UBC Theses and Dissertations

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UBC Theses and Dissertations

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

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STATISTICS OF COHERENT STRUCTURES IN TURBULENT FLUID FLOW by STUART REID LOEWEN B.Sc.,the U n i v e r s i t y Of Manitoba,1980 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE in THE FACULTY OF GRADUATE STUDIES Phy s i c s Department We accept t h i s t h e s i s as conforming to the r e q u i r e d standard THE UNIVERSITY OF BRITISH COLUMBIA December 1983 © STUART REID LOEWEN, 1983 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 requirements f o r an advanced degree a t the U n i v e r s i t y o f 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 study. 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 copying o f 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 granted by the head o f my department o r 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 understood t h a t copying 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 allowed without my w r i t t e n p e r m i s s i o n . V Department o f ftht^-s \c<>  The U n i v e r s i t y of B r i t i s h Columbia 1956 Main Mall Vancouver, Canada V6T 1Y3 Date 30 f W fern K«_r i i A b s t r a c t T h i s t h e s i s c o m p a r e s p o w e r s p e c t r a p r o d u c e d f r o m h o t -f i l m a n e m o m e t e r y m e a s u r e m e n t s i n t u r b u l e n t f l o w w i t h t h o s e g e n e r a t e d f r o m t h e d i s t r i b u t i o n o f e d d y s i z e s o b t a i n e d f r o m p h o t o g r a p h s o f t h e s a m e f l o w . T h e c o h e r e n t s t r u c t u r e s w e r e m a d e v i s i b l e b y p h o t o g r a p h i n g t h e p a t h s o f a l u m i n u m t r a c e r s o n t h e s u r f a c e o f a w a t e r f i l l e d t o w i n g t a n k . A p a r a l l e l r o w o f b a r s w a s u s e d t o g e n e r a t e t h e t u r b u l e n t f l o w . F l o w R e y n o l d ' s n u m b e r s , b a s e d o n b a r s p a c i n g , o f a b o u t 2 0 , 0 0 0 w e r e s t u d i e d . A s i m p l e a n a l y s i s o f t h e f l o w p a t t e r n s p h o t o g r a p h e d p r o v e d a d e q u a t e t o p r e d i c t p o w e r s p e c t r a c o n s i s t e n t w i t h t h o s e m e a s u r e d u s i n g a h o t - f i l m a n e m o m e t e r a n d s p e c t r u m a n a l y z e r . T h e f r e q u e n c y r a n g e a n d s h a p e o f t h e e d d y p o w e r s p e c t r a w e r e s i m i l a r t o t h e h o t - f i l m o n e s b u t w e r e c o n s i s t e n t l y l o w e r i n m a g n i t u d e . T h e i n t e g r a l l e n g t h s c a l e s o b t a i n e d f r o m t h e p o w e r s p e c t r a a g r e e w i t h i n 3 0 % a n d i t w a s f o u n d t h a t t h e a v e r a g e e d d y s i z e p r e d i c t e d t h e l e n g t h s c a l e o b t a i n e d f r o m t h e e d d y p r o d u c e d p o w e r s p e c t r a . S u c c e s s i v e p h o t o g r a p h s o f t h e f l o w a s w e l l a s t h e c h a n g e i n t h e e d d y s i z e d i s t r i b u t i o n s w i t h d i s t a n c e f r o m t h e g r i d s h o w e d e v i d e n c e o f e d d y e v o l u t i o n . T h e r e s u l t s o f t h i s t h e s i s s u g g e s t t h a t n e w k n o w l e d g e a n d u n d e r s t a n d i n g o f t h e i n t e r a c t i o n o f c o h e r e n t s t r u c t u r e s i n a t u r b u l e n t f l u i d f l o w c a n b e o b t a i n e d f r o m s u c h 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 s t u d y i s p a r t o f a n e w m o d e l f o r t u r b u l e n c e i n w h i c h t h e f l o w i s d e s c r i b e d a s a s u p e r p o s i t i o n o f c o h e r e n t l y • • • r o t a t i n g eddies and laminar flow. In t h i s model we attempt to d e r i v e macroscopic propertieso* r c t.turbulent flow from the i n t e r a c t i o n s of eddies with the f l u i d , f l u i d flow and each o t h e r . The eddy s i z e d i s t r i b u t i o n can a l s o ^ p r e d i c t e d from an understanding of the eddy i n t e r a c t i o n s . The s t a t i s t i c a l d e s c r i p t i o n of turbulence and the o f t e n used k -power spectrum approach are reviewed and compared with our new model. i v Table of Contents A b s t r a c t i i L i s t of F i g u r e s v L i s t of F i g u r e s . v i Acknowledgement v i i Chapter I INTRODUCTION 1 Chapter II STATISTICAL DESCRIPTION OF TURBULENCE 9 2.1 M o t i v a t i o n And I n t r o d u c t i o n 9 2.2 S t a t i s t i c a l Spectra 11 2.3 C o r r e l a t i o n s And Spectra 16 2.4 V e l o c i t y F l u c t u a t i o n s And The Power Spectrum 23 Chapter I I I GRID TURBULENCE EXPERIMENTS 27 3.1 I n t r o d u c t i o n 27 3.2 The Tank 29 3.3 The G r i d 32 3.4 V i s u a l i z a t i o n Apparatus 32 3.5 The Search For Coherent S t r u c t u r e s 34 3.6 H o t - f i l m Apparatus ...44 3.7 H o t - f i l m Power Spectra 48 Chapter IV EDDY-SIZE DISTRIBUTIONS AND HOT-FILM SPECTRA 56 4.1 I n t r o d u c t i o n 56 4.2 Eddy-Size Spectra 58 4.3 Generation Of E (w) From N(R) 67 4.4 Power Spectrum Obtained From U(t) 71 4.5 R e s u l t s And D i s c u s s i o n 81 Chapter V CONCLUSIONS 93 BIBLIOGRAPHY 96 APPENDIX A - HOT-FILM PROBE SENSITIVITY TO VELOCITY FLUCTUATIONS i 97 APPENDIX B - RIGID BODY EDDY VELOCITY PROFILE 98 APPENDIX C - TURB.FTN FORTRAN CODE 99 APPENDIX D - AVERAGE EDDY CHORD AND PEDDY CALCULATION ...102 V L i s t of F i g u r e s 1. C o r r e l a t i o n V e l o c i t i e s 17 2. S p a t i a l C o r r e l a t i o n Curve 18 3. Flow F i e l d Past Probe 20 4. The Towing Tank 28 5. O p t i c a l Detector and Timer 31 6. L i g h t s and Camera A c t i o n 33 7. Flow V i s u a l i z a t i o n i n G r j d and F l u i d Frames 36 8. V i s u a l i z a t i o n s at D i f f e r e n t Shutter Speeds 37 9. 20cm/sec Flow Photographs 40 10. 30cm/sec Flow Photographs 41 11. 40cm/sec Flow Photographs 42 12. 50cm/sec Flow Photographs 43 13. Hot-Film Power Spectrum Schematic 46 14. H o t - F i l m L i n e a r i z a t i o n 51 15. H o t - F i l m O s c i l l o g r a m s 52 16. Probe Support Resonance 55 17. T y p i c a l Analyzer X-Y P l o t 55 18. Power Spectra Comparison 57 19. Eddy A n a l y s i s 59 20. Eddy Angular V e l o c i t i e s 62 21. 30cm/sec Eddy-Size Spectra 63 22. 40cm/sec Eddy-Size Spectra 65 23. N(R) to E (w) Computation 68 24. u ( t ) Generated from the Eddy D i s t r i b u t i o n 70 25. Eddy Power Spectra 72 v i 26. Eddy Power S p e c t r a : V a r i a t i o n with D i s t a n c e 73 27. Spectrum. Analyzer Operation 76 28. A n a l y z e r X-Y P l o t 78 29. H o t - F i l m Power S p e c t r a : V a r i a t i o n with D i s t a n c e 7 9 30. Eddy P a i r i n g 82 31. Comparison of Length S c a l e s 89 32. Eddy Count with D i s t a n c e 89 33. V a r i a t i o n i n Occupied Area 90 34. Decay of Ti* 90 35. Comparison of Power Spectra 91 36. E q u i v a l e n t Area R e p r e s e n t a t i o n 92 v i i Acknowledgement I would l i k e to thank Paul B u r r e l who machined and designed t h i n g s t h a t worked and A l Cheuck f o r b u i l d i n g the timer and g e t t i n g s u p p l i e s "now would be b e s t " . H e l p i n g hands and minds of summer a s s i s t a n t s Norman Lo, who made the tank that " w i l l not l e a k " (and e v e n t u a l l y d i d n o t ) , Alex F i l u i k who got the v i s u a l i z a t i o n system underway and Peter Smereka of "Fine Idea" fame. The loan of the CTA u n i t from Dr.Quick of the UBC C i v i l E n g i n e e r i n g Dept. i s much a p p r e c i a t e d . Although not d i r e c t l y r e f e r e n c e d the t e a c h i n g of Dr.I.Gartshore of the UBC Mechanical E n g i n e e r i n g Dept. p r o v i d e d a s o l i d a p p r e c i a t i o n f o r what i s known about f l u i d f lows. I would a l s o l i k e to thank the UBC Plasma P h y s i c s group f o r c o n t i n u i n g to work one f l o o r beneath the tank f i l l e d with three tons of water. F i n a l l y I would l i k e to thank my s u p e r v i s o r Dr.B.Ahlborn f o r h i s e n t h u s i a s t i c help i n both the r e s e a r c h and w r i t i n g aspects of t h i s t h e s i s . 1 I. INTRODUCTION T h i s t h e s i s i n v e s t i g a t e s the r o l e of coherent s t r u c t u r e s i n a t u r b u l e n t f l u i d flow. I t i s supposed that a t u r b u l e n t flow can be d e s c r i b e d as a s u p e r p o s i t i o n of c o h e r e n t l y r o t a t i n g f l u i d elements, 'eddies', p l u s laminar flow. These eddies are taken as the b u i l d i n g b l o c k s of the f l u c t u a t i n g flow f i e l d . Experiments performed to t e s t the v a l i d i t y and v i a b i l i t y of t h i s model are presented. The i n d i v i d u a l eddies i n t e r a c t with the f l u i d , the l o c a l flow f i e l d and with each other a c c o r d i n g to the f l u i d equations of motion. We c h a r a c t e r i z e the t u r b u l e n t f l u i d flow by the eddy-size spectrum which i s the p r o b a b i l i t y of seeing an eddy of a given s i z e at a p o i n t i n space. T h i s p r o b a b i l i t y i s determined by the mechanics of the i n t e r a c t i o n p r o c e s s e s . The model i s thus d e t e r m i n i s t i c on small time and space s c a l e s while being s t o c h a s t i c on l a r g e r time and space s c a l e s . T h i s approach has the p o t e n t i a l of d e s c r i b i n g small s c a l e coherence as w e l l as l a r g e s c a l e u n p r e d i c t a b i l i t y which are s i g n i f i c a n t p r o p e r t i e s of t u r b u l e n t flows. (e.g. the weather) In order to develop an understanding of the eddy dynamics and s t a t i s t i c s the r e l a t i o n s h i p between our eddy-s i z e spectrum and the w e l l known power s p e c t r a l d e n s i t y d e s c r i p t i o n of the f l u c t u a t i n g flow f i e l d was s t u d i e d . The power s p e c t r a l d e n s i t y i s a measure of the energy content i n the f l u c t u a t i n g flow as a f u n c t i o n of temporal frequency. I t can be r e l a t e d to the wave number spectrum through the 2 c o n v e c t i o n v e l o c i t y . The wavenumber spectrum g i v e s the energy content i n the f l u c t u a t i n g flow f i e l d as a f u n c t i o n of s c a l e s i z e . How these s p e c t r a are used i n the s t a t i s t i c a l d e s c r i p t i o n of t u r b u l e n t v e l o c i t y f l u c t u a t i o n s i s d e s c r i b e d i n the second c h a p t e r . The r e l a t i o n s h i p between the eddy-size d i s t r i b u t i o n and the power s p e c t r a l d e n s i t y was s t u d i e d f o r a number of reasons. The power s p e c t r a l d e n s i t y i s c o n v e n i e n t l y obtained using a constant temperature anemometer and spectrum a n a l y z e r and has long been used to study the l e n g t h s c a l e s present i n t u r b u l e n t flows. I t s r e l a t i o n s h i p to the coherent s t r u c t u r e s i n the t u r b u l e n t flow i s n e i t h e r i n t u i t i v e nor c l e a r l y d e f i n e d . Our s t a t i s t i c a l a n a l y s i s of the t u r b u l e n t elements on the other hand i s simple i n concept, but s t i l l • somewhat s u b j e c t i v e i n the a n a l y s i s . T h i s t h e s i s w i l l show how the power s p e c t r a l d e n s i t y can be obtained from both the eddy-size d i s t r i b u t i o n and from hot-f i l m anemometery measurements. In comparing the procedures, assumptions and the r e s u l t s we hope to convince the reader of the u s e f u l n e s s of our approach. A computer code was w r i t t e n and used to generate power s p e c t r a from eddy s p e c t r a . T h i s procedure i s d e s c r i b e d i n d e t a i l . An experimental study of g r i d generated t u r b u l e n c e i n water was performed. T h i s mechanism f o r g e n e r a t i n g t u r b u l e n t flow was s e l e c t e d so as to o b t a i n a l a r g e volume of t u r b u l e n t flow with s p a t i a l l y homogeneous p r o p e r t i e s . I t was a l s o a n t i c i p a t e d that the g r i d geometery would c r e a t e a 3 flow that would evolve independently of boundary c o n d i t i o n s . A towing tank was used as i t allowed f o r easy access to both the f l u i d and the o b j e c t r e f e r e n c e frames f o r photographic and h o t - f i l m anemometry s t u d i e s . The flow was v i s u a l i z e d by photographing the motion of t r a c e r p a r t i c l e s on the s u r f a c e of the water. With the camera mounted i n the f l u i d r e f e r e n c e frame we were p l e a s a n t l y s u r p r i s e d by the p r o f u s i o n of coherent s t r u c t u r e s of a broad s i z e range. The photographs were analyzed to o b t a i n the eddy-size d i s t r i b u t i o n s as w e l l as the angular v e l o c i t i e s of the e d d i e s . Although of l a r g e u n c e r t a i n t y the angular v e l o c i t y data was c o n s i s t e n t with our t r e a t i n g the eddies as performing s o l i d body r o t a t i o n . The same flow s i t u a t i o n s were s t u d i e d by measuring one dimensional power s p e c t r a using hot wire-anemometery and a spectrum a n a l y z e r . Power s p e c t r a l d e n s i t y curves were p r e d i c t e d from the eddy s p e c t r a and compared with those more d i r e c t l y obtained using a h o t - f i l m anemometer and spectrum a n a l y z e r . I n s p i r a t i o n f o r studying coherent s t r u c t u r e s using the towing tank came from photographic r e s u l t s o b tained by F.Ahlborn. 1 While the main t o p i c of t h i s t h e s i s i s the comparison of d i f f e r e n t s t a t i s t i c a l d e s c r i p t i o n s of t u r b u l e n t flow i t i s an i n t e g r a l p a r t of a broader concept developed by 1 F.Ahlborn:"Turbulenz und Mechanismus desWiderstandes an Kugeln und Z y l i n d e r n " , Z e i t s c h r i f t f u r Technische Physik J_2, 482-491 ( 1 9 3 1 ) . 4 B.Ahlborn and m y s e l f . 1 We attempt to e x p l a i n many f a c e t s of turbule n c e based on the i n t e r a c t i o n s of i d e a l i z e d coherent elements i n f l u i d flow. We c a l l these i d e a l i z e d elements "v o r t o n s " . C e n t r a l to the vorton model i s the p r e d i c t i o n of the edd y - s i z e spectrum from the vorton dynamics. I t was t h e r e f o r e important f o r the success of the vort o n model that the coherent s t r u c t u r e s c o u l d be observed and counted by s i z e . I t was a l s o important to show- that the i n f o r m a t i o n c o n t a i n e d i n the eddy d i s t r i b u t i o n was c o n s i s t e n t with a power s p e c t r a l d e n s i t y d e s c r i p t i o n of the t u r b u l e n t f l u c t u a t i o n s . A ge n e r a l i n t r o d u c t i o n to the vorton model i s presented as i t p r o v i d e d the m o t i v a t i o n f o r performing the present s t a t i s t i c a l study. "Big whorls have l i t t l e whorls, Which feed on t h e i r v e l o c i t y ; L i t t l e whorls have s m a l l e r whorls, And so on unto v i s c o s i t y . " T h i s i n f o r m a t i o n laden poem was w r i t t e n by L.F.Richardson, (1922). I t i n t r o d u c e s s e v e r a l concepts which our vorton model addresses. F i r s t l y , t u r b u l e n t f l u i d flow i s c h a r a c t e r i z e d by the presence of a broad range i n s i z e s c a l e s of the motion. Our model i n c o r p o r a t e s t h i s as 1 UBC PLASMA PHYSICS l a b r e p o r t #89 5 the presence of d i f f e r e n t sizes of eddies in the flow. Secondly, turbulent flows are generally thought to feed energy into large eddies through shear stresses in the mean flow f i e l d . The st r a i n produced by the larger eddies in turn feeds energy into the smaller scales, t y p i c a l l y by "vortex stretching" or "tearing". The whole process i s commonly c a l l e d the "energy cascade". Our model starts with the assumption that a turbulent flow f i e l d can be described as being a composite of eddies and laminar flow. We then define rates for an eddy's interaction with the f l u i d , the laminar flow f i e l d , and other eddies. These interactions are found to dominate the eddy dynamics in the small scale or d i s s i p a t i v e regime, the large scale regime and the medium eddy size or c o l l i s i o n a l regime, respectively. We then use these rates to solve for 'the d i s t r i b u t i o n of eddy size scales. This d i s t r i b u t i o n we c a l l the eddy-size spectrum. Richardson's poem brings out that the energy transfer ultimately ends in producing heat through viscous d i s s i p a t i o n . In our model t h i s i s quantified according to the energy loss rate through viscous d i s s i p a t i o n as a function of eddy si z e . We also show how this spectrum can be used to predict macroscopic properties of a p a r t i c u l a r turbulent flow. These properties include drag forces, wake sizes, d i f f u s i o n processes, the onset of turbulence and mean ve l o c i t y p r o f i l e s . The name vorton i s chosen to indicate that we are dealing with a v o r t i c a l structure which contains energy and 6 can i n t e r a c t w i t h i t s f l o w environment. V o r t o n s a re t r e a t e d as b e i n g e x c i t e d s t a t e s of a f l u i d f l o w . The word eddy i s sometimes used i n t e r c h a n g a b l y w i t h ' v o r t o n . The e d d y - s i z e spectrum i s i n t e r p r e t e d as the p r o b a b i l i t y of o b s e r v i n g an eddy as a f u n c t i o n of eddy s i z e . T h i s spectrum i s c e n t r a l t o our d e s c r i p t i o n of t u r b u l e n c e . I t can be a f u n c t i o n of space as w e l l as t i m e . S u p p o r t i n g the e d d y - s i z e