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UBC Theses and Dissertations

Wall confinement effects for circular cylinders at low Reynolds numbers Mitry, Raafat Tawfic 1977

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WALL CONFINEMENT EFFECTS FOR CIRCULAR CYLINDERS AT LOW REYNOLDS NUMBERS by RAAFAT TAWFIC MITRY B.Eng.Sc. (Mechanical Engineering), University of Alexandria, 1970 M.Eng.Sc. (Mechanical Engineering), University of Alexandria, 1974 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF APPLIED SCIENCE i n THE FACULTY OF GRADUATE STUDIES ( M e c h a n i c a l E n g i n e e r i n g Department) We accept t h i s t h e s i s as con f o r m i n g t o the r e q u i r e d s t a n d a r d THE UNIVERSITY OF BRITISH COLUMBIA Oct o b e r , 1977 . Cc) Raafat Tawfic Mitry, 1977 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 l m e n t o f the re q u i r e m e n t s f o r an advanced degree at 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 s t u d y . I f u r t h e r agree t h a t p e r m i s s i o n f o r e x t e n s i v e c o p y i n g 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 g r a n t e d by the Head o f my Department or by h i s r e p r e s e n t a t i v e s . I t i s u n d e r s t o o d t h a t p u b l i c a t i o n , i n p a r t or i n whole, or the c o p y i n g o f t h i s t h e s i s f o r f i n a n c i a l g a i n s h a l l not be a l l o w e d w i t h o u t my w r i t t e n p e r m i s s i o n . RAAFAT TAWFIC MITRY Department of M e c h a n i c a l E n g i n e e r i n g The U n i v e r s i t y o f B r i t i s h Columbia Vancouver, B.C. Canada V6T 1W5 i i ABSTRACT F o r m a t i o n , development and i n s t a b i l i t y o f F o p p l v o r t i c e s and a s s o c i a t e d s u r f a c e p r e s s u r e d i s t r i b u t i o n are i n v e s t i g a t e d e x p e r i m e n t a l l y f o r a f a m i l y o f two d i m e n s i o n a l c i r c u l a r c y l i n d e r s i n the Reynolds number range o f 5 - 20,000 and the b lockage r a t i o o f 2 - 50%. In the b e g i n n i n g , d e s i g n and c o n s t r u c t i o n a l d e t a i l s o f a g l y c e r o l - w a t e r s o l u t i o n t u n n e l used i n the e x p e r i m e n t a l programme i s b r i e f l y d e s c r i b e d f o l l o w e d by an e x p l a n a t i o n o f the models , p r e s s u r e measur ing i n s t r u m e n t a t i o n , and t e s t p r o c e d u r e s . An approach to the data r e d u c t i o n , so c r i t i c a l at low Reynolds number, i s d i s c u s s e d and a new d e f i n i t i o n o f the p r e s s u r e c o e f f i c i e n t , which promises to be l e s s dependent on t e s t f a c i l i t i e s and p r e s s u r e g r a d i e n t s , i s e x p l a i n e d . , F i n a l l y , the t e s t da ta are a n a l y z e d as f u n c t i o n s o f the conf inement c o n d i t i o n and Reynolds number. The r e s u l t s suggest t h a t i n f l u e n c e o f the Reynolds number on the s u r f a c e p r e s s u r e d i s t r i b u t i o n i s p r i m a r i l y c o n f i n e d to the range R n < 1200. However, f o r the model w i t h the h i g h e s t b l o c k a g e r a t i o o f 50%, the p r e s s u r e c o n t i n u e s to show Reynolds number dependency f o r R^ as h i g h as 3000. In g e n e r a l , e f f e c t o f the Reynolds number i s to i n c r e a s e the minimum as w e l l as the wake p r e s s u r e s . On the o t h e r hand, the e f f e c t o f an i n c r e a s e i n the b lockage r a t i o i s j u s t the o p p o s i t e . The p r e s s u r e p r o f i l e s become extremely s e n s i t i v e to the w a l l confinement at the lower end o f the Reynolds number range under s t u d y . An e x t e n s i v e f low v i s u a l i z a t i o n s tudy u s i n g dye i n j e c t i o n i n c o n j u n c t i o n w i t h s t i l l and h i g h speed photography complements the t e s t program. Photographs suggest tha t i n f l u e n c e of the b lockage i s to r e t a r d , i n terms of the Reynolds number, e v o l u t i o n o f the near-wake. L o c a t i o n o f the s e p a r a t i n g shear l a y e r s tends to move upstream w i t h an i n c r e a s e i n the Reynolds number, however, the w a l l conf inement promotes downstream s h i f t i n the s e p a r a t i o n p o i n t , which can be as l a r g e as 2 5 ° f o r a b lockage r a t i o o f 501. The t h e s i s suggests f o r f u t u r e i n v e s t i g a t i o n s e v e r a l areas which are l i k e l y to be f r u i t f u l . In p a r t i c u l a r , aspect r a t i o e f f e c t s i n the presence o f b lockage are l i k e l y to be s i g n i f i c a n t and s h o u l d be s t u d i e d i n depth . i v TABLE OF CONTENTS Chapter Page 1. INTRODUCTION . . . . . . . . . 1 1.1 P r e l i m i n a r y Remarks 1 1.2 Drag R e d u c t i o n Due t o Long-Chain Polymers 2 1.3 S t a t i o n a r y C i r c u l a r C y l i n d e r s : P r e s s u r e D i s t r i b u t i o n and Wake. . . 9 1.4 Purpose and Scope o f the I n v e s t i g a t i o n 12 2. DESIGN AND CALIBRATION OF THE LIQUID TUNNEL 15 2.1 L i q u i d Tunnel 15 2.2 D e s i g n o f Honeycomb f o r T u r b u l e n c e C o n t r o l 22 2.3 C a l i b r a t i o n o f the L i q u i d Tunnel 27 3. TEST PROCEDURES 38 3.1 Models • • • 39 3.2 P r e s s u r e Measurements 43 3.3 Flow V i s u a l i z a t i o n 54 3.4 C r i t i c a l Reynolds Number 60 4. RESULTS AND DISCUSSION 61 4.1 C h o i c e o f R e f e r e n c e V e l o c i t y and P r e s s u r e - . . 62 V C h a p t e r Page 4.2 E f f e c t o f Reynolds Number . . . . . 69 4.3 W a l l Confinement E f f e c t s 79 4.4 Drag C o e f f i c i e n t 89 4.5 S t r o u h a l Number . . . . . . . . . . 91 4.6 Flow V i s u a l i z a t i o n and Near-Wake A n a l y s i s . . . 94 4.7 C l o s i n g Comments I l l 4 . 7 .1 C o n c l u d i n g remarks . . . . . I l l 4 .7 .2 Recommendation f o r f u t u r e s tudy 113 REFERENCES 117 APPENDIX I - CONVENTIONAL PRESSURE COEFFICIENT C IN TERMS OF MEASURED P INFORMATION 126 v i LIST OF ILLUSTRATIONS F i g u r e Page 1-1. A schemat ic diagram l i s t i n g hypotheses a t t e m p t i n g to e x p l a i n drag r e d u c t i o n due to l o n g - c h a i n polymers 4 1-2. A summary o f l i t e r a t u r e i n d i c a t i n g the scope o f recent important c o n t r i b u t i o n s i n the f i e l d o f f low pas t a c i r c u l a r c y l i n d e r 1 1- 3. A schemat ic diagram showing the p l a n of s tudy • 1 4 2- 1. A schemat ic diagram showing the g l y c e r o l -water s o l u t i o n t u n n e l 2-2. C a l i b r a t i o n p l o t f o r the V e n t u r i m e t e r • • • • 21 2-3 . A photograph showing d e t a i l s o f the power u n i t : A , a u t o t r a n s f o r m e r ; D, d r i v e motor; P, p ump 23 2-4. A photograph o f the g l y c e r o l - w a t e r t u n n e l capable o f g e n e r a t i n g Reynolds number i n the range 1 - 18,000: A , a u t o t r a n s f o r m e r ; D, d r i v e motor; E , heat exchanger; F , f low d i s t r i b u t i n g vanes ; H , p o r t h o l e s ; M, V e n t u r i meter; P, pump; S, screens and honey-comb; T , t e s t - s e c t i o n ; V , vent 24 2-5 . A photograph showing d e t a i l s o f the t u n n e l i n l e t : F , f low d i s t r i b u t i o n vanes; S^, coarse s c r e e n ; S £ , f i n e s c r e e n ; H , honeycomb 28 2-6. Arrangement f o r hot f i l m probe c a l i b r a t i o n : F , feed worm; P, p r o b e ; S, f lume; V , speed c o n t r o l gearbox 30 2-7 . C a l i b r a t i o n data f o r the hot f i l m probe ' TSI 12 39W 31 v i i F i g u r e Page •2-8. T y p i c a l p l o t s showing the e f f e c t o f screens and honeycomb i n improv ing v e l o c i t y p r o f i l e i n the t e s t - s e c t i o n 3 3 2-9 . A comparison o f v e l o c i t y p r o f i l e s at three d i f f e r e n t s t a t i o n s i n the t e s t - s e c t i o n and f o r two va lues o f mean f low v e l o c i t y . Note a l a r g e r e g i o n of e s s e n t i a l l y u n i f o r m flow which i s d e s i r a b l e f o r the p l a n n e d t e s t programme . . 34 2- 10. R e p r e s e n t a t i v e p l o t s showing v a r i a t i o n o f the t u r b u l e n c e i n t e n s i t y i n the t e s t -s e c t i o n 3 6 3- 1. T y p i c a l models used i n the f low v i s u a l i z a -t i o n s tudy 4 1 3-2. A photograph showing the models used i n s u r f a c e p r e s s u r e d i s t r i b u t i o n 44 3-3. A diagram showing c o n s t r u c t i o n a l d e t a i l s o f a t y p i c a l p r e s s u r e measurement model . . 4 5 3-4. A schemat ic diagram o f the B a r o c e l p r e s s u r e t r a n s d u c e r 47 3-5. A procedure f o r compensat ion o f the e l e c -t r o n i c d r i f t o f the p r e s s u r e measuring system 51 3-6. A l i n e drawing o f the i n s t r u m e n t a t i o n s e t -up used d u r i n g s t a t i c p r e s s u r e measurements on the s u r f a c e o f a c i r c u l a r c y l i n d e r . . . . 53 3-7. Measurement o f mean f low r a t e u s i n g ven-t u r i m e t e r : B , B a r o c e l p r e s s u r e t r a n s d u c e r ; D, d i g i t a l d . c . v o l t m e t e r ; E , heat exchanger; F , f i l t e r ; H , h e a t - s i n k ; P , power s u p p l y ; S, s i g n a l c o n d i t i o n e r ; V , V e n t u r i meter 55 3-8. A photograph showing the dye i n j e c t i o n probes ^7 3-9. A s k e t c h showing the equipment l a y o u t d u r i n g the f low v i s u a l i z a t i o n . . . . . . . 58 v i i i F i g u r e ' - ' . Page 4-1 . An i l l u s t r a t i o n showing p o s s i b l e e r r o r s i n t r o d u c e d by n o n - u n i f o r m i t y o f the v e l o c i t y p r o f i l e 66 4-2 . R e p r e s e n t a t i v e p r e s s u r e p l o t s comparing p r e s e n t r e s u l t s w i t h those by Grove et a l . 3 4 a t a low Reynolds number o f 175. Note a r e l a t i v e s e n s i t i v i t y o f the proposed p r e s s u r e c o e f f i c i e n t to d i f f e r e n c e s i n t e s t c o n d i t i o n s 70 4-3 . A comparison of p r e s e n t r e s u l t s w i t h those by E l - S h e r b i n y at a r e l a t i v e l y h i g h Reynolds number o f 15 x 1 0 3 71 4-4. S u r f a c e p r e s s u r e d i s t r i b u t i o n as a f f e c t e d by the Reynolds number f o r a g i v e n b l o c k -age r a t i o : a) S /C = 2% ; b) S / C =3.3%; c) S /C = 6.61 ; d) S /C =12.5%; e) S /C = 25% ; f) S /C = 501 . 73 4-5 . P r e s s u r e p l o t s as a f f e c t e d by b lockage at a g i v e n Reynolds number: a) R = n = 3000; b) R = n = 1200; c) R = • n = 400; d) R = n = 200; e) R = n = 30 , 4-6. E f f e c t o f w a l l confinement on the minimum and base p r e s s u r e s , 10 < R n < 18,000: a) minimum p r e s s u r e ; 87 b) base p r e s s u r e . . . . . . 88 4 - 7 . V a r i a t i o n o f the p r e s s u r e drag c o e f f i c i e n t w i t h Reynolds number and b lockage 90 i x F i g u r e . ' P a g e 4-8. V a r i a t i o n of the S t r o u h a l number w i t h b lockage at the lower end of the Reynolds number range (R < 1000) 93 4-9. A comparison o f measured S t r o u h a l data w i t h those by o ther i n v e s t i g a t o r s . . . . . . . . 95 4-10. Onset o f v o r t e x shedding as a f f e c t e d by the b lockage 96 4-11. A t y p i c a l photograph showing f o r m a t i o n o f F o p p l v o r t i c e s b e h i n d a t w o - d i m e n s i o n a l c y l i n d e r ; R n = 186, S/C = 50% 9 8 4-12. A flow v i s u a l i z a t i o n s tudy showing development and i n s t a b i l i t y o f v o r t e x r i n g w i t h Reynolds number (S /C = 15%): a) R = 4; b) R = 10; c) R = 17; J n ' J n ' J n ' d) R = 21; e) R = 31; f) R = 4 2 ; • n J n J n ' g) R = 56 ; h) R = 60 . 9 9 & J n ' J n 4-13. A photograph o f the c l a s s i c a l Karman v o r t e x s t r e e t ; R n = 60, S /C = 15% 103 4-14. P o s i t i o n o f s e p a r a t i o n as a f f e c t e d by the Reynolds number .and w a l l conf inement : a) f low v i s u a l i z a t i o n d a t a ; 104 b) based on s u r f a c e p r e s s u r e p l o t s ( F i g u r e 4-4) 105 4-15. E f f e c t o f b lockage on e v o l u t i o n o f the wake at a f i x e d Reynolds number o f 100: a) S /C = 25% ; b) S/C = 50% 107 4-16. Photographs emphas iz ing i n f l u e n c e o f w a l l confinement on e v o l u t i o n o f the F o p p l v o r t i c e s : a) R n = 85, S /C = 25%; b) R^ = 150, S /C = 50% 1 0 8 4-17. Dependence o f v o r t e x l e n g t h on the Reynolds number and b lockage 110 X T a b l e LIST OF TABLES Page 3-1 P r e s s u r e models used i n the t e s t 4 2 programme, c o r r e s p o n d i n g b l o c k a g e and a s s o c i a t e d Reynolds number range a t t a i n e d through the v a r i a t i o n o f mean f r e e stream v e l o c i t y and c o n c e n t r a t i o n o f the working f l u i d x i ACKNOWLEDGEMENT I would l i k e to take t h i s o p p o r t u n i t y to express my g r a t i t u d e and s i n c e r e thanks to P r o f e s s o r V . J . Modi f o r the e n t h u s i a s t i c guidance g i v e n throughout the r e s e a r c h programme and h e l p f u l s u g g e s t i o n s d u r i n g the p r e p a r a t i o n o f the t h e s i s . Hi s he lp and encouragement have been i n v a l u a b l e . The c h e e r f u l a s s i s t a n c e o f t h e . t e c h n i c a l s t a f f i s g r a t e f u l l y acknowledged. T h e i r s k i l l f u l a s s i s t a n c e g r e a t l y a c c e l e r a t e d the r e s e a r c h programme. F i n a l l y , s p e c i a l a p p r e c i a t i o n i s extended to my w i f e , Ragaa, f o r her p a t i e n c e and u n d e r s t a n d i n g d u r i n g d i f f i c u l t t imes and to my mother , Mary , f o r her encouragement. The i n v e s t i g a t i o n was s u p p o r t e d by the N a t i o n a l R e s e a r c h C o u n c i l o f Canada, Grant No. A-2181 . LIST OF SYMBOLS overheat r a t i o t u n n e l c r o s s - s e c t i o n a l a r e a t o t a l drag c o e f f i c i e n t s e c t i o n a l p r e s s u r e drag c o e f f i c i e n t , . D p / ( l / 2 ) p U 2 D s k i n f r i c t i o n c o e f f i c i e n t percentage c o n c e n t r a t i o n o f g l y c e r o l - w a t e r s o l u t i o n by weight ( P 0 - P r ) / ( P 0 - P r ) c o n v e n t i o n a l p r e s s u r e c o e f f i c i e n t , ( P e " P J / ( l / 2 ) p U r o 2 ( P e - P r ) / ( l / 2 ) p U 2 mean f l u c t u a t i n g p r e s s u r e c o e f f i c i e n t honeycomb c e l l - s i z e • c y l i n d e r d iameter s e c t i o n a l p r e s s u r e drag e c c e n t r i c i t y o f e l l i p t i c c y l i n d e r v o r t e x shedding f requency l e n g t h parameter i n Roshko's d e f i n i t i o n o f ' u n i v e r s a l ' S t r o u h a l number honeycomb p r e s s u r e drop c o e f f i c i e n t x i i i I l e n g t h o f honeycomb L m a c r o - l a y e r t h i c k n e s s i n polymer induced drag r e d u c t i o n h y p o t h e s i s . L l e n g t h s c a l e o f t u r b u l e n c e base p r e s s u r e P m minimum p r e s s u r e P Q s t a g n a t i o n p r e s s u r e , p r e s s u r e at 6 = 0 P r s t a t i c p r e s s u r e at r e f e r e n c e t a p , i n the p r e s e a t case r = 50 P Q s t a t i c p r e s s u r e on c y l i n d e r s u r f a c e at angle 9 from f r o n t s t a g n a t i o n p o i n t P r o s t a t i c p r e s s u r e o f u n d i s t u r b e d stream R c , R Q c o l d and o p e r a t i n g r e s i s t a n c e o f the p r o b e , r e s p e c t i v e l y R^ Reynolds number based on honeycomb c e l l - s i z e R g Reynolds number based on honeycomb c e l l - l e n g t h R n Reynolds number, UD/v R , c r i t i c a l Reynolds number c o r r e s p o n d i n g to onset o f v o r t e x shedding S d i a m e t r a l c r o s s - s e c t i o n a l a r e a S Roshko's u n i v e r s a l S t r o u h a l number, fh/U S n S t r o u h a l number, f D / U •T temperature o f s o l u t i o n u rms v a l u e o f v e l o c i t y f l u c t u a t i o n s i n downstream d i r e c t i o n x i v average v e l o c i t y i n t e s t - s e c t i o n based on f low r a t e as g i v e n by o r i f i c e meter c h a r a c t e r i s t i c v e l o c i t y as proposed by Grove et a l . * ^ s e p a r a t i o n v e l o c i t y i n . R o s h k o ' s u n i v e r s a l S t r o u h a l number u n d i s t u r b e d f r e e s tream v e l o c i t y d . c . v o l t a g e output o f c o n s t a n t temperature anemometer d i s t a n c e from end o f honeycomb i n downstream d i r e c t i o n v e r t i c a l c o - o r d i n a t e , o r i g i n at bottom of t e s t - s e c t i o n angle o f a t t a c k a n g u l a r l o c a t i o n o f p r e s s u r e tap w i t h r e f e r e n c e to f r o n t s t a g n a t i o n p o i n t a n g u l a r l o c a t i o n o f s e p a r a t i n g shear l a y e r w i t h r e s p e c t to r e a r s t a g n a t i o n p o i n t t h i c k n e s s o f l a m i n a r s u b l a y e r d i f f e r e n t i a l p r e s s u r e a c r o s s V e n t u r i meter e r r o r i n p r e s s u r e a t 9 = 0 due to n o n u n i f o r m i t y o f v e l o c i t y p r o f i l e e r r o r i n p r e s s u r e at 8 = 6 due to n o n u n i f o r m i t y o f v e l o c i t y p r o f i l e e r r o r i n p r e s s u r e at 0 due to n o n u n i f o r m i t y o f v e l o c i t y p r o f i l e dynamic v i s c o s i t y o f g l y c e r o l - w a t e r s o l u t i o n k i n e m a t i c v i s c o s i t y o f g l y c e r o l - w a t e r s o l u t i o n , y / p , k i n e m a t i c v i s c o s i t y o f water d e n s i t y o f g l y c e r o l - w a t e r s o l u t i o n 1. INTRODUCTION 1.1 P r e l i m i n a r y Remarks A l t h o u g h f low past a c i r c u l a r c y l i n d e r has been under study f o r years r e s u l t i n g i n a v a s t body o f l i t e r a t u r e , there are s e v e r a l a spec t s a s s o c i a t e d w i t h the problem which remain v i r t u a l l y u n e x p l o r e d or demand more a t t e n t i o n . One of such aspects was brought to l i g h t through problems a s s o c i a t e d w i t h low Reynolds number f lows as encountered i n : ( i ) the d e s i g n o f s t r u c t u r a l members f o r o f f - s h o r e p l a t f o r m s ; ( i i ) the submarine d e t e c t i o n system u s i n g a c a b l e supported hydrophone exposed to an ocean c u r r e n t ; ( i i i ) b i o l o g i c a l f l u i d m e c h a n i c a l systems. Even i n the absence o f any o ther c o m p l i c a t i n g f a c t o r s as the f r e e s tream t u r b u l e n c e , b l o c k a g e , p u l s a t i l e c h a r a c t e r o f the f l o w , presence of l o n g - c h a i n p o l y m e r s , e t c . , the problem as i t i s has s e v e r a l f a c e t s which are c h a l l e n g i n g . For example, p r e s s u r e d i s t r i b u t i o n on the s u r f a c e o f a c y l i n d e r as a f f e c t e d by the Reynolds number i n the range S - 40 and i t s c o r r e l a t i o n w i t h the f o r m a t i o n and i n s t a b i l i t y o f the F o p p l v o r t i c e s remains u n r e c o r d e d , a l t h o u g h i t would be q u i t e u s e f u l i n a s tudy of the above mentioned prob lems . 