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Statistics and dynamics of coherent structures on turbulent grid-flow Loewen, Stuart Reid 1987

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STATISTICS AND DYNAMICS OF COHERENT STRUCTURES ON TURBULENT GRID-FLOW by STUART R. LOEWEN B.Sc. Physics, University of Manitoba, 1980 M.Sc. Physics, University of British Columbia, 1984 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY ia THE FACULTY OF GRADUATE STUDIES (DEPARTMENT of PHYSICS) We accept this thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA January 1987 ©Stuart R. Loewen, 1987 In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make it freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the head of my department or by his or her representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission. Department of P /4V5/ C S The University of British Columbia 1956 Main Mall Vancouver, Canada V6T 1Y3 DE-6G/81) ABSTRACT T h i s t h e s i s e x a m i n e s t h e s t a t i s t i c s a n d d y n a m i c s of t u r b u l e n t flow s t r u c t u r e s g e n e r a t e d b y t o w i n g a g r i d t h r o u g h a t a n k of water. T h e s t r u c t u r e s were m ade v i s i b l e by r e c o r d i n g t h e p a t h s o f a l u m i n u m t r a c e r s m o v i n g w i t h t h e w a t e r surface. F l o w p a t t e r n s r e c o r d e d u s i n g a t i m e - e x p o s u r e m e t h o d were m a n u a l l y a n a l y z e d t o e x t r a c t i n f o r m a t i o n o n t h e s t r u c t u r e s t a t i s t i c s . T h i s t w o - d i m e n s i o n a l flow field was f o u n d t o be c o m p o s e d of closed r o t a t i n g 'surface eddies', o p e n a n d l a r g e l y t r a n s l a t i o n a l ' r i v e r ' m o t i o n a n d s t a g n a n t re-gion s . E n e r g y d i s t r i b u t i o n s o f t h e eddies a n d r i v e r s were o b t a i n e d a n d c h a r a c t e r i z e d b y B o l t z m a n n t y p e d i s t r i b u t i o n s . A n e w l y d e v e l o p e d c o m p u t e r - a u t o m a t e d s t r u c t u r e i d e n -t i f i c a t i o n a n d flow field a n a l y s i s s y s t e m was used t o s t u d y t h e s t r u c t u r e d y n a m i c s . T h e s y s t e m a n a l y z e s d i g i t a l i mages o b t a i n e d f r o m v i d e o r e c o r d i n g s of t h e t r a c e r m o t i o n . T h e p r e d o m i n a n t e v o l u t i o n processes of i n i t i a l v o r t e x p r o d u c t i o n , e d d y p a i r i n g , v i s c o u s decay a n d t h e omega decay were e x a m i n e d . F l o w R e y n o l d s n u m b e r s , based o n b a r s p a c i n g , o f a b o u t 10,000 were e x a m i n e d . T h e s t r u c t u r e s t a t i s t i c s a n d d y n a m i c s s t u d y was p e r f o r m e d i n o r d e r t o e x a m i n e the v a l i d i t y a n d v i a b i l i t y of a new m o d e l f o r t u r b u l e n c e . T h e m o d e l p r e d i c t s t h e e v o l u t i o n o f a p o p u l a t i o n of s t r u c t u r e s u s i n g r a t e e q u a t i o n s where t h e r a t e c o e f f i c i e n t s are d e t e r m i n e d b y t h e i n d i v i d u a l s t r u c t u r e d y n a m i c s . A s u m m a r y of t h e m o d e l is p r e s e n t e d a n d c o n t r a s t e d w i t h m o d e l s based t h e t h e R e y n o l d s stresses as w e l l as c o m p u t a t i o n a l models. iii T A B L E O F C O N T E N T S TITLE PAGE i A B S T R A C T ii T A B L E OF CONTENTS iii LIST OF FIGURES iv ACKNOWLEDGMENTS vi C H A P T E R 1 INTRODUCTION 1 C H A P T E R 2 TURBULENT FLOW MODELS 7 2.1 The Reynolds Equations 9 2.2 Eddy Viscosity Models 13 2.3 Reynolds Stress Models 16 2.4 Spectral Dynamics 17 2.5 Computational Models 24 2.6 The Rate Equation Approach 26 C H A P T E R 3 FLOW FIELD ANALYSIS 31 3.1 The Towing Tank 32 3.2 Visualization and Coherent Structures 34 3.3 Our Methods 40 3.3.1 Manual Analysis 40 3.3.2 Automated Analysis 41 3.4 Anemometry Based Flow Analysis 52 CHAPTER 4 STRUCTURE STATISTICS 57 4.1 Experimental Observations 58 4.2 River Flow 64 4.3 Eddy Distributions 71 4.4 Energy Decay of the Surface Flow 79 C H A P T E R 5 STRUCTURE DYNAMICS 83 5.1 Near Surface Fluid Dynamics 84 5.2 Structure Evolution 90 5.3 Initial Vortex Production 94 5.4 Spontaneous Decay 103 5.5 The Omega Decay 112 5.6 Statistics from Dynamics 115 CHAPTER 6 CONCLUSION 116 References 121 iv LIST OF FIGURES 1- 1 Time-Exposure of the Surface Motion 2 2- 1 Relationship between Navier-Stokes based turbulence models 8 2-2 Spatial Correlation Curve 18 2-3 Taylor's hypothesis 19 2-4 The three-dimensional spectrum E(k) for fully developed turbulence 22 2- 5 Structure diagram of the rate equation model 28 3- 1 The towing tank 33 3-2 Classification scheme of flow visualization techniques. 35 3-3 Flow field analysis system 42 3-4 Raw digitized image 43 3-5 Successive tracer centers 45 3-6 Tracked tracer paths 46 3-7 Coherent structure identification 48 3-8 Identified structures 50 3- 9 Contour plot of identified structures 51 4- 1 Eddy formation behind a bar grid. 59 4-2 Typical structure identification 61 4-3 Evolution of surface flow pattern 63 4-4 River-speed scatter plot 65 4-5 Evolution of river-flow speed distribution 66 4-6 River-flow energy distribution 67 4-7 "Temperature" decay 70 4-8 Surface eddy flow structure 72 4-9 Averaged peripheral speed of eddies 73 4-10 Eddy size distribution N(X,R) 76 4-11 Eddy energy distribution N{X,E) 77 4-12 Energy densities as function of time 81 4-13 Eddy size-spectra time slice 83 4- 14 Graphical determination of rate coefficients 83 5- 1 Near surface boundary layers 86 V 5-2 Bulk motion of turbulent grid-flow 89 5-3 Structure evolution types 92 5-4 Eddy amalgamation 93 5-5 Eddy pairing 94 5-6 von Karman vortex street 95 5-7 Initial vortex formation 96 5-8 Near grid video "time" exposure 97 5-9 Evolution of velocity profile 106 5-10 Variation of rate coefficient A with structure size 107 5-11 Proportionality constant determination for A = (3/R2 108 5-12 Relation between rotational energy and total energy 110 5-13 Internal flow structure I l l 5-14 The omega decay dynamics 113 5-15 Time-exposed flow photo of two-cylinder flow 114 v i A C K N O W L E D G M E N T S L i f e is m a de more e n j o y a b l e b y h a v i n g c h e e r f u l a n d c o m p e t e n t p e o p l e t o w o r k w i t h . T h e r e have b e e n m a n y s u m m e r s t u d e n t s whose energy a n d d i l i g e n c e are m u c h a p p r e c i a t e d . F o r e m o s t a m o n g s t these are A l e x F i l u k w h o t o o k a n a c t i v e i n t e r e s t i n t h e e a r l y stages of t h i s w o r k a n d N o r m a n L o w h o b u i l t t h e t a n k t h a t r a r e l y l e a ked. T e c h n i c i a n s A l C h e u c k a n d P a u l B u r r i l l have b u i l t r e l i a b l e a p p a r a t u s a n d t a u g h t me t h e v a l u e of c a r e f u l work. I b e n e f i t e d f r o m t h e v a l u a b l e w o r k of a n d d i s c u s s i o n s w i t h f e l l o w s t u d e n t A l e x i s L a u . I was p r i v i l e g e d t o have a t h e s i s c o m m i t t e e of k n o w l e d g a b l e , i n t e l l i g e n t a n d i n t e r e s t e d persons. M a t e r i a l s u p p o r t f o r m y s e l f a n d t h e e x p e r i m e n t was p r o v i d e d t h r o u g h t h e N a t u r a l Sciences a n d E n g i n e e r i n g R e s e a r c h C o u n c i l o f C a n a d a . F i n a l l y I m u s t t h a n k m y s u p e r v i s o r D r . B o y e A h l b o r n w i t h o u t w h o m t h i s w o r k w o u l d n e v e r have b e e n s t a r t e d a n d t h i s t h e s i s never f i n i s h e d . CHAPTER 1 INTRODUCTION 1 C H A P T E R 1 INTRODUCTION It is t h e b a s i c c o n t e n t i o n of t h i s t h e s i s t h a t t h e s u r f a c e flow on g r i d - g e n e r a t e d t u r b u -lence is w e l l d e s c r i b e d by t h e s t a t i s t i c s o f t h e coherent s t r u c t u r e s o f w h i c h i t is composed. It is f u r t h e r a r g u e d t h a t t h e coherent s t r u c t u r e s t a t i s t i c s are d e t e r m i n e d b y t h e d y n a m i c s of i n d i v i d u a l s t r u c t u r e s ' i n t e r a c t i o n s a n d t h a t these d y n a m i c s c a n be c h a r a c t e r i z e d b y s i m p l e i n t e r a c t i o n m o d e l s w h i c h may i n t u r n be u s e d t o p r e d i c t t h e o b s e r v e d s t a t i s t i c s . T h i s t h e s i s s u p p o r t s these c l a i m s t h r o u g h a n i n v e s t i g a t i o n of t h e s t a t i s t i c s a n d d y n a m i c s o f c o h e r e n t s t r u c t u r e s o n t u r b u l e n t g r i d - f l o w . I n t h i s t h e s i s , turbulence is c o n s i d e r e d t o be a n o n - p e r i o d i c a n d u n s t e a d y fluid flow. F i g u r e 1-1 shows a p h o t o of t h e s u r f a c e m o t i o n . F a r f r o m b e i n g a r a n d o m ve l o c -i t y field, t h e flow is seen t o be c o m p o s e d o f l o c a l r e gions o f c o h e r e n t l y m o v i n g fluid. T h e s e "coherent s t r u c t u r e s " u n d e r g o e i t h e r c l o s e d , p r e d o m i n a n t l y r o t a t i o n a l , m o t i o n (the eddies) o r are o p e n a n d c o n t a i n m a i n l y t r a n s l a t i o n a l k i n e t i c e n e r g y (the r i v e r s ) . T h e e x p e r i m e n t a l a p p a r a t u s used t o generate t h e flow was a t o w i n g t a n k p r e v i o u s l y de-s i g n e d f o r flow v i s u a l i z a t i o n s t u d i e s by t h e a u t h o r . T h e fluid m o t i o n was generated by a v e r t i c a l b a r g r i d t h a t was towed t h r o u g h t h e w a t e r a t speeds c o r r e s p o n d i n g to a mesh R e y n o l d s n u m b e r of ~ 10 4 . F o r the s t r u c t u r e s t a t i s t i c s s t u d y , t h e flow was r e c o r d e d o n t i m e - e x p o s e d p h o t o g r a p h s s h o w i n g t h e p a t h s of a l u m i n u m flakes m o v i n g w i t h t h e fluid CHAPTER 1 INTRODUCTION F i g u r e 1-1 T i m e - E x p o s u r e o f the S u r f a c e M o t i o n . s u r f a c e . F r o m t e n p h o t o series, each of 10 m e s h w i d t h s p a n a n d 40 m e s h w i d t h s l o n g , 2000 surf a c e eddies were i d e n t i f i e d a n d s t u d i e d . T h e v e l o c i t i e s a n d p o s i t i o n s of a b o u t 20,000 t r a c e r s were r e c o r d e d a n d used t o s t u d y t h e energy o f the s u r f a c e flow s t r u c t u r e s . T h e p r e d o m i n a n t s t r u c t u r e e v o l u t i o n m e c h a n i s m s were i d e n t i f i e d a n d e x a m i n e d . I n ad-d i t i o n , b o t h t h e a v a i l a b l e l i t e r a t u r e a n d s u b s u r f a c e flow v i s u a l i z a t i o n were used t o s t u d y t h e r e l a t i o n s h i p between t h e surface a n d s u b s u r f a c e flow. F o r t h e s t r u c t u r e d y n a m i c s s t u d i e s t h e same flow g e n e r a t i o n a p p a r a t u s was used. However, i n p l a c e o f a still c a mera, a v i d e o r e c o r d e r was used t o r e c o r d t h e t r a c e r m o t i o n . CHAPTER 1 INTRODUCTION S T h e r e c o r d i n g was l a t e r d i g i t i z e d a n d a n a l y z e d u s i n g a c o m p u t e r - a u t o m a t e d coherent s t r u c t u r e i d e n t i f i c a t i o n a n d flow field a n a l y s i s s y s t e m . T h e s y s t e m was d e v e l o p e d i n t h e c o u r s e of t h i s r e s e a r c h b y M . Sc. s t u d e n t A l e x i s L a u . It a l l o w e d f o r g r e a t e r o b j e c t i v i t y i n t h e i d e n t i f i c a t i o n a n d d e s c r i p t i o n o f t h e flow s t r u c t u r e s . It a l s o is m o r e s u i t e d t o t h e d y n a m i c s s t u d i e s t h a n t h e t i m e - e x p o s u r e m e t h o d . It was s o m e w h a t f o r t u i t o u s t h a t i n t h e p u r s u i t of a n M.Sc. degree t h i s a u t h o r f o u n d t h a t c o h e r e n t s t r u c t u r e s v i r t u a l l y c o v e r e d t h e s u r f a c e of t h e i n i t i a l p e r i o d of d e c a y i n g g r i d t u r b u l e n c e . G r i d - g e n e r a t e d t u r b u l e n c e has s e v e r a l p r o p e r t i e s w h i c h m a k e i t a u s e f u l flow t o s t u d y . T h e flow o n t h e s u r f a c e of a m o d e r a t e l y a g i t a t e d v o l u m e o f heavy fluid is v e r y n e a r l y t w o d i m e n s i o n a l . A s s u c h , i t a l l o w s f o r a r e l a t i v e l y s t r a i g h t f o r w a r d a n a l y s i s as c o m p a r e d w i t h a f u l l y t h r e e - d i m e n s i o n a l flow. W h i l e a t w o - d i m e n s i o n a l flow is n o t a c o m p l e t e l y g e n e r a l t u r b u l e n t fluid s y s t e m , m u c h t h a t c a n be l e a r n e d f r o m the surface flow o f g r i d - g e n e r a t e d t u r b u l e n c e is p e r t i n e n t t o m o r e c o m p l e x flows. T h e r e l a t i v e ease of flow field e x t r a c t i o n a n d c o h e r e n t s t r u c t u r e i d e n t i f i c a t i o n m ake t w o d i m e n s i o n a l flows w e l l s u i t e d t o t h e s t u d y of c o h e r e n t s t r u c t u r e d y n a m i c s a n d s t a t i s t i c s . A de s i r a b l e f e a t u r e o f t h e s p a c e - f i l l i n g g r i d - f l o w is t h a t i n d i v i d u a l s t r u c t u r e s are c l o s e l y p a c k e d a n d t h u s r e l a t i v e l y s t a g n a n t i n t h e fluid f r a m e of reference. A l s o , t h e g r i d f i l l s t h e e n t i r e cross-s e c t i o n of t h e t a n k so t h a t t h e b u l k m o t i o n o f t h e fluid is n e g l i g i b l e . T h i s allows f o r ease i n s t r u c t u r e i d e n t i f i c a t i o n a n d t r a c k i n g . A final p o i n t i n f a v o u r o f s t u d y i n g g r i d -flow is t h e r e l a t i v e s i m p l i c i t y of t h e b o u n d a r y c o n d i t i o n s . T h e g r i d serves t o generate t h e v o r t i c a l s t r u c t u r e s i n a w e l l d e f i n e d m a n n e r b u t t h e y are t h e n a l l o w e d t o i n t e r a c t free f r o m c o m p l i c a t i n g b o u n d a r y c o n d i t i o n s . It is f o r t h e a bove reasons t h a t t h e surface flow on gr i d - g e n e r a t e d t u r b u l e n c e was t h e m a i n flow s t u d i e d i n t h i s t h e s i s work. However o t h e r CHAPTER 1 INTRODUCTION 4 flows were examined in order to isolate and characterize specific, dynamical processes observed in the more complex grid-flow. The major drawback of the grid-flow system is that it is not strictly two-dimensional. This motivated the use of subsurface flow visualization to examine the near surface fluid mechanics. Part of the motivation for this thesis was to test and develop a model for turbulence based on rate equations proposed by my supervisor Dr. Boye Ahlborn. This model de-scribes how the statistics may be determined from the dynamics of coherent structures using rate coefficients to characterize the dynamical processes. A detailed description of this model can be found in reference [l]. The model uses a rate equation to predict the evolution of a population of energetic flow structures from knowledge of their local dy-namics and their probability of interaction. In order to be successful, the model requires first that coherent structures be identified in a flow, second that they be characterized by simple and meaningful parameters and finally that their evolution dynamics be described by rate coefficients. Structure has been known to be present in flows such as the von Karman vortex street for over seventy years [2]. However, the presence of coherent structures in what were previously considered to be highly disorganized flows is a fairly recent discovery. Brown and Roshko's hallmark study of coherent structures in the mixing layer [3] was published in 1974. Structures have been found to be a part, not only of flows in which they are produced by the particular vorticity generation geometry, but also of highly developed turbulence. This means that coherent flow structures are an integral part of the non-linear unsteady fluid dynamics which govern the flow. As such, understanding CHAPTER 1 INTRODUCTION 5 t h e i r p r o p e r t i e s , d y n a m i c s a n d reasons f o r o c c u r r e n c e is e s s e n t i a l t o t h e u n d e r s t a n d i n g o f t u r b u l e n t f l u i d flow. T h e c u r r e n t i n t e r e s t i n c o h e r e n t s t r u c t u r e s is c o i n c i d e n t w i t h r e n e w e d use of flow v i s u a l i z a t i o n t e c h n i q u e s . T h i s i n t u r n is at least p a r t l y a r e s u l t o f t h e d e v e l o p m e n t of h i g h speed d i g i t a l c o m p u t e r s t o t h e p o i n t w h e r e i m a g e p r o c e s s i n g a n d a n a l y s i s t e c h n i q u e s c a n b e u s e f u l l y a p p l i e d t o e x t r a c t q u a n t i t a t i v e r e s u l t s f r o m images of fluid flow. I n t h i s t h e s i s , c o m p u t e r - a u t o m a t e d flow v i s u a l i z a t i o n has been use d b o t h f o r e x t r a c t i n g t h e v e l o c i t y field a n d f o r t h e i d e n t i f i c a t i o n o f regions of coherent flow. T h e o r g a n i z a t i o n of t h i s t h e s i s is as f o l l o w s . A f t e r t h i s i n t r o d u c t i o n , t h e second c h a p t e r r e v i e w s t h e e s t a b l i s h e d m o d e l s f o r t u r b u l e n t fluid flows. T h i s s u r v e y is i n c l u d e d t o present t h e s e t t i n g i n w h i c h t h i s t h e s i s r e s e a r c h has been c o n d u c t e d a n d p r o v i d e s a bas i s f o r a c o m p a r i s o n w i t h t h e r a t e e q u a t i o n a p p r o a c h w h i c h is d e s c r i b e d at t h e e n d of t h e c h a p t e r . T h e n e x t c h a p t e r r e v i e w s e x p e r i m e n t a l m e t h o d s c o m m o n l y used t o s t u d y t u r b u l e n t flow fields as w e l l as the m a n u a l a n d a u t o m a t e d flow v i s u a l i z a t i o n m e t h o d s used f o r t h i s t h e s i s work. C h a p t e r s 4 a n d 5 r e p o r t t h e r e s u l t s of t h e w o r k p e r f o r m e d t o e x a m i n e first t h e coh e r e n t s t r u c t u r e s t a t i s t i c s o n g r i d - g e n e r a t e d t u r b u l e n c e a n d t h e n t h e d y n a m i c s of these s t r u c t u r e s a n d finally how t h e t w o are r e l a t e d . T h e r e a d e r w h o is f a m i l i a r w i t h t u r b u l e n c e r e s e a r c h w i l l find these c h a p t e r s as w e l l as t h e la s t s ections of c h a p t e r s t w o a n d three of m o s t i n t e r e s t as t h e y c o n t a i n t h e b u l k of t h i s student's o r i g i n a l c o n t r i b u t i o n s . T h e thesis ends w i t h t h e c o n c l u s i o n . CHAPTER 1 INTRODUCTION 6 M u c h o f t h e w o r k p r e s e n t e d i n t h i s t h e s i s has been p u b l i s h e d elsewhere. Reference [1] p r e s e n t s t h e r a t e e q u a t i o n a p p r o a c h t o p r e d i c t i n g t h e s t a t i s t i c a l e v o l u t i o n o f t u r b u l e n t flow fields. Some p r e l i m i n a r y s t r u c t u r e d y n a m i c s o b s e r v a t i o n s were p r e s e n t e d i n reference [4]. T h e s t r u c t u r e s t a t i s t i c s s t u d y o f t h e f o u r t h c h a p t e r was first p u b l i s h e d i n s h o r t f o r m [5] w i t h a m o r e t h o r o u g h p r e s e n t a t i o n g i v e n i n reference [6]. A p r e l i m i n a r y p r e s e n t a t i o n o f t h e c o m p u t e r - a u t o m a t e d c o h e r e n t s t r u c t u r e i d e n t i f i c a t i o n a n d flow field a n a l y s i s s y s t e m w a s p u b l i s h e d as reference [7]. A mo r e t h o r o u g h p r e s e n t a t i o n of t h e s y s t e m together w i t h r e s u l t s o f t h e s t r u c t u r e i d e n t i f i c a t i o n s t u d y p r e s e n t e d i n t h e the fifth c h a p t e r of t h i s t h e s i s h a s r e c e n t l y b e en s u b m i t t e d f o r p u b l i c a t i o n (see reference [8]). CHAPTER £ TURBULENT FLO W MODELS 1 C H A P T E R 2 T U R B U L E N T F L O W M O D E L S " I a m a firm b e l i e v e r i n l e a r n i n g t o u n d e r s t a n d t h e forces at w o r k i n s t e a d of o p e r a t i n g f r o m a set of fixed r u l e s . O b e y i n g r u l e s w i t h o u t a n u n d e r s t a n d -i n g o f t h e reasons b e h i n d t h e m creates a n a p p r o x i m a t i o n of c o m p e t e n c e w h i c h leaves one v u l n e r a b l e t o t h e e x c e p t i o n s . " f r o m Safety: The Open Coast o r All your eggs in one Kayak, b y M a t t B r o z e [9] T h e p u r p o s e of t h i s c h a p t e r is t o r e v i e w t h e m o d e l s c u r r e n t l y u s e d t o d e s c r i b e a n d p r e d i c t t h e e v o l u t i o n of t u r b u l e n t flow fields. T h e r e is n o one g e n e r a l l y a c c e p t e d m o d e l for p r e d i c t i n g p r o p e r t i e s o f t u r b u l e n t fluid flows. I n o r d e r t o p r o d u c e s a t i s f a c t o r y r e s u l t s f o r a g i v e n flow s y s t e m , m o d e l s i n v a r i a b l y i n v o k e a s s u m p t i o n s w h i c h s e v e r e l y r e s t r i c t t h e i r r a n g e of a p p l i c a b i l i t y . However, a n a p p r e c i a t i o n of t h e v a r i o u s t h e o r e t i c a l approaches w i l l a d d d e p t h t o o u r u n d e r s t a n d i n g of t h e e x p e r i m e n t a l r e s u l t s t o be presented. T h i s b a c k g r o u n d is also p r e s e n t e d t o p r o v i d e a f r a m e w o r k f r o m w h i c h t o e x a m i n e t h e r a t e e q u a t i o n a p p r o a c h . A g o o d p r e s e n t a t i o n of t h e ' s t a n d a r d ' m o d e l s f o r t u r b u l e n c e m a y a l s o be f o u n d i n reference 10. T h e c h a p t e r is d i v i d e d i n t o s i x sections. T h e f i r s t s e c t i o n discusses t h e R e y n o l d s d e c o m p o s i t i o n o f t h e N a v i e r - S t o k e s e q u a t i o n s . T h e n e x t t w o s e c t i o n s d e s c r i b e models u s e d t o close t h e r e s u l t a n t R e y n o l d s stress e q u a t i o n s s t a r t i n g w i t h t h e t h e e d dy v i s c o s i t y CHAPTER 2 TURBULENT FLOW MODELS 8 m o d e l f o r t u r b u l e n t m i x i n g a n d g o i n g o n t o m o r e c o m p l e x m o d e l s of t h e R e y n o l d s stresses. T h e s p e c t r a l e n e r g y t r a n s f e r a p p r o a c h is d i s c u s s e d i n t h e f o u r t h s e c t i o n a n d t h e r e v i e w a s p e c t o f t h i s c h a p t e r c o n c l u d e s w i t h t h e c o m p u t e r based a p p r o a c h k n o w n as la r g e e d d y s i m u l a t i o n . T h e r e l a t i o n s h i p b e t w e e n t h e m o d e l s is s h o w n i n Fig.2-1. O u r r a t e e q u a t i o n m o d e l is s u m m a r i z e d i n t h e l a s t s e c t i o n . Navier-Stokes equations + continuity condition + boundary conditions Reynolds stress equations + boundary conditions Eddy viscosity model of stresses ZT3. Reynolds stress models Mixing length models k — e models Computer simulation Large Eddy Simulation Spectral dynamics Flow properties F i g u r e 2-1 R e l a t i o n s h i p b etween N a v i e r - S t o k e s b a s e d t u r b u l e n c e models. CHAPTER 2 TURBULENT FLOW MODELS 9 2.1 T h e R e y n o l d s E q u a t i o n s T h e s t a r t i n g p o i n t w i t h m o s t t u r b u l e n t flow m o d e l s are t h e fluid e q u a t i o n s of m o t i o n . We b e g i n b y w r i t i n g these e q u a t i o n s f o r a n i n c o m p r e s s i b l e fluid, ar+->5^ = ; ^ ( 2 - l 0 ) a n d g = 0. ( 2 - U ) T h e s t a n d a r d s u m m a t i o n c o n v e n t i o n is i m p l i e d a n d t h e i n d i c e s r a n g e over t h e three s p a t i a l d i m e n s i o n s . H e r e p, t h e fluid d e n s i t y , is assumed t o be c o n s t a n t . T h e left side o f eqn.(2-la) is t h e r a t e of change of v e l o c i t y u, of a fluid e l e m ent f o l l o w i n g i t s m o t i o n . T h e r i g h t s i d e c o n t a i n s t h e stress t e n s o r cr t J w h i c h d r i v e s t h e m o t i o n . E q u a t i o n ( 2 - l b ) is t h e c o n t i n u i t y c o n d i t i o n w h i c h expresses t h e c o n s e r v a t i o n of mass for a n i n c o m p r e s s i b l e m e d i u m . Stok e s ' r e l a t i o n is u s e d t o d e t e r m i n e t h e stress cr,y f r o m t h e r a t e of s t r a i n Sgj(= | ( | ^ + a n d t h e fluid d y n a m i c pressure p i n a n i s o t r o p i c N e w t o n i a n fluid, bij = -p6{j + 2/*i„. (2 - 2) H e r e is t h e K r o n e c k e r 6 a n d fx is t he d y n a m i c v i s c o s i t y . U s i n g t h e c o n t i n u i t y c o n d i t i o n i n t h e r e s u l t we a r r i v e at t h e e q u a t i o n s of m o t i o n f o r a n i n c o m p r e s s i b l e , i s o t r o p i c a n d N e w t o n i a n fluid: CHAPTER 2 TURBULENT FLOW MODELS 10 a n d du,-— = 0 ; ( 2 - 3 6 ) t h e N a v i e r - S t o k e s a n d c o n t i n u i t y e q u a t i o n s . u(= fi/p) is c a l l e d t h e k i n e m a t i c viscos-i t y . T o g e t h e r w i t h t h e a p p r o p r i a t e b o u n d a r y c o n d i t i o n s , b o t h i n space a n d t i m e , these e q u a t i o n s are t h e s t a r t i n g p o i n t f o r a l m o s t a l l s t u d i e s of i n c o m p r e s s i b l e , i s o t r o p i c , a n d N e w t o n i a n f l u i d m o t i o n . T h e n e x t s t e p i n t h e d e v e l o p m e n t of m o s t t u r b u l e n c e m o d e l s is t o t r e a t t h e d y n a m i c a l v a r i a b l e s U i a n d p as b e i n g c o m p o s e d o f a m e a n a n d a f l u c t u a t i n g c o m p o n e n t : « t = U{ + (2 — 4a) p = P + p. ( 2 - 4 6 ) T h i s p r o c e d u r e is c a l l e d t h e R e y n o l d s d e c o m p o s i t i o n a f t e r O s b o r n e R e y n o l d s [ l l ] . Here Ui a n d P are t h e average, e i t h e r t i m e o r e n s e m b l e averaged, c o m p o n e n t s of t h e v e l o c i t y a n d p r e s s u r e fields. F o r some t i m e d e p e n d e n t q u a n t i t y a(t), t h e t i m e average a(t) is d e f i n e d as, j ft+T/2 a{t) = l i m r ^ - j a{t')dt' ( 2 - 5 ) 1 Jt'=t-T/2 I t is m e a n i n g f u l t o speak of t i m e - d e p e n d e n t t i m e averages i f , da(t)/dt < \J(da[t)/dt)2 (2 - 6) w h i c h s t a t e s t h a t t h e averages m u s t change m u c h more s l o w l y t h a n t h e average change. F l o w fields f o r w h i c h t h e time-averaged q u a n t i t i e s do not change w i t h t i m e are t e r m e d statistically steady. CHAPTER 2 TURBULENT FLOW MODELS 11 T h e str e s s t e n s o r o\y c a n a l s o b e d e c o m p o s e d i n t o m e a n a n d f l u c t u a t i n g c o m p onents, Bij: = Zij +aijt ( 2 - 7 ) w h e r e t h e t i m e i n d e p e n d e n t m e a n stress t e n s o r E t J has t h e a n a l o g o u s Stokes' r e l a t i o n , E t y = -PSij + 2/ i S y (2 - 8) a n d t h e z e r o m e a n fluctuation stresses are g i v e n by, Oij = -pSij + 2psij. (2 - 9) T h e m e a n s t r a i n r a t e t e n s o r 5,-y a n d fluctuation s t r a i n r a t e t e n s o r s t J are g i v e n b y 1 (dUi SUA s « = 2 ( ^ + ^ 7 j < 2 - 1 0 > a n d 1 / dui du,-\ r e s p e c t i v e l y . T h e e q u a t i o n s of m o t i o n f o r t h e m e a n flow m a y be o b t a i n e d b y s u b s t i t u t i n g t h e R e y n o l d s d e c o m p o s i t i o n s i n t o t h e fluid e q u a t i o n s o f m o t i o n , e q n s . ( 2 - l ) , a n d t h e n t a k i n g t h e t i m e average of a l l t e r m s i n t h e r e s u l t i n g e q u a t i o n . F i n a l l y n o t i n g t h a t , f o r p h y s i c a l l y r e a l i s t i c flow f i e l d s , a v e r a g i n g c o m m u t e s w i t h d i f f e r e n t i a t i o n t h e r e s u l t i n g e q u a t i o n s f o r t h e m e a n flow are, dUi du< Id , "'sTS'iei = (2-I2) CHAPTER 2 TURBULENT FLOW MODELS Using the continuity equation to rewrite the second term we have TT dUj 1 d . , or alternately using expression (2-8) for the mean stress dU. 1 d . U } ThT = ~P dx~ (~P6ij + 2 / i 5 , J ~ *2 ~14a) with the continuity of the mean flow a i - r 0 - { 2 ~ 1 4 b ) The contribution of the turbulent motion to the mean stress is called the Reynolds stress tensor, Tij = -puiUj. (2 - 15) We have now arrived at the starting point of the majority of the turbulence models being used and studied at the present time. The decomposition of the flow into mean and fluctuating components in the form of either eqns.