spectrum a r e o t h e r p r o p e r t i e s of the e d d i e s , such as t h e i r i n t e r n a l v e l o c i t y d i s t r i b u t i o n s , c o n v e c t i o n v e l o c i t i e s b oth i n the fl o w d i r c t i o n and normal t o i t , and i f we a r e c o n s i d e r i n g t h r e e d i m e n s i o n a l f l o w , t h e i r o r i e n t a t i o n s i n space. I t i s supposed t h a t j u s t as the e d d i e s can be i s o l a t e d and c h a r a c t e r i z e d , so can p h y s i c a l p r o c e s s e s which produce, e v o l v e , and f i n a l l y d e s t r o y the e d d i e s . In the f i r s t a p p l i c a t i o n s of the model the v o r t o n s have a c i r c u l a r c y l i n d r i c a l symmetry and are t r e a t e d as b e i n g p a r t of a two d i m e n s i o n a l f l o w . We d i v i d e the p o s s i b l e i n t e r a c t i o n s a v o r t o n can t a k e p a r t i n i n t o t h r e e main t y p e s . A v o r t o n may i n t e r a c t w i t h the f l u i d s u r r o u n d i n g i t , w i t h the f l u i d f l o w i n which i t i s embedded or w i t h o t h e r v o r t o n s . The r a t e a t which a g i v e n v o r t o n s t a t e changes, t o o t h e r s t a t e s , i s d e s c r i b e d by a r a t e e q u a t i o n w i t h c o e f f i c i e n t s q u a n t i f y i n g the v a r i o u s i n t e r a c t i o n p r o c e s s e s . We use A c o e f f i c i e n t s t o d e s c r i b e the r a t e a t which the number of e d d i e s of a g i v e n s i z e w i l l s p o n t a n e o u s l y decay t o e d d i e s of a d i f f e r e n t s i z e t h r o u g h v i s c o u s i n t e r a c t i o n w i t h the f l u i d s u r r o u n d i n g i t . B c o e f f i c i e n t s a r e used t o d e s c r i b e the r a t e a t which a 7 p o p u l a t i o n of e d d i e s of a g i v e n s i z e w i l l e i t h e r i n c r e a s e or d e c r e a s e due t o i n t e r a c t i o n w i t h the l o c a l f l u i d f l o w f i e l d . T h i s r a t e i s goverened by t e a r i n g and v o r t e x s t r e t c h i n g dynamics. C c o e f f i c i e n t s a r e used t o d e s c r i b e the p o p u l a t i o n changes due t o e d d i e s i n t e r a c t i n g w i t h o t h e r e d d i e s , i . e . c o l l i s i o n a l p r o c e s s e s . These r a t e c o e f f i c i e n t s a r e supposed d e r i v a b l e from the l o c a l p h y s i c a l p r o c e s s e s which are goverened by the f l u i d e q u a t i o n s of m o t i o n . The C c o e f f i c i e n t s i n c o r p o r a t e c o l l i s i o n a l s t a t i s t i c s as w e l l . A l t h o u g h d e r i v e d from l o c a l d e t e r m i n i s t i c p h y s i c s these c o e f f i c i e n t s are t r e a t e d as p r o b a b i l i t i e s w hich determine the p o p u l a t i o n s of energy s t a t e s . I f the r a t e s a r e known one can w r i t e a complete s e t of r a t e e q u a t i o n s , one f o r each v o r t o n s t a t e i n the system, t o p r e d i c t the e d d y - ( o r v o r t o n ) s i z e spectrum. We t h u s see our model as a q u a n t i f i c a t i o n of the eddy cascade p r o c e s s . I t i s i m p o r t a n t t o note t h a t i n a t u r b u l e n t f l u i d f l o w t h e i r w i l l i n g e n e r a l be a c o n t i n u o u s d i s t r i b u t i o n of eddy e n r g i e s and s i z e s . A l t h o u g h i n i t s i n f a n t s t a g e s the v o r t o n model o n l y d e a l s w i t h d i s c r e t e s i z e s and e n e r g i e s a c o n t i n u o u s v e r s i o n c o u l d be based on s i m i l a r p r i n c i p a l s . t h e same We use the v o r t o n d e s c r i p t i o n t o model the f l o w i n the hope t h a t our i d e a l i z a t i o n can d e s c r i b e t h e phenomenon and has some r e l a t i o n t o the p h y s i c s of a r e a l t u r b u l e n t f l o w . I t i s g e n e r a l l y a c c e p t e d t h a t d i f f e r e n t p h y s i c a l p r o c e s s e s dominate the f l o w dynamics i n d i f f e r e n t s c a l e regimes. S t a n d a r d t e x t s on t u r b u l e n c e d e s c r i b e the l a r g e 8 scale motions as being associated with energy transfer from the mean flow with these size scales making up the "production" regime. The mid-size scales which comprise the i n e r t i a l , i n v i s c i d , or Kolmogorov regime are c o l l i s i o n dominated and the small scale motions which comprise the di s s i p a t i v e regime are most e f f i c i e n t at losing energy to heat. In the l i g h t of our model we would say that the B c o e f f i c i e n t s dominate the eddy dynamics in the production regime, the C c o e f f i c i e n t s dominate in the i n e r t i a l regime and the A c o e f f i c i e n t s dominate in the d i s s i p a t i v e regime. In the interest of keeping the model simple and managable in i t s infant stage i t s applications have been confined to two dimensional flow phenomena. One can, in p r i n c i p a l , apply these ideas to three dimensional flow situ a t i o n s . The rate equation approach has been borrowed from the description of atomic ionization phenomena. This approach is a s i g n i f i c a n t departure from main stream turbulence theory. It was conceived and developed by B.Ahlborn and myself in part based on flow photographs and some ideas of F.Ahlborn. 9 I I . STATISTICAL DESCRIPTION OF TURBULENCE 2.1 M o t i v a t i o n And I n t r o d u c t i o n We a s s e r t t h a t the eddy spectrum d e s c r i p t i o n c o n t a i n s a l l t h e i n f o r m a t i o n needed t o r e c r e a t e a p h y s i c a l l y m e a n i n g f u l f l u c t u a t i n g f l o w f i e l d . The eddy spectrum based d e s c r i p t i o n s h o u l d t h u s be s u f f i c i e n t t o p r e d i c t a l l the s t a t i s t i c a l f l u c t u a t i o n s p e c t r a . A d e s c r i p t i o n of these s p e c t r a and how they a p p l y t o the study of t u r b u l e n t f l u i d f l o w i s the s u b j e c t of t h i s c h a p t e r . These s p e c t r a i n c l u d e the power s p e c t r a l d e n s i t y , t h r e e d i m e n s i o n a l wave number spectrum and s p e c t r a based on h i g h e r o r d e r moments of the time or space v a r y i n g v e l o c i t y f i e l d . The o f t e n s t u d i e d c o r r e l a t i o n c u r v e s a r e F o u r i e r t r a n s f o r m s of the above mentioned s p e c t r a and so would be e q u a l l y w e l l p r e d i c t e d by the eddy spectrum d e s c r i p t i o n . The i n t e r - e d d y c o r r e l a t i o n s may or may not be s i g n i f i c a n t , depending on the fl o w s i t u a t i o n s t u d i e d and the eddy s i z e s c a l e b e i n g c o n s i d e r e d . I f we b e l i e v e t h a t the p r i m a r y c o h e r e n t s t r u c t u r e s r e p r e s e n t a l l of the c o r r e l a t i o n s p r e s e n t i n the f l o w , m e a n i n g f u l f l o w f i e l d s s h o u l d be r e p r o d u c i b l e from the eddy spectrum d e s c r i p t i o n . T h i s d e s c r i p t i o n s h o u l d a l s o p r o v i d e p r e d i c t i o n s of the m a c r o s c o p i c f l o w p r o p e r t i e s such as d r a g , wake s i z e s , d i f f u s i o n p r o c e s s e s , t u r b u l e n c e onset and mean v e l o c i t y p r o f i l e s . A good t e s t of t h e v a l i d i t y and v i a b i l i t y of a s t a t i s t i c a l l y based c o h e r e n t s t r u c t u r e s model of t u r b u l e n t 10 f l o w would be t o see i f i t p r e d i c t s the c o r r e c t power s p e c t r a l d e n s i t y c u r v e s f o r a g i v e n f l o w . The power s p e c t r a l d e n s i t y i s a measure of the energy c o n t e n t per u n i t mass a s s o c i a t e d w i t h the t e m p o r a l f r e q u e n c y . T h i s spectrum can be o b t a i n e d by F o u r i e r a n a l y z i n g a t ime dependent v e l o c i t y s i g n a l . In an e x p e r i m e n t a l study of t u r b u l e n c e the time v a r y i n g q u a n t i t y c o u l d be the streamwise component of the f l u c t u a t i n g v e l o c i t y f i e l d measured a t a f i x e d p o s i t i o n i n r e l a t i o n t o the o b j e c t which i s c r e a t i n g the t u r b u l e n c e . T h i s d e s c r i p t i o n of the f l u c t u a t i n g v e l o c i t y f i e l d c o n t a i n s i n f o r m a t i o n on the eddy cascade dynamics and the eddy s t a t i s t i c s o n l y i m p l i c i t l y . In the words of J.L.Lumley, "An eddy, however, i s a s s o c i a t e d w i t h many F o u r i e r c o e f f i c i e n t s and the phase r e l a t i o n s among them. F o u r i e r t r a n s f o r m s a r e used because they a r e c o n v e n i e n t ( s p e c t r a can be measured e a s i l y ) ; more s o p h i s t i c a t e d t r a n s f o r m s a r e needed i f one wants t o decompose a v e l o c i t y f i e l d i n t o e d d i e s i n s t e a d of waves." 1 Our approach i s t o p r e d i c t t u r b u l e n t e f f e c t s from the d i s t r i b u t i o n of the e d d i e s . T h i s c h a p t e r d e s c r i b e s the s t a t i s t i c a l f l u c t u a t i o n s p e c t r a and how the power s p e c t r a l d e n s i t y can be o b t a i n e d from a f l u c t u a t i n g v e l o c i t y . 1 Lumley,J.L.,1970. S t o c h a s t i c t o o l s i n t u r b u l e n c e . A c a d e m i c Press,New York. 11 2.2 S t a t i s t i c a l S p e c t r a In t h e s t a t i s t i c a l s tudy of t u r b u l e n t f l u c t u a t i o n s one t r i e s t o p r e d i c t and de t e r m i n e the v e l o c i t y f l u c t u a t i o n s p e c t r a o r , e q u i v a l e n t l y , the c o r r e l a t i o n s . There a r e many t e x t s which d e s c r i b e t h i s approach i n d e t a i l . Here I w i l l p r e s e n t enough i n f o r m a t i o n t o show how the power s p e c t r a l d e n s i t y f i t s i n a more complete s t a t i s t i c a l d e s c r i p t i o n of t u r b u l e n t f l u c t u a t i o n s and how i t can be o b t a i n e d and used t o d e s c r i b e t u r b u l e n t f l o w s . The s t a r t i n g p o i n t of the s t a t i s t i c a l d e s c r i p t i o n of t u r b u l e n c e i s t o c o n s i d e r the time dependent v e l o c i t y U ( x , t ) and p r e s s u r e P ( x , t ) f i e l d s as the sum of st e a d y and unsteady components. Ui(x,t)=U k(x)+u k(x\t) i = 1,2,3 (2-1'a) P(x,t)=P(x*)+p(x\t) (2-1b) Here U^x) and TA( 5c, t ) a re the time averaged q u a n t i t i e s and u L(x,t) and p„(x,t) are the f l u c t u a t i n g components. The overba r symbol i s used t o denote a time a v e r a g e . For some time dependent q u a n t i t y a ( t ) , "a~(t) i s found a s , a ( t ) = l i m -!? (dt'a(t') (2-2) I t i s m e a n i n g f u l t o speak of time dependent time averages i f ( d / d t ) [ a ( t ) ] «^(d/dt)[a(t) ]* (2-3) which s t a t e s t h a t the averages must change much more s l o w l y than the average change. In the r e s t of t h i s t h e s i s i t w i l l be assumed t h a t the f l o w f i e l d s b e i n g d e a l t w i t h a r e 12 s t a t i s t i c a l l y s t e a d y . 1 The r e p r e s e n t a t i o n ( 2 - 1 ) f o r t h e f l u i d f l o w a n d p r e s s u r e f i e l d s i s n e x t s u b s t i t u t e d i n t o t h e a p p r o p r i a t e f l u i d e q u a t i o n s o f m o t i o n t o o b t a i n e q u a t i o n s f o r t h e mean k i n e t i c e n e r g y , t u r b u l e n t 2 k i n e t i c e n e r g y , a n d t h e R e y n o l d s n o r m a l a n d s h e a r s t r e s s e q u a t i o n s . F o r i n c o m p r e s s i b l e , i s o t h e r m a l , v i s c o u s f l u i d f l o w t h e a p p r o p r i a t e e q u a t i o n s o f m o t i o n a r e t h e w e l l known N a v i e r - S t o k e s e q u a t i o n s . I n c o m p o n e n t f o r m w i t h t h e s u m m a t i o n c o n v e n t i o n b e i n g u s e d t h e s e e q u a t i o n s a r e i J J i . Uj aUj ( - 1 /j> )± P . 9^_U- k (2 - 4 ) w i t h t h e c o n t i n u i t y c o n d i t i o n AU4 =0 ( 2 - 5 ) The t i m e a v e r a g e d N a v i e r - S t o k e s e q u a t i o n s a r e f o u n d b y s u b s t i t u t i n g e q n s . ( 2 - 1 ) i n t o e q n . ( 2 - 4 ) a n d t i m e a v e r a g i n g t h e r e s u l t i s ( 2 - 6 ) W h e r e I I U i ( _ _ A U ; . U j ^_Ut ) i s t h e t i m e r a t e o f c h a n g e o f U-L D t v~<Jt + ^ 7 f o l l o w i n g t h e f l u i d m o t i o n . M u l t i p l y i n g t h i s e q u a t i o n by U t , u s i n g t h e c o n t i n u i t y c o n d i t i o n , e q n . ( 2 - 5 ) , t o 1 S t a t i s t i c a l l y s t e a d y i s u s e d h e r e t o mean t i m e a v e r a g e d q u a n t i t i e s do n o t c h a n g e i n t i m e . 2 A l e s s m i s l e a d i n g name f o r t h i s e q u a t i o n w o u l d be t h e f l u c t u a t i o n k i n e t i c e n e r g y 13 c o n s o l i d a t e v e l o c i t y components i n s i d e the d e r i v a t i v e s 1 , and r e c o g n i z i n g t h a t Ui J^ LIj. = D(iUi U t ) we a r r i v e a t the mean Ut ot k i n e t i c energy e q u a t i o n f o r t u r b u l e n t f l o w , Dt d*i r T*Z (2-7) (A) ( B ) (C) (D) Term (D) may be r e w r i t t e n as Tx£ ex* ~ V A X K ' (2 -8) '(D)' (E) ( F ) These e q u a t i o n s s t a t e : The time r a t e of change (A) of mean k i n e t i c energy of a u n i t mass of f l u i d i s e q u a l t o (B) the p r e s s u r e work done on the u n i t mass, p l u s (C) the t r a n s p o r t of mean energy by o r t h o g o n a l v e l o c i t y f l u c t u a t i o n c o r r e l a t i o n s , p l u s (E) the change due t o v i s c o u s g r a d i e n t d i f f u s i o n , p l u s (F) the mean v i s c o u s d i s s i p a t i o n . The t u r b u l e n t k i n e t i c energy e q u a t i o n i s found by m u l t i p l y i n g the o r i g i n a l N a v i e r - S t o k e s e q u a t i o n s (2 -4) by U: , s u b s t i t u t i n g the e x p r e s s i o n s (2 -1) f o r Ui and P£ and then time a v e r a g i n g the r e s u l t . The mean k i n e t i c energy e q u a t i o n (2 -7) i s then s u b t r a c t e d from t h i s and a f t e r some c o n s o l i d a t i o n the t u r b u l e n t k i n e t i c energy e q u a t i o n r e s u l t s M Sxt \ ' /• / x i AXL ^axj ixy ( 2 - 9 ) (A) (B) (C) (D) 1 i . e . U:JLU/=> <L(U.-Uj )/2 as^ «.i=0 from c o n t i n u i t y 14 The f o l l o w i n g q u o t e , t a k e n from H i n z e , 1 d e s c r i b e s the p h y s i c a l meaning of the terms a p p e a r i n g i n t h i s e q u a t i o n . "The change (A) i n k i n e t i c energy of t u r b u l e n c e - per u n i t of mass of the f l u i d i s e q u a l t o (B) the c o n v e c t i v e d i f f u s i o n by t u r b u l e n c e of the t o t a l t u r b u l e n c e e n e r g y , p l u s (C) the energy t r a n s f e r r e d from the mean motion t h r o u g h the t u r b u l e n c e shear s t r e s s e s , o r t h e p r o d u c t i o n of t u r b u l e n c e energy, p l u s (D) the work done per u n i t of mass and of time by the v i s c o u s shear s t r e s s e s of the t u r b u l e n t m o t i o n , p l u s (E) the d i s s i p a t i o n per u n i t of mass by the t u r b u l e n t m o t i o n . " In a manner s i m i l a r t o t h a t used above, e q u a t i o n s f o r the Reynold's shear (uTuJ , - i ^ j ) and normal (u^u*., no summation) s t r e s s e s can be d e r i v e d . The v e l o c i t y f l u c t u a t i o n c o r r e l a t i o n s i n t h e s e e q u a t i o n s a r e m o s t l y of second o r d e r . E q u a t i o n s i n v o l v i n g h i g h e r o r d e r c o r r e l a t i o n s of the f l u c t u a t i n g v e l o c i t y components are seldom used due t o t h e i r m a t h e m a t i c a l c o m p l e x i t y , i n c o n v e n i e n c e i n measurement and l a c k of i n t u i t i v e p h y s i c a l meaning. One advantage of t r e a t i n g the v e l o c i t y and p r e s s u r e f i e l d s as the sum of a mean and f l u c t u a t i n g component i s t h a t the q u a n t i t i e s a p p e a r i n g i n the r e s u l t i n g e q u a t i o n s a r e 1 TURBULENCE,J.O.Hinze; McGRAW-HILL, 1959, pp 65 15 e x p e r i m e n t a l l y measureable u s i n g h o t - w i r e or l a s e r d o p p l e r anemometery t e c h n i q u e s . 1 Another advantage i s t h a t the i n d i v i d u a l terms have some p h y s i c a l meaning. A major drawback of t h i s approach i s t h a t i n some c a s e s t h i s p h y s i c a l i n t e r p r e t a t i o n can be m i s l e a d i n g . The average f l u i d v e l o c i t y a t a g i v e n p o i n t i n space may be p a r t i a l l y c o m p r i s e d of c o h e r e n t l y a d d i n g f l u c t u a t i o n s . For example, a t u r b u l e n t boundary l a y e r may c o n t a i n c o h e r e n t s t r u c t u r e s w i t h a p r e f e r r e d r o t a t i o n sense. The f l u c t u a t i o n i n v e l o c i t y measured when a s e r i e s of c o h e r e n t s t r u c t u r e s were c o n v e c t e d p a s t an anemometer would c o n t r i b u t e c o n s t r u c t i v e l y t o the measurement of the mean v e l o c i t y . Thus the i n t e r p r e t a t i o n of the mean and f l u c t u a t i n g v e l o c i t i e s must be t r e a t e d w i t h c a u t i o n . In d i s c u s s i n g the mean k i n e t i c energy e q u a t i o n (2-7) the s i g n i f i c a n c e of o r t h o g o n a l f l u c t u a t i n g v e l o c i t y c o r r e l a t i o n s t o the t r a n s p o r t of mean k i n e t i c energy was mentioned. A non-zero c o r r e l a t i o n i m p l i e s a coherence between the two f l u c t u a t i n g v e l o c i t i e s which would p o i n t t o the presence of c o h e r e n t s t r u c t u r e s i n the t u r b u l e n t f l o w . These s t r u c t u r e s , now thought t o be i m p o r t a n t t o the dynamics of most t u r b u l e n t f l o w s a re not d i r e c t l y a d d r e s s e d i n the above approach. I t s h o u l d be noted t h a t these s t r u c t u r e s a r e the s t a r t i n g p o i n t of our In the s t r e s s e q u a t i o n s the p r e s s u r e f l u c t u a t i o n - v e l o c i t y f l u c t u a t i o n c o r r e l a t i o n s a r e not g e n e r a l l y measureable and must be i n f e r r e d from the s t r e s s e q u a t i o n s and measurements of the o t h e r terms. 