2 In the p r e s e n t case the i n v e s t i g a t i o n was m o t i v a t e d by the i n t e r e s t i n the e f f e c t s o f long c h a i n polymers on the f l u i d mechanics o f b l u f f bod ies i n g e n e r a l and, i n p a r t i c u l a r , t h e i r i n f l u e n c e on the g e n e r a t i o n and d i s s i p a t i o n o f v o r t i c i t y at v e r y low Reynolds numbers. However, i t was q u i c k l y r e a l i z e d through a d e t a i l e d study of the l i t e r a t u r e t h a t the c o r r e s -ponding i n f o r m a t i o n even i n the absence o f polymers i s u n a v a i l a b l e . O b v i o u s l y , the f i r s t t a s k would be to o b t a i n the fundamental i n f o r m a t i o n which then would serve as a b a s i s f o r comparison i n e v a l u a t i n g the e f f e c t s o f l o n g - c h a i n p o l y m e r s . In t h i s Chapter the phenomenon of drag r e d u c t i o n and the a v a i l a b l e t h e o r i e s to e x p l a i n i t are b r i e f l y d e s c r i b e d . T h i s i s f o l l o w e d by a shor t i n t r o d u c t i o n to the c u r r e n t l i t e r a t u r e on f low past a c i r c u l a r c y l i n d e r at low Reynolds' numbers. T h i s v i v i d l y b r i n g s to a t t e n t i o n the gaps i n our search- f o r knowledge. F i n a l l y , i n the l i g h t o f the a v a i l a b l e i n f o r m a t i o n , a p l a n of s tudy f o r the p r e s e n t t h e s i s i s f o r m u l a t e d . 1 . 2 Drag R e d u c t i o n Due to L o n g - C h a i n Polymers F r i c t i o n drag of a body submerged i n a l i q u i d can reduce by as much as 70% i n the presence of c e r t a i n a d d i t i v e s ! A l t h o u g h there are numerous d i f f e r e n t s o l u t e s w i t h the s k i n f r i c t i o n drag r e d u c t i o n p r o p e r t y , they a l l have one t h i n g i n common: they are a l l long c h a i n po lymers . 3 The phenomenon was f i r s t r e p o r t e d by Toms 1 as e a r l y as i n 1948 when he found the f r i c t i o n l o s s i n p ipes w i t h polymer s o l u t i o n to be s u b s t a n t i a l l y l e s s than t h a t w i t h the s o l v e n t a l o n e . A l t h o u g h t h i s i s r e f e r r e d to as 'Toms Phenomenon' 2 a c t u a l l y Myse l s was the f i r s t to d i s c o v e r i t w h i l e pumping napalm d u r i n g the war y e a r s (1945) but was unable to p u b l i s h the r e s u l t s u n t i l much l a t e r . The phenomenon has been a c t i v e l y r e s e a r c h e d i n recen t y e a r s , the i n t e r e s t be ing l a r g e l y due to the p o t e n t i a l a p p l i -c a t i o n o f the concept i n s a v i n g f l u i d pumping power and improv ing the marine p r o p u l s i v e e f f i c i e n c y . A l t h o u g h everyone agrees tha t the drag i s r e d u c e d , t h e r e i s no g e n e r a l agreement as to the mechanism of the drag r e d u c t i o n . The c o n f u s i o n i s f u r t h e r a c c e n t u a t e d by the f a c t t h a t : i ) q u a n t i t a t i v e r e s u l t s by d i f f e r e n t i n v e s t i g a t o r s w i d e l y d i f f e r 3 ; i i ) even the same i n v e s t i g a t o r i s o f t e n unable to reproduce h i s da ta as polymer p r o p e r t i e s change w i t h time and shear s t r e s s 3 - ^ ; i i i ) c o n v e n t i o n a l f l u i d dynamic in s t ruments l i k e p i t o t tube , v e n t u r i meter , hot f i l m p r o b e s , e t c . show anomalous b e h a v i o u r D ~ 7 . Most i n v e s t i g a t o r s e x p l a i n the phenomenon u s i n g one of the two main hypotheses ; each s u b c l a s s i f i e d i n t o two b a s i c concepts as i n d i c a t e d i n F i g u r e 1-1. B r i e f l y they can be d e s c r i b e d as f o l l o w s : H Y P O T H E S E S E X P L A I N I N G D R A G R E D U C T I O N / T U R B U L E N C E . S U P P R E S S I O N H Y P O T H ESES W A L L L A Y E R M O D I F I C A T I O N H Y P O T H E S E S S O L U T I O N H A S V I S C O E L A S T I C P R O P E R T I E S F i g u r e 1-1 MODIFICATION IN THE T U R B U L E N C E S T R U C T U R E WALL A B S O R P T I O N OR MOBILE L A Y E R A schematic diagram l i s t i n g hypotheses a t t empt ing to e x p l a i n drag r e d u c t i o n due to l o n g - c h a i n po lymers . R E D U C T I O N IN THE WALL GENERATED T U R B U L E N C E 5 (a) T u r b u l e n c e S u p p r e s s i o n Hypotheses (a-^) S o l u t i o n has V i s c o e l a s t i c P r o p e r t i e s : Polymer s o l u -t i o n s are v i s c o e l a s t i c and hence ab le to s t o r e a p a r t o f the k i n e t i c energy , n o r m a l l y l o s t i n the f low as the p o t e n t i a l energy o f e l o n g a t i o n or more g e n e r a l d e f o r m a t i o n s . (a.2) M o d i f i c a t i o n i n the T u r b u l e n c e S t r u c t u r e : Polymer molecu les i n t e r a c t w i t h the t u r b u l e n t eddies and modify the t u r b u l e n c e c h a r a c t e r i s t i c s . O b v i o u s l y f o r t h i s to be e f f e c t i v e , d imensions o f the molecu les shou ld be comparable to Kolmogorov m i c r o s c a l e s . In p r a c t i c e even the h i g h e r m o l e c u l a r weight l o n g - c h a i n polymers do not possess these dimensions i n s o l u t i o n . Hence, to j u s t i f y the h y p o t h e s i s , shear s t r e t c h i n g of the c o i l e d m o l e c u l e s , entwined w i t h t h e i r ne ighbours to form s u p e r m o l e c u l a r aggregates has been p o s t u l a t e d . (b) W a l l M o d i f i c a t i o n Hypotheses (b-j.) W a l l A b s o r b t i o n or M o b i l e L a y e r : C h a r a c t e r i s t i c s o f the w a l l s u r f a c e are c o n s i d e r e d to be s i g n i f i c a n t l y a l t e r e d , by a b s o r b t i o n o f molecu le s making i t smoother or through o r i e n -t a t i o n o f polymer molecu les c l o s e to the w a l l , r e s u l t i n g i n a mobi le w a l l l a y e r . (b2) R e d u c t i o n i n the W a l l Generated T u r b u l e n c e : T h i s t h e o r y appears to be the most w i d e l y a c c e p t e d . S e v e r a l r e v i e w s , f o r example by Gadd 8 (1971) , L a n d a h l 9 (1972) , H o y t 1 0 (1972) and Polyuos"'"^ (1974) when taken t o g e t h e r would cover maj o r i t y 6 of the p u b l i c a t i o n s i n the f i e l d . Some o f the more i n t e r e s t i n g r e s u l t s can be b r i e f l y summarized as f o l l o w s : W a l l e s and 12 Spangter (1967) seem to suggest t h a t i n j e c t i o n o f a drag r e d u c i n g f l u i d i n t o the w a l l l a y e r has an immediate e f f e c t on drag ( r e d u c t i o n ) w h i l e i n j e c t i o n o f the same f l u i d i n t o the core r e g i o n of the b u l k f low has r e l a t i v e l y l i t t l e or no e f f e c t u n t i l the polymer has had enough t ime to d i f f u s e to the 13 w a l l r e g i o n . K o w a l s k i (1968) a r r i v e d at the same c o n c l u s i o n , 14 so d i d Van D r i e s t (1970). L a t e r ' s experiments c o n s i s t i n comparing f r i c t i o n drag f o r p i p e s o f d i f f e r e n t d iameters convey ing a d i l u t e polymer s o l u t i o n o f f i x e d c o n c e n t r a t i o n . He n o t i c e d t h a t f o r a g i v e n Reynolds number, there i s a c r i t i c a l v a l u e o f the p i p e d iameter below which drag r e d u c t i o n ceases to e x i s t . I t was conc luded t h a t there must be some o ther l e n g t h parameter c o n t r o l l i n g the drag r e d u c t i o n phenomenon. T h i s new l e n g t h parameter must be governed by the f low. He assumed a f i x e d l e n g t h L of the l a y e r ( m a c r o - l a y e r ) i n which eddying mot ion i s damped by the e n t a n g l e d long m o l e c u l e s . However, a l a m i n a r s u b l a y e r 6 s t i l l e x i s t s which may be l a r g e r or s m a l l e r than the m a c r o - l a y e r t h i c k n e s s L . I f the Reynolds number i s s m a l l enough such tha t 6 > L then the s u b l a y e r s t a b i l i t y w i l l predominate . But i f the Reynolds number i s l a r g e enough such tha t <5 < L then eddies i n the r e g i o n L-6 w i l l be damped depending on the c o n c e n t r a t i o n s o f the polymer s o l u t i o n s . 7 Arunachalam et a l . X J (1972) showed a t h i c k e r w a l l l a y e r f o r the drag r e d u c i n g s o l u t i o n than tha t f o r the s o l v e n t a l o n e . They used a n o n d i s t u r b i n g photochromic dye t r a c e t echn ique to a r r i v e at the c o n c l u s i o n v i s u a l l y . On the o ther hand, F o r t u n a 16 et a l . (1972) u s i n g spectrum a n a l y s i s i n p i p e f low observed a l s o a change i n the t u r b u l e n t s u b l a y e r . More i n t e r e s t i n g was 17 the experiment by L a t t o and E l Riedy (1976) who i n j e c t e d a dye i n the boundary l a y e r over a f l a t p l a t e and showed t h a t the t u r b u l e n t d i f f u s i o n of polymer s o l u t i o n i s suppressed compared to tha t o f water . From the above d i s c u s s i o n i t can be c o n c l u d e d tha t there i s a consensus o f o p i n i o n c o n c e r n i n g an e x i s t e n c e o f a major r e g i o n of drag r e d u c t i o n i n the d i l u t e d polymer s o l u t i o n s . However, the mechanism of drag r e d u c t i o n on a m o l e c u l a r l e v e l i s s t i l l not c l e a r . A rev iew o f the l i t e r a t u r e on the f l u i d dynamics o f bod ie s submerged i n polymer s o l u t i o n s suggests t h a t , i n s p i t e o f the e x t e n s i v e i n f o r m a t i o n on the s u b j e c t , b l u f f body geo-m e t r i e s remain v i r t u a l l y u n e x p l o r e d , except f o r a few i s o l a t e d 18 p a p e r s . Gadd (1966) was the f i r s t one to s tudy the e f f e c t o f polymer s o l u t i o n s on v o r t e x shedding beh ind a c i r c u l a r c y l i n d e r . H i s r a t i o n a l e f o r the s tudy was: as t u r b u l e n c e i s an unsteady p r o c e s s , the t ime dependent terms i n any t h e o r e t i c a l model would a t t a i n p a r t i c u l a r s i g n i f i c a n c e i n t u r b u l e n t drag 8 r e d u c t i o n . He expected drag r e d u c i n g s o l u t i o n s o f equa l e f f e c t i v e n e s s ( i n terms o f drag r e d u c t i o n ) to reduce S t r o u h a l number by the same amount, however, h i s experiments f a i l e d 3 to support t h i s c o n c l u s i o n . Kalashnikow and Kudin (1970) found the frequency of v o r t e x shedding to be dependent on the c o n d i t i o n o f the polymer s o l u t i o n : when f r e s h the f requency d e c r e a s e d , however, a r e v e r s e t r e n d was observed w i t h the ag ing o f the s o l u t i o n (POLYOX). Through a l a t e r s tudy w i t h w i r e s , 19 Kalashnikow et a l . (1976) c o n c l u d e d t h a t the c r i t i c a l Reynolds number c o r r e s p o n d i n g to the onset o f v o r t e x shedding i s lower i n a f r e s h l y mixed s o l u t i o n of p o l y e t h y l e n e o x i d e but i n c r e a s e s w i t h i t s d e g r a d a t i o n to the p o i n t , t h a t exceeds the Newtonian v a l u e o f 41. U n f o r t u n a t e l y , c o r r e s p o n d i n g experiments w i t h s o l u t i o n s o f Guar r e s i n . s h o w e d the c r i t i c a l Reynolds number to be always h i g h e r than 41, i r r e s p e c t i v e o f the age of the s o l u t i o n . The accumulated l i t e r a t u r e thus suggests t h a t the s u b j e c t o f f o r m a t i o n of the v o r t i c e s beh ind b l u f f bod ies and i t s c o r r e l a t i o n w i t h s u r f a c e p r e s s u r e d i s t r i b u t i o n i n l o n g - c h a i n polymer s o l u t i o n s has r e c e i v e d no a t t e n t i o n at a l l . T h i s i s .unfor tunate because the proces s o f g e n e r a t i o n and d i s s i p a t i o n o f v o r t i c i t y may throw some l i g h t on the c h a r a c t e r o f the boundary l a y e r and hence on the phenomenon of drag r e d u c t i o n . Of c o u r s e , to assess the i n f l u e n c e o f the presence o f polymer i t would be n e c e s s a r y to have , as a r e f e r e n c e , the c o r r e s p o n d i n g 9 i n f o r m a t i o n f o r a b l u f f body i n the s o l v e n t , n o r m a l l y a Newtonian f l u i d , most ly water . However, s u r p r i s i n g as i t may seem, such fundamental i n f o r m a t i o n i n the lower end of the Reynolds number spectrum i s v i r t u a l l y m i s s i n g . O b v i o u s l y t h e n , the f i r s t s tep shou ld be to unders tand e v o l u t i o n o f the Newtonian f low as a f f e c t e d by the Reynolds number w i t h a p a r t i c u l a r emphasis on the wake-body i n t e r a c t i o n . 1.3 S t a t i o n a r y C i r c u l a r C y l i n d e r s : P r e s s u r e D i s t r i b u t i o n  and Wake H i s t o r i c a l l y , i n t e r e s t i n the f low pas t a b l u f f body 1 2 0 goes back to the f i f t e e n t h c e n t u r y when Leonardo da V i n c i ske tched a row of v o r t i c e s i n the wake of a b l u f f body. However, i t was o n l y towards the l a t e r p a r t o f the l a s t c e n t u r y t h a t the phenomenon found some q u a n t i t a t i v e e x p r e s s i o n through 21 the p i o n e e r i n g c o n t r i b u t i o n by S t r o u h a l . The we l l -known S t r o u h a l number c o r r e l a t e s p e r i o d i c i t y o f the v o r t e x shedding w i t h the d iameter of the c i r c u l a r c y l i n d e r and v e l o c i t y o f the f l u i d s tream. Ever s i n c e , t h e o r e t i c a l and p r a c t i c a l i n t e r e s t i n the f i e l d has l e d to a cont inuous stream of important con-t r i b u t i o n s r e s u l t i n g i n a vas t body o f l i t e r a t u r e . T h i s has been reviewed r a t h e r a d e q u a t e l y by s e v e r a l authors i n c l u d i n g 22 23 20' 24 25 Rosenhead , W i l l e , M a r r i s , Mor kov i n , P a r k i n s o n and 26 Cermak . The c o l l e c t e d l i t e r a t u r e suggests that , i n g e n e r a l , 10 na ture of the s u r f a c e l o a d i n g , v o r t e x shedding f requency and wake geometry form t h r e e important parameters i n the s tudy o f b l u f f body f l u i d mechanics . The c i r c u l a r c y l i n d e r appears to have r e c e i v e d the most a t t e n t i o n , p r o b a b l y because o f i t s s imple symmetric c o n f i g u r a t i o n and the f l u i d d y n a m i c a l l y d i s t i n c t i v e boundary l a y e r type s e p a r a t i o n . U n f o r t u n a t e l y , even w i t h a s t a t i o n a r y c i r c u l a r c y l i n d e r , some o f the fundamen-t a l i n f o r m a t i o n c o n c e r n i n g p r e s s u r e d i s t r i b u t i o n , b l o c k a g e , near wake geometry, e t c . are l i m i t e d to r e l a t i v e l y h i g h e r Reynolds numbers. T h i s i s apparent from F i g u r e 1-2 which summarizes more important c o n t r i b u t i o n s i n the a r e a r e l e v a n t to the p r e s e n t t h e s i s . In the range R < 175 o n l y two c o n t r i -b u t i o n s , by Thorn"^ and Grove et a l ? \ are noteworthy . Thorn c a r r i e d out h i s exper iments i n 1933 and was no t f o r t u n a t e to have s o p h i s t i c a t e d i n s t r u m e n t a t i o n o f the modern t i m e . H i s r e s u l t s f o r the p r e s s u r e d i s t r i b u t i o n on the s u r f a c e o f a two d i m e n s i o n a l c y l i n d e r and i t s v a r i a t i o n w i t h the Reynolds number show q u e s t i o n a b l e t r e n d s . On the o ther hand , a l t h o u g h the s tudy by Grove and h i s a s s o c i a t e s appears to be more c a r e f u l , i t i s l i m i t e d to a s i n g l e Reynolds number o f 175 thus m i s s i n g a l l the i n f o r m a t i o n c o n c e r n i n g the f o r m a t i o n o f F o p p l v o r t i c e s and i n i t i a t i o n o f the v o r t e x shedd ing . F u r t h e r m o r e t h e r e are s e v e r a l u n r e s o l v e d q u e s t i o n s about the da ta r e d u c t i o n p r o -c e d u r e , so important i n the low Reynolds number s t u d i e s , and 0-1 0-23 1 0 10 1 0 3 n 40 10 S/C ., WAKE GEOMETRY 3.5 175 C p , S/C ( 1.1 - 7.6 % ) 40 WAKE GEOMETRY WAKE G E O M E T R Y 40 IC 175 2.5X10 2X10' 10 1 0 1 0 4 1.7X10 2.IX 10 4 10 6X 10 10 c p . c , S/C 2.6 F i g u r e 1 - 2 . A summary o f l i t e r a t u r e i n d i c a t i n g the scope of recent important c o n t r i b u t i o n s i n the f i e l d of f low pas t a c i r c u l a r c y l i n d e r . 