s i n the flow f i e l d ,  c o n t i n u i t y of v e l o c i t y  exact  i s not acheived at every p o i n t on 48  the  49 common boundary between two adjacent regions i n the flow field. 4. The  experimental  r e s u l t s f o r the added mass show l a r g e r  values than the t h e o r e t i c a l p r e d i c t i o n by the matching technique. The experimental  r e s u l t s a l s o show an i n c r e a s e  with frequency. T h i s can be due t o an e r r o r of  i n measurement  the phase s h i f t . Experimental v a l u e s c o u l d not be  determined  at f r e q u e n c i e s lower  sufficient  sensitivity  than  1 Hz, due to l a c k of  i n the f o r c e r e c o r d i n g equipment.  5. The experimental heave damping c o e f f i c i e n t values show the same trends as the t h e o r e t i c a l v a l u e s , but are c o n s i d e r a b l y higher than the t h e o r e t i c a l v a l u e s . T h i s can p a r t l y be due to  v i s c o u s e f f e c t s d u r i n g the c y l i n d e r motion and p a r t l y due  to  the e f f e c t s of the w a l l s of the towing  tank on the flow  field. 6. The  flow v i s u a l i s a t i o n t e s t s show the vortex  shedding  o c c u r i n g at the c o r n e r s of the c y l i n d e r . T h i s suggests the presence  of v i s c o u s damping, which i s not accounted f o r  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 deviation. 8.  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  50 The wave amplitudes measured d u r i n g o s c i l l a t i o n of the compound c y l i n d e r  model do not show good agreement with the  theoretical prediction tests  interference  by the matching technique. During the  with waves r e f l e c t e d o f f the s i d e  walls  of the towing tank was observed. T h i s may be one of the reasons f o r the l a r g e d i f f e r e n c e  between t h e o r e t i c a l and  experimental r e s u l t s . F u r t h e r , the a c t u a l magnitude of the wave h e i g h t s measured i s s m a l l .  Hence, the s e n s i t i v i t y of  the wave probe may have some e f f 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 h e i g h t s f o r the compound c y l i n d e r model decrease s i g n i f i c a n t l y as the d r a f t i n c r e a s e s due t o d e c r e a s i n g wavemaking  a c t i o n . Both  t h e o r e t i c a l and experimental r e s u l t s show t h i s t r e n d . 10. The heave e x c i t i n g compound c y l i n d e r predictions  forces  measured e x p e r i m e n t a l l y f o r the  model show good agreement with  theoretical  by the matching technique and the boundary  element method. T h i s suggests that  l i n e a r d i f f r a c t i o n theory  p r o v i d e s r e s u l t s which can be c o r r o b o r a t e d by experiment. The agreement with the r e s u l t s from the matching technique show the v a l i d i t y of the r e l 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 e x c i t i n g  force.  11. The heave e x c i t i n g  forces  measured on the s i n g l e  model show very good agreement with t h e o r e t i c a l  cylinder predictions  by the boundary element method. The agreement between theory  51 and  experiment i s not  cylinder  very good f o r the  t e s t s on the  double  model.  12. The  phases measured d u r i n g the  all  three c y l i n d e r models do not  theoretical reliable  can  on  show good agreement with  r e s u l t s . They a l s o show c o n s i d e r a b l e s c a t t e r .  inference  s c a t t e r can  heave e x c i t i n g f o r c e t e s t s  be made from these r e s u l t s .  No  The  be a t t r i b u t e d to inaccuracy in measurement of  the wave e l e v a t i o n  due  to i r r e g u l a r i t y of the  waveform  generated. 13. Though the  heave e x c i t i n g f o r c e s measured for the d i f f e r e n t  amplitude s e t t i n g s  show some s c a t t e r  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  t e s t s show that  the  e f f e c t s of v i s c o s i t y of the  medium are more important the o s c i l l a t i n g c y l i n d e r exciting  f o r c e s due  for the measurement of  fluid  forces  on  than f o r the measurement of  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  c y l i n d e r measured by McCormick /1/,  /20/.  conducted i n a towing tank at the U.S. frequency of o s c i l l a t i o n  single  These t e s t s were  Naval Academy.  i s 3 r a d i a n s / s e c . The  r e s u l t s show very much higher v a l u e s than the  The  experimental theoretical  r e s u l t s . These r e s u l t s are comparable to present comparison  52 between theory and experiment model. ANU/MH i n Fig.43 non-dimensionalized  f o r the compound c y l i n d e r  i s the added mass  by mass of the c y l i n d e r of height equal  to the depth of water.  RECOMMENDATIONS The  experimental  results  f o r 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 t h e o r e t i c a l heave e x c i t i n g  predictions  by the matching technique. The  f o r c e s computed t h e o r e t i c a l l y  damping c o e f f i c i e n t show b e t t e r experimental presence  relatively due  agreement with the  r e s u l t s . The flow v i s u a l i s a t i o n t e s t s  of vortex shedding  These suggest  from the  that  d u r i n g the c y l i n d e r  show the  oscillation.  the e f f e c t s of v i s c o s i t y may be  important. A l s o , the e f f e c t s of the i n t e r f e r e n c e  t o the presence  of the s i d e w a l l s of the towing  tank are  not assessed a t p r e s e n t . To estimate the e f f e c t s of the above i t would be d e s i r a b l e  t o do the f o l l o w i n g .  1 . Theoretically  model the vortex shedding  using r i n g  vortices  at the c o r n e r s of the c y l i n d e r . 2. Theoretically potential  model the e f f e c t s of the tank w a l l s by a  flow method such as the method of images.  3. Repeat the same t e s t s  f o r the o s c i l l a t i n g c y l i n d e r  open basin where the r e s t r a i n i n g would not be f e l t .  53  efects  i n an  of the tank w a l l s  NOMENCLATURE FOR PLOTS  HEAVE EXCITING FORCE PLOTS F  =Heave e x c i t i n g  force  RO  =Density of medium  V  =Buoyancy  g  = A c c e l e r a t i o n due to g r a v i t y  AMP  =Wave amplitude i n f t .  W  =Wave frequency  A  =Maximum radius  i n c u . f t . of c y l i n d e r model  of c y l i n d e r model  HYDRODYNAMIC COEFFICIENTS AND WAVE AMPLITUDE PLOTS a22  =Heave added mass  RHO  =Density of f r e s h water  V  =Buoyancy  g  = A c c e l e r a t i o n due to g r a v i t y  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  A  =Maximum radius  i n c u . f t . of c y l i n d e r model  frequency  of c y l i n d e r model  54  LIST OF SYMBOLS  p  = Density of water.  u>  = Circular  k , k 0  frequency of c y l i n d e r or wave motion.  = Wave number.  n  m, m  = Wave number.  <i>, <t>  = Velocity potential  J ,  J i  = Bessel  I ,  I,  = Modified  Bessel  f u n c t i o n of the f i r s t  K , K,  = Modified  Bessel  f u n c t i o n of the second kind  H , H,  = Hankel f u n c t i o n of the f i r s t  d  = Depth of water a,, a , a  0  0  0  n  0  0  f u n c t i o n of the f i r s t  2  = Cylinder d1, d , d 2  kind. kind  kind,  3  radii.  3  = Depth from c y l i n d e r , g  = A c c e l e r a t i o n due to g r a v i t y  NOTE T h i s l i s t  of symbols  i s not complete. The v a r i o u s  symbols are d e f i n e d at the p o i n t s used.  55  i n the t e x t where they are  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 c y l i n d e r s at f i n i t e depth," Ocean E n g i n e e r i n g v o l . 8 , 1981. 