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UBC Theses and Dissertations

Automated two dimensional flow visualization and coherent structure recognition Lau, Alexis Kai-Hon 1986

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AUTOMATED TWO DIMENSIONAL FLOW VISUALIZATION AND COHERENT STRUCTURE RECOGNITION by ALEXIS KAI-HON LAU B.Sc. C h i n e s e U n i v e r s i t y of Hong Kong, 1984 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE i n THE FACULTY OF GRADUATE STUDIES Department of P h y s i c s We a c c e p t t h i s t h e s i s as c o n f o r m i n g t o the r e q u i r e d s t a n d a r d THE UNIVERSITY OF J u l y © A l e x i s K a i BRITISH COLUMBIA 1986 hon Lau, 1986 In p r e s e n t i n g t h i s t h e s i s i n p a r t i a l f u l f i l m e n t of the r e q u i r e m e n t s f o r an advanced degree a t the U n i v e r s i t y of B r i t i s h Columbia, I agree t h a t the L i b r a r y s h a l l make i t f r e e l y a v a i l a b l e f o r r e f e r e n c e and s t u d y . I f u r t h e r agree t h a t p e r m i s s i o n f o r e x t e n s i v e c o p y i n g of t h i s t h e s i s f o r s c h o l a r l y purposes may be g r a n t e d by the Head of my Department or by h i s or her r e p r e s e n t a t i v e s . I t i s u n d e r s t o o d t h a t c o p y i n g or p u b l i c a t i o n of t h i s t h e s i s f o r f i n a n c i a l g a i n s h a l l not be a l l o w e d w i t h o u t my w r i t t e n p e r m i s s i o n . Department of P h y s i c s The U n i v e r s i t y of B r i t i s h Columbia 2075 Wesbrook P l a c e Vancouver, Canada V6T 1W5 Date: J u l y 31 1986 i i A b s t r a c t T h i s paper d e s c r i b e s an e f f i c i e n t method f o r e x t r a c t i n g q u a n t i t a t i v e d a t a from time sequences of f l u i d f l o w images. I t a l s o i n t r o d u c e s a computer automated method f o r the i d e n t i f i c a t i o n of 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 f l o w f i e l d s . T h i s method e l i m i n a t e s s u b j e c t i v e manual judgement i n the c r u c i a l s t a g e of c o h e r e n t s t r u c t u r e r e c o g n i t i o n . The s u r f a c e motion on t u r b u l e n t g r i d f l o w (Reynolds number 1 0 " ) was v i s u a l i z e d by r e c o r d i n g images of t r a c e r p a r t i c l e s on a v i d e o t a p e . Each v i d e o frame was then d i g i t i z e d as a b i n a r y image u s i n g a microcomputer. By t r a c k i n g and c o n n e c t i n g t h e t r a c e r p a t h s t h r o u g h s u c c e s s i v e frames u s i n g a mainframe computer the f l o w h i s t o r y was r e c o n s t r u c t e d . S t r e a k t r a j e c t o r i e s were then f i t t e d by p o l y n o m i a l s t o g i v e v a r i o u s f l o w parameters of i n t e r e s t over d e s i r e d f l o w t i m e s . I n p a r t i c u l a r the l i n e a r and 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 as s c a t t e r p o i n t s from which mesh f i e l d s were i n t e r p o l a t e d . Coherent s t r u c t u r e s were i d e n t i f i e d by t h r e s h o l d i n g the f i e l d of a n g u l a r v e l o c i t y . U s i n g the i n t e r p o l a t e d mesh f i e l d s of l i n e a r v e l o c i t i e s , each c o h e r e n t s t r u c t u r e was p a r a m e t r i z e d w i t h the p r o p e r t i e s of s i z e , average l i n e a r and a n g u l a r v e l o c i t i e s , and energy c o n t e n t . The f l o w dynamics and i n t e r a c t i o n s a r e then d i s c u s s e d u s i n g t h e s e s t r u c t u r e p r o p e r t i e s . The system was d e v e l o p e d p r i m a r i l y t o enhance d a t a r e c o g n i t i o n f o r a new model of t u r b u l e n c e based on the e n e r g e t i c s of c o h e r e n t s t r u c t u r e s . I t i s a l s o i n t e n d e d as a g e n e r a l t e c h n i q u e t o be used f o r o t h e r f l o w v i s u a l i z a t i o n and c o h e r e n t s t r u c t u r e s t u d i e s . A p p l y i n g the system t o the i n i t i a l s t a g e of g r i d t u r b u l e n c e , i t s u c c e s s f u l l y r e c o g n i z e d over 80% of a l l the c o h e r e n t s t r u c t u r e s m a n u a l l y i d e n t i f i e d . Parameter r e s u l t s u s i n g these a u t o m a t i c a l l y i d e n t i f i e d s t r u c t u r e s were compared w i t h e s t a b l i s h e d r e s u l t s and model p r e d i c t i o n s . L i m i t a t i o n and p o s s i b l e improvement on the p r e s e n t two d i m e n s i o n a l system i s d i s c u s s e d . V a r i o u s a s p e c t s i n e x t e n d i n g the system t o a t h r e e d i m e n s i o n a l environment a r e a l s o p r e s e n t e d . i v T a b l e of C o n t e n t s A b s t r a c t i i L i s t of F i g u r e s v i Acknowledgement v i i i Chapter I INTRODUCTION 1 1.1 T r a d i t i o n a l Approach 1 1.2 Coherent S t r u c t u r e s and Flow V i s u a l i z a t i o n 3 1.3 P r o s p e c t i v e f o r Computer P r o c e s s i n g 6 1 . 4 O b j e c t i v e 8 1.5 Chapter O u t l i n e s 9 Chapter I I COHERENT STRUCTURES - THE DEFINITION 12 2.1 Coherent S t r u c t u r e Model i n T u r b u l e n c e 12 2.2 D e f i n i t i o n based on Coherent V o r t i c i t y 15 2.3 V o r t i c i t y and A n g u l a r V e l o c i t y 16 2.4 The D e f i n i t i o n ....18 Chapter I I I THE EXPERIMENTAL SYSTEM 21 3.1 The Towing Tank 21 3.2 R e c o r d i n g 24 3.3 D i g i t i z a t i o n 25 3.4 Data S t o r a g e and T r a n s f e r 29 Chapter IV SYSTEM PRINCIPLES 31 4.1 N o i s e R e d u c t i o n and T r a c e r Image C e n t e r i n g 32 4.2 S t r e a k T r a c k i n g 39 4.3 G r i d I n t e r p o l a t i o n And Coherent S t r u c t u r e R e c o g n i t i o n 42 4.3.1 Time Windowing 44 4.3.2 Parameter I n t e r p o l a t i o n 45 4.3.3 P r e l i m i n a r y G r i d I n t e r p o l a t i o n 50 4.3.4 P r e l i m i n a r y Coherent S t r u c t u r e R e c o g n i t i o n 50 4.3.5 Data Refinement 52 4.3.6 F i n a l G r i d and Coherent S t r u c t u r e s 61 4.4 S t r u c t u r e P a r a m e t r i z a t i o n 64 Chapter V SYSTEM ALGORITHM 67 5.1 N o i s e R e d u c t i o n and T r a c e r Image C e n t e r i n g 67 5.2 S t r e a k T r a c k i n g 72 5.3 G r i d I n t e r p o l a t i o n and Coherent S t r u c t u r e R e c o g n i t i o n 77 5.3.1 Time Windowing and Parameter I n t e r p o l a t i o n 77 5.3.2 G r i d I n t e r p o l a t i o n 80 V 5.3.3 Coherent S t r u c t u r e R e c o g n i t i o n 80 5.3.4 Data and S t r u c t u r e Refinement 83 5.4 S t r u c t u r e P a r a m e t r i z a t i o n 84 Chapter VI HARDWARE CONSTRAINTS AND EXPERIMENTAL PARAMETERS 85 6.1 Hardware C o n s t r a i n t s and E x p e r i m e n t a l Parameters ..86 6.2 Hardware and C o n t r o l Parameters Used , 94 Chapter V I I RESULTS AND DISCUSSIONS 98 7.1 E x p e r i m e n t a l R e s u l t s on I n i t i a l G r i d T u r b u l e n c e ...98 7.1.1 Coherent S t r u c t u r e s a t P r o d u c t i o n 99 7.1.2 Spontaneous Energy Decay Rate of Coherent S t r u c t u r e s ; 115 7.2 D i s c u s s i o n s and Recommendations 117 7.2.1 L i m i t a t i o n s of the P r e s e n t System 117 7.2.2 E x t e n s i o n t o a 3D System 123 Chapter V I I I CONCLUSION 130 BIBLIOGRAPHY 132 APPENDIX A - PARAMETER EXTRACTIONS FROM FITTED TRAJECTORY 1 33 A. 1 S t a t i o n a r y Model: Vcm i s Zero 134 A.2 D r i f t i n g Model: Vcm i s Not Zero 135 APPENDIX B - USING THE PACKAGE AT UBC 136 v i L i s t of F i g u r e s 1. Flow p i c t u r e showing c o h e r e n t s t r u c t u r e s 5 2. Coherent s t r u c t u r e s o u t l i n e d by our d e f i n i t i o n i n one dimensio n 20 3. Diagrammatic drawing of the system hardware 22 4. The tow i n g tank 23 5. Numbered 8-neighbours of a p i x e l 32 6. Unprocessed c o n s e c u t i v e d i g i t i z e d images 33 7. Superimposed p l o t of t r a c e r c e n t e r s and t r a c k e d s t r e a k s 41 8. 1D s t r u c t u r e s r e c o g n i s e d from w(r) p l o t 51 9. I d e a l i z e d and m o d i f i e d V ( r ) and w(r) p l o t s f o r co h e r e n t s t r u c t u r e s 55 10. T y p i c a l p l o t of s t r e a k s t r a c k e d i n a c o h e r e n t s t r u c t u r e . 56 11. S p a t i a l i n t e r p o l a t e d w and V p l o t from s t r e a k s shown i n F i g . 10 57 12. S p a t i a l i n t e r p o l a t e d u> and V p r o f i l e from s t r e a k s shown i n F i g . 10 w i t h CM p o i n t as p o i n t of z e r o v e l o c i t y 58 13. 1D s t r u c t u r e s i d e n t i f i e d b e f o r e and a f t e r removal of s m a l l peaks . . 62 14. P l o t of f i n a l r e c o g n i z e d Coherent s t r u c t u r e s 63 15. Smoothing c o n s i d e r a t i o n s 68 16. Coherent s t r u c t u r e s a t p r o d u c t i o n 100 17. P l o t of c a l c u l a t e d r o t a t i o n a l energy vs t o t a l energy f o r r e c o g n i z e d s t r u c t u r e s 108 18. S p a t i a l p l o t s of w(r) and V ( r ) f o r d i f f e r e n t s i t u a t i o n s 109 v i i 19. S p a t i a l p l o t of V 2 ( r ) f o r d i f f e r e n t s i t u a t i o n s 110 20. L o g - l o g p l o t of i n i t i a l decay r a t e A as a f u n c t i o n of the s t r u c t u r e r a d i u s R 113 21. P l o t of i n i t i a l decay r a t e A vs 1/R2 114 v i i i Acknowledgement I want t o thank V i n c e n t Bareau f o r i n t e r f a c i n g the VCR and the microcomputer. The " d e s i g n e r s t a n d a r d " v i d e o r e c o r d e r s u p p o r t s t a n d b u i l t by P a u l B u r r e l and M a c i e j Kowalewski gave me much freedom i n s e l e c t i n g the view a r e a w h i l e r e m a i n i n g r i g i d l y i n p l a c e d u r i n g t h e e x p e r i m e n t s . Thanks s h o u l d a l s o be g i v e n t o D i r k Townsend who h e l p e d t o c o r r e c t my E n g l i s h . A l Cheuck was always t h e r e t o get the s u p p l i e s i n the s h o r t e s t p o s s i b l e time and a l s o made sure t h a t the i n s t r u m e n t a t i o n was w o r k i n g as d e s i g n e d . The plasma group must be thanked f o r s u p p o r t i n g my work here and a l s o f o r g i v i n g me the l a r g e amount of computer and r e a l d o l l a r s used t o d e v e l o p and t e s t run the programs. S p e c i a l thanks must be g i v e n t o S t u a r t Loewen who e x p l a i n e d t o me the use of a l l the equipment, s t a y e d and d i s c u s s e d w i t h me d u r i n g the e x p e r i m e n t s , and a l s o p r o o f - r e a d my d r a f t up t o the morning he had t o l e a v e f o r h i s b r o t h e r ' s wedding. F i n a l l y , I have t o thank my s u p e r v i s o r P r o f e s s o r Boye A h l b o r n who g u i d e d me throughout t h e development of the system and the w r i t i n g of t h i s t h e s i s . 1 I . INTRODUCTION In s p i t e of decades of r e s e a r c h , t u r b u l e n c e remains one of the major u n r e s o l v e d problems i n c l a s s i c a l p h y s i c s . Our l o n g -s t a n d i n g i n a b i l i t y t o g i v e r e l i a b l e p r e d i c t i o n s i n t u r b u l e n t f l o w c o n d i t i o n s c r i t i c a l l y l i m i t s t e c h n o l o g i c a l developments, and has l e d many r e s e a r c h e r s t o q u e s t i o n the c o n v e n t i o n a l method d e v e l o p e d by Reynolds i n the 19th c e n t u r y . 1.1 T r a d i t i o n a l Approach T r a d i t i o n a l l y one t r i e s t o d e f i n e a f l o w f i e l d a t a g i v e n time and p r e d i c t i t i n d e t a i l a t a l a t e r t i m e . B a s i c a l l y , t h i s method i n v o l v e s the f o l l o w i n g s t e p s : 1) w r i t e down the N a v i e r - S t o k e s e q u a t i o n 3V/3t + ( V - V ) V = - V ( P / p ) + u V 2 V , and the e q u a t i o n of C o n t i n u i t y 3p/3t + V - ( p V ) = 0 . 2) Assume i n c o m p r e s s i b i l i t y ; p = c o n s t a n t and V - V = 0 . 3) I n t r o d u c e the Reynolds number, Re Re = p V L / u , and reduce the g o v e r n i n g e q u a t i o n s t o a d i m e n s i o n l e s s form 3V/3t + ( V - V ) V = - V P + ( V 2 V ) / R e . 4) S p l i t the v e l o c i t y and p r e s s u r e f i e l d s i n t o the mean ( z e r o e t h o r d e r ) and f l u c t u a t i n g ( 1 s t o r d e r ) components: 2 V = V° + V 1 , and p _ p o + p i . t a k i n g the z e r o e t h o r d e r as st e a d y s t a t e s a t i s f y i n g the o r i g i n a l e q u a t i o n s : 3V°/9t + (V°-V)V° = -VP 0 + (V 2V°)/Re ; V-V° = 0 . 5) S u b s t i t u t e them back i n t o the o r i g i n a l e q u a t i o n s , e l i m i n a t i n g most z e r o e t h o r d e r terms t o get the f i r s t o r d e r f l o w e q u a t i o n : 3 V 1 / 3 t + (V°-V)V1 + (V 1-V)V° = -VP 1 + ( V 2 V 1 ) / R e , and V-V 1 = 0 . T h i s g i v e s two e q u a t i o n s v e r y s i m i l a r t o the f i r s t two except the (V«V)V term becomes (V°-V)V1 + (V 1-V)V°. T h i s p e r t u r b e d system of e q u a t i o n s i s then s u b j e c t e d t o the boundary c o n d i t i o n s of i n t e r e s t . However, i t s h o u l d be noted t h a t the system now becomes u n c l o s e d as we have 3 v a r i a b l e s V°, V 1 and P 1 but o n l y two e q u a t i o n s . Then the major e f f o r t i s t o lo o k f o r a g e n e r a l r e l a t i o n t o c l o s e the system of e q u a t i o n s . T h i s i s u s u a l l y r e f e r r e d t o as the c l o s u r e r e l a t i o n . There a r e o t h e r v a r i a t i o n s i n the approach, e.g. one may e l i m i n a t e P from the e q u a t i o n s and then s p l i t the f i e l d , but a c l o s u r e r e l a t i o n of some form i s s t i l l i n v o k e d . T h i s problem has been worked on f o r n e a r l y a c e n t u r y and p a r t i c u l a r r e l a t i o n s have been proposed f o r many d i f f e r e n t systems. However, t h e r e i s s t i l l l i t t l e p r o g r e s s i n the 3 s e a r c h f o r a u n i v e r s a l r e l a t i o n t o s o l v e the system (which may not i n d e e d e x i s t . ) Thus, i t appears t o many t h a t any fundamental b r e a k t h r o u g h i n t u r b u l e n c e r e s e a r c h must occur o u t s i d e the u n i v e r s a l r e l a t i o n . 1.2 Coherent S t r u c t u r e s and Flow V i s u a l i z a t i o n D u r i n g the p a s t twenty y e a r s , as a r e s u l t of the work of K l i n e et a l ( l 9 6 7 ) , Crow and Champagne(1971) and many o t h e r s , a new p e r s p e c t i v e seemed t o be emerging. The d i s c o v e r y of l a r g e - s c a l e v o r t e x motions i n t u r b u l e n t shear f l o w s i s now g e n e r a l l y agreed t o be one of t h e most i m p o r t a n t developments i n the f i e l d f o r many y e a r s . The o r i g i n a l f i n d i n g s were based on o b s e r v a t i o n s i n t u r b u l e n t boundary l a y e r f l o w but q u i c k l y a t t r a c t e d g e n e r a l i n t e r e s t as r e p o r t s of such v o r t e x motions s t a r t e d t o p i l e up i n n e a r l y every type of t u r b u l e n t shear f l o w . I t i s now r e c o g n i z e d t h a t these motions a r e i n t r i n s i c t o t u r b u l e n t f l o w s and the r e s e a r c h e r s ' work i s t o f i n d out how we can l e a r n more about t u r b u l e n c e t h r o u g h them. Such l a r g e - s c a l e v o r t e x motions a r e now c o l l e c t i v e l y known as co h e r e n t s t r u c t u r e s and s p e c u l a t i o n s are t h a t they p l a y an im p o r t a n t r o l e i n the t r a n s p o r t p r o p e r t i e s of the f l o w . A s u b s t a n t i a l amount of r e s e a r c h has been done and t h e r e i s wide a c c e p t a n c e of the importance of co h e r e n t s t r u c t u r e s t o t u r b u l e n t f l o w s . There i s , however, l i t t l e agreement on the d e f i n i t i o n of co h e r e n t s t r u c t u r e s , or on t h e i r e x a c t r o l e i n and s i g n i f i c a n c e t o t u r b u l e n t f l o w . 4 As r e c e n t l y p o i n t e d out by H u s s a i n ( 1 9 8 5 ) , the b a s i c d i f f i c u l t y stems from the v e r y method we use t o d i s c o v e r t h e s e s t r u c t u r e s . N e a r l y a l l d i s c o v e r i e s and s t u d i e s use f l o w v i s u a l i z a t i o n . There a r e many v a r i a n t s i n the method but the g e n e r a l i d e a i s t o tak e i n s t a n t a n e o u s or time - e x p o s u r e p i c t u r e s of the f l o w and then measure from them the v a r i o u s parameters of i n t e r e s t , t y p i c a l l y t he v e l o c i t y , energy and momentum. The major advantage of the method l i e s i n i t s enormous i n f o r m a t i o n d e n s i t y . A w e l l c o n c e i v e d f l o w p i c t u r e p r e s e n t s a c l e a r a n a t o m i c a l d i s s e c t i o n of the system. Another advantage which i s not so e v i d e n t i s the method of v i s u a l a n a l y s i s used by the e x p e r i m e n t a l i s t . Our v i s u a l a n a l y t i c power i s something so p o w e r f u l t h a t no modern computer can match i t . We can have a g e n e r a l i d e a of what i s meant by "coherent s t r u c t u r e " j u s t by l o o k i n g a t a flo w p i c t u r e l i k e the one shown i n F i g . 1. Coherent s t r u c t u r e s on the t o p r i g h t hand c o r n e r a r e o u t l i n e d m a n u a l l y . On c l o s e r e x a m i n a t i o n of the t r a c e r p a t h s , we can observe the s t r u c t u r e ' s v e l o c i t y p r o f i l e and a p p r e c i a t e the s t r u c t u r e e n e r g e t i c s t r e n g t h s . 5 -3 c m I 1 F i g u r e 1 - Flow p i c t u r e showing c o h e r e n t s t r u c t u r e s . T h i s v i s u a l p r e c e p t i o n i n c o r p o r a t e s many i n t r i c a t e f a c u l t i e s l i k e v i s u a l i n p u t , s i m i l a r i t y a n a l y s i s , f i l t e r i n g of i r r e l e v a n t d a t a and s u r f a c e (2D) c o r r e l a t i o n a n a l y s i s . T h i s power we i n h e r i t from our a n c e s t o r s has e n a b l e d us t o s u r v i v e i n our r a p i d l y c h a n g i n g environment. I t a l s o a l l o w s us t o d i s c o v e r the c o h e r e n t s t r u c t u r e s . The d i f f i c u l t y i s t h a t no machine can y e t p e r f o r m the r e c o g n i t i o n i n a f a s t and r e l i a b l e way. When we use f l o w v i s u a l i z a t i o n p i c t u r e s f o r a n a l y t i c p u r p o s e s , the amount of work r e q u i r e d t o e x t r a c t u s e f u l i n f o r m a t i o n i s p r o h i b i t i v e l y l a r g e . Even when t h i s i n f o r m a t i o n i s o b t a i n e d , the r e m a i n i n g problems of d a t a a n a l y s i s , s u r f a c e c o r r e l a t i o n and p a t t e r n r e c o g n i t i o n a r e 6 s t i l l d i f f i c u l t . H ussain(1983) p o i n t e d out t h a t the d o w n f a l l of t h i s t e c h n i q u e i n the e a r l y 20th c e n t u r y was because of the d i f f i c u l t i e s i n g e t t i n g hard d a t a i n the method. The i n f o r m a t i o n was t h e r e and hard d a t a c o u l d have been e x t r a c t e d . I t was j u s t the e x c e s s i v e amount of work t h a t drove away the e x p e r i m e n t a l i s t s . D u r i n g t h a t t i m e , the method of f l o w v i s u a l i z a t i o n d e v e l o p e d e a r l y i n the c e n t u r y ( A h l b o r n 1902, 1922) was g r a d u a l l y r e p l a c e d by hot w i r e or l a s e r d o p p l e r anemometers which gave f a s t , c l e a n , easy t o handle and e c o n o m i c a l d a t a . E x p e r i m e n t a l i s t s took the easy way o u t . However, t h e s e new t e c h n i q u e s s u f f e r e d from b e i n g too r e s t r i c t i v e i n i n f o r m a t i o n y i e l d i n g o n l y s i n g l e or m u l t i -p o i n t v e l o c i t y d a t a . I t i s v e r y h a r d t o i n f e r from the i n f o r m a t i o n c o l l e c t e d by a l i n e a r a r r a y of s e n s o r s the s p a t i a l c o r r e l a t i o n which i s v i t a l t o c o h e r e n t s t r u c t u r e r e c o g n i t i o n . These d i f f i c u l t i e s s i g n i f i c a n t l y l i m i t the power of t h e s e t e c h n i q u e s , and so, many r e s e a r c h e r s have t u r n e d back t o the method of f l o w v i s u a l i z a t i o n . There they f i n d the c o h e r e n t s t r u c t u r e s . Recent developments i n t h i s f i e l d were reviewed by C a n t w e l l and C o l e s ( 1 9 8 3 ) . 1.3 P r o s p e c t i v e f o r Computer P r o c e s s i n g W h i l e c o h e r e n t s t r u c t u r e s were b e i n g i d e n t i f i e d and s t u d i e d , the most i n f l u e n t i a l change s i n c e the i n d u s t r i a l r e v o l u t i o n was t a k i n g p l a c e - the development of h i g h speed computers. The a b i l i t y of computers t o escape the l i m i t a t i o n s 7 of human speed and handle e x c e s s i v e l y " l a r g e amount of d a t a w i t h unprecedented a c c u r a c y i n a most monotonous or c o m p l i c a t e d way, has pushed f o r w a r d many o t h e r w i s e d e a d l o c k e d c o r n e r s of s c i e n c e . A c t u a l l y , we f i n d the b e s t example w i t h i n the computer community i t s e l f , as t h i s p o w e r f u l t o o l i s e a s i l y a c c e s s i b l e and i s used e x t e n s i v e l y f o r i t s own advancement. The computer has become an i n d i s p e n s i b l e r e s e a r c h t o o l and c o m p u t a t i o n a l r e s e a r c h has emerged as a s e p a r a t e branch i n a d d i t i o n t o the t r a d i t i o n a l d i v i s o n of e x p e r i m e n t a l and t h e o r e t i c a l s c i e n c e . S i n c e i t s i n t r o d u c t i o n , the computer has been used e x t e n s i v e l y i n f l u i d dynamics s t u d i e s . The major e f f o r t s have been t o s o l v e the N a v i e r - S t o k e s e q u a t i o n n u m e r i c a l l y w i t h v a r i o u s boundary c o n d i t i o n s . Animated g r a p h i c s i s used t o model f l o w s as s i m u l a t e d e x p e r i m e n t s . In e x p e r i m e n t a l s t u d i e s w i t h the hot w i r e probes and LDAs ( l a s e r d o p p l e r anemometers), i t i s used e x t e n s i v e l y t o h e l p d a t a a c q u i s i t i o n , s t o r a g e , c a l c u l a t i o n and p r e s e n t a t i o n . N e v e r t h e l e s s , most v i s u a l i z a t i o n e x p e r i m e n t s s t i l l r e q u i r e c r u c i a l s u b j e c t i v e judgement of the r e s e a r c h e r s i n the d a t a a c q u i s i t i o n and c o h e r e n t s t r u c t u r e s r e c o g n i t i o n s t a g e s . The enormous amount of work i n the manual d a t a a c q u i s i t i o n p r o c e s s s t i l l l i m i t s the amount and a c c u r a c y of d a t a e x t r a c t e d and t h u s the e x t e n t and r e l i a b i l i t y of the q u a n t i t a t i v e r e s u l t s . A c c o r d i n g t o a r e c e n t computer l i t e r a t u r e s e a r c h i n t o the 8 INSPEC ( i n f o r m a t i o n s e r v i c e s f o r the p h y s i c s and e n g i n e e r i n g community), l i t t l e work has been done i n a u t o m a t i n g the d a t a a c q u i s i t i o n i n f l o w v i s u a l i z a t i o n . Manual s u b j e c t i v e p r o c e s s e s a r e s t i l l used f o r most c o h e r e n t s t r u c t u r e r e c o g n i t i o n i n f l o w v i s u a l i z a t i o n e x p e r i m e n t s . T h i s i n v o l v e s the p r e j u d i c e of the i n v e s t i g a t o r s b oth i n the d e f i n i t i o n and i d e n t i f i c a t i o n of the s t r u c t u r e s and i t r e s u l t s i n the w i d e s p r e a d d i f f e r e n c e s i n o p i n i o n s about c o h e r e n t s t r u c t u r e s . The l a c k of a g e n e r a l approach makes i t h a r d t o compare r e s u l t s and the absence of e x t e n s i v e q u a n t i t a t i v e d a t a makes i t h a r d t o d e c i d e which approach i s b e t t e r . The newness of the image p r o c e s s i n g t e c h n i q u e may be a reason f o r the u n p o p u l a r i t y of computer a n a l y s i s of f l o w v i s u a l i z a t i o n . Such an approach i s d e f i n i t e l y needed t o f u l l y u t i l i z e the power of f l o w v i s u a l i z a t i o n and t h i s t h e s i s i s a p r e l i m i n a r y attempt i n t h i s d i r e c t i o n . 1 . 4 O b j e c t i v e As w i l l be d e s c r i b e d i n more d e t a i l i n the next c h a p t e r , the immediate m o t i v a t i o n of t h i s work was t o enhance the f l o w v i s u a l i z a t i o n s tudy used by my s u p e r v i s o r , B. A h l b o r n and S. Loewen(1985). They proposed a s t a t i s t i c a l model f o r t u r b u l e n c e based on the energy and s i z e spectrum of the c o h e r e n t s t r u c t u r e s . My i n i t i a l i n t e r e s t was t o automate t h e i r manual proc e d u r e of d a t a a c q u i s i t i o n , a n a l y s i s and c o h e r e n t s t r u c t u r e r e c o g n i t i o n . A f t e r s t u d y i n g the 9 l i t e r a t u r e , the g e n e r a l need f o r such a system was a p p r e c i a t e d , and i t was d e c i d e d t o d e s i g n the system t o s e r v e as a g e n e r a l t e c h n i q u e . These two r e q u i r e m e n t s were sometimes c o n f l i c t i n g as one was aimed t o be s p e c i f i c and e f f i c i e n t w h i l e the o t h e r was i n t e n d e d f o r the g e n e r a l use. The compromise was t h a t whenever s p e c i f i c c o n s i d e r a t i o n s were f o c u s e d on the energy model, i t would be c l e a r l y noted how p o s s i b l e v a r i a t i o n s c o u l d be made i n o t h e r systems. F u r t h e r m o r e , as t h i s i s s t i l l a new method, another aim was t o e v a l u a t e i n d e t a i l the f e a s i b i l i t y of such systems i n the r e a l e x p e r i m e n t a l e n v i r o n m e n t s . The a c t u a l w o r k i n g d e s i g n s s h o u l d be i n t i m a t e l y r e l a t e d t o the ex p e r i m e n t s of i n t e r e s t . I t was hoped t h a t by comparing the r e l a t i v e importance of the param e t e r s , some g e n e r a l guidance f o r f u r t h e r d e s i g n can be e s t a b l i s h e d , t h e r e b y i n c r e a s i n g the c o s t e f f e c t i v e n e s s of the system. 1 . 5 Chapter O u t l i n e s In the next c h a p t e r , the p r i n c i p a l i d e a s i n A h l b o r n and Loewen's work a r e g i v e n , f o c u s i n g on t h e i r concept of c o h e r e n t s t r u c t u r e s . Then the d e f i n i t i o n by Hussain(1983) i s g i v e n as an example of one of the c u r r e n t c o n c e p t s of c o h e r e n t s t r u c t u r e s i n the l i t e r a t u r e . F i n a l l y , the d e f i n i t i o n used i n t h i s work i s g i v e n , and i t s r e l a t i o n t o the s t a t i s t i c a l model and t o H u s s a i n ' s d e f i n i t i o n i s d i s c u s s e d . 