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The damping of surface water waves in containers with grooved surfaces Gettel, Lorne Edward 1975

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THE DAMPING OF SURFACE WATER WAVES IN  CONTAINERS  WITH GROOVED SURFACES by  LORNE EDWARD GETTEL B.Sc.,  The U n i v e r s i t y  A THESIS SUBMITTED  of B r i t i s h  C o l u m b i a , 197^  IN PARTIAL FULFILMENT OF  THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE  in  t h e Depa r tment of PHYS ICS  We a c c e p t r e q u i red  this  thesis  as c o n f o r m i n g  to the  s tandard  THE UNIVERSITY  OF B R I T I S H  S e p t e m b e r 1975  COLUMBIA  In p r e s e n t i n g t h i s  thesis  an advanced degree at the L i b r a r y I  further  fulfilment  of  the  requirements  the 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 , I agree  s h a l l make it  agree  in p a r t i a l  freely  available  for  t h a t p e r m i s s i o n for e x t e n s i v e c o p y i n g o f  of  this  representatives. thesis for  It  financial  this  thesis  The  gain s h a l l not  of  U n i v e r s i t y of B r i t i s h Columbia  2075 Wesbrook P l a c e Vancouver, Canada V6T 1W5  or  i s understood that c o p y i n g or p u b l i c a t i o n  w r i t ten pe rm i ss i on .  Department  that  r e f e r e n c e and study.  f o r s c h o l a r l y purposes may be granted by the Head of my Department by h i s  for  be allowed without my  ABSTRACT  The damping, o f s u r f a c e waves on s h a l l o w w a t e r c a n be a f f e c t e d  by t h e r o u g h n e s s  o f t h e c o n t a i n e r w a l l s as w e l l  as by s u r f a c t a n t l a y e r s p r e s e n t on t h e a i r - w a t e r The  interface.  i n c r e a s e i n t h e damping c a u s e d by a g r o o v e d wave  base has been s t u d i e d both t h e o r e t i c a l l y In a n y e x p e r i m e n t a l i n v e s t i g a t i o n roughness  tank  and e x p e r i m e n t a l l y .  into the effect of surface  on t h e d a m p i n g , s u r f a c t a n t s c a n o b s c u r e t h e e f f e c t s  of wall roughness.  An e x p e r i m e n t a l t e c h n i q u e i s d e s c r i b e d  to e n a b l e t h e r e l a t i v e be d e t e r m i n e d  importance  unambiguously.  conducted with the groove viscous boundary  o f t h e s e two e f f e c t s t o  Initially  experiments  were  a m p l i t u d e much s m a l l e r t h a n t h e  layer thickness.  I t was f o u n d t h a t t h e  i n c r e a s e i n t h e damping c a u s e d by t h e g r o o v e d  b a s e was s m a l l  (< 5% i n c r e a s e ) . The damping has a l s o been s t u d i e d f o r l a r g e a m p l i t u d e g r o o v e s on t h e b a s e o f t h e wave t a n k ( d i m e n s i o n s much than the v i s c o u s boundary theoretically sinusoidal  layer thickness).  larger  I t i s shown  that, f o r grooves of triangular or approximately  c r o s s s e c t i o n , t h e damping  ii  i s i n c r e a s e d by an  amount p r o p o r t i o n a l surface,  and  respect  to the  total  surface  i s i n d e p e n d e n t of the  to the  grooves.  the groove spacing f l u i d d e p t h and  and  The  of the  grooved  d i r e c t i o n of flow  with  t h e o r e t i c a l model a p p l i e s  amplitude are  wavelength of the  l a r g e compared to the  area  viscous  small  when  compared to  surface  waves, but  boundary layer  the are  thickness.  E x p e r i m e n t s were c o n d u c t e d to t e s t these  predictions.  c a r r y i n g out  modes, i t i s shown  observations  on  two  surface  that the observed  increased  surface  c a n n o t be a t t r i b u t e d t o s p u r i o u s  alone  effects.  and  I t was  the  to the  increase fractional  i s due  found experimentally  independent of flow and  damping  d i r e c t i o n with  i n the  plete agreement with  the  respect  d a m p i n g was  increase  in the  area  grooved surfactant  the damping to the  strictly  theoretical  i ii  that  to the  of the  By  was  grooves,  proportional b a s e , i n com-  predictions.  TABLE OF CONTENTS  Page ABSTRACT  ii  LIST OF TABLES  vi  L I S T OF FIGURES  v i i  ACKNOWLEDGMENTS  x  NOMENCLATURE  xi  Chapter 1  INTRODUCTION  1 PART  2  A  T H E E F F E C T O F S M A L L A M P L I T U D E WALL ON T H E D A M P I N G 2.1  ROUGHNESS  O F S U R F A C E WAVES  Introduction  4  2.2  3  C a l c u l a t i o n o f the Damping Frequency f o r a Grooved Base D A M P I N G E X P E R I M E N T S FOR S M A L L R O U G H N E S S AMPLITUDE  4  8 13  3.1  Experimental Set-up  13  3.2  Experimental Results  22  iv  Chapter  Page 3.3  Discussion  36  3.4  Conclusions.  40  P A R T  4  B  T H E E F F E C T OF L A R G E G R O O V E S  ON T H E  DAMPING  OF S U R F A C E WAVES  42  4.1  Introduction  4.2  Theory  . . .  42 45  4.3  5  C a l c u l a t i o n o f /SdA f o r Grooved Surfaces 4.4 C a l c u l a t i o n o f t h e D a m p i n g Frequency, D A M P I N G E X P E R I M E N T S FOR L A R G E A M P L I T U D E GROOVES  62 68  5.1  Experimental  System  5.2  Experimental  R e s u l t s and  5.3  50  68  Discussion  77  Conclusions  86  REFERENCES  87  APPENDICES A T H E G E N E R A L R E S U L T FOR T H E FREQUENCY  v  DAMPING  89  L I S T OF  TABLES  Table  Page  1  The E x p e r i m e n t a l C o n d i t i o n s f o r t h e C i r c u l a r Wave T a n k E x p e r i m e n t s  10  2  The E x p e r i m e n t a l C o n d i t i o n s f o r t h e S q u a r e Wave T a n k E x p e r i m e n t s  71  3  V a l u e s o f t h e Damping  85  Frequency  vi  L I S T OF  FIGURES  Figure 1  Page Orientation of driver electrode with r e s p e c t to the grooved base of the wave t a n k  2  Cross-section  3  Mounting  4  Plan view of the o p t i c a l  5  Photoresistor detection circuit.  6  Damping runs f o r a smooth  7  Damping runs f o r a grooved aluminum (two smooth q u a d r a n t s )  7  of the c i r c u l a r wavetank  system f o r the d r i v e r e l e c t r o d e detection  aluminum  system.  16 . . . . 19 20  base  24  base 25  8  D a m p i n g r u n s f o r 33 r . p . m . r e c o r d base (two smooth q u a d r a n t s )  9  Damping runs f o r a g r o o v e d aluminum (" d i r t y s u r f a c e " )  10  The l o g a r i t h m o f wave a m p l i t u d e vs f o r a smooth base. .  time  11  The l o g a r i t h m o f wave a m p l i t u d e vs f o r a grooved aluminum base  time  v ii  14  u s e d as a base  26 27 28 29  Figure  Page  12  The l o g a r i t h m o f wave a m p l i t u d e vs time f o r a 33 r . p . m . r e c o r d u s e d a s a b a s e  30  13  The l o g a r i t h m o f wave a m p l i t u d e vs t i m e f o r a grooved aluminum base ( c o n t a m i n a t e d surface)  31  14  Damping f r e q u e n c y vs 0 (smooth  32  15  Damping f r e q u e n c y vs 9 ( g r o o v e d aluminum  16  Damping f r e q u e n c y vs 9 (33 r.p.m. r e c o r d used as a b a s e )  34  17  Damping f r e q u e n c y vs 9 ( g r o o v e d base, contaminated surface)  35  18  Mapping upper  base) base)  aluminum  . . 33  of a polygonal region into the half of the w plane  53  19  Features of groove geometry  20  C o o r d i n a t e system used f o r the c a l c u l a t i o n of t h e damping  55  frequency  63  21  C r o s s - s e c t i o n o f t h e s q u a r e wave t a n k .  69  22  High v o l t a g e waveform  73  23  Function generator  24  The c o n s t r u c t i o n o f an a r b i t r a r y  generation circuit  using the function generator 25  D a m p i n g r u n s f o r A = 2/3 L  26  Damping r u n s f o r A = 2L viii  74 waveform 7  6 7 9  80  Fi gure  Page  27  The l o g a r i t h m o f wave a m p l i t u d e vs t i m e ( X = 2/3 L )  81  28  The l o g a r i t h m o f wave a m p l i t u d e vs t i m e (X = 2 L )  82  ix  ACKNOWLEDGMENTS  I s i n c e r e l y w i s h t o e x p r e s s my t h a n k s t o D r . F.L. Curzon of this  f o r his excellent s u p e r v i s i o n during the  course  work. I would  in c a l c u l a t i n g boundaries.  l i k e t o thank Ralph P u d r i t z f o r h i s help  the i n v i s c i d  flow potential  over  grooved  The a s s i s t a n c e o f Jim Aazam-Zanganeh i n the  d e s i g n o f t h e c o u n t i n g l o g i c , a n d P e t e r C h e n who h e l p e d w i t h the e a r l y experiments, i sg r a t e f u l l y acknowledged.  I would  a l s o l i k e t o thank Sharon H a l l e r f o r an e x c e l l e n t t y p i n g j o b . Financial Council  h a s been  a s s i s t a n c e from the N a t i o n a l Research  gratefully received.  T h i s work was s u p p o r t e d b y a g r a n t f r o m t h e Energy C o n t r o l Board o f Canada.  x  Atomic  NOMENCLATURE  a  groove  f  frequency  i  /=T  k  s u r f a c e mode w a v e n u m b e r  I  viscous  n  u n i t v e c t o r normal tank  v  flow  v  0  amplitude o f surface  modes  boundary layer  thickness  t o a s u r f a c e o f t h e wave  velocity  inviscid  flow  velocity  w  complex v a r i a b l e  x,y,z  position  coordinates  A  amplitude  o f s u r f a c e waves  A  area  ratio  B  Beta  function  H  fluid  J  Bessel  1  depth Function  xi  L  length of the square  wave  tank  P  fluid  R  r a d i u s o f c i r c u l a r wave  S  viscous energy  n  c o o r d i n a t e a l o n g n_  8  a n g l e between t h e n o d a l l i n e o f t h e s u r f a c e wave and t h e edge o f o n e o f t h e g r o o v e q u a d r a n t s on the base o f t h e wave tank  X  wavelength  o f s u r f a c e mode  v  kinematic  viscosity  £  wavelength  o f groove  p  fluid  a  damping  frequency  <J>  inviscid  flow v e l o c i t y  co  angular  pressure tank  dissipation  per unit  spacing  density  frequency  potential  o f s u r f a c e mode  xi i  area  Chapter  1  INTRODUCTION  The  damping o f s u r f a c e waves on a f l u i d c a n be  a f f e c t e d by s u r f a c t a n t s on t h e f r e e s u r f a c e o f the f l u i d , or by the roughness o f the c o n t a i n e r wall  r o u g h n e s s has been used, with  as a c o n v e n i e n t theory [Case  The  and P a r k i n s o n ,  between  s u r f a c e waves used  i nthis  could  smaller  periodic electric  fields applied  That  i s , t h e wave  normally  than  In Part A, the case  1  the viscous  so that  amplitude  r o u g h n e s s can be d i v i d e d  larger o r smaller  layer thickness.  small  the wavelength o f the surface  dimensions o f the wall  two r e g i m e s :  examined.  I n a l l o f t h i s , work t h e wave  be used.  than  roughness  work were e x c i t e d b y s p a t i a l l y  o f t h e s u r f a c e wave was s u f f i c i e n t l y  theory  was m u c h  o fwall  s u r f a c e waves has been  to the f r e e f l u i d s u r f a c e . amplitude  waves  1957].  investigation the effect  inhomogeneous, time  The  justification,  and e x p e r i m e n t f o r the damping o f s u r f a c e  the damping o f standing  linear  little  In the past,  means o f e x p l a i n i n g d i s c r e p a n c i e s  In t h i s on  walls.  mode. into  boundary  where the roughness  2  amplitude i s s m a l l e r than the viscous boundary layer t h i c k n e s s is examined. is used.  In t h i s  investigation a cylindrical  The roughness  structure is controlled  g r o o v e d p l a t e s on t h e b a s e o f t h e wave We r e a l i z e d  wave  by  tank  inserting  tank.  through the course of these  experiments  t h a t g r o o v e a m p l i t u d e s much s m a l l e r t h a n t h e v i s c o u s b o u n d a r y l a y e r t h i c k n e s s had l i t t l e  e f f e c t on t h e d a m p i n g , c o n t r a r y  to p r e v i o u s c o n j e c t u r e [Case and P a r k i n s o n , 1957].  This  p r o m p t e d us t o e x a m i n e t h e a f f e c t on t h e d a m p i n g c a u s e d by large grooves  ( a >> £ ) , w h e r e a i s t h e r o u g h n e s s  amplitude,  and I i s t h e v i s c o u s b o u n d a r y l a y e r t h i c k n e s s .  I n p a r t B,  we e x a m i n e t h i s c a s e b o t h t h e o r e t i c a l l y a n d e x p e r i m e n t a l l y . F o r t h e s e e x p e r i m e n t s a s q u a r e wave t a n k i s used t o s i m p l i f y the a n a l y s i s .  The groove  s t r u c t u r e was c o n t r o l l e d  same m a n n e r a s i n o u r i n i t i a l was i n s e r t e d on t h e b o t t o m  experiment; a grooved  o f t h e wave  roughness  the damping f r e q u e n c y i s c a l c u l a t e d  mental  In a d d i t i o n  f o r a wave t a n k  base  Chapter 3 describes the experi-  setup and r e s u l t s f o r small groove s i z e P a r t B c o n s i s t s o f C h a p t e r 4 a n d 5.  of a general approach  Chapter  to the effects of surface  and s u r f a c t a n t l a y e r s on t h e d a m p i n g .  w i t h two g r o o v e d q u a d r a n t s .  