UBC Theses and Dissertations

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UBC Theses and Dissertations

Statistics of coherent structures in turbulent fluid flow Loewen, Stuart Reid 1983

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S T A T I S T I C S OF COHERENT STRUCTURES IN TURBULENT FLUID FLOW by STUART REID LOEWEN B.Sc.,the U n i v e r s i t y  Of M a n i t o b a , 1 9 8 0  A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE  REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE in  THE  FACULTY OF GRADUATE STUDIES Physics  We a c c e p t to  THE  this  Department  thesis  the required  as conforming standard  UNIVERSITY OF BRITISH December  ©  COLUMBIA  1983  STUART REID LOEWEN, 1983  In p r e s e n t i n g requirements  this thesis f o r an  of  British  it  freely available  agree t h a t for  understood that for  Library  shall  for reference  and  study.  I  for extensive  h i s or  her  copying or  f i n a n c i a l gain  be  shall  publication  not  be  D e p a r t m e n t o f ftht^-s \c<>  30  fW  fern  K«_r  of  Columbia  make  further this  thesis  head o f  this  my  It is thesis  a l l o w e d w i t h o u t my  V  Date  the  representatives.  permission.  The U n i v e r s i t y o f B r i t i s h 1956 Main Mall V a n c o u v e r , Canada V6T 1Y3  copying of  g r a n t e d by  the  University  the  p u r p o s e s may by  the  I agree that  permission  department or  f u l f i l m e n t of  advanced degree a t  Columbia,  scholarly  in partial  written  i i  A b s t r a c t T h i s f i l m  t h e s i s  anemometery  g e n e r a t e d  made on  of  v i s i b l e  the  row  of  were  by  bars  R e y n o l d ' s  i n  f l o w .  p h o t o g r a p h i n g  of  a  was  water  used  numbers,  of  based  s i z e s  those  o b t a i n e d  s t r u c t u r e s  the  paths  aluminum  of  towing  bar  the  tank.  of  from were  t r a c e r s  A  t u r b u l e n t  s p a c i n g ,  h o t -  w i t h  coherent  generate on  eddy  flow  from  The  f i l l e d  to  produced  t u r b u l e n t  d i s t r i b u t i o n same  s p e c t r a  p a r a l l e l  f l o w .  Flow  about  20,000  s t u d i e d . A  s i m p l e  a n a l y s i s  proved  adequate  those  measured  a n a l y z e r . s p e c t r a  o b t a i n e d found  lower the  the  from  i n  average  of  eddy the  d i s t r i b u t i o n s  evidence suggest  of that  i n t e r a c t i o n be  of  the  flow  s p e c t r a eddy  flow  as  w i t h  i s  The agree  w e l l  as  and  p a r t  d e s c r i b e d  30% the  i n  i t  was s c a l e  the  g r i d of  t u r b u l e n t  model  s c a l e s  S u c c e s s i v e  the  v i s u a l i z a t i o n  were  l e n g t h  u n d e r s t a n d i n g a  power  but  and  change  r e s u l t s  i n  spectrum  l e n g t h  s p e c t r a . the  w i t h  eddy  ones  w i t h i n  from  The  and  the  i n t e g r a l  power  d i s t a n c e  flow  of  p r e d i c t e d  s t r u c t u r e s  such  c o n s i s t e n t  h o t - f i l m  s i z e  photographed  anemometer shape  the  knowledge  from  i s  to  and  produced  c o h e r e n t  study  range  p a t t e r n s  s p e c t r a  h o t - f i l m  e v o l u t i o n .  new  o b t a i n e d  T h i s which  eddy  flow  magnitude.  power  the  the power  a  s i m i l a r  photographs s i z e  u s i n g  were  from  of  p r e d i c t  frequency  that  o b t a i n e d  to  The  c o n s i s t e n t l y  can  the the  s u r f a c e  power  measurements  from  photographs  compares  eddy showed  t h i s  t h e s i s  of f l u i d  the flow  e x p e r i m e n t s .  of  a  new  for  as  a  s u p e r p o s i t i o n  t u r b u l e n c e of  i n  c o h e r e n t l y  • • •  rotating to  eddies  derive  macroscopic  interactions other. an  The  of e d d i e s eddy  understanding The  and l a m i n a r  size  used  k -power  with  our new  with  model.  the f l u i d ,  distribution  we  fluid  flow  attempt from and  the each  c a n a l s o ^ p r e d i c t e d from  interactions.  description  spectrum  In t h i s model  propertieso*rct.turbulent flow  o f t h e eddy  statistical  flow.  of t u r b u l e n c e  approach are  reviewed  and t h e o f t e n and  compared  iv  Table  of C o n t e n t s  Abstract L i s t of F i g u r e s L i s t of F i g u r e s Acknowledgement  .  Chapter I INTRODUCTION  1  Chapter II S T A T I S T I C A L DESCRIPTION OF TURBULENCE 2.1 M o t i v a t i o n And I n t r o d u c t i o n 2.2 S t a t i s t i c a l S p e c t r a 2.3 C o r r e l a t i o n s And S p e c t r a 2.4 V e l o c i t y F l u c t u a t i o n s And The Power S p e c t r u m Chapter I I I GRID TURBULENCE EXPERIMENTS 3.1 I n t r o d u c t i o n 3.2 The Tank 3.3 The G r i d 3.4 V i s u a l i z a t i o n A p p a r a t u s 3.5 The S e a r c h F o r C o h e r e n t 3.6 H o t - f i l m A p p a r a t u s 3.7 H o t - f i l m Power S p e c t r a  Structures  C h a p t e r IV EDDY-SIZE DISTRIBUTIONS AND HOT-FILM SPECTRA 4.1 I n t r o d u c t i o n 4.2 E d d y - S i z e S p e c t r a 4.3 G e n e r a t i o n Of E (w) From N(R) 4.4 Power S p e c t r u m O b t a i n e d From U ( t ) 4.5 R e s u l t s And D i s c u s s i o n Chapter  i i v vi v i i  9 9 11 16 23 27 27 29 32 32 34 ...44 48 56 56 58 67 71 81  V  CONCLUSIONS  93  BIBLIOGRAPHY  96  APPENDIX A - HOT-FILM PROBE S E N S I T I V I T Y TO VELOCITY FLUCTUATIONS i  97  APPENDIX B - RIGID BODY EDDY VELOCITY PROFILE  98  APPENDIX C - TURB.FTN FORTRAN  99  APPENDIX D - AVERAGE  CODE  EDDY CHORD AND  PEDDY CALCULATION  ...102  V  List  1. C o r r e l a t i o n 2. S p a t i a l  3. Flow F i e l d  Past  4. The  Tank  5. O p t i c a l  Figures  Velocities  Correlation  Towing  of  17  Curve  18  Probe  20 28  Detector  and T i m e r  31  6. L i g h t s and Camera A c t i o n 7. Flow V i s u a l i z a t i o n 8. V i s u a l i z a t i o n s  33  inGrjd  and F l u i d  at Different  Shutter  Frames  36  Speeds  37  9. 20cm/sec Flow  Photographs  40  10. 30cm/sec Flow  Photographs  41  11. 40cm/sec Flow  Photographs  42  12. 50cm/sec Flow  Photographs  43  13. H o t - F i l m  Power S p e c t r u m  14. H o t - F i l m  Linearization  Schematic  46 51  15. H o t - F i l m O s c i l l o g r a m s  52  16. P r o b e  55  Support Resonance  17. T y p i c a l  A n a l y z e r X-Y  18. Power S p e c t r a  Plot  55  Comparison  57  19. Eddy  Analysis  59  20. Eddy  Angular V e l o c i t i e s  62  21.  30cm/sec E d d y - S i z e S p e c t r a  63  22.  40cm/sec E d d y - S i z e  65  23. N(R)  to E  Spectra  (w) C o m p u t a t i o n  24. u ( t ) G e n e r a t e d f r o m t h e Eddy 25. Eddy  Power S p e c t r a  68 Distribution  70 72  vi  26.  Eddy Power S p e c t r a : V a r i a t i o n  with Distance  27.  Spectrum. A n a l y z e r  28.  Analyzer  X-Y  29.  Hot-Film  Power S p e c t r a : V a r i a t i o n  30.  Eddy P a i r i n g  82  31.  Comparison  89  32.  Eddy Count w i t h D i s t a n c e  89  33.  Variation  90  34.  Decay  35.  Comparison  36.  E q u i v a l e n t Area  Operation  76  Plot  78  of L e n g t h S c a l e s  i n Occupied  73  Area  o f Ti*  with Distance  79  90  o f Power S p e c t r a Representation  91 92  vii  Acknowledgement I w o u l d l i k e t o t h a n k P a u l B u r r e l who m a c h i n e d and designed t h i n g s t h a t worked and A l Cheuck f o r b u i l d i n g t h e t i m e r and getting supplies "now would be b e s t " . H e l p i n g hands and m i n d s of summer a s s i s t a n t s Norman Lo, who made t h e t a n k t h a t " w i l l n o t l e a k " (and e v e n t u a l l y d i d n o t ) , Alex Filuik who got the v i s u a l i z a t i o n s y s t e m underway and P e t e r Smereka of " F i n e I d e a " fame. The l o a n of t h e CTA u n i t f r o m D r . Q u i c k of the UBC Civil Engineering Dept. i s much appreciated. Although not directly referenced the teaching of Dr.I.Gartshore of the UBC Mechanical Engineering Dept. p r o v i d e d a s o l i d a p p r e c i a t i o n f o r what i s known a b o u t fluid flows. I would also l i k e t o thank t h e UBC Plasma P h y s i c s g r o u p f o r c o n t i n u i n g t o work one floor beneath the tank filled with three tons of w a t e r . F i n a l l y I would l i k e t o t h a n k my s u p e r v i s o r D r . B . A h l b o r n f o r h i s enthusiastic help i n b o t h t h e r e s e a r c h and w r i t i n g a s p e c t s of t h i s t h e s i s .  1  I. This  thesis  structures turbulent  investigates  in a turbulent f l u i d flow  coherently flow.  can  be  rotating  These e d d i e s  fluctuating validity  local  flow  flow  field.  field  flow  the eddy-size  by  seeing  probability interaction  is  size  by  The model  and s p a c e  time  and  space  scales.  small  s c a l e coherence  In o r d e r dynamics size  statistics  the  fluctuating  It  can  be  are  an  to test the  fluid,  the  t o the f l u i d  turbulent  fluid  i s the p r o b a b i l i t y of  the  being  in  space.  mechanics  This of  the  deterministic  s t o c h a s t i c on  as  well  as  significant  on  larger  large  scale  properties  density flow  related  understanding  the r e l a t i o n s h i p  well  known  of the f l u c t u a t i n g  power s p e c t r a l  the  T h i s approach has t h e p o t e n t i a l of  develop  s p e c t r u m and t h e  description  laminar  of  ( e . g . the weather)  to  and  of  b l o c k s of  the  the  i s thus  scales while  which  flows.  plus  according  at a point  time  turbulent  with  spectrum which  processes.  superposition  'eddies',  each other  determined  unpredictability  a  model a r e p r e s e n t e d .  small  describing  as  characterize  an eddy o f a g i v e n  coherent  I t i s supposed t h a t a  interact  and w i t h We  of  Experiments performed  eddies  of motion.  role  as the b u i l d i n g  of t h i s  equations  flow.  elements,  a r e taken  individual  the  described  fluid  and v i a b i l i t y  The  INTRODUCTION  power  flow  field  of  the  eddy  between o u r eddyspectral  density  was s t u d i e d .  The  i s a measure o f t h e e n e r g y c o n t e n t i n  as a f u n c t i o n of  temporal  t o t h e wave number s p e c t r u m  frequency. through the  2  convection energy of  velocity.  content  scale  The  i n the  size.  How  these  description  is  i n the  The the  The  obtained  power  using  a  spectrum analyzer present  coherent  nor  turbulent  concept, This  obtained  from b o t h t h e  the  written  and  spectra. An water  was  used  This  was  our  to  performed. flow  turbulent  flow  was  with  the  hand  the  is  of  power  grid  in  as  length  the  the  neither of  simple  in  i n the  analysis. can  be  from  hot-  procedures, the  reader  A computer c o d e spectra  from  was eddy  detail.  generated  mechanism so  and  analysis  is  comparing  approach.  spatially  also a n t i c i p a t e d that  flow  hope t o c o n v i n c e  This  selected  study  d i s t r i b u t i o n and  is described  study  conveniently  power s p e c t r a l d e n s i t y  In  of  anemometer  subjective  generate  procedure  experimental  turbulent of  the  number  statistical  other  eddy-size  of  the  I t s r e l a t i o n s h i p to  Our  the  r e s u l t s we  usefulness  in  a  is  turbulent  anemometery m e a s u r e m e n t s . the  density  flows.  s t i l l • somewhat  a s s u m p t i o n s and of  on  for  been u s e d t o  the  show how  function  d i s t r i b u t i o n and  temperature  defined.  elements  a  the  velocity fluctuations  studied  long  in  as  used  eddy-size  spectral  has  clearly  but  was  in turbulent  thesis will  film  and  gives  chapter.  constant  structures  intuitive the  second  field are  turbulent  spectral density  reasons.  scales  of  spectra  r e l a t i o n s h i p between t h e  power  spectrum  f l u c t u a t i n g flow  statistical described  wavenumber  turbulence  for  to obtain  generating  a large  volume  homogeneous p r o p e r t i e s .  grid  in  geometery would c r e a t e  It a  3  flow  that  A towing the and  would e v o l v e tank  fluid  and  hot-film The  was  anemometry  particles  camera  mounted  a  on in  angular  by  spectra  were  spectra  and  1  solid  integral  body  as  large  consistent  and  one  (1931).  of t h i s  statistical  The  part  of  a  same  broader  the flow  d i m e n s i o n a l power  from  F.Ahlborn. is  analyzer. the  Inspiration  the  tank  came  1  comparison  of t u r b u l e n t  concept  eddy  obtained using  analyzer.  descriptions  angular  treating  a spectrum  thesis  to  u n c e r t a i n t y the  u s i n g the towing  o b t a i n e d by  were  structures  the  w i t h our  rotation.  spectrum  structures  results  F.Ahlborn:"Turbulenz K u g e l n und Z y l i n d e r n " , 482-491  of  we  the  were a n a l y z e d  as w e l l  by m e a s u r i n g  t h e main t o p i c  different an  photographs  of  With  frame  c u r v e s were p r e d i c t e d  anemometer  studying coherent  While  is  The  Although  studied  from p h o t o g r a p h i c  of  reference  compared w i t h t h o s e more d i r e c t l y  hot-film  the motion  t h e p r o f u s i o n of c o h e r e n t  d a t a was  density  both  for photographic  of the water.  u s i n g hot w i r e - a n e m o m e t e r y and  spectral  access to  photographing  fluid  the e d d i e s .  velocity  situations  for  by  surface  range.  e d d i e s as p e r f o r m i n g  a  the the  size  of  f o r easy  conditions.  studies.  the e d d y - s i z e d i s t r i b u t i o n s  velocities  Power  i t allowed  visualized  surprised  broad  obtain  as  the o b j e c t r e f e r e n c e frames  tracer  of  used  f l o w was  pleasantly  i n d e p e n d e n t l y of b o u n d a r y  flow i t  developed  by  und Mechanismus desWiderstandes an Z e i t s c h r i f t f u r T e c h n i s c h e P h y s i k J_2,  4  B.Ahlborn  and m y s e l f .  turbulence  b a s e d on t h e i n t e r a c t i o n s  elements  in  "vortons". the  fluid  the  spectrum  important  coherent  size.  from t h e  power  the  spectral  density  presented  i t provided  present  statistical  "Big  Little And  This  f e e d on t h e i r  that  and c o u n t e d  the  by  information  of  the  turbulent model i s  f o r performing  the  whorls,  velocity;  w h o r l s have s m a l l e r  so on u n t o  laden  poem  It  introduces  model a d d r e s s e s .  characterized  whorls,  viscosity."  information  was  Our model  PHYSICS l a b r e p o r t  #89  written  several  Firstly,  by t h e p r e s e n c e  s c a l e s of the motion.  UBC PLASMA  model  study.  which our v o r t o n  size  was  was c o n s i s t e n t w i t h a  the motivation  (1922).  is  that  It  i n t r o d u c t i o n to the vorton  L.F.Richardson,  flow  dynamics.  be o b s e r v e d  w h o r l s have l i t t l e  Which  elements  i s the p r e d i c t i o n of  description  A general  coherent  idealized  of the v o r t o n  distribution  fluctuations. as  model  t o show-  f a c e t s of  idealized  vorton  could  eddy  of these  f o r the success  structures  in  t o e x p l a i n many  We c a l l  I t was a l s o i m p o r t a n t  contained  1  flow.  C e n t r a l to the vorton  eddy-size  therefore  We a t t e m p t  1  concepts  turbulent  of a broad  incorporates  by  fluid  range i n this  as  5  the  presence  Secondly, energy  of  different  t u r b u l e n t flows  sizes  are  generally  i n t o l a r g e eddies through  flow f i e l d .  The s t r a i n produced  turn  energy  feeds  into  "vortex s t r e t c h i n g " or  the  assumption  define  rates  laminar  flow f i e l d ,  or  found  to  feed  by  the  larger  eddies  in  s m a l l e r s c a l e s , t y p i c a l l y by The  cascade".  whole  process  Our model s t a r t s  is with  that a t u r b u l e n t flow f i e l d can be d e s c r i b e d  as being a composite  are  thought  shear s t r e s s e s i n the mean  "tearing".  commonly c a l l e d the "energy the  of eddies i n the flow.  of eddies and laminar  flow.  We  then  f o r an eddy's i n t e r a c t i o n with the f l u i d , the and other  eddies.  These  interactions  to dominate the eddy dynamics i n the small s c a l e  d i s s i p a t i v e regime, the l a r g e s c a l e regime and the medium  eddy s i z e or c o l l i s i o n a l  regime, r e s p e c t i v e l y .  We then  use  these  f o r 'the d i s t r i b u t i o n of eddy  size  rates  scales.  to  T h i s d i s t r i b u t i o n we c a l l  Richardson's ultimately  poem ends  dissipation. the energy function be used  solve  In loss  of  brings in  out  the  heat  spectrum.  