@prefix vivo: . @prefix edm: . @prefix ns0: . @prefix dcterms: . @prefix skos: . vivo:departmentOrSchool "Science, Faculty of"@en, "Physics and Astronomy, Department of"@en ; edm:dataProvider "DSpace"@en ; ns0:degreeCampus "UBCV"@en ; dcterms:creator "Lau, Alexis Kai-Hon"@en ; dcterms:issued "2010-06-20T16:53:24Z"@en, "1986"@en ; vivo:relatedDegree "Master of Science - MSc"@en ; ns0:degreeGrantor "University of British Columbia"@en ; dcterms:description """This paper describes an efficient method for extracting quantitative data from time sequences of fluid flow images. It also introduces a computer automated method for the identification of coherent structures in turbulent flow fields. This method eliminates subjective manual judgement in the crucial stage of coherent structure recognition. The surface motion on turbulent grid flow (Reynolds number 10") was visualized by recording images of tracer particles on a video tape. Each video frame was then digitized as a binary image using a microcomputer. By tracking and connecting the tracer paths through successive frames using a mainframe computer the flow history was reconstructed. Streak trajectories were then fitted by polynomials to give various flow parameters of interest over desired flow times. In particular the linear and angular velocities were determined as scatter points from which mesh fields were interpolated. Coherent structures were identified by thresholding the field of angular velocity. Using the interpolated mesh fields of linear velocities, each coherent structure was parametrized with the properties of size, average linear and angular velocities, and energy content. The flow dynamics and interactions are then discussed using these structure properties. The system was developed primarily to enhance data recognition for a new model of turbulence based on the energetics of coherent structures. It is also intended as a general technique to be used for other flow visualization and coherent structure studies. Applying the system to the initial stage of grid turbulence, it successfully recognized over 80% of all the coherent structures manually identified. Parameter results using these automatically identified structures were compared with established results and model predictions. Limitation and possible improvement on the present two dimensional system is discussed. Various aspects in extending the system to a three dimensional environment are also presented."""@en ; edm:aggregatedCHO "https://circle.library.ubc.ca/rest/handle/2429/25883?expand=metadata"@en ; skos:note "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 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 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 , 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 = 34°. The same parameter was measured from the p l o t s and we found tantf> = 0.64 ± 0.08 or = 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 . "@en ; edm:hasType "Thesis/Dissertation"@en ; edm:isShownAt "10.14288/1.0096701"@en ; dcterms:language "eng"@en ; ns0:degreeDiscipline "Physics"@en ; edm:provider "Vancouver : University of British Columbia Library"@en ; dcterms:publisher "University of British Columbia"@en ; dcterms:rights "For non-commercial purposes only, such as research, private study and education. Additional conditions apply, see Terms of Use https://open.library.ubc.ca/terms_of_use."@en ; ns0:scholarLevel "Graduate"@en ; dcterms:title "Automated two dimensional flow visualization and coherent structure recognition"@en ; dcterms:type "Text"@en ; ns0:identifierURI "http://hdl.handle.net/2429/25883"@en .