10 The t h i r d c h a p t e r d e s c r i b e s the hardware p a r t of the system. Emphasis i s put on what must be done i n each s t e p r a t h e r than d e t a i l i n g the a c t u a l hardware d e s i g n s . T h i s i s p a r t l y because the a c t u a l d e s i g n s a r e v e r y system dependent and p a r t l y because my c o n t r i b u t i o n i n t h e i r development was m i n i m a l . The major work i s e l a b o r a t e d i n d e t a i l i n the next t h r e e c h a p t e r s . The s o f t w a r e of the system i s d e s c r i b e d i n c h a p t e r IV and V s e p a r a t e l y . Chapter IV p r e s e n t s the p r i n c i p l e s w i t h c o n s i d e r a t i o n s r e g a r d i n g v a r i o u s c o n s t r a i n t s of i m p l e m e n t a t i o n . The next c h a p t e r e l a b o r a t e s the a l g o r i t h m (not the programming t e c h n i q u e ) g i v i n g the v a r i o u s e f f i c i e n c y c o n s i d e r a t i o n s . T h i s s e p a r a t i o n was done i n the b e l i e f t h a t the p r i n c i p l e s of what s h o u l d be done and the a l g o r i t h m s of how t o do i t are s e p a r a b l e . M i x i n g the two t o g e t h e r would j u s t c o n f u s e the reader i n u n d e r s t a n d i n g the p r i n c i p l e s and the a l g o r i t h m fundamentals. T h i s does not mean the p r i n c i p l e s and the methods are c o m p l e t e l y s e p a r a t e . Both have t o be u n d e r s t o o d w e l l t o enable the system t o be used s u c c e s s f u l l y . Chapter VI i s devoted t o c o n s i d e r a t i o n s r e g a r d i n g the v a r i o u s hardware parameters and e x p e r i m e n t a l c o n t r o l s t h a t a f f e c t the performance of the system. T h i s s e r v e s t o c l a r i f y t h e r e l a t i o n between the system hardware and the e x p e r i m e n t a l s e t t i n g . They have t o be matched f o r the s u c c e s s f u l a p p l i c a t i o n of the s o f t w a r e package. R e s u l t s of an 11 i n v e s t i g a t i o n of coherent s t r u c t u r e i n the i n i t i a l p e r i o d of g r i d t u r b u l e n c e are g i v e n i n the s e v e n t h c h a p t e r a l o n g w i t h a d i s c u s s i o n of how our system may be extended f o r f u r t h e r s t u d i e s . A c o n c l u s i o n i s then drawn i n C hapter V I I . I n f o r m a t i o n about how t o use the package a t UBC i s g i v e n i n Appendix B. 1 2 I I . COHERENT STRUCTURES - THE DEFINITION The i n i t i a l m o t i v a t i o n of t h i s work was t o automate t h e t u r b u l e n c e s t u d y s t a r t e d by A h l b o r n and Loewen i n 1983. The f o l l o w i n g i s g i v e n as a summary of t h e i r r e c e n t paper t o connect t h i s work and t h e i r s . 2.1 Coherent S t r u c t u r e Model i n T u r b u l e n c e T h i s model d i f f e r s from t r a d i t i o n a l t u r b u l e n c e r e s e a r c h from the v e r y s t a r t i n g p o i n t . I t i s based on c o n s i d e r a t i o n of l a r g e - s c a l e energy b a l a n c e r a t h e r than l o c a l momentum b a l a n c e i n the N a v i e r - S t o k e s e q u a t i o n . I t a t t e m p t s t o d e s c r i b e " g r o s s f e a t u r e s of t u r b u l e n t f l o w from s t a t i s t i c a l l y a veraged d i s t r i b u t i o n s of i n t e r a c t i n g c o h e r e n t s t r u c t u r e s " ( A h l b o r n e t a l 1985). To a ve r y crude a p p r o x i m a t i o n , t h e s e c o h e r e n t s t r u c t u r e s a r e c o n s i d e r e d as i d e a l i z e d c y l i n d r i c a l r o t a t i n g columns of f l u i d c h a r a c t e r i z e d by the r a d i u s R and the a n g u l a r v e l o c i t y u>. They i n t e r a c t w i t h t h e s u r r o u n d i n g s i n the forms of eddy-f l u i d , e d d y - f l o w and eddy-eddy i n t e r a c t i o n s , w i t h the term "eddy" used a n a l o g o u s l y w i t h "coherent s t r u c t u r e " . The r a t e of t r a n s f e r of energy of the c o h e r e n t s t r u c t u r e s due t o t h e above p r o c e s s e s a r e denoted by A, B and C r e s p e c t i v e l y . W i t h t h e s e c o h e r e n t s t r u c t u r e s i n mind, the f l o w i s d e s c r i b e d by the s i z e spectrum N(R) or the energy spectrum N ( E ) . These d i s t r i b u t i o n s can be i n t e r p r e t e d as p r o b a b i l i t y f u n c t i o n s or 13 ensemble a v e r a g e s . System i n t e r a c t i o n s are then i n c o r p o r a t e d i n t o a s e t of r a t e e q u a t i o n s f o r the energy d i s t r i b u t i o n N(E^) a s : (2.1) d(Ni.)/dt = I (A-.jN:) + E (B^-N-) + E (C^.N^N.) where = N(E-^) and the summation i s over i n d i c e s j and k. The g r o s s p r o p e r t i e s of the f l o w a r e t o be d e r i v e d e i t h e r from the moments of the d i s t r i b u t i o n f u n c t i o n N or the i n t e r a c t i o n r a t e c o e f f i c i e n t s . W i th these major i d e a s and t h e i r crude assumption of c y l i n d r i c a l c o h e r e n t s t r u c t u r e s , they succeed i n d e r i v i n g a number of e s t a b l i s h e d t h e o r e t i c a l and e x p e r i m e n t a l r e s u l t s t o the r i g h t form and o r d e r of magnitude. These i n c l u d e 1) the Reynolds number f o r the onset of i n s t a b i l i t y b e h i n d b l u f f b o d i e s and i n boundary l a y e r s ; 2) the s i z e of the s m a l l e s t e d d i e s i n a f l o w ; 3) the l o g a r i t h m i c v e l o c i t y d i s t r i b u t i o n of t u r b u l e n t boundary l a y e r f l o w near a w a l l ; and 4) the drag c o e f f i c i e n t f o r b l u f f b o d i e s . Thus, t h i s p r o v e s t o be a u s e f u l model i n o b t a i n i n g the g r o s s f e a t u r e s of the flow and more r i g o r o u s work i s w o r t h w h i l e . However, the a c t u a l amount of work 1 r e q u i r e d t o o b t a i n e x p e r i m e n t a l data l i k e the energy d i s t r i b u t i o n s i s p r o h i b i t i v e l y l a r g e . Moreover, manual a n a l y s i s s u f f e r s from b e i n g s u b j e c t i v e and a person's In 1983, S. Loewen spent about 2 months of manual l a b o u r a n a l y z i n g over 2000 s t r u c t u r e s w i t h about 30,000 i n d i v i d u a l s t r e a k s from time exposure p i c t u r e s . 14 c o n s i s t e n c y s u f f e r s from f a t i g u e . I t appeared t h a t an automated system f o r r e c o g n i z i n g the s t r u c t u r e and e x t r a c t i n g i t s p arameters s h o u l d be d e s i g n e d . T h i s became the p r i m a r y o b j e c t i v e of t h i s work. As a computer can o n l y do e x a c t l y what i t i s programmed t o , we have t o g i v e i t a c l e a r d e f i n i t i o n f o r c o h e r e n t s t r u c t u r e s b e f o r e i t can r e c o g n i z e them. In the p r e v i o u s work by Loewen(1983), he "assumed t h a t the p e r i p h e r y of the s t r u c t u r e s h o u l d be the l a r g e s t c l o s e d s t r e a m l i n e s . " The use of s t r e a m l i n e s ( t r a c e r p a t h s ) f o r c o h e r e n t s t r u c t u r e r e c o g n i t i o n assumes the parameters of i n t e r e s t remain c o n s t a n t throughout the time of exposure. T h i s i g n o r e s the i n s t a n t a n e o u s i n f o r m a t i o n . Another d i s a d v a n t a g e i s t h a t the p h y s i c s i n the r e c o g n i t i o n p r o c e s s i s not r e a d i l y c l e a r , making i t h a r d t o match the s t r u c t u r e s so d e f i n e d w i t h an a n a l y t i c d e f i n i t i o n . F i n a l l y , t h i s method i s frame dependent and the s t r u c t u r e s so d e f i n e d w i l l be a G a l i l e a n v a r i a n t . T h i s i s not i n l i n e w i t h the u n d e r l y i n g "assumption t h a t the r o t a t i o n a l k i n e t i c energy i n the e d d i e s i s no l o n g e r dependent on the i n e r t i a l frame" i n the model ( A h l b o r n e t a l 1985). For these and o t h e r reasons g i v e n l a t e r , a n o t h e r d e f i n i t i o n based on a n g u l a r v e l o c i t y u> i s used. B e f o r e l o o k i n g i n t o i t , l e t us f i r s t t u r n our a t t e n t i o n t o the argument i n the l i t e r a t u r e on the same i s s u e . 15 2 . 2 D e f i n i t i o n based on Coherent V o r t i c i t y As p o i n t e d out i n the i n t r o d u c t i o n , the w i d e s p r e a d a c c e p t a n c e of the importance of c o h e r e n t s t r u c t u r e s i n v i t e d a s u b s t a n t i a l amount of r e s e a r c h , but has y e t t o r e s u l t i n a consensus of how a c o h e r e n t s t r u c t u r e s h o u l d be d e f i n e d . For most r e s e a r c h e r s , the a p p e a l i n g t h i n g t o do i s t o have the d e f i n i t i o n f i t t i n g o b s e r v a t i o n s ( v i s u a l concept of the c o h e r e n t s t r u c t u r e s ) and a l s o b e i n g a n a l y t i c enough f o r t h e o r e t i c a l c a l c u l a t i o n s . Coherence i s d e f i n e d e i t h e r t h r o u g h the a u t o c o r r e l a t i o n f u n c t i o n of a random v a r i a b l e or as the remains of some ensemble average. Many d i f f e r e n t d e f i n i t i o n s can be found but n e a r l y a l l of them a r e based on d y n a m i c a l q u a n t i t i e s (e.g. p r e s s u r e , momentum or v o r t i c i t y ) a s s o c i a t e d w i t h the N a v i e r - S t o k e s e q u a t i o n or i t s d e r i v a t i v e s . We w i l l choose the one by Hussain(1983) as an example: "A c o h e r e n t s t r u c t u r e i s a c o n n e c t e d , l a r g e s c a l e t u r b u l e n t f l u i d mass w i t h a phase c o r r e l a t e d v o r t i c i t y over i t s s p a t i a l e x t e n t . " Here, l a r g e s c a l e r e f e r s t o s c a l e comparable t o the e x t e n t of the shear f l o w and "phase c o r r e l a t e d " r e f e r s t o the ensemble average of s i m i l a r s t r u c t u r e s of the same phase, not the average over d i f f e r e n t phases. The phase of a c o h e r e n t s t r u c t u r e r e f e r s t o i t s stage of development ( i t s a g e ) . V o r t i c i t y t h a t s u r v i v e s such a v e r a g i n g i s c a l l e d c o h e r e n t v o r t i c i t y Oc and the o t h e r p a r t i s c a l l e d i n c o h e r e n t (random) 16 v o r t i c i t y B r . In p r e s e n t i n g t h i s d e f i n i t i o n , i t i s not my i n t e n t i o n t o argue how c o h e r e n t s t r u c t u r e s s h o u l d be d e f i n e d . I t i s beyond the scope of t h i s t h e s i s t o e n t e r t h i s f a r - r e a c h i n g argument. T h i s d e f i n i t i o n i s chosen because of the r e l a t i o n between the d e f i n i n g q u a n t i t i e s (0 and CJ) and the d e f i n i t i o n w o r d i n g s . The f i r s t p o i n t t o note i s t h a t r e c o g n i t i o n based s o l e l y on the above d e f i n i t i o n i s d i f f i c u l t as i t r e q u i r e s some i n i t i a l knowledge of the c o h e r e n t s t r u c t u r e s t o p e r f o r m the ensemble average , and a t the same time r e q u i r e s the r e s u l t of the ensemble average t o i d e n t i f y the s t r u c t u r e s . An o p e r a t i o n a l l y e a s i e r method must a l l o w us t o d e f i n e the s t r u c t u r e from an i n s t a n t a n e o u s or l o c a l time averaged parameter f i e l d ( s ) , and then r e f i n e the s t r u c t u r e s a f t e r w a r d s . There s h o u l d not be any p r e f e r r e d s p a t i a l a v e r a g i n g i n the f i r s t p l a c e . 2.3 V o r t i c i t y and Angul a r V e l o c i t y Our model i s based on energy c o n s i d e r a t i o n s , the major v a r i a b l e b e i n g the r o t a t i o n a l energy E r . In the f i r s t a p p r o x i m a t i o n of the c y l i n d r i c a l f l u i d column, Er i s a f u n c t i o n of the r a d i u s R and a n g u l a r v e l o c i t y CJ. For such a column of f l u i d , CJ i s c o n s t a n t throughout the s t r u c t u r e . T h e r e f o r e , i t would be n a t u r a l t o d e f i n e the c o h e r e n t s t r u c t u r e s t o f i t our p r e v i o u s model u s i n g CJ. Moreover, i t s h o u l d be noted t h a t f o r the above f l o w s t r u c t u r e , v o r t i c i t y 1 7 and a n g u l a r v e l o c i t y a r e a c t u a l l y p r o p o r t i o n a l (ft = 2u>) . A n a l y t i c a l l y , i f we c o n s i d e r an eddy i n i t s CM frame, we can d e f i n e u> as (2.2) V = wxR ; where R i s the r a d i a l v e c t o r from the CM. {Note t h a t the parameters a r e d e f i n e d i n the CM frame and thus a r e independent of the i n e r t i a l frame of the o b s e r v e r . } T a k i n g the c u r l of the e q u a t i o n , we have (2.3) R = VxV = Vx(wxR) = 2u> - R(V«CJ) + (R.V)w . For our r i g i d l y r o t a t i n g f l u i d , u> i s c o n s t a n t everywhere so the second and t h i r d term v a n i s h and fl = 2u>. In the g e n e r a l 2D c a s e , ft and u are both a l o n g the z - a x i s which i s p e r p e n d i c u l a r t o V and R, so the second term v a n i s h e s . Our e q u a t i o n can f u r t h e r be s i m p l i f i e d i n t o ft = (2 - ~R>V)u. In the c e n t r a l p a r t of the v o r t e x m o t i o n , R i s s m a l l and we a l s o expect the s p a t i a l r a t e of change of w t o be s m a l l , thus co i s a good r e p r e s e n t a t i o n f o r ft. Moreover, v o r t i c i t y i s d e f i n e d by p a r t i a l s p a t i a l d e r i v a t i v e s but ~£> i s d e f i n e d by the i n s t a n t a n e o u s v e c t o r s V and R; t h i s means t h a t c a l c u l a t i o n of v o r t i c i t y must i n v o l v e f i n i t e d i f f e r e n c e schemes on some i n t e r p o l a t e d f i e l d s ( u n l e s s v o r t i c i t y can be d i r e c t l y measured) which a r e secondary i n t e r p o l a t e d v a l u e s . T h i s i s bound t o be l e s s a c c u r a t e than the p r i m a r y i n t e r p o l a t e d v a l u e s of R and V which can be found d i r e c t l y from the t r a c e r p a t h s . 18 In o t h e r words, t o f i r s t o r d e r a p p r o x i m a t i o n around the c e n t e r of a v o r t e x m o t i o n , we c o u l d use the v a l u e s of co as an i n d i c a t o r f o r 0. T h i s b r i n g s the two s i d e s of the energy and the momentum approach t o g e t h e r and we hope t h a t the coherent s t r u c t u r e s so r e c o g n i z e d can be used w i t h both approaches. 2.4 The D e f i n i t i o n The above summarizes the major reasons why our r e c o g n i t i o n i s based on a n g u l a r v e l o c i t i e s . The a c t u a l d e f i n i t i o n of co h e r e n t s t r u c t u r e s i n t h i s work i s g i v e n by: "A co n n e c t e d , l a r g e - s c a l e f l u i d mass o u t l i n e d by the minimum c l o s e d c o n t o u r of a n g u l a r speed, w i t h i n which t h e r e e x i s t s one and o n l y one l o c a l maximum i n a n g u l a r speed." The use of a n g u l a r speed ( t h e magnitude) r a t h e r than co i t s e l f i n our 2D s i t u a t i o n makes l o c a l minima become l o c a l maxima so as to have u n i f o r m i t y i n the d e f i n i t i o n . A diagram showing the 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 e d u s i n g a s i m i l a r d e f i n i t i o n i n one di m e n s i o n from a s p a t i a l p l o t of co i s shown i n F i g . 2. Four c o h e r e n t s t r u c t u r e s a r e shown and numbered. In v i s u a l i z i n g the f i e l d of a n g u l a r v e l o c i t y , f i r s t c o n s i d e r our c o h e r e n t s t r u c t u r e s as i d e a l i z e d r i g i d l y r o t a t i n g f l u i d w i t h r a d i u s R and c o n s t a n t a n g u l a r v e l o c i t y co. We expect the v e l o c i t y as V(r)=cor up t o the s t r u c t u r e boundary ( f o r 0^r<R). When we i n c l u d e f l u i d i n t e r a c t i o n s , we expect 1 9 the o u t e r l a y e r of the c y l i n d r i c a l column t o slow down due t o f l u i d f r i c t i o n and the h i g h v e l o c i t y g r a d i e n t . T h i s changes the p r o f i l e s of V and co. T h i s i s t aken i n t o a c c o u n t i n our d e f i n i t i o n . Our s t r u c t u r e i s d e f i n e d by s p a t i a l t h r e s h o l d i n g t h e f i e l d of a n g u l a r v e l o c i t y . However, we do not p r e s e t the t h r e s h o l d v a l u e , r a t h e r we l e t the s o f t w a r e s e t i t s own v a l u e depending on the s i t u a t i o n e n c o u n t e r e d . To r e c o g n i z e the s t r u c t u r e s we have two t a s k s : 1 ) l o c a t e the l o c a l peaks and 2 ) f i n d the c o r r e s p o n d i n g t u r n i n g c o n t o u r t h a t c h a r a c t e r i z e s the peak. T h i s r e c o g n i t i o n i s v e r y dependent on s m a l l f l u c t u a t i o n s i n u> v a l u e s . L a t e r , the f a c t t h a t we a r e p r i m a r i l y i n t e r e s t e d i n l a r g e s c a l e s i s used t o smooth out t h e s e i r r e g u l a r i t i e s . The above g i v e s the g e n e r a l p r i n c i p l e s i n v o l v e d i n the c o h e r e n t s t r u c t u r e r e c o g n i t i o n , now we s t a r t t o l o o k a t how the hardware and s o f t w a r e i n the system were d e s i g n e d . 20 F i g u r e 2 - Coherent s t r u c t u r e s o u t l i n e d by our d e f i n i t i o n i n one d i m e n s i o n . 21 I I I . THE EXPERIMENTAL SYSTEM The f l o w a n a l y s i s system was d e v e l o p e d and t e s t e d i n the plasma l a b of UBC P h y s i c s Department. I t i s an e x t e n s i o n of the t r a d i t i o n a l f l o w v i s u a l i z a t i o n method used by my s u p e r v i s o r , B. A h l b o r n and S. Loewen i n t h e i r s t u d y . S i m i l a r e x t e n s i o n s can e a s i l y be implemented f o r o t h e r s t u d i e s which makes the work a g e n e r a l t e c h n i q u e r a t h e r than a s p e c i a l d e s i g n . In t h i s c h a p t e r , the hardware c o n f i g u r a t i o n s and some of the a s s o c i a t e d s o f t w a r e w i l l be b r i e f l y d e s c r i b e d . Changes and v a r i a t i o n s i n the hardware a r e e x p e c t e d , my major concern here i s t o h i g h l i g h t the c o n c e p t s and r e q u i r e m e n t s r a t h e r than the p a r t i c u l a r d e s i g n . A c t u a l l y , the computer hardware used f o r t h i s work i s v e r y p r i m i t i v e by today's s t a n d a r d (due t o a l a c k of s u f f i c i e n t f u n d i n g ) . The f a c t t h a t e n c o u r a g i n g r e s u l t s come out of such a system i s p r o m i s i n g . A diagrammatic showing the v a r i o u s hardware components of the system i s shown i n F i g . 3. 3.1 The Towing Tank F i g . 4 shows a s k e t c h of the t o w i n g tank where the f l o w was r e c o r d e d . The i n t e r n a l d i m e n s i o n s a r e about 1x1x5m. D e t a i l e d d e s c r i p t i o n s can be found i n the r e f e r e n c e by Loewen(1983). D u r i n g the e x p e r i m e n t s , i t i s u s u a l l y f i l l e d up w i t h water t o about 3/4 of i t s h e i g h t . 22 1 video camera 5 microcomputer with 2 video recorder V C R controller and digitizer 3 TV for video system 6 microcomputer display monitor 4 microcomputer keyboard 7 m i n j d i s k s t o r a a e d r i v e s 8 telecommunication link to mainframe computer Figure 3 - Diagrammatic d r a w i n g of the system hardware. 23 F i g u r e 4 - The towing tank. 24 Aluminium f i l i n g s w i t h a t y p i c a l l e n g t h of 0.5mm are then d i s t r i b u t e d o n t o the water s u r f a c e . A u n i f o r m g r i d , a p a i r of c y l i n d e r s or any c o n f i g u r a t i o n of i n t e r e s t i s then towed by a c a r t a l o n g the tank w i t h a p r e s e t v e l o c i t y . The f l o w i s v i s u a l i z e d by r e c o r d i n g p a t h s t r a c e d by the a luminium f i l i n g s . The t y p i c a l t o w i n g v e l o c i t y i s about 20cm/s. The l e n g t h s c a l e of the system i s about 5cm, s e t by the g r i d s e p a r a t i o n or the d i a m e t e r of the moving c y l i n d e r s . T h i s i s much s m a l l e r than the w i d t h of t h e tank so we can c o n s i d e r the c e n t e r of the f l o w as b e i n g f r e e from boundary c o n d i t i o n s o t h e r than s u r f a c e e f f e c t s . The Reynolds number of the g e n e r a t e d f l o w i s about 10". 3.2 R e c o r d i n g U s u a l l y , a v i d e o camera i s mounted d i r e c t l y above the p o s i t i o n of i n t e r e s t w i t h f o u r 150W l i g h t b u l b s p o s i t i o n e d to i l l u m i n a t e the s u r f a c e w i t h o u t g l a r e or d i s p r o p o r t i o n a t e l i g h t i n g . A support s t a n d was b u i l t t o h o l d a v i d e o and/or a p h o t o g r a p h i c camera r i g i d l y anywhere over the t a n k . T h i s g i v e s us a c c e s s t o the f l u i d frame of r e f e r e n c e which i s used most o f t e n i n our s t u d y . The o b j e c t frame can be a c h i e v e d by a t t a c h i n g the camera t o the towing c a r t . However, the m a j o r i t y of s t r u c t u r e s b e h i n d the o b j e c t are not w e l l observed i n t h i s frame. The motion of the camera overpowers the slower l o c a l m otions of the f l u i d and masks the s p a t i a l c o r r e l a t i o n . The b e s t frame f o r s t u d y i s the f l u i d r e f e r e n c e frame s i n c e 2 5 t h e g l o b a l f l u i d v e l o c i t y i s z e r o by d e f i n i t i o n . T h i s c h o i c e h a s n o t h i n g t o do w i t h t h e c o h e r e n t s t r u c t u r e s b e i n g f r a m e d e p e n d e n t a s t h e y a r e d e f i n e d by p a r a m e t e r s o f t h e i r CM f r a m e . I t i s j u s t t h a t t h e y c a n n o t be o b s e r v e d e a s i l y i n t h e m o v i n g f r a m e . The r e c o r d i n g VCR t a p e r u n s a t a r a t e o f 30 f r a m e s p e r s e c o n d a n d t h e t o t a l l e n g t h o f t i m e t a p e d i s a b o u t 10 s e c o n d s . The d i g i t i z e d image r e p r e s e n t s a two d i m e n s i o n a l p r o j e c t i o n o f t h e s u r f a c e f l o w . N o t e t h a t n o t a l l 10 s e c o n d s o f d a t a i s u s e d a s t h e f l o w c o n d i t i o n s g r a d u a l l y d e c a y o u t s i d e t h e o p e r a t i o n a l l i m i t s o f t h e s y s t e m . The d u r a t i o n t h a t c a n be u s e d d e p e n d s on t h e t o w i n g v e l o c i t y a n d t h e r a t e o f v e l o c i t y d e c a y ( e n e r g y d i s s i p a t i o n ) i n t h e f l o w . 3.3 D i g i t i z a t i o n The f l o w images r e c o r d e d on t h e VCR t a p e h a v e t o be t r a n s f o r m e d i n t o d i g i t a l f o r m b e f o r e t h e d a t a c a n be s t o r e d and p r o c e s s e d by t h e c o m p u t e r . T h i s s t e p i s done by a M i c r o -w o r k s DS-65 d i g i t i z e r r e s i d i n g on an APPLE I I m i c r o c o m p u t e r . I t t a k e s i n a n a l o g s i g n a l s f r o m t h e VCR o u t p u t , c o n v e r t s them i n t o d i g i t a l d a t a a n d s t o r e s them i n t h e c o m p u t e r RAM (random a c c e s s memory). A t i t s maximum c a p a c i t y , a d i g i t a l p i c t u r e o f 256x256 w i t h 64 g r e y l e v e l s c a n be o b t a i n e d . However, i t r e q u i r e s 2 5 6 x 2 5 6 x 6 b i t s o r 48K ( b y t e s ) o f c o m p u t e r memory t o s t o r e a l l t h i s i n f o r m a t i o n . T h i s c a n n o t be h a n d l e d by t h e RAM 26 i n t he APPLE model we have, and o n l y two images can be s t o r e d on each of the two 5.5" 128K d i s k e t t e s a t t a c h e d . Moreover, i t i s not a t a l l a d v i s a b l e t o have a development system r u n n i n g on t h i s s c a l e as i t would be v e r y i n e f f i c i e n t f o r program t e s t i n g . The d e c i s i o n was made t o use the d i g i t i z e r program b u i l t i n t o the DS-65 c a r d . The r e s u l t i n g image i s a b i n a r y (2 grey l e v e l s ) a r r a y of 256x192 p i x e l s which can be s t o r e d d i r e c t l y as H i - r e s o l u t i o n p i c t u r e s of the APPLE. D u r i n g d i g i t i z a t i o n the p l a y i n g , p a u s i n g and ad v a n c i n g of the VCR tape i s computer c o n t r o l l e d t h r o u g h an i n t e r f a c e c a r d and a d r i v e r program d e s i g n e d by V. Bareau(1985). T h i s c a r d i s c o n n e c t e d t o the remote c o n t r o l l e r of the VCR and s i m u l a t e s remote p a s s i v e ( r e s i s t i v e ) c o n t r o l s i g n a l s t h r o u g h the command of the microcomputer. The i n i t i a l d r i v e r program was w r i t t e n to o v e r l a y d i g i t i z e d images t o form time-exposed p i c t u r e s . T h i s program i s m o d i f i e d f o r the p r e s e n t system f o r c o n t i n u o u s d i g i t i z a t i o n and s t o r i n g of frames onto the two d i s k e t t e s . . In i n t e r f a c i n g w i t h o t h e r systems, t h i s c o n t r o l l e r c a r d has t o be r e d e s i g n e d but the u n d e r l y i n g p r i n c i p l e s s h o u l d be s i m i l a r . Thus, t h i s a u x i l i a r y d e v i c e s h o u l d not be a problem i n g e n e r a l i z i n g the system. F u r t h e r m o r e , some s p e c i a l i z e d computer v i d e o d i g i t i z i n g systems a l r e a d y have had s i m i l a r c i r c u i t s b u i l t i n t o t h e i r hardware. D e t a i l e d o p e r a t i o n and hardware d e s c r i p t i o n s can be found i n the r e f e r e n c e c i t e d above. 27 The b i n a r y c l i p p i n g i n the d i g i t i z a t i o n i s c o n t r o l l e d by two hardware c o n t r o l knobs on the d i g i t i z e r c a r d which must be tuned c a r e f u l l y t o ensure s a t i s f a c t o r y d i g i t i z a t i o n . Because of v a r i o u s e l e c t r o n i c and d i g i t i z a t i o n n o i s e s , i t i s found t h a t even when the same p i c t u r e i s d i g i t i z e d t w i c e under i d e n t i c a l s e t t i n g s , the r e s u l t s v ary q u i t e a b i t . T h i s randomness demands a t r a d e o f f between how much n o i s e one can remove and the amount of d a t a t h a t can be r e l i a b l y o b t a i n e d . I f the c o n t r o l i s tuned t o c l i p a l l r e c o g n i z a b l e n o i s e , too l i t t l e i n f o r m a t i o n i s r e t a i n e d . The compromise i s t o a d j u s t u n t i l most random n o i s e appears as i s o l a t e d p i x e l s whereas the t r a c e r s appear c o n s i s t e n t l y as a p a t c h of c o n n e c t e d p i x e l s . I s o l a t e d p i x e l s can be f i l t e r e d by s o f t w a r e . Sometimes the same p i c t u r e i s d i g i t i z e d more than once and o v e r l a i d t o improve the image q u a l i t y . B e s i d e s the above n o i s e i n d i g i t i z a t i o n , t h e r e a r e s t i l l some problems i n t h i s phase t h a t a r e q u i t e hardware dependent. F i r s t l y , the pause mode of the VCR i s not always steady enough f o r d i g i t i z a t i o n . J i g g l i n g of the frames f r e q u e n t l y r e s u l t e d i n d u p l i c a t i o n of images i n a d j a c e n t l o c a t i o n s . T h i s e i t h e r i n c r e a s e s the e f f e c t i v e s i z e of the t r a c e r ( i f the d u p l i c a t i o n s a r e s t i l l c o n n e c t e d ) , or r e s u l t s i n p r o d u c t i o n of two d i f f e r e n t e n t i t i e s t h a t make i t v e r y h a r d t o t r a c e the p o s i t i o n s from frame t o frame. S e c o n d l y , a f i c t i t i o u s band was found i n many images. I t s appearance i s q u i t e s y s t e m a t i c 28 and i s s u s p e c t e d t o be the i n t e r f r a m e gap which i s not a l i g n e d p r o p e r l y by t h e VCR. Both problems a r e r e l a t e d t o the frame l o c k i n g system i n the VCR pause mode. T h i s s h o u l d not be s u r p r i s i n g as our VCR was d e s i g n e d f o r g e n e r a l v i e w i n g and not f o r v i s u a l e d i t i n g . A b e t t e r VCR s h o u l d be used f o r f u r t h e r developments. To handle the above problems i n the p r e s e n t system, an o p e r a t o r must s i t i n f r o n t of the t e r m i n a l and l o o k a t each d i g i t i z e d p i c t u r e t o d e c i d e whether or not i t i s a c c e p t a b l e . T h i s manual s e l e c t i o n p r o c e s s u s u a l l y s c r e e n s out about h a l f of t h e frames d i g i t i z e d . I t t a k e s about 1.5-2 hours f o r about 4 seconds of f l o w . W i t h i n t h i s t i m e , the o p e r a t o r must not l e a v e the VCR unattended f o r more than 3-4 minutes a t one t i m e . O t h e r w i s e an i n t e r n a l mechanism i n the VCR r e l e a s e s the pause mode l o s i n g p o s i t i o n i n the image sequence. I t i s f o r t u n a t e t h a t the major problem of j i g g l i n g i s q u i t e s y s t e m a t i c and u s u a l l y appears i n a l t e r n a t e frames. We u s u a l l y d i s c a r d t h e s e frames and thus our tempor a l r e s o l u t i o n i s e f f e c t i v e l y reduced from the i d e a l 30 frames per second to about 15 frames per second. The e f f e c t s of the s p a t i a l and tem p o r a l r e s o l u t i o n s on performance w i l l be g i v e n i n Chapter V I . 