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Turbulent diffusion in a stratified fluid Grigg, Harold Russel 1960

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i  TURBULENT DIFFUSION IN A STRATIFIED FLUID by HAROLD RUSSEL GRIGG M . S c , U n i v e r s i t y o f Saskatchewan, 1957  A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY i n t h e Department of Physics  We a c c e p t t h i s t h e s i s as conforming t o t h e r e q u i r e d standard  THE UNIVERSITY OF BRITISH COLUMBIA July, i960  In p r e s e n t i n g  t h i s t h e s i s i n p a r t i a l f u l f i l m e n t of  the r e q u i r e m e n t s f o r an advanced degree a t the  University  o f B r i t i s h C o l u m b i a , I agree t h a t the L i b r a r y s h a l l make it  freely  a v a i l a b l e f o r r e f e r e n c e and  agree t h a t p e r m i s s i o n f o r e x t e n s i v e f o r s c h o l a r l y purposes may  study.  I further  c o p y i n g of t h i s  be g r a n t e d by the Head o f  Department o r by h i s r e p r e s e n t a t i v e s .  be a l l o w e d w i t h o u t my w r i t t e n  Department o f The U n i v e r s i t y of B r i t i s h Columbia, Vancouver 8, Canada. Date  9~  /?£  <Q  my  I t i s understood  t h a t c o p y i n g or p u b l i c a t i o n o f t h i s t h e s i s f o r g a i n s h a l l not  thesis  financial  permission.  tUbe Pntaersttjj of ^rtttsl] Columbia GRADUATE  FACULTY OF GRADUATE STUDIES  STUDIES  Field of Study: Physical Oceanography Synoptic Oceanography  G . L . Pickard  Fluid Dynamics  R. W. Stewart  Dynamic Oceanography  G . L . Pickard R. W. Stewart  Turbulence  PROGRAMME OF THE  R. W. Stewart  FINAL ORAL E X A M I N A T I O N  Related Studies: Waves and Tides Numerical Analysis Biological Oceanography  J. C . Savage -  -  FOR T H E D E G R E E OF  C . Froese  DOCTOR OF PHILOSOPHY  R. F . Scagel  of H A R O L D R. G R I G G B.Sc. Saskatchewan, 1949 M.Sc. Saskatchewan, 1957 I N R O O M 301, P H Y S I C S B U I L D I N G F R I D A Y , A U G U S T 5, 1960 A T 10:30 A . M .  COMMITTEE  IN CHARGE  D E A N G . M . S H R U M , Chairman R. F. R. G.  W. S T E W A R T A. KAEMPFFER W. B U R L I N G V. PARKINSON  F. R. R. R.  A. FORWARD F. S C A G E L F. H O O L E Y BARRIE  External Examiner: D R . A . P A T T E R S O N Pacific Naval Laboratory, Esquimalt  TURBULENT  DIFFUSION  IN A STRATIFIED  FLUID  ABSTRACT In natural fluids, stratification of density is common. Observations of natural turbulence in the presence of a density stratification are difficult since the stratification usually occurs in regions not readily accessible. In the laboratory, maintenance of a stratification in a shear flow presents equal difficulties. Observations were made of isolated puffs of fluid injected vertically downwards into a uniform fluid of the same density, and into a stably stratified fluid of greater density. The observations demonstrate that such isolated puffs in uniform fluid are subject to the same decay laws as is the turbulent energy in an extended turbulent fluid. This implies that a turbulent field, made up of randomly, oriented puffs of fluid of varying volumes and velocities, would display many of the - characteristics of a fluid in which the turbulence is a result of shear flow, and that observations made on such puffs can be applied with some confidence to natural turbulence. The apparatus was so constructed that the detailed mixing between the injected fluid and the surrounding fluid resulted in the formation of a finely divided precipitate which rendered the puffs visible, and permitted measurements of their path by means of moving picture photographs. The results demonstrate that mixing with the surroundings occurs throughout the puff, which retains its identity and a relatively uniform density. " Measurements of the rate of formation of precipitate permit an estimate of the rate at which the injected fluid became mixed with the entrained fluid. The density stratification had very little influence upon this rate until after the puff had reached its maximum penetration. -  The rate of horizontal spreading of the puff during its motion also showed little effect of the stratification. Measurement of the penetration and ultimate position of the center of a puff in a density stratification together with measurements of the position of the center of a puff in uniform fluid permit the calculation of a number of dimensionless ratios. These are displayed graphically as a function of a dimensionless number made up from the initial conditions of velocity, volume, density gradient, density, and the acceleration of gravity. This number resembles the reciprocal of a Richardson's number and is referred to as 1/RiThe maximum conversion of initial kinetic energy to potential energy observed was 20 percent, being greatest for small values of 1/RiThe portion of the initial energy which contributes to the breakdown of the stratification was found to be approximately 3 percent and nearly independent of 1/Rj. Some writers have thought the loss to density stratification would be much greater than this. A figure representative of the transport of properties during the history of a puff varied by a factor 35 over the possible range of initial conditions, being greater for high values of 1 / R j .  ii  ABSTRACT  In natural f l u i d s , s t r a t i f i c a t i o n of density i s common.  Observations of natural turbulence  i n t h e presence  of a density s t r a t i f i c a t i o n are d i f f i c u l t since the s t r a t i f i c a t i o n u s u a l l y occurs i n regions not r e a d i l y a c c e s s i b l e .  In  the l a b o r a t o r y , maintenance o f a s t r a t i f i c a t i o n i n a shear flow presents  equal  difficulties.  O b s e r v a t i o n s were made o f i s o l a t e d p u f f s o f f l u i d i n j e c t e d v e r t i c a l l y downwards i n t o a u n i f o r m f l u i d o f t h e same d e n s i t y , and i n t o a s t a b l y s t r a t i f i e d density.  The o b s e r v a t i o n s  f l u i d of greater  demonstrate t h a t such i s o l a t e d  p u f f s i n u n i f o r m f l u i d a r e s u b j e c t t o t h e same decay laws as i s t h e t u r b u l e n t energy i n an extended t u r b u l e n t f l u i d .  This  i m p l i e s t h a t a t u r b u l e n t f i e l d , made up o f randomly o r i e n t e d p u f f s o f f l u i d o f v a r y i n g volumes and v e l o c i t i e s , would d i s p l a y many o f t h e c h a r a c t e r i s t i c s o f a f l u i d i n which t h e t u r b u l e n c e i s a r e s u l t o f shear f l o w , and t h a t o b s e r v a t i o n s on such p u f f s can be a p p l i e d w i t h some c o n f i d e n c e  made  to natural  turbulence. The  a p p a r a t u s was so c o n s t r u c t e d  t h a t the d e t a i l e d  m i x i n g between t h e i n j e c t e d f l u i d and t h e s u r r o u n d i n g r e s u l t e d i n the formation  fluid  of a f i n e l y divided precipitate  w h i c h r e n d e r e d t h e p u f f s v i s i b l e , and p e r m i t t e d  measurements  o f t h e i r p a t h by means o f moving p i c t u r e photographs.  iii The  r e s u l t s demonstrate t h a t m i x i n g w i t h the  surround-  i n g s o c c u r s t h r o u g h o u t the p u f f , which r e t a i n s i t s i d e n t i t y and  a r e l a t i v e l y uniform  density.  Measurements of the r a t e of f o r m a t i o n o f p r e c i p i t a t e p e r m i t an e s t i m a t e  o f t h e r a t e at which the i n j e c t e d f l u i d  became mixed w i t h the e n t r a i n e d f l u i d .  The  density  stratifi-  c a t i o n had v e r y l i t t l e i n f l u e n c e upon t h i s r a t e u n t i l a f t e r the p u f f had reached i t s maximum p e n e t r a t i o n . The  r a t e of h o r i z o n t a l spreading  of the puff  i t s m o t i o n a l s o showed l i t t l e e f f e c t o f the  during  stratification.  Measurement of the p e n e t r a t i o n and u l t i m a t e p o s i t i o n o f t h e c e n t e r of a p u f f i n a d e n s i t y s t r a t i f i c a t i o n  together  w i t h measurements o f the p o s i t i o n of the c e n t e r o f a p u f f i n u n i f o r m f l u i d p e r m i t t h e c a l c u l a t i o n o f a number o f dimensionless ratios.  These are d i s p l a y e d g r a p h i c a l l y as a f u n c -  t i o n of a d i m e n s i o n l e s s  number made up from the i n i t i a l  d i t i o n s of v e l o c i t y , volume, d e n s i t y g r a d i e n t , d e n s i t y , the a c c e l e r a t i o n of g r a v i t y .  conand  T h i s number resembles the r e -  c i p r o c a l o f a R i c h a r d s o n ' s number and i s r e f e r r e d t o as l / R j _ . The maximum c o n v e r s i o n p o t e n t i a l energy observed was  o f i n i t i a l k i n e t i c energy t o 20 p e r c e n t ,  being greatest f o r  s m a l l v a l u e s o f l/Rj_.. The  p o r t i o n of t h e i n i t i a l energy which c o n t r i b u t e s  t o the breakdown o f the s t r a t i f i c a t i o n was mately 3 percent  found to be  and n e a r l y independent o f l / R j _ .  approxi  Some w r i t e r s  have thought the l o s s t o d e n s i t y s t r a t i f i c a t i o n would be much  g r e a t e r than t h i s . A f i g u r e r e p r e s e n t a t i v e of the t r a n s p o r t of propert i e s d u r i n g the h i s t o r y of a p u f f v a r i e d by a f a c t o r 35 the p o s s i b l e range of i n i t i a l  over  c o n d i t i o n s , being g r e a t e r f o r  high values of l/Rj_. An estimate  of the p o s s i b l e e f f e c t of d e n s i t y s t r a t i -  f i c a t i o n upon the p r o d u c t i o n production l/Rj_.  of t u r b u l e n t energy shows t h a t  i s reduced by as much as a f a c t o r 10 f o r s m a l l  T h i s a r i s e s from the r e v e r s a l and rebound of the  o f f l u i d i n the  stratification.  puff  V  TABLE OF CONTENTS Page TITLE PAGE ABSTRACT TABLE OF CONTENTS L i s t of Tables L i s t of Plates L i s t of Figures ACKNOWLEDGEMENTS 1 1.1 1.2 1.3 1.4 2 2.1 2.2 2.3 3 3.1 3.2 3.3 4 4.1 4.2 4.3 4.4 4.5 5 5.1 5.2  INTRODUCTION Background Purpose Method Apparatus DATA OBTAINED Detailed Mixing Center Displacement H o r i z o n t a l Spreading ANALYSIS IN NEUTRALLY BUOYANT FLUID D e t a i l e d M i x i n g i n Uniform Surroundings V e l o c i t y o f a P u f f i n Uniform F l u i d of t h e Same D e n s i t y D i s p l a c e m e n t o f a P u f f i n Uniform F l u i d o f t h e Same D e n s i t y ANALYSIS IN STABLY STRATIFIED SURROUNDINGS Motion Across a Density D i s c o n t i n u i t y Motion i n a Uniform S t r a t i f i c a t i o n P o t e n t i a l Energy o f a P u f f a t Maximum Penetration E f f e c t o f a P u f f Upon t h e S u r r o u n d i n g F l u i d Transfer Coefficient COMPARISON OF NEUTRALLY BUOYANT AND STABLE SURROUNDINGS H o r i z o n t a l Spreading V e r t i c a l Spreading  i i i v vi vi v i i viii 1 1 7 9 11 23 23 25 25 26 26 ••• 32 36 39 39 40 42 47 50 52 52 53  vi Page 5.3  Energy L o s s  53  5.4  Detailed Mixing  54  6  55  DISCUSSION  6.1 6.2  55  M i x i n g Length Momentum C o n s e r v a t i o n and R a t i o o f T r a n s f e r Coefficients T u r b u l e n t Energy D i s s i p a t i o n T u r b u l e n t Energy P r o d u c t i o n Turbulent Scales  6.3 6.4 6.5  56 60 62 64  7  -SUMMARY  66  8  APPENDIX  72 74  REFERENCES LIST OF TABLES Table 1 2  Comparison o f v i s u a l e s t i m a t e s o f d i s p l a c e m e n t with calculated values. P r o p o r t i o n a l i t y between i n p u t volume and p h o t o c e l l response.  22 24  3  Mixing f u n c t i o n k c a l c u l a t e d using slopes of mixing curves.  28  4  C a l c u l a t i o n o f m i x i n g f u n c t i o n k u s i n g an i n t e g r a l e x p r e s s i o n f o r Q.  30  Mixing function k f o r various conditions  30  5 6  initial  V a l u e s o f s l o p e c o n s t a n t i n v e l o c i t y and displacement curves  39  LIST OF PLATES Plate I II  APPARATUS REPRESENTATIVE SERIES OF PHOTOGRAPHS OF A PUFF  Page Following 75  .. v i i  LIST OF FIGURES Fig.  Page  1  APPARATUS  2  DETAILED MIXING  3  EFFECT OF SALT UPON MIXING DATA  4  CENTER DISPLACEMENT  5  HORIZONTAL SPREAD IN UNIFORM FLUID  6  HORIZONTAL SPREAD IN STRATIFIED FLUID  7  DEPENDENCE OF MIXING FUNCTION k UPON THE INITIAL CONDITIONS  8  THREE DIMENSIONAL PLOT OF k AGAINST THE INITIAL CONDITIONS  9  GRAPHICAL DETERMINATION OF POWER LAW FOR 3 VARIABLES  10  Following p  a g e  75:  , SIMILARITY IN NEUTRALLY BUOYANT SURROUNDINGS  11  VELOCITY OF A PUFF IN NEUTRALLY BUOYANT SURROUNDINGS  12  DISPLACEMENT OF A PUFF IN NEUTRALLY BUOYANT SURROUNDINGS  13  POTENTIAL ENERGY AT MAXUyiUM PENETRATION INITIAL ENERGY•  14  ALTERATION IN DENSITY STRUCTURE  15  RATIO  INITIAL VOLUME VOLUME ENTRAINED  16  MAXIMUM PENETRATION AND EQUILIBRIUM DEPTH  17  ENERGY LOST TO DENSITY STRATIFICATION INITIAL ENERGY • NET DENSITY TRANSFER COEFFICIENTS EQUILIBRIUM DISTANCE MAXIMUM PENETRATION -  18 19 20  POTENTIAL ENERGY AT MAXIMUM PENETRATION KINETIC ENERGY AT SAME TIME IN NEUTRAL SURROUNDINGS  ACKNOWLEDGEMENTS  I w i s h t o e x p r e s s my a p p r e c i a t i o n f o r the a d v i c e and s u g g e s t i o n s g i v e n t o me d u r i n g the c o u r s e o f t h i s r e s e a r c h by my s u p e r v i s o r , Dr. R. W. S t e w a r t .  I a l s o w i s h t o thank t h e I n s t i t u t e o f Oceanography, U n i v e r s i t y o f B r i t i s h Columbia, f o r p r o v i d i n g l a b o r a t o r y and o f f i c e space i n w h i c h t o c a r r y o u t t h e work.  1  1  INTRODUCTION  1.1  Background At the p r e s e n t t i m e , s e v e r a l volumes a r e a v a i l a b l e i n  w h i c h the l i t e r a t u r e on t u r b u l e n t f l o w i s summarized i n a s y s t e m a t i c manner, o r which are p a r t i c u l a r l y concerned w i t h t h e i r author's s p e c i a l f i e l d of i n t e r e s t .  In these, references to  most o f t h e o r i g i n a l p u b l i c a t i o n s can be found ( H i n z e , J . 0 . 1959,  Townsend, A. A. 1956, Regarding  and B a t c h e l o r , G. K.  1953).  the i n f l u e n c e o f d e n s i t y s t r a t i f i c a t i o n  upon t u r b u l e n t m o t i o n s and d i f f u s i o n , no such c o m p i l a t i o n s have y e t been made i n a s e p a r a t e volume.  Sverdrup et a l (1942)  c o n t a i n s a d i s c u s s i o n o f the s u b j e c t , and g i v e s r e f e r e n c e s t o some o f the o r i g i n a l papers up t o t h a t t i m e .  More r e c e n t  l i t e r a t u r e i s found i n a l a r g e number o f j o u r n a l s , which i n d i c a t e s t h a t the s u b j e c t i s i m p o r t a n t t o people  in fields  r a n g i n g from the g e o p h y s i c a l s c i e n c e s t o t h e i n d u s t r i a l handl i n g of f l u i d s .  This i s understandable  s i n c e almost a l l f l u i d s  a r e s t r a t i f i e d t o some e x t e n t u n l e s s s p e c i a l p r e c a u t i o n s are t a k e n t o produce u n i f o r m i t y . I n one o f the f i r s t s e r i e s o f papers on t h e s u b j e c t , Richardson,  (1920),  ( 1 9 2 5 ) , develops  a c r i t e r i o n f o r the  gree o f d e n s i t y s t r a t i f i c a t i o n n e c e s s a r y m o t i o n s r e s u l t i n g from a v e l o c i t y s h e a r . usual approximation  de-  t o damp out t u r b u l e n t Richardson  uses the  o f r e p r e s e n t i n g t h e t r a n s p o r t o f momentum,  and t h e t r a n s p o r t o f d e n s i t y - p r o d u c i n g p r o p e r t i e s , due t o t u r b u l e n t m o t i o n s , by a u s t a u s c h  or t r a n s f e r c o e f f i c i e n t s K  M  and  K„  2 r e s p e c t i v e l y , d e f i n e d by F l u x ? K ( G r a d i e n t ) , i n analogy w i t h a k i n e m a t i c v i s c o s i t y and d i f f u s i o n c o e f f i c i e n t . g Pd(l/g) dZ (dU/dZ) g i s t h e a c c e l e r a t i o n due t o g r a v i t y ,  He i n t r o d u c e s  the dimensionless r a t i o  ~  Rj_, where  2  ? i s the f l u i d density, Z i s the v e r t i c a l displacement, U i s t h e n o n - f l u c t u a t i n g p a r t o f t h e h o r i z o n t a l v e l o c i t y , and R^ i s now known as t h e o r d i n a r y R i c h a r d s o n ' s number. R i c h a r d s o n proposed t h a t f o r R^> 1, t u r b u l e n t motions c o u l d n o t p e r s i s t b u t would be damped o u t by t h e energy l o s s t o the d e n s i t y s t r a t i f i c a t i o n . idea that  T h i s was based upon t h e m i s t a k e n  = K^, an assumption t h a t i s a p p r o x i m a t e l y t r u e i n  a uniform f l u i d  ( E l l i s o n , 1957).  In a stably s t r a t i f i e d  fluid,  t h e r a t i o K^/K^ can assume v e r y s m a l l v a l u e s as w i l l be seen below. The u n c e r t a i n t y about t h e magnitude o f K^/K^ has l e d t o d e f i n i n g a new number, c a l l e d t h e F l u x R i c h a r d s o n ' s number, d e f i n e d by Rf = K^/K^R^) .  This represents the r a t i o _ c i S t e w a r t , ±yw) i a j .  L o s s o f mean f l o w energy t o buoyancy T o t a l l o s s o f mean f l o w energy  ( s t  w a r t  1 Q  q )  a )  T a y l o r (1931) e n l a r g e d upon R i c h a r d s o n ' s work.  Using  some o b s e r v a t i o n s t a k e n by Jacobsen i n t h e K a t t e g a t , he was a b l e t o determine R^, K^ and KJJ. , He found t h e r a t i o KJJ/K^ t o be a t a l l t i m e s l e s s than 1, and t o range from .13 where t h e s t a b i l i t y * 4-^dz  was 11.2 x K T ^ c m T  1  t o .021 where t h e  stability  was 1 0 5 . x 10  cm. .  In addition, the observations  i n d i c a t e d t h a t as Rj_ i n c r e a s e d , i t approached t h e r a t i o  K  M  /K  H  .  T a y l o r proposed t h a t t u r b u l e n t m o t i o n s c o u l d be m a i n t a i n e d provided Rf< 1. A r e c e n t argument by E l l i s o n (1957) i n d i c a t e s a c r i t i c a l v a l u e o f R f i n t h e range o f . 1 4 .  A d i s c u s s i o n o f t h i s sub-  j e c t w i t h comments from o t h e r s i n t e r e s t e d i n t h e f i e l d i s g i v e n by S t e w a r t ,  (1959) ( a ) .  The o b s e r v a t i o n i s o f t e n made (Sverdrup e t a l , 1947) t h a t t h e f a c t t h a t KJJ/K^ i s l e s s than 1 i n a s t r a t i f i e d  fluid,  a l t h o u g h t h e t r a n s f e r o f momentum and o f d e n s i t y - p r o d u c i n g p r o p e r t i e s i s caused by the same t u r b u l e n t motions, t a t i v e l y understood  as f o l l o w s :  can be q u a l i -  I f a f l u i d blob i s d i s p l a c e d  v e r t i c a l l y by t u r b u l e n t motions, momentum can be exchanged w i t h t h e s u r r o u n d i n g s by p r e s s u r e f o r c e s , w h i l e heat o r s a l t can be enchanged, t o any a p p r e c i a b l e e x t e n t , o n l y i f m i x i n g between t h e d i s p l a c e d f l u i d and i t s s u r r o u n d i n g s . o p i n i o n , t h i s statement  requires q u a l i f i c a t i o n .  occurs  I n my I t i s true  t h a t a drop o f o i l moving i n water can share i t s momentum w i t h t h e w a t e r w i t h o u t b r e a k i n g up, a l t h o u g h , i f t h e motion i s sufficiently violent,  b r e a k i n g up w i l l o c c u r .  In this  case,  s u r f a c e t e n s i o n d e f i n e s a boundary w h i c h p e r m i t s t h e d i s s i p a t i o n o f t h e energy, l o s t by momentum s h a r i n g , t o o c c u r i n each f l u i d separately.  F o r the case.of mutually m i s c i b l e f l u i d s ,  no such boundary e x i s t s . body moving i n a f l u i d  The f o r c e s which a r e e x e r t e d on a  can be broken down i n t o v i s c o u s f o r c e s  4  a t the i n t e r f a c e ( s k i n f r i c t i o n ) , and p r e s s u r e the formation  o f a wake (form d r a g ) .  f o r c e s due  to  Both o f t h e s e i m p l y  the  development of t u r b u l e n t motions which must r e s u l t i n some degree o f i n t e r m i x i n g i n the absence o f any  surface  tension.  That t h e r e need not be any s t r i c t p r o p o r t i o n a l i t y between t h e s e two phenomena, however, i s shown by the e x p e r i m e n t a l f a c t t h a t Kpj/Kjyj- assumes d i f f e r e n t v a l u e s . reasonable that  I t i s also  s h o u l d depend t o some e x t e n t upon the  f a c t o r c o n t r o l l i n g d e n s i t y , t h a t i s h e a t , s a l i n i t y , water vapor, etc.  (When i t i s n e c e s s a r y , the n o t a t i o n i n the  d i s t i n g u i s h e s between t h e s e by u s i n g KJJ, Kg, respective coefficients).  literature e t c . , as  the  I n t h e l i m i t , as t u r b u l e n t motions  become n e g l i g i b l e , each c o e f f i c i e n t must approach the ponding molecular c o e f f i c i e n t .  They may  corres-  be expected t o d i f f e r  somewhat u n t i l t u r b u l e n t t r a n s p o r t becomes l a r g e } w i t h e x c e p t i o n o f Kj^,- t h e y s h o u l d t h e n approach e q u a l i t y ,  the provided  the h y p o t h e t i c a l d i s p l a c e d b l o b of f l u i d i s r e p r e s e n t a t i v e  of  the r e g i o n f r o m which i t o r i g i n a t e d . I f the p e n e t r a t i n g f l u i d i s moving under i t s own buoyancy, as i n b u b b l e s of hot gas r i s i n g from a h e a t e d p l a t e , i t w i l l n o t , i n g e n e r a l , be r e p r e s e n t a t i v e o f i t s p o i n t o r i g i n , ( E l l i s o n , 1956).  