2. KOKKINOWRACHOS,K. et a l . "Belastungen und bewegungen gro|3volumiger seebauwerke durch w e l l e n , " Nr. 2905/ Fachgruppe Maschinenbau / V e r f a h r e n s t e c h n i k F o r s c h u n g s b e r i c h t des Landes Nordrhein-Westfalen, 1980 3. KOKKINOWRACHOS,K., HOEFELD, J . , " T h e o r e t i s c h e und e x p e r i m e n t e l l e untersuchungen des bewegungsverhaltens von Halbtauchern," Nr.2915/ Fachgruppe Umwelt / Verkehr F o r s c h u n g s b e r i c h t des Landes Nordrhein-Westfalen, 1980 4. GARRETT,C.J.R., "Wave f o r c e s on a c i r c u l a r dock," J o u r n a l 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, v o l . 3 , 1971. 6. KRITIS, I r . B., "Heaving motion of axisymmetric bodies," J o u r n a l of Ship Research, v o l . 5 , No.3, 1979. 7. RAMBERG, S.E., and NIEDZWECKI, J.M., Ocean E n g i n e e r i n g v o l . 9 , No.1,1982 8. ISAACSON, M., and SARPKAYA,T., "Mechanics of Wave Forces on S t r u c t u r e s , " Van Nostrand P u b l i c a t i o n s , 1980 9. NEWMAN, J.N., "Marine Hydrodynamics," 1980.  The MIT P r e s s ,  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 E r o s i o n Board 1954 11. CHAN, J.L.K.," Hydrodynamic c o e f f i c i e n t s f o r axisymmetric bodies,"M.A.Sc. T h e s i s , U n i v e r s i t y of B r i t i s h Columbia, 1984 12. HAVELOCK, T.H., " C o l l e c t e d papers," ONR/ACE-103, U.S. Government P r i n t i n g 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 e x c i t i n g f o r c e s on four axisymmetric ocean p l a t f o r m s , " Naval Ship Research and Development Center, Report No. SPD-670-01, 1974 14. BAI, K.J. and YEUNG, R.W., "Numerical s o l u t i o n s t o f r e e - s u r f a c e flow problems," Tenth Symposium Naval Hydrodynamics, Cambridge, Mass. 1974 56  57 15. GARRISON, C.J., "Hydrodynamics of l a r g e o b j e c t s sea, Part I I . Motion of f r e e - f l o a t i n g bodies," J . Hydronautics 9 (2), 1975  i n the  16. ISSHIKI, H. and HWANG, J.H., "An axi-symmetric dock i n waves," Seoul N a t i o n a l U n i v e r s i t y , Korea, C o l l e g e of Engineering, Dept. of Naval A r c h i t e c t u r e , Report No. 73-1, 1973 17. NEWMAN, J.N., "The i n t e r a c t i o n of s t a t i o n a r y v e s s e l s with r e g u l a r 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 f r e e s u r f a c e , " J . F l u i d Mech. 21, 1974 19. ABRAMOWITZ, M. and STEGUN, I.A., "Handbook of mathematical f u n c t i o n s , " N a t i o n a l 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 c y l i n d e r i n r e s t r i c t e d waters," U.S. Naval Academy, E n g i n e e r i n g Report EW 10-80 1980 21. WANG and SHEN, "The hydrodynamic f o r c e s and pressure d i s t r i b u t i o n s f o r an o s c i l l a t i n g sphere i n a f l u i d of f i n i t e depth,"M.I.T. Department of Naval A r c h i t e c t u r e and Marine E n g i n e e r i n g , D o c t o r a l T h e s i s 1966 22. YEUNG, R.W., "A h y b r i d i n t e g r a l - e q u a t i o n method f o r time harmonic f r e e surface flow,"1st I n t . Conf. Numer. Ship Hydrodynamics, G a i t h e r s b u r g , Maryland 1975  5. APPENDIX 1 - EVALUATION OF POTENTIAL FUNCTIONS The c o e f f i c i e n t s of the p o t e n t i a l f u n c t i o n s d e s c r i b e d in chapter  2 are obtained by s o l v i n g a system of l i n e a r  simultaeneous  equations.  F i v e systems of equations are w r i t t e n and s o l v e d to give the c o e f f i c i e n t s of the f i v e s e r i e s needed to d e s c r i b e the p o t e n t i a l s f o r the flow model. These are obtained by equating the pressures  ( p o t e n t i a l s ) and v e l o c i t i e s (normal  d e r i v a t i v e s of p o t e n t i a l s ) along the boundaries  of the  regions d e f i n e d i n chapter 2. For example, l e t $  and ^  A  be the p o t e n t i a l s i n two  a d j o i n i n g regions given by,  T *A  =  V  T, j H  -icot , *A  TT j  -icot , *B  d  e  and , *B  =  V  H  d  e  Along a common boundary f o r c o n t i n u i t y of p r e s s u r e , P  P  ° ~  A  B  ~  =  *A,t  p  =  *B t  p  f  =  1 C J P V  1  W  P  V  H  H  d  d  e  "  e  1 C J t  *A  =  +B  Or, 0  A  = 0  ...(1.1)  B  And f o r c o n t i n u i t y of r a d i a l i *A,r , *B,r  =  V  ,T H  j  d  e  ,  TT  " H  where the s u f f i x  V  d  e  -icot , ^A,r -icot , ^B,r  velocity  =  ' r ' denotes d e r i v a t i v e with r e s p e c t to  58  59 r.Or, (1.2)  A,r = 0B,r  However,the above denotes c o n t i n u i t y of pressure and v e l o c i t y a t a p o i n t along a common boundary. A c t u a l l y , i n order to separate the c o e f f i c i e n t s of the s e r i e s , the orthonormal  p r o p e r t i e s of the p o t e n t i a l  f u n c t i o n s are made  use o f . T h i s i s done by equating the i n t e g r a l of the f u n c t i o n over the depth as shown below. Using a c o n t i n u i t y of pressure and i n t e g r a t i n g as below, d  u <j> Z(z)dz u  one  o b t a i n s a set of equations.  Here, Z(z) i s an orthonormal  f u n c t i o n f o r d-^ <z<d  u  . By  s u b s t i t u t i n g the normal d e r i v a t i v e of the p o t e n t i a l f o r the p o t e n t i a l we have a s i m i l a r r e l a t i o n  f o r c o n t i n u i t y of  velocity. Because the problem i s formulated  i n t h i s manner, the  s o l u t i o n does not s a t i s f y c o n t i n u i t y of pressure and v e l o c i t y at a l l p o i n t s along the boundary e x a c t l y . But, the i n t e g r a l of the v e l o c i t y and pressure along the depth f o r one  region w i l l be equal to the r e s p e c t i v e i n t e g r a l i n the  a d j o i n i n g r e g i o n . An a l t e r n a t i v e f o r m u l a t i o n would be to equate the pressures and v e l o c i t i e s i n a d j o i n i n g regions at a number of p o i n t s along the boundary. The exact number of p o i n t s would be equal to the number of unknown c o e f f i c i e n t s  60 i n the s e r i e s f o r the p o t e n t i a l . However,the method of s o l u t i o n used here i n v o l v e s no d i s c r e t i z a t i o n of the boundary  of a d j o i n i n g r e g i o n s . A l s o , i t enables one to use  the orthonormal p r o p e r t i e s of the p o t e n t i a l f u n c t i o n s .  5.1 POTENTIALS The p o t e n t i a l s i n the v a r i o u s regions are as given  below.  5.1.1 REGION 1 *,  = V  d e~  H  [(z -r /2)/2dd,  i w t  2  + A /2  2  0  00  +  £  A  ( I (nwr/di ) / I  n  0  (n7T a,/d, ) ) c o s ( n i z / d , ) ]  0  ... (I.3)  5.1.2 REGION 2 $  = V d e" H  2  [(z -r /2)/2dd + B  i w t  2  2  2  0  OS  + £  {B  = 1  n  V  (r) + C  n  n  W  (r)} cos (nirz/d ) ]  n  2  ... ( 1 . 4 ) where, v  n  (r) =  I (n7rr/d ) 0  +( I ( n 7 r a / d ) / K ^ n T r a ^ d z )  2  1  1  I ( n 7 r a / d ) +( I,(unai/d2) 0  W  n  2  ) K (n7rr/d )  2  0  / K^njra^dj)  2  2  ) K (n7ra /d ) 0  2  2  (r) -  I (n7rr/d ) 0  +( I ( n 7 r a / d ) / K (n7ra /d )  2  0  2  2  0  2  ) K (n7rr/d )  2  0  2  I ( n i r a / d ) +( I ( n 7 r a / d ) / K (n7ra /d ) ) K (n7ra,/d ) 1  1  2  0  2  2  0  2  2  1  2  61 It  i s e a s i l y seen that V ( r ) and W ( r ) have the •* n n  following At r=a,  properties.  V ^ ( a , )  0,  =  ='mr/d  Wj(a,)  2  where the primes i n d i c a t e d e r i v a t i v e with respect At r = a , W 2  (a ) = 0  n  V  to r .  2  ( a  n  2  = 1  )  5.1.3 REGION 3 The v e l o c i t y p o t e n t i a l i n t h i s region  $  = V d e~ H  3  i s defined as:  [(z/d+g/(w d)-1)  i w t  2  00  + D X Yo + E 0  0  D  = 1  X  n  Y  n  n  ]  ...