29 3.4 Data S t o r a g e and T r a n s f e r • The image p i x e l a r r a y of each frame i s s t o r e d on two d i s k e t t e s a t t a c h e d t o the microcomputer. The t o t a l s i z e of each p i c t u r e i s 8K, the s i z e of a H i - r e s o l u t i o n page, not the 6K of the i n f o r m a t i o n s i z e (256x192 b i t s ) . Only a s m a l l number of the p i x e l s are (ON) and the s i t u a t i o n resembles a s p a r s e m a t r i x i n mathematics. The image can thus be s t o r e d i n a s i m i l a r way by r e c o r d i n g the row and column numbers of the "ON" p i x e l s , i . e . the (X,Y) c o o r d i n a t e s r a t h e r than the p i x e l a r r a y . In our b i n a r y system, we do not need t o s t o r e the v a l u e of the elements s i n c e any s t o r e d p i x e l c o r r e s p o n d s t o an (ON) v a l u e . In e x p e c t i n g a v e r y o p t i m i s t i c number of 200 t r a c e r s per frame, we w i l l have t o s t o r e 400 i n t e g e r s , t a k i n g o n l y 800 b y t e s of memory. T h i s i s a r e d u c t i o n i n s t o r a g e by a f a c t o r of 10. M a n i p u l a t i o n of such l i s t s i s a l s o much e a s i e r and f a s t e r than w i t h the o r i g i n a l p i x e l a r r a y . In the APPLE computer, t h i n g s a r e a l i t t l e more c o m p l i c a t e d . The APPLE has i t s own s p e c i a l s t r u c t u r e i n s t o r i n g the H i - r e s o l u t i o n g r a p h i c s page. We have t o t r a n s f o r m t h i s t o the p i x e l a r r a y b e f o r e we can reduce i t . T h i s p r o c e s s r e q u i r e s a l i t t l e more de c o d i n g but a r e d u c t i o n f a c t o r of 7-10 can s t i l l be e x p e c t e d . These l i s t s of d a t a must be t r a n s f e r r e d t o a more p o w e r f u l computer f o r a n a l y s i s as the APPLE l a c k s both the speed and t h e s t o r a g e r e q u i r e d . Data are t r a n s f e r r e d t o the UBC mainframe Amdahl 500 computer u s i n g the communication 30 program AMIE developed by the U n i v e r s i t y of M i c h i g a n . AMIE i s o b t a i n e d t h r o u g h the UBC Computer Center and i s d e s i g n e d f o r f i l e t r a n s f e r s between the MTS o p e r a t i n g system and APPLE I I computers. D e t a i l e d d e s c r i p t i o n s can be found i n the UBC Computer Center Documentation - UBC AMIE. As a s i m i l a r communication l i n k e x i s t s between most of the common microcomputers and o p e r a t i n g systems, t h i s i s not a r e s t r i c t i o n on the system. T h i s t r a n s f e r i s q u i t e b o r i n g and t e d i o u s i n n a t u r e as an o p e r a t o r has t o s i t and type the t r a n s f e r command f o r each f i l e one by one. The whole t r a n s f e r r i n g p r o c e s s t a k e s about 1 hour per d a t a s e t f o r an average of 70-80 f i l e s . I mproving t h i s p a r t by combining s e v e r a l f i l e s i n t o one has been t r i e d but i s not recommended. F a i l u r e i n t r a n s m i t t i n g p a r t of a l a r g e f i l e i s h a r d e r t o d i s c o v e r and r e q u i r e s more work t o r e c o v e r . Use of i n t e r n a l l y g e n e r a t e d s o u r c e commands or macros on the MTS system i s not a l l o w e d i n the p r e s e n t v e r s i o n of AMIE. There a r e s e r i o u s c o n s i d e r a t i o n s underway t o change the system t o a s o l e microcomputer environment. E v a l u a t i o n s of such p r o p o s a l s can be found i n the r e f e r e n c e s by Devan et a l ( l 9 8 5 ) and Pavan e t a l ( l 9 8 5 ) . My own s u g g e s t i o n s are g i v e n i n Chapter V I I . 31 IV. SYSTEM PRINCIPLES The s o f t w a r e package t o be d i s c u s s e d c o n s i s t s of the f o u r major phases l i s t e d below: 1. n o i s e r e d u c t i o n and t r a c e r image c e n t e r i n g ; 2. s t r e a k t r a c k i n g ; 3. g r i d i n t e r p o l a t i o n and c o h e r e n t s t r u c t u r e r e c o g n i t i o n ; 4. s t r u c t u r e p a r a m e t r i z a t i o n . For each phase, the g e n e r a l o b j e c t i v e s w i l l be s t a t e d f i r s t and the methods used t o a t t a i n these o b j e c t i v e s w i l l then be p r e s e n t e d . As s t a t e d i n the i n t r o d u c t i o n , the e x a c t a l g o r i t h m used i s d e f e r r e d t o the next c h a p t e r t o reduce c o n f u s i o n . N e v e r t h e l e s s , i t has t o be p o i n t e d out a g a i n t h a t the d i f f e r e n c e i s drawn f o r c l a r i f i c a t i o n o n l y . D u r i n g the a c t u a l development or any f u r t h e r e x t e n s i o n of the system, such d i f f e r e n c e s s h o u l d not be overemphasized or one c o u l d end up w i t h some i d e a l i z e d approach which i s i m p o s s i b l e or i m p r a c t i c a l t o implement. 32 4.1 N o i s e R e d u c t i o n and T r a c e r Image C e n t e r i n g Each phase r e p r e s e n t s a t r a n s f o r m a t i o n of i n f o r m a t i o n towards the f i n a l o b j e c t i v e of r e c o g n i z i n g and p a r a m e t r i z i n g the c o h e r e n t s t r u c t u r e s . In the f i r s t phase, we a r e g i v e n the d i g i t i z e d b i n a r y p i x e l images. Each image can be c o n s i d e r e d as a two d i m e n s i o n a l a r r a y i n which ev e r y e l e m e n t , a p i x e l , i s a s s i g n e d e i t h e r an "1" (ON) or a "0" (OFF) by t h e d i g i t i z e r . The c i r c u i t i n s i d e the d i g i t i z e r c a r d compares the i n p u t v i d e o s i g n a l w i t h a hardware chosen t h r e s h o l d . I f the i n t e n s i t y of the s i g n a l i s h i g h e r than the t h r e s h o l d , t h e c o r r e s p o n d i n g p i x e l i s s e t t o "1" (ON), i f not i t i s s e t t o "0" (OFF). In the d i g i t i z e d p i c t u r e , a t r a c e r p a r t i c l e i s i d e n t i f i e d as a group of c o n n e c t e d " d o t s " , p i x e l s t h a t a r e s e t t o " 1 " . 1 2 3 8 4 7 6 5 F i g u r e 5 - Numbered 8-neighbours of a p i x e l . 34 33 F i g u r e 6 - Unprocessed c o n s e c u t i v e d i g i t i z e d images 35 We de'fine c o n n e c t i v i t y r e c u r s i v e l y as f o l l o w s : c o n s i d e r each p i x e l t o have 8 n e i g h b o u r s l a b e l l e d from 1 t o 8 as shown i n F i g . 5 . A dot (shown a t the c e n t e r ) i s co n n e c t e d t o any and a l l d o t s t h a t r e s i d e i n i t s 8 neighbourhood. F u r t h e r m o r e , i f dot A i s conn e c t e d t o B, and B i s connected t o C, then A i s conn e c t e d t o C. T h i s e s t a b l i s h e s the m a t h e m a t i c a l i d e a of c o n n e c t i v i t y i n a d i g i t i z e d p i c t u r e . In t h i s phase, our major o b j e c t i v e i s t o e x t r a c t a c c u r a t e and c o n s i s t e n t i n f o r m a t i o n about each t r a c e r which can h e l p us i n the s t r e a k t r a c k i n g p r o c e s s . Depending on the system, t h i s i n c l u d e s the c o o r d i n a t e s , s i z e , and average i n t e n s i t y ( i f grey l e v e l e x i s t s i n the da t a s e t ) . F i g . 6 shows two d i g i t i z e d images from two c o n s e c u t i v e frames from the microcomputer. The time i n t e r v a l between them i s 1/30 of a second. The f i e l d i s a g r i d t u r b u l e n t f l o w w i t h a t o w i n g v e l o c i t y of 20cm/s, showing an a r e a of 12cmx9cm. In such a s h o r t t i m e , the t r a c e r s s h o u l d not move too f a r . However, we a c t u a l l y see c o n s i d e r a b l e d i f f e r e n c e i n the two images. Note t h a t many i s o l a t e d d o t s i n the second p i c t u r e have no c o u n t e r p a r t s i n the f i r s t one. A l s o , some w e l l d e f i n e d t r a c e r s i n the f i r s t image appear t o be s p l i t i n the second. Even i f they s t i l l appear t o be one s i n g l e e n t i t y , the shape and s i z e may v a r y . Some h o l e s or o t h e r i r r e g u l a r shapes a r e p r e s e n t which a r e u n r e a l i s t i c f o r an i d e a l image of the a c t u a l aluminium t r a c e r s . F i n a l l y , we n o t i c e a v e r y 36 i n t e n s e band of d o t s a c r o s s the second image, i t has no c o u n t e r p a r t i n the f i r s t p i c t u r e . As d e s c r i b e d i n the l a s t c h a p t e r , t h r e e major n o i s e s o u r c e s are c a u s i n g these i r r e g u l a r i t i e s . They i n c l u d e background e l e c t r o n i c impulse n o i s e s , j i g g l i n g of the VCR tape i n the pause mode and the i n a b i l i t y of the VCR t o pause r i g h t a t the frame boundary. The i s o l a t e d d o t s a r e c o n s i d e r e d t o be i m p u l s e n o i s e s ; s p l i t t i n g of t r a c e r s i s caused by the r a p i d v i b r a t i o n of the frame i n the pause mode and the n o i s e band i s due t o the i n t e r f r a m e gap. B e f o r e g e t t i n g any i n f o r m a t i o n from the p i c t u r e , these i r r e g u l a r i t i e s have t o be removed. I t i s most i m p o r t a n t to get a c l e a r image a t t h i s s t a g e . I f we cannot get good p r i m a r y d a t a , p u s h i n g t o o hard on o t h e r a s p e c t s of the system i s a waste of e f f o r t . These problems are now h a n d l e d i n 3 s t e p s . The f i r s t one i s aimed p a r t i c u l a r l y at the i s o l a t e d p i x e l s : remove them a l l from the image. T h i s may seem s i m p l e and s t r a i g h t f o r w a r d , but i t has some i m p o r t a n t i m p l i c a t i o n s . I t r e s t r i c t s the minimum r e s o l v a b l e d i s t a n c e i n the system. As we are d e l e t i n g a l l i s o l a t e d p i x e l s , t o d e t e c t the t r a c e r s c o n s i s t e n t l y they have t o have an average minimum s i z e of 2-3 p i x e l s i n t h e image. T h i s c o n d i t i o n d e t e r m i n e s the s m a l l e s t l e n g t h s c a l e t h a t our system can r e s o l v e . We w i l l a l s o show l a t e r t h a t when compounded w i t h the s p a t i a l r e s o l u t i o n of the image, t h i s d e f i n e s the maximum a r e a of the f l o w t h a t can be 3 7 s t u d i e d . Thus, t h i s s i m p l e p r o c e d u r e p l a c e s a s i g n i f i c a n t c o n s t r a i n t on the a p p l i c a t i o n s of the system. We cannot s t u d y f l o w f i e l d s of a r b i t r a r y s i z e by zooming back the v i d e o camera s i n c e t h i s may reduce the t r a c e r s i z e t o be about one p i x e l which would be e l i m i n a t e d as e l e c t r o n i c n o i s e . F i x i n g the t r a c e r s i z e t o a few p i x e l s and r e q u i r i n g a c e r t a i n t r a c e r d e n s i t y s e t s an upper l i m i t t o the f l o w a r e a t h a t can be s t u d i e d . The second s t e p i s t o smooth the d a t a , i . e . t o add d o t s i n t o the image t o remove i r r e g u l a r i t y and improve c o n n e c t i v i t y . I t i s more c o m p l i c a t e d than the p r e v i o u s one i n t h a t i t a l t e r s c o n n e c t i v i t y . The main p r i n c i p l e s a r e 1 ) t o f i l l u n r e a l i s t i c i r r e g u l a r i t i e s ; 2 ) t o improve t r a c k i n g parameter measurements ( l i k e c e n t e r of mass l o c a t i o n and s i z e ) ; and 3 ) t o add c o n n e c t i v i t y t o p a r t i c l e s "too c l o s e " t o each o t h e r , presumably s p l i t by j i g g l i n g . Because of the i n t r o d u c t i o n of c o n n e c t i v i t y , t h i s p r o c e d u r e has t o be done w i t h c a r e . The a l g o r i t h m must be s t a b l e i n the sense t h a t we do not add unwanted c o n n e c t i v i t y . The smoothing i s done on a l o c a l b a s i s w i t h t h e d e t a i l e d a l g o r i t h m and c o n n e c t i v i t y c o n s i d e r a t i o n s l i s t e d i n the next c h a p t e r . Even w i t h such n o i s e c l e a n i n g mechanisms, the n o i s e l e v e l i n many of the p i c t u r e s i s s t i l l found t o be u n a c c e p t a b l e . A j i g g l i n g frame sometimes produce t r a c e r images 3 - 4 t i m e s l a r g e r than one w i t h o u t j i g g l i n g . The average s i z e of the 38 n o i s e from the i n t e r f r a m e gap a r e 1-2 p i x e l s and are v e r y d e n s e l y packed. To compensate f o r t h e s e , we s t i l l need an o p e r a t o r t o d e c i d e whether or not an image i s too n o i s y t o be a c c e p t e d . T h i s i s the o n l y p a r t of the e n t i r e a n a l y s i s t h a t r e q u i r e s s u b j e c t i v e judgement and c o u l d be e l i m i n a t e d w i t h a b e t t e r d a t a a c q u i s i t i o n system. F i n a l l y , we come back t o our i n i t i a l problem of o b t a i n i n g t r a c e r i n f o r m a t i o n . As our p i c t u r e i s b i n a r y i n n a t u r e , the u s e f u l i n f o r m a t i o n we can a s s o c i a t e w i t h each p a r t i c l e i s i t s s i z e and CM l o c a t i o n . A f t e r i d e n t i f y i n g t h i s , each t r a c e r p a r t i c l e can be thought of as a p o i n t i n the image w i t h a s p e c i f i c s i z e and we can t r a c k i t from frame t o frame a c c o r d i n g l y . The problem of f i n d i n g the c e n t e r of mass (CM) of each t r a c e r i s s i m p l e i n concept but more c o m p l i c a t e d i n a l g o r i t h m . G i v e n the s i z e of the image, 256x192 p i x e l s , and the number of p o i n t s t h a t must be checked f o r c o n n e c t i v i t y , i t i s m o s t l y a problem of s t o r a g e and e f f i c i e n c y . A c t u a l l y , j u s t t h i s r e l a t i v e l y s i m p l e a l g o r i t h m has r e n d e r e d the APPLE computer u s e l e s s . U s i n g i t s h i g h l e v e l language BASIC, i t needs more t h a t 2 hours j u s t t o c e n t e r one s i n g l e p i c t u r e . That i s , more than a week of c o n t i n u o u s work t o p r o c e s s 90 frames (3 seconds) of f l o w . Even i f we expect a f a c t o r of 10-30 i n c r e a s e by u s i n g the machine assembly language, i . e . t r a d i n g o f f the ease of programming w i t h computer tim e , i t i s s t i l l u n a c c e p t a b l e . The whole c l e a n i n g and c e n t e r i n g p r o c e s s 39 i s s a t i s f a c t o r i l y done by the mainframe i n 3-4 seconds of e x e c u t i o n t i m e . The r e s u l t of t h i s p r o c e s s i n g i s a much reduced d a t a set w i t h q u a n t i t a t i v e l y d e f i n e d p a r a m e t e r s . As p o i n t e d out b e f o r e , s u c c e s s i n t h i s s t e p i s most i m p o r t a n t f o r any f u r t h e r n u m e r i c a l i n t e r p o l a t i o n s or q u a n t i t a t i v e a n a l y s i s . The importance of g e t t i n g a c l e a n e d CM p o s i t i o n and the s i z e of each t r a c e r f o r eve r y frame as p r i m a r y d a t a cannot be overemphasized. 4.2 S t r e a k T r a c k i n g A f t e r r e d u c i n g each frame image t o a l i s t of t r a c e r CM p o s i t i o n s and s i z e s , the next s t e p i s t o t r a c k t h e s e p o s i t i o n s from frame t o frame t o produce s t r e a k images l i k e the ones i n a time exposed p i c t u r e . I f t h i s were done m a n u a l l y , the method would be q u i t e s t r a i g h t f o r w a r d : 1 ) s t a r t w i t h the f i r s t frame, l o o k a t each t r a c e r one a t a t i m e ; 2) f i n d i t s p o s i t i o n and s i z e i n the frame; 3) proceed t o the next frame and s e a r c h f o r a t r a c e r c e n t e r near the p r e v i o u s CM p o s i t i o n w i t h a p p r o x i m a t e l y the same s i z e ; 4) connect t h e s e two p o s i t i o n s as p a r t of a s t r e a k ; 5) a f t e r we get two p o i n t s , we can repeat f o r a t h i r d p o i n t by l o o k i n g a t the neighbourhood of the l a s t one i n the t h i r d frame or we can use the v e l o c i t y o b t a i n e d from the p r e v i o u s p o i n t s t o e x t r a p o l a t e a new p r o b a b l e p o s i t i o n and do the s e a r c h from t h e r e ; and 6) the above 40 p r o c e d u r e s can be r e p e a t e d from frame to frame and from p a r t i c l e t o p a r t i c l e u n t i l a l l the t r a c e r s a r e a c c o u n t e d f o r . The fundamental r e q u i r e m e n t t o ensure t h a t such a method would work i s t h a t the average s e p a r a t i o n of t h e t r a c e r s w i t h i n a g i v e n frame must be l a r g e r than the d i s t a n c e t h a t a t r a c e r i s d i s p l a c e d i n one frame i n t e r v a l . As d e t a i l e d i n s e c t i o n ( 6 . 1 ) , t h i s i s b a s i c a l l y a r e l a t i o n between the s e e d i n g d e n s i t y , the s c a l e v e l o c i t y and the time between frames. D e f i n i t e l y , more s e e d i n g i s d e s i r a b l e i n terms of i n f o r m a t i o n d e n s i t y but t h i s means our s c a l e v e l o c i t y must be kept s m a l l . A s m a l l v e l o c i t y w i l l cause a l a r g e f r a c t i o n a l e r r o r i n parameter c a l c u l a t i o n s , r e n d e r i n g f u r t h e r c a l c u l a t i o n u n r e l i a b l e . C o n s i d e r a t i o n s l i k e t h e s e g i v e us an i d e a of what the compromise s h o u l d be. S i m p l e c a l c u l a t i o n s i n c h a p t e r VI show t h a t f o r a 20cm/s towing v e l o c i t y , we c o u l d e x pect t o t r a c k about 100 p a r t i c l e s i n an a r e a of 12x9cm v i e w i n g a r e a . E x act c o n t r o l of s e e d i n g d e n s i t y i s h a r d , i n p r a c t i c e , one i n c r e a s e s t h e s e e d i n g l i t t l e by l i t t l e u n t i l a d e s i r a b l e l e v e l i s a t t a i n e d . The a r t of t h i s s t r e a k c o n n e c t i o n p a r t i s how the p r o c e d u r e s a r e implemented e f f i c i e n t l y and e f f e c t i v e l y . With the chosen s e e d i n g d e n s i t y and t o w i n g v e l o c i t y , the a l g o r i t h m proved t o be q u i t e s a t i s f a c t o r y i n t r a c k i n g most of the s t r e a k s t h a t we can a c t u a l l y see. F i g . 7 shows a p l o t of the c e n t e r e d t r a c e r s superimposed on the t r a c k e d s t r e a k s . F i g u r e 7 - Superimposed p l o t of t r a c e r c e n t e r s and t r a c k e d s t r e a k s . 42 The r e s u l t of t h i s s t e p i s an a r r a y of s t r e a k s , each d e f i n e d as an o r d e r e d l i s t of t r a c e r CM p o s i t i o n s ( X T ? , Y T J ) w i t h s u c c e s s i v e frame numbers T? r e f l e c t i n g the time s t e p s . 4.3 G r i d I n t e r p o l a t i o n And Coherent S t r u c t u r e R e c o g n i t i o n A f t e r s t r e a k s a r e formed by t r a c k i n g the t r a c e r s , we come t o the phase of f i e l d ( g r i d ) i n t e r p o l a t i o n and coh e r e n t s t r u c t u r e r e c o g n i t i o n . As t h i s s e c t i o n i s r e l a t i v e l y l o n g , a b r i e f o v e r v i e w i s g i v e n f i r s t b e f o r e f u l l y e l a b o r a t i n g on the p r i n c i p l e s . To have the a n a l y s i s package f l e x i b l e , we w i s h t o be a b l e t o i n t e r p o l a t e any time of i n t e r e s t , and a l s o i n a s u c c e s s i o n of time s t e p s t o enable e v o l u t i o n s t u d i e s . We do t h i s by t a k i n g a f i x e d number of frames i n s u c c e s s i o n from our o r i g i n a l d a t a t o assemble a s h o r t time exposure p i c t u r e around the time of i n t e r e s t . For each s t r e a k , p o i n t s i n these frames a r e taken out t o form an o r d e r e d s e t ( X T ? , Y T ? , T T ? ) of c o o r d i n a t e s where (X T ? , Y T ? ) a r e the CM l o c a t i o n of the t r a c e r a t time T T ? d e f i n e d by the frame i t i s on. These c o o r d i n a t e s a r e f i t t e d by p o l y n o m i a l s of the form: (4.1a) X T ? = X ( T T ? ) = Z (A; T T ? L ) ; (4.1b) Y T ? = Y ( T T ? ) = Z (B; T T ? L ) ; where the summations a r e of i from z e r o t o some f i n i t e i n t e g e r K l e s s than t h e number of c o o r d i n a t e s . From t h i s approximate m a t h e m a t i c a l d e s c r i p t i o n , we can d i r e c t l y c a l c u l a t e the 43 v a r i o u s p a r a m e t e r s , namely: co, Vx, Vy, Ax, Ay and o t h e r h i g h e r o r d e r time d e r i v a t i v e s of the t r a j e c t o r y , a t any time ( w i t h i n the p e r i o d of i n t e r p o l a t i o n ) . The e x a c t d e t a i l s of g e t t i n g the parameters from the p o l y n o m i a l form of ( 4 . 1 ) i s g i v e n i n Appendix A. For reasons t h a t w i l l be g i v e n soon, a s h o r t time i n t e r v a l Ar i s s e l e c t e d around the time of i n t e r e s t . W i t h i n t h i s , each parameter i s c a l c u l a t e d a t a few p o i n t s i n time and t h e i r v a l u e s a v e r a g e d . The average v a l u e i s put back i n t o each of t h e i r o r i g i n a l p o s i t i o n s as smoothed s c a t t e r d ata p o i n t s . From the s c a t t e r e d v a l u e s of a n g u l a r v e l o c i t y co, an e q u a l l y spaced g r i d i s i n t e r p o l a t e d . T h i s forms a p r i m a r y f i e l d ( g r i d ) of i n t e r p o l a t e d a n g u l a r v e l o c i t y . As d i s c u s s e d i n s e c t i o n ( 4 . 3 . 5 ) , the i n t e r p o l a t e d a n g u l a r v e l o c i t y f i e l d and the p r i m a r y d a t a set undergo a s e r i e s of r e f i n e m e n t s d e s i g n e d t o remove n o n p h y s i c a l i r r e g u l a r i t i e s t o which the s t r u c t u r e r e c o g n i t i o n a l g o r i t h m i s s e n s i t i v e . G r i d s f o r o t h e r v a r i a b l e s : Vx, Vy and V a r e then formed from the r e f i n e d d a t a s e t . S t r u c t u r e p a r a m e t r i z a t i o n can then be done u s i n g the above i n f o r m a t i o n . These p r o c e d u r e s a r e then repeated, f o r d i f f e r e n t times of i n t e r e s t , u s u a l l y s e p a r a t e d by a s o f t w a r e c o n t r o l l e d t i m e - s t e p a p a r t , u n t i l a l l the d i g i t i z e d frames a r e a n a l y s e d . The s u c c e s s i v e l y r e f i n e d d a t a s e t s , the r e f i n e d g r i d and the f i n a l 44 r e c o g n i z e d c o h e r e n t s t r u c t u r e s a r e the r e s u l t s of t h i s phase. 4.3.1 Time Windowing C o n s i d e r the s e t of s t r e a k s formed i n the l a s t phase. L e t F be the t o t a l number of frames a c c e p t e d f o r d i g i t i z a t i o n and T 1 7 be the time of the T?th s u c c e s s f u l l y d i g i t i z e d frame. The time span of the experiment i s g i v e n by TT? w i t h T? r u n n i n g from 1 t o F. I f we denote ( X T ? , Y T ? ) as the CM p o s i t i o n f o r a g i v e n t r a c e r a t frame T?, then each s t r e a k i s a time o r d e r e d s e t of N t r i p l e s ( X T ? , Y T ? , T T ? ) r e p r e s e n t i n g the p a t h of a f l u i d p a r t i c l e around the t r a c e r , where N i s the number of frames i n which the t r a c e r has been s u c c e s s f u l l y i d e n t i f i e d . N i s l e s s than or e q u a l t o F; u s u a l l y i t i s s m a l l e r as we do not expect a s t r e a k t o be t r a c k e d i n each and every a c c e p t e d frame. To u n d e r s t a n d why we s h o u l d d i v i d e the e x p e r i m e n t a l time i n t o i n t e r v a l s f o r parameter i n t e r p o l a t i o n , l e t ' s f i r s t examine some time s c a l e c o n s i d e r a t i o n s . For a c y l i n d r i c a l eddy column w i t h d i a m e t e r e q u a l t o the s c a l e l e n g t h of the system M (5cm), and o u t e r t a n g e n t i a l v e l o c i t y Urn e q u a l t o some f r a c t i o n of the t o w i n g v e l o c i t y Ug (20cm/s), the time of r o t a t i o n i s g i v e n by TrM/Um. In the e x p e r i m e n t s t o be d e s c r i b e d , t h i s i s about 1s. T h i s time s c a l e i s s m a l l compared t o our u s u a l e x p e r i m e n t a l time of 3-4 seconds. Thus, we expect (and a l s o observe i n e x p e r i m e n t s ) t h a t the f l o w v e l o c i t i e s decay c o n s i d e r a b l y w i t h i n the whole e x p e r i m e n t a l 45 p e r i o d . A b e t t e r e s t i m a t e of the i n t e r a c t i o n time s c a l e can be made th r o u g h the r a t e of energy d i s s i p a t i o n or v e l o c i t y decay, see Chapter V I . S i n c e the i n t e r a c t i o n mechanism may a c t u a l l y change over such a time d u r a t i o n , i t i s u n r e a l i s t i c t o expect a s i n g l e f i t t i n g over the whole s t r e a k would g i v e good i n t e r p o l a t i o n r e s u l t s . A b e t t e r approach w i l l be t o model motions f o r d i f f e r e n t s e c t i o n s of a s t r e a k a t d i f f e r e n t t i m e s . Moreover, e r r o r s i n m i s t r a c k i n g of some t r a c e r s w i l l then be l o c a l i z e d i n the s e c t i o n i t i s f i t t e d and not be c a r r i e d t h r o u g h the whole c a l c u l a t i o n . In our system, the f i t t i n g i s done on s e c t i o n of s t r e a k s of time d u r a t i o n about 1 second w i t h i n which we expect the d e r i v a t i v e s t o v a r y more smoothly than when f i t t e d t o the whole s t r e a k . The f l o w f i e l d p arameters can a l s o be f i t more a c c u r a t e l y u s i n g the sub-i n t e r v a l . 4.3.2 Parameter I n t e r p o l a t i o n A f t e r d e f i n i n g the time s u b - i n t e r v a l s of f i t , p o i n t s are e x t r a c t e d from each s t r e a k t o form s e c t i o n s which become the b a s i s of our parameter c a l c u l a t i o n s . To o b t a i n q u a n t i t a t i v e d a t a from t h e s e s e c t i o n s , we can e i t h e r 1 ) c a l c u l a t e the b a s i c p a r a m e t e r s , l i k e v e l o c i t y and a c c e l e r a t i o n , u s i n g the method of f i n i t e d i f f e r e n c e ; or 2) model a m a t h e m a t i c a l d e s c r i p t i o n and then e x t r a c t v a r i o u s parameters a n a l y t i c a l l y . The f i r s t way i s s t r a i g h t f o r w a r d but t h e r e a r e a number of s h o r t c o m i n g s . The two most i m p o r t a n t drawbacks a r e t h a t i t r e q u i r e s q u i t e an 46 a c c u r a t e s e t of i n i t i a l d a t a and i s not s u i t a b l e f o r c a l c u l a t i n g i n s t a n t a n e o u s v a l u e s . T h i s i s e s p e c i a l l y t r u e when we have t o c a l c u l a t e secondary and t e r t i a r y p a rameters l i k e v e l o c i t y and a c c e l e r a t i o n . In our case the CM p o s i t i o n s a r e o n l y d e t e r m i n e d from a v e r y low r e s o l u t i o n image w i t h a c o n s i d e r a b l e amount of n o i s e , we a l s o have a l o t of d i g i t a l and e l e c t r o n i c n o i s e i n the d a t a . We need some k i n d of f i t t i n g t o smooth out t h e s e e r r o r s . On the o t h e r hand, a m a t h e m a t i c a l d e s c r i p t i o n i s p a r t i c u l a r l y u s e f u l i n c a l c u l a t i n g i n s t a n t a n e o u s v a l u e s and d e r i v a t i v e s . Moreover, the c o m p l i c a t e d r o u t i n e s r e q u i r e d t o f i t s t a n d a r d e q u a t i o n s a r e a v a i l a b l e on most mainframe computers. Thus, we chose the m o d e l l i n g method. Without d e t a i l e d knowledge of the f l u i d i n t e r a c t i o n , t h e r e i s s t i l l a problem i n d e c i d i n g which f i t t i n g a l g o r i t h m to use. One c o u l d p l o t i t w i t h a number of models f o r a s i n g l e s e t of d a t a and then judge from the d i s p l a y which model i s b e t t e r . T h i s i s out of q u e s t i o n here as the number of s t r e a k s i s l a r g e and r e p e t i t i v e f i t t i n g w i l l be too c o s t l y i n o p e r a t o r and computer t i m e . Moreover, i t would a l s o be l o c a l l y s u b j e c t i v e . From the two s t a n d a r d methods of p o l y n o m i a l and s p l i n e f i t t i n g , the f i r s t i s chosen m a i n l y because of 1) the ease i n p a r a m e t r i z a t i o n and d e r i v a t i v e c a l c u l a t i o n and 2) the assumption t h a t the c u r v e s a r e smooth enough i n the i n t e r v a l . 