I n t h i s case, the r e l a t i o n s h i p be-  tween the c o e f f i c i e n t s becomes more c o m p l i c a t e d , remarks do not a p p l y .  of  and t h e above  I n s t a b l e c o n d i t i o n s , however, i t seems  r e a s o n a b l e t o expect a d i s p l a c e d b l o b t o be a r e p r e s e n t a t i v e one,  s i n c e the motions are i n i t i a t e d by non-buoyant f o r c e s .  5 In n a t u r a l conditions, observations of the effect of d e n s i t y s t r a t i f i c a t i o n upon t h e t u r b u l e n t m o t i o n s a r e few. The e x p e r i m e n t a l instruments  technique  i sdifficult.  such as. p r e s s u r e - t u b e  Velocity-measuring  or pressure-plate  can respond t o l a r g e r s c a l e s o f t u r b u l e n c e o n l y .  anemometers  To r e c o r d  s m a l l d e t a i l s , t h e h o t w i r e anemometer i s t h e o n l y s u c c e s s f u l instrument  at t h i s time,  ( E l l i s o n , 1956) and i s q u i t e depend-  able i n clean a i r o r water.  The h o t w i r e , o r wedge,  s e n s i n g elements must be s m a l l and d e l i c a t e . t o become contaminated o r d e s t r o y e d material i n the f l u i d .  velocity-  They a r e l i a b l e  by a b r a s i o n w i t h f o r e i g n  The a m p l i f i c a t i o n , r e q u i r e d t o b u i l d  the s i g n a l t o r e c o r d a b l e v a l u e s , i n t r o d u c e s n o i s e which i s d i f f i c u l t t o s e p a r a t e from t h e n o i s e - l i k e s i g n a l . elements must be supported a m p l i f i e r and supported  The s e n s i n g  a t c o n s i d e r a b l e d i s t a n c e s from t h e  i n such a way t h a t t h e i r v e l o c i t y r e -  l a t i v e t o the surroundings  can be d e t e r m i n e d .  The d i f f i c u l t i e s a r e b e i n g overcome^ 1959), ( P a t t e r s o n , A. M. 1958, i 9 6 0 ) .  (Grant e t a l ,  Good d a t a on t h e t u r b u l -  ent components t o g e t h e r w i t h g r a d i e n t s o f mean v e l o c i t y and d e n s i t y , w i l l , c a s t l i g h t on t h e whole problem. Stewart, turbulence.  (1959) (b), d i s c u s s e s t h e q u e s t i o n o f n a t u r a l  He p o i n t s o u t t h a t s t a b i l i t y i s a t l e a s t as im-  p o r t a n t as R e y n o l d ' s number i n d e t e r m i n i n g whether o r n o t a n a t u r a l f l o w w i l l be t u r b u l e n t .  He g i v e s a b i b l i o g r a p h y o f  r e c e n t l i t e r a t u r e on n a t u r a l d i f f u s i o n and r e l a t e d s u b j e c t s . To.wnsend, (1958) d e v e l o p s a r e l a t i o n s h i p w h i c h  6 i n d i c a t e s t h a t , i n a s t a b l y - s t r a t i f i e d atmosphere, a r b i t r a r i l y s m a l l v a l u e s o f t u r b u l e n t energy are not p o s s i b l e .  He  suggests  t h a t t u r b u l e n t energy must have a c e r t a i n f i n i t e a m p l i t u d e o r d i e out c o m p l e t e l y .  S t e w a r t (1959)  (&), p o i n t s out  Townsend's a n a l y s i s r e q u i r e s a c o n s t a n t  that  c o r r e l a t i o n between  t e m p e r a t u r e f l u c t u a t i o n s and the v e r t i c a l t u r b u l e n t v e l o c i t y . Its  c o n c l u s i o n , t h e r e f o r e , i s open t o  question.  A d e n s i t y d i s c o n t i n u i t y between two m i s c i b l e f l u i d s represents  an extreme example o f t h e e f f e c t o f s t a b i l i t y  t u r b u l e n t m o t i o n s and t u r b u l e n t m i x i n g .  Such a r e g i o n  on  will  support, i n t e r n a l waves, which i n t h e m s e l v e s , r e s u l t i n l i t t l e i n c r e a s e i n m i x i n g above the m o l e c u l a r d i f f u s i o n r a t e . f o l l o w i n g two  The  e x p e r i m e n t s show t h a t , when a c r i t i c a l wave  am-  p l i t u d e i s exceeded, t u r b u l e n t motions are produced which  can  break down t h e  stratification.  K e u l e g a n , (1949) conducted e x p e r i m e n t s on the e f f e c t o f i n t e r n a l waves upon a d e n s i t y s t r a t i f i c a t i o n . t h a t t h e r e was  He  found  l i t t l e m i x i n g a c r o s s the g r a d i e n t u n t i l  i n t e r n a l waves s t a r t e d t o b r e a k .  The  the  p r i n c i p a l e f f e c t of  the  b r e a k i n g - a p p e a r e d t o be the t r a n s p o r t o f dense m a t e r i a l i n t o the upper l a y e r .  The waves were caused by a shear f l o w i n the  upper f l u i d w h i c h was f l u i d was  t u r b u l e n t i n consequence.  held stationary.  The  lower  A complementary experiment w i t h a  moving l o w e r f l u i d and a s t a t i o n a r y upper f l u i d has not been done t o my knowledge.  There i s t h e r e f o r e some doubt as t o  i n t e r p r e t a t i o n of the r e s u l t s .  the  7 In  an e a r l y experiment  by T a y l o r , (1927) a v e l o c i t y  shear was e s t a b l i s h e d by r u n n i n g a l i g h t f l u i d o v e r a s t a t i o n a r y dense f l u i d h e l d i n a s h a l l o w d e p r e s s i o n i n t h e bottom of  a channel.  L i t t l e mixing occurred u n t i l a c e r t a i n c r i t i c a l  v e l o c i t y o f t h e upper f l u i d was r e a c h e d .  Once m i x i n g commenced,  the dense f l u i d was swept o u t c o m p l e t e l y . Most o f t h e p r e c e d i n g papers and experiments  imply  t h e concept o f a d i s p l a c e d mass o f f l u i d t r a n s f e r r i n g p r o p e r t i e s o f d e n s i t y and momentum a c r o s s a d e n s i t y s t r a t i f i c a t i o n . Broadly s t a t e d , they e x p l a i n the a l t e r a t i o n s of the t u r b u l e n t m o t i o n s and t h e d e n s i t y s t r u c t u r e by means o f a r a t i o between the c o e f f i c i e n t s  and Ky[.  I f t h e concept o f a p e n e t r a t i n g f l u i d mass i s t a k e n literally,  some f u r t h e r i n s i g h t i n t o t h e n a t u r e o f t h e m i x i n g  p r o c e s s may be g a i n e d by d i r e c t o b s e r v a t i o n o f a s i n g l e mass p e n e t r a t i n g i n t o q u i e t s u r r o u n d i n g  fluid  fluid.  S c o r e r , (1957), (195$), made o b s e r v a t i o n s on d i s c r e t e volumes o f f l u i d moving under t h e i r own buoyancy i n t o q u i e t surroundings. atmosphere.  Such movements o c c u r as t h e r m a l s i n a s t i l l O b s e r v a t i o n s o f d i s p l a c e d t u r b u l e n t f l u i d moving  a g a i n s t buoyant f o r c e s , o r i n s u r r o u n d i n g s o f t h e same d e n s i t y have n o t been made t o my knowledge. 1.2  Purpose The o b j e c t o f t h i s r e s e a r c h i s t o study a volume o f  f l u i d p e n e t r a t i n g i n t o a. q u i e t s u r r o u n d i n g f l u i d .  Buoyant  f o r c e s a r e e i t h e r zero o r oppose t h e i n i t i a l m o t i o n .  The  i n i t i a l m o t i o n i s caused by i n j e c t i n g a c y l i n d e r o f f l u i d i n t o the s t i l l  fluid. The p e n e t r a t i n g f l u i d w i l l be c a l l e d a p u f f .  This  w i l l mean t h e o r i g i n a l f l u i d p l u s any i n c r e a s e i n volume o r change i n p r o p e r t i e s r e s u l t i n g f r o m t h e entrainment o f s u r rounding f l u i d .  The p u f f w i l l be r e g a r d e d as p o s s e s s i n g  a  u n i f o r m d e n s i t y as f a r as buoyant f o r c e s a r e concerned. t u r b u l e n t motions i n s i d e t h e p u f f were n o t measured. not be c o n s i d e r e d  The  They w i l l  i n d e t a i l a l t h o u g h t h e average e f f e c t o f these  m o t i o n s i n m i x i n g t h e o r i g i n a l and e n t r a i n e d f l u i d s w i l l be cons i d e r e d , and r e f e r r e d t o as t h e d e t a i l e d m i x i n g . The  density of the injected f l u i d  n e a r l y t h e same d e n s i t y as t h e s u r r o u n d i n g s injection.  When t h e s u r r o u n d i n g s  buoyancy f o r c e s e x i s t .  i s a d j u s t e d t o have at the point of  are uniform,  When the s u r r o u n d i n g s  t h e r e f o r e , no  are s t r a t i f i e d i n  d e n s i t y , buoyant f o r c e s a r e zero a t t h e onset o f m o t i o n .  They  i n c r e a s e i n a manner dependent upon t h e d e t a i l s o f t h e entrainment  process, The  and upon the d e n s i t y s t r a t i f i c a t i o n  existing.  i n f o r m a t i o n d e s i r e d from t h i s experiment i s f i r s t ,  t h e e f f e c t o f t h e d e n s i t y s t r a t i f i c a t i o n on t h e p e n e t r a t i o n and  t h e entrainment r a t e , and second, the r a t e a t which t h e en-  t r a i n e d f l u i d becomes mixed w i t h t h e o r i g i n a l  fluid.  . As t h e s e f a c t o r s can be expected t o depend upon i n i tial itial x  c o n d i t i o n s o f v e l o c i t y and volume, v a r i o u s v a l u e s o f i n c o n d i t i o n s have been used.  The d e t a i l s o f e x p e r i m e n t a l  The h o t w i r e anemometer w i l l n o t o p e r a t e i n t h e absence o f a mean f l o w ( H i n z e , 1 9 5 9 ) •  9 methods and measurements w i l l be g i v e n i n t h e n e x t s e c t i o n . Estimates  o f the c o e f f i c i e n t K  M  f o r a simple  penetra-  t i v e f l o w as d e s c r i b e d would r e q u i r e t h e presence o f a v e r t i cal  v e l o c i t y shear.  formidable.  The  experimental  complications  However, i t can be hoped t h a t the t e c h n i q u e  l i n e d above i s s u f f i c i e n t l y r e a l i s t i c nature 1.3  appear  and magnitude o f KJJ can be  out-  t h a t i n s i g h t i n t o the  obtained.  Method The  i n i t i a l m o t i o n i s caused by i n j e c t i n g a c y l i n d e r  o f f l u i d v e r t i c a l l y downwards i n t o a tank of s t i l l  fluid  t h r o u g h a n o z z l e w h i c h p r o j e c t s below the s u r f a c e . - I f a s i m p l e tube i s used as a n o z z l e , t h e r e s u l t i s almost always the f o r m a t i o n of a v o r t e x r i n g .  This i s p a r t i c u l a r l y  the  case when the l e n g t h o f t h e c y l i n d e r i n j e c t e d i s n e a r l y same as the d i a m e t e r .  the  The m o t i o n o f v o r t e x r i n g s i n a r e a l  f l u i d has been e x t e n s i v e l y s t u d i e d by N o r t h r u p , (1912) more r e c e n t l y by Turner (1957)•  and  I t i s quite possible that  such r i n g s are formed i n the ocean from b r e a k i n g waves and p l a y a p a r t i n m i x i n g the upper l a y e r .  On a s m a l l s c a l e i t i s  d i f f i c u l t to i n t r o d u c e a drop of f l u i d i n t o a q u i e t f l u i d out some v o r t e x - r i n g f o r m a t i o n .  may  However, we  with-  are concerned here  w i t h t u r b u l e n t m i x i n g and the m o t i o n r e q u i r e d i s of a n o t h e r type.  S u f f i c i e n t t u r b u l e n c e must be p r e s e n t  f l u i d t o d i s t r i b u t e the c o n c e n t r a t e d the r i n g .  T h i s was  i n the i n j e c t e d  v o r t i c i t y , and  a c c o m p l i s h e d by p l a c i n g a coarse  the end of the n o z z l e .  destroy screen  Even w i t h t h i s d e v i c e , r i n g s formed  on  10  occasionally.  When t h i s happened, t h e d a t a were d i s c a r d e d .  I n o r d e r t o study t h e d e t a i l e d m i x i n g , observable  some p h y s i c a l l y  q u a n t i t y must be i n t r o d u c e d w h i c h a l t e r s when mole-  c u l a r m i x i n g o c c u r s between t h e i n j e c t e d and e n t r a i n e d The  chemical  fluid.  r e a c t i o n between aqueous s o l u t i o n s o f sodium  carbonate i n the i n j e c t e d f l u i d ,  and c a l c i u m h y d r o x i d e  i n the  tank f l u i d , appears t o f u l f i l t h e r e q u i r e m e n t s q u i t e w e l l .  The  r e a c t i o n between i o n s i n s o l u t i o n o c c u r s v e r y r a p i d l y when t h e f l u i d s a r e t h o r o u g h l y mixed (Moelwyn-Hughes, 1 9 5 7 ) .  The r a t e  o f f o r m a t i o n o f c a l c i u m c a r b o n a t e w i l l t h e r e f o r e be c o n t r o l l e d by t h e r a t e a t which p h y s i c a l m i x i n g b r i n g s t h e r e a c t a n t s intimate  into  contact. The method o f measuring t h e amount o f c a l c i u m  carbon-  ate formed, w h i c h was by measuring t h e l i g h t s c a t t e r e d from the c a l c i u m c a r b o n a t e p r e c i p i t a t e , r e q u i r e s The  justification.  tank f l u i d i s s a t u r a t e d w i t h c a l c i u m c a r b o n a t e and i s w e l l  s u p p l i e d w i t h condensation times. scatter.  T h i s i s evidenced  n u c l e i i a f t e r b e i n g used a few by t h e i n c r e a s e o f background l i g h t  Supersaturation i s not l i a b l e t o occur.  Evidence  s u p p o r t i n g t h e assumption o f a l i n e a r r e l a t i o n s h i p between volume o f p r e c i p i t a t e and l i g h t s c a t t e r e d i s a v a i l a b l e i n t h e data obtained.  I t i s g i v e n i n s e c t i o n 2 . 1 on page 2 4 .  When a d e n s i t y g r a d i e n t was r e q u i r e d , sodium c h l o r i d e was  added t o t h e f l u i d  i n t h e l o w e r p a r t o f t h e tank, as des-  cribed i n the section 1.4.  An experiment was conducted t o  show t h a t t h e p r e s e n c e o f t h e sodium c h l o r i d e does n o t a f f e c t  11  t h e f o r m a t i o n o f c a l c i u m carbonate  significantly.  The r e s u l t s  o f t h i s experiment a r e g i v e n i n s e c t i o n 2 . 1 on page 2 5 . The c l o u d o f f i n e c a l c i u m carbonate  particles  settles  s u f f i c i e n t l y s l o w l y t h a t i t s s e t t l i n g v e l o c i t y can be n e g l e c t e d compared w i t h t h e f l u i d v e l o c i t i e s . to r e s t .  The c l o u d a p p a r e n t l y comes  I t s e t t l e s o u t o r d i s p e r s e s i n a m a t t e r o f two o r  t h r e e h o u r s so t h e tank can be used a g a i n w i t h o u t renewing t h e fluid. The p o s i t i o n o f t h e p u f f i n t h e t a n k , as shown by t h e presence o f t h e p r e c i p i t a t e , was r e c o r d e d p h o t o g r a p h i c a l l y , t o g e t h e r w i t h t h e f a c e o f t h e o s c i l l o s c o p e ( P l a t e I I ) . The method used t o measure t h e d i s p l a c e m e n t  of the center o f the  p u f f from t h e n o z z l e i s g i v e n i n t h e f o l l o w i n g s e c t i o n . 1.4  Apparatus A s k e t c h o f t h e apparatus  i s shown i n F i g . 1 and a  p h o t o g r a p h i n P l a t e I . The t i t l e s on t h e f i g u r e  correspond  to t h e d e s c r i p t i o n i n t h e t e x t below o r t o t h e b r a c k e t e d expressions.  When i t i s n e c e s s a r y ,  s p e c i f i c r e f e r e n c e w i l l be  made t o , t h e photograph. I n i t i a t i o n o f t h e f l u i d p u f f was accomplished  by i n -  j e c t i n g a s m a l l volume o f f l u i d i n t o a tank a t a depth o f 3 cm. below t h e s u r f a c e . p r o v i d e d by g r a v i t y .  The p r e s s u r e r e q u i r e d f o r t h e i n j e c t i o n was A f l u i d c o n t a i n e r ( 4 0 0 m l . beaker) was  mounted a t a h e i g h t o f 65 cm. above t h e s u r f a c e o f t h e f l u i d i n t h e t a n k . ^ A s i p h o n tube dipped i n t o t h i s c o n t a i n e r and l e d v e r t i c a l l y downwards through  a f a l l tube t o t h e n o z z l e .  The  12 s m a l l c o n t a i n e r , f a l l tube, and n o z z l e system were movable v e r t i c a l l y , so t h e n o z z l e c o u l d be removed from t h e tank between experiments,  ( P l a t e I , a.)* T h i s was n e c e s s a r y t o p r e v e n t a  s l o w r e a c t i o n between t h e i n j e c t e d and tank f l u i d s w i t h a consequent f o u l i n g o f t h e n o z z l e .  W h i l e removed from t h e t a n k ,  t h e n o z z l e was p r o t e c t e d by r a i s i n g a t e s t tube, c o n t a i n i n g some o f t h e upper f l u i d ,  t o cover the n o z z l e .  This prevented  e v a p o r a t i o n a t t h e n o z z l e and l o s s o f f l u i d from d r i p p i n g . The  s c r e e n on t h e end o f t h e n o z z l e p r o v i d e d s u f f i c i e n t  support  t h a t s u r f a c e t e n s i o n would p r e v e n t any d r i p p i n g i n t h e s h o r t t i m e i n t e r v a l between removal from t h e tank and p r o t e c t i o n by the means mentioned  above.  The o n - o f f v a l v e t o s t a r t and stop t h e f l o w was a r u b b e r hemisphere h e l d by a s p r i n g a g a i n s t t h e end o f t h e s i p h o n tube which d i p p e d i n t o t h e upper c o n t a i n e r . T h i s . s i m p l e method a v o i d e d t h e use o f p a c k i n g g l a n d s . rod, to  A vertical  tappet  which s u p p o r t e d and c o n t r o l l e d t h e rubber v a l v e , l e d up a cam. - The cam was o p e r a t e d by a s m a l l motor ( P l a t e I a )  which was so a r r a n g e d t h a t when s w i t c h e d on, i t would open t h e v a l v e once o n l y , t h e n shut o f f . A number o f cams were p r o v i d e d to g i v e a range o f v a l v e - o p e n i n g t i m e s from  .1 s e c . t o  sec.  The c o n t a c t between t h e cam and t a p p e t r o d was used t o p r o v i d e t h e t i m i n g f o r an e l e c t r i c a l p u l s e from a b a t t e r y source,  ( T i m i n g C o n t a c t ) . T h i s p u l s e gave a v e r t i c a l  deflec-  t i o n on one beam o f t h e d u a l beam o s c i l l o s c o p e and t h u s an accurate value of the valve-opening time.  13  An a d j u s t a b l e v a l v e i n t h e f a l l tube p r o v i d e d a means of  volume r e g u l a t i o n independent  of the on-off valve.  I n com-  b i n a t i o n w i t h t h e v a r i o u s cams t h i s arrangement made a l i m i t e d range o f v e l o c i t y - v o l u m e c o m b i n a t i o n s The  possible.  i n j e c t e d volume was measured by means o f a  l o o s e l y - f i t t i n g p o l y t h e n e bead (volume measuring graduated g l a s s s e c t i o n o f t h e f a l l t u b e .  bead) i n a  Polythene being  s l i g h t l y l e s s dense than w a t e r , t h e bead would f l o a t s l o w l y upwards t o a c o n s t r i c t i o n i n t h e tube when t h e o n - o f f v a l v e was closed.  When t h e v a l v e opened,.the bead f e l l w i t h t h e f l u i d  and t h e g r a d u a t i o n s p e r m i t t e d a volume d e t e r m i n a t i o n a c c u r a t e to  .02 ml.  A p i e c e o f b u r e t t e tube was used f o r t h e c a l i b r a t e d  section. The n o z z l e was a s e c t i o n o f g l a s s t u b i n g . 5 4 0 cm. i n d i a m e t e r w i t h a c o a r s e s c r e e n on t h e end. The s c r e e n c o n s i s t e d of  . 0 3 3 cm. d i a m e t e r p l a s t i c w i r e spaced a p p r o x i m a t e l y 0 . 1 cm.  apart.  The c r o s s s e c t i o n a l a r e a o f t h e n o z z l e w i t h o u t t h e 2  s c r e e n was . 2 2 9 cm.  2  The s c r e e n reduced t h i s a r e a t o . 1 9 cm.  The average i n i t i a l  v e l o c i t y of the puff w  Q  was c a l -  c u l a t e d from t h e v a l v e - o p e n i n g t i m e , t h e volume measurement, and t h e n o z z l e a r e a n o t d e c r e a s e d by t h e a r e a o f the s c r e e n . This i n i t i a l  v e l o c i t y was t a k e n t o be e q u a l t o t h e maximum  v e l o c i t y a t t a i n e d assuming c o n s t a n t a c c e l e r a t i o n , t h a t i s t o : t w i c e t h e volume d i v i d e d by t h e p r o d u c t o f t h e n o z z l e the v a l v e - o p e n i n g t i m e .  area.and  To check t h a t t h i s g i v e s a r e a s o n a b l e  f i g u r e f o r t h e average i m p u t . v e l o c i t y , e s t i m a t e s o f t h e volume  14  and v e l o c i t y o f t h e moving p u f f o f f l u i d were made from movingp i c t u r e photographs.  I t was n o t p o s s i b l e t o make these volume  measurements w i t h much p r e c i s i o n , but t h e v a l u e s o f momentum d e t e r m i n e d by t h i s means, agreed, on t h e average, w i t h t h e i n i t i a l momentum c a l c u l a t e d from t h e i n i t i a l v e l o c i t y d e t e r m i n e d as s t a t e d .  volume and t h e  I t appears t h a t t h e f l u i d w h i c h  f i r s t l e a v e s t h e n o z z l e i s a c c e l e r a t e d by t h e f l u i d which f o l l o w s so t h a t t h e e n t i r e p u f f o f f l u i d a t t a i n s t h e v e l o c i t y of the l a t t e r  part.  I n o r d e r t o f a c i l i t a t e f i l l i n g t h e s i p h o n tube, w h i c h o p e r a t i o n was c o m p l i c a t e d was  drilled  by t h e volume-measuring bead, a h o l e  i n the top of the i n v e r t e d U o f the siphon.  A  tube f i t t e d w i t h a stop cock was s o l d e r e d on t o t h e copper U tube,  ( F i l l i n g vent and v a l v e ) . Removal o f a i r a t t h i s p o i n t  w i t h t h e n o z z l e immersed i n a beaker o f t h e upper f l u i d made the o p e r a t i o n q u i t e simple, although  i t took t i m e t o g e t a l l  the a i r bubbles o u t . The  l a r g e tank was o f p l a t e g l a s s , w i t h a m e t a l frame  and an open t o p ,  ( P l a t e I , d ) . The o u t s i d e dimensions were  43 cm. on each s i d e .  