energy  transfer  through  viscous  our model t h i s i s q u a n t i f i e d a c c o r d i n g to rate  eddy s i z e .  flow.  that  producing  through  viscous  dissipation  as  a  We a l s o show how t h i s spectrum can  t o p r e d i c t macroscopic  turbulent  the eddy-size  properties  of  a  particular  These p r o p e r t i e s i n c l u d e drag f o r c e s , wake  s i z e s , d i f f u s i o n processes, the onset of t u r b u l e n c e and mean velocity The dealing  profiles. name vorton i s chosen  to  indicate  that  we  are  with a v o r t i c a l s t r u c t u r e which c o n t a i n s energy and  6  can  interact with  as b e i n g  i t s flow environment.  e x c i t e d s t a t e s of a f l u i d  sometimes  used  Vortons  flow.  interchangably with  The word  'vorton.  spectrum i s i n t e r p r e t e d as the p r o b a b i l i t y eddy to  as  a f u n c t i o n o f eddy s i z e .  as  well  as time.  Supporting  p r o p e r t i e s of the e d d i e s ,  velocity  distributions,  of  and  are  finally  a p p l i c a t i o n s of cylindrical dimensional  the  flow.  the  the  We d i v i d e  a  given  vorton  vortons  the  the v a r i o u s  state  the  size w i l l  spontaneously  through  produce,  In  the f i r s t  a  circular  p a r t o f a two  interactions A vorton  i t , with the f l u i d vortons.  with  coefficients  flow at  states, i s quantifying  We u s e A c o e f f i c i e n t s t o  r a t e a t w h i c h t h e number o f e d d i e s decay t o e d d i e s  of a given  of a d i f f e r e n t  viscous i n t e r a c t i o n with the f l u i d  B c o e f f i c i e n t s a r e used t o d e s c r i b e  a  may  The r a t e  changes, t o other  i n t e r a c t i o n processes.  describe  considering  have  possible  surrounding  d e s c r i b e d by a r a t e e q u a t i o n  i n the  which  eddies.  i n w h i c h i t i s embedded o r w i t h o t h e r which  internal  isolated  i n i n t o t h r e e main t y p e s .  with the f l u i d  spectrum  c a n be  symmetry a n d a r e t r e a t e d a s b e i n g  v o r t o n can take p a r t interact  destroy model  of  o r i e n t a t i o n s i n space.  c h a r a c t e r i z e d , so c a n p h y s i c a l p r o c e s s e s and  an  function  v e l o c i t i e s both  i s supposed t h a t j u s t as the eddies  evolve,  observing  their  f l o w d i r c t i o n a n d n o r m a l t o i t , a n d i f we  It  is  eddy-size  the eddy-size  such as  convection  flow, their  The  I t c a n be a  are other  three dimensional  eddy  This spectrum i s c e n t r a l  our d e s c r i p t i o n of t u r b u l e n c e .  space  are treated  the  size  surrounding i t .  rate  at  which  a  7  population  of e d d i e s  d e c r e a s e due This  is  goverened  dynamics.  C  population  changes  are  due  g o v e r e n e d by  to  eddies  Although  derived  coefficients  are  the p o p u l a t i o n s can  from  the  local  s t a t e i n the  s i z e spectrum.  We  fluid  t h u s see  our  eddy  infant  stages  the v o r t o n  sizes  and  energies  describe of a r e a l It processes regimes.  is  processes  of m o t i o n .  statistics  I f the  The  as  the  and  these  determine are  one  known  f o r each  be  a  Although  deals  with  the v o r t o n our  in i t s  discrete based  on  d e s c r i p t i o n to  idealization  some r e l a t i o n  in a  continuous  v e r s i o n c o u l d be  that  has  vorton)  to note that  sizes.  only  use  hope  C  well.  physics  rates  in general  a continuous  t o the  can  physics  flow.  generally  dominate the Standard  physical  m o d e l as a q u a n t i f i c a t i o n of  model  t h e phenomenon and turbulent  coefficients  rate equations,  enrgies  s i m i l a r p r i n c i p a l s . t h e same We in  These r a t e  I t i s important  flow t h e i r w i l l  of  flow  other  system, to p r e d i c t the eddy-(or  distribution  model the  interacting with  as p r o b a b i l i t i e s w h i c h  t h e eddy c a s c a d e p r o c e s s . turbulent  the  equations  of energy s t a t e s .  field.  vortex stretching  deterministic  w r i t e a c o m p l e t e s e t of  vorton  flow  or  describe  collisional  treated  fluid  to  local  fluid  incorporate  local  used  processes.  the  e i t h e r increase  t e a r i n g and are  supposed d e r i v a b l e from  coefficients  one  by  coefficients  i.e. collisional  which are  size w i l l  to i n t e r a c t i o n w i t h the  rate  eddies,  of a g i v e n  accepted flow  texts  that  dynamics on  different in  turbulence  physical  different describe  the  scale large  8  s c a l e motions as being a s s o c i a t e d with energy the  mean  flow  with  " p r o d u c t i o n " regime. inertial,inviscid, dominated  and  these  scales  Kolmogorov  small  regime  In  the  light  at  the  are  collision  losing  energy  in  the  production  C c o e f f i c i e n t s dominate i n the i n e r t i a l  and the A c o e f f i c i e n t s dominate i n the d i s s i p a t i v e In  the  managable  interest  in  of  keeping  i t s infant  apply  these  the  model  regime  regime.  simple  and  stage i t s a p p l i c a t i o n s have been  c o n f i n e d t o two dimensional principal,  to  of our model we would say that the B  c o e f f i c i e n t s dominate the eddy dynamics regime,  up the  s c a l e motions which comprise the  d i s s i p a t i v e regime are most e f f i c i e n t heat.  making  from  The mid-size s c a l e s which comprise the or  the  size  transfer  flow  ideas  phenomena. to  three  One  can, i n  dimensional  flow  situations. The  r a t e equation approach has been borrowed  description is a  significant  theory. myself  of  It  was  atomic  departure  from  conceived  and developed  i n p a r t based  F.Ahlborn.  i o n i z a t i o n phenomena. main  from  the  T h i s approach  stream  turbulence  by B.Ahlborn and  on flow photographs and some  ideas  of  9  II. 2.1  S T A T I S T I C A L DESCRIPTION OF TURBULENCE  M o t i v a t i o n And We  all  assert  the  Introduction  t h a t t h e eddy s p e c t r u m d e s c r i p t i o n  information  needed  meaningful  f l u c t u a t i n g flow  description  should  statistical  flow the  recreate  field.  spectra.  they apply  i s the subject  spectrum  and  spectra  curves  A  of t h i s c h a p t e r .  are  predict  a l l the  d e s c r i p t i o n of these of  turbulent  dimensional  field.  Fourier  physically  These s p e c t r a  b a s e d on h i g h e r  time or space v a r y i n g v e l o c i t y correlation  to  t o the study  power s p e c t r a l d e n s i t y , t h r e e  a  The eddy s p e c t r u m b a s e d  t h u s be s u f f i c i e n t  fluctuation  s p e c t r a a n d how  to  contains  order The  fluid include  wave  number  moments o f t h e often  transforms  of  studied t h e above  m e n t i o n e d s p e c t r a a n d s o w o u l d be e q u a l l y w e l l p r e d i c t e d the  eddy s p e c t r u m d e s c r i p t i o n .  may  or  may  situation If  not  be  The  significant,  inter-eddy c o r r e l a t i o n s depending  on  s t u d i e d a n d t h e eddy s i z e s c a l e b e i n g  of t h e c o r r e l a t i o n s p r e s e n t  fields  should  description.  be  description  p r e d i c t i o n s of the macroscopic wake  sizes,  velocity A  from  the  should  turbulence  flow  eddy  spectrum  also  provide  flow p r o p e r t i e s such as  d i f f u s i o n processes,  flow  represent  i n the flow, meaningful  reproducible  This  the  considered.  we b e l i e v e t h a t t h e p r i m a r y c o h e r e n t s t r u c t u r e s  all  by  drag,  o n s e t a n d mean  profiles. good  statistically  test based  of  the  validity  and  viability  c o h e r e n t s t r u c t u r e s model of  of  a  turbulent  10  f l o w w o u l d be spectral  to  see  density  spectral density  be  for  the  in  by  signal.  Fourier  I n an  information statistics  on  the  only  the  correct  flow.  be  the  The  energy content  analyzing study  This  a  streamwise  fluctuating velocity cascade  implicitly.  In the  t h e p h a s e r e l a t i o n s among them.  easily);  more  convenient  sophisticated  w a n t s t o decompose a v e l o c i t y waves." the  1  Our  This spectra  of t h e  chapter and  how  spectrum  of  turbulence.  field  dynamics  and  the  words o f J . L . L u m l e y ,  transforms  can are  eddy "An  coefficients  Fourier transforms  (spectra  field  contains  be  are  measured  needed i f  one  instead  of  into eddies  e f f e c t s from  the  statistical  power s p e c t r a l d e n s i t y c a n  fluctuation be  obtained  from a f l u c t u a t i n g v e l o c i t y .  1  the  eddies.  describes the  unit  component  approach i s to p r e d i c t t u r b u l e n t  distribution  per  of t u r b u l e n c e  which i s c r e a t i n g the  eddy  used because they are  power  time dependent  e d d y , h o w e v e r , i s a s s o c i a t e d w i t h many F o u r i e r and  power  f i e l d measured a t a f i x e d p o s i t i o n  t o the o b j e c t  T h i s d e s c r i p t i o n of t h e  given  experimental  fluctuating velocity relation  a  the  temporal frequency.  time v a r y i n g q u a n t i t y could the  predicts  i s a measure of  obtained  velocity  it  curves  mass a s s o c i a t e d w i t h can  if  Lumley,J.L.,1970. S t o c h a s t i c t o o l s i n t u r b u l e n c e . A c a d e m i c Press,New York.  11  2.2 S t a t i s t i c a l  Spectra  In t h e s t a t i s t i c a l s t u d y of t u r b u l e n t tries  to  spectra texts  predict  and  determine  which  describe  density  fits  turbulent  The  There a r e  t h i s approach i n d e t a i l . t o show how  the  power  spectral  i n a more c o m p l e t e s t a t i s t i c a l d e s c r i p t i o n o f  turbulent  starting  turbulence  many  Here I w i l l  f l u c t u a t i o n s a n d how i t c a n be o b t a i n e d  to describe  one  the v e l o c i t y f l u c t u a t i o n  o r , e q u i v a l e n t l y , the c o r r e l a t i o n s .  p r e s e n t enough i n f o r m a t i o n  and  fluctuations  and  used  flows.  point  of t h e s t a t i s t i c a l d e s c r i p t i o n of  i s to consider  t h e time dependent v e l o c i t y U ( x , t )  pressure  P(x,t)  fields  a s t h e sum o f s t e a d y a n d u n s t e a d y  components. U i ( x , t ) = U ( x ) + u ( x \ t ) i = 1,2,3  (2-1'a)  P(x,t)=P(x*)+p(x\t)  (2-1b)  k  k  H e r e U^x) a n d TA( 5c, t ) a r e t h e t i m e u (x,t) L  and  p„(x,t)  are  the  averaged  fluctuating  quantities  and  components.  The  overbar symbol i s used t o denote a time average. time dependent q u a n t i t y  (2-2)  i s m e a n i n g f u l t o speak o f time dependent time a v e r a g e s i f ( d / d t ) [ a ( t ) ] « ^ ( d / d t ) [ a ( t ) ]*  which  states that  assumed  that  (2-3)  t h e a v e r a g e s must c h a n g e much more s l o w l y  than the average change. be  some  a ( t ) , "a~(t) i s f o u n d a s ,  a ( t ) = l i m -!? (dt'a(t') It  For  the  In the r e s t of t h i s t h e s i s i t w i l l  flow  fields  being  dealt  with  are  12  statistically The  fields  equations  kinetic  and  isothermal, motion  of  turbulent stress  viscous  fluid  the  well  form w i t h  equations  the  next  motion  to  obtain  For  and  appropriate  for  the  and the  mean  Reynolds  incompressible,  appropriate  equations  of  equations.  In  convention  being  used  (2 - 4)  k  condition  time averaged  result  the  summation  4  the  energy,  flow  the  Navier-Stokes  AU  substituting  into  equations. flow  fluid  equations  kinetic  2  known  the  the  substituted  ( - 1 /j> )± P . 9^_U-  continuity  The  for  are  i J J i . Uj aUj with  (2-1)  shear  are  component these  is  energy,  normal  1  representation  pressure fluid  steady.  eqns.(2-1)  =0  (2-5)  Navier-Stokes into  equations  eqn.(2-4)  and  are  time  found  by  averaging  is (2-6)  Where  IIUi  Dt  following Ut ,  1  2  v  ( _ _ A U ; . U j ^_Ut)  ~<Jt  the  using  +  ^ 7  is  the  fluid  motion.  the  continuity  Statistically steady is q u a n t i t i e s do n o t c h a n g e i n A l e s s m i s l e a d i n g name for f l u c t u a t i o n k i n e t i c energy  time  rate  Multiplying  this  condition,  used here time. this  to  of  change  L  equation  by  eqn.(2-5),  to  mean t i m e  equation  o f U-  would  averaged be  the  13  consolidate  v e l o c i t y components i n s i d e  recognizing  that  Ui J^LIj. = D ( i U i U ) t  energy equation  Dt  d*i  (A) Term  '(D)'  (C)  ~  V  of f l u i d  unit  mass,  i s equal  plus  (D)  as (2-8)  (C)  unit  t o (B) t h e p r e s s u r e  work done on t h e  the  mean  transport  fluctuation gradient  of  correlations,  d i f f u s i o n , plus  energy plus  by  (E) t h e  (F) t h e  mean  dissipation. turbulent  multiplying  substituting  is  equation  (A)  (B)  i . e . U: LU/=>  <L(U.-Uj )/2  equations  (2-1)  from  the turbulent  /•  is  f o u n d by (2-4)  this  £  and  AXL  (C)  as^«.i=0 f r o m  ^axj  ixy  (D)  continuity  equation  after  k i n e t i c energy equation  /xi  by  f o r Ui a n d P a n d t h e n  The mean k i n e t i c e n e r g y  subtracted  Sxt \ '  J  energy  the expressions  the r e s u l t .  then  consolidation  M  kinetic  the o r i g i n a l Navier-Stokes  time averaging  1  (2-7)  state:  velocity  The  (2-7)  flow,  K  c h a n g e due t o v i s c o u s  U: ,  mean  o f c h a n g e (A) o f mean k i n e t i c e n e r g y o f a  mass  viscous  the  (F)  These e q u a t i o n s  orthogonal  and  AX '  (E)  The t i m e r a t e  at  T*Z  r  (D) may be r e w r i t t e n ex*  arrive  f o r turbulent  (B)  Tx£  we  1  ot  Ut  kinetic  the d e r i v a t i v e s ,  some  results  ( 2 - 9 )  14  The  following  physical  quote,  taken  change  - per  unit  (A) i n k i n e t i c  turbulence from  t h e mean m o t i o n  stresses,or plus  the  by  In  the  viscous  a  manner s i m i l a r  Reynold's  shear  correlations  order.  can  Equations  fluctuating  their  and  of  of  the  dissipation  per  stresses  the  shear energy,  mass  lack  advantage  f i e l d s a s t h e sum  be  involving  velocity  are  and  mostly  of  correlations  inconvenience  due in  meaning.  the v e l o c i t y fluctuating  and  pp  pressure  component  is  equations  are  i n the r e s u l t i n g  TURBULENCE,J.O.Hinze; McGRAW-HILL, 1959,  no  velocity  components are seldom used  treating  appearing  (u^u*.,  The  higher order  of i n t u i t i v e p h y s i c a l of  normal  derived.  complexity,  of a mean  that the q u a n t i t i e s  - i ^ j ) and  i n these equations  mathematical  measurement and One  transferred  t o t h a t used above, e q u a t i o n s f o r  (uTuJ ,  fluctuation  the  shear (E)  total  the t u r b u l e n c e  of  the  o f mass by t h e t u r b u l e n t m o t i o n . "  stresses  to  1  plus  summation)  of  the  p r o d u c t i o n of t u r b u l e n c e  turbulent motion, unit  of  (C) t h e e n e r g y  through  the  turbulence  i s e q u a l t o (B)  (D) t h e work done p e r u n i t  time  of  by t u r b u l e n c e  energy,plus  describes  1  in t h i s equation.  energy  of mass of t h e f l u i d  convective diffusion  second  Hinze,  meaning of the terms a p p e a r i n g  "The  the  from  65  15  experimentally anemometery individual drawback  fluid  techniques. terms  of  physical  measureable  have  this  Another  1  some  approach  interpretation  velocity  using hot-wire or laser  at  advantage  physical i s that  can  be  i s that  meaning. in  boundary l a y e r  with  preferred  a  may  rotation  the  measurement  interpretation be  treated  energy  coherent  sense.  The  equation  velocity  kinetic  energy  implies  a  mean  was  mentioned.  Thus  of  non-zero  o f mean  correlation  c o h e r e n c e b e t w e e n t h e two f l u c t u a t i n g  turbulent  flow.  These  structures,  velocities  noted that  addressed  in  the  these s t r u c t u r e s  above  in  now t h o u g h t t o be  i m p o r t a n t t o t h e d y n a m i c s o f most t u r b u l e n t directly  must  orthogonal  which would p o i n t t o t h e presence of coherent s t r u c t u r e s the  the  t h e mean k i n e t i c  to the transport A  were  velocities  significance  correlations  in  constructively  velocity.  In discussing  (2-7) the  structures  of coherent s t r u c t u r e s  the  caution.  average  fluctuation  o f t h e mean a n d f l u c t u a t i n g  with  fluctuating  of  this  F o r example, a  c o n v e c t e d p a s t an anemometer w o u l d c o n t r i b u t e to  cases  i n s p a c e may be p a r t i a l l y  contain  v e l o c i t y m e a s u r e d when a s e r i e s  the major  The  comprised of coherently adding f l u c t u a t i o n s . turbulent  A  some  misleading.  