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Numerical simulation of nonlinear waves and ship motions Fitz-Clarke, John R. 1986

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NUMERICAL SIMULATION  OF NONLINEAR WAVES AND SHIP MOTIONS  by JOHN R. FITZ-CLARKE B.A.Sc, U n i v e r s i t y of B r i t i s h Columbia  (1983)  A THESIS SUBMITTED I N PARTIAL FULFILMENT THE REQUIREMENTS FOR THE DEGREE OF MASTER OF A P P L I E D  SCIENCE  in FACULTY OF GRADUATE Department of M e c h a n i c a l  We a c c e p t  this  STUDIES Engineering  t h e s i s as conforming  to the required  standard  UNIVERSITY OF B R I T I S H COLUMBIA O c t o b e r , 1986 ©  JOHN R. F I T Z - C L A R K E , 1986  OF  In  presenting this  requirements  thesis  f o r an  in  partial  advanced  available  that permission scholarly Department  or  understood  that  financial  Library  shall  f o r r e f e r e n c e and s t u d y .  f o r extensive copying  purposes by  may  be  h i s or copying  gain s h a l l  not  granted her  allowed  Engineering  UNIVERSITY OF B R I T I S H COLUMBIA 2075 Wesbrook P l a c e V a n c o u v e r , Canada V6T 1W5  D a t e : O c t o b e r , 1986  make i t  of  this  by  the  thesis for Head  representatives.  permission.  Department of M e c h a n i c a l  the  I f u r t h e r agree  or p u b l i c a t i o n of t h i s be  of  d e g r e e a t t h e UNIVERSITY OF  B R I T I S H COLUMBIA, I a g r e e t h a t t h e freely  fulfilment  without  of It  my is  thesisfor my  written  ABSTRACT A  numerical  behaviour  a  i s presented  of l a r g e amplitude  including given  method  wave  breaking.  Various  free  surface  initial  waves  conditions are  and the subsequent s u r f a c e p r o f i l e s a r e c a l c u l a t e d  time  stepping  boundary value complex  s i m u l a t i o n . The f l o w  variable  investigated breaking  field  problem f o r the v e l o c i t y method  based  t h e o r e m . Waves o f v a r y i n g  has  nonlinear  f o r simulating the  on  shape,  i s s o l v e d as a  potential the  Cauchy  height,  and  p r o v e n t o be v e r y  fluid  accurate  velocities.  using  The  are  f o r wave technique  and s t a b l e .  The method i s e x t e n d e d t o p r e d i c t t h e m o t i o n s o f a dimensional for  floating  non-linear  presence  of  body i n l a r g e a m p l i t u d e  effects  and  fluid-body  seas  two  accounting  interaction.  The  s i n g u l a r i t i e s at the free surface i n t e r s e c t i o n  p o i n t s was f o u n d t o s e v e r e l y l i m i t  accuracy  and  problem are d i s c u s s e d .  attempts  extension described.  to  a  integral  length  t o determine the parameters necessary  and t h e r e s u l t i n g  by  to  overcome  handle  three  this  dimensional  of t h e  ships  solution  is  An  also  Table of Contents ABSTRACT  i i  L I S T OF FIGURES  iv  NOMENCLATURE  v i i  ACKNOWLEDGEMENT  x  1.  INTRODUCTION  1  1.1  1  Introduction  1.2 L i t e r a t u r e S u r v e y  3  1.2.1  Wave S o l u t i o n s  3  1.2.2  Body M o t i o n s  4  1 . 3 Objectives 2.  6  POTENTIAL FLOW SOLUTION  9  2.1  9  Introduction  2.2 C a u c h y I n t e g r a l M e t h o d  2.3 3.  10  2.2.1  Formulation  10  2.2.2  Boundary C o n d i t i o n s  13  2.2.3  Solution  14  2.2.4  Velocities  15  2.2.5  Interior  16  T e s t Case  17  WAVE SIMULATION  23  3.1  Introduction  23  3.1.1  The P r o b l e m  23  3.1.2  L i n e a r Wave T h e o r y  24  3.1.3  S t o k e s Wave T h e o r y  25  3.1.4  Waves i n N a t u r e  26  3.2 N u m e r i c a l S o l u t i o n 3.2.1  27  Formulation  27 i ii  3.2.2 C o n s t r u c t i o n  of Matrix  3.2.3 C h o i c e o f I n i t i a l  4.  29  Conditions  3.2.4 Time S t e p p i n g P r o c e d u r e  30  3.2.5 Segment S i z e a n d Time S t e p  32  3.2.6 N u m e r i c a l A d j u s t m e n t s  34  3.2.7 E n e r g y  35  3.2.8 C o m p u t e r S o l u t i o n  37  3.3 R e s u l t s  37  FLOATING BODY MOTION  59  4.1 I n t r o d u c t i o n  59  4.2 N u m e r i c a l S o l u t i o n  59  4.2.1 F o r m u l a t i o n 4.2.2 The C l o s u r e  59 Problem  62  4.2.3 Body P o s i t i o n  65  4.2.4 I n i t i a l  66  Conditions  4.2.5 I n t e r s e c t i o n S i n g u l a r i t i e s  67  4.2.6 I n t e r s e c t i o n S o l u t i o n  68  4.2.7 A N u m e r i c a l P e r t u r b a t i o n 4.3 R e s u l t s 4.4 E x t e n s i o n 5.  29  Correction  .......70 72  t o Three Dimensions  CONCLUSIONS AND RECOMMENDATIONS  76 99  REFERENCES  102  APPENDIX I  106  APPENDIX I I  1 08  APPENDIX I I I  113  APPENDIX I V  115  APPENDIX V  117  iv  APPENDIX V I APPENDIX V I I APPENDIX V I I I APPENDIX I X APPENDIX X  L I S T OF FIGURES 1. T o u r s h i p 2. G e n e r a l  being buffetted  fluid  by " f r e a k  waves"  8  domain i n c o m p l e x p l a n e  19  3. D i s t r i b u t i o n o f c o m p l e x p o t e n t i a l on e l e m e n t s 4. F i n a l c o n s t r u c t i o n solution  of complex  potential  on c o n t o u r  19  5. T e s t c a s e o f c i r c u l a r c y l i n d e r  i n uniform flow  20  f o r t e s t case  20  6. G e o m e t r y a n d b o u n d a r y c o n d i t i o n s 7. V e l o c i t y in  19  p o t e n t i a l and stream f u n c t i o n  calculated  t e s t case  21  8. C o m p a r i s o n o f n u m e r i c a l a n d t h e o r e t i c a l solutions  f o r v e l o c i t y on c y l i n d e r  surface  22  9. D e f i n i t i o n o f wave v a r i a b l e s  42  10.  Regions of v a l i d i t y  42  11.  C l a s s i f i c a t i o n o f b r e a k i n g waves  43  12.  Flow c h a r t  44  13.  D e f i n i t i o n o f wave c o n t r o l  14.  Two t y p e s o f c o n t r o l requiring  15.  o f wave s i m u l a t i o n  algorithm  volume  45  volume d i s t o r t i o n  nodal point  adjustment  45  CPU t i m e p e r s t e p v e r s u s number o f e l e m e n t s for  16.  o f wave t h e o r i e s  wave s i m u l a t i o n  Translating  fifth  46  o r d e r S t o k e s wave  with  H/L = 0.06 s i m u l a t e d f o r one p e r i o d 17.  Trajectory  of marked p a r t i c l e i n S t o k e s  o r d e r wave w i t h 18.  H/L = 0.06  Deep w a t e r s p i l l i n g from c o s i n e  breaker  i n i t i a l condition iv  47 fifth 48  simulated H/L = 0.10  49  19. Deep w a t e r p l u n g i n g b r e a k e r s i m u l a t e d from c o s i n e i n i t i a l 20. S h a l l o w w a t e r  c o n d i t i o n H/L  spilling  * from c o s i n e i n i t i a l  = 0.13  50  breaker simulated  c o n d i t i o n H/L  = 0.10  51  21. S h a l l o w water p l u n g i n g b r e a k e r s i m u l a t e d from c o s i n e i n i t i a l  c o n d i t i o n H/L  = 0.13  22. L a s t s i m u l a t e d s t e p f o r d e e p w a t e r of  waves  varying height ratios  23. Maximum f l u i d  53  p a r t i c l e v e l o c i t i e s versus time  f o r d e e p w a t e r waves f r o m c o s i n e  initial  conditions  54  24. Maximum f l u i d for  52  p a r t i c l e v e l o c i t i e s versus time  s h a l l o w w a t e r waves f r o m c o s i n e  initial  conditions  55  25. F l u i d v e l o c i t i e s a r o u n d p l u n g i n g j e t  56  26. C o n t r o l v o l u m e e n e r g y v e r s u s t i m e f o r d e e p w a t e r b r e a k i n g wave f r o m c o s i n e c o n d i t i o n H/L  initial  = 0.13  57  27. C o n t r o l v o l u m e e n e r g y v e r s u s t i m e f o r s h a l l o w w a t e r b r e a k i n g wave f r o m c o s i n e initial  c o n d i t i o n H/L  = 0.13  28. C o n t r o l v o l u m e f o r body m o t i o n 29. D e f i n i t i o n  simulation  of c o o r d i n a t e systems  30. D e c o m p o s i t i o n i n t o 31. Flow c h a r t  58  four  independent problems  f o r body m o t i o n s i m u l a t i o n  32. B o u n d a r y c o n d i t i o n s t e s t e d 33. S i m u l a t i o n o f r o l l  motion  78 72 79 80 81  i n calm water v  82  34. P l o t  of r o l l  history  i n calm water  83  3 5 . S i m u l a t i o n o f m o t i o n on wave o f H/L = 0.04  84  36. S i m u l a t i o n o f m o t i o n on wave o f H/L = 0.04  85  37. P r e s s u r e  87  38. P l o t  d i s t r i b u t i o n on h u l l H/L = 0.04  o f m o t i o n s f o r H/L = 0.04  89  39. S i m u l a t i o n o f m o t i o n on wave o f H/L = 0.08  ....90  40. S i m u l a t i o n o f m o t i o n on wave o f H/L = 0.08  91  41. P l o t  93  o f m o t i o n s f o r H/L = 0.08  42. S i m u l a t i o n o f m o t i o n on wave o f H/L = 0.12  94  4 3 . S i m u l a t i o n o f m o t i o n on wave o f H/L = 0.12  95  44. P l o t  96  o f m o t i o n s f o r H/L = 0.12  45. R e s u l t  of zero  46. E q u i v a l e n t  initial  angular v e l o c i t y  p r i s m a t i c r e p r e s e n t a t i o n of s h i p  vi  97 98  NOMENCLATURE a^,By  = body a c c e l e r a t i o n  a^j  = r e a l p a r t of m a t r i x  b^j  = imaginary  c  = phase v e l o c i t y  C  = contour of i n t e g r a t i o n  d  = water  E^  = kinetic  Ep  = potential  E  = total  fc  element  p a r t of m a t r i x  element  o f wave  depth energy energy  energy  F ,F  = body  g  = gravitation  G  = c e n t r e o f g r a v i t y o f body  H  = wave  height  h^j  = real  matrix  I  = mass moment o f i n e r t i a  k  = wave number = 2-n/L  L  = wave  m  = body mass  M  = body moment a b o u t G  n  = normal u n i t v e c t o r  N  = total  NB  = number o f b o t t o m  NH  = number o f h u l l  NS  = number o f f r e e s u r f a c e  NV  = number o f v e r t i c a l  x  forces constant  o f body  length  number o f e l e m e n t s elements  elements  vi 1  side  elements elements  P  = fluid  Q  = volume f l o w  R  = radius  s  = arc length  t  = time  At  = time s t e p  T  = wave  t„  = t i m e t o wave  u  = horizontal  fluid  = horizontal  body  u  Q  v  rate  running  surface  variable  period breaking velocity velocity velocity velocity  = n o r m a l component  n  width  interval  = v e r t i c a l body  G  per unit  from G t o body  = vertical fluid  v V  pressure  V  = volume  w  = complex  fluid  o f body  velocity  velocity  = u-iv  * w  = complex c o n j u g a t e o f w  x  = horizontal  Ax  = element  y  = vertical  z  = complex  a  = i n c l u d e d a n g l e a t node  0  = complex p o t e n t i a l  <j> .  = velocity  \jj *"ij p  = stream =  c  o  m  P l  e  = fluid  x  distance  size distance spatial coordinate  = x+iy  = <p+i\p  potential function  influence  coefficient  density  vi ii  e  = roll  V  = wave e l e v a t i o n a b o v e s t i l l  A. . 1  3  n. . 1  D  angle  = influence  coefficient  = influence  coefficient  = angular = fluid  frequency  domain  ix  =  2TI/T  water  ACKNOWLEDGEMENT The  author  would  like  supervisor  Dr.  Sander  M.  throughout  the  course  o f t h i s w o r k . The a u t h o r w o u l d  like  to  Science  t o extend sincere Calisal  express  h i s gratitude  Council  and  Atlantic, Halifax  the  to  Defence  forfinancial  possible.  x  thanks t o h i s  f o r patient  the  British  Research  guidance also  Columbia  Establishment  s u p p o r t w h i c h made t h i s  work  1. INTRODUCTION  1.1  INTRODUCTION Despite  still  advances  in  ship  no m a t c h f o r r o u g h s e a s  occur  each  year  Fundamental  with  and  the  better  understanding  the  extreme  waves and t h e r e s u l t i n g  small v e s s e l s are  capsizings  loss  t o reducing the r i s k of  design,  of  of  lives  such  and  and  to  and p r o p e r t y .  tradgedies  kinematics forces  continue  is  dynamics  response  a of  of  a  v e s s e l t o them. C a p s i z i n g o f a v e s s e l may phenomena excited  including by  a  o c c u r due  extreme  rolling  near-resonant  t o any in  encounter  in  following  or  quartering  effect),  or  broaching  or  pitch-poling  stern  waves  of  result  d o w n f l o o d i n g of deck  f i s h i n g boats c a r r i e d out  Engineering capsizing within  Centre  have  amplitude  to  impacting  B.C.  demonstrated  i n beam s e a s i s v e r y u n l i k e l y  wave  on  the  side,  1  stability. deck  and  experiments Ocean  extreme  for a vessel  roll loaded  involving  [26].  The  of the v e s s e l , forces  possibly  steep  s u c h c a p s i z i n g s h a v e been  conditions  very large a d d i t i o n a l  Mathieu  Research  that  i n t h e b a s i n under e x t e n s i v e t e s t s regular  on  roll while  involving  water  the  seas,  (the  openings. Model  at  b r e a k i n g waves i n t h e v i c i n i t y lead  in  t h e recommended l i m i t s . No  observed  seas  and l o s s o f d i r e c t i o n a l  T h e s e phenomena f r e q u e n t l y subsequent  beam  several  frequency  underway  overtaking  of  large  presence of  however,  can  from the p l u n g i n g j e t  sufficient  to  cause  2  capsizing.  Fishermen  occasionally  r e p o r t e d b e i n g h i t by " f r e a k  one  who  have  o r more waves u n u s u a l l y  survived  larger  accidents waves",  S u c h a n o m a l o u s waves h a v e  documented  1)  unwitnessed accidents.  and may The  Current  stability  quasi-static  present  and  critical  conditions large  calm  encountered i n the  are  concentrates  usually  based  the metacentric  water.  real  s t e e p waves a r e i n v o l v e d .  much  as w e l l  These  sea,  height  especially  More r e c e n t l y ,  are  however,  such  tests  frequently  where  efforts  have  i n terms of dynamic p a r a m e t e r s  p e r f o r m e d when e x p e r i e n c e w i t h a  c o n s u m i n g , and  on  on  rules  work r e m a i n s t o be d o n e . O c c a s i o n a l l y  lacking,  to  a n d become somewhat m e a n i n g l e s s i n d y n a m i c  been made t o d e f i n e s t a b i l i t y but  are  involving  r o l l angles i n  largely empirical  work  been  i n e x t r e m e beam s e a s .  criteria  definitions  is  indeed  have c o n t r i b u t e d  examining the motions of a v e s s e l  that  than those i n the normal  p r e c e d i n g sea s t a t e . (figure  have  particular  are  limited  model  design  expensive  by  tests  facility  and size  is time and  equipment. The t h e o r e t i c a l provides involved  important and  conditions  permits that  relative effects  insight  wave  and  body  motions  i n t o the fundamental processes  estimates  of t h e d i f f e r e n t  of  dynamic  governing  behaviour  in  In a d d i t i o n ,  the  parameters  can  i n t h e o r e t i c a l models whereas i n a c t u a l  the i n f l u e n c e  c a n n o t r e a d i l y be  of  c a n n o t r e a d i l y be t e s t e d .  be s e e n a n d a s s e s s e d model t e s t s  analysis  of  decomposed.  contributing  effects  usually  3 1.2  LITERATURE SURVEY  1.2.1  WAVE SOLUTIONS Analytical  developed  to  classical  solutions a  fairly  linearized  steady  c o r r e c t i o n . To calculated, coefficient to  find  f r e e s u r f a c e waves h a v e been  high  degree  to extend  waves  and  the  recently  elusive  (1847)  the  developed to  stream  solutions  Schwartz  highest  w a t e r and its  function  c n o i d a l theory  limitations  and  (1974) i n  intended a p p l i c a t i o n . an o v e r v i e w  Progress required attempt Navier  in  have  was  Stokes  most  equations  viscosity  accuracy  was  progressive  adjustments  and  including deep  E a c h method  Isaacson  (1981)  on  behaviour a  s o l u t i o n to the  c a r r i e d out  Chan a n d achieved  s o l i t a r y wave on a b e a c h up  has  computer.  has the  provide  so  far  The  first  incompressible  by H a r l o w i n 1965  method r e q u i r e d  t o compensate f o r n u m e r i c a l  poor.  attempt  a p p r o p r i a t e d e p e n d s on  wave  cell  L o s A l a m o s L a b o r a t o r i e s . The high  the  o t h e r wave t h e o r i e s .  