The o u t s i d e was p a i n t e d w i t h d u l l  p a i n t e x c e p t f o r windows on two a d j a c e n t ing  reduced t h e background l i g h t  to  This  blacken-  s c a t t e r t o a s m a l l amount  when t h e s o l u t i o n was f r e s h l y f i l t e r e d t a n k w a l l s were c l e a n .  sides.  black  and t h e i n s i d e o f t h e  The two windows were t o p e r m i t  e n t e r on one s i d e and f o r o b s e r v a t i o n a t r i g h t  light  angles.  They were made as s m a l l as was c o n s i s t e n t w i t h t h e i r p u r p o s e .  15 The c a l c i u m h y d r o x i d e s o l u t i o n i n t h e t a n k , which w i l l be desc r i b e d l a t e r , was c o v e r e d w i t h 1 cm. o f kerosene t o p r e v e n t e v a p o r a t i o n and t o p r o t e c t i t from t h e carbon d i o x i d e i n t h e air.  I t was n e c e s s a r y t o remove t h e s o l u t i o n from t h e tank  a f t e r a p e r i o d o f a few weeks, depending upon how o f t e n i t had been used, and t o c l e a n t h e w a l l s f r e e o f e n c r u s t e d c a l c i u m carbonate. acid.  T h i s was done by a d i l u t e s o l u t i o n o f s u l p h u r i c  A new s o l u t i o n o f c a l c i u m h y d r o x i d e was t h e n p r e p a r e d  and f i l t e r e d i n t o t h e t a n k . The tank f l u i d , o f which about 70 l i t r e s was r e q u i r e d , was made up by m i x i n g d i s t i l l e d water w i t h 1.28 gm./l. o r .017 m o l e / l . o f Ca(0H)2 i n a l a r g e c a r b o y .  T h i s m i x t u r e was s t i r r e d  f o r s e v e r a l h o u r s by a motor s t i r r e r and a l l o w e d t o s e t t l e f o r s e v e r a l days.  There was always s u f f i c i e n t CaCO-j formed d u r i n g  t h e h a n d l i n g o f t h i s f l u i d t o cause c l o u d i n e s s .  The s o l u t i o n  was f i l t e r e d t h r o u g h g l a s s wool w h i l e b e i n g t r a n s f e r r e d t o t h e t a n k , and covered i m m e d i a t e l y w i t h k e r o s e n e . When a d e n s i t y g r a d i e n t was r e q u i r e d i n t h e t a n k , t h e Ca(0H)2 s o l u t i o n was d i v i d e d i n t o two c a r b o y s , one c o n t a i n i n g about 45 l i t r e s and t h e o t h e r t h e r e m a i n d e r .  The f l u i d i n t h e  45 l i t r e carboy was mixed w i t h 10% by weight o f NaCl and t h o r oughly s t i r r e d f o r f o u r o r f i v e hours.  The p r e s e n c e o f NaCl  i n s o l u t i o n has a l a r g e e f f e c t upon t h e s o l u b i l i t y o f CaCO-j i n w a t e r ( R e y e l l e & F l e m i n g 1934)•  To ensure t h a t t h e m i x t u r e was  s a t u r a t e d w i t h CaCO-j, some Na C02 10H 0 i n s o l u t i o n was added 2  2  u n t i l a permanent c l o u d i n e s s appeared w i t h c o n t i n u e d s t i r r i n g .  16 When r e q u i r e d f o r use t h e s o l u t i o n was f i l t e r e d as before,  i n t o t h e tank  and the u n s a l t e d Ca(0H)2 s o l u t i o n added slowly, so  as to form a two l a y e r system.  A nearly l i n e a r density  d i e n t o f about .003 gm. cm7^ was produced by s t i r r i n g r e g i o n o f the i n t e r f a c e .  i n the  The measurement o f t h e d e n s i t y  d i e n t w i l l be d e s c r i b e d l a t e r .  gra-  gra-  The l i n e a r p o r t i o n c o u l d be  made s u f f i c i e n t l y e x t e n s i v e t o cover a depth from about 2 cm. below t h e s u r f a c e t o 12 cm. below the s u r f a c e . a l l o f t h e range covered by the experimental When t h e i n t e r f a c e was being appeared i n the tank.  This  included  puffs.  s t i r r e d , a cloudiness  T h i s has been a t t r i b u t e d t o t h e d e t a i l s  of s o l u b i l i t y o f CaCQ-j i n water c o n t a i n i n g v a r y i n g amounts o f NaCl.  T h i s c l o u d would s e t t l e out i n two days l e a v i n g the  f l u i d clear. Measurement o f the d e n s i t y g r a d i e n t without d i s t u r b i n g the f l u i d i n the tank was accomplished by means o f small g l a s s f l o a t s approximately  .5 cm. i n diameter.  Twelve o f these were  made with d e n s i t y c o v e r i n g the range i n the tank. put  into a t a l l  coarse  They were  cage with a g l a s s f r o n t and back, and with  p l a s t i c screen  sides.  The cage was s e t on the bottom  of t h e tank near one o f the windows so t h e f l o a t s could be observed from o u t s i d e . permitted ship.  Depth markings on one o f t h e g l a s s f a c e s  v i s u a l determination  o f the d e n s i t y - d e p t h  The g l a s s f l o a t s were blown from s o f t g l a s s  relationtubing.  They were c a l i b r a t e d by p l a c i n g them i n d i s t i l l e d water,  titra-  t i n g i n a 20% NaCl s o l u t i o n with constant  each  stirring until  17  f l o a t s e p a r a t e l y became n e u t r a l l y buoyant.  The temperature  was h e l d a t 20°C. which was a p p r o x i m a t e l y t h e temperature o f the  room and t a n k .  The d e n s i t y o f t h e m i x t u r e was determined  from t a b l e s (Handbook o f C h e m i s t r y and P h y s i c s ) and checked w i t h a s e n s i t i v e hydrometer.  The p h y s i c a l v a r i a t i o n s o f s i z e  and shape i n t h e d i f f e r e n t f l o a t s made i t p o s s i b l e t o d i s t i n g u i s h them from one a n o t h e r so each c o u l d be r e c o g n i z e d and assigned i t s proper density.  A f t e r b e i n g i n use f o r some t i m e ,  CaCO^ d e p o s i t e d on s e v e r a l o f t h e f l o a t s as a s c e r t a i n e d by sudden u n r e a l i s t i c v a r i a t i o n s o f t h e apparent d e n s i t y g r a d i e n t . The cage was t h e n l i f t e d o u t o f t h e t a n k , t h e f l o a t s c l e a n e d w i t h d i l u t e HgSO^ and r e - c a l i b r a t e d . The f l u i d i n t h e upper c o n t a i n e r , which was t h e f l u i d i n j e c t e d i n t o t h e t a n k , c o n t a i n e d 10. g . / l o r .035 m o l e / l . o f NagCO^lOHgO. i n d i s t i l l e d w a t e r .  I n terms o f t h e r e a c t i o n t o  form CaCO-j, t h i s f l u i d was t w i c e as c o n c e n t r a t e d as t h e tank fluid.  I t was made up i n 2 l i t r e l o t s .  stratified  F o r use i n t h e d e n s i t y  t a n k , a number o f b a t c h e s were made w i t h an i n c r e a s e d  d e n s i t y by a d d i n g NaCl and measuring t h e d e n s i t y w i t h a s e n s i t i v e hydrometer.  As f a r as p r a c t i c a l t h e d e n s i t y o f t h e i n -  j e c t e d f l u i d was k e p t c l o s e t o t h e d e n s i t y o f t h e tank a t t h e end o f t h e n o z z l e .  I t was always s l i g h t l y l i g h t e r , t o p r e v e n t  i n t e r c h a n g e o f t h e f l u i d s b e f o r e t h e i n j e c t i o n v a l v e opened. The heat d e v e l o p e d by t h e r e a c t i o n i n t h e minimum p o s s i b l e volume o f tank f l u i d c o u l d cause a d e n s i t y change o f 1 x 1 0 " ^ gm/cm?.  I n p r a c t i c e i t would be c o n s i d e r a b l y l e s s  18 t h a n t h i s s i n c e the volume r a t i o was always l a r g e r t h a n 2:1 before completion o f the r e a c t i o n .  D e n s i t y changes o f t h i s  s i z e were s m a l l e r , by an o r d e r o f magnitude,  t h a n c o u l d be  measured by the system o f f l o a t s , and have been n e g l e c t e d . The c e n t e r o f t h e tank was i l l u m i n a t e d through one o f t h e windows p r e v i o u s l y mentioned. lamp was mounted 100  A 300 w a t t i n c a n d e s c e n t  cm. from t h e c e n t e r o f the t a n k , ( P l a t e I c )  s u f f i c i e n t l y f a r t o produce r e a s o n a b l y u n i f o r m i l l u m i n a t i o n over t h e p o r t i o n o f the tank i n which t h e r e a c t i o n o c c u r r e d . T e s t s showed t h a t t h e i l l u m i n a t i o n was w i t h i n 10% o f i t s v a l u e a t t h e c e n t e r , throughout a c y l i n d e r 5 cm. i n r a d i u s e x t e n d i n g f r o m the top t o ; t h e bottom o f the t a n k .  As o n l y about h a l f o f  t h i s volume was used i n t h e experiment, the v a r i a t i o n i n l i g h t i n t e n s i t y was p r o b a b l y l e s s t h a n 5% from t h a t a t the c e n t e r . To e x c l u d e o t h e r l i g h t from t h e system, a d u l l b l a c k plywood  s h i e l d covered e v e r y t h i n g except the upper f l u i d  t a i n e r and the f a l l t u b e .  con-  The tank cover had a s m a l l h o l e  above the tank c e n t e r so t h e n o z z l e c o u l d be l e t i n t o t h e f l u i d when r e q u i r e d . A G e n e r a l E l e c t r i c PV3 p h o t o v o l t a i c c e l l was used t o measure t h e s c a t t e r e d l i g h t .  With a l o a d r e s i s t a n c e of  100  ohms, the f r e q u e n c y response shows a drop o f about 3fo a t cycles/sec,  1000  ( Z w o r y k i n , 1949)• T h i s l i n e a r i t y i s more t h a n  adequate f o r t h e purpose s i n c e t h e minimum r i s e time i s about .8 seconds.  The l i n e a r i t y o f response t o i l l u m i n a t i o n  was  checked by d e t e r m i n i n g t h e increment above t h e background  due  19 to a s i n g l e s p h e r i c a l translucent  bead (.5  cm.  d i a m e t e r ) at a  number o f v a l u e s o f i n c i d e n t i l l u m i n a t i o n c o v e r i n g used i n the e x p e r i m e n t .  The  D.C.  t o 1000  The  times.  o u t p u t o f the a m p l i f i e r was  double-beam o s c i l l o s c o p e was  t o d i s p l a y the v a l v e - t i m i n g ( P l a t e I d)»  The  f e d t o one  of  oscilloscope.  A DuMont t y p e 333  c a l l y against  a m p l i f i e d by a K i n  a m p l i f i e r g i v i n g a m p l i f i c a t i o n i n 5 steps  the v e r t i c a l i n p u t s of the  output,  i n a random manner.  o u t p u t o f t h e p h o t o c e l l was  T e l model 111BF  range  r a t i o , incremental output/back-  ground o u t p u t , v a r i e d by 10% The  the  p u l s e and  used  the  photocell-amplifier  sweep speed was  calibrated periodi-  the l i n e f r e q u e n c y which i s c l o s e l y c o n t r o l l e d  i n t h i s g e o g r a p h i c a l a r e a as i t i s s e r v e d by a l a r g e connected g r i d .  The  inter-  sweep speed of the o s c i l l o s c o p e was  t o r e q u i r e no adjustment and has  found  been t a k e n as c o r r e c t .  It  was  chosen t o d i s p l a y the v a l v e t i m i n g p u l s e on f r o m 1/4  to  3/4  o f the f a c e and  v a r i e d from .25  s e c / i n . t o .10  To ensure t h a t the v a l v e - t i m i n g on a s i n g l e s c a n , the scope was u n t i l the v a l v e c l o s e d .  .1 sec. b e f o r e the v a l v e c l o s e d , the scope was  p u l s e was  used on e x t e r n a l  A p u l s e , p r o v i d e d by an  c o n t a c t on the v a l v e - o p e r a t i n g  sec./in. displayed  sweep c o n t r o l additional  cam,, t r i g g e r e d the sweep about  s t a r t e d t o open.  When the  valve  s w i t c h e d m a n u a l l y t o a u t o m a t i c sweep t o  show the p h o t o c e l l o u t p u t and  provide a c a l i b r a t i o n f o r  the  camera f i l m speed. The  camera was  a B o l e x H. 16  s t a n d a r d model w i t h  an  20  f : 1 . 8 l e n s of f o c a l l e n g t h 2.5 at  cm.  (Plate l b ) .  I t was mounted  75 cm. from t h e c e n t e r o f the tank i n s i d e a l i g h t - t i g h t r e -  movable s h i e l d .  The f i l m speed was s e t a t 16 frames p e r second.  The e x a c t speed was d e t e r m i n e d f r o m t h e d i s p l a c e m e n t o f t h e spot on t h e o s c i l l o s c o p e and t h e known sweep speed.  A c t u a l camera  f i l m speeds v a r i e d from 18 t o 22 frames p e r second. t i o n d u r i n g any one e x p e r i m e n t was s m a l l .  The  varia-  Eastman P l u s X  P a n c h r o m a t i c n e g a t i v e f i l m was used a t f : 1 . 8 . Two p l a n e m i r r o r s were used t o r e f l e c t t h e f a c e o f the  o s c i l l o s c o p e i n t o the f i e l d o f t h e camera.  i n t h e diagram ( F i g . i n P l a t e I a.  1).  These a r e shown  The t i p s o f t h e m i r r o r s a r e v i s i b l e  Two m i r r o r s were used t o p e r m i t t h e p u f f t o  occupy t h e c e n t e r o f the f i e l d .  The image o f t h e o s c i l l o s c o p e  f a c e was s p l i t and o c c u p i e d two a d j a c e n t c o r n e r s o f t h e f i e l d , t h i s b e i n g t h e most c o n v e n i e n t p h y s i c a l  arrangement.  P l a t e I I shows a r e p r e s e n t a t i v e s e r i e s o f p h o t o g r a p h s . The upper s t r i p tank f l u i d .  (a) shows t h e p r o g r e s s o f a p u f f i n a u n i f o r m  The l o w e r s t r i p  d e n s i t y g r a d i e n t o f . 0 0 3 gm.  (b) shows a s i m i l a r p u f f when a cmT^" e x i s t e d i n t h e t a n k .  The  i n i t i a l c o n d i t i o n s o f volume and v e l o c i t y were n e a r l y the same i n each c a s e , b e i n g ,'78 cm3 s e r i e s and . 8 3 cm^  and 3 2 .  and 3 1 .  cm./sec. f o r t h e upper  cm./sec. f o r t h e l o w e r s e r i e s .  E x p e r i m e n t s were c a r r i e d out w i t h volumes r a n g i n g 3  3  f r o m . 8 cm. t o . 0 6 5 cm. tial ble.  F o r t h e l a r g e r volumes, average  ini-  v e l o c i t i e s o f from 9«4 cm./sec. t o 55 cm./sec. were p o s s i F o r t h e s m a l l e s t volume, an average i n i t i a l v e l o c i t y o f  21 9.4 cm./sec. was the g r e a t e s t t h a t c o u l d be o b t a i n e d from the apparatus.  The a c c e l e r a t i o n provided by g r a v i t y would not per-  mit h i g h e r average v e l o c i t i e s a t t h i s volume u n l e s s the apparatus were c o n s i d e r a b l y m o d i f i e d by p u t t i n g i n l a r g e r tubes throughout,  while keeping  the n o z z l e diameter  diameter the same  as b e f o r e . Up t o t e n repeat runs were made w i t h the same i n i t i a l c o n d i t i o n s , to permit an estimate o f t h e standard d e v i a t i o n . To o b t a i n t h e p o s i t i o n o f the c e n t e r o f the p u f f s from t h e f i l m r e c o r d , an enlarged image o f t h e p u f f and the t i p o f the n o z z l e was p r o j e c t e d onto a s c r e e n .  To convert  from d i s t a n c e s on t h e e n l a r g e d image to d i s t a n c e s i n the tank, a photograph was taken o f a graduated i n the tank f l u i d  v e r t i c a l s c a l e immersed  a t t h e c e n t e r o f the tank.  From t h e enlarged  image o f t h i s -scale, a g r i d was made up r e a d i n g d i r e c t l y i n cm. of  a c t u a l d i s t a n c e i n the tank.  When a d e n s i t y g r a d i e n t was  used i n the tank, f r e q u e n t photographs were taken o f the s c a l e to  compensate f o r the d i s t o r t i o n due to t h e unequal r e f r a c t i o n  of  the f l u i d . The c e n t e r s o f the p u f f s were estimated v i s u a l l y and  t h e i r displacement  from the t i p o f t h e n o z z l e recorded.  The  accuracy o f e s t i m a t i o n o f t h e c e n t e r was checked on f o u r d i s s i m i l a r p u f f s by scanning the image o f the p u f f and t h e background w i t h a p h o t o c e l l . for  a square  opening  The p h o t o c e l l was screened o f f except  2 mm. on each s i d e .  A graduated  step ex-  posure was p l a c e d on i n d i v i d u a l f i l m s u s i n g the camera s h u t t e r  22  s t o p s t o c o n t r o l t h e exposure.  For t h i s o p e r a t i o n , a white  c a r d was p l a c e d t e m p o r a r i l y i n the f i e l d o f the camera, and i l l u m i n a t e d by t h e i n c a n d e s c e n t b u l b shown i n t h e f o r e g r o u n d of P l a t e I a .  U s i n g t h e graduated exposure a t a b l e was  drawn  up f o r t h e r e l a t i v e response o f t h e p h o t o c e l l as a f u n c t i o n o f f i l m exposure.  The c e n t e r s o f t h e p u f f s were c a l c u l a t e d as a  h o r i z o n t a l l i n e about which the f i r s t moment o f t h e was  zero.  I n the f o u r cases t e s t e d , t h i s agreed w i t h t h e  v i s u a l e s t i m a t e t o w i t h i n 3 mm. ( T a b l e 1,  exposure  page 22).  The tendency  o f depth i n the t a n k , appeared  the d i s p l a c e m e n t s l i g h t l y t o o l a r g e .  t o be t o e s t i m a t e  T h i s may  have been due  t o s y s t e m a t i c d i f f e r e n c e s i n t h e appearance of the t o p and bottom o f the p u f f .  E r r o r s i n v e l o c i t i e s c a l c u l a t e d from t h e  e s t i m a t e d d i s p l a c e m e n t s may the displacements T a b l e 1.  t h u s be even l e s s than e r r o r s i n  themselves.  Comparison o f v i s u a l e s t i m a t e s o f d i s p l a c e m e n t w i t h  calculated values. Calculated  Estimated  4.6  cm.  4*7  cm.  5.1  cm.  5.3  cm.  3.8  cm.  4*1  cm.  .4.8  cm.  5.0  cm.  23 2  DATA OBTAINED  2.1  Detailed Mixing The use o f t h e o u t p u t o f t h e p h o t o c e l l as a means o f  c a l c u l a t i n g the d e t a i l e d p h y s i c a l mixing, r e q u i r e s j u s t i f i c a tion.  The q u a n t i t y o f l i g h t , s c a t t e r e d from t h e p r e c i p i t a t e ,  w i l l depend upon t h e i n c i d e n t i l l u m i n a t i o n , and upon t h e det a i l s of the s c a t t e r i n g process.  I t w i l l be p r o p o r t i o n a l t o  t h e volume o f p r e c i p i t a t e formed i f t h e f o l l o w i n g c o n d i t i o n s are met: a.  The i n c i d e n t i l l u m i n a t i o n i s c o n s t a n t .  b.  The p a r t i c l e s a r e l a r g e w i t h r e s p e c t t o t h e maximum wavelength of incident l i g h t .  c.  The d i s t r i b u t i o n o f p a r t i c l e s i z e i s u n i f o r m  so t h a t t h e  r a t i o o f s u r f a c e a r e a t o volume i s c o n s t a n t . d.  The p a r t i c l e c o n c e n t r a t i o n i s l o w so t h a t s i n g l e s c a t t e r i n g o n l y , need be c o n s i d e r e d . To show t h a t t h e s e c o n d i t i o n s a r e met  c o n s i d e r two s e r i e s o f r e s u l t s , 3  approximately,  (Table 2, page 2/j.)» I n one,  the i n i t i a l volume was .40 cnr; and t h e i n i t i a l v e l o c i t y 9.4 cm./sec.  I n t h e o t h e r t h e i n i t i a l volume was .80 cm2 and t h e  i n i t i a l v e l o c i t y 34• cm./sec.  The time f o r t h e p h o t o c e l l out-  put t o r e a c h a maximum was 7 s e c . i n t h e f i r s t c a s e , and 1 s e c . i n t h e second c a s e .  The two s e t s o f d a t a were t a k e n consecu-  t i v e l y over a p e r i o d o f s i x days.  Relative to the f i r s t set,  an a m p l i f i e r g a i n l o w e r by a f a c t o r 3/5 was used f o r t h e second s e t t o compensate f o r t h e i n c r e a s e i n volume i n j e c t e d .  24 s i g n i f i e s t h e maximum increment o f p h o t o c e l l - a m p l i f i e r o u t p u t above t h e background T a b l e 2.  value.  P r o p o r t i o n a l i t y between i n p u t volume and p h o t o c e l l  response. QjJ 3.8 3.9 3.9 , 4.05 3.0* 3.2*  Vol.  QM/VOI.  .42 .43 .42 .50 .33 .37  9.0 9.1 9.3  3.S 4.0* 3.5 3.9' 3.S 3.6  e.i  9.1 8.7  Vol.  QjJ/Vol.  Q j A o l . x 5/3  .88 .74 .77 .79 .80 .78  4.3 5.4 4.5 4.9 4.8 4.6  7.2 9.0 7.5 8.2 8.0 7.7  av. S.89 - .39 s t d . dev.  av. 7-93  t  .57 s t d . dev.  The two s e t s o f d a t a r e p r e s e n t q u i t e d i f f e r e n t  initial  c o n d i t i o n s and d i f f e r e n t m i x i n g p r o c e s s e s as i s shown by t h e f a c t t h a t t h e time t o r e a c h between t h e two s e t s .  d i f f e r s by a f a c t o r o f seven  The constancy a c h i e v e d i n O^/Vol.x g a i n  i s n o t as h i g h as c o u l d be d e s i r e d but i s not u n r e a s o n a b l e s i d e r i n g t h e n a t u r e o f t h e experiment.  con-  The r e d u c t i o n i n t h e  second s e t w i t h r e f e r e n c e t o the f i r s t , may be p a r t i a l l y  attri-  buted t o t h e i n c r e a s e o f d e p o s i t s on t h e windows w i t h a consequent r e d u c t i o n o f l i g h t  intensity.  The r e l a t i v e p r o g r e s s o f t h e r e a c t i o n i s computed as t h e r a t i o Q =r oVOn — l t o f o u t p u t over /^M ; Maximum increment n  c  r  e  m  e  n  background,  w h i c h compensates f o r v a r i a t i o n s i n i l l u m i n a t i o n and changes i n amplifier gain. Q i s p l o t t e d as a f u n c t i o n o f time i n F i g . 2.  For  * These d a t a were n o t used i n p l o t t i n g F i g . 2 . T h e i r volume and v e l o c i t y do not match c o r r e s p o n d i n g runs made i n a s t r a t i f i e d f l u i d .  s i m i l a r i n i t i a l c o n d i t i o n s o f volume and v e l o c i t y , t h e d a t a f o r a uniform surrounding f l u i d ,  and f o r one w i t h a d e n s i t y g r a d i -  ent, are p l o t t e d together. I n F i g . 3, Q i s p l o t t e d as a f u n c t i o n o f t f o r two experiments which had t h e same i n i t i a l c o n d i t i o n s and no density gradient.  