(1.5  where J (m r) 0  X  -( J  0  , ( m  0  a  3  / H,(m a ) ) H ( m r )  )  0  3  0  0  =  0  J ( m a ) -( J,(moa ) / H^moas) ) H ( m a ) 0  0  2  I (m  n  r) +(  I (m  n  a ) -(  0  X  3  I i ( m  0  a ) / 3  n  K  1  ( m  n  0  2  a ) ) K (m r) 3  0  n  =  n 0  2  I , ( m  In t h i s  formulation,  Yo = M ~  1 / 2  0  a ) / 3  n  cosh[m (z-d )] 0  3  K i ( m  n  a ) ) K (m 3  0  n  a ) 2  62 M  0  = [l+sinh{2m (d-d )}/2m (d-d3)]/2 0  ~  3  Y  n  = M  n  M  n  =  [1+sin{2mo(d-d )}/2m (d-d )3/2  1  /  cos[m  0  2  (z-d )]  n  3  3  where m  and m  0  n  0  3  are the roots of  CL) -gm tanh[m ( d - d ) ] = 0 2  0  w +gm  0  tan[m  2  n  3  (d-d )j = 0  n  3  respectively. It  i s e a s i l y seen that X  and X  n  X ' ( a ) = 0, X  p  0  have the f o l l o w i n g  properties. At r=a  3  0  and at r=a  3  ' ( a ) = 0, 3  X ( a ) = 1, X  2  0  2  The primes denote d e r i v a t i v e with  (a ) = 1 .  n  2  respect to r .  5.1.4 EXTERIOR REGION The f r e e wave and standing waves are d e f i n e d with the potential *  *_ = V E  d e  E  as:  1 U ) t  [-E (H (ko )/H, r  0  0  (k a ))Z (z) 0  2  0  H. - Z  = 1  E  n  (K (k 0  n  r)/K,(k  n  a )) Z 2  n  (z)]  ...(1.6)  63  where, Z  0  = N ~  N  0  = [1+sinh{2k d}/2k d]/2  Z  0  0  n = n "  N  cosh[k z]  1 / / 2  0  N  1  /  2 c  o  s  [  k  0  n  z  ]  = [1+sin{2k d}/2k d]/2  n  0  where k  0  and k  fi  0  are the r o o t s of  co -gk tanh[ k d ] = 0 2  0  co +qk 2  3  n  0  tan[k  n  d] = 0  respectively.  5.2 GENERATION OF SYSTEMS OF EQUATIONS FOR SOLUTION The f i v e systems of equations neccessary  to s o l v e f o r the  c o e f f i c i e n t s of the f i v e s e r i e s f o r the p o t e n t i a l s are obtained by matching the pressures and v e l o c i t i e s at common boundaries  between the regions d e f i n e d .  5.2.1 CONTINUITY OF PRESSURE BETWEEN REGIONS 1 AND 2 E x p r e s s i n g the v e l o c i t y p o t e n t i a l $ as $ = V d e <t>, f o r c o n t i n u i t y H 1 and 2, we can express l a ) t  of pressure between  regions  64 Pressure,p = -p4>  1  = ~p^  fc  > 2  t  using l i n e a r i s e d B e r n o u l l i ' s  (1.1) we can w r i t e </> = <S>  As proved i n equation c o n t i n u i t y of pressure  equation.  n  (2/d ) / 1  Using can  A  k  between the l i m i t s 0  d, a f t e r m u l t i p l y i n g by the orthogonal  2cos ( kirz/d ) /d  a,  }  function  we have,  c6 cos(k7rz/d  0  1  1  )dz = ( 2 / d ) / 1  a,  0  c6 cos(kwz/d, )dz 2  the orthonormal p r o p e r t i e s of the c o s i n e  r e w r i t e the above equation  " fi-0 k n n a  ^kn n  B  C  =  a  f u n c t i o n we  as ,  k  ...(1.7)  where, a  a  Q n  j3  a  0  Qn  k n  k n  00  = 2d [V 2  = 2d [W 2  =  2  a  =  0  n=1,2,3,..  (a, ) Isindurd, /d )/ird^n  n=1,2,3,..  2  n  2  n  = 2d!d n[W 2  ' k0  (a, ) ]sin(n7rd, /d )/7rd,n  = 2d d n[V 1  1  n  , for  at p o i n t s on the i n t e r f a c e r=a,,  0<z<di. I n t e g r a t i n g the above equation and  2  n  ( a , ) ] sinfnTrd, / d ) ( - D  /via,  ( a , ) ] sin(n7rd  /ir(d, n - d k )  2  1  /d )(-l) 2  2  2  n -d 2  2 2  2  2  2  k ) 2  2  65 a  a  0  = -d,(1-(d,/d )[1/3 -  (a,/d,) ]/d  k  k = - 2 d ( 1 - d / d ) ( - 1 ) /d  2  2  l  1  (1.7) g i v e s the f i r s t  5.2.2  UTT)  2  2  system of e q u a t i o n s .  CONTINUITY OF VELOCITY BETWEEN REGIONS 1 AND For c o n t i n u i t y of v e l o c i t i e s between regions  along the boundary  r=a, 0^z<d, , we have,  2 1 and 2  f o l l o w i n g equation  (1.2) V r  *2,r  =  M u l t i p l y i n g by c o s ( k 7 r z / d ) 2  of  d,  <t>i ,«r  and  i n t e g r a t i n g as f o l l o w s ,  d cos(k7rz/d )dz = oJ" 2  2  <t> z , ro  c o s ( k 7 r z / d )dz at r=a, 2  we have the second system of e q u a t i o n s . The second  integral  i s from 0 t o d . Hence the normal v e l o c i t y on the s u r f a c e 2  r=a,  , di^z<d  is implicitly  2  set to z e r o .  Using the orthonormal p r o p e r t i e s of cos(k7rz/d ) we 2  a r r i v e at the second system of equations as 00 C  k  " n=0  where  7  kn  =  b  k  .(1.8)  66  7  00  7  k0  7  t  n  = 0  ;  7  On  = 0 f o r n=1,2  = 0 f o r k=1,2  = 2d,d  2  n n ( - l ) sin(k7rd,/d ) I 1 (nira ,/d, )/7r 2  [d^k  b  0  b^  2  - d  2 2  n ] 2  Ioln^/d,)  f o r k=n=1,2  = 0  = -a,d  2  s i n (mrd, / d ) /d, d 2  (n7r)  2  5.2.3 CONTINUITY OF PRESSURE BETWEEN REGION 2 AND THE EXTERIOR REGION Using a procedure  s i m i l a r to the one used  f o r equation  (1.7) we equate the pressure on the i n t e r f a c e r = a , 0<z<d 2  2  between r e g i o n 2 and the e x t e r i o r r e g i o n . M u l t i p l y i n g by the f u n c t i o n cos(k7rz/d ) and i n t e g r a t i n g 2  from 0 to d  2  one  obtains:  (2/d ) / 2  for  0  r=a  </» c o s (kffz/d )dz = ( 2 / d ) J " 2  2  2  0  <S>  E  cos (k7rz/d ) dz 2  2  Using the orthonormal  p r o p e r t i e s of the c o s i n e f u n c t i o n we  can r e w r i t e the above equation  i n the form,  67  B, - £ , e, E = c, k n=l kn n k  (1.9) i s the t h i r d  e  k0  '  G  H  2  = -(K  k n  (k  0  = (2/d ) o /  Q 0  2  n  d2  2  1  /  d  /  1  d  = 2N  1  /  2  /  2  0  2  Zo c o s ( k 7 r z / d ) dz 2  d2  2  2  o  2  Z  n  dz  sin(k  n  d )/k  0  n  d / a ) o/  dz  0  0  2  < 2  Z  f o r k=0,1,2,.. ; n=1,2,  k n  sinh(k d )k d (-1 )  2  0  = (2/d ) ;  =  2  2  2  Q n  a ))G  o  1  f o r k=0,1,2,.. ; n=0  k Q  sinh(k d )/k d  2  0  = (2/d ) o/  k Q  kn  a ))G  2  = 2N  G  0  a )/H,(k  n  = 2N  G  system of equations, where,  -( o(koa )/Hi(k  =  = 2N  G  ...(1.9)  2  n  d  / [(k d ) 0  2  2  + (kvr) ] 2  2  2  Z  n  sin(k  cos(k7rz/d ) 2  d )k •n ' * n J 22  dz  d (-1) 2  U  n  / [(k n  2  d ) 2  2  - (kjr) ] 2  68 c  = -(d /d)[l/3 - (a /d ) /2] 2  0  2  2  2  k c  k  = - 2 d (-1 )  /d(k)r)  2  2  5.2.4 CONTINUITY OF PRESSURE BETWEEN REGION 3 AND THE EXTERIOR REGION We have the f o u r t h system of equations by equating the p r e s s u r e s on the boundary r=a and  2  , d ^z<d , between region 3 3  the e x t e r i o r r e g i o n . F o l l o w i n g 03  =  <t>  eqn. (1.1) we have,  E  M u l t i p l y i n g by the f u n c t i o n Y ( z - d ) and i n t e g r a t i n g 3  between d  and d , we have,  3  (1/(d-d ))  d / 0  3  d  3  Y(z-d )dz 3  3  d / 0  = d/(d-d )) 3  d  P  Y(z-d )dz  3  Using the orthonormal p r o p e r t i e s can  rewrite  3  E  of the f u n c t i o n Y ( z - d ) we  the above equation to give  3  the f o u r t h system of  equations a s ,  D  k  " kn n 5  where,  E  =  d  k  ....(1.10)  5  kO  5  "< o(koa )/Hi(ko  =  H  2))L  a  2  (k  for  Q  = -(K  L  k n  i s a s p e c i a l f u n c t i o n d e f i n e d as below.  L  Q 0  n  2  1  n  /  3  = -(M  N  Q  )  Q  0  0  N  k  )"  Q  0n  1 / 2  L  N  0  0  1  2  k  n  sin(k  N  )"  n  d  = -M  1 / 2  0  d  k  = -M  "  0  k  1 / 2  /  1 / 2  k  n  3  Z  3  (k  + m  2 0  2  R  )]  (z-d )dz  Q  3  3  Z  n  (z)Y  k  d )/[(d-d ) 3  3  2  0  d(d-d )(m 3  (z)Y  n  d )/[(d-d )  n  d(d-d )(m )  /  2  0  (k„  2  +  m )] 2  0  (z-d )dz 3  3  sin(k  3  m )]  (z-d )dz  k  3  / d  k  0  3  n  3  = -(M  Z (z)Y  -  2 0  3  = d/(d-d ))  k n  (k  3  /  1  )~ /  n  3  k sinh(k d )/[(d-d )  d  Q  (z-d )dz  0  3  3  = -(M  o  /  ( /(d-d ))  =  f o r k=0,1,2,.. ; n=1  Z (z)Y  3  3  = -(M  n=0  3  = d/(d-d ))  k Q  R n  k sinh(k d )/[(d-d )  1 / 2  d  L  2  = (l/(d-d )) d  L  a ))L  k=0,1,2,.. ;  k n  0  a )/K (k  k  k  )  2  (k  2 n  - m  R  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 r a d i a l v e l o c i t y from the e x t e r i o r region on the boundary r = a , 0<z<d , d <z<d. The normal v e l o c i t y on the s u r f a c e 2  2  3  r=a , d ^z<d 2  2  equation  <t>2 r  (2) we have,  = 4>  f o r r=a  >  E r  c6_ ^ = o, r  i s i m p l i c i t l y a s s i g n e d to be zero. Using  3  0<z<d  2  ...(1.11a)  2  ^ , f o r r=a Ei, r  d <z<d  ?  ...