simulation  a m a r k e r and  been  i n t e r m e d i a t e and  f o r shallow water.  transient  numerical  for  S a r p k a y a and  o f t h e s e and  order  an  p o s s i b l e steady  theory  the  finite  third  wave. O t h e r n o n l i n e a r t h e o r i e s h a v e been d e v e l o p e d Dean's  the  u s i n g a computer t o perform  a r i t h m e t i c a s by  the  with  solution  calculated  d a t e many h i g h e r o r d e r  more  beginning  t h e o r y of A i r y . Stokes  a p e r t u r b a t i o n expansion amplitude  to  to  reasonable the  point  the  unrealistically instability  S t r e e t (1970) i m p r o v e d t h e a  at  simulation where  the  and grid  of  a  free  4  surface  became  vertical.  Longuet-Higgins inviscid method  and  On  Cokelet  a  separate  (1976)  concentrated  s o l u t i o n using a p o t e n t i a l flow based  vertical  face.  water  breaking  Further  found i n Longuet-Higgins  (1977).  extended  by  the  solution  the complete  Vinje  on  breaking  integral  one w a v e l e n g t h  again  of t h e i r and  developing  boundary i n t e g r a l method based simulate  wave,  applications  on t h e  boundary  on G r e e n ' s t h e o r e m t o s i m u l a t e  of a p e r i o d i c d e e p  front,  a  up  how  estimate  t h e method  Brevig  theorem  wave i n f i n i t e  breaking  be  wave f o r c e s on a  water  used  fixed  to  Myrhaug  (1979),  and K j e l d s e n  to  depth  (1981a)  numerically  object  e x p e r i m e n t s t o measure such f o r c e s a r e p r e s e n t e d and  (1980)  complex v a r i a b l e  t h e Cauchy  might  a  work may be  i n c l u d i n g o v e r t u r n i n g of t h e c r e s t . V i n j e and B r e v i g describe  to  and  some  i n Kjeldsen  (1981).  1.2.2 BODY MOTIONS C a l c u l a t i o n of difficult  floating  mathematical  the  presented six  determination  of  a  very  Korvin-Kroukovsky  equations  o f body m o t i o n f o r  freedom,  forces,  c o e f f i c i e n t s were o n l y c r u d e l y p o s s i b l e i n i d e a l i z e d  cases.  developed  to  r e q u i r e d some f o r m o f l i n e a r i z a t i o n , surface  boundary  conditions  masses,  analytical damping  solutions  added  however,  and  Analytical  of  poses  equations.  the coupled  degrees  motions  problem owing t o t h e c o m p l e x i t y and  n o n l i n e a r i t y of the governing (1955)  body  date where  have  invariably  body  a r e a p p l i e d on f i x e d  and  free  surfaces  5 and  body  geometries  cylinders Ursell  or  spheres.  (1964),  Wehausen  are  Lee  (1971)  formulations,  waves  on G r e e n ' s Garrison motions  theorem  techniques  have  to  transfer  (1964),  of  used  Bhattacharyya  other  periodic  to  used  p a n e l methods  by  Kim  based  (1966),  steady  of  small  and  harmonic  so  far  the  are  frequency-domain  response  in  regular  g i v e n an  quantities  practice.  (1979) and Newman of  obtain  then  be  of  may  be  the  estimated.  l i n e a r , y e t can  Details  a  provide are  found  in  (1980).  t r a n s i e n t n o n l i n e a r motions  again  of a n u m e r i c a l t i m e - s t e p p i n g s i m u l a t i o n  initial  (1981b,c)  to  input spectrum  can  necessarily  used  i n waves t h a t a r e n o t t o o l a r g e , and  in  ( 1 9 7 7 ) , Chapman  dimensional  and  limited  the l i n e a r i z e d ,  of motions  requires application  technique to  these  (1970).  Applications  Numerical  predict  calculation  given  Ursell  shaped b o d i e s .  are  reasonable r e s u l t s  Brevig  handle  Statistical  techniques  from  MacCamy  waves. F o r t h e more g e n e r a l wave c o n d i t i o n s a  spectrum  sea s t a t e .  The  include  f u n c t i o n c a n be c o n s t r u c t e d and  frequency  widely  circular  references.  been  described  which  monochromatic  These  and m o t i o n s .  methods  such as  overview  course,  of a r b i t r a r i l y  The  an  additional  (1975)  shapes  ( 1 9 6 9 ) , and M a s k e l l and  t h e s e methods a r e , of amplitude  Examples  gives  and  simple  c o n d i t i o n s . Examples i n c l u d e F a l t i n s e n  (1979),  and  extended  include  the  Isaacson their  Vinje  and  n o n l i n e a r wave s i m u l a t i o n  presence  f l o a t i n g body and  (1982).  of  an  arbitrary  two  describe i t s application  to a  6  heaving  cylinder.  applied  the  Greenhow,  method  to  Brevig,  and  the  extreme  study  Taylor  (1982)  motions  r o t a t i n g wave e n e r g y d e v i c e , h o w e v e r , f r e e s u r f a c e near  the  body  had  to  be  i n the  work d i f f i c u l t . involving of  surface. attempts  1.3  S e v e r a l p r o b l e m s have  points  This  have  l i t e r a t u r e making comparisons w i t h  the c h o i c e of  singular  simulations  initial  where  to  c o n d i t i o n s and  the  t h e s i s addresses  t o h a n d l e them a r e  yet  a  behaviour  s p e c i f i e d e m p i r i c a l l y . Very  q u a n t i t a t i v e r e s u l t s o f body m o t i o n presented  of  few been  previous  be  overcome  the  presence  body  intersects  the  these  p r o b l e m s , and  free  various  discussed.  OBJECTIVES A t h e o r e t i c a l model f o r p r e d i c t i n g  solve exact  for  the c o u p l e d  solution  viscous  including  effects  must  tractible  within  and  r e s o u r c e s . The  usual  incompressible  and  The  be  first  made the  inviscid  fluid  behaviour.  An  fluid  motions  and  task  in  and  the  calculating  the  velocity  dimensional  fluid  complex  keep  the  various  mathematics  are  that  thus  the  satisfies  fluid  is  Laplace's  potential.  for  a  to  must  c a p a c i t y of a v a i l a b l e c o m p u t i n g  technique  using  scale  assumptions  for a velocity  numerical  small  motions  i s b e y o n d t h e s t a t e of t h e a r t and  approximations  equation  body m o t i o n s and  ship  present  solving  work i s t o d e v e l o p  Laplace's  potential  in  a  a  equation  and  general  two  domain under a r b i t r a r y boundary c o n d i t i o n s variable  boundary  i n t e g r a l method.  This  7 Laplace  solving  routine  i s then used t o c a l c u l a t e the f l u i d  v e l o c i t i e s o f a f r e e s u r f a c e wave a t d i s c r e t e t i m e and  a l l o w a t i m e - s t e p p i n g s i m u l a t i o n o f t h e b e h a v i o u r o f an  a r b i t r a r y wave up t o a n d i n c l u d i n g b r e a k i n g . The different  initial  wave  effect  detailed wave  kinematics  simulation  inclusion surface. around  of  of  t h e f l o w development.  technique  an a r b i t r a r y  Fluid  velocities  is  extended  to  and  pressures  f o r c e s . The e q u a t i o n s o f r i g i d  integrated  at  each  and  the  F i n a l l y , the permit  the  two d i m e n s i o n a l body on t h e f r e e are  t h e body a t e a c h t i m e s t e p t o d e t e r m i n e  hydrodynamic  of  c o n d i t i o n s c a n t h e n be a s s e s s e d t o  e s t a b l i s h a d o m a i n o f d e p e n d e n c e o f b r e a k i n g waves  then  intervals  time  step  h i s t o r y o f t h e n o n l i n e a r body r e s p o n s e  to  calculated  the r e s u l t i n g  body m o t i o n  are  c a l c u l a t e a time  i n t h e wave  field.  8  Figure 1.  Tour ship being buffetted by "freak waves".  2. POTENTIAL FLOW SOLUTION  2. 1 INTRODUCTION The  exact  including difficult  viscous task  Navier-Stokes cases. the  calculation and  due  problem  the  equations  and  and  permit  usual assumptions are  and  irrotational,  the  can  o n l y be  field  i s an  extremely  the  governing  of  done i n v e r y  limited simplify  a t r a c t i b l e mathematical  solution.  that the  -  fluid  is  incompressible  a c o n s e q u e n c e of n e g l e c t i n g  T h e s e two  V**  flow  that  s a t i s f i e s Laplace's  in Cartesian  effects  complexity  latter  e f f e c t s of v i s c o s i t y .  or  arbitrary  must make a s s u m p t i o n s  The  flow f i e l d  an  turbulent  to  I n g e n e r a l , one  of  the  conditions d i c t a t e that  the  equation  0  { 2  '  1 )  coordinates  +iii - o 3x2  for as  3 y  2  a s c a l a r v e l o c i t y p o t e n t i a l 0. V e l o c i t i e s a r e the g r a d i e n t  the  of  appropriate  <p. The  Analytical cases  using  variables,  perturbation  conditions  classical  (2.1)  methods  superposition  methods,  transient  problems  boundary  conditions  and  solving  the  problem.  solutions to  linear  found  p r o b l e m i s c o m p l e t e d by s p e c i f y i n g  boundary  r e s u l t i n g boundary v a l u e  then  and  such  of  conformal  involving one  are  possible as  9  separation  sources  and  many of  vortices,  mappings, however, f o r  complicated  must  in  resort  geometries to  a  and  numerical  10 procedure.  Several numerical  over  years  the  traditional which  finite  solve  interior  the  f o r the  reduced  resolution generation problems fluid flow  on  for  <p a n d the  by  one a  including the  a grid  an  has  orthogonal boundary.  turned  to  advantage  function are The p r o b l e m i s  allowing  simplified.  methods  of d i s c r e t e  have t h e p o w e r f u l  computational  dynamics  developed  element  attention  dimension  given  fluid  along  domain  i s also greatly of  been  finite  potential recently,  of  only  and  methods w h i c h  values  calculated thereby  difference  integral  have  solve Laplace's equation  p o i n t s . More  boundary that  to  methods  much  greater  effort.  Element  Furthermore,  one i s i n t e r e s t e d  in  many  only i nthe  v e l o c i t i e s a n d p r e s s u r e s a t t h e b o u n d a r y . The i n t e r i o r field  values,  c a n be f o u n d  however,  i f d e s i r e d knowing o n l y t h e boundary  this  i s usually  of  little  practical  interest. The boundary  method u s e d i n t h i s integral  method  work based  is a on  the  complex  variable  Cauchy  integral  theorem.  2.2 CAUCHY INTEGRAL METHOD  2.2.1  FORMULATION The  values  o b j e c t i v e of the numerical procedure around  insufficient differentiaton  the to  contour  determine  along  C  C,  however,  i s t o o b t a i n <j>  this  alone  is  v e l o c i t i e s a t t h e boundary s i n c e  provides  only  the  tangential  11 v e l o c i t y . The  normal  consequently  one  component  must e v a l u a t e  the  boundary as  the  c a l c u l a t i o n s f o r <p and  solves  f o r the  The  an  be  determined,  orthogonal function  along out  90/9n, w h i l e  the  current  approach  v e l o c i t y p o t e n t i a l <j> and  the  stream  function  t h e n be  found  differentiating Cauchy-Riemann  \p  along  boundary can  the  contour  since,  by  by the  property, 3£  3£  as  stream  Lapalace's  also  w e l l . Green's f u n c t i o n approaches c a r r y  normal v e l o c i t y at the  The  must  function  can  also  be  shown  to  satisfy  equation 0  V * 2  h e n c e b o t h <f> and  \p a r e  harmonic  functions.  By  defining  a  complex p o t e n t i a l  0 the  <>| + !•  p r o b l e m r e d u c e s t o one  of  domain J2 o f  f i n d i n g the  analytic  in  the  satisfies  the  boundary c o n d i t i o n s  Since 0 i s required  t o be  the  0  c  provided  z  0  i s outside  the  /3 t h a t  complex p l a n e z=x+iy, <t> o r ^  of  -  is and  given.  a n a l y t i c , i t must s a t i s f y  w e l l - k n o w n Cauchy i n t e g r a l theorem  dz  function  the  [9]  0  c o n t o u r C.  (2.2)  If z  0  is  allowed  to  1  approach C i n the l i m i t ,  t h i s equation  { - § — dz J  z  c  with  z  -  ~ o  2  becomes  1 a B(z ) 0  z  on C. D e r i v a t i o n of t h i s e x p r e s s i o n c a n be f o u n d i n  0  A p p e n d i x I . /3 must be f o u n d  that  satisfies  this  integral  equat i o n . To  solve  discretized at  (2.2)  into  N linear  e a c h e n d . The i n t e g r a l  of the i n t e g r a l s  linear  can then  each element  \l  I=zT > "  j=l  dz  Zj  the  contour  C  e l e m e n t s b o u n d e d by n o d a l  over  1  A  numerically,  1  be r e p r e s e n t e d  a  o  B ( z  points  a s t h e sum  (2.3)  )  0  variation  of  /3  i s assumed o v e r  e a c h e l e m e n t (a  higher order polynomial  d i s t r i b u t i o n c o u l d be u s e d ,  t h i s h a s n o t been f o u n d  n e c e s s a r y ) . D e f i n i n g upper and  n o d a l v a l u e s a s z . , z . ^ . , p., and 3  distribution  3  +  j+l  of  (2.3).  integrals After reduces  1  as i n f i g u r e 3  ]  +  i s then  J  the  algebraic  manipulation  kernel the  equation N  I j-1  r . B. J J  3,  lower the  J  as  3  the  D  lefthand  side  s o t h e y c a n be removed f r o m t h e  remaining  t o the l i n e a r  J+l  substituted into  |3j a r e c o n s t a n t s and  ^  however  1  o f 0 on t h e e l e m e n t c a n be e x p r e s s e d  This expression  is  0  functions integral  evaluated.  equation  (2.2)  1 3  By  letting  e a c h node t a k e  on t h e v a l u e  o b t a i n N complex e q u a t i o n s  z  0  i n turn,  one  can  f o r N unknown /3  N j=l  1 J  (2.4)  J  where t h e i n f l u e n c e c o e f f i c i e n t s a r e  D e t a i l s c a n be f o u n d i n A p p e n d i x I I . C a r e must be in  evaluating  the  term  to  m u l t i p l e - v a l u e d complex l o g a r i t h m  avoid  exercised  problems w i t h the  f u n c t i o n as  described  in  Appendix I I I .  2.2.2 BOUNDARY CONDITIONS Since  Laplace's  equation  point a f f e c t s every other be can  specified also  segments  be  is elliptic  i n nature,  p o i n t a n d b o u n d a r y c o n d i t i o n s must  on a l l b o u n d a r i e s . Due t o l i n e a r i t y , handled  are l i n e a r l y  each  where  the  boundary  r e l a t e d such as being  problems  conditions  on  p r o p o r t i o n a l or  14 equal  by  lumping  appropriately The  together  i n the f i n a l  solution  of  linear  then  the  For 0  0  solution,  i s unique  only  to  0{z)  is  a  example  s o i s 0 ( z ) + /3 w h e r e /3  unique  quantities  equations.  n u m e r i c a l method d o e s n o t g e n e r a t e fix  unknown  Laplace's equation  w i t h i n an a r b i t r a r y c o n s t a n t . solution  the  if  i s any  constant.  t h i s c o n s t a n t and  consequently  the  The  thereby boundary  c o n d i t i o n s g i v e n must i n c l u d e c o n t r i b u t i o n s f r o m b o t h tf> and t o e l i m i n a t e any  2.2.3  ambiguity.  SOLUTION Defining  r\j=a^j+ i ^ ,  equation  (2.4)  can  be w r i t t e n  as  N  w i t h the r e a l  and  It  the  part  imaginary  i s c l e a r now  (either  #j  or  part  that while there are  make  a  unknown  quantities  a t e a c h node) t h e r e a r e a c t u a l l y  e q u a t i o n s a v a i l a b l e . The must  N  choice  as  problem to  is  which  2N  real  overspecified  and  one  N equations  satisfy.  to  i  Selection  of  only  the  real parts provides  s o l u t i o n , as does s e l e c t i o n of albeit  slightly  different,  only  the  a  satisfactory  imaginary  i f one s e l e c t s t h e r e a l p a r t f o r e q u a t i o n  is  unknown  unknown.  