I n one however t h e s o l u t i o n s c o n t a i n e d no s a l t ,  w h i l e i n t h e o t h e r both upper and l o w e r s o l u t i o n s c o n t a i n e d about 6% sodium c h l o r i d e mixed t h o r o u g h l y t o a u n i f o r m d e n s i t y of  1.038 gm/cm^  From t h e f i g u r e , i t i s apparent t h a t t h e s a l t  does n o t a f f e c t t h e f o r m a t i o n o f c a l c i u m c a r b o n a t e t o any appre ciable extent. 2.k2.  Center  Displacement  The d i s p l a c e m e n t  z o f t h e c e n t e r o f t h e p u f f as a  f u n c t i o n o f t i m e i s shown i n F i g . 4 .  I t r e q u i r e s no e x p l a n a -  t i o n except t o say t h a t d i s t a n c e s were measured from t h e p o i n t of  i n j e c t i o n i n a l l c a s e s , and t h e p o s i t i o n s o f t h e c e n t e r s  were determined  as d e s c r i b e d i n s e c t i o n 1 . 4 «  Again f o r s i m i l a r  i n i t i a l c o n d i t i o n s o f volume and v e l o c i t y , t h e d a t a f o r a u n i form s u r r o u n d i n g f l u i d , and f o r one w i t h a d e n s i t y g r a d i e n t , are p l o t t e d t o g e t h e r . 2.3  H o r i z o n t a l Spreading The h o r i z o n t a l spread o f t h e p u f f i s most m e a n i n g f u l  when p l o t t e d as a f u n c t i o n o f v e r t i c a l d i s p l a c e m e n t .  For a  u n i f o r m s u r r o u n d i n g f l u i d , t h e envelope o f s u c c e s s i v e o u t l i n e s of  t h e p u f f d e f i n e s a cone.  t a i n e d by s u p e r i m p o s i n g  The o u t l i n e o f t h e cone was ob-  s u c c e s s i v e exposures  and o b s e r v i n g t h e  26  The h a l f angle of t h e cone v a r i e d from 1 5 ° -to -21?  extremeties. and  showed no s y s t e m a t i c  v e l o c i t y of the p u f f .  dependence upon the i n i t i a l volume o r  An example i s shown i n F i g . 5 .  The  s u c c e s s i v e o u t l i n e s o f t h e p u f f a r e n o t t a k e n f o r equal i n c r e ments o f t i m e . F o r t h e cases where a d e n s i t y g r a d i e n t was t h i s method c o u l d n o t be used s i n c e t h e v e r t i c a l i n c r e a s e s t o a maximum and t h e n d e c r e a s e s .  present,  displacement  The t o t a l  horizon-  t a l w i d t h o f t h e p u f f , as d e f i n e d by t h e e x t r e m i t i e s o f t h e p r e c i p i t a t e , was measured on an e n l a r g e d converted  image.  This distance,  t o a c t u a l d i s t a n c e i n the tank, i s c a l l e d 2 r , since  a f a i r degree o f c y l i n d r i c a l symmetry e x i s t s .  The c o r r e s p o n d -  i n g v e r t i c a l d i s p l a c e m e n t o f the c e n t e r z, measured as p r e v i o u s l y described  (Sec. 1 . 4 ) ,  was t h e n p l o t t e d a g a i n s t r .  For given  v a l u e s o f i n i t i a l volume, v e l o c i t y , and d e n s i t y g r a d i e n t , a l l p o i n t s were p l o t t e d on one s h e e t .  The r e s u l t s a r e shown i n  F i g . 6 where t h e o u t l i n e o n l y , o f t h e e x t r e m e t i e s  of t h e e x p e r i -  m e n t a l p o i n t s i s shown.  3. 3.1  ANALYSIS IN NEUTRALLY BUOYANT FLUID D e t a i l e d M i x i n g i n Uniform Surroundings To o b t a i n a q u a n t i t y r e p r e s e n t a t i v e of t h e average  d e t a i l e d m i x i n g , f r e e from the e f f e c t s o f c h e m i c a l  concentra-  t i o n s and g r o s s p h y s i c a l e n t r a i n m e n t , t h e f o l l o w i n g concepts are r e q u i r e d : (a)  The t o t a l momentum o f t h e p u f f i s c o n s t a n t  and e q u a l t o  p V w , t h e i n i t i a l d e n s i t y , volume, and v e l o c i t y o f t h e 0  0  0  puff. (b)  The t o t a l f i n a l volume o f p r e c i p i t a t e formed i s p r o p o r . t i o n a l t o t h e i n i t i a l volume o f f l u i d  injected,  «4~v ). 0  (c)  The i n s t a n t a n e o u s volume o f p r e c i p i t a t e i s g i v e n by Q and the r e m a i n i n g u n r e a c t e d f l u i d by  (d)  -  l  chemical i n the i n j e c t e d  Q. !  From (a) the volume o f f l u i d e n t r a i n e d w i l l be V ( w 0  0  -1)  since the density i s constant. The o r i g i n a l amount o f c h e m i c a l i n t h e e n t r a i n e d ( i n terms o f Q ) w i l l be p r o p o r t i o n a l t o QJJJ/2 ( w / w - 1 )  fluid  ra  0  s i n c e , i n terms o f t h e r e a c t i o n , t h e e n t r a i n e d f l u i d i s o n l y h a l f as c o n c e n t r a t e d as t h e tank f l u i d .  The r e m a i n i n g a c t i v e  m a t e r i a l i n t h e e n t r a i n e d f l u i d w i l l t h e n be p r o p o r t i o n a l t o Qm/2( o/ -1) ~ w  w  Q• 1  I f we l e t A r e p r e s e n t t h e i n j e c t e d f l u i d and B t h e e n t r a i n e d f l u i d , we can w r i t e f o r the r e a c t i o n r a t e : dQ'/dt-k*(Total  volume) ( C o n c e n t r a t i o n of A) ( C o n c e n t r a t i o n o f B)  where k ' i s a f u n c t i o n which r e p r e s e n t s t h e average s t a t e o f m i x i n g between two f l u i d s .  T h i s can be w r i t t e n :  dQ'/cit =k ( a c t i v e m a t e r i a l i n A) ( A c t i v e m a t e r i a l i n B) . T o t a l Volume T  (l,a)  28  P u t t i n g i n the values, dQ'/dt = k ( Q ^ - Q ) !  1  /9M/2  ( V W - 1)  t  V  D i v i d e by Qjyj, l e t Q ' / Q M " »  0  - Q' ]  I  w /w 0  substitute V  Q  0  ( ) ljb  f o r 0^  a b s o r b i n g a l l c o n s t a n t s o f p r o p o r t i o n a l i t y i n k, t o o b t a i n , dQ/dt = k ( l - Q ) [ l - w / w ( l - 2 Q ) ] () 0  2  With t h e e x c e p t i o n o f k, t h e q u a n t i t i e s i n e x p r e s s i o n (2) can be o b t a i n e d from t h e e x p e r i m e n t a l curves and from the slopes of the curves.  The s l o p e s a r e s u b j e c t t o c o n s i d e r a b l e  At the p o i n t Q =  error.  the dependence on w drops out and  dQ/dt i s observed t o v a r y s l o w l y . k can be o b t a i n e d i n t h i s range.  A f a i r l y accurate value of For a s i n g l e s e t o f i n i t i a l  c o n d i t i o n s , k i s found t o be n e a r l y t h e same a t o t h e r v a l u e s o f Q.  An example i s g i v e n i n T a b l e 3«  T a b l e 3.  M i x i n g f u n c t i o n k c a l c u l a t e d using the slopes of  the m i x i n g c u r v e s . I n i t i a l conditions:  V  0  .42  =  cm3  w =32. 0  cm ./sec.  Time (sec.)  .3  .4  .5  .6  .7  .8  Q  .52  .76  .38  .94  .98  1.0  .90  .45  .25  0.0  .083  .068  .053  0.0  dQ/dt w/w  0  k  3.1 .16  6.6  1.7 .11  6.8  7.0  7.7  12.  The l a s t two v a l u e s o f k, 7.7 and 12., a r e determined f o r Q = .94 and .98 r e s p e c t i v e l y .  I n expression (2), the  f a c t o r ( 1 - Q ) i s s e n s i t i v e t o s m a l l e r r o r s i n Q when Q i s near t o 1.  The c a l c u l a t e d v a l u e o f k i n t h i s range, must be r e -  garded w i t h s u s p i c i o n . I f we assume k t o be n e a r l y c o n s t a n t , e x p r e s s i o n (2) can be i n t e g r a t e d n u m e r i c a l l y i n the form Q = k ^*(1-Q) f l - w/w  Q  Equation  (1-2Q)J  dt.  (3)  (3) i s not so s e n s i t i v e t o s m a l l e r r o r s i n  Q and does n o t make use o f the s l o p e s o f t h e Q-t  curves.  These s l o p e s a r e i n a c c u r a t e , b e i n g o b t a i n e d by n u m e r i c a l differentiation. S i n c e w i s not known i n t h e e a r l y p a r t o f t h e range, b e f o r e two o b s e r v a t i o n s have been t a k e n , the v a l u e o f the i n t e g r a l i s known o n l y above a c o n s t a n t .  This constant  be found by assuming the v a l u e of k determined  could  above a t  Q = .5 t o be c o r r e c t , o r a l t e r n a t i v e l y t o d e t e r m i n e ,  f o r two  d i f f e r e n t t i m e s , what c o n s t a n t would make k have i d e n t i c a l values.  S i n c e t h i s i s , t o some e x t e n t , a d i s t i n c t  t i o n from the p r e v i o u s one, i t has been used.  calcula-  Calculations  are shown i n Table 4, page 30 f o r the same d a t a as used previously.  (Table; 3, page 28)  (3) i s c a l l e d B.  C signifies  The i n t e g r a n d . i n e x p r e s s i o n ; J B d t , s i n c e t = .3 i s t h e  e a r l i e s t time f o r which w, and t h e r e f o r e B, i s known.  30 C a l c u l a t i o n of mixing f u n c t i o n k u s i n g an  Table 4.  integral  e x p r e s s i o n f o r Q. Time (sec.)  .3  .4  .5  .6  •7  B  .  .48  .25  .128  .058  .021  / Bdt  c = .081  C+.036 = .117  .52  .76  Q  k = Q/yldt  c .52  two  p o i n t s at t = . 3  .081.  C+.068 = .149  .94  .98  6.5  6.45  and t = . 6 ,  = c » . 0 6 4 from which c — ..94  C+.O64 = .145  .88  was c a l c u l a t e d by l e t t i n g c + ^ B d t  c Choosing  6.5  6.5  C+.055 = .136  this  =  6.55 const.  yields  Over the range where Q  and w are a v a i l a b l e , the computed value of k i n Table 4 v a r i e s by l e s s than 1% from i t s value at Q = . 5 . of  initial  at  Q = .5  For the o t h e r s e t s  c o n d i t i o n s , k v a r i e d by l e s s than 1 0 $ from i n each  i t s value  case.  The average  v a l u e s of k computed by t h i s method are  l i s t e d i n Table 5 with the corresponding i n i t i a l c o n d i t i o n s . Table 5 .  Mixing function k f o r various i n i t i a l conditions. v  o  k  32. 9.5 55. 9.4 9.6 32. 32.  6.5 3-3 12.3 .86 .95 4.5 5.8  w 0  .42 .065 .82 .80 .42 .81 .63  31  From e q u a t i o n  (1)  i t i s apparent t h a t k has dimen-  sions t ^ since Q i s dimensionless. of the i n i t i a l  S i n c e k i s some f u n c t i o n  c o n d i t i o n s and p o s s i b l y t h e k i n e m a t i c  viscosity^  i t may be p o s s i b l e t o determine what f u n c t i o n . A l e a s t squares a p p r o x i m a t i o n o f k, V  Q  and w  0  done on t h e l o g a r i t h m s  y i e l d s the expression, k = m w l ' 5 3 V-»55  where m i s a c o n s t a n t . upon w  (4)  T h i s g i v e s t h e approximate dependence  and TI . Q  Q  A d i m e n s i o n a l l y c o r r e c t e x p r e s s i o n between k, l / , V and w is  Q J  c o n s i s t e n t w i t h t h e o b s e r v a t i o n s , can be o b t a i n e d i f V  0  i n t e r p r e t e d as a l e n g t h i n s t e a d o f a volume.  0  S i n c e o n l y one  n o z z l e d i a m e t e r was used, volume and l e n g t h a r e p r o p o r t i o n a l . I n t h i s i n t e r p r e t a t i o n , t h e r e l a t i o n s h i p k = m i/~^w^/ V~2 has 2  t h e c o r r e c t d i m e n s i o n s and f i t s  the observations  reasonably  be shown l a t e r .  I n F i g . 7, k i s p l o t t e d against  and t h e c o n s t a n t mlT"  i s found t o be . 0 2 3 . P u t t i n g i n  w e l l , as w i l l  2  the r e l a t i o n s h i p between volume and l e n g t h , we o b t a i n k = . 0 0 5 0 »~^\?J\~£  where L  Q  (5)  i s the i n i t i a l l e n g t h o f the i n j e c t e d c y l i n d e r . A simple g r a p h i c a l method o f f i n d i n g an approximate  power l a w r e l a t i o n s h i p between 3 v a r i a b l e s , i s g i v e n i n Appendix (4).  1.  T h i s method was used i n i t i a l l y  t o determine  equation  By use o f t h i s method, F i g . 8 shows t h e agreement o f t h e  data with equation  (4).  t h e d a t a w i t h an e q u a t i o n  F i g . 8 a l s o shows t h e agreement o f s i m i l a r t o (4) b u t w i t h i n d i c e s 3 / 2  32  and - J i n s t e a d o f 1.53  and  -.55.  The f u n c t i o n k i s a measure o f t h e average s t a t e o f m i x i n g which e x i s t s i n s i d e the p u f f . g r o s s entrainment  I t does not i n v o l v e  or concentration e f f e c t s .  I t must be r e -  garded as an i m p e r i c a l r e l a t i o n s h i p which s e r v e s t o e x p r e s s t h e e x p e r i m e n t a l m i x i n g d a t a i n a compact form. The a c t i o n of t u r b u l e n t motions i n m i x i n g two ble  f l u i d s can be thought of as a s t r e t c h i n g of t h e  initial  i n t e r f a c e t o produce a g r e a t e r i n t e r f a c i a l a r e a through m o l e c u l a r d i f f u s i o n can p r o c e e d .  misci-  which  M o l e c u l a r d i f f u s i o n tends t o  reduce c o n c e n t r a t i o n g r a d i e n t s a t the i n t e r f a c e w h i l e t u r b u l e n t s t r e t c h i n g tends t o sharpen t h e s e g r a d i e n t s and t o i n c r e a s e t h e i n t e r f a c i a l area. The  e x p e r i m e n t a l evidence t h a t k i s r e l a t i v e l y  con-  s t a n t o v e r t h e range i n which o b s e r v a t i o n s have been t a k e n , i n d i c a t e s t h a t , i n t h i s type of f l o w , t h e i n i t i a l c o n d i t i o n s determine  a r e l a t i o n s h i p between the i n t e r f a c i a l a r e a per u n i t  volume, and the g r a d i e n t o f t r a n s i t i o n from one f l u i d t o t h e other.  This r e l a t i o n s h i p thereafter i s r e l a t i v e l y constant.  The v a r i a t i o n s i n c h e m i c a l r e a c t i o n r a t e a r e determined t o t a l volume o c c u p i e d by the two f l u i d s , and the  by the  undepleted  a c t i v e m a t e r i a l s i n t h i s volume. 3•  2  V e l o c i t y o f a B u f f i n Uniform F l u i d o f t h e Same D e n s i t y . The s u p e r p o s i t i o n o f s u c c e s s i v e o u t l i n e s o f a p u f f ,  p e n e t r a t i n g i n t o a u n i f o r m f l u i d o f t h e same d e n s i t y , i n d i c a t e s t h a t the p u f f i s c o n t a i n e d i n a cone o f h a l f angle  approximately  33 20 d e g r e e s ,  ( F i g . 5)- The v a l u e s found v a r i e d from 15 degrees  t o 21 degrees w i t h an average v a l u e o f 19 d e g r e e s .  The v e r t e x  o f t h e cone i s l o c a t e d a t an i n d e f i n i t e d i s t a n c e from t h e t i p of the nozzle.  No s y s t e m a t i c  o f t h e cone angle,  dependence o f t h i s d i s t a n c e , o r  upon t h e i n i t i a l  c o n d i t i o n s was d e t e r m i n e d .  I f t h e shape o f t h e p u f f remains s i m i l a r , which i s roughly  t r u e , the f a c t t h a t i t s o u t l i n e i s c o n t a i n e d  i n a cone  i m p l i e s t h a t t h e t u r b u l e n t v e l o c i t i e s i n s i d e t h e p u f f must be p r o p o r t i o n a l to the v e l o c i t y of the center o f the puff with respect to the surroundings.  T h i s i s so s i n c e t h e t u r b u l e n t  v e l o c i t i e s a r e r e s p o n s i b l e f o r , and p r o p o r t i o n a l t o , t h e r a t e of  spreading. I f a l l v e l o c i t i e s a r e p r o p o r t i o n a l t o one a n o t h e r ,  and remain so, the f l o w i s o f the t y p e c a l l e d s e l f - p r e s e r v i n g , and  i s independent o f t h e Reynold's number p r o v i d i n g t h i s i s  sufficiently large.  F o r a s e l f - p r e s e r v i n g f l o w , the form o f  the s e l f - p r e s e r v i n g f u n c t i o n s a r e u n i v e r s a l f o r any one t y p e of flow,  (Townsend, 1956). I t f o l l o w s t h a t a n o n - d i m e n s i o n a l  p l o t o f any two v a r i a b l e s s h o u l d flows of a given  show the same form f o r a l l  type.  I n F i g . 10 a p l o t o f z/V^ f o r a l l the data obtained.  3  a g a i n s t t;w /V^  The c o n s i d e r a b l e  3  i s shown  scatter i s a t t r i -  buted t o two f a c t o r s : a.  The f l o w s a r e n o t s e l f - p r e s e r v i n g a t the o r i g i n  and become so o n l y a f t e r an i n t e r v a l o f time and d i s t a n c e which are n o t r e l a t e d i n the same manner as t h e y a r e a f t e r s e l f -  34 p r e s e r v a t i o n becomes e s t a b l i s h e d . b.  The  initial  values V  the  initial  c o n d i t i o n s and  has  become s e l f - p r e s e r v i n g . I f we  and w  0  s e p a r a t e l y r e l a t e to  Q  not n e c e s s a r i l y to the f l o w a f t e r i t  assume s e l f - p r e s e r v a t i o n , a dimensional argument  y i e l d s an e x p r e s s i o n  f o r the displacement and  v e l o c i t y of a  p u f f as a f u n c t i o n of time measured from a v i r t u a l o r i g i n where z = Q,  and w = o# at t = 0 . Since the d e n s i t y of the  as that of the p u f f , no  surrounding f l u i d  i s the same  buoyant f o r c e s are developed, and  the  v e r t i c a l momentum o f the p u f f , which i n c l u d e s the i n j e c t e d and entrained f l u i d ,  i s a constant. Vw  Ir  Dimensionally where L can be any choice of c^. virtual  Then  const i  i = c t , or L = c-^ t *, 7  l e n g t h t y p i c a l of the p u f f depending on  I n t e r p r e t i n g L as the displacement from  z = c  2  t  5  3  and w = c ^ t  The l i n e drawn on F i g . 10  as (Z*) ^ = 1 0 0 ( t !  t ' are dimensionless displacement and  to the o r d i n a t e the  the  origin, 1  z' and  the  and  a b s c i s s a of F i g . 1 0 ,  2.5)  where  time corresponding  i n d i c a t e s the t r e n d  of  data. I t may  be i n s t r u c t i v e to c o n s i d e r  the problem from the  p o i n t of view o f the t u r b u l e n t energy d e n s i t y E.  We  require,  i n a d d i t i o n to the concept of s e l f - p r e s e r v a t i o n , the r a t e of turbulent  energy d i s s i p a t i o n dE/dt.  In a s e l f - p r e s e r v i n g flow,  t h i s i s p r o p o r t i o n a l t o j S ^ / , where L i s a s c a l e l e n g t h 2  L  of the turbulence, therefore p r o p o r t i o n a l to V  1/3  /  typical  (Townsend,  v  }  1956).  The r a t e o f p r o d u c t i o n o f t u r b u l e n t energy p e r u n i t mass w i l l be equal t o t h e average r a t e a t which a u n i t mass o f the p u f f l o s e s energy, t h a t i s , t o  - 4- ( i w ). dt  The t u r b u l e n t energy e q u a t i o n it where B  d t  U  w  •)  becomes, p:/3  i s a p r o p o r t i o n a l i t y constant.  { 6 )  Since  turbulent  v e l o c i t i e s a r e p r o p o r t i o n a l t o t h e mean c e n t e r  velocity,  2  E «>cw. From momentum c o n s e r v a t i o n , y - l / 3 = w  1/3  (Vo> P u t t i n g i n these  l / 3  values, dw  2  _  2  -  dv/  A  dt" "  A  dt  _  10/3  R  0  [7*  w  U  1  where A and B a r e c o n s t a n t s . Subject t o w = w w" ^ 4  3  Q  at t z t  0  , the s o l u t i o n i s ,  r c t + (W34/3 - c t )  (8)  Q  where c i s a c o n s t a n t . The v a l u e s o f w o b t a i n e d n u m e r i c a l l y from t h e d i s p l a c e ment c u r v e s , F i g . 4 , i f p l o t t e d i n t h e form \f^^  vs t , show  t h i s s t r a i g h t l i n e tendency except f o r a r e g i o n where t i s small.  An example i s g i v e n i n F i g . 1 1 . The d i s t a n c e from t h e n o z z l e a t w h i c h t h e e x p e r i m e n t a l  36  p o i n t s s t a r t t o f o l l o w t h e r e l a t i o n g i v e n , i s l a r g e l y cont r o l l e d by t h e i n i t i a l l e n g t h o f t h e i n j e c t e d f l u i d c y l i n d e r , i f t h i s length i s l a r g e with respect to the r a d i u s . t h i s c r i t i c a l d i s t a n c e and L injected cylinder, z  c  8.1.  0  = f ( R ) L , where f ( R ) v a r i e s from 1 t o 2 , 0  F o r one c a s e ,  was n e a r l y equal t o t h e r a d i u s , t h e v a l u e o f f ( R ) was  I t appears t h a t i n t h i s c a s e , t h e p r o c e s s o f adjustment  to s i m i l a r i t y conditions i s a l t e r e d .  The v a l u e s o f c i n equa-  t i o n (8) a r e l i s t e d i n Table 6 , page 3 9 initial 3.3  c  i s the o r i g i n a l l e n g t h of the  0  increasing with increasing i n i t i a l v e l o c i t y . where L  If z is  }  f o r the v a r i o u s  conditions.  D i s p l a c e m e n t o f a P u f f i n U n i f o r m F l u i d o f t h e Same D e n s i t y From t h e l a s t s e c t i o n , ( S e c . 3 « 2 ) , the d i s p l a c e m e n t o f  a p u f f f o l l o w s t h e r e l a t i o n s h i p z<*t^ i f z and t a r e measured from t h e c o r r e c t o r i g i n . f i n e d by t h e cone o b t a i n e d of the puff.  T h i s o r i g i n i s n o t always w e l l deby s u p e r i m p o s i n g s u c c e s s i v e p o s i t i o n s  I t can be d e t e r m i n e d more p r e c i s e l y by u s i n g two  nearby d i s p l a c e m e n t v a l u e s o c c u r r i n g a f t e r s i m i l a r i t y  conditions  are e s t a b l i s h e d , t h a t i s a f t e r t h e v e l o c i t y b e g i n s t o f o l l o w t h e r e l a t i o n wT^^oc t .  Consider where z  b  - z  a  two d i s p l a c e m e n t s ,  z  a  at t  &  and z^ a t t ^ ,  = S , and i s s m a l l w i t h r e s p e c t t o z  a  and a l s o  w i t h r e s p e c t t o t h e d i s p l a c e m e n t from t h e v i r t u a l o r i g i n .  If  we r e p r e s e n t d i s p l a c e m e n t and time measured from t h e v i r t u a l o r i g i n by z" and t* , t h e n z£ - z * = S , and  37 From t h e v i r t u a l  z  ,4  L  - l£. t, .  origin,  w;~ '  = ct* ,  and,  Now,  «{* - 4  4  4 <*b - t*)  »  (  a  9  )  2  Expand t h e l e f t z  b  -  z  a  * *  hand s i d e , d i s c a r d a t e r m i n  , and s i m p l i f y t o  or approximately  ^  The c o n s t a n t  are known.  + Jij3 -  JL  .  (10)  c c a n be o b t a i n e d f r o m t h e = et  slope of  the  and 6- and  •*• c o n s t ,  H e n c e z ^ c a n be a p p r o x i m a t e d .  