16 model. The g a i n i n w r i t i n g the v e l o c i t y f i e l d as the sum of mean and f l u c t u a t i n g components i s t h a t t h e • c o r r e l a t i o n s and mean v a l u e s a r e q u a n t i t i e s which have a u s e f u l , a l t h o u g h p o t e n t i a l l y m i s l e a d i n g , p h y s i c a l s i g n i f i c a n c e and they a r e r e a d i l y measured w i t h h o t - w i r e or l a s e r d o p p l e r anemometry t e c h n i q u e s . I t i s i m p o r t a n t t o note t h a t i n w r i t i n g the v e l o c i t y and p r e s s u r e f i e l d s as the sum of the st e a d y and f l u c t u a t i n g components we no l o n g e r have a c l o s e d system of e q u a t i o n s . S i n c e the number of unknowns has d o u b l e d an o r i g i n a l l y c l o s e d d e s c r i p t i o n has now been become i n c o m p l e t e . C l o s u r e hypotheses a r e needed t o s o l v e the new e q u a t i o n s of mo t i o n . 2.3 C o r r e l a t i o n s And S p e c t r a The t u r b u l e n t energy e q u a t i o n c o n t a i n s o n l y mean p r o d u c t s of f l u c t u a t i n g q u a n t i t i e s a t one p o i n t i n space. To study the l e n g t h s c a l e s of the t u r b u l e n t f l u c t u a t i o n s we need t o c o n s i d e r f l u c t u a t i n g q u a n t i t i e s which a r e measured at d i f f e r e n t p o i n t s i n space. U s u a l l y no more than two-p o i n t c o r r e l a t i o n s a r e c o n s i d e r e d . The most g e n e r a l , s t a t i s t i c a l l y s t e a d y , t w o - p o i n t s p a t i a l c o r r e l a t i o n between f l u c t u a t i n g v e l o c i t y components may be w r i t t e n CijCx, r )=Uj(x,t)ui(x+? , t ) i , j = 1,2,3 (2-10) where X " i s the p o s i t i o n where the c o r r e l a t i o n i s d e f i n e d , u; i s the f l u c t u a t i n g component of the v e l o c i t y i n the i d i r e c t i o n , and x+r i s t h e p o s i t i o n where t h e j t h v e l o c i t y 17 component i s measured, see f i g u r e 1. F i g u r e 1 - C o r r e l a t i o n V e l o c i t i e s The n o n - d i m e n s i o n a l i z e d c o r r e l a t i o n i s c a l l e d the c o r r e l a t i o n c o e f f i c i e n t and i s g i v e n as R M ( x \ f )= ut (g.t)u-E (*+t ,t) (2-11) Jul (3f,t)/u> (X+ r,t) )3 forms . a second rank t e n s o r whose i n d i v i d u a l components are such t h a t -1<R y(*,f)<1 i,j=1,2,3 (2-12) T h i s a g r e e s w i t h the i n t e r p r e t a t i o n t h a t Ry i s a measure of the degree of c o r r e l a t i o n between the two v e l o c i t y components. A t y p i c a l s p a t i a l c o r r e l a t i o n c u r v e f o r i d e n t i c a l v e l o c i t y components appears i n f i g u r e 2. 18 0 0 F i g u r e 2 - S p a t i a l C o r r e l a t i o n Curve At f=0 we see from eqn. (2-11) above t h a t R:L=1 i = 1 ,2,3. I t i s a p r o p e r t y of t u r b u l e n t f l o w s t h a t f l u c t u a t i n g v e l o c i t y components are u n c o r r e l a t e d f o r s u f f i c i e n t l y l a r g e d i s t a n c e s and so we have I f the mean f l u i d v e l o c i t y measured a t # i s i n , say, the x, d i r e c t i o n then R^d^) i s c a l l e d the ' l o n g i t u d i n a l s p a t i a l c o r r e l a t i o n c o e f f i c i e n t ' and R«(r,) and R^(c,) a r e ' l a t e r a l s p a t i a l c o r r e l a t i o n c o e f f i c i e n t s ' . A s i m p l e measure of the l e n g t h scade of the energy c o n t a i n i n g f l u c t u a t i o n s i s g i v e n by c a l l e d the i n t e g r a l l e n g t h s c a l e , see f i g u r e 2. In some f l o w s i t u a t i o n s a u s e f u l measure of the l e n g t h Ry(3f , f - ) => 0 f o r l a r g e |?| (2-13) (2-14) 19 s c a l e s i n the f l u c t u a t i o n s can be found from the t e m p o r a l c o r r e l a t i o n s . The a u t o - c o r r e l a t i o n c u r v e i s the average p r o d u c t of the same q u a n t i t y measured as a f u n c t i o n of s e p a r a t i o n time T, C^ ( T ) = u a ( x , t ) u f l l ( x , t + T ) 0=1,2 or 3 (2-15) T h i s c u r v e i s much more e a s i l y o b t a i n e d from experiment than the space c o r r e l a t i o n s d i s c u s s e d above. Only one v e l o c i t y measuring probe, a c o r r e l a t o r and a s i g n a l d e l a y u n i t a re needed. The a u t o - c o r r e l a t i o n c u r v e i s o b t a i n e d by sweeping the d e l a y time T r a t h e r than p h y s i c a l l y moving a v e l o c i t y p r obe. The time s c a l e of the energy c o n t a i n i n g f l u c t u a t i o n s , the i n t e g r a l time s c a l e Te, i s d e f i n e d as Te=1/u r j c(T)dT' (2-16) The a u t o - c o r r e l a t i o n i s u s e f u l i n the study of t u r b u l e n t f l u c t u a t i o n s when i t can be r e l a t e d t o the s p a t i a l c o r r e l a t i o n s . T h i s r e l a t a t i o n s h i p and the c o n d i t i o n s of i t s a p p l i c a b i l i t y a r e d i s c u s s e d below. C o n s i d e r a f l o w f i e l d moving p a s t a v e l o c i t y probe, f i g u r e 3. 20 F i g u r e 3 - Flow F i e l d P a s t Probe Assume the c o n v e c t i o n speed i s the same f o r a l l f l u i d e lements and the c o h e r e n t s t r u c t u r e s do not change a p p r e c i a b l y i n the time i t t a k e s them t o flo w p a s t the probe. The probe measures the v e l o c i t y as a f u n c t i o n of t i m e . The space c o r r e l a t i o n i s o b t a i n e d from the time c o r r e l a t i o n of the t e m p o r a l l y v a r y i n g s i g n a l by s i m p l y m u l t i p l y i n g the time a x i s of the a u t o - c o r r e l a t i o n by the probe speed, Uc. C^(T)=C a l ) (X/Uc) a,b=1,2,3 (2-17) The i n t e g r a l time s c a l e can thus be r e l a t e d t o the i n t e g r a l l e n g t h s c a l e by the r e l a t i o n , Te=Le/Uc (2-18) T h i s c o - o r d i n a t e t r a n s f o r m a t i o n and the c o n d i t i o n s when i t can be a p p l i e d i s c a l l e d , " T a y l o r ' s h y p o t h e s i s " a f t e r G . I . T a y l o r . The i m p o r t a n t p o i n t t o note i s t h a t T a y l o r ' s h y p o t h e s i s can o n l y be a p p l i e d when the f l o w f i e l d does not 21 change a p p r e c i a b l y d u r i n g the time i t t a k e s the v e l o c i t y m easuring probe t o sample a d i s t a n c e g r e a t e r than the l e n g t h s c a l e of i n t e r e s t . In a d d i t i o n the f l u i d elements must have a c o n s t a n t c o n v e c t i o n v e l o c i t y , Uc. J u s t as a f l u c t u a t i n g f l o w f i e l d can be d e s c r i b e d by the s p a t i a l c o r r e l a t i o n s we can d e s c r i b e i t by the F o u r i e r t r a n s f o r m s of the c o r r e l a t i o n s w i t h o u t l o s s of i n f o r m a t i o n . The F o u r i e r t r a n s f o r m of the s p a t i a l c o r r e l a t i o n t e n s o r i s c a l l e d the t h r e e d i m e n s i o n a l wave v e c t o r spectrum, $i$(k,x) = [ l/(2Tf?] J,ut ( x , t ) u s (x + r , t ) e x p ( i K - r ) d r (2-19) w i t h Ikl = 2 f f / ' \ b e i n g the wave number. E x p e r i m e n t a l l y i t i s i m p r a c t i c a l t o measure a l l v e l o c i t y components needed t o d e f i n e t h i s spectrum. One spectrum which i s 1 o f t e n used i n both t h e o r y and experiment i s the k„-wave number spectrum which i s found from $tl(Tt,3n as I t i s t w i c e the t o t a l of a l l k i n e t i c e n e r g i e s per u n i t mass of the v e l o c i t y f l u c t u a t i o n s i n the X i d i r e c t i o n , h a v i n g wave number between k, -dk, /2 and k, +dk, /2. In the e x p e r i m e n t a l study of g r i d t u r b u l e n c e d e s c r i b e d i n the f o l l o w i n g c h a p t e r s the most c o n v e n i e n t q u a n t i t y t o measure was the power s p e c t r a l d e n s i t y of the u f l u c t u a t i n g component, E„(w,x). E„(w,x) i s t w i c e the u, component of the k i n e t i c energy per u n i t mass a s s o c i a t e d w i t h t e m p o r a l (2-20) 22 f r e q u e n c y w.1 I t can be o b t a i n e d by F o u r i e r a n a l y z i n g the f l u c t u a t i n g v e l o c i t y component u, ( x , t ) measured w i t h a d i r e c t i o n a l l y s e n s i t i v e probe a t a f i x e d p o s i t i o n x. C„(w) and E„(w) as d e f i n e d above form a c o s i n e t r a n s f o r m p a i r , E«. (w)=4£c,, (T)cos(2ttwT)dT ( 2 - 2 l a ) C„ (T)= j°E„(w)cos(2TfwT)dw (2-21b) The l i m i t s of i n t e g r a t i o n a r e from 0 t o o as E„ (w) =E,, (-w) and C„(T)=C„(-T). From eqn.(2-21b) we- see t h a t C„(0)= CE„(w)dw (2-22) •'o And u s i n g the e x p r e s s i o n of eqn.(2-15) f o r the a u t o c o r r e l a t i o n , uT=/flE„(w)dw (2-23) T h i s e q u a t i o n i d e n t i f i e s the t o t a l f l u c t u a t i n g k i n e t i c energy u} w i t h the ar e a under the power s p e c t r a l d e n s i t y c u r v e . A p p l y i n g T a y l o r ' s h y p o t h e s i s t o the power s p e c t r a l d e n s i t y measurement we a r r i v e a t the r e l a t i o n ©„ (£/x)=Uc/(2l7)En(w,x) (2-24) The power s p e c t r a l d e n s i t y i s r e l a t e d t o the k.,-wave number spectrum (Tt,x) t h r o u g h the l o n g i t u d i n a l c o n v e c t i o n v e l o c i t y U c a t which the probe samples the f l o w f i e l d i n the same way as the s p a t i a l and t e m p o r a l c o r r e l a t i o n s a r e . T h i s I t s h o u l d be noted t h a t the symbol w i s t o denote t h e fre q u e n c y and not the a n g u l a r f r e q u e n c y as i s commonly done. 23 i n c l u d e s the same r e s t r i c t i v e a s s u m p t i o n s , namely t h a t the c o n v e c t i o n v e l o c i t y U i s c o n s t a n t and t h a t the f l o w does not change a p p r e c i a b l y w h i l e the probe samples a d i s t a n c e g r e a t e r than the l e n g t h s c a l e s of i n t e r e s t . U s i n g eqn.(2-16) i n eqn.(2-21a) f o r w=0 we see, E M ( 0 ) = 4ujTe (2-25) And a p p l y i n g T a y l o r ' s h y p o t h e s i s i n the form of eqn.(2-18) we see t h a t the i n t e g r a l l e n g t h s c a l e can be o b t a i n e d from the power s p e c t r a l d e n s i t y E,,(w) as Le=UcE„(0)/(4u*) (2-26) 2.4 V e l o c i t y F l u c t u a t i o n s And The Power Spectrum The power spectrum E(w) which i s d e f i n e d i n terms of the t r a n s f o r m of the c o r r e l a t i o n can be o b t a i n e d d i r e c t l y from a F o u r i e r t r a n s f o r m of the f l u c t u a t i n g v e l o c i t y . 1 To see t h i s we f i r s t w r i t e the time dependent l o n g i t u d i n a l f l u c t u a t i n g v e l o c i t y u,(t) i n i t s F o u r i e r r e p r e s e n t a t i o n , u, (t)=2 { dn[a(n)cos(2TTnt)+b(n)sin(2iTnt) ] (2-27) •'-*> a(n) and b(n) a r e the F o u r i e r c o e f f i c i e n t s f o r the b a s i s f u n c t i o n s of f r e q u e n c y n, a(n)= l/rrjat u, (t)cos(2fhnt) (2-28a) b(n)= '/tfjdt u, (t)sin(2Hr»t) (2-28b) f r o m T U R B U L E N C E , J . 0 . H i n z e ; M c G R A W - H I L L , 1 9 5 9 , p p 5 4 - 5 8 24 The a u t o - c o r r e l a t i o n c o e f f i c i e n t f o r t h e l o n g i t u d i n a l v e l o c i t y f l u c t u a t i o n s i s , C„(T)=u, ( t ) u , (t+T) (2-29) I n s e r t i n g the F o u r i e r r e p r e s e n t a t i o n f o r u , ( t ) , e q n s . ( 2 - 2 8 ) , i n t o t h i s e x p r e s s i o n and p e r f o r m i n g some of t h e i n t e g r a t i o n s we have u, ( t ) u , (t+T) = |;m6m[a2'(m)+b7' (m) ]/T cos(2fTmt) (2-30) and u ^ l i n i ^ d m t a * (m)+b* (m) ]/T (2-31) where T i s the s a m p l i n g t i m e . I f we now i d e n t i f y E„(n) w i t h the c o s i n e t r a n s f o r m e d q u a n t i t y i n eqn.(2-30) f o r t=0, E ) I(n) = rr2'[ai (m)+b*(m) ]/T (2-32) we may w r i t e . R l )(T) = 1/u L fdnE„(n)cos(2rfnt) (2-33) and comparing t h i s w i t h eqn. (2-21b) we see t h a t the two e x p r e s s i o n s , (2-32) and (2-21a) f o r E,,(w) a r e e q u i v a l e n t . I t was more c o n v e n i e n t t o o b t a i n E 1 ((w) from a F o u r i e r a n a l y s i s of the v e l o c i t y f i e l d d i r e c t l y than by f i r s t o b t a i n i n g the a u t o c o r r e l a t i o n . E q u a t i o n (2-32) t e l l s us how t o do t h i s i n the c o n t i n u o u s c a s e . In t h i s t h e s i s we use the f l u c t u a t i o n s p e c t r a d e f i n e d as i n eqn.(2-32) t o compare our e d d y - s i z e s p e c t r a d e s c r i p t i o n of t u r b u l e n t f l u c t u a t i o n s w i t h the f l u c t u a t i o n s measured u s i n g a h o t - f i l m anemometer. The s t a n d a r d methods used t o p r e d i c t t h e e v o l u t i o n of a t u r b u l e n t f l o w f i e l d a r e e i t h e r based on the t u r b u l e n t k i n e t i c energy e q u a t i o n (2-9) , or s i m i l a r l y the Reynolds s t r e s s e q u a t i o n s , or k , i -wave number s p e c t r u m , e q u a t i o n (2-25 20). Both approaches need c l o s u r e hypotheses t o be s o l v e d as i n w r i t i n g the v e l o c i t y and p r e s s u r e f i e l d s as the sum of an average and a f l u c t u a t i n g component the number of unknowns has d o u b l e d w i t h o u t i n c r e a s e i n the number of e q u a t i o n s . To c l o s e the k i n e t i c energy e q u a t i o n (2-9) one needs assumptions about how the s t r e s s and v e l o c i t y d e r i v a t i v e s are r e l a t e d . In the case of the k,,-wave number spectrum one must know the energy t r a n s f e r f u n c t i o n which d e s c r i b e s how p r o c e s s e s a t one f r e q u e n c y w i l l a f f e c t the a m p l i t u d e a t a n o t h e r f r e q u e n c y . A g a i n model assumptions ar e i n v o k e d which are d i c t a t e d more by m a t h e m a t i c a l expedience than by i n s i g h t i n t o the p h y s i c a l p r o c e s s e s d i c t a t i n g the f l o w e v o l u t i o n . In both c a s e s i t i s the d e t a i l s of the c o l l e c t i v e , or c o h e r e n t , e f f e c t s which i s added t o complete the d e s c r i p t i o n . Our approach i s t o model the i n t e r a c t i o n of the c o h e r e n t s t r u c t u r e s d i r e c t l y . We seek t o d e r i v e the r a t e c o e f f i c i e n t s t h a t d e s c r i b e the eddy e v o l u t i o n from the f l u i d e q u a t i o n s of m o t i o n . These r a t e s a r e then used t o d e s c r i b e p r o p e r t i e s of the f l o w such as t h e eddy s i z e d i s t r i b u t i o n . Our model i s much l i k e the k„ - wave number approach except t h a t i t more n a t u r a l l y a c c o u n t s f o r the c o h e r e n t s t r u c t u r e dominated energy t r a n s f e r mechanisms so t h a t any assumptions can be based on the p h y s i c s of the i n t e r a c t i o n s r a t h e r than the mathematics of the wave number s p e c t r a . The aim of t h i s t h e s i s i s t o show t h a t c o h e r e n t s t r u c t u r e s e x i s t i n a t u r b u l e n t f l o w and t h a t t h e i r s i z e 2 6 d i s t r i b u t i o n can be measured and can be used t o p r e d i c t the power s p e c t r a of v e l o c i t y f l u c t u a t i o n s i n a t u r b u l e n t f l o w . T h i s study i s based on e x p e r i m e n t a l o b s e r v a t i o n s of g r i d t u r b u l e n c e . 27 I I I . GRID TURBULENCE EXPERIMENTS 3.1 I n t r o d u c t i o n For the study of the s t a t i s t i c s of c o h e r e n t s t r u c t u r e s a s i m p l e and r e l i a b l e method t o ge n e r a t e and observe t u r b u l e n t f l o w was needed. The towing tank and g r i d a p p a r a t u s shown i n f i g u r e 4 were chosen f o r s e v e r a l r e a s o n s . F i r s t , we wanted t o see i f c o h e r e n t s t r u c t u r e s c o u l d be obs e r v e d and measured w i t h a p h o t o g r a p h i c t e c h n i q u e . Second, we wanted t o see i f t h e s e s t r u c t u r e s c o u l d be m e a n i n g f u l l y d e s c r i b e d i n a s i m p l e e d d y - s i z e spectrum. T h i r d , we wanted t o compare t h i s d e s c r i p t i o n w i t h the power s p e c t r a l d e n s i t y measured u s i n g a v e l o c i t y probe. The wor k i n g f l u i d was chosen t o be water f o r v e r s a t i l i t y i n the f l o w v i s u a l i z a t i o n t e c h n i q u e . A tow i n g tank was chosen r a t h e r than a water t u n n e l f o r the f o l l o w i n g r e a s o n s . With a tow i n g tank both the f l u i d and o b j e c t r e f e r e n c e frames a r e e a s i l y a c c e s s i b l e f o r measurements. D i a g n o s t i c equiptment such as cameras can be mounted i n the l a b frame. H o t - f i l m probes and cameras can a l s o be mounted on the c a r t which moves the g r i d t h r o u g h the water. A tow i n g tank has a r b i t r a r i l y q u i e s c e n t 'upstream' c o n d i t i o n s . I t a l s o does not have problems of boundary l a y e r b u i l d u p a l o n g the w a l l s . A d i s a d v a n t a g e of any tank i s i t s f i n i t e l e n g t h which l i m i t s the o b s e r v a t i o n time T =L/U ; here L i s the u s e a b l e c a r t t r a v e l d i s t a n c e and U i s the c a r t speed. The towi n g tank b u i l t f o r t h i s s t u d y had a Figure A - The Towing Tank 29 T y p i c a l l y 10 or more runs had t o be made t o a d e q u a t e l y d e f i n e a power spectrum and i t was not p o s s i b l e t o o b t a i n h o t - f i l m d a t a a t d i s t a n c e s g r e a t e r than about 130cm from the g e n e r a t i n g g r i d . 