(2-4), or equivalently equations (2-7) through (2-9), allows for some tractable analysis and useful results, witness to this being that this flow decomposition was first introduced by Reynolds in 1895 and is still much used today. It should be emphasized at this point however that by introducing eqn.(2-4) the number of unknowns has been increased by four, three velocity components and one pressure variable, without increas-ing the number of equations. The task for the turbulence modeller is to close the system using a suitable model for the Reynolds stress tensor. In order to solve the new equations CHAPTER S TURBULENT FLO W MODELS 13 o f m o t i o n , e i t h e r p h y s i c a l l y m o t i v a t e d or e m p i r i c a l l y o b t a i n e d i n f o r m a t i o n i n t h e f o r m o f a d d i t i o n a l r e l a t i o n s m u s t be added. T h a t is t h e o b j e c t of t h e e d d y v i s c o s i t y m o d e l s s u c h as t h e m i x i n g l e n g t h a n d so c a l l e d k — e m o d e l s as w e l l as t h e m o r e g e n e r a l R e y n o l d s s t r e s s m o d els. W h i l e t h e R e y n o l d s d e c o m p o s i t i o n is r e s p o n s i b l e f o r t h e c o n s i d e r a b l e success i n t h e d e s c r i p t i o n a n d c a l c u l a t i o n of t u r b u l e n t b o u n d a r y l a y e r a n d free shear l a y e r flows i t is a l s o r e s p o n s i b l e f o r t h e l i m i t e d a p p l i c a b i l i t y o f t h e a p p r o a c h . It c a n be m i s l e a d i n g i n a p p l i c a t i o n a n d r e s u l t s , p a r t i c u l a r l y i f a p p l i e d t o flows f o r w h i c h t h e l o c a l flow s t r u c -t u r e s c o h e r e n t l y a d d t o t h e m e a n flow field. I f t h e d y n a m i c s of these l o c a l fluctuations are i n h e r e n t l y n o n - l i n e a r , e x c l u s i o n of t h e i r c o h e r e n t c o m p o n e n t w i l l r e n d e r p h y s i c a l l y m o t i v a t e d m o d e l l i n g i m p o t e n t . 2 . 2 E d d y V i s c o s i t y M o d e l s T h e t a s k is now t o p r o v i d e a n e x p r e s s i o n f o r t h e R e y n o l d s stress t e n s o r r,-y so t h a t t h e p a r t i a l d i f f e r e n t i a l e q u a t i o n s of m ean m o t i o n , eqns.(2-14), c a n be solve d . T h e s t a n d a r d a p p r o a c h is t o w r i t e u,Uj as p r o p o r t i o n a l t o t h e fluctuation stress t e n s o r s t J i n a m a n n e r s i m i l a r t o h ow t h e m e a n s t r a i n r a t e 5 t J is r e l a t e d t o t h e v i s c o u s stress. T h e r e , the v i s c o u s t e r m is uSij w i t h n q u a n t i f y i n g t h e r a t e of m o m e n t u m t r a n s p o r t due t o m o l e c u l a r m o t i o n . I n a n a n a l o g o u s m a n n e r a n eddy v i s c o s i t y is u s e d t o q u a n t i f y t h e r a t e o f m o m e n t u m t r a n s p o r t due t o t u r b u l e n t fluctuations by w r i t i n g Uiiij = —2i/rSij (2 - 16) CHAPTER 2 TURBULENT FLO W MODELS 14 where is called the eddy viscosity. This relation was first developed by J. Boussinesq 112,13] starting in 1877. While the kinematic viscosity u is an intrinsic property of the fluid the eddy viscosity is a property of the fluid flow and as such not generally a local parameter as is implied by writing eqn.(2-16). Models based on this relation are successful when applied to flows for which a local approximation for VT is not too grossly violated. These are flows for which a turbulent length scale £ is much smaller than the One of the oldest and most successful models for is obtained by pushing the analogy between turbulent and molecular transport a step further. The mixing length hypothesis was first put forward by Prandtl in 1925 [14] when he suggested using, for the eddy viscosity. Here u' and £ are appropriately chosen velocity and length scales while the constant c\ must be experimentally determined for each type of flow. An example of an appropriate choice for u' and I is for wall-bounded shear flow where u' is the velocity difference across the boundary layer and £ is the boundary layer thickness. The derivation of this relation follows the lateral motion of momentum carrying fluid lumps a distance £ across a shear layer where it is surrounded by fluid of average velocity difference u'. The reader interested in the rationale behind eqn.(2-17) will find presentations by either Tennekes and Lumley [15] or Schlichting [16] worth reading. If the velocity and length scales were known everywhere in a flow and if the mixing length model was realistic the closure problem would be solved. That this is not so is partially due to the fact that in turbulent flows the largest eddies tend to have size scales VT = c\u't (2 - 17) CHAPTER £ TURBULENT FLOW MODELS 15 c o m p a r a b l e t o t h e w i d t h of t h e flow. T h e l a r g e s t eddies are a l s o t h e m o s t efficient at e x t r a c t i n g energy f r o m t h e m e a n s t r a i n r a t e a n d t h u s c o n t r i b u t e m o s t t o t h e R e y n o l d s stre s s i n f l u e n c e o n t h e m e a n flow. A n o t h e r s i g n i f i c a n t a s p e c t of t u r b u l e n t flows f o r w h i c h eqn.(2-17) does n o t a c c o u n t is t h e u s u a l m u l t i p l i c i t y o f l e n g t h scales. M a n y m o d e l s have been p r o p o s e d t o d e s c r i b e the l e n g t h s c a l e v a r i a t i o n w i t h i n a flow. A s l i g h t e l a b o r a t i o n of t h e b o u n d a r y l a y e r m o d e l d e s c r i b e d a bove has t h e v e l o c i t y u' g i v e n by t h e l o c a l fluid speed a n d £ d e t e r m i n e d as t h e d i s t a n c e t o t h e w a l l . T h e m o r e s o p h i s t i c a t e d m odels, t h e so c a l l e d o n e - e q u a t i o n m o d e l s , use a N a v i e r -S t o k e s d e r i v e d p a r t i a l d i f f e r e n t i a l e q u a t i o n t o d e t e r m i n e t h e energy k w h i c h i n t u r n defines t h e fluctuation v e l o c i t y scale u \ T w o - e q u a t i o n m o d e l s use a l g e b r a i c e q u a t i o n s w i t h e x p e r i m e n t a l l y d e t e r m i n e d p r o p o r t i o n a l i t y c o n s t a n t s t o d e s c r i b e b o t h the l e n g t h a n d v e l o c i t y scale i n t e r m s of k a n d t h e d i s s i p a t i o n e. k a n d e themselves are d e t e r m i n e d f r o m N a v i e r - S t o k e s d e r i v e d p a r t i a l d i f f e r e n t i a l e q u a t i o n s . A c c o r d i n g t o W.C. R e y n o l d s [17], as of 1976 o n l y the z e r o - e q u a t i o n m odels, ones u s i n g o n l y a p a r t i a l d i f f e r e n t i a l e q u a t i o n f o r t h e m e a n flow, were b e i n g u s e d i n p r a c t i c e b y t h e "more s o p h i s t i c a t e d e n g i n e e r i n g i n d u s t r i e s " . I n a mo r e recent r e v i e w F e r z i n g e r [24] s t a t e s , " M i x i n g l e n g t h m o d e l s w o r k v e r y w e l l i n t w o - d i m e n s i o n a l shear flows. T h e y c a n be m o d i f i e d t o account f o r e x t r a effects s u c h as p r e s s u r e g r a d i e n t s , c u r v a t u r e , a n d t r a n s p i r a t i o n . . . . T h e m a j o r d i s a d v a n t a g e of m i x i n g l e n g t h m o d e l s is t h e d i f f i c u l t y t h e y have w i t h c o m p l e x flows." CHAPTER 2 TURBULENT FLO W MODELS 16 2.3 Reynolds Stress Models Reynolds stress models were developed as a way of avoiding the eddy viscosity as-sumption and the subsequent limitations. The basic approach is to use partial differential equations to determine the Reynolds stresses. The equations used come from the fluc-tuation counterpart of eqn.(2-14) obtained from the original Navier-Stokes equations in the following way: The mean and fluctuating decomposition of the velocity and pres-sure, eqns.(2-4), are again substituted into the Navier-Stokes equations but the mean equations of (2-14) are now subtracted to obtain equations for the fluctuating quantities. The fluctuation equations are cross-multiplied by u,- and then averaged to get equations for the Reynolds stresses, the normal components of which are the fluctuation kinetic energies. These equations have the general form duiU-j —-^L = Convection + Production — Dissipation + Redistribution (2 — 18) Whereas the production term is prescribed, the convection, dissipation, and redistribu-tion terms must be modelled. The closure problem still haunts us! Moreover the con-stants in the model are more difficult to obtain due to the lack of experimental techniques for the direct measurement of the quantities such as the pressure-strain correlations re-sponsible for the redistribution of stress components. For these reasons Ferzinger [24] has suggested using full-scale computer simulations to test the models. According to Ferzinger, Reynolds stress models nearly doubled the cost of computing a given flow without yielding results significantly better than those produced by the two-equation models mentioned in the last section. CHAPTER 2 TURBULENT FLO W MODELS 17 2.4 Spectral Dynamics D e v e l o p m e n t of m o d e l s f o r t h e R e y n o l d s stress t e n s o r r e q u i r e s k n o w l e d g e of t h e d y n a m i c s o f t u r b u l e n c e . T h e t u r b u l e n t energy e q u a t i o n s , t h e n o r m a l c o m p o n e n t s of eqns . (2-18) c o n t a i n o n l y m e a n p r o d u c t s of f l u c t u a t i n g q u a n t i t i e s at one p o i n t i n space. I n o r d e r t o s t u d y t h e l e n g t h scales of t h e t u r b u l e n t f l u c t u a t i o n s we need t o c o n s i d e r f l u c t u a t i n g q u a n t i t i e s w h i c h are m e a s u r e d at d i f f e r e n t p o i n t s i n space. T h e m o st general, s t a t i s t i c a l l y steady, t w o - p o i n t s p a t i a l c o r r e l a t i o n between f l u c t u a t i n g v e l o c i t y c o m p o n e n t s m a y b e w r i t t e n Cij ( i ,r) = Ui(x,t)xij(x + r,t) (2 - 19) w h e r e x is t h e p o s i t i o n w h e r e t h e c o r r e l a t i o n is d e f i n e d a n d x + r* is the p o s i t i o n where t h e j t h v e l o c i t y c o m p o n e n t is defined. T h e d i m e n s i o n l e s s c o r r e l a t i o n is c a l l e d t h e c o r r e l a t i o n c o e f f i c i e n t a n d is g i v e n by iE,(x-0=7lM^g±M- (2_20) yju2{x,t)sju]\x + r,t) R f o r m s a second r a n k t e n s o r whose i n d i v i d u a l c o m p o n e n t s are s u c h t h a t -\<Rij(x,r)<\ ( 2 - 2 1 ) T h i s agrees w i t h t h e i n t e r p r e t a t i o n t h a t J2,-y is a measure of the degree of c o r r e l a t i o n b e t w e e n t h e two v e l o c i t y c o m p o n e n t s . A t y p i c a l s p a t i a l c o r r e l a t i o n c u r v e f o r i d e n t i c a l v e l o c i t y c o m p o n e n t s a p p e a r s i n Fig.2 - 2 . A t f = 0 we see f r o m eqn.(2-20) t h a t Raa = 1. It is a p r o p e r t y of t u r b u l e n t flows t h a t fluctuating v e l o c i t y c o m p o n e n t s are u n c o r r e c t e d for s u f f i c i e n t l y l a r g e s e p a r a t i o n s CHAPTER 2 TURBULENT FLO W MODELS 18 F i g u r e 2-2 Spatial Correlation Curve, and so we have Rij[x,r)=>0 for large \ r\ ( 2 - 2 2 ) For homogeneous flows Rij(x,f) =» Rij(f). If the mean fluid velocity is in the x\ di-rection for example then i 2 j i ( r i ) is called the 'longitudinal correlation coefficient' and i?22 ( r i ) and Rzz,{r\) are 'lateral spatial correlation coefficients'. A simple measure of the length scale of the energy containing fluctuations is given by (2 - 23) called the integral length scale, see Fig.2-2. In some flow situations a useful measure of the length scales in the fluctuations can be found from the temporal correlations. The auto-correlation curve is the average product of the same quantity measured as a function of separation time T, Caa{x,T) = ua{x,t)ua[x,t + T) (2 - 24) CHAPTER 2 TURBULENT FLO W MODELS 19 T h i s c u r v e is m u c h more e a s i l y o b t a i n e d f r o m e x p e r i m e n t t h a n t h e space c o r r e l a t i o n s d i s c u s s e d above. O n l y one v e l o c i t y m e a s u r i n g p r o b e , a c o r r e l a t o r a n d a s i g n a l d e l a y u n i t a r e needed. T h e a u t o - c o r r e l a t i o n c u r v e is o b t a i n e d b y s w e e p i n g t h e delay t i m e T r a t h e r t h a n p h y s i c a l l y m o v i n g a v e l o c i t y p r obe. T h e t i m e scale o f t h e e n e r g y c o n t a i n i n g fluctuations, t h e i n t e g r a l t i m e s c a l e Te, is d e f i n e d as T h e a u t o - c o r r e l a t i o n is o f i n t e r e s t w h e n one w a n t s t o e x a m i n e s t r u c t u r e d y n a m i c s as i t is m o s t c l o s e l y r e l a t e d t o t h e t i m e e v o l u t i o n o f t h e flow. I t is a l s o u s e f u l i n t h e s t u d y of t h e s p a t i a l s t r u c t u r e of t u r b u l e n t fluctuations w h e n i t c a n b e r e l a t e d t o t h e s p a t i a l c o r r e l a t i o n s . T h i s r e l a t i o n s h i p a n d th e c o n d i t i o n s of i t s a p p l i c a b i l i t y are d i s c u s s e d below. oo (2 - 25) T'=0 U, probe F i g u r e 2-3 T a y l o r ' s h y p o t h e s i s . E q u i v a l e n c e o f t i m e a n d space c o o r d i n a t e s . CHAPTER 2 TURBULENT FLO W MODELS 20 C o n s i d e r a flow field m o v i n g p a s t a v e l o c i t y p r o b e , see Fig.2-3. A s s u m e t h e convec-t i o n s p e e d is t he same f o r a l l fluid e l e ments a n d t h e c o h e r e n t s t r u c t u r e s d o n o t evolve a p p r e c i a b l y i n t h e t i m e i t takes t h e m t o flow p a s t t h e p r o b e . T h e p r o b e measures t h e ve-l o c i t y as a f u n c t i o n of t i m e . T h e space c o r r e l a t i o n is o b t a i n e d f r o m t h e t i m e c o r r e l a t i o n o f t h e t e m p o r a l l y v a r y i n g s i g n a l by s i m p l y m u l t i p l y i n g t h e t i m e a x i s of t h e a u t o - c o r r e l a t i o n by t h e p r o b e speed, Uc. t h e i n t e g r a l t i m e scale c a n t h u s be r e l a t e d t o t h e i n t e g r a l l e n g t h scale b y t h e r e l a t i o n , T h i s c o o r d i n a t e t r a n s f o r m a t i o n a n d t h e c o n d i t i o n s w h e n i t c a n be a p p l i e d are c a l l e d " T a y l o r ' s h y p o t h e s i s " a f t e r G.I. T a y l o r w h o first f o r m a l i z e d i t o r a l t e r n a t e l y t h e " f r o z e n " t u r b u l e n t field h y p o t h e s i s . T h e i m p o r t a n t p o i n t t o no t e is t h a t Taylor's h y p o t h e s i s c a n o n l y be a p p l i e d w h e n t h e flow field does n o t change a p p r e c i a b l y d u r i n g t h e t i m e i t takes t h e s a m p l i n g p r o b e t o t r a v e r s e a d i s t a n c e g r e a t e r t h a n t h e l e n g t h scale o f i n t e r e s t . I n a d d i t i o n t h e fluid elements m u s t have a c o n s t a n t c o n v e c t i o n v e l o c i t y , Uc. F o r t h e p u r p o s e of s t u d y i n g coherent s t r u c t u r e s Taylor's h y p o t h e s i s m a y b e a p p l i e d w h e n t h e eddies have a l i f e t i m e t h a t is l o n g c o m p a r e d w i t h the p r o b e t r a n s i t t i m e . A d d i t i o n a l l y t h e s t r u c t u r e d r i f t v e l o c i t y m u s t be n e g l i g i b l e c o m p a r e d w i t h t h e p r o b e speed. J u s t as a fluctuating flow field c a n be d e s c r i b e d by t h e s p a t i a l c o r r e l a t i o n s we can d e s c r i b e i t by t h e F o u r i e r t r a n s f o r m s of the c o r r e l a t i o n s w i t h o u t loss of i n f o r m a t i o n . T h e CaB[T) = CaP(X/Uc) (2 - 26) Te = Le/Uc (2 - 27) CHAPTER 2 TURBULENT FLOW MODELS F o u r i e r t r a n s f o r m of t h e s p a t i a l c o r r e l a t i o n t e n s o r is c a l l e d t h e t h r e e d i m e n s i o n a l wave vector spectrum, (2n)< I Ui(x,t)uj(x + r,t)exp(ik • f)df (2-28) Jv with | k |= 2n/X being the wave number. It is impractical to measure or work with all velocity components needed to define this spectrum. Another spectrum which is simple enough to be useful is the three-dimensional spec-trum. This spectrum is obtained by removing the directional information from the wave vector spectrum by integrating $ij(k,x) over spherical shells centered at k = 0; E(k) = ± f *ii{k)do. (2-29) It is most useful when applied to isotropic flows as information about flow orientation has been removed. The factor of ^ is included to make the integral of the three-dimensional spectrum E(k) equal to the kinetic energy per unit mass r°° i / E(k)dk = -u7u7. (2 - 30) Jo 2 Discussions of spectral dynamics in isotropic flows are presented in a number of texts [15, 18]. A more rigorous presentation is given by Hinze [19]. Figure 2-4 shows the qualitative variation of the three-dimensional spectrum E(k) for fully developed flows. The energy is supplied to the turbulent spectrum at wavenumber k0 = 1/L0 where L0 corresponds to a characteristic length scale of the flow generation mechanism. For three-dimensional flows the energy continually cascades to higher wavenumber due to the action CHAPTER 2 TURBULENT FLOW MODELS E[k) energy containing wavenumbers inertia] k subrange universal equilibrium regime F i g u r e 2-4 The three-dimensional spectrum E(k) for fully developed turbulence. of vortex stretching. The integral length scale of eqn.(2-23) is shown near the center of the distribution of energy containing wavenumbers. The shape of the spectrum in this region is dependent on the flow generation mechanism and the boundary conditions. In 1941 Kolmogorov hypothesized the existence of a universal equilibrium regime where the flow is independent of the shape of the spectrum at lower wavenumbers [20]. The idea is that in the energy cascade to smaller size motion the flow loses all information about the large scale geometry. The shape of the spectrum is solely determined by the dissipation e and the kinematic viscosity v. For extremely high Reynolds numbers, Kolmogorov further hypothesized the existence of an inertial subrange where the shape of the spectrum can only depend on the rate of energy transfer through the spectrum. This transfer rate is equal to the dissipation as the inertial subrange carries all the energy that is dissipated in the viscous dominated small scales near kd- A simple dimensional analysis leads to the now famous e 2 / 3 A ; - 5 / / 3 functional form of E(k). k~5/3 inertial subranges have been found CHAPTER £ TURBULENT FLOW MODELS i n m a n y h i g h R e y n o l d s n u m b e r flows i n c l u d i n g a t i d a l c h a n n e l flow [21]. S t e w a r t a n d T o w n s e n d [53] have s h o w n t h a t t h e c o n d i t i o n s necessary f o r t h e e x i s t e n c e o f t h e i n e r t i a l s u b r a n g e a r e n o t m e t i n l a b o r a t o r y c o n d i t i o n s . T h i s does n o t , however, p r e v e n t fc~5/3 s p e c t r a f r o m b e i n g f o u n d i n t h e l a b as o f t e n happens*. W h i l e i n e r t i a l ranges are e x p e c t e d i n t w o - d i m e n s i o n a l t u r b u l e n c e , see ref. [23], t h e s p e c t r a l d y n a m i c s are q u i t e d i f f e r e n t f r o m w h a t is f o u n d i n three d i m e n s i o n s . V o r t e x s t r e t c h i n g , t h e p r e d o m i n a n t m e c h a n i s m o f e n e r g y t r a n s f e r i n t h r e e - d i m e n s i o n a l flows, is n o t p o s s i b l e f o r t w o - d i m e n s i o n a l m o t i o n . A l s o , i n t h e t w o - d i m e n s i o n a l i n e r t i a l r a n g e t h e mean-square v o r t i c i t y as w e l l as t h e k i n e t i c e n e r g y p e r u n i t mass are c o n s t a n t s of t h e m o t i o n . T h i s has t h e effect t h a t t r a n s f e r u p w a r d i n w a v e n u m b e r m u s t be a c c o m p a n i e d b y c o m p a r a b l e o r gr e a t e r d o w n w a r d t r a n s f e r t h r o u g h t h e a c t i o n o f e d d y p a i r i n g . I n two-d i m e n s i o n a l t u r b u l e n c e the energy cascade is t o lower w a v e n u m b e r w i t h i n e r t i a l ranges f o u n d a t l a r g e r size-scales t h a n t h e scale at w h i c h t h e energy is f e d i n t o t h e fluid s y stem. T u r b u l e n t flows are m o d e l l e d u s i n g t h e t h r e e - d i m e n s i o n a l s p e c t r u m b y F o u r i e r de-c o m p o s i t i o n of t h e N a v i e r - S t o k e s e q u a t i o n s . T h e e q u a t i o n s are t h e n c l o s e d i n a m a n n e r s i m i l a r t o t h a t i n w h i c h t h e R e y n o l d s stresses are e s t i m a t e d . A s p e c t r a l energy t r a n s -fer f u n c t i o n is i n t r o d u c e d w h i c h d e s c r i b e s how processes a t one w a v e n u m b e r affect t h e a m p l i t u d e s a t a n o t h e r . T h e p r o b l e m i s , however, t h a t e d d y s are n o t s t a t i o n a r y waves a n d so t h e s p a t i a l F o u r i e r d e c o m p o s i t i o n is n o t a n a t u r a l d e s c r i p t i o n o f t h e m . E d d y s are l o c a l i z e d i n space a n d so are a s s o c i a t e d w i t h m a n y wave n u m b e r s a n d t h e phase r e l a t i o n a m o n g s t t h e m i n a n o n - t r i v i a l way. S p e c t r a l energy t r a n s f e r f u n c t i o n s are t h u s e i t h e r q u i t e c o m p l e x o r v e r y r e s t r i c t e d i n a p p l i c a b i l i t y . H o w e v e r i n s i g h t i n t o s p e c t r a l d y n a m i c s * I. S. G a r t s h o r e p r i v a t e c o m m u n i c a t i o n CHAPTER 2 TURBULENT FLO W MODELS 24 o f t h e t r a c t a b l e case of i s o t r o p i c a n d homogeneous t u r b u l e n c e p r o v i d e s c o n c e p t u a l t o o l s b y w h i c h o t h e r flows c a n be a p p r e c i a t e d . A l s o , t h e s p e c t r a l d y n a m i c s o f i s o t r o p i c a n d ho m o g e n e o u s flows are t h o u g h t a p p l i c a b l e t o t h e s m a l l scale m o t i o n of a b r o a d range of fluid flow systems. 2.5 C o m p u t a t i o n a l M o d e l s T h e N a v i e r - S t o k e s e q u a t i o n s a l l o w f o r a c o m p l e t e c o m p u t e r s i m u l a t i o n o n l y at v e r y low R e y n o l d s numbers. A s th e R e y n o l d s n u m b e r increases t h e r a n g e of l e n g t h scales i n t h e flow increases. T h e L a r g e E d d y S i m u l a t i o n ( L E S ) t e c h n i q u e suggested b y L e o n a r d i n 1973 [25] h a n d l e s t h i s p r o b l e m b y s o l v i n g filtered N a v i e r - S t o k e s e q u a t i o n s f o r t h e large " e d d i e s " w h i l e m o d e l l i n g t h e a c t i o n of t h e s m a l l "eddies" u s i n g s u b g r i d scale terms. T h e t e r m eddy is now b e i n g used as i n t h e language o f F o u r i e r a n a l y s i s w here i t is t r e a t e d as s y n o n y m o u s w i t h a d i s t u r b a n c e over a na r r o w range of wavenumber. W h a t f o l l o w s is a r e p r e s e n t a t i v e o u t l i n e o f t h e filtering a n d s u b g r i d scale a n a l y s i s as p r e s e n t e d b y A u p o i x [26]. It is c o m m o n t o d e a l w i t h t h e F o u r i e r t r a n s f o r m o f the N a v i e r - S t o k e s e q u a t i o n s . T h e t r a n s f o r m e d e q u a t i o n s r e a d : j U , ( £ ) + uk7Ui(k) = -ikjiSij - ^ ) I J 6{k-p- 9 - ) u ; ( p ) u / ( 9 - ) r f 3 p dzq (2 - 31a) f o r t h e m o m e n t u m e q u a t i o n a n d Jfc,-Ui(jfc) = 0 ( 2 - 3 1 6 ) CHAPTER 2 TURBULENT FLO W MODELS 25 f o r t h e c o n t i n u i t y e q u a t i o n . H e r e u(k) is t h e F o u r i e r t r a n s f o r m of t h e v e l o c i t y f i e l d a n d 6(k) is t h e D i r a c f u n c t i o n . T h e r i g h t h a n d s i d e s t a n d s f o r b o t h t h e a d v e c t i o n a n d pressure t e r m . E a c h wave v e c t o r k i n t e r a c t s w i t h a l l wave v e c t o r s p a n d q s u c h t h a t k = p + q. T h e v e l o c i t i e s a n d p r e s s u r e are n e x t d e c o m p o s e d i n t o t w o t e r m s , a large-scale com-p o n e n t , u, a n d a s m a l l scale c o m p o n e n t , u' = u — u, u s i n g a c o n v o l u t i o n filter G. T h e filtered v a l u e of a v e l o c i t y c o m p o n e n t reads u t ( x ) = j u t ( x * ) G ( x - x * ) d 3 x (2 - 32) o r m o r e s i m p l y i n F o u r i e r space u(k) = u{k)G(k) (2 - 33) F o r c o n v e n i e n c e a s t e p - f u n c t i o n low pass filter is e m p l o y e d a r t i f i c i a l l y t o s e p a r a t e t h e l a r g e \k\ < kc a n d s m a l l \k\ > kc scales i n t h e flow. W h a t r e s u l t s is a go v e r n i n g e q u a t i o n f o r t h e large scale e d d i e s w i t h a s u b g r i d scale t e r m w h i c h m u s t be m o d e l l e d . T h e s u b g r i d scale t e r m re p r e s e n t s t h e i n t e r a c t i o n s b e t w e e n w a v e n u m b e r s above t h e filter c u t kc a n d t h o s e b e l o w i t . T h i s d e c o m p o s i t i o n is s i m i l a r t o , b u t m o r e s o p h i s t i c a t e d t h a n , t h e R e y n o l d s d e c o m p o s i t i o n o f s e c t i o n 2.1. T h e c l o s u r e p r o b l e m r e m a i n s however. T h e F o u r i e r t r a n s f o r m e d energy e q u a t i o n f o r a n i s o t r o p i c flow is (jt + 2uk2)E{k) = T{k) = lj S{k,p,q)d3pd3q. (2 - 34) H e r e E(k) is t h e energy s p e c t r u m of eqn.