a given point  doppler  flows  approach.  are the s t a r t i n g  are not  I t s h o u l d be point  of  our  In t h e s t r e s s e q u a t i o n s t h e p r e s s u r e fluctuation-velocity fluctuation correlations a r e n o t g e n e r a l l y measureable and must be i n f e r r e d f r o m t h e s t r e s s e q u a t i o n s a n d m e a s u r e m e n t s of t h e o t h e r t e r m s .  16  model. The  gain  in  writing  the v e l o c i t y  mean a n d f l u c t u a t i n g c o m p o n e n t s mean v a l u e s a r e q u a n t i t i e s potentially  misleading,  i s that  which  have  physical  f i e l d a s t h e sum o f the•correlations a  significance  r e a d i l y measured w i t h h o t - w i r e o r l a s e r techniques.  It  is  important  useful,  v e l o c i t y a n d p r e s s u r e f i e l d s a s t h e sum o f fluctuating equations.  although  and they a r e  doppler  t o note that  anemometry  i n writing the  the  steady  c o m p o n e n t s we no l o n g e r h a v e a c l o s e d S i n c e t h e number  of  description  unknowns has  and  has  and  system of  doubled  originally  closed  now  been  incomplete.  C l o s u r e h y p o t h e s e s a r e needed t o s o l v e t h e  an  become new  equations of motion. 2.3 C o r r e l a t i o n s The  And S p e c t r a  turbulent  energy  equation  p r o d u c t s of f l u c t u a t i n g q u a n t i t i e s To  study the length scales  need t o c o n s i d e r at  different  point  a t one p o i n t  of the turbulent  fluctuating quantities  points  correlations  statistically  contains  in are  space.  considered.  The  space.  are  CijCx, r )=Uj(x,t)ui(x+? , t ) where X " i s t h e p o s i t i o n is the fluctuating  no more t h a n t w o most  general,  and  x+r  i , j = 1,2,3  of  the  i s the position  between  written (2-10)  where t h e c o r r e l a t i o n  component  we  measured  steady, two-point s p a t i a l c o r r e l a t i o n  f l u c t u a t i n g v e l o c i t y c o m p o n e n t s may be  direction,  in  mean  fluctuations  which  Usually  only  i s d e f i n e d , u;  velocity  in  the  i  where t h e j t h v e l o c i t y  17  component i s m e a s u r e d , s e e f i g u r e  Figure The  1 - Correlation  non-dimensionalized  correlation  1.  Velocities  correlation  c o e f f i c i e n t and i s g i v e n  is  called  as  R ( x \ f )= ut (g.t)u-E (*+t , t ) Jul (3f,t)/u> ( X + r , t )  (2-11)  M  )3  forms . a  are  such that  s e c o n d r a n k t e n s o r whose i n d i v i d u a l  -1<R (*,f)<1 y  the  degree  components. identical  of A  correlation typical  components  i,j=1,2,3  This agrees with the i n t e r p r e t a t i o n  spatial  the  (2-12)  t h a t Ry i s a measure  between  the  correlation  v e l o c i t y components a p p e a r s i n f i g u r e  two  velocity  curve 2.  of  for  18  0  Figure  0  2 - Spatial Correlation  R:L=1  A t f = 0 we s e e f r o m e q n . ( 2 - 1 1 ) a b o v e t h a t is  a  property of t u r b u l e n t  components and s o we  are uncorrelated  the  => 0  x, d i r e c t i o n  for sufficiently  f o r large  then  correlation  'lateral measure  fluctuating  |?|  large  It  velocity distances  (2-13)  t h e mean f l u i d v e l o c i t y m e a s u r e d  spatial  i = 1 ,2,3.  have Ry(3f,f-)  If  flows that  Curve  spatial of  fluctuations  the  R^d^)  is  i n , say,  i s c a l l e d the ' l o n g i t u d i n a l  c o e f f i c i e n t ' and correlation  length  at #  scade  R«(r,)  and  coefficients'. of  the  energy  R^(c,) A  are  simple  containing  i s g i v e n by (2-14)  called In  the i n t e g r a l length scale, some f l o w s i t u a t i o n s  see f i g u r e  a useful  2.  measure  of t h e l e n g t h  19  s c a l e s i n t h e f l u c t u a t i o n s c a n be f o u n d correlations.  The  auto-correlation  p r o d u c t o f t h e same  from  the  curve  quantity  measured  (x,t+T)  0=1,2  temporal  i s the average  as  a  function  of  s e p a r a t i o n t i m e T, C^(T)=u (x,t)u a  f l l  T h i s c u r v e i s much more e a s i l y the  space  or 3  (2-15)  o b t a i n e d from experiment  c o r r e l a t i o n s d i s c u s s e d above.  O n l y one v e l o c i t y  measuring  probe, a c o r r e l a t o r  needed.  The a u t o - c o r r e l a t i o n c u r v e i s o b t a i n e d by  the d e l a y time T r a t h e r probe.  The  fluctuations,  time  and a s i g n a l  than p h y s i c a l l y scale  the i n t e g r a l  of  delay  moving  the  unit  a  energy  velocity containing  time s c a l e Te, i s d e f i n e d as (2-16)  r  auto-correlation  are  sweeping  Te=1/u j c ( T ) d T ' The  than  i s useful  in  the  study  i t can  be  related  turbulent  fluctuations  when  correlations.  T h i s r e l a t a t i o n s h i p and t h e c o n d i t i o n s of i t s  applicability  are discussed  Consider a flow figure  3.  to  of the  spatial  below.  f i e l d moving  past  a  velocity  probe,  20  Figure Assume  the  elements  convection  and  appreciably  3 - Flow F i e l d  the in  speed  coherent  the  time  is  Past  Probe  the  same  structures i t takes  The p r o b e m e a s u r e s t h e v e l o c i t y  time.  The  correlation  of  multiplying  correlation  the  the  is  temporally  time  axis  do  not  change  them t o f l o w p a s t t h e  probe.  space  fora l l fluid  as  a  obtained  varying  function  of  from t h e time  signal  by  simply  o f t h e a u t o - c o r r e l a t i o n by t h e  p r o b e s p e e d , Uc. C^(T)=C (X/Uc) a,b=1,2,3  (2-17)  al)  The  integral  time  s c a l e can thus  be r e l a t e d  t o the  integral  l e n g t h s c a l e by t h e r e l a t i o n , Te=Le/Uc This can  co-ordinate be  applied  G.I.Taylor. hypothesis  The  (2-18)  t r a n s f o r m a t i o n a n d t h e c o n d i t i o n s when i t is  called,  important  "Taylor's  hypothesis"  point t o note i s that  c a n o n l y be a p p l i e d when t h e f l o w  after  Taylor's  f i e l d does  not  21  change  appreciably  during  the  time  i t takes  m e a s u r i n g probe t o sample a d i s t a n c e g r e a t e r s c a l e of  interest.  a constant  In a d d i t i o n the  convection  velocity,  J u s t as a f l u c t u a t i n g the  spatial  transforms The  of  called  the  three dimensional  t  Ikl = 2 f f / ' \  being  t o measure a l l  define this  spectrum.  spectrum  experiment  i s the  which  the  be  describe  spatial  described  i t by  l o s s of  the  correlation  wave v e c t o r  velocity  is  1  Fourier  tensor  is  spectrum, (2-19)  Experimentally components  spectrum  i t is  needed  o f t e n used i n both  k„-wave number  by  information.  (x + r , t ) e x p ( i K - r ) d r  s  length  e l e m e n t s must h a v e  can  t h e wave number.  impractical  One  can  of the  $ i $ ( k , x ) = [ l / ( 2 T f ? ] J,u ( x , t ) u with  field  the c o r r e l a t i o n s w i t h o u t  Fourier transform  than  velocity  Uc.  flow  c o r r e l a t i o n s we  fluid  the  theory  which  is  to  and found  f r o m $ (Tt,3n a s tl  (2-20) It of  i s twice  the  the v e l o c i t y  total  of a l l k i n e t i c  fluctuations  wave number b e t w e e n k, -dk, /2 In in  the  the  experimental  f o l l o w i n g chapters  Xi  in  the  and  k, +dk,  study  energies  having  /2. turbulence  convenient  t h e power s p e c t r a l d e n s i t y o f t h e u  component,  E„(w,x).  E„(w,x)  the  energy per  u n i t mass  i s twice  t h e u,  associated  described  quantity  m e a s u r e was  kinetic  u n i t mass  direction,  of g r i d  t h e most  per  to  fluctuating component of  with  temporal  22  frequency  w.  I t c a n be o b t a i n e d  1  fluctuating  velocity  directionally and  The and  component  sensitive  by  Fourier  u, ( x , t )  of  measured  probe a t a f i x e d p o s i t i o n  E„(w) a s d e f i n e d a b o v e f o r m a c o s i n e  limits  analyzing  with  x.  transform  C„(w)  pair,  «. (w)=4£c,, ( T ) c o s ( 2 t t w T ) d T  (2-2la)  C„ (T)= j°E„(w)cos(2TfwT)dw  (2-21b)  integration  a r e f r o m 0 t o o a s E„ (w) =E,, (-w)  From e q n . ( 2 - 2 1 b ) we- s e e t h a t C„(0)= CE„(w)dw  (2-22)  •'o  using  a  E  C„(T)=C„(-T).  And  the  the  expression  of  eqn.(2-15)  for  the  autocorrelation, uT=/ E„(w)dw  (2-23)  fl  This  equation  identifies  e n e r g y u} w i t h t h e a r e a  the  under  total  the  fluctuating  power  kinetic  spectral  density  curve. Applying  Taylor's  hypothesis  d e n s i t y measurement we a r r i v e  to  the  power  at the r e l a t i o n  ©„ (£/x)=Uc/(2l7)E (w,x)  (2-24)  n  The number velocity  power s p e c t r a l spectrum  spectral  density i s related  (Tt,x) t h r o u g h  to  the  the longitudinal  convection  U a t which the probe samples the flow f i e l d c  same way a s t h e s p a t i a l a n d t e m p o r a l  It should be n o t e d that frequency and not t h e angular  correlations  k.,-wave  i n the  are.  This  t h e symbol w i s t o denote t h e f r e q u e n c y a s i s commonly d o n e .  23  includes  the  convection change  same r e s t r i c t i v e a s s u m p t i o n s , n a m e l y  v e l o c i t y U i s constant  appreciably  greater  than  while  the length  16) i n e q n . ( 2 - 2 1 a )  the  and t h a t t h e flow does n o t probe  samples  s c a l e s of i n t e r e s t .  a  distance  Using  eqn.(2-  f o r w=0 we s e e , E ( 0 ) = 4ujTe  (2-25)  M  And a p p l y i n g we  see  Taylor's  hypothesis  i n t h e form  that the i n t e g r a l length  t h e power s p e c t r a l  of  eqn.(2-18)  s c a l e c a n be o b t a i n e d  F l u c t u a t i o n s And The Power  (2-26) Spectrum  The power s p e c t r u m E(w) w h i c h i s d e f i n e d the  transform  of  the c o r r e l a t i o n  from a F o u r i e r t r a n s f o r m see  this  we  fluctuating  first  of the f l u c t u a t i n g  write  terms  To  1  longitudinal  representation,  •'-*>  coefficients  ]  (2-27) f o r the  f u n c t i o n s o f f r e q u e n c y n, a(n)=  l/rrjat u, (t)cos(2fhnt)  (2-28a)  b(n)=  '/tfjdt u, (t)sin(2Hr»t)  (2-28b)  from TURBULENCE,J.0.Hinze;McGRAW-HILL,  of  directly  velocity.  t h e time dependent  v e l o c i t y u,(t) i n i t s F o u r i e r  and b ( n ) a r e t h e F o u r i e r  in  c a n be o b t a i n e d  u, ( t ) = 2 { d n [ a ( n ) c o s ( 2 T T n t ) + b ( n ) s i n ( 2 i T n t ) a(n)  from  d e n s i t y E,,(w) a s Le=UcE„(0)/(4u*)  2.4 V e l o c i t y  that the  1959, pp  54-58  basis  24  The  auto-correlation  velocity  coefficient  for  the  longitudinal  fluctuations i s , C„(T)=u, ( t ) u , ( t + T )  Inserting  the Fourier  into this expression  representation and p e r f o r m i n g  (2-29)  for u,(t),  eqns.(2-28),  some o f t h e i n t e g r a t i o n s  we h a v e u, ( t ) u , (t+T) = |; 6m[a '(m)+b ' (m) ]/T cos(2fTmt) 2  (2-30)  7  m  and u^lini  ^ d m t a * (m)+b* (m) ]/T  where T i s t h e s a m p l i n g t i m e . the  cosine  (2-31)  I f we now i d e n t i f y E„(n)  transformed quantity  i n eqn.(2-30) f o r t=0,  E ( n ) = rr '[a (m)+b*(m) ]/T 2  (2-32)  i  )I  we may  write . R ( T ) = 1/u fdnE„(n)cos(2rfnt)  (2-33)  L  l)  and  comparing t h i s w i t h eqn.  expressions, It  was more  analysis  t o do t h i s  our  the  to  velocity  obtain field  with  f l u c t u a t i o n spectra defined spectra  E (w) 1(  from  directly  Equation  i n the continuous case.  eddy-size  that  the  ( 2 - 3 2 ) a n d ( 2 - 2 1 a ) f o r E,,(w) a r e e q u i v a l e n t convenient  of  ( 2 - 2 1 b ) we s e e  obtaining the a u t o c o r r e l a t i o n .  the  with  a  than  thesis  .  Fourier by f i r s t  (2-32) t e l l s  In t h i s  two  us how we  use  a s i n e q n . ( 2 - 3 2 ) t o compare  d e s c r i p t i o n of t u r b u l e n t  fluctuations  t h e f l u c t u a t i o n s m e a s u r e d u s i n g a h o t - f i l m anemometer. The  standard  turbulent  flow  methods u s e d t o p r e d i c t t h e e v o l u t i o n o f a field  kinetic  energy equation  stress  equations,  are  either  (2-9) , o r  based  on t h e t u r b u l e n t  similarly  the  Reynolds  o r k , i -wave number s p e c t r u m , e q u a t i o n ( 2 -  25  20).  B o t h a p p r o a c h e s need c l o s u r e  hypotheses to  be  solved  a s i n w r i t i n g t h e v e l o c i t y and p r e s s u r e f i e l d s a s t h e sum  of  an  average  and  a  fluctuating  unknowns h a s d o u b l e d equations. needs  To  close  assumptions  derivatives  without  component  how  are r e l a t e d .  how  amplitude are  of  the  number  of  energy e q u a t i o n (2-9)  one  the  processes  stress  which  expedience than dictating  the  by  a t one  velocity  insight  into  the  evolution. or  more  In  by  assumptions mathematical  physical both  coherent,  which  a f f e c t the  A g a i n model  dictated  d e t a i l s of the c o l l e c t i v e ,  function  frequency w i l l  are  flow  and  transfer  at another frequency.  invoked  in  I n t h e c a s e o f t h e k,,-wave number  s p e c t r u m one must know t h e e n e r g y describes  number  increase  the k i n e t i c  about  the  processes  cases i t i s the  effects  which  is  added t o complete the d e s c r i p t i o n . Our  approach  is  to  model  coherent s t r u c t u r e s d i r e c t l y . c o e f f i c i e n t s that describe equations  of m o t i o n .  p r o p e r t i e s of the flow  We  the  seek t o  s u c h a s t h e eddy s i z e  i s much l i k e  that  i t more n a t u r a l l y a c c o u n t s f o r t h e  from the  fluid  aim  distribution.  coherent  structures  this  exist  in  thesis a  structure assumptions  of the i n t e r a c t i o n s r a t h e r  o f t h e wave number  of  describe  t h e k„ - wave number a p p r o a c h e x c e p t  be b a s e d on t h e p h y s i c s  The  the rate  e n e r g y t r a n s f e r m e c h a n i s m s so t h a t any  the mathematics  of the  These r a t e s a r e then used t o  model  can  derive  t h e eddy e v o l u t i o n  Our  dominated  interaction  is  turbulent  than  spectra. to  show  that  f l o w and t h a t  coherent their  size  26  distribution  can  be m e a s u r e d and  power s p e c t r a of v e l o c i t y This  study  turbulence.  is  based  can  be  fluctuations  on e x p e r i m e n t a l  used to p r e d i c t in a turbulent observations  the  flow. of  grid  27  III. 3.1  GRID TURBULENCE EXPERIMENTS  Introduction For the study of the s t a t i s t i c s  a  simple  and  turbulent  flow  apparatus First, observed Second,  reliable was  needed.  we  wanted and  we  to  see  measured  wanted  The  to  a  see in  a  and  tank  simple  fluid  the f o l l o w i n g object  was c h o s e n  reasons.  s t r u c t u r e s c o u l d be technique.  s t r u c t u r e s c o u l d be eddy-size  spectrum.  using  in  a  velocity  t o be w a t e r  probe.  for versatility  reference  rather  than a water  With a towing tank frames  are  easily  D i a g n o s t i c equiptment t h e l a b frame.  It  both  the  also  H o t - f i l m probes  does  not  quiescent  have problems  'upstream' of boundary  limits  T =L/U ; h e r e L i s t h e u s e a b l e c a r t speed.  be  through the  is  which  for  and cameras can  A d i s a d v a n t a g e of  length  fluid  accessible  layer buildup along the walls. i t s finite  i n the  such as cameras can  A towing tank has a r b i t r a r i l y  conditions.  The  tunnel f o r  a l s o be mounted on t h e c a r t w h i c h moves t h e g r i d  the c a r t  grid  technique.  reasons.  measurements.  water.  observe  and  for several  photographic  i f these  A t o w i n g t a n k was c h o s e n  mounted  towing  i f coherent  with  measured  flow v i s u a l i z a t i o n  and  generate  structures  we w a n t e d t o compare t h i s d e s c r i p t i o n w i t h t h e power  spectral density working  to  shown i n f i g u r e 4 were c h o s e n  meaningfully described Third,  method  of coherent  any  tank  the o b s e r v a t i o n time  t r a v e l d i s t a n c e and U i s  The t o w i n g t a n k b u i l t  for this  s t u d y had  a  Figure A - The Towing Tank  29  Typically  10 o r more r u n s had  d e f i n e a power s p e c t r u m and h o t - f i l m data generating 3.2  The  at d i s t a n c e s greater  possible  than about  1 6 ' x 3 ' x 3 ' was t a n k of  this  tank  built  with  s i z e one  with  inside  f o r these can  reasonable  w a l l s were c o n s t r u c t e d  dimensions  s t u d i e s , see  reach  Reynold's  s i z e d m o d e l s and of  3'x4'  long  the  panels  clear plastic  the  r e s t were o f  on  top  of  which sat fitted  3/4"  t h i c k plywood.  