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Experimental investigation of the Rayleigh-Taylor instability Popil, Roman 1979

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EXPERIMENTAL  INVESTIGATION  OF THE R A Y L E I G H - T A Y L O R INSTABILITY by Roman B.Sc,  A THESIS  Popil University  SUBMITTED  OF  Regina,  FOR THE DEGREE  SCIENCE  in  THE  F A C U L T Y OF GRADUATE  Department  We  accept to  of  this  the  thesis  as  conforming  standard  OF B R I T I S H  April, @  STUDIES  Physics  required  THE U N I V E R S I T Y  ]977  IN PARTIAL. FULFILLMENT  OF THE REQUIREMENTS MASTER  of  ]979  Roman P o p i l , 1 9 7 9  COLUMBIA  OF  In p r e s e n t i n g t h i s  thesis  in p a r t i a l  f u l f i l m e n t o f the requirements f o r  an advanced degree at the U n i v e r s i t y of B r i t i s h the L i b r a r y s h a l l make i t  freely available  I f u r t h e r agree t h a t permission for scholarly by h i s of  this  written  I agree  r e f e r e n c e and  f o r e x t e n s i v e copying o f  this  It  is understood that copying or  for financial  gain s h a l l  that  study. thesis  purposes may be granted by the Head of my Department  representatives. thesis  for  Co 1umbia,  or  publication  not be allowed without my  permission.  Department of  Pti^SlC  The U n i v e r s i t y o f B r i t i s h  9  Co Iumbia  2075 Wesbrook Place Vancouver, Canada V6T 1WS  5-  ABSTRACT  The air  Rayleigh-Taylor  interface  photographic accelerates  was  investigated  methods.  produces  reproducible  standing  surface  tank  wave  The  e l e c t r i c a l measurements  the  bulk  of  the  water  features graphs  of  of  films  tank. the  the  the  phenomenon than  developed  for  of  fluid  Several  measuring  the  waves.  ii  and in  and  substantiated  a new  films  linear  and  at  were  of  and  sinusoidal amplitude.  addition on the  to  walls  other by  photo-  produced.  scaling  law  for  accelerations  devices  amplitudes  which  a pure  produced  o f these  l e d to  climbing  gravity.  are  are  that  i n s t a b i l i t i e s that  e l e c t r i c a l measurements  greater  revealed  existence  The  water-  downwards  o f known p h a s e  instability  a  described  of water  of water  The  various  is  i n s t a b i l i t i e s from  water  of  using e l e c t r i c a l  An a p p a r a t u s  a rectangular  motion,  instability  were  surface  also water  T A B L E OF  1)  INTRODUCTION  2)  CHAPTER  3)  II  CONTENTS  1  EXPERIMENTAL  A.  Tank a c c e l e r a t i o n  B.  Tank  C.  Wave e x c i t a t i o n  D.  The  E.  The I . C .  CHAPTER  and  system  Analyzing  electrical  system  A.  Capacitance  B.  Resistance  C.  "Cavitation" effects  D.  Photography  E.  Electrical  F.  Test  G.  Climbing  .  20  measuring  circuit  .  . . .  the  33 37  fluid  motion  measurements,  narrow  air  23  26  measurements  of  .  RESULTS  measurements  the  48 tank  . . .  film  63 68  films  73  I V CONCLUSION  81  4)  CHAPTER  5)  BIBLIOGRAPHY  6)  APPENDICES A. D e t a i l s B.  12 . 16  EXPERIMENTAL  for  11  Plates  capacitance  III  APPARATUS  Optical  84  of  the  waveform  Fourier  analysis  - i i i -  generator  . . . .  85 91  LIST  OF T A B L E S  I  Wave E x c i t a t i o n E l e c t r o d e s  18  II  Resistance  34  III  Fourier  Cavitation  IV  Effect  of  V  Climbing  VI  Digital  Rate  Analysis  and  Tank W i d t h  V i s c o s i t y on  Cavitation  41 Rate  .  Films Results Program  for  Wave G e n e r a t o r  -iv-  .  44 75  . . . .  90  LIST  1)  OF F I G U R E S  Photograph of a R a y l e i g h - T a y l o r I n s t a b i l i t y showing the s p i k e and t h e b u b b l e .  7  2)  Side view  13  3)  Front  A)  Horizontal  section  5)  The  harmonic  6)  Wave g e n e r a t i n g  7)  Block  8)  I.C.  9)  Response  of  view  fourth  the  of  tank  the  capacitane  Output  11)  Spectrum of  12)  Results  13)  Bridge  1A)  Capacitance mean  of  the  of  accelerator the  water  tank  .  .  .  .15  I.C.  17  assembly  . . . .  e l e c t r i c a l system  measuring to  second  the  water  output  vs.  time  .  .22 2A  changes  . . .  bridge  as  the  wave  . . .  28  dielectric  29  oscillographs  measurements,  25 27  harmonic  using  .  19  circuit  depth  impedance  the  1A  Fourier plate  the  the  10)  tank  electrode  diagram of  of  accelerator  decrease  . . .  31  in  level  32  15)  Resistance  Fourier  16)  Linear  17)  Calibration  18)  "Cavitation" rate,  19)  Cavitation rate  20)  Impedance  decrease of  of the  bridge  for  analysis tank  bridge  w i d t h 3 cm .  a 30% s u g a r  test  changes  in capacitance  21)  Initial  unperturbed  22)  First  23)  Fourth harmonic  h a r m o n i c wave  capacitance  impedance tank  35  to  . . .  39  . . .  AO  .  .  .  solution  A2 . A5  transient  surface  A7 photograph  instability  i n s t a b i l i t y shot  .  . A9 51 53  24)  Fourth harmonic  25)  Photograph  26)  Side  27)  "Wetting" flash  28)  Eigth  29)  Fourth  30)  Bridge output  31)  Resistance  32)  Validity  of  33)  Test  presence  34) 35)  on  of  flash  i n s t a b i l i t y of  the  .  .  wave  flash wave  57  for  check  growth  direction  film  Phototransistor  results  for  air  B r i d g e runs  a p.c.  board  plate films  .  level  tests  air  .  64 .  . . . .  67  film  . . .  71  tank  74  Climbing  behaviour,  37)  Logarithmic plot  38) 39)  C l i m b i n g f i l m s r e s u l t s s c a l i n g law graph Schematic o f . t h e d i g i t a l waveform generator  for  40)  Waveform g e n e r a t o r  41)  Up/down counter  42)  Set-up  graph  climbing  .  .  films  .  .  .  76  . . .  77  memory  address  optical  66  70  36)  for  62  . . . .  i n mean  an  for  tank  harmonic  change  of  60  a narrow  fourth of  58  photograph in  55 56  photographs  harmonic  capacitor  0  photographs  harmonic  for  standing  phase  -vi-  78 86 87  logic  Fourier  .  analysis  88 . . .  92  ACKNOWLEDGEMENT  This auspices stuggling found  to  of  project  Dr.F.L.Curzon  endeavours be  both  proved  to  assisted  and  be with  and  carried whose  out  keen  enthusiasm  encouraging  A.Cheuck's electronics  was  and  technical  innumerable  the  -vii  interest for  machine  i n my  science  assistance  other  the  I  inspiring. with  the  technicalities  invaluable. M. Heinrich much o f  under  and  shop  G.Auchinleuck  work.  -1-  CHAPTER  I  INTRODUCTION  Rayleigh-Taylor there  is  an  acceleration  superimposed from  the  fluids  less  dense  fluid  system  of  unstable  an  interface the  is  is  to  the  in  the  stellar  so  the  They  occur  plasma  by  unstable  most  a glass  acceleration  is  the  the  unstable. lower into  to  interface long  the  is  as  due  to  air  and  thus  spikes.  it It  a  to  two  directed  binary  condition  However,  prior  instead falls, is  the  and  the  i f  the  acceleration, with  the  time,  giving  of  physics  vacuum to  interface.  accelerate or  a  the  tokomak.  be  taken  to  be  the  upside  down.  Here,  the  gravity the of  i n plasma  z-pinch  can  nature  Rayleigh-Taylor  made  a  in  acceleration  plasma  is  in  turned  that  as  exist.  is  the  undisturbed  interest  attempt  field  water  The w a t e r ,  If  of  a common o c c u r r e n c e  common e x a m p l e of  fluid.  surfaces.  current  a magnetic  case  is  icicles,  surface  whenever  the  water  of  of  interface  exponentially  on s t e l l a r  are  whenever  Rayleigh-Taylor instability.  instability  Perhaps of  grow  occur  acceleration  denser  perturbed  called  formation  planar  the  completely  will  material  where  the  that  the  initially  instabilities  of  equilibrium will  This as  to  left  perturbation  rise  such  instabilities  and  is  directed  air-water  maintaining  will  tend  formation  to and  from  interface a nearly  be rate  is plane  deformed of  growth  -2of  these  spikes  experimental  I.  Taylor  air  Shortly  after  (1950),  the  column  above  (1959)  greater of  the  takes  accelerated and T a n k i n similar was  was  of  difference  tubing. the the  Ratafia  where  who s u b j e c t e d  pressure  fluid  heavier have  a rectangular  downward at  accelerations  lighter  (1973)  a  downward by  atmospheric  the  the Watson  i n which  At  a  in  Emmons, Chang and  direction of and  by G .  investigated  accelerated  rubber  role  paper  experiment  gravity,  the  experiments  one.  Cole  performed  tank  accelerations  being  of  water  several  times  gravity.  The by  of  (1972)  than  water.  several  investigations.  was  (1950),  a similar  i n the  accelerated  greater  the  of  theoretical  a pressure  of water  that  topic  theoretical  the  to  stretched  than air  out  tank  of  the  instability  of water  carried  force  been  by D . J . L e w i s  and b e l o w  rectangular the  has  and n u m e r o u s  experimentally short  that  first  G.I. Taylor  hydrodynamics.  order  follows The  free  theory  easily  as  from  surface  of  originally  formulated  the  conventional  the  water  is  given  by:  y=n(x,z,t)  where  y is  taken  rectangular tiating the  free  this  along  coordinates  the in  vertical the  expression with  surface  boundary  axis,  and  x,z  horizontal plane.  respect  condition:  to  time  t,  are Differen-  yields  -3-  -n  - u n  t  x  + v =  where  the  subscripts  denote  v  the  horizontal  and  are  velocity. terms  in  the  (g  vertical  v e l o c i t y gives  + g^)  interface, velocity  partial  B e r n o u l l i ' s law upon  p where  0  = p  is  p is  the  the  potential  -  0  (g  an  of  expression  for  + P*  the  order  the  pressure:  t  fluid  satisfies  u,  the  second  acceleration  of  and  neglecting  x  density which  components  + g )py  virtual  derivatives  seen  and  by  * is  Laplace's  the the  equation:  2 V * = 0 and  whose  gradient y_=  The  theory  pressure,  tial  and  two  density  and  For  upper  the  for  the  (u,v,w)  considers  pressure  Similarly,  gives  components  superimposed  fluid  fluids  (air),  the  each  velocity  with  a  poten-  :  = Aexp {-ky } f ( t ) cos {kx }  P  = P  the  velocity:  potential.  *.  x  the  = V$  velocity  are  of  0  -  lower  (g  +  fluid  gi)Pi the  v  +  p  i ( * i )  t  corresponding  are: *  2  = -Aexp{-ky}f(t)cos{kx>  P  2  = P  0  -  (g •  g!)p y 2  +  P (* 2  ) 2  t  quantities  -4where of  k = 2Tr/X  the  is  periodic The  is  that  for  conditions they on  be  the  the  adopted  standing that  tank  is  X is  the  that  the  velocity  satisfy  v e l o c i t i e s be  the  walls  for  waves  the  at  The  and  wavelength  instability.  