plate  tank.  P a r t A i s c o m p o s e d o f C h a p t e r 2 a n d 3. 2 c o n s i s t s o f an i n t r o d u c t i o n  i n the  (a <  l).  Chapter 4 consists  a p p l i c a b l e f o r d e t e r m i n i n g the damping  e n h a n c e m e n t c a u s e d by g r o o v e d c o n t a i n e r w a l l s .  Using  this  3  approach the damping frequency groove structures.  i s d e t e r m i n e d f o r some  Chapter 5 describes  system used to check the p r e d i c t i o n s experimental  the  experimental  o f C h a p t e r 4, a n d t h e  r e s u l t s are compared with  these  predictions.  In a d d i t i o n , C h a p t e r 5 a l s o c o n s i s t s o f a d i s c u s s i o n major conclusions for future  work.  of these  given  i n v e s t i g a t i o n s and  of the  suggestions  P A R T  4a  A  Chapter  2  THE E F F E C T OF S M A L L A M P L I T U D E DAMPING  2.1  Parkinson,  OF S U R F A C E WAVES  roughness h a s been used  1957]  i nthe past  to explain discrepancies  studied  s e c t i o n andfound  w a t e r the damping was a f a c t o r o f t h r e e  surface  value.  l a r g e r than  shallow  their  roughness o f t h e i r c y l i n d r i c a l v e s s e l s , even  though  (depth)  layer thickness,  a o f the roughness s t r u c t u r e o f  I.  compared t o the viscous  I tis d i f f i c u l t ,  a physical mechanism which could  i ncontact  with  i na layer o f thickness  however, t o envisage  Essentially for  the c o n t a i n e r  ~ I.  about the roughness s t r u c t u r e should  walls  i n t e r f a c e can  laminar is a t  F o r a << I n o i n f o r m a t i o n be p r o p a g a t e d  As stated previously, surfactant layers  on t h e a i r - w a t e r  boundary  produce a s i g n i f i c a n t  i n t h e d a m p i n g , i f a << I.  the f l u i d  fluid.  that, for  t o the  t h e i r v e s s e l s was s m a l l  flow,  Case a n d  They a t t r i b u t e d t h i s d i s c r e p a n c y  the amplitude  increase  theory  the damping o f water waves i n c y l i n d r i c a l  vessels o f c i r c u l a r cross  theoretical  [Case a n d  between  experiment f o r the damping o f s u r f a c e waves.  Parkinson  rest  ROUGHNESS ON THE  Introduction Wall  and  WALL  into  the  present  s i g n i f i c a n t l y increase  the  5  damping of s u r f a c e waves [see L e v i c h , 1951;  and  Pike  and  Curzon,  Surfactant the  lateral  1941,  1962;  1968].  l a y e r s on t h e  free fluid  m o b i l i t y of t h i s s u r f a c e .  On  surface  shallow  t h e d a m p i n g e s s e n t i a l l y d o u b l e s as t h e s u r f a c e laterally  immobilized  Dorrestein,  ( i . e . the a i r - w a t e r  c o m p l e t e l y c o v e r e d by a s u r f a c e  alter  fluids  becomes  interface is  f i l m ) [see  Pike  and  Curzon,  1 968] . The thin  surface  f i l m can  e s s e n t i a l l y be v i e w e d a s  e l a s t i c membrane, which i s s t r e t c h e d or  compressed  by t h e m o t i o n o f t h e f l u i d .  T h i s r e s u l t s in the  of a d d i t i o n a l f o r c e s at the  s u r f a c e w h i c h m u s t be  appearance taken  i n t o account in the boundary c o n d i t i o n s at the f r e e Surfactant  l a y e r s on t h e t o p o f t h e  surfactant residues upwards to the the  surface  effect  l e f t on  fluid  can  the v e s s e l w a l l s  f r e e s u r f a c e , and  to r e s i d u e s  from the a i r above i t .  water surface  i s c l o s e r to the c o n t a i n e r  the time r e q u i r e d  a r i s e from which t r a v e l deposited  base,  free  decreasing  f o r s u r f a c t a n t s to reach the a i r water  Hence, the enhancement i n the damping  by C a s e a n d  Parkinson could the a i r - w a t e r  be d u e  to s u r f a c t a n t  i n t e r f a c e or to s u r f a c e  I n t h i s i n v e s t i g a t i o n , we experimentally  on  the  s i n c e the  interface.  p r e s e n t on  surface.  For the f o r m e r ,  i s more s e r i o u s f o r s h a l l o w f l u i d s ,  a  set out to  reported layers roughness.  determine  t h e e f f e c t on t h e d a m p i n g o f s u r f a c e  waves  6  due  to c o n t a i n e r roughness  s a t i s f y i n g a < £.  g r e a t l y c o m p l i c a t e d by t h e p r e s e n c e can o b s c u r e  roughness  effects.  d e s i g n our experiments separated.  these e f f e c t s ,  of s u r f a c t a n t s , which  Thus,  i t i s e s s e n t i a l to  s o t h a t t h e s e two  In t h e e x p e r i m e n t a l t h e d a m p i n g was  e x c i t e d on s h a l l o w w a t e r  e f f e c t s can  i n v e s t i g a t i o n used  be  to  separate  s t u d i e d f o r a s l o s h i n g mode •  in a vessel with a l t e r n a t e l y  roughened  quadrants  roughened  s u r f a c e s o f t h e c o n t a i n e r do c a u s e  on t h e b a s e  (see Figure  then the damping f r e q u e n c y w i l l  1).  I f the  enhanced  damping,  vary s i n u s o i d a l l ywith  where 9 i s the angle between the nodal wave and  This task is  l i n e of the  t h e e d g e o f one o f t h e g r o o v e d  base of the v e s s e l (see F i g u r e  1).  quadrants  During  surface on  the  9,  the  time  r e q u i r e d f o r a s e r i e s of measurements the c o n d i t i o n of a i r - w a t e r i n t e r f a c e c a n c h a n g e due will  alter  a.  These  to s u r f a c t a n t s , which  v a r i a t i o n s i n a, however, w i l l  have any a n g u l a r dependence,  and  i n s u r f a c t a n t c o n c e n t r a t i o n s h o u l d be s m a l l . found  Hence i f roughened  f a c t o r o f t h r e e [ C a s e and p o s s i b l e to observe is not obscured  to  changes  In f a c t , i t  that a f o r a given value of 9 changes  t h a n ± 5% d u r i n g t h e t i m e r e q u i r e d t o p e r f o r m s c a n o f o.  not  i f the time r e q u i r e d to  m e a s u r e a i s s m a l l , t h e n t h e s e v a r i a t i o n s i n a due  was  the  by an  less angular  s u r f a c e s do i n c r e a s e a by a  P a r k i n s o n ] , then  i t should  a s i n u s o i d a l d e p e n d e n c e o f a on 9  by s u r f a c t a n t e f f e c t s .  be which  7  F i g u r e 1.  Orientation the grooved  of d r i v e r electrode with respect base o f t h e wave t a n k .  to  8  In  t h e next s e c t i o n t h e damping  frequency i s  c a l c u l a t e d f o r the grooved quadrant bases used mental  2.2  study.  C a l c u l a t i o n o f the Damping Frequency f o r a Grooved The.damping  frequency will  a s l o s h i n g mode f o r a s h a l l o w f l u i d  Base  now be c a l c u l a t e d f o r i n a wave tank w i t h a  p a i r o f g r o o v e d q u a d r a n t s on t h e b a s e . the  in the experi-  The nodal l i n e  of  s u r f a c e wave i s i n c l i n e d a t an a n g l e 6 t o t h e edge o f  one o f t h e g r o o v e d q u a d r a n t s . - C a s e a n d P a r k i n s o n show t h a t t h e damping  f r e q u e n c y c a n be w r i t t e n a s  ° where  o~g, a n d  =  a  W  +  a  B  +  a  MS  ( 2 > 1 )  a r e t h e c o n t r i b u t i o n s made,  respectively  by t h e w a l l s o f t h e wave t a n k , t h e b a s e , a n d a t t h e f l u i d surface, which of  i s assumed  t o be l a t e r a l l y  our e x p e r i m e n t s , however,  immobile. laterally  The term immobile  the f l u i d  mobile.  s u r f a c e was  laterally  m u s t t h e n be r e p l a c e d by o " j ^ , t h e term, where  a  T C :  = a  D  Cosh  2  (kH)  k = wave n u m b e r o f t h e s u r f a c e mode H = fluid [see  In a l l  P i k e and Curzon,  depth 1968].  (2.2)  9  For the c o n d i t i o n s used (see Table  1 ) , and  is shallow  so t h a t  The  ~ °"g f o r a s m o o t h b a s e . a  ^/ j5  damping frequency  a  1  <<c  damping frequency  inviscid 4.2)  the  2 a  =  term  i s the  can  fluid  be  neglected.  given  (2.3)  d e p e n d e n t upon (see  the Section  dA  a  inviscid  (2.4)  flow velocity  dA  i s an a r e a  t a n g e n t i a l to a  e l e m e n t on  the  (2.4)  and  be a s s u m e d t h a t t h e r o u g h n e s s  It will  f o r a < I does not a l t e r  given  the  is increased according  inviscid  surface  surface.  For a roughened surface  damping frequency  by  A)  o f the c o n t a i n e r , and  Parkinson.  1  I S  is in general  a  0  d  f l o w v e l o c i t y on t h e c o n t a i n e r w a l l s  and A p p e n d i x  where v  a n  kH <<  The  f o r a smooth base i s then  a,  The  in the experiments  to  structure  flow v e l o c i t y v .  f o r a roughened s u r f a c e could then  Case  The  be  by  o  w h e r e A'  is a constant,  a  A  1  A'  > 1  v  2 0  dA  (2.5)  10  Table 1 Experimental Conditions  Features of the water  tank  Depth, H  ( 4 . 3 3 ± 0.01 ) cm  Radius, R  ( 1 5 . 0 ± 0.1)  Kinematic  viscosity  C h a r a c t e r i s t i c s of  0.01  kR  _ 1  1 .841 2  frequency, f  D a m p i n g f r e q u e n c y OQ Boundary  2  'sioshing-mode'  D i m e n s i o n l e s s wave number, Oscillation  cm sec  cm  (smooth  (1.221 base)  l a y e r t h i c k n e s s = /v/to  Properties of Grooved  ± 0.001 ) Hz  (0.043 ± .006) 0.36  mm  Quadrants  a)  33 r . p . m . r e c o r d - g r o o v e s p a c i n g  = 0.063  b)  Aluminium discs - groove spacing  = 1 mm o r 3.1 mm (i.e. 1/8")  mm  Hz  11  The on t h e b a s e for  damping frequency,  will  a f o r the roughened  quadrants  2  now b e c a l c u l a t e d u s i n g t h e a p p r o a c h  used  (2.5) , e+Tr/2  1 + c /o2  e+TT  (A  1  v  2 0  ) r d r d<j> +  v 9+  IS  0  TT/2  (2.6)  e+TT  v  0  r d r deb R  w h e r e (r,<j>) a r e p o l a r c o - o r d i n a t e s o n t h e b a s e tank  r d r d(f>-  and v  0  o f t h e wave  i s t h e t a n g e n t i a l • v e l o c i t y f o r a s l o s h i n g mode  f o r an i n v i s c i d  fluid.  v  0  F o r a s l o s h i n g mode  (2.7)  = c o n s t a n t J i ( k r ) c o s <J>  w h e r e ^- J i ( k r ) = 0 a t r = R, t h e o u t e r e d g e o f t h e w a v e a r [see Case and P a r k i n s o n ] . (2.6)  and using e q u a t i o n a  2  1  (2.3) to eliminate  0  from  equation  yields  = 1/2 + 1 / 4 ( 1 + A ' ) - 1/2 u ( A ' - 1 ) S i n 2 0  Thus t h e damping frequency f o r t h e base  Now e l i m i n a t i n g v  tank  considered,  enhance the damping.  should vary s i n u s o i d a l l y with 6 i f roughness  s a t i s f y i n g a < I does  (2.8)  12  The tank  expressions  for  and  for a cylindrical  were d e r i v e d by Case and P a r k i n s o n .  k cosech  0 3  S  U  V  2R  1 + (s/kR) 1 - (s/kR)  They are  (2kH)  2 2  frequency  H  (2.9)  2kH " s i n h (2kH)  where R i s t h e r a d i u s o f t h e wave t a n k , o f t h e s u r f a c e mode,  wave  (2.10)  k i s t h e wave number  i s the f l u i d depth,  and  c h a r a c t e r i z i n g t h e s u r f a c e mode.  OJ«~J  i s the  The w^y's a r e  given by  oo  for  2 su  = 9 k  $  u  tanh  s u r f a c e g r a v i t y waves.* The k  J ' (k s  s u  s  (k  u  $ u  H)  satisfy  R) = 0  (2.12)  F o r t h e s l o s h i n g mode S = U = 1. T h e e x p l i c i t a^j a n d Og a r e m e n t i o n e d s i n c e a c o m p a r i s o n a ( l a t e r a l l y mobile  (2.11)  forms f o r  between p r e d i c t e d  o r immobile) and experimental  o  determines  the c o n d i t i o n o f t h e top s u r f a c e o f the f l u i d . Note t h a t e q u a t i o n ( 2 . 1 1 ) i s the g e n e r a l d i s p e r s i o n r e l a t i o n for g r a v i t y waves. F o r an unbounded s y s t e m t h e r e i s o f c o u r s e no r e s t r i c t i o n s on the v a l u e s o f the wave number.  Chapter 3  DAMPING E X P E R I M E N T S  3.1  Experimental  FOR S M A L L ROUGHNESS  Set-up  When s t u d y i n g t h e d a m p i n g temporal damping  or spatial  damping  o f s u r f a c e waves  c a n be c o n s i d e r e d .  damping.  Throughout  i n v e s t i g a t i o n s temporal damping  There are three major s t a n d i n g wave s y s t e m  advantages  either  Temporal  i n v o l v e s the use o f s t a n d i n g waves, w h i l e  waves a r e used f o r s p a t i a l mental  AMPLITUDE  travelling  our e x p e r i -  h a s been c o n s i d e r e d .  to this approach.  When a  i sconsidered, i t i spossible to drive  the system r e s o n a n t l y t o o b t a i n a pure s u r f a c e mode, w i t h an a m p l i t u d e t h a t can e a s i l y be d e t e c t e d .  T h e wave  d e t e c t o r f o r such a s y s t e m can be a t a f i x e d only onedetector i srequired.  Spatial  amplitude  position and  damping  requires  wave tank d i m e n s i o n s o f a t l e a s t the o r d e r o f the length, which s u r f a c e mode.  i s much l a r g e r t h a n t h e w a v e l e n g t h  The  o f the  This n e c e s s i t a t e s l a r g e systems, which  the case f o r temporal The  damping  geometry  i s not  damping. o f t h e wave t a n k i s shown i n F i g u r e 2.  