(2-29) a n d T(k) is t h e t o t a l energy t r a n s f e r i n t o w a v e n u m b e r k. S e v e r a l m o d e l s g i v e e x p r e s s i o n s f o r t h e d e t a i l e d energy t r a n s f e r S(k,p, q) CHAPTER 2 TURBULENT FLOW MODELS 26 a t w a v e n u m b e r s p a n d q. T h e m o r e recent t w o - p o i n t c l o s u r e s i m p o s e t h e e q u a l i t y of e n semble-averaged e n e r g y t r a n s f e r b etween w a v e n u m b e r k a n d t h e s m a l l scales. O l d e r m o d e l s s i m p l y i m p o s e d energy c o n s e r v a t i o n across t h e c u t , kc. A n i m p l e m e n t a t i o n of t h e m o d e l l i n g is p r e s e n t e d b y A u p o i x as f o l l o w s : " A t each t i m e s t e p t h e e n e r g y s p e c t r u m o f t h e large scales is c a l c u l a t e d . W i t h t h e k n o wledge of t h e e n e r g y s p e c t r u m of b o t h t h e l a r g e a n d t h e s m a l l scales, t h e E D Q N M * r o u t i n e c a n t h e n c o m p u t e on one h a n d t h e s u b g r i d scale t r a n s f e r a n d t h e e d d y v i s c o s i t y vx(k) i n t h e l a r g e scales and, o n t h e o t h e r h a n d , the energy t r a n s f e r T(k) i n t h e s m a l l scales. So t h e e v o l u t i o n of a l l scales c a n be computed." It s h o u l d be m e n t i o n e d t h a t t h e L E S m o d e l s are m o r e su c c e s s f u l w h e n a p p l i e d t o flows f o r w h i c h a p h y s i c a l scale s e p a r a t i o n o c c u r r s [28]. A n e x a m p l e b e i n g t h e flow b e h i n d a b l u f f b o d y where t h e r e l a t i v e l y homogeneous s m a l l scales p r o d u c e d by t h e s e p a r a t e d b o u n d a r y l a y e r i n t e r a c t w i t h t h e shed v o r t i c e s w h i c h are of t h e b o d y d i m e n s i o n i n size. 2.6 T h e R a t e E q u a t i o n A p p r o a c h I n s h a r p c o n t r a s t t o t h e above m o d e l s is o u r r a t e e q u a t i o n a p p r o a c h . A s m e n t i o n e d i n t h e i n t r o d u c t i o n , m u c h o f t h e m o t i v a t i o n b e h i n d t h i s s t u d y of c o h e r e n t s t r u c t u r e s o n g r i d - g e n e r a t e d t u r b u l e n c e was t o t e s t a n d d evelop a m o d e l f o r t u r b u l e n c e based o n r a t e e q u a t i o n s . T h i s m o d e l was p r o p o s e d by m y thesis a d v i s o r D r . B o y e A h l b o r n . T h e m o d e l a n d some s i m p l e a p p l i c a t i o n s of i t were d e s c r i b e d i n a recent p u b l i c a t i o n [ l ] . Its e s s e n t i a l f e a t u r e is t h a t t h e s t a t i s t i c a l e v o l u t i o n of e n e r g y - c o n t a i n i n g c o h e r e n t s t r u c t u r e s is d e s c r i b e d u s i n g a r a t e e q u a t i o n . T h e d i f f e r e n t t y p e s of e v o l u t i o n processes t h a t a * a s t a t i s t i c a l d e s c r i p t i o n of t h e s m a l l scale m o t i o n , see [27] CHAPTER 2 TURBULENT FLOW MODELS s t r u c t u r e c a n u n d e r g o are c h a r a c t e r i z e d b y r a t e c o e f f i c i e n t s i n a m a n n e r s i m i l a r t o t h a t b y w h i c h t h e e v o l u t i o n of energy states i n a n i n t e r a c t i n g s y s t e m o f e x c i t e d or i o n i z e d a t o m s ( a p l a s m a ) is m o d e l l e d . I n a sense, o u r m o d e l t a k e s t h e a n a l o g y of flow s t r u c t u r e s w i t h a t o m s a step f u r t h e r t h a n was done first b y B o u s s i n e s q i n t h e e d d y v i s c o s i t y m o d e l a n d t h e n by P r a n d t l i n h i s m i x i n g l e n g t h a p p r o a c h . T h e i n t e r a c t i o n s of a n energetic flow s t r u c t u r e w i t h i t s e n v i r o n m e n t are c l a s s i f i e d i n t o t h r e e m a i n t y p e s . A s t r u c t u r e c a n i n t e r a c t w i t h i t s fluid s u r r o u n d i n g s t h r o u g h v i s c o u s d i s s i p a t i o n , w i t h i t s flow e n v i r o n m e n t t h r o u g h t h e m e a n shear stress a n d w i t h o t h e r s t r u c t u r e s b y eddy-eddy c o l l i s i o n s . T h e r a t e s of these d y n a m i c a l processes are q u a n t i f i e d b y t h e c o e f f i c i e n t s A,B a n d C r e s p e c t i v e l y . T h e t i m e r a t e of change of t h e n u m b e r of s t r u c t u r e s of a g i v e n t y p e is t h e n d e t e r m i n e d by t h e r a t e e q u a t i o n T h e r e b e i n g one r a t e e q u a t i o n f o r each of t h e m d i s t i n c t s t r u c t u r e t y p e s . A t present we n eed n o t s p e c i f y w h a t p h y s i c a l p a r a m e t e r s d e t e r m i n e t h e s t r u c t u r e t y p e . F o r a g i v e n flow s y s t e m a s t r u c t u r e m a y be u n i q u e l y c h a r a c t e r i z e d b y i t s size a n d energy. I n the s t a t i s t i c s a n a l y s i s of s e c t i o n 4.4 we c h a r a c t e r i z e t h e s t r u c t u r e s by e i t h e r size o r energy. T h e r a t e c o e f f i c i e n t s m u s t be d e r i v e d f r o m t h e l o c a l i n t e r a c t i o n d y n a m i c s a n d t h e p r o b a b i l i t y t h a t a p a r t i c u l a r e n c o u n t e r w i l l o c c u r . T h e A c o e f f i c i e n t q u a n t i f i e s t h e r a t e at w h i c h s t r u c t u r e t y p e a is p r o d u c e d by v i s c o u s decay of s t r u c t u r e s h a v i n g g r e a t e r energy (thus a f o r m s t h e lower b o u n d i n the s u m m a t i o n ) . T h e B c o e f f i c i e n t d e s c r i b e s t h e p r o b a b i l i t y t h a t an a t y p e s t r u c t u r e w i l l be p r o d u c e d w h e n a k t y p e s t r u c t u r e i n t e r a c t s w i t h a shear stress i n t h e fluid. T h e C c o e f f i c i e n t q u a n t i f i e s t h e p r o b a b i l i t y t h a t s t r u c t u r e m ^ Aaknk + ^2 Bak^k + ]T] Cakinkm - Ckoti-k>a k k,l (2 - 35) CHAPTER £ TURBULENT FLOW MODELS 28 local interactions e d d y - f l u i d e ddy-flow e d d y - e d d y \ N a v i e r - S t o k e s — — i n t e r a c t i o n m o d e l s — 7 i n i t i a l p r o d u c t i o n m o d e l r a t e c o e f f i c i e n t s s t r u c t u r e d i s t r i b u t i o n s ~ ~ 1 mo m e n t s of d i s t r i b u t i o n s m a c r o s c o p i c flow p r o p e r t i e s F i g u r e 2-5 S t r u c t u r e d i a g r a m of t h e r a t e e q u a t i o n m o d e l . CHAPTER 2 TURBULENT FLO W MODELS 29 t y p e s k a n d £ w i l l c o l l i d e a n d p r o d u c e a s t r u c t u r e o f t y p e a. If t h e r a t e c o e f f i c i e n t s t r u l y c a p t u r e t h e essence o f t h e s t r u c t u r e e v o l u t i o n t h e n e q u a t i o n (2-35) p r o v i d e s a ge n e r a l f r a m e w o r k f o r d e s c r i b i n g t h e e v o l u t i o n o f a p o p u l a t i o n o f s t r u c t u r e s . If t h e s t r u c t u r e s are c h a r a c t e r i z e d by t h e i r s i z e a n d en e r g y t h e n t h e r a t e e q u a t i o n describes t h e e v o l u t i o n o f t h e n u m b e r of s t r u c t u r e s o f a p a r t i c u l a r s i z e a n d energy. T h i s m o d e l has t h e i n h e r e n t q u a l i t y o f a d d r e s s i n g one of t h e m o r e v e x a c i o u s p r o b l e m s i n t u r b u l e n c e m o d e l l i n g ; n a m e l y l o c a l l y d e t e r m i n i s t i c e v o l u t i o n a n d l o n g t e r m u n p r e d i c t a b i l i t y . T h e a p p r o a c h a l l o w s one n a t u r a l l y t o ac c o u n t for s t r u c t u r e d y n a m i c s . T h e r a t e c o e f f i c i e n t s m a y i n p r i n c i p l e be d e t e r m i n e d e i t h e r t h r o u g h e x p e r i m e n t a l o b s e r v a t i o n o r c a l c u l a t e d u s i n g t h e f l u i d e q u a t i o n s o f m o t i o n . F i g u r e 2-5 shows a s t r u c t u r e d i a g r a m of t h e r a t e e q u a t i o n m o d e l . In o r d e r t o e s t a b l i s h the v a l i d i t y a n d v i a b i l i t y o f t h e r a t e e q u a t i o n a p p r o a c h a n u m b e r o f f u n d a m e n t a l a n d p r a c t i c a l q u e s t i o n s need t o be answered. Some of t h e f u n d a m e n t a l q u e s t i o n s are: 1) C a n t h e flow s t r u c t u r e s be i d e n t i f i e d a n d c h a r a c t e r i z e d by s i m p l e p h y s i c a l p a r a m e t e r s ? 2) C a n t h e e v o l u t i o n of these s t r u c t u r e s be a d e q u a t e l y c h a r a c t e r i z e d b y r a t e c o e f f i c i e n t s ? 3) C a n t h e r a t e c o e f f i c i e n t s be d e t e r m i n e d by l o c a l c o n s i d e r a t i o n s ? 4) D o t h e c o h e r e n t s t r u c t u r e s a c c o u n t f o r m o st o f t h e u n s t e a d y flow i n a t u r b u l e n t fluid? 5) C a n t h e i n i t i a l p r o d u c t i o n o f s t r u c t u r e s be p r e d i c t e d ? 6) C a n t h e i d e n t i f i c a t i o n of a s t r u c t u r e be m a d e o b j e c t i v e ? 7) C a n a coh e r e n t s t r u c t u r e be i d e n t i f i e d u n a m b i g u o u s l y ? Some of the p r a c t i c a l q u e s t i o n s are: 1) A r e t h e r e g e n e r a l u n i q u e 'relaxed' s t a t e s t o t h e s t r u c t u r e s ? CHAPTER 2 TURBULENT FLOW MODELS 2) C a n a n a l y s i s of a f l u i d flow p r o v i d e s u f f i c i e n t s t a t i s t i c a l d a t a t o e v a l u a t e t he r a t e e q u a t i o n m o d e l . 3) A r e t h e r e flow s t r u c t u r e s w h i c h are c o m m o n t o m a n y t u r b u l e n t flow s y s t e m s ? 4) C a n t h e r e c o g n i t i o n of a s t r u c t u r e be a u t o m a t e d ? 5) C a n t h e p r e d i c t e d s t r u c t u r e p o p u l a t i o n s be used t o p r e d i c t flow p r o p e r t i e s of p r a c t i c a l i n t e r e s t s u c h as d r a g , m i x i n g , h e a t t r a n s f e r o r gust levels? T h i s t h e s i s a d v a n c e s t h e r a t e e q u a t i o n m o d e l b y a d d r e s s i n g m a n y of these questions i n r e l a t i o n t o g r i d - g e n e r a t e d t u r b u l e n c e . B e f o r e g o i n g o n t o d i s c u s s t h e e x p e r i m e n t a l o b s e r v a t i o n s a d e s c r i p t i o n of t h e a p p a r a t u s a n d v i s u a l i z a t i o n t e c h n i q u e s used i n t h i s r e s e a r c h is p r e s e n t e d a l o n g w i t h a r e v i e w o f o t h e r m e t h o d s used i n t h e e x p e r i m e n t a l s t u d y o f fluid flow. CHAPTER S FLOW FIELD ANALYSIS 31 C H A P T E R 3 FLOW FIELD ANALYSIS " M a n y t u r b u l e n t flows c a n be o b s e r v e d e a s i l y ; w a t c h i n g c u m u l u s c l o u d s o r t h e p l u m e of a sm o k e s t a c k is n o t t i m e w a s t e d f o r a s t u d e n t of t u r b u l e n c e . " f r o m A First Course in Turbulence: b y Tennekes a n d L u m l e y [15] T h e p u r p o s e o f t h i s c h a p t e r is t o d e s c r i b e m e t h o d s u s e d t o p r o d u c e , observe a n d a n a l y s e flow f i e l d s i n t h e s t u d y o f t u r b u l e n t fluid d y n a m i c s . T h e r e are b a s i c a l l y t wo e x p e r i m e n t a l t e c h n i q u e s a v a i l a b l e f o r t h e s t u d y o f t h e fluid m o t i o n . F l o w v i s u a l i z a t i o n a c q u i r e s two- or t h r e e - d i m e n s i o n a l flow fields at d i s c r e t e p o i n t s i n t i m e w h i l e a n e m o m e t r y p r o v i d e s c o n t i n u o u s v e l o c i t y t i m e r e c o r d s at d i s c r e t e p o i n t s i n space. B o t h m e t h o d s were u s e d i n t h i s s t u d y of g r i d - g e n e r a t e d t u r b u l e n c e a l t h o u g h t h e e m p h a s i s i n t h i s thesis is o n t h e flow v i s u a l i z a t i o n work. T h e first s e c t i o n o f t h i s c h a p t e r d e s c r i b e s t he t o w i n g t a n k w h i c h was b u i l t t o p r o d u c e a n d s t u d y t h e t u r b u l e n t grid-flow. A revi e w of flow v i s u a l i z a t i o n t e c h n i q u e s a n d t h e coherent s t r u c t u r e s d e c o m p o s i t i o n of t u r b u l e n t v e l o c i t y fields is p r e s e n t e d i n t h e s e c o n d s e c t i o n . T h e t h i r d s e c t i o n describes t h e a c q u i s i t i o n of s p e c t r a a n d c o n d i t i o n a l s a m p l i n g , m e t h o d s w h i c h a r e r o o t e d i n t h e use of single a n d m u l t i - p o i n t v e l o c i t y p r o b e s s u c h as laser a n d h o t - w i r e anemometers. F i n a l l y , the m a n u a l a n d a u t o m a t e d c o h e r e n t s t r u c t u r e s a n a l y s i s t e c h n i q u e s used i n t h i s t h e s i s are d e s c r i b e d . CHAPTER S FLO W FIELD ANALYSIS 32 3.1 The Towing Tank A turbulent flow field was generated in a water-filled towing tank using a vertical bar grid of M=5.08cm spacing and d=1.26cm bar diameter. The surface flow field was recorded using a 35mm camera fixed in the lab frame of reference, see Fig.3-1. The camera had a motor drive so that a series of photos could be taken as the flow evolved. For the structure dynamics studies an underwater cart was fitted to the tank. This permitted clearer visualization of the formation region near the towed models. The underwater cart consisted of a plastic sheet which slide between tracks of aluminum U-channel mounted near the tank bottom. The same drive system was used for both carts. The camera's shutter was triggered by an optical pick-up which detected reflective bands placed on the drive cable. The cart speed, Ug, was controlled to better than 1% using a ^ horse power constant speed motor. The speed controller setting and step pulley combination allowed for a grid speed range of from 2 to 200 cm/sec. K i l l switches were placed at either end of the cartway to guard against operator inattention. A more detailed description of the experimental apparatus may be found in reference [30]. For the surface flow investigations aluminum filing tracers of ss 0.5mm size were applied by scraping an aluminum block with a file. The tracers were illuminated with four flood lamps placed at a low angle of incidence with respect to the fluid surface in order to avoid reflection into the camera. Particular attention was paid to the water surface. Periodic skimmimg was needed to remove contaminants which could significantly alter the properties of the surface motion. CHAPTER S FLO W FIELD ANALYSIS 33 lab frame camera 5 m w a t e r s u r f a c e s e e d e d w i t h a l u m i n u m f i l i n g s s p e e d c o n t r o l F i g u r e 3-1 T h e t o w i n g t a n k . T h e t a n k was b u i l t w i t h c l e a r p l a s t i c s i d e w a l l s t o f a c i l i t a t e s u b s u r f a c e flow v i s u -a l i z a t i o n . A sl i d e p r o j e c t o r was used t o p r o d u c e a sheet of l i g h t p a r a l l e l t o e i t h e r t he w a t e r s u r f a c e o r t h e t a n k s i d e w a l l . E i t h e r w o o d c h i p s o r b i o l o g i c a l e n t i t i e s were used f o r t h e s u b s u r f a c e t r a c e r s . T h e l a t t e r were a l l o w e d t o f o r m i n t h e t a n k over t h e course of a few m o n t h s . T h e y were f o u n d t o be v e r y n e a r l y n e u t r a l l y b o u y a n t a n d reflected l i g h t a l m o s t as e f f e c t i v e l y as t h e w o o d chips. I n b o t h t h e surf a c e a n d s u b s u r f a c e studies t h e t r a c e r s p r o d u c e d c l e a r l y v i s i b l e s t r e a k s on ti m e - e x p o s e d p h o t o g r a p h s . T h e s e streaks CHAPTER S FLOW FIELD ANALYSIS i n d i c a t e d t h e l o c a l flow v e l o c i t y . F o r t h e s t r u c t u r e d y n a m i c s s t u d y t h e v i d e o - c a m e r a s i m p l y r e p l a c e d t h e s t i l l c a m e r a . 3.2 V i s u a l i z a t i o n a n d C o h e r e n t S t r u c t u r e s B e f o r e t h e d e v e l o p m e n t of t h e h o t - w i r e anemometer, flow v i s u a l i z a t i o n was the ma-j o r m e t h o d use d t o s t u d y t h e m o t i o n o f fluids. F l o w v i s u a l i z a t i o n p r o v i d e s flow field i n f o r m a t i o n over e n t i r e areas o r v o l u m e s of a fluid b u t u s u a l l y f o r d i s c r e t e p o i n t s i n t i m e . M o r e o v e r , q u a n t i t a t i v e a n a l y s i s is m u c h m o r e i n v o l v e d t h a n f o r a n e mometry. T h e a d v e n t o f h o t - w i r e a n e m o m e t r y saw t h e decreased use of flow v i s u a l i z a t i o n i n t h e s t u d y of fluid d y n a m i c s . F o r m a n y decades of res e a r c h , flows of e n g i n e e r i n g a n d s c i e n t i f i c i n t e r e s t have been e x t e n s i v e l y i n v e s t i g a t e d t h r o u g h i t s c l e a r b u t n a r r o w view. T h e r e n e w e d use of flow v i s u a l i z a t i o n is t h u s n o t o n l y due t o t h e n e w l y a c q u i r e d c a p a b i l -i t i e s i n image p r o c e s s i n g , b u t a l s o t o t h e d i m i n i s h i n g r e t u r n s i n t h e use o f p o i n t v e l o c i t y p r o b e s . W i t h t h i s new l o o k at some w e l l s t u d i e d flows t h e s e t t i n g was t h e n r i p e f o r a mo r e c o h e r e n t v i e w v i s - a - v i s t h e f r a g m e n t e d s t a t i s t i c a l c o n c e p t o f t u r b u l e n c e . T h e i n t e r e s t e d r e a d e r w i l l find a w e a l t h of v i s u a l i z a t i o n t e c h n i q u e s a n d novel v a r i a -t i o n s i n references [31,32,33] a m o n g others. A n e x c e l l e n t c o l l e c t i o n o f flow v i s u a l i z a t i o n p h o t o s has been a s s e m b l e d b y V a n D y k e [34]. F i g u r e 3-2 shows a ge n e r a l c l a s s i f i c a t i o n s cheme of flow v i s u a l i z a t i o n t e c h n i q u e s used i n t h e s t u d y of fluid flow. T h e y are c r u d e l y d i v i d e d i n t o m e t h o d s w h i c h e x p l o i t l i g h t s c a t t e r i n g f r o m p a r t i c l e s a n d t h o s e w h i c h make use o f v a r i a t i o n i n r e f r a c t i v e i n d e x of a fluid d e p e n d i n g o n t h e flow c o n d i t i o n s . CHAPTER S FLO W FIELD ANALYSIS 35 Flew Visualisation scattered light refractive index variations tracer imaging speckle photography dye techniques holography interferometry spatial filtering tomography computer techniques F i g u r e 3-2 C l a s s i f i c a t i o n scheme of flow v i s u a l i z a t i o n t e chniques. I n t e r f e r o m e t e r s y i e l d a p i c t u r e o f t h e r e f r a c t i v e i n d e x t h r o u g h a v o l u m e of compress-i b l e fluid o r m i x t u r e of fluids o f d i f f e r i n g o p t i c a l p r o p e r t i e s . S p a t i a l filtering is used i n t h e s c h l i e r e n t e c h n i q u e t o p r o d u c e a n image o f t h e average first d e r i v a t i v e of t h e i n d e x o f r e f r a c t i o n w h i l e t h e s h a d o w g r a p h t e c h n i q u e y i e l d s t h e m e a n s e c o n d d e r i v a t i v e . A n o t h e r v i s u a l i z a t i o n t e c h n i q u e is t o f o l l o w t h e m o t i o n of a c o n v e c t e d fluid w h i c h has been d y e d w i t h a m a t e r i a l t h a t is v i s u a l l y d i s t i n c t f r o m the b a c k g r o u n d m e d i u m . A c o m m o n c o m b i n a t i o n is i n k i n w a t e r . T h i s t e c h n i q u e is q u i t e e f f e c t i v e i n s h o w i n g t h e o u t l i n e s o f s t r u c t u r e s r e s p o n s i b l e f o r m i x i n g . It has been used m o s t e f f e c t i v e l y i n t he CHAPTER S FLO W FIELD ANALYSIS s t u d y o f c o h e r e n t s t r u c t u r e s i n t u r b u l e n t b o u n d a r y layers. A n ingeneous e x t e n s i o n of t h i s m e t h o d is t h e use of a f l u o r e s c i n g dye i n a flow i l l u m i n a t e d b y a sheet of laser l i g h t . T h i s m e t h o d a l l o w s f o r v i s u a l i z a t i o n i n p l a n e s t h r o u g h o u t a flow a n d a l l o w s one to 'look i n s i d e ' t h r e e d i m e n s i o n a l flows. T h e dye t e c h n i q u e does not l e n d i t s e l f t o q u a n t i t a t i v e e x t r a c t i o n of v e l o c i t y f i e l d s a n d i t s use has l a r g e l y been c o n f i n e d t o q u a l i t a t i v e studies. T h e newer (~1977) t e c h n i q u e o f s p e c k l e p h o t o g r a p h y a l l o w s f o r a c q u i s i t i o n of t h e m o t i o n i n a cross s e c t i o n o f a fluid. T h e fluid is i l l u m i n a t e d b y a m o n o c h r o m a t i c sheet of p u l s e d c o h e r e n t l i g h t . A r a n d o m p a t t e r n of s p e c k l e p a i r s w i t h v a r i a b l e d i s t a n c e a n d o r i e n t a t i o n is i m a g e d at t h e p h o t o g r a p h i c p l a n e . T h e r e s u l t i n g specklegram c a n be recon-s t r u c t e d t o p r o d u c e a s p e c k l e p a t t e r n i n w h i c h t h e s p a c i n g a n d o r i e n t a t i o n of successive s p e c k l e s are d i r e c t l y r e l a t e d t o t h e m o t i o n of the fluid. A d d i t i o n a l l y t h e speckles are m o d u l a t e d by Young's f r i n g e s w h i c h give t h e m a g n i t u d e a n d o r i e n t a t i o n of t h e in-plane c o m p o n e n t s of t h e fluid v e l o c i t y w i t h o u t t h e need for p a r t i c l e t r a c k i n g . A s L a u t e r -b o r n p o i n t s o u t [35], t h e s p e c k l e m e t h o d c a n be c o n s i d e r e d as a n e x t e n s i o n of t h e u s u a l m u l t i p l e - e x p o s u r e t e c h n i q u e b u t makes use o f F o u r i e r space f o r d i r e c t o r i e n t a t i o n deter-m i n a t i o n (the m o t i o n sense m u s t s t i l l be d e t e r m i n e d b y o t h e r m e t h o d s ) . A n e x t r a bonus w i t h t h i s t e c h n i q u e is t h e easy e x t e n s i o n t o s m a l l e r d i s p l a c e m e n t s t h r o u g h the inverse r e s o l u t i o n p r o p e r t i e s o f space a n d t h e F o u r i e r d o m a i n . H o l o g r a p h i c t e c h n i q u e s a l l o w the phase i n f o r m a t i o n f r o m phase o b j e c t s such as den-s i t y v a r i a t i o n s o r p a r t i c l e d i s t r i b u t i o n s t o be s t o r e d a n d r e c o n s t r u c t e d l a t e r . T h e recon-s t r u c t i o n c a n b e done w i t h a new s t a t e of t h e flow s y s t e m t o p r o d u c e a n interference p a t t e r n h i g h l i g h t i n g t h e differences between say a quiescent or s t e a d y flow state a n d a mo r e v i g o r o u s flow. T h i s r e c o n s t r u c t i o n c a n be p e r f o r m e d f o r a si n g l e i n s t a n t i n t i m e CHAPTER S FLOW FIELD ANALYSIS in what is known as double-exposure holographic interferometry. It can be done con-tinuously in the case of real-time interferometry. In order to overcome accuracy and processing speed limitations of holographic interferometry, heterodyne reconstruction can be used to increase the resolution by orders of magnitude. Tomography is a method of image reconstruction from multiple projections. Accord-ing to Lauterborn it shows promise but as of 1984 was just starting to be applied to the study of fluid flow. The use of flow visualization has elucidated the presence and role of structure in tur-bulent flows, however most of the reports have been confined to qualitative descriptions of the structure properties. A notable exception is the work of Hernan and Jimenez [36] who quantified the spatial extent of mixing structures in the developing mixing layer. They used a computer-automated geometric identification on cine film recordings of re-gions of active mixing highlighted with reactive gases. The identification involved fitting ellipses to the visually distinct mixing regions. They examined the growth and pairing history of the elliptical structures, and growth evolution. Because of the visualization method employed little could be said about the reasons for the observed evolution as the velocity field cannot be extracted from such data. The contouring of the hydrogen bubble wire visualization (an interesting variation of the tracer method) and picture processing data of Utami and Ueno [39] are suggestive of coherent structures. Mory and Hopfinger [40] used particle tracking to extract structure functions in a rotationally dominated turbulent flow and Sheu et al [41] used a three-dimensional measurement technique to extract bulk flow data from the motion of tracers CHAPTER S FLOW FIELD ANALYSIS 38 i n a m i x i n g c h a m b e r . T r a c e r p a r t i c l e t e c h n i q u e s i n g e n e r a l are w e l l s u i t e d t o q u a n t i t a t i v e a n a l y s i s . T h e t i m e - e x p o s u r e m e t h o d has been u s e d i n t h e l a b o r a t o r y s i n c e t h e e a r l y days of fluid d y n a m i c s r e s e a r c h . Indeed, p a r t o f t h e i n s p i r a t i o n f o r t h e present w o r k came f r o m s t u d y o f a lar g e p r i v a t e c o l l e c t i o n of p h o t o g r a p h s of fluid m o t i o n made by F. A h l b o r n i n t h e e a r l y decades o f t h i s c e n t u r y [37]. I n t h i s t e c h n i q u e , o p t i c a l l y r e f l e c t i v e p a r t i c l e s are p l a c e d e i t h e r i n a fluid f o r v o l u m e s t u d i e s o r o n a fluid f o r s u r f a c e s t u d i e s . It has been u s e d f o r q u a n t i t a t i v e s t u d i e s of t w o - d i m e n s i o n a l l a m i n a r flows f o r n e a r l y a c e n t u r y b y m e a s u r i n g s t r e a k p r o p e r t i e s f r o m t r a c e r images r e c o r d e d o n t i m e exposed p h o t o g r a p h s . A n u m b e r of v a r i a t i o n s o n t h e t i m e - e x p o s e d p a r t i c l e t r a c k i n g t e c h n i q u e have been used i n t h i s t h e s i s w ork. B e f o r e g o i n g o n t o present t h e a n a l y s i s m e t h o d s u s e d i n t h i s t h e s i s t o i d e n t i f y a n d a n a l y z e c o h e r e n t s t r u c t u r e s some o b s e r v a t i o n s of o t h e r a u t h o r s is presented. T h e t e r m 'coherent s t r u c t u r e ' is u s e d t o d e s c r i b e a r e g i o n of flow w h i c h has a more p r e d i c t a b l e s t r u c t u r e a n d e v o l u t i o n i n t e r n a l l y t h a n i t does w i t h i t s s u r r o u n d i n g s . L u m l e y [42] has s u g g e s t e d t h a t c o h e r e n t s t r u c t u r e s are s i g n i f i c a n t i n "young" flows w h i c h are s t i l l i n -fluenced by p r o d u c t i o n geometry. H e suggests t h a t t h e c u r r e n t discoveries of c o herent s t r u c t u r e s are p a r t i a l l y a r e s u l t o f researchers l o o k i n g i n e a r l y flow regimes w i t h flow g e n e r a t i o n a p p a r a t u s h a v i n g m o r e quiescent p r e c o n d i t i o n s . T h i s however, does not ex-p l a i n t h e e mergence of large scale c o herent s t r u c t u r e s f r o m an i n i t i a l l y r a n d o m v o r t i c i t y c o m p u t e r m o d e l l e d flow [43]. I t is t h i s author's b e l i e f t h a t coherent s t r u c t u r e s are i n s ome cases a p r o p e r t y of t h e i n i t i a l flow c o n d i t i o n s a n d i n o t h e r s an i n t r i n s i c p r o p e r t y CHAPTER S FLO W FIELD ANALYSIS 39 o f t h e s u b s e q u e n t f l u i d mechanics. It is q u i t e p o s s i b l e t h a t t h e m e c h a n i c s g o v e r n i n g t h e m o t i o n of f l u i d s a l l o w s f o r a n