v a l u e c a n be c h e c k e d i n (9)  let  obtain,  c o r r e s p o n d i n g v e l o c i t y c u r v e , vT^^ £  & ,  This  approximate  and s m a l l c o r r e c t i o n s made  if  necessary. The d i s p l a c e m e n t o f t h e v i r t u a l o f t h e n o z z l e i s g i v e n by z  a  -  z* .  c a l c u l a t e d f r o m the d a t a i n F i g . 4, - . 5 $ c m . t o 5»6 c m . c o n d i t i o n s c o u l d be  No s y s t e m a t i c  o r i g i n from the  These d i s p l a c e m e n t s  flows  origin,  (Hinze,  dependence on the  i n the present  case,  initial  observed.  i s a feature  1959,  were  and f o u n d t o v a r y f r o m  T h i s l a c k of system w i t h respect a virtual  tip  which often occurs i n t u r b u l e n t  page 2 1 6 ) . to the f a c t  the apparatus presents  to the p o s i t i o n of  T h i s phenomenon may be d u e , that  several lengths  in initiating  the  and v e l o c i t i e s ,  c o n s e q u e n t l y many R e y n o l d ' s n u m b e r s w h i c h become  flow, and  critical  at  38 d i f f e r e n t mean v e l o c i t i e s . i o u r of the p u f f f o r  T h i s may i n f l u e n c e t h e  a short d i s t a n c e .  s c a l e e f f e c t s have had t i m e to d e c a y , puff w i l l  be c o n t r o l l e d by t h e  A sample p l o t o f  since  time i s  straight  w  Q  (not  the  conditions i n this  s 9«5  cm./sec.  are not reached of  shown)  It  l i n e p o r t i o n of the  case were V  case,  12.  t i m e h a s b e e n made  for  Q  small values of  r .065  that  at l e a s t  dependence  z  vs t  t i o n s has been o b s e r v e d . curves  since  the  The  t. ini-  crn3, and  similarity conditions not  d u r i n g the  of the  c u r v e upon t h e  are  time  initial  straight condi-  velocity  related.  constant  c i n equations  a r e l i s t e d i n T a b l e 6, page 39,  ponding i n i t i a l  slope of the  The same h o l d s f o r t h e  two s l o p e s  The v a l u e s o f t h e  *  shown i n F i g .  observation* No s y s t e m a t i c  (10)  of i n i t i a t i o n .  equation i s not obeyed.  is possible  in this  is  the  The p o i n t s f o l l o w a  l i n e reasonably w e l l except  •In one s e t tial  t  o r i g i n of  involved l i n e a r l y .  small  the behaviour of  gross features  against  No c o r r e c t i o n f o r a v i r t u a l  A f t e r the  behav-  together  (8)  w i t h the  and corres-  conditions.  T h i s i s t h e same s e t w h i c h showed a n o m a l o u s b e h a v i o u r w i t h r e s p e c t t o t h e v e l o c i t y l a w ( E q u a t i o n 8 , page 3 5 ) . This s e t h a d a R e y n o l d ' s number (w Vo^/i>») e q u a l t o 3 8 0 , t h e lowest used. F o r o t h e r r u n s , R e y n o l d ' s numbers r a n g e d f r o m 870 t o 5 1 0 0 . 0  39 T a b l e 6.  V a l u e s o f s l o p e c o n s t a n t i n v e l o c i t y and d i s p l a c e -  ment c u r v e s . V  4  Q  cm?  w  -4/3' s e 1/3 c cm. c/  cm./sec.  Q  .42  32.  .82  55.  .82 2.5  .80  9.4  .42  9.6  .81  32.  .63  32.  .68 1.3 1.8 .90  ANALYSIS IN STABLY STRATIFIED SURROUNDINGS  4.1  Motion Across a Density D i s c o n t i n u i t y A volume V  Q  o f f l u i d o f d e n s i t y P , i f i n j e c t e d downQ  wards i n t o a u n i f o r m f l u i d o f g r e a t e r d e n s i t y P w i l l e x p e r i z  ence a c o n s t a n t buoyant f o r c e e q u a l t o V ( P 0  Z  - P ) g , as can Q  be seen by t h e f o l l o w i n g : I n c r e a s e i n mass = mass e n t r a i n e d , d(?V) = P  or  dV ; f d V + V d ? = P dV ;  Subject t o V = V i  S  v(p,-«  Q  dV/V = d P / ( p - f )  when P = ? , t h e s o l u t i o n o f t h i s Q  = ye-e). z  o  The buoyant f o r c e V ( f - P ) g i s t h e r e f o r e g i v e n by z  V (P 0  Z  - P ) g and i s c o n s t a n t . 0  This i s understandable  since  any e n t r a i n e d f l u i d c o n t r i b u t e s no buoyant f o r c e whatever. The time f o r such an i n j e c t e d p u f f t o become s t a t i o n a r y i s r e a d i l y c a l c u l a b l e by s e t t i n g t h e momentum l o s s  40 e q u a l t o t h e i n i t i a l momentum t o o b t a i n  * = (e w )/(p -p )g . 0  0  z  0  To o b t a i n the depth o f p e n e t r a t i o n would r e q u i r e a knowledge o f t h e entrainment r a t e . 4.2  Motion i n a Uniform S t r a t i f i c a t i o n Where t h e s u r r o u n d i n g  f l u i d has a d e n s i t y which i n -  c r e a s e s w i t h d e p t h , t h e buoyant f o r c e i s a l s o a f u n c t i o n o f the entrainment r a t e .  The a d d i t i o n a l n o t a t i o n used i n t h i s  s e c t i o n i s g i v e n below: P  instantaneous  mean d e n s i t y o f t h e p u f f  Z  instantaneous  depth o f t h e p u f f .  p  0  P P  z 0  m  mean d e n s i t y o f t h e p u f f a t t h e p o i n t o f i n j e c t i o n Z . Q  d e n s i t y of surroundings  at the point of i n j e c t i o n Z . Q  mean d e n s i t y o f t h e p u f f a t t h e p o i n t o f maximum penetration Z . m  P  d e n s i t y o f surroundings  Q z  d e n s i t y o f t h e soundings a t depth Z.  Pon  = P^on  at Z .  d e n s i t y o f p u f f and s u r r o u n d i n g s  a t the p o i n t  where t h e p u f f e v e n t u a l l y comes t o r e s t Z . eq A l t h o u g h \i.n r e c o r d i n g t h e d a t a ( F i g . 4), Z  was t a k e n  Q  as z e r o , and d i s p l a c e m e n t s measured f r o m t h i s p o i n t , i t i s c o n v e n i e n t i n t h i s s e c t i o n t o r e f e r t o d i s p l a c e m e n t from t h e p o i n t o f i n j e c t i o n as Z - Z . 0  The  changes which a f l u i d p u f f undergoes when i n j e c t -  ed v e r t i c a l l y downwards i n t o a s i m i l a r f l u i d w i t h d e n s i t y s t r a t i f i c a t i o n a r e as f o l l o w s :  Assuming P  Q  = P » z 0  t  n  e  volume  41  i n c r e a s e s by e n t r a i n m e n t o f s u r r o u n d i n g ,  more dense,  w i t h r e s u l t i n g i n c r e a s e o f average d e n s i t y . energy d e c r e a s e s from l o s s e s due  The  fluid  kinetic  t o e n t r a i n m e n t , and  work done a g a i n s t the buoyant f o r c e s .  The  from  p o t e n t i a l energy  o f t h e p u f f , zero a t the s t a r t , i n c r e a s e s , s i n c e i t i s necess a r i l y p o s i t i v e a t the p o i n t of maximum p e n e t r a t i o n due t h e d e n s i t y d i f f e r e n c e between the p u f f and  its  to  surroundings.  A f t e r coming t o r e s t , the p u f f r i s e s under buoyant f o r c e s , u s u a l l y o v e r s h o o t s the e q u i l i b r i u m p o s i t i o n , and comes t o r e s t a t some p o i n t below the p o i n t of The  experimental  displacements,  eventually  injection.  measured from the  p o i n t o f i n j e c t i o n , are p l o t t e d a g a i n s t t i m e i n F i g . 4 f o r the v a r i o u s i n i t i a l  conditions.  From the measured d i s p l a c e m e n t s , instantaneous  v e l o c i t y can be made.  rounding f l u i d  The  of  d e n s i t y of the  the sur-  i n c r e a s e s n e a r l y l i n e a r l y w i t h depth a t r a t e s  v a r y i n g from .0025 t o .0031 and  an e s t i m a t e  gm./cm^ ; : . I n i t i a l v a l u e s o f volume  v e l o c i t y are known. From the d a t a i t i s p o s s i b l e , i n p r i n c i p l e , to d e t e r -  mine t h e volume and d e n s i t y a t a l l t i m e s by n u m e r i c a l  methods  as f o l l o w s : Decrease o f momentum = mean buoyant f o r c e x A  (P  v n  w n  n  )  = ~^z*£~*n  J  V  n  s < A t  »  w  h  e  r  e  P  z  +  i  i s  t h e  m  time, e  d e n s i t y o f the s u r r o u n d i n g s d u r i n g t h e t i m e i n t e r v a l From t h i s we  n  ZVt.  can o b t a i n t h e momentum a t the n e x t o b s e r v e d  p o i n t (PVw) 2_. n+  a  From t h i s o b t a i n the mass at n+1  by d i v i d i n g  42  by t h e o b s e r v e d v e l o c i t y - w ^ .  The volume o f f l u i d  & V i s equal t o (mass change/mean s u r r o u n d i n g  entrained  density).  Then  T h i s p r o c e s s can be i t e r a t e d u s i n g mean v a l u e s o f v o l ume and d e n s i t y i n t h e i n t e r v a l , t o c a l c u l a t e t h e momentum loss. I n p r a c t i c e , t h e method does not work w e l l because t h e momentum i s n o t known p r e c i s e l y a t any one t i m e .  The time  o f i n j e c t i o n i s an a p p r e c i a b l e p o r t i o n o f t h e t i m e f o r t h e whole p r o c e s s .  Some entrainment o f f l u i d o c c u r s b e f o r e t h e  p u f f has a t t a i n e d i t s maximum v e l o c i t y , hence t h e i n i t i a l d i t i o n s a r e n o t known e x a c t l y .  con-  S i m i l a r o b j e c t i o n s apply t o  w o r k i n g backwards from t h e p o i n t o f zero momentum at-Z  . The  p r e c i s e t i m e a t which w i s zero cannot be d e t e r m i n e d .  Several  t r i a l c a l c u l a t i o n s were made but proved so s e n s i t i v e t o s m a l l changes i n i n i t i a l  c o n d i t i o n s t h a t no r e l i a n c e c o u l d be  p l a c e d upon them. I n t h e l a c k o f d e t a i l e d i n f o r m a t i o n on  instantaneous  volume and d e n s i t y , t h e f o l l o w i n g s e c t i o n g i v e s a method o f c a l c u l a t i n g t h e minimum v a l u e o f t h e p o t e n t i a l energy a t Z . m  4.3  P o t e n t i a l E n e r g y - o f a P u f f a t Maximum P e n e t r a t i o n To o b t a i n a minimum e s t i m a t e o f t h e p o t e n t i a l energy  of a puff at Z (a)-  m  t h e f o l l o w i n g assumptions a r e made:  Assume t h e d e n s i t y o f t h e p u f f a t Z  m  i s equal t o the  d e n s i t y o f t h e f l u i d a t t h e l e v e l where t h e p u f f e v e n t u a l l y  43  comes t o r e s t (P for  = P q)•  m  T h i s must g i v e a maximum v a l u e  z e  t h e d e n s i t y o f the p u f f a t Z  s i n c e the g r e a t e r p a r t o f  m  the e n s u i n g p a t h i s t h r o u g h f l u i d o f g r e a t e r d e n s i t y t h a n P q , and the p u f f must end up w i t h a d e n s i t y o f P ' q . z e  ze  (b)  Assume the volume o f t h e p u f f a t Z  m i x i n g o f the i n i t i a l volume V  i s d e t e r m i n e d from  m  at d e n s i t y P  Q  the maximum d e n s i t y e n c o u n t e r e d , t h a t i s P  z m  with f l u i d  Q  .  of  This y i e l d s  the s m a l l e s t volume o f the p u f f and t h e r e f o r e the  smallest  p o t e n t i a l energy c o n s i s t e n t w i t h assumption ( a ) . To examine whether assumption (a) g i v e s a minimum v a l u e f o r p o t e n t i a l energy, c o n s i d e r t h e p o t e n t i a l t h e d e n s i t y o f the p u f f at Z be g r e a t e r ) . at Z q e  s  ?za> ^  constant.  &  (P -P )  Vg  /  ?z  ?za • (  s  z  z  - > z  be  Z.  p o t e n t i a l energy i s g i v e n .zm ""za  m  z e  I n t h i s case the e q u i l i b r i u m p o i n t w i l l not  PE  e  i s l e s s t h a n P q ( i t cannot  m  but a t some h i g h e r p o i n t The  energy,if  m  d e  a  /  d Z  by, dZ  >  s  (11)  i  n  c  e  d e  /  d Z  i s  Making t h e s e s u b s t i t u t i o n s , and e v a l u a t i n g t h e  i n t e g r a l , we  obtain, PE = i Vg  (Z  m  - Z )  2  a  dP/dZ.  To o b t a i n an e q u a t i o n f o r PE i n terms of P  (12) m  and  known q u a n t i t i e s , we r e q u i r e the r e l a t i o n between depth  and  density, z  m  " a z  =  (?zm " ^m^ dP/dZ  ' 1 1 3 }  44  and a l s o an e x p r e s s i o n f o r V determined from assumption ( b ) , V  «m = o o V  P  +  Pzm <  V  " V '  f  r  o  m  W  h  i  c  h  Tp—TTT * *zm m v  v  S u b s t i t u t i n g (13) and (14)  i n t o (12) we o b t a i n a f t e r  simplification, P E  = o V  ( P m  Pp) ( f i m - P J « ' 2 dP/dZ • •'  (15)  I t i s apparent t h a t t h e minimum v a l u e o f PE w i l l be o b t a i n e d when ? observations,  has t h e g r e a t e s t v a l u e c o n s i s t e n t w i t h t h e  t h a t i s when P  m  = P q. e  The computations were made u s i n g t h e o b s e r v a t i o n s a s i n g l e p u f f , and n o t t h e averaged v a l u e s o f s e v e r a l v a t i o n s as shown i n F i g . 4 .  on  obser-  T h i s a p p l i e s t o a l l t h e compu-  t a t i o n s i n t h i s s e c t i o n where the c r i t i c a l p o i n t s were used, and accounts f o r the f a c t t h a t t h e number o f p l o t t e d p o i n t s i n F i g s . 13, 15, 1 7 , c u r v e s i n F i g . 4«  and 18, i s g r e a t e r t h a n t h e number o f  •  The s c a t t e r i n t h e i n d i v i d u a l o b s e r v a t i o n s  as i n d i c a t e d  i n F i g . 4 , i s o n l y p a r t l y due t o v a r i a t i o n s i n i n p u t c o n d i t i o n s and e r r o r s i n o b s e r v a t i o n s .  Even i n cases where t h e  i n p u t c o n d i t i o n s were n e a r l y i d e n t i c a l as f a r as c o u l d be determined,  c o n s i d e r a b l e v a r i a t i o n s i n d i s p l a c e m e n t were n o t e d .  The p r o g r e s s  o f t h e f l u i d p u f f appears t o be i n f l u e n c e d t o  some c o n s i d e r a b l e e x t e n t by c o n d i t i o n s i n the f l u i d w h i c h are  45  beyond immediate c o n t r o l , and t h e o b s e r v a t i o n s are v a l i d o n l y in  a s t a t i s t i c a l sense.  S i n c e , i n any r e a l case, t h e s e  small  p e r t u r b a t i o n s w i l l be p r e s e n t , the r e s u l t i n g s c a t t e r must be regarded  as an e s s e n t i a l p a r t o f the o b s e r v a t i o n s . C a l c u l a t i o n s have been made f r o m the d a t a u s i n g equa-  t i o n (15) w i t h P  substituted for P . m  eci  T h i s minimum v a l u e o f  p o t e n t i a l energy a t maximum p e n e t r a t i o n had been compared w i t h the i n i t i a l  energy a t t h e p o i n t o f i n j e c t i o n ( t h e sum o f i n i -  t i a l k i n e t i c and p o t e n t i a l energy when t h e l a t t e r was n o t z e r o ) , t o form the  ratio  p o t e n t i a l energy a t maximum p e n e t r a t i o n i n i t i a l t o t a l energy * In  o r d e r t o show the r e s u l t s i n the most g e n e r a l  manner p o s s i b l e , a d i m e n s i o n l e s s initial flow.  r a t i o i s made up from t h e  c o n d i t i o n s w h i c h might be expected  to influence the  These a r e P , w , g, dP/dz, and V . The k i n e m a t i c Q  Q  v i s c o s i t y has been o m i t t e d s i n c e the Reynold's number i s s u f f i c i e n t l y h i g h t h a t i t i s not expected A suitable dimensionless  t o a f f e c t the f l o w . 2 is P w , which Q ' P, 2/3 g dP/dz  combination  i n v o l v e s a comparison between k i n e t i c f o r c e s and buoyant forces.  T h i s d i m e n s i o n l e s s form has a resemblance t o t h e  r e c i p r o c a l o f the R i c h a r d s o n ' s the replacement o f ( d u / d z ) to  2  number, d i f f e r i n g from i t by  by w^/V / , and w i l l be r e f e r r e d 2  3  as l / R j _ . All  d i m e n s i o n l e s s r a t i o s o b t a i n e d are p l o t t e d as a  f u n c t i o n o f the r e c i p r o c a l o f t h i s R i c h a r d s o n ' s  number, and  46  w i l l v a r y w i t h i t i n the same g e n e r a l manner as i f i t were the u s u a l R i c h a r d s o n ' s  number.  g i v e n i n F i g s . 13  o f l/R±  The  a c t u a l numerical  values  and f o l l o w i n g f i g u r e s , bear no r e -  l a t i o n t o the v a l u e s o f r e c i p r o c a l R i c h a r d s o n ' s  number which  might be found i n p r a c t i c e . F i g . 13,  •In  the r a t i o o f the p o t e n t i a l energy a t  maximum p e n e t r a t i o n t o the i n i t i a l t o t a l energy i s p l o t t e d against l/Rj_.  The p o i n t s a r e d i s t i n g u i s h e d as t o t h e  volume, and, when p l o t t e d i n t h i s manner, show no o f the r a t i o w i t h i n i t i a l  initial  variation  volume.  The r e s u l t s i n d i c a t e t h a t as much as 20 p e r c e n t  of  the i n i t i a l energy appears as p o t e n t i a l energy f o r s m a l l v a l u e s o f l/R-j_.  F o r h i g h e r v a l u e s o f l / R ^ , the r a t i o  de-  c r e a s e s r a p i d l y , and approaches zero f o r l / R j _ a p p r o x i m a t e l y equal to  1000. For an i n i t i a l v e l o c i t y o f 54 cm./sec. the p u f f showed  no p e r c e p t i b l e r i s e a f t e r r e a c h i n g the p o i n t o f maximum penetration.  At t h i s v e l o c i t y , the p u f f d i s s i p a t e s r a p i d l y ,  cannot be observed  for long.  and  Since the e q u i l i b r i u m p o s i t i o n  i s not known, no v a l u e can be a s s i g n e d t o the energy r a t i o i n t h i s case. small value.  I t cannot be z e r o , however, but must possess some The d e n s i t y g r a d i e n t s used were a l l v e r y n e a r l y  the same a t about .003 gm./crn^ so the p r i n c i p a l p a r t of t h e change i n R^ i s due t o the v a r i a t i o n s i n i n i t i a l v e l o c i t y volume.  and  47  4.4  E f f e c t o f a P u f f Upon the S u r r o u n d i n g F l u i d R e g a r d i n g the a l t e r a t i o n o f the d e n s i t y s t r u c t u r e o f  t h e ambient f l u i d by a p e n e t r a t i n g p u f f , t h e i n f o r m a t i o n s h o u l d l i k e t o o b t a i n i s , (a) what p a r t o f the i n i t i a l i s d i s s i p a t e d i n a l t e r i n g t h e d e n s i t y s t r u c t u r e , and w i t h r e s p e c t t o t h e p o i n t o f o r i g i n , does t h i s  we  energy (b) where,  alteration  occur. E x a c t c a l c u l a t i o n of the e f f e c t o f a p u f f on the  sur-  r o u n d i n g s i s not p o s s i b l e s i n c e the d e t a i l s of e n t r a i n m e n t a l o n g the p a t h cannot be made w i t h any p r e c i s i o n . mate v a l u e f o r the energy l o s t t o the d e n s i t y  An  approxi-  stratification,  and f o r the l o c a t i o n o f t h i s d e n s i t y a l t e r a t i o n , can be o b t a i n ed u s i n g the same assumptions as were used b e f o r e , t h a t i s , t h a t the e q u i l i b r i u m d e n s i t y i s a c h i e v e d from the path extremeties  at Z  In t h i s approximation, i n g s i s as f o l l o w s . depth Z , 0  and  p n  C  H  f l u i d above Z q e  Q  The  moved from Z . m  Z. m  Z q. e  and  fluid  Z. m  t h e e f f e c t upon the  A q u a n t i t y of f l u i d V  Q  surround-  i s removed from  mixed w i t h s u f f i c i e n t f l u i d from depth Z  a density P .  from Z .  Q  by m i x i n g o f  the t o t a l volume d e p o s i t e d *  t o produce  m  at Z  .  The  eq  w i l l r i s e s l i g h t l y to r e p l a c e t h a t removed  f l u i d below Z ^ e(  w i l l f a l l to replace that r e -  A sharp g r a d i e n t w i l l be produced a t Z  0  and  at  A s h o r t r e g i o n o f u n i f o r m d e n s i t y w i l l be produced at F i g . 14 i s an exaggerated i l l u s t r a t i o n of t h e  altera-  t i o n s i n d e n s i t y s t r a t i f i c a t i o n produced by such a p u f f .  The  s o l i d l i n e r e p r e s e n t s the o r i g i n a l l i n e a r d e n s i t y g r a d i e n t .  48  The  c r o s s hatched a r e a s r e p r e s e n t , i n exaggerated  a l t e r a t i o n s produced  form, t h e  by t h e p u f f , under t h e assumption o f  m i x i n g a t t h e maximum depth o n l y .  The dashed l i n e r e p r e s e n t s  t h e p r o b a b l e a l t e r a t i o n s i n d e n s i t y s t r u c t u r e which would be produced  assuming e n t r a i n m e n t  along the e n t i r e path.  In  a c t u a l i t y , s i n c e t h e tank i s l a r g e w i t h r e s p e c t t o t h e p u f f , the d i s c o n t i n u i t i e s a t Z  and Z  Q  a r e v e r y s m a l l , as i s t h e  m  uniform p o r t i o n at Z q . e  R e f e r r i n g t o F i g . 1 4 , t h e a r e a s A and B a r e , e q u a l s i n c e t h e a r e a under t h e curve must remain unchanged.  This  can be e x p r e s s e d as f o l l o w s : (z  = ^ m - V  - o> z  e q  APa A?b  =  Z  Z  '  >  OR  m - eq eq - o 2  (16)  :Z  The r a t i o (16) i s a l s o e q u a l t o V Q / A V, where AY. i s the volume o f f l u i d removed from Z . m  T h i s can be seen by  use o f e q u a t i o n ( 1 4 ) , page 4 4 , and some a l g e b r a , s e t t i n g e  m  equal t o ?  e q  .  The r a t i o i n ( 1 6 ) , which r e p r e s e n t s t h e i n i t i a l ume d i v i d e d by t h e volume e n t r a i n e d a t Z  m  vol-  from assumption (b)  Sec. 4 * 3 , has been c a l c u l a t e d f o r t h e v a r i o u s i n i t i a l c o n d i t i o n s and i s shown g r a p h i c a l l y as a f u n c t i o n o f l / R j _ i n F i g . 15.  The r a t i o  (16) has an approximate  value of 2 f o r l / R ^  e q u a l t o 3 0 , and drops t o l e s s t h a n 1 f o r  equal t o 5 0 0 .  F o r 1/RJL e q u a l t o 1 0 0 0 . which corresponds t o an i n i t i a l  49  v e l o c i t y o f 5 5 . cm./sec. and i n i t i a l volume o f .80 cm2, p u f f was n o t observed  to r i s e .  r a t i o i s therefore zero. r i s e and a s m a l l f i n i t e Fig.  (Fig. 4 c)  the  The c a l c u l a t e d  A c t u a l l y t h e r e must be some s m a l l (16).  value f o r the r a t i o  16 i s a p l o t o f t h e e x p e r i m e n t a l v a l u e s o f m a x i -  mum p e n e t r a t i o n Z  and e q u i l i b r i u m depth, Z q , o b t a i n e d from  m  e  the same d a t a as were used t o produce F i g . 