t a n k i s made f r o m a s e c t i o n o f a n o l d g l a s s b e l l 13  j a r with  14  axis  ! electrode  wate r  F i g u r e 2.  J  i—  i I  C r o s s - s e c t i o n o f the  c i r c u l a r wave  tank.  15  the ends  ground  parallel  to each o t h e r .  The lower end o f  t h e t a n k i s s e a l e d t o a c o p p e r s h e e t by a s i l i c o n e on t h e o u t s i d e s u r f a c e o f t h e t a n k . m o u n t e d on a c o n v e n t i o n a l l e v e l l i n g adjustable measured  legs.  The depth o f water  a t any l o c a t i o n  The copper s h e e t i s table f i t t e d with three i n t h e t a n k c a n be  by a m i c r o m e t e r  depth gauge.  device, together with the three adjustable legs, the free surface of the f l u i d  adhesive  This  enables  t o be a l i g n e d p a r a l l e l  to the  b a s e o f t h e wave t a n k t o an a c c u r a c y o f a p p r o x i m a t e l y 0.04 The waves a r e e x c i t e d mounted above  the fluid  w i t h a 1.4 cm w i d e A vertical  vane  by a h o r i z o n t a l e l e c t r o d e  surface (Figure 2).  i s made f r o m a h e x a g o n a l  i s mounted below  the electrode along a  s u p p r e s s e s s p u r i o u s s l o s h i n g modes w h i c h  spring  micrometers the p i n s .  c a u s e d some  vane  problems  i s fixed across the  The s u s p e n s i o n system c o n s i s t s o f t h r e e  triangle  three identical  The  i n s i d e t h e wave tank and  from a l u c i t e frame which  loaded micrometers  equilateral  strip.  (see Discussion).  The e l e c t r o d e j u s t f i t s  top o f the tank.  board  from one d i a m e t e r .  diameter at right angles to the i n s u l a t i n g  is suspended  The e l e c t r o d e  piece of printed circuit  s t r i p o f c o p p e r removed  in our early experiments  gauge  m o u n t e d a t t h e v e r t i c e s o f an  (Figure 3).  A f t e r the tank i s l e v e l ,  p i n s a r e p l a c e d on t h e b o t t o m  are adjusted until  cm.  and the  t h e e l e c t r o d e j u s t r e s t s on  The frame and e l e c t r o d e a r e then removed  so t h a t  16  JT-  r  tJ [  Micrometer  *  r  f  .\\\\\  ?  v  W  ] IN.  k  -V  X  V  N  Y  W  W  W  M  Electrode X X _  .V...  -V  V  V  ^  N.  X  i  Wall  F i g u r e 3. M o u n t i n g s y s t e m f o r t h e d r i v e r e l e c t r o d e . (a) Plan view o f m i c r o m e t e r s u p p o r t frame. (b) The m i c r o m e t e r l e v e l l i n g s y s t e m .  17  the metal  g a u g e p i n s c a n be t a k e n o u t o f t h e t a n k .  the e l e c t r o d e and  holder are r e p l a c e d , checks  t r a v e l l i n g microscope surface are spaced The a x i s , and  When  with a  show t h a t t h e e l e c t r o d e and  water  by a c o n s t a n t d i s t a n c e o f ~ 1 cm  ± 0.1  e l e c t r o d e p l a t e c a n be r o t a t e d a b o u t  a n g u l a r p o s i t i o n can  be m e a s u r e d t o an  a  mm.  vertical  accuracy  of  ± 1° w i t h a p r o t r a c t o r s c a l e f i x e d a r o u n d  of  the wave t a n k .  to  o n e - h a l f o f t h e e l e c t r o d e s l o s h i n g m o d e s c a n be e x c i t e d  on t h e w a t e r  By a p p l y i n g a t i m e d e p e n d e n t h i g h  surface.  The  the i n s u l a t e d diameter at  any  axis.  The  nodal  edge voltage  l i n e o f the wave i s a l o n g  of the e l e c t r o d e , which  compass d i r e c t i o n  vertical  the upper  c a n be  set  by r o t a t i n g t h e p l a t e a b o u t i t s  frequency of the high v o l t a g e  waveform  i s a d j u s t e d t o r e s o n a t e w i t h t h e s l o s h i n g m o d e (~ 1.22 in our e x p e r i m e n t ) . from equation The  This result  is approximately  (2.11), where k s a t i s f i e s  s l o s h i n g mode c o n s i d e r e d i s t h e f u n d a m e n t a l  kR = 1 . 8 4 .  The  d i s p e r s i o n r e l a t i o n g i v e n by  c o m p l e t e l y c o r r e c t w h e n an e l e c t r i c to  equation  field  field)  relation  p r o p o r t i o n a l to - E  (see Ionides t h e s i s ) .  frequency  by a s m a l l a m o u n t .  (2.11).  The  (2.12). for  this  (2.11) i s not  is applied normally in the  (where E =  This shifts  the  Once the E f i e l d  however, the s u r f a c e modes o s c i l l a t e g i v e n by e q u a t i o n  2  determined  and  the s u r f a c e ; there i s a small a d d i t i o n a l term  dispersion  Hz  resonant is shut o f f ,  at the n a t u r a l  form of the high  electric  frequency  voltage  18  waveform used  i s g i v e n by V = V ( l + c o s o j t ) ^ .  The  0  H.V.  waveform generator e s s e n t i a l l y c o n s i s t s of a rotary with a p p r o p r i a t e r e s i s t o r networks to produce given above.  the waveform  Complete d e t a i l s of this generator are given  in [ I o n i d e s T h e s i s , 1972] and [ P h i l l i p s and C u r z o n , and w i l l  switch  1973]  n o t be s t a t e d h e r e . The maximum  v o l t a g e used  i s 5KV, w h i c h  produces  a w a v e a m p l i t u d e o f ~ 2 mm o n t h e w a t e r  surface.  amplitude i s observed with a microscope  ( m a g n i f i c a t i o n x35)  f o c u s e d on t h e a i r w a t e r in from the wall displaced  i n t e r f a c e a t a d i s t a n c e o f - 2 cm  o f t h e wave t a n k .  T h e m i c r o s c o p e c a n be  h o r i z o n t a l l y to ensure that i t i s never focused  on t h e w a t e r  s u r f a c e along a nodal l i n e .  the microscope  tank used to i l l u m i n a t e the tank p r o c e s s i n g o f t h e s e two s i g n a l s to a d i f f e r e n t i a l  oscilloscope  The e y e p i e c e o f  is fitted with a p h o t o r e s i s t o r .  photoresistor monitors a constant light  applied  (see Figure 4).  After  (see Figure 5 ) , they are  amplifier  (Tektronix storage  ( 5 4 9 ) e q u i p p e d w i t h a 1A2 d i f f e r e n t i a l  For complete  d e t a i l s of this measuring  C u r z o n and P h i l l i p s , The  is produced  Another  s o u r c e b e h i n d t h e wave  T h i s p r o c e d u r e e l i m i n a t e s n o i s e s i g n a l s common  1972;  T h e wave  grooved  system  amplifier).  to both i n p u t s . see [ I o n i d e s ,  1973].  s u r f a c e on t h e b a s e o f t h e wave  by i n s e r t i n g  suitably  tank  grooved d i s c s and t a p i n g  them down t o t h e b a s e o f t h e t a n k w i t h v i n y l  tape.  This  19  FLUORESCENT LAMP MICROSCOPE AMD PHOTORESISTOR WAVE TANK  F i g u r e 4.  Plan view o f the o p t i c a l  detection  system.  20  120 V  + I20_Y  PHOTORESISTOR DETECTOR  DIFFERENTIAL AMPLIFIER  PHOTORESISTOR DETECTOR  Figure  b  5. P h o t o r e s i s t o r d e t e c t i o n c i r c u i t . (a) C i r c u i t o f one o f the p h o t o r e s i s t o r d e t e c t o r s , (b) D i f f e r e n t i a l p h o t o r e s i s t o r n e t w o r k .  21  procedure  i s n e c e s s a r y i n o r d e r to p r e v e n t anomalous  damping caused  by s e e p a g e o f w a t e r  under  wave  the grooved  disc  ( s e e Di s c u s s i o n ) . A number o f d i f f e r e n t g r o o v e d in these e x p e r i m e n t s . was  d i s c s were  In a l l c a s e s t h e g r o o v e  used  amplitude  s m a l l e r o r a t l e a s t ~ b o u n d a r y l a y e r t h i c k n e s s I.  small  g r o o v e w i d t h s and a m p l i t u d e s  records  (33 r.p.m.).  The  grooves  we  used  For  phonograph  of a pair of opposite  q u a d r a n t s w e r e e l i m i n a t e d by v a r n i s h i n g t h e s e r e g i o n s . groove  widths  l a r g e r t h a n ~ 0.5  on a n a l u m i n u m p l a t e (~ 5 mm i n two  opposite quadrants  mm  we  For  cut concentric grooves  thick), using a lathe.  were removed, u s i n g a  Grooves  milling  machine. To p e r f o r m t h e e x p e r i m e n t ,  s u r f a c e waves were  e x c i t e d w i t h t h e wave g e n e r a t i o n e l e c t r o d e . a m p l i t u d e has s t a b i l i z e d ,  t h e g e n e r a t o r was  a n d t h e d e c a y o f w a v e a m p l i t u d e was oscilloscope.  Once the wave switched o f f ,  o b s e r v e d on t h e  By j u d i c i o u s l y c h a n g i n g  the gain of  storage the  o s c i l l o s c o p e a m p l i f i e r d u r i n g the course of the decay,  the  signal  c o u l d be k e p t l a r g e e n o u g h t o e n s u r e a c c u r a t e m e a s u r e -  ment.  The  d a m p i n g f r e u q e n c y was  determined  from the  slope  o f the l o g a r i t h m i c p l o t o f t h e wave a m p l i t u d e as a f u n c t i o n of  time.  22 3.2  Experimental Results E x p e r i m e n t s were i n i t i a l l y  conducted  to determine  t h e d a m p i n g f r e q u e n c y f o r a s l o s h i n g mode w i t h a s m o o t h T h i s was done t o d e t e r m i n e of the f l u i d . 1  base.  the c o n d i t i o n of the top surface  I t was f o u n d e x p e r i m e n t a l l y t h a t a  ± 0.006 s e c " ' , -where-ao  0  = 0.043  i s t h e damping f r e q u e n c y f o r a smooth  T h e r e was q u i t e a l a r g e v a r i a t i o n  run, depending  base.  i n a from run to  on t h e c o n d i t i o n o f t h e d i s t i l l e d  water.  This r e s u l t compared f a v o u r a b l y with the p r e d i c t e d l a t e r a l l y immobile r e s u l t f o r a.  Using e q u a t i o n s ( 2 . 2 ) , (2.9), and  ( 2 . 1 0 ) , t h i s p r e d i c t e d r e s u l t was a = 0 . 0 4 5 6 , f o r t h e experimental  c o n d i t i o n s g i v e n i n T a b l e 1.  Thus t h e f l u i d  s u r f a c e u s e d i n o u r e x p e r i m e n t s was l a t e r a l l y Experiments were then conducted  immobile.  to determine  t h e d a m p i n g f r e q u e n c y f o r a s l o s h i n g mode w i t h a g r o o v e d d i s c , as d e s c r i b e d i n S e c t i o n ( 3 . 1 ) , s e c u r e d on t h e b o t t o m o f t h e wave t a n k . culation  With such a base, a c c o r d i n g t o our c a l -  i n S e c t i o n ( 2 . 2 ) , the damping f r e q u e n c y should vary  s i n u s o i d a l l y w i t h 0, w h e r e 0 i s t h e a n g l e b e t w e e n t h e n o d a l l i n e o f t h e s l o s h i n g mode a n d a n e d g e o f t h e g r o o v e d  quadrant  (see F i g u r e 1 ) . I n o u r e a r l y e x p e r i m e n t s we o b s e r v e d t h a t a v a r i e d with 0 (large variations  up t o - 2 a ) , a n d t h u s we 0  thought  we h a d e s t a b l i s h e d a n e f f e c t a t t r i b u t a b l e t o s u r f a c e r o u g h n e s s . T h i s t u r n e d o u t , h o w e v e r , n o t t o be t h e c a s e .  The v a r i a t i o n  23  i n a w i t h 0 was i n f a c t c a u s e d under  the grooved  s p u r i o u s modes. of  base o r the e x c i t a t i o n The seepage  the d i s c would  by e i t h e r w a t e r g e t t i n g  result  of water  i n enhanced  and i n t e r a c t i o n o f  under  certain  d a m p i n g when t h e f l o w  v e l o c i t y was a t a m a x i m u m o v e r t h e s e r e g i o n s . d i r e c t i o n was v a r i e d , t h e e n h a n c e m e n t w o u l d resulting  our early experiments  of the e l e c t r i c  present)  t h a t t h e r e was a n  i n c r e a s e i n wave a m p l i t u d e a f t e r an i n i t i a l the removal  a  also vary,  ( w i t h o u t t h e vane  i t was n o t e d f o r c e r t a i n f l o w d i r e c t i o n s  following  When t h e f l o w  i n a.  i n an a n g u l a r v a r i a t i o n In  parts  decay  field.  period,  This  behaviour  was d u e t o t h e i n t e r a c t i o n o f s p u r i o u s m o d e s ( s e e D i s c u s s i o n ) . When t h e v a n e , in  d e s c r i b e d p r e v i o u s l y , was u s e d  t h e a m p l i t u d e o f an i n i t i a l l y  Damping runs were c o n d u c t e d  the increase  d e c a y i n g w a v e , was e l i m i n a t e d .  with d i f f e r e n t grooved  as w e l l as w i t h a smooth b a s e .  Typical  bases  o s c i l l o s c o p e outputs  d i s p l a y i n g wave a m p l i t u d e as a f u n c t i o n o f t i m e a r e shown i n Figure of are  (6-9).  In F i g u r e s (10-13)  the logarithmic plots  wave a m p l i t u d e as a f u n c t i o n o f t i m e f o r t h e v a r i o u s given.  In F i g u r e (14-17)  the damping f r e q u e n c y a i s  p l o t t e d as a f u n c t i o n o f 0 f o r each base base,  (smooth  aluminum  33 r . p . m . r e c o r d w i t h t w o s m o o t h v a r n i s h e d  and g r o o v e d  a l u m i n u m p l a t e w i t h two s m o o t h  The  properties.  not possess any d i r e c t i o n a l  We a l s o f i n d  no s i n u s o i d a l  quadrants,  quadrants).  r e s u l t s f o r the smooth aluminum base  t h e wave t a n k does  bases  show t h a t  damping  variation  i n a with 0  24a  F i g u r e 6.  