I-beams  r e s t i n g pn  float  any  untoward l e a k s or s p i l l s  the  16'  of  w h i c h was  long  s i d e s c o n s i s t e d o f an mounted a 3/4"  t a n k on  The  occurred.  the  steel  sheets  and  some  seated  while  was  bolted  foundation  catch  pan  was  edge  of  a l u m i n u m u - c h a n n e l on  top  The  upper  rod.  A cart  ran  The  aluminum a n g l e  stock.  bars  t h e o b j e c t s t o be  I t was  cart  ball-  about the c i r c u m f e r e n c e  vibration  tank.  bars  the  b i l g e pumps i n c a s e  diameter s t e e l  the  to secure  a  rods.  Rubber o - r i n g s  the wheels p r o v i d e d  The  of  w h e e l s of t h e c a r t were e q u i p p e d w i t h s e a l e d  bearings.  to  in  one  block  The  a c t i v a t e d marine type  and  frame  a concrete  i n s i d e a p l y w o o d c a t c h pan.  with  over the  The  up  placed  made of  1/2"  With a  model speeds.  floor  were  roughly  numbers  side panels  panels  of  f i g u r e 4.  Three of the  sliding  obtain  130cm f r o m  welded metal frame.  cart  to  grid.  towing  5  not  adequately  Tank  A  1x10 "  i t was  t o be made t o  isolation frame  fitted  was  between  the  constructed  of  w i t h c l a m p s and  immersed i n the  were u s e d t o p o s i t i o n t h e m o d e l s  of  in  sliding  fluid. the  The tank  30  and  t o c h a n g e t h e d i s t a n c e b e t w e e n t h e g r i d and  anemometer. shutter the The  An  trigger,  moving c a r t c a r t was  ran  in  optical  cables  sensor  two  continuous Two  umbilical  frame t o the  p u l l e d by  a  the tank. the  electrical  and  the  drive  clad  end  shaft.  pulley  d r i v e system c o n s i s t e d  set with a 4 p o s i t i o n  shaft.  The  reduction  s p e e d r a n g e f r o m a few  two  meters  1/2  H.P.  The  90V  m o t o r was  start,  stop,  switches,  one  DC  second.  operated  tank. the  cart  took  two  c e n t i m e t e r s per  IR p h o t o d i o d e flat  step  looped  reduction  stop button.  final  second to  about  p o w e r e d by 1,750  switches.  Two  These s w i t c h e s  a  rpm.  s p e e d c o n t r o l box  o f t h e t a n k , were w i r e d  with  microin series prevented  e i t h e r end  of  the  experiment  speed as w e l l  as  stop. an  a timer, figure  black  coupled  tank.  d r i v e s y s t e m was  by a c o n s t a n t  t o t h e c a r t was  of an  which  c o u p l i n g t o the d r i v e  a b o u t 30cm t o r e a c h c r u i s i n g  to trigger  a  c a b l e s were  of the  tank  50cm/sec maximum s p e e d u s e d i n t h e  used  faced  The  step-cone  The  a t e a c h end  t o come t o a c o m p l e t e Attached  cables  the  from i n a d v e r t e n t l y o v e r s h o o t i n g  At the cart  steel  of  a  forward/reverse  w i t h t h e c o n t r o l box the  of  from  i n the l a b frame.  m o t o r w i t h a maximum s p e e d o f  and  the  p u l l e y s were c h o s e n t o g i v e a  cart  per  carried  l e n g t h of e a c h s i d e of  a r o u n d i d l e r p u l l e y s a t t h e o t h e r end The  hot-film  anemometer s i g n a l s  loop along the  12" p u l l e y s on one  to  cable  instruments  plastic  the  and  adjacent  optical detector 5.  was  This detector consisted  photo  s u r f a c e about  that  1/4"  voltaic  cell.  away w h i c h r a n  It the  31  l e n g t h of trigger  the  tank.  starting  W h i t e t a p e was and  passed over the white t h a t was The  timer  was  calibrated  counter  in  the  successive d i s t a n c e , &X, determine  and  signals.  against  a  timer  box  trigger  the  desired  When t h e  detector  between the  signal  s t o p p e d by The  timer's  frequency displayed  pulses.  the c a r t speed.  f o u n d t o be  points.  send a v o l t a g e  started  detector  at  t a p e t h e p h o t o c e l l w o u l d p i c k up  r e f l e c t e d and  succesive was  stopping  placed  two The  This trigger  rising  internal  counter. the time  The  were  of  clock digital  time  between the  known to  c a r t speed t h u s measured  was  1% o v e r most of  system.  cart motion  Figure  part  RC  and  tabs  timer.  used  r e p r o d u c i b l e t o b e t t e r than  w o r k i n g r a n g e of t h e d r i v e  the  t o the  light  5 - O p t i c a l Detector  and  Timer  the  32  3.3 The G r i d The  turbulence  was g e n e r a t e d by a row o f 1/2" d i a m e t e r  r o u n d a l u m i n u m n b a r s s p a c e d 2" a p a r t , end  p l a t e was f a s t e n e d  spanned  t h e tank  vertically  i n t h e Y-Z  volume,  allow  flow  4.  A  t o t h e bottom of t h e b a r s .  cross-section  produce a t u r b u l e n t  figure  with  plane.  The  flow  boundary c o n d i t i o n s and s t i l l  The g r i d  the bars  mounted  g r i d was e x p e c t e d t o  t h a t would provide  the f l u i d  1/4x2"  a  large  sample  t o develop independently  allow  for a  two  of  dimensional  analysis. 3.4 V i s u a l i z a t i o n A p p a r a t u s The  flow  was made v i s i b l e  by p h o t o g r a p h i n g t h e m o t i o n  of a l u m i n u m f i l i n g s u s i n g a M i n o l t a X-570 35mm c a m e r a w i t h a 50mm l e n s a n d . m o t o r i z e d complete  the shutter  back. release  conducting  flapper attached  completed  the shutter  w i t h an a l u m i n u m b r a c k e t One  lead  of  flapper. along by the  t h e tank.  tank.  on  needed  to  t h e camera.  t o t h e tank, see  A  the cart  when i t made  contact  figure  6.  r e l e a s e was g r o u n d e d t o t h e t a n k  the other  was  were p l a c e d  attached  at desired  A s e r i e s of photographs could  to the locations  t h u s be made  back e q u i p e d c a m e r a a s t h e c a r t moved  A piece ensured  circuit  was  t o but i n s u l a t e d from  fixed  Aluminum b r a c k e t s  themotorized  flapper  while  trigger  release c i r c u i t  the shutter  through the c a r t  A  of e l e c t i c a l photographs  over  t a p e on t h e b a c k s i d e o f t h e were n o t t a k e n w h i l e  was r e t u r n i n g t o i t s s t a r t i n g p o s i t i o n .  the cart  33  shutter fixed release camera^^switch cart motion variac  lights water surface conducting bracket Figure Figure reference  so  photography.  appeared The  375  watt  the surface of the water.  The  power  dragged activate  spotlights to  these  The lamps were p o s i t i o n e d  i n t h e camera's  field  of view.  w a t e r s u r f a c e was s e e d e d w i t h a l u m i n u m  filings  d i a m e t e r w h i c h were p r o d u c e d by f i l i n g above  the the  the  grid  surface  through  shutter  of the  release.  the  water.  water  the  The  filing  r e c o r d e d on I l f o r d XP1-400 b l a c k a n d w h i t e exposed  fluid  no r e f l e c t i o n s o f f o f t h e w a t e r s u r f a c e o r shadows  ,1mm  aluminum  system f o r the  Four  was c o n t r o l l e d by a V a r i a c .  that  about  6 a l s o shows t h e l i g h t i n g frame  illuminated lamps  6 - L i g h t s a n d Camera A c t i o n  and  developed  f o r an  ASA  of  a p i e c e of As t h e c a r t  flapper  film  images which  r a t i n g o f 800.  would were was Local  34  v e l o c i t i e s c o u l d be o b t a i n e d by  the  exposure  were u s e d  time.  by d i v i d i n g  Exposure  the  streak  length  t i m e s o f 1/2 and 1 s e c o n d  f o r t h e 20 t o 50cm/sec g r i d  speed  range.  The f l o w p a t t e r n s were s t u d i e d o n l y a t t h e s u r f a c e . neutrally  bouyant  suspension,  inside  t h e f l u i d , was  visualization time  the  experiments  s u r f a c e of the water  were  of  the  flow  suspended  to  facilitate  p r o c e d u r e and r e s u l t s  not  formed.  provided  section'  required  on  a  the  available We  sharply  which  flow  at  defined  'cross-  filings  were  visualization.  of the photography  the  assumed t h a t t h e  aluminum  flow  for  A  appears  The  i n the next  section. 3.5 The S e a r c h F o r C o h e r e n t The the  preparation  tank.  filter fill  Structures  f o r an e x p e r i m e n t s t a r t e d  A normal c o l d  was u s e d t o g e t h e r  water  faucet with  the  water  temperature  Without the hot water  5um  filling particle  w i t h an u n f i l t e r e d h o t w a t e r t a p t o  t h e t a n k t o a 63+1cm d e p t h .  bring  a  with  i t would  The h o t w a t e r was a d d e d t o to  room t e m p e r a t u r e , 68" F.  take the i n i t i a l l y  about  62*F  w a t e r more t h a n 3 d a y s t o e q u i l i b r i a t e t o room t e m p e r a t u r e . The 62*F w a t e r h a s a b o u t a 10% g r e a t e r T h i s would given  result  model  i n a 10%  size  and  lower  speed.  kinematic  Reynold's This  is  viscosity.  number an  for  1  a  unacceptable  di f ference. After  t h e t a n k was f i l l e d  filter  system  was  used  water.  B e f o r e a r u n was  a recirculating  to clear made  the  5ym  particle  up t h e i n e v i t a b l y water  opaque  temperature  was  35  measured.  For  both  the  photographic  m e a s u r e m e n t s t h e w a t e r t e m p e r a t u r e was 68*  F.  lie  This  within  and  hot-film  w i t h i n one d e g r e e  of  corresponds t o having the k i n e m a t i c v i s c o s i t y 0.016x10'* o f  1.01x10"* m / s e c ,  an  l  ignorable  variation. The the  first  c a m e r a and  was  quickly  grid  t u r b u l e n c e p h o t o g r a p h s were t a k e n w i t h  illumination  realized  that the coherent s t r u c t u r e s  by t h e g r i d were l i k e l y frame  and  frame. for  would  stationary  thus  be  Lab chosen  an  These r e s u l t e d  i n the-most  i n the flow.  coherence  in  Figures 8  different  exposure  show  of  test  fluid  and  range of  1  1/60  l a b frame  o b s e r v a b l e and  of  the  the  traces  filing  of  measureable  the  images  for  a 35cm/sec g r i d  s p e e d w h i c h showed  the  most  grid  were  t o 1 second.  for  cm/sec g r i d  translating  second  times  cm/sec  reference  s t r u c t u r e s were o b s e r v e d  m e a s u r e d c o h e r e n t s t r u c t u r e s was  Eng.  1/2  easily  The  curvature  filings.  30  generated  i n b o t h the g r i d and  times  intitial  structures  and  the  It  speeds.  frame e x p o s u r e from  in  unobservable i n the  F i g u r e s 7 show p h o t o s  30cm/sec g r i d  shutter  s y s t e m mounted on t h e c a r t .  speeds  easily  f o u n d t o be  and  l/2sec  as  aluminum several  speed.  The  observed  and  1 sec f o r the  20  f o r t h e 40 and  50  speeds.  F l u i d Mech., R o b e r s o n / C r o w e ,  a  HOUGHTON M I F F L I N  CO.  36  a) Grid Frame 1/15 sec exposure Ug=30cm/sec  b) F l u i d Frame 1 sec exposure Ug=30cm/sec Figure  7 - Flow V i s u a l i z a t i o n  i n g r i d and F l u i d  Frames  38  The  procedure f o r o b t a i n i n g the f l u i d  i s now e x p l a i n e d . water  surface.  The c a m e r a was This  span d i r e c t i o n and Pieces  gave  80cm  mounted  a field  in  the  1.4m  of view.  The c a r t  was . p h o t o g r a p h e d .  positioned  The s h u t t e r  s o t h a t t h e g r i d was j u s t  f o r the  direction.  w e r e p l a c e d 50.0 of  the  f i r s t photograph.  creating  flow  release triggers  were  leaving  the  field  shown  in  f i g u r e . 9.  analysis this x=0.  t o x=l90. Before  A s e t of  three  such  to a field  of  more  that  of the are  statistical  observation  from  cm b e h i n d t h e g r i d .  making  any measurements t h e t i m e r and f r e q u e n c y  c o u n t e r were a l l o w e d t o warm up f o r a b o u t h a l f a s t h e RC o s c i l l a t o r  tended t o d r i f t  an  hour  or  when f i r s t t u r n e d  The c l o c k was c a l i b r a t e d a g a i n s t t h e c o u n t e r t o h a v e an  oscillating checked up t o  frequency of  periodically  f =100HZ.  This  f=0.4Hz o c c u r r e d o v e r a b o u t h a l f  resulted used.  in  calibration  was  when m e a s u r e m e n t s were made a s d r i f t s  When c h a n g i n g c a r t  was  so  photos  From t h e v i e w p o i n t o f o u r  i s equivalent  of  The o t h e r two p h o t o g r a p h s  t h r e e s h o t s p r o v i d e d an o v e r l a p p i n g t i m e h i s t o r y  f l o w beneath t h e camera.  on.  camera's  the  were t a k e n a t t i m e s T =50cm/U a n d T =1l0cm/U l a t e r the  the  s p e e d was t h u s m e a s u r e d d u r i n g t h e  same t i m e i n t e r v a l a s when t h e g r i d was which  above  longitudinal  of white tape f o r t h e t i m e r t r i g g e r  field  photographs  o f v i e w o f 60cm i n t h e  cm a p a r t w i t h t h e f i r s t t a b n e a r t h e edge  view  frame  speeds  the  step-cone  t h e h i g h e s t motor speed  This resulted  an h o u r . pulley  f o r t h e new c a r t  i n a smooth a n d  reproducible  which speed cart  39  motion.  S e v e r a l r u n s were made t o a d j u s t t h e s p e e d  resistor speed  t h e d e s i r e d c a r t v e l o c i t y was r e a c h e d .  s e t t i n g p r o c e d u r e had t h e added purpose  water  so t h a t  before or  until  any  thermal  replenished  as  frequent  needed.  had  been  twenty minutes At  the  were  of mixing the equilibriated  Aluminum f i l i n g s and  replenishment  as  the surface a g i t a t i o n  cm/sec  s i n k t o t h e tank bottom. to  speeds  runs  Once  i t ss t a r t i n g position  were a l l o w e d t o l e t t h e w a t e r cart  50  were a d d e d  40  returned  highest  This  The  made many o f t h e f i l i n g s cart  variations  t h e measurements began.  required  control  settle  the  five to down.  the strongly excited surface  waves t o o k t h e l o n g e s t t i m e t o d i s s i p a t e . Sets of photographs 50 cm/sec g r i d are  speeds.  were o b t a i n e d f o r 2 0 , 3 0 , P r i n t s of r e p r e s e n t a t i v e  shown i n f i g u r e s 9 t h r o u g h 1 2 , ,  40, and  photographs  40  Figure  9 - 20cm/sec Flow Photographs  41  Figure  10 - 30cm/sec F l o w  Photographs  42  43  To  0' X  (cm)  AO  1  '60 «  I »  t  1/2 SEC EXPOSURE 50cm  Figure  12 - 50cm/sec F l o w  jynotographs  44  3.6 H o t - f i l m A p p a r a t u s A  Thermal Systems I n c .  anemometer) s y s t e m  1  Packard  Spectrum  from  HP3582A  the  1050 CTA  (constant  was u s e d i n c o n j u n c t i o n  flow f i e l d .  Analyzer  with a  hot-film  probe type  between  is first  A flow c h a r t of t h e d i a g n o s t i c  used.  0  and  10  Once s e t up i t g e n e r a t e s a  volts  which  is  i s a thin  mounted a t t h e t i p o f t h e p r o b e . operating  of c o n v e c t i v e  temperature T .  tip.  the  A brief  follows.  s t r i p of c o n d u c t i n g  metal  I t s r e s i s t a n c e depends T  This  voltage  r e s i s t a n c e balanced  is  on  i s d e t e r m i n e d by t h e r a t e  The c o n t r o l v o l t a g e , E c , a t t h e  controlled  to  top  keep  by of arm the  w i t h t h e c o n t r o l r e s i s t a n c e , Rc.  c o n t r o l r e s i s t a n c e i s s e t so t h a t the sensor  a resistance slightly  1  to  causes a c u r r e n t t o pass through the sensor  of t h e b r i d g e .  The  in  signal  c o o l i n g by t h e f l o w a n d t h e r a t e o f h e a t i n g  a control current. bridge  proportional  flow speed a t t h e h o t - f i l m sensor  The h o t - f i l m s e n s o r  sensor  i s shown  I n a d d i t i o n i t must be c a l i b r a t e d  d e s c r i p t i o n o f t h e anemometer o p e r a t i o n  the  setup  anemometer must be l i n e a r i z e d when a new  each s e t of runs.  longitudinal  the  spectra  13.  The  before  Hewlett-  t o o b t a i n power  u s e d t o o b t a i n t h e h o t - f i l m b a s e d power s p e c t r a figure  temperature  higher  than i t s r e s i s t a n c e  The l o a n of t h i s equipment from Dr.Quick E n g i n e e r i n g Dept. i s g r a t e f u l l y acknowledged.  i s kept a t with  of  the  zero  UBC  45  control  voltage.  T h i s h o l d s the sensor element  above t h a t of t h e s u r r o u n d i n g f l u i d increases thermal  monotonically energy  with  loss rate  flow rate past the sensor temperature.  The  temperature.  