4 . are p l o t t e d against i n i t i a l v e l o c i t y .  The depths  No s y s t e m a t i c depend-  ence o f t h e s e d e p t h s upon t h e i n i t i a l volume was o b s e r v a b l e . F i g s . 15 and 16 a r e r e l a t e d t h r o u g h e q u a t i o n 1 6 , page 4 8 . The work done upon the s u r r o u n d i n g f l u i d c u l a t e d i n t h i s s i m p l i f i e d p i c t u r e as f o l l o w s : Fig.  can be c a l Referring to  1 4 , t h e work done w i l l be e q u a l t o t h e work r e q u i r e d t o  r e s t o r e the d e n s i t y g r a d i e n t t o i t s o r i g i n a l l i n e a r  form.  T h i s w i l l be t h e work r e q u i r e d t o c a r r y t h e product o f d e n s i t y d e f e c t and volume r e p r e s e n t e d by r e g i o n B t o t h e p o s i t i o n o f r e g i o n A.  This product  i s independent o f t h e s i z e o f t h e  t a n k , p r o v i d e d t h e tank i s s u f f i c i e n t l y l a r g e . case, A P  A  In the actual  and AR-^ w i l l be s m a l l , and t h e volume l a r g e .  uniform density region at Z  The  w i l l be n e g l i g i b l y small, w i t h  e q  respect t o the other d i s t a n c e s i n v o l v e d .  Let  represent the  work r e q u i r e d , t h e n E  ?  = (tank c r o s s s e c t i o n ) &P / eq - o Z  V  Z  2  •  Z  a  (  z e  q-  z 0  ) x  m " e q \ SZ  2  /  C o n s i d e r a u n i t c r o s s s e c t i o n a l tank a r e a and u n i t  (17)  50 volume i n j e c t e d . to  The f l u i d above Z  r e p l a c e the d i s p l a c e d volume.  becomes, E = df/dZ ( Z p  e q  From e q u a t i o n  - Z ) Q  (Z  m  e q  must r i s e u n i t d i s t a n c e  Then  = d /dZ and (17)  - Z )-jg , p e r u n i t V . Q  (18)  Q  (18) t h e v a l u e s o f E have been c a l c u -  l a t e d from t h e o r i g i n a l d a t a .  The r a t i o s E f / E , where E 0  Q  is  t h e i n i t i a l t o t a l energy o f the p u f f p e r u n i t volume, were computed and the r e s u l t s a r e p l o t t e d i n F i g . 17 as a f u n c t i o n of the r e c i p r o c a l Richardson's  number.  The v a r i a t i o n i n t h i s r a t i o w i t h l/R-j_ i s s m a l l .  At  l o w v a l u e s o f l / R ^ , where t h e p e n e t r a t i o n s a r e s m a l l and r e l a t i v e e r r o r s i n measurement  a r e p r o p o r t i o n a t e l y l a r g e , the  p o i n t s are badly s c a t t e r e d .  A t h i g h e r v a l u e s o f l/R-j_ the  p o i n t s a r e w e l l grouped near t o a v a l u e o f .03.  As a r e p r e -  s e n t a t i v e f i g u r e , we c o u l d say t h a t a p p r o x i m a t e l y  3 percent  o f t h e i n i t i a l energy i s used t o a l t e r the d e n s i t y c a t i o n , f o r t h e range o f R i c h a r d s o n ' s  stratifi-  number covered  i n the  experiment. 4.5  Transfer C o e f f i c i e n t S t r i c t l y speaking,  a t r a n s f e r c o e f f i c i e n t has l i t t l e  meaning f o r t h e s i n g l e d i s p l a c e d p u f f o r f l u i d .  Nevertheless,  c o m p u t a t i o n s can be made of the t r a n s f e r o f d e n s i t y d i f f e r ence d u r i n g t h e l i f e t i m e o f t h e p u f f , and t h e r e s u l t s o f these c o m p u t a t i o n s can be compared f o r the v a r i o u s i n i t i a l  condi-  t i o n s o f volume and v e l o c i t y . As has been shown i n t h e p r e c e d i n g  s e c t i o n s , and  51  making t h e same assumptions extremity Z  o n l y , the r e s u l t o f the p u f f i s t h e f o l l o w i n g .  m  A volume o f f l u i d V f l u i d at Z  i s removed from p o s i t i o n Z , mixed w i t h  Q  Q  t o make a t o t a l volume V, and the whole i s de-  m  posited at Z q. Q  to  as b e f o r e o f m i x i n g a t the p a t h  T h i s amounts t o moving a volume YQ from  Z q , and a volume A V from Z e  to  m  z e q  .  Z  Q  The product o f mass  d i f f e r e n c e and d i s t a n c e t r a n s p o r t e d w i l l be g i v e n by AM L = V  0  (f 0  - f  e q  )  (Z  e q  - Z ) • AV ( P 0  e q  - PJ  (Z -Z m  e q  ). (19)  By use o f e q u a t i o n 16, page 48, and t h e f a c t t h a t the d e n s i t y g r a d i e n t i s l i n e a r ,  Afii °  =  -  n eq- c-) z  z  *  Z  eq- o Z  ( m- eq) l Z  Z  2  ^m" eq  || • (20)  L  I f we l e t for  2  (19) can be w r i t t e n  be a n e t d e n s i t y t r a n s f e r  coefficient  t h i s p r o c e s s , such t h a t AM L = -K„ df V  then,  (21)  " o)  ^  d Z  K  H = ( e q - o> Z  Z  <m Z  Z  t 2 where KJJ has dimensions L . These n e t d e n s i t y t r a n s f e r co- . e f f i c i e n t s have been c a l c u l a t e d from the o r i g i n a l d a t a f o r s i n g l e experiments. 2/3 V  i s made d i m e n s i o n l e s s by d i v i s i o n by  and i s p l o t t e d a g a i n s t t h e d i m e n s i o n l e s s r e c i p r o c a l  R i c h a r d s o n ' s number i n F i g . 18.  The v a l u e s range from 1  where l/R^ i s a p p r o x i m a t e l y 30, t o 35 when l/R^ i s a p p r o x i m a t e l y 1100.  I n t h e range covered, K ^ A /  l i n e a r l y w i t h l/Rj_.  2  3  varies nearly  52 Another d i m e n s i o n l e s s r a t i o which can be made up from the d e n s i t y g r a d i e n t data i s the r a t i o o f the e q u i l i b r i u m d i s tance Z q - Z e  Q  t o t h e d i s t a n c e o f maximum p e n e t r a t i o n Z - Z . m  0  T h i s r a t i o i s p l o t t e d i n F i g . 19 as a f u n c t i o n o f t h e r e c i p r o c a l R i c h a r d s o n ' s number. i s a p p r o x i m a t e l y 1100.  The r a t i o approaches 1 when l/Rj_  I t f a l l s i n a r o u g h l y l i n e a r manner  to  a v a l u e o f .4 when l/Rj_  5  COMPARISON OF NEUTRALLY BUOYANT AND STABLE SURROUNDINGS  5.1  i s a p p r o x i m a t e l y 30.  H o r i z o n t a l Spreading F i g s . 5 and 6 show t h e h o r i z o n t a l spread o f a p u f f i n  b o t h u n i f o r m and s t a b l y s t r a t i f i e d s u r r o u n d i n g s as a f u n c t i o n of  vertical  displacement.  I f t h e h o r i z o n t a l e x t e n s i o n i s p l o t t e d as a f u n c t i o n of  t i m e , ( n o t shown) t h e c u r v e s o b t a i n e d f o r u n i f o r m and  s t a b l y s t r a t i f i e d surroundings f a l l very c l o s e l y together f o r the same i n i t i a l c o n d i t i o n s o f volume and v e l o c i t y .  This  f a c t can be observed i n P l a t e I I where two p u f f s w i t h n e a r l y the same i n i t i a l c o n d i t i o n s p e n e t r a t e a u n i f o r m f l u i d s t r i p ) and a s t r a t i f i e d f l u i d  (lower s t r i p ) .  (upper  D u r i n g the  a c t i v e motion o f t h e p u f f , t h e r a d i i i n t h e two cases a r e nearly i d e n t i c a l .  A f t e r t h e motion c e a s e s , t h e p u f f i n s t r a -  t i f i e d f l u i d spreads a t i t s own d e n s i t y l e v e l under t h e i n f l u ence o f g r a v i t y , i n d i c a t i n g t h a t i t s d e n s i t y i s r e l a t i v e l y uniform throughout.  T h i s can not be seen i n P l a t e I I s i n c e  t h e photographs do not c o n t i n u e t o t h a t t i m e .  53 5.2  V e r t i c a l Spreading No r e a d i n g s were made f o r v e r t i c a l s p r e a d i n g  due t o  t h e d i f f i c u l t y i n d e l i n e a t i n g t h e upper boundary o f t h e p u f f . The  l a s t few frames on P l a t e I I i n d i c a t e t h a t t h e v e r t i c a l  e x t e n t o f t h e p u f f i s reduced r e l a t i v e t o t h e p u f f i n n e u t r a l surroundings. til  T h i s r e d u c t i o n does n o t appear t o commence un-  a f t e r t h e p u f f has reached i t s maximum p e n e t r a t i o n .  Once  t h e p u f f commences t o r i s e however, t h e v e r t i c a l e x t e n t decreases r e l a t i v e t o the n e u t r a l case.  The n e t r e s u l t i s t h a t  the volume o f t h e p u f f i n s t a b l e s u r r o u n d i n g s that i n n e u t r a l surroundings  i s reduced o v e r  r o u g h l y by a f a c t o r o f 2 by t h e  time t h e e q u i l i b r i u m p o s i t i o n i s r e a c h e d . 5.3  Energy L o s s A comparison can be made between t h e energy o f a p u f f  a t t h e p o i n t o f maximum p e n e t r a t i o n , and t h e energy o f a s i m i l a r p u f f i n n e u t r a l surroundings The  time o f maximum p e n e t r a t i o n t  a s t r a t i f i e d f l u i d i n F i g . 4.  a t t h e same e l a p s e d  time.  i s read from t h e d a t a f o r  m  The v e l o c i t y o f t h e p u f f i n  u n i f o r m f l u i d a t the same time i s computed n u m e r i c a l l y from the same f i g u r e .  The r a t i o K i n e t i c Energy a t tm = tm I n i t i a l K i n e t i c Energy KE K £  is  0  simply the r a t i o of served.  Velocity at t V e l o c i t y at t  m ?  s i n c e momentum i s con-  0  D i v i d i n g t h e r a t i o p l o t t e d i n F i g . 13 by t h e r a t i o  above, f o r t h e same i n i t i a l c o n d i t i o n s , we o b t a i n m KEf ra  =  P o t e n t i a l energy a t maximum p e n e t r a t i o n K i n e t i c energy a t same time i n n e u t r a l s u r r o u n d i n g s *  54 These energy r a t i o s a r e p l o t t e d i n F i g . 20 as a f u n c t i o n o f r e c i p r o c a l R i c h a r d s o n ' s number. e q u a l t o 1100, where the r a t i o P E / K E m  ing .8.  t m  Except near l/Rj_ has a v a l u e  approach-  z e r o , t h e p o i n t s a r e s c a t t e r e d w i t h a mean v a l u e near t o As has been mentioned  p r e v i o u s l y , i n the experiment w i t h  h i g h r e c i p r o c a l R i c h a r d s o n ' s number, t h e p u f f d i s s i p a t e d r a p i d l y , and was n o t o b s e r v a b l e f o r l o n g . have had a h i g h e r v a l u e o f t h a n i s shown i n F i g . 20.  P E m  /  K E  tm  a t  "^i  I t may, t h e r e f o r e , e c  l  u a l t 0  1 1 0 0  S i n c e the v a l u e s o f p o t e n t i a l  energy were d e r i v e d from a minimum e x p r e s s i o n ( e q u a t i o n 15, page 4 4 ) , i t seems p r o b a b l e t h a t the a c t u a l v a l u e o f the r a t i o i n F i g . 20 does not d i f f e r g r e a t l y from 1 over a l a r g e of R i c h a r d s o n ' s  range  number.  T h i s approximate  e q u a l i t y o f energy c o n t e n t between a  p u f f i n n e u t r a l s u r r o u n d i n g s and one i n a d e n s i t y s t r a t i f i c a t i o n , a t t h e time when t h e l a t t e r has become s t a t i o n a r y , imp l i e s t h a t t h e y have l o s t n e a r l y the same amount o f energy t o t u r b u l e n t motions.  T h i s i s r a t h e r s u r p r i s i n g , i n view o f the  f a c t t h a t t h e v e l o c i t i e s i n t h e two cases have been c o n s i d e r a b l y d i f f e r e n t as can be seen by the r e l a t i v e d i s p l a c e m e n t s i n Fig.  4.  T h i s s u b j e c t w i l l be c o n s i d e r e d f u r t h e r i n the next  section. 5 »4  Detailed Mixing I n F i g . 2, t h e d e t a i l e d m i x i n g c u r v e s f o r a p u f f pene-  t r a t i n g n e u t r a l l y buoyant and s t a b l e s u r r o u n d i n g s a r e p l o t t e d t o g e t h e r t o p e r m i t d i r e c t comparison.  From the a c c u r a c y o f  55 observation,  as i n d i c a t e d by t h e s t a n d a r d  d e v i a t i o n marked on  the curves,  and t h e number o f o b s e r v a t i o n s  taken,  i t i s appar-  ent t h a t l i t t l e d i s t i n c t i o n can be made between the two s e t s of  results. In equation  e q u i v a l e n t V /V, 0  2, page 28, i f w/w  Q  the equation  i s r e p l a c e d by i t s  i s v a l i d f o r s t a b l e as w e l l as  n e u t r a l l y buoyant f l u i d , s i n c e we a r e no l o n g e r i m p l y i n g t h e c o n s e r v a t i o n o f momentum.  The e q u a t i o n  becomes, (23)  From t h e s i m i l a r i t y between t h e m i x i n g  curves,  ( F i g . 2)  we know t h a t dQ/dt, (1-Q), (1-2Q) a r e not g r e a t l y d i f f e r e n t i n the two c a s e s .  Also, since k i s a f u n c t i o n of the i n i t i a l  con-  d i t i o n s w h i c h a r e the same, and o f the t u r b u l e n t energy a v a i l a b l e t o mix t h e two f l u i d s , which does not d i f f e r g r e a t l y a t t h e p o i n t o f maximum p e n e t r a t i o n , s u b s t a n t i a l l y unchanged.  ( F i g . 20) k must a l s o be  T h i s i m p l i e s t h a t t h e r a t i o V /V i s D  not g r e a t l y a l t e r e d by t h e p r e s e n c e o f t h e s t r a t i f i c a t i o n . We can i n f e r from t h e s e f a c t s , t h a t the e n t r a i n m e n t i n the two c a s e s has been s u b s t a n t i a l l y t h e same up t o t h i s t i m e , and t h a t the d e t a i l s o f t h e m i x i n g p r o c e s s have n o t been g r e a t l y a l t e r e d by t h e p r e s e n c e o f t h e d e n s i t y  6 6.1  stratification.  DISCUSSION M i x i n g Length Mixing-length  t h e o r i e s of turbulent transport processes  56  assume t h a t identity  a lump o f  fluid,  for a certain distance  roundings.  They t r y t o  terms o f t h i s  volume o f f l u i d  can  to u s e f u l  we  are  than a formal  and "A  are  implying  only  that  not  theories  by H i n z e  the  length 6.2  incomplete one  semiempirical advanced  to  in  behaviour of  an  quiet  surroundings  behaviour of  among the  (1959)>  who  a  turbulent  i d e a has  first  states  transport  more  theories  the As  that  i t has  proven  considering  s u r r o u n d i n g s we  long  as  successful  pro-  can  in yielding  More modern t h e o r i e s replace  the  have  mixing-  required.  Ratio  h a v e made use i n the  s u c h a com-  transport  of Transfer  Coefficients  the motion of a moving p u f f  t o t a l momentum i s c o n s t a n t  func-  approximate."  t o a s t a g e where t h e y can  and  expected  a mixing-length theory  relationships.  answers are  be  statistical  s o l u t i o n of the  at best,  fluid, 275):  (page  p r o b l e m can  motion.  any  and,  Momentum C o n s e r v a t i o n  buoyant  process  mixing-length  believes  where p r a c t i c a l  In  the  are  turbulent  correct in detail,  yet  of the  i s complete knowledge o f  W h i l e no  useful  on  the  p l e t e knowledge i s l a c k i n g ,  be  i t s sur-  to account f o r d i f f u s i o n i n a t u r b u l e n t  tions describing  blem must be  transport  moving r e l a t i v e  s o l u t i o n of the  i f there  mixing with  significance.  discussed  complete  observations  information  Mixing-length which attempt  retain i t s  mixing-length.  isolated lead  before  account f o r the  In assuming t h a t  fluid,  when d i s p l a c e d , w i l l  of the  absence o f  fact  in neutrally that  buoyant  the  forces..  57  The  total  momentum i n v o l v e s t h e momentum o f t h e i n j e c t e d and  entrained f l u i d .  I f some o f t h e e n t r a i n e d f l u i d does n o t mix  with the i n j e c t e d f l u i d ,  i t w i l l n o t be v i s i b l e  ment, and t h e momentum o f t h e v i s i b l e  i n the e x p e r i -  p o r t i o n w i l l be r e -  duced. A s e r i e s o f measurements o f volume and v e l o c i t y o f t h e moving f l u i d were made.  Volume measurements a r e d i f f i c u l t t o  make from a s e r i e s o f photographs, so t h e r e s u l t s serve as a rough g u i d e . the v i s i b l e  only  From these measurements, t h e momentum i n  moving p u f f was c o n s t a n t w i t h i n 25 p e r c e n t  the time t h e p u f f was v i s i b l e .  during  The measurements a r e n o t i n -  c o n s i s t e n t w i t h a c o n s t a n t momentum. In determining  an e x p r e s s i o n f o r t h e d i s p l a c e m e n t o f  a puff i n n e u t r a l surroundings, momentum was u s e d . the experimental  t h e assumption o f a c o n s t a n t  The agreement between t h i s e x p r e s s i o n and  p o i n t s i s shown i n F i g . 1 2 .  While the preceding  two p a r a g r a p h s i n d i c a t e t h a t t h e  s h a r i n g of momentum i s accompanied by m i x i n g ,  a strict  c a t i o n o f t h i s p r i n c i p l e would l e a d t o d i f f i c u l t i e s .  appli* Equation  22, page 51, g i v e s an e x p r e s s i o n f o r a d e n s i t y t r a n s f e r coe f f i c i e n t f o r a displaced puff of f l u i d i n s t r a t i f i e d roundings.  I f , i n a d d i t i o n t o being s t r a t i f i e d ,  sur-  a uniform  v e l o c i t y g r a d i e n t were p r e s e n t , t h e a p p l i c a t i o n o f momentum s h a r i n g o n l y by d e t a i l e d m i x i n g , would r e s u l t i n an i d e n t i c a l e x p r e s s i o n f o r a momentum t r a n s f e r c o e f f i c i e n t .  I t appears  t h e r e f o r e , e i t h e r t h a t a moving volume o f f l u i d can l o s e some  58  momentum t o i t s s u r r o u n d i n g s w i t h o u t m i x i n g w i t h them i n anyway, for  or t h a t the d i f f e r e n c e between the t r a n s f e r  coefficients  d e n s i t y and momentum depend upon something more s u b t l e  t h a n can be shown by t h e decay o f a s i n g l e p u f f . Recent work ( E l l i s o n and Turner, I960) i n d i c a t e s t h a t , i n u n i f o r m f l u i d , t h e r a t i o of t h e t r a n s f e r K^/Kjyj has a v a l u e near t o 1.3. tification,  K /K H  M  coefficients  In a f l u i d with density stra-  f a l l s to small values.  I f we c o n s i d e r the  p o s s i b i l i t y t h a t a d i s p l a c e d volume o f f l u i d  can exchange  momentum w i t h i t s s u r r o u n d i n g s w i t h o u t m i x i n g , the v a r i a t i o n s i n Kjj/Kjyj can be understood  qualitatively.  L e t G be the i n s t a n t a n e o u s average c o n c e n t r a t i o n o f some p r o p e r t y i n a d i s p l a c e d f l u i d volume, and C  z  the concen-  t r a t i o n o f the same p r o p e r t y i n t h e s u r r o u n d i n g s a t depth Then C -C z  Z.  i s the d i f f e r e n c e i n c o n c e n t r a t i o n between the d i s -  p l a c e d f l u i d and i t s s u r r o u n d i n g s .  I f V i s the volume o f t h e  d i s p l a c e d f l u i d , the t o t a l t r a n s p o r t o f t h e p r o p e r t y by the p u f f w i l l be g i v e n by T zj In  (24)  V ( C - C ) dZ.  (24)  Z  the i n t e g r a l i s t a k e n o v e r the p a t h o f t h e p u f f t o  the p o i n t L where the p u f f comes t o t h e average v e l o c i t y o f the  surroundings. I f the c o n c e n t r a t i o n C i s determined  completely  by  m i x i n g w i t h the s u r r o u n d i n g s , then C w i l l be a f u n c t i o n o f V and C , z  and w i l l be the same f o r any p r o p e r t y i f we  molecular d i f f u s i o n .  neglect  I f , as we a r e supposing t o be the  case  w i t h momentum, some means e x i s t s o f r e d u c i n g the d i f f e r e n c e  59 C -C w i t h o u t m i x i n g , then w i t h V h a v i n g t h e same v a l u e s as z  b e f o r e , C -G w i l l be s m a l l e r and T w i l l be reduced. z  This  would g i v e a r a t i o f o r K^/K^ g r e a t e r t h a n one. I n a d e n s i t y - s t r a t i f i e d f l u i d where the d i s p l a c e d f l u i d w i l l p e n e t r a t e a c e r t a i n d i s t a n c e and then r e v e r s e d i r e c t i o n , dZ w i l l change s i g n but G -C w i l l n o t do so immediz  ately.  The s i g n o f the i n t e g r a l w i l l be r e v e r s e d f o r a d i s -  tance which w i l l be g r e a t e r as C -C i s g r e a t e r . z  to  I t i s easy  v i s u a l i z e a case i n w h i c h t h e t o t a l v a l u e o f t h e i n t e g r a l  w i l l be l a r g e r i f C -C i s r e l a t i v e l y s m a l l . z  We can suppose  t h a t t h i s i s t h e case w i t h momentum, g i v i n g s m a l l v a l u e s t o the r a t i o %/Kjyj as t h e s t r a t i f i c a t i o n becomes g r e a t and m i x i n g lengths  reduced. E x a m i n a t i o n o f e q u a t i o n (20). and c o n s i d e r i n g v e r t i c a l  momentum as t h e p r o p e r t y b e i n g t r a n s p o r t e d i n a u n i f o r m /L of  t h e same d e n s i t y , we o b t a i n  T =J  i s t h e v e l o c i t y o f t h e p u f f and w  z  V(w -w) d z , where w z  i s the v e l o c i t y of the  s u r r o u n d i n g s which i s e q u a l t o zero i n s t i l l  surroundings.  I f momentum i s conserved, t h e i n t e g r a n d i n (20) a constant.  fluid  becomes -Vw,  The p a t h o f i n t e g r a t i o n w i l l have no l i m i t and  T becomes i n f i n i t e . We must c o n c l u d e , t h e r e f o r e , t h a t v e r t i c a l momentum i s not e x a c t l y conserved i n t h e p u f f throughout  i t s history.  The  e x p r e s s i o n s f o r v e l o c i t y and d i s p l a c e m e n t o f a p u f f i n u n i f o r m f l u i d o f t h e same d e n s i t y o b t a i n e d i n s e c t i o n s 3.2 then only approximately c o r r e c t .  and 3.3 a r e  The agreement o f t h e d a t a  .60 w i t h equations  (8) and (9) as shown i n F i g s . 11 and 12, i n -  d i c a t e t h a t t h e assumption o f momentum c o n s e r v a t i o n  i n the  p u f f i s j u s t i f i e d over t h e range o f o b s e r v a t i o n , where t h e momentum i s a p p a r a n t e l y  large with respect to the l o s s e s .  