Damping runs zontal scale  f o r a smooth aluminum base 5 sec/div).  (a)  e = 0°  (vert.  0.2, 0 . 1 ' a n d 0.05  v/div)  (b)  6 = 45°  (vert.  0 . 2 , 0.1  v/div)  (c)  0 = 90°  ( v e r t . 0 . 5 , 0.2 a n d 0.1  a n d 0.05  v/div)  (hori-  2 5a  F i g u r e 7.  Damping runs f o r a grooved zontal 5 sec/div).  aluminum base  (a)  6 = 0 °  ( v e r t . 0.5,  0.2  and  0.1  v/div)  (b)  9 = 45°  (vert.  0.5,  0.2  and  0.1  v/div)  (c)  6 = 90°  (vert.  0.2,  0.1  and  0.05  v/div)  (hori-  25b  26a  F i g u r e 8.  D a m p i n g r u n s f o r 33 r . p . m . r e c o r d u s e d (horizontal 5 sec/div).  (a)  8 = 0 °  (vert.  0.2,  0.1,  0.05  v/div)  (b)  8 = 30°  (vert.  0.2,  0.1,  0.05  v/div)  (c)  8 = 60°  ( v e r t 0.5,  (d)  8 = 90°  (vert.  0.2,  0.2, 0.1,  0.1, 0.05  0.05  v/div)  v/div)  as a  base  Damping runs f o r a grooved aluminum taminated surface). ( v e r t . 0.2, 0.1, 5 sec/di v). (a)  6 =  0°  (b)  9 =  45°  (c)  0 =  90°  a n d 0.05  v/div,  base  (con-  horizontal  27b  LN  A  TIME  F i g u r e 10.  The  seconds  l o g a r i t h m o f wave a m p l i t u d e vs t i m e f o r a s m o o t h  base.  LN A  LIM  Figure  13.  A  TIME  seconds  The l o g a r i t h m o f wave a m p l i t u d e vs t i m e f o r a g r o o v e d a l i m i n u m (contaminated surface).  base  sec"1  0.0450  0.0440  0.0430 4-  0.0420 0  F i g u r e 14.  H-8  +  20  e  + 40  Damping f r e q u e n c e vs 6 ( s m o o t h  60  base).  80  CO  ro  sec  0.044 I  0.043  0.042 ±  0.041 1  0.040  0.039  e  0.038 0  90  4 5  F i g u r e 15.  Damping f r e q u e n c y  135  vs 0 ( g r o o v e d aluminum  180  base).  T  ( J  sec  0.0440 4»  0.0430  0.0420  Hi  0.0410  +  420  F i g u r e 16.  40  +  60  •f  +  e  80  D a m p i n g f r e q u e n c y vs 0 (33 r.p.m. r e c o r d u s e d as a b a s e )  sec  0.0540  M  0.0530  0.0520  e  +  0.0510 20  Figure  17  Damping f r e q u e n c y  I  40  vs f  60  (grooved  I  aluminum base,  8  \  •  80  contaminated  surface).  CO  36  f o r t h e o t h e r two g r o o v e d Figures  f i n d no s i g n i f i c a n t  o v e r t h a t f o r a smooth a with  c a n be s e e n  F o r b o t h o f t h e s e b a s e s a/i  (10-13).  t h i s r e g i m e we  b a s e s , as  enhancement  from  < 1, a n d f o r in the  b a s e , a s w e l l a s no v a r i a t i o n  damping in  6. The damping  "dirty"  f r e q u e n c y as a f u n c t i o n o f 8 f o r a  s u r f a c e u s i n g the grooved aluminum  in F i g u r e (17).  T h e w a t e r was  days and t h e n t h e s e damping  left  plate  i s shown  i n the tank f o r a  runs were performed.  The  few magni-  tude of the damping  f r e q u e n c y has i n c r e a s e d , b u t a g a i n i t  e x h i b i t s no a n g u l a r  variation.  3.3  Discussion Case and P a r k i n s o n (1957)  "roughness  w h o s e d e p t h was  claim that container  small compared  to the  boundary  l a y e r t h i c k n e s s h a d a r e m a r k a b l y l a r g e e f f e c t " on t h e of s u r f a c e waves.  They  damping  found e x p e r i m e n t a l l y values of o  t h r e e t i m e s l a r g e r t h a n t h a t p r e d i c t e d by t h e o r y f o r a d p e t h t o r a d i u s r a t i o o f < 1.5.  Case and P a r k i n s o n assumed  in  t h e i r theory a l a t e r a l l y mobile s u r f a c e l a y e r at the a i r water  interface.  With  this assumption  the energy  in the top s u r f a c e i s n e g l i g i b l e compared and w a l l s o f t h e t a n k . b a s e we  found good  to t h a t of the  In o u r e x p e r i m e n t s w i t h a  agreement  between  dissipation  e x p e r i m e n t and  base  smooth theory,  37  assuming  a laterally  immobile  top surface.  immobile  surface e s s e n t i a l l y doubles  A  laterally  the damping f o r s h a l l o w  fluids. To p r e p a r e a l a t e r a l l y m o b i l e extreme  top surface requires  c a r e and i t i s r e a s o n a b l e to assume Case  d i d n o t have  such a top s u r f a c e .  crepency between experiment  This reduces  we u s e d d e p t h  a n d we s h o u l d h a v e  certainly  t o r a d i u s r a t i o s o f 0.2  seen even  larger effects ( d i s -  crepency between t h e o r y and e x p e r i m e n t ) , based  agree w i t h Case fluids  Our e x p e r i m e n t a l  on Case a n d  r e s u l t s do n o t  and P a r k i n s o n ' s c o n t e n t i o n t h a t f o r s h a l l o w  surface roughness  increase  their d i s -  and t h e o r y t o a f a c t o r o f 1.5.  In o u r e x p e r i m e n t s  Parkinson's conjecture.  and P a r k i n s o n  ( e v e n w i t h a / £ < 1) c a n d r a s t i c a l l y  damping. We f o u n d f o r t h e g r o o v e d  variation  bases  considered that the  i n d a m p i n g w a s o n l y o f t h e o r d e r o f < ± 5%  from t h a t f o r a smooth base.  different  I f Case and P a r k i n s o n ' s  con-  tention that surface roughness  was r e s p o n s i b l e f o r t h e  d a m p i n g e n h a n c e m e n t was v a l i d ,  then f o r o u r model ( o p p o s i t e l y  grooved  quadrants), A  would  1  Substituting v a r i a t i o n o f 20% i n  this  o /oi, 2  and t h i s v a r i a t i o n would  in nature.  that found  i n our experiments  This variation  i s much l a r g e r  in a  be than  ~ < ± 7% d i f f e r e n c e b e t w e e n  base r e s u l t s .  r e c o r d and the grooved  t o 3.  i n equation (2.8) r e s u l t s  sinusoidal  smooth and groove  be e q u a l  F o r b o t h t h e 33 r . p . m .  aluminum p l a t e , a does  not possess  38  a s i n u s o i d a l dependence on 0 ( s e e F i g u r e s 15-16) a s predicted  by e q u a t i o n The  (2.8).  damping f r e q u e n c y  an. a n g u l a r v a r i a t i o n  (not  i n t h e s e runs does  possess  s i n u s o i d a l i n nature), but these  v a r i a t i o n s (~ ± 8%) a r e m u c h s m a l l e r t h a n t h a t e x p e c t e d to  Case  and P a r k i n s o n ' s  conjecture.  I t was  t h a t t h e r e was n o s y s t e m a t i c b e h a v i o u r  also discovered  to these v a r i a t i o n s .  Every angular scan o f the damping frequency would in a d i f f e r e n t angular v a r i a t i o n .  These  result  v a r i a t i o n s i n o with  0 a r e , i n f a c t , due to s u r f a c t a n t s on t h e f r e e f l u i d As m e n t i o n e d difficult  surfaces.  was e x t r e m e l y water  used  in the experiments  depending  While  The value o f the damping  caused frequency  s e n s i t i v e to the c o n d i t i o n s o f the d i s t i l l e d  present on the grooved day,  surface.  p r e v i o u s l y , s u r f a c t a n t s make i t  t o e s t a b l i s h t h e e f f e c t on wave damping  by r o u g h e n e d  due  base.  as well asto  contaminants  V a r i a t i o n s i n a from day to  on these c o n d i t i o n s , were a s high a s 30%.  the value o f the damping frequency  varied for different  r u n s , t h e damping f r e q u e n c y f o r any one r u n d i d n o t e x h i b i t any  s i n u s o i d a l angular dependence.  To determine  f u n c t i o n o f 0 (0° < 6 < 1 8 0 ° ) takes about  a as a  one hour and d u r i n g  It w i l l be n o t e d t h a t t h e l o g a r i t h m i c p l o t s of wave a m p l i t u d e v e r s u s t i m e a r e e x c e l l e n t straight lines (as g o o d a s t h o s e o b t a i n e d by l o n i d e s . T h i s i n d i c a t e s we a r e d e a l i n g w i t h a s i n g l e s u r f a c e mode a n d we a r e o p e r a t i n g in a r e g i m e where l i n e a r theory is a p p l i c a b l e .  39 t h i s time f l u c t u a t i o n s i n the v a l u e s o f a a t any g i v e n v a l u e o f 0 a r e ~ < 5%. by l e a s t  squares  The damping f r e q u e n c y  f i t ) has a t y p i c a l  so t h a t t h e s e v a r i a t i o n s c a n n o t of our experimental are " r e a l "  results.  grooved seen  error of less  quadrants  1%,  Hence these v a r i a t i o n s i n a  they cannot of the base.  than  be e x p l a i n e d by t h e u n c e r t a i n t y  v a r i a t i o n s . Since they possess  angular dependence,  (determined  no s y s t e m a t i c  be t h e r e s u l t o f t h e o p p o s i t e l y The v a r i a t i o n s i n a w i t h 9  i n F i g u r e ( 1 4 - 1 7 ) , m u s t t h e n be t h e r e s u l t o f s u r f a c t a n t s  p r e s e n t on t h e f r e e s u r f a c e o f t h e f l u i d  a n d on t h e  grooved  base. As m e n t i o n e d  p r e v i o u s l y we o b s e r v e d  a spurious  d e p e n d e n c e o f a on 9 i n a wave t a n k w h e r e t h e e l e c t r o d e had  no v e r t i c a l  vane under  t h e wave a m p l i t u d e  i t . For c e r t a i n values of 9  i n c r e a s e d f o r a c e r t a i n time a f t e r  d e c a y i n g , and then f i n a l l y behaviour  decreased  again to zero.  i s reminiscent of the energy  coupled o s c i l l a t o r s f r e q u e n c i e s , which  having results  exchange  slightly different i n damped  In a wave t a n k o f c i r c u l a r  t h e i r nodal  However, i f the tank  resonant  beats. c r o s s - s e c t i o n any sloshing  l i n e s p e r p e n d i c u l a r to each  other.  is imperfect (slightly e l l i p t i c a l ,  with a s l o p i n g base), the degeneracy modes  This  between  s l o s h i n g m o d e c a n be r e s o l v e d i n t o t w o d e g e n e r a t e modes h a v i n g  initally  or  o f t h e two r e f e r e n c e  i s removed and they become l i n e a r l y  coupled.  40  During  t h e d r i v i n g c y c l e , the r e f e r e n c e modes  are c o n s t r a i n e d such is near  phase,  energy  can  of the c o u p l i n g .  the f l u i d  be e x c h a n g e d  surface  However,  during  between the  modes  As a r e s u l t t h e d i s p l a c e m e n t  s u r f a c e e x h i b i t s t h e phenomenon o f damped  When t h e d a m p i n g f r e q u e n c y it  l i n e of the f l u i d  the c e n t r e s t r i p of the e l e c t o r d e .  the decay because  that the nodal  beats.  i s higher than the beat  frequency  i s e a s y t o o v e r l o o k t h e e x i s t e n c e o f t h e b e a t s and  the o b s e r v a t i o n s i n terms periodic fashion.  found  t h a t t h e s e damped  c o u l d be c o m p l e t e l y e l i m i n a t e d by t h e v e r t i c a l  vane  the e l e c t r o d e .  to  vane c o n s t r a i n s the system  such t h a t the nodal p e r p e n d i c u l a r to the  3.4  l i n e always  remains  along a  beats  under oscillate  diameter  vane.  Conclusions The  experimental  results  waves i n a c i r c u l a r wave tank grooved  quadrants  f o r the damping of s u r f a c e  having a base with a l t e r n a t e l y  shows t h a t the damping i s i n d e p e n d e n t  t h e d i r e c t i o n o f t h e s u r f a c e m o d e w i t h r e s p e c t t o .the for groove On  describe  of a dependence of a in 0 in a  H o w e v e r , we  The  of  amplitudes  the b a s i s of Case  frequency wall  s m a l l e r than the boundary and  roughness  1  an a n g u l a r v a r i a t i o n  d i d in f a c t g r e a t l y enhance the  base  layer thickness.  Parkinson s s u g g e s t i o n , the  s h o u l d have p o s s e s s e d  of  damping i f the  damping.  41  We  d i d however observe  (~ 3 0 % ) matic  from  day  angular  that this  f l u c t u a t i o n s i n the damping  to day,  b u t on a n y  dependence.  variation  r u n t h e r e was  These r e s u l t s  i n c i s due  strongly  to s u r f a c t a n t s .  frequency no  syste-  suggest The  ness  of the c o n t a i n e r w a l l s p l a y s a t best o n l y a minor  role  i n the  to f i n d any  i n c r e a s e o f a. systematic  base roughness, than  the v i s c o u s  In o u r e x p e r i m e n t s  we  were  i n c r e a s e in the damping caused  where the roughness amplitude boundary layer  thickness.  is  rough-  unable by  smaller  R T  42a  Chapter  4.1  4  Introduction F r o m t h e r e s u l t s o f P a r t A , we f o u n d  ness does not s i g n i f i c a n t l y w a v e s , when t h e a m p l i t u d e the  viscous  dimensions  ( a < Z) used  experiment.  ( a >> Z)  has i n the past  Wall  roughness of small  between  theory  In t h i s s e c t i o n t h e e f f e c t o f l a r g e  on t h e d a m p i n g i n a wave c h a n n e l  in Section  than  been c o n v e n i e n t l y ( b u t  to explain discrepancies  r e t i c a l l y and e x p e r i m e n t a l l y .  grooves  i s examined  theo-  The enhancement f a c t o r A  used  1  2.2 i s a n a d - h o c a p p r o a c h t o t h e p r o b l e m o f d a m p -  ing over  roughened s u r f a c e s , as i t assumes t h a t t h e flow  velocity  i s unaltered  m i g h t be j u s t i f i e d large grooves.  i n the d i s s i p a t i o n i n t e g r a l s .  