i n f i n i t e v a r i e t y of b e h a v i o u r . A t present t h e r e a p p e a r t o be as m a n y o r m o r e coherent s t r u c t u r e i d e n t i f i c a t i o n a l -g o r i t h m s as t h e r e are researchers. T h i s is d u e i n p a r t t o t h e newness of t h e research a n d i n p a r t t o t h e v a r i e t y of flows s t u d i e d a n d t o o l s u s e d i n these s t u d i e s . Researchers u s i n g a n e m o m e t e r s have been q u i c k t o s u p p l y p r e s c r i p t i o n s f o r coherent s t r u c t u r e iden-t i f i c a t i o n . H u s s a i n [44] has t r i e d t o come u p w i t h t h e c a n o n i c a l d e f i n i t i o n o f a coherent s t r u c t u r e . H e c a l l s a cohe r e n t s t r u c t u r e , "A l a r g e s c a l e c o n n e c t e d r e g i o n of f l u i d m ass h a v i n g a phase c o r r e l a t e d v o r t i c i t y o v e r i t s s p a t i a l extent." Hussain's m e t h o d r e q u i r e s a p r e v i o u s b i a s as t o w here or w h e n t h e s t r u c t u r e s may be f o u n d a n d w i l l o n l y w o r k f o r s t r u c t u r e s r e p e t i t i v e l y generated. H i s use of a n e m o m e t r y is e v i d e n t i n t h e e x p r e s s i o n , "phase c o r r e l a t e d v o r t i c i t y " . L u m l e y [42] has p r o p o s e d a r e c o g n i t i o n m e t h o d w h i c h avo i d s t h e use of c o n d i t i o n a l s a m p l i n g w h i c h he feels a l l o w s f o r t o o m u c h of t h e experimenter's b i a s t o enter i n t o t h e d a t a c o l l e c t i o n . L u m l e y ' s " e i g e n f u n c t i o n " a p p r o a c h is a n o r t h o g o n a l d e c o m p o s i t i o n o f a v e l o c i t y r e c o r d u s i n g g a u s s i a n v o r t i c i t y d i s t r i b u t i o n s as t h e b a s i s f u n c t i o n s . T h e m e t h o d is r o o t e d i n t h e s t a t i s t i c a l d e s c r i p t i o n of t u r b u l e n c e a n d t h e use of p o i n t v e l o c i t y p r o b e s . I t p r e d e f i n e s t h e basis f u n c t i o n s a n d so c a n n o t be s a i d t o be a c o m p l e t e l y "non-p r e j u d i c i a l " a p p r o a c h . T h e d e f i n i t i o n used for t h e a u t o m a t e d s t r u c t u r e r e c o g n i t i o n i n t h i s thesis is; A c o n n e c t e d , large-scale fluid mass o u t l i n e d b y t h e closed c o n t o u r of m i n i m u m a n g u l a r v e l o c i t y , w i t h i n w h i c h t h e r e e x i s t s one, a n d o n l y one, l o c a l m a x i m u m i n a n g u l a r v e l o c i t y . CHAPTER S FLOW FIELD ANALYSIS I m p l i c i t i n o u r d e f i n i t i o n is t h a t i t be a p p l i e d t o a n a l y s i s o f t w o d i m e n s i o n a l flow fields. T h e i d e n t i f i c a t i o n o f s t r u c t u r e s based o n t h i s d e f i n i t i o n is d e s c r i b e d i n s e c t i o n 3.3.2. W h i l e n o t s t r i c t l y needed, t h e c o n d i t i o n t h a t t h e s t r u c t u r e be "la r g e - s c a l e " is used t o r emove a n g u l a r v e l o c i t y peaks of s i z e t o o close t o t h e r e s o l u t i o n l i m i t s o f the a n a l y s i s s y s t e m . It has been t h e a d v e n t of t h e h i g h speed d i g i t a l c o m p u t e r a n d i m a g e a c q u i s i t i o n , p r o c e s s i n g a n d a n a l y s i s t e c h n i q u e s w h i c h has o p e n e d t h e d o o r f o r a t r u l y q u a n t i t a t i v e a n a l y s i s o f i m a g e d e r i v e d v e l o c i t y fields i n u n s t e a d y a n d t u r b u l e n t fluid flows. T h e a p p l i c a t i o n o f t h i s t e c h n i q u e is f u r t h e r m o r e n o t l i m i t e d t o t w o - d i m e n s i o n a l flows. T h e flow a n a l y s i s s y s t e m u s e d f o r t h e s t r u c t u r e d y n a m i c s s e c t i o n o f t h i s t h e s i s c o n s t i t u t e s a f u l l y a u t o m a t e d flow field a c q u i s i t i o n , p r o c e s s i n g a n d a n a l y s i s package. T h e w o r k at U B C a p p e a r s t o be t h e first t o use p a r t i c l e t r a c k i n g t o o b t a i n flow fields f o r coherent s t r u c t u r e a n a l y s i s . T h e s y s t e m has been d e s c r i b e d i n d e t a i l i n t h e M. Sc. thesis by L a u [38], A s u m m a r y of i t s p r i n c i p l e s a n d m e t h o d s a p p e a r s i n t h e n e x t s e c t i o n . 3.3 O u r M e t h o d s 3.3.1 M a n u a l A n a l y s i s F o r t h e s t r u c t u r e s t a t i s t i c s s t u d i e s of c h a p t e r f o u r a m a n u a l m e t h o d was used to i d e n t i f y t h e eddies. T h e raw d a t a f o r those e x p e r i m e n t s were a n u m b e r of t i m e exposed p h o t o g r a p h s of t r a c e r p a r t i c l e p a t h s such as Figs.4-1. T h e a n a l y s i s o f these p h o t o g r a p h s c o n s i s t e d o f i d e n t i f y i n g the v i s u a l l y o b served regions o f closed c o herent m o t i o n by t r a c i n g CHAPTER S FLOW FIELD ANALYSIS 41 the largest closed contour following the tracer paths. A n eddy radius was then defined as the radius of a circle having an enclosed area equal to that of the structure. The manual recognition of the structure geometry was completed by assigning an eddy center as the position about which the motion appeared to rotate. The internal velocity structure of the eddy was then recorded in a mechanical though tedious procedure using a digitizing table. While this method suffered from personnel subjectivity and fatigue* it was useful in obtaining a description of the structure statistics. 3.3.2 Automated Analysis The structure recognition was subsequently formalized and automated in the M.Sc. work of Alexis Lau [38] and used for the structure dynamics part of the present thesis. What follows is a summary of the method as presented in a paper recently submitted by Lau, Loewen and Ahlborn [8]. Figure 3-3 shows the computer-automated flow field analysis system developed at U B C . The flow was visualized by recording successive aluminum tracer positions on a video tape. Each video frame was then digitized as a binary 256 x 192 pixel array by a Micro-Works DS-65 digitizer residing on an Apple II microcomputer. Figure 3-4 shows one of the less noisy digitized frames. The digitized data were transferred to the U B C mainframe computer where all subsequent analysis was performed. The procedure for the noise reduction was to fill in holes and remove isolated pix-els while preserving meaningful connectivity. A dynamic tree search following adjacent points ("on" pixels) was used to create a list of the position and size of each tracer for * About 800 person-hours were needed to analyze the statistical data used below. CHAPTER $ FLOW FIELD ANALYSIS 42 1 video camera 2 video recorder 3 TV for video system U microcomputer keyboard 5 microcomputer with VCR controller and digitizer. 6 microcomputer display monitor 7 mini disk storage drives 8 telecommunication link to mainframe computer F i g u r e 3-3 F l o w field a n a l y s i s s y s t e m . CHAPTER S FLOW FIELD ANALYSIS 43 F i g u r e 3-4 Raw digitized image. Binary 256 x 192. CHAPTER S FLO W FIELD ANALYSIS 44 a l l o f t h e d i g i t i z e d frames. T h e flow h i s t o r y was t h e n r e c o n s t r u c t e d b y t r a c k i n g a n d c o n n e c t i n g t h e successive t r a c e r p o s i t i o n s s h o w n i n Fig.3-5. T h e t r a c k i n g p r o c e d u r e was a b o o t s t r a p s e a r c h u s i n g a p o l y n o m i a l fit t o p r o j e c t t h e m o s t l i k e l y n e x t p a r t i c l e p o s i t i o n . F o r each new f r a m e of d a t a , t h e p o s i t i o n a n d size o f c a n d i d a t e t r a c e r s d e t e r m i n e d the best m a t c h w i t h t h e p r e v i o u s l y t r a c k e d p a t h s . T h e final r e s u l t is a t i m e c o n n e c t e d l i s t o f t r a c e r p a r t i c l e p o s i t i o n s over t h e e x p o s u r e t i m e of t h e r u n - a streak. F i g u r e 3-6 shows a s a m p l e c o n n e c t i o n . S t r e a k t r a j e c t o r i e s were t h e n fitted by p o l y n o m i a l s t o give v a r i o u s flow p a r a m e t e r s of i n t e r e s t over d e s i r e d flow times. I n p a r t i c u l a r , t h e l i n e a r a n d a n g u l a r v e l o c i t i e s were d e t e r m i n e d f r o m the su c c e s s f u l l y t r a c k e d d a t a . T h e s c a t t e r e d set of values were t h e n i n t e r p o l a t e d o n t o an e q u a l l y spaced g r i d t o p r o v i d e a n g u l a r a n d l i n e a r v e l o c i t y fields f o r t h e subsequent a n a l y s i s . I n the a u t o m a t e d a n a l y s i s t h e a n g u l a r v e l o c i t y is d e t e r m i n e d w i t h respect t o t h e p o i n t d e t e r m i n e d by t h e l o c a l r a d i u s o f c u r v a t u r e . T h i s is d i f f e r e n t f r o m t h e m e t h o d used i n t h e m a n u a l a n a l y s i s w h e r e t h e a n g u l a r v e l o c i t y was d e t e r m i n e d w i t h respect t o t h e v i s u a l l y assigned s t r u c t u r e c e n t e r . C o h e r e n t s t r u c t u r e s were i d e n t i f i e d b y a p p l y i n g o u r d e f i n i t i o n t o t h e field of angu-l a r v e l o c i t y . F r o m t h e i n t e r p o l a t e d m e s h fields o f l i n e a r v e l o c i t i e s , each s t r u c t u r e was p a r a m e t e r i z e d w i t h p r o p e r t i e s l i k e s i z e , average l i n e a r a n d a n g u l a r v e l o c i t y a n d i n t e r n a l k i n e t i c energy. T h e flow k i n e m a t i c s , d y n a m i c s a n d i n t e r a c t i o n s c o u l d t h e n be s t u d i e d u s i n g these s t r u c t u r e p a r a m e t e r s . T h e d e f i n i t i o n of a coherent s t r u c t u r e m u s t be u n d e r s t o o d before d e c i d i n g w h i c h p a r a m e t e r s a r e t o be e x t r a c t e d f r o m t h e t r a j e c t o r i e s . D e f i n i t i o n s based o n v o r t i c i t y are CHAPTER S FLOW FIELD ANALYSIS F i g u r e 3-5 Successive t r a c e r centers. S u p e r p o s i t i o n of 15 frames. CHAPTEE S FLOW FIELD ANALYSIS 46 F i g u r e 3-6 T r a c k e d t r a c e r p aths. CHAPTER S FLOW FIELD ANALYSIS c o m m o n l y u s e d i n t h e l i t e r a t u r e . H owever o u r d i a g n o s t i c t e c h n i q u e t o g e t h e r w i t h t h e c l o s e r e l a t i o n s h i p between v o r t i c i t y a n d a n g u l a r v e l o c i t y f o r o u r t w o - d i m e n s i o n a l flow m o t i v a t e o u r use o f a d e f i n i t i o n b a s e d o n a n g u l a r v e l o c i t y . F o r t h e s t r u c t u r e d y n a m i c s s t u d i e s a co h e r e n t s t r u c t u r e -an eddy- is d e f i n e d here as: A c o n n e c t e d , large-scale fluid mass o u t l i n e d b y t h e c l o s e d c o n t o u r of m i n i m u m a n g u l a r v e l o c i t y , w i t h i n w h i c h t h e r e e x i s t s one, a n d o n l y one, l o c a l m a x i m u m i n a n g u l a r v e l o c i t y . T o e x t r a c t i n f o r m a t i o n a b o u t these coherent s t r u c t u r e s t h e s t r e a k d a t a is first pro-cessed t o p r o d u c e t h e i n t e r p o l a t e d m esh field o f a n g u l a r v e l o c i t y . F r o m t h i s g r i d , l o c a l m a x i m a are l o c a t e d a n d t h e n t h e b o u n d i n g c o n t o u r f o r each p e a k is f o u n d . A s two or m o r e a d j a c e n t p o s i t i o n s m a y have t h e same e x t r e m u m value, t h e peaks c a n n o t be l o c a t e d j u s t b y l o c a l c o n s i d e r a t i o n ( a c o m p u t a t i o n a l l y ef f i c i e n t way t o proceed). A s L a u p u t s i t i n h i s t h e s i s , 'The task is s i m i l a r t o t h e p r o b l e m of finding a m o u n t a i n t o p i n a forest w h e n one is l o s t a n d c a n n o t see t o o f a r . T h e m o t t o is "keep o n c l i m b i n g u p y o u r steepest t r a c k , do n o t descend o r t u r n back o n a level t r a c k . " If a non-decreasing t r a c k c a n no l o n g e r be f o u n d , one m u s t be at a l o c a l maximum.' F i g u r e 3-7a shows t h e r e s u l t s o f a p p l y i n g o u r d e f i n i t i o n t o a s a m p l e one d i m e n s i o n a l s c a l a r field. T h e r e c o g n i t i o n process is i m p l e m e n t e d by first " c l i m b i n g " f r o m each p o i n t o n t h e g r i d u n t i l e i t h e r a peak or a p r e v i o u s l y m a r k e d p a t h is f o u n d . I n t h e l a t t e r case t h e c l i m b is t e r m i n a t e d a n d t h e p a t h n u m b e r is r e a s s i g n e d t o e q u a l t h e one e n c o u n t e r e d , see Fig.3-7b. C o r r e s p o n d i n g t o the a c t u a l a n g u l a r v e l o c i t y g r i d , a n i d e n t i f i c a t i o n g r i d is u s e d t o r e c o r d t h e p a t h numbers. O n c e e x p e d i t i o n s have s t a r t e d f r o m e v e r y g r i d p o i n t CHAPTER S FLOW FIELD ANALYSIS 48 F i g u r e 3-7 C o h e r e n t s t r u c t u r e i d e n t i f i c a t i o n , (b) P a t h n u m b e r assignment. (a) S a m p l e one d i m e n s i o n a l s c a l a r field. CHAPTER S FLOW FIELD ANALYSIS we have e f f e c t i v e l y d e t e r m i n e d d i s t i n c t , i n t e r n a l l y c o n n e c t e d r e g i o n s w i t h i n w h i c h a non-d e c r e a s i n g p a t h e x i s t s between a n y g r i d p o i n t a n d i t s peak. E a c h of these r e g i o n s define a n extended region. T h e s e c o n d p r o b l e m of finding a m i n i m u m c l o s e d c o n t o u r a r o u n d t h e peak is now s t r a i g h t f o r w a r d . We s i m p l y find t h e m a x i m u m a n g u l a r s p e e d a m o n g a l l t h e v a l u e s of t h e b o u n d a r y p o i n t s . A n y c o n t o u r v a l u e h i g h e r t h a n t h i s w i l l d e f i n i t e l y enclose t h e peak w i t h o u t e n t e r i n g i n t o a n o t h e r r e g i o n . O n c e t h i s v a l u e is f o u n d , t h e core of t h e coherent s t r u c t u r e is d e f i n e d as t h e a r e a w i t h i n t h e e x t e n d e d r e g i o n f o r w h i c h t h e g r i d - p o i n t s have v a l u e s h i g h e r t h a n t h i s one. T h i s c o m p l e t e s t h e o p e r a t i o n a l p r o c e d u r e t o find t h e b o u n d a r y of t h e c o herent s t r u c t u r e . It s h o u l d b e n o t e d t h a t i n o u r p r o c e d u r e a l l p o i n t s i n t h e f l o w field are a s s o c i a t e d w i t h e i t h e r t h e core or e x t e n d e d r e g i o n of a coherent s t r u c t u r e . A f t e r t h e i n i t i a l i d e n t i f i c a t i o n t h e d a t a set is f u r t h e r r e f i n e d t o remove s m a l l s c a l e s t r u c t u r e s . T h i s is c o n s i s t e n t w i t h o u r i n t e r e s t i n l a r g e s c a l e s t r u c t u r e s a n d is done b y r e m o v i n g p o s s i b l y m i s t r a c k e d s t r e a k s w h i c h are i n c o n s i s t e n t w i t h t h e rest of t h e d a t a . T h e e n t i r e a n a l y s i s p r o c e d u r e is t h e n r e p e a t e d o n t h e r e f i n e d d a t a . F i g u r e 3-8 shows t h e c o h e r e n t s t r u c t u r e s i d e n t i f i e d b y a p p l y i n g t h e a n a l y s i s t o a tw o - d i m e n s i o n a l v e l o c i t y field o b t a i n e d f r o m t h e g r i d - f l o w e x p e r i m e n t s . T h e i d e n t i f i e d r e g i o n s c o r r e s p o n d c l o s e l y t o o u r i n t u i t i v e n o t i o n of a co h e r e n t s t r u c t u r e . F i g u r e 3-9 shows t h e s a m e p l o t b u t w i t h a n g u l a r v e l o c i t y c o n t o u r s o v e r l a y e d . T h e a n a l y s i s s y s t e m j u s t d e s c r i b e d was used for t h e s t r u c t u r e d y n a m i c s s t u d i e s of c h a p t e r 5. B e f o r e g o i n g on t o present t h e r e s u l t s of t h i s w ork, a s e c t i o n o n how a n e m o m e t r y based flow field a n a l y s i s is used i n t h e s t u d y of s t r u c t u r e i n flows. CHAPTER S FLOW FIELD ANALYSIS p i x e l n u m b e r ii. 64. tS. 121. ISO. 192. » U . p i x e l n u m b e r F i g u r e 3-8 I d e n t i f i e d s t r u c t u r e s . 1 se c o n d (30 frame) s t r e a k s . + i n d i c a t e s counter-c l o c k w i s e r o t a t i o n - i n d i c a t e s c l o c k w i s e r o t a t i o n CHAPTER S FLOW FIELD ANALYSIS p i x e l n u m b e r 32. 64. 98. 128. ISO. 192. 22U. p i x e l n u m b e r F i g u r e 3-9 C o n t o u r p l o t of i d e n t i f i e d s t r u c t u r e s . + i n d i c a t e s c o u n t e r - c l o c k w i s e r o t a t i o n - i n d i c a t e s c l o c k w i s e r o t a t i o n CHAPTER S FLOW FIELD ANALYSIS 52 3.4 Anemometry Based Flow Analysis It was t h e d e v e l o p m e n t of h o t - w i r e a n e m o m e t r y i n 1935, c o i n c i d e n t w i t h Taylor's s t a t i s t i c a l t h e o r y o f t u r b u l e n c e [45] w h i c h a l l o w e d researchers t o measure fluid v e l o c i t i e s a t p o i n t s i n a flow c o n t i n u o u s l y a n d a c c u r a t e l y . T h e v e r y h i g h f r e q u e n c y response a n d r e a s o n a b l e s p a t i a l r e s o l u t i o n o f these devices makes t h e m w e l l s u i t e d f o r s t u d y i n g t i m e v a r y i n g v e l o c i t i e s . T h e i m p a c t o f h o t - w i r e a n e m o m e t r y t o t h e s t u d y o f t u r b u l e n t fluid flow has b e en enormous. T h e laser d o p p l e r a n e m o m e t e r is a less i n t r u s i v e p o i n t p r o b e w h i c h p r o d u c e s d a t a s i m i l a r t o t h e h o t - w i r e . P i t o t t u b e s are u s e f u l f o r m e a s u r e m e n t o f s t e a d y or s l o w l y v a r y i n g v e l o c i t y fields. T o g e t h e r w i t h a s u i t a b l y a d j u s t e d l i n e a r i z e r a h o t - w i r e a n e m o m e t e r , i n i t s m o s t ba s i c c o n f i g u r a t i o n , p r o d u c e s a v o l t a g e s i g n a l p r o p o r t i o n a l t o t h e l o n g i t u d i n a l speed of t h e flow. T h i s s i g n a l c a n be e l e c t r o n i c a l l y a v e r a g e d or high-pass filtered t o o b t a i n t h e m e a n a n d fluctuating v e l o c i t y c o m p o n e n t s r e s p e c t i v e l y . T h e p r i n c i p l e s o f the s t a t i s t i c a l a p p r o a c h t o s t u d y i n g a n d m o d e l l i n g t u r b u l e n c e have a l r e a d y b e e n p r e s e n t e d i n t h e l a s t c h a p t e r . I n t h i s s t u d y of g r i d - g e n e r a t e d t u r b u l e n c e , F o u r i e r a n a l y s i s o f h o t - f i l m a n e m o m e t r y d e r i v e d v e l o c i t y r e c o r d s was used t o e x a m i n e t h e near - s u r f a c e fluid d y n a m i c s . T h e m o s t c o n v e n i e n t q u a n t i t y t o m e a s u r e was the s p e c t r a l d e n s i t y of t h e u j fluctuating c o m p o n e n t , Eu(f) a t a g i v e n p o s i t i o n x. E\\(/) is t w i c e t h e c o m p o n e n t o f t h e k i n e t i c energy p e r u n i t mass due t o l o n g i t u d i n a l v e l o c i t y fluctuations, t i i , at t e m p o r a l f r e q u e n c y /. T h e p o w e r s p e c t r a l d e n s i t y c a n be o b t a i n e d by F o u r i e r ana-l y z i n g t h e fluctuating v e l o c i t y c o m p o n e n t xi\(x,t) m e a s u r e d w i t h a d i r e c t i o n a l l y s e n s i t i v e CHAPTER S FLO W FIELD ANALYSIS 53 p r o b e a t a f i x e d p o s i t i o n x. T o see t h i s , we first w r i t e t h e t i m e d e p e n d e n t l o n g i t u d i n a l fluctuating v e l o c i t y U\(t) i n i t s F o u r i e r r e p r e s e n t a t i o n , u i ( t ) = 27r f df\a(f)cos{2nft) + b[f)sin[2irft)}. (3 - 1) J — oo a(f) a n d 6(/) are t h e F o u r i e r c o e f f i c i e n t s f o r t h e p e r i o d i c b a s i s f u n c t i o n s o f f r e q u e n c y /, 1 f°° a(f) = - / dtux(t)cos(2-n ft) ( 3 - 2 a ) J — oo 1 f°° b{f) = - dtu1{t)sin(2Trft). ( 3 - 2 6 ) J — oo T h e power s p e c t r u m , Eu(f), is o b t a i n e d f r o m t h e F o u r i e r c o e f f i c i e n t s as £ „ ( / ) = - 7 r 2 [ a 2 ( / ) + 6 2 ( / ) ] ( 3 - 3 ) T f o r s a m p l i n g t i m e r. T h e p o w e r s p e c t r u m m a y a l t e r n a t e l y be de f i n e d i n t e r m s of t h e a u t o c o r r e l a t i o n Cn(T) of eqn.(2-24). T h e s e t w o f u n c t i o n s f o r m a co s i n e t r a n s f o r m p a i r , En{f)= f C 1 1 ( r ) c o 5 ( 2 7 r / 7 , ) a T (3 - 4a) Jo CU{T) = 4 rEll{f)eos(2itfT)df ( 3 - 4 6 ) T h e l i m i t s o f i n t e g r a t i o n are f r o m 0 t o oo as Exl(f) = Exl(-f) a n d CU(T) = Cn(-T) f o r s t a t i s t i c a l l y s t e a d y flows. T h e two d e f i n i t i o n s of Eu(f), eqns.(3-4a) a n d (3-3), are e q u i v a l e n t . T o see t h i s we in s e r t t h e F o u r i e r r e p r e s e n t a t i o n f o r «i(<), e q n s . ( 3 - l ) , i n t o t h e d e f i n i t i o n of t h e auto-c o r r e l a t i o n , eqn.(2-24). M u l t i p l y i n g t h e v e l o c i t y r e p r e s e n t a t i o n s a n d i n t e g r a t i n g out the CHAPTER S FLO W FIELD ANALYSIS 54 c r o s s t e r m s i n a(f) a n d b(f) we have C i i ( T ) = u^Qu^t + T) = /t'm^oo- fT df[a2{f) + b2{f)]cos(27rft). ( 3 - 5 ) T Jo T h e c o s i n e t r a n s f o r m e d q u a n t i t y i n t h i s e x p r e s s i o n is e q u i v a l e n t t o t h e d e f i n i t i o n f o r En(f) i n eqn.(3-3). F r o m eqn.(3-4b) we see t h a t C u ( 0 ) = / Eu(f)df ( 3 - 6 ) Jo U s i n g eqn.(2-24) f o r t h e a u t o c o r r e l a t i o n we have Jo En(f)df ( 3 - 7 ) T h i s e q u a t i o n i d e n t i f i e s t h e t o t a l fluctuating k i n e t i c energy u\ w i t h the a r e a u n d e r t h e p o w e r s p e c t r a l d e n s i t y curve. T h e t e m p o r a l s p e c t r a at p o s i t i o n x, Eu(f,x), may be r e l a t e d t o t h e s p a t i a l s t r u c t u r e o f t h e flow. A p p l y i n g Taylor's h y p o t h e s i s t o t h e p o w e r s p e c t r a l d e n s i t y m easurement we a r r i v e at t h e r e l a t i o n * n ( * , x ) = J-tf c£„(/,i). ( 3 - 8 ) Z~n T h e p o w e r s p e c t r a l d e n s i t y is r e l a t e d t o t h e ki\-v/a.\e n u m b e r s p e c t r u m $ n ( f c , z ) t h r o u g h t h e l o n g i t u d i n a l c o n v e c t i o n v e l o c i t y , U C a t w h i c h t h e p r o b e s a m p l e s t h e flow field. T h i s is t h e F o u r i e r t r a n s f o r m e q u i v a l e n t of t h e r e l a t i o n s h i p b etween t h e s p a t i a l a n d t e m p o r a l c o r r e l a t i o n s . T h i s i n c l u d e s t h e same r e s t r i c t i v e a s s u m p t i o n s , n a m e l y t h a t t h e c o n v e c t i o n v e l o c i t y Uc is c o n s t a n t a n d t h a t t h e flow does n o t change a p p r e c i a b l y w h i l e the p robe CHAPTER S FLOW FIELD ANALYSIS s a m p l e s a d i s t a n c e g r e a t e r t h a n t h e l e n g t h scales of i n t e r e s t . U s i n g t h e d e f i n i t i o n of t h e i n t e g r a l t i m e s c a l e Te, eqn.(2-25), i n eqn.(3-4a) f o r / = 0 we see A n d a p p l y i n g T a y l o r ' s h y p o t h e s i s i n t h e f o r m of eqn.(2-27) we see t h a t t h e i n t e g r a l l e n g t h s c a l e c a n be o b t a i n e d f r o m t h e power s p e c t r a l d e n s i t y En(f) as Le is i n t e r p r e t e d as b e i n g t h e average s i z e of t h e energy c o n t a i n i n g s t r u c t u r e s . It is one of t h e m o s t b a s i c p a r a m e t e r s u s e d t o d e s c r i b e t h e s p a t i a l s t r u c t u r e of a t u r b u l e n t flow field. T h i s l e n g t h scale d e t e r m i n a t i o n is used i n t h e c o m p a r i s o n of a n e m o m e t r y a n d flow v i s u a l i z a t i o n p r e s e n t e d i n t h e near s u r f a c e fluid d y n a m i c s s t u d y of s e c t i o n 5.1. O v e r t h e l a s t t w o decades, t h e t e c h n i q u e of c o n d i t i o n a l s a m p l i n g has been developed f o r t h e s t u d y of s t r u c t u r e i n a n e m o m e t r y s i g n a l s . A c c o r d i n g t o a re v i e w by A n t o n i a [46] i t has b e en a p p l i e d t o t h e s t u d y of t h e t u r b u l e n t - n o n t u r b u l e n t i n t e r f a c e of shear flows, s h e a r l a y e r s p e r t u r b e d b y i n t e r a c t i o n w i t h a n o t h e r field o f t u r b u l e n c e , p e r i o d i c flows a n d t o t h e s t u d y of c o h e r e n t s t r u c t u r e s i n d i f f e r e n t shear flows. C o n d i t i o n a l s a m p l i n g is a pr o c e s s u s e d t o h i g h l i g h t s p e c i f i c f e a t u r e s of a flow b y o n l y s a m p l i n g t h e flow field w h e n c e r t a i n p r e v i o u s l y s e l e c t e d c o n d i t i o n s are met. It has been used by th e a n e m o m e t r y c o m m u n i t y t o p r o v i d e q u a n t i t a t i v e i n f o r m a t i o n t o c o m p l e m e n t q u a l i t a t i v e o bservations of c o h e r e n t s t r u c t u r e s o b t a i n e d f r o m flow v i s u a l i z a t i o n studies. i ; 1 1 ( 0 ) = 4 u 2 T e . ( 3 - 9 ) Le = UcEn(0)/(4ul). (3 - 10) CHAPTER S FLOW FIELD ANALYSIS T h i s c h a p t e r has p r e s e n t e d some o f t h e m a j o r e x p e r i m e n t a l t e c h n i q u e s a v a i l a b l e f o r t h e s t u d y o f t u r b u l e n t f l u i d f l o w as w e l l as t h e m e t h o d s used i n t h i s t h e s i s work. T h e n e x t c h a p t e r d e s c r i b e s t h e s t a t i s t i c a l s t u d y of t u r b u l e n t g r i d - f l o w p e r f o r m e d b y t h e a u t h o r . CHAPTER 4 STRUCTURE STATISTICS C H A P T E R 4 STRUCTURE STATISTICS "One m u s t l e a r n by d o i n g t h e t h i n g ; t h o u g h y o u t h i n k y o u k n o w i t , y o u h a v e n o c e r t a i n t y u n t i l y o u t r y . " S o p h o c l e s 495-406 B C T h e e x p e r i m e n t a l m e t h o d s have been p r e s e n t e d i n t h e l a s t c h a p t e r . T h i s c h a p t e r d e s c r i b e s a n i n v e s t i g a t i o n o f t h e s t a t i s t i c s o f coherent s t r u c t u r e s o n t h e surface of a t u r b u l e n t b o d y of f l u i d . A f t e r finding t h e s t r u c t u r e s i t was necessary t o see i f t h e y c o u l d be i d e n t i f i e d a n d c h a r a c t e r i z e d i n a s i m p l e b u t u s e f u l manner. N e x t , i t was of int e r e s t t o see if t h e s t r u c t u r e s t a t i s t i c s c o u l d be s i m p l y d e s c r i b e d . A n s w e r s t o b o t h these questions s h o u l d c o n t r i b u t e t o o u r u n d e r s t a n d i n g o f t h e ro l e t h a t coherent s t r u c t u r e s p l a y i n a t u r b u l e n t flow. T h e e x p e r i m e n t a l o b s e r v a t i o n s a r e g i v e n first, f o l l o w e d by a s t a t i s t i c a l a n a l y s i s o f t h e siz e a n d energy d i s t r i b u t i o n s o f t h e o p e n (river-flow) a n d c l o s e d (eddy-flow) c o h e r e n t m o t i o n s . I n t h e l a s t s e c t i o n of t h i s c h a p t e r , t h e m e a s u r e d eddy energy s t a t i s t i c s a r e used t o i n f e r v a lues f o r the r a t e c o e f f i c i e n t s . CHAPTER 4 STRUCTURE STATISTICS 58 4.1 Experimental Observations The experimental apparatus has already been described in chapter 3. Figures 4-l(a),(b),(c) show a series of photos taken in the lab frame of reference with the cart moving from right to left at the speed Ug = 20cm/sec. The interframe separation is 3 seconds. The left side of Fig.4-l(a) shows the receding grid while the right side of Fig.4-1(c) is at an average distance of X=200cm from the grid, corresponding to about 40 mesh widths. A non-dimensionalized distance scale in units of mesh widths (X' — X/M) is shown in Fig.4-l(a). . The exposure time of each frame is 1 sec. Hence a typical streak of 1cm path length indicates a local speed of u0 = lcm/sec. For distances greater than a few mesh widths from the grid the flow patterns did not evolve significantly during the 1 sec exposure time. The average speed thus corresponds closely to the instantaneous value. It should also be noted that for photos taken in the fluid reference frame u 0 is small compared to the cart speed of Ug = 20cm/sec. Photos taken in the cart frame of reference show practically straight streaks without clear indication of coherent structures and are not suitable for this analysis. The time evolution of the flow can be studied on subsequent frames by comparing the same region of flow, for instance, the lower right corner of each photo in the sequence of Figs.4-1. If one is interested in statistical averages, sections of the flow in some interval AX centered about the average value X can be examined. Figure 4-1 (b) shows such a sample bin. The X scale given below each photo shows some overlap, so that CHAPTER i STRUCTURE STATISTICS 0 k 8 X/M — i i \ (a) CHAPTER 4 STRUCTURE STATISTICS t h e d i s t r i b u t i o n o b t a i n e d f r o m a n i n t e r v a l a t t h e r i g h t edge of F i g . 