2X10* 7X10' 5 5 10 2.1X10 |c p , c F C P ' C F ' C D ' 9 S 1.2X10" 2.2X10 5 5 4X10 5X10 5X10 11 22 R O S E N H E A D [1930] 27 F A G E & F A L K N E R [1931] 28 F L A C H S B A R T [1932] 29 LI NKEet a l . [1932] 30 T H O M [1933] 31 G I E DT [1951] 32 R O S H K O [1953] 33 T A N E D A [1955] 34 G R O V E e t a l . [1963] 35 A C H E N B A C H [1968] 36 E L - S H E R B I N Y [1972] 37 F A R E L L et a l . [1976] 38 F A R E L L e t a l . [1977] 39 HUNER e t a l . [1977] 12 the d e f i n i t i o n o f u n d i s t u r b e d f r e e stream v e l o c i t y and p r e s s u r e employed by them. 1.4 Purpose and Scope of the I n v e s t i g a t i o n The p r o j e c t has two main o b j e c t i v e s . F i r s t l y , i t aims at the d e s i g n o f a l i q u i d t u n n e l p a r t i c u l a r l y s u i t a b l e f o r s t u d i e s i n the low Reynolds number range . C o n s i d e r a b l e t ime and e f f o r t were expended i n f i n a l i z i n g the d e s i g n , s u p e r v i s i o n o f c o n s t r u c t i o n , m o d i f i c a t i o n s , and c a l i b r a t i o n of the t u n n e l . The next stage i s concerned w i t h an i n v e s t i g a t i o n of w a l l confinement e f f e c t s on the f o r m a t i o n , e l o n g a t i o n and i n s t a b i l i t y o f the F o p p l v o r t i c e s r e s u l t i n g i n the c l a s s i c a l Ka'rman v o r t e x s t r e e t . . P r i m a r i l y the a t t e n t i o n i s f ocussed on c o r r e l a t i n g the s u r f a c e p r e s s u r e d i s t r i b u t i o n o f the v o r t i c e s i n the Reynolds number range of around 5-20,000 and the b lockage r a t i o o f 2-501. F i n a l l y , the f low v i s u a l i z a t i o n u s i n g the dye i n j e c t i o n t echn ique i n c o n j u n c t i o n w i t h s t i l l and movie photography complements the t e s t r e s u l t s . When a v a i l a b l e , r e s u l t s by o t h e r i n v e s t i g a t o r s are i n c l u d e d f o r comparison and to he lp e s t a b l i s h t r e n d s . The i n f o r m a t i o n o b t a i n e d i s fundamental i n c h a r a c t e r and shou ld prove u s e f u l as a b a s i s o f comparison f o r more c o m p l i c a t e d s t u d i e s w i t h l o n g - c h a i n p o l y m e r s , 13 p u l s a t i l e f l o w , e t c . mentioned e a r l i e r . F i g u r e 1-3 b r i e f l y summarizes the p l a n o f s t u d y . W A L L C O N F I N E M E N T E F F E C T S F O R C I R C U L A R C Y L I N D E R S A T L O W R E Y N O L D S N U M B E R S D e s i g n a n d C a l i b r a t i o n o f T e s t F a c i l i t i e s * G l y c e r o l - w a t e r s o l u t i o n t u n n e l * V e n t u r i m e t e r * H o t - f i l m anemometer ( c a l i b r a t i o n ) P r e s s u r e D i s t r i b u t i o n * P r e s s u r e d r a g * A p p r o x i m a t e l o c a t i o n o f s e p a r a t i o n D e s i g n o f M o d e l s T e s t P r o g r a m m e I R e y n o l d s N u m b e r E f f e c t 1 B l o c k a g e E f f e c t I S t r o u h a l N u m b e r * C r i t i c a l R e y n o l d s number o f o n s e t o f v o r t e x s h e d d i n g F i g u r e 1 - 3 . A schematic diagram showing the p l a n o f s t u d y . F l o w V i s u a l i z a t i o n * F o r m a t i o n , e v o l u t i o n and i n s t a b i l i t y o f E o p p l v o r t i c e s * L o c a t i o n o f s e p a r a t i n g s h e a r l a y e r * A p p r o x i m a t e e v a l u a t i o n o f t he S t r o u h a l number 15 2. DESIGN AND CALIBRATION OF THE  LIQUID TUNNEL As p o i n t e d out b e f o r e d e s i g n , s u p e r v i s i o n o f c o n s t r u c -t i o n and c a l i b r a t i o n o f a g l y c e r o l - w a t e r s o l u t i o n t u n n e l formed one o f the p r i n c i p a l o b j e c t i v e s o f the p r o j e c t . T h i s chapter aims at i n t r o d u c i n g some o f the more important aspects o f the programme. The des ign and c o n s t r u c t i o n a l d e t a i l s i n v o l v e d are o f t e n numerous and hence , though important and r e l e v a n t , cannot be c o v e r e d i n t h e i r e n t i r e t y . One i s , t h e r e f o r e , f o r c e d to c o n f i n e a t t e n t i o n to more s a l i e n t f e a t u r e s . In the b e g i n n i n g , de s ign s p e c i f i c a t i o n s and geometry o f the t u n n e l are e x p l a i n e d f o l l o w e d by a d e s c r i p -t i o n o f the power d r i v e and temperature c o n t r o l systems. Next , d e t a i l s o f the procedure f o r s e l e c t i o n o f the honey-combs to r e s t r i c t the f r e e stream t u r b u l e n c e to a p e r m i s s i b l e l e v e l are d i s c u s s e d . F i n a l l y , the c a l i b r a t i o n p l o t s are p r e s e n t e d which p r o v i d e some, a p p r e c i a t i o n as to the r e g i o n s o f u n i f o r m f low. . 2 .1 L i q u i d Tunne l S p e c i f i c a t i o n s f o r the t u n n e l were l a r g e l y d i c t a t e d by the low Reynolds number s t u d i e s p l a n n e d . For t h i s i t was n e c e s s a r y to have a f a c i l i t y w i t h the f o l l o w i n g c h a r a c t e r i s t i c s : 16 ( i ) The t u n n e l shou l d be capable o f p r o d u c i n g a maximum v e l o c i t y o f around 30 cm/s l e a d i n g to Reynolds numbers i n the range 1-20,000 based on the c y l i n d e r d i a m e t e r . ( i i ) The t u n n e l t e s t - s e c t i o n shou ld be l a r g e enough to p e r m i t wake and b l o c k a g e . s t u d i e s . ( i i i ) The t e s t - s e c t i o n w a l l s shou ld be s u f f i c i e n t l y f l a t to al low' f low v i s u a l i z a t i o n through o p t i c a l methods. ( i v ) The v e l o c i t y p r o f i l e s h o u l d be s u f f i c i e n t l y f l a t to p e r m i t u n i f o r m f low s t u d i e s . The t u r b u l e n t i n t e n s i t y s h o u l d be l e s s than 2%. (v) The t e s t - s e c t i o n s h o u l d be p r o v i d e d w i t h a s imple model support system. When mounted i n the t e s t - s e c t i o n , a model shou ld be r e a d i l y a c c e s s i b l e . ( v i ) There shou l d be a p r o v i s i o n f o r i n t r o d u c i n g p r e s s u r e , v e l o c i t y and o t h e r measuring i n s t r u m e n t a t i o n . Based on these c r i t e r i a a l i q u i d t u n n e l w i t h g l y c e r o l -water s o l u t i o n as a working f l u i d was des igned to produce the d e s i r e d range o f Reynolds number, based on c y l i n d e r d iameter and average v e l o c i t y i n the t e s t - s e c t i o n ( F i g u r e 2 -1 ) . The c h o i c e of c o n c e n t r a t i o n o f the working f l u i d p r o v i d e d a degree o f f l e x i b i l i t y , but o n l y to a c e r t a i n e x t e n t , as governed by the c h a r a c t e r i s t i c s o f the power u n i t . P r i m a r i l y the t u n n e l c o n s i s t s o f three subassembl i e s : the t e s t - s e c t i o n ; the f l u i d r e t u r n system; and the power u n i t c o n s i s t i n g o f a pump and a d r i v e motor . vent F i g u r e 2-1 . A schematic diagram showing the g l y c e r o l -water s o l u t i o n t u n n e l . 18 The t e s t - s e c t i o n i s b u i l t o f f o u r p l e x i g l a s w a l l s 2.1 m (6.9 f t . ) l o n g , 1.9 cm (3/4 i n . ) t h i c k , and wide enough to produce the i n s i d e c r o s s - s e c t i o n o f 61 cm x 15.2 cm (24 i n . x 6 i n . ) . Both the f r o n t and the r e a r w a l l s are p r o v i d e d w i t h recess -mounted o p t i c a l l y f l a t , homogeneous and thermally s t a b l e g l a s s pane l s (88 cm x 61 cm) to f a c i l i t a t e f low v i s u a l i z a t i o n . They a l s o c a r r y three p o r t s (2 .5 cm d i a . ) , each w i t h a c o l l a r s e r v i n g as a model support and an O - r i n g to prevent l e a k a g e . T h i s p r o v i s i o n o f three s t a t i o n s , 30 cm a p a r t , p e r m i t s a r e a s o n a b l e f l e x i b i l i t y i n s u p p o r t i n g a model at d i f f e r e n t l o c a t i o n s i n the t e s t - s e c t i o n . To promote u n i f o r m v e l o c i t y p r o f i l e and low t u r b u l e n c e i n t e n s i t y at the t e s t - s e c t i o n f low d i s t r i b u t i n g vanes , screens and honeycombs were i n t r o d u c e d at the entrance to the t u n n e l . In a l l 9 vanes were i n c o r p o r a t e d i n the d i v e r g i n g duct at the i n l e t to e f f e c t i v e l y d e l a y s e p a r a t i o n and c o n t r o l f l u i d d i s t r i b u t i o n . Two brass screens o f mesh-s i z e 14 and 18, l o c a t e d 3.8 cm a p a r t , h e l p e d towards a t ta inment o f u n i f o r m v e l o c i t y d i s t r i b u t i o n . T h i s was f o l l o w e d by a 15 cm long honeycomb (0.32 cm c e l l - s i z e ) l o c a t e d 7.5 cm downstream from the f i n e r mesh s c r e e n . The d imens ions of the honeycomb were governed by the maximum p e r -m i s s i b l e . t u r b u l e n c e i n t e n s i t y (<2%) i n accordance w i t h the d e s i g n procedure proposed by Lumley et a l ? ^ ' ^ L i k e s c r e e n s , 19 honeycomb a l s o tends to promote u n i f o r m i t y o f the v e l o c i t y p r o f i l e through f low r e s i s t a n c e , however, the p r o c e s s i s not p a r t i c u l a r l y e f f i c i e n t . S c r e e n s , honeycomb and models were r e a d i l y a c c e s s i b l e through two l a r g e p o r t h o l e s l o c a t e d on the top face o f the t u n n e l . In a d d i t i o n , s e v e r a l s m a l l e r p o r t h o l e s , which c o u l d take 1.6 cm p l u g s , were d r i l l e d and tapped i n the top w a l l o f the t e s t - s e c t i o n . These openings proved u s e f u l i n i n t r o d u c i n g hot f i l m p r o b e s , dye i n j e c t i o n system and p r e s s u r e c o n d u c t i n g t u b i n g s . At the o u t l e t , the r e c t a n g u l a r s e c t i o n o f 61 cm x 15.2 cm g r a d u a l l y changes to a 10.2 cm d i a . PVC duct through a c o n v e r g i n g s e c t i o n w i t h vanes (same as tha t at the i n l e t ) p r o v i d i n g f o r smooth t r a n s i t i o n . E x i t end of the t e s t - s e c t i o n a l s o c a r r i e d a vent (8.9 cm d i a . ) to accommodate f l u i d expans ion as w e l l to p r o v i d e an escape route f o r a i r b u b b l e s . I t was a l s o used as an i n l e t f o r the working f l u i d d u r i n g f i l l i n g o f the t u n n e l and served as an e f f e c t i v e check a g a i n s t over p r e s s u r i z a t i o n o f the t e s t - s e c t i o n . L o c a t e d between the end o f the p l e x i g l a s c o n v e r g i n g s e c t i o n and the power d r i v e system i s the r e t u r n s e c t i o n e s s e n t i a l l y c o m p r i s i n g o f a heat exchanger and a V e n t u r i meter. A copper p i p e , 1 . 6 m x 10.2 c m , i n c o n j u n c t i o n w i t h a 1.4 m x 17.8 cm PVC p l a s t i c p i p e formed an annu lar s i n g l e pass heat exchanger . With the c o o l a n t s u p p l i e d by a water main , 20 i t was p o s s i b l e to m a i n t a i n temperature o f the working f l u i d w i t h i n ± 0 . 1 ° C . PVC elbows and s e c t i o n s o f the r a d i a t o r hose p r o v i d e d r e l a t i v e l y easy a n t i c o r r o s i o n and v i b r a t i o n f r e e c o n n e c t i o n s . The b r a s s V e n t u r i meter w i t h a c o n t r a c t i o n r a t i o o f 4:1 was des igned a c c o r d i n g to the ASME 42 s p e c i f i c a t i o n s and was l o c a t e d 74 cm upstream o f the pump i n l e t . The l o c a t i o n was so s e l e c t e d as to make i t s performance i n s e n s i t i v e to upstream and downstream d i s t u r b -ances i n the form of e lbows, change i n s e c t i o n at the pump i n l e t , pump s u c t i o n , e t c . Before the f i n a l assembly the V e n t u r i meter and a s s o c i a t e d plumbing were c a l i b r a t e d , under s i m u l a t e d t e s t c o n d i t i o n s , by pumping water from a l a r g e c o n s t a n t head sump i n t o a tank of known volume. A gate v a l v e c o n t r o l l e d the f low r a t e . For a g iven v a l v e s e t t i n g and a f t e r the f low had a t t a i n e d the s teady s t a t e c o n d i t i o n , t ime taken to c o l l e c t 44 5 kg (1000 lb.) was r e c o r d e d t o g e t h e r w i t h the p r e s s u r e drop a c r o s s the V e n t u r i meter as g iven by the B a r o c e l p r e s s u r e t r a n s d u c e r . In g e n e r a l , three r e a d i n g s were taken f o r each of the v a l v e s e t t i n g s and t h e i r mean was used i n p r e p a r i n g the c a l i b r a t i o n c h a r t ( F i g u r e 2 -2 ) . The power u n i t c o n s i s t s of a c e n t r i f u g a l pump: P a r a -mount model 3 V / 6 , 9.6 l/s (160 gpm), 9.4 m head , 1750 rpm. It i s d r i v e n by a t h r e e horsepower v a r i a b l e speed d . c . motor . R n x l O -5 U , c m / s V P , P s i F i g u r e 2-2. C a l i b r a t i o n p l o t f o r the V e n t u r i meter . 22 The pump i m p e l l e r and hous ing are o f cas t b r a s s to guard a g a i n s t p o s s i b l e c o r r o s i o n . The motor i s e n e r g i z e d by a three phase g r i d , the v o l t a g e b e i n g a d j u s t e d through an a u t o t r a n s f o r m e r and r e c t i f i e d by se l en ium d i o d e s . No f u r t h e r smoothing o f the d . c . output was r e q u i r e d . A photograph of the power system i s shown i n F i g u r e 2 -3 . I t was' important to min imize d i r t c o n t a m i n a t i o n o f the t u n n e l f l u i d . T h i s was a c h i e v e d by i n c o r p o r a t i n g a lOu f i l t e r i n a bypass c i r c u i t a c r o s s the pump. The system f i l t e r s the e n t i r e volume at l e a s t once i n t w e n t y - f o u r hours o f o p e r a t i o n . F i g u r e 2-4 shows the e n t i r e t u n n e l assembly . 2.2 Des ign o f Honeycomb f o r T u r b u l e n c e C o n t r o l A l t h o u g h screens have been used r a t h e r r o u t i n e l y i n wind t u n n e l s to reduce t u r b u l e n c e and make f low u n i f o r m by improv ing m i x i n g , they have not proved p a r t i c u l a r l y s a t i s -f a c t o r y f o r r e d u c i n g t u r b u l e n c e i n water t u n n e l s . F u r t h e r m o r e , s c r e e n s , w i t h t h e i r wire d iameter d i c t a t e d by s t r e n g t h requirements to w i t h s t a n d i n e r t i a o f water , n o r m a l l y operate i n the Reynolds number range where v o r t e x shedding occurs and hence are s u s c e p t i b l e to resonance and a s s o c i a t e d f a i l u r e ^ On the o t h e r hand, honeycombs i n which the c e l l s have a l a r g e l e n g t h - t o - d i a m e t e r r a t i o have been used s u c c e s s f u l l y to F i g u r e 2-3. A p h o t o g r a p h showing d e t a i l s o f the power u n i t : A , a u t o t r a n s f o r m e r ; D, d r i v e motor P , pump. F i g u r e 2-4 . A photograph o f the g l y c e r o l - w a t e r t u n n e l capab le o f g e n e r a t i n g Reynolds number i n the range 1-18,000: A , a u t o t r a n s f o r m e r ; B , d r i v e motor; E , heat exchanger; F , f low d i s t r i b u t i n g vanes; H , p o r t h o l e s ; M, v e n t u r i m e t e r ; P , pump; S, s creens and honey-comb; T , t e s t - s e c t i o n ; V , v e n t . t o 25 reduce t u r b u l e n c e to an a c c e p t a b l e l e v e l . Thus screens are p a r t i c u l a r l y e f f e c t i v e i n making the f low un i form w h i l e honeycombs tend to reduce t u r b u l e n c e . As mentioned e a r l i e r , two' b r a s s s c r e e n s , no . 14 and 18 (mesh s i z e 0.18 and 0.09 cm, r e s p e c t i v e l y ) , l o c a t e d 3.8 cm apart were used to a s s i s t i n r e n d e r i n g the flow .un i form. The honeycomb c e l l s i z e and i t s l e n g t h were s e l e c t e d a c c o r d i n g to the t h e o r y proposed by L u m l e y ^ ' ^ . The procedure i n v o l v e s the use o f d e s i g n c h a r t s which r e q u i r e the-knowledge o f t u r b u l e n c e l e v e l , mean f low v e l o c i t y and l e n g t h s c a l e i n absence of the honeycomb as input i n f o r m a t i o n . Of c o u r s e , the p e r m i s s i b l e maximum f o r the t u r b u l e n c e i n t e n s i t y d u r i n g a c t u a l o p e r a t i o n ( i . e . w i t h the honeycomb) must a l s o be i d e n t i f i e d . To t h i s end the t u r b u l e n c e measurements were undertaken (without screens and honeycomb)which showed the i n t e n s i t y to be about 101 when the mean f low v e l o c i t y was 15 cm/s . The predominant l e n g t h s c a l e was e s t i m a t e d at 61 cm, the h e i g h t o f the t u n n e l t e s t - s e c t i o n . The maximum p e r m i s s i b l e t u r b u l e n c e i n t e n s i t y was l i m i t e d to 2%. From c o r r o s i o n c o n s i d e r a t i o n i t was d e c i d e d to use a paper honeycomb impreg-na ted w i t h p l a s t i c . Based on the s t a n d a r d s i z e s i n which such honeycombs are manufac tured , the one w i t h 0.32 cm c e l l s i z e and 15.25 cm l e n g t h was t e n t a t i v e l y s e l e c t e d f o r use . 26 However, i t was n e c e s s a r y to check i t s performance a g a i n s t the d e s i r e d s p e c i f i c a t i o n . In the p r e s e n t case £ / d = 48, L / £ = 4, R = 2.28 x i cl 1 0 4 , and R,. = 4 . 76 x 10 2 where: L = l e n g t h s c a l e ; I = l e n g t h o f the honeycomb; d = c e l l d i a m e t e r ; R^,R^ = Reynolds numbers based on c e l l d iameter and l e n g t h , r e s p e c t i v e l y . Wi th these d a t a , Lumley ' s d e s i g n c h a r t s gave p r e s s u r e c o e f f i c i e n t o f f r i c t i o n (K) = 7 and t u r b u l e n c e r e d u c t i o n f a c t o r = 0 . 0 9 . Thus the s e l e c t e d honeycomb can be expected to reduce the t u r b u l e n c e l e v e l from 10% to 0.9%. However, w h i l e r e d u c i n g the t u r b u l e n c e l e v e l o f the incoming f l ow , the honeycomb c r e a t e s i t s own t u r b u l e n c e . In g e n e r a l , the c o n t r i b u t i o n comes from the flow w i t h i n the c e l l s and the wake emanating from the i n d i v i d u a l c e l l . In . the p r e s e n t c a s e , as the f low i n the c e l l s i s l a m i n a r ( R n = 450 based on c e l l d i a m e t e r ) , i t s c o n t r i b u t i o n to the t u r -bu lence may be n e g l e c t e d , i . e . the t u r b u l e n c e energy at the model l o c a t i o n i s e n t i r e l y due to the wake and can be g i v e n , 41 by , u 2 / U 2 = x / d 27 Here x corresponds to the d i s t a n c e downstream from the end o f the honeycomb and d i s the c e l l - s i z e . T h i s g ive s the honeycomb generated t u r b u l e n c e i n t e n s i t y as 0.9%, i . e . the t o t a l t u r b u l e n c e i n t e n s i t y at the t e s t - s e c t i o n o f 1 .81. However, i t must be noted that i n the d e s i g n , the worse c o n d i t i o n (maximum v e l o c i t y ) has been c o n s i d e r e d , h e n c e the a c t u a l t u n n e l t u r b u l e n c e can be expected to be lower . F i g u r e 2-5 shows d e t a i l s o f the f low d i s t r i b u t i n g vanes , screens and honeycomb used i n the d e s i g n o f the t u n n e l . I t was n e c e s s a r y to c o n f i r m the above p r e d i c t i o n by a s y s t e m a t i c measurement o f v e l o c i t y p r o f i l e s and t u r b u l e n c e i n t e n s i t y . T h i s i s d e s c r i b e d i n the f o l l o w i n g s e c t i o n . 