f o r small  Any meaningful  groove amplitudes,  This  but not f o r  c a l c u l a t i o n o f t h e damping  must i n v o l v e changes i n t h e flow structure.  rough-  the damping o f s u r f a c e  of the roughness i s smaller  boundary layer thickness.  erroneously) and  increase  that wall  v e l o c i t y over  In o u r t h e o r e t i c a l s t u d y 42b  the roughness  f o r large grooves,  the  43  c a l c u l a t i o n o f t h e damping  frequency i s approached  from  this  point o f view. In d e t e r m i n i n g t h e damping it  over a roughened  i s necessary to determine the i n v i s c i d  over the roughness of the boundary  velocity  structure (see Section 4.2).  layer  (where  potential  The s t r u c t u r e  e s s e n t i a l l y a l l energy  o c c u r s ) i s t h e n d e t e r m i n e d by s o l v i n g equation s u b j e c t to the boundary  surface,  the boundary  dissipation layer  conditions that the f l u i d  i n c o n t a c t w i t h t h e c o n t a i n e r be a t r e s t . Roughened c o n t a i n e r w a l l s change potential  f r o m a v a l u e <J>o ( t h e v a l u e w i t h s m o o t h  s o m e v a l u e <\> a n d t h e y c h a n g e the boundary inviscid  the velocity  layer.  velocity  the effective  surface area of  T h e r e a r e two a p p r o a c h e s  potential  walls) to  over a roughened  to determine the surface. A  l i n e a r p e r t u r b a t i o n scheme i s the most g e n e r a l starting roughness  from the smooth  wall potential  s t r u c t u r e i s decomposed  The components  f o r which  a(£)/?  d>0.  i n changing  an e x p a n s i o n p a r a m e t e r .  T o do t h i s , t h e  into Fourier  components.  > 1» w h e r e a ( ? ) i s t h e a m p l i -  tude o f a F o u r i e r component o f wavelength greatest effect  approach,  <t>0.  £will  The term a(?)/£  F o r the F o u r i e r terms  g r e a t e s t e f f e c t on c h a n g i n g t h e p o t e n t i a l  have the i s used as  having the  t}>o, t h e t e r m  a(£)/5  i s n o t an a p p r o p r i a t e e x p a n s i o n p a r a m e t e r .  The p e r t u r b a t i o n  s c h e m e . 1 i n k i ng  certainly  linear  <> j  to  <t>0  f o r t h e s e modes w i l l  i n the expansion parameter  be  a.(5)/S i f a/£ - 0 ( 1 ) -  nonSince  44  we  are  interested in large amplitude roughness, a  approach is  inappropriate.  An  a l t e r n a t i v e a p p r o a c h i s to look  t i o n s to L a p l a c e ' s  equation  boundary condition  n_ • V<f> = 0 on  a(?)/5 = 0(1).  For  roughness structure structure  a(£)  is assumed.  the  which s a t i s f y the  and  surface  length  boundaries for  over  mode); then  t o know  s c a l e s o f ~n£  d)  0  should  should  decay to zero  (n > 1 ) , a n d  the  feature.  t e r m k£ c a n  expansion parameter replacing considered identical o f an  in our  e f f e c t i v e l y be a/?.  roughness  a cross  £.  Above  over  this  v i e w e d as  an  structure  i n v e s t i g a t i o n c o n s i s t s of a surface  p a r a l l e l grooves having  section  with  in the  shape  isosceles triangle. An  exact  s o l u t i o n f o r the  <b f o r t h i s s t r u c t u r e was  t h a t k£ <<  calculated using  The  <J>.  inviscid velocity potential  d e t e r m i n e d which i s v a l i d even i f  a ( £ ) / S > 1, p r o v i d e d  the  The  essen-  particular  the g r o o v e d s t r u c t u r e must i n c o r p o r a t e then  £)  1 where k i s  s o l u t i o n over The  the  (spacing  l i n e a r dimensions of order 0  which  calculation a certain  m o d e ( i . e . k£ <<  surface  solu-  i n v i s c i d flow  a r e many g r o o v e s  r o u g h n e s s s t r u c t u r e , dp - dp  vertical  non-flat  f o r any  If there  the wave number of the be a c o n s t a n t  for exact  this approach i t is necessary  per w a v e l e n g t h of the  tially  perturbation  1.  The  d a m p i n g was  then  r e s u l t s o f t h i s c a l c u l a t i o n show  energy d i s s i p a t i o n in the  boundary layer is  that  proportional  45  to  the s u r f a c e area of the grooved  of  the o r i e n t a t i o n o f the i n v i s c i d *  direction  4.2  of the  s u r f a c e , and with  r e s p e c t to  the  grooves.  Theory The  Navier-Stokes  motion  of a viscous f l u i d  equation.  t i o n s t h a t the f l u i d  we a r e d e a l i n g w i t h  V • v_ = 0 , w h e r e _v i s t h e f l u i d  used  in a l l of our experiments  appropriate.  fluid  from  the  incompressible,  velocity.  (water)  this  For the  fluid  is certainly handed  i s used with the x y plane on the  s u r f a c e , and  linearized Navier-Stokes  fluid  is  In t h e c a l c u l a t i o n t o f o l l o w a r i g h t  c a r t e s i a n c o - o r d i n a t e system undisturbed  is determined  I t i s assumed i n a l l of our c a l c u l a -  i.e.  The  flow  independent  the z a x i s v e r t i c a l l y  equation  upwards.  f o r an incompressible  is +  P PJ  + v V  w h e r e P = p r e s s u r e , p = d e n s i t y , and  2  (4.1 )  v  v i s the  kinematic  viscosity. The  boundary c o n d i t i o n f o r this flow is that  In t h i s c a s e t h e i n v i s c i d f l o w r e f e r r e d to is measured at s e v e r a l boundary l a y e r s above the g r o o v e d surface. It is t h e r e f o r e the f l o w w h i c h would remain i f the groove amplitude a is d e c r e a s e d to zero.  46  (4.2)  v = 0  on f i x e d b o u n d a r i e s .  Viscous  flow creates  the a d d i t i o n a l  constraint that the tangential v e l o c i t y at f i x e d is zero.  The d e c r e a s e  exclusively  boundaries  of the v e l o c i t y to zero occurs  i n a t h i n l a y e r next  to the s o l i d w a l l .  almost This  l a y e r i s c a l l e d t h e b o u n d a r y l a y e r and i s c h a r a c t e r i z e d by the presence inviscid  in i t of considerable  flow  the normal  velocity gradients.  the only c o n s t r a i n t at f i x e d boundaries  c o m p o n e n t o f t h e v e l o c i t y be z e r o .  f l o w t h e v e l o c i t y c a n be w r i t t e n i n t h e  v = Vd) +  For is that  For viscous  form  (4.3)  Vi  w h e r e y_i_ c a n b e w r i t t e n i n t e r m s o f a v e c t o r p o t e n t i a l A , i.e.  (4.4)  = V x A  The f i r s t second  term  t e r m y_i_ i s t h e r o t a t i o n a l f l o w Since  then  V<b i s t h e p o t e n t i a l f l o w t e r m , , w h i l e  i t follows  the flow  the  term.  is incompressible  (V • y_ = 0 )  that  V (b = -0 2  (4.5)  47  Substituting equation  t h e e x p r e s s i o n f o r y_ i n t o t h e Na v i e r - S t o k e s  gives  £  (V6)  This equation for of  +  £  (4.6)  (V,)  -  one  z e r o v i s c o s i t y v, Vi to  PJ  for inviscid  0.  Equation  2  (Vcj) +  (4.6)  v i )  equations,  flow.  (4.6)  then  one  In t h e  limit  integrates  yield  — + g z =-|-f-  The  + vV  i s b r o k e n d o w n i n t o two  t h e v i s c o u s f l o w and  directly  -V  (4.7)  const  +  above s o l u t i o n (4.7)  is also valid  for  the  viscous case i f  ^  Equations  (Vi)  (4.5)  = v V  and  2  (4.8)  (4.8)  Vj.  are solved subject  t h e c o n s t r a i n t t h a t t h e f l u i d be a t r e s t a t r i g i d Expressed  mathematically  these conditions  n  • V(J>  =  to  boundaries.  are  0  (n x V<}>) + n_ x (V x A ) = 0  (4.9)  (4.10)  48  w h e r e n_ i s a u n i t n o r m a l d i r e c t e d boundary to the f l u i d . varies  inwards from the  At any r i g i d  boundary the v e l o c i t y  r a p i d l y i n t h e d i r e c t i o n o f n_.  derivatives in the  A l l of the other  of the v e l o c i t y are small  direction.  Equation  St  compared  (vi).= v  harmonic time dependence  frequency of the surface Vi w i l l  with  variations  (4.11)  Vi  an'  v  waves.  2  ~ e  From  be a s s u m e d  t o be a  w h e r e <j i s t h e  1 w t  (4.11) the s o l u t i o n f o r  be o f t h e f o r m  Vi  ~  where I i s the viscous and q i s t h e c o - o r d i n a t e conditions  at the wall  n (1 + i ) /1  exp  boundary layer thickness along  n_.  Vi must  (4.12)  /2  I =  To s a t i s f y t h e  include  boundary  t h e t e r m Vcb  t h a t V<t> i n _v_j d o e s n o t a l t e r t h e s o l u t i o n o f ( 4 . 1 1 ) V <j) = 0 ) . 2  spatial  ( 4 . 8 ) c a n t h u s be s i m p l i f i e d t o  T h e t i m e v a r i a t i o n o f Vj_ w i l l simple  solid  (Nb. since  The c o m p l e t e s o l u t i o n f o r Vi i s  v  x  = n x (n  x v<|>) exp  - Jn(l  + i)/*-  /?j  (4.13)  49  w h e r e Vd> i s e v a l u a t e d tion per unit area V i , and i s given  S = -v  a t n = 0.  The v i s c o u s  of the container  walls  3n  damping  n=0  Vcp  (4.14).  i t i s necessary to integrate  /1  /2  (4.14)  dissipation  To d e t e r m i n e  S over the  the damping f r e q u e n c y i s then p r o p o r t i o n a l  d e t e r m i n e d by c a l c u l a t i n g  to evaluate  To  SdA  (grooved structure).  f o r two g r o o v e c r o s s sinusoidal)  as w e l l  sections  many g r o o v e s  scale  (spacing  SdA.  do t h i s i t i s n e c e s s a r y over the rough-  The i n v i s c i d p o t e n t i a l  ( t r i a n g u l a r and a p p r o x i m a t e l y  In t h e f o l l o w i n g  the boundary layer  any o t h e r  to  as t h e d i s s i p a t i o n i n t e g r a l s f o r e a c h ,  now c a l c u l a t e d .  assumed that  container  on t h e d a m p i n g c a n  the i n v i s c i d velocity potential  ness s t r u c t u r e  than  n=0  the energy  Hence the e f f e c t s o f a roughened s u r f a c e  are  upon  d e p e n d e n t on t h e i n v i s c i d v e l o c i t y p o t e n t i a l , a n d  surfaces;  be  i s dependent  3V j "vfTJ  •v  t h i s c a n be s e e n t o be t h e c a s e f r o m the  dissipa-  by [ s e e C a s e a n d P a r k i n s o n ]  I t was r e m a r k e d e a r l i e r t h a t is only  energy  lengths  calculations  thickness  I i s much  i tis smaller  o f the problem and there  are  £) p e r w a v e l e n g t h o f t h e waves ( X ) .  For a f l a t surface  the energy d i s s i p a t i o n S = S  0  where  S  0  = v v  2 0  / £ V2  (4.15)  50  and  v  i s the  0  If f l u i d  flows  the grooves surface from dA  dA  inviscid  flow v e l o c i t y p a r a l l e l  parallel  but  to the g r o o v e s ,  t o dA  is over  (dA  surface.  For flow transverse At the top of the grooves  above the grooves. bears  4.3  to S  0  dA  0  v  0  potential. will  The  0  and  i n the r a t i o  depth  v  0  distance  is changed.  for Grooved  Surfaces  the v e l o c i t y p o t e n t i a l over  i t is convenient m o d u l u s and  the f l o w d i r e c t i o n and  to c o n s i d e r a complex v e l o c i t y  case  the angle  t h e xy p l a n e .  For  s u r f a c e , t h e v e l o c i t y m u s t be a l o n g Hence the p r o f i l e of the  m u s t be t h e p r o f i l e o f a s t r e a m l i n e . a streamline  w h i c h makes i t d i f f i c u l t  a  argument of a complex p o t e n t i a l  i n our  to the s u r f a c e .  tures considered  0  v is changed.  v e l o c i t y some  g i v e the m a g n i t u d e o f the v e l o c i t y and  tangent  to  dA/dA .  at the bottom v <  inviscid  as the g r o o v e  flow past a s o l i d  surface,  H e n c e i t i s n o t c l e a r w h a t r e l a t i o n SdA  When d e t e r m i n i n g surface  grooved  Hence f o r flow p a r a l l e l  to the g r o o v e s ,  v > v  i s the  C a l c u l a t i o n o f /SdA  grooved  the  by  i n c r e a s i n g the d i s s i p a t i o n area  the d i s s i p a t i o n i s i n c r e a s e d  where in t h i s case  over  i s an e l e m e n t o f a r e a on t h e f l a t  0  the grooved  the grooves  surface.  is unaltered  0  the boundary l a y e r spreads  (see Figure 19(b)) 0  v  to the  For the  f o l l o w s the groove  to determine  the  between inviscid the  surface  grooved  struc-  structure,  potential.  For a l l  51  of our c a l c u l a t i o n s be i n t h e c o m p l e x  z-plane.  transformation which onto the real simplified.  s t r u c t u r e i s c o n s i d e r e d to  I f i t i s p o s s i b l e to d e t e r m i n e  maps t h e g r o o v e  a x i s of the w-plane, The  s t r u c t u r e in the  then the problem  lines parallel  to the real  axis.  a  z-plane  is greatly  s t r e a m l i n e s of the flow i n the w-plane  t h u s be s t r a i g h t problem  the groove  will  The  of flow past a given contour then reduces  to  the  d e t e r m i n a t i o n o f an a n a l y t i c f u n c t i o n w ( z )  which  v a l u e s on t h e c o n t o u r .  With  i t i s p o s s i b l e to  determine  o v e r the grooved  the p o t e n t i a l  the d i s s i p a t i o n  this approach  takes  s t r u c t u r e and  speed  thus  i n t e g r a l s c a n be s o l v e d .  