47 One o t h e r p o i n t t o note i s the u n c e r t a i n t y i n the i n t e r p o l a t e d r e s u l t s . F u l l e r r o r a n a l y s i s i s always hard i n t h i s k i n d of m o d e l i n g . No such a n a l y s i s i s performed but r a t h e r we i n f e r the u n c e r t a i n t i e s of the r e s u l t s from the s t a n d a r d d e v i a t i o n . However, c a r e must be taken d u r i n g the i n t e r p o l a t i o n t o check f o r u n r e a l i s t i c v a l u e s . T h i s i s t o p r e v e n t c r a s h i n g of the system by unexpected s i n g u l a r i t i e s from the i n t e r p o l a t e d e q u a t i o n s or m i s t r a c k e d s t r e a k d a t a . The time windowing scheme i s a c t u a l l y d e s i g n e d f o r the f i r s t d i f f i c u l t y . In s e l e c t i n g a time s u b - i n t e r v a l ' , we o n l y f i t a s m a l l p o r t i o n of the s t r e a k and most s t a n d a r d f i t t i n g a l g o r i t h m s s h o u l d work i f the v a r i a t i o n i s a c t u a l l y slow. M a t h e m a t i c a l l y , u s i n g the n o t a t i o n s i n the p r e v i o u s s e c t i o n , a s t r e a k s e c t i o n i s d e f i n e d as { ( X T J , Yrj, T T ? ) : T77 l i e s i n the chosen i n t e r v a l } . The c o o r d i n a t e s are then f i t t e d by e q u a t i o n s ( 4 . 1 ) : (4.1a) Xr? = X ( T T ? ) = L (A^ Tr?C ) ; (4.1b) Yr? = Y ( T T J ) = I (B^ T T ? C ) ; w i t h the maximum f i t t i n g o r d e r chosen by the system r o u t i n e . For t he e r r o r problem, we choose a s h o r t i n t e r v a l AT around the time of i n t e r e s t and assume p h y s i c a l c o n d i t i o n s remain c o n s t a n t w i t h i n t h i s i n t e r v a l . Then we s e l e c t a number of p o i n t s i n i t and i n t e r p o l a t e v a r i o u s p arameters. For each s t r e a k , each parameter i s then averaged and put back as d a t a 48 p o i n t s i n i t s o r i g i n a l p o s i t i o n . E f f e c t i v e l y , t h i s i s a s h o r t time average a l o n g the s t r e a m l i n e and i t i s aimed t o smear out the e f f e c t of p o s s i b l e s i n g u l a r i t i e s i n i n t e r p o l a t i o n . Moreover, i t has the a d d i t i o n a l e f f e c t of f o r c i n g a s h o r t s u r f a c e c o n t o u r a l o n g the i n t e r p o l a t e d s t r e a m l i n e . T h i s i s a c t u a l l y assumed i m p l i c i t l y i n coherent s t r u c t u r e s t u d i e s when one o u t l i n e s a s t r u c t u r e by hand. S i n c e our p r i m a r y aim i s t o automate such r e c o g n i t i o n p r o c e s s e s , i t i s p e r f e c t l y l e g i t i m a t e t o do such a v e r a g i n g . In the t e s t r u n s , t h i s p r o ves t o be v e r y i m p o r t a n t i n g e t t i n g good s u r f a c e i n t e r p o l a t i o n s . We now c o n s i d e r the t r a j e c t o r y f i t t i n g models. We have two c h o i c e s : (4.2) V = wxR ; or (4.3) V = Vcm + S5xR ; w i t h Vcm modeling a d r i f t v e l o c i t y of the s t r u c t u r e s . The d e c i s i o n of which one t o use depends on the f l o w s i t u a t i o n . The f i r s t model assume no net motion of the s t r u c t u r e (Vcm = 0) and we have a p a i r of s c a l a r e q u a t i o n s : (4.4a) X = Xc + Rcosfl (4.4b) Y = Yc + R s i n 0 ; where (Xc,Yc) i s the c e n t e r of r o t a t i o n and 8 = (coT+7) . From the p o l y n o m i a l f i t t i n g s , we can get 2 e q u a t i o n s f o r each ord e r of d e r i v a t i v e used, i . e . X ( T ) , Y ( T ) ; X ' ( T ) , Y'(T) and so on, w i t h d e c r e a s i n g a c c u r a c y as the o r d e r i n c r e a s e s . In the f i r s t 49 model, t h e r e a r e 5 unknowns (Xc, Yc, R, co and 7 ) , so we have t o use up t o second o r d e r d e r i v a t i v e s t o s o l v e f o r the p a r a m e t e r s . However, t h i s i s not a p p l i c a b l e i n s i t u a t i o n s where we e xpect t h a t t r a n s l a t i o n a l motion i s comparable t o r o t a t i o n . The second model assumes c o n s t a n t CM v e l o c i t y and we have: (4.5a) X = Xc + V x 0T + Rsini9 (4.5b) Y = Yc + V y 0T - Rcosfl ; w i t h Vcm = ( V x 0 , V y 0 ) . Now we have two more unknowns and we have t o go t o a h i g h e r (3rd) o r d e r d e r i v a t i v e t o c a l c u l a t e 00 which r e n d e r s our v a l u e s l e s s a c c u r a t e . The e x a c t d e r i v a t i o n of the v a r i o u s parameters from the t r a j e c t o r y of b o t h models i s g i v e n i n Appendix A. A l l the v a r i a b l e s can be s o l v e d a n a l y t i c a l l y a t any i n s t a n t of t i m e . A f i n a l p o i n t t o note i s the a v e r a g i n g f o r Vx and Vy, the v e l o c i t y components are d i f f e r e n t from each o t h e r as they r e f l e c t the i n s t a n t a n e o u s d i r e c t i o n of the s t r e a m l i n e . They s h o u l d be s c a l e d r a t h e r than averaged. T h i s i s done by remembering the phase a n g l e 6 r a t h e r than the v e l o c i t y components. However, we have t o c a l c u l a t e Vx and Vy each time we use i t . As an a l t e r n a t i v e , we can f i r s t d e t e r m i n e the average speed V° f o r the s e t of p o i n t s i n t e r p o l a t e d i n A T . Then at each p o i n t , the v e l o c i t y components a r e s c a l e d so as t o g i v e the same s t r e a k speed by u s i n g the f o l l o w i n g t r a n s f o r m a t i o n s : 50 ( 4 .6a ) Vx° = Vx (V/V°) ; (4 .6b ) Vy° = Vy (V/V°) . where Vx°, Vy° denote the f i n a l mean v e l o c i t y components. 4 . 3 . 3 P r e l i m i n a r y G r i d I n t e r p o l a t i o n As our co h e r e n t s t r u c t u r e i s d e f i n e d by a n g u l a r v e l o c i t y , a g r i d of i t i s f i r s t g e n e r a t e d . T h i s i s done by p u t t i n g the averaged d a t a i n t o a two d i m e n s i o n a l p l a n e and u s i n g system r o u t i n e s t o do a s u r f a c e i n t e r p o l a t i o n . These r o u t i n e s t y p i c a l l y g i v e the i n t e r p o l a t e d v a l u e s a t mesh p o i n t s on an e q u a l l y spaced g r i d . In c h o o s i n g the f i t t i n g r o u t i n e s , we have t o determine how much smoothing we want. In our c a s e , a r o u t i n e t h a t does q u i t e a b i t of smoothing i s d e s i r a b l e . T h i s i s because we s t i l l a n t i c i p a t e some e r r o r s of s t r e a k m i s t r a c k i n g t h a t we a r e g o i n g to remove. 4 . 3 . 4 P r e l i m i n a r y Coherent S t r u c t u r e R e c o g n i t i o n The above i n t e r p o l a t i o n g i v e s us a p r e l i m i n a r y a n g u l a r v e l o c i t y f i e l d . R e p e a t i n g the d e f i n i t i o n g i v e n i n Chapter I I , our c o h e r e n t s t r u c t u r e i s d e f i n e d as "A c o n n e c t e d , l a r g e s c a l e f l u i d mass o u t l i n e d by a minimum, c l o s e d c o n t o u r of a n g u l a r speed, w i t h i n which t h e r e e x i s t s one and o n l y one l o c a l maximum i n a n g u l a r speed." 51 • e x t e n d e d r e g i o n s m o n o t o n i c a l l y i n c r e a s i n g n e i g h b o u r h o o d F i g u r e 1D s t r u c t u r e s r e c o g n i s e d from o ( r ) p l o t . 52 When a f i e l d of a n g u l a r v e l o c i t y i s g i v e n , the r e c o g n i t i o n of c o h e r e n t s t r u c t u r e s d e f i n e d above i s b a s i c a l l y the same as l o c a t i n g the l o c a l maxima and then the bounding c o n t o u r s f o r each of them. The d e t a i l e d a l g o r i t h m i s g i v e n i n the next c h a p t e r . The method can be u n d e r s t o o d u s i n g the one-d i m e n s i o n a l a n a l o g y shown i n F i g . 8. To each a n g u l a r v e l o c i t y extremum we a s s o c i a t e two r e g i o n s . The " c o r e " r e g i o n i s the c o h e r e n t s t r u c t u r e as d e f i n e d above. An "extended" r e g i o n around a c o h e r e n t s t r u c t u r e A i s d e f i n e d as the "neighbourhood around A t h a t can be c o n n e c t e d t o i t w i t h p a t h s of m o n o t o n i c a l l y c h a n g i n g a n g u l a r speed." T h i s i s a l s o the same r e g i o n "around A e n c l o s e d by t h e n e i g h b o u r i n g minimum p o i n t s . " However, the extended r e g i o n s of the s t r u c t u r e s a r e not d e f i n e d t o be s p a t i a l l y e x c l u s i v e and a l s o l a c k coherence i n oo. I t i s g i v e n here as a b y - p r o d u c t of the a l g o r i t h m , but as t h e r e i s y e t no g e n e r a l l y a c c e p t e d d e f i n i t i o n f o r c o h e r e n t s t r u c t u r e s , p u t t i n g a l i t t l e e f f o r t i n s t u d y i n g what one has a l r e a d y s h o u l d be w o r t h w h i l e . 4.3.5 Data Refinement The g r i d i n t e r p o l a t i o n and the s t r u c t u r e s r e c o g n i t i o n i n the l a s t p a r t a r e q u i t e s e n s i t i v e t o s m a l l i r r e g u l a r i t i e s of the d a t a s e t . A l t h o u g h we have done q u i t e a b i t of n o i s e r e d u c t i o n and smoothing, one e r r o r s t i l l r emains. A s i n g l e m i s t r a c k e d s t r e a k c o u l d g i v e parameter v a l u e s q u i t e d i f f e r e n t from the s u r r o u n d i n g s . I f not taken c a r e o f , they c o u l d be 53 m i s i n t e r p r e t e d as l o c a l maxima or t u r n i n g p o i n t s . S i n c e our c o h e r e n t s t r u c t u r e r e c o g n i t i o n i s v e r y s e n s i t i v e t o th e s e two f e a t u r e s , we must remove t r a c k i n g e r r o r s from the d a t a s e t . One easy way would t o d i s p l a y the s t r e a k s on a t e r m i n a l and v i s u a l l y e d i t them. However, t h i s i s v e r y c o s t l y and c o n t r a d i c t s the fundamental i d e a of a u t o m a t i o n . I t r e q u i r e s manual judgement of what l o o k s l i k e a m i s t r a c k e d s t r e a k and what d o e s n ' t . S u b j e c t i v e judgements must be a v o i d e d and t h i s i d e a of v i s u a l e d i t i n g of the s t r e a k s i s r e j e c t e d . I n s t e a d , a l e s s e f f e c t i v e i t e r a t i v e a l g o r i t h m i s used t o smooth out i r r e g u l a r i t i e s . We do t h i s by c h e c k i n g the s i z e of the cor e of each s t r u c t u r e . I f the s i z e i s " s m a l l " compared t o a s o f t w a r e p r e s e t t h r e s h o l d , s c a t t e r p o i n t s i n the d a t a s e t t h a t g i v e r i s e t o t h e s e v a l u e s a r e d e l e t e d . The b a s i c s u pport f o r such smoothing i s i m p l i c i t i n our d e f i n i t i o n . The c o h e r e n t s t r u c t u r e s a re " l a r g e s c a l e " compared t o the f l o w s t r e s s l e n g t h s c a l e ( H u s s a i n 1983). S m a l l s c a l e motions a r e p r e s e n t l y not of p r i m a r y i n t e r e s t . I t i s a l s o t r u e t h a t s t r u c t u r e p a r a m e t e r s . c a l c u l a t e d f o r such s m a l l s t r u c t u r e s w i l l have a h i g h e r f r a c t i o n a l u n c e r t a i n t y and c o u l d not be taken too s e r i o u s l y . Removing these i r r e g u l a r i t i e s i s both a c l e a n i n g p r o c e s s of p o s s i b l e e r r o r s i n the d a t a s e t and a l s o an a d d i t i o n a l smoothing of s m a l l s c a l e s t r u c t u r e s . B e s i d e s the above d a t a c l e a n i n g p r o c e d u r e , some 54 a r t i f i c i a l d a t a p o i n t s a r e added i n t o the d a t a s e t . C o n s i d e r a c y l i n d r i c a l column of f l u i d w i t h r a d i u s R r o t a t i n g a t a c o n s t a n t a n g u l a r v e l o c i t y u>. The o u t e r t a n g e n t i a l v e l o c i t y Um i s g i v e n by Rw. The r a d i a l v a r i a t i o n s of a n g u l a r v e l o c i t y co(r) and V ( r ) f o r the i d e a l s t r u c t u r e a l o n g one s p a t i a l d i r e c t i o n a r e p l o t t e d as dash l i n e s i n F i g . 9. When t h i s column of f l u i d i s embedded i n a s t a t i o n a r y f l u i d medium, we expect a d i f f u s i o n of energy outward and the t a n g e n t i a l v e l o c i t i e s w i l l d e c r e ase a c c o r d i n g l y . The o u t e r ( f a s t e r ) r e g i o n i s e x p e c t e d to decay more q u i c k l y than the i n n e r r e g i o n because of the l a r g e r v e l o c i t y (energy) g r a d i e n t . So we expect a new t a n g e n t i a l v e l o c i t y V ( r ) as shown i n the s o l i d l i n e . The c o r r e s p o n d i n g co(r) = V ( r ) / r i s a l s o shown. F i g u r e 9 - I d e a l i z e d and m o d i f i e d V ( r ) and co(r) p l o t s f o r c o h e r e n t s t r u c t u r e s . 56 tracked streak r F i g u r e 10 - T y p i c a l p l o t of s t r e a k s t r a c k e d i n a coherent s t r u c t u r e . 57 key : — . ideal profile without mixing possible profile with mixing •interpolated profile data point for interpolation F i g u r e 11 - S p a t i a l i n t e r p o l a t e d u> and V p l o t from s t r e a k s shown i n F i g . 1 0 . 58 key : ideal profile without mixing possible profile with mixing — * interpolated profile 0 F i g u r e 12 - S p a t i a l i n t e r p o l a t e d CJ and V p r o f i l e from s t r e a k s shown i n F i g . 1 0 w i t h CM p o i n t as p o i n t of z e r o v e l o c i t y . 59 Suppose some s t r e a k s were t r a c k e d i n a s t r u c t u r e as shown i n F i g . 10. The s p a t i a l i n t e r p o l a t e d a n g u l a r v e l o c i t y CJ and v e l o c i t y V p r o f i l e f o r the i d e n t i f i e d s t r u c t u r e would appear as i n F i g . 11. The a n g u l a r v e l o c i t y i n t e r p o l a t i o n i s a c c e p t a b l e but the v e l o c i t y e s t i m a t i o n i s g r o s s l y o v e r e s t i m a t e d . S i n c e our s t r u c t u r e r e c o g n i t i o n i s based on co, i t s h o u l d s t i l l be a c c e p t a b l e . However, c a l c u l a t e d v a l u e s of o t h e r p a r a m e t e r s t h a t depend on V, such as the s t r u c t u r e energy, would be too l a r g e . To compensate f o r t h i s d e f e c t , some a d d i t i o n a l p o i n t s r e p r e s e n t i n g the CM or p o s i t i o n of z e r o v e l o c i t y have t o be i n t r o d u c e d a r t i f i c i a l l y i n t o the s c a t t e r d a t a . As shown i n F i g . 12, the a d d i t i o n of the CM p o i n t as z e r o v e l o c i t y improves the v e l o c i t y i n t e r p o l a t i o n s i g n i f i c a n t l y a l t h o u g h the a n g u l a r v e l o c i t y f i e l d does not change much. The a c t u a l p o i n t s t o be i n c l u d e d depend on the model we used. For the s t a t i o n a r y model of V = C J X R , where we take Vcm t o be z e r o , we j u s t need t o add the CM p o i n t t o the d a t a s e t w i t h a> = peak CJ i n the s t r u c t u r e w i t h Vx, Vy, V a l l s e t t o z e r o . For t h e second model, the CM moves w i t h a d r i f t v e l o c i t y Vcm but we do n o t have the CM v e l o c i t y of the s t r u c t u r e y e t . So we average out a l l the d r i f t v e l o c i t i e s from s t r e a k p o i n t s i n s i d e the c o r e and t a k e the average as Vcm. Moreover, to b r i n g the v e l o c i t i e s down t o z e r o , we f u r t h e r c a l c u l a t e from our s t r e a k s where t o expect a z e r o 60 v e l o c i t y . T h i s i s c a l c u l a t e d by s o l v i n g R' and 7 ' from • the equat i o n s : (4.7a) Vx = 0 = V x 0 + R'wcosr?' (4.7b) Vy = 0 = V y 0 + R'cjsincT ; where 6' = ooT+y 1 . V x 0 , V y 0 and OJ a r e a l l i n t e r p o l a t e d v a l u e s . W i t h the s o l v e d r e s u l t s , the l o c a t i o n of z e r o v e l o c i t y i s g i v e n by ( X ' , Y ' ) : (4.8a) X' = Xc + Vx 0T + R'sintf' = Xc + V x 0 T - Vy 0/c; (4.8b) Y' = Yc + Vy 0T - R'cos0' = Yc + Vy 0T + Vx 0/w u s i n g p r e v i o u s n o t a t i o n s . W i t h i n a s t r u c t u r e , such p o i n t s a re c a l c u l a t e d f o r each s t r e a k and a l l such p o i n t s a r e averaged t o a s i n g l e l o c a t i o n . B e f o r e a d d i n g any d a t a p o i n t f o r an i d e n t i f i e d c o h e r e n t s t r u c t u r e , the f o l l o w i n g p o i n t s a r e checked t o ensure we do not add m e a n i n g l e s s d a t a i n t o the system: 1) The peak i s not at the frame boundary. T h i s u s u a l l y c o r r e s p o n d s t o a peak c r e a t e d by i n s u f f i c i e n t i n f o r m a t i o n near the boundary of the frame; 2) the p o i n t t o be added l i e s i n the r e g i o n a s s o c i a t e d w i t h the s t r u c t u r e i t s e l f ; and 3) the s t r u c t u r e i s not s m a l l , compared t o a s o f t w a r e c o n t r o l t h r e s h o l d . 61 4.3.6 F i n a l G r i d and Coherent S t r u c t u r e s W i t h a l l these c o r r e c t i o n s , the r e f i n e d d a t a becomes the f i n a l b a s i s of our s u r f a c e i n t e r p o l a t i o n f o r the v e l o c i t y d a t a . The g r i d of CJ i s reformed and a r e f i n e d s e t of c o h e r e n t s t r u c t u r e s a r e g e n e r a t e d i n much the same way. From t h i s s e t of c o h e r e n t s t r u c t u r e s , a l l r e m a i n i n g s m a l l peaks are chopped o f f t o f u r t h e r remove the e f f e c t of i r r e g u l a r i t y . The e f f e c t of chopping o f f a s m a l l peak i s shown i n F i g . 13. T h i s i s a c t u a l l y an o p e r a t i o n on the a n g u l a r v e l o c i t y g r i d r a t h e r than on the s c a t t e r d a t a s e t . The f i n a l s e t of i d e n t i f i e d s t r u c t u r e s a r e d e f i n e d on t h i s r e p r o c e s s e d g r i d . F i g . 14 shows a p l o t of the f i n a l r e c o g n i z e d c o h e r e n t s t r u c t u r e s . The symbols "C" and "E" r e p r e s e n t r e s p e c t i v e l y the CM p o s i t i o n s of the c o h e r e n t s t r u c t u r e s . In p r i n c i p l e , t h i s i t e r a t i v e p r o c e s s can be c a r r i e d out a number of times u n t i l s t a b l e c o n d i t i o n s a re o b t a i n e d . However, due t o time and c o s t c o n s t r a i n t s , the above r e g r i d d i n g and data r e f i n e m e n t i s done o n l y once f o r each p i c t u r e . 62 F i g u r e 13 - 1D s t r u c t u r e s i d e n t i f i e d b e f o r e and a f t e r removal of s m a l l peaks. 63 F i g u r e 14 - P l o t of f i n a l r e c o g n i z e d Coherent s t r u c t u r e s 64 4.4 S t r u c t u r e P a r a m e t r i z a t i o n Once the s t r u c t u r e s a r e i d e n t i f i e d , we pr o c e e d t o c a l c u l a t e v a r i o u s p a r a m e t e r s . The a c t u a l parameters of i n t e r e s t a r e experiment dependent. The parameters g i v e n here a r e examples which a r e p e r t i n e n t t o the energy model. A p p l y i n g b a s i c d e f i n i t i o n s on the i n t e r p o l a t e d f i e l d s and assuming u n i f o r m mass per u n i t a r e a , we can c a l c u l a t e many param e t e r s . C o n s i d e r a r e c o g n i z e d s t r u c t u r e d e f i n e d by a co n t o u r t h a t e n c l o s e s N g r i d p o i n t s . L e t p be the mass per u n i t a r e a and a be the a r e a of a g r i d s q u a r e , then we can c a l c u l a t e the f o l l o w i n g : • a r e a : A = Na; • mass: M = Npa; • mean a n g u l a r v e l o c i t y : co0 = I (co^/N); • rms a n g u l a r v e l o c i t y : co, = /{£ ( C O J , 2 / N ) } ; • rms t a n g e n t i a l v e l o c i t y : V, = /{£ ( V L 2 / N ) } ; • rms v e l o c i t y from components: V 2 = v/{Z (Vx^ 2 +Vyc 2 ) /N}; • CM p o s i t i o n : Rem = (Xcm,Ycm) = {I (X^,Y^)}/N; • CM v e l o c i t y : Vcm = {£ ( V x ^ , V y L ) } / N ; • moment of i n e r t i a about CM: I = Z {pa(R^-Rcm) 2}; • t o t a l energy from speed: E t , = I ( p a V ^ 2 / 2 ) ; • t o t a l energy from components: E t 2 = I { p a ( V X i 2 + V y L 2 ) / 2 } ; • t r a n s l a t i o n a l energy: E t r a n = MVcm 2/2; and 65 • r o t a t i o n a l energy: Er = L { p a ( R L - R c m ) 2 c J i 2 / 2 } . Summations a r e a l l f o r index and over a l l the g r i d p o i n t s a s s i g n e d t o the s t r u c t u r e . In a d d i t i o n t o th e s e parameters c a l c u l a t e d from d e f i n i t i o n , the c y l i n d r i c a l a p p r o x i m a t i o n can be used t o c a l c u l a t e some of the parameters i n a d i f f e r e n t way. These parameters i n c l u d e s • r a d i u s : R = \/(A/ir); • moment of i n e r t i a : I' = MR 2/2; • average m o d i f i e d a n g u l a r v e l o c i t y : u' = L { V i / ( IR—Rcm] ) }/N; • r o t a t i o n a l energy: E r ' = I w , 2 / 2 . Whether the parameters agree w i t h our p r e v i o u s r e s u l t s c o u l d t e l l us i f t h e c y l i n d r i c a l a p p r o x i m a t i o n i s good. For the s e p a r a m e t e r s , we can then form d i s t r i b u t i o n f u n c t i o n s and study the f l o w dynamics. In the program, a l l the parameters a r e f i n a l l y t r a n s f o r m e d to a n o n - d i m e n s i o n a l form and s c a l e d w i t h r e s p e c t to the q u a n t i t i e s of an i d e a l eddy. The e x a c t s c a l i n g p arameters a r e g i v e n i n Chapter V I I . T h i s i s done p a r t l y t o adhere t o the u s u a l method of d a t a p r e s e n t a t i o n i n the f i e l d of f l u i d dynamics and a l s o t o e l i m i n a t e the need f o r any s p e c i f i c i n f o r m a t i o n of mass per u n i t a r e a f o r our 2D model. In summary, the f i n a l r e s u l t of the computer package i s the sequence of computer p l o t s of the i d e n t i f i e d s t r u c t u r e 66 o u t l i n e s and the v a r i o u s parameters a s s o c i a t e d w i t h each of them. From the p l o t s , we can p i c k up w e l l r e c o g n i z e d s t r u c t u r e s and study t h e i r e v o l u t i o n i n t i m e . We can a l s o l o o k a t the i n t e r a c t i o n dynamics of the s t r u c t u r e s i n a t i m e -s e r i e s of p l o t s . 67 V. SYSTEM ALGORITHM Having s t a t e d the system p r i n c i p l e s i n the l a s t c h a p t e r , we now l o o k more c l o s e l y i n t o the d e t a i l s of the c o m p u t a t i o n . As t h i s c h a p t e r i s m o s t l y on the a l g o r i t h m s , r e a d e r s i n t e r e s t e d o n l y i n the p r i n c i p l e s of the system may w e l l move on t o the next c h a p t e r . Our programming language i s FORTRAN which i s u n i v e r s a l l y s u p p o r t e d by a l l i n s t a l l a t i o n s so t h e r e i s no problem i n t r a n s p o r t i n g the codes. The g r a p h i c s d i s p l a y p a r t s i n the programs a r e w r i t t e n u s i n g IG r o u t i n e s . T h i s i s an i n s t a l l a t i o n - d e p e n d e n t f e a t u r e which i s not e s s e n t i a l t o the n u m e r i c a l p r o c e s s i n g of the d a t a . S i m i l a r g r a p h i c packages l i k e DISSIPLA and TELLAGRAF a r e w i d e l y used i n o t h e r i n s t a l l a t i o n s . In the f o l l o w i n g , v a r i a b l e s and r o u t i n e s used i n t he s o f t w a r e w i l l be r e f e r r e d by t h e i r names i n c a p i t a l l e t t e r s . 5.1 N o i s e R e d u c t i o n and T r a c e r Image C e n t e r i n g In the f i r s t phase of da t a a c q u i s i t i o n , we have t h r e e i m p o r t a n t i d e a s t o implement: 1) remove i s o l a t e d p i x e l s ; 2) smooth i r r e g u l a r i t i e s g e n e r a t e d by frame j i g g l i n g ; 3) l o c a t e the s i z e and CM of the t r a c e r s . 68 (a) • X • A B X • X ( c ) ( b ) • a • • A X • B X X X X ( d ) F i g u r e 15 - Smoothing c o n s i d e r a t i o n s . 69 There s h o u l d not be any problem i n removing the i s o l a t e d p i x e l s . In the second p a r t , smoothing i s done by s i m p l y c o u n t i n g the number of n e i g h b o u r s f o r each (OFF) p i x e l , i f the number i s e q u a l t o or g r e a t e r than a p r e s e t t h r e s h o l d SMOTHR, the p i x e l i s t u r n e d (ON). I t must be s e t h i g h enough t o pr e v e n t unwanted c o n n e c t i v i t y and low enough i n do m e a n i n g f u l smoothing. To pr e v e n t r e c u r s i v e e f f e c t s , t he neighbour c h e c k i n g s h o u l d o n l y be done on the o r i g i n a l f i l t e r e d image. To e s t i m a t e the v a l u e of the t h r e s h o l d , c o n s i d e r the s i t u a t i o n i n F i g . 15a. I t may appear t h a t we s h o u l d f i l l the c e n t e r w i t h a t h r e s h o l d of 4. However, c o n s i d e r the s i t u a t i o n i n F i g . 15b, the two patches of d o t s a r e i n i t i a l l y u nconnected. I f we add e i t h e r A or B, they s t i l l remain unconnected; but add i n g both p o i n t s w i l l connect them. T h i s i s c o n s i d e r e d as u n s t a b l e . In o t h e r words, our s t a b i l i t y c r i t e r i o n s h o u l d o n l y a l l o w two patch e s t o be connected l o c a l l y by one p o i n t when i t has a t l e a s t SMOTHR n e i g h b o u r s . No c o n n e c t i o n t h a t l o c a l l y r e q u i r e s more than one p o i n t w i l l be a l l o w e d . A t h r e s h o l d v a l u e of 5 i s the minimum. C o n s i d e r the neighbourhood of the a d j a c e n t p a i r s A and B as shown i n F i g . 1 5 c and F i g . 