(1.11b)  3  Adding (1.11a) and (I.11b) and m u l t i p l y i n g throughout by the function  (z) and i n t e g r a t i n g between the l i m i t s as shown  we have the f i f t h  (l/dk  k  d ) o/  system of e q u a t i o n s .  2  0 ,r k Z  (  z  )  d  z  +  2  ( l / d k  k d )  3  S  0  d 3,r k Z  (  z  )  d  z  d = d/dk  k  ) ; 0  <fiKir  Z  k  (  z  )  d  z  I n t e g r a t i n g using the orthonormal p r o p e r t i e s of the f u n c t i o n Z as  k  ( z ) , and r e w r i t i n g we have the f i f t h (1.11).  system of equations  -K, B - 0. C - \p. kn n kn n kn  D n  r  + E. =e. k k  ...(1.11  The terms i n (1.11) are d e f i n e d as below.  K  kn  kn  6  =  ( n 7 r  <  =  /  n 7 r  2 d k  /  k  )  v  k  = -a N  0  e  k  = -a N  R  2  2  1 / 2  1  2  )  G  f  r  ° . ' . - - ? n=1,2,..;  k =  1  2  f o r k=0,1,2,..; n=1,2,..;  '(a > F  n  2  k n  d )/2(k d) d 2  0  sin(k  2  o  k n  /dk, ) X  n  sinh(k  /  kn  2  3  0  a  >Wn(a ) G  2 d k  *kn " r<d-d )m  e  ( n  2  R  0  d )/2(k 2  2  d) d 2  k  2  v;(a ) 2  I , (una,/d2)  I^nTraz/d;)  - ( I^nn-ai/dz)  /  I (n7ra /d )  + ( I i (n7ra, / d )  / Ki(n7rai/d )  2  0  2  2  ) K,(n7ra /d ) 2  )  2  2  K (n7ra /d ) 2  0  2  w;(a ) = 2  I (n7ra /d ) 2  1  ( I (n7ra /d )  +  2  I,(n7rai/d ) 2  2  0  + (  I  0  (n7ra  2  / I (n7ra /d )  2  /d  2  2  0  )  ) K^nTraz/d;)  2  / K (n7ra /d ) 2  0  ) K^nTrai/dj)  2  J i ( m a ) + ( J , ( m a ) / H,(moa ) ) H, ( m a ) 0  2  0  3  3  0  2  Xn'(a ) = 2  J ( m a ) + ( J ( m a ) / H {viy a ) 0  0  2  1  0  3  %  0  3  ) H (m a ) 0  0  2  72 I , (in X  n  ' ( a  2  ) I  F  F  +  2  (  0  ( m  a )  n  +  2  (  3  Q  N  )  Q  1  /  k  2  N  )  k  1  /  k  2  /  3  K^nt  a )  a )  n  3  Y  /  3  3  )  3  K  0  Q  - m  2  3  *  a )  ( z - d ) Z (z)dz  Q  Q  )]  2 0  ( z - d ) Z (z)dz  0  3  k  d )/[ (d-d )(k  k  K ^ n i  3  3  +  2 k  m )] 2  0  d F „  = [l/(d-d )],  n  = (M  F  3  n  N  )  Q  kn  =  1  l  /  k  2  /  2  (  d  _  d  3  )  ]  d  1  /  2  0  Y  ( z - d ) Z (z)dz  n  3  n  d )/[ (d-d )(k 3  1  ?  3  3  n  (  Z  _  D  3 ) Z  2 Q  3  d2  0  0  2  0  0  2  - iri  2  3  Z ( z ) cos(n7rz/d )  sinh(k d )k d (-1 )  2  n  / [(k d ) 0  + m  2 n  )]  (z)dz  k  k, sin(k. d ) / [ ( d - d ) ( k ,  2  0  i  sinh(k  Q  = (2/d ) ;  Q n  = 2N  /  [  = (M„ N. )  G  1  /  2  2  a ) 2  n  K t d n  d )/[ (d-d )(k  J  d  sin(k  k  0  )  3  *  /  sinh(k  Q  3  Q  a )  I , ( m  d  = [1/(d-d )]  k Q  = (M  I , (m "  = [l/(d-d )]  Q 0  = (M  a ) n  2  )]  dz  + (nvr) ] 2  ( m  n  a ) 2  73 G  = (2/d ) 0 /  k n  d  2  2  = 2N, ~  5.2.6  1 / 2  sin(k,  Z  R  (z) cos(n7rz/d ) dz 2  d )k. d ( - D * / [ ( k . 2  2  d ) 2  2  - (nTr) ] 2  SOLUTION FOR COEFFICIENTS OF SERIES Equations  (1.7) to (1.11) form the f i v e systems of  l i n e a r equations which are s o l v e d simultaneously to o b t a i n the c o e f f i c i e n t s A  , B  , C  n n the p o t e n t i a l f u n c t i o n s .  , D n  and E n  of the s e r i e s f o r n  The above system of equations are e q u i v a l e n t to  [M]|x| = |v|  ...(1.12)  where [M] i s a complex square matrix of order  (5m-l),  |x| i s  a complex v e c t o r comprising of the unknown c o e f f i c i e n t s of the s e r i e s and |v| i s a complex v e c t o r comprising of the r i g h t hand s i d e of the system of equations The system of equations  (1.7) to (1.11).  (1.12) i s s o l v e d by a complex,  double p r e c i s i o n r o u t i n e CDSOLN a v a i l a b l e on the UBC Michigan Terminal System. The r o u t i n e s o l v e s the system by Gaussian of  e l i m i n a t i o n . CPU time r e q u i r e d f o r s o l v i n g a system  100 e q u a t i o n s , corresponding to 20 terms f o r each s e r i e s  for the p o t e n t i a l s ,  i s t y p i c a l l y 2.6 seconds on the Amdahl  470 V/8 computer. D e t a i l s of the f o r m u l a t i o n of the system  74 of equations  (1.12) are given i n the write-up on the  computer program.  5.3 INTEGRALS FOR EVALUATION OF HYDRDOYNAMIC The i n t e g r a l s I N  I N and I N  1 f  2  3  i n equation  COEFFICIENTS  (2.51) of chapter  2 are as given below.  !Ni  = o/  01 r dr a t z=d.  = [a, d,/4d -  a^/iedd,  co + ^  ( a ^ / n i r ) I , (nira , /d, ) /I (n/ra , /d, ) ]  2  + A a, /4 2  0  n  = 1  A  (-1)  n  0  ...(1.13)  IN  2  =  a /  a1  = [(a  2  2 2  0  (a  + 2  = 1  B  + ^  = 1  n  C  2  r dr at z=d  2  - a! )d /4d - ( a " - a 2  2  + B  2  0  n  2 2  2  )/-\ 6 a a  :  - a, )/4 2  (-1 )  (d /n7r) V  (-1)  (d /n7r) W ( a ) ]  2  2  n  n  (a ) 2  2  ...(1.14)  where,  I ,(nira /d ) 2  a  2  -  2  2  ) K,(n7ra /d )  2  2  I o {n.7ra /d ) + ( I ( n 7 r a / d ) / K ^ n ^ A ^ ) 2  W  / I 1 (n7ra ,/d )  ( I , (n.7ra j / d )  (a )  n  2  1  1  2  ) K (ri7ra /d )  2  0  2  2  =  2  I i ( n 7 r a / d ) + ( I ( n i r a / d ) / I ( r i 7 r a / d ) ) K, (n.7ra /d ) 2  a  2  0  2  2  0  2  2  2  2  2  I i ( n 7 r a i / d ) + ( I ( n i r a / d ) / K ( n 7 r a / d ) ) Ki(n7ra,/d ) 2  0  2  2  0  2  2  2  - a a  IN  =  3  2  </»3 r dr a t z=d  /  a 3  = [ d / d + q/co d -  1] ( a  2  3  +(  D  J i ( m J  0  M  0  0  "  0  a ; )  ( m o a  2  1  /  +  )  a  2  (  J  2  / m  - a  )  2 3  )  )  /  H , ( m  + ( J, ( m o a j )  /  H,(moa3)  1  ( m  0  2 2  0  3  a  3  0  a  3  )  )  H ^ m p a ; )  )  H  0  ( m o a  )  2  -1/2 +  (  I  1 (m  L_,  D „  M„  a  n=1 "n "n a  2  )  +  (  I  1 (in  0  ( m  n  a  / m  3  )  )  n  /  K ^ m  n  n  I  2  a  2  )  +  (  I , ( m  n  a  3  )  )  K , ( m  n a ) 3  /  K  1  ( m  n  a  2  )  ...(1.15)  n a ) 3  )  K  0  ( m  n  a ) 2  6. APPENDIX 2 - EXPERIMENTAL SET-UP  6.1 EXPERIMENTAL The  facilities  FACILITIES  of the Ocean E n g i n e e r i n g Centre a t B.C.  Research, Vancouver, facilities  were u t i l i s e d  f o r the experiments. The  used are d e s c r i b e d below.  6.1.1 THE TOWING TANK T h i s i s a 220'X12 X10' ,  tank p r i m a r i l y used f o r s h i p  model r e s i s t a n c e t e s t s . I t i s equipped with a towing carriage  fitted  with data c o l l e c t i o n equipment, which  t r a v e r s e s the l e n g t h of the tank on r a i l s . the  A photograph of  tank i s shown i n F i g . 7 . The tank i s equipped with a  hinged paddle type wavemaker at one end and a wave damping beach at the o p p o s i t e end. The tank i s a l s o equipped with underwater  windows f o r 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 f i x e d at a p o s i t i o n halfway along the  l e n g t h of the tank but capable of moving t r a n s v e r s e l y . The l a t t e r was used f o r equipment h a n d l i n g d u r i n g the t e s t s .  6.1.2 WAVEMAKER The wavemaker c o n s i s t s b a s i c a l l y of three  units.  6.1.2.1 The Wave S i g n a l Generator T h i s d e v i c e generates a time v a r y i n g v o l t a g e s i g n a l to represent the wave. The d e v i c e has the c a p a b i l i t y t o generate quasi-random  waves or r e g u l a r repeated waveforms.  The o p e r a t i o n of t h i s d e v i c e i s c o m p l i c a t e d . I n s t e a d , a 76  77 s i n u s o i d a l wave generator was used. T h i s enables easy v a r i a t i o n of amplitudes and f r e q u e n c i e s f o r s i n u s o i d a l waveforms and i s much simpler to operate. 6.1.2.2 Wave S y n t h e s i z e r The wave s y n t h e s i z e r serves the dual purpose of boosting the input s i g n a l and c o r r e c t i n g irregularities  f o r any  i n the motion. The input s i g n a l  i s boosted to  v o l t a g e and c u r r e n t l e v e l s a p p r o p r i a t e t o the h y d r a u l i c a c t u a t o r . A displacement transducer on the wave paddle p r o v i d e s the s y n t h e s i z e r with the a c t u a l p o s i t i o n of the paddle at any i n s t a n t . The wave s y n t h e s i z e r sends a c o r r e c t e d s i g n a l to the h y d r a u l i c a c t u a t o r a f t e r the  comparing  a c t u a l p o s i t i o n of the paddle to the d e s i r e d paddle  location. 