form  equal  condition  wavenumber,  interface  the  finite  at  y = 0 and  potentials boundary y = ±»,  that  y_ • n = 0  where  n_ i s  the  normal  vector  nonlinear  term  n u  in  free  neglected  n  the  to  a  wall.  surface  yielding  - v -  t  x  that  (*.)  y  = - k A f ( t ) c o s {kx} which  upon  integration n  becomes:  = -kA(  t  t  / f(t)dt)cos{kx}  o At  the  interface,  condition,  the  when u s e d  -  pressures with  the  Pj  and p  2  expression  are for  equal. n,  This  leads  to: +  -Cg -(p The point, the  i f the  fluid  dependence  2  theory  n is  +  P  is  l  velocity  is  -  2  P )kf x  )f»(t)  (t)  = 0  c o n s i d e r a b l y s i m p l e r at  requirement  is  zero.  made This  that is  at  t  the  growth  =  satisfied  sinh{nt} rate  of  the  this  = 0 and  is f(t)  where  gj) ( P  instability.  i f  the  y = 0 time  The an  explicit  condition for  expression  equal  for  the  pressures  growth  now  gives  rate:  2  n  = - Cg + g ) ( P (P  Hence, is  g i v e n by  the  the  rate  n is  effects water  of  i f  2  + Pj)  interface  as  the  a function  quantity  (g  •  gj)  positive  and  the  Bellmann  and  Peninigton  surface  tension  Pj)lc  of  time  = k A n " * c o s h { n t } c o s {kx}  and  effects  T,  is  P  " ?!  < 0 then  disturbance  tension  interface,the  Surface  2  expression: n  Therefore,  -  x  grows  (1954)  v i s c o s i t y are  introduced  through  growth  exponentially.  considered  v i s c o s i t y . For  of  the  the  an  the  air-  negligible. equal  pressures  condition:  2  +  The r e s u l t i n g  T n  xx  =  0  expression  for  the  growth  rate  b ecomes: 2 n  =  - (g  +  g )(.P 1  (pj  -  2  P )k  _  2  + P )  ^ 3  (Pj^ +  2  P ) 2  2 If the  n  > 0 the above  modes  with  surface  expression a  is that  unstable. the  It  is  surface  evident  tension  wavelength: X < 2ir,  k-Cg • gj)  (P  2  -  Pj)  from  stabilizes  -6since  are  is  n  Thus  disturbances  the  1.73  drip  and  the  spikes  velocity Emmons  theory  as  not  account  growing  into led  0.4X.  follow  initial  formulated later  conclusion that until  that  the  can  be  the  how t h e  is  in  later  the  spikes  1).  the  Lewis's  first of  the  disturbance a power stages  content  g i v e n by t h e  of  long  amplitude  surface  expanded  by G.  stages  (Figure  only  will  hang.  bubbles  of  Fourier  and  series, of  the  the distorted  series:  OC*  = I .A (t)cos{mx) m= 1 m  The  original  the  development  and  radius  rounded the  water,  wavelength  into  determined  perturbation  Curzon  the  e v o l v e . The h a r m o n i c  capacitative by  larger  underside  For  develop  assuming  disturbance  the  crests  to  about  n  to  the  a  for  to  critical  order  valid  instability  the  wavelengths  instability.  a ceiling.  first  is  al.  as  gravity,  potentials  et  such  whose  no  cling  will  where  By  droplets  is  ones  troughs  is  there  smaller  while  theory  .  of  investigations order  of  < 0  droplets  does  instability  above,  surface  that  The Taylor  the  action  cm s o  to  initial  why s m a l l  a horizontal  tend  I.  for  than  explains  under  2  such wavelengths  smaller  This of  for  purpose  with  of  of  the  time.  this  experiment  Fourier To t h i s  Fourier analyzing technique Langille  (1972)  was  used.  was  spectrum purpose, as This  of a  developed method  the  Figure  1.Photograph the  o f a Ray 1 e i g h - T a y 1 o r  arising  from  fourth  harmonic  standing  "spike"  and " b u b b l e " a r c i n d i c a t e d .  instability wave.  The  -8analyzes  each  measures  A^ft)  that  variation  of  the  the  water  capacitance variable. water, forms  sheet  is  the  from wall  equally of  number  q  that  as  an  used  is  the of  x and  to  the  be  the  the  plates  plate by  vertical  is  given  = W (1 q  q is  an  analyzed  so  essentially  is  made  and  depth  that  over  the  the  x  experiments,  increased  water  it  assumption  integral  in  first  spaced  = constant,  of  i.e.,  x d i r e c t i o n only  a parallel  K'(x) W  m. T h e  area  system  of  one  a function  where  the  separately  conductivity is  separated  of  in  calculated  plate  forms  selected  in  For the  one  number is  for  is  whose  plate,  h a r m o n i c mode  by  adding  capacitor. a thin  tank.  some  The  acid  other  polyethylene  It  strips  the  consists  whose  of  width  a  W(x)  by:  cos{qirx/L}) integer and  (1)  equal  to  the  L is  the  length  increment  dx,  the  mode of  the  tank. Hence, is  given  for  a small  capacitance  by: dC = K E ( N W ( x ) y ( x ) ) d x  (2)  o  d where of and of  K is  free  space,  N is the  Fourier  the  the  water  d i e l e c t r i c constant, d is  number surface  series  the of  thickness strips  can  expansion:  per  generally  e  of  Q  the  the  permittivity  dielectric  unit  length.  be  expressed  The by  sheet elevation a "  -9-  y  = y o  J  The is  y  Q  to  be  motion  is  purely  of  the  rth  is  the  time  is  two of  dependence.  integration  + I  Q  (1 -  over  x  x  A is  of  the  capacitance  of  the  qth  harmonic The  noting  overall change same  of  fluid  the  5^ i s  the  disturbance the  which  time  i f  amplitude and  increment  f  (t)  in  of  the  increase  of  £ E  -  of  = 1  r  c; cos{qiTx/L} r  (t)dx)  5 Lf (t)) q  the  q  cosines Thus  proportional the  change water  in capacitance  r  to  a constant. is  r  L  -  surface  s e n s i t i v i t y of  that  c: cos{rTrx/L}f (t))  cos{rirx/L}f  orthogonality and  0  =  leads  D  by  the  cos{qnx/L})dx  = My L  step  surface  Therefore,  o  last  of  independent  dimensional.  C = A(y L  the  level  is:  X  where  mean  and  the  dC = ( y  The  the  constant  harmonic  capacitance  v  term  assumed  the  + Z -5 c o s { r T r x / L } f (t) r=0 r r '  J  due  in  the  of  a qth  depth  AC h a r m o n i c  to  wave  used  time  the  =  in  the  dependence  time  dependence  disturbance.  technique  amplitude: AC w a t e r  the  capacitance  depth to  is  amount  is  determined  AC d u e h  is  to  an  twice  h a r m o n i c wave  of  the the  -10-  tce N  ~ <E  Q  b" o  N  ,  "  ( 1  C O S  ^x/Ujhdx  L  0  /  0  W (1  -  cos{qirx/L}hcos {qirx/L}dx  d = 2 A resistance along  the  same  p r i n c i p l e s . In  analyzing  plates  tank  the  with  resistance  Fourier analyzing technique  are  placed  this  along  conducting water  is  given  method, two  between  works  two F o u r i e r  sides them.  of  the  Then  water the  by:  R = p1 A where  p is  two p l a t e s strip  the  and A i s  width  and  In capacitance impedance voltage  r e s i s t i v i t y , 1 the  depth  the is  following  given  by the  the  water  as  capacitance  while  the  Details and measure  area  of  measured  bridge  across  the  by in  tank of  the  the the  is  measuring output  in  between  product  the  of  the  e q u a t i o n (2).  technique, signal  resistance  of  method  the  an the  observed.  equipment  R-T i n s t a b i l i t i e s are chapter.  distance  required  to  now p r e s e n t e d  produce in  the  -11CHAPTER  II  EXPERIMENTAL A.  Tank  acceleration  APPARATUS  system <  The interface by  a driven  from  a  that  the  the  The p i s t o n ' s travel  force  (450  supplied  a  determine  pressures.,  mm) to  with of  the  25.4  g being  air  the  normally virtual minus  throughout  a  laser  of  acceleration For  increment  any  (i.e., of  care  must  to  To  this  end,  a  the tank  the  water  is  release  to  to  an  not  tank's  of  length  inner  air linearly  bore  O.lg/p.s.i.,  The maximum the  was  tank  at  of  the  tank  2.5g. will  acceleration, t  appreciably was  pressure  maximum  perturbation  mechanism  its  constant  increase  apllied  the  of  applied  acceleration  surface  in  motion.  be  that  fall)  is  phototransistor  fall.  so  piston  the  its  found  free  release  as  air ensures  length  various  found  such  the  cylinder  free  initial under  taken  was  reservoir  the  a  supplied  compressed  a nearly  and  was the  in  is  under  25 p . s . i .  acceleration  that  beam  travel  exponentially  fashion  long  tank  grow  a  as  the  acceleration  be  during  there  acceleration  Since  is  to  that  a p l l i e d was  the  The  supplied  few p e r c e n t  pressure.  mm t h e  tank.  water-air  force  with  so  times  the  air  cylinder  By u s i n g to  and  the  the  supplied  in pressure  only  containing  downward by  which i s  reservoir  change of  tank  accelerated piston  large  order  run. of  is  Plexiglas  = 0 in  such  disturbed.  devised  by  -12-  cutting  away p a r t  remaining and  supports  onto  the  base  in  turn  no  by  longer  vertical contact  pistons valve  of  the  To  onto  rails  the  2 and which  which  shock  3).  The  are  is  relay  at  brass  ,  so  shock  connected to  90 d e g r e e s  rod which  the  the  tank  side  the  absorbers together  the  are by  tank  on  is  two into  the  spring  an  is  rod  coming of  tube  tube  slides  end  the  brass  brass  before  at  achieve  of  at  fastens  release  that  tube.The  absorbers  The  The r e l e a s i n g  tank,  either  adjusted  of  tube.  aluminum frame  the  by the  a brass  angle  tank.  the  release  supported  with  of  an  water  a solenoid  steel  (Figures  wall  a horizontal  mounted  accelerator. rotated  the  p o r t i o n subtends  axis  is  of  shaft, loaded  exhaust  air  optimum s t o p p i n g  conditions .  B.  Tank  and A n a l y z i n g  The w a t e r from The  h inch length,  used the  in  tank's  3/4  width  inch  one  walls  plate  sides  are  measurements  of  the  a capacitor  of  the  to  one  0 -  ring  is  fitted  the  tank  leakproof.  the  water  by a t h i n  sheet  are  respectively.  the  and  water tank  the  the  perimeter  The c a p a c i t o r p l a t e polyethylene sheet  in  contact  (Figure  of is of  with  4)  second p l a t e  188 x 107 mm s i d e s .  along  constructed  188 x 107 x 2 0 0 mm. When  rectangular  of  adjacent  and b a s e  Plexiglas  and h e i g h t  capacitance  largest  forms  and  Plates  is  A rubber this  to  make  insulated thickness  from  -13-  SOLENOID VALVE |  TIMING CIRCUIT  .TO  T C0MRES5ED AIR PISTON  TANK  TANK RELEASE MECHANISM SOLENOID RELAY TO | TIMING €> CIRCUIT Figure 2. Side view of the tank a c c e l e r a t o r ,  -14-  -15-  THREADED HOLES  LUCITE CAPACITOR PLATE POLYETHYLENE SHEET  Figure  4a) H o r i z o n t a l s e c t i o n o f t h e w a t e r tank. D e t a i l o f one s i d e c o n t a i n i n g t h e c a p a c i t o r and d i e l e c t r i c s h e e t .  -16-  mm p l a c e d  0.077  plate. were to  In  used be  between  experiments,  for  insulator  the  ineffective  because  To a n a l y z e  consists  of  Fourier  C.  as  q was  varied  Wave  the  standard  these  small  from  and  the  (1) to  1 to made  capacitor  epoxy  materials  copper  chosen  the  pinholes  waves, of  are  and  varnish  equation  was  q  analyzing plates  using  of  strips  W  ring  but  water  g i v e n by  experiments,  mode n u m b e r  board  the  vertical  sinusoidally the  -  0  earlier  current.  In  the  5  capacitor width  i n the  photographic  leaked plate varies  Introduction.  0 . 5 cm a n d  (Figure  from  proved  which  whose  be  resin  the  5 ) . The  printed etching  circuit technique.  Excitation  Standing surface  by  pure  a p p l y i n g an  modes  are  generated  alternating  on t h e  high voltage  of  water the  form: h V  where  V is o  frequency resonant mation waves  =  constant  of  the  the  which  Q  (  {1 -  equal ^  desired  frequency  by  V  is  to  4 . 7 K V , and  surface  wave  determined  dispersion  may b e  c o s (cot) } / 2 )  taken  relation to  be  to  in  an  for  OJ i s the  the  water.  excellent standing  :  2 u> where  h is  the  =  depth  gktanh(kh) of  the  fluid  in  the  resonant  tank.  The  approxi-  water  F i g u r e 5. The f o u r t h harmonic analyzing plate.  (q=4)  Fourier  -18-  To voltage, water  surface.  the  varied with  table  and p o s i t i o n s  mode  excited  over  of  and  high the  the  are  summarized  I  : WAVE E X C I T A T I O N  ELECTRODES  LENGTH  POSITION*  L/2  L/2  2  1  L/2  L/4  4  2  +  L/4  L/8  8  4  +  L/8  L/16  from  r i g h t ' end  of  tank  to  the  edge  of  the  electrode.  width the  were  the  positiones  NUMBER OF E L E C T R O D E S  Electrode  to  periodic  1  first  of  the  are  the  1  Distance  The  using  below:  MODE NUMBER  t  waves  The d i m e n s i o n s  TABLE  *  the  horizontal electrodes  electrodes in  excite  spacing  of  the  electrodes  particular  p o l i s h e d and  the  supported  screws  tank the  used. edges  grounded water.  electrode  plane  = electrode  of  and  the  electrode  (Figure  6).  In  water  all  the  surface,  threaded could  about  rounded  To a d j u s t  by a v e r t i c a l the  are  length.  be  rod  equal  cases, to  to  the  the  electrodes  minimize  distance the  adjusted  by  sparking  between  former  (Figure  width  2).  was The  leveling  -19-  THREADED ROD SUPPORT TO HV SUPPLY  2 0 M n  e  3  e ' /  o  o / I k  LEVELING •SCREWS  0  o  ' o '  o  H  o  >^  1  •BUBBLE I  •ELECTRODE  H /  L  ^  L/4 L/8  ;  L/8  3L/4  WATER SURFACE K  F i g u r e 6 . The e l e c t r o d e u s e d t o generate h a r m o n i c s t a n d i n g s u r f a c e wave.  the  fourth  -20-  The produced the  alternating  by a programmable  details  of which  waveform  generator  waveform  which  a  logic  direction  is  The  output  signal  of  which  (Kepco) above  to  the  circuitry  $  release that wave  n  is of is  grows  at at  the  then  the  desired  on two  standard  resistor for  together  reverses  the  16 w o r d s  of  the  converted  to  an  b y an o p e r a t i o n a l  is  voltage used  memory. analog  amplifier at  the  between  protection  of  electrodes the  the  sparking.  n of  electrostatic  the  the  of  The  System  n where  generator  An u p / d o w n c o u n t e r  sweeps  of  is  Appendix.  periodically  displacement  by t h e  the  manually  amplifier  event  waveform  waveform  a p e r i o d i c high  the  Electrical  excited  half  memory i s  A 20Mfl  i n the  The  one  amplified  water. and  in  that  count  the is  digital  I.C. chips.  produce  electrodes  D.The  the  voltage  given  entered  circuit  of  are  stores  4 x 16 RAM m e m o r y with  high  field  water-air is  of  the  interface form:  = A c o s ( u t + o>)cos(kx) phase  tank  (t  of  the  = 0)  wave.  the  a maximum, t h e n the  the  rate  given  If  phase the  by the  at is  the  time  0 or  IT s o  subsequent first  order  of  unstable theory  -21-  described at  this  the  i n the  fixed  Introduction.  phase' has  resulting  a comparator  system  is  on  output  the  when  its  wave  is  at  Fourier sides  voltage  the  ever  of  2Kft  and  the  plates,  results on t h e  the  tank  sinusoidal  in  The  the  the  and  adjusted  the This  are  of  on  so  instant done in  increase  the  that  the  connected power  to  one to  due  timing  open  the  resistance of  to  the  that  t  0.  is  of  produces  used  to  water  wave  minimum  =  series  is  the  circuit  of which  so  opposite  The  Thus,  adjusted  d i s p l a y e d at  a  water  area  two  at  supply.  sectional  fired,  the  standing  water  in voltage. was  the  by h a v i n g  acidified  of  triggers generator  area  cross  pulse  other  waveform preset  of  electrcal  circuit  the  comparator pulses,  The  the  circuit  was  simultaneous tank  an  release  signal  used.  oscilloscope. Since  a decrease  timing  is  a 50 V d . c .  on a s t o r a g e  the  instant  immersed  2 0 0 ohms  that  right  that  then at  tank  the  digital  The p l a t e s  inversely with  between  delay  operates  tank.  across  observed  is  the  advantage  The t i m i n g  maximum a m p l i t u d e .  resistance  motion  pulse  at  matches  delay  of  reproducible.  circuit  the  analyzing plates  of  varies  of  voltage  A time  release  tank  F i g u r e 7.  voltage  output  tank  the  timing  shown i n  comparator.  particular  i n s t a b i l i t i e s are  To r e l e a s e time,  the  The r e l e a s e  whenthe  two  release  solenoid valve  and  CLOCK OSCILLATOR  KEPCO OP AMP  WAVEFORM GENERATOR  5MQ  1 TANK RELEASE PULSE 500 msec.  COMPARATOR TIMING CIRCUIT  H.V. ELECTRODE  RELAY RELEASE MECHANISM  TO SCOPE EXTERNAL TRIGGER SOLENOID VALVE PULSE 50-500 msec  Figure  7.  Block  diagram  AC RELAY  of  the  SOLENOID AIR VALVE  electrical  system.  -23and  supply  pulse msec  has so  the  piston  its  width  that  the  with  adjustable  solenoid  entire  duration  of  the  pulses  are  to  switch  solenoids  E.  The  used  by means  I.C.  changes  in  measuring and  the  of  was  (10  the  when  bottom.  conductivity 18 m o l a r  Under  these  an  input set  the  tank  noting  of  the  the  depth.  chip  of  that  of  so  filled  that  achieved  of  sensitivity  calibration of  the  depth  by  For  output of  i f  9 cm the  adding water  of  was  circuit  5 0 0 Kfi  the  only  to  the  at  to  (Popil  determines  3kHz'.  1 yF and  2 liters  voltage  capacitance  essentially  of  to  The  measure  a frequency  acid  the  the  utilized  8)  increased  water. output  (Figure  is  conditions,  by  and  water  sulfuric  38 mV/mm o f  is  digital  to  a simple  adjusted  Linearity  the  to  bridge  frequency  was  for  units.  sensitivity  at  500  Circuit  circuit  best  -  The  voltage  relay  latter  50  open  tank.  developed  by means  R j was  of. the  is  water  The  C^ was  Helipot) .  4 volts  from  of  at  linearity, turn  was  be  mains  The  range be  the  time,  circuit  capacitance  to  of  state  with  integrated  conversion.  found  good  one  will  impedance  also  1979) . T h e  unknown  voltage  was  an  air.  the  Measuring  using  circuit  the  solid  capacitance  Curzon  consists  of  in  valve  motion  Capacitance  Eesides  compressed  the done  circuit  as  1 ml (Figure  9) .  circuit simply a  function  15V  1  14  13  12  11  10  9  8  5  6  7  LM2907N 100 K  1 2  3  4  1K5  100K 1K5  OUTPUT  •e 10K  Ri 500K  UNKNOWN CAPACITY (TO TANK)  CA3140  rti Figure 8. The I.C. capacitance measuring  circuit.  860  -25-  HEIGHT c.m.  Figure 9. Response of the I . e . capacitance measuring c i r c u i t to changes i n water depth.  -26-  CHAPTER  III  EXPERIMENTAL  A.  Capacitance  Measurements  Duplication set-up  d e v i s e d by  appeared the as  at  the  initially surface.  wave  with  of  wave  in  harmonic  of  jet  which  impedance  and  are  content  angular  frequency  yielded  curves  staight  lines To  electric apparatus  each  mode of  that at  versus the  eliminate  field using  of  the  the  the wave  water  an  with  an  problems, by  harmonic  However, which  11 s h o w s .  this are  where  (q = 2)  rather  rich  the  UK i s  the  analyzed,  significantly  from  12) .  problems  generator,  a  method  Plotting  mode  (Figure  noise  as  be  unperturbed  second  c o s h (UK t)  the to  wave g e n e r a t o r  deviate  times  appear runs  wter  covered  10,  e l e c t r i c a l noise  the  with  varnish  for  for  that  distilled  in Figure  particular  do n o t  early  accordance  used  i n waves  Figure  results  bridge  perturbation.  as  in  larger  than  excited  excitation results  for  be  electrostatic  initial  yielded  As shown  spurious  the  results  to  experimental  two V a r a t h a n e  surface  the  the  (1970)  arrangement  the  time  Because  air  as  from  perturbed  replaced  pulsed  The  analyzing plates. signals  original  inspection  theory.  exponential  I  the  d i e l e c t r i c between  Fourier output  of  Langille  first  general  RESULTS  caused  I rebuilt  a capacitor  plate  by  the  Langille's and  a  -27-  F i g u r e 10. O u t p u t o f t h e i m p e d a n c e b r i d g e a a function of tine. a) G r o w t h o f a p e r t u r b e d surface. b) G r o w t h o f an u n p e r t u r b e d s u r f a c e G a i n i s 0 . l v / d i v , t i m e b a s e i s 20 m s e c / d i v .  HISTOGRAM OF  SHOWING  THE  AMPLITUDES  THE FIVE HARMONICS DURING EXCITATION  F i g u r e 1 1 . S p e c t r u m o f t h e s e c o n d h a r m o n i c wave g e n e r a t e d by a p u l s e d a i r j ' e t . The h o r i z o n t a l a x i s i s number o f the q t h h a r m o n i c o f the F o u r i e r a n a l y z i n g p l a t e used to d e t e c t the waves.  -29-  Figure 12. Results using the water as a d i e l e c t r i c . The output s i g n a l s obtained by using d i f f e r e n t harmonic p l a t e s are p l o t t e d versus coshCuKt) where x ~11•*»5#  -30-  0.077mm  thick polyethylene  described were as  i n Chapter  performed  the  due  to  order  mode.  position, DQ d i a l  the  is  bridge,  the  $ = 0,  found  as  they  was u s e d  effect  of  IT o r  negligible  as  impedance is  the  fact  non-exponential the  phasing  and  14b)) .  