15d. The open squares a r e ne i g h b o u r s of A not common t o B whereas c r o s s e s a re n e i g h b o u r s of B not common to A. In the f i r s t c a s e , o n l y 3 ne i g h b o u r s of A a r e not common t o B and thus any t h r e s h o l d h i g h e r than 3 w i l l ensure 70 s t a b l e smoothing. For the second c a s e , t h e r e a r e 5 non-common n e i g h b o u r s f o r each p o i n t . However, i f a l l t h e s e p o i n t s a r e d o t s (as t o t u r n A and B on s i m u l t a n e o u s l y ) , the two p a t c h e s a r e a l r e a d y c o n n e c t e d . T h e r e f o r e a t h r e s h o l d of 5 i s enough t o ensure s t a b i l i t y and i s d e f a u l t e d t o i n the program. T h r e s h o l d s of 9 or more means no smoothing. S i m i l a r c o n s i d e r a t i o n s a r e e s p e c i a l l y handy t o extend the p r e s e n t system t o t h r e e d i m e n s i o n a l (3D) a n a l y s i s . The l a s t t a s k i s t o f i n d the CM of p o i n t s of a connected p a t c h . The major problem i s t h e n o n - l o c a l i t y of the c o n n e c t i v i t y d e f i n i t i o n . Two p o i n t s can be c o n n e c t e d even i f t hey are q u i t e f a r a p a r t . A simple-minded method i s t o t r a c e around the p e r i m e t e r by g o i n g around the p a t c h i n a f i x e d sense ( c l o c k w i s e or a n t i c l o c k w i s e ) . The c i r c u m s c r i b i n g p i x e l s a r e then t a b u l a t e d and we can go down each v e r t i c a l p a i r and check which are d o t s . The p r i n c i p l e i s s i m p l e but we have t o be c a r e f u l i n d e a l i n g w i t h d i f f e r e n t s i t u a t i o n s . F u r t h e r m o r e , t h i s 2D method becomes u s e l e s s i f we want to e x t e n d i t to 3D as the concept of c i r c u m s c r i b i n g a 3D o b j e c t i s d i f f i c u l t t o a p p r e c i a t e and i t i s hard t o d e f i n e a d e f i n i t e sense i n t r a v e r s a l . A more c o m p l i c a t e d a l g o r i t h m i s used. I t i s based d i r e c t l y on the c o n n e c t i v i t y d e f i n i t i o n and resembles a dynamic t r e e s e a r c h . C o n s i d e r t h a t we s t a r t w i t h any p o i n t A, t o check i t s c onnected p o i n t s we need t o check a l l i t s 8 71 n e i g h b o u r s . I f we number them from 1 to 8, we can s e t a s t a r t i n g d i r e c t i o n ND t o 1, a l a s t d i r e c t i o n LD t o 8 and then s e a r c h from ND t o LD. Whenever we h i t a n o t h e r dot i n the neighbourhood, we p r o c e e d t o check the n e i g h b o u r s of t h a t p o i n t . At t h i s t i m e , o n l y n e i g h b o u r s t h a t a r e not common t o the i n i t i a l p o i n t need t o be checked s i n c e o t h e r s w i l l be or have been taken c a r e of by the i n i t i a l p o i n t . We can d e f i n e a new ND and LD and s t a r t s e a r c h i n g a g a i n . I f i n the c o u r s e of the s e a r c h , a n other dot i s found, c o n t r o l i s t r a n s f e r r e d a g a i n . T h i s forms v a r i o u s l e v e l s of s e a r c h and when we complete one l e v e l , we can pop back t o the p r e v i o u s l e v e l t o c o n t i n u e s e a r c h i n g t o i t s l a s t d i r e c t i o n LD. T h i s f i n i s h e s when a l l the 8 n e i g h b o u r s of the f i r s t p i x e l a r e s e a r c h e d . The most i m p o r t a n t p o i n t i n t r y i n g the a l g o r i t h m i s t o t u r n (OFF) a p o i n t b e f o r e s t a r t i n g t o s e a r c h i t s n e i g h b o u r s , o t h e r w i s e the program w i l l be i n an i n f i n i t e l o o p . The beauty i n t h i s method i s t h a t no s o r t i n g i s i n v o l v e d and whenever the s e a r c h i s f i n i s h e d f o r a p o i n t , a l l i t s a s s o c i a t e d n e i g h b o u r s w i l l have a l r e a d y been found. We j u s t need one pass t o l o c a t e a l l the c o n n e c t e d p i x e l s as c o n t r a s t e d to o t h e r m u l t i p l e pass s h r i n k i n g a l g o r i t h m s . Moreover, e x t e n d i n g t h i s a l g o r i t h m t o the 3D case s h o u l d be t r i v i a l as we o n l y need t o modify the ND and LD i n a s u i t a b l e way. T h i s a l g o r i t h m can e a s i l y be implemented w i t h any language t h a t a l l o w s r e c u r s i o n . FORTRAN does not have t h i s advantage so 72 s t a c k s a r e d e s i g n e d t o s t o r e the v a r i a b l e s i n d i f f e r e n t l e v e l s to ensure c o r r e c t popping back i n the r e v e r s e d i r e c t i o n . 5.2 S t r e a k T r a c k i n g Each t r a c e r i n an image i s now reduced t o i t s s i z e and CM l o c a t i o n . A frame can be viewed as a l i s t of such t r a c e r i n f o r m a t i o n and the whole d a t a s e t as a c o l l e c t i o n of such frames. The t r a c k i n g c l o s e l y f o l l o w s the manual procedure o u t l i n e d i n s e c t i o n ( 4 . 2 ) . F i r s t , t o t r a c k a t r a c e r from frame t o frame, we have t o s e a r c h f o r c e n t e r s around p r o b a b l e l o c a t i o n s i n s u c c e s s i v e frames. I f N i s the average number of d a t a p o i n t s per frame and the t r a c e r s a r e s o r t e d i n (X,Y) o r d e r i n each frame. Then on t h e average we have t o go t h r o u g h h a l f of them 0(N) i n a s e q u e n t i a l s e a r c h , or 0 ( l o g 2 N ) i n a more s o p h i s t i c a t e d b i n a r y s e a r c h t o f i n d the r i g h t p o s i t i o n i n the frame l i s t . We a v o i d t h i s by s o r t i n g the whole d a t a set i n t o a "GRAND" a r r a y a t the b e g i n n i n g . T h i s i s a composite d a t a s t r u c t u r e i n c l u d i n g a number of a r r a y s (X, Y, Frame, SIZE, IndeX, Streak.Number, STreak.LINK) c o n t a i n i n g a l l i n f o r m a t i o n about the d a t a p o i n t s . We have hence both the l i s t of s o r t e d p o s i t i o n s i n each frame and i n the a s s o c i a t e d d ata s e t . Whenever we s t a r t w i t h a p o i n t i n one frame and want t o connect i t t o nearby p o i n t s i n the next frame, we use a l i n k to go from the frame l i s t t o the grand l i s t and s e a r c h around 73 i t s p o s i t i o n t h e r e , c o n c e n t r a t i n g o n l y on p o i n t s w i t h the r i g h t frame number. No a d d i t i o n a l s o r t i n g i s r e q u i r e d . The e l i m i n a t i o n of s o r t i n g m o t i v a t e d the grand s t r u c t u r e . I t has been proven t o be i m p o r t a n t i n s a v i n g s t o r a g e and e x e c u t i o n t i m e . A c t u a l l y , most of the d a t a s t r u c t u r e s a r e j u s t p o i n t e r s t o the grand a r r a y . The o r i g i n a l frame l i s t s a r e l i s t s of p o i n t e r s l i n k e d t o them and we do not need t o waste a d d i t i o n a l space f o r s t o r i n g the c o o r d i n a t e s and s i z e s . S t r e a k s a r e denoted by a l i s t of ends (HEAD and TAIL) which a r e a l s o p o i n t e r s t o the grand a r r a y . The p o i n t s of the same s t r e a k a r e i n t e r n a l l y l i n k e d i n GRAND by STreak.LINK. T h i s i s a double l i n k which a l l o w s us t o move i n e i t h e r d i r e c t i o n a l o n g a s t r e a k . T h i s s t r u c t u r e has a l l the advantages of dynamic s t o r a g e s : easy i n s e r t i o n , merging, u p d a t i n g and d e l e t i o n . The g a i n i n e f f i c i e n c y and f l e x i b i l i t y i s enough t o make the c o m p l i c a t i o n of u s i n g such a l i n k i n g s t r u c t u r e w o r t h w h i l e , e s p e c i a l l y i f we c o n s i d e r f u t u r e growth i n the d a t a s e t s i z e . We now come t o the a c t u a l t r a c k i n g a l g o r i t h m . In our b i n a r y s i t u a t i o n , the parameters t h a t can h e l p us i n t r a c k i n g the p a r t i c l e s are t h e i r s i z e and CM c o o r d i n a t e s . L a r g e r p a r t i c l e s a r e e a s i e r t o t r a c k and so we s e p a r a t e our t r a c k i n g i n t o d i f f e r e n t s i z e t h r e s h o l d s BASLVL. A base p o i n t i s a s t a r t i n g p o i n t of a s t r e a k and i n each p a s s , we s e t BASLVL so t h a t o n l y p o i n t s w i t h s i z e not l e s s than i t can be c o n s i d e r as 74 base p o i n t s . T h i s t h r e s h o l d d e c r e a s e s i n each pass by a STEP u n t i l i t i s lower than a n o t h e r t h r e s h o l d BASMIN. With each s e l e c t e d base t h r e s h o l d , we s t a r t from the l a s t frame and t r a c k backwards. T h i s i s because the p a r t i c l e s a r e moving much slowe r and t r a c k i n g w i l l be e a s i e r . Once we e s t a b l i s h a t r e n d i n the movement, i t w i l l be e a s i e r t o do f a s t e r p a r t i c l e t r a c k i n g . T r a c k i n g i s done on a frame b a s i s r a t h e r than a s t r e a k b a s i s . That i s , we f i n i s h c o n n e c t i n g a l l the s t r e a k s i n one frame b e f o r e we go t o the next one, not f i n i s h i n g a s t r e a k f i r s t and then s t a r t a n o t h e r . For each frame, t r a c k i n g i s done i n two s t e p s : 1 ) EXTEND e x i s t i n g s t r e a k s and 2 ) INCLUDE new base p o i n t s . EXTEND i s the r o u t i n e which adds a new p o i n t t o a s t r e a k . I t f i r s t d e f i n e s a p r o b a b l e p o s i t i o n (X,Y) and a maximum RANGE t o s e a r c h u s i n g the r o u t i n e LOCATE. There a r e t h r e e c h o i c e s of i n t e r p o l a t i o n f o r the p r o b a b l e p o s i t i o n c o n t r o l by the v a l u e of METHOD. I f i t i s 1 , the l a s t p o i n t i n the s t r e a k i s t a k e n as the p r o b a b l e p o i n t . 2 denotes a l i n e a r i n t e r p o l a t i o n and 3 denotes a q u a d r a t i c i n t e r p o l a t i o n . I f t h e r e a r e not enough p o i n t s t o do the i n t e r p o l a t i o n , LOCATE w i l l a d j u s t i t s e l f t o the maximum p o s s i b l e o r d e r . A l l p o i n t s i n the next frame t h a t r e s i d e w i t h i n the c i r c l e c e n t e r e d a t (X,Y) w i t h r a d i u s RANGE a r e e x t r a c t e d . These p o i n t s a r e f u r t h e r f i l t e r e d by a s i z e t h r e s h o l d i n g , i . e . , o n l y p o i n t s w i t h DSZE, s i z e d i f f e r e n c e from the average s i z e of the s t r e a k STSIZE, below a 75 t h r e s h o l d MXSZED are a l l o w e d . A t r a c k i n g f u n c t i o n DISIZE i s c a l c u l a t e d f o r each p o i n t : ( 5 . 1 ) DISIZE = DIST * WEIGHT + DSZE ; where DIST i s the d i s t a n c e from the p r o b a b l e p o i n t and WEIGHT i s a s o f t w a r e w e i g h t i n g i n d i s t a n c e and s i z e d i f f e r e n c e . P o i n t s w i t h DISIZE l a r g e r than another t h r e s h o l d MXDSZE are f u r t h e r chopped o f f and the r e m a i n i n g p o i n t s ( i f any) a r e s o r t e d by DISIZE. The p o i n t w i t h a minimum DISIZE i s conne c t e d t o the s t r e a k of i n t e r e s t . One c o n d i t i o n t o c o n s i d e r i s t h a t the p o i n t may a l r e a d y be c o n n e c t e d t o another s t r e a k . I f t h i s i s the c a s e , then the two s t r e a k s have t o compete f o r the p o i n t by c h e c k i n g the r e s p e c t i v e DISIZE v a l u e . I f the p r e s e n t s t r e a k has an e q u a l or h i g h e r DISIZE v a l u e , the second p o i n t i n the s o r t e d l i s t i s checked f o r i t s c o n n e c t i o n , i . e . i t l o s e s . O t h e r w i s e the p o i n t w i l l be con n e c t e d t o the p r e s e n t s t r e a k and the two s t r e a k s then exchange p o s i t i o n i n the s t r e a k l i s t . S t r e a k c o n n e c t i o n i s r e p e a t e d f o r the " p r e v i o u s " s t r e a k (which i s now f i r s t amongst t h e unconnected s t r e a k s i n the f r a m e ) . Here we a p p r e c i a t e the advantage of the l i n k i n g s t r u c t u r e i n exchanging the p o s i t i o n of the s t r e a k s , a swap can be done s i m p l y by cha n g i n g the s t r e a k HEAD and TAIL p o i n t e r s w i t h o u t s i g n i f i c a n t d a t a movement. A n u l l c o n d i t i o n e x i s t s i f t h e r e i s no p o i n t t o connect i n t h i s frame. In t h i s c a s e , we a l l o w a frame jump. That i s , 76 t r a c k i n g i s c o n t i n u e d f o r s t r e a k s i n the next frame even i f they a re not extended i n t h i s one. However, the maximum number of frame jumps i s c o n t r o l l e d by the t h r e s h o l d MXFRMD t o ensure t h a t our s t r e a k s w i l l not jump t o o f a r i n time w i t h o u t a d a t a p o i n t . When a n u l l c o n d i t i o n i s d e t e c t e d , the l a s t p o i n t i n the s t r e a k i s checked t o see whether i t i s go i n g t o exceed MXFRMD i n the next frame. I f t h i s i s the c a s e , the s t r e a k i s d e a c t i v a t e d , i . e . , i t i s f l a g g e d dormant and w i l l not be extended i n the next frame. I t i s s t i l l s t o r e d i n the s t r e a k a r r a y . In a n a l y z i n g a frame, any new base p o i n t s a r e appended t o the s t r e a k a r r a y by the r o u t i n e APBASE a f t e r a l l ACTIVE s t r e a k s a r e extended. For t h i s , a l l unconnected p o i n t s i n the frame a r e checked f o r s i z e g r e a t e r than BASLVL and thes e a r e added t o the s t r e a k a r r a y as new base p o i n t s . The e x t e n s i o n and appending of new base p o i n t s are r e p e a t e d from the l a s t frame t o the f i r s t one i n the "backward" d i r e c t i o n u n t i l a l l frames are ex h a u s t e d . Then the proced u r e i s re p e a t e d i n the r e v e r s e ("forward") d i r e c t i o n t o extend the s t r e a k s once more. F i n a l l y , we check a l l s t r e a k s ( a c t i v e and dormant) t o ensure t h a t e v e r y f i n a l s t r e a k w i l l have a number of dat a p o i n t s at l e a s t equal t o MINPTS. T h i s ensures t h a t our s t r e a k s have a r e a s o n a b l e amount of p r i m a r y d a t a b e f o r e we e x t r a c t any i n f o r m a t i o n from them. T h i s i s done by the CLEAN r o u t i n e where a l l f i n a l i z e d s t r e a k p o i n t s a r e p r o t e c t e d from 77 f u r t h e r c o m p e t i t i o n and d i s p o s e d s t r e a k p o i n t s r e l e a s e d back f o r use i n the next s i z e l e v e l . A l l the t h r e s h o l d s a re d e f a u l t e d t o a r e a s o n a b l e v a l u e i n the program but they can be changed by the user i n the b e g i n n i n g . The above t r a c k i n g a l g o r i t h m p r o v e d t o be v e r y e f f e c t i v e i n e x t r a c t i n g most of the s t r e a k s t h a t a p p e a l t o us v i s u a l l y . A sample p l o t of t r a c k e d s t r e a k s superimposed on c e n t e r e d t r a c e r s i s g i v e n i n F i g . 7. 5.3 G r i d I n t e r p o l a t i o n and Coherent S t r u c t u r e R e c o g n i t i o n 5.3.1 Time Windowing and Parameter I n t e r p o l a t i o n The time windowing a l g o r i t h m i s s t r a i g h t f o r w a r d . We j u s t d e f i n e a number of v a r i a b l e s f o r the time c o n t r o l s a t the b e g i n n i n g of the program. The f u l l a n a l y s i s time T, i s d e f i n e d by s e t t i n g STFRME ( s t a r t i n g frame number) and ENDFRM (end frame number). They are u s u a l l y the f i r s t and l a s t frame numbers. The l e n g t h of i n t e r p o l a t i o n AT i s d e f i n e d i n TIMDUR and t h i s i n t e r v a l resembles an exposure time of a f l o w p i c t u r e . The time of a v e r a g i n g Ar i s s e t i n DURATN and the number of average p o i n t s i n NPOINT. The a c t u a l time of i n t e r e s t i s d e f i n e d w i t h r e s p e c t t o the i n t e r v a l s t a r t i n g p o i n t and s e t i n TIMEXP, Ar i s c e n t e r e d around t h i s t i m e . In s u c c e s s i v e p i c t u r e s ( p e r i o d s of i n t e r p o l a t i o n ) , STFRME i s 78 incremented i n s t e p s of TIMEIN u n t i l i t i s "greater than ENDFRM. A f t e r d e f i n i n g t h e s e p a r a m e t e r s , we take p o i n t s w i t h time i n the s e l e c t e d i n t e r v a l from each s t r e a k t o form s t r e a k s e c t i o n s . Each of the s e s e c t i o n s i s f i t t e d by p o l y n o m i a l s of e q u a t i o n (4 .1 ) . The f i t s a r e done by a system p o l y n o m i a l f i t t i n g r o u t i n e DOLSF d e s c r i b e d i n the documentation UBC -CURVE. I t p r o v i d e s two o p t i o n s of 1) the user s p e c i f i e s the a c t u a l o r d e r of f i t or 2) the user s p e c i f i e s the maximum f i t t i n g o r d e r K and r o u t i n e chooses the a c t u a l f i t t i n g o r d e r m, w i t h m<K. The sum of squared d e v i a t i o n s i s a l s o a r e s u l t of the r o u t i n e . In our s i t u a t i o n , we make use of both f e a t u r e s . C o n s i d e r a s t r e a k s e c t i o n of N d a t a p o i n t s . We f i r s t a l l o w the r o u t i n e t o f i t up t o a maximum p r e s e t ORDER or N-2, whichever i s s m a l l e r . I f the a c t u a l o r d e r of f i t i s l e s s than the p r e s e t maximum, the sum of squares i s checked t o see i f the c u r v e d e v i a t e s too much from the o r i g i n a l d a t a p o i n t s . The r o o t mean squared d e v i a t i o n per d a t a p o i n t i s compared t o a t h r e s h o l d FITMAX, i f i t exceeds FITMAX, f i t t i n g w i l l be re p e a t e d w i t h the o r d e r i n c r e a s e d by one and the sum of squares r e c h e c k e d . T h i s c o n t i n u e s u n t i l we r e a c h the maximum a l l o w e d f i t t i n g o r d e r . I f we cannot get t h e p o i n t s f i t t e d w i t h i n FITMAX b e f o r e t h i s maximum o r d e r , the d a t a s e t i s r e j e c t e d as c o n t a i n i n g w i l d d a t a p o i n t s . 79 The r e j e c t i o n i s done on the assumption t h a t the s t r e a k s s h o u l d be smooth. W i l d d a t a a r e taken as e r r o n e o u s c o n n e c t i o n s of s t r e a k s . The f i t t e d c o e f f i c i e n t s a l l o w us t o c a l c u l a t e the time d e r i v a t i v e s and o t h e r parameters by d e f i n i t i o n r a t h e r than s o l v i n g f i n i t e d i f f e r e n c e e q u a t i o n s . However, o n l y a m i n i m a l o r d e r of d e r i v a t i v e s s h o u l d be used as they a r e i n c r e a s i n g l y l e s s a c c u r a t e . T h i s i s an i n t r i n s i c u n c e r t a i n t y of the d a t a t h a t we cannot a v o i d w i t h a f i n i t e number of d a t a p o i n t s . U s i n g the f i t t e d e q u a t i o n s , depending on the c h o i c e of f i t t i n g models: (5.2) V = uxR ; or (5.3) V = Vcm + uxR , v a r i o u s parameters can now be e x t r a c t e d from the d a t a . In p a r t i c u l a r , we are i n t e r e s t e d i n V (the t a n g e n t i a l v e l o c i t y ) , Vx, Vy ( t h e v e l o c i t y components), Ax, Ay (the a c c e l e r a t i o n components) and w ( t h e a n g u l a r v e l o c i t y ) . The e x a c t c h o i c e of model i s c o n t r o l l e d by the f l a g MODEL 1. T h i s f l a g must not be changed i n any p l a c e o t h e r than the program i n i t i a l i z a t i o n as parameter meanings would change a c c o r d i n g l y . As mentioned i n the l a s t c h a p t e r , a number of p o i n t s a l o n g the s t r e a k s a r e e x t r a c t e d and t h e i r v a l u e s c a l c u l a t e d . The average v a l u e s are put back as s c a t t e r p o i n t s f o r the next s t e p . 80 5.3.2 G r i d I n t e r p o l a t i o n The s u r f a c e i n t e r p o l a t i o n i s a l s o done by system mainframe s o f t w a r e . There a r e two r o u t i n e s a l r e a d y l i n k e d i n the package f o r t h i s purpose: 1) IQHSCV d e s c r i b e d i n the UBC -IMSL ( I n t e r n a t i o n a l S t a t i s t i c a l and M a t h e m a t i c a l L i b r a r y ) manual and 2) CGRID1 d e s c r i b e d i n UBC - SURFACE. Both t a k e i n s c a t t e r e d d a t a p o i n t s i n a 2D p l a n e as i n p u t and g i v e out an e q u a l l y spaced g r i d w i t h v a l u e s i n t e r p o l a t e d a t the mesh p o i n t s . The IQHSCV r o u t i n e r e q u i r e s a h i g h e r d a t a d e n s i t y and can g i v e s h a r p e r r i s e and f a l l i n the o u t p u t . The CGRID1 r o u t i n e works q u i t e w e l l w i t h the p r e s e n t d a t a d e n s i t y but seems l e s s s e n s i t i v e t o sharp changes. As we are assuming a l l such sharp changes as i n t e r p o l a t i o n or t r a c k i n g e r r o r s , the second r o u t i n e i s used e x c l u s i v e l y i n the p r e s e n t work. 5.3.3 Coherent S t r u c t u r e R e c o g n i t i o n As p o i n t e d out i n c h a p t e r s I I and IV, r e c o g n i t i o n of coherent s t r u c t u r e s i n our case i s the same as l o c a t i n g the l o c a l maxima and then the bounding c o n t o u r s f o r each of the peaks i n the f i e l d of a n g u l a r speed. The f i r s t problem i s not as t r i v i a l as i t may appear. We cannot judge whether or not a p o i n t i s a l o c a l maxima j u s t by l o c a l c o n s i d e r a t i o n s . The problem a r i s e s when we f i n d two or more a d j a c e n t p o i n t s w i t h the same maximum v a l u e . These p o i n t s may be the a c t u a l , maximum or they may be j u s t l o c a l p o i n t s of i n f l e c t i o n . We 81 have t o l o o k f a r t h e r t o t e l l the d i f f e r e n c e . The t a s k i s s i m i l a r t o the problem of f i n d i n g a mountain t o p i n a f o r e s t when one i s l o s t and cannot see t o o f a r . The motto i s "keep on c l i m b i n g up your s t e e p e s t t r a c k , do not descend or t u r n back on a l e v e l t r a c k . " I f one cannot f i n d a n o n - d e c r e a s i n g t r a c k anymore one must be a t the l e v e l of a l o c a l maximum. S t a r t i n g from any p o i n t on the g r i d , t h i s a l g o r i t h m w i l l f i n d a maximum p o i n t . I f a number of people a r e d o i n g t h i s from d i f f e r e n t p o i n t s and l e a v e path numbers a l o n g t h e i r t r a c k e d p o i n t s ( t o ensure they w i l l not go back a l o n g the same p a t h ) , the problem of f i n d i n g the way t o a peak becomes e i t h e r 1) f i n d i n g the peak i t s e l f ; or 2) r u n n i n g i n t o a t r a c k a t l e a s t a t your own e l e v a t i o n , s i n c e one knows f o r s u r e t h a t the l a t t e r w i l l l e a d t o a p r e v i o u s l y t r a c k e d peak. A l l the peaks are found i f t h i s i s done f o r a l l the g r i d p o i n t s . A f t e r the peaks have been c l i m b e d , we a r e i n a good p o s i t i o n t o d e f i n e the extended r e g i o n around the peaks. C o r r e s p o n d i n g t o the a c t u a l i n t e r p o l a t e d g r i d , the p r e v i o u s method used an i d e n t i f i c a t i o n g r i d IGRID t o remember the p a t h numbers. When we r e d e f i n e t h e s e numbers t o the peak number they a r e l e a d i n g t o , we e f f e c t i v e l y d e f i n e a r e g i o n i n which we can go t o the peak w i t h o u t g o i n g down. That i s , a non-d e c r e a s i n g p a t h e x i s t s between any g r i d p o i n t s i n t h i s r e g i o n and i t s peak. T h i s i s our extended r e g i o n as d e f i n e d i n the l a s t c h a p t e r . The boundary of t h i s r e g i o n i s composed of the 82 g r i d p o i n t s f o r which a t l e a s t one of t h e 8-neighbours do not b e l o n g t o the same r e g i o n . The second problem of f i n d i n g a "minimum c l o s e d c o n t o u r around the peak" i s now s t r a i g h t f o r w a r d . We j u s t have t o f i n d the maximum a n g u l a r v e l o c i t y among a l l the v a l u e s of the boundary p o i n t s . A c o n t o u r t h a t i s j u s t h i g h e r than t h i s v a l u e w i l l d e f i n i t e l y e n c l o s e the 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 . Once t h i s v a l u e i s found, the c o r e , namely the c o h e r e n t s t r u c t u r e , i s d e f i n e d as p o i n t s w i t h i n the extended r e g i o n t h a t have v a l u e s h i g h e r than t h i s one. A f t e r a s s i g n i n g t h i s e n c l o s i n g c o n t o u r v a l u e , we can s t a r t c a l c u l a t i n g p r e l i m i n a r y parameters f o r our i d e n t i f i e d s t r u c t u r e . The most i m p o r t a n t one t h a t we w i l l be u s i n g i m m e d i a t e l y i s the s i z e of the s t r u c t u r e . The c o r e and the extended r e g i o n of the same s t r u c t u r e a r e b o t h i d e n t i f i e d on IGRID w i t h the peak number. Area t h a t o n l y b e l o n g s t o the extended r e g i o n i s i d e n t i f i e d w i t h a n e g a t i v e peak number. T h i s completes the o p e r a t i o n a l p r o c e d u r e t o f i n d the c o h e r e n t s t r u c t u r e . The r e l a t i o n between the extended and the c o r e r e g i o n around a peak i s shown i n F i g . 13. 83 5.3.4 Data and S t r u c t u r e Refinement T h i s p r e l i m i n a r i l y d e t e r m i n e d c o h e r e n t s t r u c t u r e has the s i g n i f i c a n t drawback of b e i n g v e r y s e n s i t i v e t o i r r e g u l a r i t i e s i n the s c a t t e r d a t a as shown i n F i g . 13. S i n c e we are p r i m a r i l y i n t e r e s t e d i n " l a r g e s c a l e " s t r u c t u r e s , we may remove these i r r e g u l a r i t i e s by s i z e t h r e s h o l d i n g the s t r u c t u r e w i t h a s o f t w a r e v a r i a b l e NSMALL. T h i s i s done by g o i n g back i n t o the s c a t t e r d a t a and removing every d a t a p o i n t t h a t l i e s w i t h i n the s m a l l s t r u c t u r e s . At the same t i m e , p o i n t s r e p r e s e n t i n g the CM of s t r u c t u r e s and z e r o v e l o c i t i e s a r e added i n c o n s i d e r a t i o n of the reasons g i v e n i n the l a s t c h a p t e r . T h i s completes our d a t a r e f i n e m e n t on the s c a t t e r d a t a s e t . A new g r i d of a n g u l a r v e l o c i t y i s then formed from t h i s r e f i n e d d a t a set and the s t r u c t u r e r e c o g n i t i o n i s r e p e a t e d once more. Any r e m a i n i n g s m a l l s t r u c t u r e s a r e chopped. That i s , f o r such s t r u c t u r e s , a v a l u e of OJ i s c a l c u l a t e d and eve r y p o i n t i n the s m a l l s t r u c t u r e i s r e p l a c e d by such a v a l u e . T h i s v a l u e i s chosen from a p o i n t out of the 8-neighbours of t h e s t r u c t u r e boundary (but not b e l o n g i n g t o the s t r u c t u r e ) w i t h the v a l u e c l o s e s t t o the s t r u c t u r e bounding c o n t o u r v a l u e p r e v i o u s l y found. T h i s ensures e v e r y s t r u c t u r e i n our f i n a l r e s u l t i s " l a r g e " i n comparison w i t h the t h r e s h o l d NSMALL and t a k e s c a r e of most of the i r r e g u l a r i t i e s . T h i s r e f i n e d g r i d of a n g u l a r v e l o c i t y and the a s s o c i 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 g r i d 84 IGRID a r e the f i n a l r e s u l t s i n the s t r u c t u r e r e c o g n i t i o n p r o c e s s . 5.4 S t r u c t u r e P a r a m e t r i z a t i o n G r i d s of the parameters l i k e V, Vx, Vy are then f i t t e d by the i n t e r p o l a t i o n r o u t i n e s u s i n g the r e f i n e d d a t a s e t of the l a s t s e c t i o n . W ith IGRID, the v a r i o u s parameters of i n t e r e s t such as mean a n g u l a r v e l o c i t y , a r e a , mean v e l o c i t y , moment of i n e r t i a and t o t a l , t r a n s l a t i o n a l and r o t a t i o n a l energy are c a l c u l a t e d from t h e i r d e f i n i t i o n . The complete l i s t of parameters c a l c u l a t e d has been g i v e n i n s e c t i o n ( 4 . 4 ) . In p r e s e n t i n g the experiment r e s u l t s , a l l parameters a r e s c a l e d w i t h r e s p e c t t o an i d e a l eddy as d e s c r i b e d i n the l a s t c h a p t e r . The p l o t s of the f i n a l s t r u c t u r e s and the l i s t of parameters d e s c r i b i n g them i s the f i n a l r e s u l t of the package. These can now be used t o v i s u a l i z e and study the e v o l u t i o n and the i n t e r a c t i o n dynamics of the f l o w . 