I f i t were p o s s i b l e t o c o n t i n u e o b s e r v a t i o n s the e x p e r i m e n t a l  to longer  p o i n t s i n F i g s . 11 and 12 must f a l l  times,  below  the s t r a i g h t l i n e s i n d i c a t e d . I f v e r t i c a l momentum i s n o t c o n s e r v e d , i t seems reasonable  t o assume t h a t t h e h o r i z o n t a l momentum o f a p u f f  o f f l u i d p e n e t r a t i n g a r e g i o n o f v a r i a b l e mean h o r i z o n t a l v e l o c i t y s h o u l d n o t be d e t e r m i n e d e n t i r e l y by t h e h o r i z o n t a l momentum i n t h e f l u i d e n t r a i n e d .  The d i f f e r e n c e i n h o r i z o n t a l  v e l o c i t y between t h e p u f f and i t s s u r r o u n d i n g s t h a n c a l c u l a t e d s t r i c t l y by m i x i n g , i n equation  and t h e v a l u e o f G -C  6.3  z  20 t h e r e f o r e s m a l l e r when C r e f e r s t o h o r i z o n t a l  v e l o c i t y t h a n when i t r e f e r s t o a p r o p e r t y or  w i l l be l e s s  such as temperature  salinity. Turbulent  Energy D i s s i p a t i o n  Some arguments on t h e e f f e c t o f d e n s i t y upon a t u r b u l e n t f l u i d  stratification  ( B o l g i a n o , 1959) i m p l y t h a t t h e energy  t r a n s f e r from t u r b u l e n t motions t o d e n s i t y o c c u r s over a l l l e n g t h s c a l e s o f t u r b u l e n c e ,  stratification and might f o r m a  c o n s i d e r a b l e p o r t i o n o f the t o t a l l o s s o f t u r b u l e n t energy. I n t h i s case t h e t o t a l v i s c o u s d i s s i p a t i o n . w o u l d be c o n s i d e r ably reduced.  The r e s u l t s o f t h e e x p e r i m e n t a l  work r e p o r t e d  h e r e , i n p a r t i c u l a r F i g . 17, do n o t s u p p o r t t h i s view.  61 Fig.  17 i n d i c a t e s t h a t o n l y about 3 p e r c e n t o f t h e t o t a l  energy goes t o p o t e n t i a l energy o f t h e s t r a t i f i c a t i o n . The  observations  of mixing rate i n d i c a t e that the  molecular mixing, which i s c l o s e l y associated with  viscous  d i s s i p a t i o n , i s much t h e same when t h e s u r r o u n d i n g  fluid i s  s t r a t i f i e d and when i t i s n o t s t r a t i f i e d .  Observations o f the  p u f f show t h a t i t behaves as a u n i t which s u g g e s t s t h a t i t i s mixed t o a f a i r l y u n i f o r m d e n s i t y and t h a t t h e s m a l l e r t u r bulent  s c a l e s a r e n o t i n f l u e n c e d t o any g r e a t e x t e n t by t h e  stratification. I n s e c t i o n 3 . 2 i t was shown t h a t t h e assumption o f momentum c o n s e r v a t i o n the f o r m  and s i m i l a r i t y ,/ w~ / 4  3  l e a d t o an e q u a t i o n o f  0  (25)  * ct,  where w i s t h e v e l o c i t y o f t h e p u f f , t i s t h e t i m e , and c i s an e x p e r i m e n t a l  constant.  I n a t u r b u l e n t f l u i d , i f t h e R e y n o l d ' s number i s sufficiently number.  l a r g e , t h e f l o w i s independent o f Reynold's  The o n l y f a c t o r s which e n t e r i n t o t h e r a t e o f t u r -  b u l e n t energy decay w i l l be t h e t u r b u l e n t v e l o c i t i e s and t h e scale o f turbulence.  D i m e n s i o n a l homogeneity t h e n r e q u i r e s  t h a t t h e r a t e o f decay o f t u r b u l e n t d e n s i t y w i l l be g i v e n by (26)  where w now r e p r e s e n t s  t u r b u l e n t ^ v e l o c i t i e s and L i s a s c a l e  length. To demonstrate t h a t an i s o l a t e d p u f f s a t i s f i e s  equation  62 (26),  c o n s i d e r t h a t the t u r b u l e n t v e l o c i t y i s g i v e n by w i n  (25).  A s u i t a b l e s c a l e l e n g t h w i l l be the r a d i u s R o f  p u f f e q u a l t o .z t a n  where z i s p e n e t r a t i o n and ot i s the  h a l f a n g l e o f the.cone o f e x p a n s i o n o f the p u f f . w i n (25)  we  t o o b t a i n z, and  comparing dw dt 2 3 dw • 6 taneci . dt R  obtain, The  the  experimental  2  v a l u e o f <* was  makes 6 t a n ©< e q u a l t o 2 a p p r o x i m a t e l y .  with  Integrating  w£ z tan (27)  n e a r t o 19° The  which  v a l u e of  this  constant  depends upon what i s chosen as a s u i t a b l e s c a l e  length.  The  f i g u r e 2 i s of t h e same o r d e r of magnitude but  somewhat l a r g e r t h a n i s a s s o c i a t e d w i t h a d e c a y i n g t u r b u l e n t field. The  foregoing  remarks i m p l y t h a t i f a t u r b u l e n t  were b u i l t up of randomly o r i e n t e d moving p u f f s o f f l u i d  field such  as have been s t u d i e d i n t h i s e x p e r i m e n t , i t would have many o f t h e p r o p e r t i e s a c t u a l l y found i n a f i e l d o f t u r b u l e n c e s u l t i n g from a shear f l o w .  The  r e s u l t s o f a s t u d y of i s o l a t e d  p u f f s can perhaps be a p p l i e d w i t h some c o n f i d e n c e turbulent 6•4  t o an a c t u a l  flow.  Turbulent The  Energy  production  Production of t u r b u l e n t energy per u n i t mass can  be shown (Hinze I 9 6 0 , page 65) -  re-  UjUjdU^/dXj.  I n a two  to depend upon the term  dimensional  case w i t h mean f l o w U i n  t h e h o r i z o n t a l d i r e c t i o n and a v e l o c i t y g r a d i e n t du/dz i n the v e r t i c a l d i r e c t i o n , t h i s becomes  - uw c)U/dz  (28)  63 where t h e l o w e r case l e t t e r s r e f e r t o t u r b u l e n t v e l o c i t y components and t h e upper case t o mean v a l u e s . C o n s i d e r a steady s t a t e o f t u r b u l e n c e a u n i f o r m . f l u i d where the p r o d u c t i o n d i f f u s i o n , by l o s s t o p r e s s u r e viscosity.  established i n  i s c o u n t e r - b a l a n c e d by  f l u c t u a t i o n s and by l o s s t o  Now c o n s i d e r t h a t a d e n s i t y g r a d i e n t  i s s e t up.  One e f f e c t o f t h e d e n s i t y g r a d i e n t , as has been shown by t h e experimental  work ( S e c . 4)  p a r t i a l l y reversed  i s that the v e r t i c a l motion w i s  by buoyant f o r c e s .  The p a r t i a l r e v e r s a l ..of  v/ w i l l r e s u l t i n p a r t i a l r e v e r s a l o f t h e s i g n o f (28) , w h i c h w i l l be g r e a t l y r e d u c e d . but  I n a d d i t i o n , t h e r e w i l l be a s m a l l  s i g n i f i c a n t l o s s o f t u r b u l e n t energy t o t h e d e n s i t y  stra-  t i f i c a t i o n ( S e c . 4«4). A l t h o u g h t h e d i f f u s i o n and v i s c o u s d i s s i p a t i o n o f t u r b u l e n t energy w i l l  a l s o be reduced, t h e n e t  e f f e c t o f t h e d e n s i t y s t r a t i f i c a t i o n w i l l be t h a t t h e t u r b u l e n t energy w i l l  e s t a b l i s h a new e q u i l i b r i u m a t a much reduced  level. I t i s p o s s i b l e to. make a rough e s t i m a t e  of the e f f e c t  o f t h e d e n s i t y g r a d i e n t upon t h e t u r b u l e n t energy  production  i f we assume t h a t t h e s t r a t i f i c a t i o n i n f l u e n c e s o n l y t h e v e r t i c a l "motions.  A f i g u r e representative of the t o t a l  energy produced by a p u f f w i l l be - u z d t l / d z , where z n  turbulent n  i s the  n e t d i s p l a c e m e n t o f t h e p u f f , e q u a l t o t h e l a r g e s t observed d i s p l a c e m e n t i n t h e case o f a u n i f o r m s u r r o u n d i n g  f l u i d , and  e q u a l t o zeq (Sec. 4.2) i n t h e case o f a s t r a t i f i e d The r a t i o z  / z , which r e p r e s e n t s n  roughly  fluid, turbulent  64  energy p r o d u c t i o n i n ( s t r a t i f i e d f l u i d / u n i f o r m f l u i d ) v a r i e s from .1 when l/R  (Sec. 4 . 3 ) i s e q u a l t o 3 0 , t o . 8 when l / R ^  ±  i s equal t o 1 1 0 0 . 6.5  Turbulent Scales The s c a l e o f t h e t u r b u l e n t motions a r e commonly des-  cribed  (Hinze i 9 6 0 )  Eulerian scale.  by e i t h e r a L a g r a n g i a n s c a l e o r an  If ( 'j u  i s  v e l o c i t y component o f a p a r -  t n e  x  t i c u l a r p a r t i c l e o f f l u i d a t p o s i t i o n x , and u, . T  V X  » i s the 1"XJ  v e l o c i t y component o f t h e same p a r t i c l e a t some l a t e r time when t h e p a r t i c l e i s a t x ' t x , then t h e L a g r a n g i a n s c a l e i s g i v e n by  /•  0 0  u(x') u(x*+x) (^(x«)j (u?x»*x)) 2  dx (29)  4  where t h e i n t e g r a l i s t a k e n over a l a r g e number o f r e a l i z a t i o n s o f t h e product  average.  The E u l e r i a n s c a l e i s g i v e n by an e x p r e s s i o n s i m i l a r t o (29) b u t t h e averages a r e t a k e n o v e r d i f f e r e n t p a r t i c l e s a t t h e same i n s t a n t o f t i m e . We a r e concerned here w i t h p a r t i c l e movements a l o n g the a x i s o f t h e p u f f , t h a t i s , i n t h e v e r t i c a l d i r e c t i o n . I n o r d e r t o o b t a i n an e s t i m a t e o f t h e r a t i o o f L a g r a n g i a n s c a l e / E u l e r i a n s c a l e , we w i l l t a k e t h e denominator o f (29) i n 2  each case t o be e q u a l t o  u  x  t  .  From e q u a t i o n ( 2 5 ) , page 6 l , we can o b t a i n an express s i o n f o r w i n terms o f z, where a i s a c o n s t a n t .  w  = az~  3  (30)  65 S u b s t i t u t i o n o f t h i s v a l u e o f w f o r u i n (29) y i e l d s Lagrangian scale  =  z? f  z"  3  d z , which, when e v a l u a t e d i s  e q u a l t o z-j/2. The E u l e r i a n s c a l e can be approximated  by c o n s i d e r i n g  t h a t as we measure i n s t a n t a n e o u s v e l o c i t i e s i n a p u f f t h e c o r r e l a t i o n s w i l l be h i g h as l o n g as measurements a r e made i n side the puff.  When t h e boundary i s r e a c h e d , t h e c o r r e l a t i o n  w i l l drop t o z e r o .  The E u l e r i a n s c a l e w i l l t h e r e f o r e be  a p p r o x i m a t e l y e q u a l t o R o r t o z-^ t a n  where *c i s t h e h a l f  a n g l e o f t h e cone o f s u c c e s s i v e o u t l i n e s o f t h e p u f f and approximately equal t o 19°. The r a t i o L a g r a n g i a n s c a l e / E u l e r i a n s c a l e i s t h e r e f o r e a p p r o x i m a t e l y e q u a l t o 1.4. The  r e l a t i o n s h i p between t h e L a g r a n g i a n and E u l e r i a n  s c a l e s i s d i s c u s s e d by C o r r s i n (1959).  H i n z e (I960) (page 49)  s t a t e s t h a t t h e two s c a l e s a r e r o u g h l y o f t h e same magnitude. No e s t i m a t e o f t h e e x p e c t e d a c c u r a c y o f t h e r e s u l t s i n terms o f t h e e r r o r s i n t h e measurement o f t h e d e t e r m i n i n g p a r a m e t e r s has been g i v e n .  As was s t a t e d i n Sec. 4*3, page 44,  t h e e r r o r i n t h e r e s u l t s i s dominated by s t a t i s t i c a l  scatter.  A s i g n i f i c a n t improvement i n a c c u r a c y can be a c h i e v e d o n l y by g r e a t l y i n c r e a s i n g t h e number o f o b s e r v a t i o n s .  66  7  SUMMARY AND  CONCLUSIONS  T u r b u l e n t t r a n s p o r t mechanisms i n a d e n s i t y - s t r a t i f i e d f l u i d have been s t u d i e d by comparison o f o b s e r v a t i o n s  of a  p u f f o f f l u i d p r o j e c t e d v e r t i c a l l y downwards i n t o a tank cont a i n i n g a s t r a t i f i e d f l u i d , with s i m i l a r observations  of a  puff penetrating  The i n -  a u n i f o r m f l u i d o f i t s own d e n s i t y .  j e c t e d and t a n k f l u i d s c o n t a i n e d  c h e m i c a l s w h i c h produced a  f i n e l y d i v i d e d p r e c i p i t a t e when m i x i n g o c c u r r e d scale.  on a m o l e c u l a r  O b s e r v a t i o n o f t h i s p r e c i p i t a t e , b o t h as t o amount and  p o s i t i o n , permitted  estimates  of the d e t a i l e d mixing process  o c c u r r i n g between t h e i n j e c t e d and ambient f l u i d , and o f t h e p r o g r e s s o f t h e p u f f as a whole. The surrounding  r e s u l t s of the observations fluid  taken with a uniform  show t h a t t h e v e l o c i t y o f t h e c e n t e r o f t h e  p u f f obeys t h e r e l a t i o n s h i p w~^ t i a l period of orientation.  3  = c t + c o n s t , a f t e r an i n i -  T h i s r e l a t i o n s h i p can be o b t a i n e d  by assuming s e l f - p r e s e r v a t i o n o f t h e p u f f , w h i c h means t h a t a moving e q u i l i b r i u m i s s e t up i n w h i c h a l l v e l o c i t i e s remain proportional. From t h e v a r i a t i o n o f v e l o c i t y and s c a l e w i t h  time, 2 3 t h e energy decay r a t e p e r u n i t mass i s shown t o be ^ |£ dt L where L i s a l e n g t h t y p i c a l o f t h e p u f f .  The-same energy  decay r a t e p e r u n i t mass h o l d s i n a t u r b u l e n t f l u i d 1956).  (Townsend,  T h i s i m p l i e s t h a t a f l u i d w i t h a l a r g e number o f r a n -  domly o r i e n t e d p u f f s would have many f e a t u r e s i n common w i t h a n a t u r a l l y t u r b u l e n t f l u i d r e s u l t i n g from a s h e a r f l o w , and  67 t h a t c a l c u l a t i o n s made on the b a s i s o f i s o l a t e d p u f f s can be a p p l i e d w i t h some c o n f i d e n c e t o a t u r b u l e n t f l u i d . S i n c e the p u f f s can be observed  closely  t h e i r h i s t o r y , i t i s p o s s i b l e t o determine t h e y mix w i t h t h e s u r r o u n d i n g s , how  throughout  the r a t e a t which  t h i s m i x i n g proceeds i n -  s i d e the p u f f , and the d e t a i l s o f m o t i o n o f t h e p u f f as a whole.  I n a t u r b u l e n t f l u i d , the e f f e c t s o f a l l s c a l e s o f  t u r b u l e n c e are superimposed, and t u r b u l e n t v e l o c i t i e s have a l l directions.  Thus o n l y t h e i n t e g r a t e d e f f e c t s of a l l s c a l e s  and o r i e n t a t i o n s can be  observed.  For these reasons, i t i s p o s s i b l e to o b t a i n estimates o f d e t a i l e d m i x i n g and energy changes as a f u n c t i o n o f the d e t e r m i n i n g parameters which can not be o b t a i n e d from s t u d i e s of a turbulent f l u i d d i r e c t l y .  Numerical values f o r q u a n t i t i e s  u s e f u l i n t u r b u l e n c e t h e o r y have t h u s been o b t a i n e d which were p r e v i o u s l y unknown t o an o r d e r o f magnitude. I n o r d e r t o d i s p l a y t h e s e v a l u e s i n the most g e n e r a l manner, they are e x p r e s s e d i n terms o f d i m e n s i o n l e s s r a t i o s , and p l o t t e d a g a i n s t a d i m e n s i o n l e s s c o m b i n a t i o n o f t h e c o n d i t i o n s which resembles  the r e c i p r o c a l o f a R i c h a r d s o n ' s P  number.  T h i s number i s g i v e n by 1/Rj_  -  2  • ^o o  .  w  2  e  be thought o f as t h e r a t i o o f i n e r t i a l f o r c e s t o buoyant f o r c e s .  initial  & dx  I n the experiment  /  I t can  3  v  o  l / R ^ v a r i e s from  37 to.1140, changes i n t h e i n i t i a l v e l o c i t y and volume b e i n g the p r i n c i p a l cause o f v a r i a t i o n . One  q u a n t i t y determined  was  the r a t i o of p o t e n t i a l  68 energy o f a p u f f a t maximum p e n e t r a t i o n t o t h e i n i t i a l  energy,  w h i c h was f o u n d t o vary from .2 a t s m a l l v a l u e s o f l / R j _ t o z e r o a t l / R ^ equal t o 1100 as is^shown i n F i g . 1 3 .  The o r d e r  o f magnitude o f t h i s r a t i o was n o t known p r e v i o u s l y . Since, i n a turbulent f i e l d ,  i t i s often possible to  d e t e r m i n e the t u r b u l e n t energy, t h e knowledge o f t h e energy r a t i o i n the preceding  p a r a g r a p h p e r m i t s an e s t i m a t e o f t h e  d i s t a n c e a d i s p l a c e d volume o f f l u i d may be from i t s e q u i l i b rium  position. Another q u a n t i t y i s t h e f r a c t i o n o f t h e i n i t i a l •  energy which goes t o a l t e r i n g t h e d e n s i t y g r a d i e n t , shown i n Fig.  17 as a f u n c t i o n o f l / R . i  Over a l a r g e range o f l / R ^ ,  t h i s q u a n t i t y shows no s y s t e m a t i c dependence on 1/Rj_ and has a mean v a l u e o f .03. No e s t i m a t e o f i t s magnitude has been o b t a i n e d from s t u d i e s o f t u r b u l e n t f l u i d . gested  I t has been sug-  t h a t a l a r g e p o r t i o n o f t h e energy i n l a r g e r t u r b u l e n t  s c a l e s would be l o s t t o t h e d e n s i t y s t r u c t u r e s i n c e energy would be l o s t by a l l stages o f t h e decay cascade. dence p r e s e n t e d  The e v i -  h e r e demonstrates t h a t t h i s i s n o t t h e c a s e .  V a l u e s have a l s o been o b t a i n e d f o r the v a r i a t i o n o f d e n s i t y t r a n s p o r t and t u r b u l e n t energy p r o d u c t i o n w i t h t h e determining  parameters.  This v a r i a t i o n agrees i n k i n d w i t h  what i s known from s t u d i e s o f t u r b u l e n t f i e l d s . i n t h e s e cases g i v e s added c o n f i d e n c e i s o l a t e d puff observations  The agreement  to the application of  to a turbulent  field.  A number Ku r e p r e s e n t a t i v e o f t h e d e n s i t y t r a n s p o r t  69 due t o a s i n g l e d i s p l a c e d b l o b o f f l u i d was computed. made d i m e n s i o n l e s s by d i v i s i o n by V / , 2  function of l / R  ±  i n F i g . 13.  %/V / 2  3  3  was  and i s p l o t t e d as a  v a r i e s from 1 f o r  e q u a l t o 30,to 35 f o r l/Rj_ e q u a l t o 1100.  l/R  ±  To t h e a c c u r a c y  o b t a i n e d , the v a r i a t i o n i s l i n e a r over t h i s range.  Density  t r a n s p o r t e s t i m a t i o n s f o r a t u r b u l e n t f l u i d a r e known t o v a r y i n a s i m i l a r manner. An e s t i m a t e was o b t a i n e d o f the e f f e c t upon t h e p r o d u c t i o n o f t u r b u l e n t energy o f t h e sudden a p p l i c a t i o n o f a s t r a t i f i c a t i o n o f d e n s i t y upon a t u r b u l e n t f i e l d .  A  son o f t h e e q u i l i b r i u m depth o f a p u f f i n a s t r a t i f i e d  comparifluid  w i t h t h e maximum p e n e t r a t i o n o f a s i m i l a r p u f f i n u n i f o r m f l u i d g i v e s a measure o f the r e l a t i v e p r o d u c t i o n . v a r i e s l i n e a r l y w i t h l/Rj_ from at l / R i e q u a l t o 1200.  The r a t i o  .1 a t l / R j _ e q u a l t o 35 t o .3  I t i s known t h a t t h e presence o f a  d e n s i t y s t r a t i f i c a t i o n w i l l cause a g r e a t r e d u c t i o n o f t u r b u l e n t energy d e n s i t y i n a g i v e n shear f l o w . An e s t i m a t e o f t h e d e t a i l e d m i x i n g between t h e i n j e c t ed and s u r r o u n d i n g u n i f o r m f l u i d was o b t a i n e d by removing volume and c o n c e n t r a t i o n e f f e c t s from t h e observed r a t e o f formation of p r e c i p i t a t e .  F o r each s e t o f i n i t i a l c o n d i t i o n s ,  t h e d e t a i l e d m i x i n g p e r u n i t volume o f t h e p u f f can be r e p r e sented by a s i n g l e number k, ( e q . 2, p. 28) which v a r i e s by l e s s t h a n 1 0 % o f i t s mean v a l u e from t h e t i m e o f until sec).  t h e time t h e c h e m i c a l r e a g e n t s a r e exhausted  initiation (1 t o 7  The v a l u e s o f k were found t o depend upon t h e i n i t i a l  70 c o n d i t i o n s o f v e l o c i t y , l e n g t h and v i s c o s i t y a c c o r d i n g t o t h e r e l a t i o n s h i p k = .0050 The  d e t a i l e d m i x i n g . i n a s t r a t i f i e d f l u i d was found t o  be unchanged from t h a t i n a u n i f o r m f l u i d w i t h i n t h e experiment a l error. Comparison o f t h e h o r i z o n t a l w i d t h o f a p u f f i n a s t r a t i f i e d f l u i d w i t h t h e width of a s i m i l a r p u f f i n uniform f l u i d shows t h a t t h e r e i s no s i g n i f i c a n t d i f f e r e n c e i n t h e spreading  r a t e i n the two c a s e s .  The v e r t i c a l k i n e t i c energy  which t h e p u f f l o s e s t o p o t e n t i a l energy i s r e t u r n e d t o v e r t i c a l energy o r t o t h e a l t e r a t i o n o f t h e d e n s i t y s t r u c t u r e and does n o t enhance t h e h o r i z o n t a l s p r e a d i n g  rate.  I t has been  suggested, ( P a r r , 1936) t h a t h o r i z o n t a l t u r b u l e n c e by t h e presence o f a s t r a t i f i c a t i o n .  i s increased  The o b s e r v a t i o n s do n o t  support t h i s view. A comparison has been made between t h e p o t e n t i a l energy o f . a p u f f a t maximum p e n e t r a t i o n i n a s t r a t i f i e d  fluid  and t h e k i n e t i c energy o f a s i m i l a r p u f f a t t h e same time a f t e r i n i t i a t i o n , but i n a n e u t r a l l y buoyant f l u i d .  The r e -  s u l t s i n F i g . 20 show t h a t over t h e range o f l / R j _ from 30 t o 500  t h i s r a t i o has a mean v a l u e o f 0.8, n o t g r e a t l y d i f f e r e n t  f r o m 1.  The v a l u e o f zero shown a t l / R j _ equal t o 1100 must be  regarded with s u s p i c i o n .  