6.1.2.3 Wave Paddle A photograph of the wave paddle i s shown i n Fig.10. T h i s i s an aluminium  paddle which spans the width of the  towing tank. The paddle i s hinged approximately four below the water s u r f a c e . The paddle i s o s c i l l a t e d  feet  by a  h y d r a u l i c p i s t o n c o n t r o l l e d by the h y d r a u l i c a c t u a t o r which in turn i s c o n t r o l l e d by the wave s y n t h e s i z e r . Any  irregularities  i n the waves generated and  propagating along the tank may•be due t o one or more of the f o l l o w i n g reasons. The wave paddle has a n a t u r a l , undriven o s c i l l a t i o n . The amplitude of t h i s o s c i l l a t i o n v a r i e s with time and and  78 becomes more pronounced oscillation  as the a c t u a t o r heats up. The  i s approximately 0.4" a t a frequency of 2.3Hz.  Except f o r experiments  near t h i s frequency, the e f f e c t s can  be i s o l a t e d by s p e c t r a l a n a l y s i s . T h i s o s c i l l a t i o n  decreased  c o n s i d e r a b l y on i n s t a l l a t i o n of a new h y d r a u l i c a c t u a t o r v a l v e . So, only a few experiments  were a f f e c t e d by t h i 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 i r r e g u l a r waveform. At c e r t a i n  f r e q u e n c i e s , beats were observed  i n the  waveform. T h i s can be d i r e c t l y a t t r i b u t e d to the presence of a l a r g e 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 i n the waveforms a t c e r t a i n f r e q u e n c i e s , normally higher than 1Hz. The frequency of the i n t e r f e r i n g d i f f e r s by about  waveforms  1Hz. from the d r i v e n frequency. The beats  in the waveforms may p o s s i b l y have been generated by a c r o s s - f l o w between the tanks. Other  iregularities  i n the waveform may be induced by  r e f l e c t i o n s from the tank w a l l s or d i s t u r b a n c e s due to a i r c u r r e n t s or gusts d i s t u r b i n g the water s u r f a c e .  6.2 MOTION GENERATOR The motion  generator i s a h y d r a u l i c a l l y d r i v e n s c o t c h yoke  mechanism. F i g . 8 shows a photograph  of the unit.' The u n i t  was redesigned s i n c e i t d i d not p r o v i d e a smooth s i n u s o i d a l oscillation  in i t soriginal  form.  79 At present i t c o n s i s t s of a h o r i z o n t a l f i x e d frame 12'x4' made up of two I-beams. A v e r t i c a l  frame i s mounted  on t h i s frame and holds the moving frame which p r o v i d e s 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 s c o t c h yoke mechanism which d r i v e s i t . The moving frame i s a l s o connected  to the c y l i n d e r model through a  connecting b l o c k . The s c o t c h yoke i t s e l f  i s d r i v e n by a r a d i a l  h y d r a u l i c motor having a displacement  piston  of 12.7 c u .  in./  r e v o l u t i o n . The motor i s capable of an output torque of 500 f t . - l b s . A c l o s e d loop v a r i a b l e displacement transmission c i r c u i t  hydrostatic  i s used to ensure t i g h t c o n t r o l of  angular v e l o c i t y over the speed range of 3 t o 150 rpm. Owing to l i m i t a t i o n s  i n the a v a i l a b l e e l e c t r i c a l power supply the  h y d r a u l i c motor can only d e l i v e r r a t e d torque upto a frequency of 0.83 Hz, with torque c a p a c i t y d i m i n i s h i n g to 167 f t . - l b s a t a frequency of 2.5 Hz. The h y d r a u l i c power u n i t c o n s i s t s of a v a r i a b l e displacement a x i a l p i s t o n pump having a maximum displacement of 2.5 c u . i n . / r e v o l u t i o n , d r i v e n by a 1750 rpm 5 H.P. e l e c t r i c motor. The pump displacement mechanical  i s l i m i t e d by e x t e r n a l  stops to a maximum of 1.25 c u . i n . / r e v o l u t i o n to  prevent o v e r l o a d i n g of the motor. A c r o s s - p o r t r e l e i f v a l v e set at 3000 p s i i s used t o prevent e x c e s s i v e c i r c u i t pressures and t o l i m i t motor torque t o 500 f t - l b s . The s c o t c h yoke mechanism and the h y d r a u l i c  system  were  redesigned by F r a s e r E l h o r n , a graduate of the Mechanical  80 E n g i n e e r i n g Department at UBC. The moving frame was constructed  at the Machine Shop at the Mechanical  E n g i n e e r i n g Department. The h y d r a u l i c were s u p p l i e d  by Fleck  Hydraulics  motor and power u n i t  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. T h i s  consisted  of a MINC-11 computer and a m p l i f i e r s and a m p l i f i e r s and signal conditioners  mounted on the towing  carriage.  6.3.1 AMPLIFIERS AND SIGNAL CONDITIONERS The  towing c a r r i a g e  Internal  c a r r i e s ten s i g n a l  conditioners.  r e g i s t e r s can be set t o high,low or band-pass.  E x t e r n a l l y , d i a l s allow a m p l i f i c a t i o n of the input  voltage  s i g n a l by a f a c t o r of 10. A few of the a m p l i f i e r s are d i f f e r e n t i a l a m p l i f i e r s and permit a m p l i f i c a t i o n s For  upto 1000.  these experiments, low pass f i l t e r i n g was chosen with  varying  a m p l i f i c a t i o n . The a m p l i f i e r s  induce a phase s h i f t  i n the s i g n a l which was analyzed and c o r r e c t e d  for.  6.3.2 MINC-11 COMPUTER The  hardware f o r the system comprises of the f o l l o w i n g .  6.3.2.1 Main Console This  incorporates  analog and d i g i t a l  input  ports  with  c l o c k i n g c a p a b i l i t i e s . A maximum of 16 analog p o r t s can be used. Each port w i l l accept as input  ±5.12 v o l t s with a  r e s o l u t i o n of 2.5mV. V o l t a g e s i n excess of the l i m i t s are  81 removed by the MINC-11. Analog  v o l t a g e s are converted to  i n t e g e r v a l u e s f o r i n t e r n a l use, the v o l t a g e range being t r a n s l a t e d from 0 to 4096. Time d i f f e r e n c e between the sampling  of two channels  so was not of concern  i s of the order of microseconds  f o r these  and  experiments.  6.3.2.2 Dual Floppy Disk D r i v e System System and o f t e n used programs are s t o r e d on one d i s k and data and development programs are s t o r e d on the o t h e r . 6.3.2.3 VT105 Video The  Terminal  t e r m i n a l has graphic d i s p l a y c a p a b i l i t i e s . I t can  d i s p l a y upto a maximum of two s i n g l e valued f u n c t i o n s . T h i s enabled v i s u a l o b s e r v a t i o n of the recorded s i g n a l and comparison of two d i f f e r e n t  channels.  6.3.2.4 L i n e P r i n t e r A high speed v a r i a b l e c h a r a c t e r s i z e l i n e p r i n t e r i s used to o b t a i n hard c o p i e s of data and program  listings.  6.3.2.5 T e k t r o n i x Screen Dump P r i n t e r T h i s p r i n t e r generates a d u p l i c a t e of the video screen contents on heat s e n s i t i v e paper. T h i s i s u s e f u l f o r o b t a i n i n g p l o t s of the recorded data. Software  used  f o r the experiments  were p a r t l y  existing  on the system and p a r t l y w r i t t e n or m o d i f i e d . These a r e described  later.  82 6.4 MODELS Three types of models were used f o r the experiments. T h e i r dimensions and shapes are as shown i n F i g s . 4, 5, and 6.The models were c o n s t r u c t e d by the machine shop a t the Department of Mechanical E n g i n e e r i n g . The  s i n g l e c y l i n d e r model was made of PVC tubing 15"  O.D. The ends were sealed  with aluminium p l a t i n g . The  c y l i n d e r was connected t o the block on the motion generator by means of four  threaded rods f i x e d on the bottom p l a t e .  