8 5 V I . HARDWARE CONSTRAINTS AND EXPERIMENTAL PARAMETERS In t h i s c h a p t e r , we w i l l l o o k i n t o the system environment and c o n s i d e r v a r i o u s c o n s t r a i n t s on the system performance. For s u c c e s s f u l a p p l i c a t i o n , the e x p e r i m e n t a l s e t u p and system d e s i g n must match. There a r e 3 major p a r t s t o c o n s i d e r : 1) the hardware which we cannot change e a s i l y ; 2) t h e e x p e r i m e n t a l c o n d i t i o n s t h a t must be matched t o the above c o n s t r a i n t s ; and 3 ) the c r i t i c a l s o f t w a r e parameters used t o c o n t r o l the method and a c c u r a c y of a n a l y s i s . S o f t w a r e parameters d i d not a f f e c t the major system and e x p e r i m e n t a l c o n d i t i o n s . They were p r e s e n t e d a l o n g w i t h the system s o f t w a r e i n the l a s t two c h a p t e r s . System parameters and e x p e r i m e n t a l c o n t r o l s a r e i n t i m a t e l y r e l a t e d . We cannot d i s c u s s them s e p a r a t e l y . Thus, we w i l l f i r s t l i s t them b e f o r e g o i n g i n t o the d e t a i l s . 1. Hardware c o n s t r a i n t s : • s p a t i a l r e s o l u t i o n : Nx * Ny; • t e m p o r a l r e s o l u t i o n : 1/At; • number of d i f f e r e n t s i g n a t u r e s of a t r a c e r , i n c l u d i n g s i z e , c o l o r , grey l e v e l , e t c . : S. 2. E x p e r i m e n t a l p a r a m e t e r s : • t o t a l time of e x p e r i m e n t : T; • average t r a c e r s i z e : 5; • v i e w i n g a r e a c o n t r o l l e d by the zoom: Dx * Dy; • s c a l e l e n g t h of system: M; 86 • average r a t e of change of s t r u c t u r e d i a m e t e r : dDm/dt; • towing v e l o c i t y : Ug; • r a t e of change of maximum v e l o c i t y : dUm/dt; • s e e d i n g d e n s i t y : D. A f t e r d e f i n i n g t h e s e p a r a m e t e r s , we w i l l e s t a b l i s h some q u a n t i t a t i v e r e l a t i o n s between them t o show how they are r e l a t e d and what v a l u e s s h o u l d be used f o r each of them. 6.1 Hardware C o n s t r a i n t s and E x p e r i m e n t a l Parameters In u s i n g the p r e s e n t system, one must ensure t h a t the a s p e c t r a t i o of the p i c t u r e (image) and the camera's f i e l d of view a r e the same, i . e . the p i c t u r e i s f r e e of d i s t o r t i o n : (6.1) Nx : Ny = Dx : Dy . T h i s i s assumed i n the s o f t w a r e but changes would be t r i v i a l . We may e i t h e r r e s c a l e the p r i m a r y d a t a from the image f i l e or we can code a s c a l i n g parameter f o r the X or Y d i r e c t i o n s . However, an e q u a l a s p e c t r a t i o a s s u r e s u n i f o r m r e s o l u t i o n i n both d i r e c t i o n s and e n s u r e s t h a t t r a c e r images w i l l have the same s i z e i n e i t h e r d i r e c t i o n . Moreover, our v i s u a l a p p r e c i a t i o n w i l l be d i s t o r t e d i f the p i c t u r e does not have an e q u a l m e t r i c i n both d i r e c t i o n s . To t e s t f o r d i s t o r t i o n , a s q u a r e - g r i d d e d paper i s p l a c e d under the VCR and the d i g i t i z e r " w i d t h " c o n t r o l knob i s a d j u s t e d u n t i l the squares i n the image agree w i t h a square 87 g r i d i n t e r n a l l y g e n e r a t e d by the computer. T h i s p r o c e d u r e t a k e s c a r e of the d i s t o r t i o n problem and a l s o e n a b l e s us t o dete r m i n e the a c t u a l d i m e n s i o n , Dx * Dy of the f i e l d of view. I t i s a l s o u s e f u l f o r f o c u s i n g onto the f l u i d s u r f a c e . I t must be noted t h a t the d i s p l a y p i c t u r e s i n CRTs are u s u a l l y d i s t o r t e d by t h e m s e l v e s . T h i s i s why we must compare the p i c t u r e w i t h a computer g e n e r a t e d g r i d t o ensure a c o r r e c t a s p e c t r a t i o , not r e l y i n g on the p r i n t e r o u t p u t or the s c r e e n d i s p l a y . A good method i s t o o v e r l a y t r a n s p a r e n c i e s of computer g e n e r a t e d g r i d s onto the d i g i t i z e d p i c t u r e of the t e s t g r i d , t h e r e b y e n s u r i n g the c o r r e c t a s p e c t r a t i o and d e t e r m i n i n g the image d i m e n s i o n s . When th e image i s f r e e of d i s t o r t i o n , the minimum r e s o l v a b l e d i s t a n c e i n both X and Y d i r e c t i o n s i s g i v e n by: (6.2) A = Dx/Nx = Dy/Ny . We expect t h a t a n y t h i n g under t h i s d i m e n s i o n cannot be d i g i t i z e d a c c u r a t e l y and c o n s i s t e n t l y . I f such f e a t u r e s a r e d i g i t i z e d , t h ey w i l l appear as i s o l a t e d p i x e l s and w i l l be removed from the image as n o i s e . T h i s i n t u r n i m p l i e s t h a t our t r a c e r s i z e s h o u l d be g r e a t e r than t h i s v a l u e , i . e . (6.3a) 6 > A = Dx/Nx . We may r e l a x the c o n s t r a i n t a l i t t l e s i n c e the o r i e n t a t i o n of the t r a c e r s seldom f i t e x a c t l y under a p i x e l a r e a and u s u a l l y become l a r g e r , so: (6.3b) 5 =* A. 88 S i n c e our t r a c e r s i z e s a r e somewhat f i x e d and so i s the r e s o l u t i o n . We can i n t e r p r e t t h i s as an e q u a t i o n f o r the maximum view s i z e : (6.4a) Dx = NxA «* Nx6 ; (6.4b) Dy = NyA « Ny5 . I mmediately, t h i s d e s t r o y s the p o s s i b i l i t y of d e s i g n i n g a system v e r s a t i l e enough t o s t u d y any g i v e n environment. I f we want t o view a l a r g e r a r e a , we must e i t h e r use b i g g e r t r a c e r s or improve the s p a t i a l r e s o l u t i o n of the image. To i n c r e a s e s p a t i a l r e s o l u t i o n of the image we must buy a new d i g i t i z e r , t h i s i s beyond our p r e s e n t means. We a r e , t h e r e f o r e f o r c e d t o use l a r g e r t r a c e r s . However, one cannot c o n t r o l the s i z e too w e l l and a l s o one cannot use a r b i t r a r i l y l a r g e t r a c e r s s i n c e t h e i r i n e r t i a w i l l i n c r e a s e w i t h the cube of t h e i r s i z e . T h i s w i l l r a i s e the q u e s t i o n of whether the t r a c e r s a r e f o l l o w i n g the f l o w . For the case of a l uminium f i l i n g s , l a r g e r p a r t i c l e s w i l l s i n k r a p i d l y and cannot be used. As a consequence, we have t o f i t the a r e a of i n t e r e s t i n t o a l i m i t e d f i e l d of view. L i m i t i n g the f i e l d of view i s the same as c o n s t r a i n i n g the l e n g t h s c a l e i n the e x p e r i m e n t s . For c o h e r e n t s t r u c t u r e r e c o g n i t i o n , i t w i l l d e f i n i t e l y be i m p o s s i b l e t o study any s t r u c t u r e of l e n g t h s c a l e around 5, but i t i s a l s o a waste of e f f o r t t o study s c a l e s comparable t o the f i e l d s i z e . To have r e l i a b l e s t a t i s t i c a l d a t a , we p r e f e r t o have a t l e a s t a few 89 (3-4) complete s t r u c t u r e s i n each p i c t u r e . Remembering t h a t our i n t e r p o l a t i o n s a r e r e c u r s i v e l y u s i n g the CM of a r e c o g n i z e d s t r u c t u r e , t h o s e t h a t a r e p a r t l y chopped by the b o u n d a r i e s a r e hard t o e s t i m a t e and thus u s e l e s s i n the a n a l y s i s . S p e c i f i c a l l y we want: (6.5a) M >> A — 5 ; (6.5b) D * nM ; where D = min(Dx,Dy) and n i s a s m a l l i n t e g e r (say 2-3). Moreover, we know t h a t the s t r u c t u r e s c a l e s o f t e n i n c r e a s e w i t h t i m e , so the e v o l u t i o n time d u r i n g which we can s t u d y i s a l s o l i m i t e d . The average r a t e of change i n s t r u c t u r e d i a m e t e r , dDm/dt, depends on the e x p e r i m e n t . For a h i g h l y homogeneous f i e l d l i k e the c a s e of g r i d t u r b u l e n c e , t h i s r a t e i s seen t o be s m a l l e r than f o r v o r t i c e s shed b e h i n d b l u f f b o d i e s . When the average c o h e r e n t s t r u c t u r e s i z e approaches the d i m e n s i o n of the f i e l d of view, the a b i l i t y t o i d e n t i f y and s t u d y them w i l l drop s i g n i f i c a n t l y . T h i s l i m i t s the t i m e , T , t h a t can be used f o r a n a l y s i s . P r e l i m i n a r y e x p e r i m e n t s s h o u l d be run t o e s t i m a t e t h i s r a t e and the experiment time i s bounded by: (6.6) T < (D-M)/(dDm/dt) . A l l the above c o n s i d e r a t i o n s show t h a t a l a r g e v i e w i n g a r ea i s v e r y i m p o r t a n t t o the v e r s a t i l i t y and a p p l i c a b i l i t y of the system. Major hardware enhancement s h o u l d be pushed i n t h i s d i r e c t i o n to i n c r e a s e the image r e s o l u t i o n Nx * Ny as 90 much as p o s s i b l e . N e v e r t h e l e s s , ' t h i s w i l l i n e v i t a b l y i n c r e a s e the computing l o a d , e s p e c i a l l y i n the i n i t i a l p a r t of the system and e f f i c i e n t codes become even more i m p o r t a n t . In c o n t r a s t t o the p r e s e n t image r e s o l u t i o n of 256x192, an IBM-PC w i t h r e s o l u t i o n 512x512 i s an improvement by a f a c t o r of 5 i n a r e a . T h i s s h o u l d be a minimum c o n s i d e r a t i o n f o r f u t u r e systems. Another s o l u t i o n i s t o go back t o the p h o t o g r a p h i c t e c h n i q u e s of h i g h speed or s t r o b i n g cameras. D i g i t i z a t i o n of d i f f e r e n t s e c t i o n s of a s i n g l e p i c t u r e can then be done s e p a r a t e l y and the r e s u l t i n g images c o u l d be combined t o form a l a r g e r p i c t u r e . Moreover, the q u a l i t y of a p h o t o g r a p h i c image i s u s u a l l y much b e t t e r than a v i d e o image, so one would have l e s s problems w i t h n o i s e and so be a b l e t o o b t a i n h i g h e r p r e c i s i o n i n the f i t t i n g . These as w e l l as o t h e r s u g g e s t i o n s w i l l be d i s c u s s e d i n the next c h a p t e r . A f t e r c o n s i d e r i n g the v a r i o u s c o n s t r a i n t s a s s o c i a t e d w i t h the s p a t i a l r e s o l u t i o n , we come t o the t e m p o r a l r e s o l u t i o n , 1/At. I t i s the s a m p l i n g r a t e of the VCR. L e t Um(t) be the maximum v e l o c i t y i n the f l u i d . T h i s i s u s u a l l y the o u t e r t a n g e n t i a l v e l o c i t y of the most e n e r g e t i c s t r u c t u r e . C o n s i d e r the c a r t moving a t a v e l o c i t y Ug, the i n i t i a l range of v e l o c i t y we can expect f o r the system i s from 0 t o Um o=Um(t=0), a f r a c t i o n of Ug. The maximum d i s p l a c e m e n t e x p e c t e d i n t h i s s tage i s around Um 0At. F i r s t , t h i s d i s t a n c e must be s m a l l compared t o M o t h e r w i s e we cannot e x t r a c t 9 1 a c c u r a t e i n s t a n t a n e o u s parameters of the s t r u c t u r e s , and a l s o i t must be l a r g e compared t o A or 6 so t h a t we can c a l c u l a t e the v e l o c i t i e s w i t h enough p r e c i s i o n . That i s : ( 6 . 7 ) 6 « Um 0At < UgAt < M. A g a i n , we have t o do some p r e l i m i n a r y e x p e r i m e n t s t o determine the decay r a t e dUm/dt b e f o r e we can f u l l y a p p r e c i a t e t h i s c o n s t r a i n t . However, the v e l o c i t y i n e v i t a b l y decays and our system i s pushed towards the R.H.S. of the e q u a t i o n . T h i s g i v e s us an e q u a t i o n s i m i l a r t o ( 6 . 6 ) a s : ( 6 . 8 ) T < (Um 0 - 6/At) / (dUm/dt) . I n c r e a s i n g t e m p o r a l r e s o l u t i o n ( d e c r e a s i n g A t ) does not do any good. Any i n c r e a s e i n t e m p o r a l r e s o l u t i o n must be accompanied by a t l e a s t the same improvement i n s p a t i a l r e s o l u t i o n . T h i s i s e s p e c i a l l y t r u e f o r any system based on f i n i t e d i f f e r e n c e methods r a t h e r than i n t e r p o l a t i o n f i t t i n g s ; the l a t t e r have g r e a t e r f l e x i b i l i t i e s t o smooth d i g i t i z a t i o n e r r o r s . I f a system w i t h h i g h e r t e m p o r a l r e s o l u t i o n c o u l d be used, one would s e l e c t b e t t e r p i c t u r e s and not t r y t o use each and eve r y frame a v a i l a b l e . Next, we come t o the problem of s e e d i n g d e n s i t y . S u c c e s s f u l t r a c k i n g can be e x p e c t e d o n l y i f the s e e d i n g d e n s i t y i s not too h i g h . To be s p e c i f i c , the average s e p a r a t i o n between the p a r t i c l e s has t o be g r e a t e r than the p r o b a b l e d i s t a n c e t h a t a t r a c e r can move between two c o n s e c u t i v e frames. In o t h e r words, each p a r t i c l e " d e f i n e s " a 92 neighbourhood a r e a of the o r d e r (UmAt) 2 i n t o which no o t h e r p a r t i c l e s h o u l d e n t e r . From t h i s , we can e s t i m a t e the maximum a l l o w a b l e s e e d i n g d e n s i t y D i n the number of t r a c e r s per u n i t a r e a a s : (6.9) D = l/{7r(UmAt) 2} ; where a c i r c u l a r neighbourhood i s assumed. T h i s r e p r e s e n t s the i n f o r m a t i o n d e n s i t y t h a t we can c o m f o r t a b l y e x t r a c t f o r a g i v e n system. I t i s i n t e r e s t i n g t o note t h a t t h i s i s independent of the zoom. T h i s i s because when the s p a t i a l r e s o l u t i o n i s changed by zooming, the range i n which we expect the p a r t i c l e s w i l l move i s a l s o s c a l e d , c a n c e l l i n g the apparent change i n i n f o r m a t i o n d e n s i t y . I t s h o u l d a l s o be noted t h a t D i s p r o p o r t i o n a l t o t h e square of the t e m p o r a l r e s o l u t i o n ( 1 / A t ) . We cannot d e c r e a s e our s a m p l i n g r a t e too much t o s a t i s f y the e q u a t i o n 6 << UmAt and so we come back t o the o l d demand i n i n c r e a s i n g s p a t i a l r e s o l u t i o n ( 1 / 5 ) . E l i m i n a t i n g UmAt from e q u a t i o n s (6.7) and ( 6 . 9 ) , we get an u n c e r t a i n t y r e l a t i o n between the s p a t i a l r e s o l u t i o n (1/5) and the t r a c e r d e n s i t y D t o be used: (6.10) St/UD) « 1 Another e f f e c t i v e method t o i n c r e a s e the i n f o r m a t i o n d e n s i t y i s t o work on the d i g i t i z e r and the t r a c e r s . I f the d i g i t i z e r has c o l o u r or d i f f e r e n t grey l e v e l a b i l i t i e s , our t r a c k i n g f u n c t i o n can be much more e f f e c t i v e . The whole i d e a of u s i n g t r a c e r s w i t h d i f f e r e n t s i g n a t u r e s i s t o h e l p 93 d i f f e r e n t i a t e amongst them. One can expect a t h e o r e t i c a l i n c r e a s e i n maximum i n f o r m a t i o n d e n s i t y by a f a c t o r of S, the number of d i f f e r e n t s i g n a t u r e s t h a t one c o u l d get i n p r i m a r y i d e n t i f i c a t i o n . T h i s can be improved much more e a s i l y than the s p a t i a l r e s o l u t i o n which may e v e n t u a l o v e r l o a d the system w i t h g i g a n t i c image s i z e . The c o m f o r t a b l e number of t r a c e r s N 0 f o r a g i v e n f i e l d of view i s g i v e n by: (6.11a) N 0 < SDxDyD = SDxDy / {7r(UmAt) 2 } ; and the v e l o c i t y e x p e c t e d f o r a g i v e n N 0 i s g i v e n by: (6.11b) Um < i/(SDxDy/7rN 0) / At . The above g i v e s the l i m i t s w i t h i n which the system i s e x p e c t e d t o work p r o p e r l y . W i t h i n these l i m i t s , the c h o i c e s of a c t u a l parameters a r e s t i l l s u b j e c t e d t o v a r i o u s e x p e r i m e n t a l c o n d i t i o n s . We have t o run p r e l i m i n a r y e x p e r i m e n t s t o see t h a t what we are l o o k i n g f o r i s a c t u a l l y seen under the f i e l d of view b e f o r e u s i n g the system t o do any a n a l y s i s or r e c o g n i t i o n . T h i s system i s d e s i g n e d t o i n c r e a s e the speed and remove the s u b j e c t i v i t y i n manual p r o c e s s . At the p r e s e n t s t a g e , i t s h o u l d not be c o n s i d e r e d or used as one c a p a b l e of r e c o g n i z i n g s t r u c t u r e s t h a t are not v i s u a l l y a p p e a l i n g . 94 6.2 Hardware and C o n t r o l Parameters Used A study of t h e s u r f a c e f l o w on g r i d t u r b u l e n c e was used t o t e s t t h e system. T h i s s e c t i o n g i v e s an example of how the hardware c o n s t r a i n t s and e x p e r i m e n t a l c o n d i t i o n s a f f e c t the c h o i c e of p a r a m e t e r s . The s p a t i a l r e s o l u t i o n , Nx * Ny, of the t e s t system i s 256 * 192 as f i x e d by the d i g i t i z e r . The a s p e c t r a t i o i s 4 : 3 . The i d e a l t e m p o r a l r e s o l u t i o n i s 30 frames per seconds which g i v e s At = 1/30 s e c . A f t e r the manual s e l e c t i o n of d i g i t i z e d frames, the a c t u a l t e m p o r a l r e s o l u t i o n i s u s u a l l y lowered t o about 15 frames per second. S i n c e we o n l y have b i n a r y images and not much v a r i a t i o n i n t r a c e r s i z e , the number of d i f f e r e n t s i g n a t u r e s f o r the t r a c e r s , S, i s 1. Under t h e s e hardware c o n s t r a i n t s , t h e e x p e r i m e n t a l parameters a r e chosen. The average aluminium t r a c e r s i z e , 6, was 0.05 cm. From e q u a t i o n ( 6 . 4 ) , the maximum s i z e of the f i e l d of v i e w , Dx * Dy, i s g i v e n by: (6.12a) Dx = Nx8 = 256 * 0.05 = 12.8 cm ; (6.12b) Dy = NyS = 192 * 0.05 = 9.6 cm . T h i s c o r r e s p o n d s t o a s p a t i a l r e s o l u t i o n of 20 p i x e l s / c m . The a c t u a l r e s o l u t i o n used i s 22.6 p i x e l s / c m w i t h the c o r r e s p o n d i n g f i e l d s i z e of 11.3cm * 8.5 cm. The a s p e c t r a t i o was m a i n t a i n e d . From e q u a t i o n ( 6 . 5 ) , the l e n g t h s c a l e of t h e system i s t o 95 be chosen w i t h i n the l i m i t s , D = min(Dx,Dy) 9cm, and 5 = 0.05cm. The s c a l e l e n g t h i s the s e p a r a t i o n of the g r i d and i s chosen t o be 5.08 cm. The diameter of t h e g r i d i s 1.26 cm. T h i s i s the same g r i d used by A h l b o r n and Loewen i n t h e i r p r e v i o u s work. I f a new g r i d i s t o be b u i l t , a s m a l l g r i d s e p a r a t i o n of about 3 cm w i t h s i m i l a r s p a c i n g t o d i a m e t e r r a t i o (4 : 1) i s recommended as i t would a l l o w a l a r g e r number of s t r u c t u r e s t o be viewed i n the same a r e a . In s e t t i n g the towing speed, Ug, we c o n s i d e r e q u a t i o n ( 6 . 7 ) . W i t h a t e m p o r a l r e s o l u t i o n of 15 frames per second, we have (6.13a) 6 « Um 0At < UgAt < M ; or (6.13b) 0.75 « Ug < 75 (cm/s) w i t h chosen v a l u e s s u b s t i t u t e d . The speeds of 10, 15, 20, 30 and 40 cm/s were t r i e d . By l o o k i n g a t the VCR r e c o r d i n g s , w e l l d e f i n e d s t r u c t u r e s i n the v i e w i n g area were seen f o r the f i r s t 3 speeds. I t was a l s o found t h a t s t r u c t u r e s c r e a t e d w i t h a h i g h e r towing speed had more i n i t i a l energy and took l o n g e r t o decay. T h i s was more d e s i r a b l e as we want t o study t h e i r e v o l u t i o n as l o n g as p o s s i b l e . Most e x p e r i m e n t s were run w i t h a to w i n g speed of 20 cm/s and a few samples w i t h 15 cm/s t o w i n g speed were a l s o t a k e n . Each run was r e c o r d e d on VCR tape f o r about 10 seconds. W i t h i n t h i s t i m e , most c o h e r e n t s t r u c t u r e s had e v o l v e d o u t s i d e the l i m i t s of the system and a r e not s u i t a b l e f o r a n a l y s i s . 96 T h i s was a l s o the time at which s u r f a c e waves r e f l e c t e d from the b o u n d a r i e s of the tank were seen t o be a f f e c t i n g the s u r f a c e m o t i o n . The run time t h a t c o u l d be s t u d i e d by the system was governed by (6.6) and ( 6 . 8 ) . The r a t e of change of s t r u c t u r e d i a m e t e r , dDm/dt, i s not measured i n the p r e l i m i n a r y e x p e r i m e n t s . T h i s parameter was d e t e r m i n e d as around 0.8 cm/s from t i m e - e x p o s u r e p i c t u r e s t a k e n by Loewen 6 under s i m i l a r c o n d i t i o n s . P u t t i n g t h i s i n t o ( 6 . 6 ) , we had: (6.14) T < (D-M) / (dDm/dt) = 5 s. The decay r a t e of the v e l o c i t y was found t o be about 2 cm/s 2. The maximum i n i t i a l v e l o c i t y , Um0, was d e t e r m i n e d t o be around 30% of the t o w i n g speed. For a towing speed of 20 cm/s, t h i s i s about 6 cm/s. P u t t i n g t h i s and o t h e r q u a n t i t i e s i n ( 6 . 8 ) , the upper bound of T was found a s : (6.15) T < (Um 0-5/At) / (dUm/dt) = 2.68 s. The time of f l o w d i g i t i z e d and t r a n s f e r r e d t o the mainframe computer i s about 4 seconds f o r each r u n . In the f i r s t 2 seconds the f l o w speed d e c r e a s e d so much t h a t i t was hard t o e x t r a c t the a n g u l a r v e l o c i t y a c c u r a t e l y . T h i s r e s u l t e d i n a much l e s s e f f e c t i v e r e c o g n i t i o n by the package. F i n a l l y , we c o n s i d e r the s e e d i n g d e n s i t y , D. A p p l y i n g (6.9) and s e t t i n g Um0 = 6cm/s, the maximum s e e d i n g d e n s i t y was: (6.16) D = 1 /{7r (6/1 5 ) 2 } = 2 p a r t i c l e s per cm 2. The number of t r a c e r p a r t i c l e s w i t h i n the f i e l d of view i s 97 g i v e n by (6.11a) a s : (6.17) N 0 < SDxDyD = 218 p a r t i c l e s . T h i s i s a the upper bound of s e e d i n g and we assumed the t r a c e r s t o be e v e n l y d i s t r i b u t e d . The a c t u a l t r a c e r d e n s i t y used i s about h a l f of t h i s maximum v a l u e . The number of t r a c e r s i n the f i e l d of view i s about 100. With t h i s s e e d i n g d e n s i t y , the program p r o v e d t o be v e r y s u c c e s s f u l i n the t r a c k i n g of the t r a c e r p a t h s . 98 *' V I I . RESULTS AND DISCUSSIONS 7.1 E x p e r i m e n t a l R e s u l t s on I n i t i a l G r i d T u r b u l e n c e The system was used t o study e x p e r i m e n t s performed i n the to w i n g tank shown i n F i g . 4. T u r b u l e n c e was g e n e r a t e d by towing a g r i d a t a p r e s e t speed. The mesh s e p a r a t i o n M i s 5.08 cm and the g r i d d i a m e t e r d i s 1.26 cm. Most e x p e r i m e n t s were c a r r i e d out w i t h a tow i n g speed of 20 cm/s w i t h a few a t 15 cm/s. I n the e x p e r i m e n t s , we l o o k e d at a v i e w i n g a r e a of 11.3x8.5cm (2.3x1.7M). On the a v e r a g e , we c o u l d f i n d 3-4 complete s t r u c t u r e s i n each frame. A n a l y s i s was f o c u s e d on the 20 cm/s runs and u n l e s s o t h e r w i s e s p e c i f i e d , the r e s u l t s quoted i n t h i s c h a p t e r r e f e r t o them. For each r u n , about 3-4 seconds of flow was d i g i t i z e d and a n a l y z e d . T h i s c o r r e s p o n d e d t o a d i s t a n c e of 12-16 mesh w i d t h s i n t h e t r a d i t i o n a l f l o w v i s u a l i z a t i o n p i c t u r e s . In g r i d t u r b u l e n c e c l a s s i f i c a t i o n , t h i s i s the i n i t i a l p e r i o d d u r i n g which "the energy decay of the system i s d e t e r m i n e d by the energy c o n t a i n i n g e d d i e s . " (Hinze 1959) 99 7.1.1 Coherent S t r u c t u r e s a t P r o d u c t i o n With the p r e s e n t system, we a r e a b l e t o l o o k a t the v e r y e a r l y p r o d u c t i o n regime of g r i d t u r b u l e n c e t h a t i s u s u a l l y i n a c c e s s i b l e by p h o t o g r a p h i c f l o w v i s u a l i z a t i o n methods. T y p i c a l l y , the system was a b l e t o r e c o g n i z e c o h e r e n t s t r u c t u r e s s t a r t i n g a t around t = (6/30) sec a f t e r the g r i d p assed o u t s i d e the f i e l d of view. T h i s c o r r e s p o n d s t o 0.8M from the towing g r i d . The image d i m e n s i o n i n the d i r e c t i o n of the moving c a r t i s 11.33 cm (2.27M). E f f e c t i v e l y , t h i s g i v e s us a r e s o l v i n g power of 2.27M i n w i d t h s t a r t i n g a t x = 0.8M from the moving g r i d . As w i l l be d i s c u s s e d i n the next s e c t i o n , the average i n i t i a l d i a m e t e r of the s t r u c t u r e s D was found t o be 0.8±0.1M f o r most i n i t i a l l y r e c o g n i z e d s t r u c t u r e s . T h e r e f o r e , we can s t a r t s t u d y i n g them r i g h t a t t h e i r p r o d u c t i o n . 100 F i g u r e 16 - Coherent s t r u c t u r e s at p r o d u c t i o n . 101 S t r u c t u r e P r o d u c t i o n Rate From p l o t s of r e c o g n i z e d s t r u c t u r e s , the s i m p l e s t parameter t h a t we can c a l c u l a t e i s the i n i t i a l s t r u c t u r e p r o d u c t i o n r a t e . C o n s i d e r the s i m p l i f i e d p l o t drawn i n F i g . 16, the number of s t r u c t u r e s produced per u n i t time i s g i v e n by Ug/X. I f we c o n s i d e r a Von Rarman type of v o r t e x p r o d u c t i o n , the v o r t e x shedding r a t e i s f = (Ug/2X). I t i s w e l l e s t a b l i s h e d 2 t h a t the n o n - d i m e n s i o n a l i z e d p r o d u c t i o n r a t e , or the S t r o u h a l number S = (fd/Ug) i s 0.2 f o r c y l i n d e r s w i t h Reynolds number between 10 2 t o 1 0 5 . From the p l o t s , t h i s v a l u e was found t o be i n good agreement as (7.1) S = 0.20 ± 0.01 . Another v o r t e x shedding model was r e c e n t l y proposed by Loewen, A h l b o r n and F i l u k ( l 9 8 6 ) i n which they c a l c u l a t e d the i n i t i a l s t r u c t u r e c o n c e n t r a t i o n C a s : (7.2) C = T T R 2 / M(2R) = 7rR / 2M . Under i d e n t i c a l e x p e r i m e n t a l c o n d i t i o n s , C was found t o be 0.59 a t 6M from the towing g r i d . To r e l a t e t h i s w i t h the S t r o u h a l number, l e t a = (M/d), the mesh t o d iameter r a t i o . From geometry, 'we have (7.3) X = 2Rcos0 , assuming the s t r u c t u r e s touch each o t h e r . E l i m i n a t e a l l 2 see f o r i n s t a n c e , R o b e r t s o n J.A. and Crowe C.T. 1975, E n g i n e e r i n g F l u i d M e chanics, Houghton M i f f l i n . 1 02 l e n g t h parameters and e x p r e s s S i n term o f C and <t>, we get (7.4) S = 7r / (8Cacostf>) . E q u i v a l e n t l y , upon f i n d i n g S, C and a, we can e x p r e s s t h i s as an e q u a t i o n f o r co s 0 : (7.5) cos0 = Tr / (8SCa) . T a k i n g S = 0.2, C = 0.59 and a = 4, we f i n d cos<£ = 0.83 or <j> = 34°. The same parameter was measured from the p l o t s and we found tantf> = 0.64 ± 0.08 or <j> = 33 ± 4°. The r e s u l t i s a l s o i n good agreement w i t h the proposed model. T h e r e f o r e , j u d g i n g from t h e s e r e s u l t s , we cannot t e l l whether the i n i t i a l p r o d u c t i o n i s a Von Karman type or as proposed. To t e s t t h i s , we must e i t h e r l o o k at the p r o d u c t i o n on slow v i d e o p l a y b a c k or check the c o n d i t i o n s w i t h a d i f f e r e n t a. The VCR v i e w i n g l e d us t o b e l i e v e the proposed model. N e v e r t h e l e s s , t h e r e must be a t r a n s i t i o n between the two mechanisms even i f the proposed model a c c u r a t e l y d e s c r i b e s the p r e s e n t g r i d system. T h i s i s because when a approaches i n f i n i t y , we r e t u r n t o the s i n g l e bar s i t u a t i o n and the Von Karman mechanism s h o u l d p r e v a i l . To g a i n a deeper u n d e r s t a n d i n g , the model s h o u l d be t e s t e d w i t h d i f f e r e n t v a l u e s of a t o see the r e l a t i o n of the s t r u c t u r e p r o d u c t i o n mechanism and the geometry. 