Ec,  The  sensor's  water  required  density  and  to maintain the  i s thus a  unique  function  the flow v e l o c i t y a t the sensor t i p . The  signal one  resistance  f u n c t i o n of the f l u i d  f o r constant  sensor a t a constant temperature of  as t h e sensor  i s a unique  voltage,  temperature  linearizer  unit  is  used  to convert t h i s voltage  from t h i s n o n - l i n e a r f u n c t i o n of t h e f l o w v e l o c i t y t o  which  signal  i s . The l i n e a r i z e d  conditioner  which  components lower than  2  signal  was Hz  i s then  used  and  fed  t o remove  greater  than  into  the  fluctuating 1  kHz  in  frequency. When  the  largest  velocity  than the c o n v e c t i o n v e l o c i t y output s i g n a l the of  probe  contributions flow.  less  fluctuating  the  part  of  i s s i m p l y due t o t h e l o n g i t u d i n a l component o f  fluctuating the  the  f l u c t u a t i o n s a r e much  velocity, is  to  designed cooling  see Appendix to from  A.  further lateral  The wedge shape suppress  the  components of t h e  46  sensor UCCt)  (M7ZZH3=  hot-film probe  (wedge shaped)  li nearizer  signal conditioner e1  ^\J \^A/ A  V W  7* oscilloscope & camera  spectrum analyzer  •VE X-Yplotter Figure  13  - Hot-Film  Power  Spectrum  w  Schematic  47  The l i n e a r i z e d a n d f i l t e r e d Tektronics  454A  analyzer.  oscilloscope  A T e k t r o n i c s C-31  signal and  i s d i s p l a y e d on  fed  polaroid  into  the spectrum  camera  was  used  o b t a i n some t y p i c a l p h o t o s o f t h e f l u c t u a t i n g v o l t a g e The  spectrum  analyzer  c o u l d be u s e d t o s t a r t duration  Ts=2.5sec  had an e x t e r n a l t r i g g e r the sampling  for  the  f e a t u r e was u s e d t o e n s u r e d a t a  was t a k e n  f i l m p r o b e was i n t h e t u r b u l e n t  flow  trigger  the  was  used  to  trigger  loading.  I f the analyzer  sequence  a  detector  new  time  record  first  two p i e c e s o f t a p e were p l a c e d  signals  from  analyzer. before  the the  The  the  strategic  timer optical  analyzer  cart  was  positions  and  the  returned  a  sufficient  number  of  span.  This  The  data  Tape  was  the tank.  The  analyzer.  had  x-y  using  facilities  at  digitized the  UBC  were o n l y u s e d by t h e to  be  disconnected  t o the s t a r t i n g p o s i t i o n or the  previous  spectra, typically  for  obtained.  subsequent  computing c e n t e r .  s e c t i o n g i v e s a d e t a i l e d a c c o u n t o f how was  These  Subsequent  p l o t t e r which i n t e r f a c e d to the spectrum  T h i s h a r d c o p y was t h e n  data  loading  tape.  a v e r a g e d t h e r e s u l t a n t power s p e c t r u m was p l o t t e d HP  timer's  X=50.0cm a p a r t .  m e a n i n g l e s s s p e c t r a w o u l d be a d d e d t o When  time  when t h e o p t i c a l  along  detector  trigger  of  spectrum a n a l y z e r  of white  placed  t r i g g e r e d both  field.  would s t a r t  thus  which  o n l y when t h e h o t -  was r e a d y f o r a new  next encountered a p i e c e at  frequency  to  trace.  input  time i n t e r v a l ,  100Hz  the  the  data. 2 0 , were  using  an  analyzer. analysis The  hot-film  next data  48  3.7 H o t - f i l m Power S p e c t r a In was  preparation  f o r t h e h o t - f i l m experiments t h e tank  f i l l e d and f i l t e r e d  as f o r the photography  e v e n more i m p o r t a n t t h a t t h e w a t e r constant was  temperature  Before  t o both  probe.  followed.  described here. 30'  coaxial  element  element,  Only  taken t h e  model  "MODEL  dirt.  hot-film 1050/1050A  manual  were  s e t u p p r o c e d u r e w i l l be  The p r o b e arm o f t h e b r i d g e c o n s i s t e d o f a with  and Rp,  resistance  probe  t h e probe at  mount,  Rs,  together  itself,  the t i p of see f i g u r e  having  resistance  less  w h i c h was t h e s e n s o r 13.  I t was  first  sensor r e s i s t a n c e , Rs, and  by t h e o v e r h e a t r a t i o o f 1.06 t o o b t a i n t h e s e n s o r  o p e r a t i n g r e s i s t a n c e , R o p . The resistance,  probe  support  R c a b , was m e a s u r e d by n u l l i n g  s h o r t i n g wire i n p l a c e of balancing  the  probe  minus  sensor's  t h e probe.  arm  r e s i s t a n c e , R c , w h i c h formed probe  a  r a t e was  and  be  instruction  the basic  n e c e s s a r y t o measure t h e unheated multiply  could  i n t h e TSI  Anemometer"  cable  r e s i s t a n c e Rcab, sensor  measurements  instructions  Constant Temperature closely  The s e n s o r c o o l i n g  h a d t o be s e t up f o r t h e 1232W  The  and have  temperature - v a r i a t i o n s  a n y anemometer  linearizer  clean  I t was  f o r t h e h o t - f i l m measurements than i t  f o r the flow v i s u a l i z a t i o n .  very s e n s i t i v e  be  work.  against  plus  cable  thebridge with a This  a  was  done  variable  the opposite bridge  by  decade  arm.  The  r e s i s t a n c e , Rp, i s t h e n a d d e d t o t h e  decade r e s i s t a n c e and t h e decade r e s i s t a n c e b a l a n c e d a g a i n s t the  ZERO  OHMS  resistor.  The  ZERO  OHMS  resistor  thus  49  adjusted  t o t h e p r o b e arm m i n u s s e n s o r  r e s i s t a n c e , Rcab+Rp,  i s c o n n e c t e d i n s e r i e s w i t h t h e decade r e s i s t o r . r e s i s t a n c e would t h e r e f o r e read  only the sensor  Rs  ,when  The  p r o b e c o u l d now be i n s e r t e d i n t o t h e p r o b e  immersed  t h e b r i d g e was b a l a n c e d  in  t h e 68 F q u i e s c e n t  r e s i s t a n c e , R s , was r e a d  off  resistance,  resistance, switched  Rc.  to  RUN  the  the  the  sensor  sensor  overheat  ratio  The u n h e a t e d  nulled  bridge  dialed  bridge  into  control  would  be  and sensor  and  3.92  ohms  the  the The  decade  circuit  maintained  For the data presented  r e s i s t a n c e Rs o f  support  Rop=Rsxoverheat r a t i o .  was  When  e l e v a t e d temperature. the  Rop,  resistance,  with the probe i n p l a c e .  water.  operating r e s i s t a n c e determined, operating  The d e c a d e  was at the  in this thesis  multiplied  by  the  o f 1.06 d e t e r m i n e d t h e o p e r a t i n g r e s i s t a n c e  Rop t o be 4.15 ohms. The  1050  different  anemometer  control  power r e q u i r e m e n t environment.  bridges.  d e t e r m i n e d by  probe  avoid the voltage number  as  choice  of  three  u s e d d e p e n d s on t h e  probe  type  and  flow  i s g e n e r a l l y used f o r low as  probes  in  a i r or  The number 2 b r i d g e was u s e d f o r t h e  i t s higher clipping  output  which  c u r r e n t was n e e d e d t o  was  observed  when  the  1 b r i d g e was u s e d .  After  the  bridge  was  bridge voltage, Ec, versus 14a.  the  a p p l i c a t i o n s such  small probes i n water.  a  The b r i d g e  The number 1 b r i d g e  power r e q u i r e m e n t  hot-film  provided  This  non-linear  set  up a c a l i b r a t i o n c u r v e o f  f l o w s p e e d was  curve  was  needed  measured, to  figure  adjust  the  50  linearizer bridge  settings.  output  It  was  voltage using a d i g i t a l  cart  moved t h e p r o b e t h r o u g h  For  these  measurements  fluctuation about  levels  7.5cm  parallel  obtained  the  the  t o the d i r e c t i o n  grid  measuring  a  known  speed.  was removed s o t h a t t h e  l o w . The p r o b e was water  the  v o l t meter w h i l e t h e  the water a t  were v e r y  beneath  by  surface  of t h e c a r t ' s  positioned  and  was a l i g n e d  motion.  The  cart  s p e e d was m e a s u r e d u s i n g t h e t i m e r . The process on  bridge of s e t t i n g  the  model  corresponding were  signal  was l i n e a r i z e d  9 interdependent  1055  with  bridge  so  that  the  voltages interest  A reasonable the  l i n e a r i z a t i o n could  resistor  turned  from t h i s p o s i t i o n  the  experimental  checks runs.  completely  curve were  average value  in  made  only  be  achieved  clockwise.  A slight  figure during  The c a l i b r a t i o n  curve  14. and  constant  curve.  Minor  change appears  Subsequent after  the  i s determined variations  i n t h e 40cm/sec r e g i o n were o b s e r v e d of t h e t h r e e c a l i b r a t i o n s  The  f o r i t s4th slope  The l i n e a r i z e d  as t h e s l o p e o f t h e l i n e a r i z e d slope  would  produced a d i s c o n t i n u o u s  response.  calibration  linearization  linearizer  v o l t a g e s o f 0.00 a n d 10.00 v o l t s .  l i n e a r i z e r had a broken r e s i s t o r  in the l i n e a r i z e r ' s  this  resistors  t o z e r o a n d t h e maximum f l o w s p e e d o f  corresponding  variation  over  The  tedious  s u p p l i e d t o t h e l i n e a r i z e r a n d t h e l i n e a r i z e r ZERO a n d  available point.  slope changing  linearizer.  SPAN c o n t r o l s were a d j u s t e d output  by a r a t h e r  was u s e d . .  of  and so t h e  51  o linearized • calibration o o  • o  20  Figure  40 60 V E L O C I T Y (cm/sec) 14 - H o t - F i l m  Linearization  52  The  linearized  signal  was  f e d through  the  signal  c o n d i t i o n e r w i t h a p a s s band o f f r o m 2hz t o 1Khz a n d t h e n t o the  monitor scope and spectrum a n a l y z e r .  Some o s c i l l o g r a m s  o f t h e l i n e a r i z e d f l u c t u a t i n g anemometer  signal  appear  in  f i g u r e 15.  Ug=40cm/sec X POSN=30cm calibration constant = 8. (cm/secVvolt  8 (cm/sec)/div  1 v/div  r  0  J  Figure  The  f  .2  Oscillograms  HP3582A S p e c t r u m A n a l y z e r  following  control  was u s e d t o o b t a i n power  anemometer  settings.  "chassis isolated" position.  A sec  .3  15 - H o t - F i l m  s p e c t r a from t h e f l u c t u a t i n g  as  — r — T  f  The  signal  using  i n p u t s w i t c h was i n t h e  The DC c o u p l i n g mode was  t h e s i g n a l c o n d i t i o n e r had a l r e a d y  filtered  t h e s i g n a l was f i l t e r e d .  sensitivity position  selector  possible  indicator  light  required  a lower  c h o s e n t o be f r o m showed  no  up.  still  The  to  higher  significant  100Hz.  t h e most  not having  sensitive  the data  turbulence  input s e n s i t i v i t y . 0  end  The i n p u t s e c t i o n h a d an i n p u t  w h i c h was s e t a t while  used  the signal.  T h i s c o u p l i n g e n s u r e d t h a t no more o f t h e l o w f r e q u e n c y of  the  intensities  The f r e q u e n c y  Preliminary  energy content  overload  s p a n was  measurements  a t frequency  greater  53  than  100Hz  for  experiments. shapes  The  depending  resolution used  the  as  was a  successive  power  signal  was  up  the  loading  The  used  in  so  to  the  the  and  or  cart  speed  trigger  two  frequency  data  average  loading  trigger  l o a d d a t a on t h e e x t e r n a l  trigger  timer.  Enough t i m e The  was  given  separation,  for  s p e c t r a a v e r a g i n g was  o u t p u t was The  the  water  r e s e t and  the  connected to the analyzer's data d r i v e motor  was  s t a r t e d and  the  t h e g r i d and h o t - f i l m p r o b e t h r o u g h t h e w a t e r .  to four analyzer  depended  RMS  would  t i m e r box c l o c k e d t h e 50.0cm d i s t a n c e and a l s o  f r o m one  was  extremes.  analyzer  the  pass  were s e t up t h e c a r t was moved t o i t s  trigger.input.  towed  these  band p a s s shape  the a n a l y z e r , h o t - f i l m probe t o g r i d  motions to subside. timer's  Hanning  between  spectra  set  range  amplitude  The  selected  starting position.  cart  whether  s u p p l i e d by t h e  After and  on  compromise was  speed  a n a l y z e r had a c h o i c e o f t h r e e band  desired.  averaging  section  grid  on t h e c a r t  periods.  This  number  speed as t h e d a t a l o a d i n g took a  amount o f t i m e , T s = 2 . 5 s e c .  After  the c a r t  at  cart  returned  cart  speed, from f i v e t o about twenty minutes e l a p s e d b e f o r e  the  waves  The  position.  Depending  on  i n t h e t a n k d i e d down and a n o t h e r r u n was  found  mechanical  to i t s s t a r t i n g  d i s c o n e c t e d and  the  of  was  t a n k t h e t i m e r t r i g g e r was  stopped  fixed  end  It  the  sampling  triggered  that  the  hot-film  probe  mounting  r e s o n a n c e w h i c h showed up i n t h e power  mounting  system  was  reinforced  and  the  the the  made. had  a  spectra. resonance  54  diminished  i n a m p l i t u d e and s l i g h t l y  increased  i n frequency.  To d i s t i n g u i s h b e t w e e n t u r b u l e n t s i g n a l s a n d t h i s resonance  a  quiescent. of  the  power  spectra  were  obtained in figure  obtained  u s i n g an HP X-Y p l o t t e r .  digitized analysis.  and  stored  in  in  These  computer  figure  17.  ignored.  The  w i t h the water 16.  from  Hard the  plots  files  When t h e s p e c t r a were d i g i t i z e d  r e s o n a n c e c o n t r i b u t i o n was shown  was  This spectrum appears  power  analyzer  spectrum  mechanical  spectrum were  for  X-Y  then  susequent  the probe  A typical  copy  support plot  is  d i g i t i z e d c u r v e h a s been drawn  through the spectrum. Power s p e c t r a were o b t a i n e d probe-grid  separations.  grid  changed  was  brackets. the  The d i s t a n c e o f t h e p r o b e sliding  The e x p e r i m e n t a l  hot-film  predicted  by  power  for various grid  the  grid  in  would  from the f l o w v i s u a l i z a t i o n  and  from  the  i t s mounting  c o n d i t i o n s were c h o s e n  spectra  speeds  correspond photographs.  so to  that those  55  80 m'v-i PRO BE S U P P O R T RESONANCE 1 5avgs  PM)  04  0  W Figure  100Hz  16 - P r o b e S u p p o r t  Resonance  40 mvi R M S Power Spectrum XPOSN=90cm VELOCITY=A0cm/sec 1 0 avgs  P(W)  0 100Hz Figure  17 - T y p i c a l  A n a l y z e r X-Y  Plot  56  IV. 4.1  EDDY-SIZE DISTRIBUTIONS AND  Introduction A major aim  spectrum  with  viability  statistical  description  s e e n as a t e s t  18  the  of  the  author  was  to  i s a flow two  chart  of  descriptions.  the  To  structures.  The  density.  Assumptions about the d i s t r i b u t i o n s  generate  and  their  a  time  fluctuation. a  manner  analyzer power  The  spectra  record  of  identical hot-film  from  how  the  power  anemometer s i g n a l . to  o b t a i n the  kinetic and  described  The  integral  energy.  the  The  i n the  to  eddy  size  eddies  needed  u s e d by The  the  prediction  Section  of the 4.4  from  the  power s p e c t r a were c o m p a r e d and  used  length scale r e s u l t s of  last  and  obtained  in  spectrum  d i s t r i b u t i o n s and  were  to  velocity  time record  i n s e c t i o n 4.3.  spectra  spectral  longitudinal  that  the  i s then  of t h e  were  the  to  eddy-spectra  power  velocity  experiment.  assumptions used are d e s c r i b e d describes  the  computer code a n a l y z e s  nearly i n the  convection  flows.  used  distribution  predict  the  i n t e r m s of  computer  space  to  how  analysis  o b t a i n the  eddy-size  and  turbulence.  learn  used i n a  in  code  turbulent  turbulent  f l o w v i s u a l i z a t i o n p h o t o s were a n a l y z e d  coherent  of  c o h e r e n t s t r u c t u r e s t o model  aim  eddy-size  f o r the v a l i d i t y  d e s c r i p t i o n i s used t o d e s c r i b e  Figure  the  t h e s i s i s t o compare t h e  statistical  T h i s was  of u s i n g  personal  compare  of t h i s the  fluctuations.  A  HOT-FILM SPECTRA  total  fluctuating  t h i s a n a l y s i s are  s e c t i o n of t h i s  chapter.  presented  57  VISUALIZATION  ANEMOMETER  'N(R),JUR)J XPOSN Ug  /bataf i le :DDY40.30  plot eddyspectrum  N  JL 1  y  calculate le.PEDDY FRAC 7URB.FTN NKR)-t€,iVv)  Tdatafile 'POW4030  F i g u r e 18 - Power S p e c t r a  P(w)-*E,[ ) W  fdataf ile SQR40.30 /  Comparison  58  4.2 E d d y - S i z e  Spectra  Eddy-size the  s p e c t r a were o b t a i n e d  f l o w p a t t e r n s as f o l l o w s .  negatives their  Lb,  size.  This  corresponds  of w h i t e  19a.  b i n to the average p o s i t i o n  referenced  The  bins  could  by t h e a v e r a g e t i m e  time,  the g r i d  patterns  on  the  paper t o h a l f 1OcmFS  The d i s t a n c e  t o t h e d i s t a n c e between t h e  exposure time.  This  flow  p a p e r was d i v i d e d i n t o  s c a l e ) wide b i n s , see f i g u r e  sampling  by  The  were p r o j e c t e d o n t o a s h e e t  original  (full  from t h e photographs of  middle  label, of  the  of the g r i d d u r i n g the  equally  well  have  been  s i n c e t h e g r i d had p a s s e d by.  