These rods a l s o served as a mounting f o r l e a d weights used to b a l l a s t the c y l i n d e r  to achieve n e u t r a l  buoyancy. The  d r a f t T, was v a r i e d d u r i n g the experiments. The  double c y l i n d e r model was s i m i l a r i n c o n s t r u c t i o n  except f o r the f a c t that  the top p l a t e was r e p l a c e d  by an  aluminium u n i t comprising of an 8.625" O.D. aluminium c y l i n d e r mounted on a p l a t e . The dimensions of the model are shown i n F i g . 5. The  compound c y l i n d e r model had, i n a d d i t i o n ,  another  u n i t mounted on the bottom, s i m i l a r to the aluminium u n i t mounted on the top f o r the double c y l i n d e r . The dimensions are as shown i n F i g . 6.  6.5 EQUIPMENT USED E l e c t r o n i c equipment used i n a s s o c i a t i o n c o l l e c t i o n system are d e s c r i b e d  below.  with the data  83 6.5.1  STRAIN INDICATORS Strain  i n d i c a t o r s were use i n a s s o c i a t i o n with the wire  r e s i s t a n c e wave probe and the dynamometers used t o measure f o r c e s . These were model P-350A s t r a i n manufactured  by Vishay Instruments  the i n d i c a t o r e x c i t e s the connected  indicators,  Inc. The output p o r t s of bridge c i r c u i t s with a  1.5 v o l t s R.M.S., 1000Hz square wave. The output  is a  maximum of ± 250 mV DC. T h i s i s too low t o be used i n a s s o c i a t i o n with the MINC-11, and hence had t o be a m p l i f i e d . The Accuracy  s p e c i f i c a t i o n s are as f o l l o w s . : ±0.5% of reading or 5ye (whichever  i s greater.)  Sensitivity : 0.2 t o 20Me/mV Output  : L i n e a r range ±250 mV DC  Noise & R i p p l e : 3/xe, 1mV.  6.5.2  SONAR LEVEL MONITOR T h i s i s a s o l i d s t a t e d e v i c e used f o r measurement of  displacement. The d e v i c e was manufactured  by Wesmar Marine  E l e c t r o n i c s Inc.,Seattle,USA. The d e v i c e measures the time r e q u i r e d f o r a sonar pulse t o t r a v e l t o and from an o b j e c t . V a r i o u s range  s e t t i n g s a r e a v a i l a b l e with a manual  m u l t i p l i e r d i a l which g i v e s a continuous c o n t r o l of the s e n s i t i v i t y . The s p e c i f i c a t i o n s are as f o l l o w s . Output r i p p l e Output D r i f t  : <0.1% of f u l l s c a l e  stability  84 : B e t t e r than 0.25% of f u l l scale/hour after  warm-up.  Resolution  B e t t e r than 0.5% of measured range.  Linearity  B e t t e r than 0.5% of f u l l  Operating  scale.  temperature 45° t o 140°F  Beam emitted Pulse r e p e t i t i o n  3° in  span.  rate 160 Hz at 7  to 30  range.  6.5.3 WAVE PROBE T h i s i s a simple device used t o measure the water s u r f a c e e l e v a t i o n and hence the wave p r o f i l e . The device b a s i c a l l y c o n s i s t s of two separated wires i n the water, which a c t as the f o u r t h arm of a Wheatston  b r i d g e . The  changing water l e v e l p r o v i d e s a r e c o r d a b l e change i n c i r c u i t r e s i s t a n c e . A Vishay i n d i c a t o r  i s used t o input a 1.5 V o l t s  R.M.S. 1000 Hz square wave.  6.5.4 FORCE RECORDING  EQUIPMENT  These c o n s i s t e d p r i m a r i l y of an 80 l b . dynamometer and two 500 l b . U n i v e r s a l Shear beams used t o 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 f o r c e , a h o r i z o n t a l f o r c e and a moment. 80 l b s . i s the maximum range f o r the v e r t i c a l  f o r c e . The e f f e c t s of any one  85 of  the f o r c e s on the o t h e r s are excluded by a s u i t a b l y  designed c i r c u i t . The measuring d e v i c e s are f o i l strain  gauges Type CEA-06- 125-OT-350. A f u l l  bridge arrangement  bonded  Wheatstone  i s used f o r each parameter  the bridge  c i r c u i t s were e x c i t e d by v o l t a g e s from Vishay  strain  i n d i c a t o r s . The dynamometer was designed and c o n s t r u c t e d by the  Ocean E n g i n e e r i n g Centre at B.C.Research.  bridges have s e l f compensating  gauges to p r o v i d e s e l f  temperature c o r r e c t i o n . T h i s dynamometer was wave f o r c e t e s t s on s t a t i c  A l l the  used f o r the  cylinders.  6.5.4.2 U n i v e r s a l Shear Beam Two the  U n i v e r s a l Shear Beams (USB) were used to measure  vertical  f o r c e s on the o s c i l l a t i n g c y l i n d e r model. The  USBs were manufactured Inc. the  by H o t t i n g e r Baldwin Measurements  of Framingham, MA.  The s p e c i f i c a t i o n s as s u p p l i e d by  manufacturer are as f o l l o w s .  Rated C a p a c i t y  5001bs  Rated Output  1.9925  Non-linearity  -0.02% of rated ouptut.  Hysteresis  0.01%  of r a t e d output.  Non  0.01%  of r a t e d output  repeatability  Excitation  voltage  Side Load  rejection  Max.  Load, Safe  mV/V  18V DC or AC RMS  max.  500:1 150%  The USBs were mounted below the two v e r t i c a l  s h a f t s of the  moving frame on the motion generator as shown i n F i g . 8a. They were p i n - j o i n t e d to exclude moments and measure only  86 vertical  forces.  6.5.5 CALIBRATION The sonar l e v e l recorder and the wave probe were calibrated  before each set of experiments by moving them  known d i s t a n c e s and r e c o r d i n g the output v o l t a g e s . A calibration  program  existing  on the O.E.C. data c o l l e c t i o n  system was used to o b t a i n the slope and i n t e r c e p t of the c a l i b r a t i o n data. The 80 l b . dynamometer and the USBs were statically at  calibrated  i n compression on the U n i v e r s a l T e s t i n g Machine  the Mechanical E n g i n e e r i n g Department,  and a l s o  dynamically using the motion generator. They were a l s o calibrated  i n t e n s i o n 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 f o r the experiments. One set  was used f o r data c o l l e c t i o n on the O.E.C. MINC-11  system.  The other s e t was used f o r data a n a l y s i s on the PDP-11 system at the Department of Mechanical E n g i n e e r i n g at UBC, which was s i m i l a r  to the MINC-11. Part of the software  neccessary was a l r e a d y e x i s t i n g  at the O.E.C. Some of i t had  to be adapted f o r these experiments. However, some of the software was developed s p e c i f i c a l l y f o r 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 s t o r e d on d i s k and t r a n s f e r r e d to the PDP-11 at the Mechanical  Engineering  Department at UBC. 6.6.1.1 ADCAL T h i s 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 . T h i s program reads  input v o l t a g e on a s p e c i f i e d  channel which can be indexed to any d e s i r e d v a l u e . By v a r y i n g the range of measurements f o r a d e v i c e with the corresponding  input of such v a r i a t i o n s a s e r i e s of  c a l i b r a t i o n p o i n t s i s e s t a b l i s h e d . At l e a s t  f i v e p o i n t s are  r e q u i r e d . The program then e s t a b l i s h e s the best f i t s t r a i g h t l i n e through  these p o i n t s by l e a s t squares m i n i m i z a t i o n . The  s l o p e , i n t e r c e p t , v a r i a n c e and d e l t a of the data l i n e i s output. T h i s i n f o r m a t i o n i s a l s o s t o r e d i n a matrix on a user s p e c i f i e d several If  f i l e . The f i l e c o n t a i n s c a l i b r a t i o n data f o r  channels. c a l i b r a t i o n cannnot be c a 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 c r e a t e d t o  run the d e m u l t i p l e x i n g program. 