2.3 C a l i b r a t i o n o f the L i q u i d Tunne l The items o f utmost importance were the u n i f o r m i t y o f the v e l o c i t y d i s t r i b u t i o n at v a r i o u s s t a t i o n s i n the t e s t -s e c t i o n at d i f f e r e n t f low r a t e s and the a s s o c i a t e d t u r b u l e n c e i n t e n s i t y d i s t r i b u t i o n . T h i s was a c h i e v e d through the h o t - f i l m anemometry.A Thermo-Systems h e m i s p h e r i c a l probe (model TSI 1239W) w i t h the c o l d r e s i s t a n c e o f 7.20, was used at the overheat r a t i o o f 1.0972 ( i . e . , o p e r a t i n g r e s i s t a n c e o f 7.9P-). However, the probe must be c a l i b r a t e d f i r s t to use i t i n the c a l i b r a t i o n o f the t u n n e l . For t h i s i t was mounted on 29 the t o o l h o l d e r o f a l a t h e and towed at c o n t r o l l e d speeds i n a s l o t t e d flume employing the feed mechanism. The m u l t i s p e e d gear box a l l o w e d s a t i s f a c t o r y d e t e r m i n a t i o n o f the tow v e l o c i t i e s . The probe was immersed i n the f l u i d to a depth o f at l e a s t t en t imes i t s d i a m e t e r . The arrangement worked s a t i s f a c t o r i l y up to a v e l o c i t y o f around 16 cm/s beyond which n o i s e caused by s p u r i o u s v i b r a t i o n s s u b s t a n -t i a l l y a f f e c t e d the s i g n a l . F u r t h e r m o r e , at h i g h e r v e l o c i t i e s the towing time became so s h o r t , due to the l i m i t e d l e n g t h o f the flume (1.52 m), that the output s i g n a l from the probe c o u l d not r each a s teady s t a t e c o n d i t i o n . The t e s t a r r a n g e -ment i s shown i n F i g u r e 2-6 w h i l e the c a l i b r a t i o n p l o t i s p r e s e n t e d i n F i g u r e 2-7 . For c h a r t i n g o f the v e l o c i t y p r o f i l e , the probe was supported by a t r a v e r s i n g gear w i t h a p o s i t i o n a l a c c u r a c y o f ± 0 . 1 cm. I t would be u s e f u l to ment ion two problems en-c o u n t e r e d d u r i n g the measurements: f o r m a t i o n o f a i r bubbles on the probe s u r f a c e and dust c o n t a m i n a t i o n . The presence o f a i r bubbles changes p r o b e ' s c o l d r e s i s t a n c e thus a f f e c t i n g i t s s e n s i t i v i t y and c a l i b r a t i o n . F u r t h e r m o r e , i t promotes c o r r o s i o n ' o f the probe c o a t i n g thus r e d u c i n g i t s l i f e . The problem was e s s e n t i a l l y e l i m i n a t e d through the a d d i t i o n o f a "wet t ing agent" (Kodak P h o t o - F l o 200) which has a p r o p e r t y o f r e d u c i n g s u r f a c e t e n s i o n o f the s o l v e n t . 31 3 0 0 2 5 0 2 2 V . v o l t 2 0 0 A / / A 5 0 • / / / • /• v P r o b e : T S I 1 2 3 9 W R c - 7 - 2 n R 0 = 7 - 9 A a = 1 - 0 9 7 T - 2 0 ° C ,1/2 d • 1/2 U , ( c m / s ) F i g u r e 2 - 7 . C a l i b r a t i o n data f o r the hot f i l m probe TSI 12 39W. ' 32 C o n t a m i n a t i o n of the probe by dust p a r t i c l e s and o ther d e p o s i t s posed a s e r i o u s c h a l l e n g e . To minimize d i r t c o n t a m i -n a t i o n the s u r f a c e o f the l i q u i d was always kept s h i e l d e d . F u r t h e r m o r e , a p a r t o f the c i r c u l a t i n g f l u i d was c o n t i n u o u s l y f i l t e r e d . In s p i t e o f these p r e c a u t i o n s i t was n e c e s s a r y to c l e a n the probe p e r i o d i c a l l y u s i n g methyl a l c o h o l . To assess e f f e c t i v e n e s s o f the screens and the honey-comb i n r e n d e r i n g the f low u n i f o r m , v e l o c i t y measurements at a t y p i c a l s t a t i o n (x = 90 cm) were c a r r i e d out wi thout and w i t h the s t r a i g h t e n i n g d e v i c e s i n p o s i t i o n . The r e s u l t s are compared i n F i g u r e 2-8. Rather s p e c t a c u l a r improvement i n the v e l o c i t y p r o f i l e i s a p p a r e n t . Note, the v e l o c i t y i s e s s e n t i a l l y u n i f o r m over c e n t r a l 30 cm o f the t e s t -s e c t i o n beyond which there i s a tendency to i n c r e a s e , f i n a l l y d i m i n i s h i n g to zero at the top and bottom w a l l s . Even d u r i n g the r i s e i n v e l o c i t y , the maximum d e v i a t i o n from the mean was found to be 8%. Thus the t u n n e l has an adequate r e g i o n o f u n i f o r m v e l o c i t y s u i t a b l e f o r the p lanned t e s t s . F i g u r e 2-9 shows v e l o c i t y p r o f i l e s at t h r e e d i f f e r e n t s t a t i o n s and f o r two va lues o f the mean f low v e l o c i t y , U = 7 cm/s and 12 cm/s . These v a l u e s a p p r o x i m a t e l y e s t a b l i s h the range o f v e l o c i t i e s used d u r i n g the p r e s s u r e measurements. E s s e n t i a l c h a r a c t e r o f the p l o t s remains the same as tha t observed i n the p r e v i o u s d iagram. There c o n t i n u e s to be a + 30 o H - 15 H -30 -I Y , c m 33 • • • • • • • • • • without s t ra igh ten ing devi • • • • with straightening device • • • • • • • 10 20 U , c m / s F i g u r e 2-8. T y p i c a l p l o t s showing the e f f e c t o f screens and honeycomb i n improv ing v e l o c i t y p r o f i l e i n the t e s t - sect i o n . 34 Y , c m x = • 3 0 « 9 0 • 1 3 0 • B 4 U = 7 c m / s ' " • 1 1 . 9 c m / s • "4* 5 1 0 U , c m / s gure 2-9. A c o m p a r i s o n ' o f v e l o c i t y p r o f i l e s at three d i f f e r e n t s t a t i o n s i n the t e s t - s e c t i o n and f o r two v a l u e s of- mean f low v e l o c i t y . Note a l a r g e r e g i o n of e s s e n t i a l l y u n i f o r m flow which i s d e s i r a b l e f o r the p lanned t e s t programme'. 35 l a r g e r e g i o n ( ± 1 5 cm) s y m m e t r i c a l l y l o c a t e d around the t u n n e l c e n t e r l i n e where the v e l o c i t y p r o f i l e i s e s s e n t i a l l y u n i f o r m . The n o n u n i f o r m i t y beyond t h i s i s p r o b a b l y due to a l a r g e change i n the c r o s s - s e c t i o n a l a r e a , by a f a c t o r o f 11.5;, at the entrance to the t e s t - s e c t i o n . A l t h o u g h guide vanes tend to d i s t r i b u t e the f low even ly over the t e s t -s e c t i o n because o f t h e i r c a r e f u l l y arranged s e t t i n g , a p p a r e n t l y 'they are not e n t i r e l y e f f e c t i v e , p r o b a b l y due to non-u n i f o r m i t y o f the f low at the e n t r a n c e . A l t h o u g h the t e s t programme d i d not aim at a s y s t e m a t i c s tudy o f the t u r b u l e n c e e f f e c t s on the f l u i d mechanics o f b l u f f b o d i e s , i t was d e s i r a b l e to have some a p p r e c i t a t i o n as to the average l e v e l o f t u r b u l e n c e i n the 40 41 t e s t - sect i o n . Based on the t h e o r y proposed by Lumley ' , the average t u r b u l e n c e l e v e l was expected to be around 1.8%, however, t h i s has to be v e r i f i e d . Measurement o f t u r b u l e n c e i n t e n s i t y a c r o s s the t u n n e l t e s t - s e c t i o n at d i f f e r e n t s t a t i o n s and f low r a t e s showed i t to be e s s e n t i a l l y i n d e -pendent o f the l o c a t i o n (over the s h o r t span o f 40 cm}, however, as can be e x p e c t e d , the t u r b u l e n c e l e v e l i n c r e a s e d w i t h the f low r a t e . F i g u r e 2-10(a) shows the case when the mean f low v e l o c i t y was about 12 cm/s . The t u r b u l e n c e l e v e l i s around 1% i n the m i d d l e - h a l f o f the t e s t - s e c t i o n and g e n e r a l l y l e s s than 3 6 3 0 -15 -0-1 -15-- 3 0 -Y , c m ( a ) o o U = "11 • 9 c m / s o x = 9 0 c m o o o o o o 0 I 2 ( b ) n U - 7 c m / s n X ; 9 0 c m • • a 4 0 i 2 u / u A • ( c ) > U = 7 c m / s > x - 1 3 0 c m 0 > 4 0 i 2 F i g u r e 2-10. R e p r e s e n t a t i v e p l o t s showing v a r i a t i o n of the t u r b u l e n c e i n t e n s i t y i n the t e s t - s e c t i o n , i 4 37 21 except f o r the lower q u a r t e r . With a decrease i n the f low r a t e or a change i n s t a t i o n ( F i g u r e 2 - 1 0 b , c ) , the c h a r a c t e r o f the p l o t remains e s s e n t i a l l y the same except f o r a r e d u c t i o n i n the v a l u e o f the t u r b u l e n c e i n t e n s i t y . With the d e s i g n and c a l i b r a t i o n o f the , tunnel s a t i s -f a c t o r i l y accompl i shed the a t t e n t i o n was focussed on the d e s i g n o f t e s t models , s e l e c t i o n o f p r e s s u r e measur ing i n s t r u m e n t a t i o n and c o n s t r u c t i o n o f a dye i n j e c t i o n system f o r f low v i s u a l i z a t i o n . These are 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 . 3. TEST PROCEDURES Be fore p r o c e e d i n g to p r e s e n t the t e s t r e s u l t s and t h e i r d i s c u s s i o n i t would be a p p r o p r i a t e to b r i e f l y d e s c r i b e some o f the important t e s t p r o c e d u r e s . In g e n e r a l , the i n s t r u m e n t a t i o n employed c o n s t i t u t e s a s t a n d a r d e q u i p -ment i n any w e l l equipped f l u i d mechanics l a b o r a t o r y and hence needs no e l a b o r a t i o n . On the o t h e r hand , d e s i g n , c o n s t r u c t i o n a l and o p e r a t i o n a l f e a t u r e s i n v o l v e d w i t h r e f e r e n c e to a s p e c i f i c equipment are o f t e n too numerous to account c o m p l e t e l y . The a t t e n t i o n i s , t h e r e f o r e , focussed on more s i g n i f i c a n t aspec t s o f the i n s t r u m e n t a t i o n r e l e v a n t to the s tudy i n hand. The t e s t procedures employed are c o n c e p t u a l l y w e l l known but t h e i r implementa t ion o f t e n a t t a i n s c o m p l e x i t y o f a h i g h e r o r d e r , m a i n l y because o f the c h a r a c t e r o f the working f l u i d ( g l y c e r o l - w a t e r s o l u t i o n ) . Of ten p e c u l i a r -i t i e s o f s p e c i f i c experiments make c e r t a i n measurements q u i t e d i f f i c u l t . T h r o u g h o u t , t h e emphasis i s on p r a c t i c a l c o n s i d e r a t i o n s i n v o l v e d i n e x e c u t i n g the e x p e r i m e n t a l programme. At t imes the f a c t o r s i n v o l v e d a r e , s eeming ly , so t r i v i a l tha t one would seldom g ive them a second l o o k . However, i t i s a common e x p e r i e n c e o f most exper imenters t h a t r e s o l u t i o n o f a p p a r e n t l y s imple problems o c c a s i o n a l l y 39 takes days , i f not weeks or months. T h i s i s p a r t i c u l a r l y t rue i n the case where l i q u i d i s the working f l u i d . The g l y c e r o l - w a t e r s o l u t i o n t u n n e l r e p r e s e n t i n g a fundamental f a c i l i t y f o r the e n t i r e t e s t programme and i t s c a l i b r a t i o n u s i n g the h o t - f i l m anemometry was d e s c r i b e d e a r l i e r , In t h i s chapter models used i n the t e s t programme are i n t r o d u c e d f i r s t . T h i s i s f o l l o w e d by a d i s c u s s i o n of the h i g h l y s e n s i t i v e p r e s s u r e t r a n s d u c i n g system used i n the measurement o f the s u r f a c e p r e s s u r e d i s t r i b u t i o n . F i n a l l y , d e t a i l s o f the f low v i s u a l i z a t i o n p r o c e d u r e , which proved ex tremely u s e f u l i n o b t a i n i n g p h y s i c a l a p p r e c i a t i o n as to the c h a r a c t e r o f the f l o w , are p r e s e n t e d . 3.1 Models Two se t s o f c i r c u l a r c y l i n d r i c a l models , one f o r the s u r f a c e p r e s s u r e measurements and the o ther f o r f low v i s u a l -i z a t i o n s t u d y , were used i n the t e s t programme. The p r e s s u r e models were r e l a t i v e l y more e l a b o r a t e because of the presence o f the p r e s s u r e c o n d u c t i n g l i n e and the e x t e r n a l l y l o c a t e d mechanism f o r changing r e l a t i v e o r i e n t a t i o n o f the p r e s s u r e tap w i t h r e s p e c t to the f r e e s t ream. T h i s r e q u i r e d d r a i n i n g o f the t u n n e l f l u i d d u r i n g changing o f the p r e s s u r e models . Flow v i s u a l i z a t i o n models were s i m p l e r (no p r e s s u r e 40 t a p , models r i g i d l y f i x e d i n p o s i t i o n ) and hence avo ided t h i s t ime consuming and n e c e s s a r i l y messy o p e r a t i o n , i . e . , these models can be i n t r o d u c e d wi thout emptying the t u n n e l . The ends were p r o v i d e d w i t h so f t b l a c k rubber pads , which when p r e s s e d a g a i n s t the w a l l h e l d the c y l i n d e r i n p o s i t i o n ( F i g u r e 3 - 1 ) . For p r e s s u r e measurements, a f a m i l y o f seven two-d i m e n s i o n a l , c y l i n d r i c a l models , r a n g i n g i n d iameter from 0.5-30 cm and 15.2 cm l o n g , were c a r e f u l l y machined e i t h e r from s t a i n l e s s s t e e l or p l e x i g l a s t u b e s . D e v i a t i o n from c i r c u l a r geometry ,checked u s i n g m i c r o m e t e r , was found to be l e s s than 0.2%. The a c c u r a c y was c o n s i d e r e d q u i t e adequate f o r the t e s t programme. The range o f d iameter was so chosen as to cover the b lockage r a t i o v a r i a t i o n from 2-50%, and the Reynolds number range of 7-20,000 through the c o n c e n t r a t i o n c o n t r o l o f the working f l u i d (Tab le 3 - 1 ) . A model was suppor ted by two-end s l eeves w i t h 0 - r i n g s to guard a g a i n s t leakage o f the working f l u i d . A s t a t i c p r e s s u r e t a p , 0.5 mm i n d i a m e t e r , was l o c a t e d on the c e n t r a l s t a t i o n and connected to a p o l y e t h y l e n e tube , r u n -n i n g i n s i d e the mode l , which conveyed the p r e s s u r e to an e x t e r n a l l y l o c a t e d t r a n s d u c e r . S i z e o f the p r e s s u r e conduct -i n g l i n e was c a r e f u l l y chosen (1 .7 - 5 mm i n s i d e d i a m e t e r , 110 cm length) so that the r e s u l t i n g t ime cons tant has a F i g u r e 3 - 1 . T y p i c a l models used i n the f l o w v i s u a l i z a t i o n s t u d y . T a b l e 3-1 Pressure models used i n the t e s t programme, c o r r e s p o n d i n g b lockage and a s s o c i a t e d Reynolds number range a t t a i n e d through the v a r i a t i o n o f mean f r e e stream v e l o c i t y and c o n c e n t r a t i o n o f the working f l u i d NO. S/C % \ C , °"o D ,cm . 0 (v/v = 1) w 41 (v/v = 4) w 66 (v/v = 15) w 76 (v/v = 40) w 1 0.8 0.5 200 - 600 7 5 - 200 . 30 - 60 7 - 13 2 2 1.2 600 - 1500 150 - 400 40 - 150 16 - 30 3 3.3 2 1200 - 3000 300 - 600 60 - 300 30 - 50 4 6.6 5 3000 - 7300 800 - 2500 150 - 750 75 - 200 5 12. 5 7.6 4500 - 11000 3000 - 11000 250 - 1100 120 - 250 6 25 15 9000 - 15000 2000 - 6000 450 - 2200 200 - 500 7 50 30 12000 - 30000 3000 - 8000 300 - 3000 150 - 300 43 r e a s o n a b l e v a l u e . A s y s t e m a t i c r o t a t i o n o f the c y l i n d e r i n s m a l l increments through 1 8 0 ° p r o v i d e d d e s i r e d p r e s s u r e d i s t r i b u t i o n over the c y l i n d e r s u r f a c e . F i g u r e 3-2 shows the set of p r e s s u r e models used i n the t e s t programme w h i l e F i g u r e 3-3 p r e s e n t s c o n s t r u c t i o n a l d e t a i l s o f a t y p i c a l model . 3.2 P r e s s u r e Measurements The mean p r e s s u r e component b e i n g extremely s m a l l 2 (of the o r d e r o f 0.7 N/m ) demanded a h i g h l y s e n s i t i v e i n s t r u m e n t a t i o n f o r i t s measurement. T h i s was accompl i shed u s i n g the " B a r o c e l Modular P r e s s u r e T r a n s d u c i n g System" deve loped by D a t a m e t r i c s Inc . o f Watertown, M a s s a c h u s e t t s . The type 550-5 B a r o c e l sensor i s des igned to operate w i t h 2 f l u i d s over the p r e s s u r e range o f 0 - 10 p s i a (68.98 kN/m ) . The U n i t i s a h i g h p r e c i s i o n , s t a b l e c a p a c i t i v e v o l t a g e d i v i d e r , the v a r i a b l e element o f which i s a t h i n p r e -s t r e s s e d s t e e l diaphragm p o s i t i o n e d between f i x e d c a p a c i t o r p l a t e s . The diaphragm d e f l e c t s p r o p o r t i o n a l l y to the magnitude o f the a p p l i e d p r e s s u r e . To i s o l a t e the e x t e r n a l p r e s s u r e medium from the sensor d i a p h r a g m - c a p a c i t a n c e sys tem, the u n i t uses h i g h l y s e n s i t i v e m e t a l l i c b e l l o w s . The volume between the b e l l o w s , i s o l a t o r and sensor diaphragm i s f i l l e d wi th degassed s i l i c o n e o i l which serves both as 3 - 2 . A photograph showing the models used i n s u r f a c e p r e s s u r e d i s t r i b u t i o n . To pressure transducer End sleeve End sleeve F i g u r e 3 - 3 . A diagram showing c o n s t r u c t i o n a l d e t a i l s of a t y p i c a l p r e s s u r e measurement model . 