I n t h i s case t h e p u f f s were n o t ob-  served t o r i s e from t h e p o i n t o f maximum p e n e t r a t i o n .  A slight  unobserved r i s e would y i e l d a v a l u e f o r t h e energy r a t i o . S i n c e , f o r l a r g e l / R j _ due t o a v e r y s m a l l d e n s i t y g r a d i e n t , t h e  71 surroundings would be n e a r l y i d e n t i c a l i n the two energies  should The  cases,  the  a l s o approach e q u a l i t y .  expression  used to deduce the p o t e n t i a l energy  y i e l d e d a minimum v a l u e .  I t t h e r e f o r e seems probable t h a t  the  r a t i o does not d i f f e r s i g n i f i c a n t l y from 1 over the whole range. has  T h i s i n d i c a t e s that the l o s s o f energy to  not been g r e a t l y a l t e r e d by the  time i n the h i s t o r y of the The  turbulence  s t r a t i f i c a t i o n up to  this  puff.  f a c t that a puff i n s t r a t i f i e d f l u i d retains a  d e f i n i t e shape, does not break up i n t o separate p o r t i o n s ,  and  p a r t i c u l a r l y , the r e d u c t i o n i n v e r t i c a l extent when i t f i n a l l y stops,  i n d i c a t e t h a t i t has  compared w i t h the  of d e n s i t y s t r u c t u r e i n s i d e the  puff  w i t h the observed s i m i l a r i t y i n the d e t a i l e d mixing  o f a p u f f w i t h uniform and  stratified  surroundings, i n d i c a t e s  t h a t a l l c o n d i t i o n s i n s i d e the p u f f are l i t t l e the  structure  surroundings.  This uniformity together  a n e a r l y uniform d e n s i t y  stratification.  r a t e and  The  influenced  e q u a l i t y i n the h o r i z o n t a l  by  spreading  the e q u a l i t y o f the energy of the p u f f as a whole i n  uniform and  i n s t r a t i f i e d f l u i d are c o n s i s t e n t with t h i s con-  cept. We  can conclude t h e r e f o r e t h a t the e f f e c t o f the den-  s i t y s t r a t i f i c a t i o n i s p r i n c i p a l l y upon the i n i t i a l turbulent  s c a l e s i n which t u r b u l e n t  only s l i g h t l y upon the decays.  (large)  energy i s produced,  small s c a l e s i n which t u r b u l e n t  and energy  72  8  APPENDIX A g r a p h i c a l method o f f i n d i n g an approximate power  l a w r e l a t i o n s h i p between 3 v a r i a b l e s was used i n i t i a l l y on the data f o r V , w , Q  and k.  Q  I t may be o f i n t e r e s t , and w i l l  be d e s c r i b e d below. I f t h r e e v a r i a b l e s A, B, and C a r e r e l a t e d by A  =  mB  Log A  =  l o g m t v l o g B + £ l o g G, o r  A  =  m' + <*B* • @cV,  a logarithm.  C,  where m i s a c o n s t a n t , t h e n  where t h e prime i n d i c a t e s  The l i n e a r r e l a t i o n s h i p between A , B , and C , T  f  f  d e f i n e s a p l a n e , and t h e power l a w r e l a t i o n s h i p between A, B, and C, a l s o d e f i n e s a p l a n e i n space where the d i s t a n c e s a r e p r o p o r t i o n a l " t o logarithms of the v a r i a b l e s . to  I t i s desired  f i n d t h e t r a c e s o f t h i s p l a n e on two o r t h o g o n a l p l a n e s , one  where B i s c o n s t a n t and o t h e r where C i s c o n s t a n t .  The s l o p e s  o f these t r a c e s w i l l g i v e t h e v a l u e s o f @ and <>C r e s p e c t i v e l y . Use i s made o f t h e p r i n c i p l e s o r o r t h o g r a p h i c p r o j e c t i o n . Any two s e t s o f e x p e r i m e n t a l p o i n t s d e t e r m i n e one p o i n t on each t r a c e .  I f t h e e x p e r i m e n t a l d a t a obey a power l a w , a l l  t h e t r a c e p o i n t s w i l l l i e on a l i n e , o r n e a r l y s o . F o r two s e t s o f e x p e r i m e n t a l p o i n t s , t h e method o f f i n d i n g the two t r a c e p o i n t s i s i l l u s t r a t e d i n F i g . 9« L e t t h e e x p e r i m e n t a l p o i n t s be A-^, B^, C-^ and A^, B^, C , 2  r e p r e s e n t e d by open c i r c l e s i n F i g . 9.  The l i n e x — x i n -  d i c a t e s t h e i n t e r s e c t i o n o f t h e B = c o n s t a n t p l a n e and t h e C - constant plane.  I t can be chosen i n any c o n v e n i e n t  position.  73  The d i r e c t i o n s o f i n c r e a s i n g A, B, and C, as shown a r e convenient  so t h a t o r d i n a r y l o g - l o g paper can be used.  o r i g i n f o r A, B, C, = 1 may be chosen. incident i f desired.  Any  B and C = 1 may be c o -  The marked p o i n t s , (open c i r c l e s ) a r e  the p r o j e c t i o n s o f t h e d a t a p o i n t s on t h e s e two o r t h o g o n a l planes.  I t i s n e c e s s a r y t o know w h i c h p l a n e ,  therefore the  p r o j e c t i o n s must be l a b e l l e d as A^B^ i f on t h e B = c o n s t a n t plane,  o r as A C 1  1  i f on t h e C = c o n s t a n t p l a n e .  t r a c e o f t h e l i n e d e f i n e d by A^B^C-^ and k^2^2  n  t  n  e  ^  =!  ~  con  j o i n A-^B-^ and A2B2 and produce t o i n t e r s e c t t h e  stant plane, x—x  o  To f i n e t h e  l i n e at 0 .  s e c t AT_C-J_, A C 2  2  Erect a perpendicular produced a t  P.  P,  represented  c i r c l e , i s the required trace point. C = constant,  to x — x at 0 to interby a s o l i d  The t r a c e p o i n t Q, on  i s found s i m i l a r l y , as shown on t h e diagram.  I t i s represented  by an open t r i a n g l e .  For t h e e x p e r i m e n t a l  d a t a i n T a b l e 6 , page 3 9 , a p l o t  o f t h e p o i n t s made i n t h i s manner, and t h e t r a c e s o f t h e p l a n e s d e t e r m i n e d by a l l p o s s i b l e p a i r s o f p o i n t s i s shown i n Fig. 8.  (Some o f t h e t r a c e p o i n t s l i e o f f t h e paper, as do  some o f t h e e x p e r i m e n t a l  points).  The r e l a t i o n k = m w^*53 V~*55 ^  s  shown by t h e dashed  line.  The r e l a t i o n k " m w^*5 V £ * 5 , i s shown by t h e s o l i d  line.  I n each case t h e i n t e r s e c t i o n o f t h e two t r a c e s must  l i e on t h e x — x l i n e . the best  fit.  T h i s p o i n t was chosen by eye t o g i v e  74 REFERENCES  Batchelor,  G. K . , lence.  Bolgiano,  R. J r . , tified  The t h e o r y o f homogeneous Cambridge U n i v e r s i t y P r e s s .  turbu-  (1953).  (1959^« T u r b u l e n t s p e c t r a i n a s t a b l y atmosphere. J o u r . G e o p h y s . R e s . , 6^.,  stra12.  Corrsin,  S . , (1959). L a g r a n g i a n c o r r e l a t i o n c u l t i e s i n d i f f u s i o n experiments. G e o p h y s i c s , 6.  Ellison,  T . H.„ (1956). A t m o s p h e r i c T u r b u l e n c e . m e c h a n i c s . - Camb. U n i v . P r e s s .  Ellison,  T . H . , (1957). T u r b u l e n t t r a n s p o r t o f h e a t and momentum f r o m an i n f i n i t e r o u g h p l a n e . Jour, of F l u i d M e c h . , 2, 5, p p . 456.  Ellison,  T . H . and T u r n e r J . F l u i d Mech.  Grant,  S.,  (I960) .  and some d i f f i Advances i n  In press  Surveys  Jour.  H . L . , M o i l l i e t , A . , S t e w a r t , R . W . , (1959). A s p e c trum of t u r b u l e n c e at very h i g h R e y n o l d ' s number. N a t u r e 811, p p . 8 0 8 .  Handbook o f C h e m i s t r y and P h y s i c s , (1943). Chemical Publishing Co., Cleveland, Ohio. Hinze,  in  J.  Keulegan,  0., (1959).  Turbulence.  Rubber  McGraw-Hill.  G . H . , (1949). I n t e r f a c i a l i n s t a b i l i t y and m i x i n g in stratified flows. Jour, of Research, 437-  500.  M o e l w y n - H u g h e s , (1957) • P h y s i c a l C h e m i s t r y . p 1123 and 1260. Northrup, Parr,  E . F . , (1911). E x p e r i m e n t a l in liquids. Franklin Inst.  Pergamon  Press,  study of vortex motions J o u r . , 172, pp 211-216.  A . E . , (1936). On t h e p o s s i b l e r e l a t i o n s h i p b e t w e e n v e r t i c a l s t a b i l i t y and l a t e r a l m i x i n g p r o c e s s e s . C o n s e i l P e r m . I n t e r n a t . p . I ' E x p l o r . de l a M e r , J o u r n a l d u C o n s e i l , v . 1 1 , p . -303-313.  Patterson,  A . M . , (1953). T u r b u l e n c e s p e c t r u m s t u d i e s i n t h e sea w i t h hot w i r e s . L i m n o l o g y and O c e a n o g r a p h y  III,  2, p . 1 7 1 - 1 3 0 .  75  P a t t e r s o n , A. M., (I960). Hot w i r e t u r b u l e n c e measurements i n a t i d a l channel. (P.C.C. D12-95-20-08). P a c i f i c N a v a l L a b o r a t o r y , E s q u i m a l t , B. C. R e v e l l e , R. and F l e m i n g , R. H., (1934) as d i s c u s s e d i n The Oceans, P r e n t i c e H a l l . 1942. R i c h a r d s o n , L. F., ( 1 9 2 0 ) . The s u p p l y o f energy from and t o atmospheric e d d i e s . P r o c . Roy. Soc. London, A, 97:  354-373.  R i c h a r d s o n , L. F., ( 1 9 2 5 ) . Turbulence and v e r t i c a l temperat u r e d i f f e r e n c e near t r e e s . P h i l . Mag., 49: 81-90. S c o r e r , R. S., ( 1 9 5 7 ) . Experiments on c o n v e c t i o n o f i s o l a t e d masses of.buoyant f l u i d . J o u r . F l u i d Mech. 2, 533.  S c o r e r , R. S.,  (1958).  N a t u r a l Aerodynamics, Pergamon P r e s s .  S t e w a r t , R. W., (1959) ( a ) . The problem o f d i f f u s i o n i n a stratified.fluid. Atmospheric D i f f u s i o n and p o l l u t i o n , Advances i n G e o p h y s i c s , 6 , 3 0 3 - 3 1 1 . S t e w a r t , R. W., (1959) ( b ) . The n a t u r a l o c c u r r e n c e o f t u r b u l e n c e . J o u r . Geophys. Res., 6 4 , 1 2 , 2 1 1 2 - 2 1 1 5 . S v e r d r u p , H. U., Johnson, M. W. and F l e m i n g , R. H., The Oceans, P r e n t i c e H a l l .  (1942).  T a y l o r , G. I . , ( 1 9 2 7 ) . An experiment on t h e s t a b i l i t y o f superposed streams o f f l u i d . P r o c . Camb. P h i l . Soc,  23_, 1 9 2 7 ,  pp.  730-731.  T a y l o r , G. I . , ( 1 9 3 1 ) . I n t e r n a l waves and t u r b u l e n c e i n a f l u i d o f v a r i a b l e d e n s i t y . C o n s e i l . Perm. I n t e r n , pour l ' E x p l . mer, Rapp et P r o c . Verb., £ 6 : 3 5 - 4 3 . Townsend, A. A., ( 1 9 5 6 ) . The s t r u c t u r e o f t u r b u l e n t shear f l o w . Cambridge U n i v . P r e s s . Townsend, A. A., ( 1 9 5 8 ) . T u r b u l e n t f l o w i n a s t a b l y s t r a t i f i e d atmosphere. J o u r . F l u i d Mech., 2, 4, pp. 3 6 1 372.  T u r n e r , J . S.,  Soc.  (1957). A.,  22£,  Buoyant v o r t e x r i n g s . pp.  61-75.  Z w o r y k i n , V. K. and Ramberg, E. G., ( 1 9 4 9 ) . John W i l e y & Sons, New T o r k .  Proc.  Roy.  Photoelectricity.  PLATE  I  4 4.5cmr-  T  61 cm.  £ o  E  5  Shield  Ta n k  (£>  •1 Oscilloscope E u tO CM  *-23cmr*  Cam  Timing Contact Toppe t  Va I ve  Graduated Tube  PLAN  VIEW  Filling  Vent and Valve  — — Siphon (on—of f)  Tube  400 ml. Beaker Volume Measuring Valve  Bead  (adjustable)  Fall Tube Light Camera  ELEVATION Fig I  APPARATUS  Source  1.0  0.5 " Standard  V  Deviation  W„  0  4  cm.  cm/sec.-gm./cm.  0.42 0.41 J  I  1  1  0.5  i  Obs.  *>p/bz  3  32  I  I  I  0 .0025  I  I  10 3 I  i  1.0  t (sec.)  F i g. 2 a I.or  0.5 Standard  cm.  Wo  ~bp/dz  cm./sec. gm./cmf  3  .06 5  9. 5  0.5  0  1.0 t  Fig.2b  Deviation  DETAILED  (sec.)  MIXING  Obs. 5  |  l.Or  2 C  0.8  0.6  Standard  Deviation  0.4 3/D/9 z Obs.  cm./sec gm./cm  0.2  55  0  7 4  .0031  55 0.2  4  0.4  t (sec.)  0.6  l.Or  0.8  Standard  Q6  Deviation  0.4  0.2  Wo  2p/dz  cm./sec  gm./cm  9.4  .0030  9.7 2.0 Fig. 2  t (sec.)  DETAILED  o  4.0 MIXING  Obs. 4  4 5 6.0  Ol  0  .  I  I  '  •  I  i  1  1-0  t Fig.2  I  0.5  DETAILED  (sec.) MIXING  1  L :  1  i.or 2 9.  Standard  0.5  Deviation  dp/3r  Obs.  gm7cm? 0  4  .0 025 -J  .2  Fig. 2  J  .4  DETAILED  I  .6  •  '  t (sec.)  .8  '  I  1.0  4 I  I  1.2  L  MIXING.  I.or  O  Standard  0.5  •  .2  Fig. 3  .4  EFFECT  6  Deviation  In  Light  Uniform  Fluid  In  Dense  Uniform  Fluid  i  L  i  I  8  1.0  OF SALT UPON  MIXING  t (sec.)  I—i—I  1.2  DATA  1  T  T  r  1  1  r-—-i  1  Wo  Obs.  3/>/3z  cm.  cm./sec.  0.42  32  gm./cm.  4  0  31  0.41  r  1  10  .0025  3  2h E o  M Standard  •  i  05  i i i i i—i—i—i—i—i—i—i—i—i—i—«2.5 1.0 15 2.0 t (sec.)  1  i  r  T  Deviation  1  1  1  ~*  1  r  W  dp/dz  cm./sec.  gm./cm.  9.5  4  0  Obs. 5  E u N  Standard  Deviation  4b j  Fig.4  I  i  i  i  0.5  CENTER  i  t  1  1  u  1.0  (sec.) DISPLACEMENT  _i  L  Fig. 4  CENTER  DISPLACEMENT  1  ~1  1  1  1  T  1  1  1  i  r  i  **•  Vo  V E o  Vs.  5  Wo  Obs. .  dp/dz  J-"  cm.  cm./sec . gm./cm  V  081  32  0  4  0.82  32  .0025  3  3  T  X.  4  -  1  Standard Deviation  i  4f. 0  1  •  •  i  i  2.0  Fig. 4  CENTER  i  t  •  (sec.)  i  f  4.0  DISPLACEMENT  •  6.0  0  T  T  T  IStandard  Deviation  E  u  Vo  N  Wo  dp/dz  Obs.  cm.  cm./sec. gm./cm  0.63  32  0  4  0.62  32  .0025  4  3  4  4 g, 2.0 t  Fig. 4  6.0  4.0  CENTER  (sec.)  DISPLACEMENT  Nozzle Successive of Cone  Fig. 5  HORIZONTAL  SPREAD FLUID  of  IN  Outlines Puff Expansion  UNIFORM  r  r  (cm.)  (cm.)  E o.  M  a Ve  cm  Wo  cm./sec.  dp/3z  gm./cm ,0025 3  SPREAD  0.41  0.82  31  55  4  Obs. HORIZONTAL  3  IN  b  STRATIFIED  Extremeties  of  Poinls  .0031 4 FLUID  Fitted  Line  Plotted  r. {cm)  Points Fitted  Line  cm.  6  0.84  0.42 9.5  Wo  cm./sec.  97  Bp/dz  gm./cm  .0030  Obs.  Fig.  d  c  Extremeties of Plotted  HORIZONTAL  4  5 SPREAD  IN  . 0 0 30  5  STRATIFIED  FLUID  Fig.6  HORIZONTAL  SPREAD  IN  STRATIFIED  FLUID  400  M IO  u Q) to  •s. E u  CM \  O  1  >  CM •V IO  o 200  J -  k  3  Fig. 7  DEPENDENCE UPON  OF THE  _L  (sec-')  MIXING INITIAL  _l_  (0  FUNCTION CONDITIONS  T—i—i—T—i  Fig. 8  r  3-DIMENSIONAL THE  INITIAL  PLOT  OF  K  CONDITIONS  AGAINST  / i / i /  c  • •  /  /  •  k  z  z  z  \  /  1  , I  .1  1 '  ' . k  •  '  1  P'A, C.  I 1  i X  B I  Fig.9  * ^ '  1  A  A  1  2  -  B  X  1 J  A  2  i .° A,B, 0  GRAPHICAL OF  POWER 3  DETERMINATION LAW  VARIABLES  FOR  SURROUNDINGS  4000-  E o  NJ  2 000 -  2  4 t I sec.)  Fig. 12  DISPLACEMENT NEUTRALLY  OF A PUFF IN BUOYANT  SURROUNDINGS  ~i  1  "  r  0.3  o bJ  V  o o 0.2  E  0  (cm. ) 3  o  0 8  •  0. 6  X  0.4  X  Iii  X  a.  X X  o  I-  0.1  o  O _l  V  O  x  ,  Xx  l_  I  Ri  Fig. 13  o  POTENTIAL  500 g "3/>/3z ENERGY  AT  INITIAL  Vo  •  1000  2 / 3  MAXIMUM  PENETRATION  ENERGY  Z„  Original Density  A —B  Simplified  Case  P ro f i I e Possible Q- Zeql  o  ZmL  Fig.  14  ALTERATION  IN  STRUCTURE  DENSITY  Real  Case  1 3.  I  1  1  r  --  2  ~  _o X  0  >  o  ,  1  V  ( cm. ) 3  0  °  0.8  •  0,6  X  0.4  X  o  1_o o  O  ,  X X X  >  ,  r  i  o  1  1  1  •  •  1 500  1  F i g . 15  o o  RATIO  r  T  1  /i  1  i  y^z  1  1  Q'b/o/'hz  ,000 Vo  2 / 3  ED V O INITIAL LUME E NVTORLAUI M NE 1  1  1  1  r  X »-  Q. UJ Q  20 Fig. 16  Wo ( cm./sec. )  MAXIMUM AND  40  PENETRATION  EQUILIBRIUM  DEPTH  (Z  m  (Z  ) e q  .)  1  o o  1  1  1  1  1  1  1  1  1  1  o  r  V (cm. ) 3  0  o UJ  •  0.8 0.6  x  0.4  o  0.1  UJ^  o  4  o o  oo  _L  J_  R,  Fig. 17  ENERGY  500  1000  —  g y/'azVo  LOST  TO  INITIAL  2 7 3  DENSITY  STRATIFICATION  ENERGY  O >  J_  ~  g ^ / V 8 z Vo  DENSITY  TRANSFER  Ri  Fig. 18  NET  1000 z/3  COEFFICIENTS  1.0  1  1  • -1  :—j  6  —  i  o  •  o o  o 0 . 5 "o cr o a>  1  1  r—  i  ••  X  1  o  0.8  •  0 . 6  X  0 . 4  X  I  1  i  1  .  — Fig.19  i  1  I/R,  EQUILIBRIUM  1  1  1  i , 0 0  i  °  DISTANCE  MAXIMUM  PENETRATION  1  1  •  1  1  1  :  A/VO  o  —1  3  o  o  •  0.6  X  0.4  uj 1.0 " \ XX  UJ Q.  o  o  o  o  o  • • • •  1  (cm. )  ° X  \ E  'o« b o  X  °x  N  '  1  0.8  X X x  oo 1  0 F  ,  '  1  1  500 Q  '  2  0  j  /  R  '  1  1000  POTENTIAL ENERGY AT MAXIMUM PENETRATION KINETIC ENERGY AT SAME TIME IN NEUTRAL SURROUNDINGS  4 4.5cmr  T  £ u E ro u in to  61 cm.  1  Shield  Ta n k  Oscilloscope  E u  Mirrors  u> CM  Photocell Shield C a mer a *-23cm.-*  Timing Contact Tappet Graduated Tube  Cam Va I ve  PLAN  VIEW  Filling  Vent and Valve  Siphon (on—of f )  Tube  4 0 0 ml. Beaker Volume Measuring Valve  Bead  (adjustable)  Fall Tube Light  Source  Camera Shield  ELEVATION Fig. I  APPARATUS  1.0  0.5 Standard  b/>/bz Obs.  W  •  •  I  »  •  0.5  Deviation  4  c m.  0.42  cm/sec. gm./cm. 32 0  0.41  31  I  I  ;  I  I  .0025 I  1.0  t (sec.)  10 3  1  I  1  1  Fig. 2 a  i.or  0.5 Standard  Deviation  7>/0/3z Obs.  W,  ,4  cm./sec. gm./cm 9. 5  •  «  I  i  •  L  0.5  t Fig.2b  DETAILED  '  L  (sec.) MIXING  0  L  1.0  U  5  J  I  J  i  0  I  i  —  i  i  i  i  J  1  0.5  1.0  t Fig.2  —I  DETAILED  (sec.) MIXING  1——i—i  1  i.or 2 9.  0.5  Standard  Deviation  dp/dz Obs. grh/cmf  .2  Fig.2  _l  l_ .4  .6  DETAILED  j  . .8 t (sec.)  i  0  4  .0025  4  I—  1.0  L  1.2  MIXING.  I.Or  O  Standard  0.5 In  j  0  1  .2  Fig. 3  -I  1_  .4  EFFECT  .6  Deviation  Light  Uniform  Fluid  In Dense  Uniform  Fluid  J  L  I  .8  1.0  OF SALT UPON  MIXING  t (sec.)  I  I  1.2  DATA  I  1  1  I  i  I  i  Vo cm.  -  *  l  i  l  "i •  —  0.42  cm./sec. 3-2'  — -—  0.41  31  1  i  1  dp/dz  Wo  3  1  gm./cm.  Obs. 4  0  10  .0025  r  "  3  \  \ V Standard  Deviation  4a 0  i  i  i  i  •  i  •  i  Q5  1  1  1  '. I  i  I  "i  i  i  1.0  i  i  t  i _ i  .  I  .  I  1.5  •  i  i  >  i  i  2.0  (sec.)  1  1—  1  1  1  1  1  I  I  I  I  _J  2.5  :—r  -  r  4b 1 0  0.5 Fig.4  CENTER  t  (sec.)  1.0  DISPLACEMENT  i  J  t  1  T  1  1  T  r  1  Wo  r  Obs.  dp/Bz  cm./sec. gm./cm.  55  0  7  55  .0031  4  Jstondard  0  i  i.  i  i  i  i  0.5  i  i  i  i  1  (sec.)  1  1  1—:  t (sec.) Fig. 4  Deviation  i.O t  T  4  CENTER  DISPLACEMENT  ~r  i  i  i  ~r  1  1  i  I  - 1  1  S  ]  Standard  Deviation  Wo  dp/dz  cm./sec  gm./cm 4  Vo cm.  4e. •  •  i  i  0.42  9.6  0  3  0.42  9.5  .0030  5  I  I  t  «  1  1  1  3  Obs.  1  •L  1  •  1  (sec.)  1  •  4  1  1 "  -  -  \ \  x  Vi-"' V  >sT  Vo  Wo  cm.  cm./sec. g m . / c m  3  32  0  4  0.82  3 2  .0025  3  Deviation  Standard  -  4f.  0  2.0  4.0 t  Fig.4  .  4  081  •  Obs.  dp/dz  CENTER  •  i  (sec.) DISPLACEMENT  i  • 6.0  1 "•  1  ,  1-  • .  ]Standard  Deviation  j  •\ \  V  / r  y •  '  Vo  Wo  Bp/dz  cm.  cm./sec.  gm./cm  0.63  32  0  4  0.62  32  .0025  4  3  -  •  ,  Obs. 4  4g,  0  1  2.0  CENTER  M  Fig. 5  I  4.0 t  Fig. 4  1  HORIZONTAL  6.0  (sec.)  DISPLACEMENT  Nozzle  Successive of  Outlines Puff  Cone  Expansion  SPREAD FLUID  of  IN  UNIFORM  -  r.  (cm.)  r  (cm.)  E  M  b  a Vo  cm.  Wo  cm./sec. 4  3  Obs. 6  HORIZONTAL  SPREAD  3 li  g m . / c m .0025  dp/dz Fig.  0.41*:  3  IN  STRATIFIED  0.82?. 55 .0031 4  FLUID  Extremeties  of  Points Fitted  Line  Plotted  Extremities  of Plotted  Points Fitted  Line  Vo  cm.  Wo  cm./sec.  Bp/dz  gm./cm  Obs. Fig.  6  HORIZONTAL  SPREAD  4  IN  STRATIFIED  FLUID  r.  (cm.)  r  e cm.  Wo  cm./sec  "dp/dZ  .4  gm./cm?  Obs.  Fig.  6  HORIZONTAL  SPREAD  f  0.8 2  Vo  IN  0.6 2  32  32  .0025  .0025  3  (cm.)  4  STRATIFIED  Extremeties  of  Points - F itted FLUID  Line  Plotted  Fig.7  DEPENDENCE UPON  OF THE  MIXING INITIAL  FUNCTION CONDITIONS  k  1—I—i—i—i  Fig,8  i  1  3-DIMENSIONAL THE  INITIAL  1  PLOT  1  1  OF  1—i—i—r  K  CONDITIONS  AGAINST  • / i / i 1  /  c  • • /  X  .  A  ,  C  i • ,  ,  '  2  B I  Fig.9  X  J  1  A  2  i.'°A|B, 0  GRAPHICAL OF  POWER 3  DETERMINATION LAW  VARIABLES  FOR  Line  (Z')  4  = 100 { t'- 2.5 )  f  0  F i g . 10  I  I  20  SIMILARITY  I tWo / V  IN  L_  40 0  I  I / 3  NEUTRALLY  SURROUNDINGS  BUOYANT  l  60  1 —  1  —1  je  i  1  3IO  m./sec  1_  2 v = 0.80 o  1  IO  cm?  W = 9-4 cm./sec0  1  0  •  _  «  L  t (sec.)  Fig.  VELOCITY  OF  PUFF  BUOYANT  IN  NEUTRALLY  SURROUNDINGS T  1  1  4000-  £  V 0.80 o  2000  s  cm  Wo = 9-4 cm./sec.  2  4  6  t (sec.)  Fig. 12  DISPLACEMENT NEUTRALLY  OF A PUFF IN BUOYANT  SURROUNDINGS  T  1  1  1  1  1  1  r  1  0.3r ~ - Vo ( c m . ) 3  o o o UJ  0.2  £  o  0.8  •  0.6  X  0.4  x x xx  UJ  a.  o  0.1 o  o o o 1  -I  1  X  1 500  X X  At  I  w g 3/V3z  F i g . 13  POTENTIAL  ENERGY AT INITIAL  Original Density  A —B  _1  I  Vo ' 2  L  1000  3  MAXIMUM ENERGY  Simplified  PENETRATION  Case  Profile Possible 0- Zeqf.  UJ  Q  '  ZmL-  Fig.  14  ALTERATION  IN  DENSITY  STRUCTURE  0%  Real  Case  1  .  1  1  1  1  f  1  1  1  •  —  i  X  ~  X X  Vo ( cm. ) 3  oX  0  >  I  °  0.8  •  0.6  x  0.4  X  o o _D O  ;  o  1  1  1  i  1  j_  i  500  R,  F i g . 15  * R  A  T  l  x  |  0  r  i  x •  i  1  i  1000  /9> Wo  q'b/O/'hz Vo'  INITIAL VOLUME VOLUME ENTRAINED  I  1  r=—i  r  £ u  I 0.  O  20 Fig. 16  Wo ( cm./sec. )  MAXIMUM AND  40  PENETRATION  EQUILIBRIUM  DEPTH  ( Z  )  m  (Z q,) e  o°o  1  1  1  1  1  1  1  1  1  1  r  o  Vo (cm. ) 3  o UJ  •  0.8 0.6  x  0.4  o  0.1  o o  J_  oo  01  500  1000  -  Ri  Fig. 17  •  ENERGY  g *a/>/^z V o  LOST  TO  DENSITY  INITIAL  5  Ri  Fig.  1 8  NET  0  JO  0  —  DENSITY  .  2 / 3  U I  STRATIFICATION  ENERGY  '000  2  "°  gfc/V8z Vo TRANSFER  z/3  COEFFICIENTS  1  '  —  — i  1  1  1  1  '  1  o<b  l  o  9  ° o O  o "o 0.5 o  0  X  •  O  X X  •*  v  / v V o (cm. ) 3  °x  o  %X  •  0.8 0. 6  X  0.4  »  0  i  •  1  1  1  500  •  1 / R, EQUILIBRIUM  Fig. 19  1  1  1  i  1  1  1  1  ^  o  °x  •  1.0 " \  • •  XX o o  o  i  I  r"  PENETRATION  1  o  i  1000  DISTANCE  MAXIMUM J  i  Vo  (cm. ) 3  o  0.8  • X  0.6 0.4  X  •  X x  o o  0  <  0  1 1  1 L_  1  J  1  1  1  500  . .  1 1 D  1I  oo 1•  '  1000  POTENTIAL ENERGY AT MAXIMUM PENETRATION KINETIC ENERGY AT SAME TIME IN NEUTRAL SURROUNDINGS  

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