4 - l ( b ) s h o u l d be s t a t i s t i c a l l y e q u a l t o t h e d i s t r i b u t i o n o b t a i n e d f r o m t h e left edge of F i g . 4 - l ( c ) . F o r t h e s t r u c t u r e s t a t i s t i c s a n a l y s i s t e n p h o t o g r a p h i c series c o n t a i n i n g a b o u t 20,000 i n d i v i d u a l s t r e a k s were t r a c e d b y h a n d a n d t h e i r e n d p o i n t p o s i t i o n s were d i g i t i z e d a n d t r a n s f e r r e d t o c o m p u t e r memory. S t r e a k s c o n t a i n e d w i t h i n a m a n u a l l y i d e n t i f i e d s t r u c -t u r e b o u n d a r y were g r o u p e d w h i l e o t h e r s t r e a k s were c o n s i d e r e d t o be p a r t o f t h e i n t e r -e d d y ( r i v e r ) flow. T h e flow field was a n a l y z e d o n l y i n t h e cen t e r s e c t i o n o f t h e t o w i n g t a n k w h e r e i t is n o t a f f e c t e d b y t h e side w a l l s . T h i s t h e s i s r e p o r t s a n a l y s i s r e s u l t s f o r Ug = 20cm/sec (a m e s h R e y n o l d s n u m b e r of 10,000) a l t h o u g h q u a l i t a t i v e l y s i m i l a r r e s u l t s w ere f o u n d f o r 30 a n d 40cm/sec g r i d speeds [30]. T h e flow p h o t o s show c l e a r e v i d e n c e o f o r g a n i z e d fluid m o t i o n . O n e c a n d i s t i n g u i s h r e l a t i v e l y s t a g n a n t areas, w h e r e t h e filings p r o d u c e p o i n t s i n s t e a d o f t r a c e s , a n d m o v i n g s e c t i o n s , w h e r e t h e st r e a k l e n g t h is p r o p o r t i o n a l t o t h e l o c a l fluid speed. T h e m o v i n g fluid a p p e a r s t o u n d e r g o e i t h e r l o c a l l y r o t a t i n g m o t i o n w i t h c l o s e d s t r e a m l i n e s (surface eddies), w h e r e a l o c a l a n g u l a r speed u c a n be d e f i n e d , o r i t shows p r e d o m i n a n t l y t r a n s l a t i o n a l m o t i o n t h a t m a y be c h a r a c t e r i z e d b y an average r i v e r s p e ed u c . A s m a l l n u m b e r of r a p i d l y d e c a y i n g eddies were obs e r v e d , b u t f o r t h e s t a t i s t i c a l a n a l y s i s t h e surf a c e flow is co n s i d e r e d t o be c o m p o s e d of e i t h e r eddies w i t h a n g u l a r speed u>, r i v e r s w i t h t r a n s l a t i o n a l s p e e d u D , o r s t a g n a n t fluid w h e r e uD a n d w are n e g l i g i b l e . B o t h eddies a n d r i v e r s c an s t o r e k i n e t i c energy t h a t c o n t r i b u t e s t o t h e i n t e r n a l energy o f t h e u n s t e a d y surface flow. F o r t h e p u r p o s e of o b t a i n i n g a s t a t i s t i c a l d e s c r i p t i o n of t h e eddies, t h e i r b o u n d a r y is d e t e r m i n e d by e n c l o s i n g t h e lar g e s t a r e a w h i l e d r a w i n g a clo s e d l i n e f o l l o w i n g the CHAPTER 4 STRUCTURE STATISTICS v i s i b l e t r a c e r s . T h i s d e f i n i t i o n is s i m i l a r t o t h a t o f "the m i n i m u m c l o s e d a n g u l a r v e l o c i t y c o n t o u r " f o r t h e a u t o m a t e d r e c o g n i t i o n o f s t r u c t u r e cores d e s c r i b e d i n t h e las t c h a p t e r . T y p i c a l e d d y i d e n t i f i c a t i o n s u s i n g t h e m a n u a l m e t h o d are s h o w n i n F i g . 4 - 2 . A f t e r t h e s t r u c t u r e is o u t l i n e d , a n e d d y r a d i u s is a s s i g n e d as t h e r a d i u s o f a c i r c l e e n c l o s i n g a n a r e a e q u a l t o t h a t c o v e r e d b y t h e s t r u c t u r e . It was f o u n d t h a t t h r e e o b s e r v e r s i n d e p e n d e n t l y a s s i g n e d r a d i i t h a t d i f f e r e d b y less t h a n 5%. S t r u c t u r e c e n t e r s were as s i g n e d as t h e p o s i t i o n a b o u t w h i c h t h e s u r f a c e eddies a p p e a r e d t o r o t a t e . F i g u r e 4-2 T y p i c a l s t r u c t u r e i d e n t i f i c a t i o n CHAPTER 4 STRUCTURE STATISTICS 62 For the 20cm/sec grid speeds vertical displacement of the fluid surface was observed to be less than a few millimeters. Significant near-surface vertical motion must therefore be accompanied by converging or diverging flow at the surface. This would appear as relatively stagnant regions of the material surface. If the accompanying stress was strong enough to overcome the surface incompressibility, one would record either convergent or divergent tracer motion. For cart speeds less than 50cm/sec (grid Re w 25,000) no signs of subsurface vertical motion were found on the flow photos. One can therefore consider the surface motion for Ug = 20cm/sec as being two-dimensional. The relative abundance of open and closed structures is easily determined. One has to measure the surface area fraction or concentration ce = Se/Si, cr = Sr/Sb, and Cst — Sst/Sb of eddies, rivers and stagnant areas. For this purpose the total surface areas of all eddies Se, of all the rivers Sr, and of all the stagnant areas Set were measured and divided by the total area analyzed, the bin area Sb-Figure 4-3(a) shows the concentration of these flow components as a function of posi-tion X . The statistical analysis was performed in AX = 20cm wide bins. The distance X is the center of the bin. The downstream distance can alternately be quoted in terms of the number of mesh widths (X'). Between X=0 and X=20cm (X ' = 4) the motion and evolution is too rapid to be analyzed from photos such as Figs.4-1. The near grid eddy concentration, c 0 , is obtained from a simple initial vortex production model described in the next chapter. After the initial production the eddy area fraction c e decays while the fractions of river area cr and the stagnant area c8 increase. About 20 mesh widths downstream, the river area fraction starts to decrease as well. This follows the ultimate dissipation of the grid induced motion to a quiescent state. When the eddy area fraction CHAPTER 4 STRUCTURE STATISTICS F i g u r e 4-3 E v o l u t i o n of surf a c e flow p a t t e r n , (a) F r a c t i o n o f s u r f a c e are S covered by eddies Se, r i v e r s 5 r , a n d s t a g n a n t fluid St. (b) E d d y n u m b e r d e n s i t y n e , average e ddy a r e a a7, a n d energy d e n s i t y r a t i o e r / e e . CHAPTER 4 STRUCTURE STATISTICS ce is h i g h one e x p e c t s s i g n i f i c a n t i n t e r a c t i o n b e tween eddies, so t h a t t h e e d d y d i s t r i b u t i o n s h o u l d b e g o v e r n e d by e d d y " c o l l i s i o n " processes. W h e n t h e e d d y f r a c t i o n becomes low, t h e i r d i s t r i b u t i o n w i l l m a i n l y c hange t h r o u g h (viscous) e d d y - f l u i d a n d eddy-flow i n t e r -a c t i o n s , hence Fig.4-3(a) is d i v i d e d i n t o t h e p r o d u c t i o n r a n g e 0 < X' < 6, a c o l l i s i o n a l o r e q u i l i b r i u m r e g i m e w i t h 6 < X' < 23 a n d a (c o l l i s i o n l e s s ) decay r a n g e w i t h 23 < X'. T h e p r o d u c t i o n r e g i o n is d e f i n e d as t h e r a n g e i n w h i c h t h e i n i t i a l v o r t e x g e n e r a t i o n is o b s e r v e d t o o c c u r . T h e c o l l i s i o n a l o r e q u i l i b r i u m r e g i o n is w here frequent s t r u c t u r e i n t e r -a c t i o n r e s u l t s i n a n e q u i l i b r i u m energy d i s t r i b u t i o n . I n t h e decay r e g i o n , t h e d y n a m i c s of t h e w i d e l y s p a c e d flow s t r u c t u r e s a p p e a r t o b e v i s c o s i t y d o m i n a t e d . S h o w n i n Fig.4-3(b) is t h e e d d y n u m b e r d e n s i t y n e = Ne/Ss (where Ne is t h e t o t a l n u m b e r of surface eddies) a n d t h e average e d d y a r e a ae = l/ne. T h e l a t t e r c u r v e shows a st e a d y increase of t h e average s t r u c t u r e size w i t h d o w n s t r e a m d i s t a n c e . T h e energy d e n s i t y r a t i o of t h e t w o flow t y p e s , e r / e e , is a l s o s h o w n as a f u n c t i o n o f d i s t a n c e . Here, c r a n d ee are the average k i n e t i c energies p e r u n i t mass of t h e r i v e r a n d e d d y flow s t r u c t u r e s . T h e d e t e r m i n a t i o n of these energy d e n s i t i e s is d i s c u s s e d i n t h e f o l l o w i n g t w o s e c t i o n s . 4.2 R i v e r F l o w T h e o p e n s t r u c t u r e s ( r i v e r s ) a p p e a r t o have t h e i r o r i g i n i n the m o m e n t u m defect "shadow" b e h i n d t h e g r i d b a rs. T h e r e s u l t i n g a n i s o t r o p y i n t h e r i v e r o r i e n t a t i o n decays d u r i n g t h e o b s e r v a t i o n p e r i o d a n d becomes n e g l i g i b l e a b o u t X = 2 0 0 c m d o w n s t r e a m of the g r i d . T h i s is i n agreement w i t h s t u d i e s of t h e b u l k flow i n w h i c h g r i d - g e n e r a t e d t u r b u l e n c e CHAPTER 4 STRUCTURE STATISTICS Figure 4-4 River-speed scatter plot. Bin averaged river speed u0 as solid line. is found to be approximately isotropic within 40 or 50 mesh widths downstream (about 200cm for our experiment) [47]. The speed and positions of a large number of tracers in the rivers were determined from the digitized data. Figure 4-4 shows a scatter plot of these speeds for the grid speed Ug = 20cm/sec. The average speed, u, is drawn as a heavy line through the point constellation. These raw data have again been divided into 20cm wide bins. The number of points in these bins N(u,X) can then be used to generate energy and speed distribution CHAPTER 4 STRUCTURE STATISTICS F i g u r e 4 - 5 E v o l u t i o n o f r i v e r - f l o w s p e ed d i s t r i b u t i o n . f u n c t i o n s . T h e net t r a n s l a t i o n a l m o t i o n is n e g l i g i b l e f o r t h i s flow as t h e fluid is c o n f i n e d t o t h e t a n k a n d t h e g r i d f i l l s t h e tank's c r o s s - s e c t i o n . F i g u r e 4-5 shows t h e speed d i s t r i b u t i o n N ( u , X ) . F i g u r e 4-6 shows t h e d i s t r i b u t i o n of l o c a l k i n e t i c energy Er = ^mru20, (4 - la) w h i c h c a n be e x t r a c t e d f r o m t h e m e a s u r e d d a t a as fo l l o w s : E a c h b i n i n Fig.4-4 c a n be c h a r a c t e r i z e d by a n average v a l u e f o r t h e speed, u 0 . I n a d d i t i o n , one m u s t assign a va l u e f o r t h e average mass m r = plar a s s o c i a t e d w i t h each s t r e a k , because each streak CHAPTER 4 STRUCTURE STATISTICS 0 5 erg/cm 10 — — E F i g u r e 4-6 R i v e r - f l o w energy d i s t r i b u t i o n . r e p r e s e n t s a l o c a l p a t c h , of a r e a aT, of c o h e r e n t l y m o v i n g fluid. H ere / is t h e d e p t h of t h e o b s e r v e d s u r f a c e p a t t e r n . T h e v a l u e o f t h e d e p t h w h i c h is r e p r e s e n t e d b y t h e surface flow w i l l d i f f e r d e p e n d i n g o n d i s t a n c e f r o m t h e g r i d . I n p a r t i c u l a r , t h e d e p t h w i l l grow w i t h d i s t a n c e d o w n s t r e a m , see s e c t i o n 5.1. I n t h e a n a l y s i s t h a t f o l l o w s £ s h o u l d be c o n s i d e r e d t o be t h e v e r y s h a l l o w d e p t h of t h e fluid surface. A s s u m i n g t h a t t h e s t r e a k s are evenly d i s t r i b u t e d one c a n e s t i m a t e t h e average a r e a p e r s t r e a k , a r , by d i v i d i n g t h e t o t a l r i v e r a r e a 5 r = crSb by t h e t o t a l n u m b e r of s t r e a k s , Nr, i n t h e A X range. T h e n m = plar was d e t e r m i n e d at n i n e X p o s i t i o n s a n d i t was f o u n d t h a t t h e average s u r f a c e a r e a represented CHAPTER 4 STRUCTURE STATISTICS 68 by e a c h s t r e a k v a r i e d by n o t more t h a n 1 0 % . T h e average v a l u e is ar = 7.1cm 2 so t h a t t h e k i n e t i c e nergy r e p r e s e n t e d b y each s t r e a k becomes a p p r o x i m a t e l y ( i n cgs u n i t s ) E = l/2plaru20 = ( 7 . l / 2 ) p / u 2 . (4 - 16) I t s h o u l d b e m e n t i o n e d t h a t t h e se e d i n g d e n s i t y was m u c h h i g h e r t h a n t h e v a l u e of ar i n d i c a t e s as o n l y a r e p r e s e n t a t i v e s a m p l i n g o f t h e r i v e r speed was p e r f o r m e d . N o t k n o w i n g how f a r t h e m o t i o n e x t e n d s i n t o t h e f l u i d t h e e nergy p e r u n i t d e p t h , E = E/l, is d e f i n e d . A n o t h e r u s e f u l q u a n t i t y a s s o c i a t e d w i t h t h e r i v e r m o t i o n is t h e energy d e n s i t y e r = Er/mr = u 2 / 2 (4 - l c ) H e r e e r c o n t r i b u t e s t o t h e t o t a l energy d e n s i t y o f t h e flow w i t h t h e f r a c t i o n e r c r . A s m e n t i o n e d i n t h e i n t r o d u c t i o n t o t h i s c h a p t e r i t was of i n t e r e s t t o see i f t h e e n e r g y d i s t r i b u t i o n s of t h e s t r u c t u r e p o p u l a t i o n s c o u l d be c h a r a c t e r i z e d i n a s i m p l e way. F i g u r e 4-6 shows t h e energy d i s t r i b u t i o n s i n s e m i l o g p l o t s . T h e high-energy t a i l o f the d i s t r i b u t i o n c a n be re p r e s e n t e d by a n e x p o n e n t i a l c h a r a c t e r i z e d b y a p a r a m e t e r 6r. T h e c u r v e has a h a l f - w i d t h A 0 r . I n a d d i t i o n , t h e low-energy p a r t o f t h e d i s t r i b u t i o n s were p l o t t e d i n l o g f a s h i o n a n d i t was f o u n d t h a t logN grows a p p r o x i m a t e l y p r o p o r t i o n a l t o logEr. T h e d a t a are t h e r e f o r e a p p r o x i m a t e d by N(Er) = const E?exp{-Er/6) (4 - 2a) w h e r e N(Er) is t h e n u m b e r of s t r u c t u r e s p e r u n i t energy i n t e r v a l h a v i n g energy Er. T h e e x p e r i m e n t a l c u r v e s a l l have a w e l l defined m a x i m u m Nm at th e a b s c i s s a Em = E(Nm). T h e f u n c t i o n (4-2a) has i t s m a x i m u m at Em = a6r, so t h a t a = Em/6T c a n be e x t r a c t e d CHAPTER 4 STRUCTURE STATISTICS f r o m t h e m e a s u r e d p o s i t i o n o f t h e m a x i m u m . T h e e x p e r i m e n t a l r e s u l t f r o m t h e r i v e r e n e r g y c u r v e s is a = 0.8 ±0.1. ( 4 - 2 6 ) E q u a t i o n (4-2a) is s i m i l a r t o t h e M a x w e l l - B o l t z m a n n f u n c t i o n u s e d t o de s c r i b e t h e ene r g y d i s t r i b u t i o n s of m a n y i n t e r a c t i n g systems. E x a m p l e s are t h e e x c i t a t i o n , r a d i a t i o n , a n d k i n e t i c e nergy i n p l a s m a s , as w e l l as t h e k i n e t i c energy o f gases. B o l t z m a n n f u n c t i o n s a r e e q u i l i b r i u m d i s t r i b u t i o n s a n d so are o n l y f o u n d i f t h e i n t e r a c t i o n t i m e scale o f t h e system's c o n s t i t u e n t s is s m a l l c o m p a r e d t o t h e system's e v o l u t i o n t i m e . I n su c h cases a t e m p e r a t u r e c a n be d e f i n e d f o r each energy m o d e b y 6, t h e s l o p e o f t h e g r a p h o f N(E) at large E, o r AO, t h e w i d t h of t h e d i s t r i b u t i o n a t h a l f t h e peak energy. I n a p l a s m a , t h e t e m p e r a t u r e s o f e l e c t r o n s , h e avy p a r t i c l e s , a n d e l e c t r o n i c e x c i t a t i o n o r v i b r a t i o n , o f t e n d i f f e r . We m a y e x p e c t a s i m i l a r d i f f e r e n c e b e t w e e n t h e e d d y a n d r i v e r d i s t r i b u t i o n s . I n t h e e q u i l i b r i u m r e g i o n o f t h e r i v e r e n e r g y d i s t r i b u t i o n we c a n i n t e r p r e t t h e param-e t e r 6r as a r i v e r " t e m p e r a t u r e " t h a t d e s c r i b e s t h e s p r e a d i n t h e r i v e r energy d i s t r i b u t i o n . K n o w i n g t h e c o n s t a n t a, 6T d e t e r m i n e s t h e m o s t p r o b a b l e energy s t a t e Em(= a6r). T h e d i s t r i b u t i o n f u n c t i o n s , Fig.4-6(a), r e m a i n B o l t z m a n n t y p e f r o m X = 3 0 t o X = 1 3 0 c m . T h e r i v e r e n e r g y m o d e m u s t t h e r e f o r e b e i n a f o r m o f e q u i l i b r i u m t h a t is m a i n t a i n e d b y a h i g h i n t e r a c t i o n r a t e b etween t h e s t r u c t u r e s i n t h e flow. F i g u r e 4-7(a) shows t h e r i v e r t e m p e r a t u r e 0r, t h e energy h a l f - w i d t h s A6r a n d t h e energy Em a t t h e m a x i m u m of t h e d i s t r i b u t i o n J V m as a f u n c t i o n of d i s t a n c e f r o m t h e g r i d . T h e s t a t i s t i c a l u n c e r t a i n t y of any 6 v a l u e is less t h a n 1 0 % . It is s u r p r i s i n g how m u c h o r d e r e x i s t s i n t h e d i s t r i b u t i o n CHAPTER 4 STRUCTURE STATISTICS 70 F i g u r e 4-7 " T e m p e r a t u r e " decay, (a) R i v e r t e m p e r a t u r e Qr. (b) E d d y t e m p e r a t u r e 0e. CHAPTER 4 STRUCTURE STATISTICS 71 o f t h e s e o p e n fl o w s t r u c t u r e s . It w i l l be i n t e r e s t i n g t o see w h e t h e r t h e r e is a s i m i l a r e q u i l i b r i u m f o r t h e energies of t h e c l o s e d s u r f a c e s t r u c t u r e s . 4.3 E d d y D i s t r i b u t i o n s T h e c l o s e d s u r f a c e flow s t r u c t u r e s (surface eddies) are c o n n e c t e d r e g i o n s i n w h i c h t h e fluid e l e m e n t s a p p e a r t o r o t a t e a b o u t a c o m m o n a x i s . A n i m p o r t a n t f e a t u r e of our flow v i s u a l i z a t i o n a n a l y s i s is t h e c h a r a c t e r i z a t i o n of t h e i n t e r n a l flow of the eddies. T y p i c a l l y , e i g h t s t r e a k s p e r e d d y were r e c o r d e d . T h e average a n g u l a r speed f o r e a c h s t r u c t u r e was t h e n c a l c u l a t e d a n d averaged f o r a l l e ddies i n a p a r t i c u l a r size a n d d i s t a n c e range as f o l l o w s . We s i m p l i f y t h e t r a c e r p a t h i n f o r m a t i o n a n d define t h e a n g u l a r displacement a of a t r a c e r as t h e ang l e w h i c h t h e t w o e n d p o i n t s o f t h e s t r e a k s u b t e n d w i t h respect t o t h e e s t i m a t e d e d d y center. T h e a n g u l a r speed u; is t h e n f o u n d by d i v i d i n g a by t h e e x p o s u r e t i m e . T h i s a n g u l a r speed at r a d i u s R f r o m t h e e d d y center defines a t a n g e n t i a l s p e e d Up = uR. A t y p i c a l e d d y v e l o c i t y p r o f i l e is s h o w n i n Fig.4-8(a). We f u r t h e r s i m p l i f y each eddy's v e l o c i t y s t r u c t u r e b y least squares fitting a r i g i d b o d y p r o f i l e ( t h e h e a v y l i n e ) t o t h e i n t e r n a l speeds. A n a n g u l a r s p e e d is t h u s assigned t o each s t r u c t u r e . T h i s c r u d e s i m p l i f i c a t i o n o f t h e i n t e r n a l v e l o c i t y s t r u c t u r e is p e r f o r m e d i n o r d e r t o have a s i m p l e c h a r a c t e r i z a t i o n f o r used i n the s t r u c t u r e s t a t i s t i c s a n a l y s i s . M o r e r e a l i s t i c p r o f i l e s w o u l d have a s m o o t h v a r i a t i o n i n v e l o c i t y a t t h e s t r u c t u r e b o u n d a r y s u c h as s h o w n i n Fig.5-9. F i g u r e 4-8(b) shows how these v a l u e s are d i s t r i b u t e d f o r a l l e d d i e s i n t h e r a d i u s range 1.4 < R < 1.5cm w h i c h are l o c a t e d i n a b i n ( A X = 20cm) c e n t e r e d at X = 7 0 c m . A n average a n g u l a r speed was assigned t o t h i s s a m p l e range as F i g u r e 4-8 S u r f a c e e d d y flow s t r u c t u r e , (a) V e l o c i t y p r o f i l e f o r a n e d d y at X = 71 cm. (b) V e l o c i t y p r o f i l e s f o r a l l 25 eddies at X = 70 ± 10 cm. (c) A n g u l a r speed d i s t r i b u t i o n o f s u r f a c e eddies. CHAPTER 4 STRUCTURE STATISTICS 73 i n d i c a t e d b y t h e c i r c l e d l i n e . F i g u r e 4-8(c) shows t h e a n g u l a r speeds of a l l s u r f a c e eddies as a f u n c t i o n of s t r u c t u r e s i z e R a n d d o w n s t r e a m d i s t a n c e X. T h e g e n e r a l t r e n d is of l i n e a r l y d e c r e a s i n g uJ w i t h i n c r e a s i n g R a n d g e n e r a l l y d e c r e a s i n g w w i t h d o w n s t r e a m d i s t a n c e X. T h e v a r i a t i o n i n uJ f o r t h e e x t r e m e v a l u e s of R is due t o low s a m p l e number. x F i g u r e 4-9 A v e r a g e d p e r i p h e r a l speed of eddies. P l o t t e d i n Fig.4-9 are t h e averaged v e l o c i t i e s UR = RQ at t h e edge of eddies of r a d i u s R fo r t h e d i s t a n c e s a n a l y z e d . F o r c o m p a r i s o n t h e average r i v e r v e l o c i t i e s are also i n d i c a t e d . It is seen t h a t t h e s m a l l eddies move slower at t h e i r edge t h a n t h e r i v e r s a n d CHAPTER 4 STRUCTURE STATISTICS 74 some l a r g e e d d i e s move fas t e r . However, o n average t h e edge speed UR is close t o t h e r i v e r s p e e d u0 UR = Rjujj « u 0 . (4 - 3) T h i s r e s u l t c a n be i n t e r p r e t e d as s h o w i n g t h a t o n average t h e a n g u l a r s p e e d is i n v e r s e l y p r o p o r t i o n a l t o s t r u c t u r e size. T h i s e x p e r i m e n t a l o b s e r v a t i o n i n d i c a t e s a n effective cou-p l i n g e x i s t s b e t w e e n t h e eddies a n d t h e r i v e r s . T h e e n e r g y c o n t e n t of edd i e s c a n b e d e s c r i b e d b y t h r e e q u a n t i t i e s : t h e t o t a l e d d y e n e r g y ( e x t r a c t e d f r o m o u r m o d e l o f r i g i d b o d y r o t a t i o n a n d c y l i n d r i c a l eddies of s m a l l d e p t h £) Ei = (1/2)/©? = ( l / 4 ) m * ? 0 ? = (n/4)pl Rj Q2 (erg) (4 - 4) (where I is t h e m o m e n t of i n e r t i a a n d p is t h e d e n s i t y o f t h e f l u i d ) , t h e surf a c e e d d y en e r g y p e r u n i t e d d y d e p t h E, = (*/4)pRJ u) (erg/cm), (4 - 5) a n d t h e e d d y e n e r g y d e n s i t y , o r s p e c i f i c e n e r g y tj = Ej/mj = (1/4) R] Q) (erg/g). (4 - 6) A l l t h r e e are f u n c t i o n s o f t h e average a n g u l a r speed <Z> t h a t is k n o w n f r o m t h e s t a t i s t i c a l a n a l y s i s . It w o u l d be p r e f e r a b l e t o c o n s i d e r t h e m e a n sq u a r e a n g u l a r speed u;2 f o r t h i s e n e r g y a n a l y s i s as o p p o s e d t o t h e sq u a r e m e a n u>2 used here. I n f u r t h e r s t u d y of the s t r u c t u r e ' s s p e e d p r o f i l e , s e c t i o n 5.4, i t is f o u n d t h a t these t w o q u a n t i t i e s a r e p r o p o r t i o n a l t o each o t h e r . O u r use of Q2 w i l l t h u s affect t h e m a g n i t u d e o f o u r i n f e r r e d energy CHAPTER 4 STRUCTURE STATISTICS quantities by a constant factor and not undermine the analysis to follow. With the experimental observation of eqn.(4-3) one can give these energies as a function of the river speed u 0 « RjQj e , - s H l , ^ S (n/4)PRy0. (4-7) The total energy density of the turbulent motion is the sum of the river and the eddy energies, e = e r + ee = c re r + ^ cjei « ( l / 2 W l c r + (l/2)c e], (4 - 8a) j where cy is the concentration of eddies of radius Rj, namely CJ = N{Rj)nR]/SB,ce = J2 e,- (4 - 86) j Equation 4-8(a) indicates that the rivers may contain a significant fraction of the total kinetic energy of the surface motion. The eddy size spectrum N(X,R) can be used to characterize the spatial structure of the rotating coherent motion. The measured data for the 20cm/sec flow are shown in Fig.4-10. Comparing the distributions at various distances from the grid it is seen that the center of the eddy distribution shifts to larger size as the turbulence decays. The increase of the number of large structures stops after about six seconds (X = 110cm downstream), when the eddy concentration has fallen below ce « 30%, and the population becomes dominated by viscous (spontaneous) decay processes. The eddy flow represents a reservoir of rotational kinetic energy, which may be de-scribed by an energy density ee that is averaged over the total surface area. More inter-CHAPTER 4 STRUCTURE STATISTICS F i g u r e 4-10 E d d y s i z e d i s t r i b u t i o n N(X,R). e s t i n g a t t h i s t i m e is t o c o n s i d e r t h e ed d i e s as i n d i v i d u a l e n t i t i e s t h a t have t h e energy E p e r u n i t l e n g t h s t o r e d w i t h i n t h e o b s e r v e d sur f a c e l a y e r o f t h e flow. F o r a n y e d d y of g i v e n r a d i u s a n d k n o w n a n g u l a r speed Q one c a n c a l c u l a t e a n energy E. S i n c e t h e e d d y s i z e d i s t r i b u t i o n N ( X , R ) is al s o k n o w n one c a n d e t e r m i n e t h e eddy e n e r g y d i s t r i b u t i o n N\E(R)}. A g a i n , we are i n t e r e s t e d i n the f o r m o f t h e c u r v e N[E] v e r s u s E as c h a r a c t e r i z e d by t h e energy width a n d t h e slope of t h e h i g h energy t a i l . F o r t h a t p u r p o s e t h e d i s t r i b u t i o n is p l o t t e d i n s e m i l o g f a s h i o n i n F i g . 