3.2 The Tank A t o w i n g tank w i t h i n s i d e d i m e n s i o n s of r o u g h l y 16'x3'x3' was b u i l t f o r t h e s e s t u d i e s , see f i g u r e 4. W i t h a tank of t h i s s i z e one can r e a c h Reynold's numbers up t o 1x105" w i t h r e a s o n a b l e s i z e d models and model speeds. The w a l l s were c o n s t r u c t e d of 3'x4' l o n g p a n e l s p l a c e d i n a welded m e t a l frame. Three of the s i d e p a n e l s and one of the f l o o r p a n e l s were made of 1/2" c l e a r p l a s t i c s h e e t s w h i l e the r e s t were of 3/4" t h i c k plywood. The frame was b o l t e d on t o p of I-beams r e s t i n g pn a c o n c r e t e b l o c k f o u n d a t i o n which s a t i n s i d e a plywood c a t c h pan. The c a t c h pan was f i t t e d w i t h f l o a t a c t i v a t e d marine type b i l g e pumps i n case any untoward l e a k s or s p i l l s o c c u r r e d . The upper edge of the 16' l o n g s i d e s c o n s i s t e d of an aluminum u-channel on t o p of which was mounted a 3/4" d i a m e t e r s t e e l r o d . A c a r t ran over the tank on the s t e e l r o d s . The wheels of the c a r t were equipped w i t h s e a l e d b a l l -b e a r i n g s . Rubber o - r i n g s s e a t e d about the c i r c u m f e r e n c e of the wheels p r o v i d e d some v i b r a t i o n i s o l a t i o n between the c a r t and the t a n k . The c a r t frame was c o n s t r u c t e d of aluminum a n g l e s t o c k . I t was f i t t e d w i t h clamps and s l i d i n g b a r s t o s e c u r e the o b j e c t s t o be immersed i n the f l u i d . The s l i d i n g b a r s were used t o p o s i t i o n the models i n the tank 30 and t o change the d i s t a n c e between the g r i d and the h o t - f i l m anemometer. An e l e c t r i c a l u m b i l i c a l c a b l e c a r r i e d the s h u t t e r t r i g g e r , o p t i c a l sensor and anemometer s i g n a l s from the moving c a r t frame t o the i n s t r u m e n t s i n the l a b frame. The c a r t was p u l l e d by two p l a s t i c c l a d s t e e l c a b l e s which ran i n a c o n t i n u o u s l o o p a l o n g the l e n g t h of each s i d e of the t a n k . Two 12" p u l l e y s on one end of the tank c o u p l e d the c a b l e s t o the d r i v e s h a f t . The c a b l e s were loop e d around i d l e r p u l l e y s a t the o t h e r end of the t a n k . The d r i v e system c o n s i s t e d of a two s t e p r e d u c t i o n p u l l e y s e t w i t h a 4 p o s i t i o n step-cone c o u p l i n g t o the d r i v e s h a f t . The r e d u c t i o n p u l l e y s were chosen t o g i v e a f i n a l c a r t speed range from a few c e n t i m e t e r s per second t o about two meters per second. The d r i v e system was powered by a 1/2 H.P. 90V DC motor w i t h a maximum speed of 1,750 rpm. The motor was o p e r a t e d by a c o n s t a n t speed c o n t r o l box w i t h s t a r t , s t o p , and f o r w a r d / r e v e r s e s w i t c h e s . Two m i c r o -s w i t c h e s , one a t each end of the t a n k , were w i r e d i n s e r i e s w i t h the c o n t r o l box s t o p b u t t o n . These s w i t c h e s p r e v e n t e d the c a r t from i n a d v e r t e n t l y o v e r s h o o t i n g e i t h e r end of the t a n k . At the 50cm/sec maximum speed used i n the experiment the c a r t took about 30cm t o r e a c h c r u i s i n g speed as w e l l as t o come t o a complete s t o p . A t t a c h e d t o t h e c a r t was an o p t i c a l d e t e c t o r t h a t was used t o t r i g g e r a t i m e r , f i g u r e 5. T h i s d e t e c t o r c o n s i s t e d of an IR photo d i o d e and a d j a c e n t photo v o l t a i c c e l l . I t f a c e d a f l a t b l a c k s u r f a c e about 1/4" away which ran the 31 l e n g t h of the t a n k . White tape was p l a c e d a t the d e s i r e d t r i g g e r s t a r t i n g and s t o p p i n g p o i n t s . When the d e t e c t o r passed over the w h i t e tape the p h o t o c e l l would p i c k up l i g h t t h a t was r e f l e c t e d and send a v o l t a g e s i g n a l t o the t i m e r . The t i m e r was s t a r t e d and stopped by t h e r i s i n g p a r t of s u c c e s i v e d e t e c t o r s i g n a l s . The t i m e r ' s i n t e r n a l RC c l o c k was c a l i b r a t e d a g a i n s t a f r e q u e n c y c o u n t e r . The d i g i t a l c o u n t e r i n the t i m e r box d i s p l a y e d the time between s u c c e s s i v e t r i g g e r p u l s e s . T h i s time and the known d i s t a n c e , &X, between the two t r i g g e r t a b s were used t o d e t e r m i n e the c a r t speed. The c a r t speed thus measured was found t o be r e p r o d u c i b l e t o b e t t e r than 1% over most of the w o r k i n g range of the d r i v e system. cart motion F i g u r e 5 - O p t i c a l D e t e c t o r and Timer 32 3.3 The G r i d The t u r b u l e n c e was g e n e r a t e d by a row of 1/2" di a m e t e r round aluminumn b a r s spaced 2" a p a r t , f i g u r e 4. A 1/4x2" end p l a t e was f a s t e n e d t o the bottom of the b a r s . The g r i d spanned the tank c r o s s - s e c t i o n w i t h t he b a r s mounted v e r t i c a l l y i n the Y-Z p l a n e . The g r i d was ex p e c t e d t o produce a t u r b u l e n t f l o w t h a t would p r o v i d e a l a r g e sample volume, a l l o w the f l u i d f l o w t o d e v e l o p i n d e p e n d e n t l y of boundary c o n d i t i o n s and s t i l l a l l o w f o r a two d i m e n s i o n a l a n a l y s i s . 3.4 V i s u a l i z a t i o n A pparatus The f l o w was made v i s i b l e by p h o t o g r a p h i n g the motion of aluminum f i l i n g s u s i n g a M i n o l t a X-570 35mm camera w i t h a 50mm l e n s and .motorized back. A t r i g g e r was needed t o complete the s h u t t e r r e l e a s e c i r c u i t on the camera. A c o n d u c t i n g f l a p p e r a t t a c h e d t o but i n s u l a t e d from the c a r t completed the s h u t t e r r e l e a s e c i r c u i t when i t made c o n t a c t w i t h an aluminum b r a c k e t f i x e d t o the ta n k , see f i g u r e 6. One l e a d of the s h u t t e r r e l e a s e was grounded t o the tank t h r o u g h the c a r t w h i l e the o t h e r was a t t a c h e d t o the f l a p p e r . Aluminum b r a c k e t s were p l a c e d a t d e s i r e d l o c a t i o n s a l o n g t he tank. A s e r i e s of photographs c o u l d t h u s be made by the m o t o r i z e d back e q u i p e d camera as the c a r t moved over the t a n k . A p i e c e of e l e c t i c a l tape on the back s i d e of the f l a p p e r e nsured photographs were not taken w h i l e the c a r t was r e t u r n i n g t o i t s s t a r t i n g p o s i t i o n . 33 shutter fixed release camera^^switch cart motion variac lights water surface conducting bracket F i g u r e 6 - L i g h t s and Camera A c t i o n F i g u r e 6 a l s o shows the l i g h t i n g system f o r the f l u i d r e f e r e n c e frame photography. Four 375 watt s p o t l i g h t s i l l u m i n a t e d the s u r f a c e of the water . The power t o these lamps was c o n t r o l l e d by a V a r i a c . The lamps were p o s i t i o n e d so t h a t no r e f l e c t i o n s o f f of the water s u r f a c e or shadows appeared i n the camera's f i e l d of view. The water s u r f a c e was seeded w i t h aluminum f i l i n g s of about ,1mm d i a m e t e r which were produced by f i l i n g a p i e c e of aluminum above the s u r f a c e of the water. As the c a r t dragged the g r i d t h r o u g h the water the f l a p p e r would a c t i v a t e the s h u t t e r r e l e a s e . The f i l i n g images were r e c o r d e d on I l f o r d XP1-400 b l a c k and w h i t e f i l m which was exposed and de v e l o p e d f o r an ASA r a t i n g of 800. L o c a l 34 v e l o c i t i e s c o u l d be o b t a i n e d by d i v i d i n g the s t r e a k l e n g t h by the exposure t i m e . Exposure t i m e s of 1/2 and 1 second were used f o r the 20 t o 50cm/sec g r i d speed range. The f l o w p a t t e r n s were s t u d i e d o n l y a t the s u r f a c e . A n e u t r a l l y bouyant s u s p e n s i o n , r e q u i r e d f o r f l o w v i s u a l i z a t i o n i n s i d e the f l u i d , was not a v a i l a b l e a t the time the e x p e r i m e n t s were formed. We assumed t h a t the s u r f a c e of the water p r o v i d e d a s h a r p l y d e f i n e d ' c r o s s -s e c t i o n ' of the f l o w on which aluminum f i l i n g s were suspended t o f a c i l i t a t e the f l o w v i s u a l i z a t i o n . The pr o c e d u r e and r e s u l t s of the photography appears i n the next s e c t i o n . 3.5 The Se a r c h For Coherent S t r u c t u r e s The p r e p a r a t i o n f o r an experiment s t a r t e d w i t h f i l l i n g the t a n k . A normal c o l d water f a u c e t w i t h a 5um p a r t i c l e f i l t e r was used t o g e t h e r w i t h an u n f i l t e r e d hot water t a p t o f i l l the tank t o a 63+1cm d e p t h . The hot water was added t o b r i n g the water temperature t o room t e m p e r a t u r e , 68" F. With o u t the hot water i t would t a k e the i n i t i a l l y about 62*F water more than 3 days t o e q u i l i b r i a t e t o room t e m p e r a t u r e . The 62*F water has about a 10% g r e a t e r k i n e m a t i c v i s c o s i t y . 1 T h i s would r e s u l t i n a 10% lower Reynold's number f o r a g i v e n model s i z e and speed. T h i s i s an u n a c c e p t a b l e d i f f e r e n c e . A f t e r the tank was f i l l e d a r e c i r c u l a t i n g 5ym p a r t i c l e f i l t e r system was used t o c l e a r up the i n e v i t a b l y opaque w a t e r . B e f o r e a run was made the water temperature was 35 measured. For both the p h o t o g r a p h i c and h o t - f i l m measurements the water t e m p e r a t u r e was w i t h i n one degree of 68* F. T h i s c o r r e s p o n d s t o h a v i n g the k i n e m a t i c v i s c o s i t y l i e w i t h i n 0.016x10'* of 1.01x10"* m l/sec, an i g n o r a b l e v a r i a t i o n . The f i r s t g r i d t u r b u l e n c e photographs were taken w i t h the camera and i l l u m i n a t i o n system mounted on the c a r t . I t was q u i c k l y r e a l i z e d t h a t the c o h e r e n t s t r u c t u r e s g e n e r a t e d by the g r i d were l i k e l y s t a t i o n a r y i n the f l u i d r e f e r e n c e frame and would thus be u n o b s e r v a b l e i n the t r a n s l a t i n g frame. F i g u r e s 7 show photos i n both the g r i d and l a b frame f o r 30cm/sec g r i d speeds. Lab frame exposure t i m e s of 1/2 and 1 second were chosen from an i n t i t i a l t e s t range of 1/60 t o 1 second. These r e s u l t e d i n the-most e a s i l y o b s e r v a b l e and measureable s t r u c t u r e s i n the f l o w . The s t r u c t u r e s were o b s e r v e d as a coherence i n c u r v a t u r e of the t r a c e s of the aluminum f i l i n g s . F i g u r e s 8 show the f i l i n g images f o r s e v e r a l d i f f e r e n t exposure t i m e s f o r a 35cm/sec g r i d speed. The s h u t t e r speed which showed the most e a s i l y o b s e r v e d and measured c o h e r e n t s t r u c t u r e s was found t o be 1 sec f o r the 20 and 30 cm/sec g r i d speeds and l / 2 s e c f o r the 40 and 50 cm/sec g r i d speeds. Eng. F l u i d Mech., Roberson/Crowe, HOUGHTON MIFFLIN CO. 36 a) Grid Frame 1/15 sec exposure Ug=30cm/sec b) F l u i d Frame 1 sec exposure Ug=30cm/sec F i g u r e 7 - Flow V i s u a l i z a t i o n i n grid and F l u i d Frames 38 The p r o c e d u r e f o r o b t a i n i n g the f l u i d frame photographs i s now e x p l a i n e d . The camera was mounted 1.4m above the water s u r f a c e . T h i s gave a f i e l d of view of 60cm i n the span d i r e c t i o n and 80cm i n the l o n g i t u d i n a l d i r e c t i o n . P i e c e s of w h i t e tape f o r the t i m e r t r i g g e r were p l a c e d 50.0 cm a p a r t w i t h the f i r s t t a b near the edge of the camera's f i e l d of view. The c a r t speed was thus measured d u r i n g the same time i n t e r v a l as when the g r i d was c r e a t i n g the f l o w which was .photographed. The s h u t t e r r e l e a s e t r i g g e r s were p o s i t i o n e d so t h a t the g r i d was j u s t l e a v i n g the f i e l d of view f o r the f i r s t p h o t ograph. The o t h e r two photographs were taken a t times T =50cm/U and T =1l0cm/U l a t e r so t h a t the t h r e e s h o t s p r o v i d e d an o v e r l a p p i n g time h i s t o r y of the f l o w beneath the camera. A s e t of t h r e e such photos a re shown i n f i g u r e . 9. From the v i e w p o i n t of our s t a t i s t i c a l a n a l y s i s t h i s i s e q u i v a l e n t t o a f i e l d of o b s e r v a t i o n from x=0. t o x=l90. cm b e h i n d the g r i d . B e f o r e making any measurements the t i m e r and frequency c o u n t e r were a l l o w e d t o warm up f o r about h a l f an hour or more as the RC o s c i l l a t o r tended t o d r i f t when f i r s t t u r n e d on. The c l o c k was c a l i b r a t e d a g a i n s t the c o u n t e r t o have an o s c i l l a t i n g f r e q u e n c y of f =100HZ. T h i s c a l i b r a t i o n was checked p e r i o d i c a l l y when measurements were made as d r i f t s up t o f=0.4Hz o c c u r r e d over about h a l f an hour. When cha n g i n g c a r t speeds the step-cone p u l l e y which r e s u l t e d i n the h i g h e s t motor speed f o r the new c a r t speed was used. T h i s r e s u l t e d i n a smooth and r e p r o d u c i b l e c a r t 39 mo t i o n . S e v e r a l runs were made t o a d j u s t the speed c o n t r o l r e s i s t o r u n t i l t he d e s i r e d c a r t v e l o c i t y was reached. T h i s speed s e t t i n g p r o c e d u r e had the added purpose of m i x i n g the water so t h a t any t h e r m a l v a r i a t i o n s were e q u i l i b r i a t e d b e f o r e the measurements began. Aluminum f i l i n g s were added or r e p l e n i s h e d as needed. The 40 and 50 cm/sec runs r e q u i r e d f r e q u e n t r e p l e n i s h m e n t as the s u r f a c e a g i t a t i o n made many of the f i l i n g s s i n k t o the tank bottom. Once the c a r t had been r e t u r n e d t o i t s s t a r t i n g p o s i t i o n f i v e t o twenty minutes were a l l o w e d t o l e t the water s e t t l e down. At the h i g h e s t c a r t speeds the s t r o n g l y e x c i t e d s u r f a c e waves took the l o n g e s t time t o d i s s i p a t e . S e t s of photographs were o b t a i n e d f o r 20, 30, 40, and 50 cm/sec g r i d speeds. P r i n t s of r e p r e s e n t a t i v e photographs are shown i n f i g u r e s 9 through 12,, 40 Figu r e 9 - 20cm/sec Flow Photographs 41 F i g u r e 10 - 30cm/sec Flow Photographs 4 2 43 0' X (cm) To AO 1 '60 I « » t 1/2 SEC EXPOSURE 50cm F i g u r e 12 - 50cm/sec Flow jynotographs 44 3.6 H o t - f i l m A pparatus A Thermal Systems I n c . 1050 CTA ( c o n s t a n t t e m p e r a t u r e anemometer) syste m 1 was used i n c o n j u n c t i o n w i t h a H e w l e t t -P a c k a r d HP3582A Spectrum A n a l y z e r t o o b t a i n power s p e c t r a from the f l o w f i e l d . A f l o w c h a r t of the d i a g n o s t i c s e t u p used t o o b t a i n the h o t - f i l m based power s p e c t r a i s shown i n f i g u r e 13. The h o t - f i l m anemometer must be l i n e a r i z e d when a new probe type i s f i r s t used. In a d d i t i o n i t must be c a l i b r a t e d b e f o r e each s e t of r u n s . Once s e t up i t g e n e r a t e s a s i g n a l between 0 and 10 v o l t s w hich i s p r o p o r t i o n a l t o the l o n g i t u d i n a l f l o w speed a t the h o t - f i l m sensor t i p . A b r i e f d e s c r i p t i o n of the anemometer o p e r a t i o n f o l l o w s . The h o t - f i l m sensor i s a t h i n s t r i p of c o n d u c t i n g m e t a l mounted a t the t i p of the probe. I t s r e s i s t a n c e depends on the o p e r a t i n g t e m p e r a t u r e T . T i s d e t e r m i n e d by the r a t e of c o n v e c t i v e c o o l i n g by the f l o w and the r a t e of h e a t i n g by a c o n t r o l c u r r e n t . The c o n t r o l v o l t a g e , Ec, a t the t o p of the b r i d g e causes a c u r r e n t t o pass t h r o u g h the sensor arm of the b r i d g e . T h i s v o l t a g e i s c o n t r o l l e d t o keep the sensor r e s i s t a n c e b a l a n c e d w i t h the c o n t r o l r e s i s t a n c e , Rc. The c o n t r o l r e s i s t a n c e i s s e t so t h a t the sens o r i s kept a t a r e s i s t a n c e s l i g h t l y h i g h e r than i t s r e s i s t a n c e w i t h z e r o 1 The l o a n of t h i s equipment from Dr.Quick of the UBC E n g i n e e r i n g Dept. i s g r a t e f u l l y acknowledged. 