46 p r e s s u r e t r a n s m i t t i n g f l u i d and as a d i e l e c t r i c . A p r e s s u r e s i g n a l from the e x t e r n a l l i q u i d medium i s t r a n s -m i t t e d by the be l lows to the s i l i c o n e o i l which i n t u r n d e f l e c t s the diaphragm to produce the r e q u i r e d change i n c a p a c i t a n c e . An a . c . c a r r i e r v o l t a g e at 10 Hz i s a p p l i e d to the s t a t i o n a r y c a p a c i t o r p l a t e s , and a b r i d g e c i r c u i t determines an output v o l t a g e dependent on the r a t i o o f the c a p a c i t a n c e o f the diaphragm to each of the s t a t i o n a r y p l a t e s . The c a r r i e r v o l t a g e i s t h e r e f o r e modulated a c c o r d -i n g to the input p r e s s u r e . The u n i t s e n s i t i v i t y i s 10 ^ 2 p s i (0.07 N/m ) p r o v i d e d the p r e s s u r e sensor i s f u l l y i s o l a t e d from e x t e r n a l sources o f v i b r a t i o n and n o i s e . I t was i m p e r a t i v e to ensure removal of a l l t r a c e s o f a i r pockets from the p r e s s u r e d u c t i n g f o r s t a t i s f a c t o r y o p e r a t i o n . B a r o c e l i s a c c u r a t e l y c a l i b r a t e d f o r s teady p r e s s u r e s . F i g u r e 3-4 p r e s e n t s a schemat ic d iagram of the p r e s s u r e t r a n s d u c e r . I t was important to minimize the e f f e c t o f ambient temperature e x c u r s i o n s on the B a r o c e l ' s per formance . T h i s was a c h i e v e d by mounting the t r a n s d u c e r on a heat s i n k , a l a r g e aluminum b l o c k w i t h working f l u i d c i r c u l a t i n g i n s i d e . The arrangement v i r t u a l l y e l i m i n a t e d the i n f l u e n c e o f . t emperature t r a n s i e n t s . 47 Power input Stationary Diaphragm c a p c i t o r plates F i g u r e 3-4 . A schemat ic diagram of the B a r o c e l . p r e s s u r e t r a n s d u c e r . 48 A f t e r a complete removal o f a i r bubbles from the f l u i d , a model was p o s i t i o n e d i n the t e s t s e c t i o n w i t h i t s c e n t e r 90 cm downstream of the end of the honeycomb. Next , the p r e s s u r e d u c t i n g was f i l l e d w i t h the t e s t f l u i d and was connected to a B a r o c e l p r e s s u r e t r a n s d u c e r v i a a set o f p o l y e t h y l e n e and M y l a r t u b i n g s a f t e r removal o f a i r pockets from the l i n e . The p r e s s u r e s e n s i n g u n i t was b a l a n c e d to read zero output i n the n o - f l o w c o n d i t i o n . With the pump o p e r a t i n g at a p r e s e l e c t e d speed to g ive a d e s i r e d Reynolds number and the t e s t f l u i d h e l d at a c o n s t a n t t e m p e r a t u r e , the mean p r e s s u r e d i s t r i b u t i o n around the v e r t i c a l m e r i d i o n a l c r o s s - s e c t i o n was measured. A p o i n t c o n c e r n i n g an a p p r o p r i a t e c h o i c e o f the s i z e o f the p r e s s u r e tub ings must be emphasized h e r e . A s y s t e m a t i c s tudy w i t h tubes o f d i f f e r e n t s i z e and a s s o c i a t e d time to reach s teady s t a t e p r e s s u r e showed the tubes w i t h i n t e r n a l d iameter l e s s than 1.6 mm-to have an e x c e s s i v e l y l a r g e time cons tant (>20 m i n . ) . Of c o u r s e , as suggested by s e v e r a l t h e o r e t i c a l and e x p e r i m e n t a l s t u d i e s on the dyna-mic response o f f l u i d l i n e s 4 ^ the t ime c o n s t a n t would depend on -a number o f parameters i n c l u d i n g the d iameter and l e n g t h o f the t u b i n g s , v i s c o s i t y o f the f l u i d , i n l i n e volume i n c l u d i n g the t r a n s d u c e r ' s c a v i t y , c h a r a c t e r o f p r e s s u r e s i g n a l s , e t c . For the mean p r e s s u r e measurements 49 under c o n s i d e r a t i o n , i t was conven ient to use f l u i d l i n e s o f 1.7.-5 mm d i a . r e s u l t i n g i n the t ime cons tant o f around 3 minutes . To i n s u r e a c c u r a c y as w e l l as r e p e a t a b i l i t y o f the measured d a t a , i t was o f utmost importance to min imize and compensate f o r any d r i f t o f the p r e s s u r e t r a n s d u c e r and a s s o c i a t e d e l e c t r o n i c c i r c u i t r y . Minute c h a r a c t e r o f the -4 p r e s s u r e s i g n a l s (10 p s i ) t o g e t h e r w i t h the r e l a t i v e l y long time i n v o l v e d i n r e a c h i n g the s teady s t a t e made t h i s a l l the more n e c e s s a r y . Chart r e c o r d i n g s o f the d r i f t over p e r i o d s o f 24-48 hours showed them to be q u i t e s i g n i f i c a n t , at t imes as l a r g e as 50% o f the a c t u a l s i g n a l , but o f no w e l l d e f i n e d p a t t e r n . The d r i f t compensation procedure i n v o l v e d t h r e e s u c c e s s i v e measurements at equa l i n t e r v a l s o f time c o r r e s p o n d i n g to the time cons tant o f the system. T h i s i s e x p l a i n e d i n d e t a i l below. Let the o b j e c t i v e be to measure a d i f f e r e n t i a l p r e s s u r e P - P , where P r e p r e s e n t s p r e s s u r e on the s u r -a r a face o f the c y l i n d e r at p o i n t ' a ' and P .^ corresponds to the p r e s s u r e at a r e f e r e n c e p o i n t . In g e n e r a l , a known e x t e r n a l p r e s s u r e source ( e . g . a l i q u i d column) or p r e s s u r e at a g i v e n l o c a t i o n i n the t u n n e l can serve t h i s purpose . ,In the presen t case p r e s s u r e at a p o i n t on the bottom w a l l o f the t u n n e l was used to serve t h i s end. Let the a r b i t r a r y zero d r i f t o f the e l e c t r o n i c system be as i n d i c a t e d i n F i g u r e 3-5 . The diagram a l s o shows the c o r r e s p o n d i n g d r i f t o f the d i f f e r e n t i a l p r e s s u r e s A-Pa and AP^, where AP = P - P and AP = P - P . Here P r e p r e s e n t s a a w r r w w 1 p r e s s u r e at a s u i t a b l e l o c a t i o n , taken to be on the t u n n e l w a l l i n the p r e s e n t case . T h u s , the d e s i r e d P - P cl T* AP„ - AP.. . a r Now, from F i g u r e 3-5 : AP a + 6 1 + 6 2 (AP ) + (AP ) „ ^ rJ 1 . ^ r 3 (AP^. +. 6 J + (AP + 6 +• 6 + 6 ) 2 2 = AP r + 6 1 Hence , 51 Drift < * f V ^ ^ — <A pr '3 (Ap r), ( ^ P r ^ - r" " I \ b< ! ' ! h ? " i 05 3 1 | 1 V1 I I | — T *j j Ti T2 T 3 T 4 T 5 F i g u r e 3 - 5 . A pr o c e d u r e f o r compensation o f the e l e c t r o n i c d r i f t o f the p r e s s u r e measuring system. 52 Assuming tha t the e l e c t r o n i c zero s e t t i n g d r i f t s l i n e a r l y d u r i n g the i n t e r v a l marked by the p r e s s u r e measure ments (AP ) (AP ) „ then v r 1 r 2 S 2 " « 3 i . e . , (AP ) + (AP ) ( A P a ) 2 ^ I _ i . = P a - P r . (3.1) Thus d e t e r m i n a t i o n o f the d i f f e r e n t i a l p r e s s u r e i n v o l v e d the measurement o f (AP ) , (AP ) and (AP ) i n that o r d e r . The procedure gave da ta t h a t can be reproduced w i t h i n an a c c u r a c y o f ±2%. Note , the e v a l u a t i o n o f the d i f f e r e n t i a l p r e s s u r e P^ - P ,^ at some d i f f e r e n t l o c a t i o n on the c y l i n d e r would f o l l o w the same p r o c e d u r e . Thus (AP ) + (AP ) P u - P =• ( A P , ) , - r 5 ^ — i • b r b '4 4 6 47 Modi et a l . ' have used s i m i l a r procedure f o r d r i f t compensat ion i n t h e i r s t u d i e s w i t h an a o r t i c h e a r t v a l v e and s p h e r i c a l models . F i g u r e 3-6 shows i n s t r u m e n t a t i o n l a y o u t d u r i n g the s t a t i c p r e s s u r e measurments. 53 rss/sssj?sssssssssss/ssssssssss\ fo fo fo fo fo: re ference \ p r e s s u r e Jj Baroce l manifold air supply tof lush liquid in line s ignal condi t ioner d.c. d ig i ta l v o l t m e t e r o s c i Hoscope u- v. recorder f i l t e r 1 i F i g u r e 3-6. A l i n e drawing of the i n s t r u m e n t a t i o n se t -up used d u r i n g s t a t i c p r e s s u r e measurements on the s u r f a c e o f a c i r c u l a r c y l i n d e r . 54 The mean f low r a t e through the t u n n e l was m a i n t a i n e d cons tant d u r i n g the p r e s s u r e measurements. T h i s was a c h i e v e d by a cont inuous m o n i t o r i n g o f the V e n t u r i meter output as i n d i c a t e d by the d i f f e r e n t i a l p r e s s u r e t r a n s d u c e r ( B a r o c e l ) and c o r r e c t i n g f o r any f l u c t u a t i o n s i n the pump speed due to changes i n the l i n e v o l t a g e . Temperature o f the working f l u i d was h e l d cons tant ( ± 0 . 1 ° C ) u s i n g the heat exchanger system d e s c r i b e d b e f o r e . F i g u r e 3-7 shows a photograph o f the mean f low measurement system. 3 . 3 . Flow V i s u a l i z a t i o n To b e t t e r a p p r e c i a t e the p h y s i c a l c h a r a c t e r o f the f l u i d f i e l d a s s o c i a t e d w i t h c i r c u l a r c y l i n d r i c a l models under c o n f i n e d c o n d i t i o n , f low v i s u a l i z a t i o n was u n d e r t a k e n . The dyed g l y c e r o l - w a t e r s o l u t i o n o f the same c o n c e n t r a t i o n as t h a t o f the t e s t f l u i d was i n j e c t e d a p p r o x i m a t e l y 10 cm upstream o f the model . The dye employed was an i m i t a t i o n c o c h i n e a l food c o l o u r . A p p r o p r i a t e volumes o f the dye and pure g l y c e r i n were mixed to produce a g l y c e r o l - w a t e r s o l u t i o n o f the same d e n s i t y as t h a t o f the t e s t f l u i d . A dye i n j e c t i n g p r o b e , c o n s i s t i n g o f seven #23 s y r i n g e need le s (0.38 mm) p l a c e d 0.5 - 1.0 cm apar t on a s t r e a m l i n e d s u p p o r t , was c o n s t r u c t e d . O c c a s i o n a l l y a l ong s t a i n l e s s s t e e l Measurement o f mean f low r a t e u s i n g v e n t u r i m e t e r : B , B a r o c e l p r e s s u r e t r a n s d u c e r ; D, d i g i t a l d . c . v o l t m e t e r ; E, heat exchanger ; F , f i l t e r ; H , h e a t - s i n k ; P , power s u p p l y ; S, s i g n a l c o n d i t i o n e r ; V , v e n t u r i meter . 56 t u b i n g ( 3 mm d i a . , 30 cm long) , bent at r i g h t angle near the t i p , was used i n c o n j u n c t i o n w i t h a c l i n i c a l s y r i n g e . I t proved to be p a r t i c u l a r l y u s e f u l i n i n j e c t i n g dye i n the near-wake r e g i o n ( F i g u r e 3 - 8 ) . "Intramedic" t u b i n g s (0.6 mm i n s i d e d iameter) were used to connect the needles to a m a n i f o l d . The. r a t e o f i n j e c t i o n was c o n t r o l l e d w i t h b r a s s need le v a l v e s . To ensure adequate f low through each n e e d l e , i . e . , to p r o v i d e s u f f i c i e n t head, the s u p p l y b o t t l e was suspended from the c e i l i n g 4 m above the i n j e c t i o n l e v e l . A schemat ic d iagram of the complete s e t -up i s shown i n F i g u r e 3 -9 . A f t e r a s c e r t a i n i n g s u c c e s s f u l o p e r a t i o n of the e n t i r e assembly , f low p a t t e r n s were photographed , u s i n g s t i l l and 16 mm movie cameras , over a range o f Reynolds number and b l o c k a g e . The a t t e n t i o n was p r i m a r i l y focussed on c a p t u r i n g the f o r m a t i o n , e v o l u t i o n and onset o f i n s t a b i l i t y o f the Fopp l v o r t i c e s as a f f e c t e d by these parameters . I t would be a p p r o p r i a t e to p o i n t out here the type o f l i g h t i n g system used as i t p l a y e d a c r i t i c a l r o l e i n the p h o t o g r a p h i n g p r o c e s s . A combinat ion o f three v a r i a b l e i n t e n s i t y photo f l o o d s (maximum 500 w a t t s , 3 4 0 0 ° K ) back-i l l u m i n a t e d the s u b j e c t through the t u n n e l g l a s s window. To e l i m i n a t e hot s p o t s , the l i g h t beam was e v e n l y d i f f u s e d by masking the t e s t s e c t i o n w a l l w i t h a t r a c i n g p a p e r . A set Cn c a m e r a F i g u r e 3-9 . A ske tch showing the equipment layout d u r i n g the f low v i s u a l i z a t i o n . 59 of t r i a l runs he lped a r r i v e at the a p p r o p r i a t e a p e r t u r e s e t t i n g and exposure t ime f o r the type o f f i l m used (Kodak h i g h speed Ektachrome type EHB-135 ( s t i l l ) or EF-7242 (mov ie ) , t u n g s t e n , 3 2 0 0 ° K , ASA 125, f i l t e r 81A) . D u r i n g the course o f v i s u a l i z a t i o n s t u d y , i t was d i s c o v e r e d tha t i n s p i t e o f the l a r g e volume of the t e s t f l u i d (==265&) , a r e l a t i v e l y s m a l l amount o f dye (200 cc) was s u f f i c i e n t to p o l l u t e the working f l u i d to the p o i n t tha t no c l e a r photographs c o u l d be t a k e n . T h i s p r e s e n t e d a r a t h e r s e r i o u s problem i n terms o f t i m e , e f f o r t and cos t i n v o l v e d i n r e p l e n i s h i n g the working f l u i d . C l e a r l y , i t was n e c e s s a r y to f i n d an agent t h a t would n e u t r a l i z e the dye wi thout a t t a c k i n g the t u n n e l m a t e r i a l or i t s c i r c u l a t i n g system and which does not a l t e r the p h y s i c a l p r o p e r t i e s o f the t e s t f l u i d . U n f o r t u n a t e l y , no such agent has been r e p o r t e d i n the l i t e r a t u r e . A c o n s i d e r a b l e amount o f p a t i e n t t e s t i n g w i t h numerous o x i d i z i n g agents l e d ' t o sodium h y p o c h l o r i t e which has a l l the d e s i r a b l e a t t r i b u t e s . Only 300 cc o f the agent was s u f f i c i e n t to" c o m p l e t e l y n e u t r a l i z e the dye . To keep the c o n c e n t r a t i o n o f the t e s t f l u i d c o n s t a n t , s u f f i c i e n t amount o f g l y c e r i n was p e r i o d i c a l l y added thus o f f s e t t i n g the d i l u t i n g e f f e c t of the dye removing agent . 60 3.4 C r i t i c a l Reynolds Number The f low v i s u a l i z a t i o n models were a l s o used f o r d e t e r m i n a t i o n o f the c r i t i c a l Reynolds number as a f f e c t e d by the b l o c k a g e . In the low Reynolds number s t u d i e s , ' c r i t i c a l ' r e f e r s to the onset o f i n s t a b i l i t y o f the F o p p l v o r t i c e s . The wake bubble beg ins to o s c i l l a t e and w i t h any i n c r e a s e i n the Reynolds number beyond the c r i t i c a l v a l u e r e s u l t s i n shedding o f v o r t i c e s l e a d i n g to the o r g a n i z e d Karman v o r t e x s t r e e t . D e t e r m i n a t i o n o f the c r i t i c a l Reynolds number f o r a g i v e n c y l i n d e r was accompl i shed u s i n g a hot f i l m anemometer system i n c o n j u n c t i o n w i t h a h i g h f requency f i l t e r to e l i m i n a t e t u r b u l e n c e generated n o i s e . The hot f i l m probe was s u i t a b l y l o c a t e d a few d iameters downstream to y i e l d s t r o n g c l e a r s i g n a l s . The Reynolds number o f the f low was g r a d u a l l y i n c r e a s e d by i n c r e a s i n g the pump rpm. E v o l u t i o n o f the F b p p l v o r t i c e s was t r a c e d through dye i n j e c t i o n and was c o r r e l a t e d w i t h the hot f i l m response as d i s p l a y e d on an o s c i l l o s c o p e . The onset o f i n s t a b i l i t y was r e a s o n a b l y sharp and was i n d i c a t e d by a s t r o n g s i n u s o i d a l s i g n a l from the hot f i l m p r o b e . 4. RESULTS AND DISCUSSION With some a p p r e c i a t i o n o f background to the p r o b l e m , i n s t r u m e n t a t i o n used and the e x p e r i m e n t a l procedures adopted , we are ready to look i n t o the t e s t r e s u l t s and t h e i r i n t e r p r e t a t i o n . The amount o f e x p e r i m e n t a l data o b t a i n e d i s r a t h e r e x t e n s i v e , however, the g u i d i n g p r i n c i p l e has been to i n c l u d e o n l y those r e s u l t s which have immediate r e l e v a n c e to the s tudy i n hand and he lp i n e s t a b l i s h i n g d e f i n i t e t r e n d s . In g e n e r a l , the sequence i n which the r e s u l t s are p r e s e n t e d a l s o denotes the c h r o n o l o g i c a l o r d e r of the t e s t s . To b e g i n w i t h , an approach to da ta r e d u c t i o n , so c r i t i c a l at low Reynolds number, i s d i s c u s s e d . T h i s i s f o l l o w e d by p r e s e n t a t i o n o f the s u r f a c e p r e s s u r e d i s -t r i b u t i o n r e s u l t s as f u n c t i o n s o f Reynolds number and b l o c k a g e . Next , the a t t e n t i o n i s focussed on the p r e s s u r e i n t e g r a t e d drag data and the S t r o u h a l number as a f f e c t e d by the w a l l conf inement . F i n a l l y , n e a r - w a k e s t r u c t u r e i s s t u d i e d u s i n g f low v i s u a l i z a t i o n i n c o n j u n c t i o n w i t h s t i l l and 16 mm movie photography . A v a i l a b l e r e s u l t s from l i t e r a t u r e are i n c l u d e d where a p p r o p r i a t e f o r comparison and to a s s i s t i n emphas iz ing the i n f l u e n c e o f b l o c k a g e . 