of  that  made  the  (on t h e using  growths  were  " C " and s  no  set-up it  was u s e d feature  capacitance all  cases  the  seen  tank  that  when  has  Figure  13. I.C.  in place of  scale)  where  was  the  in  the  longer  c o n f i r m e d when t h e  striking  initial  as  9 o'clock  impedance  <f>)cos(kx)) ,  growths  were  in  •  the  " l o w D"  the  Langille's  circuit  fashion  experiments  at  < 0.1  order  lower  set  was  (i.e.,releasing  the  a  i n these  dielectric. Also,  results  that  used  of  switch  the  on t h e s e  bridge.The  growth  higher  a the  i r / 2 i n n = A c o s (wt  measuring  the  of  waves  the  set  in  "phasing"  These  standing  aim o f  as  experiments  b r i d g e was  were  were  the  all  F o r optimum r e s u l t s ,  at  that  the  effect  capacitance  set  When r u n s  was  exponential  bridge  CRL s e l e c t o r  dielectric  development  1500-A.  typically  it  the  the  the  harmonic  since  d i s t o r t i o n of  impedance  as  B. I n i t i a l l y ,  first  follow  control of  balance.  water  the  General Radio  sensitivity  for  to  the  The the  part  perturbation  was  modes  is  using  initial  experiment  II  sheet  these  of  the  results  decreases  in  irrespective  perturbation, (Figures  a of  14a>  -31-  a)  b)  c)  F i g u r e 13. Inpedance b r i d g e o u t p u t versus time showing that the growth rate of the i n s t a b i l i t y is independent of the phase,*. a ) P h a s e <> I =0 b)  Cain  * - T T / 2  C) < J > = TT i s O . l v / d i v Time  base  is  20  msec/div.  -32-  time  decreasing capacitance  F i g u r e 14. C a p a c i t a n c e d e c r e a s e i n mean l e v e l instabilities. a) I . C . c i r c u i t time base is b) I.C. circuit time base is c) C a p a c i t a n c e O . l v / d i v and  measurements showing c a u s e d by t h e growth  the of  output,gain is 0.2v/div, 20 m s e c / d i v . output,gain is 0.5v/div 0.2sec/div. bridge output,gain is the time base is 20msec/div.  -33-  The using  the  impedance  reference that of  of  course,  tank  than  the  the  null  to  signal  has  diminished  further to  to  level that  reduction in  grow w i t h  time,  circuit  level  of  water  tank.  it and  B.  the  would  the  increase  contact  of the  Because  voltage since  it  the  the  mean  level  the  below  water  in  it.  that  is  appreciably-  When t h e  seen  to  This  tank  was  decrease  capacitance  of  the  tank  impedance b r i d g e . A then  causes  the  output  impedance b r i d g e  there  is  water  a change with  capacitance water  is  the  and  i n the walls  decreases  separating  mean  of  with  from  the time,  the  walls  decreases.  area  the  t  voltage  signal  immersed i n i o n i z e d  results of  when t h e contact  tank  with  to  the  growth of  between  the  plates  is  setting  Measurements  plates  due  is  the  the  that  Observing  Since  from  that  surprising  signal.  verified  ( F i g u r e 14c)) .  seem  analyzing  signal  on t h e  suggest  Resistance  by  impedance b r i d g e  output once  later  done  unperturbed  set  then  hence  was  capacitance  The r e s u l t s I.C.  This  balance  downward, the  null  g r o w t h was  i n an o u t p u t  accelerated the  the  on t h e  with  results  of  bridge.  capacitance  the  greater  direction  was  the the  between  water  two  yielded  accelerated  walls  is  downward.  area.  to  the  should correspondingly the  rather  expected  instability,  i n v e r s e l y p r o p o r t i o n a l to  Fourier  decrease  However,  -34-  as  seen  in  Figure  15,  the  exponential  fashion.  the  initial  perturbation,  the  amplitude  slopes using the  then the  same  rate.  NUMBER  of  rate. plates  linear  graphs  It  found  was  produced  The r e s u l t s  are  presented  GROWTH  2 3.4  ± 15%  4  25.4  ± 17%  5  19.8  ± 18%  20.7  ± 11%  sheet  expected  that  the  voltage can  area  be of  indicate. Since a rising  as  to  signals  explained contact  l i n e a r l y with  is  below:  RATE n ( s e c  3  the  that  approximately  ± 14%  of  whose  : RESISTANCE FOURIER ANALYSIS  (q)  proportional  appear  semi-logarithmic plots  23.4  function are  as  2  copper  an  wave  yielded  harmonic  ± 13%  decreases tions  1st  in  23.5  that  again  increases  1  The g r o w t h of  the  growth  harmonic  TABLE III  plain  time  the  various growth  HARMONIC  Using  versus  gave  voltage  time  the  as  inverse  p o s i t i v e l y going  the  opposite  water  a  with  linearly  voltage the  the  walls  observadecreasing  signals  contact  exponentials.  direction  once  capacitance  of  the of  the  by a s s u m i n g  the  inverse  hyperbola, the  of  in  )  area  which then  -35-  Figure 1 5 . The r e s i s t a n c e across t h e t a n k as a f u n c t Gain is 0 . 0 5 v / d i v and t h e  Fourier analysis. Voltage ion of time. time b a s e i s 50msec/div.  -36-  The v o l t a g e used  to  interpret  observed  signals  the  when t h e  observed  exponential  water  is  used  growths  as  the  between  two F o u r i e r a n a l y z i n g p l a t e s .  ration,  ( Curzon  able and as  leakage the  of  current  Thus,  plates  that  bridge  and hence  Langille  is  one tank  there null  is  as  of  change with  tank in  the  yield  the  run  tank  balance  than  Thus,  of  the  the  impedance  are  obtained. in  and  low  H o w e v e r , when (see  F i g u r e 4) ,  to  obtain  a  the  The f a c t  that  water  measurements  container.  capacitance  capacitance  indicates  of  «  water  ineffective  by the  and  results.  instability  the  the  between  bridge  resistance  and  the  are  constant  next.  the  consider-  1 mV r . m . s .  the  level  is  difficulty  capacitor  unexpected  mean  the  the  configu-  since  growths is  be are  this  resistance  signal  to  that  there  plates  there  accelerated  walls.  walls  the  a proper  increases  is  of  is  balance  summary,  the  indication at  as  less  resistance the  capacitor  that  no p r o b l e m t o  measurements  the  exponential  null  used  In  the  the  experimental  signal  across  it  notes  the  is  1972),  can  dielectric In  a c t u a l l y being measured  (1970)  maintaining  the  Langille  v a r n i s h on t h e  insulators.  from  and  here  but  that  decreases  there is  do n o t  describe  that  the  is  in  a contact  give  an  situation  -37-  C.  "Cavitation"  Effects  Irrespective surface,  the  indicate  a linear  S ince of  the  the  signals  of  the  mode  from  the  impedance  growth  capacitance  due  to  measured  initially  bridge  a decrease is  on  always  in  averaged  the  capacitance.  over  the  area  plate C = / ( e / d ) dA  the  only  the  separation  This so  means  that  sheet, to  way t h a t  decrease.  compared  to  i f  gets  d will  capacitance  between  that  air  the  the  the  the  the  and  away f r o m  water  and  the  of  the  the  the  air  i f  d,  increases.  the  walls  polyethylene  capacitance of  is  plates  pulls  thickness  thickness  decrease  two c a p a c i t o r  water  between  increase If  the  can  will  film  be  is  seen  large  d i e l e c t r i c sheet,  then  C = / ( e / d ) dA» A' where  A' is  the  area  of  dielectric  sheet.  If  is  sheet,  then  a metal  that  C will  be  the  the  other  the  water  which  For  given  applied virtual  dC/dt  that  this  the  touches  (i.e.  dD"/dt)  phenomenon  bubbles  of  the  is  the  in  contact  plate  above  p r o p o r t i o n a l to  the  that  water  capacitor  expression  indicates  average  d i e l e c t r i c of  acceleration,  is the  constant. same  instability  the  the  the  of  with  as  The the  c a n be  it  depth~D~ o f the was  capacitor. observed  possibility formation  eliminated  of since  -38-  others  (Emmons e t  velocity  is  virtual  acceleration.  called  of water  affected were  baffle with  its  due  plane of  is  to  by  the  placed  baffle  there  that  the  square  root  of  For convenience,  this  apparent  determine  to  an  i n the  parallel  the was  where  air  film  water  an  bubble the  henceforth  to  varied  from  of  be  the  capacitor  plate.  capacitor  cm t o  curvature  of  initial  were  vertical, The  plate  and  cm. A t 0.5  water  water  performed  cm  surface  the  slopes  signals  the of  tank water  (Figure  bridge  and  was  The  16) .  The  established  observing  depth.  excited  cavitation  the  capacitance  no  The  from  bridge  with  surface.  s i m p l y by m e a s u r i n g  function  10.5  the  experi-  Plexiglas was  determined  filling  surface,  The b a f f l e  the  0.5  phenomenon  meniscus.  on t h e  the  the  adjustable  between  considerable  the  shape  were  of  will  cavitation  tank.  Investigations waves  the  how t h e  initial  performed  was  distance the  found  by  have  "cavitation". To  ments  1959) ,  proportional  replacenent  is  al  the  output  rates  obtained  sensitivity by  slowly  voltage  s e n s i t i v i t y was  as  a  6.67cm/V p —p  and of  was  observed  water  bridge (Figure  provided  is  to  be  that  maintained  17) . A typical  at  linear  to  a net  the  sensitivity  one  quarter  graph  of  the  of  change  of  control a turn  ^ measured  or  on  9cm the less  cavitation  -39-  Figure 16. O s c i l l o g r a p h s h o w i n g t h e l i n e a r decrease t a n k c a p a c i t a n c e w i t h time.The tank width was  of  2 cm i n t h i s i n s t a n c e a n d t h e v i r t u a l a c c e l e r a t i o n 1 . 0 g's.Gain i s 0 . 2 v / d i v , t i m e base 20msec/div, The c a l i b r a t i o n i s 6.67cm w a t e r / V  P"P  -40-  F i g u r e 1 7 . C a l i b r a t i o n o f t h e GR - 1 5 0 0 A i m p e d a n c e b r i d g e . Water h e i g h t i s measured upwards from the equilibrium level.  -41-  velocity  (dtT/dt)  is  in Figure  shown  accelerations considerably in  of  the  noise  tank  WIDTH  The the  w i d t h are  III  error  times  for  of virtual  and  is  investigation of  the  effect  RATE  37  1  30  2  38  3  32  4  39  6  27  8  34  10.5  32  net  mean  of  result  level  is  the is  related dfT/dt  where  g is  the  III,  AND TANK  virtual  the to  there  velocity  the  = v =  WIDTH  (cm/sec-g)  cavitation velocity that  increase bridge  0.5  dependence  an the  : CAVITATION  table  larger  of  below:  from  the  for  also  envelope  RATE  seen  is  presented  (cm)  for  observed  there  i n the  the  acceleration  increases  growth  signals  The r e s u l t s  As  the  18.  because  TABLE  TANK  a function  shorter  spurious  output.  as  virtual  no  on t h e  of  the  observable tank  or  width;  decreasing  acceleration  (34cm/sec-s  acceleration  is  by:  ± 4cm/sec-g) excess  x g  acceleration  -42-  Figurc  3cm.  18.  "Cavitation"  rate  f o r a tank  width  of  -43-  in  terms  of  free  fall  acceleration  i.e.,  g =  ag , Q  2 g  Q  = 980  cm/sec  , a >  Dimensional between to  the  scaling  cavitation deduced. ality and  rate  In  was  inspection  velocity laws,  and  on  various  considered the  as  result  that  inversely  proportional dD~  dT is  Q  ranging  v  check  were  from  to  the  60  viscosities  of  Handbook  Chemistry  are  of  listed  results:  in  these  table  being  constant  a function  of  of of  analysis  the  the may  be  proportionthe  yields  rate  leads  density the  should  be  viscosity:  2  — ° and  n the  viscosity of  sugar  solutions and  at  Physics,  together  viscosity.  effect,  different  percent  IV  acceleration  quantities  cavitation  pgL  proportionality  dependence  the  length  prepared  20  the  when  to =  the  physical  the  a scaling  To solutions  =  of  applied  dimensional  plausible  L  the  from which  particular,  viscosity,  where  0.  weight.  C is  43rd  with  sugar  concentrations by  25  aqueous  the  given  edition  The by  the  and  experimental  -44  TABLE  IV  : EFFECT  OF V I S C O S I T Y  CAVITATION  VISCOSITY  PERCENT SUGAR (by  (cP)  CAVITATION  1 .695  32  30  2.735  31  40  5 . 164  46  50  12.40  35  60  44.03  77  the  of  44  cm/sec-g  are  the  as  so  The solutions  is  to  large due  than  1.5  g.  best  fit  line  through  Nonetheless,  even  dtermination  of  viscosity.  for  the  the  that  the  in  viscous  is  no  a  mean  of solutions  water.  the  data  for  the  sugar  irregularities  accelerations  as  the  origin  with  yield  deviation  the  points of  apparent  inversely  above  determined  cavitation there  of  pure  at  were  vary  rates  erratic  exclusion if  not  standard  rates  appear  rates  the  remains  the  that  The  a  scatter  to  necessitated  still  those  immediately  The  with the  is  does  supposed.  that  signals  it  rate  cm/sec-g  comparable  bridge  table  cavitation  viscosity  value  RATE  (cm/sec-g)  20  that the  THE  RATE  weight)  From  17  ON  is  in  by  drawing  at  included  inverse  the  the  greater the  F i g u r e 19 .  origin  rates,  of  This  0 acceleration, in  the  conclusion  variation  with  -45-  VIRTUAL ACCERATION  (g's) 0.5  1.5  2.5  F i g u r e 19. C a v i t a t i o n r a t e v e r s u s t h e v i r t u a l a c c e l e r a t i o n f o r a 30% s u g a r s o l u t i o n .  -46Th e r e s p o n s e in  capacitance  growths  was  obtained  of  the  physical  of  the  bridge One  of  checked  in  the  to  follow  and  and  other  capacitor  set-up,  the  capacitor  was  maximum  plates the  tank,  when  two  plates  the  are  not  was  against  plate  in  the  was  had  to  linear  some  inability  capacitance. to  the  tank  dielectric  water 320  one  capacitance longer  due in  the  the  changes  representative  fixed  overlapped  no  that  changes  plate  transient  truly  are  capacitance  the  to  ensure  and  held  fully  dropping  to  fast  capacitor  frame  bridge  runs  situation  support the  the  sheet  tank.In  pF  when  the  another.  By  reduced  to  was  a common  this  area  two  250  pF  of  overlap. Calibration gave  a conversion  amplitude the so  and  capacitor that  the  observed against  the  factor  plates.  The  a function  the  time  of  relative  capacitance  as  excellent  the  of  output  for  mV/cm b e t w e e n  tank  of  approximation  11  vertical  of  squared  signal  was  this the  displacement then  allowed  the  two p l a t e s  time.  Plotting  yield  a staight  (Figure  20).The  could the  test output  of to  fall,  be  amplitude  line slope  to  an  yields  2 the  value  Thus, indeed  one  of may  547  safely  accurately In  electrical  cm/sec  view  as  the  conclude  effective  that  follow  abrupt  of  unexpected  the  measurements,  I  the  changes  decided  acceleration.  bridge in  results  does  capacitance. from  the  to photograph  the  -47-  Figure 20. changes in  Test of the impedance capacitance.  bridge  to  transient  -48-  fluid sort  motion of  exists the  D.  in  between  the  Photography  as  occur  to  and  discover  if  capacitor plates  the  a standard  an a i r  and  what film  the water  in  near  experiments  the  camera  before  the  (500 in  where  msec their  they  the motion  of  the  performed held  the  of  instabili-  traverse, at  total  the  triggered  and  the  duration  i n s t a b i l i t i e s were  stages  of  growth  maximum  is  one  tank.  darkness  throughout  l e s s ) . The  tank  was  located alongside  aimed  in  reached  instability  tank's  open  later  have of  switch  flashgun  or  presence  resulting  end  was  the  e x c i t e d e l e c t r o s t a t i c a l l y on  and t h e  were  shutter  photographed instant  visually  photographic  run  Motion  A mechanical  The  the  Fluid  modes w e r e  surface  rails  the  determine  various  water  of  photographed.  of  so  tank.  ties,  of  ways  i n s t a b i l i t i e s did  To  the  various  at  the  amplitude  stopped  by  all  the  just  shock  t absorbers. EL  The  photographs  were  all  t a k e n by  camera w i t h  a 50 mm f / 2  lens  using  a  Ilford  Nikormat  PANF  50  A .S . A . f i lm . When t h e the  tank  growth  in  first  accelerated, it the  harmonic was  wave  found  that  v e r t i c a l d i r e c t i o n and  the  disturbance  appeared  The  photographs  do  across  show h o w e v e r ,  the  was  excited  there  instead, tank  that  was  there  is  negligible  most  (Figure a  and  of  22a)) . climbing  F i g u r e 2 1 . The i n i t i a l l u n p e r t u r b e d a c c e l e r a t i o n o f 1.5 g's.  surface  at  an  -50-  film  of  edges. tank in  water This  but  the  that  film  is  bulges  exists  mostly  seen  photograph.  in  the  form  of  drops  at  on  all  four  sides  of  the  of  on  the  side  walls  of  the  tank  There  is  also  the  appearance  "bubbles"  at  the  corners  of  the  tank  as  by  et  a l . Figures  21  and  22a)  show t h a t  Emmons  disturbance first the data the  of  the  harmonic  same.  unperturbed  e x c i t e d wave  Therefore,  obtained first  for  harmonic  unperturbed  the  the  second  Langille  also  as  well.  for  the  harmonic  (1970) first  some g r o w t h the  of  show  transverse  the  harmonic  previously in  Chapter  II  was  Figure part  D,  effect  phasing  The  tank  results (Figure  appears  Using  obtained was  a phase  the  on  also  of  essentially  arise  from  initially  was  as  used  those  the  obtained is  once  again  the  achieved  circuit  4th  harmonic  run  (Figure  instant 0 and  when  electrode  successful  IT o r  by  dominate.  timing  each  from  22b)) . T h e r e  to  reproducible  a c c e l e r a t e d at  wave h a d  which  excited using 5.  The  standing  to  the  i n s t a b i l i t i e s were  were  was  to  the  resistance  v e r t i c a l d i r e c t i o n , but  instabilities of  and  of  i n s t a b i l i t y evolving  wave  disturbance  wave  in  the  similar  Successful 4th  apply  e x c i t e d wave  harmonic  in  capacitance  noted  the  that are  the  of  first  and  i n s t a b i l i t y thought  Photographs the  surface  was  respectively,  wave  surface  the  in  when the  the  shown described  23) .  this  case.  the  excited  results  Figure  22. F l a s h p h o t o g r a p h s o f t h e w a t e r surface a) The f i r s t h a r m o n i c e x c i t e d wave a t 1.5 g's  virtual  acceleration.  Figure  22. b)  T  n  e  1.5  second g' s .  harmonic  instability  at  -53-  F i g u r e 23. The f o u r t h harmonic i n s t a b i l i t y . The h o r i z o n t a l l i n e i n the photographs i n d i c a t e s the i n i t i a l water l e v e l . T h e two c o n s e c u t i v e shots i n d i c a t e the degree o f r c p r o d u c i b i l t y o f the i n s t a b i l i t y . The i n i t i a l phase f o r these photographs was * = T T .  -54-  are  as  expected  indicate  that  (Figures  t h e maximum  spikes  as  measured  12  In  contrast,  cm.  amplitude  from  In  Figure  23,  spikes  perpendicular  photograph on  of  it  a "cross-tank"  harmonic (Figure  along 26  tendency of  the  the  a)  of  adhesion  to  the and  the  with  the  to  the  walls  the  front  wall  with  a thin  sheet  of  cases,  the  being  to  about  an  water  the  photograph. with  tank.  in  motion  of  this  excitation 4th  Side-on  in  A  standing  a source  there  is  the  the  desired  outward  its  of  resonant  the  that  disturbance  top  that  that  some  the  some  the  bulge  lags  of  upon  of  with  of  evidence  half  dark  is  is  photographs a  the  net  middle  contact  because  of  with its  walls.  is  surface  the  level  seen  surface  b) ) , i n d i c a t e  instability  blasted  also  plane  along  length  water  across  due  restricted in  Further water  as  the  is  mode  whereas  is  is  25) , i n d i c a t e s  the water  tank  walls  well  disturbance  a t t a i n e d by  mm.  there  to  photographs  p e r t u r b a t i o n has  the unacce1erated  (Figure  transverse of  as  initial  initial  t y p i c a l l y 1.5  surface  24) . T h e  amplitude  of  the  and  the  the  across  wave  23  provided  is  that behind  by of  the  contact  w e t t e d by  sandblasted  the  side  growth  of  has  been  the  coloured that  the  left  fitted The  water  and  water.In the  of  the  the  Plexiglas.  with  shows  area  27. H e r e ,  tank  sandblasted  in  the  Figure the  contact  sandbecomes all  wetting  due  F i g u r e 24. The same f o u r t h harmonic i n s t a b i l i t y i n f i g u r e 23, except that the tank was r e l e a s e d when the phase o f the s t a n d i n g wave was < J > = 0.  as  -56-  F i a u r e 25. F l a s h p h o t o g r a p h o f t h e e x c i t e d fourtl harmonic standing w a v e a t p h a s e $ = TT i n f r e e fall.  -57-  F i g u r e 26. Side on f l a s h photographs o f the f o u r t h harmonic i n s t a b i l i t y showing the upward b u l g i n g of the f l u i d i n the t r a n s v e r s e d i r e c t i o n a c r o s s the t ank.  -58-  F i g u r e 2 7 . The f o u r t h harmonic i n s t a b i l i t y with l e f t h a l f o f the tank showing the w e t t i n g o f a translucent screen. a) at 1.5 g's a c c e l e r a t i o n b) at 1.0 g's.  the  -59-  to  the  than  growth  the  the  of  spike  the  which  is  appears  narrower on  the  of  width  and  right  smaller hand  side  of  photographs. Hence,  been  seen  that  amplitudes does The  not  the  ments  the  water  follow  then  in  tank  although  the  tendency  must as  spike  the  of  the  that  in  contact  development  necessarily  of  surface  pull  in  water with growth  of  only  a wall, the  apparently  measure they  do n o t  decrease  to  of  with  bulge  contact  the  walls  upward  from  the  area  of  the the  indicate  contact  walls  measure-  represent  instead the  high  electrical  truly  has  instability.  