6.6.1.2 ADMAIN T h i s i s the p r i n c i p a l program f o r data c o l l e c t i o n . The program c o l l e c t s data on s e v e r a l channels, upto a maximum of 16 s i m u l t a n e o u s l y . The sampled data  i s stored in a  m u l t i p l e x e d group f o r memory m i n i m i z a t i o n . The program a l s o  88 allows r e a l time viewing of any channel p r i o r t o sampling. T h i s i s u s e f u l as a check t o ensure  that r e c o r d i n g i s  occuring. 6.6.1.3 ADMUX T h i s program i s used t o separate each channel's  unique  s i g n a l and s t o r e the data p o i n t s as numeric v a l u e s on user specified  f i l e s . The program d e m u l t i p l e x e s the s t o r e d d a t a .  T h i s program i s a v a i l a b l e on the MINC-11  system.  6.6.1.4 GRAPH T h i s program d i s p l a y s upto two graphs on the video t e r m i n a l . The program has the a b i l i t y t o d i s p l a y e i t h e r an x-y graph or r e a l time p l o t of a channel. I t can a l s o shade from v a r i o u s p o r t i o n s of the graph  field  t o the data p o i n t s .  F u r t h e r , the program can c a l c u l a t e and d i s p l a y a c u b i c moving s p l i n e f i t t o the data. Comments, t i t l e and axes l a b e l s can be i n p u t . The program can a l s o d i s p l a y v a r i o u s p o r t i o n s of a curve, i e , s c a l i n g can be changed t o magnify some p o r t i o n of a curve.  6.6.2 DATA ANALYSIS SOFTWARE The data a n a l y s i s software was used on the PDP-11 a t the Mechanical  E n g i n e e r i n g Department. They were mostly  w r i t t e n f o r the s p e c i f i c purpose of these  experiments.  6.6.2.1 DEMUX T h i s program demultiplexes one or more s i g n a l s of the c o l l e c t e d data, c o n v e r t s the d i g i t i z e d v o l t a g e l e v e l s to  89 user values and s t o r e s the time record on a user file.  specified  I f 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 d e m u l t i p l e x i n g .  6.6.2.2 AMP T h i s program takes as input a time record of some measured data and c a l c u l a t e s the F o u r i e r spectrum of the r e c o r d using the Fast F o u r i e r Transform a l g o r i t h m  (FFT). The  program outputs the frequency vs amplitude and the frequency vs phase ( i n radians)  to two separate  data  f i l e s . The phase  depends on the s t a r t i n g time of the experiment. I t can be used t o judge the the r e l a t i v e phases of two  simultaneously  sampled channels. 6.6.2.3 PHAMP T h i s program removes phase s h i f t s caused by the O.E.C. s i g n a l c o n d i t i o n e r s . The a m p l i f i e r s were c a l i b r a t e d p r i o r to the experiments to determine t h e i r phase s h i f t s . In g e n e r a l , a l l a m p l i f i e r s were set up as low pass f i l t e r s high  t o remove  frequency noise and i n t e r f e r e n c e . The program takes as  input a phase f i l e and the slope and i n t e r c e p t of the phase s h i f t caused by the a m p l i f i e r used. The output f i l e  contains  the c o r r e c t e d phase v a l u e s . 6.6.2.4 DELAY T h i s program a d j u s t s the phase of the wave probe record to account f o r the d i s t a n c e between the c e n t r e l i n e of the c y l i n d e r and the wave probe. I t takes as input the phase  90 record of the wave and the d i s t a n c e from the wave probe to the c y l i n d e r c e n t r e l i n e . The output i s the a d j u s t e d phase of the wave. 6.6.2.5 SPECADD T h i s program adds or s u b t r a c t s s p e c t r a . For the purpose of these experiments  i t was used to remove a r e f e r e n c e phase  from one or more phase s p e c t r a . Thus the phase of one r e c o r d can be expressed r e l a t i v e t o another. 6.6.2.6 ZERO T h i s i s a program used t o zero the spectrum  of some  r e c o r d . Here, i t was used t o zero the r e f e r e n c e phase spectrum. 6.6.2.7 TABLE T h i s program scans an amplitude spectrum  for local  maxima. A c u t o f f amplitude value can be input and the user can s e l e c t i v e l y chose the frequency/amplitude p o i n t to output.This way d r i v i n g The harmonics at d e f i n i t e  frequency harmonics  can be ignored.  are a r e s u l t of a f i n i t e time r e c o r d sampled  i n t e r v a l s . The program outputs the amplitude and  phase of the r e c o r d f o r the chosen  frequency. The 90° and  180° components of the v e c t o r can a l s o be output i f required. 6.6.2.8 FINAL T h i s program c a l c u l a t e s the added mass and damping c o e f f i c e n t s from the 90° and 180° components of the v e r t i c a l  91 f o r c e . The c y l i n d e r cylinder,  mass, the c r o s s s e c t i o n a l area of the  and the buoyancy  a r e input f o r  non-dimensionalisation. 6.6.2.9 STAT T h i s program  performs the same f u n c t i o n  as FINAL, but  f o r the c a l c u l a t i o n of the non-dimensional e x c i t i n g from the v e r t i c a l e x c i t i n g records.  force  f o r c e and the wave amplitude  92  Fig.l  Definition  of  motions  Fig.2  Compound c y l i n d e r  geometry  ® EXTERIOR REGION  EXTERIOR REGION  i  Fig.3  REGION  S u b d i v i s i o n of flow  field  95  Fig.4  Compound  cylinder  model  Fig.5  Double  cylinder  model  Fig.6  Single  cylinder  model  Fig.7  Towing  tank  at  B.C.  Research  ID  to  Fig.8  Motion  generator  100  Fig.8a  Positioning  of  load  cells  101  Pig.9  Data  collection  equipment  Fig.10  Wave  paddle  o  Fig.11  Flow  visualization  o w  Legend -  1  1  1  l  I  1.75  1  1  1  J  2.33  L  Displacement  I  i  I  i  2.92  Time, sees. Fig.12  Displacement  record  Compound c y l i n d e r  T=35.5"  f=2.5Hz  1  1  I  3.50  Legend Disp. spectrum  0.70 _ 0.61 0.52£0.44 0.35 0.26 0.17 0.09 0.00 T 0.00  I  r  3.33  • • i  6.67  i  i  i  i  I  i  10.00  i  t  i  '-4-  13.33  16.67  J  L J  20.00  Freq. , Hz Fig.12a  Disp„  a m p l i t u d e s p e c t r u m . Compound c y l i n d e r  T=35.5" f = 2 5 H z 0  Legend °-  3 0  0 26 0.22  rr H  -  Theory-B.E.M.  "  Present theory  n  t  K  0.19 -Jl 0.15 t. f-  6,1 i j i 0.08 f l 0.04  L U L  O.OOQL j  0.00  |_ j ..I.  I . J . - l . ..I  1.17  I  J . L .. I  l  2.33  I  t  I  I—I —J—I — I 1 1 1 1 1  I. I.I  3.50  4.67  5.83  T=42 625"  D'=13.125"  WxWxA/G F i g . 1 3 Heave added mass. Compound  cylinder  0  7.00  Legend -  0.20  Theory-B.E.M. -  Present theory  0.17 0.15  £ rU  0.12 t-  E  > X  O I  GC x CM CM CO  0.10  \\  "  *.\  0.08 0.05 0.03  V  0.00  j  0.00  i I  1.17  .i.'S.' t * T T r . ? i u ; . n a . . L z i z - i . . u ^ - L - . 1 . . - -  2.33  3.50  4.67  ... J. .-I  rtr-.L- I  5.83  r- J  7.00  WxWxA/G F i g . 1 4 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  Legend Present theory A Expt. amp. - 0.5in. -I- Expt. amp. - 0.39in.  0.80 0.70  r F  0.60 ^r  0.5oE-  0.40 nt  0.30 E0.2010.10o.oo  r. i  0.00  i i i J  1 . 1 i . i . J i. i i i i  1.17  2.33  i i i.i  3.50  .!  4.67  i__L-^_i_J  5.83  WxWxA/G F i g . 1 5 Heave added mass. Compound  cylinder  T=35.5" D*=6"  7.00  Legend Present theory ^ Expt. amp. - 0 . 5 i n . + Expt. amp. - 0 . 3 9 i n .  i  -I  i.  i  5.83 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 = 3 5 5 " D'=6" 0  L  L-  7.00  Legend Present theory A Expt. amp. - 0.5in. + Expt. amp. - 0.39in.  0.05  °" « 0  fnn  °-00  "  1  1  1  1.17  '  '  '  1  '  2.33  '  l  '  '  A  3.50  -  '  1  -  4.67  - 1 ^ - 1 ^  5.83  WxWxA/G F i g . 1 7 Wave a m p l i t u d e .  Compound c y l i n d e r  T=35.5" D'=6"  7.00  0.80 0.70  Legend _ Present theory  r  A  Expt. amp. - 0 . 5 i n .  b r  0.60  r  0.50 r-  > X  O I  DC  t 0.40 r rc  0.30  |r  0.20  fT  CM CM <  0.10 r w  i o.oo  E-  1  0.00  .1.... I. ...  