4 - l l ( a ) . I n t h e range 5 0 c m < X < 110cm, t h e h i g h energy t a i l is w e l l a p p r o x i m a t e d by a s t r a i g h t l i n e . W h e n CHAPTER 4 STRUCTURE STATISTICS F i g u r e 4-11 E d d y energy d i s t r i b u t i o n N(X,E). (a) E v o l u t i o n w i t h d i s t a n c e , (b) C u r v e -f i t f o r X = 7 0 c m u s i n g eqn.(4-9a). 0e = 2.8erg/cm a = 0.86 CHAPTER 4 STRUCTURE STATISTICS 78 p l o t t i n g t h e lower energy p a r t of t h e d i s t r i b u t i o n i n log-log f a s h i o n i t was observed t h a t N(E) rises a p p r o x i m a t e l y p r o p o r t i o n a l t o EQ. T h e e x p e r i m e n t a l d a t a c a n therefore be r e p r e s e n t e d b y N(E) = constEaexp(-E/Oe) (4 - 9c ) w h e r e N(E) is t h e n u m b e r of s t r u c t u r e s p e r u n i t energy i n t e r v a l h a v i n g t h e energy E. T h i s energy d i s t r i b u t i o n is a g a i n s i m i l a r t o a B o l t z m a n n f u n c t i o n . I t is a n i n d i c a t i o n t h a t f r e q u e n t i n t e r a c t i o n s are c o u p l i n g t h e e d d y energies. F i g u r e 4-11 (b) shows t h e c u r v e fit p l o t t e d over t h e s i n g l e d i s t a n c e of X = 70cm. J u s t as f o r t h e r i v e r s , t h e p o s i t i o n of t h e d i s t r i b u t i o n ' s m a x i m u m has been measured i n o r d e r t o d e t e r m i n e t h e e x p o n e n t a(= Em/0e) w i t h t h e r e s u l t a = 0.9±.l. ( 4 - 9 6 ) T h e r i v e r a n d e d d y energy d i s t r i b u t i o n ' s v alues f o r a , eqns.(4-2) a n d (4-9) are seen t o be s i m i l a r i n v a l u e . T h e s l o p e of t h e t a i l o f t h e d i s t r i b u t i o n is c h a r a c t e r i z e d by t h e p a r a m e t e r 0e, w h i c h m a y be c a l l e d the "eddy t e m p e r a t u r e " . T h e energy h a l f - w i d t h A0e a n d t h e e d d y tem-p e r a t u r e 0e have been m e a s u r e d i n t h e e q u i l i b r i u m range f o r v a r i o u s d i s t a n c e s X. T h e y are s h o w n t o g e t h e r w i t h Em i n Fig.4-6(b). T h e n u m b e r of eddies i n t h e decay range X > 1 2 0 c m is t o o s m a l l t o a s s i g n m e a n i n g f u l values t o these s t a t i s t i c a l p a r a m e t e r s . It is i n t e r e s t i n g t o see t h a t t h e e d d y a n d r i v e r t e m p e r a t u r e s i n i t i a l l y have t h e same m a g n i t u d e a n d decay a t a b o u t t h e same r a t e . CHAPTER 4 STRUCTURE STATISTICS We i n v e s t i g a t e d w h e t h e r t h e e d d y energy d e n s i t y c o u l d be o b t a i n e d as t h e p r o d u c t o f e d d y n u m b e r a n d e d d y t e m p e r a t u r e m u l t i p l i e d b y some s u i t a b l e c o n s t a n t a n d f o u n d t h a t t h e d a t a i n t h e e q u i l i b r i u m range c o u l d be a p p r o x i m a t e d b y t h e r e l a t i o n e e = KOnNe/Se = Kne6e, (4 - 10) w h e r e t h e n u m e r i c a l c o n s t a n t has t h e v a l u e K 0.65. T h e e m p i r i c a l r e l a t i o n (4-10) is a n a l o g o u s t o t h e s t a n d a r d r e l a t i o n f o r t h e i n t e r n a l e n e r g y (e = const x NKT, where K is t h e B o l t z m a n n c o n s t a n t ) of a s y s t e m of p a r t i c l e s i n t h e r m a l e q u i l i b r i u m . We have s h o w n t h a t t h e s u r f a c e f l o w s t r u c t u r e s o n g r i d - g e n e r a t e d t u r b u l e n c e c a n be i d e n t i f i e d a n d d e s c r i b e d b y s i m p l e p a r a m e t e r s . F u r t h e r m o r e , we have f o u n d t h a t t h e d i s t r i b u t i o n s of these s t r u c t u r e p r o p e r t i e s f o l l o w m e a n i n g f u l t r e n d s as t h e flow evolves. A l s o , t o g o o d a p p r o x i m a t i o n t h e d i s t r i b u t i o n s t h e m s e l v e s m a y be c h a r a c t e r i z e d by s i m p l e p a r a m e t e r s . B e f o r e g o i n g o n t o d i s c u s s t h e s t r u c t u r e d y n a m i c s i t w i l l be of i n t e r e s t t o a n a l y z e t h e e v o l u t i o n o f t h e s t r u c t u r e s t a t i s t i c s i n t e r m s o f t h e r a t e e q u a t i o n m o d e l p r e s e n t e d i n the s e c o n d c h a p t e r . 4.4 E n e r g y D e c a y o f t h e S u r f a c e F l o w T h e v e l o c i t y fields o f eddies a n d r i v e r s were r e c o r d e d as a f u n c t i o n of d i s t a n c e X f r o m t h e g r i d , a n d f r o m these d a t a we d e r i v e d t h e size, a n g u l a r speed a n d e n e r g y d i s t r i b u t i o n s . T h e d i s t a n c e f r o m the g r i d m a y a l t e r n a t e l y be c o n s i d e r e d as a measure o f t h e age of the t u r b u l e n t m o t i o n , n a m e l y t h e t i m e s i n c e t h e g r i d has passed t h e l a b f r a m e p o s i t i o n CHAPTER 4 STRUCTURE STATISTICS t = X/Ug. Thus, study of the structure statistics as a function of distance allows one to investigate some aspects of structure dynamics. The total energy density and the partial values for the eddy and the river modes are given in Fig.4-12. The observation period has been divided into the production, the equilibrium, and the decay range. In the equilibrium range there is strong coupling between and within the two modes. Eddies and rivers have about the same temperature as shown in Figs.4-7, and the eddy edge speed UR is about equal to the average river speed u 0 . It is noted, however, that the energy density of the two structure types decay at different rates. The energy losses occur more readily in the rotational (eddy) motion than in the translational (river) motion. The eddy-eddy interaction should contribute most significantly to the evolution of the initial distribution, as frequent occurrence of such events requires a high number density. The time variation of the distributions allow one to estimate whether eddy-eddy interactions are important. For this purpose we write a rate equation for the change of numbers of eddies N of some radius R where the A, B, and C coefficients are in general dependent on the particular eddy radius and flow environment. dN —- = (A + B)N + CN2 (4 - 11) dt The rate coefficients A, B, and C quantify the evolution of the population of structures due to interactions with the fluid (frictional decay), with the fluid flow surrounding the eddies (pumping and tearing) and with other eddies (collisions) respectively. Equation (4-11) is a simplification of the more detailed rate equation (2-35) where each of the terms on the right-hand side is replaced by a summation of contributions to a particular eddy size's population change from all other eddies in the distribution. Equation (4-11) CHAPTER 4 STRUCTURE STATISTICS 81 c o n t a i n s o n l y t h e d o m i n a n t t e r m s t h a t are o f i n t e r e s t here. T h e A a n d B c o e f f i c i e n t s are l o c a l t o t h e d i s t r i b u t i o n so e q n . ( 4 - l l ) is a first o r d e r m o d e l f o r t h e s u m of these rates. F o r o u r s m o o t h l y v a r y i n g d i s t r i b u t i o n a d j a c e n t s t r u c t u r e n u m b e r s are n e a r l y e q u a l . T h e c o e f f i c i e n t C as w r i t t e n here is a gross s i m p l i f i c a t i o n of t h e p a i r i n g process w h i c h w o u l d i n v o l v e i n t e r a c t i o n between s t r u c t u r e s of s m a l l e r size. T h e v a l u e f o u n d here s h o u l d o n l y be t a k e n as a n i n d i c a t o r of t h e i n f l u e n c e of c o l l i s i o n a l processes (i.e. n o n - l o c a l w i t h i n t h e d i s t r i b u t i o n ) . A t t h i s p o i n t , i t is n o t necessary t o have any d e t a i l e d k n o wledge of t h e r a t e c o e f f i c i e n t s . However, a q u a d r a t i c t e r m (co e f f i c i e n t C ^  0) m u s t be present i f the CHAPTER 4 STRUCTURE STATISTICS 82 eddy concentration has any influence on the interaction dynamics. Here C is obtained from a numerical evaluation of the size spectrum decay. For that purpose eqn.(4-ll) and the differentials are replaced by differences Y = ± A £ = (A + B) + CN. ( 4 - 1 2 ) The left-hand side of eqn.(4-12) contains numbers that can be obtained from the data by slicing through the N(R,t) distributions in the time direction, see Fig.4-13. When Y is plotted as a function of N a straight line should result. The slope of this line is C and the intercept gives the sum of A and B . To demonstrate this approach an analysis was carried out for eddies of radius R= 1.4cm, and the results are shown in Fig.4-14. For R= 1.4cm the linear least squares fit has the intercept A + B = —0.70\sec -1,<7 = 0 . 5 5 s e c - 1 , and the slope C = +4.6 x 1 0 _ 3 s e c _ 1 ,<r = 6.3 x 10~ 3 sec _ 1 . A model estimate of the rate coefficient A will be presented in chapter five. The line intersects the N axis at N=152. Here C is positive, implying the collisional processes of two or more structures combining will increase the number of eddies of ra-dius R = 1.4cm. For large N the production of this size eddy by merging of smaller ones would tend to compensate for the two loss mechanisms. For small N one would expect a rapid decay of eddies of this radius. This agrees with observation. Similar results were obtained for data taken at other structure sizes. The statistical properties of the eddies and rivers arise from the interaction of the structures with their surroundings. The next step is to see if the dynamics of these structures can be identified and quantified and then used in the rate equation model to predict the observed statistics. CHAPTER 4 STRUCTURE STATISTICS F i g u r e 4-14 G r a p h i c a l d e t e r m i n a t i o n of r a t e c o e f f i c i e n t s . Y = (l/N){AN f At). E x p e r i m e n t a l N values f o r R = 1.4cm are t a k e n f o r t i m e i n t e r v a l s i n the range 1.5 < t < 9.5 sec a n d averaged w i t h adjacent values. CHAPTER 5 STRUCTURE DYNAMICS C H A P T E R 5 STRUCTURE DYNAMICS A c r u c i a l q u e s t i o n f o r t h e r a t e e q u a t i o n a p p r o a c h is w h e t h e r t h e d y n a m i c s of i n -d i v i d u a l s t r u c t u r e s c a n be c h a r a c t e r i z e d a n d c a t e g o r i z e d . T h e present c h a p t e r r e p o r t s a n e x p e r i m e n t a l e x a m i n a t i o n o f t h e d y n a m i c s o f t h e r o t a t i n g flow-structures f o u n d o n g r i d - g e n e r a t e d t u r b u l e n c e . I n t h i s c h a p t e r t h e r i v e r - f l o w is t r e a t e d as b e l o n g i n g t o t h e b a c k g r o u n d fluid flow i n w h i c h t h e eddies evolve. T h e r i v e r - f l o w w i l l p r e d o m i n a n t l y e v o l v e t h r o u g h v i s c o u s decay a l t h o u g h i t is p o s s i b l e t h a t e d d i e s c o u l d be p r o d u c e d by a n in v e r s e o m e g a decay process. T h e a n a l y s i s t o o l s d e v e l o p e d f o r t h i s s t u d y have a l r e a d y b e e n p r e s e n t e d i n c h a p t e r 3. T h e first s e c t i o n e x a m i n e s t h e nea r s u r f a c e fluid d y n a m i c s i n o r d e r t o p r o v i d e a m o r e i n - d e p t h u n d e r s t a n d i n g of t h e o b s e r v e d s u r f a c e m o t i o n . T h i s is f o l l o w e d b y a s u r v e y o f t h e way t h e s t r u c t u r e s are o b s e r v e d t o i n t e r a c t . S e c t i o n t h r e e d e s c r i b e s t h e i n i t i a l v o r t e x p r o d u c t i o n a n d presents a s i m p l e k i n e m a t i c m o d e l w h i c h pre-d i c t s t h e i n i t i a l s t r u c t u r e p o p u l a t i o n . A m o d e l f o r t h e v i s c o u s decay r a t e is g i v e n n e x t a n d is c o m p a r e d w i t h e x p e r i m e n t a l r e s u l t s f r o m t h e g r i d - f l o w . O b s e r v a t i o n s o f t h e omega decay a n d e d d y p a i r i n g a n d t e a r i n g are t h e n presented. T h e c h a p t e r ends by d i s c u s s i n g t h e p r e d i c t i o n o f s t a t i s t i c s f r o m t h e o b s e r v e d d y n a m i c s . CHAPTER 5 STRUCTURE DYNAMICS 85 5.1 Near Surface Fluid Dynamics T h e s u r f a c e s t r u c t u r e s s t u d i e d i n t h i s t h e s i s are c o n s i d e r e d t o b e t h e " f o o t p r i n t s " * o f m o r e e x o t i c c r e a t u r e s r e s i d i n g i n t h e b o d y of t h e fluid. I t is t h e r e f o r e i n s t r u c t i v e t o i n v e s t i g a t e t h e d y n a m i c s o f t h e b u l k a n d near-surface m o t i o n . C o n t a m i n a t e d surfaces are k n o w n t o h a v e v i s c o s i t i e s o r d e r s o f m a g n i t u d e h i g h e r t h a n c l e a n ones w h i c h i n t u r n have a v i s c o s i t y m u c h h i g h e r t h a n t h e same fluid i n b u l k [48]. W h i l e n o t c o m p l e t e l y c l e a n , t h e s u r f a c e flows s t u d i e d i n t h i s t h e s i s s howed l i t t l e effects of c o n t a m i n a t i o n . T h e e l a s t i c i t y o f t h e film o f s u r f a c e c o n t a m i n a t i o n w i l l s uppress v e r t i c a l d i s p l a c e m e n t a n d make t h e s u r f a c e a c t s o m e t h i n g l i k e a r i g i d w a l l i n r e l a t i o n t o t h e s u b s u r f a c e flow. T h e t h e o r y o f homogeneous t u r b u l e n c e i m p i n g i n g u p o n a r i g i d w a l l has been exam-i n e d b y H u n t a n d G r a h a m [49]. T h e y f o u n d a g r o w i n g v i s c o u s b o u n d a r y layer a d j a c e n t t o t h e w a l l a n d a la r g e r i n v i s c i d "source l a y e r " , see Fig.5-1. I n t h e source layer t h e i s o t r o p i c b u l k flow r e o r i e n t s t o e l i m i n a t e v e r t i c a l m o t i o n s . T h e v i s c o u s b o u n d a r y l a y e r h a s a t h i c k n e s s 6V ~ 4 ( f X / C / s ) 1 / 2 (« 0.5cm f o r Ug = 20cm/sec at X = 3 0 c m ) . T h e l a r g e r i n v i s c i d s o u r c e l a y e r has a t h i c k n e s s S8 ~ Lxoo, w h e r e L x o o is t h e str e a m w i s e i n t e g r a l l e n g t h s c a l e i n t h e b u l k o f t h e flow, see s e c t i o n 3.4. I n reference [30], L x o o was f o u n d t o s t e a d i l y i n c r e a s e f r o m a v a l u e of 1.3cm a t X = 3 0 c m f o r Ug = 20cm/sec. A t t h e edge of t h e v i s c o u s s u r f a c e l a y e r (z = 0) t h e k i n e t i c energy o f t h e t u r b u l e n t m o t i o n (vx0 + v20) is t h e s a m e as i n t h e b u l k o f t h e fluid w h e r e i t is e q u a l t o ( v f ^ + t ; 2 ^ + u 2 oo), b u t the v e l o c i t y c o m p o n e n t n o r m a l t o t h e s u r f a c e has v a n i s h e d (vzo = 0). T h e energy i n t h i s c o m p o n e n t h a s b e en p a r t i t i o n e d ( e q u a l l y f o r a n i s o t r o p i c b u l k flow) t o t h e s t r e a m w i s e VXQ a n d l a t e r a l vyo m o t i o n s . A s i m i l a r p h e n o m e n o n is o b s e r v e d i n s t r a t i f i e d flow i n t h e l a b o r a t o r y [50, * A s a p t l y d e s c r i b e d b y A. K. M . F. H u s s a i n ( p r i v a t e c o m m u n i c a t i o n ) . CHAPTER 5 STRUCTURE DYNAMICS 86 Ut = 20cm /etc F i g u r e 5-1 N e a r s u r f a c e b o u n d a r y layers. 51], t h e o c e a n , i n m a g n e t o h y d r o d y n a m i c s [52] a n d t h e a t m o s p h e r e [53] where layers of c o n s t a n t d e n s i t y l o c k t h e t u r b u l e n t flow i n t o q u a s i - t w o - d i m e n s i o n a l m o t i o n . O n e m a y t h i n k of a s u r f a c e f l o w as a h i g h l y s t r a t i f i e d f l u i d l a yer. It is also i n t e r e s t i n g t o note t h a t a s t r o n g a n a l o g y e x i s t s b etween r o t a t i n g a n d s t r a t i f i e d u n s t e a d y flows [54] so t h a t t h e r e s u l t s r e p o r t e d here s h o u l d be p e r t i n e n t t o g e o s t r o p h i c t u r b u l e n c e . Indeed a s t r i k i n g s i m i l a r i t y is seen b e t w e e n t h e flow fields o n g r i d - g e n e r a t e d t u r b u l e n c e a n d t h a t observed w i t h i n a r a p i d l y r o t a t i n g v o l u m e of fluid, see t h e p h o t o s of M o r y a n d H o p f i n g e r [40]. CHAPTER 5 STRUCTURE D YNAMICS 87 T h a t we m a y e x p e c t r e s u l t s s i m i l a r t o those p r e d i c t e d b y H u n t a n d G r a h a m is sup-p o r t e d b y t h e e x p e r i m e n t a l o b s e r v a t i o n b y B r u m l e y [55] o f t h e r e o r i e n t i n g "source" r e g i o n f o r v e r t i c a l l y o s c i l l a t i n g g r i d - g e n e r a t e d t u r b u l e n c e i m p i n g i n g u p o n a shea r free s u r f a c e o f w a t e r . B r u m l e y a l s o f o u n d e v i d e n c e of t h e v i s c o u s b o u n d a r y l a y e r i n d i c a t i n g (to h i m ) t h a t h i s f l u i d s u r f a c e m a y n o t have been p e r f e c t l y c l e a n e d . A c o m p a r i s o n of e x p e r i m e n t a l l y d e t e r m i n e d s u r f a c e a n d b u l k i n t e g r a l l e n g t h scales, L x o a n d L x o o r e s p e c t i v e l y , is r e p o r t e d i n [30] f o r t h e present a p p a r a t u s a n d a 40cm/sec g r i d speed. T h e p o w e r s p e c t r a l density, E\\(f) of eqn.(3-3), was o b t a i n e d t h r o u g h F o u r i e r a n a l y s i s of v e l o c i t y r e c o r d s m e a s u r e d w i t h a h o t - f i l m a nemometer. T h e i n t e g r a l l e n g t h s c a l e of t h e b u l k flow, L x o o , was t h e n c a l c u l a t e d f r o m a n e x t r a p o l a t e d v a l u e f o r En(0) u s i n g eqn.(3-9). T h e s u r f a c e l e n g t h s c a l e Lxo was c a l c u l a t e d i n a s i m i l a r m a n n e r f r o m v e l o c i t y r e c o r d s o f s u r f a c e flow g e n e r a t e d b y c o m p u t e r u s i n g t h e o b s e r v e d s t r u c t u r e s t a t i s -t i c s N ( X , R ) a n d u>[X, R). A n a l t e r n a t e m e t h o d of d i r e c t l y a v e r a g i n g t h e m e a n eddy c h o r d o v e r t h e N(R) d i s t r i b u t i o n was f o u n d t o g i v e a n e a r l y i d e n t i c a l l e n g t h s c a l e value. N e a r t h e g r i d t h e m a g n i t u d e s o f these l e n g t h scale measures were e q u a l w i t h i n t h e e x p e r i m e n -t a l u n c e r t a i n t i e s . A s t h e flow e v o l v e d , t h e b u l k m o t i o n l e n g t h s c a l e L x o o i n c r e a s e d m o r e r a p i d l y t h a n Lxo. I n t h e s a me r e p o r t , t h e r e l a t i o n s h i p b etween the l o n g i t u d i n a l c o m p o n e n t of t h e t u r -b u l e n t k i n e t i c energy f o r t h e s u r f a c e a n d b u l k m o t i o n s were c o m p a r e d f o r a 40cm/sec g r i d speed. T h e e n e r g y decay c u r v e s were s u r p r i s i n g l y s i m i l a r i n s h a p e w i t h t h e b u l k m o t i o n b e i n g c o n s i s t e n t l y 20 t i m e s m o r e energetic t h a n t h e c o r r e s p o n d i n g s u r f a c e e d d y energy density. T h i s f a c t o r w i l l d e p e n d on t h e i s o t r o p y of t h e a c t u a l s t r u c t u r e o r i e n t a t i o n , t h e CHAPTER 5 STRUCTURE DYNAMICS 88 e d d y - r i v e r e n e r g y r a t i o a n d t h e degree o f c o n t a m i n a t i o n of t h e fluid s u r f a c e . T h e i n f l u -e n c e of s u r f a c e c o n t a m i n a n t i n d u c e d v i s c o s i t y o n t h e sur f a c e m o t i o n was e x a m i n e d . It w a s f o u n d t h a t m o t i o n o n a c o n t a m i n a t e d s u r f a c e was i n i t i a l l y less v i g o r o u s , decayed m o r e q u i c k l y a n d grew i n si z e scale m o r e r a p i d l y t h a n t h e flow of a f r e s h l y s k i m m e d s u r f a c e . A d i r e c t c o m p a r i s o n b e t w e e n t h e s u r f a c e a n d s u b s u r f a c e c o h e r e n t s t r u c t u r e p r o p e r t i e s is n o t p o s s i b l e as s u b s u r f a c e s t r u c t u r e s have y e t t o be s t u d i e d f o r t h i s flow. However, s o m e q u a l i t a t i v e s u b s u r f a c e flow v i s u a l i z a t i o n was a c c o m p l i s h e d u s i n g a sheet l i g h t i n g t e c h n i q u e . T h e r e s u l t s s howed t w o - d i m e n s i o n a l flow i m m e d i a t e l y b e h i n d t h e g r i d a n d j u s t b e n e a t h t h e s u r f a c e , see Figs.5-2. F i g u r e 5-2a is a flow p h o t o of t h e m o t i o n i n a 2 c m w i d e sheet p a r a l l e l t o a n d 3 c m b e n e a t h t h e fluid s u r face. T h e i m a g e of the m o v i n g g r i d c a n be seen a t t h e t o p . T h e p r e d o m i n a n t l y t w o - d i m e n s i o n a l s t r u c t u r e s i m m e d i a t e l y b e h i n d t h e g r i d s o o n a p p e a r d i s o r g a n i z e d . T h i s i n d i c a t e s t h a t a large f r a c t i o n of the m o t i o n is o c c u r r i n g n o r m a l t o t h e sheet of l i g h t . T h e flow is b e c o m i n g i s o t r o p i c . Fig.5-2 b shows t h e m o t i o n i n a few c e n t i m e t e r w i d e sheet of l i g h t p a r a l l e l t o t h e side w a l l s of t h e t a n k . I n m o s t of t h e p h o t o as m u c h m o t i o n a p p e a r s t o be o c c u r r i n g i n t h e v e r t i c a l as i n t h e h o r i z o n t a l d i r e c t i o n . P r e d o m i n a n t l y h o r i z o n t a l m o t i o n c a n be seen v e r y near t h e s u r f a c e a n d p e r h a p s a t t h e v e r y lef t s i d e of t h e l i g h t sheet were t h e g r i d has j u s t d i s a p p e a r e d . T h e b u l k of t h e flow is c h a o t i c a n d t h r e e - d i m e n s i o n a l a n d n o t a menable t o o u r present t w o - d i m e n s i o n a l v i s u a l i z a t i o n a n d coherent s t r u c t u r e a n a l y s i s methods. S i m i l a r i t y b e t w e e n power s p e c t r a of l o n g i t u d i n a l v e l o c i t y fluctuations g e n e r a t e d f r o m the s u r f a c e s t r u c t u r e s t a t i s t i c s a n d those m e a s u r e d w i t h a h o t - f i l m a n e m o m e t e r [30] suggest a s i m i l a r v e l o c i t y s t r u c t u r e over large scales m a y e x i s t . A l s o , t h e close agreement i n CHAPTER 5 STRUCTURE DYNAMICS Figure 5-2 Bulk motion of turbulent grid-flow. Sheet lighting: (a) parallel to and 3 c m beneath the surface, (interpretation shown alongside photo) (b) parallel to side walls. CHAPTER 5 STRUCTURE DYNAMICS t h e l e n g t h scales i n d i c a t e s t h a t a s i m i l a r s i z e d i s t r i b u t i o n m a y b e f o u n d . T h e b u l k flow s t r u c t u r e s are t h r e e - d i m e n s i o n a l i n n a t u r e , have a m o r e c o m p l e x i n t e r n a l flow s t r u c t u r e , a n d w i l l i n t e r a c t w i t h t h e i r e n v i r o n m e n t i n t h e a d d i t i o n a l m a n n e r of energy p u m p i n g t h r o u g h v o r t e x s t r e t c h i n g . It is r e a s o n a b l e t o e x p e c t t h a t t h e m o r e c o m p l e x subsurface flow w i l l e v o l v e t h r o u g h some o f t h e processes f o u n d i n t h e s u r f a c e flow. A s a final p o i n t i t s h o u l d be n o t e d t h a t i m p l i c i t t o t h e e n t i r e flow v i s u a l i z a t i o n a n a l y s i s has been t h e a s s u m p t i o n t h a t t h e m o t i o n of t h e t r a c e r p a r t i c l e s a c c u r a t e l y represents t h e m o t i o n of t h e u n d e r l y i n g fluid. T h e d i f f e r e n c e b e t w e e n t h e t r a c e r a n d fluid m o t i o n was c a l c u l a t e d f o l l o w i n g Z a l u t s k i i [56] a n d f o u n d t o b e less t h a n 1% f o r t h e ~ 0.4mm a l u m i n u m filings used. T h e d i f f e r e n c e i n r a d i u s of c u r v a t u r e was s i m i l a r l y d e t e r m i n e d t o be less t h a n 1 % f o r t h e flow c o n d i t i o n s generated. T h e s e c a l c u l a t i o n s a p p l y t o t h e i n t e r i o r of t h e fluid. W e ex p e c t even closer c o u p l i n g between t h e filings a n d the surface m o t i o n due t o t h e a c t i o n of s u r f a c e t e n s i o n a n d s u r f a c e v i s c o s i t y . 5.2 S t r u c t u r e E v o l u t i o n F i g u r e 5-3 shows a c l a s s i f i c a t i o n scheme of t h e e d d y decay d y n a m i c s observed i n t i m e - e x p o s e d p h o t o g r a p h s a n d v i d e o sequences of t h e s u r f a c e flow o n grid-generated t u r b u l e n c e . F i v e d i s t i n c t t y p e s of e v o l u t i o n processes were d i s t i n g u i s h e d . T h e s e were t h e i n i t i a l v o r t e x p r o d u c t i o n , e d d y t e a r i n g , e d d y p a i r i n g , v i s c o u s decay a n d the omega decay. T h e i n i t i a l v o r t e x p r o d u c t i o n is d e s c r i b e d i n s e c t i o n 5.3. E d d y t e a r i n g is a v e r y r a r e o c c u r r e n c e i n t h i s g r i d - f l o w a n d was o n l y o b s e r v e d once a m o n g s t t h e m a n y s t r u c t u r e e v o l u t i o n s f o l l o w e d . T h a t i n s t a n c e o c c u r r e d a few m e s h w i d t h s f r o m t h e g r i d i n the i n i t i a l CHAPTER 5 STRUCTURE DYNAMICS 91 p r o d u c t i o n r e g i o n w here t h e shear stresses are hi g h e s t . A s i m p l e m o d e l f o r t h i s process h a s b e e n p r o p o s e d [l] a n d a r a t e c o e f f i c i e n t d e t e r m i n e d . T h i s p r o c e s s w i l l be s i g n i f i c a n t f o r flows w i t h h i g h v e l o c i t y g r a d i e n t s over areas l a r g e c o m p a r e d w i t h t h e s t r u c t u r e s i z e b u t is n o t a s i g n i f i c a n t f a c t o r i n t h e d y n a m i c s o f g r i d - g e n e r a t e d t u r b u l e n c e . E d d y p a i r i n g is a f r e q u e n t l y o b s e r v e d c o l l i s i o n a l process w h i c h is t h e d o m i n a n t mech-a n i s m b y w h i c h s t r u c t u r e size is i n c r e a s e d i n t h e e q u i l i b r i u m r e g i o n o f t h e flow. F i g u r e 5-4 shows a sequence of flow p h o t o s d e p i c t i n g t h e a m a l g a m a t i o n o f a n u m b e r of s t r u c -t u r e s . E d d y p a i r i n g is a w e l l k n o w n p h e n o m e n o n i n w h i c h l i k e - s i g n e d v o r t i c e s c o m b i n e t o f o r m a s i n g l e l a r g e r s t r u c t u r e . T h i s a u t h o r first o b s e r v e d s t r u c t u r e p a i r i n g as the g r o w t h of a s e p a r a t e d peak i n t h e siz e s p e c t r a . F i g u r e 5-5 d e p i c t s t h i s e v o l u t i o n . I n g r i d - g e n e r a t e d flow, s t r u c t u r e s o f l i k e r o t a t i o n sense are o c c a s i o n a l l y b r o u g h t t o -g e t h e r t h r o u g h t h e a c t i o n of t h e s u r r o u n d i n g flow. A s t a g n a t i o n zone of h i g h p r e s s u r e w i l l a p p e a r i n t h e h i g h stress c o n t a c t p a t c h , see Fig.5-5. G r a d u a l l y m o r e a n d more of t h e flow f r o m each s t r u c t u r e is r e d i r e c t e d a r o u n d b o t h s t r u c t u r e s . W h e n a l l t h e flow has b e en r e d i r e c t e d t h e o r i g i n a l s t r u c t u r e s have ceased t o e x i s t , l e a v i n g i n t h e i r p l a c e a si n g l e e d d y w i t h s o m e w h a t less energy t h a n t h a t o f t h e two p a r e n t s t r u c t u r e s c o m b i n e d . S i m i l a r l o c a l d y n a m i c s are r e s p o n s i b l e f o r t h e o m e g a decay d i s c u s s e d i n s e c t i o n 5.5. T h e u l t i m a t e f a t e of a l l t h e fluid m o t i o n is v i s c o u s decay t o a quiescent state. T h e flow becomes v i s c o u s d o m i n a t e d at s m a l l scales a n d i n t h e final p e r i o d . S e c t i o n 5.4 discusses t h e v i s c o u s decay of the s t r u c t u r e s i n t h e near g r i d r e g i o n . CHAPTER 5 STRUCTURE DYNAMICS 92 Initial Vortex Production F i g u r e 5-3 S t r u c t u r e e v o l u t i o n t y p e s . CHAPTER 5 STRUCTURE DYNAMICS F i g u r e 5-4 E d d y a m a l g a m a t i o n , (seen i n c e n t e r o f p h o t o series) (a) t = 3 sec, (b) t = 6 sec, a n d (c) t — 9 sec. CHAPTER 5 STRUCTURE DYNAMICS F i g u r e 5-5 E d d y p a i r i n g . 