Such  inhomogeneous  q u a n t i t y and t h e i m a g i n a r y selection  integral  c a s e a n d one w o u l d result. Recently, utilizing  will  ensure  equations  expect  a  that  are being  more  squares sense, r e s u l t i n g  which  stable  are  i when <p^  p a r t when  is  only  the  chosen i n each solution  S c h u l t z e t a l (1986) f o r m u l a t e d  a l l 2N e q u a t i o n s  parts,  h o w e v e r an i m p r o v e d s o l u t i o n i s  possible the  5  solved  as  a  a solution  in  a  least  i n a f u r t h e r improvement.  The N s e l e c t e d e q u a t i o n s  c o n t a i n t h e unknown v a l u e s  Xj  as w e l l a s t e r m s i n v o l v i n g t h e known b o u n d a r y c o n d i t i o n s . I f these  latter  summed  one  terms are transposed  to the righthand  obtains  N  a  set  unknowns w h i c h c a n be s o l v e d N I h j-1  The  complete  solution  X 3  of  i n matrix  -  equations  and for N  form as  g 2  3  /3j  linear  side  i s constructed  by c o m b i n i n g t h e  c a l c u l a t e d X j w i t h t h e known b o u n d a r y c o n d i t i o n s a s shown i n figure  5.  2.2.4 V E L O C I T I E S Once  /3  has  been  calculated  v e l o c i t i t e s c a n be d e t e r m i n e d a s  on  the  boundary,  the  16 w  The  derivative  direction the  of  (2.5)  a  complex  function  so t h i s d i f f e r e n t i a t i o n  contour  30/3Z i s  C.  S i n c e 0 has  discontinuous  can  i s independent  be c a r r i e d  out  been assumed p i e c e w i s e  at  the  nodal  points.  Appendix  2.2.5  be  along linear,  A  central  d i f f e r e n c e scheme i s u s e d b a s e d on a T a y l o r s e r i e s o f /3 a b o u t t h e p o i n t o f e v a l u a t i o n . D e t a i l s may  of  expansion found  in  IV.  INTERIOR The  method  presented  t h e b o u n d a r y C. sufficient  For  since  practical  often  v e l o c i t i e s and  pressures  interior  field  flow  allows calculation  one  is  on  can  k  C  this  is  intetrested  bounding  found  on  usually i n the  surfaces.  The  h o w e v e r , i f d e s i r e d , by  as  B  B(z )  only  the  be  r e a p p l y i n g Cauchy's theorem  purposes  o f /3 o n l y  z-z, k  dz  and  The  i n t e g r a l s h e r e can a g a i n  linear  be  evaluated  sums i n v o l v i n g t h e known n o d a l v a l u e s  f a s h i o n as b e f o r e  resulting  i n equations  numerically /3j i n a  of t h e  form  as  similar  17 N  j=l  2  3  and N W  where  the  contour  -  (z)  influence  are  functions  g e o m e t r y z... I n t e r i o r v a l u e s c a l c u l a t e d  of  the  as such  are  work.  TEST CASE The  Cauchy  technique cast  To  integral  for solving  into  involving  the  form  an of  as  i s well  mixed  boundary  powerful  that  can  value  be  problem  in  figure  known. The  5,  the  flow  analytical  velocity potential  can  be -  B(z)  the  a  field  a  t h e method an e x a m p l e i s c h o s e n o f u n i f o r m  of which  shown t o  with  provides  <j> a n d i//.  test  solution  method  i n t e r i o r flow  past a c i r c u l a r c y l i n d e r  be  B.  coefficients  not used i n the p r e s e n t  2.3  n  i  origin  at  U(z +  the  |^)  centre  of the c y l i n d e r ,  and  the  velocity w(z)  For in  -  U(l -  the purpose of n u m e r i c a l the f l u i d  and  R  2  —)  *2  solution,  boundaries  assumed f a r e n o u g h away f r o m  the  are  placed  cylinder  18 that their solution  effect can  q u a d r a n t as a  be  set  as  such t h a t t h e i r streamlines  = line  to vary  are  placed  calculated  the  numerical  above  are  plotted  u p p e r b o u n d a r y i s assumed  the  The  flow  with  \p  chosen  2  between  righthand  side  <j> c h o s e n  the  is  an  l e f t h a n d s i d e i s assumed f a r enough  flow conditions p r e v a i l  been r u n w i t h R=1, on  L=5,  and  as  */>  t h e c y l i n d e r a t 5° the  H=5,  an  can  be  p r i m a r i l y to the  agree  well  solution.  a  with  stream the  function  analytical  the c y l i n d e r s u r f a c e , c a l c u l a t e d  differentiation f i g u r e 8 along  finite  U=1.  contour.  v e l o c i t y p o t e n t i a l and  in  and  intervals with  technique w i t h the  v a l u e s . Agreement i s g e n e r a l l y good, w i t h the  the n u m e r i c a l  left  The  V e l o c i t i e s along  using  due  the upper  symmetry,  shown i n f i g u r e 7 and  solution.  only  numerical  by  o f 80 e l e m e n t s on The  the  linearly.  A t e s t c a s e has  total  The  equals  ^\~^2-  constant.  are  symmetry,  considering  difference  upstream that uniform  Elements  up  By  i s t h e m i d l i n e a x i s w i t h i//, and  Q  equipotent i a l  considered  small.  shown i n f i g u r e 6.  streamline  arbitrary  is  far f i e l d  described theoretical  discrepencies  b o u n d a r i e s imposed i n  19  C Figure 2.  General f l u i d domain i n complex plane.  Figure 3 .  D i s t r i b u t i o n of complex potential on elements.  Figure 4.  Final construction of complex potential s o l u t i o n on contour.  Figure 5.  Test case of c i r c u l a r cylinder  in uniform  flow.  0=10  0 linear variation  <t> = 0  0=0 Figure 6 .  Geometry and boundary conditions for test case.  Figure 7.  V e l o c i t y potential and stream function calculated i n test case.  Figure 8.  Comparison of numerical and t h e o r e t i c a l solutions for v e l o c i t y on c y l i n d e r surface. Angle i s measured from forward stagnation p o i n t .  3. WAVE SIMULATION  3.1  INTRODUCTION  3.1.1 THE PROBLEM A wave i s c o n s i d e r e d and  surface  level.  of w a v e l e n g t h L, 7 j ( x , t ) measured  elevation  The wave i s p e r i o d i c o f p e r i o d  a' p h a s e  speed  c  incompressible Laplace's  as  and  water  from t h e s t i l l  and  d,  water  T, a n d t r a n s l a t e s  i n f i g u r e 9. The f l u i d irrotational  depth  with  i s considered  again  satisfies  equation  ili +iii = o subject  t o the f o l l o w i n g boundary  impermeable  and t h e r e f o r e  c a n h a v e no n o r m a l  || = The  free surface  must  B e r n o u l l i ' s equation !F  If  the  satisfy  a  dynamic  is  velocity  y = -  T [<|£> + ( f ^ ) ] + s n - f ( t )  +  2  particles  profile  0  The s e a b e d  (3.1)  d  condition  that  i s obeyed  as w e l l as a kinematic fluid  conditions.  have  2  condition  which states  velocities  identical  y = V  (3.2)  that  surface  to  the  wave  velocities  wave  i s considered  periodic  23  i n space then e x p l i c i t  24 b o u n d a r y c o n d i t i o n s on t h e v e r t i c a l  c o n t r o l volume  segments  are unnecessary.  3.1.2  LINEAR WAVE THEORY The  problem  because  the  nonlinear priori.  in  defined  two  free  <j> a n d  The c l a s s i c a l  amplitude  i s small  sinusoidal  surface  above  surface  difficult  boundary  t o solve  conditions  are  s u r f a c e e l e v a t i o n 17 i s unknown a  the  solution so  i s very  the  of  Airy  problem  assumes  the  wave  c a n be l i n e a r i z e d . A  i s assumed j] » -j cos(kx-o)t)  and t h e s l o p e  i s considered  s m a l l so t h e n o n l i n e a r  terms a r e  negligible. The  boundary c o n d i t i o n s then reduce t o |£ + gn -  0  |i 3y  0  y =  0  and  Solution  of  the  variables  yields:  3t  boundary  *  =  - | J L  value  •nH cosh k(y+d)  =  y =  KT slnh kd  0  p r o b l e m by s e p a r a t i o n o f  .  k X  •inC**-"*) (3.4)  C  2  -  tanh(kd)  25 3.1.3  STOKES WAVE THEORY Early  solution in  work  Stokes  (1847) extended  the  analytical  t o i n c l u d e n o n l i n e a r e f f e c t s by e x p a n d i n g  a p e r t u r b a t i o n s e r i e s . V a r i a b l e s are expressed  where e i s  a  expressions and  by  $  =  e o> + E  n  =  e rij + e  2  ri  2  parameter  are s u b s t i t u t e d  •••  + 2  of  the  order  into equations  terms of l i k e o r d e r of magnitude are  s u c c e s s i v e l y higher order s o l u t i o n s , the f i r s t third  approximation.  order  involved. Skjelbreia Stokes  fifth  Stokes  solution,  as  and  as  4> + .. •  2  1  small  variables  the  o r d e r s o l u t i o n of the  (3.2)  (3.3),  yielding  theory p r o v i d i n g  originally algebra  These  and  gathered  linear  Hendrickson  H/L.  calculated  quickly  (1960)  a  becomes  presented  a  form:  5 n  =  I n=l  n  cos(nKx) n  5 4>  The and and  =  I <J> n=l  sin(nKx) n  e f f e c t of h i g h e r o r d e r terms i s t o flatten  t h e t r o u g h s . The  and  Isaacson  (1981).  the  r e g i o n s of v a l i d i t y  h i g h e r o r d e r s o l u t i o n s c a n be  Sarpkaya  sharpen  seen  in  figure  crests  of 10  linear from  26 3.1.4  WAVES I N NATURE M o s t o c e a n waves a r e g e n e r a t e d  on  the  sea  surface  grow a s a r e s u l t  by w i n d s e x e r t i n g  forming s m a l l d i s t u r b a n c e s which  o f work done by a e r o d y n a m i c  p e r i o d i c p r o g r e s s i v e waves have been w e l l but p r o b a b l y e x i s t sea,  by  o n l y under  contrast,  ideal  forces.  conditions.  i s very unsteady.  to  addition,  amplitude the  intersecting and  and  real  frequency  ocean  wave t r a i n s  from changes i n wind  is  speed  The  some  critical  when d e e p surface  limit.  water to  energy  real time  effects. by  in different  In many  locations  and d i r e c t i o n . that can develop  d e n s i t y o f t h e wave f i e l d  Examples  waves  real  over  characterized  Wave b r e a k i n g i s a f o r m o f i n s t a b i l i t y whenever t h e l o c a l  Steady  More g e n e r a l l y ,  dispersion  from storms  then  studied i n theory,  wave t r a i n s a r e i r r e g u l a r a n d u n d e r g o d i s t o r t i o n due  shear  exceeds  o f when t h i s may o c c u r a r e  overtake  or  collide  raising  the  u n s t a b l e h e i g h t s , o r due t o s h o a l i n g i n s h a l l o w  w a t e r . W i n d s may  also  induce  shear  forces  at  the  wave  crests. B r e a k i n g waves  classified  as  in  figure  11.  b r e a k e r s u s u a l l y o c c u r o n l y on s t e e p l y  sloped  beaches  and  are  work.  Spilling  and  water,  the  spilling,  not  or  are  plunging  considered  commonly as  in  the  p l u n g i n g b r e a k e r s may o c c u r resulting  type depending  Spilling the  wave  shown  present  i n deep o r  shallow  on t h e a v a i l a b l e  surging, Surging  energy.  b r e a k e r s a r e c h a r a c t e r i z e d by a s h a r p e n i n g  crest until  the forward face begins t o c u r l  of  over.  27 The  e j e c t e d j e t i s weak a n d i m m e d i a t e l y  simply  f l o w i n g down t h e f o r w a r d  viscous  turbulence.  dramatic  are  This  plunging  succumbs t o g r a v i t y ,  face d i s s i p a t i n g  is  breakers  the  classic  which  energy  white  contain  in  c a p . More much  more  energy and a r e a b l e t o e j e c t a w e l l d e f i n e d j e t ahead of the forward on  face r e s u l t i n g  itself.  The  i n the surface completely  momentum  of  the plunging  overturning  j e t c a n be v e r y  large.  3.2 NUMERICAL SOLUTION  3.2.1  FORMULATION The  severe  analytical  restriction  theories  described  t h a t they can handle  above only  s y m m e t r i c w a v e s . To o v e r c o m e t h i s l i m i t a t i o n to  a numerical  given  initial  time-stepping  potential  consisting vertical  can  side  time  from  and  a  initial  (3.2) and (3.3) a r e i n t e g r a t e d  a boundary v a l u e problem f o r t h e  i s s o l v e d a t each time  step t o  provide  the  p a r a m e t e r s . A c o n t r o l volume i s c o n s i d e r e d  o f a segment o f t h e s e a s u r f a c e , t h e s e a b e d ,  boundaries  considered necessary  in  state  one must r e s o r t  c o n d i t i o n . The p r o b l e m i s s o l v e d a s an  field  right-hand  steady  s i m u l a t i o n o f t h e wave  v a l u e p r o b l e m where e q u a t i o n s numerically  s u f f e r the  through  and  the water column. T h i s r e g i o n i s  t o be p e r i o d i c i n s p a c e f u r n i s h i n g t h e  remaining  condition.  The  seabed i m p e r m e a b i l i t y  be  rewritten  as  the  condition,  seabed  being  equation a  (3.1),  s t r e a m l i n e of  28 a r b i t r a r y constant  The  free surface  the  material  value  kinematic  their  that fluid  ( 3 . 2 ) , can  the  =  v e l o c i t i e s , and  be  rewritten  s e e n by  the  Dft D t  Equations  (3.5)  and  determine  the  new  surface  particles the  move  according  dynamic c o n d i t i o n ,  to  equation  as ww* 2  fluid  ww*  8 7  particles,  _8 y  (3.6)  2  (3.6)  are  positions  nodes. V e l o c i t i e s are  evaluated and  at each time step  potentials  then determined  as  3B_ 3z  f o l l o w i n g the  of  (3.5)  w*  free surface  3<fr 3t  o r , as  i n terms  d e r i v a t i v e , becomes  £|  stating  c o n d i t i o n , expressed  procedure described  i n Chapter  2.  of  the  to  free  29 A in  flow  figure  3.2.2  c h a r t f o r the time  12.  CONSTRUCTION OF The  nodal  numbered  s u r f a c e nodes,  n o d e s , N3  t o N4,  are  equations  in  vertical  be  used is  for  on  here. as  Similarly, the  on  unknown  the  e q u a t i o n s are s e l e c t e d here. A set of the  N  unknowns. An  r e c o g n i z i n g t h a t 1 and N4  allowing  equations  3.2.3  used are g i v e n  INITIAL  Initialization  The  for  latter  potential  N2  of  of  two  i n Appendix  4>(x)  as  the r e a l  <p  0  is  imaginary  equations  results  be made h o w e v e r  unknowns.  is  equation  side  the  the  The  N3  by and  actual  V.  CONDITIONS the  simulation  the s u r f a c e p o s i t i o n i s achieved  When  a r e t h e same p o i n t a s a r e  elimination  CHOICE OF  values  improvement can  imaginary  boundary,  and N  seabed  h o w e v e r , have  righthand  quantity  real  imaginary  points.  lefthand  free  the  t h e r e a l and  these  the  the  boundaries,  t o be t h e unknown q u a n t i t y and  considered  for  and  with  13. The  are s e l e c t e d here, w h i l e the  The  0  wavelength,  figure  0 a s unknowns, h e n c e b o t h  collocation point z  i s taken  shown  one  have known v a l u e s of \p  must  considered  of  h a v e known v a l u e s of <t> and  (2.4)  chosen.  b o t h <t> and  as  1 t o N2,  of e q u a t i o n  parts  MATRIX  c o n t r o l volume c o n s i s t s  points  parts  s t e p p i n g a l g o r i t h m i s shown  indirectly  there  is  a  T J ( X ) and by  requires  starting  surface  velocity.  s p e c i f y i n g the  one  t o one  velocity  correspondence  30 between  <j> a n d  w on t h e s u r f a c e . S e l e c t i o n o f <t> t o m a t c h a  given v e l o c i t y d i s t r i b u t i o n , problem  requiring  v e l o c i t y or t r i a l  either  however, would integration  of  an  the  tangential  PROCEDURE  Several standard numerical procedures equations  were c o m p a r e d f o r  (3.5) and (3.6) w i t h r e s p e c t t o time.  The p r e l i m i n a r y v e r s i o n o f t h e s i m u l a t i o n p r o g r a m single  step  E u l e r method w h i c h  at the present simple  scheme  step to predict yielded  methods were t h e n second  order  satisfactory  method  of  the  values  derivative actual  are  again  values  beginning  [ 2 ] . This  of  step forward  + *  =  A  '  =  2  Z  n+1  <t> r  n+l  -  An  l  At  n  4> + [*' *n  At  + \»  n  *n+l  *  a  calculates to  make  an  average  of  these  At i s used t o take t h e  Specifically,  *'  them  with  ^ i  and a f t e r  i n time.  method  order  a n d <t>. a f t e r A t where t h e  z.  calculated.  