away  the  the  i n s t a b i l i t y , but  linear  the  water  i n s t a b i l i t y grows. S i n c e employed  cm i t  i n s t a b i l i t i e s reach  is  the water  10.5  area  the with  time. Lastly, the  resultant  for  exciting  approaches  case  does  however that  It  complex  for  this do  high  used was an  by  by  the  is  mode was  that  amplitude  the  4th  the  of  it i t .  motivation  its  wavelength  investigators  was  wave. of  the  (Figure  is  about  In  considered  this 28) , half  view  fluid  and  mentioned  i n s t a b i l i t y in  achieved  harmonic  excited  The  amplitude  and n a t u r e  p a r t i c u l a r mode, studies  that  other  appreciable  structure  further  harmonic  mode  found  maximum  achieved  the  to  this  reach  the  8th  i n s t a b i l i t y observed.  that  previously.  the  of  surface  unprofitable  F i g u r e 28. The e i g h t h i n i t i a l p h a s e * = ir.  harmonic  instability  with  the  -61-  In narrowest was  order  dimension  constructed  cut-off  to  X For  of  such  wavelength  suppress the  that  for  water  water its  this  across  surface  the  another  w i d t h was  tank  less  than  T is  75  the  mode:  <J/gp)  = 2u  c  t h e mode  h  the  surface  tension  dynes/cm  3  and  the  density  acceleration 14  of  p is 1.5  reduced  cross  amplitudes less  than  tank. was  of  for  1.5  mm o b t a i n e d  9 cm i n  a  typical is  about  the  same  this  case.  The  photographs  is  due  the  When t h e  part  tank  the w a l l s  film's In  same h e i g h t  to  of  is of  as  the  narrow the  thickness  as  of  in  tank  the of  smaller  width,  is  In  about  tank,  spikes.The  throughout  the  half film  except  instability tank q=4 show  dimensional. the  meniscus  unacce1erated  the  film  wider  29),  two  tank.  the  the  (Figure  initial  much  wide  much m o r e  the  the  are  the  accelerated this  amplitude the  in  amplitude  narrow  motion  a considerable  uniform  previously  the  tank,  waves  about  However,  instability.  driven  t o be  the  to  narrow  final  resulting  the  the  due  the  the  c l i m b up  damping  of  the  that  tank,  area  electrically  Nevertheless,  surface.  large  the  instability in  forms  the  sectional  mode  of  that  c u t - o f f wavelength  of  observed  about  the  g the  so  mm. Because  to  1 g/cm  film  is  wide that now  (10.5 of  the  reaches  appears near  seen  its  to  be edge  cm)  F i g u r e 29. The f o u r t h harmonic i n s t a b i l i t y i n the narrow tank o f 1.5cm. Here the motion appears to be two d i m e n s i o n a l .  -63-  where the  it  bulges.  film,  the  photograph that and  the  were  troughs of  regions  of  compared  (Figure  29) . I t  was  of  the  (or  bubbles)  the  bridge  measurements  appears  the  bridge  the  unaccelerated  observed  to  14c) . T h i s  the  various to  be  no  two  in  the found  intial  level  balanced  the  to  an  accelerations  other  once level modes, time. g,  d i f f e r e n c e between  the  runs  surface  unperturbed The  (Figure  saturation  characteristic capacitance  feature bridge.  of  those  setting that from  shown that  signal  there  which  which  before  with  Runs w e r e  yet  growths done  at  appears  had  had  the  the  decrease the  of  the  the  initial  30) . the  common The  as  the  capacitance  By  than  again  3.0  e x c i t e d and  the  signals  zero  to  surface  up  lower  through  Tank  i n s t a b i l i t y in  resulting  t h e mean of  Narrow  contrary.  a value  the go  the  dimensional,  linearly with  initial  the  the  it  harmonic  indicates  case  change  to  tank,  time.  to  below  indicate  and hence  appear  be  capacitance  in  above  using  4th  to  capacitance As  spikes  Measurements  tank  figure  is  various  narrow  in  fluid  of  Although  are  how much  10%.  Electrical  bridge  estimate  weights  the weights  accuracy  E.  To  signal  to  level  all at  growths  runs which  done the  is  a  by  using  saturation  -64-  F i g u r e 30. C a p a c i t a n c e b r i d g e s i g n a l s v e r s u s time f o r the f o u r t h harmonic i n s t a b i l i t y i n the narrow tank. C a i n i s 0 . 2 v / d i v , t i m e base i s 20 msec/div. a) u n p e r t u r b e d s u r f a c e b) e x c i t e d wave s u r f a c e A c c e l e r a t i o n f o r both runs was 1.5 g's.  -65-  of  the  in  capacitance  5-6  growth  cm. T h i s  occurs  corresponding  is  not  an  terictics  as  the  linear  to  a depth  the  up  the  capacitance  the  results  the  water  imply  and  diminished  to The  again  using  plates walls at  of  drop  the  about  in  the  tank  bias  of in  the  with on  initial  the  the  Upon  was  and  mean  level  verify as  this  that the  a siphon  water and  a storage water  level  is  checked  bias  across  oscilloscope.  voltage this  signal  voltage  drop  increase  capacitance with the  tank  the  does  tank  was  was  Figure  cm m e a s u r e d  bridge  time.  voltage In  tank  31) . I n  falls,  output  larger  the  d.c.  the  copper  the  a positive  level  22  between  area  the  in  decrease  the  was  of  the  confirms  voltage  time,  film  was  (Figure  the  is  of  Two p l a i n  a rise  shows  does  established  accelerating  represents  there  air  be  obtaining  contact  sides  observed  hence  to  is  level  method. inner  charac-  centimeters.  mean  acceleration, tank  the  of  v e r i f i e d to  it  an  change  depth  a function is  that  a  bridge  cm p r i o r  as  there  is in  the  Since  several  tank.  voltage,  increase  displayed the  such  oscillograph  To  drained  wall  on  to  11  decreases  5 volts. Since  that  indeed  30.  the  narrow  the  the  of  Figure  resistance  g virtual  across  test  the  of  decrease  a change  due  change  a depth  there  c a l i b r a t i o n was  that  placed  across  case,  the  the  were  2.0  in  that  to  effect  bridge  oscillographs  that  indicates  from  32  a) ,  the  -66-  F i g u r e 31. C h e c k o f c h a n g e o f mean l e v e l u s i n g the resistance method.First l i n e in the osci11ogranh is the d . c . v o l t a g e a c r o s s the tank d u r i n g a run at 2.0 g*s a c c e l e r a t i o n . T h i r d l i n e i s t h e g r o u n d reference. G a i n i s 2 v / d i v a n d t h e t i m e base i s 50 msec/div.  -67-  a)  b)  F i ! ; u r e 32.  Check  of  v a l i d i t y of growth d i r e c t i o n tests. m e t h o d . V o l t a g e a c r o s s t h e tank as w a t e r i s d r a i n e d t h r o u g h a s i p h o n . C a i n 2v/div,time 5sec/div. C a p a c i t a n c e b r i d g e . The w a t e r a p p r o a c h e s the b a l a n c e l e v e l a n d r e c e d e s b e y o n d i t , t h u s the " p i n c h i n g " o f t h e e n v e l o p e o c c u r s a t the balance l e v e l . Cain O.lv/div, time 5sec/div.  a) R e s i s t a n c e  b)  -68-  bottom  corresponding  to  a n d was  drained  to  voltage  drop  about  of  The the  a  same  level 6  null  level  was  signal  3 cm b e l o w verify the  this  that  hyperbolic decreases  in  the  the  as  the  in  was  water  about  2  in  volts  a  performed  Figure  The  the  to  "balance" decrease  oscillographs level  in  method  the  31  linearly,  increases  approaches  or  figure  bridge  the  to  decreases  capacitance  level  using  32 b ) . H e r e ,  the  allowed  mean w a t e r  and  was  1 cm a b o v e  resistance  fashion  of  3 cm r e s u l t i n g  check  shown  it  level.  when  voltage  and  of  of  initially  level  voltage  volts.  sort  capacitance bridge  water  a bias  in  a  output  balance  level. Therefore, harmonic  i n s t a b i l i t y show a n  waveform, seem  to  level of is  F.  the  electrical  indicate  as  the  doing  that  tank  is  a Fourier  photographs apparently  measurements  there  is  two  a decrease  using  the  4th  dimensional  described  a c c e l e r a t e d and  analysis  of  thus  the  in  the  the  given  would mean  feasibility  techniques  nullified.  Test  suggest the  although  for  the  Air  Film  S ince  the  capacitance bridge  air  film  that  walls  of  an the  water  must  tank,  get  signals  between  a test  for  the  the  (Figure water  presence  30)  and of  -69-  this  film  was  devised.  a helium-neon air  or  water-air  Snell's  law  internal and  laser  is  show  internal  tank  of  laser  there  was  containing by  to  critical  the  and  same f o r  proper  total  air  on  the  level  was  masked The  prism  are and  its  total  geometry  for  total  internal the  side  water  to  of  the  placed  in  front  a diameter the  of  of  output  detect  50°  of  face  incidence  the  level,  was  the the  and  face  about of  whenever  was  wall  detected  monitored  air  film  prism  the  adjusted  Plexiglas  reflected light  whose To  was  cm l o n g  r e f l e c t i o n occurred  other The  20  the  onto  on  that  above  a exist  this  off. are  shown  solely  due  to  tank  sec)  prism  angle  as  In  34  it  was  in  stray  appears  to  Figure  34.  the  In  reflections  accelerated past  b) , a s h a r p  occurs. However,  0.004  for  using  interfaces  beam  results  phototransistor. reflection  Plexiglas-  two  onto  incident  prism.  initial  the  the  33) , a n d  the  pulses  use  the  33) . A l e n s  oscilloscope.  the  glued  spread  below  (about  the  the  a phototransistor  storage  the  angle  beam was  the  off  to  that  made  (Figure  that  reflect  was  calculations  (Figure  cm. The  principle  interface. Simple  obtain  was  narrow  so  to  reflection a Plexiglas  2 cm w i d e  prism  basic  42°.  To  1.5  beam  reflection is  about  the  The  pulse  duration  b e much  due of  shorter  34  a),  from the  to  internal  this  pulse  than  expected  F i g u r e 33. S c h e m a t i c o f the t e s t f o r the p r e s e n c e o f an a i r f i l m u s i n g t o t a l i n t e r n a l r e f l e c t i o n f r o m a Plexiglas-air interface.  -71-  F i g u r e 34. T e s t f o r t h e p r e s e n c e o f a n a i r f i l m . Output from the p h o t o t r a n s i s t o r . a) P u l s e s d u e t o s t r a y r e f l e c t i o n o f f the tank a c c e l e r a t e d at 2 . 