1  1.1  1.17  F i g . 1 8 Heave added  -I  .1  I—L  2.33  .J—J.  I  I.  I-  3.50  J  .J  I  4.67  1—1  1  A—I  5.83  WxWxA/G mass. Compound  cylinder  T = 3 9 „ 5 " D'=10"  1  L  7.00  Legend  0.10  F  0.09 0.08  C.  0.06  P-  -  Present theory  *  Expt. amp. - 0.5in.  0.05 0.04  i-  0.03  r_  0.01 0.00 J = 4 i  °-  0 0  i i I i i i , i ,  1-17  F i g . 1 9 Damping  coeffi  2.33  - > WxWxA/G  cient.  3  5(  4.67  Compound c y l i n d e r T »  39.5"  5.83  D'=10"  7.00  Legend - Present theory A Expt. amp. - 0 . 5 i n .  0.20  0.00  1.17  2.33  3.50  4.67  WxWxA/G F i g . 2 0 Wave a m p l i t u d e .  Compound c y l i n d e r  T=39.5" D'=10"  5.83  7.00  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. . . . » • • • • 0.00  1.17  i tiiti iiiii ii»ii i' 2.33  3.50  4.67  5.83  WxWxA/G F i g . 2 1 Heave added mass. Compound c y l i n d e r  T=43.5" D'=14"  i 7.00  Legend Present theory A Expt. amp. - 0.5in.  0.50 _ 0.44 0.37 0.31 X  > X  O  p.25  I  0.19 X  CM CM  0.12  CD  0.06 0.00 0.00  J  1.17  2.33  L  J  I  3.50  I  I  1  1  L  J  4.67  I  I  I  5.83  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  1  L J  7.00  Legend Present theory A Expt. amp. - 0.5in. 0.17  r  0.15 1 0.12  L  7.00  Legend Present theory * Expt. amp. - 6.5in.  0.80 0.70 0.60 0.50  >  0.40  X  O x  0.30  CM CM <  0.20 0.10 0.00  (l J  0.00  I  I  I  1  I  1.17  1  I  I  I  I  2.33  1  I  I  I  I  I  I  3.50  I  I  L  4.67  J  I  5.83  WxWxA/G F i g . 2 4 Heave added mass. Compound c y l i n d e r  I  T=47.5" D ^ I B "  I  I  L  7.00  Legend Present theory A Expt. amp. - 0 . 5 i n .  0.70 0.61 0.52 0.44 0.35 0.26 0.17 0.09 0.00  \—A  0.00  i  1  1  1  1.17  F i g . 2 5 Damping  1  1  1  ~  j  i  i  l  1  1  3.50  2.33 WxWxA/G coefficient.  Compound  i—i—1—'—'—'  '•-  4.67  cylinder  T=47.5»  5.83  D'=18»  ' ^  7.00  Legend Present theory A Expt. amp. - 0.5in.  .  0.00  1.17  1 • . • > ' • • •  2.33  3.50  • i i i  4.67  1  5.83  WxWxA/G F i g . 2 6 Wave a m p l i t u d e .  Compound c y l i n d e r  T-47.5" D'=18"  i i i '  1  7.00  Legend © Expt. amp., - ° - 1 2 i n . A Expt. amp. - 0.2in. + Expt. amp. ^ 0 . 4 i n . Theory-B.E.M.  2.00  a E < x O x > X  O DC  u. 0.25 0.00 LJ_ j 0.00  i_a  IT83  1.67  J  I  L  2.50  i  I i  4.17  3.33  i  i  i—I  5.00  WxWxA/G F i g » 2 7 Heave e x c i t i n g  force  on s i n g l e  cylinder.  T-7"  M o  Legend ©  Expt. amp. - 0.12in. Expt. amp. - 0.2in. Expt. amp. - 0.4in. Theory-B.E.M.  +  -4.50 -6.00  - i i i i i i i i—i—i—i—i—i—i 0.00 0.83 1.67 2.50  3.33  j  i  WxWxA/G F i g . 2 8 Heave e x c i t i n g  force  phase. S i n g l e  c y l i n d e r T=7"  I i—i—i—i—I  4.17  5.00  Legend © Expt. amp. - 0.35in. & Expt. amp. - 0.5in. — Theory-B.E.M.  2.00 • 1.75 1.50 1.25 Q.  E < x O x >  1.00 0.75  Li  \  X  O GC  0.50  • CD  0.25 0.00 J 0 .00  I  1  L  I  I I I L  0.83  I | • •i—JI I  1.67  2.50  3.33  4.17  1  1  L_J.  5.00  WxWxA/G Fig„29 Heave e x c i t i n g  force  on s i n g l e  cylinder.  T-10.5"  rO  6.00  Legend o Expt. amp. -o.35in. Expt. amp. - 0.5in. Theory-B.E.M. !  _  A  <D  4.50  CD  3.00 1.50 0.00  L  -1.50  L  -3.00 -4.50 -6.00  0.00  i  i t i  LJ  0.83  A  l  1.67  i i i *  J  2.50  L  1 i i • • 1 • i i '1  3.33  4.17  WxWxA/G F i g . 3 0 Heave e x c i t i n g  force  phase.  Single  c y l i n d e r T=10.5"  5.00  Legend o Expt. amp. - 0 . 2 i n . A Expt. amp. ~0.4in. + Expt. amp. - 0 . 6 i n . x Expt. amp. - 1.0in. Theory-B.E.M.  0.40 _  0.00 j i i i I i i i—i—I—i—i—i—i I i t i I I I 3.33 0.83 1.67 2.50 0.00  I  I  1  1  WxWxA/G F i g . 3 1 Heave e x c i t i n g  force  on d o u b l e  1  4.17  c y l i n d e r . T-23.5"  1  1  L  5.00  Legend © Expt. amp. - 0.2in. A Expt. amp. - 0.4in. 4Expt. amp. - 0.6in. X Expt. amp. - 0.7in. Theory-B.E.M.  5.00 CO  3.75  CO CD > CO  I  <D O  2.50 1.25 0.00  •o <D CO CO SI Q.  - 1.25 2.50 3.75  V  S  CD  -5.0.0 r t i i a> I i i i i I J 0.00 0.83 1.67  I  I  L  J  I  I  2.50  L  J  1  3.33  I  L  1  J  4.17  1  1  L  5.00  WxWxA/G Fig.32  Heave e x c i t i n g  force  phase.  Double  cylinder.  T=23.5" H  1  in  Legend © Expt. amp. - O.zin. A Expt. amp. - o.4in. + Expt. amp. - 0.6in. x Expt. amp. - 0.7in, - Theory-B.E.M.  0.50 0.44  o  0.37 0.31 _ X  0.25  0.00  0.00  0.83  1.67  2.50  3.33  4.17  WxWxA/G Fig.33  Heave e x c i t i n g  force  on d o u b l e  cylinder.  T=27.5"  5.00  Legend © * Expt. amp. - 0.2in. ^ Expt. amp. - 0.4in + Expt. amp. - 0.6in. X Expt. amp. - 0.7'm. - Theory-B.E.M.  CD  J  0.00  0.83  I  I  L  1  1.67  I  I  I  J  I  I  2.50  L  1  J  3.33  L  J  4.17  WxWxA/G F i g . 3 4 Heave e x c i t i n g  force  phase.  Double  cylinder.  T-27.5"  I  1  L  5.00  Legend - Theory-B.E.M. — Present theory + Expt. amp. - 0.12in. x E x p t . amp. "0.30in. 4>Expt. amp. - 0.5in.l 4 Expt. amp. - 0 . 7 i n . XExpt. amp. - ^ i n .  a.  E < x O x > X  O DC  0.00  0.83  1.67  2.50  3.33  4.17  5.00  WxWxA/G Fig.35  Heave e x c i t i n g  force  on compound c y l i n d e r .  T=35.5"  to 00  Legend © Expt. amp. - 0.12in. Expt. amp. - 0 . 3 0 i n . •+• Expt. amp. - 0 . 5 m . x Expt. amp. - 0 . 7 i n . «> Expt. amp. " L Q i n . - Theory-B.E.M.  5.00  A  •5.00 t_i—i—i—i—I—i—i—i—i—I—i  0.00  0.83  1.67  i i i I i i i i l i i i i I i i i i l  2.50  3.33  4.17  WxWxA/G F i g . 3 6 Heave e x c i t i n g  force  p h a s e . Compound c y l i n d e r .  T-35.5"  5.00  0.50  -+ x * *  0.44 0.37  Legend Theory-B.E.M. Present theory Expt. amp. - 0 . 2 i n . Expt. amp. - 0.5'mJ Expt. amp. - 0 . 7 S i n . Expt. amp. - I.Oin. ;  0.31 fj.25 0.19  J I  0.00  I—I—l—J—I—I—I I , , i i I* i  0.83  1.67  2.50  iffLLtU—1 3.33  4.17  WxWxA/G F i g . 3 7 Heave e x c i t i n g  force  mpound c y l i n d e r . on co  T=38.375"  5.00  CO  Legend o Expt. amp. - 0 . 2 i n . Expt. amp. - 0 . 5 i n . 4- Expt. amp. -  CO  X  5.00  0  7  5  j  Expt. amp. -join. Theory-B.E.M.  CD > CO  I CD O  CD CO CO  JC  Q.  '^•^Jj  0.00  i i t 1 i i i i I i i i i 1 i » i i I i • i i I i i . ,  0.83  1.67  2.50  3.33  4.17  WxWxA/G F i g . 3 8 Heave e x c i t i n g  force  p h a s e . Compound  cylinder.  T=38.375"  5.00  n  0.20 _  0.00  o  0.83  1.67  2.50  3.33  Legend Expt. amp. - 0.5in. Expt. amp. - I.Oin. Theory-B.E.M. Present theory  4.17  WxWxA/G 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  5.00 ,  © A -  3.75  Legend Expt. amp. -0.5in. Expt. amp. - 1 . 0 i n . Theory-B.E.M.  2.50 1.25 0.00  tb  -1.25 -2.50 -3.75 '5.00  t_j—i—i—i—I—i—i—i—i—I—i—i  0.00  0.83  1.67  i i  | i  i  2:50  i  i  | i  i i  3.33  i  I i  i  i  4.17  WxWxA/G F i g . 4 0 Heave e x c i t i n g  force  p h a s e . Compound  cylinder  T=42.625"  i l  5.00  Legend - Theory-B.E.M. -- Present theory + Expt. amp. -0.12in. x Expt. amp. - 0 . 5 i n . <> Expt. amp. -0.7inJ 4 Expt. amp. - 0 . 9 0 i n .  0.40 0.35 ~  0.30  Q  E < x O x  0.25  X  0.20  >  O  - X  0.15 0.10 0.05 0.00 _ t 0.00  J—i—i  I  i  i  i  0.83  i  1  1.67  ' . I i i—i—i  2.50  I i  3.33  • • • t • i  4.17  i  i i  5.00  WxWxA/G Fig 41 0  Heave e x c i t i n g  force  on compound  cylinder.  T=49.5« W  Legend  •fi.OOJ-  .  0.00  •  i  i  I  i  i  i  0.83  i_J I  I  1.67  I  I  I  © A +  Expt. amp. - 0.12in. Expt. amp. _ 0.5in. Expt. amp. _0.7in.  x  Expt. amp. ~o.90in  -  Theory-B.E.M.  L _ l 1 1 1 1 1—I 1 1 L _  2.50  3.33  4.17  WxWxA/G Fig.42  Heave e x c i t i n g  force  p h a s e . Compound  cylinder.  T=49.5"  J  L  5.00  136  Adctoa mms ratio for dfott» T = J . f T , T = J 5 F T 5 •  0  Eip«nmtntol (McCormick)  O  Triii tritory  Omtgo = 3 0 rod /tec  J  !_  1  L  J  L  »» «* Dtprh /rodus  Fig.43  Hydrodynamic c o e f f i c i e n t s f o r s i n g l e  cylinder.  

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