Tb, w o u l d s i m p l y be t h e b i n d i s t a n c e L b d i v i d e d s p e e d Ug.  The p e r s o n a n a l y z i n g t h e f l o w p a t t e r n s drew t h e o u t l i n e o f what he c o n s i d e r e d paper.  This  process  t o be t h e c o h e r e n t i s of c o u r s e  40cm/sec p h o t o s were a n a l y s e d indoctrination like.  Generally  structure 19a very  should  we  the  highly subjective.  The  person  assumed  that  who  had  little  s t r u c t u r e should  the  periphery  of  be t h e l a r g e s t c l o s e d s t r e a m l i n e s .  crude although,  flow  on  t o what a c o h e r e n t  a l s o shows t h e s t r u c t u r e s  y i e l d s useable the  as  by a  structures  circled.  This  f i e l d according  the  Figure  analysis  a s i t t u r n s o u t , i t was b o t h q u i c k  eddy s p e c t r a .  look  is and  An a u t o m a t e d d e c o m p o s i t i o n  of  to rigorously defined c r i t e r i a  was  beyond t h e scope of t h i s t h e s i s .  We d e c i d e d  the consequences of our simple a n a l y s i s .  to  learn  from  Figure  19 - Eddy A n a l y s i s  60  Having  outlined  the  contours  of the eddies a c i r c l e  t e m p l a t e was u s e d  t o determine  structures  t r e a t e d as c i r c u l a r  area.  Eddy r a d i i  For each after  were  their  radii.  Non-circular  e d d i e s h a v i n g t h e same  were e s t i m a t e d w i t h an u n c e r t a i n t y o f  b i n t h e eddy s i z e d i s t r i b u t i o n , N ( R ) , was  the r a d i i  h a d been d e t e r m i n e d ,  the  eddy d i s t r i b u t i o n s  one  flow  speed  distribution, The  19b.  recorded The sum o f  from a l l t e n s e t s of photographs f o r  provided  a  statistically  significant  f i g u r e 19c.  eddy's  internal  using a clear p l a s t i c centered  figure  1mm.  over  a n g u l a r v e l o c i t i e s were measured  angle  template.  The  template  was  t h e s t r u c t u r e a n d t h e a n g l e t h a t a t r a c e made  w i t h r e s p e c t t o t h e c e n t e r o f t h e s t r u c t u r e was c o n v e n i e n t l y read o f f the angle s c a l e . for  the  40cm/sec  measurement  The r e s u l t s o f t h e s e m e a s u r e m e n t s  data  proved  appear  much  in  figure  more d i f f i c u l t  20a.  and u n c e r t a i n than  e s t i m a t i n g an eddy s i z e a s t h e i m a g e s o f t e n d i d n o t cicularly  symmetric  trace densities  needed  characterization very  difficult.  assumption  about  choose a r i g i d velocity  to  distribution.  avoid  trace  It the  was  thus  necessary  and  have  overlapping  to  choice  convenience.  was  made  distribution make  v e l o c i t y p r o f i l e w i t h i n an e d d y .  This  a  A l s o t h e low  o f an e d d y ' s a n g u l a r v e l o c i t y  body r o t a t i o n , no i n t e r n a l  profile.  observation  velocity  This  stress,  some We  f o r the  made on t h e b a s i s o f  61  The  assumption that  m o t i o n was r e a s o n a b l y The  eddy-size  particular  cart  into  a  eddies  consistent with  spectra speed  a v e r a g e eddy a n g u l a r written  the  undergo  f o r t e n sets of photographs f o r a  velocity  These s p e c t r a and t h e  as a f u n c t i o n of d i s t a n c e  file.  The p o p u l a t i o n  r a d i u s was a v e r a g e d w i t h  the populations  size  size  greater  analysis bias eddy-size 30  and  one  i n using  spectra  body  observation.  were summed.  computer  rigid  smaller  the c i r c l e  of the  were  at a  given  eddies  one  t o remove a p o s s i b l e  template.  The  smoothed  summed o v e r a l l t e n s e t s o f p h o t o s f o r t h e  a n d 40cm/sec c a r t s p e e d a p p e a r s i n f i g u r e s 21 a n d 2 2 .  q u a l i t a t i v e d i s c u s s i o n of these spectra discussion  s e c t i o n a t t h e end of t h i s  i s presented chapter.  A  i n the  62  F i g u r e 20 - Eddy A n g u l a r  Velocities  63  E D D Y  S P E C T R U M  E D D Y  VELOCITY* 30 Ctl/SEC X POSN* 70 CM  8-  4 + 44  03  CD  444 +-H  44 „ „ 00  '  1  1  1  0.6  44 44-  +44  r-—i 1 1 1.2 16 2.4 RRDIUS(CM)  1  1  E D D Y  S P E C T R U M  VELOCITY* 30 CM/SEC X POSN* 80 CM  1  1  3.0  3.6  O.O  ~i  S P E C T R U M  1  1  0.6  1  E D D Y  VELOCITY* 30 CM/SEC X POSN= 90 CM  1  1  0.6  1  1  1  3.0  1  1  3.6  44 44  1  1  1.2 1.8 RRDIUS(CM)  E D D Y  1  S P E C T R U M  44 I I I I I I I  44 1  1  VELOCITY* 30 CM/SEC X POSN* 100 Ctl  + ++  !  1  1.2 1.8 2.4 RRD1USICM)  1  1  2.4  1  3.0  3.6  0.0  -i  S P E C T R U M  1  1  1  0.6  E D D Y  VELOCITY* 30 CM/SEC X POSN* 110 CM  1  1  1  1  1  1.2 18 2.4 RFOJ.U5ICM)  1  1  3.0  1  3.6  S P E C T R U M  VELOCITY* 30 CM/SEC X POSN* 120 CM  4 4 4  I I I I I I I I I I I I I  44 0.0  ~1  I  0.6  I  1  1  1  1  1  1.2 1.8 2.4 RRDIUS(CM!  1  1  3.0  1  444 44444  1  3.6  0.0  ~i  i 0.6  F i g u r e 21 - 30cm/sec E d d y - S i z e  i  444 4 I I I I I I I i i i i i .2 18 2.4 RRDIUS(CM)  Spectra  i  i 3.0  i  i 3.6  64  E D D Y  S P E C T R U M  E D D Y  VEL0CITY=30 CM/SEC X POSN= 130 CM  4  S P E C T R U M  VELOCITY=30 CM/SEC X POSN= 140 CM  4 + +  .-H44-.  44444 4444 „„ ~ T »•»  1  0.6  i — i — i — i — i — i — i — i  i  1.2 I S 2.4 RHD1U5(CH)  E D D Y  3 0  0.0  ~i  4 +44  I I I 1I 1I 1I1I1I1 I1 I I I I I I I I I •H I I I I I I I I I I I I I I I I I I I I I i 1 1 1 1 1 1 1 1 1 1 0.6 1.2 18 2.4 3.0 3.6  RRDIUSICM)  3 6  S P E C T R U M  E D D Y  VELOCITY* 30 CM/SEC X POSN= 150 CM  S P E C T R U M  VEL0CITY=30 CM/SEC X POSN= 160 CM  +4 444  0.0  -i  1 1 0.6  E D D Y  44 444  44 4  •Hill  +4H  1 1 1 1++1 I1 I I 1 III1 II I1 I 1.2 ] .8 2.4 3.0 3.6 RADII'SICM1  0.0  S P E C T R U M  -i  1 1 0.6  E D D Y  VEL0CITY=30 CM/SEC X POSN= 170 CM  1 1 1 1 1 1.2 1.8 2.4 RADIUS(CM)  -1— 0.6  r  3.6  S P E C T R U M  44 44 4444 +444  I I III I  +4  44  VEL0CITY=30 CM/SEC X POSN= 180 CM  444 I IIIIIIIIII  4  IIIIIIIII II I I  444  4  II I I I I I I I I I I I I I I I I I I I -i 1 1 1 1 r — i — 1.2 18 2.4 3.0 RflDIUS(CM)  -I 3.6  0.0  —I— 0.6  F i g u r e 2 1 b - 30cm/sec E d d y - S i z e  4  II III I IIIIII I I "T" T T" 1.2 18 2.4 RRDIUSICM1  Spectra  -1 3.6  65  E D D Y  S P E C T R U M  E D D Y S P E C T R U M VEL0CITY=40 CM/SEC X POSN* 3 0 CM  Ctt/SEC  VEL0CITY*40  R-  x POSN= 20 cn  ++ +++ +++ ++ +++ +++++ +++++ ++++++ ++++++++ ++++++++ ++++++++  "1 00  I I 0.4  I 1 0.8  1  1  1  1.2 1.6 RntuusitHi  1  1 1 2.0  1 1 1 2 . 4 2.1  00  i i 0.4  i  1 1 0.8  1 1 1 1.2 |.8  r  RHDIUS(CM)  E D D Y S P E C T R U M VELOCITY*4O CM/SEC X POSN* 5 0 CM  E D D Y S P E C T R U M VELOCITY*40 CH/SEC X POSN= 40 CM  +++++++ ++++++++ +++++++++  0.0  — i 1.2  - 1 — I — I — I — I — I 1 — I — I — 1 — I — I — 0.4 0.3 1.2 1.6 2.0 2.4 RHDIUS1CM1  E D D Y S P E C T R U M VELOCITY=40 CM/SEC  s-  X POSN* 60  1  1 1.6  RflDIUSICM)  E D D Y S P E C T R U M VELOCITY* 40 CM/SEC X POSN* 70 CM  cn  •oi.  —'Z— — — 1  1  RH't1lUSlC MI ,  Figure  r  ~i r - i — r -  2.4  0.0  -1  1 — 0.4  RliUIUSIcWi  2.0  22 - 4 0 c m / s e c  Eddy-Size  Spectra  1  1  1  1  r  66  EDDY SPECTRUM  EDDY SPECTRUM  VEL0CITr=40 CM/SEC x POSN= 80 cn  VELOCITY=4O cn/src x POSN= 90 cn  -1— o.o  -1— ?.4  RrtDIUSICM)  EDDY SPECTRUM  1  1"  1  0"  1  1  0.8  RHUIUIilCMI  —I— ?.4  VELOCITT-40 CM/SEC x POSN= n o cn  ++ +  ++  + + 4  >•«  i — • — i — i — i — i — i — i — i — i  1.2  —I 1 1— 1.2 1.6  EDDY SPECTRUM  vELOcnr=<to cn/SEc x POSN= ioo cn  +  +  ++  ++  1.6  RI1UIUS1CHI  2.0  24  "1  + H- + +  +-(•+-(•+  I 1 1 I 1 1 1 1 1 1 1 1 1 0.4 n.a i.:> i.i; :.n ?.4 2.e  2 1  RlirjHJSICMl  EDDY SPECTRUM VEL0CITr=40 CH/SEC x POSN= 120 cn  CO  RRtllUSIEHl  Figure  22b- 40cm/sec E d d y - S i z e  Spectra  67  4.3  G e n e r a t i o n Of E (w) From The  program  eddy-size  N(R)  distribution  was  used  t o g e n e r a t e t h e power s p e c t r a l  by  a  computer  d e n s i t y c u r v e s E (w).  F i g u r e 23 i s a f l o w c h a r t o f t h e p r o g r a m and d e t a i l s o f  the  Fortran  C.  The  IV computer  fluctuating  moving  c o d e TURB.FTN a r e g i v e n  velocity  t h a t w o u l d be m e a s u r e d  with constant v e l o c i t y  and  then  fast  the  power  fourier  spectrum.  i n Appendix by  t h r o u g h t h e e d d i e s was  transform Assuming  probe modeled  (FFT) a n a l y z e d t o p r o d u c e an  eddy  rotates  internal  s t r e s s e s t h e s t r e a m w i s e component o f t h e  velocity  i s simply, u=U  a  without rotational  b/Rm  (4-1)  where b i s t h e i m p a c t p a r a m e t e r a t w h i c h t h e p r o b e meets t h e eddy,  U  radius,  is  the  peripheral  see Appendix  observed  in  the  B. lab  velocity,  The  eddies  reference  v e l o c i t y , Uc, u s e d i n t h e c o m p u t e r the  cart  and Rm  of  s p e e d , Ug, a s t h e p r o b e was  19  figure  frame. model  i s the  The  were  convection  i s t h u s t a k e n t o be  mounted t o  the  cart.  From o b s e r v a t i o n s o f t h e 40 cm/sec f l o w p h o t o g r a p h s one that  t h e eddy  compared  speed measured  w i t h the c a r t  convection  velocity,  predictions.  Thus,  and  structures  were  indication  of  using  such  lengths  ignored. significant  or The  as  this  constant  vertical  b e c a u s e we e x p e c t e d t h e f l o w  model's  different  non-cylindrical photos  did  not  eddy  coherent show  velocities. to  sees  i s negligible  Uc=Ug, s h o u l d n o t d i s t o r t t h e  Complications  orientations  surprising  i n the l a b frame  speed.  eddy  quickly  This  any was  become  68  three  d i m e n s i o n a l .  f l u c t u a t i o n  If  s t a t i s t i c s  the  eddies  should  not  were be  a c t u a l l y  s i g n i f i c a n t l y  TURB.FTN READ  N ( R )  P E D D Y . J U R )  I  NUMBER  I SET UP  PSUM(I)  0  ~ ^ INITIALIZE  FRAND  by  d*y  tl»e  F i g u r e  of  23  -  N(R)  to  E  PSUM  (w)  t w i s t e d  Computation  the  a f f e c t e d .  69  The  computer code g i v e s e q u a l w e i g h t i n g t o a l l p o s s i b l e  eddy impact p a r a m e t e r s  b as  homogenous  sample  randomly  within  flow  bin.  was The  assumed  to  chosen  w i t h t h e random number g e n e r a t o r i n i t i a l i z e d  to the  Thus i n t e r - e d d y c o r r e l a t i o n s were i g n o r e d .  As t h e e d d i e s o n l y t a k e up some f r a c t i o n o f t h e space  a  O.Ocm/sec  fluctuating  flow v e l o c i t y  s e l e c t e d along w i t h the p r o b a b i l i t y The  probability  eddy was  chosen  zero v e l o c i t y bin  area  so t h a t t h e f r a c t i o n  would  as e d d i e s .  sampling  required,  see Appendix above  incident  on  longitudinal convecting  in  probe  velocity  at  1)  constant  generate  was  of  This figure  chosen  probability  length,  convection  of n o n - z e r o  samples  eddies which  a  speed  being 2)  randomly  measures  velocity were  fluctuating  and  used  the 3)  i n the  velocity  time  a n a l y z e d u s i n g a UBC The  t i m e r e c o r d c a n be c o m p a r e d w i t h t h e 15.  to result  100Hz f r e q u e n c y s p a n .  DF0UR2.  total  D.  a  to  24.  oscillograms  to  f r a c t i o n of the  A t y p i c a l computer g e n e r a t e d time r e c o r d i s graphed  figure  samples  non-  fraction  of  it  an  with  eddy  assumptions  past  eddy.  r a t h e r than  Determining t h i s  and  randomly  an  samples  component o f t h e f l u c t u a t i n g  computer model record.  rate  sample  was  choosing  of  equal the observed  counted  velocity,  of  of s a m p l i n g a z e r o v e l o c i t y  r e q u i r e s knowledge of t h e a v e r a g e  The  be  eddies are  t i m e of day.  a  the  The  time  interval  i n a n o n - a l i a s e d spectrum  T h i s t i m e r e c o r d was  Computing Center l i b r a r y  Fourier  between  coefficients  were  then  Fourier  routine then  of 0  called  used  to  70  calculate  t h e E„(w) a c c o r d i n g t o e q n . ( 2 - 3 2 ) .  4.0n 2.0. fcrrVsac  u  -2.0-  T  0.5  1.5  t (sec)  -4JOJ  F i g u r e 24 - u,(t) G e n e r a t e d H a v i n g o b t a i n e d an E, (w)  spectrum  t  g e n e r a t e d and a n a l y z e d . the  previous  value.  f r o m t h e Eddy D i s t r i b u t i o n a  new  The new E„(w) This  time  record  i s then averaged  process  continues  is with  until  the  r e q u e s t e d number o f a v e r a g e s h a s been made.  For  the  eddy  spectra  averages  were  analyzed  sufficient  i t was  found  that  200  t o d e f i n e a m e a n i n g f u l power s p e c t r u m .  Appendix  C describes t h i s entire process i n d e t a i l . The  power  s p e c t r a l d e n s i t y c u r v e was t h e n p l o t t e d .  t y p i c a l eddy power s p e c t r u m eddy s p e c t r u m f r o m w h i c h 25a. 25b.  Figure  25c  The r e s c a l i n g  obtained could  As  be  figure  25b.  The  i t was g e n e r a t e d i s shown i n f i g u r e  i s discussed power  later. spectra  spectrum  F i g u r e 26 shows t h e as  a  function  eddy a n g u l a r v e l o c i t y m e a s u r e m e n t s were  f o r t h e 40cm/sec p h o t o g r a p h s ,  only  in  i s a r e s c a l e d p l o t o f t h e power  40cm/sec eddy g e n e r a t e d distance.  i s shown  A  generated  for  eddy  t h a t speed.  power  of only  spectra  These d a t a a r e  71  discussed  i n the l a s t  s e c t i o n of t h i s  chapter.  4.4 Power S p e c t r u m O b t a i n e d From U ( t ) In t h e p r e v i o u s spectrum  s e c t i o n s we have d e s c r i b e d  c a n be e x t r a c t e d f r o m t h e eddy s i z e  how a  distribution.  T h e s e r e s u l t s a r e t o be c o m p a r e d w i t h power s p e c t r a using and  t h e h o t - f i l m CTA, spectrum  voltage  analyzer  analyzer.  signal  component.  (constant The  proportional  This  signal  to  produces the  power s p e c t r a o f  obtained  anemometer) a DC  streamwise  i s Fourier analyzed  t o p r o d u c e t h e RMS  fluctuations.  temperature  CTA  power  filtered velocity  by t h e s p e c t r u m these  voltage  The power s p e c t r a l d e n s i t y c u r v e i s o b t a i n e d  from t h i s  by  multiplying  constant,  squaring  analyzer's  band-width.  the  by result  the and  anemometer then  calibration  d i v i d i n g by t h e  72  EDDY SPECTRUM V E L 0 C I T Y = 4 0 CM/SEC X POSN= 3 D C M  ++  —|— T 0.6  a)  ++ I  1.2  I  I  1.8  I  I  3*  2.4  RODIU3ICM) EDDY  POUER SPECTRUM VELOCITY= 40 CM/SEC XPOSN= 30 CM  b)  0 0  EDDY  ~1  I  2n.o  REPRESENTATION)  VELOCITY= X  I  40.n  40CM/SEC  P O S N = 3 0 CM  c)  3.0  i  i  1  1  1  1  1 — i  - 2 . 0 - 1 . 0 0 . 0 1.0 2 . 0 3 . 0 4 . 0 L O G i W.i  1  FRLOUEHO  POWER SPECTRUM (RRER  I  5.0  F i g u r e 25 - Eddy Power  Spectra  -1 ri.o iHZI  1—'—I ep  n  " - i  !  inn n  73  EDDY  POWER SPECTRUM VRRIRTION WITH  POSITION  + 3  *  »  0.0  -  X=30cm 50 70 90  t  20.0  i  40.0  I  FREQUENCY  i — i — — i — n  60.0  80.0  r  100.0  (HZ)  F i g u r e 26 - Eddy Power S p e c t r a : V a r i a t i o n w i t h  Distance  74  Figure  27 i s a f l o w c h a r t o f how E (w) i s o b t a i n e d  u ( t ) by t h e s p e c t r u m a n a l y z e r . subtracted voltage  from  signal  the  The mean  anemometer  voltage  signal.  from  has  been  The f l u c t u a t i n g  i s then d i g i t i z e d a t a r a t e , F s ,  four  times  t h a t o f t h e maximum f r e q u e n c y  of i n t e r e s t .  F o r a 0 t o 100Hz  span  F s i s 400Hz.  I f u(t)  contains  than Fs these  w o u l d be  the  sampling  frequency  fluctuations with frequencies aliased times  t o lower higher  amplitudes  being  frequencies.  frequency to  fluctuation  be  signal  the  greater  However s a m p l i n g  allows  the  eliminated.  isu  total  The  time.  four  potentially  aliased  digitized  velocity  j=1,2,..,N-1 where  sampling  at this  In  N=T/Fs  with  practice  N  p r e d e t e r m i n e d due t o l i m i t e d a v a i l a b l e memory s p a c e a n d binary  operations  transform  i n t h e FFT p r o c e s s .  (DFT) o f t h e u  T is the  The d i s c r e t e F o u r i e r  i s d e f i n e d as  #4 A ( k ) = C u . exp[ ( 2 r r i / N ) ( - j k ) ]  (4-2)  0=o  The  A(k) a r e the,  the  sinusoidal  i n g e n e r a l , complex F o u r i e r a m p l i t u d e s f o r basis  functions  0, l / ( N A t ) , ... (N-1 )/(Nt»t) interval first The then  between s u c c e s s i v e  1/4 o f t h e s e  absolute squared  previous  where  value as  of t h e f i r s t in  frequencies  t i ( = 1/Fs)  samples.  amplitudes  with  i s the  In p r a c t i c e  only  the  a r e used as mentioned above. N/4 A ( k ) c o m p l e x v a l u e s  eqn.