C o n s i d e r <f> = w = f ( z ) t o b e t h e c o m p l e x potential  real  t r a n s v e r s e to the g r o o v e s .  u at the grooved  The  magnitude  s u r f a c e i s then given  velocity of  the  by  u = y |V<i | = u dw dz This velocity above  the groove  i . e . u -> v this  potential  0  as  in  the p r o p e r t y t h a t f a r  structure, their effect  I m ( z ) -* °°.  i s the case.  results  must have  Using  (4  is  negligible,  The  constant y i s chosen  so t h a t  (4.16)  i n the d i s s i p a t i o n  integral  52  SdA  L/2  dw dz  V  d z | j dL  w h e r e dL i s a l e n g t h e l e m e n t on a g r o o v e considered. ture.  The i n t e g r a l over  dz i s a l o n g  axis of the w-plane.  transforms  the groove  to a line  s t r u c t u r e i s mapped Then t h e l i n e  i n t e g r a l o v e r dw.  be w r i t t e n i n a m o r e c o n v e n i e n t  SdA  The form  the area  being struc-  As s t a t e d p r e v i o u s l y , t h e e v a l u a t i o n o f t h i s i n t e g r a l  is s i m p l i f i e d i f the groove real  over  (4.17)  case  =  u  •/I  of grooves  dw dz  To d e t e r m i n e  this case,  For t h i s (4.17) can  V  with  a cross section i n the considered  mapping the groove  the r e a l axis o f the w-plane i s r e q u i r e d . t h a t t h e i n t e r i o r o f any polygon mapped i n t o t h e upper h a l f D  The  (4.18)  (see Figure  the p o t e n t i a l and d i s s i p a t i o n i n t e g r a l f o r  a transformation  Christoffel  dz  dw  o f an i s o s c e l e s t r i a n g l e i s f i r s t  19a).  i n t e g r a l over  form.  V  :  into the  transformation most general  1  structure into  It i s well  D i n the z-plane  o f t h e w - p l a n e by a  known  c a n be Schwartz-  (see Figure 18a). form  of the Schwartz-Christoffel  transformation is  ^  = A(w-w )^ 1  / 7 r  -  1  (w-w ) 2  a 2 / 7 r  -  ]  ...  (w-w^n/*-  1  (4.19)  53  Figure  18.  Mapping of a polygonal region into the upper of the w-plane. (a) A r b i t r a r y p o l y g o n a l region. (b) R e g i o n b o u n d e d by a t r i a n g u l a r g r o o v e structure.  half  54  and 00  .  t h i s c a n be used f o r p o l y g o n a l  regions  T h u s i t i s p o s s i b l e t o map r e g i o n  with  vertices at  S bounded by a g r o o v e d  s t r u c t u r e i n t o t h e u p p e r h a l f p l a n e S' ( F i g u r e 1 8 b ) . T h e g r o o v e s t r u c t u r e t h a t h a s been c o n s i d e r e d a superposition The  c a n be v i e w e d a s  of individual triangles.  Schwartz-Christoffel  single triangle i sf i r s t  transformation  calculated.  The points  fora z = -b, i, a  +b a r e m a p p e d i n t o t h e p o i n t s w = - 1 , 0, + 1 r e s p e c t i v e l y i n the w p l a n e  (Figure 19a).  Then  .j s  /Tr+2g  A(w+l) 7 r  a / 7 T  = A(w -1 ) 2  -  1  w  ( 7 r _ a / 7 T )  A w (w -l)  2 a / 7 T  2  The  constant  the  isosceles triangle.  a / 7 r  *  "  (w-l)^-  1 w  (  7 T + 2 a  /  _ K w ' (l-w )  T r  )  ~  0 1  ^-  1  1  2 a / T r  2  ( 4 a /  ^  K i s chosen t o f i x the length o f one side o f Integrating  (4.19) y i e l d s  55  F i g u r e 19:  Features of the groove  geometry.  (a)  Mapping contours f o r the potential .  (b)  Flow of water  (c)  G e o m e t r y o f g r o o v e d a r e a dA, w i t h p g r o o v e s p e r u n i t l e n g t h (p = 3 in Figure 19c).  along the  velocity grooves.  a  55b  56  K  w  air  ,2  -  (1  ^  duV + B  ij,2)a/Tr  F o r w = 0, z = a i a n d f o r w = 1, z = b . given  (4.21 )  Using t h i s ,  K is  by  • .,.U L  <->  a  a/it  (1 The structure single  S-C T r a n s f o r m a t i o n  is calculated  triangle.  i t maps v e r t i c e s and (see  maps v e r t i c e s  f o r the complete  i n a s i m i l a r manner t o that  The t r a n s f o r m a t i o n at height  is calculated  groove fora so  a i i n t o 0, ± 2 ( ^ - ) , ± 4 ( ^ - ) ,  on t h e r e a l  axis  into ±  ± —,  that •••  •••  1 9 a ) w h e r e TT - 26 i s t h e a p e x a n g l e o f t h e g r o o v e s  Figure The  product  42 2  SC t r a n s f o r m a t i o n  f o r this i s the i n f i n i t e  5?  ^dw - A ... (w-rr)  A w  TT+29 T  TT-29  ,  (W-TT/2)  w  0  29/TT  4 [p  w  16(f)  W'  2  36(f)  w  n  - (f)  2  2  w  2  - 9(f)  2  n  1 -  r e s u l t s from form  n  IF  K = A const. = AC.  o f (4.23) which  enables  i n thefinal  (4.23  29/TT  2  The constant C  the factoring of the bracketed  be e x p r e s s e d  20/TT  —2—T  4w (2n-l  1 -  n=l  Here K i s a c o n s t a n t ,  2  29/TT  w n n=l  to  terms  i n the  t h e S.C. t r a n s f o r m a t i o n  simple  form.  Now s i n w a n d c o s w h a v e t h e f o l l o w i n g product  )  25(f) 2  w  00  second  ^ ,  26/ir  29/TT  w  {  n  infinite  r e p r e s e n t a t i o n s [see Abramswitz and Stegun].  sln w =w n n= CO  COS w  n  4w (2n-l ) T T 2  n  n=  w'  —2—2" TT  2  1  (4 .24)  58  Thus (4.23)  becomes  K (tan w)  dw The c o n s t a n t  K i s determined  for the s i n g l e t r i a n g l e case  (4.25)  2 0 / 7 r  i n t h e s a m e way  done p r e v i o u s l y .  as  Integrating  (4.25) r e s u l t s i n  z =  W  (tan x )  2 9 / 7 r  d  X  (4.26)  + B  F o r W = 0, z = a i s o t h a t B = a i , a n d f o r W = IT/2, Thus K i s given  by  TT/2  b - ai (tan  The c o n s t a n t using  y (equation  the c o n d i t i o n  u -* v  w)  (4.27) 2 6 / 7 T  ( 4 . 1 6 ) c a n now 0  be  TT  determined This  =  requires  that  (4.28)  V  clearly y = K v . 0  Now This  dw  a s I m ( z ) -* °°.  1 1 im y (Im z-*°°) K ( t a n w) 2 0 /  and  Z = b.  t h e d i s s i p a t i o n i n t e g r a l c a n be  i n t e g r a l i s c a l c u l a t e d over  the length  calculated.  o f one s i d e  of  59  an  isosceles  structure, to any  triangle. the  2p  to the  dissipation  other.  t h e n be  Due  The  total  times the  symmetry of the  o v e r any  one  dissipation  dissipation  side  in length  for a single  isosceles  t r i a n g l e , where p i s t h e n umber o f  traversed  in a distance  dx  In t h e  w-plane i n t e g r a t i n g  valent  to i n t e g r a t i n g  stituting  (4.25),  across  the  o v e r one  over the  ( 4 . 2 7 ) , and  is  equivalent dx  will  side  of  the  grooves  grooves  side  groove  (see  Figure  of a groove i s  r a n g e ( 0 - TT/2). (4.28) into  NOW  19c) equi-  sub-  (4.18) r e s u l t s  in  TT/2  v  2p SdA  /z  =  v  a  2 0  (cot w )  c dl_  2  Now and  dw (4.29)  TT/2  (tan w )  where c = ( a  2 9 / T r  + b )  .  2  the  S t e g u n , 1965]  dw  2 9 / l T  general  form f o r a Beta Function  [Abramowitz  is TT/2  B(m,n) = 2  jo  <  s i n  t) "" 2  (cost) ""  1  2  TT/2  Hence  (cot  w)  2 6 / T r  (tan  w)  2 0 / l T  dw  TT/2  dw  2  IT '  n +1 2  2  i  TT'  2  TT  1  dt  (4.30)  60  The  beta-function  is symmetric  B ( n , m ) ] so t h a t ( 4 . 2 9 ) r e d u c e s  SdA  But  =  2 p c dL = dA  to the  f/2  v  2 0  in i t s argument, to  p c d L = 2 S  (see Figure 19c).  0  p c d L  The  Hence f o r flow  = S  n  d  transverse  d A d A  Ai  d i s s i p a t i o n i s thus  (4.32)  increased  in the  dA/dAo c o m p a r e d t o the d i s s i p a t i o n e x p e c t e d  discussed  previously.  ratio  for a flat  T h i s r e s u l t i s t h e same as f o r f l o w p a r a l l e l  with  (4.31)  grooves.  SdA  a s was  [B(m,n)  surface  to the  grooves  Hence f o r a roughened  surface  grooves  c o n s i s t i n g of i d e n t i c a l i s o s c e l e s t r i a n g l e s the d i s s i p a t i o n i s p r o p o r t i o n a l t o t h e e x p r e s s i o n S f dA 1 d A, d A i r r e s p e c t i v e of the d i r e c t i o n of the f l o w w i t h r e s p e c t to the 0  grooves. The  streamlines  t u r e h a v e an a p p r o x i m a t e l y s t r u c t u r e of t h i s form, would of course (4.25) along in the z-plane  above the t r i a n g u l a r groove s i n u s o i d a l shape.  the s t r e a m l i n e  be s i n u s o i d a l .  the l i n e  Im(w)  Using  groove  at the groove the  = Y generates  w i t h an a p p r o x i m a t e l y  For a  struc-  surface  transformation a groove  structure  sinusoidal cross  section  For t h i s type of s t r u c t u r e the d i s s i p a t i o n i n t e g r a l becomes  61  SdA  where  =  v  (f)  p K dL  2 0  cot(x + i y ) |  dx  2 6 / l T  w = x + iy. The  integral  i n (4.33) can  be w r i t t e n  in the  I =  cot(x + iy) |  the  arc length  f o r such dz dw  J The  integral  can  be s e e n u s i n g  dw  in (4.36),  dissipation  = K |  •  =  V  L/2  I-  vo  2  TT  (4.34)  x  (4.35)  cot(x + i y ) \ ~  however, i s the  (4.35).  integral  = - tan  TJO  a groove structure is given  Thus equation  c  SdA  dx  71  identity  •  The  + |cot(x + iy) |  2 9 / 7 T  c o t (x +  The  form  29  IT/2  using  (4.33)  is  p dL  1  =  2  Q  ^  by  dx  same as  (4.36)  (4.34),  which  (4.36) reduces  to  (4.37)  KI  then  c  Sn  dA  S  0  d  A  f d 0  d A,  Al  (4.38)  62  Hence f o r a groove sinusoidal dA  cross-section  s t r u c t u r e of approximately  the damping  i s enhanced  i n the  j ft ' w i t h r e s p e c t t o a f l a t s u r f a c e ; a n d t h e d a m p i n g  ratio  enhance-  1  ment i s i n d e p e n d e n t of f l o w d i r e c t i o n w i t h r e s p e c t to the grooves . 4.4  Calculation  of the Damping F r e q u e n c y  It has been hence to  the damping  SdA  shown p r e v i o u s l y t h a t the damping  i n t e g r a t e d over the c o n t a i n e r s u r f a c e s . frequency a is given i  1  f/(v<i>)T dA / (Vcf)) d r z  w h e r e (V<i>) is t h e t a n g e n t i a l T  complete  by  inviscid  + (SF)  element  (4.39)  v e l o c i t y , dA  e l e m e n t o f a r e a on t h e i n t e r f a c e b e t w e e n wave t a n k , dx i s a volume  The  2  ° = 2  i s an  the water and  i n the f l u i d  r e s u l t is derived in Appendix (4.39)  the  a n d Sp a l l o w s  energy d i s s i p a t i o n at the a i r - w a t e r i n t e r f a c e .  Using  (This  A).  the damping  frequency for a  square  wave tank o f l e n g t h L and d e p t h H w i t h a g r o o v e d base calculated.  The  equilibrium fluid The to  bottom  is  o f t h e t a n k i s a t z = -H w h i l e  now  the  s u r f a c e i s at z = 0 (see F i g u r e 20).  inviscid  a pair of v e r t i c a l  andLiftshitz].  (and  f r e q u e n c y ) f o r s u r f a c e waves i s p r o p o r t i o n a l  e x p r e s s i o n f o r the damping  for  o  velocity  potential  for flow  parallel  w a l l s i n the wave tank i s [ s e e  Landau  *  F i g u r e 20.  C o o r d i n a t e system used f o r the c a l c u l a t i o n the damping f r e q u e n c y .  X  of  64  <f>  A c o s h k ( z + H) c o s kx c o s wt  =  (4.40)  and 3d) v  = x  3x"  A  =  3 (b v  z  =  8z  =  " Y ' S1  A k~  s  i  n  n  n  k x  (  k  z  c o s h  +  H  )  k  c o s  (z  +  k x  H) c o s wt  c o s  A/k = A i  (4.41)  Using these results the integrals  i n (4.39) can  now be c a l c u l a t e d , w h e r e t h e i n t e g r a l s a r e a v e r a g e d oscillation  p e r i o d o f t h e water wave.  (V<t>)2  dx =  1 2  , L  A 4  2  L  i  A 4  2  i  2  o v e r one  Now  | cosh  2  k ( z + H) d z  sinh  k ( z + H) d z  2  Ax  2  2  kLAi2  c o s h 2 k ( z + H)  s i n h 2kH 2k  (4.42)  65  The  ( (Vdp)  2  base term  d A )g = A i  cos  2  for  2  (Vcj))2  cot s i n  The  end  wall  Ar  L A  The  cos  sin  2  2  L Ax  Combining the side walls  and  end  i s  2  k ( z + H)  are  sinh  k ( z + H) d y d z  cosh  2k(z  terms  are  walls  kx s i n h  kx c o s h  dx dy  (4.4.  terms  2  H  L  + H) - 1  x  side wall  Ai  A  kxc o s h  2  4  d  2  cosh  2  k ( z + H) d x d z  k ( z + H) d x d z  2k(z  (4.45)  + H) d z  contributions to yields  (4 .44)  dz  (Vcb)T  dA f r o m  the  66  L A  :  2 c o s h 2 k ( z + H) - 1  L A  sinh  dz  2kH  (4.46)  F o r a l a t e r a l l y m o b i l e a i r - w a t e r s u r f a c e Sp i s g i v e n by  2 v k  and f o r a l a t e r a l l y  immobile  surface  Sp = o-g  where 1968].  = damping The  surfaces. damping  due  (4.48)  2  to the base  r e s u l t s c a l c u l a t e d above  [ P i k e and  cross section  the base term  0  (4.43)  is directly  triangular pro-  f o r a grooved  base,  becomes  A'  W h e r e A' = a r e a  Thus  smooth  i n the  t o a g r o o v e s t r u c t u r e w i t h an i s o s c e l e s  to the a r e a r a t i o d A/d A .  Curzon,  are of course f o r  shown t h a t t h e enhancement  or a p p r o x i m a t e l y s i n u s o i d a l portional  cosh (kH)  f r e q u e n c y due  I t has been  (4.47)  :  ratio.  L  2  (4.49)  67  Using  (4.42),  l a t e r a l l y mobile  (4 . 4 4 ) - ( 4 . 