0 g ' s . T h e w a t e r level w a s 3 c m a b o v e t h e m a s k e d p o r t i o n o f the Plexiglas prism. b) P u l s e s due t o s t r a y r e f l e c t i o n s p l u s a s h a r p l a r g e peak due t o i n t e r n a l r e f l e c t i o n s due t o t h e p r e s e n c e o f an a i r f i l m . T h e w a t e r l e v e l w a s 3mm a b o v e t h e m a s k e d h a l f o f t h e prism. Gain f o r both o s c i l l o g r a p h s i s 2mV/div,time 20msec/div.  -72-  if  an  air  film  centimeters  penetrates  below  the  Another is  considerably  sharp  pulse  initial  water  indication that  less  than, e x p e c t e d  disappeared  the  air  film  amplitude  is  the  fact  that  initial  the  i n t e r n a l r e l e c t i o n r e o c c u r r e d when  was  lowered  the  air  film  for  the  amplitude  (Figure to  is  that  large  water it  signals  from the  the  sheet  This  effective  the  the  is  the  level  of  the  prism;  initial  level  amplitude low  now m o r e  voltage  less  than  2 cm,  suggested  figure consistent  resistance d.c.  of  technique corresponds considerably  by  the  runs.  is  and  internal  some o t h e r  from  that  the  the  causes  under  the  sheet  problem  the  resulting of  the  in  air  that  is  to an and  to  causing  air the  space  the away between  capacitor of  a decreased was  the  tension,  be p u l l e d  thickness  film  results  bridge. Since  adhesion  latter  creating  reflection  considerable  water's  increases  dielectric  effect  impedance  polyethylene  process  The  porion  in  7 centimeters  capacitor plate  dielectric  plate.  of  water  7 mm. T h e  the  change  level  that  film  using  acceleration is  suspected  dielectric  the  off  than  air  resistance  there  under  was  the  obtained  bridge  The  the  of  6 or  shows  greater  i n mean  the  capacitance  imply  not  31) . T h e r e ,  than  masked  5 mm. T h i s  result  a change  less  to  the  the  several  level.  set  the  7 mm a b o v e  when  of  was  with  at  a distance  the capacitance.  finally  resolved  -73-  when  a new  circuit  capacitor  board  layer  and  board  formed  sheet  the  comparative  the  thickness  885 p F  instead  the  growth  Similar show  tests  that  the  consistent walls  G.  of  the  the  dielectric of  the  Due t o  the  dielectric,  with  water with  was  the  the  now  poly-  excess  output  described  those  the  w  a  s  acceleration  of  1.5  now s h o w e d  a strikingly  not  (Figure  linear  in  Chapter  increases  movement  of  with  water  3 part  time,  layers  a  up  35) . A,  result the  tank.  Films  of  the  versus  wall  observation  board  printed  side  tank.  pF o b t a i n e d  to  indicates tank  the  fiber  that  Plots  This  bridge  accelerated  Climbing  of  envelope  with  capacitance  upon  capacitance  the  The p l a t e d  a  dielectric.  capacitance  smooth  the  2,200  Nonetheless, g the  wall  of  from  now p r o v i d e d b o t h  when h a l f - f i l l e d  of  sheet  fashioned  plate.  outside  capacitance  was  which  conducting  tank's  ethylene  plate  time  that at  was  square  the  by  of  the  yielded  straight  edge  the  a constant made  root  of  change lines  water  film  acceleration.  Emmons e t  al  A  using  in (Figure  moves  36) .  up  similar photographic  observations. S everal acceleration  from  runs 1.7  were to  4.7  made g.  varying Plots  of  the  virtual  the  square  root  Figure 35. Capacitance bridge runs usinp, a printed c i r c u i t board as the d i e l e c t r i c and c a p a c i t o r plate. a) A run with the bridge i n i t i a l l y balanced at the water l e v e l capacitance. b) Here the bridge was balanced below the i n i t i a l water l e v e l . c) In t h i s run the bridge was balanced above the i n i t i a l l e v e l capacitance. Gain for a l l shots i s 0.2v/div,time i s 20nsec/div.  -75-  of  the  that  amplitude  the  virtual  about  1.7  0.3  at  When  straight  surface the  ordinary  results water  of  the  /s  TABLE  FLUID  Water (viscosity  V  or  are  made  1 cP,  than  the  ratio  of  using  sugar  an  the  presented  in  : CLIMBING FILMS  to  that  table  • 75  soap determine In  for  RESULTS  dynes  CORRELATION COEFFICIENT  _j cm" )  1.526  0.9929 _j  surface  0.00820  tension  ~ 35  dynes  cm  )  2.398  0.9866  0.569  0.9976  40% S u g a r s o l u t i o n ( v i s c o s i t y 5.164 cP) 0.00740  all  V.  INTERCEPT . h -U (cm ' ' s e c )  tension  from  obtained  effect.  as  film  graphs  were  aqueous  an  indicate  accelerations  solution  same  the  t  origin  about  0.00792 S oap s o l u t i o n (viscosity 1 cP,  the  has  surface  less  versus  viscosity  SLOPE • -% . (cm sec)  t  applied virtual  through  a 40%  time  acceleration varied  slopes the  the  36) . T h e  g.  are  and  is  4.7  as  tension  film  at  passing  well  versus  0.6  38) . R u n s w e r e as  the  virtual  g to  the  lines  solution  cases  the  film  (Figure  plotted against  (Figure  the  acceleration to  if  of  a c c e l e r a t i o n of  acceleration  were  s  TIME (msec) _i 20  i  i 60  '  •—»100  Figure 36. The climbing f i l m , The square root of the s i g n a l amplitude of f i g u r e 3 5 a) i s p l o t t e d as a function of time.  -77-  F i g u r e 37. The c l i m b i n g f i l m . The l o g a r i t h m of the s i g n a l a m p l i t u d e o f f i g u r e 35 a) i s p l o t t e d a g a i n s t the t i m e . The g r a p h above i n d i c a t e s t h e the s i g n a l g r o w t h i s n o t due t o an i n s t a b i l i t y .  F i g u r e 3 S . The c l i m b i n g f i l m s r e s u l t s . The s q u a r e r o o t o f the a c c e l e r a t i o n o f the f i l m i s plotted v e r s u s t h e a p p l i e d v i r t u a l a c c e l e r a t i o n . The results for different fluids are shown.  -79-  Thus, behaviour  from  of  table  the  V,  the  climbing  scaling  films  with  law  for  the  applied  acceleration  is  B  g... film  =  Using  the  dielectric  and  the  previous  the  water  the  can  be  the  the  still  climbing the  narrow  and  actually  In  the  behind film  wide the  can  be  Indeed, to  be  seen  to  and  stays  Fourier  cannot  be  films.  tank  becomes  where  This the same  tank,  the  films  instability seen  to  and  reach  because  plate  of  decrease  the  the  the  quickly. away  of  virtual  earlier  are  amplitude  capacitance The  from  the  of  the  the  quite  capacitance  presence  of  the  in  case  predominant  as  the  instability.  seen  to  lag  somewhat  Figure  23  shows,  are as  using  particularly  films  the  in  there.  is  have  gets  applied  so  by  the  the  pulled  analysis  done  from  that  rate  saturate  caused  capacitor  to  that  proportional  proportional  are  simply  away  air  why  the  were  is  the  the  explains  water  of  and  also  limit  sheet  water  amount  as  2  conclusively  signals  This  However, bridge  the  sheet  board  shows  be p r o p o r t i o n a l  seen  sheet  its  on  (g . „ .) ^virtuar  2  circuit  bridge  the  well.  was  signals  to  to  as  acceleration.  1  dielectric  dielectric  capacitance  wall  the  tension  expected  dielectric  4  capacitor plate  pulling  acceleration  bridge  10" cm" sec  printed  acceleration,  between  x  capacitance  wall. Since to  1.25  about  half  the  a  thin  amplitude  of  -80-  the  spike.  the  rounded  of  In  this  bubbles,  distinguishing  runs  with  were  varies  the  medium,  the  and  to  be  should  water.  technique  analyzing of  the  the  capacitor  same.  narrow  side  of  be  to  amount  saturable  independent from  the  and  Fourier  should the  then  of  of  the  instability.  in  the  Appendix.  optical  the  that  would  be  is  of  exposed  light  its to  intensity.  should  Details  of  then the  The give  light  the with  in  absorbing the  darkened the  negative  on one on  can  of  replacing  the  face  of  other  photodetector  output light  for  analyzer  of  along  incident  The  that  films  a photographic  plate such  signals  Fourier  noticeable  negative.  possibility  absorption  appear  in  surfaces.  thickness  consist  with  light  the  some  water  is  photodetector  of  of  be  thicker  bridge  a photodetector  area  the  may  climbing an  much  initial  Since  not  plate  Parallel  tank  the  if  water  instability This  the  used.  of  there  unexcited of  are  capacitance  exponential  films  films  thus  altogether  technique as  and  problem  eliminated  the  between  excited The  be  case,  is  and  proportional is  largely  signal the  technique  output  development are  discussed  -81-  CHAPTER  IV  CONCLUSION  developed  In  the  an  apparatus  reproducible sinusoidal such  that  shape  of  This of  the the  phase  of  produces  resulting the  theory.  previous  experiments resulting  large  wave  of  the  is  in  the  be  of  time  contrast  initial  i n s t a b i l i t y were  a  pure  developed  thus set  amplitude  a function  where  amplitude  and  i n s t a b i l i t y could  This  have  c i r c u i t r y was  standing  measurement  with  I  i n s t a b i l i t i e s from  Timing  the  i n s t a b i l i t i e s as  and  my e x p e r i m e n t s ,  which  wave.  directly  phase  of  Rayleigh-Taylor  standing  allows  the  course  the  exactly.  variation  to to  be  compared  all  other  perturbation,  largely  left  to  chance. I techniques simplest  developed for  is  across  the  plates  as  a simple  the  the tank  the  resistance is  to  capacitance  of  solar  to measure  methods waves  have  of  detected.  the  of  the  amplitudes  of  where  the  Fourier  converter  tank.  The  light  Fourier  waves.  analyzing method  to  1 mm o r  plate.  that less  uses  measure  transmitted  analyzing  The  voltage  t h i r d method  sensitivity about  measurement  water  contacts.Another  the  sufficient  standing  using  voltage  water  electrical  method  measured  electrical  a transparency  novel  d e t e c t i o n of  frequency  cells  three  small are  the  uses  through All  these  standing readily  -82-  The  electrical  Rayleigh-Taylor instability of  the  instabilities  that  were  experiments.  indicated  a convex  the  of  width  mean  level  bulging of  the  of  measurements  not  bulging  the of  This  the  of  the water  Photographs  of  the  the  water  surface,  films  These water  films  unfortunately  to  distinguish  However, used  the  to  do  function of  the  the  the  studies  of  of  which  is  water  tank.  the  the  this  water  versus passing  behaviour  of  work  of the  g  excess  bridge  of  moves  the  the  downward  square  applied the  been  film  m  root  -  the  origin.  2  5  x  the a  capactative  is  unable  now  be  films  easily as  a  Observations  indicate  the  that  acceleration  acceleration  film  of  the  accele-  yielded  scaling  to  this  instability.  