62 4.1 Cho ice o f Reference V e l o c i t y and P r e s s u r e Before p r o c e e d i n g w i t h p r e s e n t a t i o n and a n a l y s i s o f the t e s t r e s u l t s , one must address to s e v e r a l fundamental q u e s t i o n s which are p a r t i c u l a r l y s i g n i f i c a n t i n the low Reynolds number f low s t u d i e s . C l e a r l y , w i t h a model immersed i n an unbounded u n i f o r m stream there i s no a m b i g u i t y c o n c e r n i n g r e f e r e n c e or c h a r a c t e r i s t i c v e l o c i t y and p r e s s u r e : It i s the cons tant v e l o c i t y and p r e s s u r e o f the s tream f a r away from the model . For low Reynolds number f low i n a t u n n e l , however, the f l u i d v e l o c i t y and p r e s s u r e v a r y s i g n i f i c a n t l y a l on g the a x i s o f the t e s t . s e c t i o n , even i n absence o f the model due to boundary l a y e r growth a long the w a l l s . Presence o f the model and a s s o c i a t e d wake would o n l y accentuate the prob lem. O b v i o u s l y some compromise i s i n d i c a t e d i n s e l e c t i o n o f these parameters . A c r i v o s et a l ? ^ ' ^ have suggested use of the p r e s s u r e d i r e c t l y below the c e n t e r l i n e o f t h e i r model as the r e f e r -ence s t a t i c p r e s s u r e and the c e n t e r l i n e v e l o c i t y , w i t h the model absent but at the same s e t t i n g o f the pump, as the c h a r a c t e r i s t i c v e l o c i t y . For models w i t h a s m a l l b lockage t h i s c h o i c e o f r e f e r e n c e p r e s s u r e may prove to be adequate but w i t h a l a r g e r b l o c k a g e , due to a c c e l e r a t i o n o f the f low at the model l o c a t i o n , the r e f e r e n c e p r e s s u r e i s indeed a f f e c t e d and becomes a f u n c t i o n o f w a l l conf inement (bes ides 63 other p a r a m e t e r s ) . To put i t d i f f e r e n t l y , the c h o i c e of r e f e r e n c e p r e s s u r e as suggested above has a degree o f opt imism i m p l i c i t i n i t . It assumes t h a t e f f e c t s o f the upstream adverse p r e s s u r e g r a d i e n t c r e a t e d by presence o f the model e x a c t l y c a n c e l s the i n f l u e n c e o f a c c e l e r a t i o n i n gaps at the model l o c a t i o n thus g i v i n g the d e s i r e d P ^ . One p o s s i b l e improvement i n the c h o i c e o f P o t would be to take i t as the p r e s s u r e at the model l o c a t i o n (but wi thout the model) w i t h o p e r a t i n g c o n d i t i o n o f the t u n n e l kept the same as tha t used w i t h the model i n p o s i t i o n . However, t h i s s t i l l cannot account f o r the changes i n v e l o c i t y p r o f i l e from s e c t i o n to s e c t i o n i n a g i v e n t u n n e l , and between t u n n e l s used by d i f f e r e n t i n v e s t i g a t o r s . U s e f u l n e s s o f the c e n t e r l i n e v e l o c i t y as a c h a r a c -t e r i s t i c v e l o c i t y a l s o poses s e v e r a l q u e s t i o n s . In g e n e r a l , the v e l o c i t y p r o f i l e s are s u b s t a n t i a l l y a f f e c t e d by l o c a t i o n , boundary l a y e r growth , s c r e e n ' s mesh s i z e , b l o c k a g e , pump speed and the t o t a l c i r c u i t r e s i s t a n c e . Hence , the c h a r a c t e r i s t i c v e l o c i t y U c proposed by Grove et a l , can h a r d l y be c o n s i d e r e d a s u i t a b l e r e f e r e n c e . Another p o s s i b l e compromise would be to take u n i f o r m p o r t i o n of the v e l o c i t y p r o f i l e f a r upstram and use i t , -as a c h a r a c t e r i s t i c v e l o c i t y . However, the d i s t a n c e i n v o l v e d to account f o r boundary l a y e r e f f e c t s would , i n g e n e r a l , depend upon the t u n n e l u s e d , model and i t s l o c a t i o n . 64 A r a t h e r s i g n i f i c a n t p o i n t to keep i n mind i n p r e s e n t i n g data i s to ensure i t s r e p e a t a b i l i t y by o ther i n v e s t i g a t o r s , u s i n g d i f f e r e n t t e s t f a c i l i t i e s , to permi t compar i son . With t h i s i n mind and a f t e r c a r e f u l c o n s i d e r a t i o n o f the a l t e r n a t i v e methods d i s c u s s e d above a compromise c h a r a c -t e r i s t i c v e l o c i t y , average v e l o c i t y i n the t e s t - s e c t i o n based on the mean f low r a t e ( U ) , was adopted . T h i s approach has s e v e r a l obvious advantages . I t e l i m i n a t e s most o f the problems mentioned above. O b v i o u s l y , t e s t s conducted w i t h and wi thout model (but at the same meter s e t t i n g as w i t h the model) would l eave the average v e l o c i t y i n the t e s t - s e c t i o n unchanged. T h u s , not o n l y does i t e l i m i n a t e the q u e s t i o n o f model l o c a t i o n , type of t u n n e l , f low s t r a i g h t e n e r s used and s i z e o f the t e s t s e c t i o n but a l s o overcomes problems o f p r e s s u r e g r a d i e n t and b l o c k a g e . The c h o i c e would f a c i l i t a t e d u p l i c a t i o n of R , r e f e r e n c e v e l o c i t y b e i n g more p r e c i s e l y d e f i n e d . F u r t h e r m o r e , i t s measurement i s q u i t e s imple and i n v o l v e s o n l y c o n v e n t i o n a l i n s t r u m e n t a t i o n . However, i t must be emphasized t h a t t h i s does not c o r r e c t f o r changes i n v e l o c i t y p r o f i l e w i t h d i s t a n c e and hence the r e s u l t i n g p r e s s u r e e f f e c t s due to l o c a t i o n of the model . T h i s b r i n g s us to tha t e l u s i v e task o f s e l e c t i n g P , 48 As d i s c u s s e d e a r l i e r , the P^ advocated by A c r i v o s et a l . 65 has l i t t l e meaning here i n view of the l a r g e b lockage p r e s e n t e d by the model . From the p o i n t o f view of r e p e a t -a b i l i t y and comparison of d a t a , the use o f p r e s s u r e at a s p e c i f i e d tap on the s u r f a c e o f the model as r e f e r e n c e appears q u i t e a t t r a c t i v e . A l t h o u g h t h i s cannot account f o r l o c a l v a r i a t i o n s due to b lockage e f f e c t s (from p o i n t to p o i n t at the s u r f a c e o f the m o d e l ) , i t c o u l d e f f e c t i v e l y compensate f o r i t i n an average f a s h i o n . Thus one way to p r e s e n t p r e s s u r e data i n c o e f f i c i e n t 2 form would be as C p = (P Q - P r ) / ( p U /2) where P r c o r r e s -ponds to the p r e s s u r e at a s p e c i f i e d tap on the s u r f a c e o f the c y l i n d e r and U as c a l c u l a t e d from the average f low r a t e (average f low r a t e / t e s t - s e c t i o n a r e a o f 61 cm x 15 cm). However, t h i s d e f i n i t i o n i s s t i l l s u s c e p t i b l e to e r r o r s , i n t r o d u c e d by n o n - u n i f o r m i t y o f the v e l o c i t y p r o f i l e (at a p r e s s u r e tap and the r e f e r e n c e l o c a t i o n ) p a r t i c u l a r l y because the denominator remains u n a f f e c t e d by t h i s change. One way to v i r t u a l l y e l i m i n a t e t h i s shortcoming i s to express p r e s s u r e c o e f f i c i e n t as e x p l a i n e d below ( F i g u r e 4 - 1 ) . L e t e r r o r s i n p r e s s u r e due to n o n - u n i f o r m i t y o f the v e l o c i t y p r o f i l e be at P^, e Q at P Q and at P^. E x p r e s s -ing p r e s s u r e c o e f f i c i e n t as a r a t i o o f the d i f f e r e n t i a l p r e s s u r e s , between t h a t at a tap i n q u e s t i o n and the s t a g -n a t i o n p o i n t w i t h r e s p e c t to the r e f e r e n c e p r e s s u r e , g ives 66 F i g u r e 4 -1 . An i l l u s t r a t i o n showing p o s s i b l e e r r o r s i n t r o d u c e d by n o n - u n i f o r m i t y o f the v e l o c i t y p r o f i l e c = CP e + e 9 ) - ( P r + S ) 5 where P Q , P^, p- c o r r e s p o n d to p r e s s u r e s w i t h u n i f o r m v e l o c i t y p r o f i l e . Thus P . - P 0 r j 1 1 + ( e 0 " e r ) / ( P 0 - - P r ) Note t h a t eQ - and - are l i k e l y to be v e r y s m a l l . On the o t h e r hand, P N - P and P . - P r e p r e s e n t r e l a t i v e l y ' 8 r O r ' l a r g e q u a n t i t i e s compared to the r e s p e c t i v e e r r o r d i f -f e r e n t i a l s . T h e r e f o r e , £ - £ F - £ = 19 _x_ , = ^0 ^r e 9 r P N - P a n d e 0 r P N - P O r O r are l i k e l y to be v a n i s h i n g l y s m a l l . C o n s e q u e n t l y , the term _o r 0 r Both numerator and denominator b e i n g s e n s i t i v e , the proposed d e f i n i t i o n o f the p r e s s u r e c o e f f i c i e n t promises to p r o v i d e adequate compensat ion f o r e r r o r s i n t r o d u c e d by non-u n i f o r m i t y o f the v e l o c i t y p r o f i l e . The r e f e r e n c e l o c a t i o n was taken to be at 6 = 5 0 ° . The cho ice was prompted by the t e s t da ta which showed C^ to reach zero i n the g e n e r a l v i c i n i t y o f 8 = 5 0 ° , i . e . , P ^ Q O ± P ^ . Of c o u r s e , i n g e n e r a l , l o c a t i o n of the r e f e r e n c e p r e s s u r e i s e n t i r e l y a r b i t r a r y . The p r e s s u r e d a t a p r e s e n t e d i n t h i s c h a p t e r use the d e f i n i t i o n of p r e s s u r e c o e f f i c i e n t as 1 + £ 9r 1 + £ and Or 6 8 P 0 - P 5 0 ° C P P - P 0 5 0 ° I t i s easy to r e c o g n i z e the term PQ - P ^ Q O as an a p p r o -x i m a t i o n to (1 /2)pU^. However, now we are l i k e l y to account f o r the e r r o r s i t t t roduced'by n o n - u n i f o r m i t y o f the v e l o c i t y p r o f i l e . Thus , i n summary, t h i s c o e f f i c i e n t has s e v e r a l advantages: i t tends to compensate f o r the p r e s s u r e g r a d i e n t , b lockage e f f e c t s , i r r e g u l a r i t y of the v e l o c i t y p r o f i l e and p o s s i b l e e r r o r s i n p r e s s u r e measure-ments caused by e l e c t r i c a l d r i f t s o f the p r e s s u r e s e n s i n g system ( the e l e c t r i c a l d r i f t was d i s c u s s e d i n Chapter 3 ) . F u r t h e r m o r e , i n c o n j u n c t i o n w i t h the Reynolds number (based on average f low v e l o c i t y and sphere d i a m e t e r ) , i t promises to a s s i s t i n comparison w i t h s i m i l a r data by o ther i n v e s -t i g a t o r s u s i n g d i f f e r e n t t e s t f a c i l i t i e s . 1 A q u e s t i o n may a r i s e as to the p o s s i b l e d i f f i c u l t y t h i s new d e f i n i t i o n may cause i n comparing t e s t d a t a w i t h o ther p u b l i s h e d i n f o r m a t i o n . F o r t u n a t e l y , t h i s does not p r e s e n t any prob lem. As shown i n Appendix I , c o n v e n t i o n a l p r e s s u r e c o e f f i c i e n t C can be w r i t t e n i n terms o f measured P ' i n f o r m a t i o n as w i t h an e r r o r of <3% i n the Reynolds number range i n v e s t i -gated h e r e . I t i s of i n t e r e s t to p o i n t out here tha t s i m i l a r data r e d u c t i o n procedure was f i r s t employed by Modi and 46 Aminzadeh . i n t h e i r s t u d i e s w i t h a o r t i c hear t v a l v e s where the Reynolds number range o f i n t e r e s t was 100-1200. 49 More r e c e n t l y , e x t e n s i v e t e s t s w i t h spheres by Akutsu have s u b s t a n t i a t e d r e l a t i v e i n s e n s i t i v i t y o f the proposed p r e s s u r e c o e f f i c i e n t to t e s t - s e c t i o n geometry, v e l o c i t y p r o f i l e , Reynolds number and b l o c k a g e . 4.2 E f f e c t o f Reynolds Number F i g u r e 4-2 through 4-4 summarize a r a t h e r comprehensive set o f d a t a on the s u r f a c e p r e s s u r e d i s t r i b u t i o n f o r two d i m e n s i o n a l c i r c u l a r c y l i n d e r s as a f f e c t e d by the Reynolds number f o r a g i v e n b lockage i n the range 2-50%. The i n f o r m a t i o n i s p r e s e n t e d u s i n g the new d e f i n i t i o n o f the p r e s s u r e c o e f f i c i e n t d i s c u s s e d b e f o r e . ' A l t h o u g h the use o f c o n v e n t i o n a l d e f i n i t i o n (CT) showed e s s e n t i a l l y the same t r e n d s , i t m a g n i f i e d the i n f l u e n c e o f Reynolds number and 70 1.0 0.8 -1-5 * 0.6 C. -1.0\ 1 \ R n - 1 7 5 A G R O V E e ta l • P R E S E N T 34 S/C 10 % 1 2 . 5 % 0.4 L 0 5 0.2 0 0 U - 0 . 5 c p ,Grove etal 0-2 U - 1 0 • • - 04 L - 1 - 5 A - 0 . 6 0 3 0 .6 0 9 0 120 e 150 180 F i g u r e 4 .2 . R e p r e s e n t a t i v e p r e s s u r e p l o t s comparing p r e s e n t r e s u l t s w i t h those by Grove et a l . at a low Reynolds number o f 175. Note a r e l a t i v e i n s e n s i t i v i t y o f the proposed p r e s s u r e c o e f f i c i e n t to d i f f e r e n c e s i n t e s t c o n d i t i o n s . 71 1.0 .1-0 0.8 >0.5 \f 36 - H 3 A EL-SHERBINY 15x10 3 • PRESENT 15x10 0.6 0.4 - 0 C P - 0 - 5 0.2 L - 1 0 Cp , El-Sherbiny 0 _1.5 0-2 - - 2 . 0 A * AV A • 0.4 0 3 0 6 0 90 120 150 9 F i g u r e 4-3 . A comparison of p r e s e n t r e s u l t s w i t h those by E l - S h e r b i n y at a r e l a t i v e l y h i g h Reynolds number o f 15 x 10^. ,72 w a l l confinement ( F i g u r e s 4-2 and 4 -3 ) . Akutsu came to the same c o n c l u s i o n d u r i n g h i s s t u d i e s w i t h s p h e r i c a l models . Note the new d e f i n i t i o n c o r r e l a t e s d i f f e r e n t se t s o f r e s u l t s q u i t e w e l l a l though they c o r r e s p o n d to i n v e s t i g a t i o n s conducted i n d i f f e r e n t t e s t f a c i l i t i e s . At the out se t one can say t h a t the e f f e c t o f Reynolds number i s e s s e n t i a l l y c o n f i n e d to the r e g i o n downstream of the zero p r e s s u r e p o i n t ( F i g u r e 4-4) and even here i t i s l i m i t e d to R^ of up to around 1200, except f o r the h i g h e r b lockage r a t i o s o f 25% and 501. In g e n e r a l , an i n c r e a s e i n the Reynolds number leads to a c o r r e s p o n d i n g i n c r e a s e i n the minimum as w e l l as the wake p r e s s u r e s . F u r t h e r m o r e , l o c a t i o n o f the minimum p r e s s u r e p o i n t t o g e t h e r w i t h the approximate l o c a t i o n o f the s e p a r a t i o n p o i n t (as i n d i c a t e d by the b e g i n n i n g o f the u n i f o r m p r e s s u r e r e g i o n of the wake) tend to s h i f t a l i t t l e upstream. At h i g h e r b lockage r a t i o s ( F i g u r e s 4-4 e, f) e s s e n t i a l l y the same t r e n d i s mainta ined. , however, the Reynolds number dependency i s extended to around R n = 3000. It i s o f i n t e r e s t 36 to note here tha t S h e r b i n y a l s o observed the same t r e n d i n h i s s tudy w i t h a c i r c u l a r c y l i n d e r of 35.5 % blockage i n 4 4 ' the Reynolds number range of 5 x 10 - 12 x 10 . 73 1-0 0.8 0.6 0.4 0.2 S / C =2% [D=1.2cm] Rn= 18 • 34 o 160 A 400 A 720 • 1200 * 1560 o C. 0 - 0.2 0-4 -0-6 Km*-• • • A A \ A A A a \ o ^ ° \ o o o A A a-o J3' O Dennis and Chang 50 0.8 -1.0 0 F i g u r e 4-4 30 60 90 0° 120 150 180 S u r f a c e p r e s s u r e d i s t r i b u t i o n as a f f e c t e d by the Reynolds number f o r a g iven b lockage r a t i o : a) S /C = 21. 74 1.0 S/C =3.3'%CD= 2 cm] 0.8 0.6 R n = 30 55 . 300 • 12 00 • 3000 < 0.4 0.2 T c p 0 -0.2 - 0-4 - 0-6 D n D • - 0 8 -1.0 0 30 60 90 o 0 120 150 F i g u r e 4-4 S u r f a c e p r e s s u r e d i s t r i b u t i o n as a f f e c t e d by the Reynolds number f o r a g iven b lockage r a t i o : b) S /C =3.3%. 75 to 0-8 0 -6 0-4 S/C= 6.6% CD = 4cm] R : n 60 80 130 600 120 0 3000 6000 D O A • < c 0-2 0 - 0 - 2 •0-4 -0-6 l «1 e < • A 2 i f 8 A O A A A O ° o o ^ o o • • • a A O • • D A O • A a o A D O -0-8 0 30 60 90 120 150 0 F i g u r e 4-4. Sur face p r e s s u r e d i s t r i b u t i o n as a f f e c t e d by the Reynolds number f o r a g i v e n b lockage r a t i o : c) S /C = 6 . 6 % . 76 1 0 0-8 0-6 0-4 Rn= S/C =12.5 % [ • =7.6 cm] 110 230 120 0 3000 61 50 11000 • o o 0- 2 a c 0 p u -0-2 - 0 - 4 ® • . J o n ° O o O ° o o • • • • • o o D o o a - 0-6 -0-8 -1.0 30 60 90 120 150 1 8 0 0 F i g u r e 4-4 S u r f a c e p r e s s u r e d i s t r i b u t i o n as a f f e c t e d by the Reynolds number f o r a g i v e n b lockage r a t i o : d) S /C = 12.5%. 77 1-0 0-8 0-6 0.4 C p 0.2 0 -0-2 -0-4 -0-6 -0-8 S/C= 25 % [ D=15 cm J R = 190 0 360 800 120 0 2000 3000 5000 30 4 * O O o • 8~o<> o o 0 o v V V v D • O n a V o o 6 o 60 90 1 20 1 50 e F i g u r e 4-4 S u r f a c e p r e s s u r e d i s t r i b u t i o n as a f f e c t e d by the Reynolds number f o r a g i v e n b lockage r a t i o : e) S /C = 25%. 78 1.0 0-8 0.6 0-4 0-2 S/Cz50 % CD= 30 cm] RR= 350 730 1200 240 0 3000 18000 V o o C 0 P 0.2 - 0,4 A A - 0-6 0-8 - 1.0 • T 0 30 60 90 120 150 180 6 ° F i g u r e 4-4 Sur face p r e s s u r e d i s t r i b u t i o n as a f f e c t e d by the Reynolds number f o r a g i v e n b lockage r a t i o : f) S /C = 50%. 79 It must be emphasized tha t such d e t a i l e d measurements o f p r e s s u r e d i s t r i b u t i o n o n . t h e s u r f a c e o f a c i r c u l a r c y l i n d e r (even i n absence of b lockage e f f e c t s ) i n the i n d i c a t e d range o f the Reynolds number (R < 20,000) have not been r e c o r d e d i n the l i t e r a t u r e , except f o r the attempts by Thom*^ and Grove et a l . 3 4 U n f o r t u n a t e l y , Thorn d i d not have access to the modern s o p h i s t i c a t e d i n s t r u m e n t a t i o n . H i s e x p e r i m e n t s , c a r r i e d out w i t h o i l or water as a working f l u i d and i n the Reynolds number range o f 3-174, r e q u i r e d , f o r p r e s s u r e measurements, r e a d i n g o f a manometer w i t h d i f f e r e n t i a l column o f l e s s than 1 mm! And tha t too w i t h the system h a v i n g a time cons tant o f one hour! H i s r e s u l t s show q u e s t i o n a b l e trends and are g e n e r a l l y c o n s i d e r e d m i s l e a d i n g . On the o t h e r hand , r e s u l t s by Grove and h i s a s s o c i a t e s are indeed r e l i a b l e , however, they are l i m i t e d to a s i n g l e Reynolds number o f 145. Hence they f a i l to throw any l i g h t on c o r r e -l a t i o n between s u r f a c e p r e s s u r e d i s t r i b u t i o n and e v o l u t i o n of the Fopp l v o r t i c e s l e a d i n g to i n s t a b i l i t y and v o r t e x shedd ing . R e c e n t l y , Dennis and C h a n g ^ have deve loped a f i n i t e d i f f e r e n c e scheme f o r s o l u t i o n o f equat ions o f m o t i o n . Present r e s u l t s t end to c o n f i r m t h e i r p r e d i c t i o n up to R^ = 100 ( F i g u r e 4 -4a ) . 