before  values  r e s u l t s . Higher  I  derivatives  the  t h e new f u n c t i o n v a l u e s . T h i s  t e s t e d f o r comparison  Heun  used  uses time d e r i v a t i v e  d e r i v a t i v e s a t t h e c u r r e n t time and u s e s estimate  inverse  and e r r o r .  3.2.4 TIME STEPPING  integrating  be  + I1 *n  J  *  2  At  31 where  s u b s c r i p t n i n d i c a t e s c u r r e n t v a l u e s and s u b s c r i p t n+1  refers  to  the  new  values  scheme p r o v i d e s a s l i g h t Euler  method  of both  improvement over  and r e s u l t s  the  single  i n a very stable solution as  step  i n terms discussed  section.  To t e s t then  t h e a d e q u a c y o f t h e two s t e p scheme carried  out  using a fourth order  p r e d i c t o r - c o r r e c t o r method steps  surface nodal p o i n t s . This  s m o o t h n e s s and s t a t i o n a r i t y o f e n e r g y  in a later  were  of  are  [ 2 ] , Values  from  test  Adams-Moulton  three  previous  r e q u i r e d a n d new v a l u e s a r e p r e d i c t e d a s a  approximation. correction  New  made  to step forward.  *n+l  =  < *n 55  n-  (55w  24  *n *n+l  =  +  z  n+1  n  + w'  a  9w  n  n+1.  =  T ()>  z'.,  =  z  n  +Y | i>  + 37  *n-2  . + 37w  n-1  " *n-3> 9  , n-^  9w  n J-)  At  n+1  h ^ n+l  o>' T  n+1  and  At  9  =  Vl  5 9  59w  *n+l - 1 4 <*n+l n+l  calculated  t o o b t a i n the a c t u a l time d e r i v a t i v e s used  14  z'  w  then  first  Specifically,  n+1  *n+l  d e r i v a t i v e s are  runs  *n ' V l  + 19  +  19w  5  +  •«-2  )  n ' n-l n-2> 5w  +  At  n+w n At  The r e s u l t s o f t h e s e c o n d o r d e r method have been f o u n d  t o be  32 virtually  indistinguishible  m e t h o d . The only  latter  possible  be  a d v a n t a g e of  two  step  are  s i m u l a t i o n as  3.2.5  stability  computational  criterion.  not  necessary.  s u c h r e q u i r e s two  SEGMENT S I Z E AND  f o r t h e e l e m e n t s . Too  the  time  As  higher  the  order  in  the  matrix s o l u t i o n s .  TIME STEP  few  the  segment s i z e s e l e c t e d  elements r e s u l t  i n poor r e s o l u t i o n  t i m e and  the  risk  of  numerical  in excessive instability.  optimum number of e l e m e n t s s e l e c t e d d e p e n d s somewhat  on  situation. A  non-breaking  c u r v a t u r e can surface  be  points  wave  simulated f o r one  notably  at  the  wave  calculated velocities breakdown. B r e a k i n g  with  relatively  reasonably  proceeds,  migrate  surface few  this  as  crest.  Excessive  errors  f o l l o w w i t h subsequent  30  results  r e g i o n s of h i g h c u r v a t u r e ,  most  in  the  instability  and  waves h a v e r e g i o n s of h i g h c u r v a t u r e  and  t h e r e f o r e r e q u i r e more e l e m e n t s . F o r t u n a t e l y , to  low  w e l l w i t h as  w a v e l e n g t h . Fewer t h a n  in u n d e s i r a b l e cusps i n the  tend  time  that  step  l a r g e e r r o r s w h i l e t o o many e l e m e n t s r e s u l t  computational The  and  the  is dictated  I t i s concluded  Each  P r o g r a m p e r f o m a n c e d e p e n d s on  and  order  time.  s e c t i o n , h o w e v e r , At  i n t e g r a t i o n scheme i s s u f f i c i e n t  techniques  fourth  i n c r e a s i n g the p e r m i s s i b l e  reducing  shown i n t h e n e x t  by a n u m e r i c a l  of the  r e q u i r e s more p r o g r a m m i n g e f f o r t w i t h  step i n t e r v a l At, thereby will  from those  into  automatically  the  nodal  points  c r e s t r e g i o n as the s i m u l a t i o n  providing  better  resolution  here  33 where i t i s n e e d e d . surface  elements.  especially  resolved  can  due  density over  may  migration  require  i n the c r e s t region  problem  up  a  would  crest  t o 100  t i p was The  the  more  be u s e f u l  being  number  becomes  sparse  the  poorly  this  area.  surface  elements.  elements,  however,  in  nodes  fluid  element crossing  domain  solution  trough  and  to  this  density  region.  This  i f the d e t a i l e d s t r u c t u r e of the  examined. of  elements  recommended f o r t h e  vertical  d e p e n d s on t h e w a t e r d e p t h . F o r d e e p w a t e r waves a t  15 s h o u l d  20 f o r h i g h e r using  increases.  be u s e d  i f elements are uniformly  waves. T h i s  number c o u l d  progressively On  sufficient  the  for  larger  seabed  deep  water  20  elements  are  required  with  number o f e l e m e n t s ,  large  amplitude shallow  up t o  180.  be  waves.  30  In  become being  therefore,  water breaking  spaced, or  reduced  elements  elements  h o w e v e r , where s e a b e d v e l o c i t i e s  greatest  and  as the i n c r e a s i n g  result  60  needed,  t o remove e l e m e n t s f r o m t h e h i g h  to  would  be  away f r o m  multiply-connected  be  region  boundary  may  may  about  jet i s thin  breakdown of the s i m u l a t i o n . A  procedure  by  more  wave  a d d i t i o n a l problems  creating  least  the  to nodal point  to  immediate  jet  under  require  i n c r e a s i n g t h e number o f s u r f a c e  lead  crest  Occasionally  surface  situations  Simply  waves  i n c a s e s where t h e p l u n g i n g  elliptical  Such  Most b r e a k i n g  as  slightly the  depth  usually  proves  shallower  water,  significant,  more  used t y p i c a l l y . are  required  waves where N may  The for be  34 Once e l e m e n t  s i z e s have been  i n t e r v a l At i s s e l e c t e d criterion  which  determined the time  a c c o r d i n g t o the Courant  states  that  a  particle  step  stability  should  not  p e r m i t t e d t o move a d i s t a n c e g r e a t e r t h a n a p p r o x i m a t e l y element  be the  s i z e . T h i s c o n d i t i o n c a n be e x p r e s s e d r o u g h l y a s  if <  c  or .  At  where Ax i s t h e e l e m e n t t h e wave.  A  Ax  s i z e and c i s t h e phase  convenient  time  following this criterion  based  One f u r t h e r d i s a d v a n t a g e  3.2.6  increase  interval  on t h e i n i t i a l  o f t o o many  s u r f a c e a p p a r e n t now i s t h e the r e s u l t i n g  step  velocity is  of  selected  element  size.  e l e m e n t s on t h e  free  s m a l l e r time s t e p r e q u i r e d  and  i n computation time.  NUMERICAL ADJUSTMENTS S e v e r a l c h e c k s a n d a d j u s t m e n t s o f e l e m e n t s must be made  at  each  time  step  to  s i m u l a t i o n . The v e r t i c a l  ensure  smooth  nodes a r e  execution  fixed points,  of  however,  t h e s u r f a c e c o r n e r n o d e s 1 a n d N2 a r e f r e e t o move a n d t e n d t o s t r e t c h and compress resulting problem  i n poor a c c u r a c y i n t h i s  the  e l e m e n t s on  r e d i s t r i b u t e d at boundary  t h e uppermost  length  each time by  s t e p by  t h e number  of  vertical  boundary dividing  are  the  e l e m e n t s on  will  element  r e g i o n . To o v e r c o m e  the v e r t i c a l  the  this evenly  vertical  the  side.  35 Element will  s i z e here  usually  t h e n becomes (d+rj)/NV.  result  F a i l u r e t o do  i n "sawtooth" i n s t a b i l i t i e s  so  developing  on t h e s u r f a c e n e a r t h e e d g e s . A similar excursion of  problem  results  the surface  from  excessive  corner points.  horizontal  The s i t u a t i o n  is  compounded by n o n l i n e a r e f f e c t s c a u s i n g a g r a d u a l d o w n s t r e a m migration  of  surface  c o n t r o l volume w i l l of t h e r e g i o n readjusted closest  particles.  d i s t o r t as  i s made a t  If  left  shown i n f i g u r e  each s t e p and  the  vertical  boundary.  simulation has  not  i s very been  3.2.7  rare unstable  stable  found  Occasionally, altering correct  check  the corner  Surface  i n d i c e s a r e incremented i f such a s h i f t  surface  14. A  the  points  i f necessary t o ensure that they a r e always those  to  The  unchecked  initial  nodal  point  i s required.  and smoothing  necessary element  of the  i n most size  cases.  slightly  will  situations.  ENERGY A wave c o n t a i n s k i n e t i c  potential Under t h e  e n e r g y due  e n e r g y due t o f l u i d m o t i o n  t o displacement of  assumption of  zero v i s c o s i t y  the free  surface.  i n potential  t h e r e i s no m e c h a n i s m f o r e n e r g y d i s s i p a t i o n ,  and  and t h e  flow total  c o n t r o l volume energy  E  must  in  theory  a p p r o x i m a t i o n s and  t "K E  +  Ep  remain computer  constant. roundoff  Due  to  numerical  e r r o r s , however,  one  36 would  expect  a  slight in  "numerical  cause  solution  should  changes  exhibit  no  such change, t h e r e f o r e , s t a t i o n a r i t y  energy  provides  excellent  perfect  to  artificial  an  energy. A  viscosity"  assessment  of  of  the  total  simulation  accuracy. Kinetic  e n e r g y can  »  be  « K  which,  by  invoking  numerically  expressed  £.  / / 2 V  a  1  Green's  as  dV  theorem,  can  be  evaluated  as N2 1=1  P o t e n t i a l energy i s given  by  (3.8)  or n u m e r i c a l l y  as N2  h  -  • i i x  +  D e r i v a t i o n of t h e s e e q u a t i o n s Linear are  theory  exactly equal,  predicts while  i+i i  may  y  be  kinetic  nonlinear  k i n e t i c component i s s l i g h t l y a shift  ) ( y  +  y  i+i  2  +  y  2  )  found i n Appendix and  potential  VI.  energies  theories predict that  l a r g e r . Breaking  from p o t e n t i a l t o k i n e t i c  i  energy as  the  waves e x h i b i t  time  proceeds.  37 3.2.8 COMPUTER SOLUTION The  wave  simulation  library  developed  consists  p r e p r o c e s s i n g programs f o r g e n e r a t i n g the v a r i o u s wave  initial  conditions  which a r e then passed  s i m u l a t i o n p r o g r a m . The p r i m a r y numerical suitable for  values  f o r each  output  time  output  of  t o t h e main  provides a l l  step i n formatted  f o r a n a l y s i s . A secondary  hard-copy p l o t t i n g  file  types  of  file  can  tables  be  used  o f t h e waves o r a n i m a t e d d i s p l a y on a  graphics terminal. All  p r o g r a m s were w r i t t e n i n FORTRAN, a n d c o m p i l e d  r u n on a VAX 11/750. M a t r i x s o l u t i o n s standard  Gaussian  elimination  v a r i a b l e s used throughout. numbers single  of  were  with  Required  obtained double  CPU t i m e s  e l e m e n t s a r e shown i n f i g u r e  for  and  using  precision different  15. CPU t i m e  fora  s t e p i n c r e a s e s r o u g h l y as N , however, as N i n c r e a s e s 2  a s m a l l e r A t i s r e q u i r e d so CPU t i m e a c t u a l l y  i n c r e a s e s by a  power a p p r o a c h i n g  breaking  N . S i m u l a t i o n of a t y p i c a l 3  w i t h N=120 r e q u i r e s a b o u t f o u r  wave  hours.  3.3 RESULTS The tool  s i m u l a t i o n procedure  for analysing  the  described provides a  behaviour  of  arbitrary  powerful nonlinear  waves u n d e r t h e a s s u m p t i o n s d e s c r i b e d p r e v i o u s l y . To t e s t order  wave  the accuracy was  chosen.  of the s i m u l a t i o n a Linear  fifth  wave t h e o r y p r e d i c t s t h a t  f l u i d m o t i o n s s h o u l d be n e g l i g i b l e b e l o w a d/L = 0 . 5  Stokes  which i s considered the t r a n s i t i o n  depth  of  between  about shallow  38  and  deep  water. Since  Stokes theory  i s valid  only  w a t e r , t h e s e a b e d was p l a c e d a t a d e p t h o f d/L negligible  fluid  motions  were c o n f i r m e d .  f o r deep  =  0.6  The h e i g h t  ratio  was s e l e c t e d t o be H/L = 0.06 w h i c h c a n be s e e n f r o m 10  to  l i e well  theory.  within  the  As s u c h , one w o u l d  steadily  expect  w i t h no d e f o r m a t i o n  r e s u l t s of a s i m u l a t i o n  r e g i o n of v a l i d i t y the  wave  over time.  carried  16 shows t h e NV=15,  NB=20,  L=100 f e e t , a n d A t = 0.05 s e c . The r e s u l t i n g  surface  profile  after  on  initial  wave.  The  accurate  showing l i t t l e  using  translate  NS=60,  one wave p e r i o d  out  figure  of Stokes  to  Figure  and  i s shown s u p e r i m p o s e d  simulation  in  t h i s case i s remarkably  distinguishible difference after  p e r i o d . F o l l o w i n g t h r e e c o m p l e t e wave p e r i o d s  t h e wave  showed n e g l i g i b l e d i s t o r t i o n . T o t a l  volume  exhibited the  fluctuations  trajectory  over  of  time  of  marked f l u i d  a  a  drift  i n a net migration  d i r e c t i o n . Net e x c u r s i o n  control  l e s s t h a n 0.1%. F i g u r e  representing results  the  typical  particle.  17 shows  nodal  at the surface  in  the  still energy  point  The n o n l i n e a r  of f l u i d  one  Stokes  downwave  i n t h i s c a s e was x/L  = 0.036 a n d i n c r e a s e d w i t h wave h e i g h t . Generation condition  that  selection  is  satisfy  this  particular with Such  of  a  breaking  is  unstable.  somewhat  wave As  arbitrary,  criterion.  To  be  requires  discussed and  many  specific,  c l a s s has been chosen o f a c o s i n e  <j> f r o m l i n e a r waves  cannot  theory  steady  in  form  initial  earlier  this  waves  would  however, surface  a p p l i e d a t the exact  remain  an  a  profile  free  surface.  and  w o u l d be  39 expected  t o break i f given  sufficient  initial  height.  A d e e p w a t e r c o s i n e wave was r u n w i t h H/L this  case  0.06.  In  n o n l i n e a r e f f e c t s were q u i t e s m a l l a n d a f t e r  two  wave p e r i o d s increase initial  the  only  to  horizontal  the  initial  the  simply  0.10  produced  of the f l u i d  a  spilling  h e r e was a b o u t 0.04 g  s m a l l and  fluid  in  down t h e f o r w a r d  little  i n the forward  the  incipient  jet  f a c e o f t h e wave u n d e r  computation  i s not p o s s i b l e as the nodal  i n s i d e t h e c o n t r o l volume  changed  Acceleration  i n f l u e n c e of g r a v i t y . Further  point  slight  18. F l u i d p a r t i c l e v e l o c i t i e s a t  condition.  flow  a  surface. Increasing the  a c c e l e r a t i o n -0.22 g, h a v i n g  d i r e c t i o n was v e r y would  =  was  t h e p h a s e s p e e d a r o u n d t h i s t i m e . The  acceleration  the v e r t i c a l  from  H/L  a s shown i n F i g u r e  t h e wave c r e s t r e a c h e d  and  change  i n t h e slope of the forward wave h e i g h t  breaker  discernible  =  beyond  this  points at the crest  producing  a  multiply  fall  connected  region. The are  r e s u l t s o f a d e e p w a t e r c o s i n e wave w i t h H/L = 0.13  shown  i n Figure  19. The i n i t i a l  energy  much h i g h e r a n d a w e l l d e f i n e d p l u n g i n g simulation  in  that presented initial in  this  breaker  r e s u l t s . The  case looked q u a l i t a t i v e l y  identical to  i n Vinje  and  Brevig  (1980)  c o n d i t i o n , h o w e v e r , no n u m e r i c a l  t h e i r work. At t h e time  vertical  the  i n t h i s case i s  horizontal  of fluid  the  the  same  r e s u l t s were  given  forward  for  face  a c c e l e r a t i o n near the c r e s t  was 0.58 g w h i l e t h e v e r t i c a l a c c e l e r a t i o n was -0.76 the  j e t became w e l l d e v e l o p e d ,  becoming  g.  