4 7 ) a n d  air-water  cov  sinh  For a l a t e r a l l y  cov  A'  kH  immobile  1  interface  +  2H  kH  is correspondingly 2  will  be  by  s i n h 2kH - 1 kH  + 2vk  (4.50)  2  surface  A' + c o s h ( k H ) + 2H  If the surface i s only p a r t l y covered,  (1 - e ) 2 v k  is given  2  sinh  a for a  (4.49),  decreased,  s i n h 2kH kH  the f a c t o r c o s h  i . e . a term  e cosh  i n c o r p o r a t e d , where 0 < e <  2  kH 1).  (4.51 )  2  and  kH  Chapter 5  DAMPING E X P E R I M E N T S  5.1  Experimental In  this experimental investigation,  t h ecase o f l a r g e  i s c o n s i d e r e d a >> I.  approach  t h e s t u d y o f wave damping  been  used  ( i . e . temporal As h a s been  (mono-1ayers) alter  Section two  5.2).  deep  fluid  previously, contaminants  in the  roughness  s u r f a c e modes a r e c o n -  i t i sp o s s i b l e t o separate thec o n t r i b u t i o n s  from the free s u r f a c e and t h ebase ( s e e I n a l l o f o u r e x p e r i m e n t a l w o r k ( f o r a >>  case  been  used, where one s a t i s f i e s t h e  ( k H << 1 ) a n d t h e o t h e r s a t i s f i e s t h e  c a s e kH > 1 . The  in  remarked  i s used t h r o u g h o u t ) .  When t w o d i f f e r e n t  s u r f a c e modes have  shallow fluid  damping  o f s u r f a c e waves, c a u s i n g problems  s i d e r e d , however, the damping  a s g i v e n i n S e c t i o n 3.1 h a s  investigation o f the effect o f wall  on t h e d a m p i n g .  to  T h e same  p r e s e n t on t h ef r e e s u r f a c e c a n s i g n i f i c a n t l y  t h e damping  experimental  GROOVES  System  groove dimensions to  FOR L A R G E A M P L I T U D E  geometry  o f t h e s q u a r e l u c i t e wave tank  this experimental investigation 68  used  i s shown i n F i g u r e 2 1 .  l),  electrode  wate  g u r e 21.  Cross-section  o f the s q u a r e wave  tank.  70  Essentially,  t h e same e x p e r i m e n t a l  S e c t i o n 3.1 h a s b e e n u s e d .  A grooved  m o u n t e d on t h e b o t t o m o f t h e t a n k a s i l i c o n e adhesive was n e c e s s a r y damping caused plate. to  (General  i n order  The grooves  on i t w i t h  This  anomalously  procedure  l a r g e wave grooved  having  a cross s e c t i o n i n the form dimensions  o f an  are given i n  2) . The  waves a r e a g a i n e x c i t e d by a h o r i z o n t a l  mounted above the f l u i d previously  i s used.  s u r f a c e ; t h e same e l e c t r o d e  the top of the tank.  of the e l e c t r o d e above the f l u i d four levelling  series of experiments separated  experiment  The o u t p u t  by s e q u e n t i a l  surface i s adjusted  v o l t a g e wave g e n e r a t o r  incorporates  timing  d i f f e r e n t from two n o v e l  i s determined  used  by  In t h i s  for this  t h a t used  features.  previously.  A l l of  by a c r y s t a l c o n t r o l l e d m a s t e r  waveform of the generator  scanning  height  d i s t a n c e o f ( 1 . 9 ± 0 . 1 ) cm.  was c o m p l e t e l y  generator  generator  high  a  the e l e c t r o d e and water s u r f a c e were  by a c o n s t a n t The  The  screws a t the ends o f the c r o s s .  electrode  described  The e l e c t r o d e i s suspended from  lucite cross fixed across  clock.  RTV).  be  on t h e p l a t e w e r e i d e n t i c a l a n d p a r a l l e l  i s o s c e l e s t r i a n g l e ( t h e groove  The  aluminum p l a t e could  a n d was s e c u r e d  Electric  to prevent  described in  by s e e p a g e o f t h e w a t e r u n d e r t h e  one another,  Table  procedure,  (with pulses from  of a series of voltage r e g i s t e r s .  i s constructed  the master c l o c k ) ,  71  Table 2 Experimental  Features o f the Water  Conditions  Tank  Depth Length,  L  Kinematic  Properties  of Grooved  Groove Apex A  1  viscosity  ± 0 . 0 1 ) cm  (26.7  ± 0 . 1 ) cm  0.01 c m  2  sec-  1  Base  Spacing  (3.2  Angle  (Area  (3.01  ± 0.1)  mm  90°  Ratio)  /2  C h a r a c t e r i s t i c s o f Modes Mode 1 ( S l o s h i n g Mode) Wave n u m b e r , k Oscillation Boundary  Mode  0.1178  Frequency,  f  Layer Thickness =  cm"  1  ( 1 . 0 2 0 ± 0 . 0 0 1 ) Hz /v/2irf  0.40  mm  2 Wave n u m b e r , k Oscillation Boundary  0.3533  Frequency,  Layer  f  Thickness  cm  - 1  ( 2 . 6 3 0 ± 0 . 0 0 1 ) Hz 0.25  mm  72  A schematic generation circuit the o s c i l l a t o r 800  in stages  delayed  i s shown i n F i g u r e 22.  (crystal B and  D occurs  B, a n d  C respectively.  the j i t t e r 0.5  msec).  C.  and  from  C is  The  output  of the  pulses  Since the j i t t e r  in the  the i n t e r v a l  B, t h e t i m i n g a c c u r a c y  between  i s c o n t r o l l e d by  B.  This j i t t e r  is less  D) p l u s t h e o s c i l l a t o r  variable frequency  output  pulses.  E (programmable d i v i d e r National  7520) c o n s t i t u t e the c l o c k p u l s e s which  r a t e at which generator, generator  are scanned.  A schematic  i s shown i n F i g u r e 23.  corresponding  Each  r e g i s t e r can  5 volts.  output  the  in the f u n c t i o n of the f u n c t i o n  The memory c o n s i s t s o f 128  level  l e v e l s , with  z e r o to 0  be p r o g r a m m e d t o a n y  To c o n s t r u c t a n y  The  determine  diagram  r e g i s t e r has  to 5 v o l t s and  produces  Semi-Conductor  the v o l t a g e r e g i s t e r s c o n t a i n e d  registers.  Thus each to  from  usee.  pulses from  128  - 2.5  of the pulses coming from  high s t a b i l i t y  storage  pulses  output  i n c o i n c i d e n c e w i t h one  T h i s u n i t ( B , C, a n d  DM  The  pulse i s l e s s than  pulses coming from  The  waveform  c o n t r o l l e d ) a r e d i v i d e d by 10  a l s o r e s e t s B and  delay unit output  than  of the high v o l t a g e  by D ( v a r i a b l e d e l a y 0.5  p u l s e from from  diagram  DC  16  level  volts.  level  from  0  p e r i o d i c p o s i t i v e waveform,  o n e - h a l f of the waveform i s c o n s i d e r e d .  This h a l f waveform  i s t h e n d i v i d e d i n t o 16 e q u a l  The  of  sections.  desired  t h e w a v e f o r m i s t h e n d e p o s i t e d by t h e u s e o f t h e  address  in a successive storage r e g i s t e r .  To  value manual  i11ustrate  this  I MHZ  DIVIDE  OSCILLATOR  DIVIDE  BY IO  COINCIDENCE  BY 8 0 0  CIRCUIT  B RESET  PROGRAMMABLE 1  FUNCTION GENERATOR  DIVIDER  IE  PROGRAMMABLE POWER SUPPLY  F  FREQUENCY COUNTER H  F i g u r e 22.  High v o l t a g e  w a v e f o r rn g e n e r a t i o n  circuit  INPUT ADDRESS  count U P COUNTER LOGIC  L  MEMORY  UD count down,  COUNTER  DIGITAL TO ANALOGUE CONVERTER  T  MANUAL ADDRESS DATA ENTRY  Figure 23.  Function  generator.  INTEGRATOR  75  procedure,  the programmed v o l t a g e l e v e l s o f t h e r e g i s t e r s  to produce a simple output  p u l s e ( s e e F i g u r e 2 4 c ) a r e shown  in Figure 24a. The  output of the function generator i s derived  from the storage r e g i s t e r s . to ( t h i s f r e q u e n c y  When a c l o c k p u l s e o f f r e q u e n c y  b e i n g c o n t r o l l e d by t h e o u t p u t  E) i s a p p l i e d t o t h e i n p u t a d d r e s s , an o u t p u t  pulses  from  signal of  to/32 i s p r o d u c e d b y s w e e p i n g u p a n d d o w n t h r o u g h  t h e 16  s u c c e s s i v e r e g i s t e r s o f t h e memory.  l o g i c and  The counter  t h e UP/DOWN C o u n t e r c o n t r o l t h i s s w e e p i n g t h r o u g h The 24b.  scanning cycle f o r this  the registers.  i s shown s c h e m a t i c a l l y i n F i g u r e  The o u t p u t f r o m t h e memory ( s t o r a g e r e g i s t e r s ) i s a  digital  s i g n a l and t h i s  i n t h e D/A c o n v e r t e r .  i s converted  t o an a n a l o g u e  result  The b u f f e r and i n t e g r a t o r a r e used  to smooth o u t t h e r e s u l t i n g waveform.  For the register  levels  given i n Figure 24a, the r e s u l t a n t output o f the f u n c t i o n g e n e r a t o r w o u l d be a s i n F i g u r e 2 4 c , w i t h t h e f r e q u e n c y o f this output  p u l s e b e i n g c o n t r o l l e d by t h e p u l s e s f r o m  The  output of the function generator  a KEPCO p r o g r a m m a b l e h i g h v o l t a g e power s u p p l y . operating  E.  i s applied to This unit  i n t h e p r o g r a m m a b l e mode a c t s a s a l o w f r e q u e n c y  a m p l i f i e r with a gain o f 1000. is applied d i r e c t l y  The o u t p u t o f t h e Kepco u n i t  to the electrode.  76a  Figure  24.  The c o n s t r u c t i o n o f an a r b i t r a r y waveform the f u n c t i o n generator. (a)  R e g i s t e r v o l t a g e ( d e p o s i t e d by m a n u a l a d d r e s s s y s t e m ) vs r e g i s t e r number  (b)  S c a n n i n g c y c l e ( d e t e r m i n e d by p u l s e s from E ( F i g u r e 22) and l o g i c o f up-down c o u n t e r  (c)  Resultant  waveform  using  clock switching  76b  77  5.2  E x p e r i m e n t a l R e s u l t s and D i s c u s s i o n E x p e r i m e n t s were c o n d u c t e d to d e t e r m i n e the  f r e q u e n c y f o r both a smooth  l u c i t e base and a g r o o v e d  p r e s e n t i n t h e wave t a n k .  The e x p e r i m e n t s p e r f o r m e d  i n v e s t i g a t i o n were p e r f o r m e d described  i n S e c t i o n 3.1.  i n an i d e n t i c a l  As was  remarked  s u r f a c e modes were c o n s i d e r e d i n t h i s a p p l y i n g the high v o l t a g e waveform s u r f a c e modes o f w a v e l e n g t h  considered  b a s e was 90°).  ( A ) , 2L a n d  manner to those  p r e v i o u s l y , two By  ( 2 / 3 ) L c a n be The  excited,  frequency of the  i s a d j u s t e d t o r e s o n a t e w i t h t h e mode  the damping  d e t e r m i n e d f o r two  T h i s was  in this  to o n e - h a l f of the e l e c t r o d e  ( 1 . 0 2 Hz f o r 2L a n d 2 . 6 3 Initially  base  investigation.  where L i s the l e n g t h o f the wave t a n k . high v o l t a g e waveform  damping  for (2/3)L) .  frequency f o r the  flow directions  smooth  ( s e p a r a t e d by  done to e n s u r e t h a t t h e wave tank d i d n o t  p o s s e s s any d i r e c t i o n a l  damping  been c h e c k e d , any d i r e c t i o n a l  properties. damping  base p r e s e n t m i g h t e r r o n e o u s l y have  I f t h i s had  e f f e c t s with the been a t t r i b u t e d  not grooved  to the  grooves. For the smooth properties.  b a s e we  With the grooved base p r e s e n t , damping  conducted for flow parallel g r o o v e s , f o r both modes.  The damping  damping  runs  were  as w e l l as t r a n s v e r s e to t h e  In o u r i n i t i a l  had d i s c o v e r e d a d i r e c t i o n a l base p r e s e n t .  f o u n d no d i r e c t i o n a l  damping  r u n s , we  thought  e f f e c t with the  f o r flow parallel  to the  we  grooved grooves  78  was  consistently larger  grooves.  than f o r f l o w t r a n s v e r s e to  T h i s r e s u l t was  d a m p i n g f r e q u e n c y was  bottom  o f t h e t a n k , we  allowed water  r e p r o d u c i b l e ; the value of  not.  s i l i c o n e adhesive, which  the  On c l o s e e x a m i n a t i o n  secures the grooved  of  base  the the  to  the  found small cracks in the s e a l e r which  to get under  the base.  These  c r a c k s were  l o c a t e d mid-way a l o n g one w a l l , where t h e p l a n e o f t h i s w a l l was  t r a n s v e r s e to the g r o o v e s .  directional  damping e f f e c t s .  This accounted  For flow along the grooves  v e l o c i t y a t the c r a c k s i n the s e a l e r would t h e r e would  to f l o w  For f l o w t r a n s v e r s e to the grooves,  i s a minimum.  The  for this case parallel s e a l e d , no  s e a l e r would  ( f l o w t r a n s v e r s e to grooves)  to the g r o o v e s . directional  d a m p i n g e f f e c t was  than f o r flow  observed. The  Typical  straight  these graphs  ( F i g u r e 27-28) a r e shown i n F i g u r e 2 5 - 2 6 . t h a t the a m p l i t u d e s f o r F i g u r e s 25a, T h i s i s due  the time a f t e r  t h u s be s m a l l e r  lines  by a l e a s t s q u a r e s f i t t o t h e d a t a p o i n t s .  The o s c i l l o s c o p e t r a c e s from which  different.  caused  A f t e r these c r a c k s were p r o p e r l y  d a m p i n g r u n s a r e shown i n F i g u r e 2 7 - 2 8 . are determined  flow  enhancement i n the damping  by t h e c r a c k s i n t h e s i l i c o n e  into  the  c r a c k s a r e e s s e n t i a l l y on t h e n o d a l l i n e w h e r e t h e velocity  the  be l a r g e , a n d  be e n h a n c e m e n t o f t h e d a m p i n g d u e  these cracks.  f o r the  25b  to the p o s i t i o n  s w i t c h - o f f of the f i e l d  are d e r i v e d  It will and 26a,  be 26b  noted are  o f the d e t e c t o r and at which  the r e c o r d i n g  79a  F i g u r e 25.  