4 5 c o n t r o l v o l t a g e . T h i s h o l d s the sens o r element temperature above t h a t of the s u r r o u n d i n g f l u i d as the sens o r r e s i s t a n c e i n c r e a s e s m o n o t o n i c a l l y w i t h t e m p e r a t u r e . The s e n s o r ' s t h e r m a l energy l o s s r a t e i s a unique f u n c t i o n of the f l u i d f l o w r a t e p a s t the sensor f o r c o n s t a n t water d e n s i t y and t e m p e r a t u r e . The v o l t a g e , Ec, r e q u i r e d t o m a i n t a i n the sensor a t a c o n s t a n t t e m p e r a t u r e i s thus a unique f u n c t i o n of the f l o w v e l o c i t y a t the sensor t i p . The l i n e a r i z e r u n i t i s used t o c o n v e r t t h i s v o l t a g e s i g n a l from t h i s n o n - l i n e a r f u n c t i o n of the f l o w v e l o c i t y t o one which i s . The l i n e a r i z e d s i g n a l i s then f e d i n t o the s i g n a l c o n d i t i o n e r which was used t o remove f l u c t u a t i n g components lower than 2 Hz and g r e a t e r than 1 kHz i n f r e q u e n c y . When the l a r g e s t v e l o c i t y f l u c t u a t i o n s a r e much l e s s than the c o n v e c t i o n v e l o c i t y the f l u c t u a t i n g p a r t of the out p u t s i g n a l i s s i m p l y due t o the l o n g i t u d i n a l component of the f l u c t u a t i n g v e l o c i t y , see Appendix A. The wedge shape of the probe i s d e s i g n e d t o f u r t h e r s u p p r e s s the c o n t r i b u t i o n s t o c o o l i n g from l a t e r a l components of the f l o w . 4 6 sensor UCCt) (M7ZZH3= hot-film probe (wedge shaped) li nearizer e1 ^ \ J A \ ^ A / V W 7* oscilloscope & camera signal conditioner spectrum analyzer •VE X-Yplotterw F i g u r e 13 - H o t - F i l m Power S p e c t r u m S c h e m a t i c 47 The l i n e a r i z e d and f i l t e r e d s i g n a l i s d i s p l a y e d on the T e k t r o n i c s 454A o s c i l l o s c o p e and f e d i n t o the spectrum a n a l y z e r . A T e k t r o n i c s C-31 p o l a r o i d camera was used t o o b t a i n some t y p i c a l photos of the f l u c t u a t i n g v o l t a g e t r a c e . The spectrum a n a l y z e r had an e x t e r n a l t r i g g e r i n p u t which c o u l d be used t o s t a r t the s a m p l i n g time i n t e r v a l , of time d u r a t i o n Ts=2.5sec f o r the 100Hz f r e q u e n c y span. T h i s f e a t u r e was used t o ensure d a t a was taken o n l y when the h o t -f i l m probe was i n the t u r b u l e n t f l o w f i e l d . The t i m e r ' s t r i g g e r was used t o t r i g g e r the spectrum a n a l y z e r d a t a l o a d i n g . I f the a n a l y z e r was ready f o r a new data l o a d i n g sequence a new time r e c o r d would s t a r t when the o p t i c a l d e t e c t o r next e n c o u n t e r e d a p i e c e of w h i t e t a p e . Tape was thus p l a c e d a t s t r a t e g i c p o s i t i o n s a l o n g the tank . The f i r s t two p i e c e s of tape were p l a c e d X=50.0cm a p a r t . These t r i g g e r e d b o t h the t i m e r and the a n a l y z e r . Subsequent s i g n a l s from the o p t i c a l d e t e c t o r were o n l y used by the a n a l y z e r . The a n a l y z e r t r i g g e r had t o be d i s c o n n e c t e d b e f o r e the c a r t was r e t u r n e d t o the s t a r t i n g p o s i t i o n or m e a n i n g l e s s s p e c t r a would be added t o the p r e v i o u s d a t a . When a s u f f i c i e n t number of s p e c t r a , t y p i c a l l y 20, were averaged the r e s u l t a n t power spectrum was p l o t t e d u s i n g an HP x-y p l o t t e r which i n t e r f a c e d t o the spectrum a n a l y z e r . T h i s hardcopy was then d i g i t i z e d f o r subsequent a n a l y s i s u s i n g f a c i l i t i e s a t the UBC computing c e n t e r . The next s e c t i o n g i v e s a d e t a i l e d account of how the h o t - f i l m d a t a was o b t a i n e d . 48 3.7 H o t - f i l m Power S p e c t r a In p r e p a r a t i o n f o r the h o t - f i l m e x p e r i m e n t s the tank was f i l l e d and f i l t e r e d as f o r the photography work. I t was even more i m p o r t a n t t h a t t he water be c l e a n and have a c o n s t a n t t e m p e r a t u r e f o r the h o t - f i l m measurements than i t was f o r the f l o w v i s u a l i z a t i o n . The senso r c o o l i n g r a t e was v e r y s e n s i t i v e t o bo t h t e m p e r a t u r e - v a r i a t i o n s and d i r t . B e f o r e any anemometer measurements c o u l d be taken the l i n e a r i z e r had t o be s e t up f o r the 1232W model h o t - f i l m probe. The i n s t r u c t i o n s i n the TSI "MODEL 1050/1050A Co n s t a n t Temperature Anemometer" i n s t r u c t i o n manual were c l o s e l y f o l l o w e d . Only the b a s i c s e t u p p r o c e d u r e w i l l be d e s c r i b e d h e r e . The probe arm of the b r i d g e c o n s i s t e d of a 30' c o a x i a l c a b l e w i t h probe mount, t o g e t h e r h a v i n g r e s i s t a n c e Rcab, and the probe i t s e l f , r e s i s t a n c e l e s s s ensor element Rp, a t the t i p of which was the sensor element, r e s i s t a n c e Rs, see f i g u r e 13. I t was f i r s t n e c e s s a r y t o measure the unheated sensor r e s i s t a n c e , Rs, and m u l t i p l y by the o v e r h e a t r a t i o of 1.06 t o o b t a i n the sensor o p e r a t i n g r e s i s t a n c e , Rop. The probe s u p p o r t p l u s c a b l e r e s i s t a n c e , Rcab, was measured by n u l l i n g the b r i d g e w i t h a s h o r t i n g w i r e i n p l a c e of the probe. T h i s was done by b a l a n c i n g t he probe arm a g a i n s t a v a r i a b l e decade r e s i s t a n c e , Rc, which formed the o p p o s i t e b r i d g e arm. The probe minus s e n s o r ' s r e s i s t a n c e , Rp, i s then added t o the decade r e s i s t a n c e and the decade r e s i s t a n c e b a l a n c e d a g a i n s t the ZERO OHMS r e s i s t o r . The ZERO OHMS r e s i s t o r t h u s 49 a d j u s t e d t o the probe arm minus sensor r e s i s t a n c e , Rcab+Rp, i s c o n n e c t e d i n s e r i e s w i t h the decade r e s i s t o r . The decade r e s i s t a n c e would t h e r e f o r e read o n l y the se n s o r r e s i s t a n c e , Rs ,when the b r i d g e was b a l a n c e d w i t h the probe i n p l a c e . The probe c o u l d now be i n s e r t e d i n t o the probe support and immersed i n the 68 F q u i e s c e n t water. The unheated sensor r e s i s t a n c e , Rs, was re a d o f f the n u l l e d b r i d g e and the o p e r a t i n g r e s i s t a n c e d e t e r m i n e d , Rop=Rsxoverheat r a t i o . The o p e r a t i n g r e s i s t a n c e , Rop, was d i a l e d i n t o the decade r e s i s t a n c e , Rc. When the b r i d g e c o n t r o l c i r c u i t was s w i t c h e d t o RUN the sensor would be m a i n t a i n e d a t the e l e v a t e d t e m p e r a t u r e . For the d a t a p r e s e n t e d i n t h i s t h e s i s the sensor r e s i s t a n c e Rs of 3.92 ohms m u l t i p l i e d by the ov e r h e a t r a t i o of 1.06 d e t e r m i n e d the o p e r a t i n g r e s i s t a n c e Rop t o be 4.15 ohms. The 1050 anemometer p r o v i d e d a c h o i c e of t h r e e d i f f e r e n t c o n t r o l b r i d g e s . The b r i d g e used depends on the power re q u i r e m e n t d e t e r m i n e d by the probe type and fl o w environment. The number 1 b r i d g e i s g e n e r a l l y used f o r low power re q u i r e m e n t a p p l i c a t i o n s such as probes i n a i r or s m a l l probes i n water . The number 2 b r i d g e was used f o r the h o t - f i l m probe as i t s h i g h e r output c u r r e n t was needed t o a v o i d the v o l t a g e c l i p p i n g which was o b s e r v e d when the number 1 b r i d g e was used. A f t e r the b r i d g e was s e t up a c a l i b r a t i o n c u r v e of b r i d g e v o l t a g e , Ec, v e r s u s f l o w speed was measured, f i g u r e 14a. T h i s n o n - l i n e a r c u r v e was needed t o a d j u s t the 50 l i n e a r i z e r s e t t i n g s . I t was o b t a i n e d by measuring the b r i d g e output v o l t a g e u s i n g a d i g i t a l v o l t meter w h i l e the c a r t moved the probe t h r o u g h the water a t a known speed. For t h e s e measurements the g r i d was removed so t h a t the f l u c t u a t i o n l e v e l s were v e r y low. The probe was p o s i t i o n e d about 7.5cm beneath the water s u r f a c e and was a l i g n e d p a r a l l e l t o the d i r e c t i o n of the c a r t ' s m o t i o n . The c a r t speed was measured u s i n g the t i m e r . The b r i d g e s i g n a l was l i n e a r i z e d by a r a t h e r t e d i o u s p r o c e s s of s e t t i n g 9 i n t e r d e p e n d e n t s l o p e changing r e s i s t o r s on the model 1055 l i n e a r i z e r . The b r i d g e v o l t a g e s c o r r e s p o n d i n g t o z e r o and the maximum f l o w speed of i n t e r e s t were s u p p l i e d t o the l i n e a r i z e r and the l i n e a r i z e r ZERO and SPAN c o n t r o l s were a d j u s t e d so t h a t the l i n e a r i z e r would output c o r r e s p o n d i n g v o l t a g e s of 0.00 and 10.00 v o l t s . The a v a i l a b l e l i n e a r i z e r had a broken r e s i s t o r f o r i t s 4th s l o p e p o i n t . A r e a s o n a b l e l i n e a r i z a t i o n c o u l d o n l y be a c h i e v e d w i t h the r e s i s t o r t u r n e d c o m p l e t e l y c l o c k w i s e . A s l i g h t v a r i a t i o n from t h i s p o s i t i o n produced a d i s c o n t i n u o u s change i n the l i n e a r i z e r ' s r e s p o n s e . The l i n e a r i z e d c u r v e appears over the c a l i b r a t i o n c u r v e i n f i g u r e 14. Subsequent l i n e a r i z a t i o n checks were made d u r i n g and a f t e r the e x p e r i m e n t a l r u n s . The c a l i b r a t i o n c o n s t a n t i s determined as the s l o p e of the l i n e a r i z e d c u r v e . Minor v a r i a t i o n s of t h i s s l o p e i n the 40cm/sec r e g i o n were ob s e r v e d and so the average v a l u e of the t h r e e c a l i b r a t i o n s was used.. 51 o linearized • calibration o o • o 20 40 60 VELOCITY (cm/sec) F i g u r e 14 - H o t - F i l m L i n e a r i z a t i o n 52 The l i n e a r i z e d s i g n a l was f e d th r o u g h the s i g n a l c o n d i t i o n e r w i t h a pass band of from 2hz t o 1Khz and then t o the monitor scope and spectrum a n a l y z e r . Some o s c i l l o g r a m s of the l i n e a r i z e d f l u c t u a t i n g anemometer s i g n a l appear i n f i g u r e 15. The HP3582A Spectrum A n a l y z e r was used t o o b t a i n power s p e c t r a from the f l u c t u a t i n g anemometer s i g n a l u s i n g the f o l l o w i n g c o n t r o l s e t t i n g s . The i n p u t s w i t c h was i n the " c h a s s i s i s o l a t e d " p o s i t i o n . The DC c o u p l i n g mode was used as the s i g n a l c o n d i t i o n e r had a l r e a d y f i l t e r e d t he s i g n a l . T h i s c o u p l i n g e nsured t h a t no more of the low f r e q u e n c y end of the s i g n a l was f i l t e r e d . The i n p u t s e c t i o n had an i n p u t s e n s i t i v i t y s e l e c t o r which was s e t a t the most s e n s i t i v e p o s i t i o n p o s s i b l e w h i l e s t i l l not h a v i n g the d a t a o v e r l o a d i n d i c a t o r l i g h t up. The h i g h e r t u r b u l e n c e i n t e n s i t i e s r e q u i r e d a lower i n p u t s e n s i t i v i t y . The f r e q u e n c y span was chosen t o be from 0 t o 100Hz. P r e l i m i n a r y measurements showed no s i g n i f i c a n t energy c o n t e n t a t f r e q u e n c y g r e a t e r Ug=40cm/sec X POSN=30cm calibration constant = 8. (cm/secVvolt 1 v/div r f — r — T 0 J f .2 .3 A sec 8 (cm/sec)/div F i g u r e 15 - H o t - F i l m O s c i l l o g r a m s 53 than 100Hz f o r the g r i d speed range used i n t h e s e e x p e r i m e n t s . The a n a l y z e r had a c h o i c e of t h r e e band pass shapes depending on whether a m p l i t u d e or f r e q u e n c y r e s o l u t i o n was d e s i r e d . The Hanning band pass shape was used as a compromise between the two extremes. RMS a v e r a g i n g was s e l e c t e d so the a n a l y z e r would average s u c c e s s i v e power s p e c t r a and the d a t a l o a d i n g t r i g g e r s e c t i o n was s e t up t o l o a d d a t a on the e x t e r n a l t r i g g e r s i g n a l s u p p l i e d by the t i m e r . A f t e r the a n a l y z e r , h o t - f i l m probe t o g r i d s e p a r a t i o n , and the c a r t speed were s e t up the c a r t was moved t o i t s s t a r t i n g p o s i t i o n . Enough time was g i v e n f o r the water motions t o s u b s i d e . The s p e c t r a a v e r a g i n g was r e s e t and the t i m e r ' s t r i g g e r output was c o n n e c t e d t o the a n a l y z e r ' s d a t a l o a d i n g t r i g g e r . i n p u t . The d r i v e motor was s t a r t e d and the c a r t towed the g r i d and h o t - f i l m probe t h r o u g h the w a t e r . The t i m e r box c l o c k e d the 50.0cm d i s t a n c e and a l s o t r i g g e r e d from one t o f o u r a n a l y z e r s a m p l i n g p e r i o d s . T h i s number depended on the c a r t speed as the data l o a d i n g took a f i x e d amount of t i m e , Ts=2.5sec. A f t e r the c a r t stopped a t the end of the tank the t i m e r t r i g g e r was d i s c o n e c t e d and the c a r t r e t u r n e d t o i t s s t a r t i n g p o s i t i o n . Depending on the c a r t speed, from f i v e t o about twenty minutes e l a p s e d b e f o r e the waves i n the tank d i e d down and a n o t h e r run was made. I t was found t h a t the h o t - f i l m probe mounting had a m e c h a n i c a l resonance which showed up i n the power s p e c t r a . The mounting system was r e i n f o r c e d and the resonance 54 d i m i n i s h e d i n a m p l i t u d e and s l i g h t l y i n c r e a s e d i n f r e q u e n c y . To d i s t i n g u i s h between t u r b u l e n t s i g n a l s and t h i s m e c h a n i c a l resonance a power spectrum was o b t a i n e d w i t h the water q u i e s c e n t . T h i s spectrum appears i n f i g u r e 16. Hard copy of the power s p e c t r a were o b t a i n e d from the spectrum a n a l y z e r u s i n g an HP X-Y p l o t t e r . These p l o t s were then d i g i t i z e d and s t o r e d i n computer f i l e s f o r susequent a n a l y s i s . When the s p e c t r a were d i g i t i z e d the probe support resonance c o n t r i b u t i o n was i g n o r e d . A t y p i c a l X-Y p l o t i s shown i n f i g u r e 17. The d i g i t i z e d c u r v e has been drawn t h r o u g h the spectrum. Power s p e c t r a were o b t a i n e d f o r v a r i o u s g r i d speeds and p r o b e - g r i d s e p a r a t i o n s . The d i s t a n c e of the probe from the g r i d was changed by s l i d i n g the g r i d i n i t s mounting b r a c k e t s . The e x p e r i m e n t a l c o n d i t i o n s were chosen so t h a t t h e h o t - f i l m power s p e c t r a would c o r r e s p o n d t o those p r e d i c t e d from the f l o w v i s u a l i z a t i o n p h o t o g r a p h s . 5 5 80 m'v-i PM) 04 PRO BE SUPPORT R E S O N A N C E 1 5avgs 0 W 100Hz F i g u r e 16 - Probe Support Resonance 40 mvi P(W) 0 R M S Power Spectrum X P O S N = 9 0 c m VELOCITY=A0cm/sec 1 0 avgs 100Hz F i g u r e 17 - T y p i c a l A n a l y z e r X-Y P l o t 56 IV. EDDY-SIZE DISTRIBUTIONS AND HOT-FILM SPECTRA 4.1 I n t r o d u c t i o n A major aim of t h i s t h e s i s i s t o compare the e d d y - s i z e spectrum w i t h the s t a t i s t i c a l d e s c r i p t i o n of t u r b u l e n t f l u c t u a t i o n s . T h i s was seen as a t e s t f o r the v a l i d i t y and v i a b i l i t y of u s i n g c o h e r e n t s t r u c t u r e s t o model t u r b u l e n c e . A p e r s o n a l aim of the a u t h o r was t o l e a r n how the s t a t i s t i c a l d e s c r i p t i o n i s used t o d e s c r i b e t u r b u l e n t f l o w s . F i g u r e 18 i s a f l o w c h a r t of the a n a l y s i s used t o compare the two d e s c r i p t i o n s . To o b t a i n the e d d y - s p e c t r a the f l o w v i s u a l i z a t i o n photos were a n a l y z e d i n terms of the c o h e r e n t s t r u c t u r e s . The e d d y - s i z e d i s t r i b u t i o n i s then used i n a computer code t o p r e d i c t the power s p e c t r a l d e n s i t y . Assumptions about the d i s t r i b u t i o n s of the e d d i e s i n space and t h e i r c o n v e c t i o n v e l o c i t y were needed t o g e n e r a t e a time r e c o r d of the l o n g i t u d i n a l v e l o c i t y f l u c t u a t i o n . The computer code a n a l y z e s the time r e c o r d i n a manner n e a r l y i d e n t i c a l t o t h a t used by the spectrum a n a l y z e r i n the h o t - f i l m e x p e r i m e n t . The p r e d i c t i o n of power s p e c t r a from the eddy s i z e d i s t r i b u t i o n s and the a ssumptions used a r e d e s c r i b e d i n s e c t i o n 4.3. S e c t i o n 4.4 d e s c r i b e s how power s p e c t r a were o b t a i n e d from the anemometer s i g n a l . The power s p e c t r a were compared and used t o o b t a i n the i n t e g r a l l e n g t h s c a l e and t o t a l f l u c t u a t i n g k i n e t i c energy. The r e s u l t s of t h i s a n a l y s i s a r e p r e s e n t e d and d e s c r i b e d i n t h e l a s t s e c t i o n of t h i s c h a p t e r . 