5.3 I n i t i a l V o r t e x P r o d u c t i o n T h e flow field becomes a m e n a b l e t o t h e p h o t o g r a p h i c a n a l y s i s m e t h o d at a b o u t X = 3 0 c m (X' « 6) w h e n t h e s t r u c t u r e s are f u l l y f o r m e d a n d r e l a t i v e l y s t a t i o n a r y i n t h e t a n k f r a m e o f reference. I n o r d e r t o s t u d y t h e i n i t i a l v o r t e x p r o d u c t i o n t h e flow was r e c o r d e d o n a 30 f r a m e p e r s e c o n d (fps) v i d e o cassette r e c o r d e r . A q u a l i t a t i v e a n a l y s i s of t h e v i d e o sequence was t h e n p e r f o r m e d i n o r d e r t o o b t a i n a d e s c r i p t i o n o f t h e p r o d u c t i o n d y n a m i c s . T h e c o m p u t e r - a u t o m a t e d coherent s t r u c t u r e i d e n t i f i c a t i o n s y s t e m d e s c r i b e d i n t h e t h i r d c h a p t e r of t h i s t h e s i s was t h e n used t o e x t r a c t q u a n t i t a t i v e i n f o r m a t i o n . T h e s a me flow c o n d i t i o n s were u s e d as f o r t h e s t r u c t u r e s t a t i s t i c s s t u d i e s , n a m e l y : a 4:1 c y l i n d e r s p a c i n g t o d i a m e t e r r a t i o g r i d m o v i n g w i t h a R e y n o l d s n u m b e r based o n b a r s e p a r a t i o n of 20,000. A t t h i s g r i d speed of Ug = 20cm/sec t h e i n i t a l v o r t e x p r o d u c t i o n d i d not resemble i s o l a t e d v o n K a r m a n v o r t e x s t r e e t s of i n d i v i d u a l r o d s as m i g h t be e x p e c t e d ; t h e m a j o r CHAPTER 5 STRUCTURE DYNAMICS F i g u r e 5-6 v o n K a r m a n v o r t e x street. F l o w p h o t o f r o m F. A h l b o r n « 1905. d i f f e r e n c e b e i n g t h e m u c h m o r e r a p i d g r o w t h i n size o f t h e v o r t i c e s . T h e a l t e r n a t e s h e d d i n g of v o r t i c e s f r o m c y l i n d e r s , see Fig.5-6, over a w i d e range of R e y n o l d s n u m b e r s has b e en k n o w n for m a n y decades a n d was first e x a m i n e d by v o n K a r m a n i n 1911 [2]. F i g u r e 5-7 d e p i c t s t h e f o r m a t i o n d y n a m i c s d e t e r m i n e d by o b s e r v i n g 13 sets of v i d e o sequences o f the flow between ad j a c e n t bars. F i g u r e 5-8(a) shows a s u p e r p o s i t i o n of 20 f r a m e s o f raw d i g i t i z e d d a t a t a k e n w i t h t h e g r i d b a r s l e a v i n g t h e left h a n d side of the first f r a m e . F i g u r e 5-8(b) shows t h e same " t i m e e x p o s u r e " b u t w i t h t h e s h e d v o r t i c e s t r a c e d i n b y h a n d . T h i s figure c l o s e l y resembles t h e i d e a l i z e d flow of Fig.5-7. T h e p i c t u r e of the CHAPTER 5 STRUCTURE DYNAMICS 0 I X F i g u r e 5-7 I n i t i a l v o r t e x f o r m a t i o n . f o r m a t i o n d y n a m i c s is as f o l l o w s : T h e v o r t i c e s have t h e i r o r i g i n i n t h e v o r t e x s h e d d i n g r e g i o n a t i n d i v i d u a l c y l i n d e r s a n d t h e i r p r o d u c t i o n r a t e is f o u n d t o be e q u a l t o t h a t o f a n i s o l a t e d c y l i n d e r o f d i a m e t e r d. T h e s h e d d i n g f r e q u e n c y / is d e t e r m i n e d b y t h e n o n - d i m e n s i o n a l S t r o u h a l n u m b e r ( 5 - 1 ) T h e S t r o u h a l n u m b e r is w e l l k n o w n [16] t o be e q u a l t o 0.2 f o r c i r c u l a r c y l i n d e r s over a w i d e range o f R e y n o l d s n u m b e r s , i n c l u d i n g t h e present v a l u e of Re=2,500 b a s e d o n c y l i n d e r d i a m e t e r . A c u r i o u s a d d i t i o n a l f e a t u r e is t h a t t h e f o r m a t i o n d y n a m i c s a p p e a r t o be n CHAPTER 5 STRUCTURE DYNAMICS F i g u r e 5-8 Near grid video \ ime" exposure, (a) Raw digitized data, (b) Identification superimposed CHAPTER 5 STRUCTURE DYNAMICS r a d i a n s o u t o f phase i m m e d i a t e l y b e h i n d a d j a c e n t bars. T h e v o r t i c e s r a p i d l y ( w i t h i n ~ 1-2 M ) grow t o f i l l i n t h e r e g i o n b etween t h e m o m e n t u m defect "shadows" of t h e g r i d b a r s a n d s i m u l t a n e o u s l y s h i f t t o t h e t a n d e m a r r a n g e m e n t s h o w n i n t h e d o w n s t r e a m e n d of Fig.5-7. F r o m these o b s e r v a t i o n s a s i m p l e k i n e m a t i c m o d e l was a d v a n c e d i n a recent pub-l i c a t i o n [6]. It was o b s e r v e d t h a t i n t h e r e s u l t i n g w a v y c o n f i g u r a t i o n t h e v o r t i c e s (or eddi e s ) are ne s t e d i n a m a n n e r w h i c h reduces t h e o v e r a l l s t r a i n i n t h e fluid. T h e re-s u l t i n g flow p a t t e r n e x h i b i t s phase l o c k i n g b e t w e e n a d j a c e n t v o r t e x s t r e e t s . A l t e r n a t e e d d i e s r o t a t e i n o p p o s i t e d i r e c t i o n s so t h a t p o i n t s of c o m m o n t a n g e n t s e x p e r i e n c e l i t t l e s h e a r stress. A l s o , t h e i r d i a m e t e r is s l i g h t l y s m a l l e r t h a n t h e free s p ace between t h e rod s : R < (M — d)/2 = 1.9cm, wh e r e M is t h e mesh w i d t h a n d d is t h e b a r diam e t e r . T h e eddies are s l i g h t l y offset t o t h e side where t h e i r edge v e l o c i t y is p a r a l l e l t o the flow i n t h e m o m e n t u m defect r e g i o n , l e a v i n g a s t a g n a n t r e g i o n t o s e p a r a t e flows o f o p p o s i n g d i r e c t i o n ( t h e s h a d e d areas of Fig.5-7). T h e r e s u l t i n g i n i t i a l e d d y s i z e at X' w 2 M is i n r e a s o n a b l e agreement w i t h t h e o b s e r v e d p o p u l a t i o n p e a k of 1.6cm r a d i u s . T h e i n i t i a l c o n c e n t r a t i o n w o u l d be C0 = {TCR2/M2R) = 0.59, (5 - 2) w h i c h is c o n s i s t e n t w i t h t h e m e a s u r e d v a l u e s s h o w n i n F i g . 4-3a. A s s u m i n g these s t r u c t u r e s r o t a t e i n a s t r a i n free m a n n e r w i t h edge v e l o c i t y e q u a l t o t h a t o f the a d j a c e n t m o m e n t defect r e g i o n (U0 « 5cm/sec f r o m o b s e r v a t i o n ) , the i n i t i a l i n t e r n a l energy of t h e s u r f a c e eddies w i l l be a p p r o x i m a t e l y Ex = {7r/4)pR2U2 « 70(e r o / c m ) (5 - 3) CHAPTER 5 STRUCTURE DYNAMICS and the energy density becomes eD « 6erg/g as indicated in Fig.4-12. The above scenario was the most predominant evolution observed. On occasion however, the vortices were observed to suddenly reorganized into a different geometry. A likely reason is that the flow field cannot simultaneously satisfy the constraints of the structure production rate, grid speed and preferred structure size. The relationship between these effects was studied using the computer-automated analysis system described in chapter three. The system was used to study the properties of the structures when they were first identifiable. Typically 3 or 4 complete structures were found within the viewing area of 11.3 x 8.5cm (2.3 x 1.7M). For each run, about 3-4 seconds of the video recording was digitized and analyzed. This corresponds to a distance of X' = 12 — 16 mesh widths. Part of the power of the automated system is that it extracts instantaneous informa-tion about the velocity field from the polynomial fits to the tracer paths. The system thus allows for the study of rapidly evolving flows which enables us to examine the production regime, something that we cannot do with the photographic flow visualization. Typically, the system was able to recognize structures starting from 6 video frames (t = 6/30\sec) after the grid passed out of the field of view. This corresponds to 0.8M from the moving grid. The image dimension in the direction of the moving grid is 11.4cm (2.3M). From contour plots of the recognized structures we can calculate the initial structure production rate. Referring back to Fig.5-7, the number of structures with a given rotation sense, produced per unit time by an individual rod is equal to the vortex shedding frequency f=Ug/X ( 5 - 4 ) CHAPTER 5 STRUCTURE DYNAMICS 100 T h i s f r e q u e n c y was d e t e r m i n e d f r o m a n u m b e r of c o n t o u r p l o t s o f r e c o g n i z e d s t r u c -t u r e s . C a s t i n t o i t s n o n - d i m e n s i o n a l f o r m u s i n g eqn.(5-l) t h e e x p e r i m e n t a l r e s u l t was 5 = 0.20 ±0.01, ( 5 - 5 ) w h i c h agrees w e l l w i t h t h e S t r o u h a l n u m b e r f o r i s o l a t e d c i r c u l a r c y l i n d e r s u n d e r t h e same flow c o n d i t i o n s . A p r e d i c t i o n o f t h e angle between edge a d j a c e n t eddy centers i n t h e f u l l y f o r m e d w a v y r e g i o n m a y be o b t a i n e d . A g a i n , r e f e r r i n g t o Fig.5-7 we observe t h e g e o m e t r i c r e l a t i o n b etween t h e a n g l e <j>, e d d y r a d i u s R a n d l o n g i t u d i n a l s e p a r a t i o n A, - = 2Rcos<t>. (5 - 6) 2 A s t h e s t r u c t u r e s are s t a t i o n a r y i n t h e t a n k f r a m e of reference A c a n be used t o d e t e r m i n e t h e s h e d d i n g f r e q u e n c y (the r a t e at w h i c h l i k e s i g n e d v o r t i c e s are p r o d u c e d ) u s i n g eqn.(5-4) t o get VJL = _U1_ . . 1 A 4Rcos<f>' y ' I f we a s s u m e t h a t t h e r e s u l t a n t s t r u c t u r e s grow t o c o m p l e t e l y f i l l t h e r e g i o n between t h e m o m e n t u m defect shadows we have R = (M-d)/2 w h i c h m a y b e u s e d i n eqn.(5-7) t o get ( 5 - 8 ) CHAPTER 5 STRUCTURE DYNAMICS 101 If we now assume that the initial eddy creation is determined by the dynamics of the individual cylinder vortex shedding we have S = 0.2 and using eqn.(5-l) for the Strouhal number we find c _ fd _ d Ug~2(M- d)cos<t> ( 5 ~ 1 0 ) Calling rj = M/d, the cylinder spacing to diameter ratio, and rewriting the above equation to form a prediction for <f> we have * = j 4 r " M ( 2 S ( ^ T ) ) = 3 4 ° <5-n> upon inserting the values n = 4 and S = 0.2. This equation describes a relationship between the mesh ratio rj, the Strouhal number 5, and the angle preferred by adjacent eddies. The angle <f> was measured from the plots of identified structures in the thesis of Lau [38] and reported to be <j> = 33 ± 4° (5 - 12) for well formed structures. The result is in good agreement with the proposed model. Further experimental testing of eqn.(5-ll) and the proposed model by varying n would be worthwhile. It should be mentioned that eqn.(5-ll) implies a compression of the vortex street in order to accomodate all new vortices produced between two bars. For a finite span grid competition for space must occasionally destroy the orderliness of this locally preferred pattern. In the same experimental series the eddy diameters D were also measured. By choosing well recognized structures at the initial times (t < 12/30sec or X' < 1.6M), CHAPTER 5 STRUCTURE D YNAMICS 102 i t was f o u n d t h a t t h e average i n i t i a l d i a m e t e r ( c a l c u l a t e d f r o m t h e a r e a b y c i r c u l a r a p p r o x i m a t i o n ) of t h e s t r u c t u r e was D = 0.8 ± 0.1M (5 - 13) U s i n g t h e a s s u m p t i o n t h a t t h e s t r u c t u r e s grow t o f i l l t h e space between t h e "shadows" of t h e c y l i n d e r s , eqn.(5-8), t o d e t e r m i n e D we have D = 2R = ( M - d) = (1 - ^ ) M = 0.75M, (5 - 14) i n g o o d agreement w i t h t h e m e a s u r e d r e s u l t above. A s a final s t u d y o f t h e i n i t i a l v o r t e x p r o d u c t i o n t h e i n i t i a l b o u n d a r y speed o f t h e s t r u c t u r e s was d e t e r m i n e d . T h i s was done by c o m p a r i n g t h e m e a n v e l o c i t y o f the rec-o g n i z e d s t r u c t u r e w i t h t h a t of a n i d e a l i z e d r i g i d l y r o t a t i n g one. T h e m e a n speed i n t h e i d e a l s t r u c t u r e is | £ / m w h e r e Um is i t s b o u n d a r y speed. T h e r e f o r e , t h e b o u n d a r y speed of t h e s t r u c t u r e c a n be e s t i m a t e d as Um = \u' (5 - 15) whe r e U' denotes t h e m e a s u r e d average speed w i t h i n t h e s t r u c t u r e . S i m i l a r c o n s i d e r a -t i o n s o n r o o t m e a n s q u a r e d s p e e d also p r o v i d e s a n e s t i m a t e o f Um. M o r e o v e r , t h e speed c a n be c a l c u l a t e d f r o m e i t h e r t h e g r i d of t a n g e n t i a l s p e e d o r f r o m t h e g r i d o f v e l o c i t y c o m p o n e n t s . T h e s e f o u r d i f f e r e n t e s t i m a t e s o f Um t u r n e d o u t t o b e i n agreement w i t h each o t h e r a n d i t was f o u n d t h a t Um = ( 0 . 2 8 ±0.08)U g , ( 5 - 1 6 ) CHAPTER 5 STRUCTURE DYNAMICS 103 a value of U m = 5.6 ± 1.6cm/sec for our 20cm/sec grid speed. This value is in good agreement with the observed momentum defect velocity of Ua « 5cm/sec, thus support-ing the no slip assumption used to obtain eqn.(5-3). As a final point it should be noted that a consistent difference was found between the mean square and square mean angular velocity averagings. This was interpreted as evidence of departure from the ideal rigidly rotating eddies. More will be said about these results in the next section. 5.4 S p o n t a n e o u s D e c a y The simplest decay archetype that can be characterized is the viscous decay of an isolated rotating structure. In the introductory paper "A model for turbulence based on rate equations" [l] a simple model was presented for the spontaneous decay rate. The derivation is reproduced here. A cylindrical eddy of length £, radius R and angular velocity u contains at a given time the total energy E = \lJ (5 - 17) where / = ^mR? = ^irplR4 is its moment of inertia. If JR remains approximately constant the energy will be lost at the rate ^ = %plRAu)Co, where u> is a negative quantity. A typical time r0 for the energy loss is used to define the energy decay rate A = l / r 0 . The energy loss from the assumed rigid body structure is due to the frictional drag acting on its surface du u(R) Ffr = {area)pu— « 2%lRpv-\r1 (5 - 18) J or 6 CHAPTER 5 STRUCTURE DYNAMICS 104 g i v i n g a d i s s i p a t i o n power Fjru(R). H e r e is t h e diff e r e n c e between the eddy pe-r i p h e r a l v e l o c i t y a n d t h e s t a g n a n t s u r r o u n d i n g s d i v i d e d b y t h e w i d t h 6 over w h i c h t h i s v e l o c i t y c h a n g e o c c u r s . T h e w i d t h 6 of t h e b o u n d a r y l a y e r i n w h i c h t h i s d i s s i p a t i o n o c c u r s s t e a d i l y increases. T h e b o u n d a r y l a y e r g r o w t h r a t e is s i m i l a r t o t h e R a y l e i g h p r o b l e m o f a n i n f i n i t e flat p l a t e s t a r t i n g s u d d e n l y f r o m r e s t . T h e r e 6 = 4y/vi, so t h a t t h e d i s s i p a t i o n p o w e r c a n be w r i t t e n as FfrU{R) = 2i:puR2u}2l(4\fui). (5 - 19) T h i s p o w e r d i s s i p a t i o n m u s t b e e q u a l t o t h e power loss of t h e r o t a t i n g s t r u c t u r e . T h i s l e a d s t o du) , i — y/t/u = i/$R. ( 5 - 2 0 ) T h i s r e l a t i o n c a n be i n t e g r a t e d t o y i e l d t h e i n s t a n t a n e o u s a n g u l a r v e l o c i t y u = u0exp(-2\fui/R) (5 - 21) a n d t h e e d d y e n e r g y becomes E = const u2 = E0exp(-4y/ui/R). (5 - 22) A s i g n i f i c a n t a m o u n t of t h e eddy energy w i l l be d i s s i p a t e d d u r i n g t h e e - f o l d i n g t i m e r 0 d e f i n e d b y eqn.(5-22), n a m e l y T0 = R2/I6u. (5 - 23) T h e energy decay r a t e is hence A = 1/T0 = 16u/R2. ( 5 - 2 4 ) CHAPTER 5 STRUCTURE DYNAMICS 105 This result shows that smaller structures have a higher frictional decay rate. The interaction with the fluid (the spontaneous decay) therefore dominates the low end of the eddy size distribution, while large structures suffer relatively little energy loss due to frictional decay. The above crude model motivated a more extensive study of the viscous decay of two-dimensional axisymmetric flow structures [57]. The computer model was an attempt to obtain rate coefficients from more physical velocity profiles than a rigid body. The Navier-Stokes equations in cylindrical coordinates were used as a starting point. For v = 0ve(r,t) only (0 being the unit polar vector) and (Vp)tf = 0 by symmetry, the Navier-Stokes equations reduce to dve _ v d . dve. _ ~dt ~ rdr^~dr' ~ U d2Vg 1 dv$ dr2 r dr (5 - 25) since (v • v)v = 0 for t; = vg(r,t). Equation (5-25) is the simple diffusion equation for fluid momentum. A discrete version of this equation was used to compute the evolution of a given velocity profile v$(r). At the end of the specified evolution time, plots were made of the eddy kinetic energy, boundary layer thickness, boundary layer to core energy ratio, and the rate coefficient, all as functions of time. The total eddy kinetic energy per unit mass was found by summing up the contributions from the discrete radii. The evolution of this energy was then used to determine the rate coefficient = - | f . (5-26) CHAPTER 5 STRUCTURE DYNAMICS 106 Radius (cm) F i g u r e 5-9 E v o l u t i o n o f v e l o c i t y p r o f i l e . C o m p u t e r m o d e l l e d p r o f i l e s a t t w o dif f e r e n t t i m e s . F i g u r e 5-9 shows t h e e v o l u t i o n o f t h e v e l o c i t y p r o f i l e b e t w e e n t w o t i m e s . T h e r a t e c o e f f i c i e n t c u r v e s show t h a t A d e p e n d s o n t i m e . However, i f one a l l o w s t h e o r i g i n a l l y u n r e a l i s t i c v e l o c i t y p r o f i l e t o e v o l v e t o a mo r e p h y s i c a l l y m e a n i n g f u l s t a t e , a r e l a t i v e l y c o n s t a n t decay r a t e is f o u n d . F i g u r e 5-10 shows how t h i s A v a l u e v a r i e s w i t h 1/R2. T h e r a n g e b a r s f o r t h e A values r e f l e c t t h e weak t i m e d e p e n d e n c e over the 10 sec e v o l u t i o n i n t e r v a l i n w h i c h the s t a b l e v a l u e was d e t e r m i n e d . A n e s t i m a t e d l i n e t h r o u g h t h e b a r s y i e l d e d A ~ P/Rl (/? « 4.2(5) x 1 0 ~ 2 c m 2 / s e c ) . D i m e n s i o n a l l y one m u s t have A oc v/R2 i m p l y i n g t h a t A ~ 4vjR2 as v was t a k e n as 1.0 x 1 0 _ 2 c m 2 / s e c i n CHAPTER 5 STRUCTURE DYNAMICS 107 t h e c o m p u t e r m o d e l l i n g . T h e d i f f e r e n c e between t h e c o n s t a n t a bove a n d t h e v a l u e of 16 i n eqn.(5-24) b e i n g t h e a r b i t r a r y c h o i c e o f t h e e - f o l d i n g t i m e t o c h a r a c t e r i z e t h e decay r a t e . F i g u r e 5-10 V a r i a t i o n o f r a t e c o e f f i c i e n t A w i t h s t r u c t u r e s i z e . C o m p u t e r m o d e l l e d . T h e v i d e o - c o m p u t e r a n a l y s i s s y s t e m was u s e d t o s t u d y , e x p e r i m e n t a l l y , t h e decay of s t a b l e s u r f a c e s t r u c t u r e s i n t h e i n i t i a l p e r i o d of t h e g r i d flow. D u e t o t h e low r e s o l u t i o n o f t h e present h a r d w a r e t h e a n a l y s i s s y s t e m is i n a c c u r a t e i n d e t e r m i n i n g t h e l o c a t i o n of t h e s t r u c t u r e o u t l i n e , a n d hence t h e r a d i u s . T h e s t r u c t u r e r e c o g n i t i o n p l o t s were CHAPTER 5 STRUCTURE D YNAMICS 108 t h u s o v e r l a y e d o n t h e s t r e a k d a t a t o j u d g e w h e t h e r t h e b o u n d a r y r e g i o n has s u f f i c i e n t i n f o r m a t i o n d e n s i t y f o r a c c u r a t e s i z e d e t e r m i n a t i o n . A v i s u a l check was a l s o m a de t o e n s u r e a s t e a d y v i s c o u s decay was t h e d o m i n a n t i n f l u e n c e o n t h e s t r u c t u r e . T h u s , o n l y w e l l d e f i n e d s t r u c t u r e s w h i c h u n d e r w e n t a s t e a d y decay, free f r o m i n t e r f e r e n c e f r o m o t h e r s t r u c t u r e s , were chosen f o r t h i s a n a l y s i s . 0.2 0.4 7^R*0.6 0.8 F i g u r e 5 - 1 1 P r o p o r t i o n a l i t y c o n s t a n t d e t e r m i n a t i o n f o r A = P/R2. F i g u r e 5-11 shows a log-log p l o t of t h e decay r a t e a g a i n s t t h e e d d y siz e R. T h e p l o t shows a g o o d p o w e r law r e l a t i o n s h i p w i t h a n e x p o n e n t o f - 2 as p r e d i c t e d by eqn.(5-24). CHAPTER 5 STRUCTURE D YNAMICS 109 H o w e v e r , t h e p r o p o r t i o n a l i t y c o n s t a n t changes b e t w e e n d i f f e r e n t r u n s . F o r t h e left-most c u r v e t h e v a l u e f o r u, a s s u m i n g /? = 16i/ f r o m eqn.(5-24), was 0.10 ± 0.05cm 2/sec. T h i s is a n o r d e r o f m a g n i t u d e h i g h e r t h a n t h e e s t a b l i s h e d b u l k v i s c o s i t y of w a t e r o f 0.01cm 2/sec. It was also f o u n d t h a t t h e c a l c u l a t e d v i s c o s i t y v a l u e s i n c r e a s e d d u r i n g the c o u r s e of t h e day l o n g e x p e r i m e n t a t i o n . T h i s was i n t e r p r e t e d as t h e effect of i n c r e a s i n g s u r f a c e c o n t a m i n a t i o n . T h e o r d e r of m a g n i t u d e d i f f e r e n c e has a l r e a d y b e e n discussed i n s e c t i o n 5.1 o n su r f a c e fluid m e c h a n i c s . W i t h t h i s u v a l u e a n d t h e R = 1.4cm s t r u c t u r e s i z e o f s e c t i o n 4.4 a r a t e c o e f f i c i e n t f o r t h e s p o n t a n e o u s decay m a y b e o b t a i n e d f r o m eqn.(5-24), A = —1.6sec~i. T h i s v a l u e is of t h e same o r d e r of m a g n i t u d e as t h e v a l u e A + B = — 0 . 7 s e c _ 1 , a = 0 . 5 5 s e c - 1 o b t a i n e d f r o m t h e s t r u c t u r e s t a t i s t i c s a n a l y s i s o f c h a p t e r f o u r . A decay v a l u e o b t a i n e d f r o m a mo r e t h o r o u g h s t u d y o f t h e s t a t i s t i c s e v o l u t i o n w o u l d a l l o w f o r a mo r e c o n c l u s i v e c o m p a r i s o n . B e f o r e g o i n g o n t o d i s c u s s o t h e r decay t y p e s some o b s e r v a t i o n s o n t h e eddy's i n t e r n a l v e l o c i t y p r o f i l e are i n order. I n t h e c o u r s e of t e s t i n g t h e self c o n s i s t e n c y of the a n a l y s i s s y s t e m i t was n o t e d t h a t k i n e t i c energies d e t e r m i n e d by a r i g i d b o d y fit t o the a n g u l a r v e l o c i t y d e f i n e d as u'=jf E V9/\R-RCM\ ( 5 - 2 7 ) structure were c o n s i s t e n t l y l a r g e r t h a n w h a t was d i r e c t l y d e t e r m i n e d f o r t h e r e c o g n i z e d s t r u c t u r e s f r o m t h e i n t e r p o l a t e d v e l o c i t y g r i d . T h e r o t a t i o n a l energy of t h e s t r u c t u r e is c a l c u l a t e d as Er = \lu'2 (5 - 28) CHAPTER 5 STRUCTURE D YNAMICS 110 F i g u r e 5-12 R e l a t i o n b etween r o t a t i o n a l energy a n d t o t a l energy. N o n - u n i t y p r o p o r -t i o n a l i t y i n d i c a t e s s y s t e m a t i c d e v i a t i o n f r o m w = const p r o f i l e . F i g u r e 5-12 shows a p l o t o f t h i s energy a g a i n s t t h e k i n e t i c energy c a l c u l a t e d d i r e c t l y f r o m t h e v e l o c i t y g r i d (Et) f o r a l l t h e r e c o g n i z e d s t r u c t u r e s . A c o n s i s t e n t d e v i a t i o n between t h e average s q u a r e d d a t a i n t h e f o r m of Er a n d t h e s q u a r e averaged d a t a of Et, Er = (1.44 ± .17) xEt (5 - 29) was f o u n d . T h i s i n d i c a t e s t h e a c t u a l p r o f i l e s were a c o n s i s t e n t d e v i a t i o n , o r m e mbers of a f a m i l y o f d e v i a n t s , f r o m t h e assumed r i g i d b o d y p r o f i l e . CHAPTER 5 STRUCTURE DYNAMICS 111 tU* is defined to that the two shaded parts have same area (a) ideal profile without mixing possible profile with mixing Et by definition rigid body profile E r by lfa)'/2 (b) F i g u r e 5-13 I n t e r n a l flow s t r u c t u r e , (a) A n g u l a r v e l o c i t y p r o f i l e s , (b) C o r r e s p o n d i n g v e l o c i t y p r o f i l e s . CHAPTER 5 STRUCTURE DYNAMICS 112 Considering eqn.(5-29) we realize that for the deviation from a rigid body profile to produce a larger square average than average square the angular velocity profile must be skewed with lower u at larger radius as shown in Fig.5-13a. This is consistent with both our understanding of what viscosity will do to the profile and direct observation of angular velocity plots. Unfortunately, at the present time, the data density available from our analysis hardware is too low to merit a quantitative study of an individual structure's internal velocity profile. 5.5 T h e O m e g a D e c a y The omega decay was first observed in the time exposed flow photos such as Figs.4-1. These short lived structures comprise about one in twenty of the flow structures in the equilibrium and decay flow regions. Figure 5-14 shows the omega decay dynamics. The process is similar to that of eddy pairing described in section 5.2 except one of the structures is infinitely large. In the present analysis the river motion is considered to be part of the background flow environment. The omega decay is thus a type of structure-fluid interaction and so would be characterized by a B coefficient. In the course of a study of staggered cylinder flow dynamics a method was found to produce, predictably, a flow situation which resulted in the omega decay. Its structure is similar to that shown in Fig.5-7 of the initial vortex production. In this instance however the structures are pulled into the oppositely moving momentum defect "river" by the elastic nature of the fluid being pulled by the cylinder and dragged through the surrounding fluid. Figures 5-15 show surface and subsurface time exposed flow photos of CHAPTER 5 STRUCTURE D YNAMICS 113 Figure 5-14 The omega decay dynamics. this flow for an object speed of 20cm/sec. An omega decay structure can be seen very near (~ 1M) to the two cylinders' starting position in Fig.5-15a. The photo does not show the phenomenon as clearly as can be observed by stepping through the video frames. In the time exposure the rapid structure evolution and non-negligible drift velocity tend to smear out the flow features. Figure 5-15b is taken with a 3.5cm wide sheet of light parallel to the water surface at a depth of 14cm. The pattern is similar to the surface photo in structure and velocity. CHAPTER 5 STRUCTURE DYNAMICS 114 F i g u r e 5-15 T i m e - e x p o s e d flow p h o t o of t w o - c y l i n d e r flow, (a) s u r f a c e flow showing o m e g a decay (b) s h e e t - l i g h t i n g (20cm depth) p a r a l l e l t o sur f a c e s h o w i n g p r e d o m i n a n t l y 2-D flow. CHAPTER 5 STRUCTURE DYNAMICS 115 H o w e v e r , e v i d e n c e o f th r e e d i m e n s i o n a l m o t i o n is v i s i b l e i n t h e f o r m of i r r e g u l a r t r a c e r p a t h s . A n a t t e m p t was m a d e t o use t h e c o m p u t e r - a u t o m a t e d a n a l y s i s s y s t e m t o s t u d y t h e o m e g a decay i n t h i s flow. T h a t a t t e m p t was un s u c c e s s f u l due t o t h e r e s o l u t i o n l i m i t s a n d l a c k o f c o m p e n s a t i o n f o r s t r u c t u r e d r i f t i n the present a n a l y s i s s y s t e m . A more p o w e r f u l a n a l y s i s s y s t e m needs t o be d e v e l o p e d i n o r d e r t o s t u d y t h e o m e g a decay. 