As  the h o r i z o n t a l a c c e l e r a t i o n  40 at  the  t i p dropped  to  near  zero  while  the  a c c e l e r a t i o n a p p r o a c h e d -0.98 g  characteristic  gravity  t h e same i n i t i a l  j e t . For  comparison,  were r u n i n s h a l l o w w a t e r  using  breaker  is  of  remarkably  H/L similar  spilling  is  water depth. water  =  a  0.10  d/L  shown  phenomenon  .water c o s i n e  breaker,  initial  a  nodal  sharpening  initial  21  cannot  A  spilling 20  and i s  for  that  sensitive to the  shallow fluid  small  amplitude  of the c r e s t over  height  22.  of  ratio  is  increased,  to plunging  Simulations  several  behaviour  terminate  when  be r e s o l v e d i n t h e j e t t i p r e g i o n due t o  point crossover  o r when t h e p l u n g i n g  jet  touches  the  face.  F i g u r e 23 velocities  height  shallow  water  velocities  shows  the  time  course  r a t i o s . F i g u r e 24 shows t h e waves  of  depth  ratio  maximum  height  ratios.  Breaking  fluid  plunging  breakers  =  thing 0.25.  greater  for Fluid  f o r the  wave j e t v e l o c i t i e s c a n  approach t w i c e the phase speed of t h e the  same  d/L  i n t h e s h a l l o w w a t e r waves a r e  same i n i t i a l  For  of  on t h e s u r f a c e f o r d e e p w a t e r waves o f i n c r e a s i n g  initial  wave.  conditions  velocity.  from s p i l l i n g  a s shown i n f i g u r e  forward  pure  the j e t r e s u l t e d i n a greater  transition  velocities  figure  c o n d i t i o n s . Waves  wave p e r i o d s . As t h e  occurs  in  a  wave s i m u l a t i o n s h a v e been r u n u s i n g t h e d e e p  only a gradual  however,  0.25.  and not v e r y  volume e j e c t e d a t a s l i g h t l y h i g h e r  exhibit  =  As c a n be s e e n i n f i g u r e  Numerous  of  t o t h e deep water case d e m o n s t r a t i n g  local  plunging  vertical  corresponding  t h e s e maximum  linear  velocities  41 tend jet  to  occur  on t h e a d v a n c i n g t o p s u r f a c e j u s t a b o v e t h e  t i p and a r e d i r e c t e d  figure  almost  2 5 . F o r t h e c l a s s o f waves s t u d i e d ,  indicated  in  indicate  figures  limits  23 a n d 24 a r e n o t r e a l l y w e l l  defined  where f u r t h e r  27.  component o n l y s l i g h t l y As t h e wave potential as  breaks,  initial larger  however,  i s not p o s s i b l e  breakers  is  increase  a  near t e r m i n a t i o n  in the j e t region.  due  H/L  a  =  26 a n d kinetic  component.  transfer  from  increasingly  rapid  j e t f o r m s . The t o t a l e n e r g y  c o n s t a n t t h r o u g h o u t most o f t h e s i m u l a t i o n slight  has  than the p o t e n t i a l there  of  in figures  condition  t o k i n e t i c e n e r g y w h i c h becomes  the plunging  jet  breakers reaching the  d e e p a n d s h a l l o w w a t e r a r e shown  At t = 0 the c o s i n e  plunging  resolution.  The e n e r g y h i s t o r i e s f o r p l u n g i n g in  The  computation  to problems i n s p a t i a l or temporal  0.13  in  breaking  period.  r o u g h l y a t r a n s i t i o n between t h e  point  seen  usually  t o u c h i n g t h e f o r w a r d f a c e and s p i l l i n g critical  as  breaking  o c c u r s i n l e s s t h a n one wave  but  horizontally  remains  nearly  showing a t y p i c a l  due t o i m p e r f e c t  resolution  Figure 9 .  D e f i n i t i o n o f wave v a r i a b l e s .  Figure 10. Regions o f v a l i d i t y o f wave theories (27).  SURGING  Figure 11.  C l a s s i f i c a t i o n of breaking waves.  44  GENERATE ELEMENTS  INITIAL CONDITION  SET UP BOUNDARY CONDITIONS  SOLVE MATRIX  CALCULATE VELOCITIES  *•¥•  TIME STEP t • t + At  it  CALCULATE Dt  Dt  CALCULATE NEW . SURFACE  z  *  STOP  Figure 12. Flow chart o f wave simulation algorithm.  45  $ known  N3  N4 i|)  Figure 13.  »o  Definition of wave control volume.  ( a )  (b)  Figure 14.  Two types of control volume distortion requiring nodal point adjustment: (a) horizontal d r i f t , and (b) vertical stretch.  200  Ld  ±1  100  0  0  50  100  150  NUMBER OF ELEMENTS  Figure 15.  CPU time per step versus number of elements for wave s i m u l a t i o n .  200  F i g u r e 16.  T r a n s l a t i n g S t o k e s f i f t h o r d e r wave o f H/L = 0.06 s i m u l a t e d f o r one p e r i o d and s u p e r p o s e d on I n i t i a l c o n d i t i o n . V e r t i c a l s c a l e has been d o u b l e d f o r c l a r i t y .  48  0.10  <J1 o  F i g u r e 20.  S h a l l o w water s p i l l i n g b r e a k e r s i m u l a t e d from c o s i n e i n i t i a l c o n d i t i o n H/L = 0.10. Depth r a t i o d/L = 0 . 2 5 .  0.10  r  0.00  -0.10 0.5  X/L  F i g u r e 21.  S h a l l o w water p l u n g i n g b r e a k e r s i m u l a t e d from c o s i n e i n i t i a l c o n d i t i o n H/L = 0.13. Depth r a t i o d/L = 0.25.  Figure 22.  Last simulated step f o r deep water waves o f varying height r a t i o s showing t r a n s i t i o n from s p i l l i n g to plunging breakers.  2.5  0.4  0.5  TIME F i g u r e 23.  o.6  0.7  0.8  0.9  t/T -pi  Maximum f l u i d p a r t i c l e v e l o c i t i e s v e r s u s time f o r deep water waves from c o s i n e i n i t i a l c o n d i t i o n s . V a r i a b l e s a r e n o r m a l i z e d w i t h v a l u e s from l i n e a r t h e o r y .  2.5  2 .  TIME  t/T tn tn  F i g u r e 24.  Maximum f l u i d p a r t i c l e v e l o c i t i e s v e r s u s time f o r s h a l l o w water waves from c o s i n e i n i t i a l c o n d i t i o n s . V a r i a b l e s a r e n o r m a l i z e d w i t h deep w a t e r l i n e a r t h e o r y .  F i g u r e 25.  F l u i d v e l o c i t i e s around deep water p l u n g i n g j e t f o r c a s e H/L = 0.13. CTi  57  1.2  P  0  0.1  0.2  0.3  TIME  F i g u r e 26.  0.4  0.5  0.6  0.7  0.8  t/T  C o n t r o l volume e n e r g y v e r s u s t i m e f o r deep water b r e a k i n g wave from c o s i n e i n i t i a l c o n d i t i o n H/L = 0.13.  58  1.2  r  TIME  F i g u r e 27.  t/T  C o n t r o l volume e n e r g y v e r s u s t i m e f o r s h a l l o w water b r e a k i n g wave from c o s i n e i n i t i a l c o n d i t i o n H/L = 0.13. Depth r a t i o d/L = 0.25.  4. FLOATING BODY MOTION  4.1 INTRODUCTION The  formulation  stepping  of  a  general  and  in  arbitrary  capsizing could  would  dimensional  body m o t i o n s i m u l a t i o n i s p r e s e n t e d .  such a s i m u l a t i o n would permit assessed  two  allow  extreme s h i p  steep  be  Development of motions  to  be  n o n l i n e a r w a v e s . W a t e r on d e c k  modelled,  breaking  time  wave  and  forces  further  against  extensions  the h u l l  t o be  included.  4.2 NUMERICAL SOLUTION  4.2.1  FORMULATION The  modified piercing figure  control  volume  used  to include  an  arbitrary  body  in  the  contour  28. The d o m a i n i s a g a i n  free surface discussed  f o r t h e wave s i m u l a t i o n i s  satisfying  two-dimensional  surface  o f i n t e g r a t i o n a s shown i n  considered  the kinematic  periodic with  and dynamic  the  conditions  previously  3t  *  +  <3x' U'  ~  C  f  [  (  | i  )  2  +  (  3y"  | ±  )  (4.1) 2  ]  +  g  n  =  f  (  t  )  (4.2)  The  stream f u n c t i o n d i s t r i b u t i o n  integrating  on t h e body c a n be f o u n d by  the normal v e l o c i t i e s along  59  the surface g i v i n g  60 t  Details  =  -  u (y-y )  ~2~+  G  are described  -  G  in  vix-x )  Appendix  *  (4.3)  o  VII.  The  of  is arbitrarily  set  \jj c a n  t o z e r o . The  seabed i s a g a i n a s t r e a m l i n e , however, i n  the  0  value  cannot  v a l u e and  constant  integration  case  t a k e on any  +  G  be c h o s e n a r b i t r a r i l y  f i x e d a b o v e and  \p h e r e must be d e t e r m i n e d  solution.  matrix  constructed lumped single  The such  together  part  was of  the  the complex p o t e n t i a l  t h a t t e r m s i n v o l v i n g \p on appropriately  since  and  the  seabed  is are  4> h e r e i s c o n s i d e r e d  a  unknown.  Solution  of t h e r e s u l t i n g b o u n d a r y v a l u e p r o b l e m  t h e C a u c h y method which  for solving  as  this  are  provides  velocities  used t o s t e p the  along  the  using  contour  f r e e s u r f a c e n o d a l p o i n t s as  in  t h e wave s i m u l a t i o n  D<ft Dt  The  pressure  Bernoulli's  ww* 2  gy  d i s t r i b u t i o n on t h e h u l l  i s found  by  applying  equation  p  where  _ ~  =  +  b<t>/dt r e m a i n s t o be d e t e r m i n e d .  t o use a b a c k w a r d s f i n i t e  gy)  The  (4.4)  obvious  d i f f e r e n c e approximation,  choice i s however,  61 V i n j e and B r e v i g  (1981b) c l a i m  s u c h a scheme i s u n s t a b l e a n d  a n o t h e r t e c h n i q u e must be u s e d . An a l t e r n a t i v e m e t h o d c a n be o b t a i n e d by r e c o g n i z i n g t h a t functions,  then  <j> a n d  since  3/3/3t i s a l s o an a n a l y t i c  function  Cauchy  i s again  method  f a s h i o n a s f o r /3 u s i n g the f r e e  surface  applied  d<j>/bt i s f o u n d  harmonic  consequently,  i n the f l u i d  the time d e r i v a t i v e s  3$ 3t  and  are  must be d<f>/dt a n d b\p/dt,  so  integral  ^  domain.  The  i n an  identical  in this  c a s e . On  from  WW*  2  8  7  on t h e body 3\/>/3t may be f o u n d by d i f f e r e n t i a t i n g  (4.3)  giving  It  =  ( y  " G y  ) a  x -  ( x  " G y x  ) a  "  + u v - v u + [(x-x )(u -u) + ( y - y ) ( v - v ) ] e ( 4 . 5 ) G  The from  seabed  G  G  G  G  h a s c o n s t a n t b u t unknown 3i///3t  t h e s o l u t i o n . The r e s u l t  which the pressure d i s t r i b u t i o n Body f o r c e s a n d moment a b o u t  F  =  yields can  - / P n ds  -  J  s  out  3#/3t on t h e body  from  found  G a r e found a s  8  M =  comes  be  P rxn ds  which  G  using  (4.4)  62 which a r e evaluated F  -  n u m e r i c a l l y as -  N6-1 P -P Z 1( , ) i=N5 1  (z,  +  1  -0  i Nj =  D e r i v a t i o n s a r e found i n Appendix then determined  VIII.  Accelerations  are  from the equations of motion F _x m  _ x  F a  y  "  e  =  =  m  M T  which can then be i n t e g r a t e d heave, surge and r o l l  twice  in  time  to  find  the  motions.  4.2.2 THE CLOSURE PROBLEM The  technique o u t l i n e d above s u f f e r s a c l o s u r e  problem  in that the a c c e l e r a t i o n s a , a^ and 69 i n (4.5) a r e unknown x  a  priori  and  Fortunately considered  must  Cauchy's as  components as  being  come  out  theorem composed  as is of  part linear four  of  the s o l u t i o n .  and 3/3/9t can be p h y s i c a l l y unreal  63 The  problem  solutions  is for  schematically surface  then  decomposed  the  four  i n f i g u r e 30.  into  four  contributing  independent  terms  as  The b o u n d a r y c o n d i t i o n s  shown on  the  are  • !£>2  -  (  0  w h i l e t h o s e on t h e body a r e  ~  V U+  (!£>!  -  (|t)  - (y-y )  ,3K  _  2  G  V  9R  p  where  +  P  [(^-^ )(v  u)  G  +  (y-y )(v -v)]e G  2 x a  +  2  2  ;,  The body p r e s s u r e d i s t r i b u t i o n  P - i  G  G  at»  v  U  P  3 y a  +  (4.4) c a n be w r i t t e n a s  V  G  64 F o r c e s a n d moments a r e t h e n c a l c u l a t e d problem  using  equations  e q u a t i o n s of motion F i  F yi M  These  three  (4.7) which +  x  1  (4.5),  x  for  and  each  individual  substituted  into the  t a k e on t h e f o r m  F a + F a + F 6 = m a 3 y <* x  x  x  x  2  + F a y  F a + F 6 - W y y y*  x  2  =  ma y  3  + M„a + M a + M 6 2 x 3 y »•  =  3  equations  I e  c a n be s o l v e d f o r t h e t h r e e unknown  a c c e l e r a t i o n s a , a a n d 6, t h e r e b y c l o s i n g t h e p r o b l e m . x y Given motions finite  accelerations  are determined difference  a t each  time s t e p , v e l o c i t i e s and  by s u c c e s s i v e i n t e g r a t i o n s . A Euler  scheme  is  used  integrations U  n+1  =  U  n  +  a  x  A  t  n + ay - v_ _ At  e  - e + e At  n+1 x  y  e  n a  = x + u At + [At] n+1 n n 2 ^  n+l  n+l  ;  y  =  =  e  +  V  n  n  +  n  A  °  t  +  A n  t  +  f (  A t  )  2  2  simple  f o r these  65 A f l o w c h a r t f o r t h e body m o t i o n s i m u l a t i o n i s p r e s e n t e d figure  4.2.3  31.  BODY POSITION The  in  body g e o m e t r y i s s t o r e d a s a s e t o f p o i n t s  a ship coordinate  s y s t e m (x , y  w i t h the o r i g i n a t G as  ) attached  shown i n f i g u r e 29.  to  defined  the  the  roll  8  angle  measured c o u n t e r c l o c k w i s e  vertical.  These t h r e e v a r i a b l e s a r e  allowing  the l o c a t i o n of s h i p n o d a l  in  t h e g l o b a l frame u s i n g t h e  x = x  y = x  The  intersection  s  cosB - y  sin6 + y  points  body a r e d i f f i c u l t of  s  s  s  on to  e a c h s i d e and keep  wetted on  the  the  the  p o i n t may  enter  the  to the p o s s i b l e presence purposes  in a later  approximate  section.  l o c a t e d , nodes a r e p l a c e d  s u r f a c e changes throughout the  surface nodal  step  f r e e s u r f a c e and  same number of e l e m e n t s NH  fluid  the  x„ G  for practical  of  from  p o i n t s t o be c a l c u l a t e d  o t h e r s a d d e d o r removed a t e a c h  direction  y)  cos6 + y_ G  between  i n t e r s e c t i o n p o i n t s are  (x,  known a t e a c h t i m e  sin6 +  m e t h o d s must be u s e d a s d i s c u s s e d the  is  following transformation  t o d e t e r m i n e due  s i n g u l a r i t i e s h e r e and  body  Body p o s i t i o n  s p e c i f i e d by t h e l o c a t i o n o f G i n t h e g l o b a l f r a m e and  in  motion  on  Once there  time  step  the h u l l as  simulation.  the  Depending  near the h u l l ,  the body. T h i s c o n d i t i o n  a free must  66  be c h e c k e d a t e a c h t i m e s t e p a n d i f n e c e s s a r y t h e point point  replaced  outside  the  migrates excessively  large  free  surface  body.  offending  Conversely,  i f a nodal  f a r from the body, the  resulting  element  should  be  subdivided  appropriately.  4.2.4  I N I T I A L CONDITIONS Initialization  of  the  requires  surface  profile  T?(X)  position  (x , y ,  e) and v e l o c i t i e s ( u , v , 8).  Q  Q  and  simulation  a  free  p o t e n t i a l 0 ( x ) , a s w e l l a s body G  There  Q  are  c o n s e q u e n t l y s i x f r e e p a r a m e t e r s t o s p e c i f y when p l a c i n g t h e body on t h e wave. U n l i k e initial surface body  conditions  t h e wave s i m u l a t i o n ,  cannot  be c h o s e n a r b i t r a r i l y .  v e l o c i t y i s s p e c i f i e d by <p(s) velocity  erroneously condition  at  the  forcing fluid  free  must  match  the  points  to  and  intersection  i n t o o r away f r o m t h e  i s e n s u r e d by a d j u s t i n g  |t3s the  intersection  along the free quadratic Lc the  point  This  the p o t e n t i a l d i s t r i b u t i o n the r e l a t i o n  • % n P,  where  extrapolation  % i s the u n i t  over a region  on e i t h e r s i d e o f t h e body t o m a t c h intersection points.  avoid  body.  The <t> d i s t r i b u t i o n i s a d j u s t e d  surface.  polynomial  v  these  The  i n t h e n e i g h b o u r h o o d o f t h e body t o s a t i s f y  at  however,  the  of  vector by  a  length  derivatives  at  67 4.2.5  INTERSECTION SINGULARITIES The  intersection  surface w(z)  p o i n t s between the  present special  may  be  problems because the  singular  analytical  solution  downward on  the  free  here.  to  Yim  (1985)  a semi-infinite  surface.  square root s i n g u l a r i t y at  The  the  plate  Greenhow and  Lin  solution  impulsively  moved  predicts to  the  an  a logarithmically w a l l . T h e s e two  h e a v e and  surge  motions  is  Singularity general  of  obviously  a  behaviour  case  is  at  less  vertical surface  more  an  impulsively  the  free  clear.  