1 03 S t r u c t u r e S i z e and E n e r g e t i c s A l l s t r u c t u r e parameters a r e n o r m a l i z e d t o non-d i m e n s i o n a l forms. T h i s i s done by u s i n g the i d e a l s t r u c t u r e which i s a r i g i d l y c i r c u l a r r o t a t i n g eddy w i t h o u t e r t a n g e n t i a l speed Ug e q u a l t o the t o w i n g speed and d i a m e t e r D e q u a l t o the mesh w i d t h . No t r a n s l a t i o n i s a s s o c i a t e d t o t h i s s t r u c t u r e . W i t h the e x c e p t i o n of CM v e l o c i t i e s and the t r a n s l a t i o n a l energy, every s t r u c t u r e parameter i s compared w i t h i t s c o u n t e r p a r t i n the i d e a l s t r u c t u r e . The CM v e l o c i t i e s and the t r a n s l a t i o n a l energy a r e compared r e s p e c t i v e l y w i t h the towi n g speed and t o t a l energy of the i d e a l s t r u c t u r e . Amongst a l l the s t r u c t u r e parameters d i r e c t l y c a l c u l a t e d , the major u n c e r t a i n t y i s the s i z e of the s t r u c t u r e . T h i s parameter i s h i g h l y dependent on the a c t u a l t r a c e r d e n s i t y around a s t r u c t u r e . For p l a c e s where t h i s d e n s i t y i s h i g h , the boundary can be d e f i n e d c l e a r l y . However, f o r p l a c e s where t h i s d e n s i t y i s low, the boundary d e f i n e d i s l e s s p r e d i c t a b l e and may be q u i t e d i f f e r e n t from what would have been drawn by hand. T h i s i s not j u s t a problem of the package as the f l o w f i e l d a t such p l a c e s i s d o u b t f u l . The manual r e c o g n i t i o n p r o c e s s of assuming the l o n g e s t v i s i b l e s t r e a k as the boundary a l s o cannot be c o m p l e t e l y j u s t i f i e d i n such s i t u a t i o n s . However, i n u s i n g p h o t o g r a p h i c t e c h n i q u e s , much 1 0 4 h i g h e r t r a c e r d e n s i t i e s c o u l d be used and so the problem i s not a p p a r e n t . In the new system, we cannot use a v e r y h i g h t r a c e r c o n c e n t r a t i o n and i t i s hard t o d i s t r i b u t e the t r a c e r s e v e n l y over the v i e w i n g a r e a . T h e r e f o r e , we a r e bound t o have p l a c e s where the i n f o r m a t i o n d e n s i t y i s low and have u n c e r t a i n t i e s i n i n t e r p o l a t i o n . For s t u d i e s where s i z e i s a c r i t i c a l p a r a meter, we have t o go back t o the p l o t s and check whether or not t h i s parameter i s r e a s o n a b l y e s t i m a t e d . We cannot take the r e c o g n i t i o n f o r g r a n t e d . T h i s i n a b i l i t y t o c a l c u l a t e the s i z e of the s t r u c t u r e a c c u r a t e l y f o r e v e r y r e c o g n i z e d s t r u c t u r e a l s o poses o t h e r problems. I n our s t u d y , we were m o s t l y i n t e r e s t e d i n the e v o l u t i o n of average v e l o c i t i e s and e n e r g i e s . These v a l u e s a r e summed and average over the s t r u c t u r e a r e a and they would v a r y i f we used d i f f e r e n t a r e a s . T h i s may seem v e r y d i s s a t i s f a c t o r y . However, the s i t u a t i o n i s not so bad i f we are more i n t e r e s t e d i n the r a t e of change of the parameters r a t h e r than the e x a c t v a l u e . T h i s i s because we expect t h e l o c a l t r a c e r d e n s i t i e s t o remain r o u g h l y c o n s t a n t d u r i a g our experiment and the s t r u c t u r e s r e c o g n i z e d under s i m i l a r c o n d i t i o n s s h o u l d not v a r y . E q u i v a l e n t l y , we c o n s i s t e n t l y l o o k a t an a r e a of a f i x e d r a t i o t o t h a t of the " a c t u a l " s t r u c t u r e . Assuming the r a t e s of change of the v a r i o u s parameters are n e a r l y u n i f o r m and can be d e s c r i b e d by an average v a l u e , we can s t i l l e s t i m a t e the r a t e s of 105 e v o l u t i o n . By l o o k i n g a t the p l o t s and s e l e c t i n g w e l l r e c o g n i z e d s t r u c t u r e s a t the i n i t i a l t i mes ( t ^  12/30 s or x < 1.6M), we found the 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 from area by c i r c u l a r a p p r o x i m a t i o n ) of the s t r u c t u r e , D, as (7.6) D = 0.8 ± 0.1 mesh w i d t h s . The i n i t i a l boundary speed, Um0, of the s t r u c t u r e s was a l s o found. T h i s i s c a l c u l a t e d by comparing the mean v e l o c i t y of the r e c 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 the i d e a l eddy. The mean speed over the i d e a l s t r u c t u r e i s 2U/3 where U i s i t s boundary speed. T h e r e f o r e , we e s t i m a t e the boundary speed of the s t r u c t u r e by t a k i n g (7.7) Urn = 3/2 * U' ; where U' denotes the average speed of the 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 on r o o t mean squared speed a l s o g i v e s an e s t i m a t e of Urn. Moreover, we can c a l c u l a t e the speed e i t h e r from the g r i d of t a n g e n t i a l speed or from the g r i d of v e l o c i t y components. T h e r e f o r e , we have f o u r d i f f e r e n t e s t i m a t e s of the t a n g e n t i a l speed of the s t r u c t u r e . They a l l t u r n e d out be i n agreement w i t h each o t h e r and we found (7.8) Um0 = (0.28 ± 0.08) Ug i n a l l samples. The agreement between v a l u e s g e n e r a t e d from the t a n g e n t i a l speed and v e l o c i t y component g r i d i s p a r t i c u l a r l y good ( u s u a l l y w i t h i n 5 % ) . T h i s was i n t e r p r e t e d as showing the v e l o c i t y i n t e r p o l a t i o n i s s t a b l e w i t h the 106 p r e s e n t r o u t i n e and d a t a d e n s i t y . The l a r g e r d i s c r e p a n c y (about 15%) between r e s u l t s from the average speeds and the r o o t mean squared speeds was i n t e r p r e t e d as the d e p a r t u r e from the i d e a l r i g i d l y r o t a t i n g s i t u a t i o n . A l t h o u g h i t was not done i n the package, c a l c u l a t i n g the boundary v e l o c i t y d i r e c t l y from the v e l o c i t y g r i d s s h o u l d be t r i v i a l . T h i s v a l u e c o u l d then be used as a d i r e c t e s t i m a t e o f Um. The t r a n s l a t i o n a l energy of the s t r u c t u r e s was found t o be 3-4 o r d e r s of magnitude l e s s than the r o t a t i o n a l or t o t a l energy. The c a l c u l a t e d t r a n s l a t i o n a l energy i s much s m a l l e r than the u n c e r t a i n t y of the c a l c u l a t i o n . T h i s can be i n t e r p r e t e d i n two ways. F i r s t l y , i t i s the b a s i c assumption i n the s t a t i o n a r y f i t t i n g model t h a t Vcm=0. There s h o u l d be no s u r p r i s e f o r an i n t e r p o l a t e d g r i d g e n e r a t e d under t h i s model t o g i v e us back the as s u m p t i o n . T h i s can be c o n s i d e r e d as j u s t an i n t e r n a l c o n s i s t e n c y check and the a c t u a l t r a n s l a t i o n may be smeared out i n the f i t t i n g . However, based on o b s e r v a t i o n s we b e l i e v e d t h a t the t r a n s l a t i o n s a r e a c t u a l l y s m a l l . C o n s i d e r i n g the s t r u c t u r e s t o be m u t u a l l y e x c l u s i v e ( H u s s a i n 1983), i t was found t h a t they a r e q u i t e c l o s e l y packed a t p r o d u c t i o n (not i n the sense of c l o s e - p a c k i n s o l i d s t a t e p h y s i c s ) . There j u s t i s not much space a v a i l a b l e f o r t r a n s l a t i o n . From the p l o t s of r e c o g n i z e d s t r u c t u r e s , we l o o k e d a t the CM p o s i t i o n s a t s u c c e s s i v e t i m e s . We found t h a t 107 most s t r u c t u r e CM's a r e p r a c t i c a l l y s t a t i o n a r y when compared w i t h the s t r e a k m o t i o n s . T h i s s u p p o r t s our a n a l y s i s t h a t r o t a t i o n a l m o t ions as shown by the s t r e a k s a r e much g r e a t e r than the s t r u c t u r e t r a n s l a t i o n s shown by the s t r u c t u r e c e n t e r s . We expect t h e t o t a l energy t o be r o t a t i o n a l w i t h i n the l i m i t s of u n c e r t a i n t i e s . A p l o t of r o t a t i o n a l energy a g a i n s t t o t a l energy f o r the r e c o g n i z e d s t r u c t u r e s i s shown i n F i g . 17. The t o t a l energy i s c a l c u l a t e d from d e f i n i t i o n by summing the energy c o n t e n t a t each g r i d p o i n t . The r o t a t i o n a l energy i s c a l c u l a t e d by a r i g i d body c i r c u l a r a p p r o x i m a t i o n . As shown i n s e c t i o n ( 4 . 4 ) , we f i r s t c a l c u l a t e d the s t r u c t u r e moment of i n e r t i a I by d e f i n i t i o n . W i th a r e v i s e d average a n g u l a r v e l o c i t y 3 d e f i n e d as (7.9) a)' = I {VL/( |R-L-Rcm| ) } / N . The r o t a t i o n a l energy of the s t r u c t u r e i s c a l c u l a t e d as : (7. 10) Er = (ICJ' 2 ) / 2 . 3 T h i s i s not the average a n g u l a r v e l o c i t y c a l c u l a t e d from the a n g u l a r v e l o c i t y g r i d , we have done so many smooth o p e r a t i o n s on the u> g r i d t h a t i t i s not a d v i s a b l e t o use i t f o r a n y t h i n g o t h e r than s t r u c t u r e r e c o g n i t i o n . 108 F i g u r e 17 - P l o t of c a l c u l a t e d r o t a t i o n a l energy vs t o t a l energy f o r r e c o g n i z e d s t r u c t u r e s . 109 key : — — — ideal profile without mixing possible profile with mixing E{ by definition UJ' is defined so that the two shaded parts have same area F i g u r e 18 - S p a t i a l p l o t s of w(r) and V ( r ) f o r d i f f e r e n t s i t u a t i o n s . 110 key : ideal profile without mixing possible profile with mixing E{ by definition rigid body profile E r by I(jJ'2/2 F i g u r e 19 - S p a t i a l p l o t of V 2 ( r ) f o r d i f f e r e n t s i t u a t i o n s . 111 E x p e r i m e n t a l l y , we found t h a t the r o t a t i o n a l energy so d e t e r m i n e d i s c o n s i s t e n t l y g r e a t e r than the c a l c u l a t e d t o t a l energy. However, t h e r e i s a s t r o n g l i n e a r r e l a t i o n between them. To i n t e r p r e t t h i s d i f f e r e n c e , we denote the c a l c u l a t e d t o t a l energy by E, and the r o t a t i o n a l energy by E 2 . C o n s i d e r a g a i n the 1D s p a t i a l V and oo p l o t of the i d e a l and p o s s i b l e s i t u a t i o n shown i n F i g . 18. The s o l i d l i n e shows a more r e a l i s t i c p r o f i l e w i t h m i x i n g . F i r s t , we n o t i c e t h a t the e f f e c t i v e r a d i u s of the s t r u c t u r e i s i n c r e a s e d ( R 0 t o R') as we i n t r o d u c e m i x i n g of the i d e a l s t r u c t u r e w i t h the f l u i d . We can c a l c u l a t e the new cj(r) from the V ( r ) p r o f i l e by d e f i n i n g OJ( r ) =V( r )/r . An average w' d e f i n e d under t h i s d e f i n i t i o n can be s k e t c h e d by drawing a l i n e a c r o s s the w(r) p l o t as i n the f i g u r e so t h a t the two shaded a r e a s are a p p r o x i m a t e l y the same. W i t h t h i s m o d i f i e d r a d i u s and f u r t h e r assuming r a d i a l symmetry, E 2 would be the t o t a l energy of a r i g i d l y r o t a t i n g c i r c u l a r eddy w i t h r a d i u s R' and boundary t a n g e n t i a l speed R'CL)'. The V ( r ) p r o f i l e of such an i d e a l eddy i s a l s o shown i n the f i g u r e . To c l a r i f y the r e l a t i o n between E, and E 2 , a p l o t of V 2 ( r ) i s shown i n F i g . 1 9 . The d i s c r e p a n c y between the two c a l c u l a t e d e n e r g i e s would be the d i f f e r e n c e i n the two shaded a r e a s i n the p l o t . Note t h a t the r o t a t i o n a l energy i s p r o p o r t i o n a l t o the f o u r t h power of r a d i u s f o r an i d e a l eddy. When the m i x i n g l a y e r has grown t o a c e r t a i n l e n g t h , E 2 w i l l be g r e a t e r than E,. T h i s e x p l a i n s the d i f f e r e n c e between the two p arameters v a l u e s . 1 12 From the above c o n s i d e r a t i o n , we note t h a t a l l the v a r i a t i o n s o r i g i n a t e from the change of v e l o c i t y p r o f i l e V ( r ) . T h i s i s caused by outward energy d i f f u s i o n or m i x i n g . As shown i n the f i g u r e , the two e n e r g i e s are v e r y much l i n e a r l y r e l a t e d . T h i s i s e v i d e n c e t h a t t h e r e e x i s t s a V ( r ) p r o f i l e (or a f a m i l y of p r o f i l e s ) common t o most r e c o g n i z e d s t r u c t u r e s . T h i s s h o u l d a l s o be t r u e as we expect the p h y s i c a l m i x i n g p r o c e s s t o be the same. On the o t h e r hand, the above r e l a t i o n between the two c a l c u l a t e d e n e r g i e s can be used t o t e s t t h e o r e t i c a l models of m i x i n g i n the c o h e r e n t s t r u c t u r e s . T h i s s h o u l d be done as the next s t e p i n t e s t i n g the system and a l s o t o v e r i f y d i f f e r e n t m i x i n g models. 113 C?2 O'A f ^ - 016 F i g u r e 20 - L o g - l o g p l o t of i n i t i a l decay r a t e A as a f u n c t i o n of the s t r u c t u r e r a d i u s R 1 14 0,2 0,4 0,6 0,8 F i g u r e 21 - P l o t of i n i t i a l decay r a t e A vs 1/R2 1 1 5 7.1.2 Spontaneous Energy Decay Rate of Coherent S t r u c t u r e s A c c o r d i n g t o the model proposed by A h l b o r n and Loewen, the spontaneous energy decay r a t e A of a c o h e r e n t s t r u c t u r e i s g i v e n by : (7.11) A = 16u / R 2 ; where v i s the k i n e m a t i c v i s c o s i t y , R i s t h e r a d i u s of the s t r u c t u r e and A i s d e f i n e d t h r o u g h the e q u a t i o n : (7.12) E ( t ) = E ( t = 0 ) e x p ( - A t ) . T h i s i s based on a s i m p l e d e r i v a t i o n from a c i r c u l a r s t r u c t u r e u s i n g c o n s i d e r a t i o n s of power d i s s i p a t i o n . I t i s a n o t h e r o b j e c t i v e of the e x p e r i m e n t s t o study t h i s r e l a t i o n . "Spontaneous" here r e f e r s t o s t r u c t u r e - f l u i d i n t e r a c t i o n as c o n t r a s t e d t o i n t e r a c t i o n w i t h f l o w and w i t h o t h e r s t r u c t u r e s . In the i n i t i a l p e r i o d of the system when we have l i t t l e t r a n s l a t i o n , we e xpect t h i s mechanism t o dominate the energy decay of the c o h e r e n t s t r u c t u r e s . As d i s c u s s e d i n the l a s t s e c t i o n , we s t i l l have d i f f i c u l t i e s i n d e t e r m i n i n g the s i z e (hence r a d i u s ) f o r each and every s t r u c t u r e . We have t o check the r e s u l t w i t h the r e c o g n i z e d p l o t o v e r l a y e d on the s t r e a k d a t a t o judge whether the boundary 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 . We can a l s o check whether t h e r e a r e o t h e r i n t e r a c t i o n s by j u d g i n g the l o c a l c o n d i t i o n s of the s t r u c t u r e . W i t h t h e s e c o n s i d e r a t i o n s , we can s e l e c t w e l l r e c o g n i z e d s t r u c t u r e s f r e e of v i s i b l e i n t e r f e r e n c e by o t h e r 1 16 s t r u c t u r e s or the f l o w , i . e . s t r u c t u r e s t h a t decay w h i l e m a i n t a i n i n g more or l e s s the same geometry. A l o g - l o g p l o t of the s t r u c t u r e s ' decay r a t e a g a i n s t t h e i r r a d i u s (by c i r c u l a r a p p r o x i m a t i o n ) i s g i v e n i n F i g . 20. We found t h a t p o i n t s from s t r u c t u r e s i n t h e same run f a l l on a s i n g l e s t r a i g h t l i n e . Moreover, t h e s e l i n e s a r e found t o of the same s l o p e around -2 a l t h o u g h t h e i n t e r c e p t s a r e d i f f e r e n t . T h i s shows t h a t the energy decay r a t e A i s i n v e r s e l y p r o p o r t i o n a l t o the square of the r a d i u s of the s t r u c t u r e s w i t h i n the same run as p r e d i c t by ( 7 . 1 1 ) . However, the p r o p o r t i o n a l i t y c o n s t a n t changes over d i f f e r e n t r u n s . To c a l c u l a t e d the p r o p o r t i o n a l i t y c o n s t a n t s , a n o t h e r p l o t of the decay r a t e a g a i n s t 1/R2 i s shown i n F i g . 21. From the model, t h i s v a l u e s h o u l d g i v e an e s t i m a t e of the k i n e m a t i c v i s c o s i t y v of the system. The n u m e r i c a l v a l u e found was about 0.2±0.1 cm 2/s which i s an o r d e r of magnitude h i g h e r than the e s t a b l i s h e d water b u l k v i s c o s i t y of 0.01 cm 2/s. We a l s o found t h a t the c a l c u l a t e d u's were h i g h e r f o r e x p e r i m e n t s t h a t were done l a t e r . T h i s i s i n t e r p r e t e d as the combined e f f e c t of s u r f a c e c o n t a m i n a n t s and i n h e r e n t s u r f a c e v i s c o s i t y . I t i s known t h a t a f r e e s u r f a c e w i t h c o n t a m i n a t i o n can have a s u r f a c e v i s c o s i t y s e v e r a l o r d e r of magnitude g r e a t e r than the c o r r e s p o n d i n g b u l k v i s c o s i t y ( C r i d d l e 1960) S u r f a c e c o n t a m i n a n t s were not c o n s i d e r e d a t the time of the e x p e r i m e n t s but the p r o p o r t i o n a l i t y of the decay r a t e w i t h 1 17 r e s p e c t t o 1/R2 i s q u i t e e v i d e n t from the p l o t s . F u r t h e r e x p e r i m e n t s w i t h b e t t e r c o n t r o l l e d s u r f a c e c o n t a m i n a n t s (e.g. skimming the s u r f a c e b e f o r e each experiment) s h o u l d be run t o t e s t the h y p o t h e s i s and a l s o t o st u d y the r e l a t i o n of the decay r a t e as a f u n c t i o n of v i s c o s i t y . 7.2 D i s c u s s i o n s and Recommendations 7.2.1 L i m i t a t i o n s of the P r e s e n t System The g r i d i n t e r p o l a t i o n r o u t i n e i s c o n s i d e r e d as the major s o f t w a r e weakness. A l t h o u g h the p r e s e n t r o u t i n e CGRID1 has proven t o be adequate i n most s i t u a t i o n s , t h e r e a r e s t i l l some p i c t u r e s t h a t produce i n t e r p o l a t e d f i e l d s much d i f f e r e n t from what we expect p h y s i c a l l y . T h i s can be improved by p u t t i n g more p h y s i c a l i n f o r m a t i o n i n t h i s i n t e r p o l a t i o n . I n c l u s i o n of p h y s i c a l c o n s t r a i n t s such as i n c o m p r e s s i b i l i t y and o t h e r d i f f e r e n t i a l e q u a t i o n s g o v e r n i n g the r a t e s of change of the v a r i a b l e s can be used t o i n t e r p o l a t e more r e a l i s t i c f i e l d s . The l i m i t i n time of a n a l y s i s may be eased by s o f t w a r e s e l e c t i n g a lower sampling r a t e . The package r e q u i r e s s u f f i c i e n t a c c u r a c y and c u r v a t u r e t o d e f i n e the a n g u l a r v e l o c i t y . These can be o b t a i n e d by i n c r e a s i n g the exposure time and d e c r e a s i n g the s a m p l i n g r a t e a t l a t e r t i m e s i n the e x p e r i m e n t . The a b i l i t y t o c o n t r o l the exposure time by s o f t w a r e i s a l s o one of the major advantages over the method 1 1 8 of p h o t o g r a p h i c f l o w v i s u a l i z a t i o n . T h e o r e t i c a l l y , we can study s t r u c t u r e s w i t h s c a l e t i m e s much l a r g e r than t h a t of the f l o w p i c t u r e s . The d r i f t i n g model i n the package has been t r i e d i n c a l c u l a t i n g the g r i d system w i t h l i t t l e s u c c e s s . T h i s i s i n t e r p r e t e d as b e i n g caused by the l a r g e u n c e r t a i n t y i n the p r i m a r y d a t a . The a p p l i c a b i l i t y of t h i s model i n s i t u a t i o n s where Vcm i s not z e r o s h o u l d be f u r t h e r e v a l u a t e d w i t h d a t a of h i g h e r p r e c i s i o n . The major l i m i t a t i o n s i n our p r e s e n t system a r e i n hardware. We have shown t h a t the s i z e of v iew, the d u r a t i o n of a n a l y s i s , the range of towing v e l o c i t y and the a c c u r a c y of the r e s u l t s , a r e a l l dependent on the s p a t i a l r e s o l u t i o n of the p r i m a r y images. The n o i s e l e v e l of our d i g i t i z e d p i c t u r e s c r e a t e d much h a z a r d i n e x t r a c t i n g t h i s p r i m a r y i n f o r m a t i o n . Moreover, the l a r g e amount of computer r e s o u r c e s r e q u i r e d i n the p r e s e n t system i s a s u b s t a n t i a l c o s t f o r the u s e r s . As t h i s was f i r s t d e s i g n e d t o be g e n e r a l , p o s s i b l e o p t i m i z a t i o n i n the package can be done i f the experiment of i n t e r e s t i s s h a r p l y d e f i n e d . At UBC, r u n n i n g the package a t normal p r i o r i t y over a complete s e t of d a t a (3-4 seconds of fl o w ) r e q u i r e s more than 250 computer d o l l a r s , c o r r e s p o n d i n g t o around 160 r e a l d o l l a r s . Running the system e x t e n s i v e l y i n the p r e s e n t environment would not be recommended. A working v e r s i o n of the package has t o be implemented i n a much lower 119 c o s t e n v i r o n m e n t . These a r e a s s h o u l d be g i v e n h i g h e s t p r i o r i t y f o r any improvement over the p r e s e n t system. There have been two proposed ways t o a d d r e s s the data a c q u i s i t i o n problem. The s t r a i g h t f o r w a r d way i s t o r e p l a c e the equipment, namely the v i d e o r e c o r d e r and the microcomputer, w i t h more up - t o - d a t e equipment. T h i s i n v o l v e s a l a r g e r amount of c a p i t a l i n v e s t m e n t but would enable the system t o be workable i n the s h o r t e s t p o s s i b l e t i m e . Two e v a l u a t i o n s (Dewan e t a l , Paven e t a l 1985) on how the system c o u l d be improved w i t h i n a d e f i n e d budget have been c a r r i e d out and the recommended microcomputer and VCR system a r e q u i t e a t t r a c t i v e . T h e i r o t h e r recommendations i n v o l v i n g s e t t i n g up a s t a n d a l o n e microcomputer environment w i l l be d i s c u s s e d i n t h i s s e c t i o n . J u d g i n g on the l i m i t e d view s i z e of the p r e s e n t system and the n o i s e l e v e l i n v i d e o d i g i t i z a t i o n , a r e t u r n t o the p h o t o g r a p h i c t e c h n i q u e has been proposed by S. Loewen. In t h i s t e c h n i q u e , t i m e - e x p o s u r e d p i c t u r e s of the system a r e taken w i t h a dim background l i g h t and a s t r o n g s t r o b i n g l i g h t s e t a t a known f r e q u e n c y . The r e s u l t a n t p i c t u r e s would c o n s i s t of s t r e a k s superimposed by i n t e n s e s p o t s showing the t r a c e r p o s i t i o n s as a f u n c t i o n of t i m e . W i t h such a system, we can have a h i g h e r t r a c e r d e n s i t y s i n c e the problem of s t r e a k t r a c k i n g i s n e a r l y s o l v e d by the background l i g h t n i n g . 1 20 There a r e many advantages i n such a system. The v i e w i n g a r e a i s v i r t u a l l y u n l i m i t e d because the s p a t i a l r e s o l u t i o n of the p h o t o g r a p h i c image i s s e v e r a l o r d e r s of magnitude h i g h e r than any e x i s t i n g v i d e o d i g i t i z e r . We can use s m a l l e r t r a c e r s f o r the same a r e a , t h e r e b y d e f i n i n g the parameters more s h a r p l y than a t p r e s e n t . Moreover, we can a l s o d i g i t i z e d i f f e r e n t p o r t i o n s of a s i n g l e p i c t u r e and then superimpose them back i n t o computer memory. We can a l s o v i s u a l l y choose a r e a s of good t r a c e r d e n s i t y and s t r u c t u r e s f o r study r a t h e r than depending on the l u c k of the outcome i n a s m a l l v i e w i n g a r e a . Such d i g i t i z a t i o n s a re supposed t o be done by hardware i n the computing c e n t e r and the r e s u l t s t r a n s f e r r e d t o MTS, tapes or m i n i d i s k s . In c o n s i d e r i n g the d i f f i c u l t i e s , the e x p e r i m e n t a l time d u r i n g which such a system can be used t o study may be l i m i t e d more s e v e r e l y than the p r e s e n t system. The p r i m a r y reason i s t h a t the t r a c e r s have t o move over d i s t a n c e s g r e a t e r than t h e i r own s i z e s t o be t e m p o r a l l y r e s o l v a b l e . I t would be h a r d t o d i f f e r e n t i a t e o v e r l a p p e d t r a c e r s w i t h l a r g e r ones. U s i n g s t r o b e s of d i f f e r e n t c o l o r s may h e l p a b i t but t h i s assumption has to be r i g o r o u s l y t e s t e d by p r e l i m i n a r y e x p e r i m e n t s . We s t i l l do not know how superimposed c o l o r s t u r n out upon d i g i t i z a t i o n . The s o f t w a r e t h a t has t o be d e v e l o p e d t o i n c o r p o r a t e t h i s new method i n t o the p r e s e n t one a l s o p r e s e n t s a major d i s a d v a n t a g e . The a u t h o r e x p e c t s such a l i n k between 121 the two systems would t a k e months t o be f u l l y o p e r a t i o n a l . A more d e t a i l e d e v a l u a t i o n w i t h p r e l i m i n a r y a n a l y s i s of the a n t i c i p a t e d problems and subsequent g a i n s i n such system s h o u l d be c a r r i e d out b e f o r e c o m m i t t i n g o u r s e l v e s . I t i s my b e l i e f t h a t the p r e s e n t system has been d e v e l o p e d t o the p o i n t where we can do e l e g a n t 2D i s o t r o p i c t u r b u l e n t f l o w e x p e r i m e n t s w i t h minor u p g r a d i n g of the data a c q u i s i t i o n equipment. We s h o u l d s t a r t u s i n g the system r a t h e r than c o n t i n u a l l y t r y i n g t o improve on the t e c h n i q u e . F u r t h e r development w i t h the Apple computer i s not r e a l i s t i c and a microcomputer w i t h h i g h e r s p a t i a l r e s o l u t i o n i s d e f i n i t e l y needed. I recommend a m i n i m a l upgrade of the microcomputer and i f p o s s i b l e , the v i d e o r e c o r d e r , and then s t a r t d o i n g more e x p e r i m e n t s . Always t r y i n g t o push t o the s t a t e - o f - a r t may e v e n t u a l l y render t h i s system o b s o l e t e b e f o r e f u l l y u t i l i z i n g i t . Both p r o p o s a l s r e q u i r e the e x t e n s i v e a n a l y s i s system t h a t has been d e v e l o p e d i n the p r e s e n t system. We have t o d e a l w i t h the a n a l y s i s c o s t problem s e p a r a t e l y . I t was recommended t h a t the system s h o u l d be implemented i n a s t a n d a l o n e microcomputer environment(Dewan et a l , Pavan et a l 1985). However, at t h e time of e v a l u a t i o n , the MTS mainframe computer was c o n s i d e r e d as the o n l y a l t e r n a t i v e t o the microcomputer p r o p o s a l . S i n c e UBC i n s t a l l e d the FPS A r r a y P r o c e s s o r ( A P ) , 1 22 t h i s s h o u l d be a d i f f e r e n t s t o r y . The major c o n c e r n over the c o s t of r u n n i n g the system can be s i g n i f i c a n t l y l o w e r e d i f we change the system t o run on the a r r a y p r o c e s s o r . In the MTS environment, one hour of e x e c u t i o n time c o s t s 480 computer d o l l a r s as c o n t r a s t e d t o the 24 d o l l a r s f o r the AP. In our system, the memory usage c o n s t i t u t e d more than o n e - t h i r d of the o v e r a l l computing charges and t h i s i s f r e e on the AP. C o n v e r t i n g the system t o use the AP seems t o be an i d e a l s o l u t i o n t o the c o s t problem. As the p r e s e n t package i s w r i t t e n m o s t l y i n s t a n d a r d FORTRAN w i t h a few MTS FTN e x t e n s i o n s f o r f i l e m a n i p u l a t i o n , t r a n s l a t i n g i t to the APFTN64 language used i n the AP s h o u l d be t r i v i a l . Except f o r the g r a p h i c s package, most system r o u t i n e s c a l l e d a r e a l s o w r i t t e n i n FORTRAN, s i m i l a r c o n v e r s i o n can a l s o be done. The package was w r i t t e n i n such a way t h a t the n u m e r i c a l d a t a p r o c e s s i n g i s independent of the g r a p h i c a l a n a l y s i s . We can s e p a r a t e the two p a r t s c o m p l e t e l y w i t h o u t v i s i b l e changes seen from the o u t s i d e . We can use the AP t o do a l l the n u m e r i c a l p r o c e s s i n g and pass the r e s u l t out to MTS f o r g r a p h i c s p l o t s . W ith such a c o m b i n a t i o n , we s t i l l have the e x t e n s i v e g r a p h i c s support of the mainframe w h i l e r e d u c i n g most c o s t i n the image p r o c e s s i n g . A summer of an APSC 459 type p r o j e c t i n UBC s h o u l d be a b l e t o f i n i s h the whole c o n v e r s i o n and make the system f i n a n c i a l l y f e a s i b l e . The a u t h o r p e r s o n a l l y d i s l i k e s the i d e a of downloading 123 the system t o a microcomputer. In such an environment, f u r t h e r development of the system i s h a r d . A l t h o u g h most microcomputer systems a r e upgraded c o n s t a n t l y , i t would be a c o n s i d e r a b l e i nvestment f o r each improvement i n b o t h s o f t w a r e and hardware. In u s i n g a system t h a t i s d i r e c t l y s u p p o r t e d by the computer c e n t e r , we can always b e n e f i t from t h e i r immediate a t t e n t i o n i f a n y t h i n g goes wrong and a l s o from f r e q u e n t i n t e r a c t i o n w i t h these knowledgeable p e o p l e . A s t a n d a l o n e system may a l s o make us s t a n d a l o n e i n i n f o r m a t i o n exchange. The speed and l i b r a r y s u p p o r t i n a mainframe system a l s o cannot be matched by any microcomputer and t h e s e may t u r n out t o be a g r e a t h i n d r a n c e f o r development. 