a constant  of  the  instability.  time  The  straight  law  for  be:  10- sec cm- • ( * 4  in  presence  that  can  acceleration  determined J  with  water  at  across  confirm  method  vs.  course  walls.  acceleration.  signals  film  through has  climbing  the  the  indicate  from  the  measurements  the  as  the  of  a decrease  pace  as  observe  surface  with  transducer  the  water  in  well  keep  films  applied  a fraction  Plots  lines  not  capacitative  capacitance  edge  ration  will  that  as  water  analysis  to  instabilities  climbing  Fourier  prior  water  results  to  features  electrical  contact  of  used  revealed  considered  Namely,  tank.  I  2  1  v  i  r  t  U  B  l  )  *  .  -83-  and  was  found  to  and  the  viscosity  be  Attempts the  f l u i d by  (less  than  independent  of  the  to  length  the  of  width  the  accelerations)  of  i n s t a b i l i t y show t h a t  than  the  same  centimeters that and of  the are  the  dyed  optical the  climbing  water  that  water  in  technique  is  been it  of  is  light  films  should  rest  analysis used  the  as  not  factor, now  be  tank  to  of  1.5  wavelength  cm  at  is  two  much m o r e  are  that  of  of  the  is  is  indicate  uniform to  tank.  feasible This  thickness  the  bulk  This if  result  an  technique  c a p a c i t a t i v e and  the  dimensional  several  also  compared  instead. the  bulging  Photographs  difference being  the major  tension  successful.  a tank  films  the  same p r i n c i p l e s the major  of  same p h o t o g r a p h s  Fourier  methods,  water  The  convex  critical  r e l a t i v e l y transparent  suggests  on  have  i n s t a b i l i t y in  wide.  surface  fluid.  modest the  the  e l i m i n a t e the  decreasing  the  of  that  since  comparatively  detectable.  works  resistive absortion thin  -84-  BI BLIOGRAPHY  Bellmann 12_,  151  Cole  R.,  Pennington  R .S . ,  F . L . , Langille  Emmons H . W. , Mech.  1_,  177  Chang  Columbia  (19 70)  Lewis  D . J . , Proc.  Popil  R. ,  B. L . ,  Fluids  Can.  and  M . , Phys.  J.J.,  G.I.,  J.  1810  Phys.  Watson  Thesis,  (1972)  50_,  B.C., J.  Roy. S o c .  University  A 202,  Curzon F . L . , Rev. S c i .  App1i cat i o n s ,  Taylor  16_,  Math.  73  (1972)  Fluid  (1960)  B.L., M.Sc  Stoker  Phys.  C.T.,  Langille  Ratafia  J. A p p l .  (19 5 4)  R . L . , Tankin  Curzon  R . H . , Quart.  Water  Fluids  16_,  Waves;  Interscience  Proc.  81  Ins.  1207  of  (1950)  50_.  70  Publishers  A 201,  (1979)  (1973)  The M a t h e m a t i c a l  Roy. S o c .  British  Theory  (1957)  192  (1950)  with  -85-  APPENDIX A  : Details  To p r o d u c e water  surface,  an V  is  used.  Initially,  to produce  then  a m p l i f i e d by  of  1000.  The  presented  (Figure advance the of  There  are  a)) . An  of  each word  is  excite  standing  labelled  enter  the  count  requires to  outputs  wave  the  generator signal  of  the  with  a  of  its  the  pulses  memories  are  operation.  to  counter  generator which  that  scans  (RAM) . T h e  converted  is  gain  generator  sections  is  is  which  to  a m p l i f i e d and  an  used  JK  the  a set  circuit  of  output  analog to  frequency  through  The  clock  counter  through  the  OR g a t e s  input  This  into  from  INPUT".  the  C  pulses  "CLOCK  from  the  peak  an u p / d o w n  t h e memory  upwards  of  the  waves.  up/down  be h i g h .  on  form  amplifier  main  two 4 - b i t  entered  input  to  explanation  consequently  Square are  peak  three  waves  the  waveform  diagrams  an  of  Generator  c o s (u)t)}/2)  inverting  on  the  from  which  -  Waveform  o s c i l l a t o r provides  address  signal  To  an  schematic  16 w o r d s  dividers  ({1  a 5 volt  with  the  q  the  standing  field  a digital  along  39  mode  electric = V  used  pure  of  0 to  15,  t h e up/down  t o be p u l s e d  condition flip-flop  is  and  provided  3062 w h i c h  are  pulses of  3003.  counter  the by  the  0^  the set  o w n  input  Q and high  Q  -86-  OSCILLATOR  UP/DOWN COUNTER ADDRESS LOGIC  b)  MEMORY DAC  2(4x16) RAM  DIGITAL FROM  _0  OUTPUT  RAMs  +15V 10  v 13  11  y  OUTPUT  w  *  -I5V  * iL  B  H  16  12  DAC 02  14  100K 17  15 47„f  10 K  (GAIN)  68K  Figure 39. Schematic of the d i g i t a l waveform generator, a) Block diagran b) The d i g i t a l to analog c o n v e r t e r and output stage.  87-  DEPOSIT SWITCH DEBOUNCE  DPST  FROM ADDRESS LOGIC  11  3002  4064  IS  A B C ADDRESS  WE  14  13  [D A B (: _ADDRESS. MEM O U T i D A T A IN E F G H i E: F G H  4064  MEM OUT DATA IN A B C D _,_ A B C D  wx_  11 9  220fi  DISPLAY DRIVER DATA ENTRY SWITCHES  13 f  {  ,16 TO LED * :32 DISPLAY ^ J 4 4  TO DISPLAY 4_  330  rti  330  n  1  -128  n  A B C D  E FG H  TO D A CONVERTER F i g u r e 40. section.  The  waveform  generator:  memory  and  display  -88-  ADRS  3002  TO RAM  A BC D  Figure logic.  41.  Detail  of  the  up/down counter  address  -89-  and  low r e s p e c t i v e l y  output  of  the  S ET i n p u t sets  the  count  of  Cf = 1.  by t h e  count  The  backwards  which  to  address output  to  an  (I.C.  To generator's and  the  switch  be  so  the  is  the  at  8 bit  by  8 bit  the  the  clock to  the  counter  the  up/down  to  clock  is to  the  a  4-bit  pulse.  This  which  then  at  that  finally analog  converted converter  into  the  P  L  into  the  o s c i l l a t o r is  sets  and  is  and  operations  function  zero  position  function  Q = 0  the  generated.  sets  desired  to  any w a v e f o r m e n t e r e d  reset  sent  pulse  produces  word  digital  and  low p u l s e  above  is  clock  word s t o r e d  position  a  the  the  RAM m e m o r i e s  count-up  of  state  every  two  simultaneously  sets  to  counter  a periodic  memory,  Mien  that  itself.  that  enter  asynchronous  flip-flop  now make  periodically  address that  its  and  This  signal  DAC 02) ,  memory w i l l  the  incremented  address.  analog  1111,  applied  the  a LO i n p u t  at  will  a corresponding  particular  receives  When  the  upwards.  up/down  word reaches  0000,  count  0000  the  upwards.  the  pulses  repeats  is  is  now r e v e r s e s  Thus, word  for  LO s i g n a l  clock  automatically  count  counter  terminates  CP, which  the  JK f l i p - f l o p  flip-flop  triggered input  up/down the  upward  for  by the  a push  counter input.  approximated  by  disconnected button  flip-flop  flip-flop to  to  to  the  0000 by  One h a l f 16  waveform  steps  of in  reset the count-up means the digital  -90-  form To  and  deposit  write  entered  a word  enable  switch  and  address manually is  are  into  (WE)  the  manually  input  word i s  repeated  the  next  until  all  The p r o g r a m standing  waves  TABLE  is  memory by  the  is  activated  by  entered  :  into  the  entry  the  is  sequential  address  and  for  the in  DIGITAL  have  program table  (ENTERED  the  to  VI.  PROGRAM FOR WAVE  GENERATOR  ADDRESS  WORD  ON S W I T C H E S )  0 0 0 0  1 0  0 0 0 1  1 0 0 0 1 1 0 0  0 0 1 0  1 0 0 0 1 0 0 1  0 0 1 1  1 0 0 0 0 1 1 0  0 1 0 0  1 0 0 0 0 0 0 1  0 1 0 1  0 1 1 1 1 0 1 0  0 1 1 0  0 1 1 1 0 0 1 0  0 1 1 1  0 1 1 0 1 0 0 1  1 0 0 0  0 1 1 0 0 0 0 1  1 0 0 1  0 1 0 1 0 0 0 1  1 0 1 0  0 1 0 0 1 0 1 1  1 0 1 1  0 0 1 1 1 1 1 1  1 1 0 0  0 0 1 1 0 1 0 0  1 1 0 1  0 0 1 0 1 0 1 0  1 1 1 0  0 0 1 0 0 0 0 1  1 1 1 1  0 0 0 1 1 1 1 0  0 0 1 1 0 1  at  then  been  used  the  deposit  memory  address  16 a d d r e s s e s  switches.  switches,  The  included  VI  data  the  location displayed. to  on t h e  the incremented procedure  filled. excite  -91-  APPENDIX  The analysis The  by  experimental  an  suggested  struction  of  the  tank  Fourier  optical  any  several  mounted  analyzing  be  kept  linearity this  is  connected  end,  converter the  the  as  To easiest  such  as  from  the  no  need  liters  is  light  a virtual  is  shown i n  of  in  of  order  terminal ground,  to  solar  of  the  solar  current  to  waves  or  d.c.  obtain  parallel  incident  to  have  a point  placed  a suitable  front  face  of  water  mounting  the  5cc  of  water  tank.  water  adequately  soluble  darkens  instabilities,  Thus, onto  the  source.  light,  light  ink  a  voltage  source  distance  o p t i c a l components  About  changes  using  variable  is  cells  observe  e l i m i n a t e d by  output.  voltage  a m p l i f i e r anda  solution  cells  achieve current  surface  uniform  of  the  the  to  c a n be  are  side  of  the  To  voltage  cells  voltage-current  figure.  due  withstand  the  the  voltage  to  41.  con-  and  arc,  of  of  their  a d.c.  of  little  negative  of  Figure  solar  face  intensity  positive  in  very  terminals  voltage  Fourier  designed  outside  output  the  shown  A panel  Because  to  bias  differential  g.  plate.  ground  the  output  d.c.  the  at  is  requires and  the  for  a photographic  the  between  To  in  to  containing  characteristics, must  technique  apparatus  of  FOURIER ANALYSIS  set-up  configuration  accelerations rigorously  B : OPTICAL  away  there the  mixed  water.  is  tank. into  The  two  resulting  FOURIER ANALYZER SCREEN SOLAR CELLS  POINT LIGHT SOURCE  i ID  I  DARKENED WATER  +V.  LM 308H  SCOPE  i—  PLUG-IN UNIT  F i g u r e 42. E x p e r i m e n t a l s e t - u p f o r the o p t i c a l F o u r i e r a n a l y s i s o f R-T i n s t a b i l i t i e s . T h e f i g u r e a l s o shows t l i e c u r r e n t t o v o l t a g e converter.  -93-  films upon the  of  water which  its  The and  adjusted  op  amp  of  is  to  respect mm 4 t h  of  circuit  to  the  60  wave  v  On t h e Fourier to  be  analyzing  quite  solar  such the  sides  now  of  the  interfere  that  area  exposed  standing was  in  of  tank with  as  on  II  the  the  50  been ft  resistor  saturation  of  is  For  indicating  above  signal  that the  this I.C.  E. observations,  the  whole.  as  linear  the  a sinusoidal  part  the  conditions.  light.  sensitive  using  has  arrangement to  obtained  technique  promising  the  waves  chapter  basis  the  The  experimental  that  p-p  converter cells.  show  detection is  described  not  voltage  the  under  harmonic  amplitude *  method  to  a value  avoided  the  instability.  current  tentative tests  with 1.5  the  tested using  is  The  at  a c c e l e r a t i o n should  d e t e c t i o n of  made  appear  solar  cells  the appears  

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