Lin  s i n g u l a r i t y may Photographs  actually  show  perpendicularly peculiar  as  may  a  small  paddle at an  and  flow  next  problem. the  more  (1984) d e s c r i b e behaviour  jet  next  that  how  regions,  the  interpretation. being  ejected  intersection. of  the  general  suggesting  the  example  for  in  al  fluid  that  analogous to  surface et  a  series  elevation  physical  b r e a k down i n c e r t a i n  further  be  is  has  This  potential where  s u r f a c e t e n s i o n keep the  flow  effects  real  fluid  finite.  even i n the i>  have  moving  boundary  which  wavemaker,  very  from the  viscosity  behaviour One  a  phenomenon  m o d e l l i n g can such  moved  the  velocity  complicated  e x p e r i m e n t s which c l o s e l y examined the to  describes  edge where t h e  body,  much  velocity  (1985) d e s c r i b e a  infinite  floating  fluid  free  velocity  i d e a l i z e d cases are  a  the  flate plate  complex  becomes i n f i n i t e . to  body and  problem at  absence of  rapid  close  the  intersection  s i n g u l a r i t i e s , the t o the  points  is  v a r i a t i o n of  b o d y . M a s k e l l and  Ursell  that 4>  or  ( 1970)  68 give  a  second  shows a s t e e p  order  s o l u t i o n for a heaving c y l i n d e r which  gradient  of 0 and  high  b o d y . H i g h e l e m e n t d e n s i t y w o u l d be good r e s o l u t i o n , though i t i s not necessary  for  a  stable  element d e n s i t y can in  computational  reducing  the  numerical  stability,  c o s t . One  may  and  near the  number o f  this  further  though  sacrificed  really  excessive  problems  both  complication  b o d y . I t has  been f o u n d  elements often  resolution  the  achieve  is  Furthermore,  free surface  have t o be  is  f o r the  of that  improves  obviously  the  other.  INTERSECTION SOLUTION Several  techniques  solution  in  separate  simulation  the  were t r i e d  intersection program.  t h r e e main v e r s i o n s a r e the  same  as  node i s p l a c e d function  specified  potential  with  as  to  be  this  a  the  often  c o n d i t i o n at the  requiring  conditions  Brevig point  free surface  resolved d i d not  satisfy  and  numerical  I  was  the  as  stream velocity  such  does  boundary c o n d i t i o n . the  the the  i n t e r s e c t i o n . The  l a r g e . E x a m i n a t i o n of the  the  ( 1 9 8 1 c ) where a  showed near  a  for  Version  s o l u t i o n and  formulation  poorly  numerical  b o u n d a r y c o n d i t i o n . The  from the  v e l o c i t y p o t e n t i a l indeed boundary  each  Boundary  intersection  necessarily satisfy  Experience  region,  t h a t u s e d by V i n j e and at the  to handle the  shown i n f i g u r e 32.  p o t e n t i a l h e r e comes o u t not  clear i f  solution.  stability  near  r e q u i r e d here to  cause a d d i t i o n a l numerical  element r e d i s t r i b u t i o n  4.2.6  velocities  complex  b o d y , and free  surface  discrepency  solution  the  at  was each  69 time step  revealed  calculated  in  the  unrealistically correct  velocities  intersection  large values  regions,  that  were  were  often  not  poorly yielding  even  of  I I attempted t o reduce the s i n g u l a r i t y  displacing  true  fluid  the  sign.  Version by  that the  nodes a s m a l l d i s t a n c e  intersection  points  and  problem  on e i t h e r s i d e o f t h e  thereby  avoid  integrating  t h r o u g h t h e s i n g u l a r i t y . T h i s method d i d n o t a p p e a r t o o f f e r any  improvement.  elements  are  formulation  Since  the  included  i s really  in  small the  diagonal  contour  equivalent  integration, this  to version I with  s h i f t e d a p p r o p r i a t e l y . In the usual  formulation  considered  t o have c o n t r i b u t i o n s f r o m e a c h o f  elements.  To  avoid  integrating  however, these s m a l l contour.  elements  As s u c h , t h e m a t r i x  Version a  modelling  must  be  and  adjacent  out  of  the  f o r these s p e c i a l  the c o n t r i b u t i o n  to  0  I I I a t t e m p t s t o i m p r o v e t h e s o l u t i o n by p l a c i n g  a  wave  al  ( 1984)  maker  the matrix  equations  formulation fluid  left  coefficients  in  used a  this  basin  i m p r o v e d r e s u l t s . The number o f unknowns  actual  the  only.  ii h e r e . L i n e t  less  e a c h node i s  d i r e c t l y a t e a c h i n t e r s e c t i o n and s p e c i f y i n g b o t h <j>  node  and  t h e nodes  through the s i n g l a r i t i e s ,  p o i n t s must be r e d e r i v e d c o n s i d e r i n g f r o m one e l e m e n t  connecting  and c l a i m e d  are  given  in  greatly  accordingly.  Appendix  IX.  r e s u l t e d i n a more s t a b l e s o l u t i o n , a l t h o u g h  velocities  near  the  for  i n t h i s c a s e i s two  must be r e c o n s t r u c t e d  used  formulation  intersections  were  The This the still  questionable. In  a l l t h e above f o r m u l a t i o n s  intersection numerical  the  p r o c e d u r e and t h e v a l u e s c a l c u l a t e d a t  free surface position determined.  This  in achieving a  at  the  good  time  simulation.  For  extrapolation  immediately  adjacent  by t h e  these  two  however, t h e  step  cannot  practical  be  purposes,  by o t h e r means u s i n g  The most o b v i o u s  polynomial  the  t h e most c h a l l e n g i n g p r o b l e m  v e l o c i t i e s must be e s t i m a t e d  singularities  velocites,  next  i s certainly  s o r t of a p p r o x i m a t i o n .  the  at  p o i n t s c o u l d n o t be d e t e r m i n e d d i r e c t l y  p o i n t s were d i s c a r d e d . W i t h o u t t h e s e  these  velocities  choice  i s to  some  use  a  i n t o t h e body, however, s i n c e t h e  p o i n t s a r e a l s o under t h e i n f l u e n c e  t h e i r p o s i t i o n s are questionable  as  of  well.  F u r t h e r m o r e , s u r f a c e i r r e g u a r i t i e s can produce l a r g e e r r o r s , especially  if  the element nearest  e x t r a p o l a t i o n . A more t e n a b l e the  adjacent  nodes well  t h e body r e q u i r e s a l o n g  solution  horizontally provided  i s to into  the  simply the  extend  body.  alternative  works  excessively  s t e e p and a l l o w s t h e s i m u l a t i o n t o p r o c e e d .  steeper  waves t h r e e p o i n t s m o o t h i n g  region  next  to  must  surface  be  is  used  t h e body. R e s u l t s a r e p r e s e n t e d  in  in a  This not For the later  section.  4.2.7  A NUMERICAL PERTURBATION CORRECTION Throughout the s i m u l a t i o n s both  total  s y s t e m e n e r g y were m o n i t o r e d  stability.  the  fluid  volume  as a measure of  and  numerical  The v o l u m e was u s u a l l y q u i t e s t a b l e a l t h o u g h  the  71  total  energy often  doubt  due  largely  intersection values  the  two  of  the  e n e r g y can  be  are  the E  the are  c o n t o u r geometry linked  solution  to  i s of  (x,y)  obviously  undesirable  two  correcting  Conservation  equations  for  (3.8)  the  i t can  be  An  cast  unknown i n t e r s e c t i o n  numerically.  An  two  and the  volume  seen  the  complex  closed  positions potential  iterative  constraints  into a  trial  might  and  be  since (<t>,\p) error  possible, with  no  convergence.  alternative  perturbation"  the  form  h o w e v e r , c o m p u t a t i o n a l t i m e w o u l d become p r o h i b i t i v e g u a r a n t e e of  of  fc  and  the  for  +  (x,y)  satisfy  the  artificial  points.  ( 3 . 7 ) and  fluid  = E  f o r the  in  impose  U n f o r t u n a t e l y , t h i s e x p r e s s i o n c a n n o t be solution  no  locations.  P  form  were  accuracy  means  c o n s i d e r e d as  to equations  e n e r g y of  of to  intersection  unknown i n t e r s e c t i o n  the  need  these errors  the  These e r r o r s  problems  however, p r o v i d e a p o s s i b l e  Referring that  the  and  While  positions  mass and  to  regions  here.  t h e y do,  grew s i g n i f i c a n t l y .  is  to  consider  method where s m a l l c h a n g e s can  be  intersection  d e r i v e d as  positions.  functions  That i s ,  AV  = f ( A y , Ay ,  J^,  J^)  AE  =  * ,  i )  L  f(Ay , L  R  Ay , R  L  R  in of  a  "numerical  system  energy  s m a l l changes  in  72  where A y the  L  left  and A y  R  represent  and r i g h t  upward v e r t i c a l d i s p l a c e m e n t s  i n t e r s e c t i o n p o i n t s and l a n d 1 L  l e n g t h s of the elements next  t o t h e body. T h e s e  R  of  are the  expressions  t a k e on t h e f o r m  AV  = A(Ay ) +  AE  = C(Ay ) + D(Ay ) + E(Ay^)  L  B(Ay ) R  L  R  where t h e c o e f f i c i e n t s a r e f u n c t i o n s These e q u a t i o n s  can then  2  +  F(Ay )  of  be s o l v e d f o r A y  2  R  known  quantities.  and Ay_. D e t a i l e d  r  Li  d e r i v a t i o n s may be f o u n d procedure,  a  i n Appendix  preliminary  solution  X.  K  To  is  obtained  p r e v i o u s l y d e s c r i b e d method a n d t h e c h a n g e s energy  noted.  These a r e then  in  only  involved. adjacent  A  more  points  segments,  is  Only  the  volume  and  involving  a s w e l l w o u l d be p o s s i b l e u s i n g  polynomial  the the  correction  t o t h e body a r e  scheme  considered  self-correcting  by  constraints are  two e l e m e n t s a d j a c e n t  elaborate  however,  complicated. above  the  the  used t o c a l c u l a t e c o r r e c t i o n s  t o t h e i n t e r s e c t i o n p o s i t i o n s . A s o n l y two available,  implement  derivation first in  would  be  much  order approximation the  present  more  described  work.  s i m u l a t i o n i s r e f e r r e d t o as v e r s i o n IV.  This  73 4.3  RESULTS For  is  the purpose  examined f i r s t  water.  Upon  roll  o f a r e c t a n g u l a r body h e e l e d o v e r  release  oscillations removing  of t e s t i n g the s i m u l a t i o n a s i m p l e case  while  energy  period  the  body  should  disturbance  waves  this  undergo  the  c a s e c a n be c a l c u l a t e d  as Comstock  lb/ft,  = 30°. For  The  r a d i u s o f g y r a t i o n = 2.5 theoretical  roll  period  well-behaved  additional  complications.  and  NS=25,  avoid  following  = 3 f e e t , weight =  f e e t , and  initial  creating  III  f i g u r e r e v e a l s v e r y low d i s t u r b a n c e s  as  the  t o keep  unnecessary  with  L=100  feet,  radiating  in  this  b e h a v i o u r was  i n the i n t e r s e c t i o n  actual the  roll  seconds  during  which  increase  in roll  a m p l i t u d e of about  c y c l e . The  r e g i o n , however, as  s i m p l e c a s e were p r i m a r i l y h y d r o s t a t i c  g o o d . The  period  system gained energy five  c a l c u l a t e d motions are p l o t t e d  second a  low  was  about  over  in figure of  a m p l i t u d e wave. F i g u r e s 35 and  were the the 2.6  c a u s i n g an  degrees  t e s t case examines the motion  of  outward  body o s c i l l a t e s . As e x p e c t e d , f l u i d v e l o c i t i e s  poorly resolved  on  sec.  sec. Close inspection  the  The  angle  F i g u r e 33 shows t h e r e s u l t s o f a  NH=15, N=100, a n d At = 0.05  forces  text  f o r t h i s body i s 2.5  s i m u l a t i o n c a r r i e d out u s i n g v e r s i o n  body  i n any  p r e l i m i n a r y t e s t s t h e s h a r p c o r n e r s were r o u n d e d  velocities  The  theoretically  (1967). For the present case the  c o n d i t i o n s a p p l i e d : beam = 10 f e e t , d r a f t 1900  outward  motion.  from the s t a n d a r d f o r m u l a f o r s m a l l a n g l e s found such  roll  radiate  f r o m t h e body and d a m p i n g  in  i n calm  one  34. the  same  36 show t h e  74 results  of a motion s i m u l a t i o n  on a c o s i n e wave o f l e n g t h  = 100 f e e t a n d h e i g h t r a t i o H/L = 0.04 The  initial  placing  potential  both sides  were  modification imparted and  to  water  on  surface are equal horizontal  a n g l e was c h o s e n  arbitrarily  the  the  figure  wave  the  one  cycle.  shown i n f i g u r e  i n figure  nonlinear  considerable  body.  Energy roll  Hull  was angle  pressure  37 f o r v a r i o u s  t h e wave a n d t h e m o t i o n s f o r sway, h e a v e ,  plotted  the  underwent  passed  after  the  initial  in  crest  where  body r e s u l t i n g i n an i n c r e a s e d  deck are  roll  seen and  the  the  distributions on  be  large as  trough  g r a d i e n t s on t h e f r e e  body v e l o c i t y . The i n i t i a l  effects  the  o f t h e body a n d m a t c h t h e  t o be 15°. A s c a n  version I I I .  i n t h i s c a s e were c h o s e n c a r e f u l l y by  t h e body a t t h e m i d p o i n t o f  velocity on  conditions  using  L  and  positions roll  are  38. Of p a r t i c u l a r i n t e r e s t a r e t h e r a t h e r  large  horizontal  displacements. Accelerations  in  this  were  a s much a s 0.5 g. T h e s e c h a r a c t e r i s t i c s h a v e a l s o  case been  n o t e d i n model e x p e r i m e n t s [ 2 6 ] . The further examine the  third  t e s t case involved  t o H/L = 0.08 t o i n v o k e l a r g e the  limitations  trough with  velocity  potential  and  velocity  eventually  rolled  too  t h e wave h e i g h t  nonlinear  of t h e model. P l a c i n g  an i n i t i a l  horizontal  gradient  w a t e r on d e c k . By g i v i n g angular  increasing  the  an  and  t h e body i n  v e l o c i t y t o match  resulted body  effects  the  i n almost  immediate  additional  negative  o f -0.15 r a d / s e c d e c k w e t t i n g was d e l a y e d  came a b o u t f r o m t h e downwave s i d e f a r back.  Figures  39  a s t h e body  a n d 40 show t h e a c t u a l  75 s i m u l a t i o n . The r e s u l t i n g m o t i o n s a r e p l o t t e d One f i n a l wave  of  H/L  i n f i g u r e 41.  t e s t case  i s t h a t o f t h e body i n a v e r y  =  Undisturbed,  0.12.  this  steep  i n i t i a l condition  would r e s u l t  i n a d i s t i n c t plunging breaker  in  3, h o w e v e r , a s c a n be s e e n i n f i g u r e s 42 and 43  chapter  the presence breaking  o f t h e body m o d i f i e d t h e wave c o n s i d e r a b l y  did  not occur  and  a l l resulted  c o n d i t i o n s were t r i e d  in  w a t e r on deck  s e c o n d . The m o t i o n s f o r t h i s c a s e For  steep  waves  one  floating  converted keep next  breakwater.  This  for  expect  one  i n f i g u r e 44.  the  body t o a c t  r e m o v i n g e n e r g y much  wave  t o body m o t i o n a n d , a s a  energy, result,  like  of course, i s would  tend  t h e wave f r o m b r e a k i n g . A c h i e v e m e n t o f a b r e a k i n g t o t h e body w i l l  this  i n l e s s than  are plotted  would  somewhat a s a wave d a m p i n g d e v i c e a  and  b e f o r e w a t e r on d e c k t e r m i n a t e d t h e  simulation. Several i n i t i a l wave  a s shown e a r l i e r  r e q u i r e some  other  type  of  to wave  initial  wave c o n f i g u r a t i o n . Version  IV  self-correction  of  the  scheme  met  simulation with  u s e f u l only f o r n e a r l y calm water.  