(2-32)  and  RMS  E„(w ) = A ( k ) A * ( k ) A f  are  averaged t o the  spectra.  K  time  (4-3)  75  Enough s u c h d e t e r m i n a t i o n s o f E„(w*) k = 0 , . . ( N - 1 ) / 4 a r e and  averaged  significantly  so  that  averaging  change t h e spectrum.  in  made  a new E„(w) d o e s n o t  76  h o t - f i Im anemometer signal  A/D Converter  F i g u r e 27 - S p e c t r u m A n a l y z e r  Operation  77  An  HP  X-Y  p l o t t e r was  u s e d t o o b t a i n a c o p y o f t h e RMS  spectra.  T h i s p l o t was  Center.  In the d i g i t i z i n g  and  the probe support  t h e n d i g i t i z e d a t t h e UBC process  resonance  values  were a l s o m u l t i p l i e d  divided  by  The  the  square r o o t of  written  into  squared to  a  the  the  file.  curve.  analyzer  o u t p u t f r o m w h i c h i t was  s t u d i e d by  distance. of  The  curve  was  values  measured  thus  were t h e n  power  spectral  a p p e a r s i n f i g u r e 28b. calculated  and  process.  density  These  RMS  is  shown  The in  28a.  Properties be  eliminated.  spectral  density  figure  were s m o o t h e d  i n the d i g i t i z a t i o n  hot-film  A typical  Computing  the c a l i b r a t i o n constant  power  computer  obtain  was  by  band p a s s w i d t h  the curves  power  of  e x a m i n i n g t h e c h a n g e i n t h e power s p e c t r a  Figure  distance  t h e d e c a y of t u r b u l e n t f l u c t u a t i o n s c a n  29  from  appears i n the  shows t h e E„(w) the  distribution  anemometer  one  fluctuating  s i g n a l was  should  designed  power  spectra  with that obtained remember  that  En(w)  from  obtained using  spectra  the  t h e a n a l y s i s of Indeed  averaging  spectrum a n a l y s e r ' s .  3582A s p e c t r u m a n a l y z e r the  from  voltage  the  the h o t - f i l m  operationally equivalent.  after  h o t - f i l m a n a l y s i s an HP generate  function  A d i s c u s s i o n of t h e s e  c o m p u t e r m o d e l ' s method of c o m p u t i n g and s p e c t r a was  a  next s e c t i o n .  When c o m p a r i n g t h e eddy-size  grid.  s p e c t r a as  with  was  fluctuations  the the  succesive For  the  used  to  while  the  78  0 mv-i RMS  Power Spectrum XPOSN=90cm VELOCITY=40cm/sec 1 Oavgs  P(w)  0 100Hz typical  xy p l o t  HOT-FILM  POWER S P E C T R I N VELOCITY=40 CM/SEC XP0SN= 90 c n  LU  0 0  1 20.0  1 — v "" r ' 'i * ' r " -| ' "i - - r 40.0  FREQUENCY  60.0  IHZ)  80.0  100 0  F i g u r e 28 - A n a l y z e r X-Y P l o t  79  o  HOT-riL  POU  D  VARIATION  WITH  'ECTRUM  VELOCITY=40  POSITION  CM/SEC  X=30 50 70 90  IE CJ—' LU  CO \ 21 " O oo •—-o"  IS LU  o "  Hurt n a  0.0  20 .0  40. n  1  FREQUENT  r  « «  60 .0  «  80.0  100 .0  1 H Z )  F i g u r e 29 - H o t - F i l m Power S p e c t r a : V a r i a t i o n Distance  with  80  c o m p u t e r m o d e l u s e d UBC achieve eddy-size  the  same end  spectrum.  Fast Fourier Transform  software  to  s t a r t i n g w i t h u ( t ) g e n e r a t e d from the  81  4.5 R e s u l t s And D i s c u s s i o n The  40cm/sec e d d y - s i z e  Several observations of  20cm  behind  not  as peaked  completely the  The  bar  converted  The  size  and  i s both lower and energy  has n o t  to rotational  kinetic  energy of  30cm  from  the  separation determine t h i s  we s e e t h a t  and  the  the  total size  grid  geometery  total  number o f e d d i e s  distribution  number o f e d d i e s  structures  dominated  small s t a t i s t i c a l  have  peak. 23  depicted  eddy  sizes  decreases.  Both  from  the  increase i n due t o  sample u s e d .  increase  seen t h a t a d j a c e n t  in d i f f e r e n t  of  As t h e  a t t h e 60cm p o s i t i o n i s l i k e l y  POSN=130  t e r m s o f eddy p a i r i n g . was  the  eddy-size  spectra  i s the  r a d i u s than the  See f i g u r e 22 f o r X POSN=100 a n d 110 cm a n d  for X  discontinuous  size.  The  a p p e a r a n c e o f a s e c o n d hump a t a l a r g e r s i z e  figure  grid  evolved  distribution.  A prominent f e a t u r e of t h e  initial  distance  stress  structures.  l a r g e r and s m a l l e r  the  30cm.  At a  i s q u i t e s h a r p l y p e a k e d a b o u t t h e 1.4cm r a d i u s .  flow evolves broadens  at  been  distribution  c a n be made f r o m t h e s e .  the g r i d the d i s t r i b u t i o n  as  coherent  s p e c t r a were shown i n f i g u r e 2 2 .  stages  through  190cm.  This  i n size  i s convincingly explained i n  Looking  back t o t h e f l o w p a t t e r n s i t  c o - r o t a t i n g eddies  o f an e v o l u t i o n  i n f i g u r e 30.  to  to  would o f t e n a  single  appear  eddy  as  82  1  2 Figure  This  can  present two  in  understood  the  viscously  eddies.  The  interaction  9  The  is  well  through  worth  12. persist  supports  our  use  of  of  eddies  flow for  the  the  The from the  compared w i t h the  eddy  eddy  eddy  soon  spectra  is  the  thus flow  flow  individually is  seen  the  around  the  composite  pbserved. patterns  that  show  of  many  figures coherent  This  hypothesis.  patterns  pressure  between  the  photographs.  Taylor's  technic  of  realizes  successive  further  Imaginative the  evolution  and o r g a n i z a t i o n a l l y . to  be a n  effective  Our tool  dynamics. chord  length  distributions, the  at  higher  redirected  part  size  the  boundary  be  become  of  both  average eddy  would  succesive  visualization studying  in  considering  qazing  One  structures  scrutiny  in  Pairing  interacting  to  jump  Eddy  by  flow  region  structure. It  30 -  be  U  3  integral and the  of  an eddy  see Appendix  length  scale  c a n be D.  This  derived  h o t - f i l m power  spectra  calculated length from in  is  both figure  83  31.  A d i s c u s s i o n of t h e s e  section.  The  of d i s t a n c e The  total  results  number o f e d d i e s  i s plotted in figure eddy  corresponding uncertainty  produced hot-film  in  uncertainty  appears  the  32  eddy  observed  f o r the  spectra  measured  i n the a n g u l a r  is  The  uncertainty  to  drift  i s a b o u t 15%.  insufficient the  effect  statistical  spectra averaging of  averaging  a q u i s i t i o n process. range,  The  An  can  due  uncertainty  increasing with decreasing  their  and  The  to  the  This i s  hot-film  curve's mechanical  u n c e r t a i n t y due  be e s t i m a t e d  succesive  data.  with  v e l o c i t y of the e d d i e s .  t o be a b o u t 25%.  resonance  40cm/sec  largely  calibration  this  i n f i g u r e 35.  s e e n f r o m f i g u r e 20 due  in  as a f u n c t i o n  appear  curves  curves  later  by  spectra  of  to  observing  during  10%  for  frequency  is a  the  the  25Hz  reasonable  estimate. The the  eddy c u r v e s  are  corresponding  s e e n t o be  hot-film  c o n s i s t e n t l y lower  spectra.  The  frequencies  c o n t a i n i n g a p p r e c i a b l e energy are  also  for  P o s s i b l e reasons f o r  the  eddy g e n e r a t e d  are  several.  1/5  of t h e  eddies The  I n s p e c t i o n of f i g u r e  t o t a l area  while  local  about as  of the  flow v e l o c i t y  19a  f l o w has  a l m o s t a l l of the i n the  consistently  shows t h a t o n l y  lower this about  been c h a r a c t e r i z e d  as  f l o w seems t o be a g i t a t e d .  r e g i o n s between the e d d i e s  l a r g e as t h e r o t a t i o n a l v e l o c i t y a t t h e edqe o f  eddies,  hence the  kinetic  energy  the  spectra.  than  total  inter-eddy f l u i d per  eddy a r e a  contains  about  u n i t mass a s t h e e d d i e s . to the b i n a r e a ,  figure  33,  as  is the  much  The  ratio  of  is  therfore  84  a  good measure f o r t h e  of  the  f r a c t i o n of the t o t a l  flow s t o r e d i n the  The  figure  changes. 15  as  accounted in the  This  well.  simple  eddy i n t e r a c t i o n  substantially  is  of  flow  occur  field  analysis.  the higher  i s the  b o t h t h e v i s u a l i z a t i o n and photographic coherent  print  structures  projection  system  a l s o c a u s e us  of  f o r a l l but  the  the  velocity  often  A Fourier  regions  frequencies  of t h e  than  scale resolution l i m i t  were  not  The  smaller  into  l a c k of energy i n  A  low  counted  of  large  f l o w showed some s m a l l e r  used.  been  changes  distances  eddy s i z e .  of  scale  .when  the  tracer density  could  scale eddies  f l o w p h o t o g r a p h s i t was  50cm/sec c a r t s p e e d t h e  however  from  concluded  that  spatial  resolution  adequate. In  generating  the  eddy  power  i n t e r - e d d y c o r r e l a t i o n s were i g n o r e d . flow  not  a n a l y s i s methods used.  which was  The  higher  size  the  to miss the  observation  was  of  have  t a k i n g these  Another e x p l a n a t i o n  frequencies  substantial  the o s c i l l o g r a m s  over  typical  a c c o u n t w o u l d show more e n e r g y i n our  in  analysis.  regions  the  contains  fluctuations  s h o r t e r than the  decomposition  was  seen  These  f o r i n our  energy  eddies.  f l o w between the e d d i e s  velocity  kinetic  patterns  separated  by  rotating  eddies  showed  "rivers"  the  presence  of  would c o n t r i b u t e to the Electrical  diagnostic  well  as  as  any  explicit  Observation clumps  of r e l a m i n a r i z e d f l o w .  the F o u r i e r a n a l y s i s . system  spectra  noise  low in  of of  the eddies  Adjacent frequencies the  coof  hot-film  unaccounted f o r mechanical  85  v i b r a t i o n s would c o n t r i b u t e t o constant signal  low frequency would  distance  also  or  hot-film  non-turbulent  noise  spectrum.  counting  regime.  Increasing  the c i r c u l a r  core  with  intereddy  o f a l a r g e non-  circular  structure could also explain t h i s .  There,  enough  information  attribute  discrepancy  Suffice  fluctuating  regime  as  to  kinetic  to  more t h a n  energy  in  the  eddies.  the to  f r a c t i o n of  proper  frequency  eddy d e s c r i p t i o n o f t h e  I t w o u l d be u s e f u l t o a n a l y z e  incomplete  i s not  enough  say t h a t a s i g n i f i c a n t  a n d t r e a t lumps o f f l u i d  field  what  i s p r e d i c t e d from our simple  fluctuations. again  decide  i n the spectra to although  speculate. the  to  A  i n t h e anemometer  explain the increasing discrepancy  i n t h e low frequency  correlations  the  the flow  w i t h a common  photos  rotation  axis  T h i s would account f o r a l l t h e flow  i n a s t r a i g h t f o r w a r d manner. . The  t o t a l mean  fluctuations  is  square  found  by  of  the  longitudinal  integrating  velocity  E ( w ) , a s shown i n  eqn.(2-23).  Figure  34 shows t h e d e c a y o f u f w i t h d i s t a n c e X f o r b o t h t h e  fluctuations  generated  fluctuations  measured  "eddy" u  values  facilitate  a comparison  spectrum  fluctuation  from  the  eddy  the  spectrum  and  h o t - f i l m anemometer.  the The  have been m u l t i p l i e d by a f a c t o r o f 10.0 t o  remarkable s i m i l a r i t y eddy  by  in  energy  i n t h e decay t r e n d s  description  dynamics.  the  is  decay  trend.  The  i n d i c a t e s that our  intimately  l i n k e d with the  86  The  integral  length scale  power s p e c t r a u s i n g eqn  can  be  obtained  from  the  (2-26).  Le=UcE„ ( 0 ) / ( 4 u ? ) This  length  scale  eddies.  Figure  against  distance  scales obtained  should  31 shows  be  the  related average  X together  The s i m i l a r i t y  s c a l e i s not r e m a r k a b l e e x c e p t t h a t length scale should  a v e r a g e eddy s i z e .  end  integral  i t tells  us t h a t  associated  with  between t h e h o t - f i l m  integral  description  the  length  has  This  scales tells  characterized  us the  s c a l e of the energy c o n t a i n i n g f l u c t u a t i o n s q u i t e w e l l  s p i t e of our crude a n a l y s i s .  film to  length  between  q u i t e good f o r t h e f i r s t t h r e e d i s t a n c e s .  t h a t t h e eddy s p e c t r u m  in  be  The s i m i l a r i t y  m e a s u r e d a n d eddy s p e c t r u m p r o d u c e d  size  plotted  a n d t h e eddy p r o d u c e d s p e c t r u m ' s  twice the i n t e g r a l  is  chord  f r o m t h e eddy p r o d u c e d a n d h o t - f i l m m e a s u r e d  a v e r a g e eddy c h o r d  the  eddy  with twice the i n t e g r a l  power s p e c t r a l d e n s i t y c u r v e s .  length  t o the s i z e of t h e  integral  The d i v e r g e n c e  l e n g t h s c a l e a t t h e 90cm d i s t a n c e  the anomalously high energy content of  the  constant  90cm  curve  background  of  noise  figure or  Although  study  contributions  in different  frequency  to  of  wE«, (w)  against  of  inter-eddy  i t i s also quite digitizing.  fluctuating  kinetic  r e g i m e s t h e power s p e c t r a were  r e s c a l e d t o expand the frequency plotting  the  frequency  Speculations  t h a t t h i s a n o m o l y i s due t o e r r o n e o u s  To energy  35.  hot-  i s related  i n t h e low  influence  c o r r e l a t i o n s h a v e been m e n t i o n e d . likeky  of t h e  axis.  log(w).  This  was  The new c u r v e  done has  by  equal  87  areas  representing  turbulent  kinetic  that  the  integral  two  logged  simplifies  The  compares  total Te  as  the  of  the  point  near  to  occurs.  This in  the  can  hot-film curve.  and  log(w^)  between  was  distributions  were  over  from the  velocities  is  the  to  the  used.  used  study  Figure  of  of  the  format.  integeral  energy  is  see  different new  that  the  eddy  to  the  highly  peaked  angular  velocity velocity  constant  measured the  than  simplified  intra-eddy  of  when  the  sharply  instead  the  In  more  be a  the  to  used  among  is  Had t h e  time  containing  best  energy  The  36  interpretation•of  i d e a l i z e d and the  grid.  the  b e e n m a r k e d on  our  clearly  due  (4-4)  new  the  has  s i m p l i f i e d to  been  w/wdw=dw  interest.  density  highly  distance  of  This  that  in  with  scale  more  were  angular  between  regimes.  representation  distributions  in  observing  maximum c o n t r i b u t i o n  agrees  are  spectral  given  total  wE,,(w)  facilitates  It  distribution  generated  analysis  scale  the  time  area  regimes  power  wd{log(w)}=  log(Te).  the  we  eddy  by  E„(w)  frequency  where  representation  the  under  of  now p l o t t e d  This  a measure of  frequency  seen  log(w,)  corresponding  w=l/Te a t  fluctuations.  as  different  axis  is  differences  points  frequency  same s p e c t r a  energy  area  the  frequencies,  where and  to  c a n be  the  integral  in  the  is  figures  representing  to  frequency  scale  This  d { l o g ( w ) } = ) E , » (w)dw  trends  The  energy.  axis  expansion  of  contributions  frequency  corresponding  wEn(w)  equal  for  a  distribution average  value  88  a  broader  distribution  s c a l e w o u l d be e x p e c t e d .  of energy about the i n t e g r a l  length  89  P  eddy Le  A  eddy  •  hot film 2T -U,  2T 'U e  e  A  4J  • Al  1  50 cm Figure  .  1  70  90  r  1 1 0  31 - C o m p a r i s o n o f . L e n g t h  Scales  Total Eddy Count vs. Distance  40 c m Figure  60  80  32 - Eddy C o u n t w i t h  100  Distance  g  90  EF  1  0.20-  I Eddy Fraction of Flow Area E F vs Distance X  0.15-1 U = 40cm/s g  0.10-1 • •  I 005-  20  40 cm  Figure  33  -  60  80  100  Variation  •  \ 77*  in  120  Occupied  hot-film  A eddy  10  \  \  \  w  0430  50 cm  7  0  Figure-34  X  A  90  110  Decay of u  z  Area  91  EDDY HOT-FILM POWER S P E C T R U M VELOCITY=40 CM/SEC XPOSN= 30 cn  EDDY HOT-FILM POWER S P E C T R U M VEL0CITY=40 CH/SEC XPOSN= 50 cn  <_>°' UJ  in ~v  40.0 en o FREQUENCY (HZ)  20-0  80.0  .  IOO.O o.o  "Tn—-i 20.0  " f  40.0  FREQUENCY  "I -  60.0  IHZ)  80.0  100  + eddy - hot-f i Im  EDDY  EDDY HOT-FILM POWER S P E C T R U M VEL0CITY=40 CM/SEC XP0SN= 90 CM  HOT-FILM  P O W E R  S P E C T R U M  VELOCITY=40 CM/SEC XPOSN- 70 CM  <_)•= LU  in  4U  0  FREQUENCY  Figure  60.0 IHZi  100.0  0.0  20.0  40.0  60.0  FREQUENCY (HZ)  3 5 - C o m p a r i s o n o f Power S p e c t r a  80.  