57 VISUALIZATION ANEMOMETER 'N(R),JUR)J XPOSN Ug /bataf i le :DDY40.30 y calculate le.PEDDY FRAC 7URB.FTN NKR)-t€,iVv) Tdatafile 'POW4030 plot N eddy- JL spectrum 1 P(w)-*E,[W) /fdataf ile SQR40.30 F i g u r e 18 - Power S p e c t r a Comparison 58 4.2 E d d y - S i z e S p e c t r a E d d y - s i z e s p e c t r a were o b t a i n e d from the photographs of the f l o w p a t t e r n s as f o l l o w s . The f l o w p a t t e r n s on the n e g a t i v e s were p r o j e c t e d onto a sheet of w h i t e paper t o h a l f t h e i r o r i g i n a l s i z e . T h i s paper was d i v i d e d i n t o 1OcmFS ( f u l l s c a l e ) wide b i n s , see f i g u r e 19a. The d i s t a n c e l a b e l , Lb, c o r r e s p o n d s t o the d i s t a n c e between the m i d d l e of the sa m p l i n g b i n t o the average p o s i t i o n of the g r i d d u r i n g the exposure t i m e . The b i n s c o u l d e q u a l l y w e l l have been r e f e r e n c e d by the average time s i n c e the g r i d had passed by. T h i s t i m e , Tb, would s i m p l y be the b i n d i s t a n c e Lb d i v i d e d by the g r i d speed Ug. The per s o n a n a l y z i n g the f l o w p a t t e r n s drew the o u t l i n e of what he c o n s i d e r e d t o be the c o h e r e n t s t r u c t u r e s on the paper. T h i s p r o c e s s i s of c o u r s e h i g h l y s u b j e c t i v e . The 40cm/sec photos were a n a l y s e d by a person who had l i t t l e i n d o c t r i n a t i o n as t o what a c o h e r e n t s t r u c t u r e s h o u l d l o o k l i k e . G e n e r a l l y we 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 . F i g u r e 19a a l s o shows the s t r u c t u r e s c i r c l e d . T h i s a n a l y s i s i s v e r y c r u d e a l t h o u g h , as i t t u r n s o u t , i t was b o t h q u i c k and y i e l d s u s e a b l e eddy s p e c t r a . An automated d e c o m p o s i t i o n of the f l o w f i e l d a c c o r d i n g t o r i g o r o u s l y d e f i n e d c r i t e r i a was beyond the scope of t h i s t h e s i s . We d e c i d e d t o l e a r n from the consequences of our s i m p l e a n a l y s i s . F i g u r e 19 - Eddy A n a l y s i s 60 Having o u t l i n e d the c o n t o u r s of the e d d i e s a c i r c l e t e m p l a t e was used t o determine t h e i r r a d i i . N o n - c i r c u l a r s t r u c t u r e s were t r e a t e d as c i r c u l a r e d d i e s h a v i n g the same a r e a . Eddy r a d i i were e s t i m a t e d w i t h an u n c e r t a i n t y of 1mm. For each b i n the eddy s i z e d i s t r i b u t i o n , N ( R), was r e c o r d e d a f t e r the r a d i i had been d e t e r m i n e d , f i g u r e 19b. The sum of the eddy d i s t r i b u t i o n s from a l l t en s e t s of photographs f o r one f l o w speed p r o v i d e d a s t a t i s t i c a l l y s i g n i f i c a n t d i s t r i b u t i o n , f i g u r e 19c. The eddy's i n t e r n a l a n g u l a r v e l o c i t i e s were measured u s i n g a c l e a r p l a s t i c a n g l e t e m p l a t e . The t e m p l a t e was c e n t e r e d over the s t r u c t u r e and the a n g l e t h a t a t r a c e made w i t h r e s p e c t t o the c e n t e r of the s t r u c t u r e was c o n v e n i e n t l y read o f f the a n g l e s c a l e . The r e s u l t s of t h e s e measurements f o r the 40cm/sec d a t a appear i n f i g u r e 20a. T h i s measurement proved much more d i f f i c u l t and u n c e r t a i n than e s t i m a t i n g an eddy s i z e as the images o f t e n d i d not have a c i c u l a r l y symmetric v e l o c i t y d i s t r i b u t i o n . A l s o the low t r a c e d e n s i t i e s needed t o a v o i d t r a c e o v e r l a p p i n g made c h a r a c t e r i z a t i o n of an eddy's a n g u l a r v e l o c i t y d i s t r i b u t i o n v e r y d i f f i c u l t . I t was thus n e c e s s a r y t o make some assumption about the v e l o c i t y p r o f i l e w i t h i n an eddy. We choose a r i g i d body r o t a t i o n , no i n t e r n a l s t r e s s , f o r the v e l o c i t y p r o f i l e . T h i s c h o i c e was made on the b a s i s of o b s e r v a t i o n and c o n v e n i e n c e . 61 The assumption t h a t the e d d i e s undergo r i g i d body motion was r e a s o n a b l y c o n s i s t e n t w i t h o b s e r v a t i o n . The e d d y - s i z e s p e c t r a f o r t e n s e t s of photographs f o r a p a r t i c u l a r c a r t speed were summed. These s p e c t r a and the average eddy a n g u l a r v e l o c i t y as a f u n c t i o n of d i s t a n c e were w r i t t e n i n t o a computer f i l e . The p o p u l a t i o n a t a g i v e n r a d i u s was averaged w i t h the p o p u l a t i o n s of the e d d i e s one s i z e g r e a t e r and one s i z e s m a l l e r t o remove a p o s s i b l e a n a l y s i s b i a s i n u s i n g the c i r c l e t e m p l a t e . The smoothed e d d y - s i z e s p e c t r a summed over a l l t en s e t s of photos f o r the 30 and 40cm/sec c a r t speed appears i n f i g u r e s 21 and 22. A q u a l i t a t i v e d i s c u s s i o n of t h e s e s p e c t r a i s p r e s e n t e d i n the d i s c u s s i o n s e c t i o n a t the end of t h i s c h a p t e r . 62 F i g u r e 20 - Eddy A n g u l a r V e l o c i t i e s 63 8-0 3 E D D Y S P E C T R U M VELOCITY* 30 Ctl/SEC X POSN* 70 CM 4 + 44 44 444 +-H „ „ ' 1 1 1 1 r - — i 1 1 1 1 1 00 0.6 1.2 16 2.4 3.0 3.6 RRDIUS(CM) C D E D D Y S P E C T R U M VELOCITY* 30 CM/SEC X POSN* 80 CM +44 44 44-~i 1 1 1 1 1 1 1 1 1 1 1 O.O 0.6 1.2 1.8 2.4 3.0 3.6 RRD1USICM) E D D Y S P E C T R U M VELOCITY* 30 CM/SEC X POSN= 90 CM + + + ! 1 1 1 1 1 1 1 1 0.6 1.2 1.8 2.4 3.0 RRDIUS(CM) 3.6 E D D Y S P E C T R U M VELOCITY* 30 CM/SEC X POSN* 100 Ctl 44 44 44 44 I I I I I I I - i 1 1 1 1 1 1 1 1 1 1 1 0.0 0.6 1.2 18 2.4 3.0 3.6 RFOJ.U5ICM) E D D Y S P E C T R U M VELOCITY* 30 CM/SEC X POSN* 110 CM 44 I I I I I I I I I I I I I ~ 1 I I 1 1 1 1 1 1 1 1 1 0.0 0.6 1.2 1.8 2.4 3.0 3.6 RRDIUS(CM! E D D Y S P E C T R U M VELOCITY* 30 CM/SEC X POSN* 120 CM 4 4 4 444 44444 444 4 I I I I I I I ~i i i i i i i i i i i i 0.0 0.6 .2 18 2.4 RRDIUS(CM) 3.0 3.6 F i g u r e 21 - 30cm/sec E d d y - S i z e S p e c t r a 64 E D D Y S P E C T R U M VEL0CITY=30 CM/SEC X POSN= 130 CM 4 + 4 + 44444 4444 „ „ ~ T 1 i i — i — i — i — i — i — i — i »•» 0.6 1.2 I S 2.4 3 0 3 6 RHD1U5(CH) E D D Y S P E C T R U M VELOCITY* 30 CM/SEC X POSN= 150 CM +4H • H i l l - i 1 1 1 1 1 0.0 0.6 1.2 ] RADII' 44 4 I I I I I I I I I I ++ 1 1 1 1 1 1 .8 2.4 3.0 3.6 SICM1 E D D Y S P E C T R U M VELOCITY=30 CM/SEC X POSN= 140 CM .-H44-. 1 1 1 1 1 1 1 4 +44 I I I I I I I I I I I I I I I I I I •H I I I I I I I I I I I I I I I I I I I I I ~i i 1 1 1 1 1 1 1 1 1 1 0.0 0.6 1.2 18 2.4 3.0 3.6 RRDIUSICM) E D D Y S P E C T R U M VEL0CITY=30 CM/SEC X POSN= 160 CM +4 444 44 444 4 44 I I I I I I I I I I I I I - i 1 1 1 1 1 1 1 r 0.0 0.6 1.2 1.8 2.4 RADIUS(CM) 3.6 E D D Y S P E C T R U M VEL0CITY=30 CM/SEC X POSN= 170 CM I I I I I I +4 I I I I I I I I I I I 444 4 II I I I I I I I I I I I I I I I I I I I - 1 — 0 . 6 - i 1 1 1 1 r 1.2 18 2.4 RflDIUS(CM) — i — 3.0 - I 3.6 E D D Y S P E C T R U M VEL0CITY=30 CM/SEC X POSN= 180 CM 444 4 44 44 4444 +444 I I I I I I I I I I I I I I 0.0 — I — 0.6 "T" T 1.2 18 2.4 RRDIUSICM1 T" -1 3.6 F i g u r e 21b- 30cm/sec E d d y - S i z e S p e c t r a 65 E D D Y S P E C T R U M VEL0CITY*40 Ctt/SEC x POSN= 20 cn "1 I I I 1 1 1 1 1 1 1 1 1 1 0 0 0 . 4 0 . 8 1.2 1 . 6 2 . 0 2 . 4 2.1 RntuusitHi E D D Y S P E C T R U M VELOCITY*40 CH/SEC X POSN= 40 CM +++++++ ++++++++ +++++++++ - 1 — I — I — I — I — I 1 — I — I — 1 — I — I — 0 . 0 0 . 4 0 . 3 1 . 2 1 . 6 2 . 0 2 . 4 RHDIUS1CM1 R-E D D Y S P E C T R U M VEL0CITY=40 CM/SEC X POSN* 30 CM + + + + + + + + + + + + + + + + + + + + + + + ++++++ ++++++++ ++++++++ ++++++++ i i i 1 1 1 1 1 r 0 0 0 . 4 0 . 8 1 . 2 | . 8 RHDIUS(CM) E D D Y S P E C T R U M VELOCITY*4O CM/SEC X POSN* 50 CM — i 1 1 1 1 1 1 r 1 . 2 1.6 RflDIUSICM) E D D Y S P E C T R U M s-VELOCITY=40 CM/SEC X POSN* 60 cn • o i . —'Z—1—1—r RH't1lUSlC,MI ~i r-2 . 0 - i — r 2 . 4 E D D Y S P E C T R U M VELOCITY* 40 CM/SEC X POSN* 70 CM - 1 1 — 0 . 0 0 . 4 RliUIUSIcWi F i g u r e 22 - 40cm/sec E d d y - S i z e S p e c t r a 66 EDDY SPECTRUM VEL0CITr=40 CM/SEC x POSN= 80 cn ++ RrtDIUSICM) -1— ?.4 EDDY SPECTRUM VELOCITY=4O cn/src x POSN= 90 cn ++ + -1— o.o — I 1 1— 1.2 1.6 RHUIUIilCMI — I — ?.4 EDDY SPECTRUM vELOcnr=<to cn/SEc x POSN= ioo cn + + + + + + 1 1 1 1 i — • — i — i — i — i — i — i — i — i 1" 0" 0.8 1.2 1.6 2.0 24 2 1 RI1UIUS1CHI EDDY SPECTRUM VELOCITT-40 CM/SEC x POSN= no cn + + 4 + H- + + +-(•+-(•+ "1 I 1 1 I 1 1 1 1 1 1 1 1 1 >•« 0.4 n.a i.:> i.i; :.n ?.4 2.e RlirjHJSICMl CO EDDY SPECTRUM VEL0CITr=40 CH/SEC x POSN= 120 cn RRtllUSIEHl F i g u r e 22b- 40cm/sec Eddy-Size Spectra 67 4.3 G e n e r a t i o n Of E (w) From N(R) The e d d y - s i z e d i s t r i b u t i o n was used by a computer program t o g e n e r a t e the power s p e c t r a l d e n s i t y c u r v e s E (w). F i g u r e 23 i s a f l o w c h a r t of the program and d e t a i l s of the F o r t r a n IV computer code TURB.FTN a r e g i v e n i n Appendix C. The f l u c t u a t i n g v e l o c i t y t h a t would be measured by a probe moving w i t h c o n s t a n t v e l o c i t y t h r o u g h the e d d i e s was modeled and then f a s t f o u r i e r t r a n s f o r m (FFT) a n a l y z e d t o produce the power spectrum. Assuming an eddy r o t a t e s w i t h o u t i n t e r n a l s t r e s s e s the streamwise component of the r o t a t i o n a l v e l o c i t y i s s i m p l y , u=U b/Rm (4-1) where b i s the impact parameter a t which the probe meets the eddy, U i s the p e r i p h e r a l v e l o c i t y , and Rm i s the eddy r a d i u s , see Appendix B. The e d d i e s of f i g u r e 19 were obser v e d i n the l a b r e f e r e n c e frame. The c o n v e c t i o n v e l o c i t y , Uc, used i n the computer model i s t h u s taken t o be the c a r t speed, Ug, as the probe was mounted t o the c a r t . From o b s e r v a t i o n s of the 40 cm/sec f l o w photographs one sees t h a t the eddy speed measured i n the l a b frame i s n e g l i g i b l e compared w i t h the c a r t speed. Thus, u s i n g t h i s c o n s t a n t c o n v e c t i o n v e l o c i t y , Uc=Ug, s h o u l d not d i s t o r t the model's p r e d i c t i o n s . C o m p l i c a t i o n s such as d i f f e r e n t eddy o r i e n t a t i o n s and l e n g t h s or n o n - 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 were i g n o r e d . The photos d i d not show any i n d i c a t i o n of s i g n i f i c a n t v e r t i c a l v e l o c i t i e s . T h i s was s u r p r i s i n g because we e x p e c t e d the f l o w t o q u i c k l y become 68 t h r e e d i m e n s i o n a l . I f t h e e d d i e s w e r e a c t u a l l y t w i s t e d t h e f l u c t u a t i o n s t a t i s t i c s s h o u l d n o t b e s i g n i f i c a n t l y a f f e c t e d . T U R B . F T N R E A D N ( R ) P E D D Y . J U R ) I N U M B E R I SET UP PSUM(I) ~ ^ 0 PSUM I N I T I A L I Z E F R A N D b y t l » e o f d * y F i g u r e 2 3 - N ( R ) t o E ( w ) C o m p u t a t i o n 69 The computer code g i v e s e q u a l w e i g h t i n g t o a l l p o s s i b l e eddy impact parameters b as the f l o w was assumed t o be homogenous w i t h i n a sample b i n . The e d d i e s are chosen randomly w i t h the random number g e n e r a t o r i n i t i a l i z e d t o the time of day. Thus i n t e r - e d d y c o r r e l a t i o n s were i g n o r e d . As the e d d i e s o n l y take up some f r a c t i o n of the sample space a O.Ocm/sec f l u c t u a t i n g f l o w v e l o c i t y was randomly s e l e c t e d a l o n g w i t h the p r o b a b i l i t y of c h o o s i n g an eddy. The p r o b a b i l i t y of s a m p l i n g a z e r o v e l o c i t y r a t h e r than an eddy was chosen so t h a t the f r a c t i o n of samples w i t h non-z e r o v e l o c i t y would e q u a l the o b s e r v e d f r a c t i o n of the t o t a l b i n a r e a counted as e d d i e s . D e t e r m i n i n g t h i s p r o b a b i l i t y r e q u i r e s knowledge of the average eddy l e n g t h , c o n v e c t i o n v e l o c i t y , s a m p l i n g r a t e and f r a c t i o n of non-zero samples r e q u i r e d , see Appendix D. The above as s u m p t i o n s of 1) e d d i e s b e i n g randomly i n c i d e n t on a v e l o c i t y probe which 2) measures the l o n g i t u d i n a l component of the f l u c t u a t i n g v e l o c i t y and 3) c o n v e c t i n g p a s t i t a t c o n s t a n t speed were used i n the computer model t o g e n e r a t e a f l u c t u a t i n g v e l o c i t y time r e c o r d . A t y p i c a l computer g e n e r a t e d time r e c o r d i s graphed i n f i g u r e 24. T h i s time r e c o r d can be compared w i t h the o s c i l l o g r a m s of f i g u r e 15. The time i n t e r v a l between samples was chosen t o r e s u l t i n a n o n - a l i a s e d spectrum of 0 t o 100Hz f r e q u e n c y span. T h i s time r e c o r d was then F o u r i e r a n a l y z e d u s i n g a UBC Computing C e n t e r l i b r a r y r o u t i n e c a l l e d DF0UR2. The F o u r i e r c o e f f i c i e n t s were then used t o 70 c a l c u l a t e the E„(w) a c c o r d i n g t o e q n . ( 2 - 3 2 ) . 4.0n 2.0. fcrrVsac u 0.5 T 1.5 -2.0- t (sec) - 4 J O J F i g u r e 24 - u,(t) Generated from the Eddy D i s t r i b u t i o n H a ving o b t a i n e d an E,t(w) spectrum a new time r e c o r d i s g e n e r a t e d and a n a l y z e d . The new E„(w) i s then averaged w i t h the p r e v i o u s v a l u e . T h i s p r o c e s s c o n t i n u e s u n t i l the r e q u e s t e d number of averages has been made. For the eddy s p e c t r a a n a l y z e d i t was found t h a t 200 averages were s u f f i c i e n t t o d e f i n e a m e a n i n g f u l power spectrum. Appendix C d e s c r i b e s t h i s e n t i r e p r o c e s s i n d e t a i l . The power s p e c t r a l d e n s i t y c u r v e was then p l o t t e d . A t y p i c a l eddy power spectrum i s shown i n f i g u r e 25b. The eddy spectrum from which i t was g e n e r a t e d i s shown i n f i g u r e 25a. F i g u r e 25c i s a r e s c a l e d p l o t of the power spectrum 25b. The r e s c a l i n g i s d i s c u s s e d l a t e r . F i g u r e 26 shows the 40cm/sec eddy g e n e r a t e d power s p e c t r a as a f u n c t i o n of d i s t a n c e . As eddy a n g u l a r v e l o c i t y measurements were o n l y o b t a i n e d f o r the 40cm/sec ph o t o g r a p h s , eddy power s p e c t r a c o u l d o n l y be g e n e r a t e d f o r t h a t speed. These d a t a a r e 71 d i s c u s s e d i n the l a s t s e c t i o n of t h i s c h a p t e r . 4.4 Power Spectrum O b t a i n e d From U ( t ) In the p r e v i o u s s e c t i o n s we have d e s c r i b e d how a power spectrum can be e x t r a c t e d from the eddy s i z e d i s t r i b u t i o n . These r e s u l t s a r e t o be compared w i t h power s p e c t r a o b t a i n e d u s i n g the h o t - f i l m CTA, ( c o n s t a n t t emperature anemometer) and spectrum a n a l y z e r . The CTA produces a DC f i l t e r e d v o l t a g e s i g n a l p r o p o r t i o n a l t o the streamwise v e l o c i t y component. T h i s s i g n a l i s F o u r i e r a n a l y z e d by the spectrum a n a l y z e r t o produce the RMS power s p e c t r a of th e s e v o l t a g e f l u c t u a t i o n s . The power s p e c t r a l d e n s i t y c u r v e i s o b t a i n e d from t h i s by m u l t i p l y i n g by the anemometer c a l i b r a t i o n c o n s t a n t , s q u a r i n g the r e s u l t and then d i v i d i n g by the a n a l y z e r ' s band-width. 72 EDDY SPECTRUM V E L 0 C I T Y = 4 0 CM/SEC X POSN= 3D CM ++ ++ —|— 0.6 T I I I I I 1 . 2 1 . 8 2 . 4 RODIU3ICM) 3 * a) EDDY POWER SPECTRUM (RRER R E P R E S E N T A T I O N ) V E L O C I T Y = 4 0 C M / S E C X P O S N = 3 0 CM b) i i 1 1 1 1 1 — i 3.0 -2.0 -1.0 0.0 1.0 2.0 3.0 4.0 5.0 L O G i W.i EDDY POUER SPECTRUM VELOCITY= 40 CM/SEC XPOSN= 30 CM ~1 I I I 1 -1 1—'—I " - i ! 0 0 2n.o 40.n r i . o ep n inn n F R L O U E H O i H Z I c) F i g u r e 25 - Eddy Power S p e c t r a 73 EDDY POWER SPECTRUM VRRIRTION WITH POSITION + X=30cm 50 3 70 * 90 » - t i I i — i — — i — n r 0.0 20.0 40.0 60.0 80.0 100.0 FREQUENCY (HZ) F i g u r e 26 - Eddy Power S p e c t r a : V a r i a t i o n w i t h D i s t a n c e 74 F i g u r e 27 i s a f l o w c h a r t of how E (w) i s o b t a i n e d from u ( t ) by the spectrum a n a l y z e r . The mean v o l t a g e has been s u b t r a c t e d from the anemometer s i g n a l . The f l u c t u a t i n g v o l t a g e s i g n a l i s then d i g i t i z e d a t a r a t e , F s , f o u r t i m e s t h a t of the maximum fr e q u e n c y of i n t e r e s t . For a 0 t o 100Hz span the sampl i n g f r e q u e n c y Fs i s 400Hz. I f u ( t ) c o n t a i n s f l u c t u a t i o n s w i t h f r e q u e n c i e s g r e a t e r than Fs t h e s e would be a l i a s e d t o lower f r e q u e n c i e s . However s a m p l i n g a t t h i s f o u r t i m e s h i g h e r f r e q u e n c y a l l o w s the p o t e n t i a l l y a l i a s e d a m p l i t u d e s t o be e l i m i n a t e d . The d i g i t i z e d v e l o c i t y f l u c t u a t i o n s i g n a l i s u j=1,2,..,N-1 where N=T/Fs w i t h T b e i n g the t o t a l s a m p l i n g t i m e . In p r a c t i c e N i s p r e d e t e r m i n e d due t o l i m i t e d a v a i l a b l e memory space and the b i n a r y o p e r a t i o n s i n the FFT p r o c e s s . The d i s c r e t e F o u r i e r t r a n s f o r m (DFT) of the u i s d e f i n e d as #4 A(k) = C u . exp[ (2rri/N) (-jk) ] (4-2) 0=o The A ( k) a r e t h e , i n g e n e r a l , complex F o u r i e r a m p l i t u d e s f o r the s i n u s o i d a l b a s i s f u n c t i o n s w i t h f r e q u e n c i e s 0, l / ( N A t ) , ... (N-1 )/(Nt»t) where t i ( = 1/Fs) i s the time i n t e r v a l between s u c c e s s i v e samples. In p r a c t i c e o n l y the f i r s t 1/4 of thes e a m p l i t u d e s a r e used as mentioned above. The a b s o l u t e v a l u e of the f i r s t N/4 A(k) complex v a l u e s a re then squared as i n eqn.(2-32) and RMS averaged t o the p r e v i o u s s p e c t r a . E„(w K )=A(k)A* ( k ) A f (4-3) 7 5 Enough such d e t e r m i n a t i o n s of E„(w*) k=0,..(N-1)/4 are made and averaged so t h a t a v e r a g i n g i n a new E„(w) does not s i g n i f i c a n t l y change the spectrum. 