4.3 W a l l Confinement E f f e c t s F i g u r e 4-5 summarizes r e s u l t s on the i n f l u e n c e o f b lockage 80 f o r c i r c u l a r c y l i n d e r s at a g iven Reynolds number. I t must be r e c o g n i z e d tha t the minimum and maximum a t t a i n a b l e speeds i n any l i q u i d t u n n e l are l i m i t e d by des ign c o n s i d e r a t i o n s . For the p r e s e n t f a c i l i t y , they were 0.5 cm/s and 15 cm/s , r e s p e c t i v e l y . However, the u s e f u l range i s f u r t h e r r e s t r i c t e d by the s e n s i t i v i t y o f the p r e s s u r e t r a n s d u c e r and a d d i t i o n a l l o a d imposed on the d r i v e system d u r i n g the h i g h b lockage c o n d i t i o n . Hence f o r a g i v e n b l o c k a g e , i t was not always p o s s i b l e to cover the d e s i r e d range o f Reynolds number (10 - 3000). T h i s has l e d to u n a v o i d a b l e gaps i n the r e s u l t s p r e s e n t e d h e r e . However, the t rends are r e a s o n a b l y w e l l e s t a b l i s h e d . From F i g u r e 4-5 i t i s apparent t h a t , as b e f o r e , the w a l l confinement e f f e c t s at a g i v e n Reynolds number are s i g n i f i c a n t o n l y over the r e g i o n downstream of the zero p r e s s u r e p o i n t . The b lockage has a d e f i n i t e tendency to reduce the minimum and base p r e s s u r e s . The minimum p r e s s u r e p o i n t shows a d i s t i n c t rearward s h i f t w i t h an i n c r e a s e i n the b l o c k a g e . S i m i l a r downstream movement o f the s e p a r a t i o n p o i n t can a l s o be d i s c e r n e d , a l though i t i s not always q u i t e d i s t i n c t . A f low v i s u a l i z a t i o n s tudy d e s c r i b e d l a t e r ( s e c t i o n 4.6) conf i rmed t h i s t r e n d . As can be expected from the p r e v i o u s d i s c u s s i o n , the b lockage e f f e c t s remain e s s e n t i a l l y the same f o r R n > 3000. However, what i s more s i g n i f i c a n t i s an i n c r e a s i n g dependence 81 1.0 .0,8 0-6 0 4 0.2 0 0.2 .0.4 - 0 - 6 - 0 -8 C - 1-0 0 R n S/C % = 3 0 0 0 3.3 . 6.6 o 12.5 A 25 • 5 0 . o a • o • o • • o • • o • a o • • • o D A A a a 2 • 2 3 0 6 0 9 0 0° 1 2 0 150 • 180 F i g u r e 4-5 . P r e s s u r e p l o t s as a f f e c t e d by b lockage at a g i v e n Reynolds number: a) R = 3000. . n 82 c 1-0 0.8 0-6 0-4 0.2 0 .0-2 -0-4 .0-6 0-8 -1.0 • R p z1200 S/C % , 3.3 6.6 12.5 25 50 o o • o • • o • V r t • O * •V A • O A '* * °• • V V • • a • <5 • v 0 30 60 90 120 150 180 9 ° F i g u r e 4-5 . P r e s s u r e p l o t s as a f f e c t e d by b lockage at a g iven Reynolds number: b) R = 1200. 83 to 0.8 C. 0-6 'P 0-4 0-2 0 0-2 - 0-4 0-6 0.8 1-0 R = 4 0 0 S / c L 2 . 3.3 o 12.5 -25 D 50 T • 9 • • • o • o • o • o • o • o • • P • o 0 3 0 60 90 120 150 180 0° F i g u r e 4-5 . P r e s s u r e p l o t s as. a f f e c t e d by b lockage at a g i v e n Reynolds number: c) R = 400. J n 84 i-o 0-8 0-6 • T R n . - 2 0 0 S / C ; 2 °/o T 12.5 °/o a 25 I. • 0-4 0-2 C 0 - 0-2 r a T • • n • • • • • • • • r D • • - 0 - 4 0.6 0.8 • -1-0 0 3 0 6 0 9 0 o 12 0 6 150 180 F i g u r e 4-5 . P r e s s u r e p l o t s as a f f e c t e d by b lockage at a g i v e n Reynold* number: d) R = 200. n 85 1.0, 0.8 R n = 3 0 0.6 S / C = 2 /Q 3-3 °/0 r C 0.4 0.2 0 - 0.2 T - 0 . 4 - 0 . 6 T • • T T • - 0 . 8 - 1 - 0 0 3 0 6 0 90 120 150 180 e F i g u r e 4-5 . P r e s s u r e p l o t s as a f f e c t e d by b lockage at a g i v e n Reynolds number e) R = 30 J n 86 of the s u r f a c e p r e s s u r e on b lockage at lower Reynolds numbers. The p o i n t i s w e l l emphasized by F i g u r e 4-5e, which shows a s u b s t a n t i a l m o d i f i c a t i o n i n the p r e s s u r e d i s t r i b u t i o n f o r a change i n b lockage by l e s s than 1.5%. In g e n e r a l , a n a l y t i c a l approaches i n t h i s range o f the Reynolds number have tended to be u n r e l i a b l e because o f the d r a s t i c assumptions i n v o l v e d and breakdown i n the schemes o f n u m e r i c a l i n t e g r a t i o n ^ . On _the o ther hand, t h e o r e t i c a l p r e d i c t i o n s on b lockage e f f e c t s are indeed none. In these c i r c u m s t a n c e s , r e s u l t s p r e s e n t e d i n F i g u r e 4-5 r e p r e s e n t an important p i e c e o f i n f o r m a t i o n . F i g u r e 4-6 shows v a r i a t i o n o f the minimum p r e s s u r e and the average base p r e s s u r e w i t h Reynolds number and b l o c k a g e . In g e n e r a l , f o r a cons tant b lockage r a t i o , both the minimum and base p r e s s u r e s i n c r e a s e w i t h the Reynolds number a t t a i n i n g n e a r l y u n i f o r m va lues around R = 3000. Note , the r a t e o f r i s e i n both the cases i s somewhat g r a d u a l at lower b lockage but becomes r a t h e r s teep under the h i g h e r confinement c o n d i -t ions of 25%and 50%. For a g i v e n Reynolds number, the e f f e c t of b lockage i s to reduce these p a r a m e t e r s , under c e r t a i n con-d i t i o n s r a t h e r d r a m a t i c a l l y . I n t e r e s t i n g l y , ~ C p m 1 S n o t 49 independent o f b l o c k a g e , as observed by A k u t s u i n h i s s t u d i e s w i t h s p h e r e s , but i s s t r o n g l y dependent on i t (not shown). Numer ica l r e s u l t s by Dennis and Chang*^ compare r a t h e r w e l l . U n f o r t u n a t e l y , t h e i r f i n i t e d i f f e r e n c e procedure i s v a l i d on ly up to R n = 100 and tha t too i n absence o f b l o c k a g e . c. ml 0 0.2 S/C % * 2 o 3-3 * 6-6 a 12.5 * 25 • 50 o A • o A D 0-4 0-6 0.8 10 F i g u r e 4-6 10' 10v 10 R n E f f e c t o f w a l l confinement on the minimum and base pressures ' , 10 < R < 18,000 a) minimum p r e s s u r e . . n CO c I S / C 7 0  P b r . 9 _ D e n n i s a n d C h a n g 5 0 0 0-2 0-4 0-6 0-8 • 2 o 3-3 A 6-6 • 1 2 5 • 25 • 50 A • O A A A A • D A A 10 10 2 i o 3 i o 4 R n F i g u r e 4-6. E f f e c t of wal l . conf inement on the minimum and base p r e s s u r e s , ' 10 <. R < 18 ,000: b) base p r e s s u r e . n ' 8 9 4.4 Drag C o e f f i c i e n t P r e s s u r e d i s t r i b u t i o n on the s u r f a c e o f a c y l i n d e r h a v i n g been e s t a b l i s h e d , the next l o g i c a l s tep was to o b t a i n p r e s s u r e i n t e g r a t e d v a l u e o f drag and ass,ess i t s dependence on b l o c k a g e . T h i s , of c o u r s e , cannot account f o r the s k i n f r i c t i o n c o n t r i b u t i o n . On the o ther hand, the d i r e c t measure-ment o f s k i n f r i c t i o n i s c o n f i n e d to the h i g h Reynolds number range o f 6 x 1 0 4 - 5 x 10^ by Achenbach*^ and the low Reynolds number range o f 10 - 300 by Thom^"*". As p o i n t e d out b e f o r e , Thorn's r e s u l t s are g e n e r a l l y c o n s i d e r e d u n r e l i a b l e , and Achenbach's are not r e l e v a n t to the p r e s e n t s tudy . One would expect p r e s s u r e drag c o e f f i c i e n t to be p r i m a r i l y governed by magnitude and l o c a t i o n o f the minimum p r e s s u r e p o i n t and the p r e s s u r e d i s t r i b u t i o n downstream of i t . S ince the p r e s s u r e p r o f i l e s do not change s u b s t a n t i a l l y beyond R n = 1000 - 3000 (R = 1000 f o r lower b lockage r a t i o s and R n = 3000 at the h i g h e r confinement c o n d i t i o n ) , the p r e s s u r e drag c o e f f i c i e n t f o r a g i v e n b lockage i s expected to remain e s s e n t i a l l y cons tant beyond these c r i t i c a l va lues o f the Reynolds number ( F i g u r e 4 - 7 ) . As can be a n t i c i p a t e d , the e f f e c t o f b lockage i s to i n c r e a s e the drag c o e f f i c i e n t p r i -m a r i l y because o f the l o c a l i n c r e a s e i n the f r e e s tream v e l o c i t y . Note , over the b lockage r a t i o range o f 50%, the drag c o e f f i c i e n t can i n c r e a s e by as much as 125%. Numer ica l c D, S / C % . 2 - 3 . 3 12.5 A - 25 50 Present 54 Rosenhead Dennis and Ghang 50 • A I I I I I I I I I 10 10' 10 v R n 10 F i g u r e 4-7 . V a r i a t i o n o f the p r e s s u r e drag c o e f f i c i e n t w i t h Reynolds number and b l o c k a g e . 91 r e s u l t s by Dennis and C h a n g J U , f o r < 100, compare f a v o r -a b l y w i t h the measured d a t a . The s o l i d l i n e corresponds to 5 2 5 3 e x p e r i m e n t a l r e s u l t s by R e l f and P r a n d t l et a l . as quoted by Rosenhead^ 4 . 4.5 S t r o u h a l Number The f a c t o r s govern ing the v o r t e x f o r m a t i o n and shedding frequency f o r b l u f f bodies are not yet c l e a r l y u n d e r s t o o d . S e v e r a l l e n g t h parameters have been proposed which are thought to have d i r e c t r e l a t i o n w i t h the shedding f r e q u e n c y . One of t h i s i s the p r o j e c t e d h e i g h t o f a model (normal to the f ree stream) which w i t h i n c i d e n t v e l o c i t y and v o r t e x frequency forms the S t r o u h a l number, S n . Importance o f the S t r o u h a l number l i e s i n the f a c t t h a t , f o r a g iven o r i e n t a t i o n o f a body, i t tends to be u n i f o r m i n the s u b c r i t i c a l range o f the Reynolds number, 10 4 < R < R . Attempts have been made' J ' - n . - n , c r ' r to d e f i n e a more g e n e r a l form o f the S t r o u h a l number tha t does not depend on the body shape or i t s o r i e n t a t i o n . The 55 most w e l l known i s that due to Roshko where the normal d i s -tance between the s e p a r a t e d shear - l a y e r s a f t e r they become p a r a l l e l , h , and the s e p a r a t i o n v e l o c i t y u"s are used as c h a r a c t e r i s t i c parameters to g ive 92 * = fh b U s ' The ' u n i v e r s a l ' S t r o u h a l number was found to have the va lue o f 0.181 f o r c i r c u l a r c y l i n d e r s , 9 0 ° wedges and normal f l a t p l a t e s . T h i s suggests tha t the d i s t a n c e between the shear l a y e r s r e p r e s e n t s one o f the important parameters a s s o c i a t e d w i t h the v o r t e x s h e d d i n g . In t h i s contex t i t i s p e r t i n e n t to mention here e x p e r i -ments w i t h a f a m i l y of e l l i p t i c c y l i n d e r s conducted by Modi and D i k s h i t " ^ . T h e i r e f f o r t at c o r r e l a t i n g r e s u l t s over a wide r a n g e , o f the c y l i n d e r e c c e n t r i c i t y and angle o f a t t a c k showed S to f a i r p o o r l y f o r s l e n d e r e l l i p s e s (e = 0 .92 , 0.98) at a < 5 0 ° . However, the c o r r e l a t i o n improved sub-s t a n t i a l l y w i t h the normal d i s t a n c e between s e p a r a t i o n p o i n t s as the c h a r a c t e r i s t i c l e n g t h . V a r i a t i o n of the S t r o u h a l number (based on the c y l i n d e r d iameter) w i t h the Reynolds number and b l u f f n e s s i s p r e s e n t e d i n F i g u r e 4-8. In g e n e r a l , f o r a g i v e n b l o c k a g e , the S t r o u h a l number i n c r e a s e s w i t h the Reynolds number t e n d i n g to approach a u n i f o r m va lue around R n = 1000. Comparing these r e s u l t s 32 5 7 w i t h those by Roshko , R e l f and Simmons , and Modi and 5 8 ' E l - S h e r b i n y ( f o r n e g l i g i b l e b lockage) i n the Reynolds number 4 range o f 40 < R^ < 10 suggests t h a t t h i s a p p a r e n t l y cons tant va lue r e p r e s e n t s a peak f o l l o w e d by a drop i n the S t r o u h a l S/C % 0-4 A 5 • 1 5 • 25 0-3 'n 0-2 0-1 A A * 0 20 A • J L I J I L». 100 • A J - I I R 1000 n F i g u r e 4-8 . V a r i a t i o n of the S t r o u h a l number wi th b lockage at the lower end o f the Reynolds number range (R < 1000). 94 number f o r 1 0 J < R '5 10 H ( F i g u r e 4 -9 ) . As a n t i c i p a t e d , the e f f e c t o f b lockage i s to i n c r e a s e the S t r o u h a l number . because o f the l o c a l i n c r e a s e i n the f r e e s t r e a m v e l o c i t y . For the b lockage r a t i o s c o n s i d e r e d , t h i s t r e n d i s apparent f o r R n > 2000. However, at lower Reynolds number, the t r e n d i s r e v e r s e d because o f the d e l a y i n the onset o f v o r t e x shedding at h i g h e r b l o c k a g e . T h u s , at lower Reynolds numbers, r a t e o f g e n e r a t i o n o f v o r t i c i t y d i m i n i s h e s w i t h an i n c r e a s e i n the b l o c k a g e . I t i s important to emphasize here tha t the o n l y o t h e r r e l e v a n t 5 8 i n f o r m a t i o n on the s u b j e c t i s due to Modi and S h e r b i n y i n 4 5 the Reynolds number range o f 10 - 1.2 x 10 . T h e i r r e s u l t s show s i m i l a r t r e n d and are p r e s e n t e d f o r comparison i n F i g u r e 4 - 9. F i g u r e 4-10 shows v a r i a t i o n o f the Reynolds number c o r r e s p o n d i n g to the i n i t i a t i o n o f the v o r t e x shedding as a f u n c t i o n o f w a l l conf inement . Note , i n absence o f b lockage the v o r t e x shedding i n i t i a t e s at R = 41. However, i t i s de layed to R n = 300 f o r the b lockage r a t i o o f 501! 4.6 Flow V i s u a l i z a t i o n and Near-Wake A n a l y s i s To p r o v i d e b e t t e r a p p r e c i a t i o n as w e l l as s u b s t a n t i a t i o n of the c e r t a i n b e h a v i o u r e x h i b i t e d by the measured d a t a , i t was d e c i d e d to undertake an e x t e n s i v e f low v i s u a l i z a t i o n 0.4 'n 0.3 S/C°A " 5 ) • 15 / Present • 25 0.2 Roshko 32 59 Nishioka and Sato 35.5 % . . - - - 2 6 . 5 ?Q 14.8 % S/C = 4-5 \ Modi and £ 8 El -Sherbiny 0.1 10 F i g u r e 4-9 10 10" R n 10V to on A comparison o f measured S t r o u h a l data w i t h those by o t h e r i n v e s t i g a t o r s 300 n R n .cr. 2 0 0 -100 -Flow visualization Hot film probe measurements 0 10 20 3 0 40 5 0 S/C % F i g u r e 4-10. Onset of v o r t e x shedding as a f f e c t e d by the b lockage <o O N 97 programme.A set o f c y l i n d e r s r a n g i n g i n d iameter from 1-10 cm were used i n the g l y c e r o l - w a t e r s o l u t i o n o f 761 c o n c e n t r a t i o n by we ight . The main o b j e c t i v e was to observe the f o r m a t i o n , development and i n s t a b i l i t y o f the F o p p l v o r t i c e s and the a s s o c i a t e d i n f l u e n c e on the measured p r e s s u r e d a t a . I t was a l s o hoped tha t t h i s would p r o v i d e some i n d i c a t i o n c o n c e r n i n g l o c a t i o n of the s e p a r a t i o n p o s i t i o n and i t s movement. The use o f dye i n j e c t i o n p r o c e d u r e , e x p l a i n e d i n d e t a i l e a r l i e r , proved to be q u i t e e f f e c t i v e i n a c h i e v i n g these o b j e c t i v e s . I t showed f o r m a t i o n of the v o r t i c e s i n a r a t h e r s p e c t a c u l a r f a s h i o n as p r e s e n t e d i n F i g u r e 4-11. Numerous photographs were taken at s y s t e m a t i c increments o f the Reynolds number. Only a few of the t y p i c a l p i c t u r e s i l l u s t r a t i n g f o r m a t i o n , symmetric e l o n g a t i o n , onset o f asymmetry and i n s t a b i l i t y f o l l o w e d by t u r b u l e n t shedding are p r e s e n t e d i n F i g u r e 4-12. The e x i s t e n c e o f an a x i s y m m e t r i c , s t a b l e v o r t e x system f o r low Reynolds number i n a s t ream, e s s e n t i a l l y f r e e o f macro-s c o p i c t u r b u l e n c e i s shown i n F i g u r e 4-12(a) through 4 - 1 2 ( f ) . For the Reynolds number above a c r i t i c a l v a l u e ( c o r r e s p o n d i n g to the f i r s t f o r m a t i o n of a s t a b l e bubble or F o p p l v o r t i c e s , R^ - 2 ) , the s t r e a m l i n e s s eparate from the s u r f a c e and form a c l o s e d r e g i o n immediate ly b e h i n d the c y l i n d e r . A s i n g l e stream emerges from the v e r t e x o f the c l o s e d r e g i o n e x t e n d i n g to a l o n g d i s t a n c e b e h i n d the c y l i n d e r . The s i z e o f the bubble i s such as to m a i n t a i n an e q u i l i b r i u m between the r a t e at 98 99 F i g u r e 4-12 A f low v i s u a l i z a t i o n s tudy showing development and i n s t a b i l i t y o f v o r t e x r i n g w i t h Reynolds number (S/C = 15%): (a) R = 4; (b) R = 10; (c) R n = 17; (d) R n = 21 n n 100 F i g u r e 4-12 ( c e n t . ) A f low v i s u a l i z a t i o n s tudy showing development and i n s t a b i l i t y o f v o r t e x r i n g m \ R e y n ? J d s n u m b e r ( S / C = 15%): (e) R = 31-( f ) R n = 42; (g) R R = 5 6 ; (h) R ^ ' = 6 0 N 101 which v o r t i c i t y i s generated and d i s s i p a t e d i n t o the main s tream. As the Reynolds number i s i n c r e a s e d , the F o p p l v o r t i c e s become e l o n g a t e d i n the f low d i r e c t i o n to m a i n t a i n t h i s e q u i l i b r i u m , and the s e p a r a t i o n p o i n t s move upstream towards the f r o n t s t a g n a t i o n p o i n t . T h i s forward movement o f the s e p a r a t i o n p o i n t s was a l s o suggested by the p r e s s u r e p l o t s p r e s e n t e d e a r l i e r ( F i g u r e 4 - 4 ) . For Reynolds number between 42-300 (depending on the w a l l confinement c o n d i t i o n , see F i g u r e 4 -10) , an asymmetry i n the c i r c u l a t o r y motion w i t h i n the v o r t e x sheet produces a c o r r e s p o n d i n g asymmetry i n the c i r c u l a t o r y mot ion i n the bub-b l e i t s e l f and a r e s u l t a n t s h i f t from the c e n t e r l i n e . Note the b e g i n n i n g o f a s m a l l v o r t e x i n F i g u r e 4-12 (g) which p r o -g r e s s i v e l y e n l a r g e s a c c e n t u a t i n g the asymmetry as i n F i g u r e 4 -12(h) . The s t a t e o f unsymmetr ica l but s teady wake i s d i s t u r b e d by f u r t h e r i n c r e a s e i n the Reynolds number. The r a t e at which v o r t i c i t y i s d i f f u s e d from the sheet i n t o the main body o f the f l u i d appears to remain p r a c t i c a l l y c o n s t a n t , but the i n c r e a s e d r a t e at which i t i s t r a n s f e r r e d to the v o r t i c e s c r e a t e s u n s t a b l e c o n d i t i o n w i t h i n the v o r t e x shee t . B a s i c a l l y , the p r o c e s s i s one o f b u i l d - u p and r e l e a s e , but no s i z e a b l e p o r t i o n o f the bubble escapes through an opening i n the end of the v o r t e x sheet d u r i n g the c y c l e . T h i s i n t u r n causes the o s c i l l a t i o n 102 of the a symmetr i ca l wake about the ax i s o f symmetry ( F i g u r e s 4-12 g , h) . When the v o r t e x s t r e n g t h o f the bubble reaches a c r i t i c a l v a l u e , a sudden motion of the Foppl v o r t i c e s d i s -turbs the s h e e t , which i n t u r n i s r e s p o n s i b l e f o r r e l e a s e o f the v o r t i c i t y and a consequent r e t u r n o f the bubble to i t s o r i g i n a l p o s i t i o n and shape. In the c y c l e o f b u i l d - u p and r e l e a s e , the v o r t i c i t y generated i n the boundary l a y e r becomes c o n c e n t r a t e d on d i a m e t r i c a l l y o p p o s i t e s i d e s o f the f low a x i s w i t h i n the v o r t e x shee t . The s e c t i o n s i n which the v o r t e x s t r e n g t h i s the g r e a t e s t are a l t e r n a t e l y d i s c h a r g e d i n t o the main body o f the f l u i d . With each e j e c t i o n , a p o r t i o n o f the sheet i s c a r r i e d away. The v o r t e x element d i s c h a r g e d i n t o the s tream i n t e r a c t s w i t h the d i s p e r s e d l i q u i d to form a r e g u l a r wake p a t t e r n , f a m i l i a r l y known as the Karman v o r t e x s t r e e t ( F i g u r e 4 -13) . As mentioned b e f o r e , the flow v i s u a l i z a t i o n r e s u l t s p r o -v i d e u s e f u l i n f o r m a t i o n c o n c e r n i n g l o c a t i o n o f the s e p a r a t i n g shear l a y e r . To t h i s end, the photographs were a n a l y z e d s y s t e m a t i c a l l y and the s e p a r a t i o n p o s i t i o n p l o t t e d as a f u n c -t i o n o f as shown i n F i g u r e 4 -14 (a ) . The c o r r e s p o n d i n g r e s u l t s as g i v e n by p r e s s u r e p l o t s are shown i n F i g u r e 4 -14 (b) . In the common range o f Reynolds number (R n - 10 •-. 