employing  limited  success  I t became  the a n d was  apparent  that  c o r r e c t i n g t h e p o s i t i o n s and boundary c o n d i t i o n s of o n l y t h e i n t e r s e c t i o n n o d a l p o i n t s was i n s u f f i c i e n t , a n d method  should  be  used  in  segments t o a l l o w t h e a d j a c e n t c o r r e c t e d as w e l l , both Having body  conjunction free  with  surface  ability  to  to  be  potential.  simulate  m o t i o n s i n s t e e p n o n l i n e a r waves, t h e next  the  polynomial  points  i n p o s i t i o n and v e l o c i t y  demonstrated the  really  extreme  s t e p was t o  76  use  the  simulation  program  e x p e r i m e n t s under v a r y i n g versions  have  numerical  problems at the  therefore  been  really  simulation required to if  [3]  initial  found  only  Furthermore, previous has  to  to  this  becomes  very  are  shown t h a t  for  a r b i t r a r y , however, the initial great  velocities  f i g u r e 45  More  simpler two  the  shows t h e  likely,  a  ship  the  of  the  subsequent  present  ask  version of  of time.  conditions  three  initial  this point  be  corresponding  For  have a  example,  angular v e l o c i t y .  to r o t a t i o n a l i n e r t i a . would have a  negative  depend  on  the  to  water  on  must  now  unsteady response h i s t o r y . It  deck  body  Therefore,  initial  angular v e l o c i t y , a l t h o u g h t h i s would r e a l l y prior  are  the  period  motions.  i m m e d i a t e due at  allow  r e l a t i v e phases w i l l  e f f e c t of z e r o  deck i s almost  linearized  p o s i t i o n placement can  of  their  periods.  wave p e r i o d s  i n a short  Initial  choice  time  i n a wave t r a i n .  selection  and  i n f l u e n c e on  W a t e r on  energy  short  roughly  induced  important.  current  i n t e r s e c t i o n s i n g u l a r i t i e s and  experience with a  i n mind, the  The  to  useful  i t must be  numerical  due  i s t o be m o d e l l e d w i t h  simulation  •Bearing  conditions.  f o r t r a n s i e n t e f f e c t s t o d e c a y and  capsizing  out  accumulate  acheive steady s t a t e r o l l i n g  the  carry  or  i s c l e a r now  c a p s i z i n g can  whether or not  to occur  that conditions  in  nature.  the  i n d e e d be  leading  simulated  chosen i n i t i a l  and  conditions  one are  likely  77 4.4  EXTENSION TO  THREE DIMENSIONS  Comparisons  made  with  a  constant  f u n c t i o n m e t h o d u s e d by o t h e r i n v e s t i g a t o r s a  breaking  Cauchy  wave  produced  Green's  [3], to  simulate  somewhat d i f f e r i n g  results.  i n t e g r a l method r e m a i n e d r e m a r k a b l y c o n s t a n t i n  energy  and  volume  and  therefore  surface problems appears Unfortunately,  because  c a n n o t be e x t e n d e d t o however,  one  may  to  .effects  would  three  be  especially  for  capsizing.  If  be  be  in  preferred  are dominated to  time  be  technique. variables i t  For  some  cases,  the  beam/length  2  2  ship"  end  significance,  involved  "slender  t e r m d <j>/b z  by t h e m i d b o d y ,  of minor  periods  addition  crossflow  both  a b l e t o make an a p p r o x i m a t i o n . S i n c e  r e a s o n a b l y low one c a n u s e a the  the  dimensions.  expected  short  The  f o r two d i m e n s i o n a l f r e e  the method uses complex  s h i p m o t i o n s and s t a b i l i t y  where  element  in ratio  ship is  approximation  in Laplace's equation i s  assumed s m a l l enough t o n e g l e c t and t h e f l o w  i s locally  two  d i m e n s i o n a l . S t r i p t h e o r y c a n t h e n be u s e d where t h e s h i p i s considered prismatic  equivalent sections  as  simulations are c a r r i e d can  then  to  a  shown  composite in figure  o u t f o r e a c h . The  summed o v e r e a c h s e c t i o n  a c c e l e r a t i o n s and  resulting  motions.  of  representative  46 and  independent  f o r c e s and moments  to obtain  the t o t a l  body  78  •  known  $  F i g u r e 28.  - constant  C o n t r o l volume f o r body m o t i o n s i m u l a t i o n .  79  Figure 30.  Decomposition of time d e r i v a t i v e s into four independent problems.  80  SET UP BOUNDARY CONDITIONS  GENERATE ELEMENTS  INITIAL CONDITION  • U  SOLVE MATRIX  V  G G V  e CALCULATE VELOCITIES  CALCULATE  Dz Dt  CALCULATE  D£ Dt  it  CALCULATE NEW SURFACE  St  CALCULATE  TIKE STEP t = t + At  CALCULATE  ' l 2 3 '» l 2 », , F  M  F  M  M  SOLVE FOR s  x  a  y  e  DETERMINE NEW BODY POSITION  STOP  gure 31.  Flow c h a r t f o r body m o t i o n s i m u l a t i o n a l g o r i t h m .  F i g u r e 32.  Boundary c o n d i t i o n s t e s t e d .  0.0  Figure 33.  Simulation of r o l l motion i n calm water.  sec  0.6  sec  1.2  sec  1.8  sec  2.4  sec  -40  TIME  Figure 34.  (sec)  Roll motion f o r calm water case.  F i g u r e 35.  S i m u l a t i o n o f m o t i o n i n wave H/L = 0.04. CD  F i g u r e 36.  S i m u l a t i o n o f m o t i o n i n wave H/L = 0.04.  86  F i g u r e 37.  H u l l p r e s s u r e d i s t r i b u t i o n s f o r H/L =  0.04.  88  TIME  F i g u r e -38.  (sec)  M o t i o n s f o r c a s e H/L = 0.04.  F i g u r e 39.  S i m u l a t i o n o f m o t i o n i n wave H/L = 0.08.  Figure 40.  S i m u l a t i o n o f m o t i o n i n wave H/L = 0.08.  92  40  -5 • -10. 101  -5' -10. 0  0.5  1  1.5  2 TIME  F i g u r e 41.  2.5  3  (sec)  M o t i o n s f o r c a s e H/L = 0.08.  3.5  F i g u r e 42.  S i m u l a t i o n o f m o t i o n i n wave H/L = 0.12.  95  F i g u r e 43.  S i m u l a t i o n o f m o t i o n i n wave H/L = 0.12.  40  F i g u r e 44.  M o t i o n s f o r c a s e H/L = 0.12.  Figure 45.  Result of zero i n i t i a l angular  velocity.  F i g u r e 46.  Equivalent prismatic representation of ship.  5. CONCLUSIONS AND A  complex  variable  used t o n u m e r i c a l l y surface  been s i m u l a t e d has  b o u n d a r y i n t e g r a l method h a s been  simulate  waves. B r e a k i n g  to  be  powerful  The  method  and  be  robust.  simulated  free  water  v e l o c i t i e s d e t e r m i n e d . The  c o n t i n u o u s smooth wave c a n c o n d i t i o n s c a n be  the b e h a v i o u r of n o n l i n e a r  waves i n d e e p a n d s h a l l o w  and p r o f i l e  proven  RECOMMENDATIONS  method  Virtually  provided  initial  extended  to  include  the  nonlinear  motions  o f a body on t h e f r e e s u r f a c e , a n d s i m u l a t i o n s  carried  out  singularities  several  at  the  prevented the d i r e c t surrounding accurate  A  step to  presence  intersection  of  velocities  of  points in  the  intersection  problem  No  have  been  perturbation  locating  the  immediately  questionable  the  This  procedure  was  adjacent  and  Local  first  order  correction  was  used  two i n t e r s e c t i o n p o i n t s . The a c c u r a c y o f  work  p r o c e d u r e more r o b u s t  points, is  needed  to allow  however, to  was  also  make t h e c o r r e c t i o n  t h e s e a d d i t i o n a l p o i n t s t o be  s m o o t h i n g o f f u n c t i o n d i s t r i b u t i o n s may  r e q u i r e d . Free surface care  correction  f o r c e t h e a d d i t i o n a l c o n s t r a i n t s o f mass a n d  for  surface  surface  The  date.  two  included.  cases.  were  l i m i t i n g a c c u r a c y of the s i m u l a t i o n .  energy c o n s e r v a t i o n .  however,  free  solutions to this  introduced  test  determination  regions,  developed to  any  assigned. was  for  have  must  s m o o t h i n g may be t a k e n t h a t  behaviour are not l o s t 99  in  be  helpful  important the  as  be  well,  d e t a i l s of the  process.  Empirical  100 information  w o u l d be u s e f u l h e r e a s a  guide.  T h e r e i s w i d e s c o p e f o r f u r t h e r work on t h e body m o t i o n problem. so  One  p o s s i b l e a l t e r n a t i v e s o l u t i o n method t h a t h a s  f a r n o t been a t t e m p t e d  an  inner  and  outer  i s to reformulate  solution.  The  i n v o l v e t h e u s u a l Cauchy i n t e g r a l clear  of  the  outer  around  problem  s o l u t i o n would  a  path  s o l u t i o n s on e a c h s i d e o f t h e body c o u l d u t i l i z e such as c o n s e r v a t i o n  energy while and  ensuring  velocities  along  the  body p r o v i d e d  common  be  means o f a p p l y i n g  idealized  j e t whose f o r c e on a f l a t  j e t to  be  momentum c o n s i d e r a t i o n s flow  Empirical surface  potential Polynomial  close  to the  the  side  of  a  the s a f e t y of ships by  an e x p l i c i t  wave  is  on  estimated  breaking  effective  complex  assumed  t h e r e c i p i e n t s u r f a c e , o r more s i m p l y  from  additional  boundaries.  i n t e r e s t i n studying  a t sea. These f o r c e s c o u l d  on  be  of  f o r c e s due t o a wave b r e a k i n g  by  inner  t h e i r c o e f f i c i e n t s c o u l d be d e t e r m i n e d .  v e s s e l a r e of great  either  is  o f mass, momentum f l u x , a n d  matched v a l u e s  d i s t r i b u t i o n s of f u n c t i o n s c o u l d  The  that  t h e s i n g u l a r p o i n t s and w e l l behaved, w h i l e  information  as  a  simulation  boundary c o n d i t i o n by c o n s i d e r i n g  approximately  equivalent  p l a t e c a n be  knowing t h e f l u i d  the t o an  calculated  v e l o c i t y and  r a t e as a f u n c t i o n of time. input  is still  required to define  the free  b e h a v i o u r near t h e i n t e r s e c t i o n s , but i s l a c k i n g . I t  recommended  dimensional  that  experiments  b o d i e s i n a wave  basin  be  carried  under  both  out  on two  small  and  l a r g e a m p l i t u d e waves s o t h a t d e t a i l e d e m p i r i c a l r e s u l t s may  101 be  obtained  as a b a s e l i n e f o r n u m e r i c a l  p h o t o g r a p h s and accurately  floating  obtain  the  velocities.  This  establishing  realistic  progress are  free  particles  surface  information  of n u m e r i c a l  initial  are  to  and  be  useful  for  assessing  the  c o n d i t i o n s and as  needed  profiles  would  simulations  Strobe  various  fluid  formulations  tried. The  layer  i m p o r t a n c e of development  d e t e r m i n e d and where  the  studies  provide  may  hull  should  elucidate  by  marker  experiments.  the  viscous and  effects  vortex  shedding  be q u i t e s i g n i f i c a n t , geometry i s not  be flow  carried  such  using  component.  to  be  e s p e c i a l l y i n cases  dye  visualization injection  to  s t r u c t u r e . Such e x p e r i m e n t s would a l s o  a u s e f u l a s s e s s m e n t of t h e v a l i d i t y  allowing  boundary  remain  smooth. Flow  out  as  observation  of  the  of  strip  longitudinal  theory  crossflow  REFERENCES  1.  Bhattacharyya,  R.  (1979)  J o h n W i l e y & S o n s , New 2.  C a r n a h a n , B.  3.  Chan,  J.L.K.  of Marine  Vehicles.  York.  (1969)  W i l e y & S o n s , New  Dynamics  Applied  Numerical  Methods.  York.  (1986)  Unpublished  results.  Dep't  M e c h a n i c a l E n g i n e e r i n g , U n i v e r s i t y of B r i t i s h Vancouver, 4.  Chan,  5.  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MIT  Press,  1 04 25. Rawson, K . J . & T u p p e r , E.C.  (1968)  Basic  Ship  Theory.  Longmans, L o n d o n . 26. R o h l i n g ,  G.F.  Fishing  (1986)  Vessel  M.A.Sc.  "Experimental  Stability  Thesis,  Dept.  Investigation  in  a  Transverse  of  Mechanical  of  Seaway",  Engineering,  U n i v e r s i t y of B r i t i s h C o l u m b i a , V a n c o u v e r , Canada. 27. S a r p k a y a , T. & I s a a c s o n , M. Forces  on  Co., New  Offshore  Structures.  Mechanics Van  of  Nostrand  Wave  Reinhold  York.  28. S c h u l t z , W.W., "Steep Symp.  (1981)  Ramberg,  and  Breaking  Naval  S.E. Deep  Hydrodynamics,  29. S c h w a r t z , L.W.  &  Griffin,  O.M.  (1986)  W a t e r Waves", P r e p r i n t Berkeley,  (1974) "Computer  16i h  California.  Extension  and  Analytic  C o n t i n u a t i o n o f S t o k e ' s E x p a n s i o n f o r G r a v i t y Waves", J. Fluid  Mech.  V o l . 62,  pp.  553-578.  30. S k j e l b r e i a , L. & H e n d r i c k s o n , Gravity  Wave T h e o r y " , Proc.  Hague,  J.  7th Coastal  G.G.  (1847)  Waves", Trans.  32. U r s e l l ,  Camb.  (1964)  Floating  "On  Water  the  Phil.  "The  Soc.  Decay  Body", /. F l u i d Mech.  33. V i n j e , T. & B r e v i g , P.  Eng.  Theory  of  V o l . 8,  pp.  the  Free  of  V o l . 19, pp.  Order  Conf.,  The  Hydrodynamic  M o t i o n of a 305-319.  Report,  Finite  Norwegian  (1981a) " N u m e r i c a l C a l c u l a t i o n of  F o r c e s f r o m B r e a k i n g Waves", Int.  Symp. Hydrodynamics  N o r w e g i a n I n s t . T e c h . , pp.  35. V i n j e , T. & B r e v i g , P. Int.  441-455.  Laboratories.  34. V i n j e , T. & B r e v i g , P. Eng.,  Oscillatory  ( 1 9 8 0 ) " B r e a k i n g Waves on  D e p t h - A N u m e r i c a l S t u d y " , SIS  Ocean  "Fifth  pp.184-196.  31. S t o k e s ,  3rd  (1960)  Conf.  in  547-565.  (1981b) " N o n l i n e a r S h i p M o t i o n s " ,  Numerical  Ship  Hydrodynamics,  Paris,  pp.  257-266. 36. V i n j e ,  T.  &  Brevig,  P.  (1981c)  D i m e n s i o n a l S h i p M o t i o n s " , S t u d y " , SIS Hydrodynamic Rev.  38. W i e g e l ,  Report,  Two  Norwegian  Laboratories.  37. Wehausen, J.V. Ann.  "Nonlinear  Fluid  R.L.  ( 1 9 7 1 ) "The M o t i o n o f Mech.  V o l . 3,  (1964)  Floating  Bodies",  pp.237-268.  Oceanographical  Engineering.  105 P r e n t i c e - H a l l , Englewood C l i f f s , Yim,  B.  N.J.  (1985) " N u m e r i c a l S o l u t i o n  f o r Two  Dimensional  Wedge Slamming W i t h a N o n l i n e a r F r e e S u r f a c e C o n d i t i o n " , Preprint,  Washington,  4th  D.C.  Int.  Conf.  Numerical  Ship  Hydrodynamics,  APPENDIX I THE CAUCHY INTEGRAL THEOREM According analytic  t o t h e C a u c h y Theorem t h e p a t h  f u n c t i o n around a c l o s e d contour  i n t e g r a l o f an  i s zero  f(z) dz  therefore -  so l o n g a s z For  0  (A1.1)  0  i s outside the enclosed  t h e purpose of numerical  region  solution z  t h e v a l u e s o f t h e n o d a l p o i n t s on t h e c o n t o u r be a l l o w e d t o a p p r o a c h C. The c o n t o u r composed  of  C, a n d C  2  where C  2  0  must  C c a n be c o n s i d e r e d a s  subtends z  0  with a c i r c u l a r  C a u c h y ' s Theorem c a n be w r i t t e n a s 106  on  a n d h e n c e must  a r c o f r a d i u s e.  In t h i s case  take  107  '  C  z-z 0  On C  d z  + c' ^ 0" Z  Z  2  1  d  " °  z  Z  (A1  ' > 2  2  2  1 6  z + ee 0  dz  so e q u a t i o n  =  16 ,„ i e e do  (A1.2) r e d u c e s t o  f -2— dz + f" — J L y i e e C,  where  a  is  Z  " 0  o  Z  the  de - o  1 6  i n t e r i o r angle at z  smooth a n d ir/2 a t a e —**• 0,  c e  1 6  corner.  Evaluating  becomes C a n d 8  |^z7  dz  =  0  1  a  ^ o> z  e q u a l t o n when C i s in  the  limit  as  APPENDIX I I FORMULATION OF INTEGRAL EQUATION The  contour  i s discretized  N elements  and t h e  values of 0(z) are c a l c u l a t e d  a t t h e nodes j o i n i n g  e a c h . The  integral  equation  t h e n be  a  equation  f o r numerical solution,  (2.2) can  As o u t l i n e d  in  written  Chapter  < £z --2— dz z  <J  c  o  N  Z  1  f /  Cauchy's  theorem  can  n  u  be  (A2. 1 )  1 a B(z )  d b -  =  8 J  distribution then  linear  j+1 d  z  *  "  i « B  1  °  B  (  z  o  )  ( A 2  - > 2  4  J  To e v a l u a t e t h e i n t e g r a l  /3(z) c a n  to  as d e s c r i b e d below.  