5.6 Statistics from Dynamics A n i n i t i a l v o r t e x p r o d u c t i o n m o d e l has been p r e s e n t e d w h i c h agrees w i t h t h e exper-i m e n t a l l y d e t e r m i n e d s t r u c t u r e p o p u l a t i o n . A l s o , a s i m p l e m o d e l has been p r o p o s e d f o r t h e s p o n t a n e o u s decay r a t e A. I t was also f o u n d t o agree w i t h e x p e r i m e n t a l o b s e r v a t i o n . T h e r e m a i n i n g s t r u c t u r e d y n a m i c s of eddy p a i r i n g a n d t h e omega decay have been ex-a m i n e d b u t as ye t no m o d e l has been d e v e l o p e d f o r these processes. T h e r e a l test o f the r a t e e q u a t i o n a p p r o a c h w o u l d be t o use r a t e coefficients t o p r e d i c t t h e s i z e , a n g u l a r speed a n d e n e r g y d i s t r i b u t i o n s m e a s u r e d i n t h e las t c h a p t e r . T h i s w o u l d p r o c e e d b y u s i n g the r a t e e q u a t i o n , (2-35), on t h e i n i t i a l d i s t r i b u t i o n d e t e r m i n e d b y t h e v o r t e x p r o d u c t i o n of s e c t i o n 5.3. S u c h a c o m p a r i s o n w o u l d be a po w e r f u l t e s t o f t h e r a t e e q u a t i o n a p p r o a c h t o d e s c r i b i n g t h e e v o l u t i o n o f t h e ensemble of coherent flow s t r u c t u r e s i n t u r b u l e n t fluid flow. T h e d e v e l o p m e n t o f s u c h a p o w e r f u l p r e d i c t i v e t o o l f o r t u r b u l e n t flows s h o u l d be th e p r i m a r y o b j e c t i v e o f any research a r i s i n g f r o m t h i s t h e s i s w ork. CHAPTER 6 CONCLUSION 116 C H A P T E R 6 CONCLUSION w It was on a dr e a r y n i g h t o f N o v e m b e r t h a t I b e h e l d t h e a c c o m p l i s h m e n t of my t o i l s . " i n F r a n k e n s t e i n by M a r y S h e l l y (1818) T h i s t h e s i s has e x a m i n e d t h e v a l i d i t y a n d v i a b i l i t y o f a m o d e l f o r t u r b u l e n c e based o n r a t e e q u a t i o n s by s t u d y i n g t h e s t a t i s t i c s a n d d y n a m i c s of c o h e r e n t s t r u c t u r e s o n g r i d - g e n e r a t e d t u r b u l e n c e . T h e e x p e r i m e n t a l s t u d y focussed o n t h e s u r f a c e m o t i o n o f g r i d - f l o w p r o d u c e d i n a t o w i n g t a n k . F o r t h e s t r u c t u r e s t a t i s t i c s i n v e s t i g a t i o n surface flow p a t t e r n s were r e c o r d e d as t i m e - e x p o s e d p h o t o g r a p h s o f t r a c e r p a t h s f o l l o w i n g t h e m o t i o n o f t h e fluid. T h e flow p a t t e r n s were m a n u a l l y a n a l y z e d t o d e t e r m i n e the size a n d v e l o c i t y s t r u c t u r e o f t h e s u r f a c e eddies. S t r u c t u r e d y n a m i c s were i n v e s t i g a t e d u s i n g a c o m p u t e r - a u t o m a t e d s t r u c t u r e i d e n t i f i c a t i o n a n d flow field a n a l y s i s package. T h i s s y s t e m a n a l y z e d v i d e o r e c o r d i n g s of t h e t r a c e r m o t i o n . R e s u l t s f r o m h o t - f i l m a n e m o m e t r y a n d s u b s u r f a c e flow v i s u a l i z a t i o n were used t o c o m p a r e t h e s u r f a c e a n d s u b s u r f a c e flows. T h e flow p h o t o s revealed a p r o f u s i o n o f coherent s u r f a c e s t r u c t u r e s t h a t e i t h e r r o t a t e ( s u r f a c e e d d i e s ) , t r a n s l a t e ( r i v e r flow), o r are r e l a t i v e l y s t a g n a n t . T h e s i z e a n d a n g u l a r v e l o c i t i e s o f t h e s u r f a c e eddies were d e t e r m i n e d a n d used t o c a l c u l a t e t h e e d d y energy di s -t r i b u t i o n . K i n e t i c energy d i s t r i b u t i o n s o f t h e r i v e r m o t i o n were also d e t e r m i n e d . W h e n CHAPTER 6 CONCLUSION 117 fitting e m p i r i c a l f o r m u l a s t o these d i s t r i b u t i o n s , e q u a t i o n s s i m i l a r t o a B o l t z m a n n e n e r g y d i s t r i b u t i o n f o r a s y s t e m of p a r t i c l e s i n t h e r m a l e q u i l i b r i u m w e r e f o u n d . T h i s r e s u l t has t w o consequences: first, i t i m p l i e s t h a t " c o l l i s i o n a l " i n t e r a c t i o n s m u s t d o m i n a t e , a n d s e c o n d , i t is p o s s i b l e t o c h a r a c t e r i z e a n energy d i s t r i b u t i o n b y a t e m p e r a t u r e p a r a m e t e r . E d d y a n d r i v e r " t e m p e r a t u r e s " were d e t e r m i n e d a n d i t w a s f o u n d t h a t 6e a n d 6T are a b o u t t h e s a m e i n t h e e q u i l i b r i u m r e g i o n . T h e s e t e m p e r a t u r e s s l o w l y decay a n d t h e d i s t r i b u t i o n s r e m a i n B o l t z m a n n - l i k e t h r o u g h o u t t h e e q u i l i b r i u m r e g i o n . A s w e l l , i n t e r -a c t i o n a n d d e c a y r a t e s of t h e s u r f a c e e d d i e s were e x t r a c t e d f r o m t h e e v o l u t i o n of t h e s i z e s p e c t r u m N(R, X). T h e s u r f a c e flow is p r e d o m i n a n t l y two d i m e n s i o n a l a n d n o s o u r c e o r s i n k s of fluid were o b s e r v e d t o p e n e t r a t e t h e m a t e r i a l surface. T h i s is i n s h a r p c o n t r a s t t o t h e s u b s u r f a c e m o t i o n w h i c h r a p i d l y becomes t h r e e d i m e n s i o n a l . S t u d y o f v i d e o sequences of t h e n e a r - g r i d flow l e d t o a m o d e l f o r t h e i n i t i a l v o r t e x p r o d u c t i o n . T h e m o d e l assumes t h a t t h e s t r u c t u r e s are p r o d u c e d a t t h e S t r o u h a l fre -q u e n c y o f t h e i n d i v i d u a l c y l i n d e r s b u t are c o n s t r a i n e d i n s i z e a n d s p a t i a l o r i e n t a t i o n b y t h e m e s h - w i d t h . T h e m o d e l p r e d i c t s v a l u e s f o r s t r u c t u r e s i z e , i n i t i a l p o p u l a t i o n d e n s i t y a n d a n g l e b e t w e e n a d j a c e n t s t r u c t u r e c e n t e r s , a l l i n g o o d a g r e e m e n t w i t h o b s e r v a t i o n . F u r t h e r e x p e r i m e n t a l s t u d y o f t h i s m o d e l u s i n g d i f f e r e n t g r i d speeds a n d m e s h - w i d t h t o b a r - d i a m e t e r r a t i o s w o u l d be b o t h w o r t h w h i l e a n d p o s s i b l e u s i n g t h e e x i s t i n g a n a l y s i s s y s t e m . I n t h e r a t e e q u a t i o n m o d e l t h e c hange o f t h e e d d y d i s t r i b u t i o n is a t t r i b u t e d t o t h r e e d i s t i n c t i n t e r a c t i o n s t h a t are c h a r a c t e r i z e d b y r a t e c o e f f i c i e n t s A, B a n d C. S t r u c t u r e i n t e r a c t i o n w i t h t h e s u r r o u n d i n g fluid is t e r m e d v i s c o u s o r s p o n t a n e o u s decay a n d is q u a n t i f i e d b y t h e r a t e c o e f f i c i e n t A. I n t e r a c t i o n w i t h t h e fluid flow is q u a n t i f i e d b y r a t e CHAPTER 6 CONCLUSION 118 c o e f f i c i e n t B. S t r u c t u r e s m a y g a i n e n e r g y f r o m t h e flow e n v i r o n m e n t as i n t h e i n i t i a l v o r t e x p r o d u c t i o n r e g i o n of t h e g r i d - f l o w o r lose energy t o t h e r i v e r - f l o w as i n t h e o m e g a decay. T h e y m a y a l s o be t o r n a p a r t i n s u f f i c i e n t l y h i g h s t r a i n - f i e l d s . T h e process o f v o r t e x s t r e t c h i n g is n o t f o u n d i n o u r t w o - d i m e n s i o n a l s u r f a c e flow. F i n a l l y , s t r u c t u r e s m a y i n t e r a c t w i t h o t h e r s t r u c t u r e s as seen i n t h e e d d y p a i r i n g p r o c e s s (C c o e f f i c i e n t ) . S t u d y of v i d e o r e c o r d i n g s o f t h e s u r f a c e m o t i o n showed t h e d o m i n a n t e v o l u t i o n pro-cesses t o be v i s c o u s d i s s i p a t i o n , e d d y p a i r i n g a n d t h e o m e g a decay. R e l a t i v e l y few e d d y s p l i t t i n g events were f o u n d . A n a l y s i s of t h e v i s c o u s decay o f s t e a d i l y e v o l v i n g s t r u c t u r e s w as p r e s e n t e d . T h e s e r e s u l t s w ere c o m p a r e d w i t h c o m p u t e r a n d a n a l y t i c m o d e l s of iso-l a t e d v o r t e x e v o l u t i o n . A g r e e m e n t o n t h e dependence o f t h e decay r a t e A o n s t r u c t u r e s i z e (A a 1/R2) was f o u n d a l t h o u g h t h e r e was some a m b i g u i t y a b o u t t h e v a l u e of t h e p r o p o r t i o n a l i t y c o n s t a n t . T h e o m e g a decay, a s u d d e n r e l a m i n a r i z a t i o n o f a r o t a t i n g s t r u c t u r e , was f o u n d t o be a f r e q u e n t o c c u r r e n c e i n t h e p r o d u c t i o n a n d e q u i l i b r i u m r e g i o n s o f t h e flow. T h i s t y p e of e v o l u t i o n was f o u n d t o o c c u r w h e n a n e d d y c o n t a c t e d a r i v e r m o t i o n w i t h d i r e c t i o n o p p o s i t e t o t h a t o f t h e e d d y c o n t a c t p o i n t . S t u d y o f t h e o m e g a decay r a t e w o u l d be a g o o d first a p p l i c a t i o n o f t h e p r e s e n t a n a l y s i s s y s t e m once fit w i t h m o r e p o w e r f u l image a c q u i s i t i o n h a r d w a r e because o f p r a c t i c a l l i m i t a t i o n s o f t h e e x i s t i n g a n a l y s i s system. T h e r e e x i s t s t h e p o s s i b i l i t y o f e d d y p r o d u c t i o n b y a n inv e r s e o m e g a decay process. A sear c h f o r t h i s r i v e r e v o l u t i o n m a y p r o v e r e w a r d i n g . M a n y eddy p a i r i n g s were o b s e r v e d i n t h e flow p h o t o s a n d v i d e o sequences. Q uan-t i f i c a t i o n of t h e e n c o u n t e r s t a t i s t i c s a n d p a i r i n g d y n a m i c s o f t h i s decay process is t he CHAPTER 6 CONCLUSION 119 l a s t m a j o r s t e p needed before t h e r a t e e q u a t i o n c a n be u s e d f o r a m e a n i n g f u l p r e d i c t i o n o f t h e s t r u c t u r e p o p u l a t i o n . T h e success i n t h e s t a t i s t i c a l a n a l y s i s a n d d e s c r i p t i o n a n d i n t h e i s o l a t i o n of t h e s i g n i f i c a n t s t r u c t u r e d y n a m i c s show t h e g r e a t p r o m i s e of t h e r a t e e q u a t i o n m o d e l f o r t h e s t a t i s t i c s o f t h e l a r g e scale s t r u c t u r e s i n a t u r b u l e n t fluid flow. T h i s t h e s i s has p r e s e n t e d a n u m b e r of my o r i g i n a l c o n t r i b u t i o n s t o t h e u n d e r s t a n d i n g o f t u r b u l e n t fluid d y n a m i c s . T h e d i s c o v e r y of s u r f a c e flow s t r u c t u r e s o n grid-generated t u r b u l e n c e has a l r e a d y b e en r e p o r t e d i n m y M. Sc. t h e s i s [59]. T h e s t a t i s t i c a l a n a l y s i s o f t h e s i z e , a n g u l a r speed a n d e n e r g y d i s t r i b u t i o n s of t h e eddies a n d r i v e r s (closed a n d o p e n flow s t r u c t u r e s ) d e s c r i b e d i n c h a p t e r f o u r has b e en p r e s e n t e d elsewhere [5]. T h e s i z e a n d a n g u l a r s p e ed d i s t r i b u t i o n s were us e d t o p r e d i c t a n i n t e g r a l l e n g t h scale w h i c h was c o m p a r e d w i t h one o b t a i n e d f r o m F o u r i e r a n a l y s i s of h o t - f i l m a n e m o m e t r y sig n a l s m e a s u r e d i n t h e b u l k of t h e flow [30]. T h e r e s u l t s of t h i s c o m p a r i s o n were used i n c h a p t e r five of t h i s t h e s i s i n a s t u d y of t h e r e l a t i o n s h i p b e t w e e n t h e s u r f a c e a n d subsurface flow. T h e r a t e e q u a t i o n m o d e l , o r i g i n a l l y p r o p o s e d by D r . B. A h l b o r n [ l ] , l e d t o t h e d i s c u s s i o n o f t h e energy d i s t r i b u t i o n s i n t e r m s o f a t e m p e r a t u r e . M y c o n t r i b u t i o n s t o t h i s a p p r o a c h a r e c o n t a i n e d i n a recent p u b l i c a t i o n [6] as w e l l as i n c h a p t e r f o u r of t h i s thesis. T h e n e e d t o a u t o m a t e t h e r e c o g n i t i o n a n d a n a l y s i s of a c o h e r e n t s t r u c t u r e l e d t o the M. Sc. w o r k of A. L a u [38] w h i c h was u s e d i n t h e s t r u c t u r e d y n a m i c s s t u d y o f c h a p t e r five. A l t h o u g h some e a r l i e r w o r k o n s t r u c t u r e d y n a m i c s was p u b l i s h e d [4], t h e i n i t i a l v o r t e x p r o d u c t i o n m o d e l , c o m p a r i s o n of t h e s p o n t a n e o u s decay e x p e r i m e n t a n d theory, as w e l l as t h e i s o l a t i o n of t h e s t r u c t u r e decay t y p e s o n t h e s u r f a c e flow o f g r i d - g e n e r a t e d t u r b u l e n c e are o r i g i n a l t o t h i s t h e s i s . CHAPTER 6 CONCLUSION 120 I t is h o p e d t h a t t h e r e s u l t s a n d a p p r o a c h p r e s e n t e d i n t h i s t h e s i s w i l l s t i m u l a t e f u r t h e r r e s e a r c h . I n o r d e r t o c o m p l e t e t h e d e m o n s t r a t i o n of t h e r a t e e q u a t i o n a p p r o a c h , m o d e l s n e e d t o b e d e v e l o p e d t o q u a n t i f y t h e o m e g a decay ( B rat e ) a n d e d d y p a i r i n g ( C rat e ) processes. O n c e t h i s h a s b e e n a c c o m p l i s h e d , t h e i n i t i a l v o r t e x p r o d u c t i o n m o d e l p r o v i d e s a n i n i t i a l s t r u c t u r e d i s t r i b u t i o n w h i c h s h o u l d e v o l v e a c c o r d i n g t o t h e r a t e e q u a t i o n . T h e o b s e r v e d s t a t i s t i c s c a n t h e n be c o m p a r e d w i t h those p r e d i c t e d . A s i n d i c a t e d i n Fig.2-5, t h e u l t i m a t e t e s t of t h e m o d e l is t o use m o m e n t s of the p r e d i c t e d s t r u c t u r e s t a t i s t i c s t o d e t e r m i n e s u c h m a c r o s c o p i c flow p r o p e r t i e s as d r a g , m i x i n g r a t e s , a n d gust levels. I n f a c t , a f t e r f u r t h e r v e r i f i c a t i o n of t h e i n i t i a l v o r t e x p r o d u c t i o n m o d e l presented i n t h i s t h e s i s i t s h o u l d b e s t r a i g h t f o r w a r d t o p r e d i c t t h e f o r m d r a g a c t i n g o n p a r a l l e l b a r g r i d s . W h i l e a n a t t e m p t has been m a d e t o u n d e r s t a n d the c o u p l i n g between t h e s u r f a c e a n d s u b s u r f a c e m o t i o n s i t w o u l d be d e s i r a b l e t o r e p e a t t h e s t a t i s t i c s a n d d y n a m i c s s t u d y f o r a s y s t e m w h i c h is m o r e c l o s e l y t w o - d i m e n s i o n a l . S t r u c t u r e i d e n t i f i c a t i o n a n d a n a l y s i s c o u l d t h e n p r o c e e d u s i n g a n a u t o m a t e d a n a l y s i s s y s t e m such as t h e one d e s c r i b e d i n s e c t i o n 3.3.2. E x t e n s i o n o f t h e m o d e l t o t h r e e - d i m e n s i o n a l flows w o u l d r e q u i r e t h e i n c l u s i o n o f v o r t e x s t r e t c h i n g i n t o t h e B co e f f i c i e n t i n a d d i t i o n t o i n c l u d i n g t h e a d d e d d i m e n s i o n i n t o t h e i n t e r a c t i o n s t a t i s t i c s . W h o m e v e r e m b a r k s o n such a n a m b i t i o u s task faces m a n y yea r s o f c h a l l e n g i n g a n d i n t e r e s t i n g work. References 121 References [l] B. Ahlborn, F. Ahlborn and S. Loewen, "A model for turbulence based on rate equations", Applied Physics 18, pp. 2127-41 (1985) [2] Th. von Karman , "Uber den mechanismus des widerstandes, den ein bewegter korper in einer fiussigkeit erzeugt", Nachr. Ges. Wiss. Gottigen, Math. Phys. Klasst pp. 509-517 (1911) and pp. 547-556 (1912); see also Coll. Works J, pp. 324-338 [3] G. L. Brown and A. Roshko, "On density effects and large structures in turbulent mixing layers", J. Fluid Mech. 64, pp. 775-816 (1974) [4] B. Ahlborn, A. Filuk and S. Loewen, "Eddy formation and break-up in a turbulent flow", in Proc. Fifth Eng. Mech. Conf., U. of Wyoming (1984), Ed by A. P. Boresi and K. P. Chong, Am. Soc. Civ. Eng., New York [5] S. Loewen and B. Ahlborn, "Empirical energy distribution functions of decaying grid turbulence", pp. 196-99, in Seventh Symposium on Turbulence and Diffusion, pub. by the American Meteorological Society, Boston, Mass. (1985) [6] S. Loewen, B. Ahlborn, and A. B. Filuk, "Statistics of surface flow structures on decaying grid turbulence", Physics of Fluids 29 (8), pp. 2388-97 (1986) (7] A. Lau, S. Loewen, B. Ahlborn, and V. Bareau, "Automated recognition of internal structures in 2D fluid flow", Bui. Am. Phys. Soc. 30, pp. 1729 (1985) [8] A. Lau, S. Loewen and B. Ahlborn, "Automated two-dimensional flow visualization and coherent structure recognition", submitted to Experiments in Fluids (August 1986) [9] M. Broze, "Safety: The open coast or all your eggs in one kayak", pp. 27, Sea Kayaker 3 No.l 1986 [10] B. E. Launder and D. B. Spalding, Mathematical Models of Turbulence, Academic Press (1972) [11] O. Reynolds, "On the dynamical theory of incompressible viscous fluids and the determination of the criterion", Philosophical Transactions of the Royal Society of London, Series A, 186, pp. 123 (1895) [12] J . Boussinesq, "Essai sur la theorie des eaux courantes.", Mem. pres. Aca. Sci. XXIII, 46, Paris (1877) [13] J . Boussinesq, "Theorie de l'ecoulement tourbillonant et tumultueux des liquides dans les lits rectilignes a grande section (tuyaux de conduite et canaux decouverts), quand cet ecoulement s'est regularise en un regime uniforme, c'est-a-dire, moyennement References 122 pareil a travers toutes les sections normales du lit." Comptes Rendus de l'Academie des Sciences CXXII, pp. 1290-95 (1896) [14] L. Prandtl, "Uber die ausgebildete turbulenz", Z A M M 5, pp. 136-39 (1925). Also, S. Goldstein "Modern Developments in Fluid Dynamics", 1 p205, Oxford University Press, New York (1938) [15] H. Tennekes and J. L . Lumley, A First Course in Turbulence , MIT Press (1972) [16] H. Schlichting, Boundary Layer Theory, McGraw-Hill, New York (1979) [17] W. C. Reynolds, "Computation of turbulent flows", Ann. Rev. Fluid Mech., 8 pp. 183-208 (1976) [18] M . T. Landahl and E. Mollo-Christensen, Turbulence and random processes in fluid mechanics , Cambridge University Press (1986) [19] J. O. Hinze, Turbulence, McGraw-Hill (1975) [20] A. N . Kolmogorov, "The local structure of turbulence in incompressible viscous fluid for very large Reynolds numbers", Dokl. Akad. Nauk. SSSR 30, pp. 299-303, (1941). see also G. K. Batchelor, "Kolmogorov's theory of locally isotropic turbulence", Proc. Camb. Phil. Soc, 43, pp. 533-59, (1947) [21] H. L. Grant, R. W. Stewart and A. Moillet, "Turbulence spectra from a tidal channel", J. Fluid Mech. 12, pp. 241-63 (1962) [22] R. W. Stewart, "Turbulence and waves in a stratified atmosphere", Radio Sci., 4, pp. 1289 (1969) [23] R. H. Kraichnan, "Inertial ranges in two-dimensional turbulence", Phys. Fluids, 10 (7) pp. 1417-23 (1967) [24] J . H. Ferzinger, "Simulation as an aid to phenomenological modelling", in Macro-scopic Modelling of Turbulent Flows, edited by U. Frisch et al, Proceedings, Sophia-Antipolis, France 1984, pp. 263-76, Springer-Verlag (1985) [25] A. Leonard, "Energy cascades in large eddy simulations of turbulent fluid flow", Advances in Geophysics 1 8 A , pp. 237-48 (1973) [26] B. Aupoix, "Eddy viscosity subgrid scale models for homogeneous turbulence", in Macroscopic Modelling of Turbulent Flows, ed. U. Frisch et al, Proceedings, Sophia-Antipolis, France 1984, pp. 45-64, Springer-Verlag (1985) [27] S. Orszag, "Analytic theories of turbulence", J. Fluid Mech. 41, pp. 363-86 (1970) [28] P. Perrier, "Large and small structures in the computation of transition to fully developed turbulent flows", in Macroscopic Modelling of Turbulent Flows, ed. U. Frisch et al, Proceedings, Sophia-Antipolis, France 1984, pp. 32-44, Springer-Verlag (1985) [29] Brian J. Cantwell, "Organized motion in turbulent flow", Ann. Rev. Fluid Mech. 13, pp. 457-515 (1981) [30] S. Loewen, "Statistics of Coherent Structures in Turbulent Fluid Flow", M . Sc. thesis, The University of British Columbia, Vancouver, Canada (1983) (unpublished) [31] W. Merzkirch, Flow Visualization, Academic Press (1974) References 123 W. Merzkirch, Flow Visualization II, Proc. Second Int. Symp. Flow Vis., 1980 Bochum, West Germany, McGraw-Hill (1980) T. Asanuma, Flow Visualization, Proc. Int. Symp. Flow Vis., 1977 Tokyo, Japan, McGraw-Hill (1979) M . Van Dyke, An Album of Fluid Motion, The Parabolic Press, Stanford (1982) W. Lauterborn and A. Vogel, "Modern optical techniques in fluid mechanics", Ann. Rev. Fluid Mech. 16 , pp. 223-44 (1984) M . A. Hernan and J . Jimenez, "Computer analysis of high-speed film in the plane turbulent mixing layer", J . Fluid Mech., 119 pp. 323, June (1982) F. Ahlborn, "Uber den Mechanismus des hydrodynamischen Widerstandes", Abhand-lungen aus dem Gebiet der Naturwissenschaften", Naturwiss. Verein, Hamburg Bd XVII, L. Friedrichsen & Co. (1902) A. Lau, "Automated Two Dimensional Flow Visualization and Coherent Structure Recognition", M . Sc. thesis, The University of British Columbia, Vancouver Canada (1986) (unpublished) T. Utami and T. Ueno, "Visualization and picture processing of turbulent flow", Experiments in Fluids 2, pp. 25-32 (1984) M . Mory and E. J . Hopfinger, "Structure functions in a rotationally dominated trubu-lent flow", Physics of Fluids, 29 pp. 2140-47 (1986) Y. -H . E. Sheu, T. P. K. Chang, G. B. Tatterson and D. S. Dickey, "A three-dimensional measurement technique for turbulent flows", Chem. Eng. Commun., 17, pp. 67-83 (1982) J . L. Lumley, "Coherent structures in turbulence", in Transitions and Turbulence, Academic Press, pp. 215-42 (1981) James C. McWilliams, "The emergence of isolated coherent structures in turbulent fluid flow", J . Fluid Mech. 146, pp. 21-43 (1984) A. K. M . F. Hussain, "Coherent structures - reality and myth", Phys. Fluids, 26 (10) pp. 2816-50 (1983) G. I. Taylor, "Statistical theory of turbulence", Proceedings of the Royal Society of London, Series A , 151, pp. 421 (1935) R. A. Antonia, "Conditional sampling in turbulence measurement", Ann. Rev. Fluid Mech. 13 , pp. 131-56 (1981) G. Comte-Bellot and S. Corrsin, "The use of a contraction to improve the isotropy of grid-generated turbulence", J. Fluid Mech. 25, pp. 657 (1966) Dean W. Criddle, "The viscosity and elasticity of interfaces", in Rheology Volume 3, edited by F. R. Eirich, pp. 429-42, Academic Press (1960) J . C. R. Hunt and J. M . R. Graham, "Free stream turbulence near plane boundaries", J. Fluid Mech., 84 (2), pp. 209-35 (1978) E. C. Itsweire and K. N . Helland, "Turbulent mixing and energy transfer in sta-bly stratified turbulence", pp. 172-175 in Seventh Symposium on Turbulence and Diffusion, pub. by the American Meteorological Society, Boston, Mass. (1985) References 124 J . T. L i n a n d Y. H. P a o , "Wakes i n s t r a t i f i e d fluids", Annual Review of Fluid Mechanics, 1 1 , pp. 317 (1979) J . S o m m e r i a , " T w o - d i m e n s i o n a l b e h a v i o u r of M H D f u l l y d e v e l o p e d t u r b u l e n c e (Rm > 1 ) " , J. Mecanique Theor. App. I. Supp. 1, 1 6 9 , ed. b y R. M o r e a u (1983) R. W. S t e w a r t , " T u r b u l e n c e a n d waves i n a s t r a t i f i e d a t m o s p h e r e " , Radio Set"., 4, pp. 1289 (1969) G. V e r o n i s , " T h e a n a l o g y b e t w e e n r o t a t i n g a n d s t r a t i f i e d fluids", Annual Review of Fluid Mechanics, 2, pp. 37-66 (1970) B l a i r B r u m l e y , " T u r b u l e n c e m e a s u r e m e n t s n e a r t h e free s u r f a c e i n s t i r r e d g r i d exper-i m e n t s " , i n G a s Transfer a t W a t e r Surfaces, pp. 83-92, e d i t e d b y W. B r u t s a e r t a n d G. H . J i r k a , D. R e i d e l P u b . C o . (1984) E. V. Z a l u t s k i i , "Some e s t i m a t e s o f t h e a c c u r a c y o f d e f i n i n g flow t u r b u l e n c e char-a c t e r i s t i c s b y t h e flow v i s u a l i z a t i o n m e t h o d " , Fluid Mechanics-Soviet Research, 2, N o . l - F e b r u a r y (1973) A. B. F i l u k , " C o m p u t e r m o d e l l i n g f o r s p o n t a n e o u s v o r t o n t r a n s i t i o n r a t e s " , UBC Plasma Physics Group Lab Report#95 (1985) ( u n p u b l i s h e d ) A. R o s h k o , " S t r u c t u r e o f t u r b u l e n t s h e a r flows: A new l o o k " , A I A A J o u r n a l , 1 4 No.10, pp. 1349-57 (1976) S. L o e w e n a n d B. A h l b o r n , " S t a t i s t i c a l a n a l y s i s o f c o h e r e n t s t r u c t u r e s i n t u r b u l e n t g r i d - f l o w " , P l a s m a G r o u p L a b R e p o r t # 103, U.B.C. P h y s i c s D e p a r t m e n t , V a n c o u v e r , C a n a d a (1984) ( u n p u b l i s h e d ) I. A. H a n n o u n , H. J . S. F e r n a n d o a n d E. J . L i s t , " T h e n a t u r e o f t u r b u l e n c e near a de n s i t y i n t e r f a c e " , B u i . A m . P h y s . Soc. 3 0 , pp. 1735 (1985) A. Hasegawa, " S e l f o r g a n i z a t i o n processes i n c o n t i n u o u s m e d i a " , A d v . P h y s . 34, 1 (1985) E. H o p f i n g e r , M. G r i f f i t h s a n d M. M o r y , "The s t r u c t u r e o f t u r b u l e n c e i n homogeneous a n d s t r a t i f i e d r o t a t i n g fluids", J. Mecanique Theor. App. I. Supp. 1, 2 1 , ed. by R. M o r e a u (1983) T h . v o n K a r m a n , a n d H. R u b a c h , " U b e r d e n m e c h a n i s m u s des flussigkeits- u n d l u f t w i d e r s t a n d e s " , Phys. Z. 1 3 , pp. 49-59 (1912); see a l s o C o l l . W o r k s / pp. 339-358 Macroscopic Modelling of Turbulent Flows, ed. U. F r i s c h et a l , P r o c e e d i n g s , Sophia-A n t i p o l i s , F r a n c e 1984, S p r i n g e r - V e r l a g (1985) 

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