260) , the c o r r e l a t i o n between the two se t s o f r e s u l t s i s r a t h e r good. Note that the s e p a r a t i o n p o i n t moves forward by about 1 0 ° f o r u 100 200 300 F i g u r e 4-14. P o s i t i o n o f s e p a r a t i o n as a f f e c t e d by the Reynolds number and w a l l confinement: ! a) flow v i s u a l i z a t i o n d a t a . • 150 100 50 o »o • • T o • • o S/C 2% - 6 . 6 % . 12.5 % 25 % 5 0 % o 10 10' 1Cf R n 1CT F i g u r e 4-14. P o s i t i o n o £ s e p a r a t i o n as a f f e c t e d by the Reynolds number and w a l l confinement •u ^ 1 ^ „ J 1 _ J. _ /" ^ 106 the b lockage r a t i o o f 5% over the s m a l l Reynolds number range o f 9-28, and can move upstream by as much as 45° over the range R n = 30 - 6000 ( F i g u r e 4-14 b ) . For c o m p a r i s o n , a v a i l a b l e r e s u l t s by o t h e r i n v e s t i g a t o r s are a l s o i n c l u d e d . Here the l i n e a t t r i b u t e d to Dennis and C h a n g ^ r e p r e s e n t s r e s u l t s o f the n u m e r i c a l i n t e g r a t i o n .o f the govern ing equa-t i o n s o f mot ion u s i n g a f i n i t e - d i f f e r e n c e scheme. The f i g u r e a l s o shows e f f e c t o f b lockage on p o s i t i o n o f the s e p a r a t i n g v o r t e x shee t . In g e n e r a l , the e f f e c t o f w a l l conf inement i s to move l o c a t i o n o f the shear l a y e r s e p a r a t i o n downstream, by as much as 2 5 ° over the b lockage r a t i o range o f 2-50%. I t must be emphasized tha t the v i s u a l d e t e r m i n a t i o n o f s e p a r a t i o n p o i n t i s , at b e s t , approx imate . C o n s i d e r i n g t h i s and the u n s t a b l e c h a r a c t e r , o f the p r o c e s s , s c a t t e r i n the experimen-t a l r e s u l t s i s s u r p r i s i n g l y s m a l l . T y p i c a l photographs o f the wake a s s o c i a t e d w i t h c y l i n d e r s o f f e r i n g d i f f e r e n t b lockage are p r e s e n t e d i n F i g u r e 4-15 and 4-16. E f f e c t o f b lockage on e v o l u t i o n o f the wake at a f i x e d Reynolds number o f 100 i s v i v i d l y d e p i c t e d i n F i g u r e 4-15. At a lower b lockage o f 25%, i n s t a b i l i t y o f the wake has j u s t set i n ( F i g u r e 4-15 a) , however, at the same Reynolds number, the wake beh ind a c y l i n d e r o f 50% b lockage i s s t i l l d e v e l o p i n g and i s q u i t e s t a b l e ( F i g u r e 4-15 b) . Photographs i n F i g u r e 4-16 f u r t h e r emphasize t h i s p o i n t . Note , the wake bubble 108 (b) R n = 150 , S/C= 50% F i g u r e 4-16 Photographs emphas iz ing i n f l u e n c e o f w a l l confinement on e v o l u t i o n o f the F o p p l v o r t i c e s : (a) R n = 85, S/C = 25%; (b) R n = 150, S/C = 50% 109 a s s o c i a t e d w i t h the c y l i n d e r o f 50% blockage i s s m a l l e r and s t a b l e even at a Reynolds number as h i g h as 150. On the o ther hand, the F o p p l v o r t i c e s have evo lved f u r t h e r and wake i s approach ing i n s t a b i l i t y at R n = 85 f o r the c y l i n d e r w i t h b lockage o f 25%. F i g u r e 4-17 attempts to a s s e s s , q u a n t i t a t i v e l y , e f f e c t of the Reynolds number and b lockage on the development o f the Fopp l v o r t i c e s by u s i n g l e n g t h o f the c a v i t y i n the d i r e c t i o n of the f low as a measure f o r compar i son . E x p e r i m e n t a l r e s u l t s 33 by Taneda and n u m e r i c a l va lues as o b t a i n e d by Dennis and C h a n g ^ , i n absence o f b l o c k a g e , are a l s o i n c l u d e d . As e x p e c t e d , the c a v i t y l e n g t h i n c r e a s e s w i t h the Reynolds number. However, w a l l confinement d r a s t i c a l l y d i m i n i s h e s i t s r a t e o f growth. Note , there i s c o n s i d e r a b l e d i s c r e p a n c y between the 3 3 p r e s e n t r e s u l t s f o r S /C = 5% and those o f Taneda as w e l l as Dennis and Chang*^. Taneda c a r r i e d out h i s experiments i n 1956 towing c y l i n d e r s coa ted w i t h condensed m i l k . A l t h o u g h Taneda's e x c e l l e n c e i n e x p e r i m e n t a l t echn ique i s q u i t e w e l l known, the procedure i s l i k e l y to g ive da ta o f l i m i t e d a c c u r a c y . Dennis and Chang's r e s u l t s are through n u m e r i c a l a n a l y s i s of the N a v i e r - S t o k e s equat ions as mentioned e a r l i e r . As p o i n t e d out by the authors themse lves , the procedure i s s u s c e p t i b l e to i n s t a b i l i t y and has o c c a s i o n a l l y shown v o r t e x l e n g t h to d i m i n i s h wi th an i n c r e a s e i n the Reynolds number! Taneda 33 Dennis&Chang 50 » o T • s / c % 5 15 25 50 R n F i g u r e 4-17 100 200 Dependence o f v o r t e x l e n g t h on the Reynolds number and b l o c k a g e 300 4.7 C l o s i n g Comments Before c l o s i n g i t would be. a p p r o p r i a t e to review some of the more s i g n i f i c a n t r e s u l t s and express a few thoughts on p o s s i b l e avenues f o r f u t u r e e x p l o r a t i o n which are l i k e l y to be p r o f i t a b l e . 4 .7 .1 C o n c l u d i n g remarks Important c o n c l u s i o n s based on the e x p e r i m e n t a l r e s u l t s may be summarized as f o l l o w s : ( i ) The use o f average v e l o c i t y i n the t e s t - s e c t i o n (based on the mean f low r a t e ) as a r e f e r e n c e v e l o c i t y t o g e t h e r w i t h the p r e s s u r e c o e f f i c i e n t d e f i n e d as Pn - P C = J 1 P promises to promote r e p e a t a b i l i t y and comparison o f d a t a by o t h e r i n v e s t i g a t o r s r e g a r d l e s s o f the t e s t f a c i l i t i e s used . T h i s approach tends to compensate f o r i r r e g u l a r i t y o f the v e l o c i t y p r o f i l e and p o s s i b l e e r r o r s i n p r e s s u r e measurements caused by e l e c t r i c a l d r i f t o f the p r e s s u r e measur ing system. 112 ( i i ) For p r e s s u r e d i s t r i b u t i o n on the s u r f a c e o f a c y l i n d e r , the e f f e c t o f Reynolds number i s e s s e n t i a l l y c o n f i n e d to the r e g i o n downstream of zero p r e s s u r e p o i n t and even here i t i s l i m i t e d to R n < 3000. In g e n e r a l , the e f f e c t o f Reynolds number i s to i n c r e a s e the minimum as w e l l as the wake p r e s s u r e s . F u r t h e r m o r e , l o c a t i o n s o f the minimum p r e s s u r e and s e p a r a t i o n tend to s h i f t l i t t l e upstream. ( i i i ) P r e s s u r e d i s t r i b u t i o n on the s u r f a c e o f a c y l i n d e r i s s u b s t a n t i a l l y a f f e c t e d by the b l o c k a g e . P r e s s u r e p r o f i l e s become ex tremely s e n s i t i v e to the w a l l confinement p a r t i c u l a r l y at the lower end of the Reynolds number range . In g e n e r a l , there i s a d e f i n i t e tendency to reduce the minimum and base p r e s s u r e s . The minimum p r e s s u r e p o i n t shows a d i s t i n c t rearward s h i f t w i t h an i n c r e a s e i n the b l o c k a g e . ( i v ) Drag c o e f f i c i e n t o b t a i n e d by i n t e g r a t i n g p r e s s u r e d i s t r i b u t i o n data agrees r a t h e r w e l l w i t h the r e s u l t s , at s m a l l b l o c k a g e , by o t h e r i n v e s t i g a t o r s s u b s t a n t i a t i n g r e l i a b i l i t y o f the measuring i n s t r u m e n t a t i o n . In g e n e r a l , the drag c o e f f i c i e n t i n c r e a s e s w i t h b lockage because o f the l o c a l r i s e i n the f r e e s tream v e l o c i t y . 113 (v) Flow v i s u a l i z a t i o n p r o v i d e d b e t t e r a p p r e c i a t i o n as to the p h y s i c a l c h a r a c t e r o f the f low i n terms o f f o r m a t i o n , e l o n g a t i o n and i n s t a b i l i t y o f the F o p p l v o r t i c e s . I t showed the s e p a r a t i o n l o c a t i o n to move downstream w i t h an i n c r e a s e i n b l o c k a g e . ( i v ) R e s u l t s o b t a i n e d here sh ou ld serve as a u s e f u l r e f e r e n c e i n comparing c o r r e s p o n d i n g data o b t a i n e d i n : (a) t u r b u l e n t , shear and p u l s a t i l e f l o w s ; (b) l o n g c h a i n polymer s o l u t i o n s . 4 .7 .2 Recommendation f o r f u t u r e s tudy As p o i n t e d out b e f o r e , the p r e s e n t e f f o r t s at o b t a i n i n g some a p p r e c i a t i o n as to the p h y s i c s o f the w a l l confinement e f f e c t s at low Reynolds numbers r e p r e s e n t o n l y a modest b e g i n n i n g . There are numerous avenues a lon g which the r e s e a r c h programme may p r o g r e s s i n f u t u r e . Some o f the more important a s p e c t s , recommended f o r f u t u r e s t u d i e s , are summarized below: ( i ) In the p r e s e n t se t o f e x p e r i m e n t s , b lockage e f f e c t s on the s u r f a c e p r e s s u r e d i s t r i b u t i o n c o u l d not be s t u d i e d 114 over the e n t i r e range o f R n o f i n t e r e s t i n a l l the cases . L i m i t i n g f a c t o r s were power o f the d r i v e system and sen-s i t i v i t y o f the p r e s s u r e measuring i n s t r u m e n t a t i o n . The s u r f a c e p r e s s u r e at lower Reynolds numberswas found to be so s m a l l [0(10 ^)ps i ] tha t i t p r e s e n t e d a problem of measurement w i t h an a c c e p t a b l e degree o f a c c u r a c y and r e p e a t a b i l i t y . T h e r e f o r e , i t i s sugges ted t h a t p r e s s u r e , measumrenents at lower Reynolds numbers(and h i g h e r b lockage ) shou l d be undertaken to p r o v i d e a comprehensive p i c t u r e o f w a l l confinement e f f e c t s . T h i s can be accompl i shed u s i n g : (a) more s e n s i t i v e and s t a b l e p r e s s u r e t r a n s d u c e r ( e . g . , D i g i q u a r t z p r e s s u r e t r a n s d u c e r s ) ; (b) a m o d i f i e d d r i v e and the pump system so t h a t h i g h e r c o n c e n t r a t i o n o f g l y c e r o l - w a t e r s o l u t i o n can be h a n d l e d . ( i i ) No e f f o r t has been made to e v a l u a t e p r e s s u r e d i s t r i b u t i o n , t u r b u l e n c e c h a r a c t e r and shear s t r e s s i n the wake. In f a c t shear s t r e s s on the s u r f a c e o f a c y l i n d e r , even i n absence o f b l o c k a g e , i n t h i s range o f Reynolds number remains u n r e c o r d e d . The , i n f o r m a t i o n i s q u i t e important i n e v a l u a t i n g performance o f l o n g - c h a i n p o l y m e r s . 115 ( i i i ) A d i r e c t measurement o f the s k i n f r i c t i o n though d e s i r a b l e p r e s e n t s s e v e r a l c h a l l e n g i n g problems . On the o t h e r hand, measurement o f the t o t a l drag as a f u n c t i o n o f the Reynolds number and b lockage i s r e l a t i v e l y s i m p l e . I t i s suggested t h a t a s e n s i t i v e drag ba lance be des igned f o r the purpose and a s y s t e m a t i c programme o f t o t a l drag measurement i n i t i a t e d . Wi th the p r e s s u r e drag d a t a i n hand , t h i s shou l d y i e l d r a t h e r u s e f u l i n f o r m a t i o n on s k i n f r i c t i o n v a r i a t i o n w i t h b l o c k a g e near the lower end o f the Reynolds number range . ( i v ) T e s t s s h o u l d be c a r r i e d out w i t h c y l i n d r i c a l models under d i v e r s e c o n d i t i o n s o f v e l o c i t y p r o f i l e , b l o c k a g e , t u r b u l e n c e and p r e s s u r e g r a d i e n t to f i r m l y e s t a b l i s h u n i v e r s a l c h a r a c t e r o f the p r e s s u r e d i s t r i b u t i o n u s i n g proposed d e f i n i t i o n o f the p r e s s u r e c o e f f i c i e n t . (v) B lockage c o r r e c t i o n s f o r b l u f f bod ies such as c i r c u l a r c y l i n d e r , f l a t p l a t e , s p h e r e , e t c . i n shear f low s h o u l d p r o v i d e u s e f u l i n f o r m a t i o n . ( v i ) An important area o f i n t e r e s t , which i s p a r t i c u l a r l y s i g n i f i c a n t i n b i o l o g i c a l f l u i d mechanics , would be the s tudy o f p u l s a t i l e f low past b l u f f bod ies under w a l l conf inement s i m u l a t i n g a t y p i c a l c a r d i a c c y c l e . T h i s may a l s o i n v o l v e model ing o f t u r b u l e n c e c h a r a c t e r o f the b i o l o g i c a l f l u i d f low and e l a s t i c i t y o f the r e g i o n . The f i e l d i s r e l a t i v e l y new and remains v i r t u a l l y u n e x p l o r e d to d a t e . ( v i i ) I t would b e . u s e f u l to s tudy e f f e c t o f aspec t r a t i o i n the presence o f w a l l conf inement . 117 REFERENCES Toms, B . A . , "Some O b s e r v a t i o n s on the Flow of L i n e a r Polymer S o l u t i o n s Through S t r a i g h t Tubes at Large Reynolds Numbers", P r o c e e d i n g s o f the 1st I n t e r n a t i o n a l Congress on Rheology , V o l . I I , North H o l l a n d P u b l i s h i n g C o . , Amsterdam, 1949, p . 135. M y s e l s , K . J . , "Flow o f T h i c k e n e d F l u i d s " , U . S . P a t e n t , 2 ,492 ,173 , December 1949. Ka lashn ikow, V . N . , a n d K u d i n , A . 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J . , a n d E l - S h e r b i n y , S . , " E f f e c t o f W a l l C o n f i n e -ment on Aerodynamics o f S t a t i o n a r y C i r c u l a r C y l i n d e r s " , Proceed ings o f the 3rd I n t e r n a t i o n a l Conference on Wind E f f e c t s on B u i l d i n g s and S t r u c t u r e s , E d i t o r : Japanese O r g a n i z i n g Committee, Sa ikon Shuppan C o . , L t d . , Tokyo 1971, pp . 365-375. 59. N i s h i o k a , M . , and S a t o , M . , "Measurements o f V e l o c i t y D i s t r i b u t i o n i n the Wake o f a C i r c u l a r C y l i n d e r at Low Reynolds Numbers," J o u r n a l o f F l u i d  M e c h a n i c s , V o l / 65, P a r t 1, 1974, pp . 97-112. 126 APPENDIX-I CONVENTIONAL PRESSURE COEFFICIENT C~ IN TERMS p OF MEASURED INFORMATION In a low R e y n o l d s number e x p e r i m e n t u s i n g a l i q u i d t u n n e l , d i f f i c u l t i e s i n e s t a b l i s h i n g c h a r a c t e r i s t i c r e f e r e n c e v e l o c i t y and p r e s s u r e , P and U , were d i s c u s s e d •* c ' CO CO ' b e f o r e . However, f o r c o m p a r i s o n and t o e s t a b l i s h r e l a t i v e e f f e c t i v e n e s s o f t h e new d e f i n i t i o n o f p r e s s u r e c o e f f i c i e n t , one c a n c a l c u l a t e t h e c o n v e n t i o n a l p r e s s u r e c o e f f i c i e n t — 2 Cp = (Pg - P^) ( 1 / 2 ) p U ^ q u i t e r e a d i l y u s i n g d i f f e r e n t i a l p r e s s u r e d a t a m e asured d u r i n g t h e e x p e r i m e n t . The x component o f N a v i e r - S t o k e s e q u a t i o n a l o n g t h e s t a g n a t i o n s t r e a m l i n e y = 0 c a n be w r i t t e n as 3u 1 3P , r 3 2 u , 3 2u, U TT- ~ — — "TT— + V [ T + o ] 3x p 3x . 2 2 3x 3y-I n t e g r a t i n g f r o m f r o n t s t a g n a t i o n p o i n t to, minus i n f i n i t y u p s t r e a m o f t h e s p h e r e y i e l d s , 3u , = - — . — dx + v [ n + — s H dx u -r— dx p 3x . 2 2 J , 3x • ' -co J-co a x 3y J —CO J 2 u CO 2 127 u + + V 2 2 r 9 u . 9 u , , [ — j + — 2 " ^ 3x* 3y 0 0 0 1 / 2 P u ; 1 + 1 / 2 u " r3 U , 3 U n , -6 3x dy + r d U , 3 U , , L — o + — t - J d x 3x' •3y w h e r e 6 i s t h e b o u n d a r y - l a y e r t h i c k n e s s . T h e s e c o n d i n t e g r a l v a n i s h e s ' b e c a u s e o f i r r o t a t i o n a l i t y o f t h e o u t e r f l o w w h i l e i n t h e f i r s t i n t e g r a l u s u a l b o u n d a r y l a y e r a p p r o x i m a t i o n 2 2 3 u > > 3 u 3x* 3y c a n b e i n t r o d u c e d . S i n c e 3u 3x 3v 3y a t x = 0 0 °o = 1 - 3u x = 6 y = 0 = 1 + i + 1 / 2 P ^ 1 / 2 u f 3 x w h e r e A i s a c o n s t a n t a n d R i s t h e R e y n o l d s n u m b e r . 128 H e r e t h e n u m e r i c a l v a l u e o f A f o l l o w s d i r e c t l y f r o m t h e o u t e r f l o w s o l u t i o n . U s i n g t h e p o t e n t i a l f l o w a n a l y s i s , H o n 1 3 1 1 1 1 a s w e l l a s G r o v e e t a l . 3 4 h a v e s h o w n , i n d e p e n d e n t l y , t h e v a l u e o f A t o be 8 . T h u s , P - P 0 °° _ , . 8 . — _[_ -\- — -J- • • • • 1/2 p U 2 R i . e . , P - P 5 0 ° = ( P 0 - P 5 0 ^ ~ + • • " ) l / 2 P U 2 Now p e - < p « , - p 5 t f > = p e - { ( p o - p 5 o o ) - ( 1 + l + - - - * ) 1 / 2 p U c » } > p e p » _ r e "5 0° '<d r 5 ( f ± „ j ± 2" ~ 2 ~ 2 R 1/2 p Vt 1/2 p u : 1/2 p ui R e c o g n i z i n g t h a t f o r a R e y n o l d s n u m b e r as l o w a s 300 c o n -t r i b u t i o n o f 8 / R t e r m i s l e s s t h a n 3%, t h e a b o v e e x p r e s s i o n r e d u c e s t o P 6 - P ~ = ( P 8 - P 5 ( ^ - < P 0 - P 5 ^ + x V 2 p l £ 1/2 p uj 

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