2,  =  reduced  n  N E y j=l  a linear  into  i n ( A 2 . 2 ) , a n d hence c a l c u l a t e  of 0  i s assumed  be e x p r e s s e d  by  the  over each linear  Yj y  element.  interpolation  formula Z-Z.  j+1  which  i s then s u b s t i t u t e d  carried  Z  j  ^  into  out as .  108  4+1~  J+1  Z  J  3+1  3  (A2.2). I n t e g r a t i o n  is  then  109 B .  Z  _—1—_ B  j+1  Z...-Z  /  B,.,  r-Ji^-i dz +  J  r  j+1  z-z  > /  +1  (—i) dz  B  i  I, +  where  i  = /  j + 1  1  Z  litl z-z  J  j  n  - f  dz  Z  Z  (  Substituting  /  Zj+1  I , and I  2  «  2  «JH-«j> -  ( z  dz 0  n  j  "Z  o  2 j  z-z  J  0  Z q  j - o ) in z  V o z  back i n t o  (A2.4) g i v e s  d = B rrV " ) 1  Z  rV " ] - n  20  1  l n  >1  z  0  j 0 z  z ., , - z J  ^  X  Z  j+1  Z  0  z  j  z  r  0  R e t u r n i n g t o (A2.2) and u s i n g t h e above r e s u l t one  1 a B(z ) =  Jj  Q  z-z-  N  1 a B  where  =  I v j=l  z  Z q  e  obtains  1 10  z  i,J  j r i z  +  J  J-1  J-l  i  J+l  J  Z  J  _  Z  I  (A2.3) This expression for  /3j a t any  represents  Z j away f r o m  the  the c o n t r o l point  examined  of Z j  coefficient  z^. I t can  be  when t h e c o n t r o l p o i n t  is  and t h e s e s p e c i a l c a s e s must  be  s e e n however t h a t p r o b l e m s a r i s e in the neighbourhood  influence  separately.  i=i ~ 1  I n t h i s c a s e , ^j-i j e q u a l s t h e e x p r e s s i o n limit  as z . ^ z ^ _ . 1  The  first  (A2. ) i n the  t e r m becomes  z-z Z Z  l* j-l  j+r±  z  J" J-1 Z  Z  z  w h i c h i s o f t h e f o r m l i m [ z l n ( 1 / z ) ] = 0, a n d h e n c e v a n i s h e s z-~o  leaving only  the second  Vl.J r  , J  -  term , -z,  f ^J ^ ) 1  Z  j+1  Z  Z  j j-1 2  i ="  The  quantities premultiplying  reduce t o 1 i n the l i m i t  2 , 3  i+j  i  the In  terms  j leaving .  Z  j-1  Z  i  Z  j  Z  i  in  (A2. )  111  J-l  i  Z  j-l j z  i=j+ 1  Here T  j e q u a l s (A2.3) i n  the l i m i t  a s i _ * . j + 1.  The  s e c o n d t e r m becomes  z  Z  i* j+l z  J" 4  Vl J  =  ^^^^  to prevent solution  j  z  side of  c a r e o f when t h e l i m i t  z - z  J  ;  J  1  vanishes  1»  j-l  z  j - l j+1 2  (A2.3) h a s  resulting  . N  where  v  was t a k e n o f i  duplication,  j=l  L n  term  z  The l e f t h a n d  •  [ z l n ( z ) ] = 0, and h e n c e  2-0  only the f i r s t  J  1  which i s of the form l i m leaving  ,  .—z  j,  1 J  J  in  actually  been  taken  j and i s s e t t o z e r o the f i n a l  form  for  1 12  z-z z  ri z  i = j-1 J+1 j 2  Z  j  Z  j-1  [IXLll)  in Z  1=  j  J-1" J Z  i=j+ 1 j  j-l  Z  j-1  Z  j+1  APPENDIX I I I CALCULATION OF r..  TERM  C a l c u l a t i o n of the T^j terms i s s t r a i g h t the  Cauchy  function. attention as  principle Calculating  for  the  however,  complex  using  logarithm  requires  special  s i n c e i t c o n t a i n s t h e term - i a from e q u a t i o n (2.3)  i t s imaginary  written  values  forward  p a r t . To  clarify  the problem  can  be  i n p o l a r form as  where ( 0 . - d O  The f i g u r e  should equal  on  the l e f t  r e s u l t s when t h e b r a n c h  cut  -a.  above  a t -it i s i n s i d e  e? ) e r r o n e o u s l y y i e l d s t h e e x t e r i o r B  i s o u t s i d e a, as i n t h e no p r o b l e m .  In  general,  shows t h e  a. H e r e  that (#  A  angle. I f the branch c u t  f i g u r e on t h e r i g h t , this error 1 13  angle  occurs  then  there  whenever  is the  11 4 branch  cut  of  the  log  function lies  a n g l e a. I n t h e s e c a s e s 2 i r i must imaginary p a r t of Alternatively,  be  inside the included  subtracted  from  the  . since a  number,  an  equivalent  subtract  2ni  t o ensure  is  rule that  by  is  definition that  a  i f Im(r^)  i t i s equal to - i a .  positive < 0 then  APPENDIX I V CALCULATION OF  VELOCITIES  The v e l o c i t y o f e a c h p o i n t on t h e c o n t o u r  i s found as  36  w  —  —— 3z  The c o m p l e x p o t e n t i a l /J has been assumed p i e c e w i s e l i n e a r s o 3/3/3z i s d i s c o n t i n u o u s a t e a c h n o d e . P r o v i d e d small,  3/3/3z c a n be a p p r o x i m a t e d (  fj }  elements  by  ~ j-l J-l V j ' j f l V l 3  6  +  +  (A4.1)  where t h e c o e f f i c i e n t s a r e t o be d e t e r m i n e d . c a n be w r i t t e n  a n < + 1  3 ^j-1  a s T a y l o r s e r ii e ss e x p a n s i o n s away f r o m /3j (n) j-l  B  B  "  j+1  '-it-  Z n  = n  = 0  =  j n!  0  which are then s u b s t i t u t e d  +  +  l  M  a  (  z  ( z  Z  M- j  )  (A4.1) g i v i n g  z  V i  +  i[ j-i( j-i- j a  f  z  j + l " j>  into  z  j-l "  n ( z  z  115  are  ) 2  +  (  V i  v r  (  z  z  j  )  ^ i "  ]  z  B  j +  j 5 )2]e  1 16 Coefficients  can  be  equated g i v i n g  f o r t h e t h r e e unknown  Solution  quantities  a  j-l j j + l  a  j - l j - l • j>  a  j-i j-l- j  + a  + a  (z  z  ( z  z  °  =  Vl j+1 ~ j  +  (z  ) 2 +  a  z  } =  1  jfl j+l- j  ) 2 =  ( z  z  yields  ,i+1 - i 2  Z  " l j-l ~ jJ ^ j+1 " j - J z  Z  2  =  '*1  a  j  =  I^  1-l  Z  -  Z  2  i  " jJ l ^ i " V l J 2  " j-i" V i a  three l i n e a r  °  equations  APPENDIX V EQUATIONS FOR WAVE SIMULATION The s e t o f l i n e a r e q u a t i o n s t o be s o l v e d t o o b t a i n unknown q u a n t i t i e s  X j which  <f>^ o r  are either  0^, c a n  the be  w r i t t e n as  There a r e N nodal p o i n t s  i n t h e c o n t o u r , however, t h e r e  a c t u a l l y N-2 unknowns s i n c e p o i n t s a r e N3 a n d N4. The also  1 a n d N2 a r e i d e n t i c a l a s  number o f l i n e a r e q u a t i o n s  N-2. The a c t u a l e q u a t i o n s a r e :  1 < i <  N2  N4 < i <  N  N2-1  j-1  N3-1 i j  j  N4-1  j=N2+l  i J  l k  2  N +  E  j=N4+l  j=N3 N2  ^ik^ii^i l k  1 J  2  =  " j-1 V j 1  J  117  are  J  1 J  2  is  therefore  118 N2 < i <  N4  N2~l  N3-1  J-1  J  m  j=N2+l  J  l  k  J  _  ±  j = N 3  N2  N  "*V»J  j=N4+l  " " £  V j  where k r e p r e s e n t s t h e n o d a l p o i n t on t h e v e r t i c a l e x a c t l y o p p o s i t e j and i s equal t o k=N+NS+2-j By p e r i o d i c i t y  the f o l l o w i n g  *  = k  *k  =  •j *j  relations also hold  *N2  N4  <t>  =  =  +1  *N3  boundary  APPENDIX V I CALCULATION OF ENERGY The  kinetic  energy f o r a f l u i d  \  -  J  ! J v  volume i s  dV  2  a In  order t o evaluate t h i s  must  be  rewritten  expression  as a contour  integral  v a l u e s on t h e b o u n d a r y . The f l u i d be  the  volume  integral  involving  function  velocity  a t any p o i n t  expressed as  V2  Vd) • Vd>  m  w h i c h c a n be s u b s t i t u t e d  J  /  i n t o Green's f i r s t  [4>V<|> + vi|> • 2  =  V$]dv  v  / [d>Vo>] • n c  yeilding  -  f  / •  (|t)  C  c  -  f  /  •  d *  c  119  as  identity  d s  can  120  Since the normal v e l o c i t y and  cancels  component i s z e r o  on  on t h e v e r t i c a l b o u n d a r i e s , t h i s  o n l y be e v a l u a t e d  on  the  s u r f a c e t o be p i e c e w i s e  linear  N2 =  surface.  the i n t e g r a l  seabed  integral  need  Considering  the  becomes  4. A*  J  £.  free  the  i=l  N2 "  2  w h i c h by p e r i o d i c i t y  *  1  +  *.  2  (  1=1  ) (  V  *i+l -  reduces t o N2  The  potential  energy f o r the f l u i d  volume i s  L  By c o n s i d e r i n g t h e volume  to  be  made  up  s e c t i o n s under each element t h i s e x p r e s s i o n  N2  x  I  |*  i=l  ££. 2  ^  T  i+l / y2  dx  Ti  l  rlli 3 x  i  J  ±  +  1  £x Ay  of  trapezoidal  becomes  121  yielding  finally  E  Again only The  P  Z  ( X  1  " l+l X  free surface floating  translational can  6*  =  values  body  be m e a s u r e d f r o m any level  being  energy  i s therefore  -  < G U  2  y  W  +  +  V  G  +  y  i  2  )  are required. energy  includes  datum  with  the  f o r c o n v e n i e n c e . The t o t a l  2 )  both  c o n t r i b u t i o n s . P o t e n t i a l energy  arbitrary  chosen  f  i+l i  kinetic  and r o t a t i o n a l  water  ) ( y  +  1  1 1  +  m  g  y  G  still body  APPENDIX V I I CALCULATION OF BODY BOUNDARY CONDITIONS The p o s i t i o n  and v e l o c i t i e s  o f t h e body a r e  assumed  known a t e a c h t i m e s t e p h e n c e t h e v e l o c i t y component  normal  t o t h e body c a n be w r i t t e n a s  n  This expression  0  3n  3s  i s then i n t e g r a t e d n u m e r i c a l l y t o  determine  as ij>  = /v J  6  The n o r m a l v e l o c i t y velocity  n  ds +  component  V  n  il>  o  i s found as f o l l o w s .  o f a p o i n t P on t h e h u l l i s  122  The  123  where *G  =  V  v , = P/G  +  V k x R  6  R = (x-x )i + (y-y )J G  G  The u n i t n o r m a l v e c t o r n = k x s w h e r e _ dx 1 + dy j ds  a s  The v e l o c i t y a t P i s t h e r e f o r e  \ =G {U  1 +  G l * [-(y-y  V  J  +  )1+ ( X  G  -VN  (A?. 1)  and t h e n o r m a l component i s  3*  -  Tr = v • n p 9 s  The s t r e a m f u n c t i o n i s f o u n d by i n t e g r a t i n g d+ - u  to f i n a l l y  dy - v  G  dx - 6[(y-y )dy + (x-x >dx]  G  G  G  obtain 6R +  =  " —  2  +  U  G  I  Y  ~ G Y  )  "  V(  - ~ G> X  X  +  *  0  (A7.2)  124 where  R  For time  = ( -x )  2  X  +  2  G  (y-y )  2 ;  G  s t e p p i n g purposes t h e time d e r i v a t i v e  needed. E q u a t i o n  (A7.2) r e p r e s e n t s \p on  t h e moving  boundary  s o t h e m a t e r i a l d e r i v a t i v e must be t a k e n  the f l u i d  particles. 6R =  (  y  _  y  G  )  a  x  '  (  "  x  G  X  )  "  a y  (  )  V  fluid  following  2  —  G h ^ G "G 57 " G  + U  3<^/8t i s  (X  X  }  where  v_ , Dt  ( X  ~V  " DF  ^ ( y - y ^  The  last  since  Dx  s  " G  =  U  = |f -  - <y-y > e  G  v  G  »  6(x-x ) G  s t e p was o b t a i n e d by r e f e r r i n g  Dx/Dt  a n d Dy/Dt a r e j u s t  to  equation  thevelocity  (A7.1)  components o f  p o i n t s on t h e b o u n d a r y V . T h e r e f o r e , p  2£  -  (y-y )a G  +  x  -  6[u (x-x ) G  G  (x-x )a G  +  y  -  -  r  v (y-y )] G  G  For a p p l y i n g a boundary c o n d i t i o n i n t h e n u m e r i c a l  solution,  125 however,  the  derivative  is  needed  as  seen  by  to  motion  the  i n s t a n t a n e o u s boundary. That i s , Dili  8iJ>  97 - DF  "  '  V  V  *  where t h e a d v e c t i v e t e r m i n t h i s c a s e i s due the  rigid  body and i t s v e l o c i t y must be u s e d f o r v =  G  the  giving  + <|*)j = -v i + u j  v - [u -6(y-y )]i +  Completing  [v +9(x-x )]j  G  algebra  G  G  yields  for  the  final  boundary  condition If  -  of  (y-y )a G  x  -  (  „  6  ) .  -I f i  y  + u v - v u + [(x-x )(u -u) G  G  G  G  +  (y-y )(v -v)]6 G  G  APPENDIX  VIII  CALCULATION OF BODY FORCES AND MOMENTS The b o d y f o r c e v e c t o r  i s d e t e r m i n e d by i n t e g r a t i n g  the  pressures over the wetted h u l l as  F  or,  for  numerical  =  - / P n-ds s  evaluation,  c o n t r i b u t i o n s over each  the  of  the  I P n As s  as t h e average of  bounding n o d a l p o i n t s and a c t s  the values at the  i n t h e d i r e c t o n normal t o t h e  hull.  The u n i t n o r m a l v e c t o r  sum  element.  F = -  The p r e s s u r e i s t a k e n  as  i s f o u n d n = i s where  s  Az As  126  127 The  total  force F  + i F c a n t h e r e f o r e be c o m p u t e d a s y  x  F  . _  N6-1  Tf  -If  i(^±L-i) (  E  )  Z  i=N5  The  moment a c t i n g on t h e h u l l  about G i s equal t o  M = - / P rxri ds s where t h e p r e s s u r e  over  each  average of t h e bounding nodal at  the  centroid  radius vector pressure  of t h e  from  element values  pressure  G to  i s again  taken  and c o n s i d e r e d  distribution.  t h e body s u r f a c e  R"  as  the  to act i s the  p o i n t where  the  i s acting  M = - Z P(r x fi)As s  The  cross product  i s evaluated  by c o n s i d e r i n g R" a n d n t o  c o m p l e x numbers a n d u s i n g t h e i d e n t i t y The  Z  ft  X Z  fi  =  I m  ^ A  moment i s t h e r e f o r e c o m p u t e d a s  N 6 - 1  p 1  M  =  " i=  (  N5  +i  - ^  z  _ p 1  }  X  J  i+r i  1 ( J i  z  H"  z  G>< i-r i>*} z  z  Z  Z  be  B^*  APPENDIX I X EQUATIONS FOR BODY MOTION SIMULATION The  s e t of l i n e a r e q u a t i o n s used i n v e r s i o n  body m o t i o n s i m u l a t i o n <t> a n d hence  are given  below. In t h i s v e r s i o n  $ are s p e c i f i e d at the i n t e r s e c t i o n points there  are  simulation. constant  two  less  unknowns  The s t r e a m f u n c t i o n  evaluated  at  N3.  I I I of t h e  than  in  N5 and N6 the  There  are  consequently  The a c t u a l e q u a t i o n s a r e :  i < N5  N5 < i <: N2 N4 < i ^  N  i = N3  N5-1  N6-1 Z  N2-1  2  j=N5+l  j-1  j=N6+l N4-1  N3-1  +  I  (  j=N2+l N5  W*j  N  j=N3 N2  N6  -  j=N5  j=N4+l ik I  +  I  j=N6  128  wave  on t h e s e a b e d i s an unknown  equat i o n s .  1 ^  both  va (a  +a  N4-1  I ij^j ,_, )il> +  j=N3  N-3  129 N5 < i < N6 N2 < i < N3 i < N4  N3  N5-1 j-l  N6-1 i j  +  j  N2-1  j=N5+l  ± J  j=N6+l  3  N3-1 N4-1 I (a. .+a..H,+ I a j=N2+l j=N3 1:1  i  k  3  i j  N5 N6 = " £ a A. + Z b j=l ^ j=N5 3  where  1 3  j  1 3  A  j  N2 i|> - E j=N6  J  N I j=N4+l  (b i  k  i J  +b.)^, " j=N3 3  N4-1 I b 1 3  ad> 1 J  3  k r e p r e s e n t s t h e n o d a l p o i n t on t h e v e r t i c a l b o u n d a r y  e x a c t l y o p p o s i t e j and i n t h i s case  i s equal t o  k=N+NS+NH+2-j  By p e r i o d i c i t y  the following *k "  *j  *k *j =  relations also hold *H2  *m  =  =  *1  *N3  *  1 30 The N3  s t r e a m f u n c t i o n on  the  seabed i s a constant  so *j  =  *N3  j  =  N  3  ,  N  4  evaluated  at  APPENDIX X NUMERICAL PERTURBATION The  control  solution points  energy  i s E and t h e volume a r e then d i s p l a c e d  resulting V.  volume  CORRECTION  f o r the f i r s t  i s V.  The  two  approximate intersection  upward on t h e body by A y  L  and A y  i n new v a l u e s f o r t h e e n e r g y a n d v o l u m e o f E'  The t o t a l  change i n energy  including  R  and  t h e known c h a n g e i n  body e n e r g y i s AE  =  AE  + AE  B  R  +  AEp  w h i l e t h e change i n volume i s  AV  The p r o b l e m required  =  V  - V  i s to find the p o s i t i o n corrections  to offset  the errors  from t h e n u m e r i c a l s o l u t i o n  Ay  i n energy and volume at  each  time  e l e m e n t s a d j a c e n t t o t h e body a r e a d j u s t e d  131  step.  L  and  Ay  R  resulting The  accordingly.  two  1 32 R e f e r r i n g t o t h e above f i g u r e the change i n a r e a o f t h e two  L  and  i s the  first  •p  1  R  and  + £  L  r i g h t element l e n g t h s . T h i s  - W Wi (  and  AE  the  Ay )  L  c o n s t r a i n t . Using equation  =  W  +  subtracted  p  (3.8)  +  f o r the p r e l i m i n a r y  d i s p l a c e d c o n f i g u r a t i o n . The  Similarly,  Ay  R  are the l e f t  R  "  expanded  \U  =  the p o t e n t i a l energy be  is  triangles  AV  where 1  volume  f  {(3  £  using equation  R  from  y  N 5  )(Ay ) + R  2  )  configuration  the  difference  i  y  expansion  can  f o r the  i s then  * (Ay ) R  2  R  (3.7) N2  \  the k i n e t i c the  energy  expression  "  i  ±  l  ±  <*i*i+l '  •i+lV  before displacement  resulting  f o l l o w i n g Cauchy Riemann  *a ~ *N5  is  subtracted  a f t e r displacement. approximations  *N5 ~  *N5-1  By u s i n g  from the  *N5-1 " *N5 A y  •b " *N6 A y  and  *N6+1  L  +b " *N6 A y  *R  R  £  L  *N6+1 "  K  L  carrying out thealgebra,  *N6  t h e d i f f e r e n c e works out t o  R  +  IT^N6-*N6 l +  + (  Wl-*N6  ) 2  ]}  

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