100.0  92  EDDY  HOT-FILM  EDDY  POWER SPECTRUM  HOT-FILM  POWER SPECTRUM  (RRER REPRESENTATION) VELOCITY= 40CM/SEC X POSN= 30CM  (RRER REPRESENTATION) VELOCITY- 40CM/SEC X POSN= 50CM  UJo 3c  1  -3.0  -2.0  1 —  1  -1.0  0.0  1  1.0 LOGIWi  2  -°  3.0  4.0  5.0  -3.0  "i -2.0  1  r  1  -1.0  o.n  T  111  r.O  LOG'i'W  +Te  1  1  1  3.0  4.0  5.0  +Te -Te + eddy - hot - f i l m  EDDY  HOT-FILM  POWER SPECTRUM  EDDY  (RRER REPRESENTRTIONI VELOCITY* 40CM/SEC X POSN= 70 CM  3.0  1  -2.0  1  -l.o  1  o.o  IfiREfl REPRESENTRTION). VELOCITY= 40CM/SEC X POSN- 90CM  1—  T—,  1.0 LOG IV | 2.0  -Te •Te  Figure  HOT-FILM .  POWER SPECTRUM  • *  1 '  3.0  '  '  4.0  I  I  5.0  36 - E q u i v a l e n t  •3.0  T- 2". 0  - 11. 0  *  1 * *****  1 — 0.0  < l.U<S\  * * 4 »«* *  ^***m*tm  V ^' 0  LOG(W)\  3.0  W s  Area  Representation  +  Te  4.0  5.0  93  V.  CONCLUSIONS  A model which t r e a t s a t u r b u l e n t of  coherent r o t a t i o n s  plus laminar  E x p e r i m e n t s were p e r f o r m e d t o  f l o w h a s been  study  how  well  turbulent  flow.  structures  by p h o t o g r a p h i n g t h e m o t i o n s o f a l u m i n u m  towing  for  tank  searched  constructed  easy access t o e i t h e r  frame. from  both  frames  reference  coherent tracers.  f o r these experiments fluid  or  object  found with  that  the  allowed  reference  i s available  the  camera  model  generated  found  v e l o c i t y information  i t was  t e c h n i q u e was o n l y u s e f u l  a grid  f o r and  the  Although the f l u i d  in  this  the  We  fluctuations  composed  presented.  describes  The  velocity  flow as being  visualization in  the  fluid  frame.  The d i s t r i b u t i o n o f s i z e s a n d a n g u l a r v e l o c i t i e s o f t h e e d d i e s were m e a s u r e d f r o m t h e s e p h o t o g r a p h s . spectrum the  was u s e d t o g e n e r a t e t h e power  longitudinal velocity A  hot-film  anemometer  analyzer  these f l u c t u a t i o n s . eddy  spectrum  was  used  10%  the turbulent of  the  power  of  measure  the  spectral  density  spectral  A  from and  densities  we have  the size  scales  conclusions:  fluctuations.  total  to  By c o m p a r i n g t h e h o t - f i l m m e a s u r e d  The e d d y - s p e c t r u m d e s c r i p t i o n of  density  g e n e r a t e d by t h e g r i d .  p r o d u c e d t h e power  generated  drawn t h e f o l l o w i n g  spectral  eddy-size  fluctuations.  longitudinal velocity fluctuations spectrum  This  predicts  I t also predicted  k i n e t i c energy of v e l o c i t y  m e a s u r e d by t h e h o t - f i l m p r o b e .  The eddy  model  a constant fluctuations was  least  94  successful low  frequency  partly  due  coherent not the  at describing  the k i n e t i c  regimes. to  the  We  fact  energy i n t h e h i g h and  believe that  this  inter-eddy  discrepancy  c o r r e l a t i o n s and  f l u c t u a t i o n s i n t h e space between t h e  accounted f o r i n our a n a l y s i s .  is  eddies  were  Our s i m p l e t r e a t m e n t o f  v e l o c i t y p r o f i l e w i t h i n an eddy may have c o n t r i b u t e d  as  well. The the  eddy-size spectrum proved u s e f u l  dynamics of the e v o l u t i o n  The  evolution  flow  photographs.  co-rotating evolution  of t u r b u l e n t  of the s t r u c t u r e s could  length  both  i n the flow  in  scales.  be s e e n i n s u c c e s s i v e  Evidence of e d d y - p a i r i n g  structures and  f o r understanding  the  was o b s e r v e d f o r  eddy-size We l o o k  forward t o  l e a r n i n g more a b o u t eddy i n t e r a c t i o n s by s t u d y i n g  sequential  photographs of f l u i d Our  model  statistical shown  has  to  to  studying  the  turbulence  understand  takes  into  structures.  Also,  predicting  properties  influence  can of in  and  be  observed  terms  of  usual  in  fluid eddy  and a r e flow.  spectra  more p h y s i c a l l y m o t i v a t e d  fluctuations". account  of  the  As t h i s t h e s i s h a s  turbulent  the  t h e eddy-dynamics  of turbulence  over  of t u r b u l e n c e .  dynamics  "statistical  naturally  advantages  structures  d e s c r i p t i o n of simpler  patterns.  many  description  coherent  important  flow  photographs.  spectrum's  This  A is  than  description  mechanics  of coherent  could  useful  turbulent  on mean v e l o c i t y  flows  be  for  such  as  the  profiles,  on  the  95  drag  of  a  b o d y , and  on m i x i n g  processes.  k n o w l e d g e o f t h e eddy d y n a m i c s c a n  be  eddy  can  statistics  velocity  which  fluctuations  in  turn  in a turbulent  used be  a  coherent  understanding arise.  structures  from  its  machinery  not  better c r i t e r i a  three dimensional  the  and  w o u l d be  An  of  is  in  that  may  a e s t h e t i c value understanding  the  could lead  to  more  More e f f i c i e n t  this  aircraft  canoes  and  flow  field  phenomenon  of  quite  sailboats  developing flow  would  done by a n a l y z i n g an  entire  on  the  T h i s a n a l y s i s would partly  straightforward  Many f l o w s c o u l d t h e n  fashion.  PLASMA PHYSICS l a b r e p o r t  of  c o u l d be p o s s i b l e .  concentrating  velocity.  decay  and  thesis indicate that  c o u l d be  efficient  formed, or  local address  fractional,  automated a n a l y s i s u s i n g a d i g i t i z e d video  definition.  UBC  turbulence  viewpoint  This  angular  observed  eddis.  studying  f o r a n a l y z i n g s t r u c t u r e s i n the  worthwhile.  curvature  to predict  new  t o m e n t i o n f r e i g h t e r s and results  the  used  i n f l u e n c i n g the p r o d u c t i o n  structures.  The  be  by  course,  predict  of t h e m e c h a n i c s of t u r b u l e n t f l o w s  Aside  coherent  to  of  the  m e c h a n i c s o f eddy p r o d u c t i o n fluid  And,  flow.  Perhaps the g r e a t e s t advantage of from  1  #89  to  implement  be a n a l y z e d  using in a  signal this  rigorous  96  BIBLIOGRAPHY B . A h l b o r n , F . A h l b o r n a n d S.Loewen: UBC PLASMA PHYSICS l a b r e p o r t #89, ( 1 9 8 3 ) F.Ahlborn: Z e i t s c h r i f t (1931 )  f u r Technische  J.O.HINZE: TURBULENCE, McGRAW-HILL,  P h y s i k _1_2 , 482-491 (1959)  J.L.LUMELY: S t o c h a s t i c T o o l s i n T u r b u l e n c e , A c a d e m i c (1970) R o b e r s o n a n d Crowe: E n g i n e e r i n g F l u i d M i f f l i n Co., ( 1 9 7 5 )  Mechanics,  Press,  Houghton  H . T e n n e k e s a n d J . L . L u m l e y : A F i r s t C o u r s e i n T u r b u l e n c e , MIT Press,(1972)  97  APPENDIX A - HOT-FILM PROBE S E N S I T I V I T Y TO FLUCTUATIONS  VELOCITY  I g n o r i n g t h i s i n f l u e n c e the probe w i l l respond to the speed of f l o w p a s t the s e n s o r . For a l i n e a r i z e d probe the flow s p e e d S i s r e l a t e d t o t h e l i n e a r i z e d anemometer v o l t a g e s i g n a l , E , by t h e r e l a t i o n S=KE (A-1) where K i s t h e c a l i b r a t i o n c o n s t a n t . The f l o w s p e e d S i s r e l a t e d t o t h e f l o w v e l o c i t y c o m p o n e n t s by t h e r e l a t i o n S = (U+u) +v' +WV=W=0 (A-2) where u,v,w a r e the. l o n g i t u d i n a l and two l a t e r a l f l u c t u a t i n g v e l o c i t y c o m p o n e n t s and U i s t h e l o n g i t u d i n a l mean v e l o c i t y c o m p o n e n t . W r i t i n g out t h e (U+u) t e r m we h a v e S* =U + 2Uu+u +v +w' (A-3) I f t h e t u r b u l e n c e i n t e n s i t y i s s u f f i c i e n t l y low so t h a t u,v,w<<U t h e s q u a r e d t e r m s i n e q n . ( 3 - 3 ) may be n e g l e c t e d . A f t e r d i v i d i n g b o t h s i d e s by U and t a k i n g t h e i r s q u a r e r o o t we h a v e a  1  l  1  l  l  S/U=/1+2U/U  L  (A-4)  Now u s i n g a M a c l a u r i n s e r i e s e x p a n s i o n f o r t h e s q u a r e r o o t w i t h u<<U and m u l t i p l y i n g by U we have S=U+u (A-5) T r e a t i n g the l i n e a r i z e d v o l t a g e E as t h e sum of a s t e a d y , E , p l u s a f l u c t u a t i n g c o m p o n e n t , e, and substituting e q n . ( A - 5 ) i n t o e q n . ( A - 1 ) we see that U+u=KE +Ke (A-6) The f l u c t u a t i n g p a r t of t h e anemometer s i g n a l i s p r o p o r t i o n a l t o t h e f l u c t u a t i n g p a r t of l o n g i t u d i n a l velocity. The c o n s t a n t of p r o p o r t i o n a l i t y b e i n g t h e same as t h e c a l i b r a t i o n c o n s t a n t f o r t h e mean c o m p o n e n t . The wedge s h a p e o f t h e p r o b e t e n d s t o s u p p r e s s t h e v and w v e l o c i t y component c o n t r i b u t i o n s t o t h e c o o l i n g t h u s f u r t h e r r e i n f o r c i n g equation (A-6).  98  APPENDIX B - R I G I D BODY EDDY VELOCITY  PROFILE  A r i g i d body c i r c u l a r l y c y l i n d r i c a l eddy o f r a d i u s Rm r o t a t i n g a b o u t i t s a x i s o f symmetry h a s a v e l o c i t y p r o f i l e g i v e n by u (r)=Um(r/Rm) e  0<r<Rm  (B-1)  where Um i s t h e t a n g e n t i a l v e l o c i t y a t t h e p e r i p h e r y o f t h e e d d y a n d r i s t h e r a d i u s , s e e f i g u r e b e l o w . The u, component o f t h e v e l o c i t y i s g i v e n b y , u,=u„rsine and  (B-2)  u s i n g e q u a t i o n B-1 becomes u,=(Um/Rm)rsine  (B-3)  I f a u component v e l o c i t y p r o b e moves t h r o u g h t h e eddy a t c o n s t a n t s p e e d i n t h e x d i r e c t i o n , a s shown, w i t h i m p a c t p a r a m e t e r b we have rsine=b so  (B-4)  t h a t e q u a t i o n B-3 becomes u,=(Umb/Rm)  (B-5)  s h o w i n g t h a t t h e f l u c t u a t i n g component o f t h e v e l o c i t y f o r a r i g i d body e d d i e s i s a c o n s t a n t . A l i t t l e study of the f i g u r e b e l o w shows t h a t t h e s a m p l i n g t i m e t h r o u g h t h e eddy w o u l d be g i v e n by t=(2/Uc){Rm - b } x  (B-6)  w i t h Uc b e i n g t h e p r o b e s p e e d r e l a t i v e t o t h e e d d y .  99  A P P E N D I X C ~ T U R B . F T N F O R T R A N CODE  1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58  59 60  C C C C C C C C C  C C C C C C C  C C  TURB.FTN INPUT DATA ON UNIT 11 OUTPUT DATA ON UNIT 6 PROGRAM CHANGES EDDY SPECTRUM TO POWER  SPECTRUM  EDDY CHARACTERIZING DISTRIBUTIONS ARE READ IN INTEGER NLNGTH, NVEL, NCONV, NPTS, N1PTS, MPTS, NRMAX, NRNO'T INTEGER J , I , CHOOSE, M, NUMBER, NDATA, NUX, SUMNUX INTEGER LOGSPC(37) / I , 2, 3, 4, 5, 6, 7, 8, 9, 10, 11,16, 21j 1 27,32,37,42,47,52,57,62,67,73,78,83,88,93,97,103, 155, XOt 2 257,308,359,411,462,513/ REAL NR(500), RNOT(500), RMAX(500), V E L ( 5 0 0 ) , RAD(500) REAL CONVEL(500),PSUM(500) ,ENERGY/0./,AREA/0./,TE/0./ REAL LEBAR, RCHOSE, RMAXO, RNOTO, B, CONVEC, TSTEP, TRATE REAL N L ( 5 0 0 ) , L ( 5 0 0 ) , UX(6000), UMAX(500), RATIO, TEMP REAL POWERO025) /1025*0./, F R E Q O 0 2 5 ) , REAL I REAL*8 DATA(2048) COMPLEX* 16 TRANO025) EQUIVALENCE (DATA(1),TRAN(1)) READ F I L E NUMBER READ (11,230) SPECN READ SAMPLING FREQUENCY READ (11,260) TRATE TRATE=FREQUENCY RESOLUTION/NYQUIST CRITERIA NUMBER IS NUMBER OF RMS AVERAGES TO BE MADE READ (11,210) NUMBER READ PEDDY ET AL READ(11,320)PEDDY,AREA,LEBAR,NUMED READ IN EDDY SIZE SPECTRUM READ (11,210) NRMAX DO 10 J = 1, NRMAX READ (11,270) N R ( J ) , RMAX(J), RNOT(J), UMAX(J) 10 CONTINUE READ IN CONVECTION VELOCITIES 60 READ (11,210) NCONV IF (NCONV .EQ. 1) GO TO 80 DO 70 J = 1, NCONV READ (11,230) CONVEL(J) 70 CONTINUE 80 READ (11,230) CONVEC  C C C C  C C  PREPARE SIZE DISTRIBUTION FOR RANDOM  SAMPLING  I N I T I A L I Z E FRAND BY TIME-OF-DAY B «= RAND(SCLOCK(0. )) PSUM(1) = 0. MPTS = NRMAX - 1 DO 90 I = 1, MPTS PSUM(I + 1) = PSUM(I) + NR(I)*2*RMAX(I) 90 CONTINUE DO 100 I = 1, NRMAX PSUM(I) = PSUM(I) / PSUM(NRMAX) 100 CONTINUE SAMPLING LOOP STARTS  100  61 62 63 64 65 66 67 68 69 70 71 72 73 74 75' 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 97 98 99 100 101 102 103 104 105 106 107 108 109 110  111  112 113 1 14 114.5 114.6 114.7 114.75 114.8 1 15 116  C  C C C C  C  C C  C C C C C C  PBLANK=1.-PEDDY DO 1 90 K = 1 , NUMBER USQBAR=0. SUMNUX = 0 . NDATA IS NUMBER OF VELOCITY SAMPLES TAKEN PER FOURIER ANALYZED RECORD NDATA = 2048 TIME RECORD BEING MADE CHOOSE UX=0. OR SAMPLE AN EDDY 105 X=FRAND(0.) I F (X.GT.PBLANK) GOTO 110 SUMNUX=SUMNUX+1 DATA(SUMNUX)=0. IF (SUMNUX.GT.NDATA)GOTO 170 GOTO 105 CHOOSE EDDY RADIUS 110 IRAD = CHOOSE(M,PSUM,NRMAX,RCHOSE) RMAXO = RMAX(IRAD) RNOTO = RNOT(IRAD) CHOOSE IMPACT PARAMETER 130 B = 2. * (FRAND(0.) - .5) * RMAXO FIND CONVECTION VELOCITY IF (NCONV .GT. 1) CONVEC = CONVEL(I RAD) 140 CALL SAMPLE(TRATE, RNOTO, RMAXO, UMAX(IRAD), B, CONVEC, NUX.&of) DO 150 I = 1, NUX IF (I + SUMNUX .GT. NDATA) GO TO 170 DATA(I + SUMNUX) = UX(I) USQBAR=USQBAR+UX(I)*UX(I) 150 CONTINUE 160 SUMNUX = SUMNUX + NUX GO TO 105 WRITE OUT TIME RECORD IF WANTED 170 WRITE(9,370)(DATA(II),11=1,2000) 170 ENERGY=ENERGY+USQBAR/NDATA/NUMBER DFOUR2 FOURIER ANALYZES THE TIME RECORD NDIM = NDATA NODIM= # OF DIMENSIONS OF TIME RECORD NODIM = 1 ISIGN=-1 FOR DFT +1 FOR IDFT ISIGN = -1 IFORM=0 FOR REAL DATA IFORM = 0 CALL DFOUR2(DATA, NDIM, NODIM, ISIGN, I FORM) DO 180 I = 1, 1025 POWER(I) = POWER(I) + ((CDABS(TRAN(I))/NDATA)**2)/NUMBER 180 CONTINUE 190 CONTINUE TSTEP = 1. / TRATE DO 200 1 = 1 , 1025 REALI = FLOAT(I) FREQ(I) = (REALI - 1.) / (NDATA*TSTEP) 200 CONTINUE AREA=QINT4P(FREQ,POWER,513,1,513) TE=POWER(1)/(4.*AREA) DO 202 KK=1,513 POWER(KK)=POWER(KK)*9.8696 202 CONTINUE N1PTS = 35 WRITE (6,280) SPECN, CONVEC, N1PTS, ENERGY,TE  101  120 122 123 C 124 125 126 127 128 129 130 1 31 132 133 134 135 136 137 138 139 C 140 C 141 C 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 C 165 166 167 168 170 173 174 174.5 175 176 177 178 179 End of f i l e  WRITE (6,300) (POWER(LOGSPC(I)),FREQ(LOGSPC(I)),I=1,N1PTS) STOP FORMAT CODES 210 FORMAT (13) 220 FORMAT (213) 230 FORMAT (F7.3) 240 FORMAT (2F7.3) 250 FORMAT (3F7.3) 260 FORMAT (F5.0) 270 FORMAT (4F7.3) 280 FORMAT (F7.3,F7.2, 13,F7.4,E10.3) 290 FORMAT (3(D16.9,F5.0)) 300. FORMAT (3(F10.3, F9.3)) 310 FORMAT (3(F10.3,F9.3)) 320 FORMAT (F7.5,2F7.3,13) 370 FORMAT(10F9.3) END INTEGER FUNCTION CHOOSE(M,PSUM,NRMAX,RCHOSE) THIS PROGRAM RANDOMLY CHOOSES AN INTERVAL FROM A PREDETERMINED DISTRIBUTION M I S THE INTEGER NUMBER OF THE INTERVAL CHOSEN REAL PSUM(500), RCHOSE INTEGER M, NPTS, SCOPE, MPTS LOGICAL GT, LT MPTS=NRMAX-1 M = NRMAX / 2 SCOPE = M + 1 RCHOSE = FRAND(0.) 10 GT = .FALSE. LT = .FALSE. I F (RCHOSE .GT. PSUM(M)) GT = .TRUE. I F (RCHOSE .LT. PSUM(M + 1)) LT = .TRUE. I F (LT .AND. GT) GO TO 20 SCOPE = SCOPE / 2 + 1 M = M + SCOPE I F (LT) M = M - (2*SCOPE) + 1 IF (M.LT.1) M=1 IF (M.GT.MPTS) M=MPTS GO TO 10 20 CHOOSE = M RETURN END SUBROUTINE SAMPLE(TRATE, RNOTO, RMAXO, UMAX, B, CONV, NUX, UX) SAMPLES UX FOR GIVEN EDDY INTEGER NCORE, NUX, I REAL TIME, TMAX, RMAXO, UMAX, B, CONV, UX(6000) REAL TRATE, TSTEP TSTEP = 1. / TRATE TMAX = SQRT(RMAXO*RMAXO - B*B) / CONV UCONST = UMAX * B / RMAXO NCORE =2*TMAX / TSTEP NUX=NCORE DO 30 I = 1, NCORE U X ( I ) = UCONST 30 CONTINUE 40 RETURN END  102  APPENDIX D ~ AVERAGE EDDY CHORD AND PEDDY CALCULATION F o r a c i r c u l a r l y c y l i n d r i c a l eddy h a v i n g a random i m p a c t p a r a m e t e r b w i t h a p r o b e t h e a v e r a g e c h o r d , L, i s t h e w i d t h of a r e c t a n g l e w i t h l e n g t h 2Rm h a v i n g an a r e a e q u a l t o t h a t of t h e e d d y ' s c i r c u l a r c r o s s s e c t i o n . L{2Rm}=TTRm  (D-1)  L=(TT/2)Rm  (D-2)  so t h a t  W i t h a d i s t r i b u t i o n o f eddy s i z e s N(Rm) t h e a v e r a g e a l l e d d i e s r a n d o m l y i n c i d e n t on t h e p r o b e w i l l be L e d = [ ^ N ( R m ) {Rmr^2} ] / t ^ N(Rm) ]  chord of  (D-3)  To r e c r e a t e a r a n d o m l y d i s t r i b u t e d v e l o c i t y r e c o r d composed o f z e r o a n d eddy c o n t r i b u t i o n s c h o o s e 0 . v e l o c i t y o r s a m p l e an eddy v e l o c i t y p r o f i l e a c c o r d i n g t o P e = p r o b a b i l i t y o f c h o o s i n g an eddy P o = p r o b a b i l i t y of c h o o s i n g a z e r o v e l o c i t y L s = a v e r a g e number o f v e l o c i t y s a m p l e s f o r t h e eddies' Lb=number o f v e l o c i t y s a m p l e s when a z e r o i s • c h o s e n (=1) F r a c = f r a c t i o n o f s a m p l e a r e a t a k e n by e d d i e s The number o f v e l o c i t y s a m p l e s f o r a l e n g t h X a n d v e l o c i t y p r o b e s p e e d Uc i s g i v e n by ( X / U c ) F s where F s i s t h e s a m p l i n g frequency. We f i r s t h a v e Pe+Po=1 (D-4) and LePe LePe+LoPo  Frac  (D-5)  so t h a t Pe=  LoxFrac {Le(1-Frac)+LoxFrac}  i s t h e p r o b a b i l i t y o f c h o o s i n g an eddy w h i c h r e c r e a t e o b s e r v e d eddy f r a c t i o n F r a c .  (D-6) will  randomly  

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