76 hot - f i Im anemometer s i g n a l A/D Converter F i g u r e 27 - Spectrum A n a l y z e r O p e r a t i o n 77 An HP X-Y p l o t t e r was used t o o b t a i n a copy of the RMS power s p e c t r a . T h i s p l o t was then d i g i t i z e d a t the UBC Computing C e n t e r . In the d i g i t i z i n g p r o c e s s the c u r v e s were smoothed and the probe support resonance was e l i m i n a t e d . The RMS v a l u e s were a l s o m u l t i p l i e d by the c a l i b r a t i o n c o n s t a n t and d i v i d e d by the band pass w i d t h i n the d i g i t i z a t i o n p r o c e s s . The square r o o t of the power s p e c t r a l d e n s i t y was thus w r i t t e n i n t o a computer f i l e . These v a l u e s were then squared t o o b t a i n the h o t - f i l m measured power s p e c t r a l d e n s i t y c u r v e . A t y p i c a l c u r v e appears i n f i g u r e 28b. The a n a l y z e r output from which i t was c a l c u l a t e d i s shown i n f i g u r e 28a. P r o p e r t i e s of the decay of t u r b u l e n t f l u c t u a t i o n s can be s t u d i e d by e xamining the change i n the power s p e c t r a w i t h d i s t a n c e . F i g u r e 29 shows the E„(w) s p e c t r a as a f u n c t i o n of d i s t a n c e from the g r i d . A d i s c u s s i o n of t h e s e s p e c t r a appears i n the next s e c t i o n . When comparing the power s p e c t r a o b t a i n e d from the e d d y - s i z e d i s t r i b u t i o n w i t h t h a t o b t a i n e d u s i n g the h o t - f i l m anemometer one s h o u l d remember t h a t t h e a n a l y s i s of the f l u c t u a t i n g s i g n a l was o p e r a t i o n a l l y e q u i v a l e n t . Indeed the computer model's method of computing and a v e r a g i n g s u c c e s i v e s p e c t r a was d e s i g n e d a f t e r the spectrum a n a l y s e r ' s . For the h o t - f i l m a n a l y s i s an HP 3582A spectrum a n a l y z e r was used t o g e n e r a t e En(w) from the v o l t a g e f l u c t u a t i o n s w h i l e the 78 0 mv-i P(w) R M S Power Spectrum X P O S N = 9 0 c m VELOCITY=40cm/sec 1 Oavgs 0 100Hz t y p i c a l xy p l o t HOT-FILM POWER SPECTRIN VELOCITY=40 CM/SEC XP0SN= 90 cn L U 1 1—v "" r ' 'i * ' r " -| ' "i - - r 0 0 2 0 . 0 4 0 . 0 6 0 . 0 8 0 . 0 100 0 FREQUENCY IHZ) F i g u r e 28 - A n a l y z e r X-Y P l o t 79 o IE C J — ' LU CO \ 21 " O oo • — - o " IS LU HOT-riL POU D V A R I A T I O N W I T H P O S I T I O N 'ECTRUM V E L O C I T Y = 4 0 C M / S E C X=30 50 70 90 o " Hurt n a « « « 0.0 20 .0 1 40. n FREQUENT r 60 .0 1 H Z ) 80.0 100 .0 F i g u r e 29 - H o t - F i l m Power S p e c t r a : V a r i a t i o n w i t h D i s t a n c e 80 computer model used UBC F a s t F o u r i e r T r a n s f o r m s o f t w a r e t o a c h i e v e the same end s t a r t i n g w i t h u ( t ) g e n e r a t e d from the e d d y - s i z e spectrum. 81 4.5 R e s u l t s And D i s c u s s i o n The 40cm/sec e d d y - s i z e s p e c t r a were shown i n f i g u r e 22. S e v e r a l o b s e r v a t i o n s can be made from t h e s e . At a d i s t a n c e of 20cm b e h i n d the g r i d the d i s t r i b u t i o n i s both lower and not as peaked as a t 30cm. The s t r e s s energy has not c o m p l e t e l y been c o n v e r t e d t o r o t a t i o n a l k i n e t i c energy of the c o h e r e n t s t r u c t u r e s . 30cm from the g r i d the d i s t r i b u t i o n i s q u i t e s h a r p l y peaked about the 1.4cm r a d i u s . The bar s i z e and s e p a r a t i o n d etermine t h i s s i z e . As the f l o w e v o l v e s we see t h a t the d i s t r i b u t i o n of eddy s i z e s broadens and the t o t a l number of e d d i e s d e c r e a s e s . Both l a r g e r and s m a l l e r s i z e s t r u c t u r e s have e v o l v e d from the g r i d geometery dominated d i s t r i b u t i o n . The i n c r e a s e i n t o t a l number of e d d i e s a t the 60cm p o s i t i o n i s l i k e l y due t o the s m a l l s t a t i s t i c a l sample used. A prominent f e a t u r e of the e d d y - s i z e s p e c t r a i s the appearance of a second hump a t a l a r g e r s i z e r a d i u s than the i n i t i a l peak. See f i g u r e 22 f o r X POSN=100 and 110 cm and f i g u r e 23 f o r X POSN=130 th r o u g h t o 190cm. T h i s d i s c o n t i n u o u s i n c r e a s e i n s i z e i s c o n v i n c i n g l y e x p l a i n e d i n terms of eddy p a i r i n g . L o o k i n g back t o the f l o w p a t t e r n s i t was seen t h a t a d j a c e n t c o - r o t a t i n g e d d i e s would o f t e n appear i n d i f f e r e n t s t a g e s of an e v o l u t i o n t o a s i n g l e eddy as d e p i c t e d i n f i g u r e 30. 82 1 2 3 U F i g u r e 30 - E d d y P a i r i n g T h i s c a n be u n d e r s t o o d by c o n s i d e r i n g t h e h i g h e r p r e s s u r e p r e s e n t i n t h e v i s c o u s l y i n t e r a c t i n g b o u n d a r y b e t w e e n t h e t w o e d d i e s . The f l o w w o u l d be r e d i r e c t e d a r o u n d t h e i n t e r a c t i o n r e g i o n t o become p a r t o f t h e c o m p o s i t e s t r u c t u r e . The jump i n e d d y s i z e i s t h u s p b s e r v e d . I t i s w e l l w o r t h q a z i n g a t t h e f l o w p a t t e r n s o f f i g u r e s 9 t h r o u g h 1 2 . One s o o n r e a l i z e s t h a t many c o h e r e n t s t r u c t u r e s p e r s i s t i n s u c c e s s i v e p h o t o g r a p h s . T h i s f u r t h e r s u p p o r t s o u r u s e o f T a y l o r ' s h y p o t h e s i s . I m a g i n a t i v e s c r u t i n y o f t h e s u c c e s i v e f l o w p a t t e r n s show t h e e v o l u t i o n o f t h e e d d i e s b o t h i n d i v i d u a l l y a n d o r g a n i z a t i o n a l l y . Our f l o w v i s u a l i z a t i o n t e c h n i c i s s e e n t o be a n e f f e c t i v e t o o l f o r s t u d y i n g e d d y d y n a m i c s . The a v e r a g e c h o r d l e n g t h o f a n e d d y c a n be c a l c u l a t e d f r o m t h e e d d y d i s t r i b u t i o n s , s e e A p p e n d i x D . T h i s l e n g t h i s c o m p a r e d w i t h t h e i n t e g r a l l e n g t h s c a l e d e r i v e d f r o m b o t h t h e e d d y s p e c t r a a n d t h e h o t - f i l m p o w e r s p e c t r a i n f i g u r e 83 31. A d i s c u s s i o n of these r e s u l t s appears l a t e r i n t h i s s e c t i o n . The t o t a l number of e d d i e s o b s e r v e d as a f u n c t i o n of d i s t a n c e i s p l o t t e d i n f i g u r e 32 f o r the 40cm/sec d a t a . The eddy produced s p e c t r a appear w i t h t h e i r c o r r e s p o n d i n g h o t - f i l m measured c u r v e s i n f i g u r e 35. The u n c e r t a i n t y i n the eddy c u r v e s i s l a r g e l y due t o the u n c e r t a i n t y i n the a n g u l a r v e l o c i t y of the e d d i e s . T h i s i s seen from f i g u r e 20 t o be about 25%. The h o t - f i l m c u r v e ' s u n c e r t a i n t y due t o c a l i b r a t i o n d r i f t and m e c h a n i c a l resonance i s about 15%. The s t a t i s t i c a l u n c e r t a i n t y due t o i n s u f f i c i e n t s p e c t r a a v e r a g i n g can be e s t i m a t e d by o b s e r v i n g the e f f e c t of a v e r a g i n g s u c c e s i v e s p e c t r a d u r i n g the a q u i s i t i o n p r o c e s s . An u n c e r t a i n t y of 10% f o r the 25Hz range, i n c r e a s i n g w i t h d e c r e a s i n g f r e q u e n c y i s a r e a s o n a b l e e s t i m a t e . The eddy c u r v e s a r e seen t o be c o n s i s t e n t l y lower than the c o r r e s p o n d i n g h o t - f i l m s p e c t r a . The f r e q u e n c i e s c o n t a i n i n g a p p r e c i a b l e energy a r e a l s o c o n s i s t e n t l y lower f o r the eddy g e n e r a t e d s p e c t r a . P o s s i b l e reasons f o r t h i s a r e s e v e r a l . I n s p e c t i o n of f i g u r e 19a shows t h a t o n l y about 1/5 of the t o t a l a r e a of the f l o w has been c h a r a c t e r i z e d as e d d i e s w h i l e almost a l l of the f l o w seems t o be a g i t a t e d . The l o c a l f l o w v e l o c i t y i n the r e g i o n s between the e d d i e s i s about as l a r g e as the r o t a t i o n a l v e l o c i t y a t the edqe of the e d d i e s , hence the i n t e r - e d d y f l u i d c o n t a i n s about as much k i n e t i c energy per u n i t mass as the e d d i e s . The r a t i o of the t o t a l eddy a r e a t o the b i n a r e a , f i g u r e 33, i s t h e r f o r e 84 a good measure f o r the f r a c t i o n of the t o t a l k i n e t i c energy of the f l o w s t o r e d i n the e d d i e s . The f l o w between the e d d i e s was c o n t a i n s s u b s t a n t i a l v e l o c i t y changes. T h i s i s seen i n the o s c i l l o g r a m s of f i g u r e 15 as w e l l . These f l u c t u a t i o n s have not been a c c o u n t e d f o r i n our s i m p l e a n a l y s i s . The v e l o c i t y changes i n the eddy i n t e r a c t i o n r e g i o n s occur over d i s t a n c e s o f t e n s u b s t a n t i a l l y s h o r t e r than the t y p i c a l eddy s i z e . A F o u r i e r d e c o m p o s i t i o n of the f l o w f i e l d t a k i n g t h e s e r e g i o n s i n t o account would show more energy i n h i g h e r f r e q u e n c i e s than our a n a l y s i s . Another e x p l a n a t i o n of the l a c k of energy i n the h i g h e r f r e q u e n c i e s i s the s i z e s c a l e r e s o l u t i o n l i m i t of b o th the v i s u a l i z a t i o n and a n a l y s i s methods used. A l a r g e p h o t o g r a p h i c p r i n t of the f l o w showed some s m a l l e r s c a l e c o h e r e n t s t r u c t u r e s which were not c o u n t e d .when the p r o j e c t i o n system was used. The low t r a c e r d e n s i t y c o u l d a l s o cause us t o miss the s m a l l e r s c a l e e d d i e s however from o b s e r v a t i o n of the f l o w photographs i t was c o n c l u d e d t h a t f o r a l l but the 50cm/sec c a r t speed the s p a t i a l r e s o l u t i o n was adequate. In g e n e r a t i n g the eddy power s p e c t r a any e x p l i c i t i n t e r - e d d y c o r r e l a t i o n s were i g n o r e d . O b s e r v a t i o n of the f l o w p a t t e r n s showed the p r e s ence of clumps of e d d i e s s e p a r a t e d by " r i v e r s " of r e l a m i n a r i z e d f l o w . A d j a c e n t c o -r o t a t i n g e d d i e s would c o n t r i b u t e t o the low f r e q u e n c i e s of the F o u r i e r a n a l y s i s . E l e c t r i c a l n o i s e i n the h o t - f i l m d i a g n o s t i c system as w e l l as unaccounted f o r m e c h a n i c a l 8 5 v i b r a t i o n s would c o n t r i b u t e t o the h o t - f i l m spectrum. A c o n s t a n t low fr e q u e n c y n o n - t u r b u l e n t n o i s e i n the anemometer s i g n a l would a l s o e x p l a i n the i n c r e a s i n g d i s c r e p a n c y w i t h d i s t a n c e i n the low fre q u e n c y regime. I n c r e a s i n g i n t e r e d d y c o r r e l a t i o n s or c o u n t i n g the c i r c u l a r c o r e of a l a r g e non-c i r c u l a r s t r u c t u r e c o u l d a l s o e x p l a i n t h i s . There, i s not enough i n f o r m a t i o n t o d e c i d e what t o a t t r i b u t e the d i s c r e p a n c y i n the s p e c t r a t o a l t h o u g h more than enough t o s p e c u l a t e . S u f f i c e t o say t h a t a s i g n i f i c a n t f r a c t i o n of the f l u c t u a t i n g k i n e t i c energy i n the pr o p e r fre q u e n c y regime i s p r e d i c t e d from our s i m p l e eddy d e s c r i p t i o n of the f l u c t u a t i o n s . I t would be u s e f u l t o a n a l y z e the f l o w photos a g a i n and t r e a t lumps of f l u i d w i t h a common r o t a t i o n a x i s as i n c o m p l e t e e d d i e s . T h i s would account f o r a l l the f l o w f i e l d i n a s t r a i g h t f o r w a r d manner. . The t o t a l mean square of the l o n g i t u d i n a l v e l o c i t y f l u c t u a t i o n s i s found by i n t e g r a t i n g E (w), as shown i n eqn . ( 2 - 2 3 ) . F i g u r e 34 shows the decay of uf w i t h d i s t a n c e X f o r both the f l u c t u a t i o n s g e n e r a t e d by the eddy spectrum and the f l u c t u a t i o n s measured from the h o t - f i l m anemometer. The "eddy" u v a l u e s have been m u l t i p l i e d by a f a c t o r of 10.0 t o f a c i l i t a t e a comparison i n the energy decay t r e n d . The remark a b l e s i m i l a r i t y i n the decay t r e n d s i n d i c a t e s t h a t our eddy spectrum d e s c r i p t i o n i s i n t i m a t e l y l i n k e d w i t h the f l u c t u a t i o n dynamics. 86 The i n t e g r a l l e n g t h s c a l e can be o b t a i n e d from the power s p e c t r a u s i n g eqn ( 2 - 2 6 ) . Le=UcE„ (0)/(4u? ) T h i s l e n g t h s c a l e s h o u l d be r e l a t e d t o the s i z e of the e d d i e s . F i g u r e 31 shows the average eddy c h o r d p l o t t e d a g a i n s t d i s t a n c e X t o g e t h e r w i t h t w i c e the i n t e g r a l l e n g t h s c a l e s o b t a i n e d from the eddy produced and h o t - f i l m measured power s p e c t r a l d e n s i t y c u r v e s . The s i m i l a r i t y between the average eddy c h o r d and the eddy produced spectrum's i n t e g r a l l e n g t h s c a l e i s not remarkable except t h a t i t t e l l s us t h a t t w i c e the i n t e g r a l l e n g t h s c a l e s h o u l d be a s s o c i a t e d w i t h the average eddy s i z e . The s i m i l a r i t y between the h o t - f i l m measured and eddy spectrum produced i n t e g r a l l e n g t h s c a l e s i s q u i t e good f o r the f i r s t t h r e e d i s t a n c e s . T h i s t e l l s us t h a t the eddy spectrum d e s c r i p t i o n has c h a r a c t e r i z e d the s i z e s c a l e of the energy c o n t a i n i n g f l u c t u a t i o n s q u i t e w e l l i n s p i t e of our crude a n a l y s i s . The d i v e r g e n c e of the h o t -f i l m i n t e g r a l l e n g t h s c a l e a t the 90cm d i s t a n c e i s r e l a t e d t o the anomalously h i g h energy c o n t e n t i n the low fr e q u e n c y end of the 90cm c u r v e of f i g u r e 35. S p e c u l a t i o n s of c o n s t a n t background n o i s e or i n f l u e n c e of i n t e r - e d d y c o r r e l a t i o n s have been mentioned. A l t h o u g h i t i s a l s o q u i t e l i k e k y t h a t t h i s anomoly i s due t o e r r o n e o u s d i g i t i z i n g . To s t u d y c o n t r i b u t i o n s t o the f l u c t u a t i n g k i n e t i c energy i n d i f f e r e n t f r e q u e n c y regimes the power s p e c t r a were r e s c a l e d t o expand the f r e q u e n c y a x i s . T h i s was done by p l o t t i n g wE«, (w) a g a i n s t l o g ( w ) . The new c u r v e has e q u a l 87 a r e a s r e p r e s e n t i n g e q u a l c o n t r i b u t i o n s t o t h e t o t a l t u r b u l e n t k i n e t i c e n e r g y . T h i s c a n be s e e n by o b s e r v i n g t h a t t h e i n t e g r a l r e p r e s e n t i n g t h e a r e a u n d e r wE, , (w) b e t w e e n t w o l o g g e d f r e q u e n c y a x i s p o i n t s l o g ( w , ) a n d l o g ( w ^ ) s i m p l i f i e s t o t h e i n t e g r a l o f E „ ( w ) b e t w e e n t h e c o r r e s p o n d i n g f r e q u e n c i e s , w E n ( w ) d { l o g ( w ) } = )E , » (w)dw a s w d { l o g ( w ) } = w/wdw=dw ( 4 - 4 ) The e x p a n s i o n o f t h e f r e q u e n c y s c a l e f a c i l i t a t e s t h e s t u d y o f t r e n d s i n d i f f e r e n t f r e q u e n c y r e g i m e s . F i g u r e 36 c o m p a r e s t h e same s p e c t r a now p l o t t e d i n t h e new f o r m a t . The f r e q u e n c y a x i s p o i n t c o r r e s p o n d i n g t o t h e i n t e g e r a l t i m e s c a l e i s w h e r e w = l / T e a t l o g ( T e ) . I t h a s b e e n m a r k e d on t h e f i g u r e s a n d i s n e a r t o w h e r e t h e maximum c o n t r i b u t i o n t o t h e t o t a l e n e r g y o c c u r s . T h i s a g r e e s w i t h o u r i n t e r p r e t a t i o n • o f Te a s a m e a s u r e o f t h e t i m e s c a l e o f t h e e n e r g y c o n t a i n i n g f l u c t u a t i o n s . T h i s a r e a r e p r e s e n t a t i o n i s b e s t u s e d when d i f f e r e n c e s i n t h e d i s t r i b u t i o n o f e n e r g y among d i f f e r e n t f r e q u e n c y r e g i m e s a r e o f i n t e r e s t . I n t h e new r e p r e s e n t a t i o n we c a n m o r e c l e a r l y s e e t h a t t h e e d d y g e n e r a t e d p o w e r s p e c t r a l d e n s i t y i s m o r e s h a r p l y p e a k e d t h a n t h e h o t - f i l m c u r v e . T h i s i s due t o t h e h i g h l y s i m p l i f i e d e d d y a n a l y s i s t h a t was u s e d . The i n t r a - e d d y v e l o c i t y d i s t r i b u t i o n s w e r e h i g h l y i d e a l i z e d a n d t h e a n g u l a r v e l o c i t y d i s t r i b u t i o n s w e r e o v e r s i m p l i f i e d t o be a c o n s t a n t f o r a g i v e n d i s t a n c e f r o m t h e g r i d . H a d t h e m e a s u r e d d i s t r i b u t i o n i n a n g u l a r v e l o c i t i e s b e e n u s e d i n s t e a d o f t h e a v e r a g e v a l u e 88 a broader d i s t r i b u t i o n of energy about the i n t e g r a l l e n g t h s c a l e would be e x p e c t e d . 89 P eddy Le A eddy 2T e'U g • hot film 2Te-U, A • 4J Al 1 . 1 r  50 cm 70 90 1 1 0 F i g u r e 31 - Comparison o f . L e n g t h S c a l e s Total Eddy Count vs. Distance 40 c m 60 80 100 F i g u r e 32 - Eddy Count w i t h D i s t a n c e 90 EF 1 0.20-I Eddy Fraction of Flow Area E F vs Distance X 0.15-1 U g = 40cm/s 0.10-1 • • I 005-20 40 cm 60 80 100 120 F i g u r e 33 - V a r i a t i o n i n O c c u p i e d A r e a 10 \ 77* \ \ \ w • hot-film A eddy 0 4 -30 X A 50 cm 7 0 90 110 F i g u r e - 3 4 Decay of u z 91 EDDY HOT-FILM POWER S P E C T R U M VELOCITY=40 CM/SEC XPOSN= 30 cn <_>°' UJ i n ~v . 