D a m p i n g r u n s f o r X = -| L ( h o r i z o n t a l s c a l e 0.05 v / d i v , v e r t i c a l s c a l e 2 s e c / d i v ) . (a)  smooth  base  (b)  smooth base w i t h f l o w d i r e c t i o n r o t a t e d 90° w i t h r e s p e c t to (a)  (c)  grooved base with flow p a r a l l e l the g r o o v e s  (d)  grooved base with flow t r a n s v e r s e to the g r o o v e s  to  79b  80 a  F i g u r e 26.  Damping runs 5 s e c / d i v ).  f o r A = 2L  (horizontal scale  (a)  smooth base  ( v e r t . 0.05  v/div)  (b)  smooth base with flow d i r e c t i o n r o t a t e d 9 0 ° w i t h r e s p e c t t o ( a ) ( v e r t . 0.1 v / d i v and 0.05 v / d i v )  (c)  grooved grooves  (d)  grooved base with flow t r a n s v e r s e to t h e g r o o v e s ( v e r t . 0 . 2 , 0.1 a n d 0.05 v/div)  base with flow p a r a l l e l to the ( v e r t . 0.1 v / d i v a n d 0.05 v / d i v )  81 a  F i g u r e 27.  The l o g a r i t h m o f wave a m p l i t u d e vs U - f L).  time  (a)  smooth  base  (b)  smooth base w i t h f l o w d i r e c t i o n r o t a t e d 90° w i t h r e s p e c t to (a)  (c)  grooved base with flow p a r a l l e l grooves  (d)  grooved base with flow t r a n s v e r s e to grooves  to  82a  F i g u r e 28.  The  l o g a r i t h m o f wave a m p l i t u d e vs t i m e  (X =  (a)  smooth  (b)  smooth base w i t h f l o w d i r e c t i o n r o t a t e d by 9 0 ° w i t h r e s p e c t t o ( a )  (c)  grooved base with flow p a r a l l e l  (d)  grooved base w i t h flow t r a n s v e r s e to grooves  2L)  base  to  grooves  LN A  0  5  10  15  20 TIME  seconds  25  30  35  a?  83  of  the amplitude started.  From t h e s e g r a p h s , i t c a n be s e e n  that the damping frequency i s independent o f the d i r e c t i o n of  the flow v e l o c i t y with r e s p e c t to the grooves, f o r both  modesconsidered. In o u r e a r l i e r e x p e r i m e n t s been a p r o b l e m . distillation  r e p r o d u c i b i l i t y had  In t h e s e r u n s d i s t i l l e d  s y s t e m had been  conductivity distilled  water  used.  water  from a copper  In o u r l a t e r runs low  f r o m a g l a s s s y s t e m was  used.  B e f o r e a s e r i e s o f r u n s , t h e wave t a n k was r i n s e d o u t w i t h this d i s t i l l e d water before f i l l i n g  the tank.  With  these  p r e c a u t i o n s , r e p r o d u c i b l e r e s u l t s were o b t a i n e d f o r t h e damping  frequency. According to the theoretical  S e c t i o n 4.2, t h e damping o v e r g r o o v e d  model d e v e l o p e d i n s u r f a c e s s h o u l d be  independent o f the flow d i r e c t i o n , and the enhancement i n the d a m p i n g s h o u l d be p r o p o r t i o n a l t o t h e f r a c t i o n a l of  the surface.  When c o m p a r i n g  this theory, i t i s essential experimental  our experimental results  of  The s h o r t e s t wavelength  ~ 18 cm, w h i c h  (~ 0.3 c m ) . fluid  depth  the theoretical  grooved aluminum base used  s a t i s f i e s a l l of the assumptions model.  with  t o c o n s i d e r how w e l l t h e  conditions approximate  The  area increase  used  in these  model. experiments  i n the theoretical  s u r f a c e mode h a s a l e n g t h  i s much l a r g e r  than the groove  spacing  T h e g r o o v e d e p t h i s a l s o much s m a l l e r t h a n t h e (~ 3 c m ) .  Finally  the t h i c k e s t boundary  layer  84  used  ( f o r t h e l o w e s t f r e q u e n c y mode) i s ~ 10% o f t h e  spacing  (see Table Another  3). e f f e c t which  between the experimental  could cause  c o n d i t i o n s and  a discrepency  t h a t of the  i s the c o n d i t i o n of the a i r - w a t e r i n t e r f a c e . the damping frequency and  laterally  water  theory,  In S e c t i o n  is calculated for a laterally  immobile  surface.  i n t e r f a c e may,  The  might  cover  c o n d i t i o n of the a i r -  only a part of this  In T a b l e I I I t h e p r e d i c t i o n s o f e q u a t i o n the damping frequency  equation for  (4.51)  for a laterally  for a laterally  immobile  mobile  The  experimental  also tabulated.  The  u n c e r t a i n t y i n a i s ~ 1%.  mobile  by t h e p r e d i c t i o n s o f e q u a t i o n  values for a laterally the observed  immobile  values.  t h e o r y and a laterally good  and  given  As  Table  r e s u l t s are well (4.50)  (the  laterally  v a l u e s o f a.  There  experiment mobile  would  that the  theoretical  s u r f a c e are too l a r g e to If a monolayer  o n l y a p a r t of the s u r f a c e a would mobile  surface  surface layer thoery). F r o m T a b l e 3, i t c a n be s e e n  for  surface.  a from F i g u r e 27-28 are  shows, f o r both modes the e x p e r i m e n t a l  described  a  (4.50)  surface are  both modes.  3  4.3  mobile  h o w e v e r , be n e i t h e r o f t h e a b o v e ;  s u r f a c t a n t monolayer  for  groove  were p r e s e n t  be l a r g e r t h a n  n o t be g o o d  for this case.  the  agreement  H e n c e , we  on  laterally  between  are d e a l i n g with  surface layer in these experiments.  agreement between our experimental  account  r e s u l t s and  the  The results.  Table 3 V a l u e s o f t h e Damping Frequency, a ( s e c  Mode 1  Experimental  (wavelength = 2 L) smooth  base  grooved  base  (±  V/o)  - 1  )  Theory  Theory  ( L a t e r a l l y mobi1e air-water interface)  ( L a t e r a l l y immobile air-water interface)  0.035  0. 0 3 4  0.064  0. 0 4 5  0. 0 4 5  0.075  0.039  0.041  0.105  0.050  0. 0 5 2  0.115  Mode 2 (wavelength = 2L/3) smooth grooved  base base  CO  86  predicted  by e q u a t i o n  (4.50) confirm  that  the enhancement  t h e d a m p i n g c a u s e d by a g r o o v e d b a s e i s p r o p o r t i o n a l fractional  5.3  area  increase  of the  two  t h e o r e t i c a l l y and  increase  sinusoidal  groove cross  This  in Part B not  t o be  I t has the v a r i a b i l i t y vations  simple  sections).  of high  damping  then  low  be  to the  grooves;  to  the  approximately  former r e s u l t is  i t w o u l d seem t h a t be d e p e n d e n t o n  i n t u i t i v e p r e d i c t i o n has  the  the  flow been  shown  in Part A for a < %  i n t h e d a m p i n g i s due  provide  dynamic p r o p e r t i e s  frequency  The  intuitively  been e s t a b l i s h e d  Non-uniform surface high  respect  is  valid.  of damping then  in time) should  both tank  at a rate proportional  damping f o r a grooved base should direction.  shown  t h a t the damping  o f the b a s e ( f o r t r i a n g u l a r and  more s u r p r i s i n g , s i n c e  the  l),  d i r e c t i o n with  the damping i s i n c r e a s e d  I t was  results  ( f o r a s q u a r e wave  a g r o o v e d b a s e s a t i s f y i n g a >> flow  surprising)  i n P a r t B.  experimentally  i n d e p e n d e n t of the  area  base.  s i g n i f i c a n t (and  which have been e s t a b l i s h e d  and  the  Conclusions There are  with  to  in  a very  convenient  of two-dimensional  field  be  regions.  studied  method of physical  Obser-  of surfactants)  than  studying systems.  by a p p l y i n g  to the w a t e r s u r f a c e .  (large concentration  amplitude  to s u r f a c t a n t s .  ( p a r t i c u l a r l y in space, rather  films could  electric  that  a Regions would  REFERENCES  Abramowitz and I.A. Stegun ( 1 9 6 5 ) . Handbook o f mathematical f u n c t i o n s (Dover P u b l i c a t i o n s , Inc New Y o r k . K.M.  Case  a n d W.C.  F.L. Curzon  P a r k i n s o n , J . F l u i d M e c h . , 2,  172  a n d R . L . P i k e , C a n . J . P h y s . , '46, 2001  F . L . C u r z o n a n d M.G.R. P h i l l i p s , 21 75 ( 1 9 7 3 ) . •  Can. J . Phys.,  R. D o r r e s t e i n , P r o c . K o n i n k l . N e d . A k a d . BB4 , 2 6 0 a n d 3 5 0 ( 1 951 ) .  (1 968).  51, —  Wetenschap,  G.N.  I o n i d e s and F.L. C u r z o n , (1971).  Can. J . Phys.,  49, 2733 —  G.N.  I o n i d e s and F.L. Curzon, (1972).  Can. J . Phys.,  50, 2698  G.N.  Ionides (1972). Ph.D. d i s s e r t a t i o n British Columbia).  L . D . L a n d a u a n d E.M. L i f s h i t s - (1959). ( P e r g a m o n P r e s s , L o n d , New Y o r k ) .  (1957).  —  (University of Fluid  Mechanics  V . G . L e v i c h , A c t a . P h y s i c o c h i m. , U . S . S . R . , 1_4, 3 0 7 a n d 321 ( 1 9 4 1 ) . V.G. L e v i c h , P h y s i o c h e m i c a l H y d r o d y n a i m c s , ( P r e n t i c e Hal 1 , N . J . , 1 968) .  87  Chapter  11  88  12.  N. L e v i n s o n a n d R.M. (Holden-Day  Redheffer (1970).  Complex V a r i a b l e s  I n c . , San F r a n c i s c o , L o n d o n ) .  13.  R.L. P i k e and F . L . C u r z o n ,  14.  R.L. P i k e ( 1 9 6 7 ) . Ph.D. d i s s e r t a t i o n ( U n i v e r s i t y o f B r i t i s h Columbia. M.R. S p i e g e l . C o m p l e x V a r i a b l e s ( S c h a u m P u b l i s h i n g C o . , N.Y. ) .  15.  Can. J . Phys.,  46, 2009  (1968).  APPENDIX A  THE  GENERAL  RESULT  FOR THE  a wave tank  damping frequency  z  geometry as given  by F i g u r e 20. The  d u e t o t h e w a l l s o f t h e wave tank  now b e c a l c u l a t e d . surface will  Without  a n d the form  surfaces.  of this result will  In p a r t i c u l a r , t h i s will  be d o n e f o r t h e b o t t o m s u r f a c e o f t h e w a v e equation  will  loss o f g e n e r a l i t y , a due t o one  be d e t e r m i n e d  be a p p l i e d t o t h e o t h e r  The  FREQUENCY  7 ( V < b ) f dA / (Vcb) dx  cov  Derivation o f a =  Consider  DAMPING  o f motion  tank.  f o r a viscous fluid,  given  previously i s 3v p  Jt  "  =  V  P  1A.  +  "  p  g  +  v  P  y  2  1  (  A  where  1 Equation  =  v_i_; y_o_ = Vd>  ( A . l ) i s now m u l t i p l i e d b y y_ a n d t h e e q u a t i o n i s  integrated over  the fluid  volume. 89  (A  90  a  3t  f  ,  v P  l + v • ( V P + p g ) dx =  2  "2".  Iii t h e a b s e n c e zero.  V  V  V  o f v i s c o s i t y t h e RHS  2  v  dx  of (A.3) i s  T h e t e r m _y_ • (VP + g ) m u s t t h e n b e t h e r a t e o f  change  P  of the p o t e n t i a l  e n e r g y , i . e . y_ • ( V P + p g ) = —• ( P . E . ) .  T h e LHS  i s then the time rate  total  of (A.3)  energy E  of change  of the  T  _3_ 3t  E dx  Pv v • V  T  a caused by the bottom  Since calculated  (A.3)  Viwill  have t  V<j)(t)  v1  the  v dx  (A.4)  surface  i s to be  form  - z ( l + i )  exp  Y  (A.5)  + i o ) t  where y = /Z  I  Now  in equation  (A.4)  are  the real parts  the q u a n t i t i e s  v)  t .N o t e bottom  surface  0  in  of the v e l o c i t y terms, i . e .  (Re  (Re  we a r e i n t e r e s t e d  (Re V  2  v)  v i)  •  (Re V  2  v ) x  dx  d x + (Re v ! ) ( R e V  t h a t Vcj) i s e v a l u a t e d o f the tank.  on  the  2  v,)dx  boundary,  (A.6) i.e  91 Now  f o r v i i n t h e f o r m g i v e n by  "(Re V i M R e V where the bar above  z V l  )  (A.5)  = Re(v  term of (A.6) r R e H v T U e V"  v j  0  V  2  dx = Re  Vi  v * 2  of the s u r f a c e .  2i V  2  + i )  v! (1  2  The  2  Y  2  v* 2  dx  dx  (A.8)  dx = Re  8v 8Z  (v ) 0  i s a s u r f a c e e l e m e n t on t h e b o t t o m The boundary  are  indicates a  term of (A.6) i s  Re v  w h e r e dA  (A.7)  i s then  Re The f i r s t  v,)*)  2  t h e e x p r e s s i o n s o n t h e LHS  t i m e a v e r a g e o v e r many o s c i l l a t i o n s second  (V  x  t h a t y_ = 0.  -Y  (1+1)  then  total  tank.  energy  tank  becomes  V*(t)  V4>(t)  The  of the  (A.9)  c o n d i t i o n s on t h e s u r f a c e s o f t h e  E q u a t i o n (A.9)  Re  dA  dA  ^ 2  i s g i v e n by  dA  (A.10)  92  VcD(t)  Equation  ( A . 4 ) i s t h e n g i v e n by _8_  V(j)(t)  at  For  convenience  a time dependent  9_ ^r-r  at  Vcj)  For driving will  T  = -v y P  V<j>(t) c a n b e b r o k e n  dA  into a spatial  t)  Vcj)  (A.n)  part and  f(t)  (A.12)  = 0.  a l i n e a r v i s c o u s f l u i d which  m e c h a n i s m h a s been  i s damping ( i . e .  turned off) the inviscid  decay e x p o n e n t i a l l y with time.  f(t)  velocity  Hence f ( t ) i s g i v e n by  ~ e-at  where a i s t h e damping f r e q u e n c y . becomes  V<j>(t)  part  Vd>(  where  d  (A.13) Equation  (A.11)  then  93  _9_  3t f ( t )  -v  V<J)  mil  at  -  (A.15) i s completely  to any s u r f a c e o f the tank, over  the surface considered.  of a from  f(t)  -v Y f ( t ) 2 7  i  The r e s u l t  Y  / ( V  Vcb  ^S (vcb)  / (V(b) 7 (V(b)  general  2  z  dA  d A z  (A.14)  dx  dA dr  (A.15)  a n d c a n be a p p l i e d  the surface i n t e g r a t i o n being The t o t a l  a i s then  a l l o f t h e s u r f a c e s o f t h e wave  tank.  the  sum  

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