7.2.2 E x t e n s i o n t o a 3D System A l t h o u g h c o n s i d e r a t i o n s f o r p o s s i b l e e x t e n s i o n t o a 3D system were always made d u r i n g the development of the system, such e x t e n s i o n s a r e by no means t r i v i a l . The major i d e a s i n t h i s system can be c a r r i e d o v e r , but the i n c r e a s e demand i n d a t a d e n s i t y , a c c u r a c y and memory s i z e w i l l d e f i n i t e l y exceed the c a p a c i t y of most of our e x i s t i n g equipment. Without equipment u p g r a d i n g , s u c c e s s f u l e x t e n s i o n w i l l be v e r y i m p r o b a b l e . F u n d a m e n t a l l y , the d e f i n i t i o n of c o h e r e n t s t r u c t u r e s used i n the 2D system must be m o d i f i e d . T h i s r e l a t e s back t o the v e r y a r c h i t e c t u r e of the p r e s e n t system. G e n e r a l l y s p e a k i n g , 1 24 we s t a r t e d by c h o o s i n g a p r o p e r t y t h a t i s c o n s t a n t throughout our i d e a l c o h e r e n t s t r u c t u r e . We then i n f e r v a r i a t i o n of the parameter from p h y s i c a l c o n s i d e r a t i o n s . Through such e s t i m a t i o n s , we proposed t h a t the r e a l o p e r a t i o n of i d e n t i f y i n g a c o h e r e n t s t r u c t u r e i n v o l v e s the two s t e p s of 1) f i n d i n g the peak and 2) o u t l i n i n g the boundary. The f i r s t s t e p t e l l s us where we s h o u l d l o o k and the second one i s the d e f i n i n g o p e r a t i o n . In 2D, a n g u l a r v e l o c i t y i s found t o be a s u i t a b l e c a n d i d a t e . I n 3D, we have t o f i n d a s i m i l a r v a r i a b l e . Because of t h e i n c r e a s e d v a r i e t y and c o m p l e x i t y of 3D f l o w s , we may not be a b l e t o f i n d a s u i t a b l e c a n d i d a t e f o r the g e n e r a l c a s e . We w i l l p r o b a b l y have t o l o o k f o r some s i m p l i f i c a t i o n s . For example, we c o u l d c o n s i d e r o n l y f l o w s t h a t a r e n e a r l y two d i m e n s i o n a l , i . e . z-component v e l o c i t y i s much s m a l l e r than the component on the x y - p l a n e . Such systems e x i s t and a r e of g e n e r a l i n t e r e s t f o r many d i f f e r e n t d i s c i p l i n e s and so t h e r e s h o u l d be no doubt as t o the u s e f u l n e s s of such s t u d i e s . The d e f i n i n g parameter need not be a n g u l a r v e l o c i t y , however i t must be a parameter t h a t can be c a l c u l a t e d d i r e c t l y from the t r a j e c t o r y . For the same reason t h a t we have i n 2D, v o r t i c i t y i s not recommended. One p r o b a b l e c h o i c e would be the f u l l a n g u l a r v e l o c i t y v e c t o r . The magnitude of the a n g u l a r v e l o c i t y v e c t o r seems t o be a r e a s o n a b l e c a n d i d a t e as 1 25 the d e f i n i n g parameter. When we run the e x p e r i m e n t , we are s u b j e c t e d t o s i m i l a r but s t r i c t e r c o n s t r a i n t s as d i s c u s s e d i n l a s t c h a p t e r . Assuming t h a t we have a more p o w e r f u l microcomputer and a b e t t e r d i g i t i z e r , we can use the s t e r e o s c o p i c t e c h n i q u e d e v e l o p e d by Sheu et a l ( l 9 8 2 ) t o c a p t u r e 3 d i m e n s i o n a l c o o r d i n a t e d a t a . Another p o s s i b l e method proposed by S. Loewen i s t o p r o j e c t d i f f e r e n t c o l o r l i g h t a c r o s s the f l u i d a t d i f f e r e n t l e v e l s . The depths can then be i d e n t i f i e d by c o l o r . In e i t h e r c a s e , we w i l l have t o put a n e u t r a l d e n s i t y t r a c e r i n the f l u i d t o t r a c e i t s i n t e r n a l m o t i o n . The r e s o l u t i o n of the image and the t r a c e r s i z e w i l l a l s o d e t e r m i n e the volume of v i e w, and hence n e a r l y e v e r y t h i n g e l s e . I t would be even h a r d e r t o c o n t r o l the d i s t r i b u t i o n of the t r a c e r s and we c o u l d expect a number d e n s i t y much s m a l l e r than the one c a l c u l a t e d by the methods of the l a s t c h a p t e r . We have t o c o n s i d e r image d i s t o r t i o n and a l s o p o s s i b l e o v e r l a p of t r a c e r s a t d i f f e r e n t l e v e l s when viewed by one camera. The l a t t e r w i l l f u r t h e r l i m i t the number d e n s i t y of t r a c e r s . F i n d i n g a n e u t r a l d e n s i t y t r a c e r , p u t t i n g enough of them i n the f l u i d t o o b t a i n a h i g h enough i n f o r m a t i o n d e n s i t y and s t i l l s u c c e s s f u l l y g e t t i n g t h e i r c o o r d i n a t e s i s the f i r s t major problem t h a t must be s o l v e d . Once we can get such d a t a , we w i l l have t o f a c e the 1 26 s t o r a g e problem. I f we cannot t a k e d a t a d i r e c t l y i n a l i s t form, we s h o u l d t r a n s f o r m them t o t h i s form as soon as p o s s i b l e . A t h r e e d i m e n s i o n a l image p i x e l a r r a y w i t h r e s o l u t i o n 256x256x256 and a f u r t h e r 64 grey l e v e l s at presumably 30 frames per second c o n s t i t u t e s a g i g a n t i c d a t a s t r u c t u r e . T h i s s h o u l d be t r a n s f o r m e d t o a l i s t of p i x e l c o o r d i n a t e s and a t t r i b u t e s (X,Y,Z,T,grey l e v e l ) as e a r l y as p o s s i b l e . W i t h such a l i s t s t r u c t u r e , we c o u l d s o r t i t and use a s u b r o u t i n e t o s i m u l a t e a v i r t u a l a r r a y f o r any m a n i p u l a t i o n t h a t i s best done i n a p i x e l a r r a y environment. T h i s k i n d of d a t a r e d u c t i o n would be i m p o r t a n t f o r s a v i n g s t o r a g e and co m p u t a t i o n t i m e . H o p e f u l l y we would not have t o do much smoothing f o r the 3D d a t a w i t h b e t t e r equipment, but e x t e n s i o n of the p r e s e n t smoothing a l g o r i t h m s h o u l d be easy. L o c a t i n g the t r a c e r c e n t e r s and a l s o j o i n i n g the s t r e a k s can a l s o be d i r e c t e x t e n s i o n s of the p r e s e n t system. W i t h good p r i m a r y d a t a , we c o u l d expect the p r e s e n t system t o be extended up t o the end of the s t r e a k c o n n e c t i o n phase w i t h o u t many d i f f i c u l t i e s . We then come back t o where the o r i g i n a l c o h e rent s t r u c t u r e d e f i n i n g parameter has t o be e x t r a c t e d . In the 2D c a s e , i t was found t h a t the parameter e x t r a c t i o n w i t h the d r i f t i n g model a r e l e s s a c c u r a t e than the s t a t i o n a r y model. For a 3D model, we have t o l o o k a t the f l o w a t d i f f e r e n t depths a t the same time and the s t a t i o n a r y model may not be 1 27 a b l e t o d e s c r i b e the whole f l u i d f i e l d a d e q u a t e l y . In u s i n g s i m i l a r f i t t i n g methods, the d r i f t i n g model seems more a p p r o p r i a t e . However, t h i s demands q u i t e an a c c u r a t e data s e t as we have t o use the t h i r d o r d e r time d e r i v a t i v e s of the t r a j e c t o r y . (See Appendix A f o r the d i f f e r e n c e i n the two models.) I f t h i s p r o v e s t o be too h a r d , we may have t o f i n d d i f f e r e n t methods t o e x t r a c t the v a r i o u s parameters ( i . e . e i t h e r not f i t t i n g by p o l y n o m i a l s or u s i n g another model f o r the f i t t e d t r a j e c t o r y ) . T h i s i s c o n s i d e r e d as the second major d i f f i c u l t y i n the e x t e n s i o n . I f we can a l s o s o l v e t h i s problem and be a b l e t o e x t r a c t the p h y s i c a l parameters w i t h c o n s i d e r a b l e a c c u r a c y , we come t o the t h i r d major problem of space (3D) i n t e r p o l a t i o n . I t appears t o me t h a t we cannot expect a h i g h d a t a d e n s i t y . Space i n t e r p o l a t i o n under such s i t u a t i o n s would be d i f f i c u l t . There i s no e x i s t i n g l i b r a r y r o u t i n e a t UBC t h a t does t h i s k i n d of i n t e r p o l a t i o n . Even i f one does e x i s t , i t would p r o b a b l y be of l i t t l e use. We need p h y s i c a l i n p u t t o d i r e c t the i n t e r p o l a t i o n . The r o u t i n e s h o u l d i n c o r p o r a t e v a r i o u s c o n s t r a i n t s g i v e n under the d i f f e r e n t p h y s i c a l c o n d i t i o n s . A s u i t a b l y m o d i f i e d s e t of g o v e r n i n g e q u a t i o n s i n c l u d i n g the e q u a t i o n of C o n t i n u i t y , i n c o m p r e s s i b i l i t y and the N a v i e r - S t o k e s e q u a t i o n s h o u l d be coded i n t o the i n t e r p o l a t i o n r o u t i n e . Under t h i s c o n s i d e r a t i o n , our space i n t e r p o l a t i o n would p r o b a b l y be 1 28 w o r k i n g on a number of 3D g r i d s r a t h e r than j u s t one. Such i n t e r p o l a t i o n s are j u s t l i k e s o l v i n g the a c t u a l g o v e r n i n g d i f f e r e n t i a l e q u a t i o n s w i t h the i n i t i a l d a t a as boundary d a t a . T h i s would l e a d us i n t o a n o t h e r heated t o p i c of n u m e r i c a l l y m o d e l i n g the f u l l s e t of f l u i d g o v e r n i n g e q u a t i o n s . D e v e l o p i n g such r o u t i n e s would be the major c h a l l e n g e i n e x t e n d i n g the p r e s e n t system t o 3 D . Rather than w r i t i n g such a r o u t i n e o u r s e l v e s , i t i s w o r t h w h i l e t o spend some time s e a r c h i n g f o r s i m i l a r r o u t i n e s from e x t e r n a l s o u r c e s . Because of the s i m i l a r i n t e r e s t i n o c e a n i c and a t m o s p h e r i c s c i e n c e s , i t i s my b e l i e f t h a t such r a t h e r s p e c i a l purpose r o u t i n e s a l r e a d y e x i s t . I n t e r a c t i o n w i t h o t h e r i n s t i t u t i o n s may a l s o g i v e us new i n s i g h t i n t o the t h e o r e t i c a l and c o m p u t a t i o n a l a s p e c t s of the problem. I f a l l the above o u t l i n e d problems a r e s o l v e d , the next p a r t of c o h e r e n t s t r u c t u r e r e c o g n i t i o n can f o l l o w v e r y much the same method d e s c r i b e d p r e v i o u s l y and the r e m a i n i n g p a r t of s t r u c t u r e p a r a m e t r i z a t i o n s h o u l d a l s o be t r i v i a l . The above gave a s i m p l e o u t l i n e on a p o s s i b l e e x t e n s i o n t o a 3D system. With the p r e s e n t system as the main framework, more d e t a i l e d e v a l u a t i o n has t o be done b e f o r e a l l the problems can be i d e n t i f i e d . T h i s i n v e s t i g a t i o n i s a d e f i n i t e n e c e s s i t y t o d e t e r m i n e the f e a s i b i l i t y of the e x t e n s i o n and must be done b e f o r e c o m m i t t i n g s u b s t a n t i a l r e s o u r c e s . In c h o o s i n g between u s i n g the p r e s e n t system t o do 1 29 more e x p e r i m e n t s , e.g. t o study a s u b s u r f a c e l a y e r i n a near two d i m e n s i o n a l f l o w , and a c t i v e l y engaging i n t h e development of a 3D system, I would recommend t h e f i r s t one. 130 V I I I . CONCLUSION A computer package has been d e v e l o p e d t o automate f i e l d i n t e r p o l a t i o n and c o h e r e n t s t r u c t u r e r e c o g n i t i o n i n 2D t u r b u l e n t f l u i d f l o w . I t was f i r s t aimed as an e f f i c i e n t and o b j e c t i v e method t o p r o v i d e s u f f i c i e n t q u a n t i t a t i v e i n f o r m a t i o n from v i d e o f l o w v i s u a l i z a t i o n . T h i s i n f o r m a t i o n i s f u r t h e r used t o e x t r a c t c o h e r e n t s t r u c t u r e s from the t u r b u l e n t f l o w w i t h o u t s u b j e c t i v e manual judgements. The r e s u l t s e s t a b l i s h e d u s i n g t h i s p r e l i m i n a r y system prove i t t o be a p o w e r f u l one. We have been a b l e t o reproduce some e s t a b l i s h e d r e s u l t s and a l s o g a i n new i n s i g h t i n t o the i n i t i a l p e r i o d of g r i d t u r b u l e n c e . U s i n g the p r e s e n t system, we i d e n t i f i e d over 80% of a l l of the 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 s . Such a system i s a d e f i n i t e improvement over the t r a d i t i o n a l f l o w v i s u a l i z a t i o n methods i n o b j e c t i v i t y , t e m p o r a l r e s o l u t i o n and most i m p o r t a n t l y , the volume of r e a d i l y a v a i l a b l 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 h i s system e n a b l e s us f o r the f i r s t time t o f o l l o w the e v o l u t i o n of i n d i v i d u a l f l o w s t r u c t u r e s w h i l e a l s o b e i n g e f f i c i e n t enough t o p r o v i d e s u f f i c i e n t c o h e r e n t s t r u c t u r e d a t a f o r s t a t i s t i c a l a n a l y s i s w i t h i n a r e a s o n a b l e amount of t i m e . W i t h i n i t s f i e l d of view, the p l o t s of t r a c k e d s t r e a k s p r o v i d e us w i t h no l e s s i n f o r m a t i o n than we can get from an o r d i n a r y f l o w p i c t u r e . E v e r y t h i n g a f t e r w a r d s i s a g a i n . I f 131 d y n a m i c a l p r o p e r t i e s such as v e l o c i t y , energy, momentum and v o r t i c i t y a r e of p r i m a r y i n t e r e s t r a t h e r than the c o h e r e n t s t r u c t u r e s , they can be e x t r a c t e d from the t r a c k e d s t r e a k s w i t h o u t much d i f f i c u l t y . Such s t u d i e s a l l o w us the g l o b a l c h a r a c t e r i s t i c s of the flo w and i t s v a r i o u s parameter f i e l d s as a f u n c t i o n of t i m e . T h i s would be a t r i v i a l e x t e n s i o n of the system and c o u l d be used t o f u r t h e r c o n t r a s t the system w i t h e s t a b l i s h e d r e s u l t s . The major work here i s the c o h e r e n t s t r u c t u r e r e c o g n i t i o n p a r t . With the growing i n t e r e s t i n co h e r e n t s t r u c t u r e s of t u r b u l e n t f l o w , r e s e a r c h e r s s h o u l d f i n d such a system a v a l u a b l e t o o l . W i th t h i s package as an e f f i c i e n t d a t a a c q u i s i t i o n and a n a l y s i s system, i t i s hoped t h a t we can e s t a b l i s h enough 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 h e l p our u n d e r s t a n d i n g i n the exact r o l e and s i g n i f i c a n c e of the co h e r e n t s t r u c t u r e , and thus t u r b u l e n t f l u i d f l o w s i n g e n e r a l . T h i s p a r t i c u l a r l y a d d r e s s e s the c a l l by major r e s e a r c h e r s " f o r a combined f l o w v i s u a l i z a t i o n , image p r o c e s s i n g and c o h e r e n t s t r u c t u r e r e c o g n i t i o n system. An e x t e n s i o n o u t l i n e f o r 3D c o h e r e n t s t r u c t u r e r e c o g n i t i o n i s g i v e n w i t h t h e major d i f f i c u l t i e s i d e n t i f i e d and a d d r e s s e d . The problems a r e by not t r i v i a l but we b e l i e v e once we know the problem, the s o l u t i o n w i l l not be too f a r away. S i m i l a r systems w i t h such e x t e n s i o n s c o u l d t u r n out t o be i n d i s p e n s i b l e f o r f u t u r e f l u i d v i s u a l i z a t i o n s t u d i e s . 132 BIBLIOGRAPHY A h l b o r n B., A h l b o r n F. and Loewen S. 1985, J . Phys. D: A p p l . Phys. 18, 2127. A h l b o r n F. 1902, "Uber den Mechanismus des Hydrodynamischen W i d e r s t a u d e s " , Abhandlungen aus deur G e b i e t der N a t u r w i s s e n S c h a f t e n , N a t u r s w i s s . V e r e i n Hamburg P u b l . L. F r i e d r i c h s e n & Co. A h l b o r n F. 1922, Phys. Z. 23, 57-65. Bareau V. 1985, UBC P h y s i c s 459 p r o j e c t r e p o r t C a n t w e l l , B. and C o l e s , D. 1983, J . F l u i d Mech., 136, 321. C r i d d i e W. 1960, Rheology Volume 3, Academic P r e s s . Crow S.J. and Campagne F.H. 1971, J . F l u i d Mech. 48, 547. Dewan e t a l 1985, UBC P h y s i c s 459 p r o j e c t r e p o r t . H i n z e J.O. 1959, T u r b u l e n c e , M c G r a w - H i l l Book Company. H u s s a i n A.K.M.F. 1983, Phys. F l u i d s 26, 237. H u s s a i n A.K.M.F. 1985, "Forum on Unsteady Flows i n B i o l o g i c a l Systems", ASME. K l i n e e t a l 1967, J . F l u i d Mech. 30, 741. Loewen S. 1983, M a s t e r s t h e s i s , UBC. Loewen S., A h l b o r n B. and F i l u k A.B. 1986, t o be p u b l i s h e d Phys. F l u i d s . August 1986. Pavan e t a l 1985, UBC P h y s i c s 459 p r o j e c t r e p o r t . R o b e r t s o n J.A. and Crowe C T . 1975, E n g i n e e r i n g F l u i d M e c h a n i c s , Houghton M i f f l i n . Sheu e t a l 1982, Chem. Eng. Commun. 17, 67. 1 33 APPENDIX A - PARAMETER EXTRACTIONS FROM FITTED TRAJECTORY In the package, the s t r e a k c o o r d i n a t e s a r e f i t t e d as p o l y n o m i a l f u n c t i o n s of t i m e : ( A . 1 a ) X(T) = Z A.T 1 ; ( A . 1 b ) Y(T) = I B L T L ; where the sum i s over A. from 0 t o some i n t e g e r K. From t h i s r e p r e s e n t a t i o n of X(T) and Y ( T ) , we can d e r i v e the i n s t a n t a n e o u s p a r a m e t e r s , most i m p o r t a n t l y the l i n e a r and a n g u l a r v e l o c i t i e s . Depending on the c o n d i t i o n of the e x p e r i m e n t , the f i t t i n g models w i l l be d i f f e r e n t . Two s i t u a t i o n s are d i s c u s s e d h e r e . F i r s t , when we expect the t r a n s l a t i o n a l motion of the s t r u c t u r e t o be n e g l i g i b l e (as i n the case of g r i d t u r b u l e n c e ) , we w i l l f i t (A.2) V = wxR ; i . e . assuming pure r o t a t i o n a l m o t i o n . T h i s i s r e f e r r e d t o here as the s t a t i o n a r y model. A l t e r n a t i v e l y , f o r system or a t t i m e s where we expect the c o h e r e n t s t r u c t u r e s t r a n s l a t i o n t o be comparable t o r o t a t i o n , we have t o i n c l u d e the c e n t e r of mass motio n . In such s i t u a t i o n s , the t r a j e c t o r y i s f i t t e d by (A. 3) V = Vcm + ZJXR . T h i s i s r e f e r r e d t o here as the d r i f t i n g model. Parameters of i n t e r e s t a r e c a l c u l a t e d from t h e s e f i t t i n g e q u a t i o n s . From the p o l y n o m i a l a p p r o x i m a t i o n of t r a j e c t o r y , t h e o r e t i c a l l y we have an i n f i n i t e number of e q u a t i o n s t h a t can be used t o s o l v e any unknown parameters i n any f i t t i n g e q u a t i o n . They a r e the v a r i o u s time d e r i v a t i v e s : X 1 ( T ) , Y ' ( T ) , X ' ' ( T ) , Y''(T) up t o any o r d e r we want. However, the a c c u r a c y of the d e r i v a t i v e s d e c r e a s e s w i t h i n c r e a s i n g o r d e r . T h i s i s because of the u n c e r t a i n t y of the c o e f f i c i e n t s u s u a l l y i n c r e a s e w i t h o r d e r of the term. On d i f f e r e n t i a t i n g , the low o r d e r c o e f f i c i e n t s a r e s u c c e s s i v e l y e l i m i n a t e d , hence l e a v i n g the h i g h e r o r d e r d e r i v a t i v e l e s s a c c u r a t e . So as a g e n e r a l r u l e , we s h o u l d t r y t o m i n i m i z e the o r d e r of d i f f e r e n t i a t i o n s used and a l s o t r y t o m i n i m i z e the e f f e c t of the h i g h e r o r d e r d e r i v a t i v e s even i f they a r e i n v o k e d . 134 A.1 S t a t i o n a r y Model: Vcm i s Zero In t he s t a t i o n a r y model, the b a s i c f i t t i n g e q u a t i o n i s (A.2) V = wxR ; and a l l parameters of i n t e r e s t s a r e d e f i n e d i n the e q u a t i o n s (A.4a) X(T) = Xc + R C O S ( C J T + 7 ) ; and (A.4b) Y (T) = Yc + Rsin(coT+ 7 ) . (Xc,Yc) i s the c e n t e r of r o t a t i o n , R i s the r a d i a l v e c t o r i n the CM frame, u i s the a n g u l a r v e l o c i t y and 7 i s t h e i n i t i a l phase a n g l e a t T=0. There are f i v e unknowns, Xc, Yc, R, CJ and 7 . We r e q u i r e at l e a s t f i v e e q u a t i o n s t o s o l v e them. As our e q u a t i o n s always come i n p a i r s , we would be u s i n g s i x e q u a t i o n s up t o the a c c e l e r a t i o n terms. D i f f e r e n t i a t i n g (A.2) w i t h r e s p e c t t o tim e , assuming the parameters t o be c o n s t a n t , we have (A.5) A = uxV . C o n s i d e r the c r o s s p r o d u c t of V and A and expanding A by the above e q u a t i o n , we have VxA = V X ( C J X V ) = o>V2 . T h i s i s an e q u a t i o n f o r CJ: (A.6) CJ = (VxA) / V 2 . i — » S i n c e CJ and R are p e r p e n d i c u l a r i n our 2D system, we o b t a i n from e q u a t i o n (A.2) (A. 7) R = V / C J . D i f f e r e n t i a t i n g e q u a t i o n ( A . 4 ) , we have Vx = -Rcjsin (cjT+7) and Vy = R C J C O S (cjT+7) . T h i s g i v e s us an e q u a t i o n f o r 7 : (A. 8 ) tan(cjT+ 7 ) = -Vx/Vy . By c a r e f u l l y c o n s i d e r i n g the d i r e c t i o n of the v e l o c i t y components, we can be s o l v e the i n i t i a l phase a n g l e 7 . F i n a l l y we can c a l c u l a t e the c e n t e r of r o t a t i o n as (A. 9a) Xc = X - R C O S ( C J T + 7 ) = X - Vy/cj ; and (A.9b) Yc = Y - R s i n ( c j T+ 7 ) = Y + V X / C J . 1 35 A. 2 D r i f t i n g Model: Vcm i s Not Zero In t h i s model, the b a s i c f i t t i n g e q u a t i o n i s (A.3) V = Vcm + u>xR ; w i t h a l l parameters of i n t e r e s t s d e f i n e d i n the e q u a t i o n s (A.10a) X(T) = Xc + Vx 0T + R s i n ( w T + 7 ) ; and (A. 10b) Y(T) = Yc + Vy 0T - Rcos(uT+7) . The a d d i t i o n a l parameter ( V x 0 , V y 0 ) denotes the d r i f t i n g v e l o c i t y of the r o t a t i o n c e n t e r . As we i n t r o d u c e two more unknowns, we expect t o use up t o A', the time d e r i v a t i v e of a c c e l e r a t i o n A. D i f f e r e n t i a t i n g the above e q u a t i o n s once, we have: (A. 11a) Vx(T) = V x 0 + R W C O S ( O J T + 7 ) ; and (A.11b) Vy(T) = V y 0 + Rwsin(wT+ 7 ) . T h i s p a i r of e q u a t i o n s i s v e r y s i m i l a r t o e q u a t i o n ( A . 4 ) . Moreover, i f we d i f f e r e n t i a t e the f i t t i n g e q u a t i o n once, we would have (A.12) A = wxV which i s a l s o v e r y s i m i l a r t o ( A . 2 ) . C o n t r a s t i n g the p r e v i o u s model w i t h the new e q u a t i o n s , i t i s not hard t o i n f e r w i t h o u t anymore d e r i v a t i o n t h a t (A. 1 3) u = ("AxA' ) / A 2 ; (A. 14) Ru = A / C J or R = A/CJ2 ; (A.15) tan(wT+ 7 ) = -Ax/Ay ; (A. 16a) V x 0 = Vx - R C J C O S (coT+7) = Vx - Ay/w ; and (A. 16b) V y 0 = Vy - R u s i n (coT+7) = Vy + A X / C J . The o n l y r e m a i n i n g problem i s t o s o l v e the c e n t e r of r o t a t i o n ( X c , Y c ) . T h i s i s done by s u b s t i t u t i n g the parameters back i n t o (A.10) and s i m p l i f i n g the r e s u l t i n g e q u a t i o n s . We f i n a l l y get (A. 17a) Xc = X - Vx 0T + AX/OJ2 ; and (A.17b) Yc = Y - Vy 0T + Ay/w 2 . 1 36 APPENDIX B - USING THE PACKAGE AT UBC The package i s s t o r e d under the CCID "LKHA". There a r e 6 f i l e s p e r m i t t e d t o p u b l i c f o r those whose want t o t e s t run the system. They are • LKHA:RUN.LOG - a t e r m i n a l l o g f i l e c o n t a i n i n g sample runs of the v a r i o u s phases of the package. Comments d e s c r i b i n g the v a r i o u s s t a g e s are a l s o i n c l u d e d . • LKHA:O.LIB - L i b r a r y o b j e c t f i l e t h a t c o n t a i n s v a r i o u s r o u t i n e s r e q u i r e d by most phases of the package. T h i s f i l e must be i n c l u d e d f o r l i b r a r y s e a r c h ( l i n k e d ) b e f o r e r u n n i n g the package. • LKHA:0.CNTR - O b j e c t f i l e f o r the n o i s e r e d u c t i o n and t r a c e r c e n t e r i n g phase. • LKHA:0.STK - O b j e c t f i l e f o r the s t r e a k c o n n e c t i o n phase. • LKHA:0.ANA - O b j e c t f i l e f o r the f i e l d i n t e r p o l a t i o n and r e c o g n i t i o n phase. The p a r a m e t r i z a t i o n phase i s i n c l u d e s as a s u b r o u t i n e i n t h i s f i l e as we merged the r e c o g n i t i o n and p a r a m e t r i z a t i o n p a r t i n the package. U s e r s can s e l e c t whether they want the p a r a m e t r i z a t i o n a n a l y s i s d u r i n g the r u n . • LKHA:PRIMARY - Sample d a t a f i l e t h a t i s t r a n s f e r r e d t o MTS from the microcomputer by AMIE. T h i s i s the i n p u t f i l e f o r LKHA:0.CNTR. Moreover, s e v e r a l system r o u t i n e s must a l s o be l i n k e d b e f o r e r u n n i n g the package, these i n c l u d e the g r a p h i c s package *IG, the IMSL double p r e c i s i o n l i b r a r y IMSL:0.9D and a l s o the main l i b r a r y *LIBRARY ( u s u a l l y l i n k e d a u t o m a t i c a l l y by MTS). For r e a d e r s i n t e r e s t e d i n the sou r c e code of the package, they s h o u l d r e f e r t o P r o f e s s o r Boye A h l b o r n of the P h y s i c s Department. The c o s t s of r u n n i n g the v a r i o u s p a r t s of the package and t h e i r 10 assignments can be found i n the l o g f i l e . 

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