2 0 - 0 4 0 . 0 en o FREQUENCY (HZ) 80.0 IOO.O o.o EDDY HOT-FILM POWER S P E C T R U M VEL0CITY=40 CH/SEC XPOSN= 50 cn " T n — - i " f "I -20.0 40.0 60.0 FREQUENCY IHZ) 80.0 100 + eddy - hot-f i Im EDDY HOT-FILM P O W E R S P E C T R U M VELOCITY=40 CM/SEC XPOSN- 70 CM <_)•= L U in EDDY HOT-FILM POWER S P E C T R U M VEL0CITY=40 CM/SEC XP0SN= 90 CM 4 U 0 FREQUENCY 60.0 IHZi 100.0 0 . 0 20 .0 40.0 FREQUENCY 60.0 (HZ) 8 0 . 100.0 F i g u r e 35 - Comparison of Power S p e c t r a 92 EDDY HOT-FILM POWER SPECTRUM (RRER REPRESENTATION) VELOCITY= 40CM/SEC X POSN= 30CM 1 1 1 — 1 -3.0 -2.0 -1.0 0.0 1.0 LOGIWi UJo 3c EDDY HOT-FILM POWER SPECTRUM (RRER REPRESENTATION) VELOCITY- 40CM/SEC X POSN= 50CM "i 1 1 r 2-° 3.0 4.0 5.0 -3.0 -2.0 -1.0 o.n 111 LOG'i'W +Te T 1 1 1 r.O 3.0 4.0 5.0 +Te -Te + eddy - ho t - f i l m EDDY HOT-FILM POWER SPECTRUM (RRER REPRESENTRTIONI VELOCITY* 40CM/SEC X POSN= 70 CM 1 1 1 1 — 3.0 -2.0 -l.o o.o 1.0 LOG IV T—, 1 ' ' ' I I | 2.0 3.0 4.0 5.0 -Te •Te EDDY HOT-FILM . POWER SPECTRUM IfiREfl REPRESENTRTION). VELOCITY= 40CM/SEC X POSN- 90CM • * * 4 »«* * * 1 * ***** * ^***m*tm T " 1 1 — U < V ' •3.0 - 2 . 0 - 1 . 0 0.0 l.<S\ ^ 0 3.0 4.0 5.0 LOG(W)\ W s + Te F i g u r e 36 - E q u i v a l e n t Area R e p r e s e n t a t i o n 93 V. CONCLUSIONS A model which t r e a t s a t u r b u l e n t f l o w as b e i n g composed of c o h e r e n t r o t a t i o n s p l u s l a m i n a r f l o w has been p r e s e n t e d . E x p e r i m e n t s were performed t o study how w e l l t h i s model d e s c r i b e s the v e l o c i t y f l u c t u a t i o n s i n a g r i d g e n e r a t e d t u r b u l e n t f l o w . We se a r c h e d f o r and found c o h e r e n t s t r u c t u r e s by p h o t o g r a p h i n g the motions of aluminum t r a c e r s . The to w i n g tank c o n s t r u c t e d f o r the s e e x p e r i m e n t s a l l o w e d f o r easy a c c e s s t o e i t h e r the f l u i d or o b j e c t r e f e r e n c e frame. A l t h o u g h the f l u i d v e l o c i t y i n f o r m a t i o n i s a v a i l a b l e from both frames i t was found t h a t the v i s u a l i z a t i o n t e c h n i q u e was o n l y u s e f u l w i t h the camera i n the f l u i d r e f e r e n c e frame. The d i s t r i b u t i o n of s i z e s and a n g u l a r v e l o c i t i e s of the e d d i e s were measured from t h e s e p h o t o g r a p h s . T h i s e d d y - s i z e spectrum was used t o g e n e r a t e the power s p e c t r a l d e n s i t y of the l o n g i t u d i n a l v e l o c i t y f l u c t u a t i o n s . A h o t - f i l m anemometer was used t o measure the l o n g i t u d i n a l v e l o c i t y f l u c t u a t i o n s g e n e r a t e d by the g r i d . A spectrum a n a l y z e r produced the power s p e c t r a l d e n s i t y from t h e s e f l u c t u a t i o n s . By comparing the h o t - f i l m measured and eddy spectrum g e n e r a t e d power s p e c t r a l d e n s i t i e s we have drawn the f o l l o w i n g c o n c l u s i o n s : The eddy-spectrum d e s c r i p t i o n p r e d i c t s the s i z e s c a l e s of t h e t u r b u l e n t f l u c t u a t i o n s . I t a l s o p r e d i c t e d a c o n s t a n t 10% of the t o t a l k i n e t i c energy of v e l o c i t y f l u c t u a t i o n s measured by the h o t - f i l m probe. The eddy model was l e a s t 94 s u c c e s s f u l a t d e s c r i b i n g the k i n e t i c energy i n the h i g h and low f r e q u e n c y regimes. We b e l i e v e t h i s d i s c r e p a n c y i s p a r t l y due t o the f a c t t h a t i n t e r - e d d y c o r r e l a t i o n s and co h e r e n t f l u c t u a t i o n s i n the space between the e d d i e s were not a c c o u n t e d f o r i n our a n a l y s i s . Our s i m p l e t r e a t m e n t of the v e l o c i t y p r o f i l e w i t h i n an eddy may have c o n t r i b u t e d as w e l l . The e d d y - s i z e spectrum p r o v e d u s e f u l f o r u n d e r s t a n d i n g the dynamics of the e v o l u t i o n of t u r b u l e n t l e n g t h s c a l e s . The e v o l u t i o n of the s t r u c t u r e s c o u l d be seen i n s u c c e s s i v e f l o w p h o t o g r a p h s . E v i d e n c e of e d d y - p a i r i n g was obser v e d f o r c o - r o t a t i n g s t r u c t u r e s b o t h i n the e d d y - s i z e spectrum's e v o l u t i o n and i n the f l o w p h otographs. We l o o k f o r w a r d t o l e a r n i n g more about eddy i n t e r a c t i o n s by s t u d y i n g s e q u e n t i a l p hotographs of f l u i d f l o w p a t t e r n s . Our model has many advantages over the u s u a l s t a t i s t i c a l d e s c r i p t i o n of t u r b u l e n c e . As t h i s t h e s i s has shown c o h e r e n t s t r u c t u r e s can be observed i n and are im p o r t a n t t o the dynamics of t u r b u l e n t f l u i d f l o w . A d e s c r i p t i o n of t u r b u l e n c e i n terms of eddy s p e c t r a i s s i m p l e r t o u n d e r s t a n d and more p h y s i c a l l y m o t i v a t e d than s t u d y i n g " s t a t i s t i c a l f l u c t u a t i o n s " . T h i s d e s c r i p t i o n n a t u r a l l y t a k e s i n t o account the mechanics of coherent s t r u c t u r e s . A l s o , the eddy-dynamics c o u l d be u s e f u l f o r p r e d i c t i n g p r o p e r t i e s of t u r b u l e n t f l o w s such as the i n f l u e n c e of t u r b u l e n c e on mean v e l o c i t y p r o f i l e s , on the 95 d r a g of a body, and on m i x i n g p r o c e s s e s . 1 And, of c o u r s e , knowledge of the eddy dynamics can be used t o p r e d i c t the eddy s t a t i s t i c s which i n t u r n can be used t o p r e d i c t v e l o c i t y f l u c t u a t i o n s i n a t u r b u l e n t f l o w . Perhaps the g r e a t e s t advantage of s t u d y i n g t u r b u l e n c e from a c o h e r e n t s t r u c t u r e s v i e w p o i n t i s i n the new u n d e r s t a n d i n g of the mechanics of t u r b u l e n t f l o w s t h a t may a r i s e . A s i d e from i t s a e s t h e t i c v a l u e u n d e r s t a n d i n g the mechanics of eddy p r o d u c t i o n c o u l d l e a d t o more e f f i c i e n t f l u i d machinery by i n f l u e n c i n g the p r o d u c t i o n and decay of c o h e r e n t s t r u c t u r e s . More e f f i c i e n t canoes and s a i l b o a t s not t o mention f r e i g h t e r s and a i r c r a f t c o u l d be p o s s i b l e . The r e s u l t s of t h i s t h e s i s i n d i c a t e t h a t d e v e l o p i n g b e t t e r c r i t e r i a f o r a n a l y z i n g s t r u c t u r e s i n the f l o w would be w o r t h w h i l e . T h i s c o u l d be done by a n a l y z i n g an e n t i r e t h r e e d i m e n s i o n a l f l o w f i e l d c o n c e n t r a t i n g on the l o c a l c u r v a t u r e and a n g u l a r v e l o c i t y . T h i s a n a l y s i s would a d d r e s s the o b s e r v e d phenomenon of p a r t l y formed, or f r a c t i o n a l , e d d i s . An automated a n a l y s i s u s i n g a d i g i t i z e d v i d e o s i g n a l would be q u i t e s t r a i g h t f o r w a r d t o implement u s i n g t h i s d e f i n i t i o n . Many f l o w s c o u l d then be a n a l y z e d i n a r i g o r o u s f a s h i o n . UBC PLASMA PHYSICS l a b r e p o r t #89 96 BIBLIOGRAPHY B . A h l b o r n , F . A h l b o r n and S.Loewen: UBC PLASMA PHYSICS l a b r e p o r t #89, (1983) F . A h l b o r n : Z e i t s c h r i f t f u r T e c h n i s c h e P h y s i k _1_2 , 482-491 (1931 ) J.O.HINZE: TURBULENCE, McGRAW-HILL, (1959) J.L.LUMELY: S t o c h a s t i c T o o l s i n T u r b u l e n c e , Academic P r e s s , (1970) Roberson and Crowe: 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 Co., (1975) H.Tennekes and J.L.Lumley: A F i r s t Course i n T u r b u l e n c e , MIT Pr e s s , ( 1 9 7 2 ) 97 APPENDIX A - HOT-FILM PROBE SENSITIVITY TO VELOCITY FLUCTUATIONS I g n o r i n g t h i s i n f l u e n c e the probe w i l l respond t o the speed of f l o w p a s t the s e n s o r . For a l i n e a r i z e d probe the f l o w speed S i s r e l a t e d t o the l i n e a r i z e d anemometer v o l t a g e s i g n a l , E , by the r e l a t i o n S=KE (A-1) where K i s the c a l i b r a t i o n c o n s t a n t . The f l o w speed S i s r e l a t e d t o the f l o w v e l o c i t y components by the r e l a t i o n S a = (U+u) 1 +v' +W1- V=W=0 (A-2) where u,v,w a r e the. l o n g i t u d i n a l and two l a t e r a l f l u c t u a t i n g v e l o c i t y components and U i s the l o n g i t u d i n a l mean v e l o c i t y component. W r i t i n g out the (U+u) term we have S* =Ul + 2Uu+ul +vl+w'L (A-3) I f the t u r b u l e n c e i n t e n s i t y i s s u f f i c i e n t l y low so t h a t u,v,w<<U the squared terms i n eqn.(3-3) may be n e g l e c t e d . A f t e r d i v i d i n g b o th s i d e s by U and t a k i n g t h e i r square r o o t we have S / U=/1+2U / U (A-4) Now u s i n g a M a c l a u r i n s e r i e s e x p a n s i o n f o r t h e square r o o t w i t h u<<U and m u l t i p l y i n g by U we have S=U+u (A-5) T r e a t i n g the l i n e a r i z e d v o l t a g e E as the sum of a s t e a d y , E , p l u s a f l u c t u a t i n g component, e, and s u b s t i t u t i n g eqn.(A-5) i n t o eqn.(A-1) we see t h a t U+u=KE +Ke (A-6) The f l u c t u a t i n g p a r t of the anemometer s i g n a l i s p r o p o r t i o n a l t o the f l u c t u a t i n g p a r t of l o n g i t u d i n a l v e l o c i t y . The c o n s t a n t of p r o p o r t i o n a l i t y b e i n g the same as the c a l i b r a t i o n c o n s t a n t f o r the mean component. The wedge shape of the probe tends t o s u p p r e s s the v and w v e l o c i t y component c o n t r i b u t i o n s t o the c o o l i n g thus f u r t h e r r e i n f o r c i n g e q u a t i o n ( A - 6 ) . 98 APPENDIX B - RIGID BODY EDDY VELOCITY PROFILE A r i g i d body c i r c u l a r l y c y l i n d r i c a l eddy of r a d i u s Rm r o t a t i n g about i t s a x i s of symmetry has a v e l o c i t y p r o f i l e g i v e n by ue(r)=Um(r/Rm) 0<r<Rm (B-1) where Um i s the t a n g e n t i a l v e l o c i t y a t the p e r i p h e r y of the eddy and r i s the r a d i u s , see f i g u r e below. The u, component of the v e l o c i t y i s g i v e n by, u,=u„rsine (B-2) and u s i n g e q u a t i o n B-1 becomes u,=(Um/Rm)rsine (B-3) I f a u component v e l o c i t y probe moves th r o u g h the eddy a t c o n s t a n t speed i n the x d i r e c t i o n , as shown, w i t h impact parameter b we have r s i n e = b (B-4) so t h a t e q u a t i o n B-3 becomes u,=(Umb/Rm) (B-5) showing t h a t the f l u c t u a t i n g component of the v e l o c i t y f o r a r i g i d body e d d i e s i s a c o n s t a n t . A l i t t l e s t u d y of the f i g u r e below shows t h a t the sa m p l i n g time t h r o u g h the eddy would be g i v e n by t=(2/Uc){Rm -b x} (B-6) w i t h Uc b e i n g the probe speed r e l a t i v e t o the eddy. 99 A P P E N D I X C ~ T U R B . F T N FORTRAN CODE 1 C TURB.FTN 2 C INPUT DATA ON UNIT 11 3 C OUTPUT DATA ON UNIT 6 4 C 5 C PROGRAM CHANGES EDDY SPECTRUM TO POWER SPECTRUM 6 C 7 C EDDY CHARACTERIZING DISTRIBUTIONS ARE READ IN 8 C 9 C 10 INTEGER NLNGTH, NVEL, NCONV, NPTS, N1PTS, MPTS, NRMAX, NRNO'T 11 INTEGER J, I, CHOOSE, M, NUMBER, NDATA, NUX, SUMNUX 12 INTEGER LOGSPC(37) / I , 2, 3, 4, 5, 6, 7, 8, 9, 10, 11,16, 21j 13 1 27,32,37,42,47,52,57,62,67,73,78,83,88,93,97,103, 155, XOt 14 2 257,308,359,411,462,513/ 15 REAL NR(500), RNOT(500), RMAX(500), VEL(500), RAD(500) 16 REAL CONVEL(500),PSUM(500) ,ENERGY/0./,AREA/0./,TE/0./ 17 REAL LEBAR, RCHOSE, RMAXO, RNOTO, B, CONVEC, TSTEP, TRATE 18 REAL NL(500), L(500), UX(6000), UMAX(500), RATIO, TEMP 19 REAL POWERO025) /1025*0./, FREQO025), REAL I 20 REAL*8 DATA(2048) 21 COMPLEX* 16 TRANO025) 22 EQUIVALENCE (DATA(1),TRAN(1)) 23 C READ FILE NUMBER 24 READ (11,230) SPECN 25 C READ SAMPLING FREQUENCY 26 READ (11,260) TRATE 27 C TRATE=FREQUENCY RESOLUTION/NYQUIST CRITERIA 28 C 29 C NUMBER IS NUMBER OF RMS AVERAGES TO BE MADE 30 READ (11,210) NUMBER 31 C READ PEDDY ET AL 32 READ(11,320)PEDDY,AREA,LEBAR,NUMED 33 C READ IN EDDY SIZE SPECTRUM 34 READ (11,210) NRMAX 35 DO 10 J = 1, NRMAX 36 READ (11,270) NR(J), RMAX(J), RNOT(J), UMAX(J) 37 10 CONTINUE 38 C READ IN CONVECTION VELOCITIES 39 C 40 60 READ (11,210) NCONV 41 IF (NCONV .EQ. 1) GO TO 80 42 DO 70 J = 1, NCONV 43 READ (11,230) CONVEL(J) 44 70 CONTINUE 45 80 READ (11,230) CONVEC 46 C 47 C PREPARE SIZE DISTRIBUTION FOR RANDOM SAMPLING 48 C 49 C INITIALIZE FRAND BY TIME-OF-DAY 50 B «= RAND(SCLOCK(0. )) 51 PSUM(1) = 0. 52 MPTS = NRMAX - 1 53 DO 90 I = 1, MPTS 54 PSUM(I + 1) = PSUM(I) + NR(I)*2*RMAX(I) 55 90 CONTINUE 56 DO 100 I = 1, NRMAX 57 PSUM(I) = PSUM(I) / PSUM(NRMAX) 58 100 CONTINUE 59 C 60 C SAMPLING LOOP STARTS 100 61 C 62 PBLANK=1.-PEDDY 63 DO 1 90 K = 1 , NUMBER 64 USQBAR=0. 65 SUMNUX =0. 66 C NDATA IS NUMBER OF VELOCITY SAMPLES TAKEN 67 C PER FOURIER ANALYZED RECORD 68 NDATA = 2048 69 C TIME RECORD BEING MADE 70 C CHOOSE UX=0. OR SAMPLE AN EDDY 71 105 X=FRAND(0.) 72 IF (X.GT.PBLANK) GOTO 110 73 SUMNUX=SUMNUX+1 74 DATA(SUMNUX)=0. 75' IF (SUMNUX.GT.NDATA)GOTO 170 76 GOTO 105 77 C CHOOSE EDDY RADIUS 78 110 IRAD = CHOOSE(M,PSUM,NRMAX,RCHOSE) 79 RMAXO = RMAX(IRAD) 80 RNOTO = RNOT(IRAD) 81 C CHOOSE IMPACT PARAMETER 82 130 B = 2. * (FRAND(0.) - .5) * RMAXO 83 C FIND CONVECTION VELOCITY 84 IF (NCONV .GT. 1) CONVEC = CONVEL(I RAD) 85 140 CALL SAMPLE(TRATE, RNOTO, RMAXO, UMAX(IRAD), B, CONVEC, NUX.&of) 86 DO 150 I = 1, NUX 87 IF (I + SUMNUX .GT. NDATA) GO TO 170 88 DATA(I + SUMNUX) = UX(I) 89 USQBAR=USQBAR+UX(I)*UX(I) 90 150 CONTINUE 91 160 SUMNUX = SUMNUX + NUX 92 GO TO 105 93 C WRITE OUT TIME RECORD IF WANTED 94 C 170 WRITE(9,370)(DATA(II),11=1,2000) 95 170 ENERGY=ENERGY+USQBAR/NDATA/NUMBER 97 C DFOUR2 FOURIER ANALYZES THE TIME RECORD 98 NDIM = NDATA 99 C NODIM= # OF DIMENSIONS OF TIME RECORD 100 NODIM = 1 101 C ISIGN=-1 FOR DFT +1 FOR IDFT 102 ISIGN = -1 103 C IFORM=0 FOR REAL DATA 104 IFORM = 0 105 CALL DFOUR2(DATA, NDIM, NODIM, ISIGN, I FORM) 106 DO 180 I = 1, 1025 107 POWER(I) = POWER(I) + ((CDABS(TRAN(I))/NDATA)**2)/NUMBER 108 180 CONTINUE 109 190 CONTINUE 110 TSTEP = 1. / TRATE 111 DO 200 1 = 1 , 1025 112 REALI = FLOAT(I) 113 FREQ(I) = (REALI - 1.) / (NDATA*TSTEP) 1 14 200 CONTINUE 114.5 AREA=QINT4P(FREQ,POWER,513,1,513) 114.6 TE=POWER(1)/(4.*AREA) 114.7 DO 202 KK=1,513 114.75 POWER(KK)=POWER(KK)*9.8696 114.8 202 CONTINUE 1 1 5 N1PTS = 35 116 WRITE (6,280) SPECN, CONVEC, N1PTS, ENERGY,TE 101 120 WRITE (6,300) (POWER(LOGSPC(I)),FREQ(LOGSPC(I)),I=1,N1PTS) 122 STOP 123 C FORMAT CODES 124 210 FORMAT (13) 125 220 FORMAT (213) 126 230 FORMAT (F7.3) 127 240 FORMAT (2F7.3) 128 250 FORMAT (3F7.3) 129 260 FORMAT (F5.0) 130 270 FORMAT (4F7.3) 1 31 280 FORMAT (F7.3,F7.2, 13,F7.4,E10.3) 132 290 FORMAT (3(D16.9,F5.0)) 133 300. FORMAT (3(F10.3, F9.3)) 134 310 FORMAT (3(F10.3,F9.3)) 135 320 FORMAT (F7.5,2F7.3,13) 136 370 FORMAT(10F9.3) 137 END 138 INTEGER FUNCTION CHOOSE(M,PSUM,NRMAX,RCHOSE) 139 C THIS PROGRAM RANDOMLY CHOOSES AN INTERVAL FROM A 140 C PREDETERMINED DISTRIBUTION 141 C M I S THE INTEGER NUMBER OF THE INTERVAL CHOSEN 142 REAL PSUM(500), RCHOSE 143 INTEGER M, NPTS, SCOPE, MPTS 144 LOGICAL GT, LT 145 MPTS=NRMAX-1 146 M = NRMAX / 2 147 SCOPE = M + 1 148 RCHOSE = FRAND(0.) 149 10 GT = .FALSE. 150 LT = .FALSE. 151 IF (RCHOSE .GT. PSUM(M)) GT = .TRUE. 152 IF (RCHOSE .LT. PSUM(M + 1)) LT = .TRUE. 153 IF (LT .AND. GT) GO TO 20 154 SCOPE = SCOPE / 2 + 1 155 M = M + SCOPE 156 IF (LT) M = M - (2*SCOPE) + 1 157 IF (M.LT.1) M=1 158 IF (M.GT.MPTS) M=MPTS 159 GO TO 10 160 20 CHOOSE = M 161 RETURN 162 END 163 SUBROUTINE SAMPLE(TRATE, RNOTO, RMAXO, UMAX, B, CONV, NUX, UX) 164 C SAMPLES UX FOR GIVEN EDDY 165 INTEGER NCORE, NUX, I 166 REAL TIME, TMAX, RMAXO, UMAX, B, CONV, UX(6000) 167 REAL TRATE, TSTEP 168 TSTEP = 1. / TRATE 170 TMAX = SQRT(RMAXO*RMAXO - B*B) / CONV 173 UCONST = UMAX * B / RMAXO 174 NCORE =2*TMAX / TSTEP 174.5 NUX=NCORE 175 DO 30 I = 1, NCORE 176 UX(I) = UCONST 177 30 CONTINUE 178 40 RETURN 179 END End of f i l e 102 APPENDIX D ~ AVERAGE EDDY CHORD AND PEDDY CALCULATION For a c i r c u l a r l y c y l i n d r i c a l eddy h a v i n g a random impact parameter b w i t h a probe the average c h o r d , L, i s the w i d t h of a r e c t a n g l e w i t h l e n g t h 2Rm h a v i n g an a r e a e q u a l t o t h a t of the eddy's c i r c u l a r c r o s s s e c t i o n . L{2Rm}=TTRm (D-1) so t h a t L=(TT/2)Rm (D-2) W i t h a d i s t r i b u t i o n of eddy s i z e s N(Rm) the average c h o r d of a l l e d d i e s randomly i n c i d e n t on the probe w i l l be Led=[^N(Rm) {Rmr^2} ] / t ^ N(Rm) ] (D-3) To r e c r e a t e a randomly d i s t r i b u t e d v e l o c i t y r e c o r d composed of z e r o and eddy c o n t r i b u t i o n s choose 0 . v e l o c i t y or sample an eddy v e l o c i t y p r o f i l e a c c o r d i n g t o P e = p r o b a b i l i t y of c h o o s i n g an eddy P o = p r o b a b i l i t y of c h o o s i n g a z e r o v e l o c i t y Ls=average number of v e l o c i t y samples f o r the eddies' Lb=number of v e l o c i t y samples when a z e r o i s • chosen (=1) F r a c = f r a c t i o n of sample area taken by e d d i e s The number of v e l o c i t y samples f o r a l e n g t h X and v e l o c i t y probe speed Uc i s g i v e n by (X/Uc)Fs where Fs i s the s a m p l i n g f r e q u e n c y . We f i r s t have Pe+Po=1 (D-4) and LePe F r a c (D-5) LePe+LoPo so t h a t Pe= L o x F r a c { L e ( 1 - F r a c ) + L o x F r a c } (D-6) i s the p r o b a b i l i t y of c h o o s i n g an eddy which w i l l randomly r e c r e a t e o b s e r v e d eddy f r a c t i o n F r a c . 

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